Blog » John Coltrane’s Tone Circle

Reading Time: 11 minutesJanuary 22, 2016

JOHN COLTRANE’S TONE CIRCLE

My music is the spiritual expression of what I am: my faith, my knowledge, my being.
– John Coltrane

FOREWORD

I do like to mention that I am no “authority” or “expert” when it comes to Coltrane’s work, or the music theory behind it and the compositions themselves. And as sax player, well, I’m still miles away from even standing in the giant shadow he cast … not to mention his giant footsteps. Anyway, as admirer of Coltrane’s work I could not resist to write this article. I wrote this article because I am fascinated by his music and have an interest in the relationship between music and math / geometry.

For an expert opinion on Coltrane you should listen to what musicians who played with him or extensively studied his work have/had to say about it.

This blog article is an addition to the article “Music and Geometry” and contains only the information about the Coltrane Tone Circle and the relationship between some of his music and geometry. Do read the mentioned article for general information about the relationship between music and geometry.

Thelonious Monk once said “All musicians are subconsciously mathematicians“. Musicians like John Coltrane though have been very much aware of the mathematics of music and consciously applied it to their works. The “Coltrane Circle” is (to me) proof of it in Coltrane’s case …

SHORT INTRODUCTION ABOUT 12-TONE CIRCLES

Circle of Fourths - Clockwise — “Circle of Fourths” (counterclockwise the “Circle of Fifths”)

A Tone Circle is is a geometrical representation of relationships among the 12 pitch classes (or pitch intervals) of the chromatic scale in pitch class space (circle). The most common tone circles in Western music are the “Chromatic Circle” and the “Circle of Fifths / Fourths“.

In Western music theory there are 13 intervals from Tonic (unison) to Octave. These intervals are the: Unison, Minor Second, Major Second, Minor Third, Major Third, Fourth, Tritone, Fifth, Minor Sixth, Major Sixth, Minor Seventh, Major Seventh and Octave. When we look at these intervals (or pitch classes) and how they relate to one another in the musical tone circles, some nice geometric shapes appear.

Note: If you are interested in a more esoteric-philosophical perspective on the intervals, then read the article: “The Function of the Intervals” on Roel’s World.

COLTRANE’S TONE CIRCLE

An interesting variant to the ‘Circle of Fourths / Fifths’ is the ‘Coltrane Circle’, created by saxophonist John Coltrane (perhaps influenced and based on the Joseph Schillinger System of Musical Composition and/or Nicolas Slominksy’s Thesaurus of scales and musical patterns?) and was used by Yusef Lateef for his work “Repository of Scales and Melodic Patterns” (1981).

Stephon Alexander wrote in his book “The Jazz of Physics: The Secret Link Between Music and the Structure of the Universe” that he tried calling Yusef Lateef and asked when he was told that Yusef Lateef was not availabe: “Could I leave him a message about the diagram that John Coltrane gave him as a birthday gift in ’61?“

The year 1961 would “date” the Coltrane tone circle one year after the release of the groundbreaking album “Giant Steps” (1960), but in the same year as the release of the albums: “Coltrane Jazz”, “My Favorite Things”, “Olé Coltrane” and “Africa/Brass” and several years before unique albums like for example “A Love Supreme” (1965) and “Ascension” (1966).

According to Lucas Gonze Yusef Lateef mentioned: “Coltrane was always drawing things like this. This particular drawing was something Coltrane did between set breaks at a gig they did together. Coltrane gave it to Lateef at that gig.” This is an intriguing thought, if Coltrane was always drawing “things like that”, could that mean that there might be more versions of this tone circle (or other) somewhere in a box or folder in a museum, at the residence of one of his relatives or in the archive of musicians John Coltrane worked with as well?

Did John Coltrane drew it to work out a particular composition? Did he try to find a new approach for his solos in that period? I haven’t been able to find any clear sources that can provide a clear answer to those questions.

A Pentagram & Pentagon appears between the same tones in the Coltrane Circle” (in the original and reproduction with the tone C) when connected by a line.

Click on the Coltrane drawing to enlarge it.

CIRCLE VERSIONS

There are two versions of the circle shared online. A “detailed” version and a “clean” version with only the circled tones. What is good to point out is that these are two are separately drawn circles, as you can see in the “overlay” in the middle below. I have aligned the letter “C” of both drawings. The blue-overlay is the “clean” version (first one displayed below):

When you look closer, you can see two more differences:

(img.1) On the left a cut from the “clean” version, on the right a cut from the “detailed version”.

(img.1): In both versions the [A] (tone center) has been “squared”. In the “clean” version A♭ is notated, in the “detailed” version G♯ is notated. The “detailed” version also shows a mistake. Instead of circling both tones siding the [A] chromatically, the tones siding the G♯ chromatically have been circled.

(img.2) On the left a cut from the “clean” version, on the right a cut from the “detailed version”.

(img.2): In both versions the [E] has been “squared”. In the “clean” version E♭ is notated, in the “detailed” version D♯ is notated.

You might wonder, which circle was drawn first?
Well, most logically is to presume the more “detailed” version was draw first. Why? It contains a mistake in the circling of the neighboring tones of tone center [A] and it seems logical that this mistake would have been corrected in a next version, thus the “clean” version (without the mistake) would have come second, only displaying the most important aspect of the circle, the 12 “tone centers” and circled neighbor tones. But, I am guessing here.

What about those numbers and lines?
There has been some speculating going on about if the lines and numbers drawn in the Coltrane Circle “detailed” version were drawn by Coltrane himself or perhaps if they were added later by someone else?. We could compare the numbers drawn in the Coltrane Circle with those from copies of his scores. For this comparison I have used the score of Love Supreme and several scores displayed at recordmecca.com

Below you see the numbers found in various scores side by side with the numbers of the Coltrane Circle image:

John Coltrane - handwriting - numbers compared.

Now, I’m no expert in graphoanalysis, so I will just describe what I noticed:
In all scores as well in the tone circle we see a certain inconsistency in the writing of the numbers. The “1” is sometimes written as a single line, sometimes with additional horizontal lines. The 7 is sometimes written with a horizontal line in the center, sometimes without. The “4” is open sometimes and closed at the top at other times. The “2” has a little “loop” in some cases but others not. It seems though that the writing in the scores was done quicker, more like scribbling then seems to be the case with the Coltrane Circle. This is not a surprise though, specially with last minutes arrangements scores often look like scribbles.

Below links to the used scores to compare with the Coltrane Tone Circle:

ohn Coltrane – Handwritten Musical Manuscript 1

ohn Coltrane – Handwritten Musical Manuscript 2

John Coltrane – Handwritten Manuscript for Stablemates, etc.

One thought shared among musicians online is that the writing of the numbers (and lines) in the “detailed” version of the tone circle could perhaps have been drawn by someone else. Yusef Lateef seems to be the #1 “suspect”, after all, he shared the drawing in his book “Repository of Scales and Melodic Patterns“. I have not been able though to find any handwritten music sheets by Yusef Lateef to compare his handwriting.

In my search for answers I send a message to the Yusef Lateef Facebook page, hoping someone could shed some light on it. I am grateful I got a reply, Ayesha Lateef wrote:

“Brother John gifted the “circle” to Yusef Lateef while the content of both repositories is the result of Yusef’s own research.“

When I specifically asked if the numbers and lines in the circle might have been drawn by Yusef Lateef, she replied:

“From what I understand the whole thing is from Brother John.“

More about the numbers and their possible meaning/function later on in this article.

Below on the left you see a scanned copy of an original drawing of the “Coltrane Circle”. On the right an better readable (by Roel modified) image by Corey Mwamba from his article “Coltrane’s Way Of Seeing“:

In the drawing (on the left) there are a couple of sharps notated, they have been replaced by Corey Mwamba with their enharmonic equivalents (C♯ = D♭ and F♯ = G♭) in his drawings.

The circles above might seem a bit odd, but if we “simplify” the circle things become a lot clearer.

What we see is a circle with two concentric rings.

The outer ring displays the “Hexatonic” (6-Tone) or “Whole Tone” Scale of C (C – D – E – G♭ – A♭ – B♭ – C).

The inner ring displays the Hexatonic scale of B
(B – D♭ – E♭ – F – G – A – B).

When you “zig-zag” clockwise between the tones of these Hexatonic scales of the concentric rings (the 12 “Tone Centers”) it turns out to be the “Circle of Fourths” (and thus counterclockwise the “Circle of Fifths“).:

C – F – B♭ – E♭ – A♭ – D♭ – G♭ – B – E – A – D – G – C

WHAT ABOUT ALL THOSE TONES IN BETWEEN?

The smaller spaces (light grey) between the larger (“main”) “Tone Center” spaces (darker grey) of the Hexatonic scale of C (outer ring): C–D–E–G♭–A♭–B♭–C) and B (inner ring): B–D♭–E♭–F–G-A-B contain 4 tones that – when combined with the “Tone Center” spaces (pitch classes) – form 6x the same Hexatonic scale within the same ring, just each shifting a tone.

All Hexatonic scales within the same ring use exactly the same 6 tones but any of these tones could be used as the tonic of a hexatonic scale.

“Double Power”

We know Coltrane had an investigative mind and a deep interest in mathematics, philosophy, the occult and religion.

Earlier in this article you probably noticed the Pentagram Coltrane drew in his circle. Now we have looked at the Hexatonic nature of the tone circle, we can also draw a Hexagram between the tones of the Hexatonic scale of the outer ring.

What appears when we combine the Pentagram and Hexagram, is the symbol of “Double Power“. As symbol of ‘double power’ or the unity of the Pentagram and Hexagram, it symbolizes the “mystical marriage” of the micro and macrocosms. The Inner and Outer Worlds. “As Above, So Below“.

Below you see on the left the Coltrane Circle. In this case instead of single tones I approach it as chords. Those of you familiar with Coltrane’s music will directly notice that the colored sections of the circle form the chord progressions of Giant Steps (in “concert pitch“).

When you merge the triangles formed with the Minor 7th and Dominant 7th chords (the II‘s and V‘s), a Hexagram is formed. The Hexagram can be seen as a 2D version of the 3D Star Tetrahedron, also known as “Merkaba“.

HEXAGRAM (GIANT STEPS)

HEXAGRAM

STAR TETRAHEDRON (MERKABA)

“Mer-ka-ba” means “light-spirit-body”. The Merkaba represent the innermost law of the physical world: the inseparable relationship between the two complementary halves – the positive and negative, the manifest and the unmanifest – which form a perfect equilibrium. In creation they rule as two opposite laws: the law of spirit and the law of matter. The Merkaba is also been called the “divine light vehicle” allegedly used by ascended masters to connect with and reach those in tune with the higher realms, the spirit/body surrounded by counter-rotating fields of light, (wheels within wheels).

I understand for those of you reading this with no interest in mathematics, philosophy, the occult and religion this might all seem a bit far-fetched. But, if you look at some of the titles of his compositions (“Ascension“, “Ascent“, “Sun Ship“, “Cosmos”, “Interstellar Space“, “Spiritual”, et cetera) then is seems more then clear that music, the occult / religion and geometry / math were all connected for Coltrane.

More about Giant Steps you can read in the article “The Geometry of John Coltrane’s Music“.

WHY HAVE TONES BEEN CIRCLED?

It’s not completely clear why Coltrane circled those tones, he never made note of it. The tones that have been circled are the Major 7th or “Leading Tone“, the Tonic and the Minor 2nd or “Supertonic” (see image below).

Perhaps Coltrane wanted to visualize how chromatic neighbor tones lead to adjacent neighbor tones / Tone Centers?

Every Major 7th (mentioned above) is the Major Third of the key (tone center) a Fifth higher (next tone center counterclockwise) as well. [suggestion by Mark Rossi]

Example: the B circled along with the C (tone center) is the Major Third of G (next tone center counterclockwise in the Coltrane Circle).

Every Minor 2nd is also the Major Third of the parallel Major of the Relative Minor key of the by circle connected tone center. [suggestion by Mark Rossi]

Example: the D♭ circled along with the C (tone center) is the Major Third of A Major, the parallel Major key of A Minor, the relative minor key of C Major (tone center).

DIMINISHED SCALES

Perhaps the circled tones outline the relationship between Diminished 7th Chords within the Diminished Scale? An example:

The C Diminished 7th Chord is C – E♭ – G♭ – A. To turn this into a Diminished scale, you need to add another Diminished 7th Chord a semitone higher: D♭ – E – G – B♭ or lower: B – D – F – A♭. Results:

C – D♭ – E♭ – E – G♭ – G – A – B♭ – C
&
C – D – E♭ – F – G♭ – A♭ – A – B – C

It is commonly known that Coltrane did like using the Diminished Scale (or “Double Diminished” as it was called because it is build from two Diminished 7th Chords). An example of that is his solo in “Moment’s Notice” (in measure 74 where he plays a Bb7 diminished scale pattern). Another example is his solo in “Epistrophy” during the live perfomance at Carnegie Hall with Thelonious Monk.

ALTERED DOMINANT (ALT DOM) CHORDS

Jazz guitarist, composer and music theorist Mark Rossi shared another way of looking at the circled tones.

An Alt Dom chord is a dominant chord (centered around the 5th of the key) but with a minor 7th on top (hereby creating a Dominant 7th) and the 5th and 9th of the chord either lowered or raised by one half step. This in turn gives us either a ^b5 or a ^#5 instead of a natural 5 as well as a ^b9 and ^#9.

When you add the 3 Diminished 7th Chords to a table you get the following result:

G	A♭	A	B♭	B	C	D♭	D	E^b	E	F	G♭
B♭	B	C	D♭	D	E♭	E	F	G♭	G	A♭	A
D♭	D	E♭	E	F	G♭	G	A♭	A	B♭	B	C
E	F	G♭	G	A♭	A	B♭	B	C	D♭	D	E♭

5	←	♭6 or 7	→	8	3	←	4 or 2	→	♭3
B♭	←	B	→	C	B♭	←	B	→	C
D♭	←	D	→	E♭	D♭	←	D	→	E♭
E	←	F	→	G^b	E	←	F	→	G♭
G	←	A♭	→	A	G	←	A♭	→	A

NATABHAIRAVI-CHARUKESI (NATURAL MINOR + MELODIC MAJOR) “COMPOUND” SCALE

Corey Mwamba shared an alternative interpretation about the meaning of the circled tones, he thinks they might form what he calls a “compound scale”. This compound scale is formed my combining the “Natural Minor” scale (Natabhairavi) and the “Melodic Major” scale (Charukesi) a semitone lower, characteristic for North Indian music (something Coltrane developed an interest for in the 60s (see “John Coltrane and the integration of Indian concepts in Jazz improvisation” by Carl Clements).

Corey writes: “We can see that the two scales have two enharmonic points; one at the third degree of each scale, and one at the sixth. If we transliterate Natabhairavi to d♭ and combine it with Charukesi mapped from c, we can see an intersection that contains e and a♭. Natabhairavi is the top line, circled in blue; Charukesi is circled in red.“

He continues: “Arranged in chromatic order, the first, fourth and seventh degrees of Natabhairavi are aligned with the degrees from Charukesi in a way that matches the segment 3–4 on the original diagram.” With the “original diagram” Corey referes to the Coltrane Circle with the Pentagram drawn into it. In that version the Circle the 5 segments are numbered.

For additional information and images, read Corey’s article “Way of Seeing Coltrane (IV)“.

ALL-INTERVAL TETRACHORD

Stephon Alexander wrote in his book “The Jazz of Physics: The Secret Link Between Music and the Structure of the Universe that it has been argued by Australian pianist Sean Wayland that the All-Interval Tetrachord can be used as a method to play through the chord changes of “Giant Steps” (see video: https://www.youtube.com/watch?v=sQGWAnYd7Iw by Marc Hannaford).

An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords when expressed in prime form. In set theory notation, these are {0146} and {0137} (their inversions: {0256} and {0467}).

From the Tonic C we would get: C-Db-E-Gb {0146} and C-Db-Eb-G {0137} (their inversions: C-D-F-Gb {0256} and C-E-Gb-G {0467}). As you can see, the {0146} sets contain only tones circled on the Coltrane Circle (C-Db-E-Gb) if you follow the Circle clockwise from C.

ALL-TRICHORD HEXACHORD

This though made me wonder if another “tone-series” would align better with or include more tones of the series of circled tones: the All-Trichord Hexachord.

The all-trichord hexachord is a unique hexachord that contains all twelve trichords, or from which all twelve possible trichords may be derived. The prime form of this set class is {012478}

From the Tonic C we would get: C-Db-D-E-F-Gb. All but the 2nd pitch class (D) used in this All-Trichord Hexachord are circled at the Coltrane Circle if you follow the Circle clockwise from C.

WHAT DO THE NUMBERS IN THE DRAWING MEAN?

“What do those numbers mean?” is a question I have received via mail several times.

Well, the 5 numbers outside the circle 1-5 are the easiest to explain. They mark the 5 octaves this tone circle covers.

Not per say related or intended, but 5 octaves = 5 x 12 tones = 60 tones. There are 60 seconds in a minute and 60 minutes in an hour.

Perhaps that’s why some would refer to this circle as a “clock”. There is nothing in this drawing though that suggests this to be one of the reasons for the design of this circle.

Inside the circle you notice a sequence of numbers 7-6-5-4-3-2-1-2-3-4-5-6-7 and reversed 1-2-3-4-5-6-7-6-5-4-3-2-1, apparently showing you the chromatic (semitone) relationship between the tones listed in both inner and outer ring when combined in one. The 1’s (C) and 7’s (F#) are a Tritone (six “spaces” between the lines) apart from each other. This might suggest a so called “Tritone Substitution“.

A Tritone substitution is one of the most common chord substitutions used in Jazz and is the foundation for more complex substitution patterns like Coltrane changes. Other examples of the tritone substitution (known in the classical world as an augmented sixth chord) can be found in classical music since the Renaissance period. The Tritone substitution can be performed by exchanging a dominant seven chord for another dominant seven chord which is a Tritone away from it.

In the Coltrane Circle you see a sequence from 1-7 starting from C (top of the Circle) to F# both clockwise and counterclockwise. Could that suggest a substitution of C⁷ by F#⁷?

If you have another (perhaps better) idea about this sequence, please do contact me.

FLOWER OF LIFE (61)

As mentioned above, the Coltrane Circle covers 5 octaves = 5 x 12 tones = 60 tones within 1 circle. That number reminded met of (an extended version of) the Flower of Life, that contains 60 circles drawn around/over 1 circle in the center (61 in total).

The Flower of Life is a geometric pattern grid of repeating, overlapping circles of an equal radius in two-dimensional space. Commonly, designs are based on circles centered on triangles (with the simple, two circle form named vesica piscis) or on the square lattice pattern of points. The Flower Of Life symbol is one of the most known and recognized geometric Sacred Geometry symbols. This special symbol represents the cycle of life. It visualizes that all consciousness arises from one source (the first, center circle). The 5 platonic solids are found within Flower Of Life, as well as many others including the Seed Of Life, Tree Of Life, and Metatron’s Cube just to name a few. These shapes act as building blocks for all living things, starting with the very first circle. There are many variations of the Flower Of Life, some having as little as only seven circles.

When you place the Flower of Life over the Coltrane Circle you can see the fit nicely together. The outer circles and crossings of circles align with the trigons C-E-Ab & D-Db-Gb, als well as with the trigons G-B-Eb & F-A-Db, 12 tones that together form 2 Hexagons.

Simplified Coltrane Circle + Flower of Life (overlay)

Coltrane Circle + Flower of Life (overlay)

Perhaps it is “coincidence” that the 60 around 1 circle (61 circles) Flower of Life aligns with Coltrane’s tone circle with 5 x 12 = 60 tones within 1 circle (61 circles), but as mentioned before, Coltrane’s interest in mathematics, philosophy and the occult might have played a role here too … perhaps not, we will never know for sure.

Another funny coincidence is that Coltrane drew his tone circle in 1961 as mentioned earlier in this article.

JOHN COLTRANE’S MUSIC & GEOMETRY

If you find this article interesting, you might like to read the Roel’s World article “John Coltrane’s Music & Geometry” as well. In this article I write a bit more about the relationship between Coltrane’s music and it’s mathematical / geometrical interpretation.

To finish this article with I like to share a “music video” of Coltrane’s piece “11383” with the Coltrane Tone Circle used as base/inspiration for the visualization. Note: the visualization of the Coltrane Circle does not accurately follows the music – as becomes obvious later on in the video – but is nonetheless a nice ‘work of art’.

You can watch this video on Facebook as well.

REFERENCES MENTIONED IN THIS ARTICLE:

“Coltrane’s Way Of Seeing” by Corey Mwamba
“Yusef Lateef official website“

Blog » The Astrological Zodiac & Musical Tonality – Tone Zodiac – Roel’s Interpretation

Reading Time: 16 minutesJanuary 17, 2016

The Astrological Zodiac & Musical Tonality – Roel’s Interpretation

This article is an addition / continuation of the article ‘The Astrological Zodiac & Musical Tonality – Tone Zodiac‘ and contains my personal interpretation of the relationship between the Astrological Zodiac and Musical Tonality …

I am not an astrologer or astronomer (I am but a musician-composer and sound engineer with a keen interest in this subject). This article is in the first place written for musicians that – like myself – are intriqued by this topic and perhaps for some astrologers it might turn out to be an interesting read as well.

FOOTNOTES:

The seasons mentioned in this article are as the are at the Northern hemisphere (North of the Equator).
The exact dates that cover the periods of the Zodiac Signs might differ, that depends on the system used.
All dates, numbers and frequencies mentioned in this blog article are an approximation.

With some concepts (such as Kepler’s Tone Zodiac) the astrological, astronomical and/or mathematical values have been the starting point to build a Tone Zodiac. This sometimes mend that music came ‘second’ in order to make a concept work. Important to me though is to be sure the functions of the intervals, the meaning of tones, scales and tonalities are preserved and not “re-interpreted” to prove an otherwise mathematical correct concept with.

In this next section of this article I will go more into detail about the tone-sign combination that I think is the most cogent.

WHAT CIRCLE TO USE AND WHAT TONE TO START WITH?

THE CHROMATIC CIRCLE OR THE CIRCLE OF FIFTHS?

WHAT CIRCLE TO USE AND

The Chromatic Circle (CC) and the Circle of Fifths (CoF) have a different following order of tones so the result when aligning them with the Zodiac circle will differ partly.

The Chromatic Circle is related to the 12-Tone Equal Temperament. This Temperament is the modern standard and most modern instruments were build for it. An equal-tempered chromatic scale is a nondiatonic scale having no tonic because of the symmetry of its equally spaced notes. Each tone represents 100 cents (1/12th) of the total of 1200 cents between Tonic and Octave. The Circle of the Zodiac Signs is divided into 12 sections of each 30 degrees (1/12th of 360 degrees).

The Circle of Fifths is related to the Pythagorean Temperament. This Temperament is based on the “Perfect Fifth” (the 3rd Harmonic of the Harmonic Series and a consonant interval), an important tone in music that leads you (the listener) to the Tonic (root). The tone G leads to C, like D leads to G.

From an esoteric-philosophical point of the view we can imagine the Fourth and Fifth to be the border between the Earthly (tonic, seconds and thirds) and the Cosmic or spiritual (sixth, seventh and octave) realms. The “function” of the Perfect Fifth – being on the Cosmic side of the “border” – is to bring the Cosmic into the Earthly realm. You can read more about the function of the intervals in my blog article “The Function Of The Intervals” that includes fractions (quotes) of Rudolf Steiner‘s lectures given on the 7th and 8th of March 1923 (Stuttgard), the 16th of March 1923 and 2nd of December 1923 (Dornach).

WHAT TONE TO START WITH?

To compare the CC with the CoF, we need to start from some tone.

In most cultures Spring and “springtime” refer to ideas of rebirth, rejuvenation, renewal, resurrection, and regrowth. It is the starting point for what still has to become. So, what we are looking for is a tonality that is “neutral”. The tonalities without sharps or flats are C Major and it’s parallel A minor. Since we now focus on the Major tonalities, C is the tone we relate to Aries and thus start with.

So, lets see what we get when we start with both Circles from C.

Chromatic Circle:

SIGN

TONE

C♯/D♭

D♯/E♭

F♯/G♭

G♯/A♭

A♯/B♭

♯ & ♭

7♯/5♭

2♯/10♭

9♯/3♭

4♯/8♭

11♯/1♭

6♯/6♭

1♯/11♭

8♯/4♭

3♯/9♭

10♯/2♭

5♯/7♭

Circle of Fifths:

SIGN

TONE

F♯/G♭

C♯/D♭

G♯/A♭

D♯/E♭

A♯/B♭

♯ & ♭

1♯/11♭

2♯/10♭

3♯/9♭

4♯/8♭

5♯/7♭

6♯/6♭

7♯/5♭

8♯/4♭

9♯/3♭

10♯/2♭

11♯/1♭

Some tones correspond to particular signs (Aries, Gemini, Leo, Libra, Sagittarius and Aquarius = the “Masculine” signs) in both circles. The Hexatonic Scale or Whole Tone Scale of C is what both circles have in common: C – D – E – G♭ (F♯) – A♭ (G♯) – B♭ (A♯) – C.

The other 6 signs (Taurus, Cancer, Virgo, Scorpio, Capricorn and Pisces = the “Feminine” signs) correspond to the other 6 tones (B – D♭ – E♭ – F – G – A – B) but the tone-sign relationship is different with both circles..

Interesting to note is that these two scales belong to signs that are in “opposition” a subject (polarities) that I will write a bit more about later in this article. In the example above, the Hexatonic Scale of G belongs to Taurus at the CoF and Scorpio at the CC, while the Hexatonic Scale of D♭ belongs to Scorpio at the CoF and to Taurus at the CC.

Now we have looked at both Tone Circles, we still can’t say with certainty which one is the right one to use when relating the Zodicac Signs to Tonalities. We need to “broaden our horizon” and look to the “bigger picture” to get an answer to that question.

THE TROPICAL YEAR AND THE CYCLE OF THE SEASONS.

Before we continue with the tone circles, lets take a closer look at the Tropical Year and the cycle of the seasons first.

GENERAL INFORMATION
A Tropical Year (also known as a Solar year) is the time that the Sun takes to return to the same position in the cycle of seasons, as seen from Earth; for example, the time from vernal equinox to vernal equinox, or from summer solstice to summer solstice. The solar year is divided in 4 seasons. A season is a division of the year, marked by changes in weather ecology and hours of daylight. During May, June, and July, the northern hemisphere is exposed to more direct sunlight because the hemisphere faces the sun. However, due to seasonal lag, June, July, and August are the hottest months in the northern hemisphere.

ARE THERE ANY SIMILARITIES BETWEEN THE TONE CIRLCE AND SEASONS?

In music the addition of sharps and flats is generally seen (and experienced) as an intensification of emotion or the “mood” of the music. Can the changes in weather, ecology and hours of daylight (and how this effect life on Earth) be represented by the key signature (sharps and flats) in music?

! NOTE: I am not the first person who looks at this subject in this manner. Prof. Dr. Hermann Beckh has done so long before me in his work “Das geistige Wesen der Tonarten” (The Spiritual Character of Musical Keys – 1923) and “Die Sprache der Tonart in der Musik von Bach bis Bruckner…” (The Language of the Art of Keys in Music from Bach to Bruckner… – 1937), as referred to by Sigismund von Gleich in his work “Über die Wirkung der Tonarten in der Musik” (About the Functioning of the Tonalities in Music – 1949).

When following that idea, the next sign Taurus (following Aries) should have an intensification of “emotion” relatively to the our starting point C Major, and we do so by adding one sharp. This sets Taurus to the key of G Major (1♯) a Fifth above the C of Aries. If we would use the Chromatic Circle, then C♯/D♭ with 7♯/5♭ should be connected with Taurus, an illogical step up from C.

You might ask, but what if we intensify “the mood” by adding 1 flat instead of 1 sharp? Well, the the next tone after C would then be the F Major (1♭). The interval between the C and (to) the F is a Fourth. The Major tonalities with flats though, are generally perceived as more “pessimistic” then Major tonalities with sharps, that are perceived as “optimistic“. It seems thus more logical to choose for G Major (1♯) as the next step from Spring (when everything starts to grow and blossom) towards Summer then F Major (1♭).

Now our question about what circle to use has been answered! The Circle of Fifths!

I will keep going up in Fifths, increasing the number of sharps, we pass D (2♯), A (3♯), E (4♯), B(5♯) and finish you increase of sharps at F♯. F♯(with 6 sharps) is the enharmonic equivalent of G♭ (with 6 flats), 6 semitones or a “Tritone” above the C. The Zodiac Sign that represents both (F♯ and G♭) is Libra (in English translated as “Scales”).

Is it coincidence that Libra, the Zodiac sign that features a “desire for balance” aligns exactly with the tonic of the ONLY tonality that has an equal number of sharps and flats? I think not!

Libra is the first sign in Autumn from where the signs are changing from sharp to flat and with it changing it’s “mood” and “livelyness”. Now with the tone for tone decrease of flats – D♭ (5♭), A♭ (4♭), E♭ (3♭), B♭ (2♭) and F (1♭) – the intension decreases, returning to C “natural” to start the cycle over again.

COMBINING THE CIRCLE OF FIFTHS AND THE ZODIAC CIRCLE

When we look at the Earth from far above the North Pole, the Earth moves counterclockwise around the Sun. Thus, if we look at the circles bellow, we should start with Aries in the “Circle of Zodiac Signs” and go counterclockwise around, with Taurus coming next. Naturally if you would look at Earth from below the South Pole (Southern Hemisphere), then Earth moves clockwise around the Sun. Then we start with Aries and find Taurus coming next to the right. Many Zodiac circles you find online are “drawn” clockwise. But, since I am blogging from the Northern Hemisphere, I have decided to describe the Zodiac circle as it would be correct in my position.

With the “Circle of Fifths” we start at the tone “C” but go clockwise around (as is common for tone-circles), with “G” next up after “C”.

So, in relation to one another, we follow these two circles in opposite direction!

Now if we like to display the circles below completely accurate (to what is common in Astrology), then we should turn both circles 90°counterclockwise so both Aries and “C” will start from the left-side of the screen (eastern horizon). But since most musicians are used to the Circle of Fifths with the “C” on top, I have chosen for this article to “align” Aries with “C”, instead of “C” with Aries.

Just some “fun-facts” for number-fetishists: if we would start at “C” and go counterclockwise round the “Circle of Fifths“, we would actually use the “Circle of Fours“. The number 4 is a number connected to the Zodiac as well. The Zodiac is divided into 4 groups of 3 signs, that belong to the 4 elements (Fire, Water, Air & Earth). The interval “The Fourth” sounds at the 4th Harmonic and the 3rd Overtone of the Harmonic Series.

Below the Zodiac Circle and Cycle of Fifths together in one table:

Symbol	Long.	Name	Period (Tropical)	Season	Key
	0°	Aries	21 Mar. – 20 Apr.	20.03 Sping Equinox*	C	0
	30°	Taurus	21 Apr. – 21 May		G	1♯
	60°	Gemini	22 May – 21 Jun.	21.06 Summer Solstice*	D	2♯
	90°	Cancer	22 Jun. – 22 Jul.	04/06.07 Aphelion	A	3♯
	120°	Leo	23 Jul. – 22 Aug.		E	4♯
	150°	Virgo	23 Aug. – 23 Sep.	23.09 Autumn Equinox*	B	5♯
	180°	Libra	24 Sep. – 23 Oct.		F♯/G♭	6♯/6♭
	210°	Scorpio	24 Oct. – 22 Nov.		Db	5♭
	240°	Sagittarius	23 Nov. – 21 Dec.	21.12 Winter Solstice*	Ab	4♭
	270°	Capricorn	22 Dec. – 20 Jan.	04/06.01 Perihelion	Eb	3♭
	300°	Aquarius	21 Jan. – 19 Feb.		Bb	2♭
	330°	Pisces	20 Feb. – 20 Mar.		F	1♭

* In a leap year the solstice moves back one day, in Spring from the 21st to the 20th of March and in Winter from the 22rd to the 21th of December. The Autumn Equinox also switches between the 23rd and the 22nd of September. The Spring Equinox always takes place on the 20th.

MINOR TONALITIES, the CICLE OF FIFTHS and the SOLAR YEAR.

In music all Major tonalities have a parallel tonality. The parallel tonality of C Major is A minor, so it is most logical to relate both C Major and A minor with Aries. But, is there a way we can explain this based upon the position of Earth in relationship to the path it takes around the Sun during the Solar Year?

For this we should look at the moment that the Earth if closest to the Sun (Perihelion) and furthest away (Aphelion). The word “perihelion” stems from the Ancient Greek words “peri”, meaning “near”, and “helios“, meaning “the Sun”. “Aphelion” derives from the preposition “apo”, meaning “away, off, apart”.

Opposite to the distance from the Sun is the intensity of the sunlight on the Northern hemisphere. When Earth is furthest away from the Sun, the intensity on the Northern hemisphere is highest.

Due to the earth’s tilted position (on it’s axis) the Northern Hemisphere “leans” towards the Sun during summertime. The Northern Hemisphere is therefor largely exposed to the sunlight while the Southern Hemisphere is only partly exposed to the sunlight. In the winter we see the opposite. The Northern Hemisphere “leans” away from the Sun. The Southern Hemisphere is then for the greater part exposed, while the Northern Hemisphere is only partly exposed to the sunlight.

Northern Hemisphere Summer Southern Hemisphere Winter	Autumnal (Fall) Equinox Vernal (Spring) Equinox	Northern Hemisphere Winter Southern Hemisphere Summer

As you can see from the images above the place where the Sunlight shines upon Earth most directly (the “overhead sun”) changes: from the Equator (in Autumn and Spring) to the Tropic of Cancer (June – Aphelion) and the Tropic of Capricorn (January – Perihelion). If you wish you could watch this “movement” in an youtube video.

If we place the tonic of the most “intense” Major tonality (the F♯) and place it at the Aphelion (Cancer), and place the tonic of the least “intense” Major tonality (the C) at the Perihelion (Capricorn), then we find the A at Aries.

Below the whole cycle put together in one table:

Symbol	Long.	Name	Period (Tropical)	Season	Key
	0°	Aries	21 Mar. – 20 Apr.	20.03 Sping Equinox*	Am	0
	30°	Taurus	21 Apr. – 21 May		Em	1♯
	60°	Gemini	22 May – 21 Jun.	21.06 Summer Solstice*	Bm	2♯
	90°	Cancer	22 Jun. – 22 Jul.	04/06.07 Aphelion	F♯m	3♯
	120°	Leo	23 Jul. – 22 Aug.		C♯m	4♯
	150°	Virgo	23 Aug. – 23 Sep.	23.09 Autumn Equinox*	G♯	5♯
	180°	Libra	24 Sep. – 23 Oct.		D♯m E♭m	6♯/6♭
	210°	Scorpio	24 Oct. – 22 Nov.		B♭m	5♭
	240°	Sagittarius	23 Nov. – 21 Dec.	21.12 Winter Solstice*	Fm	4♭
	270°	Capricorn	22 Dec. – 20 Jan.	04/06.01 Perihelion	Cm	3♭
	300°	Aquarius	21 Jan. – 19 Feb.		Gm	2♭
	330°	Pisces	20 Feb. – 20 Mar.		Dm	1♭

THE 12 ZODIAC SIGNS WITH THEIR MAJOR & MINOR TONALITIES


C	G	D	A	E	B	G♭	D♭	A♭	E♭	B♭	F
Am	Em	Bm	F♯m	C♯m	G♯m	E♭m	B♭m	Fm	Cm	Gm	Dm

TONE ZODIACS STEINER & VON GLEICH

I am not the first that comes with the particular combination of tones and Zodiac signs as described above be most suitable. If you have read the historical time-line about Tone Zodiacs in the beginning of this article then you have already seen two Tone Zodiacs that match the tone-sign combination as I have described it above.

RUDOLF STEINER	SIGISMUND VON GLEICH

I would not call myself an “anthroposophist” (if I would have to label myself, then “Pantheist” would be the term I could use), but some of the work of Rudolf Steiner (and other anthroposophists) I have read usually turn out to be a pretty good match with how I see and experience music, the world and the universe.

COLOR SPECTRUM, TONE/TONALITY (CHROMATICALLY) AND THE ZODIAC SIGNS

On the left you see the combination of Tones, Zodiac Signs and colors all in one Tone-Zodiac. This Tone-Zodiac combines the Chromatic Circle with the Zodiac Signs placed as with the Circle of Fifth tone-sign combination to maintain the natural spectrum of the visible light and wavelengths, from approx. 6870 Ångström (Red) to 4090 Ångström (violet).

The order of the Zodiac Signs around the circle might be a bit “uncommon” with this Tone-Zodiac. The difference with the common Zodiac Circle is that two opposing signs of each square have swapped places (Taurus & Scorpio, Capricorn & Cancer and Virgo and Pisces). The Earth signs swapped places with the Water signs.

When “drawing lines” between the consecutive Zodiac Signs, a Dodecagram _✹ appears. Within this Dodecagram the “grouping” of the signs (in Polarities, Triplicities, Quadruplicities and Hexagon) applies.

The Hexagon does not represent Fifths and Fourths in this configuration (as it does when it is applied to the Circle of Fifths), but whole tone distances. The Whole-Tone Scale is also known as the “Hexatonic Scale“. The Dodecagon visualized the Chromatic Circle in this Tone-Zodiac, but would represent the Fifths and Fourths in the Circle of Fifths. The Fifths and Fourths are “linked” in this Tone-Zodiac by the lines between the points of the Dodecagram.

For this Tone-Zodiac I have used the same color-circle twice, shifted 90 degrees (a “trigon” or minor third). The “character” of Major and minor tonalities is different, so Major and minor tonalities should be represented by a different color. The “tonic” or “root tone” is the first scale degree of a tonality and determines the actual color. The Major tonality relates to the extraversion and the minor tonality to the introversion of the human personality in correspondence with the related Zodiac Sign.

Now, some of you (reading this blog) that are “into” Astrology might say: “Hey, isn’t the color for Aries Red instead of Green?” Yes, Red is the color that traditionally represent the character of Aries. So, why is Aries listed above as Green then?

Well, the tones above follow the color spectrum as we see it in Nature. In nature: the bigger the wavelength = the lower the frequency. We know that the color of the lowest frequency is Red. When we convert color to sound frequency we find the tone G in Red. If G = Red and we align the Circle of Fifths with G = Red, then Green aligns with Aries.

MAJOR COLORS (appearance to the “outside world” or “extraversion“):
G	G♯	A	A♯	B	C	C♯	D	D♯	E	F	F♯
MINOR COLORS (“inner world” or “introversion“):
Gm	A♭m	Am	B♭m	Bm	Cm	D♭m	Dm	E♭m	Em	Fm	G♭m
_{(ALL COLORS WERE CALCULATED USING A4=432HZ AT 16 DEGREES TEMPERATURE AND 45% RELATIVE HUMIDITY)} Read more about sound and color in the article “Light & Sound, Colour & Music”.

Red isn’t the only color mentioned in various books and blogs that belong to Aries. Orange is often suggested as well. The “outer world” of Aries could be represented with C Major (Green) while the “inner world” of Aries could be represented with A minor (Red-Orange). Most colors used in the Tone Zodiac will not match those traditionally associated with Zodiac Sings in most books and blogs. Taurus (Pink:Red+White/brightness = G Major and Blue = E minor) and Sagittarius (Red = G♯/A♭ Major and Violet = F minor) do match as listed. Virgo (Moss Green = B Major), Aries (Orange = A minor), Gemini (Green = B minor) and Libra (Blue = D♯/E♭ minor) do only partly match. The rest does not match with the traditionally ascribed colors.

Various colors traditionally associated with some Zodiac Signs are not part of the natural color spectrum. Capricorn and Scorpio for example are associated with Dark Brown and Black, Cancer with White, Grey and Silver, and Virgo with Grey. Another reason why some of the colors don’t match as well is because some colors are contributed to the Zodiac Signs based upon the Planet that “rules” the sign. With other words colors associated with the Planet. A “perfect match” between Zodiac Sign, tone/tonality and color seems to be impossible.

For me personally though the natural color-tone connection is a logical (scientifically substantiate) one and the Zodiac Signs-tone connection as I described in this article is logical as well. The partial color mismatch with the traditional color-sign combination is – for me personally – of lesser importance.

THE ASTROLOGICAL ZODIAC & MUSIC TONALITY – ROEL’S INTERPRETATION

This is the main article about “The Astrological Zodiac & Music Tonality” – Tone Zodiacs.

ASTRO-MUSIC COMPOSITION

This is an article with some ideas about how to create a “tone-collection” based on your astrological (birth) chart that can be used for composing music: “Astro-Music Composition (chart to music)“.

THE ASTRONOMICAL ZODIAC SCALE

If you like my article about the Astrological Zodiac & Tonality and you are interested in Micro-tonal music, then you might also like the article “The Astronomical Zodiac Scale (Temperament)“.

THE FUNCTIONS OF THE INTERVALS

Another article I wrote for my blog named “The Functions Of The Intervals” (referred to in this article) describes might be an interesting read as well?

SOURCES / REFERENCES:

Über die Wirkung der Tonarten in der Musik” (by Sigismund von Gleich)
www.flutopedia.com – Color of Sound calculator
http://wiki.astro.com/astrowiki/ (AstroWiki)
http://en.wikipedia.org/wiki/Zodiac (Wikipedia)
http://en.wikipedia.org/wiki/Seasons (Wikipedia)
http://en.wikipedia.org/wiki/Planets_in_astrology (Wikipedia)

Blog » Tempo & Pitch – Bpm & Cps – Rhythm & Harmony

Reading Time: 4 minutesJanuary 9, 2015

TEMPO & PITCH – RHYTHM & HARMONY

The idea of an assumed relationship between Bpm (Beats Per Minute) and Cps (Cycles Per Second, a once-common English name for the unit of frequency now known as the hertz) is – in my opinion – a valid one.

Both Bpm and Cps are a: “particular number of occurrences + repeating event + time unit” (you could call this “periodicity”).

Some composers / producers do actually use a tempo that ‘synchronizes’ with the tonality of the composition, this idea isn’t something “new”. Temperament does have an effect as well …

TEMPO & PITCH: BPM & CPS (HZ)

In the case of Bpm the time unit is minutes, in the case of Cps the time unit is seconds.

From Cps to Bpm we need to multiply the number of cycles by 60.
From Bpm to Cps we need to devide the number of beats by 60.

EXAMPLE CPS TO BPM:
The tone C4=256Hz has 256 Cycles Per Second. In order to find a matching tempo (Bpm) we have to multiply 256 by 60 (turning second into minutes). 256 Cps · 60 = 15360 Cpm. Naturally 15360 Bpm would be too fast to use for any musical piece, so we have to halve the tempo until we get a usable tempo.

15360 /2/2/2/2/2/2/2 = 120 Cpm or Bpm
15360 /2/2/2/2/2/2/2/2 = 60 Cpm or Bpm

What if you find 60 Bpm or 120 Bpm too slow or fast for your composition?

Well, perhaps it would be worth exploring other tone frequencies of importance in your composition, for example the 3rd or 5th. Depending on the tonality (Major or minor) and Temperament used you create the following tempi with the 3rd and 5th, based in this example on Concert Pitch C4=256Hz:

Scale Degree	CPS (Hz)	BPM	rounded-up
Equal Temp. minor Third	304.4370214406993	142,7048538 or 71,3524269	143 or 71
Just Intonation minor Third	307,2	144 or 72	144 or 72
Equal Temp. Major Third	322.5397887730913	151,190525987 or 75,595262994	151 or 76
Just Intonation Major Third	320	150 or 75	150 or 75
Just Fifth	384	90	90

Naturally you could select any tones you feel is important in your composition. If your melody has one tone that is used more often then any other, then you could experiment with adjusting the tempo to that tone as well.

EXAMPLE BPM TO CPS:
If we like to use 130 bpm as tempo for a piece of (dance) music, what tone frequency would be “in sync”?. In order to find this tone, we have to divide 130 by 60 (turning 130 beats per minute into beats per second).

130 Bpm / 60 = 2,1666666667 Bps.

This tone of 2,1666666667 Cps (Hz) is too low to use as reference pitch (concert pitch) so we have to double it untill we reach a frequency between C4 and C5.

2,1666666667 ·2·2·2·2·2·2·2 = 277,333333338Hz

The tone in this example is actually pretty much the C♯4 or D♭4 (277.183Hz) when A4=440Hz and Equal Temperament is used.

If you would like to use the Bpm to Cps method, do keep in mind that you might have to use a different Concert Pitch then the standard 440Hz. This is important to know, because not all instruments are capable of changing pitch.

RHYTHM, TEMPO & PITCH

If you take a basic “4 on the floor” kick drum pattern and speed it up, a tone will appear. The higher the tempo, the higher the pitch. Adam Neely shows this in his youtube video (below).

In the article so far we only looked at converting a single tone to tempo (Cps to Bpm). Melody (and harmony) – generally only looked at from the point of pitch and ratio – does have a rhythmic aspect as well. Rhythm, melody and harmony are kind of the same thing, just on a different scale.

THE POLYRHYTHMIC ASPECTS OF HARMONY

Harmony can be seen as complex interlocking polyrhythm. When you play two different rhythmic patterns, for example 5 equally spaced rhythmic elements in one measure together with 4 equally spaced rhythmic elements and speed it up, a Just Major Third appears. This way you can actually create complete chords (see video below).

You might wonder why I specifically mention Equal Temperament and Just Intonation? Well, there is a good reason for that. The difference between 12-TET and Just Intonation is more then just pitch. As mentioned in other articles on Roel’s World, with 12-Tone Equal Temperament all intervals are slightly “off pitch” in comparison with the harmonic series (except for the octave of course). I will not go into detail about ratios and temperament in this article, but it is important to “address” this.

DIFFERENCE E.T. AND J.I.

In the video on the left Adam Neely let us hear the difference between a Just and an Equal Tempered Major triad. With the Just triad the “beats” of the tones align. The 12-TET chord though does not align perfectly, creating a phaser-like effect.

In an example given by Adam Neely of the Major Thirds, the ratio of the Equal Tempered Major Thirds is 2^4/12, while that of the Just Intonated Major Third is 5:4. and the ratio of the Equal Tempered Minor Thirds is 2^3/12, while that of the Just Intonated Minor Third is 6:5.

The blue tracks are (from top down) the Tonic, 3rd and 5th of the Equal Tempered Major chord, the green tracks those of the Just Intonated Major chord.

All waves of the tones of the Just Intonated chord (green tracks) align (get in sync) at particular moments (where the white line is drawn). This is though not the case with the waves of the tones of the Equal Tempered chord. The 3rd end too early and the 5th a tiny bit too late relatively to the Tonic. They never get completely “in sync” with one another.

I suggest you take some time to watch the complete video:

CONCLUSION:

Generally speaking: the more “synchronized” all “elements” within the music are, the more “harmonious” and “natural” it sounds to the listener.

If you like to create music that is rhythmically, harmonically and melodically in “sync” with one another, then you might like to consider using Just Intonation, rather then Equal Temperament.

With other words: calculate a tempo to “match” the tonality of the composition AND make sure that the harmony and melody themselves synchronize with the rhythm and tempo of the piece as well.

If you like to covert Cps to Bpm yourself, then try the online Cps to Bpm converter at www.convertworld.com

Blog » The Function Of The Musical Intervals

Reading Time: 12 minutesOctober 5, 2014

THE FUNCTIONS OF THE INTERVALS (A PHILOSOPHICAL VIEW ON)

“I interweave my earthly existence with the divine-spiritual world, and, on the wings of the tone structure, the gods move through the world. I participate in their moving when I perceive the tones.” ~ Rudolf Steiner

FORWORD
There have been various composers and musicians as well as some philosophers (like Rudolf Steiner), that recognized a relationship between the intervals and how the development of mankind in a changing world is being perceived.

“The facts of human evolution are expressed in musical development more clearly than anywhere else.”
(Rudolf Steiner)

So, how do we “relate” music to the world (universe) around us and the experiences we go through ourselves?

I think most of you reading the article do feel the difference between the Major and minor thirds. The minor third feels more “introvert” (Outer World to Inner World movement), while the Major third feels more “extrovert” (Inner World to Outer World movement). We do experience this as well when listening to pieces in minor and Major tonalities, although the level of “internalization” or “outgoingness” might differ. But, what about the other intervals? How do they relate to what we experience?

I the past I always wondered how to formulate my thoughts and feelings about it. Several years ago my mother gave me a book that contained various lectures by Rudolf Steiner. In these lectures I found the answers I was searching for.

I will share my thoughts (personal interpretation) about my experiences and what I have read, supported by fractions (quotes) of the explanation of this given by Rudolf Steiner from lectures given by Steiner on 7 and 8 March 1923 (Stuttgard), 16 March 1923 and 2 December 1923 (Dornach).

FUNCTIONS OF THE DEGREES OF TONES IN THE DIATONIC SCALES

In common music theory the function of each degree in diatonic scales is – in relationship to the Tonic – described as following:

1st: Tonic
2nd: Supertonic
3rd: Mediant
4rd: Subdominant
5th: Dominant
6th: Submediant
♭7th: Subtonic
7th: Leading Tone
8th: Octave

Most musicians and composers know (or should know) this. In case you don’t, then click on the functions listed above to read more about then on Wikipedia.

A PHILOSOPHICAL VIEW ON THE INTERVALS

From a more esoteric-philosophical point of view we can look at the divisions within the octave as a representation of the both physical and spiritual nature of man, man’s “Inner World” and “Outer World“, as well as the Earthly or physical and Cosmic or spiritual relationship. Perhaps even the expression “AS ABOVE, SO BELOW” could “represent” this division.

“THE ABOVE“

THE OCTAVE:

The octave represents the “higher self”. The Higher self is generally described as the eternal, omnipotent, conscious, and intelligent being, who is one’s real non-corporeal self.

Rudolf Steiner about the Octave:

“Every time the octave appears in a musical composition, man will have a feeling that I can only describe with the words: “I have found my ‘I’ anew; I am uplifted in my humanity by the feeling for the octave.” We do not know it fully, however, we know it only through feeling. Finally, the feeling for the octave brings us to find our own self on a higher level.

THE 7TH’S:

The ♭7^th(minor 7^th) is called the Subtonic. The Subtonic is found a whole tone below the Octave (and Tonic, thus the reason for being called Sub-Tonic). The Subtonic and Supertonic (Major 2nd) “flank” the Tonic (and Octave, although in this case the Supertonic is called a “9th”) at equal distance, both two semitones apart from the Tonic.

The (Major) 7th is the leading tone to the octave, the “higher self”.

Rudolf Steiner about the 7ths:

“One who fully experiences sevenths knows what intuition is. What I mean is that in the experience of the seventh the form of the soul’s composition is the same as clairvoyantly with intuition.

This musical experience, which was based on an experience of the seventh through the full range of octaves, always consisted of man feeling completely transported [entrückt]. He felt free of his earthbound existence and transported into another world in this experience of the seventh. At that time he could just as well have said, “I experience music,” as “I feel myself in the spiritual world.” This was the predominant experience of the seventh.

As the human being wished to incarnate more deeply into this physical body and take possession of it, the experience of the seventh became faintly painful.”

THE 6THS:

The 6ths do play a similar role as the 3rds, but then (mirrored) on the “other side of the border” and relate to the cosmic or “spiritual” realm.

Rudolf Steiner about the 6th:

“One who experiences sixths knows what inspiration is. The form of the soul’s composition during the experience of the sixth is that of inspiration with clairvoyance.“

THE 5TH:

The 5^th is the first interval (when stated “counting” from the tonic going up) that does not “belong” to the “earthly realm”.

My personal experience with it is that the Descending Fifth brings the cosmic realm into the earthly realm. The Ascending Fifth “lifts” a human out of it’s lower self, as Steiner mentioned in his lectures. I come to that conclusion based on the sensation that the descending 5^th generates with it’s attraction towards the 1^{s t}(Tonic). It “brings us (back) to our “lower self”, crossing the “border” between the “Cosmic realm” and the “Earthly realm”. The descending 5th can be seen as the “descension” or the “expansion” from the “spirit” into the “earthly” as well.

Rudolf Steiner about the 5th:

“The interval of the fifth is a real experience of imagination. He who can experience fifths correctly is actually in a position to know on the subjective level what imagination is like. The experience of the fifth is a real imaginative experience. The same composition of soul need only be filled with vision. Such a composition of soul is definitely present in the case of music.

In the music of the fifths [Quintenmusik], a human being felt lifted out of himself.

The experience of the fifth brings awareness of man within the divine world order. The experience of the (add: ascending) fifth is, as it were, an expansion into the vast universe.

What does this really mean in relation to the whole musical experience? It means that within the experience of the (add: ascending) fifth, man with his “I” is in motion outside his physical organization. He paces the seven scales (add: upwards) in twelve steps, as it were. He is therefore in motion outside his physical organization through the experience of the (add: ascending) fifth.

This becomes more obvious if we take the scales through the range of seven octaves — from the contra-tones up to the tones above C — and consider that it is possible for the fifth to occur twelve times within these seven scales. In the sequence of the seven musical scales, we discover hidden, as it were, an additional twelve-part scale with the interval of the fifth.

In the age when the fifths predominated, it was impossible to color music in a subjective direction. Subjectivity only came into play in that the subjective felt transported, lifted into objectivity.”

“THE BORDER“

There are two intervals (sonically one) that – depending on the interpretation – could be seen as “the border” between the “The Above” and The Below”.

THE TRITONE, THE “IN BETWEEN”.

The augmented 4rd or diminished 5th, also called “Tritone” (TT or 3T), is (to me) the border between the Cosmic and the Earthly, the Octave (higher self) and Tonic (lower self).

It is surrounded with some “mystery” and has been called “Diabolus in Musica” (“the devil in music”). It is often presumed that it has received this name due to it being the most dissonant, least-harmonic interval in the 12-Tone system.

Most intervals “lead” you from one tone to the other. The most obvious example is the 5th that leads you to the Tonic. With all (but two intervals) the direction you are led to (ascending or descending) is clear. Two intervals though do not seem to “point” in any direction when played simultaniously are the Octave and the TriTone. The Tonic and Octave when played together sounds to most people as a single tone. The TriTone does sound as two different tones when played simultaneously, but most people feel like there is no movement … it sounds kind of “dead in space”.

You might expect that this most dissonant c.q. least harmonic interval is not very “useful” for mainstream music? The opposite is true. One of the most used chords, the Dominant 7th Chord, contains an embedded Tritone: the interval between the 3rd and the 7th of the chord. Most people (besides musicians, composers, studio engineers) are probably not even aware they are listening to a chord with the “Devil’s Interval” embedded within.

Another chord – but less common in mainstream (pop) music – with a Tritone embedded within, is the Diminished Triad. The Diminished triad contain two Minor Thirds stacked with the tonic and 5th a TriTone (6 semitones) apart. With this chord you do hear the dissonance created by the TriTone interval well though. The interesting thing about this chord, is that the tones of this chord do form half (two legs of) a Square between tones of the Circle of Fifth as well as the Chromatic Circle. If you like to know more about Geometric Shapes and music, then you might like to read “Music & Geometry“.

“DIABOLUS IN MUSICA” (Devil in the music).
Many presume the interval got this name due to it’s dissonance. Some historians claim the interval was “forbidden” to be used by the Church. But why? My theory (if the historians are right):

As the Tritone is the border (and a point of reference) between the “heavenly” and “earthly” realms, it sets these realms “apart” from one another. If you want do “differentiate” between these realms, to determine and experience both your physical and spiritual self, you need to be aware of this border.

Everyone with a “open mind” knows that – throughout history – religious institutes have always been “powerhouses” that – even though proclaiming only to be there for the “spiritual well-being” of it’s followers – had many “interests” in the physical world as well (from political relations to property and earthly wealth). By “demonizing” the border (fear tactics) and “banning” it, the only clear “reference” was no longer “available” to “the people”. The oldest “trick in the book” to “control” people is to create dependency. If people no longer can determine themselves what “belongs” to the earthly realm (to the people themselves) and what belongs to the heavenly realm (to God), then you depend on the Church to “guide” you. Without a clear “border” (Tritone), everything could “belong to God”, including your “id” (lower self – Tonic), “ego” (2nds) and “emotional life” (3rds) as well.

Of course this is just a theory I have …

THE 4TH:

The 4th can be seen as the outer “border” of the physical (corporeal / earthly) self. What is interesting to make note of, is that the 4th relates to the Octave as the 5th does to the Tonic, as if the ascending 4th is “reaching out” for what is there above the border, beyond the earthly realm, it “moves” you in opposite direction as that of the descending 5th. The ascending 4th can be seen as the “ascension” from the “earthly” into the “spiritual” as well.

Rudolf Steiner about the 4rd:

“The experience of the fourth is perhaps one of the most interesting for one who wishes to penetrate the secrets of the musical element. This is not because the experience of the fourth in itself is the most interesting but because it arises at the dividing line between the experience of the fifth of the outer world and the experience of the third in man’s inner being. The experience of the fourth lies right at the border, as it were, of the human organism.

With the experience of the fourth, man moves about, as it were, in the divine world; he stands precisely at the border of his humanness, retaining it, yet viewing it from the other side.“

“THE BELOW”

THE 3RDS:

The 3rd’s represent the (earthly) emotions and is related to the physical world. The Major generate a more “extrovert” feeling (toward the fourth), minor the more “introvert” feelings (toward the “I”). While earlier Major tonalities were more “favored” by composers, we see in the last century more composers favoring minor tonalities. In particular in the last couple of decades with the rise of Electronic Music. That we “move” from Major to minor does follow the way we “function” in the world today, in the last couple of centuries we have been moving closer to the “ego”, the individualization.

Art-wise terms like Expressionism (Major 3rd) and Impressionism (Minor 3rd) could be used as well.

Rudolf Steiner about the 3rd:

“This transition to the experience of the third signifies at the same time that man feels music in relation to his own physical organization. For the first time, man feels that he is an earthly being when he plays music.

The third guides us to our inner being.

The “I” is, so to speak, within the confines of the human organism; man experiences the interval of the third inwardly. In a manner of speaking, man withdraws his experience of the world from the cosmos and unites it with himself.

Not until man could experience the third did the subjective element feel that it rested within itself. Man began to relate the feeling for his destiny and ordinary life to the musical element.

The difference between major and minor keys appears; the subjective soul element relates itself to the musical element. Man can color the musical element in various ways. He is in himself, then outside himself; his soul swings back and forth between self-awareness and self-surrender. One therefore can say that in the case of the experience of the third the mood is one of consolidation of the inner being, of man’s becoming aware of the human being within himself.”

THE 2ND:

The Major 2nd (also called “Super Tonic”) represents what Freud called the “Super-Ego” (critical and moralizing role), the minor second represents what Freud called the “ego” (realistic part that mediates between the desires of the “id” and the “super-ego“).

The 2^nd’s (the “ego” and “super-ego”) are the connection between the “id” (Tonic) and the emotions (3rds). To me the minor 2^nd sensation it gives is similar to that of the Major 7^th, the “Leading Tone”, in opposite direction. As the Major 7^th leads us to the Octave, so does the minor 2^nd to the Tonic.

TONIC:

The tonic represents the “physicalness” of the “lower self” and the “instincts”, something Sigmund Freud called the “id” (a set of uncoordinated instinctual trends).

MANKIND, INTERVALS AND PERCEPTION

ADDITIONAL THOUGHTS SHARED BY RUDOLF STEINER:

“Even in our age, in which intellectualism wishes to tackle everything, there is a feeling that intellectualism cannot reach the musical element, because it can deal only with something for which there are outer subjects. This explains the strange fact that nowhere in the well-meant instruction of music appreciation does tone physiology (acoustics) have anything to say about the musical element. It is widely admitted that there is a tone physiology only for sounds; there is none for tones. With the means customary today one cannot grasp the element of music. If one does begin to speak about the musical element, it is thus necessary to avoid the ordinary concepts that otherwise we use to grasp our world.

By the beginning form the first tone of the octave, we have begun from the inner “I,” the physical, living, inner “I”. When I first experience my ‘I’ as it is on earth, in the prime, and then experience it a second time the way it is in spirit, then this is inner proof of God’s existence.

In primeval times, when the experience of the seventh existed — and therefore, in fact, the experience of the entire scale — man felt that he was a unified being standing on earth; at that time when he heard the seventh, he also experienced himself outside his body. He therefore felt himself in the world. Music was for him the possibility of feeling himself in the world. The human being could receive religious instruction by being taught the music of that time. He could readily understand that through music man is not only an earthly being but also a transported being.

The experience of the fifth arose, and during this time man still felt united with what lived in his breath. He said to himself — though he did not say it, he felt it; in order to express it, we must word it like that — “I breath in, I breath out”.

The musical element, however, does not live in me at all; it lives in inhalation and exhalation.” Man felt always as if he were leaving and returning to himself in the musical experience. The fifth comprised both inhalation and exhalation; the seventh comprised only exhalation. The third enabled man to experience the continuation of the breathing process within.

All this is extraordinarily important when one is faced with the task of guiding the evolution of the human being regarding the musical element. You see, up to about the age of nine, the child does not yet possess a proper grasp of major and minor moods, though one can approach the child with them. When entering school, the child can experience major and minor moods in preparation for what is to come later, but the child has neither one nor the other. Though it is not readily admitted, the child essentially dwells in moods of fifths. Naturally, one can resort in school to examples already containing thirds, but if one really wishes to reach the child, musical appreciation must be based on the appreciation of the fifths; this is what is important.

Based on all this, you find a specific explanation for the advancement from the pure singing-with-accompaniment that existed in ancient times of human evolution to independent singing.

Originally, singing was always produced along with some outer tone, an outer tone structure. [Tongebilde]. Emancipated singing actually came about later; emancipated instrumental music is connected with that. One can now say that in the musical experience man experienced himself as being at one with the world. He experienced himself neither within nor outside himself.

This whole experience naturally penetrates today into everything musical. On the one hand, music occupies a special position in the world, because, as yet, man cannot find the link to the world in the musical experience. This link to the world will be discovered one day when the experience of the octave comes into being in the manner previously outlined.

The great progress made by humanity in the musical element is that the human being is not just possessed by God but takes hold of his own self as well, that man feels the musical scale as himself, but himself as existing in both worlds.

We must say, as it were: man actually lives in us in all seven tones, but we do not know it.

You must take all these concepts that I use only as substitutes and in each case resort to feelings. Then you will be able to see how the musical experience really strives to lead man back to what he lost in primeval times.

You see, however, how matters proceed from the seventh to the fifth, from the fifth to the third, and from the third down to the prime, the single tone, and so forth. What was once the loss of the divine must transform itself for human evolution if humanity on earth is not to perish but to continue its development. The loss must transform itself for earthly humanity into a rediscovery of the divine.”

Rudolf Steiner’s fragments were taken from: The Inner Nature of Music and the Experience of Tone

Blog » The Astrological Zodiac & Musical Tonality

Reading Time: 25 minutesMarch 3, 2014

FOREWORD

This article is an ongoing “work-in-progress” … and because I am no “expert” in astrology I do welcome additional info, suggestions and corrections (in case of mistakes made in this article). Thank you!

FOOTNOTES:

The seasons mentioned in this article are as the are at the Northern hemisphere (North of the Equator).
The exact dates that cover the periods of the Zodiac Signs might differ, that depends on the system used.
All dates, numbers and frequencies mentioned in this blog article are an approximation.

THE ZODIAC (INTRODUCTION)

The zodiac is the circle of constellations traversed by the sun, moon and planets in their yearly cycle, as seen from Earth. Traditionally his circle is divided into 12 sections of 30 degrees each, the Zodiac signs. The Zodiac is an ecliptic coordinate system, which takes the ecliptic as the origin of latitude, and the position of the Sun at vernal equinox as the origin of longitude.

Astrologers track the planets as they continue on in the sky after birth, moving through the houses of the natal chart as they progress through the zodiac.

A SHORT HISTORICAL TIME-LINE:

During the early 1st millennium BC the Babylonian astronomers divided the ecliptic into twelve equal zones of celestial longitude to create the first known celestial coordinate system. The Babylonian calendar as it stood in the 7th century BC assigned each month to a sign, beginning with the position of the Sun at vernal equinox, which at the time was depicted as the Aries constellation (“Age of Aries“). This is the reason why the “first sign” in Astrology is still called “Aries“, even after the vernal equinox has moved away from the Aries constellation due to the slow precession of the Earth’s axis of rotation.

Knowledge of the Babylonian zodiac is also reflected in the Tanakh (Jewish Bible), and is the first recorded astrological division into 12 constellations, elaborated on in the Talmuds (= “learning”, a collection of writings named specifically the Babylonian Talmud), Midrash Rabba (books of the Tanakh), and other minor works.

The Babylonian star catalogs entered Greek astronomy in the 4th century BC. Horoscopic astrology first appeared in Ptolemaic Egypt. The Dendera zodiac, a relief dating to ca. 50 BC, is the first known depiction of the classical zodiac of twelve signs.

Particularly important in the development of Western horoscopic astrology was the astrologer and astronomer Ptolemy, whose work Tetrabiblos laid the basis of the Western astrological tradition. Ptolemy explained the theoretical basis of the western zodiac as being a tropical coordinate system, by which the zodiac is aligned to the equinoxes and solstices, rather than the visible constellations that bear the same names as the zodiac signs.

THE ZODIAC SIGNS (SYMBOLS)

Not all visitors of this blog have experience with astrology, therefor first the symbols of and the names of the Zodiac signs they represent used in this article:


Aries	Taurus	Gemini	Cancer	Leo	Virgo	Libra	Scorpio	Sagittarius	Capricorn	Aquarius	Pisces

A HISTORICAL PERSPECTIVE

Before we come to the practical implementation it would be good to place this topic in historical perspective. The relationship between tonality and the signs of the Zodiac has been a subject that intrigued and inspired various astronomers, astrologers, philosophers, composers and musicians. After all, there are 12 tones in Western music and there are 12 Zodiac signs, all with their own “characteristics” and “color”.

When it comes to the relationship between western Astrology and western music, you can’t write an article without at least shortly mentioning Pythagoras (a name you might have come across in various other tuning-related articles on my blog) and his theory Musica Universalis (Music Of The Spheres).

PYTHAGORAS (c. 570 – c. 495 BC) – MUSIC OF THE SPHERES

Musica Universalis (literally universal music), also called Music of the Spheres or Harmony of the Spheres, is an ancient philosophical concept that regards proportions in the movements of celestial bodies — the Sun, Moon, and planets — as a form of musica (the Medieval Latin term for music).

This “music” is not usually thought to be literally audible, but a harmonic, mathematical or religious concept. The idea continued to appeal to thinkers about music until the end of the Renaissance, influencing scholars of many kinds, including humanists.

The “Music of the Spheres” incorporates the metaphysical principle that mathematical relationships express qualities or “tones” of energy which manifest in numbers, visual angles, shapes and sounds – all connected within a pattern of proportion.

Pythagoras first identified that the pitch of a musical note is in proportion to the length of the string that produces it, and that intervals between harmonious sound frequencies form simple numerical ratios.

The Pythagoreans were mainly interested in the mathematical aspects of the movements of the stars and less in the “images” of the constellations.

“Sounds are the echo of the “Harmony of the Spheres” which man took into himself when he came down from the divine-spiritual world into the physical world.“
Rudolf Steiner

Fellow blogger Joe Dubs created for his article about Musical Geometry a nice visualization of the planetary relationships (distances according to NASA) as written about in an article of Keplerstern Verlag. A couple of these planetary relationships represent musical ratios. Read this article to learn more about the Semi-Minor Axes Geometry …

Left to right: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto.

In the image above you see the following musical ratios: 2:1 = Octave, 3:2 = Perfect Fifths and 4:1 = 15th (or 2 Octaves). The other ratios (8:3, 25:1 and 25:4) aren’t common intervals in musical tuning, but for those who enjoy microtonal music those macrocosmic ratios might be worth experimenting with.

Another article by Joe Dubs with nice visualizations on this subject is his article about the orbital patterns of the planets in our solar system reveals other musical ratios such as the tonic (1:1) and perfect fourths (4:3). “The Music of the Spheres – Musica Universalis” (Joe Dubs) might be a nice read for you as well …

It is interesting to note that the most important interval in music, the 5th (the interval Pythagoras based his musical tuning system on), represents the relationship between Earth and Mars. Astrology wasn’t something Pythagoreans were much interested in though, unlike Astronomy … but that aside:

In astrology, Mars is the planet of determination, ambition and competition, energy, action, desire, passion, aggression, anger. Mars, the God of War, is the ruler of Aries. It is the survival instinct, and can be thought of as the “leftover” animal nature of man. Our sexual desires come under the rule of Mars. Mars’s energy can be constructive or destructive. It takes nearly two years for Mars to complete its orbit through the zodiac.

Many great thinkers such asPlato, Nicomachus of Gerasa, Galileo, Martin Mersenne and Athanasius Kirchner continued developing what Pythagoras started.

A different more literal, creative, modern scientific approach to “Music of the Spheres” is the work “Wheel of Stars” by Jim Bumgardner, where Bumgardner relates star position, parallax, magnitude, and color information to sound pitch.

In the more recent history Music of the Spheres has inspired various (contemporary) composers and artists such as: Rued Langaard (“Music of the Spheres“), Mike Oldfield (“Music of the Spheres“), Björk (“Cosmogony“) and others …

Even though there is a lot more to share and write about Music Of The Spheres, I will proceed with “The Tone-Zodiac” (a historical time-line), a topic that has a more direct connection to the practical implementation of Astrology and music.

THE TONE ZODIAC (A HISTORICAL TIME-LINE)

It is only half a millennium after Pythagoras that there is evidence of a “Tone-Zodiac”, in Ptolemy’s writings. At that basis of the connection between music and the Zodiac lies the Pythagorean-Platonic vision of heavenly harmony. The “Tone-Zodiac” is a system that combines a Tone Circle with the Zodiac Circle and is by far the easiest method to visualize the connection between the Zodiac and musical tonalities.

PTOLEMY (c. 100 – c. 168 CE)

(img. Godwin, 1993: 30)

Ptolemy explains the astrological aspects by invoking music as an analogy, following Pythagorean ideas about the role of the ratios.

He presents his tone-zodiac in chapter 8 of book III of “The Harmonics“, written near the end of his life, around 168 CE.

He lays out a two-octave Pythagorean scale on a zodiac circle, noting that the rotating movements of the stars are all circular and regular and similar to the movements within the tone-system.

In this form of the tone-zodiac, the octave (2:1) comes opposite in the circle, and so cuts it in two (1:2), which he considers ‘a great mystery’. To the modern sense of consonance it seems strange to portray opposition by the octave and conjunction by the double octave.

This Tone-Zodiac has the following Tone-Sign combination:

This Tone-Zodiac starts at Aries with A and goes around the Zodiac circle counterclockwise (as is common with Astrology).

Ptolemy continues his argument in “The Harmonics” with a comparison of the rising and falling motion of stars with the genera in music. This comparison, based on the tropics, connects the Seven Greek Modes with the twelve star signs (image below: Godwin, 1993, p. 36):

This comparison, based on the tropics, connects the Seven Greek Modes with the twelve star signs:

Gemini () & Cancer (): Mixolydian (B)
Taurus () & Leo (): Lydian (C)
Aries (): Phrygian (D)
Pisces () & Virgo (): Dorian (E)
Aquarius () & Libra (): Hypolydian (F)
Scorpio (): Hypophrygian (G)
Capricorn () & Sagittarius (): Hypodorian (A)

This Tone-Zodiac has the following Tone-Sign combination:


D	C	B	B	C	E	F	G	A	A	F	E

The Circle with the Zodiac Signs is drawn clockwise, (instead counterclockwise as is common in Astrology) and starts with Aries at D. The way the Signs are grouped in the image (on what “line” the signs are placed) is not logical. There are several ways to “group” Zodiac signs (traditionally grouped in: Polarities, Triplicities, Quadruplicities) but the “grouping” seen in the picture does not match any. Later in this article more about these groups.

(Godwin, 1993, p. 36)

Ptolemy also presented a tone-zodiac of two octaves showing the course of the moon as well, exemplifying that of all the planets, from its conjunction with the sun at the dark of the moon.

As this chapter survives only in very corrupt manuscripts it is not very useful not go into detailhere (source: Godwin 1993, p. 36, 416). But it does show that the concept of the tone-zodiac, in different forms, was well-known to Ptolemy.

The used musical scale in the image on the left is the same as in the image with the first shared Ptolemy Tone Zodiacs. This Tone-Zodiac has the following Tone-Sign combination:


E	D	C	B/A	G	F	E	D	C	B/A	C	D

Above we see the “Perfect Immutable System” (diatonic genus) that combines the “Greater Perfect System” and the “Lesser Perfect System. The “Perfect Immutable System” is divided into 4 “tetrachords“, each diatonic tetrachord consists of two intervals of a tone and one of a semitone.

The “Tetrachord Diezeugmenon” contains the tones E (Nete diezeugmenon at Aries), D, C and B.
The “Tetrachord Meson” contains E (Hypate Meson at Libra), F, G and A.
The “Tetrachord Hypaton” contains the tones E (Hypate Meson at Libra), D, C and B.
The “Tetrachord Hyperbolaion” contain the tones E (Nete diezeugmenon at Aries), D, C and B.

In this Tone Zodiac the Cardinal Directions were added as well. If the Tetrachords were implemented correctly, we find E at both North (Aries) and South (Libra). The East (Capricorn) is “sidelined” by “Hypate Hypaton” (B) and “Proslambanomenos” (A) / “Nete Hyperbolaion” (A), the West (Cancer) is “sidelined” by “Mese” (A) and “Paramese” (B). These “sidelines” connect diametrically to the same tone on the other side of the Tone Zodiac.

The Tone Zodiac starts at Aries with E and goes around counterclockwise (as is common in astrology).

In the Ptolemaic Tone-Zodiac on the left (Godwin, 1993) only the 7 tones of the Diatonic Scale (white key tones on the piano) of A-Minor (A-B-C-D-E-F-G) have been noted, covering two octaves to complete the circle counterclockwise (as is common in Astrology). What is interesting though about this Tone-Zodiac is that not all tones align with the cusps of the signs. In this Tone-Zodiac the C, D, and E are placed in the center of the sign sector. That might “suggests” that we are looking at a 12-Tone Equal Tempered (Chromatic) circle with only some tones high-lighted.

What is interesting though about this Tone-Zodiac is that not all tones align with the cusps of the signs. In this Tone-Zodiac the C, D, and E are placed in the center of the sign sector. That might “suggests” that we are looking at a 12-Tone Equal Tempered (Chromatic) circle with only some tones high-lighted.

Or, perhaps we are looking at a Tone-Zodiac that shows a single scale. When we write the tones only (not the tonalities) that are positioned at the cusps of each sector we get the following scale: A-B-C♯-D♯-F-G, a Hexatonic or Whole-Tone scale of A.

This Tone-Zodiac has the following Tone-Sign combination:

A/A1

This Tone-Zodiac starts at Aries with A and goes around the Zodiac circle counterclockwise (as is common with Astrology).

After Ptolemy ancient and medieval scholars in Western Europe discussed the connections between the planetary movements to the musical scale, but not to the zodiac.

In the Italian Renaissance the tone-zodiac surfaces again.

MARSILIO FICINO (1433-1499)

Marsilio Ficino, standing on Ptolemy’s shoulders, adds something new. In his letter to the musician-philosopher Domenico Benivieni he allows for a connection between the astrological houses and the musical intervals on the basis of the astrological aspects (source: Godwin, 1993, p. 168, Voss 2006: 186).

In his tone-zodiac he uses a one-octave scale and places the major seventh as the opposite aspect. However, he manages to keep the Ptolemaic consonances of whole-tone (sextile), fourth (square) and fifth (trine) in the same place (source: J. Godwin 1993, p. 32).

Voss argues that Ficino is advocating in his letter a tuning system in accordance with contemporary practice and the requirements of musicians, and which correlates more exactly with astrological law (source: A. Voss 2006, p. 187). As Ficino was a practising astrologer and a musician himself he may have chosen to adapt the Ptolemaic tone-zodiac to fit his own practice.

This Tone-Zodiac has the following Tone-Sign combination:


C	D	E	F/A	G	A	B	C	G	F/A	E	D

If Ficino was using the “Movable-Do” system, this Tone-Zodiac would display the Major scales with “Do” reprecenting the “Tonic” of the scale. If Ficino was using the “Fixed-Do” system, then this Tone-Zodiac would represent the scale of C Major. Ficino follows the circle diatonically (7-tone white note scale) counterclockwise (as is common in Astrology).

In the “Age of Enlightenment” no new tone-zodiacs appear.

The renewed interest in the esoteric wisdom of antiquity in the romantic times of the later nineteenth century reintroduces the idea, which flowers in the Theosophical and Anthroposophical movements.

RUDOLF STEINER (1861–1925)

(image: Godwin 1987, 147)

The Anthroposophical Tone-Zodiac, based on the ideas of Rudolf Steiner (the founder of Antroposophy) does not only connected the star signs with the major and minor keys, but represented the twelve creative figurations of the Archangels, the Seraphim and Cherubim, so that each tone existed as sound and as inner experience on the Etheric and Astral levels.

“All spiritual experience resulted from the awareness that the number of planets was contained in the seven scales, and the number of signs in the Zodiac was contained in the twelve fifths within the seven scales. The great mystery of man was revealed in the Circle of Fifths.” (Rudolf Steiner)

This Tone-Zodiac has the following Tone-Sign combination:


C	G	D	A	E	B	F♯ G♭	D♭	A♭	E♭	B♭	F
Am	Em	Bm	F♯m	C♯m	G♯m	D♯m E♭m	B♭m	Fm	Cm	Gm	Dm

Rudolf Steiner’s Tone-Zodiac starts at Aries with C (and A minor) and goes around the Circle of Fifths counterclockwise (as is common with Astrology).

MAX HEINDEL (1865-1919)

(image: Godwin)

Max Heindel (born Carl Louis von Grasshoff) was a Danish-American Christian occultist, astrologer, mystic and Rosicrucian Institute.

Max Heindel and Rudolf Steiner met in 1907 in Berlin and Heindel sat-in on several lectures of Steiner and spoke with him several times. Heindel did develop his own teachings Western Wisdom Teachings, very different from that of Steiner.

This Tone-Zodiac has the following Tone-Sign combination:


C♯/D♭	D♯/E♭	F♯/G♭	G♯/A♭	A♯/B♭	C	D	E	F	G/A	A	B

His Tone-Zodiac starts at Aries with C♯ and goes counterclockwise around the Zodiac Circle.

There is though an “odd thing” going on with this Tone-Zodiac. Most sections of this Zodiac covers 2 semitones per 30°. But, the section that belongs to Taurus (from D♯/E♭ to F♯/G♭) covers 3 semitones and the section that belongs to Scorpio(from E to F) only covers 1 semitone.

This means that the drawing of Max Heindel’s Tone Zodiac is incorrect. In the image on the left you can see how the Tone-Zodiac should have been divided based on the number of semitones as based on the noted tones.

If you analyze the tones used, we can form two Hexatonic (whole tone) scales with it:
Gb-Ab-Bb-C-D-E and F-G-A-B-C#-D#. Is that what Max Heindel had in mind?

The “problem” with the Hexatonic scale is that it closes a circle in 6 steps, not in 12 steps, as with the Chromatic Circle and the Circle of Fifths. This could explain why two of the sections (Taurus and Scorpio) are different in size, to be able to “close the circe” with two Hexatonic scales in the same circle. This is though a “fix” that is not logical or correct not from a musical standpoint, nor from a mathematical or geometric point of view. The circle (when drawn with sections equal in size to the intervals between the tones) does not look very “aesthetic” (something common for Taurus), the largest section in the circle.

Depending on the source the Taurus-Scorpio axis could represent one of various things such as: desires (in the sense of ownership, possession), the intensification and stabilization of the self and its environment, “Axis of True Being” (the confrontation of the self with its shadow), et cetera. Perhaps the axis of Taurus and Scorpio had special meaning to Max Heindel and that is why he chose to start the Hexatonic scales from these two signs?

There is only one “right way” to use Hexatonic scales as basis for a tone circle though. You can only use the Hexatonic scales as basis for a mathematically and musically correct tone circle if you use a tone circle with two concentric rings. Saxophonist John Coltrane drew the only functioning Hexatonic Tone Circle in existence. You can read more about that in the article “The Coltrane Tone Circle“.

MAX HEINDEL TONE-ZODIAC “REPRODUCTION”

This Tone Zodiac image is one of the most shared Tone Zodiacs on the internet. I had difficulties finding out who made it (and I still don’t know), but when I analyzed this Tone-Zodiac I recognized the discrepancies in the tone-spacing.

This Tone-Zodiac is a “reproduction” of the Tone-Zodiac made by Max Heindel shown above.

ANNY VON LANGE (1887-1959)

Rudolf Steiner’s Student Anny von Lange worked her ideas out for her book “Man, Music and Cosmos” (1956).

On her tone-zodiac she writes: “Twelve heavenly spheres are portals for the incoming cosmic forming forces which work upon man, each in its own appointed way.” (von Lange 1992, p. 309)

This Tone-Zodiac has the following Tone-Sign combination:


F	C	G	D	A	E	B	F♯	C♯	A♭	E♭	B♭
Dm	Gm	Cm	Fm	B♭m	E♭m	G♯m	C♯m	F♯m	Bm	Em	Am

Anny von Lange’s Tone-Zodiac starts at Aries with F (and D minor) and goes around the Circle of Fifths clockwise (as is common in music with the Circle of Fifths). It is important to note that the minor tonalities listed at von Lange’s Tone-Zodiac are NOT the minor relative tonalities of the listed Major tonalities! The tonic of the minor tonalities are a descending Fourth apart from the tonic of the Major tonalities, instead of a descending minor Third.

SIGISMUND VON GLEICH (1896-1953)

In the book “Über die Wirkung der Tonarten in der Musik “(About the Effect of Tonalities in Music)” by Sigismund von Gleich we find a rather original drawing of the Tone-Zodiac, in the shape of a lemniscate.

This concept is based on the idea that there is a “mirror-relationship” between the Major and minor tonalities.

The 12 tones are divided into two groups, a group of connected Major tonalities (C-G-D-A-E-B) directed outwards going “up-around-and-down” the left part of the lemniscate, followed by a group of connected Major tonalities (Gb-Db-Ab-Eb-Bb-F) directed inwards going “down-around-and-up” the right part of the lemniscate. The minor tonalities follow the same movement in opposite direction. You could say that the Major and minor tonalities “mirror” each others movement. Libra – the only tonality with an equal number of sharps (#) and flats (♭) is located at the crossing between both sides of the lemniscate.

Within this Tone-Zodiac design you can see a Lenmiscate Ouroboros, (the tail-devouring snake).

This Tone Zodiac Lemniscate has the following Tone-Sign combination:


C	G	D	A	E	B	F♯ G♭	D♭	A♭	E♭	B♭	F
Am	Em	Bm	F♯m	C♯m	G♯m	D♯m E♭m	B♭m	Fm	Cm	Gm	Dm

This Tone Zodiac follows the Circle of Fifths and starts with Aries at C Major (on the left) and A minor (on the right) of the lemniscate and goes around in both clockwise and counterclockwise movements.

This Tone-Sign combination is the same as Rudolf Steiner’s.

MARIUS SCHNEIDER (1903-1982)

TONE ZODIAC

_(Godwin)

SOLAR-LUNAR TONE ZODIAC

_(Godwin)

TONE ZODIAC SPIRAL

_{(Godwin 1987, p. 151)}

Marius Schneider was a musicologist who created an entirely original system that traces its origin to the world-wide spiritual culture of the Megalithic era (Godwin 1987, p. 149). His work is full of tone-zodiacs, some very creative and original, when looking at the Tone-Zodiacs by Marius Schneider it is obvious we are looking at the work of a unique individual.

SCHNEIDER’S TONE ZODIAC
(1st Zodiac above)
This Tone-Zodiac starts at Aries on C and go around the circle counterclockwise. The intervals between the tones differ though.

Most of the sections cover Major Thirds (4 semitones) except for Virgo (from A♭ to D) covering 6 semitones, Sagittarius (from B♭ to E♭) covering 5 semitones and Pisces (from B to C) – covering 1 semitone.

This means that the drawing of Marius Schneiders’ Tone Zodiac (as provided by Godwin) isn’t mathematically drawn correct. In the image on the right you can see how the Tone-Zodiac should have been divided based on the number of semitones as based on the noted tones.

This Tone-Zodiac has the following Tone-Sign combination:
C	E	A♭	D♭	F	A	D	F♯	B♭	E♭	G	B

SOLAR-LUNAR TONE ZODIAC (2nd Zodiac above)

This Tone-Zodiac is probably the most “complicated” in design that I have come across. It combines a “Solar Zodiac” (circle) and what Schneider called a “Lunar Zodiac” (ellipse).

The intervals between the tones placed on the Tone Zodiacs and the size of the “sectors” do not match and that means that the drawing of Marius Schneiders Solar-Lunar Tone Zodiac (as provided by Godwin) is mathematically incorrect. To make a Tone-Zodiac for this concept that is mathematically drawn correct and still “pretty-looking” is probably out of the question.

But, we can appreciate it for it’s creative design, originality and the astrological idea behind it!

This Tone-Zodiac has the following Tone-Sign combination:
E♭	E	B♭	C	F	A	D	G	G♯	C♯	F♯	B

TONE ZODIAC SPIRAL
(3rd Zodiac above)

That Schneider was a creative thinker becomes very clear with the “Tone-Spiral” – called the “Form of Cosmic Evolution” – the 12 tones are placed on a Spiral according to the astrological quadruplicities.

Within this sign-group the Zodiac Signs are divided in 3 Modes. The Signs from the same Mode form a Square or a Cross.These Modes are: CARDINAL (Aries, Cancer, Libra and Capricorn), FIXED (Taurus, Leo, Scorpio and Aquarius) and MUTABLE (Gemini, Virgo, Sagittarius and Pisces).

This Tone-Zodiac has the following Tone-Sign combination:
Eb	E	Bb	C	F	A	D	G	G#	C#	F#	B

Later in this article you will find more information about the “Zodiac sign-groups” and an example with squares.

ALAIN DANIÉLOU (1907-1994)

This Tone Zodiac shows the correspondences of the 12 tones with the Zodiac signs, months and seasons. The names for the notes are in Indian, though the Tone Zodiac is ancient Chinese (from the book “Music and the Power of Sound” by Alain Daniélou).

The order of succession of the Fifths is pointed-out by the points of the dodecagram (12-point star) and numbered I-XII (Roman numerals). Each tone is connected by a dotted lines to its upper Fifth and its lower Fifth.

The notes that correspond to the angles of the hexagon are masculine (C–D–E–F#–Ab–Bb), the notes that correspond to the middle of the sides are feminine (Db–Eb–F–G–A–B), forming two Hexatonic or Whole-Tone scales.

This Tone-Zodiac has the following Tone-Sign combination:
E♭	E	F	F♯	G	A♭	A	B♭	B	C	D♭	E

This Tone-Zodiac is uses the Chromatic circle and goes round clockwise. C relates to Capricorn as Aries (the Western starting point) relates to Eb.

STOCKHAUSEN (1928-2007)

1975 – German composer Karlheinz Stockhausen wrote a musical composition in 1975 called “Tierkreis” (Zodiac in German). Stockhausen was a student of Schönberg, abandoning the seven uneven steps of the major scale to the twelve equal notes of dodecaphony and serialism.

The Tierkreis melodies are constructed on several series that rule pitch and duration. The “zentraltöne” of the melodies form a chromatic scale. The tempos of melodies are ordered like a chromatic scale.

This Tone-Zodiac has the following Tone-Sign combination:
F	F♯	G	G♯	A	A♯	B	C	C♯	D	D♯	E

Karlheinz Stockhausen used the Chromatic Circle, starting at Aries with F, following the circle clockwise (as is common in music with tone circles) in semitone-steps.

HENSCHEL & HENSCHEL-FISHER

1954 – Tone-Zodiac by Joan Henschel and Mary Henschel-Fischer (musicologist). They see their own times as chaotic and urge change, expecting the Age of Aquarius to bring about the joining of music, mathematics and astrology in the true universal religion.

Joscelyn Godwin has pointed out their “poor astrology” and “dubious musical results“, such as calling the perfect fifth and fourth “dissonant” intervals (source: Godwin 1987, p. 142).

This Tone-Zodiac has the following Tone-Sign combination:
C	C♯	D	D♯	E	F	F♯	G	G♯	A	A♯	B

The Henschels use the Chromatic Circle, starting at Aries with “C”, following the circle counterclock-wise (as is common in Astrology) in semitone-steps.

MCMULLIN

(image: Godwin)

The Irish astrologer and musicologist Michael McMullin sees Carl Jung as one of the great teachers of our age. His Tone-Zodiac dates from (1984).

“One can say that the voice of God has come to the West not through organized religion but through music, and that music’s great ‘bodhisattvas’, or avatars have been the messengers of a higher reality.” (McMullin 2006)

This Tone-Zodiac has the following Tone-Sign combination:
B	B♭	A	A♭	G	F♯/G♭	F	E	E♭	D	C♯/D♭	C
G♯m/A♭	Gm	F♯m	Fm	Cm	E♭m	Dm	C♯m	Cm	Bm	B♭m	Am

Michael McMullin used the Chromatic Circle, starting at Aries with B, following the circle counterclockwise in semitone-steps down the chromatic scale.

ASTROLOGICAL ASPECTS, SIGN-GROUPS, MODES AND DEGREES

“Opposition“, “Trine” and “Square” are commonly used terms by astrologers to define important relative positions of planets. Every aspect relates to a particular number of degrees. Opposition = 180°, the Trine = 120° and the Square = 90°. More information about some of these “Zodiac sign-groups” will follow later in this article. For now I will focus on the degrees and how they relate to the Tone-Zodiac.

Kepler is said to have been impressed – as Ptolemy before him – by the fact that these three most powerful aspects are expressed by the same ratios as the three perfect consonances in music.

ASPECT	DEGREES	RATIO CIRCLE	RATIO INTERVAL
Opposition	180°	360:180 = whole circle to half	2:1 = Octave
Trine	120°	360:240 = whole to greater part	3:2 = Perfect Fifth
Square	90°	360:270 = whole to greater part	4:3 = Perfect Fourth

JOHANNES KEPLER (1571-1630)

In his Tone-Zodiac (image: Godwin) Johannes Kepler places the Unison (Tonic) and Octave across (opposite) each other with all aspects in between, as if the Zodiac represented a hypothetical monochord string, in other words a logarithmic scale. The musicians among the readers of this article will undoubtedly frown looking at it, as did Rudolf Haase (see below).

Even though quantitatively perfect, it still did not match the astrological qualities with the qualities of the intervals. The most obvious example is the “opposition“, one of the more tension-producing (disharmonious) aspects in Astrology. The great challenge of the opposition is to integrate the two poles. At Kepler’s Tone-Zodiac the Unison and the Octave have been placed in opposition with each other.

The distance (interval) between the Unison and Octave in music though, is the most consonant (tension-free or “harmonious“) of all intervals. They are so harmonious, that when you play them together, it becomes very difficult to separate them from each other. They instantaneously “become one” to the listener.

What we see in Kepler’s Tone-Zodiac, is the incompatibility of musical and astronomical distances, the incomparability between the logarithmic scale of the monochord (on which equal divisions produce increasing intervals) with the geometry of a circle divided into 360 equal degrees.

The astrological website www.astro.com offers a free “Sound Horoscope” based on Kepler’s concept. Personally I would not recommend using this “Sound Horoscope”. If they – like Kepler – use the Unison and Octave (most harmonious interval, tension-free) as “opposition” (much tension), then they obviously do not know much about sound and music theory!

Personal footnote: At first I was thrilled when I came across Kepler’s concept, I thought “I stroke gold” and had found the answers I had been looking for. But when comparing the functions of the intervals and their particularities to the aspects related to the degrees, I was a bit disappointed, they didn’t truly match. Not much later I saw I had not been the only person that felt that way, Rudolf Haase has done so way before me. Rudolf Haase thus suggested an alternative scheme for the degree-based Tone-Zodiac:

RUDOLF HAASE (1920–2013)

German musicologist Rudolf Haase aligned the placement of the 12 tones with the standard division of the Zodiac circle, where each sector of the circle is 30°, much alike 12-Tone Equal Temperament, that is represented by a circle of 1200 cents, divided into 12 Equal sectors of 100 cents per semitone.

The opposition is now the dissonant Tritone while the Unison and Octave coincide. The Major Third (an interval with an “optimistic” character) aligns with the (positive) Trine (120°) while the Minor Third (an interval with a somewhat “pessimistic” character) aligns with the (negative) Square(90°). Haase also mentions that in modern times even the semitone and Major Seventh are treated as consonant just as their aspects the Sextile (60°) and Semi-Sextile (30°).

Not all aspects can be used in this circle The “Decile” (36°), “Semisquare” (45°), “Quintile” (72°), “Tridecile” (108°), “Sesquiquartile” (135°) and Biquintile (144°) are not part of the scheme.

You might also have noticed that no Zodiac Signs have been drawn into this Tone-Zodiac. The focus point of this scheme is not the Tones, but the Intervals, as it is with the “Movable Do” system. This scheme can be used for every tonality.

SUMMERY HISTORICAL TIME-LINE

After reading some books, lectures and online articles and “studying” various Tone-Zodiacs, I noticed that the Circles (Scales) and tonalities used differs a lot as you can see in this summery:

KEY→ ↓CIRCLE	C	C♯ D♭	D	D♯ E♭	E	F	F♯ G♭	G	G♯ A♭	A	A♯ B♭	B
CC	1			1		1						1
C.O.F	2					1
VARIOUS	2	2	1	2	1					4

This means that looking into the past did not give us an unambiguous answer to the question: “What tonality relates to what Zodiac Sign”?

Later in this article I will share my own view on it. I will first proceed with some additional information about Astrological Modes and the Zodiac Signs & Colors.

GEOMETRY & GROUPS: POLARITIES, TRIPLICITIES, QUADRUPLICITIES & DUALITIES

I mentioned “grouping” of the Zodiac Signs during the historical time-line. So what are those “groups”?

POLARITIES (opposition – 180°)	TRIPLICITIES (trine – 120°)	QUADRUPLICITIES (square – 90°)

NOTE: the colors used in these examples are just to differentiate between the sign-groups based on the mode-division. The proper tone-color combination can be found later in this article!

CIRCLE OF FIFTHS & THE ZODIAC CIRCLE and their SHAPES, DEGREES, INTERVALS:

LINE: POLARITIES

Opposition (180°), 6{2}

The opposition causes exaggeration as it is not unifying.

The “connected” (opposite) tones in the Circle of Fifths are a “Tritone“ apart from one another: C-G♭, G-D♭, D-A♭, A-E♭, E-B♭ & B-F.

TRIGON: TRIPLICITIES (THE 4 ELEMENTS)
Trine (120°), 4{3}

The trigon, triangle or trine indicates harmony and ease.

FIRE: Aries, Leo and Sagittarius
EARTH: Taurus, Virgo and Capricorn
AIR: Gemini, Libra and Aquarius
WATER: Cancer, Scorpio and Pisces

The clockwise “connected” tones in the Circle of Fifths are a Major Third apart from one another:
C-E, E-A♭ & A♭-C / G-B, B-E♭ & E♭-G D-G♭, G♭-B♭ B♭-D / A-D♭, D♭-F F-A. Counterclockwise the “connected” tones in the Circle of Fifths are a Minor Sixth apart.

SQUARE: QUADRUPLICITIES (THE MODES)
Square (90°), 3{4}

CARDINAL: Aries, Cancer, Libra and Capricorn
FIXED: Taurus, Leo, Scorpio and Aquarius
MUTABLE: Gemini, Virgo, Sagittarius and Pisces

The counterclockwise “connected” tones in the Circle of Fifths are a Minor Third apart from one another: C-E♭, E♭-G♭, G♭-A & A-C / B♭-D♭, D♭-E, G-B♭ / F-A♭, A♭-B, B-D & D-F. Clockwise the “connected” tones in the Circle of Fifths are a Major Sixth apart.

HEXAGON: DUALITIES (♂ & ♀)

Sextile (60°), 2{6}

Representing “masculine and feminine” or “positive and negative” (+ & –).

The Sextile represents “ease of communication” between the elements involved, with compatibility and harmony between them. The Zodiac signs can be divided into two groups of 6 signs:

MASCULINE SIGNS: Fire + Air signs
FEMININE SIGNS: Earth + Water signs

The “connected” signs form two Hexatonic or Whole Tone Scales when combined: C–D–E–F♯–G♯–A♯–C and G–A–B–D♭–E♭–F–G. The tone distances up the scale, represent the Whole Tones. Naturally going down a whole tone (Major Second) is “equal to” going up the Scale a Minor Seventh (10 Semitones). The Hexagon thus represents both the Major Second (Whole Tone) and Minor Seventh.

DODECAGON
Semi-Sextile (30°), t{6}

The Dodecagon is not one of the traditional 4 “groups” but does represent an aspect called “Semi-Sextile“. The Dodecagon follows the circle in both directions.

Clockwise in Fifths, counterclockwise in Fourths.

DODECAGRAM
Quincunx (150°), t{6/5}={12/5}

The Dodecagram is – like the Dodecagon – not one of the traditional 3 “groups” but does represent an aspect called the “Quincunx“.

When you follow the lines of the Dodecagram across, you “traverse” chromatically up or down the Chromatic Scale. Naturally going a Minor Second (semitone) down the Chromatic Scale is “equal to” going a Major Seventh (11 semitones) up the scale. The Dodecagram thus represents both the Minor Second (Semitone) and the Major Seventh.

OTHER ASPECTS
The “Decile” (36°), “Semisquare” (45°), “Quintile” (72°), “Tridecile” (108°), “Sesquiquartile” (135°) and Biquintile (144°) are not part of this scheme. These aspects fall in between the intervals we use in the 12-Tone (Equal Tempered) musical tuning.

Below all aspects that relate to musical intervals used:

	LINE	TRIGON	SQUARE	HEXAGON	DODECAGON	DODECAGRAM

	6{2}	4{3}	3{4}	2{6}	t{6}	t{6/5}={12/5}
C C	TRITONES	MAJOR THIRDS	MINOR THIRDS	MAJOR SECOND	MINOR SECOND MAJOR SEVENTH	FIFTHS FOURTHS
C o F	TRITONES	MINOR SIXTHS	MAJOR SIXTHS	MINOR SEVENTH	FIFTHS FOURTHS	MINOR SECOND MAJOR SEVENTH
	Opposition	Trine	Square	Sextile	Semi-Sextile	Quincunx

	180°	120°	90°	60°	30°	150°
	“POLARITIES“ Opposition	“TRIPLICITIES“ Fire Earth Air Water	“QUADRUPLICITIES“ Cardinal Fixed Mutable	“DUALITIES“ Masculine Feminine

The degrees listed above relate to the angles of 2 dots relatively to each other from the center point of the geometric shapes. With these geometric “shapes” combined, all possible tone-connections can be made, as is clear from the superimposed graph. Naturally this counts only for 12-Tone systems.

CHROMATIC CIRLCE
If you use the Chromatic Circle instead of the Circle of Fifths (as used for the examples above), then the sign-groups (Polarities, Triplicities and Quadruplicities) and the Hexagon align with the Signs and Tones in a similar manner as they do with the Circle of Fifths.

There is one difference though. With the Circle of Fifths the Dodecagon connects the Fifths/Fourths and the Dodecagram visualizes the Chromatic Scale, but if the Chromatic Circle is used, the Dodecagon would visualize the Chromatic Scale and the Dodecagram would connect the Fifths/Fourths instead.

OTHER MUSICAL INTERVAL SYSTEMS
Not all Musical Interval Systems contain 12 tones. With a different number of tones different geometric “shapes” need to be used. Some of the Tone-Zodiacs presented earlier in this article (like those by Ptolemy) use for example only 7 tones (Greek modes). For Tone-Zodiacs using 7 tones or scale degrees you could use the Heptagram and Heptagon.

THE CHROMATIC SCALE, THE PIANO AND THE MONTHS OF THE YEAR

There is yet another way to connect keys / tonalities with Zodiac signs. The Western tone system has 12 tones, as there are 12 Zodiac signs and 12 months. The 12 months do not contain the same numbers of days. There are 7 months with 31 days, and 5 months with less (30 or 28/29), as there are 7 white keys and 5 black keys.

If we examine the following order of keys of the piano (and the Chromatic Scale), then there are 2 places where on the piano two white keys are next to each other, between B and C and E and F.

Much alike the division of the months, December and January and July and August are the only two moments in the year that 31-day (white key) months follow each other directly.

Even more interesting is that both piano keys and months follow the same pattern:
Starting from January 31:less = White:Black = 7:5 = 4:3 + 3:2
Starting from tone C White:Black = 31:less = 7:5 = 3:2 + 4:3

There is though one thing to keep in mind … the Zodiac Signs do not align with the months of our calendar. Each sign starts and ends around the 20th (18-23th) of each month. There are thus 2 variants possible.

The first variation aligns to key / tonality with the start of the Zodiac signs (Decan 1) and is as shown in the image above (by Mark Peltier). The second variation aligns with the a sign with the largest coverage of a month (Decans 2 & 3).

Every Zodiac Sign covers 30 Degrees of the Zodiac Circle. Every Zodiac Sign is divided in 3 Decans, each of 10 degrees. It is said that the 1st Decan (0-9 degrees) corresponds to the physical side and relates to the body, the 2nd Decan (10-19 degrees) corresponds to the mental or the Mind side and the 3rd Decan (20-29 degrees) corresponds to the spiritual aspects of a person. Astrologers could base this decision also on the actual position of the planets within the Zodiac Sign’s Decans.

VARIANT1 – STARTING WITH ARIES (THE FIRST SIGN):

G♯
A♭

A♯
B♭

C♯
D♭

D♯
E♭

F♯
G♭

Mar.

Apr.

May

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

Jan.

Feb.

VARIANT 2 – STARTING WITH ARIES (THE FIRST SIGN):

F♯
G♭

G♯
A♭

A♯
B♭

C♯
D♭

D♯
E♭

Apr.

May

Jun.

Jul.

Aug.

Sep.

Oct.

Nov.

Dec.

Jan.

Feb.

Mar.

What variant should you use? Well, that’s up to you to decide.

Alternatively, for those who do not read astrological charts a choice for each variant could very well depend on the type of music you are making. I would suggest using variant 1 for music related to dance and other physical expressions of music and variant 2 for listening and meditative purposes.

MUSIC & GEOMETRY

If you are interested in more information about the geometry in music then read the article “Music & Geometry“.

! ROEL’S WORLD INTERPRETATION & IMPLEMENTATION !

In the article “The Astrological Zodiac & Musical Tonality – Roel’s Interpretation” I wrote in more detail about the tone-sign combination that I think is the most cogent.

Continue reading …

WORKS BY MUSICIANS, COMPOSERS & PRODUCERS

Various composers and musicians have been inspired by the Zodiac signs. Note that not all of them listed below followed a concrete Tone-Zodiac. Some of these care:

Nat Adderley (“Soul Zodiac“), Cannonball Adderley (‘Love, Sex and the Zodiac‘), John Coltrane (“Interstellar Space“), Mary Lou Williams (“Zodiac Suite“), Marion Brown (“Capricorn Moon“), Mort Garson (“The Zodiac – Cosmic Sounds“), Dick Jacobs & Rod McKuen (“Written in the Stars“), Nathan Hall (“Earth, Sea, Sky“), Aleksandar Simić (“Zodiac“), Barry Cockcroft (“Zodiac – 12 pieces“), Bright Sheng (“Zodiac Tales – Concerto for Orchestra“), Key Poulan (“Zodiac“), Charles Matthews (“Signs of the Zodiac – Study method for Flute and Piano“) and others.

I also came across a series of albums by “Zodiac Tribe“. The music on these albums is “Ambient”, for “meditative and visualization”. You can listen to it on Spotify, Deezer and other online stream services and online stores. ! NOTE: The creator(s) of “Music Of The Zodiac” did not pay much attention to the tonality and Zodiac sign relationships! Perhaps this is because of their choice of instruments, some “ethnic” instruments used (like a flute) might not be able to play in all 12 tonalities.

Each episode is specific to a music key and a vocal narration by Pietropaolo discusses the personality traits of that key, with accompanying music written in that key, taken from music written over the centuries. In 2013, Pietropaolo won the Prix Italia in the category of best work for music for the program. Listen on Soundcloud!

THE ASTROLOGICAL ZODIAC & MUSIC TONALITY – ROEL’S INTERPRETATION

This is an addition to this article, with Roel’s personal interpretation of “The Astrological Zodiac & Music Tonality“.

ASTRO-MUSIC COMPOSITION

THE ASTRONOMICAL ZODIAC SCALE

THE FUNCTIONS OF THE INTERVALS

Another article I wrote for my blog named “The Functions Of The Intervals” (referred to in this article) describes might be an interesting read as well?

AND NOW … TO CONCLUDE THIS ARTICLE WITH:

A youtube playlist with a collection of Zodiac music by some of the artists mentioned earlier in this article:

CREDITS

For the historical “time-line” about the Tone Zodiacs I have quoted Eduard Heyning from his essay “The Tone-Zodiac Symbol“. I did make a few changes and added some footnotes. If you like that part of this blog article, then you might like his essay about the Tone Zodiacs (and more) on academia.edu as well, or visit www.tierkreis.nu (about Stockhausen).

The images used are mostly copies from “Harmonies of Heaven and Earth: Mysticism in Music from Antiquity to the Avant-Garde” by Joscelyn Godwin, completed with images from the work of Sigismund von Gleich, Alain Daniélou, Karlheinz Stockhausen and Henschel. I have modified several tone-circle images that were shared online and added a few of my own.

OTHER SOURCES / REFERENCES:

“Harmonies of Heaven and Earth: Mysticism in Music from Antiquity to the Avant-Garde” (by Joscelyn Godwin)
“Über die Wirkung der Tonarten in der Musik” (by Sigismund von Gleich)
“Writings on Music” (by John Wallis)
“Mensch, Musik und Kosmos” (by Anny von Lange)
www.tonalsoft.com – Encyclopedia of Microtonal Music Theory
www.flutopedia.com – Color of Sound calculator
http://wiki.astro.com/astrowiki/ (AstroWiki)
http://en.wikipedia.org/wiki/Zodiac (Wikipedia)
http://en.wikipedia.org/wiki/Seasons (Wikipedia)
http://en.wikipedia.org/wiki/Planets_in_astrology (Wikipedia)
http://en.wikipedia.org/wiki/Tierkreis_(Stockhausen) (Wikipedia)

Blog » Sound and Light / Colour

Reading Time: 16 minutesMarch 1, 2014

LIGHT & SOUND, COLOUR & MUSIC

Is there a relationship between sound and light?

You would not be the first to ask this question, in fact, this question was put “on the table” by some of great composers in the past, such as Hector Berlioz, Claude Debussy, Richard Wagner, Alexander Nikolayevich Scriabin, Nikolai Rimsky-Korsakov, and others.

In this blog article I will share some information about the proclaimed relationship between Sound and Color from various sources throughout history. This blog article is not a scientific thesis or theory, but more like a compilation of various ideas, thoughts and theories (some scientific, others philosophical) throughout history.

Even though both light and sound can be visually “represented” by waves and the unit Hertz can be used for both, there are nonetheless at least as many differences as there are similarities. It is therefor very important to mention that sound and light are not the same “thing” and the effect they have when “interacting” with the world around us (as well as the effect they have on use beings) differs thus as well. More about this later. Because of this difference you can not simply “replace” one with the other. This said, for some people sound and light/color though can be “complimentary” and/or “enhance” the experience of both sound and light.

Note: The text quoted and shared images used belong to the authors I quote. I have provided links to their web sites, where you can find a lot more interesting information about this and other subjects.

EARLY RESEARCH

In the 13th century, Roger Bacon theorized that rainbows were produced by a similar process to the passage of light through glass or crystal.

In the 17th century, Isaac Newton discovered that prisms could disassemble and reassemble white light, and described the phenomenon in his book Opticks (1704).

Newton divided the spectrum into seven named colors: red, orange, yellow, green, blue, indigo, and violet. He chose seven colors out of a belief, derived from the ancient Greek sophists, of there being a connection between the colors, the musical notes, the known objects in the solar system, and the days of the week.Newton’s color circle (from Opticks) shows the colors correlated with musical notes. The spectral colors from red to violet are divided by the notes of the musical scale, starting at D. The circle completes a full octave, from D to D. Newton’s circle places red, at one end of the spectrum, next to violet, at the other. This reflects the fact that non-spectral purple colors are observed when red and violet light are mixed.

In the 18th century, Goethe wrote about optical spectra in his Theory of Colours. Goethe used the word spectrum (Spektrum) to designate a ghostly optical afterimage, as did Schopenhauer in On Vision and Colors. Goethe did create his own color-wheel, but did not specify a particular color-tone combination as Newton did (see above).

SENSATION OF SOUND AND COLOUR AND THEIR “DIMENSIONS”

“If our ears contained just a few individual sensing elements, each tuned to one particular absolute frequency, we might all be able to recognize the absolute “color” of audible tones just as well as we can recognize absolute red. However, the ear needs to respond over a much larger range of frequencies, and the dimensionality of the “space” of audible sensation is much greater, we can distinguish a much greater variety of spectral characteristics of sound than we can of light.

Roughly speaking, the coiled cochlea of the human ear has a varying elasticity along its length, so it can be regarded as a series of oscillators of different resonant frequencies, and these perform a fairly detailed spectral analysis of incoming sound waves, transmitting to the brain something a 3000 point spectral profile. The detailed mechanics of how the cochlea responds to stimuli are very complicated, and the study of this function is hampered by the fact that the mechanical properties change significantly if a cochlea is removed for study. Nevertheless, it seems clear that whereas the spectral analysis of optical stimuli has only three dimensions, the spectral analysis of aural stimuli has at least 3000 dimensions. It is not surprising that we (most of us) don’t memorize the absolute sensations associated with tones over ten octaves.”
(Source: www.mathpages.com)

WHAT DO SOUND AND LIGHT HAVE IN COMMON?

– Sound and light both exhibit oscillatory wavelike characteristics with various frequencies, wavelengths, and amplitudes.
– The sensed intensity is dependent on the amplitude of the wave.
– Both propagate* at a finite speed.
– Both exhibit Doppler shifts toward higher frequencies when the source of the wave is approaching us.
– Both undergo refraction, diffraction and interference.

* Some scientist, theorists and natural philosophers such as Walter Russell state that waves of light do not propagate but ‘reproduce’ each other from wave field to wave field of space. The planes of zero curvature, which bound all wave fields, act as mirrors to reflect light from one field into another. This sets up an appearance of light as traveling, which is pure illusion.

WHAT IS THE DIFFERENCE BETWEEN SOUND AND LIGHT?

– The frequencies of visible light and audible sound differ from each other by more than ten orders of magnitude. Audible acoustic range: roughly 20 Hz to 20,000 Hz vs. visible optical range: roughly 380 trillion Hz to 760 trillion Hz.
– Sound is a form of mechanical energy caused by vibrations of matter. Light is electromagnetic energy caused by interacting electric and magnetic fields.
– Light waves are composed of transverse waves (meaning that the oscillations are perpendicular to its direction of “propagation”) in an electromagnetic field, while sound waves are mechanical longitudinal waves (meaning that the oscillations occur in the direction of propagation) causing alternating compression and expansion of matter. Regardless of the source of the sound wave – whether it is a vibrating string or the vibrating tines of a tuning fork – sound waves traveling through air are longitudinal waves. Through solids, however, sound can be transmitted as both longitudinal waves and transverse waves.
– Sound requires a medium to propagate, light does not. Therefor while light does “propagate” through a vacuum (absence of a medium), sound does not.
– The denser the medium, the greater the speed of sound. The opposite is true of light.
– Electromagnetic waves (including light) is a “stream of particles” (photons). Sound does not consist out of particles. With other words, sound is only a wave, whereas light exhibits both wave and particle properties.
– Light waves can be polarized, but sound waves cannot.
– It’s also worth noting that while the frequency range of audible sound covers a factor of 1000, (about 10 octaves), the range of visible light covers only a factor of two (just one octave).

“IN THE MIX”

Another interesting difference between sound and light / color is what happens when you use more then one color or tone.

LIGHT & COLOUR: 1 + 1 = 1 (equals 1)
When you take two or more colors and mix them, you get a “new” color, a unique color of it’s own. You will not be able to see anymore what exact colors were used to get to this color by just watching this new color. When you use light as medium the a color will become lighter (closer to white, additive – RGB) with every added color, when you use pigments (paint/ink) as medium the new color will become darker (closer to black, subtractive – CMYK).Results obtained when mixing additive colors are often counter-intuitive for people accustomed to the subtractive color system of pigments, dyes, inks and other substances that present color to the eye by reflection rather than emission. Additive color is a result of the way the eye detects color, and is not a property of light. There is a vast difference between a pure spectral yellow light, with a wavelength of approximately 580 nm, and a mixture of red and green light. However, both stimulate our eyes in a similar manner, so we do not detect that difference, and both are yellow light to the human eye (color vision).

SOUND/TONE: 1 + 1 ≥ 2 (equals or is greater then 2)
When you play two or more tones together, those tones will not become a “new” single tone. You will still be able to differentiate the tones used. Combined tones could even create additional “tones” when played together due to the possible interaction between harmonics or interference between the played tones. With other words, when mixing sound/tones you can create a ‘poly-tone’ or ‘multicolored’ work of art, but when mixing light/colors you will end up with a ‘mono-colored’ or ‘monotone’ work.

Another good visualization of additive and subtracted light was made by Clay Taylor for his article “Light & Color Theory“. I do suggest you visit his page for more information!

SINGLE FREQUENCY (LIGHT/COLOR) vs. HARMONIC SERIES (TONE)

A musical tone is actually a whole series of frequencies (theoretically never ending) and thus not just one color (one frequency), but a whole series of colors. If you would mix the colors of all the harmonics of a tone together with paint, then you would end up with a “black-ish” color. Just as when you would mix all colors of light you would end up with white.

But, when we hear a tone, most of us predominantly hear the 1st harmonic (fundamental) and it’s multiples (2, 4, 8, 16, et cetera). Only very few people can clearly differentiate between the individual harmonics within the spiral of harmonics of that tone. With other words, there is a particular “tone” (“boosted” by it’s multiples) that “stands out”, the fundamental. And that is the frequency you can “base” the color on.

CONVERTING AUDIO TONES TO THE VISIBLE SPECTRUM OF LIGHT – COLOR

Nick Anthony Fiorenza writes at his website www.lunarplanner.com:

“The octave of visible light, extending from the color red to the color violet, is forty octaves higher than the middle audio octave, that which you would hear on a piano keyboard. Light, however, is measured by its wavelength, whereas sound in measured by its frequency.”

“Waves of light are quite short. For example, the center frequency of the color green has a wavelength that is 0.0000005132 meters long (0.5132 x 10-6 meters). To make this easy, we measure visible light in a unit called the Ångstrom (Å) (that is a capital Å with a little circle on top). One Ångstrom = 1 x 10-10 meters (that is 0.1 nanometers). The colors of the visible spectrum are measured in thousands of Ångstroms. As show in the following chart, the visible spectrum of light extends from about 7000 Å (red) to about 4000 Å (violet). Also shown in the chart are the center wavelengths for each of the seven basic colors; their corresponding audio frequencies; and the location of the musical notes of an audio octave translated to the the visible spectrum.”

“Thus, when we raise each note in middle audio octave by forty octaves we find its corresponding color harmonic. As shown in the chart below, the note “G” lies in the red area of the color spectrum. The note “A” raised forty octaves lies in the orange part of the spectrum. The note “B” lies in the lemon (yellow-green) part of the spectrum. The note “C” in the green band; the note “D” in the turquoise-blue band; and the note “E” lies in the violet band. Notice that the note “F” lies in the far violet area of the visible spectrum. This is near where the human eye range of color perception begins to drop off (although unique to each person). Also notice that the note F# lies even further from violet, in the near-UV (ultra-violet) area of the spectrum. Thus (when raised 39 octaves rather than forty octaves), it also it resides in the far-red (or near infra-red). Because of this, the note F# embraces the visible spectrum, and thus has some red and some violet, a combination that produces more of a purple color.”

Image by Nick Antony Fiorenza (www.lunarplanner.com)

Image by Nick Anthony Fiorenza (www.lunarplanner.com)

Very similar, but slightly different is the tone-color combination below (another great visualization of the color-tone relationship made by Clay Taylor for his article “Light & Color Theory“. I do suggest you visit his page for more information!).

The difference with the color-tone combination of Nick Anthony Fiorenza is approx. a semitone. Cyan for example is represented by the tone “D” at Clay Taylor‘s concept and a “D♭” in Nick Fiorenza‘s concept.

Another interesting color-tone combination is mentioned in “Interference – A Grand Scientific Musical Theory” by Richard Merrick.The difference between his color-tone combination with that of Nick Anthony Fiorenza and Clay Taylor displayed above. Rich Merrick‘s “C” is “kind of blue”, like the “D” of Nick Fiorenza and the D#/Eb of Clay Taylor.Excerpt from “Synesthetic Coupling”

“There have been many attempts through history to establish an association between color and pitch, though none have been universally accepted. Composers like Berlioz, Debussy, Wagner and Scriabin all had ideas about which colors matched which tones. The Rosicrucian Order developed their own color mapping and even Charles Fourier suggested in his 1846 Theorie de l’Unite Universelle an alchemical connection between certain pitches, colors and metals.

One of the more recent proposals suggests that we should reduce light frequencies down to the speed of sound in order to produce a color mapping. While this last theory is a reasonable approach, physicists would argue against this, pointing out that that sound and light waves are not the same kind of energy. Science requires some other causal link or coupling.”

“Rather than use any of the above methods, we will construct our synesthetic model from Isaac Newton’s popular 12-step tertiary color wheel containing three primary colors, three secondary colors and six tertiary colors. Taken as two groups of six colors, the even group of primary and secondary colors can mix adjacently to produce an odd group of tertiary colors in much the same way as one wholetone scale mixes to the other. This suggests coherent light is perceived to mix harmonically just like coherent sound mixes into music harmony. This is without a doubt due to the fact that the visible light spectrum frequency doubles to form an octave of light just like an octave of sound.”

“Since the visible color spectrum ranges from about 375 terahertz on the low end to about 750 terahertz on the high end, the visible color spectrum naturally forms a 2:1 octave doubling of light frequencies like that of a musical octave. From this, we can proportionally map twelve colors to twelve tones by starting just below human visibility at 370 terahertz and then calculating twelve color frequencies by multiplying each preceding color by 2^(1/12), making sure to balance around the center of the visible spectrum. Doing this creates a logarithmic color scale that perfectly matches an equal-tempered musical octave. It also places each color within its corresponding spectral color band for the three cone photoreceptors on the retina of the human eye.”

Another excellent image, displaying the Color, Tone and Chakra relationship was made by Rich Merrick from his work “Interference Theory“, a “must read” if you are interested in this and related subjects!

Richard relates the middle Chakra (Green) to “B” instead of “C” though.

440, 432 (and 256) IN TONE AND COLOR I have chosen a different color for the Major tonalities and their Minor parallel tonalities. I have chosen to base the color of the tonality on the tonic (unison). The reason I have done so is that the character and function of the tones and chords of the Major and Minor parallel tonalities differ, no matter that they contain the same set of tones. The Major tonalities/colors represent the appearance to the “outside world” (extraversion) and the Minor tonalities/colors represent the “inner world” (introversion).

The first thing you might have noticed, is that the difference in Concert Pitch results in a different colors. Not only does a difference in Concert Pitch change the color, what Temperament is used does play a role as well. Below a table with several Concert Pitches and Temperaments more often mentioned at Roel’s World.

NOTE: All colors were calculated at 16 degrees temperature and 45% relative humidity. When the temperature and humidity (as well as air pressure – PSI) changes, the pitch – and thus color – will change slightly as well. The same “tone” played on an instrument at sea level or on a mountain top can actually be “different” tones with www.flutopedia.com

Keep in mind that the differences optically (in color) are far less obvious then the aural differences (in sound). This has to do with the difference of the number of “dimensions” (3 optical “dimensions” vs. at least 3000 aural “dimensions” as mentioned earlier in this article). The aural differences are pretty obvious for most listeners.

WHY SOME RELATED RED TO “C”

If you have browsed the internet, you might have come across various articles and images that set C as Red. There are a couple of reasons why they might have done so:

The first Diatonic (Greek) Mode in modern times called “Ionian” is that of C. Ionian (fist mode) / C = Red (first color of the spectrum). According to various sources though the Aeolian and Ionian were introduced by Heinrich Glarean in 1547. Originally the first mode is said to have been the Dorian mode (tonic on D) for it’s mathematical symmetry and not the Ionian scale. In that case Dorian mode / D = Red instead.
C is generally on top (first) in the Circle of Fifths and Chromatic Circle, red is the first of the visible colors thus C = Red.
Some composers might have placed C at red due to the relationship between the Zodiac sign Aries and the tone C (see Roel’s World article about Tone Zodiacs and Roel’s interpertation / implementation article). The tone C is generally seen as the starting point of a tone circle, as Aries is the first sign in the Zodiac Circle. Aries is “ruled” by the planet Mars and the color related to Mars is red. Thus: Red = Mars / Aries = C.

Even though some of the these reasons mentioned for using Red with C do make sense up to a certain degree, the actual frequency light-to-sound relationship tells us that in reality (nature) C is Green, not Red, as evident from work of Nick Anthony Fiorenza.

WHY DO I SUGGEST C TO BE GREEN?

From a music theoretic point of view we do find “Middle C” in the middle between the treble and bass cleft. Thus, relating the note / tone in the middle to the color in the middle of the color spectrum (green) makes sense.

You could also look at it from a “spiritual” point of view.In Sanskrit, Anahata (the 4th and “Middle” Chakra) is associated with balance, calmness, and serenity. Anahata means “pure” or “clean, stainless”, “unhurt, unstruck, and unbeaten”.

That seems to me like a good match with the tonality of C:
no “accidentals“ (sharps or flats).The color traditionally used for the 4th Chakra is Green.

This is how the Chakra Tones sound if the color-tone conversion of Nick Anthony Fiorenza is used:

MY PREFERED COMBINATION: middle C, with the middle Color (green) and middle Chakra.

VARIOUS OTHER COLOUR-TONE COMBINATIONS

WALTER BOWMAN RUSSELL

Walter Russell, an American painter of the Boston School and a sculptor, an illuminate, a natural philosopher, a musician, an author and a builder connects tone and color in his book “The Universal One” (1929) on page 249 as follows:

Walter Russels - from the book "The Universal One" (1926)

It might be worth your time to read the works of Walter Russels, he provides a very interesting view point on many different subjects concerning natural philosophy and more. His color-tone combination is though very different from those mentioned earlier in the article!

Most common is the combination low pitch (tone frequency) with high wavelength (light) and visa versa. That would result from left to right (low pitch to high pitch) on the piano with Red via Orange, Yellow and Green to Blue and Violet.

What you might have noticed, is that unlike most of the tone-color combinations Walter Russell has “reversed” the following order of the spectrum in comparison to the tone pitches. That to me does not look very “scientific”, if one ignores the fact that wavelength and frequency are not the same and can not be converted one-on-one.Another interesting difference is that Russell added infrared to the spectrum, infrared is not visible to the human eye and thus a “color” we can’t see. If we on one hand only use tones / sound we can hear in a “concept”, then why would we relate / combine one of these tones we experience with one of your senses (ears) with a color / light we can not see with the “naked eye” (another one of our senses)? This does not make sense to me.

That Walter Russell has some “odd” views on the relationship between tone and color also becomes clear when we look at the following drawing:

What we see in this image is the color spectrum and its relationship to elements as well as tones. But, not all tones of the scale, only a part: Do, Re, Mi, Fa and Sol. Thus no La and Ti/Si.

What makes it a rather “odd” drawing, is that only 2 of the 5 tones mentioned are represented by one color each (Do / C by White and Sol / G by Yellow), but the other 3 tones are represented by 2 different colors each: Re (D) is represented by both Magenta (or as Russell called it in the previous image “Infra Red”) and Violet, Mi (E) is represented by Red and Blue and Fa (F) is represented by both Orange and Green.

The lack of “consistency” (in this case some tones are represented by one color, others by two) by itself makes a theory questionable.

More important: if the full visible color/light spectrum represents only Do – Sol, then what colors/light are/is “La” (A) and Ti (B) represented by? And why is C represented by White? White is generally “seen” as either all colors together (light) or the absence of color (paint).

This drawing of Walter Russell’s work just does not make sense, it is incomplete and inconsistent, probably one of the worst tone-color combinations suggested in history. If one thing, then it proofs that great minds might not be right about everything that they share with the world. 😉

ALEXANDER NIKOLAYEVICH SCRIABIN

Alexander Nikolayevich Scriabin was a Russian composer and pianist. Though Scriabin’s late works are often considered to be influenced by synesthesia, a condition wherein one experiences sensation in one sense in response to stimulus in another. His colour system, unlike most synesthetic experience, accords with the circle of fifths: it was a thought-out system based on Sir Isaac Newton‘s Opticks.

Note that Scriabin did not, for his theory, recognize a difference between a major and a minor tonality of the same name.

In his autobiographical recollections, Sergei Rachmaninoff recorded a conversation he had had with Scriabin and Nikolai Rimsky-Korsakov about Scriabin’s association of colour and music. Rachmaninoff was surprised to find that Rimsky-Korsakov agreed with Scriabin on associations of musical keys with colors; himself skeptical, Rachmaninoff made the obvious objection that the two composers did not always agree on the colours involved. Both maintained that the key of D major was golden-brown; but Scriabin linked E-flat major with red-purple, while Rimsky-Korsakov favored blue.

(Source: Wikipedia)

THE ROSICRUCIAN ORDER

The Rosicrucian Order based their theories on Just intonation and suggest that note names, frequencies, and colours should be:

THREE CENTURIES OF COLOR SCALES

image by Fred Collopy (www.rhythmiclight.com)

Image made by Fred Collopy (rhytgmiclight.com)

NEIL HARBISSON: “I LISTEN TO COLOR”

An interview for BBC and a TEDTALKS presentation by Neil Harbisson, born with achromatism (he does not see color), about how the implementation of technology can make him experience color through sound. He colors the C Major scale (starting in his interview with Red = F) as follows:

CONCLUSION

I personally find the tone-colour relationship as described by Nick Anthony Fiorenza, Clay Taylor or Richard Merrick as most likely to be accurate, there is mathematical and scientific reasoning that validates it. This is how nature seems to work and thus the most natural implementation.

For tone-color combinations in various articles on Roel’s World I have used the tone-colour relationship as provided by Nick Anthony Fiorenza.

This does not mean that if you have come to a different tone-colour combination that you are ‘wrong’!

In fact, the “highlighted” concepts by Clay Taylor and Richard Merrick for example are equally well created.

As unique beings the ‘data’ our brain receives from our senses and the interpretation by the brain of this “data” can vary from person to person. The shape of ones ears, in particular the “cochlea” (inner ear), effects the way sound is perceived, the same tone can sound ‘different’ (in timbre) from person to person … also in the perception of light / colour there can be differences, there are even ‘conditions’ mentioned in the literature on this subject where the relationship of wavelength/frequency and colour differ, for example the hypothetical case with the ‘inverted spectrum’ (someone with an inverted spectrum might experience green while seeing ‘red’ (700 nm) light, and experience red while seeing ‘green’ (530 nm) light), or for example a condition called Synesthesia. Synesthesia provides some atypical but illuminating examples of subjective colour experience triggered by input that is not even light, such as sounds or shapes.

IN OTHER WORDS:
The possibility of a clean dissociation between colour experience from properties of the world reveals that colour is a subjective psychological phenomenon.

REFERENCES & CREDITS:

Nick Anthony Fiorenza (www.lunarplanner.com) / R.I.P. Nick (Obituary)
INTERFERENCE: A Grand Scientific Musical Theory by Richard Merrick
Licht & Color Theory by Clay Taylor
Fred Collopy (www.rhythmiclight.com)
www.mathpages.com
www.wikipedia.com
www.musicalcolors.com

Blog » ⚠ 432Hz and “the Factor 9”

Reading Time: 7 minutes

March 1, 2014

432 & THE “FACTOR 9”

The Factor 9 concept is mentioned in the video called “Sonic Geometry: The Language of Frequency and Form” (by Eric Rankin and Alanna Luna). In this article we will take a closer look at the Factor 9 “formula”, it’s relationship to the Harmonic Series and the difference between the Factor 9 and “standard” temperamental implementation.

Official Website: www.sonicgeometry.com

SECOND VIDEO (2016)

FIRST VIDEO (2015)

NOTE: The number grid shown in the first video (12:52) contains some mistakes, more about this later.

FACTOR 9 AND THE HARMONIC SERIES OF D=9HZ

The “base frequency” or “Fundamental” of this grid is 9Hz. The other tones are the result of the implementation of the Harmonic Series. Depending on the Harmonic used as “starting point” of the a scale a different Musical Interval System is formed.

The number of tones in the scale formed from the Harmonic Series relates directly to the number of the Harmonic you start with. If you start from the 15th Harmonic, you would end up with a 15-Tone scale. This is not only the case when you use 9Hz as base frequency for this concept, this is simply the “nature” of the Harmonic Series.

For example:

12TH HARMONIC (TONIC A): 12-TONE SYSTEM

When we use A2=108Hz the first degree of the scale (12th-24th Harmonic of 9Hz) to it’s octave A3=216Hz, we would end up with a 12-Tone Temperament. On octave above 216Hz we find A4=432Hz.

14TH HARMONIC (TONIC C): 14-TONE SYSTEM

When we use C3=126Hz the first degree of the scale (14th-28th Harmonic of 9Hz) to it’s octave C4=252Hz, we would end up with a 14-Tone Temperament.

16TH HARMONIC (TONIC D): 16-TONE SYSTEM

When we use D3=144Hz as the first degree of the scale (16th-32nd Harmonic of 9Hz) to it’s octave D=288Hz, we would end up with a 16-Tone Temperament. D4=288Hz is a pretty familiar frequency for those who have been exploring 432-Tuning. We find the D4 at 288Hz with various 12-Tone Temperaments, such as the Pythagorean Temperament (in combination with Concert Pitch C4=256Hz or A4=432Hz) and Maria Renold’s “Scale of Fifths“. If you would like to use the 16-EDO system with Concert Pitch A4=432Hz, all frequencies (listed in the 3rd column) need to be pitched up with approximately 62 cents, the 16-EDO scale has “A4” at 416.8Hz, instead of 432Hz.

FACTOR 9 12T vs. 12-EDO vs. FIBONACCI-8 vs. RENOLD-I

The most “common” 12-tone scale belongs to 12-TET (12-Tone Equal Temperament) or 12-EDO (Equal Divisions of the Octave). I will try to “visualize” the difference between the “Factor 9 Temperament” and 12-TET in this article. If you do not know what a tuning system is made of, then please do read this article on my blog: Tuning Basics.

ABOUT 12-TET or 12-EDO

“12–EDO, perhaps better known as 12TET since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division which can seriously claim to represent 5-limit harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma) meantone, it represents meantone.

It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.“

Source: Xenharmonic

Below a table with cents, ratios and more, comparing 12-EDO with FACTOR 9 12T, Fibonacci-8 and Renold-I, based on A=108Hz. I have transposed the complete scale in order to include 432Hz in the table so we start from A3=216Hz.

Degree	Tone	12-EDO	Factor 9 12T	Fibonacci-8	Renold-I
0	A₃	216	216	216	216
1	A#/Bb	228.8	234	230,4	229,1
2	B	242.5	252	240 or 243	243
3	C	256.9	270	259,2	256,4
4	C#/Db	272.1	288	270	273,4
5	D	288.3	306	288	257,7
6	D#/Eb	305.5	324	303,8	305,4
7	E	323.6	342	324	324
8	F	342.9	360	345,6	343,6
9	F#/Gb	363.3	378	360	364,5
10	G	384.9	396	384 or 388,8	386,5
11	G#/Ab	407.8	414	405	410
12	A₄	432	432	432	432

All frequencies above are rounded up to 1 digit behind the decimal point.

Below an example of a piece using the Factor 9 12-Tone scale (as listed above) by Derrick Scott van Heerden:

FACTOR 9 16-TONE By Derrick Scott van Heerden

“The factor 9 scale is really a harmonic scale, a one octave portion of the harmonic series repeated over octaves and played as a music scale on a multi-tonal instrument.

The long vertical column on the right (of the chart) shows the harmonic series for 9 Hz, while the bottom half shows the portion that makes the factor 9 scale repeated over 4 octaves to the right and left. There are a few missing notes in the factor 9 scale chart in the movie, here you can see that there are really 16 tones in each octave (16th to 31st harmonic).

Image on the right:
“This scale / slice of the harmonic series contains many intervals found in ‘pure’ versions of our 12 tone equal temperament scale and makes it seem obvious that our modern day 12 tone scale must have it’s roots in the harmonic series.“

More information at the source article: http://mathemagicalmusic.weebly.com/

DISADVANTAGES OF THE FACTOR 9 TEMPERAMENT

The most significant disadvantage of the Factor 9 Temperament is the specifications of some instruments, in particular acoustic instruments. Some acoustic instruments without temperamental issues are for example: the human voice, fret-less string instruments (like the Violin family), Trombone (a wind instrument without valves or tone-holes), the Harp and percussive instruments.

Naturally one could compose and produce music with modern Synthesizers, software with micro tuning capabilities or design / invent a new instrument based on this Temperament.

For instruments without micro-tonal tuning capabilities would be difficult to use the Factor 9 system, even in a 12-Tone scale.

NOTATION (SHEET MUSIC)

Up to a 14-tone scale the present notation system would work fine (adding a B#/Cb and E#/Fb to the scale). Notation of a 15-tone scale (or larger) using the traditional notation system can be a bit “tricky” and would require some time to study to be able to read it “prima vista“.

A 15-tone scale would look something like this (Easley Blackwood‘s notation system for 15 Equal Temperament):

A 24-tone scale would already require 3 variations in sharps and flats (24 Equal Temperament “Arab Tone System”):

SONIC GEOMETRY FIRST VIDEO: MISTAKES IN THE FACTOR 9 GRID

IMPORTANT FOOTNOTES ABOUT THE MOVIE “SONIC GEOMETRY:

Some numbers listed are simple miscalculations, but a more crucial mistake is that tones are missing in their grid!!! The scale/temperament displayed in the movie is a 13-tone system. But, after comparing the grid with the Harmonic Series we can conclude the grid shown in this video (12:52) is incorrect. Instead of a 13-tone system it would generate a 12 / 14 / 16-tone system if the implementation is done properly.

Also the “build-up” of the grid starts wrong, the “Factor 9” temperament is based on the tone D being 9Hz. If we set D as the first tone of the scale, then the first two frequencies listed in the 1st column belong to the “great octave” (stacking of 4.5Hz), while the rest of the tones listed in the first column belong to the “small octave” (above the “great octave”, stacking of 9Hz).

Movie screenshot:

Missing frequencies:

130.5 (between C and C# in column 1) – “great octave” (stacking of 4.5Hz)
139.5 (between C# and D in column 1) – “great octave” (stacking of 4.5Hz)
243 (between 234 in column 1 and 525 in column 2) – “small octave ” (stacking of 9Hz)
486 (between 456 in column 1 and 504 – missing – column 3) – “1-line octave” (stacking of 18Hz)
504 & 522 (between 486 – missing – and 540 in column 3) – “1-line octave” (stacking of 18Hz)
972 (between 936 in column 3 and 1008 in column 4) – “2-line octave” (stacking of 36Hz)
1044 (between 1008 in column 4 and 1080 in column 4) – “2-line octave” (stacking of 36Hz)
1944 (between 1872 in column 4 and 2016 in column 5) – “3-line octave” (stacking of 72Hz)
2088 (between 2016 in column 5 and 2160 in column 5) – “3-line octave” (stacking of 72Hz)

Miscalculations:
B in column 2 has to be 468 instead of 456, 643 in column 3 has to be 648 and 3755 in column 5 has to be 3744.

Another footnote to make is related to what is being said in the movie about Concert Pitch and instruments. In the movie Eric Rankin mentions that most modern musical instruments have been tuned to 432 for decades (until A4=440Hz became the International Standard). This is not correct, 432Hz has never been a standard, and only some old instruments seem to / might have been build for (or close to) 432Hz as Concert Pitch such as 435Hz (Diapason Normal). There are many old instruments in museums, as well as old Pitchpipe (Church) organs with various pitches ranging between A4=360Hz up to A4=460Hz. Instruments for Baroque music (1600-1750) for example, were designed for a Concert Pitch 415Hz.

OTHER REFERENCES:

Blog » Renold-I Temperament (“The Scale of Fifths”) by Maria Renold

Reading Time: 8 minutes

January 21, 2014

RENOLD-I TEMPERAMENT (“THE SCALE OF FIFTHS”)

I came across the “Twelve True-Fifths Tuning” concept (by Maria Renold) through a facebook friend, Brain T. Collins, who mentioned this tuning concept on his web site, the Omega432.com. Brian T. Collins refers to the page of Graham H Jackson where the “Twelve True-Fifths Tuning” concept is explained. This concept is a great alternative implementation of the Pythagorean Temperament.

NOTES:

The frequencies listed in this blog article are the result of the original “Scale of Fifths” tuning method (discovered in 1962), also referred to as “Renold I” Temperament. Maria Renold later found out that Henricus Grammateus had constructed a similar scale in 1518 (source: Jorgensen, 1991, p.332).
In her work Maria Renold used alternative names for the black keys of the piano. C#/Db = “delis“, D#/Eb =”elis“, F#/Gb = “gelis“, G#/Ab = “alis” and A#/Bb = “belis“. For this article I will use the common tone names instead.
Maria Renold did continue experimenting and developing her concept, with the “Renold II” (also known as “Middle Tuning” temperament as result. This Renold-II temperament is described in the article Renold-II Temperament.

WHY AN ALTERNATIVE FOR THE PYTHAGOREAN TEMPERAMENT?

Well, the Pythagorean Temperament works with the stacking of perfect fifths. But, when we stack 12 perfect fifths, we do not end up at a the same note we started, but approximately a quarter-tone above. With other words, we don’t end up with a circle but a spiral. This actually happens with any perfect (pure, natural) interval (except for the octave) when stacked (and that does not work very well with a closed musical interval system). In order to close the circle, one or more 5th’s should be slightly decreased to end up with a perfect circle. Maria Renold though came up with an tempered version of the Pythagorean Temperament, using mostly Perfect Fifths and still create a working closed circle.

Graham H. Jackson explains on his site:

“For the “twelve true-5ths tuning”: you first set C at 256 Hz. Then you tune the 7 “white keys” by the circle of 5ths, using however natural 5ths. Then you divide the octave at C exactly in half (which can be done handily with a special tuning fork), and tune the 5 “black keys” by natural 5ths to that F#.

You end up with two series of natural 5ths: one of 7 notes, and one of 5 notes, linked by an “unnatural” interval of an augmented 4th (which is actually the same augmented 4th found in the equal-tempered system).“

SO, HOW DOES THAT WORK?

We know that the perfect fifth uses the ratio 3:2. We use the following formula for calculating the fifth:

Frequency • 3 / 2 = Frequency of the Fifth above.

Now, let’s have a look at the Circle of Fifths:

When we follow the Circle clockwise, we go up a fifth. The outer “ring” with the capital letters is the Major Circle of Fifth, with inside the parallel minor Circle of Fifths.

As suggested by Graham H. Jackson, we start with from the tonic of the the C Major scale (the white keys on the piano). Now, the C Major scale contains the following tones: C – D – E – F – G – A – B.

When we start with C3=128Hz (as Renold suggested) and implement the formula we get the following:

128 (C3) • 3 / 2 = 192 (G3) • 3 / 2 = 288 (D4) • 3 / 2 = 432 (A4) • 3 / 2 = 648 (E5) • 3 / 2 = 972 (B5).

In this article I will use the tone frequencies between C4=256Hz and C5=512Hz so it will be easier to “compare” to the
tone frequencies on the modern standard.

256 (C4) • 3 / 2 = 384 (G4) • 3 / 2 = 576 (D5) • 3 / 2 = 864 (A5) • 3 / 2 = 1296 (E6) • 3 / 2 = 1944 (B6).

!IMPORTANT:

If we continue stacking 5ths (• 3 / 2) after B6 (1944Hz), we pass Gb, Db, Ab, Eb and Bb, ending up with F10=22143.375Hz. If we would stack another 5th on top, we end up at C11=33215.0625Hz.

!!! A perfect C11 relative to C4=256Hz would be C11=32768Hz, not C11=33215.0625Hz !!!

With other words, the fifth between F10 and C11 would be imperfect. If we bring that F10=22143.375Hz back to F4, we will end up with 345^_.990234375Hz. This is about 4Hz higher then the “harmonic mean”. So, we have to use a different method, to get the harmonic mean right. This is where mathematical formulas can help us out.

CALCULATING THE ARITHMETIC, GEOMETRIC AND HARMONIC MEANS

When we calculate the means in between C4 = 256Hz and C5 = 512Hz we get the following

The arithmetical mean formula: (256 + 512) / 2 = 384Hz (G, the 5th above C).
The geometrical mean formula: √ (256 • 512) = 362^_.038671968Hz (F# or Gb, the dim. 5th above C).
The harmonic mean formula: (2 • 256 • 512) / (256 + 512) = 341.^_333∞Hz (F, the 4th above C).

The arithmetical mean G = 384Hz was already part of the scale (the first perfect 5th we stacked on the C). Now we have calculated the harmonic mean as well, we can add F = 341.^_333∞Hz to the scale and we have completed the C Major scale.

C	D	E	F	G	A	B	C
256	288	324	341^_._333∞	384	432	486	512

If we start from the geometrical mean between C4 and C5, the F# (or Gb), stack 5 perfect fifths, and bring them back in between C4-C5, we get the following Major Pentatonic Scale of F# (or Gb):

F#/Gb	C#/Db	G#/Ab	D#/Eb	A#/Bb
362^_.039	271^_.529	407^_.294	305^_.470	458^_.205

Note: the frequencies noted above are rounded off on 3 digits behind the dot.

When we combine the C Major Scale and the F#/Gb Major Pentatonic Scale, we get the following Chromatic Scale from C:

C#/Db

D#/Eb

F#/Gb

G#/Ab

A#/Bb

256

271^_.5

288

305^_.5

324

341^_._333∞

362

384

407^_.3

432

458^_.2

486

512

In modern music we use only 12 tones. The consequence of this choice is that it is impossible to create a perfect system (a perfect circle of intervals using only perfect intervals), the natural “movement” sound makes when only using perfect intervals is spirally. And you can’t expect to see a spiral if you are drawing a perfect circle right? When we use Maria Renold’s “Scale of Fifths” concept and like to use a closed circle, then we end up with “just another” tempered version of the Pythagorean Temperament.

When we use the tone frequencies as listed above and calculate the amount of cents (rounded off) in between the Fifths, you notice that most intervals are about 0.1 cent off (lower or higher) from the 702 cents of the Perfect Fifth (Just Intonation). There are though two intervals – the Fifths between B and Gb/F# and between Bb/A# and F – that are smaller then the other 10 intervals:

C	G	D	A	E	B	Gb F#	Db C#	Ab G#	Eb D#	Bb A#	F	C
701.9	701.9	701.9	701.9	701.9	690	701.9	702.1	702.1	701.7	647.5	701.9

When we rearrange the circle chromatically we end up with semitones of 101.8-102.1 cents between every semitone, except in between between E-F and B-C, the only two semitone intervals of the Diatonic Scale:

C	C# Db	D	D# Eb	E	F	F# Gb	G	G# Ab	A	A# Bb	B	C
101.8	102.1	102.1	101.9	90.2	101.8	102.1	101.9	101.9	101.9	101.9	90.2

Below a list with the differences in cents and ratio between the Equal Temperament (present standard) and Maria Renold’s “Scale of Fifths” Renold-I Temperament, both using the Scientific Concert Pitch C4=256Hz.

TONE	12-TET C4=256Hz	M. R. SCALE OF FIFTHS	CENTS DIFFERENCE	RATIO f₂ / f₁DIFFERENCE
C4	256 Hz	256 Hz	0	0
C#/Db	271.2 Hz	271.5 Hz	1.9140234664560511	1.0011061946902655
D	287.4 Hz	288 Hz	3.6104998468059137	1.0020876826722338
D#/Eb	304.4 Hz	305.5 Hz	6.244825443055806	1.0036136662286466
E	322.5 Hz	324 Hz	8.033583088810076	1.0046511627906978
F	341.7 Hz	341.333∞ Hz	-1.858725915674287	-0.9989269339576626
F#/Gb	362 Hz	362 Hz	0	0
G	383.6 Hz	384 Hz	1.8043087084650533	1.0010427528675703
G#/Ab	406.4 Hz	407.3 Hz	3.8296946774990657	1.002214566929134
A	430.5 Hz	432 Hz	6.021689719949426	1.0034843205574913
A#/Bb	456.1 Hz	458.2 Hz	7.952746136304442	1.00460425345319
B	483.3 Hz	486 Hz	9.644782878502031	1.005586592178771
C5	512 Hz	512 Hz	0	0

For tuning by ear three tuning fork are needed: C4=256Hz, A4=432Hz and “gelis” (F#/Gb)=362,04Hz (Renold 1). How to tune by ear is described (scheme included) in Maria Renold’s book

Here an example how the Renold I temperament sounds (in comparison with Equal Temperament):

RENOLD-II

An alternative for the Renold-I Temperament (in particular for acoustic instruments) is the Renold-II Temperament.

DOWNLOADS (MIDI & SOFTSYNTH SCRIPTS)

I have added the “Scale of Fifths” to Scala (a software tool for creating scales, temperaments and more) and exported it in various formats. If you need a different format then provided, download Scala as well as this: SoF – Scala scale.scl scala-file and export it in the format you require.

Note: I do not guarantee the files bellow will work with your hardware and software setup. In case it does not, then try creating and exporting the “Scale of Fifths” yourself with Scala. Please, do feel free to send me a message if the provided files do not work for you, preferably with some info about the hardware, operating system (version) and software you have used.

DOWNLOAD (right-click mouse button: “Safe as”):

THE BOOK BY MARIA RENOLD

“The Scale Of Twelve Fifths” (and more) can be found in the book “Intervals, Scales, Tones and the Concert Pitch C = 128Hz“. by Maria Renold. Information about “The Scale Of Twelve Fifths” can be found at Chapter 13, 21, 24 & 25 of this book.

ISBN 9781902636467
www.templelodge.com

Why is it that certain intervals, scales and tones sound genuine and others false? Is the modern person able to experience a qualitative difference in a tone’s pitch? If so, what are the implications for modern concert pitch and how instruments of fixed tuning are tuned?

Maria Renold tackles these and many other questions, providing a wealth of scientific data. Her pioneering work is the result of a lifetime’s research into Western music’s Classical Greek origins, as well as a search for new developments in modern times. She strives to deepen musical understanding through Rudolf Steiner’s spiritual-scientific research, and she also elucidates many of Steiner’s often puzzling statements about music.

The results of her work include the following discoveries: that the octave has two sizes (a ‘genuine’ sounding octave is bigger than the ‘perfect’ octave); that there are three sizes of ‘perfect’ fifths; that an underlying ‘form principle’ for all scales can be found; and, most importantly, the discovery of a method of tuning the piano which is more satisfactory than equal temperament. She also gives foundation to some of Rudolf Steiner’s statements such as: ‘c is always prime’ and ‘c = 128 Hz = Sun’.

MARIA RENOLD (1917-2003) spent her childhood in the United States, where her parents emigrated to found a eurythmy school in New York. She studied eurythmy and later violin and viola and toured with the Bush Chamber Orchestra and the Bush String Quartet. One of Maria Renold’s deeply-felt questions concerned the correct concert pitch. When she heard of Rudolf Steiner’s concert pitch suggestion of c = 128 Hz she put it into practice immediately, and experimented with it over many years in America and Europe. She also discovered a new method of tuning the piano, closer to the tuning of stringed instruments, arriving at the concert pitch of a4=432 Hz. First published in German in 1985, her book has become a modern classic of musical research.

REFERENCES:

Graham H. Jackson (Official web site)
Omega432.com (a website by Brain T. Collins)

Special thanks to Sławek Kędra providing additional information and suggestions.

Blog » A Horological and Mathematical defense of the “Philosophical Pitch”

Reading Time: 6 minutesJanuary 20, 2014

A HOROLOGICAL AND MATHEMATICAL DEFENSE OF THE “PHILOSOPHICAL PITCH”

In this article Brendan Bombaci will shortly introduce his work: “A Horological and Mathematical Defense of Philosophical Pitch”. Even though I can not use his concept for my own music productions (since my saxophones are “stuck” in 12-TET at 440Hz), I do think it is worth the read for those interested in micro-tuning and temperament.

Roel

INTRODUCTION

I propose an alteration of the concert pitch standard outlined in ISO 16. As of now, it is set to A440 (A=440Hz), which has been chosen subjectively (rather than empirically as based upon the mathematical or geometrical values of art composition), as most all other concert pitch standards have been chosen throughout history. I have sought out various ways to make a compositionally cogent concert pitch standard, and I have succeeded at finding one that is perfectly tailored to synchronize with both the sexagesimal timekeeping system upon which all music is measured, and the 5 Limit Tuning system. It is well-known that this form of just intonation is the most consonant of all tuning systems, including that of equal temperament (whether or not equal temperament mostly corrects for the arguably noticeable near-Wolf fifths of just intonation). In as much, it is perfectly suited to be the model tuning system for this innovative new pitch standard, especially when one considers its fractional values for deriving each note of the chromatic scale. I will now explain both of my justifications in detail with some corroborative horological references.

TIME IN

It should be imagined that Western music, with an original meter basis of 4/4 that originally hinged upon the second hand of the clock for metering rhythm (a la the 120bpm Roman standard for marches) even before the second was academically identified [7], should have a pitch frequency that is similarly correlated. When tuning music to A440, most of the pitch frequencies are not whole numbers; the first octave of B (B1), for example, is 61.74Hz. If this were set to 60Hz instead, being the only note of the chromatic scale which comes close to synchronizing with the clock as a fractal continuance of the sexagesimal system, we would find the middle C note, C256, at the “scientific” or “philosophical” pitch of Joseph Sauveur (a mathematician, physicist, and music theorist) [1] and Ernst Chladni [1, 2], “the father of acoustics.” At the first octave of C, we would have the value of 1Hz, perfectly matching the second hand complication (movement).

Using 5 Limit Tuning set to C256, the frequencies of notes C4 (256), G4 (384), E4 (320), D4 (288), and B4 (240) are reducible to, respectively: 1, 3, 5, 9, and 15. You may notice that these notes, C, E, G, B, and D respectively, rearrange to a set of “stacking thirds,” in perfect chordal harmony. With the lowest C also standing in for its multiples of 2, 4, 8, 16, and 32, all of the numbers which are member to that set of stacking thirds are the very same numbers which comprise the numerators and denominators by which every chromatic note is derived (except 45, but this is still a harmonic of 15). This makes for more mellifluous tonal vibrations. In addition, the numbers 1, 2, 3, 4 and 5 represent the most commonly used values for meter in classical and modern music (with the 3 also standing in for its multiple of 6). There are important historical implications to this system, making it more geometrically, and even astronomically, intrinsic.

The helek (helakim, pl.) is an ancient and still used unit of time in Hebrew horology [4], which the second was extrapolated from. Further preceding helakim were the Babylonian names barleycorn or she, but no matter which name is used, all effectively mark the passage of 1/72nd of one degree of celestial rotation in a day. There are 1080 helakim per hour, and therefore 25920 helakim per day (and that many years in one astronomical Precession of the Equinoxes). This gives a discrete measurement unit that relates each minute to a visibly interesting astronomical cycle that has captured the imaginations of many cultures worldwide. Half of a day is akin to half of a precession of equinoxes, thereby; and likewise, periods of 2160 helakim are similar to the 2160 years of one astrological Age, meaning there are 12 Signs that pass in one day. Many historical European clock towers, such as the Torre dell’ Orologio in Venice, graphically purvey this along with the 24 hour segments. The conversion between helakim and seconds is this: 1 helakim = 3.333 seconds, or 60 seconds to every 18 helakim. 72 helakim, like the 72 years that pass in one degree of celestial precession, are equal to 4 minutes. 4 minutes multiplied by the whole 360 degrees equals 1440, the amount of minutes in one day. This is also the frequency in Hertz of the F# (the 7^thinterval, or perfect chromatic center) when tuned with the Philosophical Pitch and 5-Limit Tuning System.

Making the transition from helakim to seconds would only be a matter of deciding that the sexagesimal Babylonian calendar and navigational system should apply to a momentary measure for better precision. Musicians of the Middle Ages would have noticed that the divisionally attractive twelve factors of that system (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60) are perfectly coherent with four of the five stacking thirds frequencies of the new 5 Limit Tuning system which was designed to fix Pythagorean tuning dissonance in thirds intervals. With the addition of the fifth stacking third (9/18/36hz, etc.: the 2nd interval D note), they altogether cross-correlate with all aforementioned time measurement references within the Precession of Equinoxes, paying ultimate homage to the more prolific origins of timekeeping.

On a more esoteric note, the contemporary system also corresponds in some cases to culturally relevant “sacred” geometrical figures, whether or not any ancient musicians played note values that represented the same cosmic motions their timing system held to. Some of the latter include the conversions: 1440/3.333 = 432.0432 (considered by some to be a “spiritually” correct concert pitch value), 360/3.333 = 108.0108 (roughly a quarter of 432), and 72/3.333 = 21.60216 (representing a figure resonant with half of the first solution, 432.0432). These are all numbers of Biblical, Gematria (Hebrew numerological), Buddhist, and Hindu reference, with the latter two being angle degrees within a pentagram that reference the phi ratio (and Fibonacci sequence) – a fundamentally common pattern which all biological matter utilizes for efficient growth – and the faces of the dodecahedral Cosmic Microwave Background itself [6]. Interesting as they are, these solutions are not the note values we should make standard, but rather intriguing sign posts that show the astro-horological bases for certain compositional conventions in both secular and religious visual (including architectural) and sonic art.

For the sake of remaining true to horology in sonic form, harking back to but making better sense than the “Music of the Spheres,” the usefulness and the intricate aesthetics of tuning to C256 is inarguably better than any other standard. It also becomes far more intuitive to explain, due to whole number relationships, how various notes interact with one another and with tempo bases. Any “brighter” compositional sound, such as desired by proponents of A440, can be manifested by simply transposing a song. Although doing so alters interval relationships (because just intonation is not equally tempered), just as playing in any key other than C256 generally will within this system, it offers a new way to realize music in the same way that modes within a key provide mood and depth. Many Western composers prefer this and use just intonation specifically to achieve enhanced dramatic effect; some people who do so are: John Luther Adams, Glenn Branca, Martin Bresnick, Wendy Carlos, Lawrence Chandler, Tony Conrad, Fabio Costa, Stuart Dempster, David B. Doty, Arnold Dreyblatt, Kyle Gann, Kraig Grady, Lou Harrison, Michael Harrison, Ben Johnston, Elodie Lauten, György Ligeti, Douglas Leedy, Pauline Oliveros, Harry Partch, Robert Rich, Terry Riley, Marc Sabat, Wolfgang von Schweinitz, Adam Silverman, James Tenney, Michael Waller, Daniel James Wolf, and La Monte Young. Perhaps, with the rationality I provide in this article, many more yet will.

REFERENCES

Bruce Haynes. History of Performing Pitch: The Story of “A,” pp 42,53 (Lanham, Maryland: Scarecrow Press, 2002).
Ernst Florens Friedrich Chladni. Traitéd’acoustique, pp 363 (Paris, France: Chez Courcier, 1809)
Hebra, Alex. Measure for Measure: The Story of Imperial, Metric, and Other Units, pp 53 (The John Hopkins University Press, 2003)
Mackey, Damien F. The Sothic Star Theory of the Egyptian Calendar: A Critical Evaluation, abr. ed. (Sydney, New South Wales, Australia: University of Sydney, 1995).
Luminet, Jean-Pierre, Jeffrey R. Weeks, Alain Riazuelo, Roland Lehoucq, and Jean-Phillipe Uzan. Dodecahedral Space Typology as an Explanation for Weak Wide-Angle Temperature Correlations in the Cosmic Microwave Background. Nature 425:593-595.
Sachau, Edward C. The Chronology of Ancient Nations. Kessinger Publishing.

ONLINE ARTICLE: http://kairologic.blogspot.com/2015/08/a-horological-and-mathematical-defense.html

ISBN: 9781304362308

Copyright: Brendan Bombaci
(Standard Copyright License)

Published: August 24, 2013
Language: English
File Formate: PubFile
Size8.34 KB

Purchase a copy of:
A Horological and Mathematical Defense of Philosophical Pitch

You might also like to visit, like and or share A Horological Defense of Philosophical Pitch C = 256Hz on Facebook.

FREQUENCIES
(Source: Facebook Note | Facebook Page by Brendan Bombaci)

TONE	RATIO	FREQUENCIES
C	1:1	1, 2, 4, 8,16, 32, 64, 128, 256, 512, 1024
C# / Db	16:15	273.066
D	9:8	9, 18, 36,72, 144, 288, 576, 1152
D# / Eb	6:5	307.2
E	5:4	5, 10, 20,40, 80, 160, 320, 640, 1280
F	4:3	341.xxx
F# / Gb	45:32	1440
G	3:2	3, 6, 12,24, 48, 96, 192, 384, 768
G# / Ab	8:5	409.6
A	5:3	426.666
A# / Bb	16:9	455.111
B	15:8	15, 30, 60,120, 240, 480, 960

SCALE C4 – C5

C	C# Db	D	D# Eb	E	F	F# Gb	G	G# Ab	A	A# Bb	B
256	273	288	307	320	341	360	384	409.6	426.6	455	480

Blog » Myth ⚠ 432Hz & The Speed of Light

Reading Time: 4 minutes

November 20, 2013

MYTH ⚠ 432Hz & THE SPEED OF LIGHT

“432 IS THE SQUARE ROOT (√) OF THE “SPEED OF LIGHT”.
(PSEUDOSCIENCE)

Some people like to suggest that 432Hz is the most natural concert pitch due to a presumed relation to the speed of light:

“432 is the square root of the Speed of Light”.

432 = √186624

The calculation above is correct … but does NOT make the statement right.

The actual speed of light is not exactly 186624, but (depending on the source) 299792458 meters/s. = 186282.397 miles/s. That is about 341.603 miles per second off. If you would use that number instead to calculate the square root, you get:

√186282.397 = 431,60444506515453634375

Some sources (nasa.gov) mention 186000 miles/s. If we use this number to calculate the square root:

√186000 = 431,277173057

In both examples above we speak about the Speed of Light in a vacuum.

The difference between 432 and 431.6 or 431.2 might look insignificant, but I am pretty sure a math or physics teacher would not except “432” as answer, if the “assignment” was to calculate the square root of the Speed of Light (186282.397 or 186000 miles/s).

Besides that, only if we use miles as unit of length for the Speed of Light, we come close to 432 when calculating the square root. Usage of the mile as an official unit of measurement is largely confined to the United Kingdom, the United States, and Canada. If we would use other units, like meters (√299792458 = 17314,51581766) or yards (√327857018.81 = 18106,822438241), or other miles, we would not end up with ≈432 as the square root.

USING UNITS …

Another thing we can say about this theory, is that – if you want to compare or “relate” two “things” with one another – you have to use values that work for/with both ‘things’.

But we are not!

We are trying to relate the unit Hz (a particular number of occurrences of a repeating event per unit time) to the unit of speed “miles per second” (the magnitude of the velocity – the rate of change of its position)

After all, we are not talking about 186282.397Hz, but 186282.397 miles/s. when we speak about the Speed of Light.

The unit “speed” and “Hz” can not simple be related, compared or exchanged 1 on 1 by one another!

CONDITIONS & DIFFERENCES SOUND AND LIGHT WAVES

Only when electromagnetic waves (radio, microwave, infrared, the visible spectrum we perceive as: visible light, ultraviolet, X-rays, and/or gamma rays) propagate* through a VACUUM, the electromagnetic waves “propagates” with the Speed of Light, at 186282.397 miles/second. Under other conditions (through a medium) light propagates slower.

The properties of a sound wave always depend on the properties of the medium it travels through: no medium (vacuum) = no Sound. Sure, space isn’t a complete vacuum (specially gas clouds could be counted as “medium”), nonetheless … The speed of sound in air (c) = 343 meters/s at 20°C. The speed of sound through air could vary slightly, depending on the temperature, pressure and humidity. The speed of sound in water is about 4 times faster than this.

Do you see the first “problem” we run into?

We practically try to relate a particular frequency through a medium (air) to the Speed of Light (186282.397 m/s.) in a “vacuum” (= no medium) in which the speed of sound waves would be pretty much “0” (nonexistent)!!!

OTHER DIFFERENCES:

The frequencies of visible light and audible sound differ from each other by more than ten orders of magnitude. Audible acoustic range: roughly 20 Hz to 20,000 Hz vs. visible optical range: roughly 380 trillion Hz to 760 trillion Hz.
Light waves are composed of transverse waves in an electromagnetic field, while sound waves are mechanical longitudinal waves (alternate compression and expansion of matter).
The denser the medium, the greater the speed of sound. The opposite is true of light.
The speed of light in a medium is constant. The velocity of sound waves can change.
Electromagnetic waves, including light is a “stream of particles” (photons). Sound does not consist out of particles.
Light waves can be polarized, but sound waves cannot.

CONCLUSION?

When simply looking at the numbers “432” and “186624” (186282.397) without proper context, one could easily draw wrong conclusions.

No matter how we look at it, there is no real direct relationship between the tone / frequency of 432Hz and the Speed of Light. The “432Hz & the Speed of Light” story is a pseudo-scientific fantasy.

IS THERE NO OTHER RELATIONSHIP BETWEEN SOUND AND LIGHT?

The answer to this question is … yes there is!
You can read more about it in blog articles: “Sound & Light (colour)” and “Sound (256Hz & 432Hz) and Colors“.

REFERENCES:

NOTE: The wavelenght Calculator linked to above, uses meters/s not miles/s, so you first have to convert miles to meters.

JOHN COLTRANE’S TONE CIRCLE

FOREWORD

SHORT INTRODUCTION ABOUT 12-TONE CIRCLES

COLTRANE’S TONE CIRCLE

CIRCLE VERSIONS

WHAT ABOUT ALL THOSE TONES IN BETWEEN?

“Double Power”

WHY HAVE TONES BEEN CIRCLED?

DIMINISHED SCALES

ALTERED DOMINANT (ALT DOM) CHORDS

NATABHAIRAVI-CHARUKESI (NATURAL MINOR + MELODIC MAJOR) “COMPOUND” SCALE

ALL-INTERVAL TETRACHORD

ALL-TRICHORD HEXACHORD

WHAT DO THE NUMBERS IN THE DRAWING MEAN?

FLOWER OF LIFE (61)

JOHN COLTRANE’S MUSIC & GEOMETRY

REFERENCES MENTIONED IN THIS ARTICLE:

The Astrological Zodiac & Musical Tonality – Roel’s Interpretation

WHAT CIRCLE TO USE AND WHAT TONE TO START WITH?

THE CHROMATIC CIRCLE OR THE CIRCLE OF FIFTHS?

THE TROPICAL YEAR AND THE CYCLE OF THE SEASONS.

ARE THERE ANY SIMILARITIES BETWEEN THE TONE CIRLCE AND SEASONS?

COMBINING THE CIRCLE OF FIFTHS AND THE ZODIAC CIRCLE

MINOR TONALITIES, the CICLE OF FIFTHS and the SOLAR YEAR.

TONE ZODIACS STEINER & VON GLEICH

COLOR SPECTRUM, TONE/TONALITY (CHROMATICALLY) AND THE ZODIAC SIGNS

RELATED ARTICLES:

THE ASTROLOGICAL ZODIAC & MUSIC TONALITY – ROEL’S INTERPRETATION

ASTRO-MUSIC COMPOSITION

THE ASTRONOMICAL ZODIAC SCALE

THE FUNCTIONS OF THE INTERVALS

TEMPO & PITCH – RHYTHM & HARMONY

TEMPO & PITCH: BPM & CPS (HZ)

RHYTHM, TEMPO & PITCH

THE POLYRHYTHMIC ASPECTS OF HARMONY

DIFFERENCE E.T. AND J.I.

CONCLUSION:

THE FUNCTIONS OF THE INTERVALS (A PHILOSOPHICAL VIEW ON)

FUNCTIONS OF THE DEGREES OF TONES IN THE DIATONIC SCALES

A PHILOSOPHICAL VIEW ON THE INTERVALS

“THE ABOVE“

THE OCTAVE:

THE 7TH’S:

THE 6THS:

THE 5TH:

“THE BORDER“

THE TRITONE, THE “IN BETWEEN”.

THE 4TH:

“THE BELOW”

THE 3RDS:

THE 2ND:

TONIC:

MANKIND, INTERVALS AND PERCEPTION

FOREWORD

THE ZODIAC (INTRODUCTION)

A SHORT HISTORICAL TIME-LINE:

THE ZODIAC SIGNS (SYMBOLS)

A HISTORICAL PERSPECTIVE

PYTHAGORAS (c. 570 – c. 495 BC) – MUSIC OF THE SPHERES

THE TONE ZODIAC (A HISTORICAL TIME-LINE)

PTOLEMY (c. 100 – c. 168 CE)

MARSILIO FICINO (1433-1499)

RUDOLF STEINER (1861–1925)

MAX HEINDEL (1865-1919)

MAX HEINDEL TONE-ZODIAC “REPRODUCTION”

ANNY VON LANGE (1887-1959)

SIGISMUND VON GLEICH (1896-1953)

MARIUS SCHNEIDER (1903-1982)

ALAIN DANIÉLOU (1907-1994)

STOCKHAUSEN (1928-2007)

HENSCHEL & HENSCHEL-FISHER

MCMULLIN

ASTROLOGICAL ASPECTS, SIGN-GROUPS, MODES AND DEGREES

JOHANNES KEPLER (1571-1630)

RUDOLF HAASE (1920–2013)

SUMMERY HISTORICAL TIME-LINE

GEOMETRY & GROUPS: POLARITIES, TRIPLICITIES, QUADRUPLICITIES & DUALITIES

CIRCLE OF FIFTHS & THE ZODIAC CIRCLE and their SHAPES, DEGREES, INTERVALS:

THE CHROMATIC SCALE, THE PIANO AND THE MONTHS OF THE YEAR

MUSIC & GEOMETRY