Blog » Music Composition & Math (Numbers & Sequences)

Reading Time: 7 minutesJuly 1, 2016

MUSIC COMPOSITION & MATH (NUMBERS/SEQUENCES)

“May not music be described as the mathematics of the sense, mathematics as music of the reason?
The musician feels mathematics, the mathematician thinks music:
music the dream, mathematics the working life.“

~ James Joseph Sylvester

Most people that have visited this blog before have come across various music and math related stuff already, like for example Music & Geometry. In this article I will share some interesting and fun-stuff about music composition and math. Let me start in this article with something I wrote in another article:

MUSIC = MATH
Music can be explained with math formulas.

How?

We use ratios – for example 2:1 (Octave) and 3:2 (Perfect 5th) – to “size-up” the intervals between 2 tones (more about that in the article “Music & Geometry” at Roel’s World).
A system of combined interval ratios generates a musical Temperament (Roel’s World).
With 3 or more tones a geometric relationship can be formed (Roel’s World article: “Music & Geometry“).
By using number sequences (consecutive intervals and/or patterns/cycles).

Temperaments and Tuning Systems, Geometry … all cool and so. But what about composing music with Mathematical sequences and patterns?

NUMBER SEQUENCES

PI ( $π)$

$The number π (Pi) is a mathematical constant, the ratio of a circle’s circumference to its diameter, commonly approximated as 3.14159… Being an irrational number, π cannot be expressed exactly as a fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Here you can find the first one million digits. There are various musicians and composers that have experimented with using the digits of Pi as melody for their composition. Here’s one of the examples, a implementation by David Macdonald:$

PHI (φ)

Next up is Phi (φ) = 1.61803398874989484820458683436563811772… Phi is used as a symbol for the golden ratio and on other occasions in math and science.

Phi has been a source of inspiration for some musicians and composers. Here is an example by Michael Blake:

FIBONACCI SEQUENCE

The Fibonacci Sequence (0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …) is named after Italian mathematician Leonardo of Pisa, known as Fibonacci. His 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier as Virahanka numbers in Indian mathematics. Here you can find the first 1000 digits.

In the example below Radomir Nowotarski has “related” the Fibonacci Sequence to the Lydian Mode (scale) and made the following assignment:

1=C, 2=D, 3=E, 4=F#, 5=G, 6=A, 7=B, 8=C, 9=D, 10=E, 11=F#, 12=G, 13=A et cetera. (see video).

1-8 are the intervals of the Scale (tonic-octave). The following numbers 9, 10, 11 (et cetera) represent intervals greater then the octave and have been added to the table to complete it with all possible intervals related to the scale used. Numbers like 4, 6, 7, 9, 10, 11, 12 (et cetera) are not part of the Fibonacci frequencies and haven’t been used for constructing the melody. This does not mean the tones related to those intervals are not used. “6=A” (not used) as well as “13=A“, as well as for example 34=A, 55=A, while 89=G (et cetera, see video).

This assignment can be applied with any other 7-tone scale (by replacing the 7 tones listed in the first column) or you could use the concept to build a table for a scale with a different number of tones.

At Roels World you can find two more articles about Fibonacci: Fibonacci Temperament and Fibonacci Tones.

NUMBER SEQUENCES & PROGRESSIONS

There are many more sequences you could use. For example:

Arithmetic progression (a sequence of numbers such that the difference between the consecutive terms is constant, in this example the difference is “+3“: 1, 4, 7, 10, 13, 16, 19, 22, 25, …)

Geometric progression (a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio, in this example “x3“: 1, 3, 9, 27, 81, 243, …).

Triangular number sequence (counts the objects that can form an equilateral triangle):
0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, …

Square number sequence (the product of some integer with itself. 9 is a square number, since it can be written as “ $3\times3 “$ . – Sequence example: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, …)

Cube number sequence (the cube of a number is its third power: 1³=1, 2³=8, 3³=27, et cetera – the result of the number multiplied by itself twice. It is also the number multiplied by its square. – Sequence example: 1, 8, 27, 64, 125, 216, 343, 512, 729, …)

There are many more sequences, a complete list of sequences can be found at the “On-Line Encyclopedia of Integer Sequences® (OEIS®)“. This sequence database was begun by Neil J. A. Sloane (henceforth, “NJAS”) in early 1964 when he was a graduate student at Cornell University in Ithaca, NY. He noticed that although several books in the Cornell library contained sequences. In order to keep track of the sequences in these books, NJAS started recording them on file cards.

Of course instead of using an existing irrational number or known sequence you could also use a “personal” number, like your telephone number or your social security number or the grades on your diploma (or what ever inspires you) to compose a piece of music. 😉

DODECAPHONY (“Twelve-tone technique” or “twelve-note composition”)

Dodecaphony is a method of musical composition devised by Austrian composer Arnold Schoenberg (1874–1951). The technique is a means of ensuring that all 12 notes of the chromatic scale are sounded as often as one another in a piece of music while preventing the emphasis of any one note through the use of tone rows, orderings of the 12 pitch classes. All 12 notes are thus given more or less equal importance, and the music avoids being in a key.

Now, I could continue writing about it, but instead I’ll post a lovely video about it by Vi Hart. This lovely girl has made many cool and funny math-related videos in a truly original manner (something I highly appreciate and admire in people). She does talk a lot and up-tempo in her videos though, brace yourself! 😉 Hihihi

If you liked what you saw and heard, then you might enjoy many of her other videos as well …

RETROGRADE, INVERSION AND RETROGRADE-INVERSION

You can allow yourself also some creative liberties and use the sequences also reversed or “retrograde” [ R ](backward), inversion [ I ] (vertically mirrored) and retrograde-inversion [ IR ] (a combination of both).

Here an examples with a Requiem Canticles by Igor Stravinsky:

INTERESTING VIDEO ABOUT MUSIC & MATH (SYMMETRY):

“From Pythagoras’ observations of the fundamental mathematical relationship between vibrating strings and harmony to the digitized musical world we enjoy today, The Majesty of Music and Mathematics with the Santa Fe Symphony and the Santa Fe Institute will explore the remarkable interweaving of the languages of music and mathematics.”

USING THE NUMEROLOGIC VALUE OF LETTERS TO CREATE A TONE-SEQUENCE

NOTE: Not everyone is equally “interested” in this topic, so I have left it for the end of the article. If this is not “your thing”, then feel free to return to the top (menu) of Roel’s World.

Numerology is any belief in the divine, mystical relationship between a number and one or more coinciding events. It is also the study of the numerical value of the letters in words, names and ideas.

There are various numerology systems which assign numerical value to the letters of an alphabet, including the Abjad numerals in Arabic, the Hebrew numerals, Armenian numerals, Cyrillic numerals, and Greek numerals.

There are also non-alphabetical systems, such as Chinese Numerology where numbers are associated with particular connotations (based on its sound).

Different methods of calculation exist, including Chaldean, Pythagorean, Hebraic, Helyn Hitchcock‘s method, Phonetic, Japanese, Arabic and Indian.

One of the more common assignments is the Pythagorean Method (1-9):

1	2	3	4	5	6	7	8	9
☉	☽	♃		☿	♀		♄	♂
A	B	C	D	E	F	G	H	I
J	K	L	M	N	O	P	Q	R
S	T	U	V	W	X	Y	Z

Another polular assignments is the Chaldean (Indian) Method (1-8):

1	2	3	4	5	6	7	8
☉	☽	♃		☿	♀		♄
A	B	C	D	E	U	O	F
I	K	G	M	H	V	Z	P
J	R	L	T	N	W
Q		S		X
Y

And for the Bulgarian visitors of my blog also the Bulgarian Cyrillic versio:

1	2	3	4	5	6	7	8	9
А	Б	В	Г	Д	Е	Ж	З	И
Й	К	Л	М	Н	О	П	Р	С
Т	У	Ф	Х	Ц	Ч	Ш	Щ	Ъ
Ь	Ю	Я

Some Numerologers (like Kathleen Roquemore) also add “Master Numbers” to the table:

1	2	3	4	5	6	7	8	9	11	22
A	B	C	D	E	F	G	H	I
J		L	M	N	O	P	Q	R	K
S	T	U		W	X	Y	Z			V

THE IMPLEMENTATION

This is an original concept and has not been used/implemented as far as I know. If you can show me a composition made with this concept predating June 1 2016, then I will provide the credits.

As example I will use the name of this blog: Roel’s World for the calculations. For the first row I have used the Pythagorean method, for the second row the Chaldean method.

R	O	E	L	S	(SUM)	W	O	R	L	D	(SUM)
9	6	5	3	1	9+6+5+3+1=24 2+4=6	5	6	9	3	4	5+6+9+3+4=27 2+7=9
2	7	5	3	3	2+7+5+3+3=20 2+0=2	6	7	2	3	4	6+7+2+3+4=22 2+2=4

I will not write about the Numerological interpretation (meaning of the sum), there are enough websites and blogs about that.

How to relate the tones to numbers related to the numbers they belong with? For this example I will use the scale of C Major, but you could use any scale you with. Let’s “map” the scale.

1	2	3	4	5	6	7	8	9
C	D	E	F	G	A	B	C	D

When using the Pythagorean method the melody would progress as follows:

D – A – G – E – C G – A – D – E – F

When using the Chaldean method the melody would progress as follows:

D – B – G – E – E A – B – D – E – F

You could turn these tones also into a chord progression if you like, in the case of the Pythagorean Method the chord progression could be something like:

Dm – Am – G(7) – Em – C G(7) – Am – Bdim – Dm – Em – F

Another option is to give every word of a lyrics it’s own chord, based on the tones related to the letters of the words. This way you can create a word-compatible chord progression. With words longer then 5 or 6 letters one could devide the tones into two progressive chords or use the additional tones for creating a bass-line and or melody.

ROELS: C Major 6,9 (C-E-G-A-D) WORLD: D minor 9,11 (D-F-A-E-G)

These are just a few different ideas for implementing this concept. A combination of these concepts might also be an option and perhaps you come up with yet another way to implement it.

You like to read something about music and Geometry? Then check the article “Music & Geometry“.

Blog » Music & Geometry

Reading Time: 2 minutes July 1, 2016

MUSIC & GEOMETRY (The Sacred Geometry Of Music)

“Geometry is knowledge that appears to be produced by human beings, yet whose meaning is totally independent of them.”
– Rudolf Steiner

A note to start this article with:

Perfect geometric shapes only occur when using Equal Temperament. Sonically though the intervals of Equal Tempered instruments/music aren’t perfect (accept for the Octave). They are always a bit sharp or flat in comparison with the Harmonic Series (how sound occurs in nature) and Just Intonation.

The most harmonious sounding (Just) intervals (the octave excluded) do NOT create perfect symmetrical geometric shapes!
If you “stack” one or more “type” of Just Intervals you will always end up with a spiral!

In this article all geometrically perfect shapes (trigone, square, pentagon, et cetera) thus relate to 12-Tone Equal Temperament as do any and all perfect geometrical shapes used on this blog as well as on any other blog, website and book available.

“Music theory has no self-evident foundation in modern mathematics yet the basis of musical sound can be described mathematically (in acoustics) and exhibits “a remarkable array of number properties”.
In its form, rhythm and metre, the pitches of its notes (intervals) and the tempo of its pulse music can be related to the mathematical measurement of time and frequency, offering ready analogies in geometry.”

Now, of course tones and chords are not the same “things” as for example polygons and polygrams. Units like Hertz and Degrees are not the same, they have their own function and use. But in the math behind many “things” there are formulas and ratios (relationships) that are very similar, if not identical. Would it thus be too ‘far-fetched’ to say that the same “rules” apply for many – if not all – things in the universe?

This article will “zoom-in” into geometry in particular.

Over time this article grew so big that the time it took to load started to have a negative effect SEO-wise. I thus decided to split it up in 4 smaller articles: Intervals & Scales, Harmony, Rhythm and Various.

Intervals &
Scales

Chords & Progressions

Rhythm &
Grooves

Various &
Related

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

Reading Time: 4 minutes

April 20, 2016

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

(by Maria Renold, revised by Peter Stevens & Paul Davis)

In an earlier article on this blog I have written about and explained the “Renold-I” temperament, the first version of the “Scale of Fifths” that Maria Renold made in 1962. If you have not read that article yet, then I suggest you do so, just to be able to place everything in perspective.

After Maria Renold created – what later would be known as the “Renold-I” temperament – she continue experimenting and developing her concept. The result is what is known as the “Renold-II” (also known as “Middle Tuning” temperament.

In this article I will share more information about how the Renold-II temperament works. How this temperament works was explained by Peter Stevens in “A Handbook on the Middle Tuning” (2nd edition). This work is shared (in PDF format) online at www.eurythmy.co.nz. You can download it there to have more information then I will provide in this article.

WHAT IS DIFFERENT ABOUT RENOLD II?

The most important difference between “Renold-I” and “Renold-II” is the type of “Musical Interval System” (the range of notes within the system / the number of actual tones available to use). In the case of “Renold I” a closed system is used, the “Renold-II” uses an “open system“.

What does that mean?

CLOSED SYSTEM (RENOLD-I)

Well, generally with a “closed system” the octaves are perfect doublings: C4=256Hz thus C5=512Hz, a difference of 1200 cents. In this tuning system there are 12 tones per octave and every octave includes a perfect doubling of those 12 tones.

Let’s look at the piano to visualize it. A standard acoustic piano has 88 keys. This covers 7 octaves with 12 tones (=84), +4 semitones.

From from every tone of the 12-tone scale there are 7 (or 8) exact doubles/halves. So, if we take for example the tone C4=256Hz as reference, then we have a perfect doubling at:

C1=32Hz, C2=64Hz, C3=128Hz, C4=256Hz, C5=512Hz, C6=1024HZ, C7=2048Hz and C8=4096Hz.

All these “C’s” are “identical”, they are the same frequencies doubled or halved. When you would play two or more perfect octaves (1200 cents precisely) of for example those “C’s” simultaneously, most people would no longer differentiate between the “C’s” but hear them as one.

Another reason why I call them “identical” is that they are all part of the same harmonic series. If we take for example C1 as the “fundamental” (the first harmonic), then C2 aligns perfectly with the 2nd harmonic of C1, as do all other perfect octaves of C1: C3=4th harmonic, C4=8th harmonic, C5=16th harmonic, C6=32nd harmonic, C7=64th harmonic and C8=128th harmonic. From C2 we could do exactly the same and those “C’s” above C2 would align in the same relative fasion to C2 as the “C’s” above C1 did to C1.

To visualize a closed system you could draw it as a circle.

OPEN SYSTEM (RENOLD-II)

In the case of this “open system” the octave is not a perfect doubling but slightly larger, (1203.35 cents to be precise). The consequence of this is that the 12 tones of the scale no longer “double”. If we set C4=256Hz as reference, then we have the following:

C1=31.815Hz, C2=63.753Hz, C3=127.753Hz, C4=256Hz, C5=512.992Hz, C6=1027.971Hz, C7=2059.923 and C8=4127.927Hz.

The consequence of this is that we end up having 88 “different” tones on the piano. To visualize an open system, you could draw it as a spiral.

RENOLD-II @ A4=432Hz

Below a part of the Renold-II temperament. To add the complete range (C0-C9) would be too much for this page. You can though download the full range of tones and frequencies in a PDF file.

A4=432Hz has been used as reference pitch for the example table in this article and the downloadable PDF file because this temperament is mostly used by members of the 432-community. If you wish to use another concert pitch you have to do “the math” yourself. This is all you need to know for it:

In the 1st column you see “-1203.35 down” and “1203.35 up“. This is the amount of cents up or down from every tone from C4 to B4.

Example: A4=432Hz -1203.35 cents = 215.5824363807 Hz (A3)

In the 2nd column you see a series of cent corrections for the tones C4 – B4 compared to Equal Temperament.
The 3rd column containes the tone names.
The frequencies in the 4th column are C4 to B4 in 432Hz 12-TET
In column 5 you see the corrected tone frequencies (Renold-II).

TONE CORRECTION		TONE	FREQUENCIES
-1203.35 down 1203.35 up		…	…	…
		G♯3 / A♭3	203.88	203.3163932923
		A3	216	215.5824363807
		A♯3 / B♭3	228.84	228.5873823048
		B♭3	242.45	242.3715877002
	-5.874	C4	256.87	255.99992842560002
	-4.47	C♯4 / D♭4	272.14	271.4382481425206
	-3.07	D4	288.33	287.819157041
	-1.68	D♯4 / E♭4	305.47	305.1737138452
	-0.28	E4	323.63	323.5776621474
	-5.59	F4	342.88	341.7746566158
	-4.19	F♯4 / G♭4	363.27	362.3918628923
	-2.79	G4	384.87	384.2502556477
	-1.4	G♯4 / A♭4	407.75	407.4203973981
	0	A4	432	432
	1.4	A♯4 / B♭4	457.69	458.0602706488
	2.79	B♭4	484.9	485.6820789499
		C5	513.74	512.9915541197
		C♯5 / D♭5	544.29	543.9279988027
		…	…	…

IMPORTANT NOTES

The recommended concert pitch for this temperament is C3=128Hz (C4=256Hz & A4=432Hz). Reason of it is explained in Maria Renold’s book on page 69.
The frequencies in the table on in this article are only a “guideline“. For proper results fine tuning should be done by ear (according to scheme included in Maria Renold’s book). For tuning by ear three tuning fork are required: C4=256Hz, A4=432Hz and “gelis” (F#/Gb)=362,40Hz.
It might be good to mention that the Renold-II temperament doesn’t sound (as well as Renold-I) on synthesizers and other electronic devices! The Renold-II temperament does work well for acoustic instruments.

DOWNLOADS (MIDI & SOFTSYNTH SCRIPTS)

Making a script for MIDI & SOFTSYNTH with Scala turns out to be a bit tricky … have not found time yet to create it … or forgot about it …

REFERENCES:

“Intervals, Scales, Tones and the Concert Pitch C=128Hz” by Maria Renold
“A Handbook on the Middle Tuning” by Peter Stevens

Special thanks to Sławek Kędra providing additional information.

Blog » The Astronomical Constellational Zodiac Scale

Reading Time: 9 minutesMarch 17, 2016

THE ASTRONOMICAL (CONSTELLATIONAL) ZODIAC SCALE

The “Astronomical Zodiac Scale” (musical temperament) is a creation by Roel Hollander.

Feel free to use it for your musical compositions or quote from it for own work, but please, do place a reference to this article when you do (see licence at the bottom of this page).

13

The Zodiac represents the constellations that the Sun passes through in its apparent path across Earth’s sky. Because the Sun (and the planets) are all on about the same plane in the Solar System, they pass through the same constellations. While traditionally the zodiac is considered to have 12 constellations, technically the Sun passes through 13 at present time.

The reason that there are now 13 constellations the Sun passes through according to various sources is that the axis of the Earth has changed over the millennia. What is important to understand, is that the The zodiac is a belt of space which extends 9° either side of the ecliptic. All the observable planetary activity in our solar system takes place within this narrow celestial belt. Due to a “wobble” of the Earth (one full turn with the “Great Year“) Ophiuchus isn’t always in between Earth and the Sun. As a result of this “wobble” the direction of north has changed with respect to the sky. Vega was the North Star (Pole Star) several thousand years ago, and will become it again in about 13,000 years, according to NASA. Today, the North Star is Polaris.

IMPORTANT: The tropical and sidereal zodiac (and the division of the ecliptic into a number of “signs” named after constellations) is based on the path of the Sun when it’s in between Earth and the Zodiac Constellations, as well as in the case of Tropical Astrology the 4 seasons on Earth (with the Equinoxes and the line of the Solstices forming a cross) while the sidereal system defines the signs based on the fixed stars. The 12 signs used in tropical and sidereal astrology are the so called “Sun signs”, Ophiuchus (the 13th) is not.

13 CONSTELLATIONAL vs 12 ASTROLOGICAL (TROPIC/SIDEREAL) ZODIAC SIGNS
	The Astrological or Tropical (aligns with the equinoxes and the solstices) / Sidereal (alignes with the constellations) Zodiac divides the year into 12 Equal periods (this correlates with the 12-Tone Equal Temperament). The Astronomical or Constellational Zodiac depicts the number of days of the Earth’s orbit when the Sun is between Earth and any one of the zodiacal constellations. The Astronomical Zodiac counts 13 constellations, the 13th is named “Ophiuchus“
Since each constellation is of different size, and because the ecliptic (the sun’s path) passes through larger or smaller portions of each constellation and the speed of the Earth around the Sun varies along its orbit, the Sun is between the Earth and each zodiacal constellation for varying periods. Varying (unequal) periods = unequal sectors = unequal interval size!
Another example of the difference between the mapping of the Tropical Zodiac and the actual time that the Sun travels through the Constellations can be seen on the image on the right I found on the internet.
The red dashed line shows the path of the Sun (the Ecliptic) crossing the Constellations. NOTE: In the image on the right the Sun moves from right to left! The “time-line” in this image starts at the Spring Equinox with Aries (Astrology) and Pisces (Astronomically) and ends after the Winter with Pisces (Astrology) and Aquarius (Astronomy).

ROEL’S WORLD IMPLEMENTATION:

For the length of the days per sign I have used calculations made by Juan Anthony Revilla with data from an article by Pierre Bacchus in “L’Astronomie” June 1979, pp.297-99.

The exact number of days per Zodiac sign could differ a bit (that depends on the source used and the “type” of year you use for your calculations). A 24-hour difference will effect the exact outcome, but I estimate those differences to be up to approx. 1 day per sign. The frequencies thus mentioned in this article are therefor an approximation of the tone frequencies and might differ up to 1Hz.

To calculate tone-span in cents per Zodiac signs, you have to divide 1200 cents (the distance between a Tonic and the Octave) by the number of days per year and multiply that with the number of days as was calculated. Now, I could use the round number of 365 days for it, but I have decided to use the Anomalistic Year for my calculations.

The Anomalistic year is the time taken for the Earth to complete one revolution with respect to its Apsides. The orbit of the Earth is elliptical; the extreme points, called apsides, are the Perihelion, where the Earth is closest to the Sun (January 3 in 2011), and the Aphelion, where the Earth is farthest from the Sun (July 4 in 2011). The Anomalistic year is usually defined as the time between Perihelion passages. Its average duration is 365.259636 days (365 d 6 h 13 min 52.6 s) at the epoch J2011.0).

1200 (cents) : 365.259636 (days of the Anomalistic year) = 3.285334271 cents per day (24 hours).

You could also use the Sidereal or Tropical years. In comparison:

1200 (cents) : 365.25636 (days of the Sidereal year) = 3.285363737 cents per day (24 hours).
1200 (cents) : 365.24219 (days of the Tropical year) = 3.285491197 cents per day (24 hours).

Sign	Constellation	Date Span	Days	Tone-span (cents)
	Aries	19 Apr – 13 May	25.48	83.710317225
	Taurus	14 May – 20 Jun	38.31	125.861155922
	Gemini	21 Jun – 20 Jul	29.20	95.931760713
	Cancer	21 Jul – 10 Aug	20.96	68.86060632
	Leo	11 Aug – 15 Sep	37.05	121.721634741
	Virgo	17 Sep – 30 Oct	44.51	146.230228402
	Libra	31 Oct – 22 Nov	23.10	75.89122166
	Scorpio	23 Nov – 29 Nov	6.52	21.420379447	= 81.574849949
	Ophiuchus	30 Nov – 17 Dec	18.31	60.154470502	= 81.574849949
	Sagittarius	18 Dec – 19 Jan	32.55	106.937630521
	Capricorn	20 Jan – 15 Feb	27.44	90.149572396
	Aquarius	16 Feb – 11 Mar	24.06	79.04514256
	Pisces	12 Mar – 18 Apr	37.51	123.232888505

This is how the division of the circles look like with 13 & 12 signs::

⇣ 13 SIGNS THE ASTRONOMICAL TONE ZODIAC 12 SIGNS ⇣
	Because we use a 12-tone scale in Western music, the 13th sign (Ophiuchus) and Scorpio (who in the Astronomical Zodiac only has span of 7 days) will be “merged“.

The interval created by just Scorpio alone would be too small (approximately 23 cents) to be of use anyway, while combined with Ophiuchus we end up with a interval of approximately 82 cents, in between a “Hard Semitone” and a “Septimal chromatic semitone“.

I have chosen to start with C4=256Hz (the “Scientific Pitch“) for this temperament, but you are naturally free to use another pitch for C4 to start with. When you add the results from the tables above (combined with the Circle Of Fifth based Tone Zodiac”) to the chromatic scale, we get the following result:

Sign	Constellation	Tone	Cents To Next Tone	Frequency (Hz)
	Aries	C		256	256
			83.710317225
&	Scorpio & Ophiuchus	C♯/D♭		268.6825093184329	269
			81.574849949
	Gemini	D		281.6457007820792	282
			95.931760713
	Capricorn	D♯/E♭		297.69285251441477	298
			90.149572396
	Leo	E		313.60514470726037	314
			121.721634741
	Pisces	F		336.44808806172546	336
			123.232888505
	Libra	F♯/G♭		361.2701348994703	361
			75.89122166
	Taurus	G		377.45919239188333	377
			125.861155922
	Sagittarius	G♯/A♭		405.92268794831494	406
			106.937630521
	Cancer	A		431.7869586242998	432
			68.86060632
	Aquarius	A♯/B♭		449.30761351210754	449
			79.04514256
	Virgo	B		470.29775997051314	470
			146.230228402

THE ASTRONOMICAL (SIDEREAL) ZODIAC SCALE:

SIGN
TONE	C	D♭	D	E♭	E	F	F♯/G♭	G	A♭	A	B♭	B	C
HZ	256	269	282	298	314	336	361	377	406	432	449	470	512

IMPORTANT NOTE: Now, it seems fair to warn those who have no “experience” with micro-tonal music that the temperament created with this concept might sound “strange” to your ears!

This temperament might not work well in all tonalities or for music with complex harmonic structures. The implementation of this temperament works best for Ambient music and for music with “meditative” purposes.

In case you are fairly new to musical tuning systems and temperaments, perhaps you should read this short explanation first.

WHAT DOES THE ASTRONOMICAL ZODIAC SCALE SOUND LIKE?

Image used for the article banner and Soundcloud player by: whybecausescience.com

Implementation (with Scala) and music by Derrick Scott van Heerden.

Tuning files (256/432Hz): .scl file (Scala) | .tun file
Tuning files (440Hz): .scl file (Scala) | .tun file

THIS TEMPERAMENT & 432HZ

Those of you who are interested in 432-Tuning might have already noticed that we find A4 rounded-up at 432Hz (431.7869586242998Hz). Just to place this number in perspective:

24 hours = 3.285334271 cents, 1 hour = 0,136888928 cents, one minute = 0,002281482 cents. 432Hz – 431.7869586242998Hz = 0,213041376 cents.

This means we speak about a difference of:

1 hour and 33,3 minutes (0.02%) between 431.786… and 432Hz.

A4=432Hz and C4=256Hz are though the only two tone-frequencies that the Astronomical Zodiac Scale has in common with the standard 432-Tuning (Pythagorean Temperament). With 12-TET@432 there are two other tones that are not too far off F♯/G♭ (363.3Hz) and G♯/A♭ (407.7Hz).

In other words, this is a unique temperament, an other “cosmic connection” to 432Hz then what you have seen and heard so far.

BIRTH-TONE PROJECT (ONLINE FREQUENCY GENERATOR)

The Birth-Tone Project is an interactive birth chart frequency generator created by David Sutrin (Podcast Producer and Mix Engineer). Planetary location data is used to produce a unique tone based on the birthday, time, and location of the user. Each frequency is produced using a conversion system that assigns a frequency(Hz) value to each of the 12 zodiac signs.

This is an original concept by Roel Hollander and falls under the following license:

Blog » John Coltrane’s Tone Circle

Reading Time: 16 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). <- You might need to read that twice. 😉

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: 15 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:

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.

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:

THE 12 ZODIAC SIGNS WITH THEIR MAJOR & MINOR TONALITIES

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.

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: 11 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 sources claim the interval was forbidden to be used by the Church. This is though just a myth.

“THE BELOW”

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 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: 3 minutesMarch 3, 2014

The Astrological Zodiac & Musical Tonality

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:

Part 3: Aspects, Grouping, Modes & Degrees

OTHER SOURCES / REFERENCES: PART 1-4

“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


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

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.		D	2♯
	90°	Cancer	22 Jun. – 22 Jul.	21.06 Summer Solstice*	A	3♯
	120°	Leo	23 Jul. – 22 Aug.	04/06.07 Aphelion	E	4♯
	150°	Virgo	23 Aug. – 23 Sep.		B	5♯
	180°	Libra	24 Sep. – 23 Oct.	23.09 Autumn Equinox*	F♯/G♭	6♯/6♭
	210°	Scorpio	24 Oct. – 22 Nov.		Db	5♭
	240°	Sagittarius	23 Nov. – 21 Dec.		Ab	4♭
	270°	Capricorn	22 Dec. – 20 Jan.	21.12 Winter Solstice*	Eb	3♭
	300°	Aquarius	21 Jan. – 19 Feb.	04/06.01 Perihelion	Bb	2♭
	330°	Pisces	20 Feb. – 20 Mar.		F	1♭

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.		Bm	2♯
	90°	Cancer	22 Jun. – 22 Jul.	21.06 Summer Solstice*	F♯m	3♯
	120°	Leo	23 Jul. – 22 Aug.	04/06.07 Aphelion	C♯m	4♯
	150°	Virgo	23 Aug. – 23 Sep.		G♯	5♯
	180°	Libra	24 Sep. – 23 Oct.	23.09 Autumn Equinox*	D♯m E♭m	6♯/6♭
	210°	Scorpio	24 Oct. – 22 Nov.		B♭m	5♭
	240°	Sagittarius	23 Nov. – 21 Dec.		Fm	4♭
	270°	Capricorn	22 Dec. – 20 Jan.	21.12 Winter Solstice*	Cm	3♭
	300°	Aquarius	21 Jan. – 19 Feb.	04/06.01 Perihelion	Gm	2♭
	330°	Pisces	20 Feb. – 20 Mar.		Dm	1♭

MUSIC COMPOSITION & MATH (NUMBERS/SEQUENCES)

NUMBER SEQUENCES

USING THE NUMEROLOGIC VALUE OF LETTERS TO CREATE A TONE-SEQUENCE

THE IMPLEMENTATION

MUSIC & GEOMETRY (The Sacred Geometry Of Music)

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

WHAT IS DIFFERENT ABOUT RENOLD II?

CLOSED SYSTEM (RENOLD-I)

OPEN SYSTEM (RENOLD-II)

RENOLD-II @ A4=432Hz

DOWNLOADS (MIDI & SOFTSYNTH SCRIPTS)

THE ASTRONOMICAL (CONSTELLATIONAL) ZODIAC SCALE

13

ROEL’S WORLD IMPLEMENTATION:

THE ASTRONOMICAL (SIDEREAL) ZODIAC SCALE:

THIS TEMPERAMENT & 432HZ

BIRTH-TONE PROJECT (ONLINE FREQUENCY GENERATOR)

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.

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 BELOW”

THE 4TH:

THE 3RDS:

THE 2ND:

TONIC:

MANKIND, INTERVALS AND PERCEPTION

The Astrological Zodiac & Musical Tonality

FOREWORD

THE ZODIAC (INTRODUCTION)

A SHORT HISTORICAL TIME-LINE:

THE ZODIAC SIGNS (SYMBOLS)

LIGHT & SOUND, COLOUR & MUSIC

EARLY RESEARCH

SENSATION OF SOUND AND COLOUR AND THEIR “DIMENSIONS”

WHAT DO SOUND AND LIGHT HAVE IN COMMON?

WHAT IS THE DIFFERENCE BETWEEN SOUND AND LIGHT?

“IN THE MIX”