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October 27, 2013

FIBONACCI 432Hz TEMPERAMENT

This is the 432Hz related abstract of the blog article “Fibonacci & Tuning“. For more information and alternative temperaments do read the full article!

8-FIBONACCI (TEMPERAMENT)

Ratios made with the first 5 (unique) numbers of the Fibonacci series (1, 2, 3, 5, 8) are related to key intervals of musical temperaments:

  • 1/1 = Tonic
  • 2/1 = Octave
  • 2/3 = Just Fourth
  • 3/2 = Just Fifth
  • 3/5 = Just minor Third
  • 5/3 = Just Major Sixth
  • 5/8 = Just Major Third
  • 8/5 = Just minor Sixth

Some of the 13 tone interval ratios of 12-Tone scale contain numbers not found in the Fibonacci series, like the Perfect Fourth with ratio: 4/3. But you can use alternative mathematical formulas to replace those with using only Fibonacci numbers, in the case of the Perfect Fourth you could use 2/3·2 or 2/3 (8va). 8va = ‘ottava’ = transpose an octave up.

For my 12-Tone “8-Fibonacci” I will use the numbers 1, 2, 3, 5 and 8 for the ratio formulas.

WHEN USING A4=432HZ AS BASE:

Ratios with
Fibonacci Numbers
 Note in
Scale		 Musical
Interval		 When
A4=432Hz
1/1 A3 Tonic (1/1) 216Hz
28/35 A#/Bb Pythagorean Minor Second (256/243) 230.4Hz

5·232

32/8

 B

Pythagorean Major Second (10/9)

Just Major Second (9/8)

240Hz

243Hz
3/5 (8va) C Just Minor Third (6/5) 259,2Hz
5/8 (8va) C#/Db Just Major Third (5/4) 270Hz
2/3  (8va) D Just Fourth (4/3) 288Hz

32·5/25

√2/1

D#/Eb

Just Tritone (45/32)

Equal-Tempered Tritone (26/12)

303,75Hz

305,470…Hz
3/2 E Just / Pythgorean Fifth (3/2) 324Hz
8/5 F Just Minor Sixt (8/5) 345,6Hz
5/3 F#/Gb Just Major Sixt (5/3) 360Hz

24/32

32/5

 G

Pythagorean Minor Seventh (16/9)

Just Minor Seventh (9/5)

384Hz

388,8Hz
·5/8 G#/Ab Just Major Seventh (15:8) 405Hz
2/1 A4 Octave (2/1) 432Hz

WHAT ABOUT THE OTHER FIBONACCI NUMBERS?

Well if we use the next 2 numbers of the sequence for ratios we can add several more tones. The ORANGE bars are the tones related to number 13, the PURPLE bars to number 21. Here are a few …

Ratios with
Fibonacci Numbers		 Note in
Scale		 Musical
Interval		 When
A4=432Hz
21/5 (-15ma) A#↓/Bb↑ ? 226,8Hz

13/3 (-15ma)

 B↓ ?

235Hz
3/21 (22ma) B↑ ? 246,857…hz
8/13 (8va) C↑ ? 265,846…Hz
13/21 (8va) C#↓ ? 267,428…Hz
13/5 (8va) Db↑ ? 280,8Hz
21/8 (-8va) D↓
 ? 283,5Hz
8/21 E↑ ? 329,142…Hz
5/13 (15ma) F↓ ? 332,307…Hz
21/13 F↑ ? 348,923Hz
13/8 F#↓ ? 351Hz

21/3 (22ma)

 G↓ ?

378
3/13 (22ma) Gb↑/G#↓ ? 398,769…
5/21 (22ma) G#↑ ? 411,428…Hz

The arrows behind the tones (in column 2) tell you if the tones are a bit sharper (↑) or flatter (↓) in relationship to “8-Fibonacci”.

Naturally if you continue using more numbers of the Fibonacci Sequence (34, 55, …) you will be able to add many more tones in between those listed above … FIBONACCI TEMPERAMENTS:

  • 8-Fibonacci (1, 2, 3, 5, 8)
  • 13-Fibonacci (1, 2, 3, 5, 8, 13)
  • 21-Fibonacci (1, 2, 3, 5, 8, 13, 21)
  • et cetera.

This was the 432Hz related abstract of the blog article “Fibonacci & Tuning“. For more information and alternative temperaments do read the full article!

REFERENCES & CREDITS:

Banner images “Fibonacci Spiral” by Rahzizzle

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