FIBONACCI TEMPERAMENTS
WHO WAS FIBONACCI?
The name Fibonacci refers to the Italian Mathematician Leonardo Bigollo, he is famous for introducing the Hindu-Arabic form of numbers to the western world in his book Liber Abaci. Although Fibonacci did not originate or develop the sequence he would later become famous for, as the sequence had been discussed earlier in Indian mathematics since the 6th century, he is cited as having used it in an example within the third section of his book. In his example, Fibonacci illustrates the growth of a group of rabbits in an ideal situation, which is where the Fibonacci Sequence had its beginnings.
Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ... et cetera.
The clearest demonstration of Fibonacci being represented in music is seen in scales. 13: the Octave is made of 12 chromatic tones plus 1 the octave. A (basic) scale is composed of 8 notes. The 5th and 3rd notes create basic foundation of chords, based off a whole tone that is 2 steps above root tone, which is the 1st note of scale.
The Fibonacci Sequence, the Golden Ratio, and the Pascal Triangle are closely related.
HOW TO CALCULATE?
Example, when we like to calculate a Just Minor Third (3/5)·2 of for example Concert Pitch 440Hz, we get the following:
(440·3/5)·2 = 528
"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 intervals 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, -8va = transpose an octave down.
For my 12-Tone "8-Fibonacci" Temperament I will use the numbers 1, 2, 3, 5 and 8 for the ratio formulas.
Below I will use C4=256Hz (the "Scientific" or "Philosophic" Pitch), but you could use any other Concert Pitch as well of course. You can find the 432Hz-based version here.
Ratios |
Note in Scale |
Musical Interval |
When C4=256Hz |
1/1 | C^{4} | Tonic (1/1) | 256Hz |
2^{8}/3^{5}^{} | C# / Db | Pythagorean Minor Second (256/243) | 269,695...Hz |
5·2/3^{2}3^{2}/8 | D | Just Major Second (10/9) Pythagorean Major Second (9/8) |
284.444∞Hz 288Hz |
3/5 ^{(8va)}^{} | D# / Eb | Just Minor Third (6/5) | 307,2Hz |
5/8 ^{(8va)}^{} | E | Just Major Third (5/4) | 320Hz |
2/3 ^{(8va)}^{} | F | Just Fourth (4/3) | 341,333∞Hz |
3^{2}·5/2^{5}√2/1 | F# / Gb | Just Tritone (45/32) Equal-Tempered Tritone (2^{6/12}) |
360Hz 362,038...Hz |
3/2 | G | Just / Pythagorean Fifth (3/2) | 384Hz |
8/5^{} | G# / Ab | Just Minor Sixt (8/5) | 409,6Hz |
5/3 | A | Just Major Sixt (5/3) | 426.666∞Hz |
2^{4}/3^{2}3^{2}/5 | A# / Bb | Pythagorean Minor Seventh (16/9) Just Minor Seventh (9/5) |
455,111∞Hz 460,8Hz |
3·5/8 | B | Just Major Seventh (15:8) | 496,875Hz |
2/1^{} | C^{5} | Octave (2/1) | 512Hz |
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 |
Note in Scale |
Musical Interval |
When A4=256Hz |
21/5 ^{(-15ma)} |
C#↓ or Db↓ |
? | 268,8Hz |
13/3 ^{(-15ma)} |
C#↑ or D↓ |
? |
277,333∞Hz |
3/21 ^{(22ma)} |
D↑ or Eb↓ |
? | 292,571...hz |
8/13 ^{(8va)} |
D#↑ or E↓ |
? | 315,076...Hz |
13/21 ^{(8va)} |
D#↑ or E↓ |
? | 316,952...Hz |
13/5 ^{(-8va)} |
E↑ or F↓ |
? | 332,8Hz |
21/8 ^{(-8va)} |
E↑ or F↓ |
? | 336Hz |
8/21 |
G↑ or Ab↓ |
? | 390,095...Hz |
5/13 ^{(15ma)} |
G↑ or Ab↓ |
? | 393,846...Hz |
21/13 |
G#↑ or A↓ |
? | 413,538...Hz |
13/8 |
G#↑ or A↓ |
? | 416Hz |
21/3 ^{(-15ma)} |
A↑ or Bb↓ |
? | 448Hz |
3/13 ^{(22ma)} |
A#↑ or B↓ |
? | 472,615...Hz |
5/21 ^{(22ma)} |
B↑ or C↓ |
? | 487,619...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".
FIBONACCI TEMPERAMENTS:
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.
- 8-Fibonacci (1, 2, 3, 5, 8)
- 13-Fibonacci (1, 2, 3, 5, 8, 13)
- 21-Fibonacci (1, 2, 3, 5, 8, 13, 21)
- 34-Fibonacci (1, 2, 3, 5, 8, 13, 21, 34)
- et cetera ...
REFERENCES:
Banner images "Fibonacci Spiral" by Rahzizzle
WIKIPEDIA
- Fibonacci numbers in Music: http://en.wikipedia.org/wiki/Fibonacci_numbers_in_popular_culture#Music
- Golden Ratio: http://en.wikipedia.org/wiki/Golden_ratio
- Pascal Triangle: http://en.wikipedia.org/wiki/Pascal's_triangle
OTHER
- Fibonacci Sequence: http://oeis.org/A000045
- The Golden Number: http://www.goldennumber.net/music/
- Math 2033 (1): http://math2033.uark.edu/wiki/index.php/Fibonacci_Sequence_and_Music%3F
- Math 2033 (2): http://math2033.uark.edu/wiki/index.php/Fibonacci_Musical_Compositon#Fibonacci_Music
- Golden Ratio: http://math2033.uark.edu/wiki/index.php/Golden_ratio