FIBONACCI "TONES"
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.
NUMBERS TO TONE FREQUENCIES
To be able to compare the Fibonacci numbers to tone frequencies of existing Temperaments we are going to "bring the numbers back" in between 256 and 512 (Hz).
From this list of "frequencies" below you could create many different temperaments (scales):
n | F(n) | DOUBLING (OCTAVES) DIVIDING | TONES |
1, 2 | 1 | 1-2-3-4-8-16-32-64-128-256 | 256Hz |
3 | 2 | 1-2-3-4-8-16-32-64-128-256 | 256Hz |
4 | 3 | 3-6-12-24-48-96-192-384 | 384Hz |
5 | 5 | 5-10-15-30-60-120-240-480 | 480Hz |
6 | 8 |
1-2-3-4-8-16-32-64-128-256 |
256Hz |
7 | 13 |
13-26-52-104-208-416 |
416Hz |
8 | 21 | 21-42-84-168-336 | 335Hz |
9 | 34 | 34-68-136-272 | 272Hz |
10 | 55 | 55-110-220-440 | 440Hz |
11 | 89 | 89-178-365 | 365Hz |
12 | 144 | 288 | 288Hz |
13 | 233 | 233-466 | 466Hz |
14 | 377 | 377 | 377Hz |
15 | 610 | 610-305 | 305Hz |
16 | 987 | 987-493,5 | 493,5Hz |
17 | 1597 | 1597-798,5-399,25 | 399,2..Hz |
18 | 2584 | 2584-1292-646-323 | 323Hz |
19 | 4181 | et cetera | 261,3..Hz |
20 | 6765 |
... |
422,8..Hz |
21 | 10946 |
... |
342,0...Hz |
22 | 17711 |
... |
276,7...Hz |
23 | 28657 |
... |
447,7...Hz |
24 | 46368 |
... |
362,25Hz |
25 | 75025 |
... |
293,0...Hz |
26 | 121393 |
... |
474,1...Hz |
27 | 196418 |
... |
383,6...Hz |
28 | 317811 |
... |
310,3...Hz |
29 | 514229 |
... |
502,1...Hz |
30 | 832040 |
... |
406,2...Hz |
31 | 1346269 |
... |
328,6...Hz |
32 | 2178309 |
... |
265,9...Hz |
33 | 3524578 |
... |
430,2...Hz |
34 | 5702887 |
... |
348,0...Hz |
35 | 9227465 |
... |
281,5...Hz |
36 | 14930352 |
... |
455,6...Hz |
37 | 24157817 |
... |
368,6...Hz |
38 | 39088169 |
... |
298,2...Hz |
39 |
63245986 |
... |
482,528...Hz |
40 |
102334155 |
... |
390,373...Hz |
41 | 165580141 |
... |
315,819...Hz |
42 | 267914296 |
... |
511,005...Hz |
43 | 433494437 |
... |
413,412...Hz |
44 | 701408733 |
... |
334,457...Hz |
45 | 1134903170 |
... |
270,582...Hz |
46 | 1836311903 |
... |
437,810...Hz |
47 | 2971215073 |
... |
354,196...Hz |
48 | 4807526976 |
... |
286,550...Hz |
49 | 7778742049 |
... |
463,649...Hz |
50 | 12586269025 |
... |
375,100...Hz |
... |
... |
... |
... |
51 | 20365011074 |
... |
303,462...Hz |
54 | 86267571272 |
... |
321,371...Hz |
57 | 365435296162 |
... |
340,338...Hz |
64 | 10610209857723 |
... |
308,797...Hz |
... | et cetera |
... |
... |
Now I will compare the frequencies listed above with 4 common 12-Tone Temperaments:
- Pythagorean Tuning + C4=256Hz (A4=432Hz)
- 12-Tone Equal Temperament + C4=256Hz
- 12-Tone Equal Temperament + A4=432Hz
- 12-Tone Equal Temperament + A4=440Hz
C | ^{C#/Db} | D | ^{D#/Eb} | E | F |
^{F#/Gb} | G | ^{G#/Ab} |
A | ^{A#/Bb} | B | C |
256 | 288 | 308,8 | 323 | 362,3 | 384 | 512 | ||||||
270,6 | 286,5 |
303,5 | 321,4 | 340,3 | 383,6 | 430,2 | ||||||
272 | 288 | 305 | 323 | 365 | 384 | |||||||
276,7 | 310,3 | 328,6 | 368,6 | 390,4 | 416 | 440 | 466 | 493,5 | ||||
335 |
The tones with a colored background are a "spot-on" match. The colored frequencies differ up to about 1Hz with these temperaments mentioned above.
I have only compared the frequencies generated with the first 64 Fibonacci numbers to 4 temperaments that I have been blogging about before. If you continue the "conversion" beyond the 64th Fibonacci number more empy spaces in the table above will be filled. There are many more temperaments and concert pitches that might "match" with some of the Fibonacci Numbers Tone Frequencies.
Feel free to complete the list yourself: http://www.fullbooks.com/The-first-1001-Fibonacci-Numbers.html
SCALE ASSIGNMENT OF THE FIBONACCI SEQUENCE
You could also use the Fibonacci Sequence itself to create a melody by assigning subsequent Fibonacci numbers to tone from a scale of choice. Radomir Nowotarski for example 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).
FIBONACCI TEMPERAMENTS
You can also use the Fibonacci numbers to create interval ratios with. Read more about that in the article "Fibonacci Temperaments".
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