Blog » Fibonacci “Tones”

Reading Time: 4 minutesOctober 27, 2013

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) OCTAVES  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

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