Blog » Fibonacci "Tones" - ROEL'S WORLD (blog)

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#/Db

^_D#/Eb

^_F#/Gb

^_G#/Ab

^_A#/Bb

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).