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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):

nF(n)OCTAVES TONES
1, 211-2-3-4-8-16-32-64-128-256256Hz
321-2-3-4-8-16-32-64-128-256256Hz
433-6-12-24-48-96-192-384384Hz
555-10-15-30-60-120-240-480480Hz
68

1-2-3-4-8-16-32-64-128-256

256Hz
713

13-26-52-104-208-416

416Hz
82121-42-84-168-336335Hz
93434-68-136-272272Hz
105555-110-220-440440Hz
118989-178-365365Hz
12144288288Hz
13233233-466466Hz
14377377377Hz
15610610-305305Hz
16987987-493,5493,5Hz
1715971597-798,5-399,25399,2..Hz
1825842584-1292-646-323323Hz
194181et cetera261,3..Hz
206765

422,8..Hz
2110946

342,0…Hz
2217711

276,7…Hz
2328657

447,7…Hz
2446368

362,25Hz
2575025 

 293,0…Hz
26121393

474,1…Hz
27196418

383,6…Hz
28317811

310,3…Hz
29514229

502,1…Hz
30832040

406,2…Hz
311346269

328,6…Hz
322178309

265,9…Hz
333524578

430,2…Hz
345702887

348,0…Hz
359227465

281,5…Hz
3614930352

455,6…Hz
3724157817

368,6…Hz
3839088169

298,2…Hz

39

63245986

482,528…Hz

40

102334155

390,373…Hz
41165580141

315,819…Hz
42267914296

511,005…Hz
43433494437

413,412…Hz
44701408733

334,457…Hz
451134903170

270,582…Hz
461836311903

437,810…Hz
472971215073

354,196…Hz
484807526976

286,550…Hz
497778742049

463,649…Hz
5012586269025

375,100…Hz

 …

… 

5120365011074

303,462…Hz
5486267571272

321,371…Hz
57365435296162

340,338…Hz
6410610209857723

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#/DbDD#/EbEF
F#/GbGG#/Ab
AA#/BbBC
256 288308,8323 362,3384    512
 270,6286,5
303,5321,4340,3 383,6 430,2   
 272288305323 365384     
 276,7 310,3328,6 368,6390,4416440466493,5 
     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

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