Fibonacci Scale

If the Fibonacci sequence (1,1,2,3,5,8,13,21…) were translated into music, how would it sound?  The answer – surprisingly – is, pretty good.

Creating the Fibonacci Scale

To play the Fibonacci sequence on a piano, one must assign a number value for every note of the keyboard; A=1, B=2, etc.  As every octave has seven notes, every eighth note starts over at A.  Therefore, 8=A, 9=B, 10=C, etc.  Because there are only 7 possible notes, determining where a given Fibonacci number falls on a scale essentially deals with remainders:

Scale Note = Mod (F,7)

The next, and trickiest step for playing the Fibonacci sequence is finding a piano keyboard that extends into infinity.   Playing only the Fibonacci numbers on your infinite keyboard, one discovers the repeating sixteen note group A-A-B-C-E-A-F-G-F-F-E-D-B-F-A-G (repeat). Expressed numerically (Fibonacci sequence, Modulo 7), the sequence is 1-1-2-3-5-1-6-7-6-6-5-4-2-6-1-7, repeatedly infinitely.

fibonacci-sequence

It’s pretty nifty to find an infinite recursive sequence yielding a repeating finite group under modulo 7. But then, the universe is a nifty place.

Playing the Fibonacci sequence on a regular piano (for instance, in one octave of A major) is not unpleasant as the sixteen notes fit squarely into four measures. The sequence begins on the tonic note and ends on the leading tone, musically resolving when the sequence repeats.

It sounds like this: Fibonacci Sequence MP3. Take a listen!

Proving the Finite Group

To prove the Fibonacci sequence under modulo seven has a finite order of sixteen, we can use mathematical induction…  Special thanks to Kiri Wagstaff for helping me with my math!

Since it’s known that Fn+1 = Fn + Fn-1, let’s assume that:

Mod (Fn+1, 7) = Mod (Fn + Fn-1, 7)

Or in other words:

Mod (Fn+1, 7) = Mod (Mod (Fn, 7) + Mod (Fn-1, 7), 7)

This holds true for our base cases, n = 1 and n = 2.

For our inductive step (to prove the sequence repeats every sixteenth natural number) let’s assume that:

Mod (F(n+1)+16, 7) = Mod (F(n+16) + F(n-1)+16 ,7)

OR

Mod (F n+17, 7) = Mod (Mod (Fn+16, 7) + Mod (F n+15, 7), 7)

OR

Mod (2584, 7) = Mod (Mod (1597, 7) + Mod (987, 7), 7)

OR

1 = 1

AND

Mod (Fn, 7) = Mod (Fn+16, 7)

Fibonacci Fugue

Without further ado, here is a fugue I whipped up using the 16 note Fibonacci Scale as the main theme: Fibonacci Fugue MP3. Take a listen!

Order of Group for Non-Western Scales

What does the Fibonacci sequence sound like if played on non-Western scales? Repeating sequences again occur on scales of any length (excepting base 10-divisible scales which spit out gigantic Pisano periods before yielding an identity).  For instance, playing the Fibonacci sequence on a pentatonic scale yields the repeating pattern: A-A-B-C-E-C-C-A-D-E-D-D-C-B-E-B-B-D-A-E.   Playing in a nine note scale also yields a twenty note repeating pattern: A-A-B-C-E-H-D-C-G-A-H-I-H-H-G-F-D-A-E-F-B-H-A-I.

Fibonacci Scales in Bases 2 -15

Examining the modulo sequences, distinct patterns emerge.   The longer the modulo sequence, the more it approaches the Fibonacci sequence.   For instance, all of the sequences begin with 1-1-2-3-5, or A-A-B-C-E.   Furthermore, the sequences all end with the highest possible note in the scale.   Melodically, this means that every sequence begins on the tonic and ends on a leading tone, which is harmonically pleasing.  The second to last note is always A.  The third to last is always the second highest note in the scale.   The fourth to last note is always B, etc.   As larger sequences are generated, greater patterns emerge (see chart above).   Although it must be a mathematical coincidence, these patterns create harmonic consistencies that are not half-bad to listen to.

Perhaps most bizarrely, these Fibonacci scales all appear to obey the laws of musical phrasing.   Simply stated, the peak of a musical phrase is very often the highest note of a musical phrase, and tends to occur around 2/3rds (or maybe .618…) through the length of the phrase.   Furthermore, if a musical phrase begins and ends in the tonic key, the peak of the phrase will often be in the dominant key.   This pattern is bizarrely exhibited in the Fibonacci scales (chart above), which all contain a middle sequence of A-M-N-M-M-L-K, where N is the highest note in the scale, and M is the second highest, L is the third highest, and K is the fourth highest.

Square Scales and Cube Scales

Recurring sequences do not appear to be endemic to the Fibonacci sequence.  Playing the squares (1,4,9, etc) on the infinite piano yields a recurring order of seven {A,D,B,B,D,A,G}.  Playing the cubes (1,8,27) on the infinite piano yields a different recurring group order of seven {A-A-F-A-F-F-G}.   Perhaps these math-based melodies will one day provide inspiration for modern composers.

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