A curious connection between the golden ratio rhythms and the golden ratio pitch interval Golden ratio polyrhythm with golden ratio pitch interval

Notice how the beats nearly coincide when they reach successive Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,

Nate DeBelle recently asked on the Xenharmonic alliance on facebook, what''s special about the golden ratio pitch interval, as a musical interval. "I know some people have done it, but I don''t really understand it. Why actually use the ratio of phi in your tunings? What is the point?" - here is the original post

So anyway that intrigued me, and as a result of following that up have found an intriguing connection between the beating partials of harmonic notes played at the golden ratio interval and the golden ratio polyrhhythm.

The golden ratio interval can sound "out there"

This depends on context but sometimes at least the golden ratio interval can seem somehow "out there" not part of the usual harmonic relations - for instance in this video:

- in that video you seem to be getting further and further from pure ratio type pitch relationships.

And indeed, in a certain mathematical sense it is "as out of tune as it is possible to get" - as far as possible from closely approximating ratios.

In other ways it is a rather ordinary interval

In many ways it really isn''t that exotic an interval. It is a somewhat sharp minor sixth - in C major you would play it as an Ab almost exactly a third of a semitone sharp - between an 8/5 and a 5/3. It''s inversion is a major third flat by a third of a semitone. So, at first glance it seems no more than one of many interesting intervals between a minor and a major sixth, close to 13/8.

First let''s be clear what we are talking about - there are other ways to use the golden ratio

Let''s make it clear we are talking about the linear golden ratio interval, which means that the ratio of the frequencies is exactly the golden ratio. There are good reasons for using golden ratios in other ways, e.g. you might make the whole tone size in cents a golden ratio multiple of the semitone, or the octave a golden ratio multiple of (a rather approximate) fifth, or whatever. If you do that you normally use the size of the interval in cents (say) - so based on the way we hear intervals (we hear interval sizes logarithmically). That''s easier to understand, as you can imagine intervals like that sounding well proportioned for the same reason that geometrical shapes look well proportioned, because we have an aesthetic preference for golden ratio proportions for whatever reason.

But - the linear golden ratio - we don''t hear frequencies linearly so the linear golden ratio pitch interval can''t have any direct connection to the way we use golden ratio proprotions visually.

What I found is an interesting feature of the beating partials

One interesting thing about the linear phi: with a rich harmonic timbre, the beating partials include 2, 3, 5, 8,13, 21, 34, 54, 89 with none of them picked out particularly strongly, and these beats are much slower than the other faster beats similar in pitch in the complex texture of the interval. But - it all blends together so well, often you might be hard pressed to hear many of the beats individually.

However there''s an even stronger relation. If you look more closely at the beating partials, the predicted tempo of the beats for 3 beating with 5 is slower than the tempo for the beats for the 2 beating with 3 by the golden ratio. Ditto, for 5 beating with 8 that''s even slower, again by the golden ratio, and so it goes on. That''s an interesting texture - especially since usually beats get faster as you go up the harmonic series - but here they are getting slower if you focus on the fibonacci partials - and even more interesting the partials themselves are playing the golden ratio polyrhythm - as polyrhythmic, or as "out of time" with each other as it is possible to get. So that helps to create an interesting very smooth type of texture.

Again I don''t expect to hear many of those beats individually - but the texture of them all playing together is perhaps something you could expect to influence the sound of the itnerval.

To help you hear it more clearly I have boosted the fibonacci partials here in this collection of audio clips at various frequencies for the 1/1.

I first observed this golden ratio rhythm in the beats, after reading Nate BeDelle''s question at the Xenharmonic Alliance facebook group, and inspecting the predicted numbers of beats for each partial. Then I proved it. See the end of this blog post for the proof for those interested.

I made these clips using the harmonic interval tester / trainer in Bounce Metronome Pro. However if you try to do the same yourself you will find it won''t let you enter the interval as g for the golden ratio you just have to enter its calculated value in cents 833.0902963567, - but it''s an easy fix, it''s fixed for next upload just didnt'' think to permit mathematical formulae input for that text area.

Here iis the proof for those interested, rather simple actually.

Result to prove - the fibonacci partials for the golden ratio pitch interval play beats at slower and slower tempi as you go up the fibonacci series, and the tempo of each set of beats is slower than the previous one by exactly the golden ratio.

So - putting that more mathematically:

Theorem

Let''s use g for the golden ratio and F(n) for the nth Fibonacci number.

If the partials for the two notes are

a 2a 3a 5a 8a ...
a*g 2a*g 3a*g 5a*g ...

then the number of beats for the nth pair of partials is a/g^n, for example
(5a-3a*g)= a / g^4

Proof -

It is easy to prove using the identity F(n) - F(n-1) * g = 1/ g^n, so a*F(n) - a*F(n-1) * g = 1=a/ g^n

So it just remains to prove that identity. You do that by mathematical induction:

true up to n = 2 (easily checked)
if true for m < n, then
F(n) - F(n-1)*g
= F(n-1) + F(n-2) - (F(n-2) + F(n-3)) * g
= F(n-1) - F(n-2) * g
+ F(n-2) - F(n-3) * g
= 1/g^(n-1) + 1/g^(n-2)
= 1/g^n

Lets just check it for n = 1 and n = 2 to complete the proof:

case n = 1
1 - 0*g = g^0
case n = 2
2 - g = 1/g