Re: Irrational time signature and tuplets

[Hans Aberg]

On 12 Jun 2014, at 22:30, Mark Stephen Mrotek <address@hidden> wrote:

> (1+sqrt 5)/2 = 1.618... is the golden ratio, phi. 
> https://en.wikipedia.org/wiki/Golden_ratio
> 
> Do you know of other instances of this ratio in music?

The WP [1] mentions one other case where "irrational" in music is irrational in 
also the mathematical sense, and it is also a square root. However, I only 
found it after making this example:

The original meter [2] is written 12 = 3+2+2+3+2 subject to interpretation of 
the exact ratios, with duplets or quadruplets on the 3s, and one can also have 
triplets on the 2s, as in the example I posted. Write, as in dance notation, s 
= slow, q = quick; then the meter is s q q s q, with the original, written 
ratio s/q = 3/2.

What I did was setting s/q = x so that also (s + q)/s = x; this gives x + 1 = 
1/x, which is the defining property of the golden ration, as you can see in the 
upper right hand box in your reference [3].

Then, as LilyPond does not handle these irrational time values, the next step 
is to find rational approximations, which can be done via continued fractions 
[4]. To get the denominators, as in this reference, take the integral part of 
the number, invert the fractional part, and repeat. For the golden ratio x the 
formula 1/x = 1 + x will show the it is a sequence of 1s: 1, 1, 1, ... One can 
can see that this leads to the successive quotients of the Fibonacci series 
[5], 1, 1, 2, 3, 5, 8, 13, ..., where the next integer in the series is the sum 
of the two immediate preceding integers. This gives the approximations 1/1, 
2/1, 3/2, 5/3, 8/5, 13/8, ...

But the continued fractions above work with any irrational number. Another idea 
I used was making y = s/q equal to q/(s/2) = 2/y, because of the typical rhythm 
 s/2 s/2 q q s/2 s/2 q. This gives y = sqrt 2, and the continued fractions 
numbers are 1, 2, 2, 2, ..., giving rational approximations 1, 3/2, 7/5, ...

The traditional written value s/q = 3/2, x = (1+sqrt 5)/2 = 1.618..., and y = 
sqrt 2 =
1.414..., but in reality there is a lot of variation in the interpretation.

So one can play around with any mathematically irrational number. But the 
usability in music is another question.


1. https://en.wikipedia.org/wiki/Time_signature#Irrational_meters
2. https://en.wikipedia.org/wiki/Leventikos
3. https://en.wikipedia.org/wiki/Golden_ratio
4. https://en.wikipedia.org/wiki/Continued_fraction
5. https://en.wikipedia.org/wiki/Fibonacci_number