RE: Irrational time signature and tuplets

[Mark Stephen Mrotek]

Hans,

Thank you for your detailed and informative response. I shall take time to
study it and the references that you provide.

Mark

-----Original Message-----
From: Hans Aberg [mailto:address@hidden 
Sent: Thursday, June 12, 2014 2:20 PM
To: Mark Stephen Mrotek
Cc: Malte Meyn; address@hidden
Subject: Re: Irrational time signature and tuplets

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