[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
Re: Irrational time signature and tuplets
From: |
Hans Aberg |
Subject: |
Re: Irrational time signature and tuplets |
Date: |
Thu, 12 Jun 2014 23:19:57 +0200 |
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
Re: Irrational time signature and tuplets, Hans Aberg, 2014/06/12