On 12 Jun 2014, at 22:30, Mark Stephen Mrotek <carsonm...@ca.rr.com> 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 _______________________________________________ lilypond-user mailing list lilypond-user@gnu.org https://lists.gnu.org/mailman/listinfo/lilypond-user