Tuning Methods

Prime Limit Otonal Tunings

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5-limit Full6.5133714
5-limit Exact9.2182227
7-limit Full8.5133637
7-limit Exact8.51822710
11-limit Full10.9151539
11-limit Exact10.91824912
13-limit Full11.81514410
13-limit Exact13.41825311
17-limit Full14.61514411
17-limit Exact16.31826312
19-limit Full17.61516312
19-limit Exact18.11516812
23-limit Full21.21656812
23-limit Exact19.61397412
12-"tat"Two pieces of music (mp3s) in a slightly arbitrary PLOT.

Prime-limit tunings are a kind of rational tuning. The otonal variety, or PLOTs, are constructed by placing in the numerators of the ratios a series of numbers containing prime factors no greater than the stated limit, while the denominators are the all same. That may sound far more complicated than it really is. To construct a 5-prime-limit otonal tuning, for example, simply take any set of numbers that factor wholly to 2s, 3s & 5s, and place them over the same denominator. Since octave equivalency is assumed, "same" includes all numbers related by powers of two alone. Thus, the numbers 5 and 9 may placed over 4 and 8, respectively; 4 and 8 are "the same" in terms of placement within the octave because with all factors of two removed, they are equal.

The pages below address only 12-tone, octave-repeating versions of prime limit otonal tunings. For simplicity's sake, I have spelled them out using powers of two in the denominators.

Tuning Names

For the Full versions, the ratios chosen for each tuning are the simplest available near each key on a standard 7+5 keyboard; therefore, they are not always the 12 absolute simplest—often a few get skipped over because they most closely approximate a chromatic tone that is already "taken" by a simpler ratio.

For the Exact versions, the twelve absolute simplest ratios are used regardless of how inconveniently the result maps to a 7+5 keyboard. For both types, tuning tables are offset to average 0 cents difference from 12 EDO. Where no distinction is made, the exact version maps as well as is feasible. Above 23, mapping the 12 absolute simplest ratios would exclude the higher primes, so one or more of the simpler ratios are omitted to make room, usually the cubes and squares of the lowest primes. Since 37 is the 11th prime (starting with 3), that's the highest limit that will fit in 12 tones without omitting some.

Harmonic Complexity

I rate rational scales by Harmonic Complexity, as I understand it. In tuning theory, the harmonic complexity of a set of ratios, such as a tuning, is the complexity of the Least Common Multiples of the set. There are different ways to measure complexity. For this purpose, I prefer LOG(LCM), the natural logarithm of the least common multiple of the prime factors. For purposes of comparing tunings that assume octave equivalence, I believe that powers of two can be removed. While that's not true when comparing individual ratios, for tunings that assume octave equivalence, to fail to remove powers of two provides different ratings for tunings that are sonically and musically identical, differing only in what portion of the harmonic lattice they cover. Therefore, for my purposes here, I have removed powers of two from the ratios of each tuning before calculating their HC.

The Harmonic Complexity rating should not be taken to indicate "better" or "worse," of course; it only gives some relative idea of the overall complexity of the scale members' harmonic relationships, for whatever that's worth.

Copyright © 2005 by David J. Finnamore
All rights reserved.