Kite's thoughts on pergens: Difference between revisions

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Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.

This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.

The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most "pergen-friendly" edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).

__**EDO-pair names**__

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&lt;u&gt;&lt;span style="font-size: 110%;"&gt;Mizarian Porcupine Overture by Herman Miller (P8, P4/3)&lt;/span&gt;&lt;/u&gt;&lt;br /&gt;

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Pergen squares can be generalized to any prime subgroup by representing the notes as dots. Below are the first 32 rank-2 pergens in a completely JI-agnostic format. A is the interval of equivalence, the period of the unsplit pergen. B is the generator of the unsplit pergen. For 2.3 pergens, A = 8ve and B = 5th. The (A, (A-B)/2) square corresponds to (P8, P4/2). In the 2.5 subgroup, B = 5/4. In Bohlen-Peirce, A = 3/1 and B = 5/3. True doubles are in red. The true/false property of a pergen is independent of the prime subgroup. Imperfect multigens are in green. Imperfect is generalized to other subgroups as requiring multiples of B in the pergen.&lt;br /&gt;

A similar chart could be made for all rank-3 pergens, using pergen cubes.&lt;br /&gt;

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Given an edo, a period, and a generator, what is the pergen? There is usually more than one right answer. For 10edo with P = 5\10 and G = 2\10, it could be either (P8/2, P4/2) or (P8/2, P5/3). Every coprime period/generator pair results in a valid pergen. It isn't yet known if there are period/generator pairs that require a true double pergen, or if all such pairs can result from either a false double or single-split pergen.&lt;br /&gt;

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This table lists all pergens up to quarter-splits, with all edos that support them. Partial support is indicated with an asterisk. The generator's keyspan depends on the multigen's keyspan, and thus on the 5th's keyspan. The latter is occasionally ambiguous, as in 13-edo and 18-edo. Both of these edos are incompatible with heptatonic notation, and 13edo's half-5th pergen is actually notated as a half-upfifth. 13b-edo and 18b-edo are listed as well. 6-edo, 11-edo and 23-edo could also be considered ambiguous.&lt;br /&gt;

The edos that support the fewest pergens are prime-number edos like 11edo or 13edo. The most &amp;quot;pergen-friendly&amp;quot; edos tend to be ones in which the circle of 5ths doesn't reach every edostep. For example, 24edo supports all half-split pergens, since both P8 and P5 map to an even number of edosteps. 72edo supports all half-splits and all third-splits. 15, 21 and 36 edo support many but not all third-splits (not those with m = 2 or n = 2).&lt;br /&gt;

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&lt;u&gt;&lt;strong&gt;EDO-pair names&lt;/strong&gt;&lt;/u&gt;&lt;br /&gt;

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Screenshots of the first 2 pages:&lt;br /&gt;

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Alt-pergenLister lists out thousands of rank-2 pergens, and suggests periods, generators and enharmonics for each one. Alternate enharmonics are not listed, but single-pair notation for false-double pergens is. It can also list only those pergens supported by a specific edo or edo pair. Written in Jesusonic, runs inside Reaper.&lt;br /&gt;

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The first section (PERGEN and Per/Gen cents) describes each pergen without regard to notational issues. The period and generator's cents are given, assuming a 5th of 700¢ + c. The generator is reduced, e.g. (P8/2, P5) has a generator of 100¢ + c, not 700¢ + c. The next two sections show a possible notation for P and G. The last section shows the unreduced pergen, and for false doubles, a possible single-pair notation. Horizontal lines group the pergens into blocks (half-splits, third-splits, etc). Red indicates problems. Generators of 50¢ or less are in red. Enharmonics of a 3rd or more are in red.&lt;br /&gt;

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Screenshots of the first 69 pergens:&lt;br /&gt;

The first 29 pergens supported by 12edo:&lt;br /&gt;

Some of the pergens supported by 15edo. A red asterisk means partial support.&lt;br /&gt;

Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block.&lt;br /&gt;

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Listing all valid pergens is not a trivial task, like listing all valid edos or all valid MOS scales. Not all combinations of octave fractions and multigen fractions make a valid pergen. The search for rank-2 pergens can be done by looping through all possible square mappings [(x, y), (0, z)], and using the formula (P8/x, (i·z - y, x) / xz). While x is always positive and z is always nonzero, y can take on any value. For any x and z, y can be constrained to produce a reasonable cents value for 3/1. Let T be the tempered twefth 3/1. The mapping says T = y·P + z·G = y·P8/x + z·G. Thus y = x·(T/P8 - z·G/P8). We adopt the convention that G is less than half an octave. We constrain T so that the 5th is between 600¢ and 800¢, which certainly includes anything that sounds like a 5th. Thus T is between 3/2 and 5/3 of an octave. We assume that if the octave is stretched, the ranges of T and G will be stretched along with it. The outer ranges of y can now be computed, using the floor function to round down to the nearest integer, and the ceiling function to round up:&lt;br /&gt;