Kite's thoughts on pergens: Difference between revisions
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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. EUs of a 3rd or more are in red. | 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. EUs of a 3rd or more are in red. | ||
Screenshots of the first | Screenshots of the first 170 pergens: | ||
[[File:alt-pergenLister_1.png| | [[File:alt-pergenLister_1.png|852x852px|alt-pergenLister 1.png]] | ||
[[File: | [[File:Alt-pergenLister 2a.png|frameless|852x852px]] | ||
[[File:Alt-pergenLister 3.png|frameless|854x854px]] | |||
The first 39 pergens supported by 12edo: | |||
[[File:alt- | [[File:alt-pergenLister_12edo.png|857x857px|alt-pergenLister 12edo.png]] | ||
Some of the pergens supported by 15edo. A red asterisk means partial support, e.g. (P8, P5) only uses a 5edo subset of 15edo. | |||
[[File:alt-pergenLister_15edo.png|854x854px|alt-pergenLister 15edo.png]] | |||
Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block. | Pergens supported by 19edo. Edos that are a prime number support only 1 pergen per block. | ||
[[File:alt-pergenLister_19edo.png| | [[File:alt-pergenLister_19edo.png|857x857px|alt-pergenLister 19edo.png]] | ||
The first 54 imperfect pergens: | |||
[[File:Imperfect pergens.png|frameless|863x863px]] | |||
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: | 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: | ||