Kite's thoughts on pergens: Difference between revisions

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==Ratio and cents of the accidentals==

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, the up symbol ~~often~~ equals only a few ~~ratios~~. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 11~~-limit~~ temperaments use either 33/32 or 729/704. These '''mapping commas''' are ~~used to~~ map ~~higher primes~~ to 3-limit ~~intervals, and~~ are essential for notation~~. They~~ also ~~determine~~ where a ratio ~~"lands"~~ on a keyboard. By definition ~~they are~~ a P1, and the only intervals that map to P1 are ~~these~~ commas and combinations of them.

The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2x3yPz, where P is a higher prime and z = ±1. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for P = 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 are the currently employed commas and combinations of them.

If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80.

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See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because those scales are contained in 5edo. Periods of 1 or 2 edosteps are excluded as too trivial, because the scale is the entire edo. Every step of the edo appears in the genchains, even if the chains are only one step long.

To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator.

To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. It appears, but has yet to be formally proven, that the number of P/G combinations that N-edo supports is (N-1)/2 rounded down. If we include the trivial combination where P = 1 edostep, the number of combinations would be N/2 rounded up.

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 ==Ratio and cents of the accidentals==
-The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, the up symbol often equals only a few ratios. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 11-limit temperaments use either 33/32 or 729/704. These '''mapping commas''' are used to map higher primes to 3-limit intervals, and are essential for notation. They also determine where a ratio "lands" on a keyboard. By definition they are a P1, and the only intervals that map to P1 are these commas and combinations of them.
+The sharp symbol's ratio is always (-11,7) = 2187/2048, by definition. Looking at the table in the Applications section, and the ratios that the up symbol equals, only a few commas account for most entries. For most 5-limit temperaments, ^1 = 81/80. For most 2.3.7 temperaments, ^1 = 64/63. Most 2.3.11 temperaments use either 33/32 or 729/704. These are all '''mapping commas''', which is a comma of the form 2<sup>x</sup>3<sup>y</sup>P<sup>z</sup>, where P is a higher prime and z = ±1. They are called mapping commas because they equate or map P/1 to a 3-limit interval. They are essential for notation, and also for determining where a ratio is placed on a keyboard. Potential mapping commas for P = 5 include 81/80, 135/128, and the schisma = Ly-2 = 2¢. Only one of these at a time is actually used in notation, e.g. 5/4 is either a M3 or a m3 or a d4. By definition the currently employed mapping comma is a P1, and the only intervals that map to P1 are the currently employed commas and combinations of them.
 If a single-comma temperament uses double-pair notation, neither accidental will equal the mapping comma. A double-comma temperament using double-pair notation may use the difference between two mapping commas, as in lemba, where ^1 equals 64/63 minus 81/80.
@@ Line 3,016: / Line 3,016: @@
 See the screenshots in the Supplemental Materials section for a complete list of which pergens are supported by 12, 15 and 19 edo. The list includes multiple pergens per period/generator combination. The next table shows only the simplest pergen for each combination. Combinations are excluded if the period and generator are not coprime, because the scale is contained in a smaller edo. For example, 15edo has no unsplit pergen, because those scales are contained in 5edo. Periods of 1 or 2 edosteps are excluded as too trivial, because the scale is the entire edo. Every step of the edo appears in the genchains, even if the chains are only one step long.
-To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator.
+To find the pergen, find the edo, then find the row that corresponds to the period, then find the column that corresponds to the generator. It appears, but has yet to be formally proven, that the number of P/G combinations that N-edo supports is (N-1)/2 rounded down. If we include the trivial combination where P = 1 edostep, the number of combinations would be N/2 rounded up.
 {| class="wikitable"