Kite's color notation/Temperament names: Difference between revisions
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== Explanation == | == Explanation == | ||
[[Color notation]] can name every regular temperament. The name is the same as that of the comma(s) tempered out, but without the degree (unison, 2nd, etc.). The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. Each segment contains 7 ratios. The comma that is tempered out is assumed to be the smallest in cents of those 7. | [[Color notation]] can name every regular temperament. The name is the same as that of the comma(s) tempered out, but without the degree (unison, 2nd, etc.). For example, Semaphore is the zozo temperament, abbreviated zzT. The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. Each segment contains 7 ratios. The comma that is tempered out is assumed to be the smallest in cents of those 7. | ||
Words like large, small, double, etc. are abbreviated, to make the names a reasonable length. | Words like large, small, double, etc. are abbreviated, to make the names a reasonable length. | ||
* Double = '''bi-''' ("bee"), triple = '''tri-''' ("tree"), quadruple = '''quad-''', quintuple = '''quin-''', septuple = '''sep-''' | * Double = '''bi-''' ("bee"), triple = '''tri-''' ("tree"), quadruple = '''quad-''', quintuple = '''quin-''', septuple = '''sep-''' | ||
* -bi or -tri at the end of a name indicates that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Layobi = | * -bi or -tri at the end of a name indicates that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Mavila = Layobi = Ly#2T | ||
* Large = '''la-''', small = '''sa-''', double large = lala-, triple small = trisa-, etc. | * Large = '''la-''', small = '''sa-''', double large = lala-, triple small = trisa-, etc. | ||
Some 5-limit examples, sorted by color depth. Many more examples can be found on the comma pages [[Comma|here]], [[Medium commas|here]], [[Large commas|here]] and [[Unnoticeable comma|here]]. | Some 5-limit examples, sorted by color depth. Many more examples can be found on the comma pages [[Comma|here]], [[Medium commas|here]], [[Large commas|here]] and [[Unnoticeable comma|here]]. | ||
# [[Schismatic]] = Layo = LyT, [[Mavila]] = Layobi = Ly | # [[Schismatic]] = Layo = LyT, [[Mavila]] = Layobi = Ly#2T, [[Superpyth]] = Sayo = syT. [[Meantone]] = Gu = gT, [[Father]] = Gubi = g#2T. | ||
# [[Dicot]] = Yoyo = yyT, [[Immunity family|Immunity]] = Sasa-yoyo = ssyyT. [[Bug]] = Gugu = ggT, [[Diaschismic]] = Sagugu = sggT, [[Beatles]] = Sasa-gugu = ssggT. | # [[Dicot]] = Yoyo = yyT, [[Immunity family|Immunity]] = Sasa-yoyo = ssyyT. [[Bug]] = Gugu = ggT, [[Diaschismic]] = Sagugu = sggT, [[Beatles]] = Sasa-gugu = ssggT. | ||
# [[Porcupine]] = Triyo = y<sup>3</sup>T. [[Augmented]] = Trigu = g<sup>3</sup>T, [[Laconic family|Laconic]] = Latrigu = Lg<sup>3</sup>T, [[Misty comma|Misty]] = Sasa-trigu = ssg<sup>3</sup>T. | # [[Porcupine]] = Triyo = y<sup>3</sup>T. [[Augmented]] = Trigu = g<sup>3</sup>T, [[Laconic family|Laconic]] = Latrigu = Lg<sup>3</sup>T, [[Misty comma|Misty]] = Sasa-trigu = ssg<sup>3</sup>T. | ||
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Sometimes the smallest ratio in a segment is a multiple of another comma. For example, the smallest ratio in the central segment of the zozogugu row is 441/400. But since this is (21/20)<sup>2</sup>, tempering it out would simply result in zgT. Thus there is no bizogu temperament, although there is a bizogubi one. | Sometimes the smallest ratio in a segment is a multiple of another comma. For example, the smallest ratio in the central segment of the zozogugu row is 441/400. But since this is (21/20)<sup>2</sup>, tempering it out would simply result in zgT. Thus there is no bizogu temperament, although there is a bizogubi one. | ||
La means large and | La means both large and 11-all, and sa means both small and 17-all. To avoid confusion, large and small should never be abbreviated unless part of a longer word. La is also the La note in solfege, and Sa is the tonic in saregam. The meaning will always be clear from context. Notes are never large or small. | ||
La is also the La note in solfege, and Sa is the tonic in saregam. The meaning will always be clear from context. Notes are never large or small | |||
Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is triyo & ru = y<sup>3</sup>&rT. There are three obvious ways to choose the comma set for multi-comma temperaments. The '''odd name''' minimizes the [[Odd limit|double odd limit]] of the comma set. The '''subgroup name''' minimizes the number and size of the primes used by each comma, but not their depth. Assuming no wa commas, the 1st comma is ya, the 2nd za, the 3rd ila, etc. The '''prime/odd name''' minimizes the prime-limit for each comma, and for each prime limit uses the comma of least double odd limit. The 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. (There is also the '''hermite name''', formed by hermite reduction, which minimizes the prime-limit of each comma, and for each prime limit, tends to minimize the color depth. Finally the '''LLL name''' uses Lenstra–Lenstra–Lovász reduction, and tends to minimize the integer limit of each comma.) | Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is triyo & ru = y<sup>3</sup>&rT. There are three obvious ways to choose the comma set for multi-comma temperaments. The '''odd name''' minimizes the [[Odd limit|double odd limit]] of the comma set. The '''subgroup name''' minimizes the number and size of the primes used by each comma, but not their depth. Assuming no wa commas, the 1st comma is ya, the 2nd za, the 3rd ila, etc. The '''prime/odd name''' minimizes the prime-limit for each comma, and for each prime limit uses the comma of least double odd limit. The 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. (There is also the '''hermite name''', formed by hermite reduction, which minimizes the prime-limit of each comma, and for each prime limit, tends to minimize the color depth. Finally the '''LLL name''' uses Lenstra–Lenstra–Lovász reduction, and tends to minimize the integer limit of each comma.) | ||
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1u<sup>3</sup>z<sup>3</sup>g = triluzo-agu | 1u<sup>3</sup>z<sup>3</sup>g = triluzo-agu | ||
If the 2nd color could be merged with either the 1st color or the 3rd color, but not with both, it merges with whichever one has a larger exponent. Thus in | If the 2nd color could be merged with either the 1st color or the 3rd color, but not with both, it merges with whichever one has a larger exponent. Thus in 1uuz<sup>6</sup>g<sup>3</sup>, zo merges with the tripled color, not the doubled color, to make lulu-trizozogu, not bilutrizo-atrigu. | ||
1uuz<sup>6</sup>g<sup>3</sup> | |||
=== Quadricolored examples === | === Quadricolored examples === | ||