Kite's color notation/Temperament names: Difference between revisions

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[[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. Sometimes the smallest ratio is a multiple of another comma. For example, the smallest ratio in the central segment of the zozogugu row is 441/440. But since this is (21/20)2, tempering it out would simply result in zgT. Thus there is no zozogugu temperament.

[[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.

If the comma is the 2nd largest ratio in the row segment, the temperament name ends with "2". For example, [[Schismatic]] is LyT and [[Mavila]] is LyT2. Likewise, LyT3 would temper out the 3rd largest ratio. Any comma ≤ 90¢ is guaranteed to be the smallest ratio in its segment.

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-'''

* -bi, -tri~~, etc.~~ at the end of a name ~~means~~ 2nd/3rd~~/etc.~~ largest ratio, e.g. ~~Schismatic~~ = ~~Layo and~~ Mavila ~~= 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. Layobi = Mavila

* 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|Commas]] page.

# [[Schismatic]] = Layo = LyT, [[Mavila]] = Layobi = ~~LyT2~~, [[Superpyth]] = Sayo = syT. [[Meantone]] = Gu = gT, [[Father]] = Gubi = ~~gT2~~.

# [[Schismatic]] = Layo = LyT, [[Mavila]] = Layobi = Ly2T, [[Superpyth]] = Sayo = syT. [[Meantone]] = Gu = gT, [[Father]] = Gubi = g2T.

# [[Dicot]] = Yoyo = yyT, [[Immunity family|Immunity]] = Sasa-yoyo = ssyyT. [[Bug]] = Gugu = ggT, [[Diaschismic]] = Sagugu = sggT, [[Beatles]] = Sasa-gugu = ssggT.

# [[Porcupine]] = Triyo = y3T. [[Augmented]] = Trigu = g3T, [[Laconic family|Laconic]] = Latrigu = Lg3T, [[Misty comma|Misty]] = Sasa-trigu = ssg3T.

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Multipliers like bi-, tri-, etc. can be combined: 6-fold = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. Higher primes use their color word, but with the suffix '''-e''' for exponent:

* 11-fold = '''le-''' ("leh"), 13-fold = '''the-''' (unvoiced "th"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc.

~~Even with these abbreviations~~, ~~temperament names can get quite long~~. If the comma~~'s ratio has N digits~~, the ~~temperament~~ name ~~will usually have N, N~~-1, ~~N+1~~ or ~~occasionally N+~~2 ~~syllables~~.

To make the names easier for non-Anglophones, the 5 vowels are the basic vowels found in Spanish, ah eh ee oh oo.

Any comma < 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma > 9/8 = 204¢ is guaranteed to not be the smallest, and -bi must be appended to the name. If a comma is 90-204¢, let X = the sum of all the exponents of the monzo except the two exponent. If and only if X mod 7 is 4 or 5, 256/243 can be subtracted without changing the magnitude, and the comma is the 2nd smallest ratio.

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 441/440. But since this is (21/20)2, tempering it out would simply result in zgT. Thus there is no bizogu temperament, although there is a bizogubi one.

La means large and also 11-all. The meaning will almost always be clear from context, however "this piece uses la notes" is ambiguous. To clarify, one should say either "large notes" (fifthward notes) or "ila notes" (11-limit notes). Likewise, sa also means 17-all, and "sa notes" should become either "small notes" or "isa notes".

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. In fixed-do countries, the chord ALw (81/64 3rd) is "La lawa".

Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is triyo & ru = y3&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. Assuming no wa commas, the 1st comma is ya, the 2nd za, the 3rd ila, etc. The '''prime 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. The odd name for 7-limit [[Pajara]] is rryy&rT, the subgroup name is sgg&rT, and the prime name is sgg&rryyT. Often the three names are identical, e.g. y3&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.

If the comma is wa, an edo is implied. The temperament is named after the edo, not the wa comma. 2.3.5.7 and the pyth comma and 225/224 is 12edo&ryyT. If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with '''plus''': [[Blackwood]] is 5edo+yT = 5-edo plus ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is g+z1aT = gu plus zala.

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The temperament name indicates the prime subgroup and the rank of the temperament. For example, ryyT ([[Marvel]]) is rank-3 because it has 2 explicit colors ru and yo and 2 implicit colors wa and clear, and 4 colors minus 1 comma = rank-3. Edos count as commas, but plusses don't. Both 12edo&ryyT and 5edo+yT are rank-2. Primes 2 and 3 are assumed present in the temperament even if they are not present in the comma. Biruyo is rank-3 and biruyo nowa is rank-2.

There are three obvious ways to name 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. Thus the ~~1st~~ comma ~~is ya, the 2nd za~~, the ~~3rd ila, etc. The prime-limit~~ name minimizes the prime-limit for each comma, and for each prime limit uses the comma of least double odd limit. Thus the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. The odd name for 7-limit [[Pajara]] is rryy&rT, and the prime-subgroup name is sgg&rT, ~~and the prime~~-~~limit name is sgg&rryyT. Often the three names are identical~~, ~~e.g. y3&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-~~2 ~~temperaments better. The question of which name to use is not yet fully resolved~~.

If the comma's ratio has N digits, the temperament name will usually have N, N-1, N+1 or occasionally N+2 syllables.

== Rules for naming remote colors and extreme magnitudes ==

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z9g = tritrizo-agu

''Possible solution to the GCD problem:''

''Possible solution to the GCD problem:'' ''bi- + -a- = double-all,'' ''affects the whole name''

''bi- + -a- = double-all ~~-->~~ affects the whole name''

z6g4 = tribizo-aquadgu* = biatrizo-agugu?

''z6g4 = tribizo-aquadgu* = biatrizo-agugu?

z6gg = tribizo-agugu* = biatrizo-agu?

z6g8 = tribizo-aquadbigu* = biatrizo-aquadgu?

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z8gg = quadbizo-agugu* = biaquadzo-agu?

z9g6 = tritrizo-atribigu* = triatrizo-agugu?

z9g3 = tritrizo-atrigu* = triatrizo-agu?

z9g3 = tritrizo-atrigu* = triatrizo-agu?''

== Tricolored examples ==

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-[[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. Sometimes the smallest ratio is a multiple of another comma. For example, the smallest ratio in the central segment of the zozogugu row is 441/440. But since this is (21/20)<sup>2</sup>, tempering it out would simply result in zgT. Thus there is no zozogugu temperament.
+[[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.
-If the comma is the 2nd largest ratio in the row segment, the temperament name ends with "2". For example, [[Schismatic]] is LyT and [[Mavila]] is LyT2. Likewise, LyT3 would temper out the 3rd largest ratio. Any comma ≤ 90¢ is guaranteed to be the smallest ratio in its segment.
 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-'''
-* -bi, -tri, etc. at the end of a name means 2nd/3rd/etc. largest ratio, e.g. Schismatic = Layo and Mavila = 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. Layobi = Mavila
 * 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|Commas]] page.
-# [[Schismatic]] = Layo = LyT, [[Mavila]] = Layobi = LyT2, [[Superpyth]] = Sayo = syT. [[Meantone]] = Gu = gT, [[Father]] = Gubi = gT2.
+# [[Schismatic]] = Layo = LyT, [[Mavila]] = Layobi = Ly2T, [[Superpyth]] = Sayo = syT. [[Meantone]] = Gu = gT, [[Father]] = Gubi = g2T.
 # [[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.
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 Multipliers like bi-, tri-, etc. can be combined: 6-fold = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. Higher primes use their color word, but with the suffix '''-e''' for exponent:
 * 11-fold = '''le-''' ("leh"), 13-fold = '''the-''' (unvoiced "th"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc.
-Even with these abbreviations, temperament names can get quite long. If the comma's ratio has N digits, the temperament name will usually have N, N-1, N+1 or occasionally N+2 syllables.
+To make the names easier for non-Anglophones, the 5 vowels are the basic vowels found in Spanish, ah eh ee oh oo.
+Any comma < 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma > 9/8 = 204¢ is guaranteed to <u>not</u> be the smallest, and -bi must be appended to the name. If a comma is 90-204¢, let X = the sum of all the exponents of the monzo except the two exponent. If and only if X mod 7 is 4 or 5, 256/243 can be subtracted without changing the magnitude, and the comma is the 2nd smallest ratio.
+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 441/440. 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 also 11-all. The meaning will almost always be clear from context, however "this piece uses la notes" is ambiguous. To clarify, one should say either "large notes" (fifthward notes) or "ila notes" (11-limit notes). Likewise, sa also means 17-all, and "sa notes" should become either "small notes" or "isa notes".
 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. In fixed-do countries, the chord ALw (81/64 3rd) is "La lawa".
+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. Assuming no wa commas, the 1st comma is ya, the 2nd za, the 3rd ila, etc. The '''prime 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. The odd name for 7-limit [[Pajara]] is rryy&rT, the subgroup name is sgg&rT, and the prime name is sgg&rryyT. Often the three names are identical, e.g. y<sup>3</sup>&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.
 If the comma is wa, an edo is implied. The temperament is named after the edo, not the wa comma. 2.3.5.7 and the pyth comma and 225/224 is 12edo&ryyT. If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with '''plus''': [[Blackwood]] is 5edo+yT = 5-edo plus ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is g+z1aT = gu plus zala.
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 The temperament name indicates the prime subgroup and the rank of the temperament. For example, ryyT ([[Marvel]]) is rank-3 because it has 2 explicit colors ru and yo and 2 implicit colors wa and clear, and 4 colors minus 1 comma = rank-3. <u>Edos count as commas, but plusses don't</u>. Both 12edo&ryyT and 5edo+yT are rank-2. Primes 2 and 3 are assumed present in the temperament even if they are not present in the comma. Biruyo is rank-3 and biruyo nowa is rank-2.
-There are three obvious ways to name 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. Thus the 1st comma is ya, the 2nd za, the 3rd ila, etc. The prime-limit name minimizes the prime-limit for each comma, and for each prime limit uses the comma of least double odd limit. Thus the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. The odd name for 7-limit [[Pajara]] is rryy&rT, and the prime-subgroup name is sgg&rT, and the prime-limit name is sgg&rryyT. Often the three names are identical, e.g. y<sup>3</sup>&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The subgroup name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.
+If the comma's ratio has N digits, the temperament name will usually have N, N-1, N+1 or occasionally N+2 syllables.
 == Rules for naming remote colors and extreme magnitudes ==
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 z<sup>9</sup>g = tritrizo-agu
-''<u>Possible solution to the GCD problem</u>:''
+''<u>Possible solution to the GCD problem</u>:'' ''bi- + -a- = double-all,'' ''affects the whole name''
-''bi- + -a- = double-all --> affects the whole name''
-z<sup>6</sup>g<sup>4</sup> = tribizo-aquadgu* = biatrizo-agugu?<br />
+''z<sup>6</sup>g<sup>4</sup> = tribizo-aquadgu* = biatrizo-agugu?<br />
 z<sup>6</sup>gg = tribizo-agugu* = biatrizo-agu?<br />
 z<sup>6</sup>g<sup>8</sup> = tribizo-aquadbigu* = biatrizo-aquadgu?<br />
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 z<sup>8</sup>gg = quadbizo-agugu* = biaquadzo-agu?<br />
 z<sup>9</sup>g<sup>6</sup> = tritrizo-atribigu* = triatrizo-agugu?<br />
-z<sup>9</sup>g<sup>3</sup> = tritrizo-atrigu* = triatrizo-agu?
+z<sup>9</sup>g<sup>3</sup> = tritrizo-atrigu* = triatrizo-agu?''
 == Tricolored examples ==