Kite's color notation/Temperament names

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.

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 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 pages here, here and here.

1. Schismatic = Layo = LyT, Mavila = Layobi = Ly₂T, Superpyth = Sayo = syT. Meantone = Gu = gT, Father = Gubi = g₂T.
2. Dicot = Yoyo = yyT, Immunity = Sasa-yoyo = ssyyT. Bug = Gugu = ggT, Diaschismic = Sagugu = sggT, Beatles = Sasa-gugu = ssggT.
3. Porcupine = Triyo = y³T. Augmented = Trigu = g³T, Laconic = Latrigu = Lg³T, Misty = Sasa-trigu = ssg³T.
4. Negri = Laquadyo = Ly⁴T, Tetracot = Saquadyo = sy⁴T, Vulture = Sasa-quadyo = ssy⁴T. Diminished = Quadgu = g⁴T.

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.

To make the names easier for non-Anglophones, and to make the names quicker to say, the 5 vowels are the basic vowels found in Spanish, ah eh ee oh oo. Quin is an exception. In Spanish and many other languages, "th" would become "tr". See Color Notation/Translations. Number words like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-gugu, biruyo, bizozogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: sasa-gugu, zozotrigu.

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 S = the sum of all the numbers in the monzo except the first one. If and only if S mod 7 is 4 or 5, 256/243 can be subtracted without changing the magnitude, and the comma is the 2nd smallest ratio. Any 204-294¢ comma is -bi, and any 408-498¢ comma is -tri.

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)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 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.)

The odd name for 7-limit Pajara is rryy&rT, the subgroup name is sgg&rT, and the prime/odd 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. See User:TallKite/Catalog of seven-limit rank two temperaments with Color names for further discussion.

If the comma is wa, an edo is implied. The temperament is named after the edo, not the wa comma. Tempering out the pythagorean comma and 225/224 makes 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.

A non-wa comma can also imply an edo, but the temperament name doesn't use edos. Tempering out 128/125 from 2.3.5 makes Trigu, not 3edo plus wa.

Advantages

Color names are fairly concise. If the comma's numerator has N digits, the temperament name will usually have N, N-1, N+1 or occasionally N+2 syllables. Thus the spoken color name is generally much shorter than the spoken ratio.

The color name indicates the prime subgroup. For example, Ruyoyo (225/224, Marvel) is yaza (2.3.5.7) because it contains 2 explicit colors ru and yo and 2 implicit colors wa and clear. For explicit colors, each color pair (yo/gu, zo/ru, ilo/lu etc.) indicates a single prime. For example, Sagugu & Biruyo has only 2 explicit color pairs, and is yaza.

The color name also indicates the rank of the temperament. Ruyoyo is rank-3 because 4 colors minus 1 comma = rank-3. Sagugu & Biruyo is 4 color pairs minus 2 commas = rank-2. Subtract edos, but not plusses. Both 12edo&ryyT (4 colors minus 1 edo and 1 comma) and 5edo+yT (3 colors minus 1 edo) are rank-2. Primes 2 and 3 are assumed present in the temperament even if they are not present in the comma. Biruyo is yaza and rank-3, and Biruyo Nowa is yaza nowa and rank-2.

The color name also hints at the pergen. The name only indicates the amount of splitting, not which wa interval is split. Because Sagugu has gu twice, it halves something, in this case the 8ve. Zozo halves the 4th, Bizozogu halves the 5th, and Latrizo splits the 5th into three parts. A name with a tribi color either splits something into six parts, or splits something into two and something else into three. (This is the rationale for using tribi and not hexa, to show the possibilities.) A strong extension of a temperament has the same pergen, and a weak extension has a different one. Thus Gu & Biruyo must be a weak extension of Gu, and a strong extension of Biruyo. The commas in a multi-comma temperament are chosen to indicate strong & weak extensions.

The color name also indicates splitting of colors other than wa. For example, Ruyoyo equates every zo ratio with a yoyo ratio. Every other yoyo ratio is some yo ratio doubled, so every other zo ratio is halved. The zo ratio may need to be widened by an 8ve, so actually every other voicing of every other zo ratio is halved. Likewise every other ru ratio equals two gu ratios. For example, two yo 3rds equals a zo 6th, and two gu 2nds equals a ru 2nd.

The length of the color name is a rough indication of the comma's taxicab distance in the lattice. Each la- or sa- adds on average 7 steps on the three-axis. Each yo or gu adds a step on the five-axis, each zo/ru adds a seven-axis step, etc. If triangularized taxicab distance is desired, let over-colors (yo, zo, ilo, etc.) cancel under-colors of smaller primes (gu, ru, etc.), and let under-colors cancel smaller over-colors.

The color name indicates the cents of the comma only very loosely. Without an ending -bi, the comma is 0-204¢. If ending with -bi, the comma is 90-408¢, if with -tri, it's 294-612¢, and if with -quad it's 498-702¢.

The taxicab distance and the cents together roughly indicate the damage of the temperament. Gubi is > 90¢ and not far away, and thus high damage. Layobi is medium damage, and Sasa-quadyo is low damage.

It's fairly easy to find the color name for a temperament, except for multi-comma temperaments (see above). If there's only one comma, and it's < 90¢, the name can be found directly from the monzo. The color is obvious. The magnitude is the sum of all the exponents except the 2-exponent, divided by 7 and rounded off.

It's a little harder to find the comma(s) from the color name. The 3-exponent can be found by summing commas. For example, to find the sagugu comma, start by adding two gu commas. This makes |-8 8 -2>, which is unfortunately large, not small. Correct the magnitude by adding or subtracting a centswise-small wa interval. Since we want to traverse two segments, the pythagorean comma is ideal, because it's double large. Subtracting it makes 2*g1 - LLw-2 = |11 -4 -2>, which is indeed small. These commas are all under 25¢, so two of one minus another must be < 90¢, and this must be the smallest ratio in the sagugu segment, and the one we're looking for.

The Triyo comma can be found by subtracting three gu commas from some wa interval. The pythagorean comma is too small at 24¢, so try the large wa unison Lw1 = |-11 7>, aka the apotome. This makes |1 -5 3>, which is indeed central. The cents of Lw1 - 3*g1 is a semitone minus 3 small commas, roughly a quartertone. Again, this is < 90¢, so it must be the smallest ratio in the segment.

One last advantage: Color names are very flowing, and fun to say out loud. :)

Rules for naming remote colors

There can be more than one way to name a comma. To avoid duplicate names, there are naming rules.

• Colors are always listed highest primes first.
• Multipliers affect all subsequent syllables until the -a- delimiter occurs: trizogu = z³g³, but trizo-agu = z³g.
• The "a" in la and sa acts as a delimiter: trilayo = L³y and trila-triyo = L³y³.
• Avoid using the -a- delimiter if possible: z⁴gg = bizozogu, not quadzo-agugu.

Therefore if the name (minus the magnitude) starts with a multiplier word, and there's no -a- delimiter, that first multiplier word usually indicates the color GCD and thus the pergen's split(s). e.g. bizozogu = (P8, P5/2, /1). In the list of colors below, an asterisk marks cases where this isn't possible, and the GCD is not obvious.

Hyphens are used to make the name easier to parse. There are strict rules for hyphenation, to ensure uniformity. Hyphens are inserted before every -a- delimiter and after the magnitude (after the final la- or sa-). However, the hyphen after the magnitude is omitted if it would create a subunit of 1 syllable. Thus Layo, Lalagu and Sagugu are unhyphenated.

Gugu is generally preferred over bigu (zogugu not zobigu, and zozotrigu not bizo-atrigu). But bizo is preferred over zozo sometimes to indicate the GCD, e.g. bizogugu not zozoquadgu. Likewise, tribigu is preferred over trigugu, as is quadbigu over quadgugu, etc.

There follows examples of remote colors, for illustration. These examples don't all correspond to actual temperaments.

Bicolored examples

gg = gugu (Bug)
zgg = zogugu
zzgg = bizogu
zzg = zozogu

g³ = trigu (Augmented)
zg³ = zotrigu (Starling)
zzg³ = zozotrigu
z³g³ = trizogu
z³gg = trizo-agugu
z³g = trizo-agu

g⁴ = quadgu (Diminished)
zg⁴ = zoquadgu
zzg⁴ = bizogugu
z³g⁴ = trizo-aquadgu
z⁴g⁴ = quadzogu
z⁴g³ = quadzo-atrigu
z⁴gg = bizozogu (Breedsmic)
z⁴g = quadzo-agu

g⁵ = quingu
zg⁵ = zoquingu
zzg⁵ = zozoquingu
z³g⁵ = trizo-aquingu
z⁴g⁵ = quadzo-aquingu
z⁵g⁵ = quinzogu
z⁵g⁴ = quinzo-aquadgu
z⁵g³ = quinzo-atrigu
z⁵gg = quinzo-agugu
z⁵g = quinzo-agu

g⁶ = tribigu
zg⁶ = zotribigu
zzg⁶ = bizotrigu
z³g⁶ = trizogugu
z⁴g⁶ = bizozotrigu
z⁵g⁶ = quinzo-atribigu
z⁶g⁶ = tribizogu
z⁶g⁵ = tribizo-aquingu
z⁶g⁴ = tribizo-aquadgu*
z⁶g³ = trizozogu
z⁶gg = tribizo-agugu*
z⁶g = tribizo-agu

g⁷ = sepgu
zg⁷ = zosepgu
zzg⁷ = zozosepgu
z³g⁷ = trizo-asepgu
z⁴g⁷ = quadzo-asepgu
z⁵g⁷ = quinzo-asepgu
z⁶g⁷ = tribizo-asepgu
z⁷g⁷ = sepzogu
z⁷g⁶ = sepzo-atribigu
z⁷g⁵ = sepzo-aquingu
z⁷g⁴ = sepzo-aquadgu
z⁷g³ = sepzo-atrigu
z⁷gg = sepzo-agugu
z⁷g = sepzo-agu

g⁸ = quadbigu
zg⁸ = zoquadbigu
zzg⁸ = bizoquadgu
z³g⁸ = trizo-aquadbigu
z⁴g⁸ = quadzogugu
z⁵g⁸ = quinzo-aquadbigu
z⁶g⁸ = tribizo-aquadbigu*
z⁷g⁸ = sepzo-aquadbigu
z⁸g⁸ = quadbizogu
z⁸g⁷ = quadbizo-asepgu
z⁸g⁶ = quadbizo-atribigu*
z⁸g⁵ = quadbizo-aquingu
z⁸g⁴ = quadzozogu
z⁸g³ = quadbizo-atrigu
z⁸gg = quadbizo-agugu*
z⁸g = quadbizo-agu

g⁹ = tritrigu
zg⁹ = zotritrigu
zzg⁹ = zozotritrigu
z³g⁹ = trizotrigu
z⁴g⁹ = quadzo-atritrigu
z⁵g⁹ = quinzo-atritrigu
z⁶g⁹ = trizozotrigu
z⁷g⁹ = sepzo-atritrigu
z⁸g⁹ = quadbizo-atritrigu
z⁹g⁹ = tritrizogu
z⁹g⁸ = tritrizo-aquadbigu
z⁹g⁷ = tritrizo-asepgu
z⁹g⁶ = tritrizo-atribigu*
z⁹g⁵ = tritrizo-aquingu
z⁹g⁴ = tritrizo-aquadgu
z⁹g³ = tritrizo-atrigu*
z⁹gg = tritrizo-agugu
z⁹g = tritrizo-agu

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

z⁶g⁴ = tribizo-aquadgu* = biatrizo-agugu?
z⁶gg = tribizo-agugu* = biatrizo-agu?
z⁶g⁸ = tribizo-aquadbigu* = biatrizo-aquadgu?
z⁸g⁶ = quadbizo-atribigu* = biaquadzo-atrigu?
z⁸gg = quadbizo-agugu* = biaquadzo-agu?
z⁹g⁶ = tritrizo-atribigu* = triatrizo-agugu?
z⁹g³ = tritrizo-atrigu* = triatrizo-agu?

Tricolored examples

if lu is not doubled or tripled, it just gets tacked onto the beginning:

1uzgg = luzogugu
1uzzgg = lubizogu
1uzzg = luzozogu
1uzg³ = luzotrigu
etc.

1uuzg = luluzogu
1uugg = bilugu
1uuzgg = luluzogugu
1uuzzzgg = biluzogu
1uuzzg = biluzo-agu

1uuzg³ = luluzotrigu
1uuzzg³ = biluzo-atrigu
1uuz³g³ = lulutrizogu
1uuz³gg = lulutrizo-agugu
1uuz³g = lulutrizo-agu

1u³zg = trilu-azogu
1u³zgg = trilu-azogugu
1u³zzgg = trilu-abizogu
1u³zzg = trilu-azozogu

1u³zg³ = trilu-azotrigu
1u³zzg³ = trilu-azozotrigu
1u³z³g³ = triluzogu
1u³z³gg = triluzo-agugu
1u³z³g = 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 these examples, zo merges with the tripled color, not the doubled color:

1uuz⁶g³ = lulu-trizozogu, not bilutrizo-atrigu
1u³z⁶gg = triluzozo-agugu, not trilu-atribizo-agugu

Quadricolored examples

if tho is not doubled or tripled, it just gets tacked onto the beginning:

3o1uzg = tholuzogu
3o1uzgg = tholuzogugu
3o1uzzgg = tholubizogu
3o1uuzzg = thobiluzo-agu
etc.

3oo1uzg = thotholuzogu
3oo1uzgg = thotholuzogugu
3oo1uzzg = thotholuzozogu
3oo1uzzgg = thotholubizogu
3oo1uuzg = bitholu-azogu
3oo1uuzgg = bitholu-azogugu
3oo1uuzzg = bitholuzo-agu
3oo1uuzzgg = bitholuzogu
etc.