Kite's color notation/Temperament names

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, excluding any ratio that 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 zzggT refers to tempering out the next largest ratio, 784/675 = 259¢.

If the comma is the 2nd largest valid 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-
• 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 Commas page.

1. Schismatic = Layo = LyT, Mavila = Layo-two = LyT2, Superpyth = Sayo = syT. Meantone = Gu = gT, Father = Gu-two = gT2.
2. Dicot = Yoyo = yyT, Immunity = Sasayoyo = ssyyT. Bug = Gugu = ggT, Diaschismic = Sagugu = sggT, Beatles = Sasagugu = ssggT.
3. Porcupine = Triyo = y³T. Augmented = Trigu = g³T, Laconic = Latrigu = Lg³T, Misty = Sasatrigu = ssg³T.
4. Negri = Laquadyo = Ly⁴T, Tetracot = Saquadyo = sy⁴T, Vulture = Sasaquadyo = 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.

Even with these abbreviations, temperament names can get quite long. To make them quicker to say, "bee" and "tree" are preferred over "bye" and "try". If the comma's ratio has N digits, the temperament name will usually have N, N-1, N+1 or occasionally N+2 syllables.

La also means 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".

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.

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 two obvious ways to name multi-comma temperaments. The odd name minimizes the double odd limit of the comma set, and the prime name minimizes the number and size of the primes used by each comma. The odd name for 7-limit Pajara is rryy&rT, and the prime name is sgg&rT. Often the two names are identical, e.g. y³&rT. The odd name is often shorter, and usually indicates commas more likely to be pumped. The prime name shows relationships between bicolored rank-2 temperaments better. The question of which name to use is not yet fully resolved.

Rules for naming remote colors and extreme magnitudes

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 trizoagu = z³g.
• The "a" in la and sa acts as a delimiter: trilayo = L³y and trilatriyo = L³y³
• [Is this rule needed?] Multiplier syllables fuse into one multiplier word: tribizogu = (tribi) x (zogu) = z⁶g⁶, not (tri) x (bizogu).
• Avoid using the -a- delimiter if possible: z⁴gg = bizozogu, not quadzoagugu

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

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

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

Bicolored examples

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

g³ = trigu (Augmented)
zg³ = zotrigu (Starling)
zzg³ = zozotrigu
z³g³ = trizogu
z³gg = trizoagugu
z³g = trizoagu

g⁴ = quadgu (Diminished)
zg⁴ = zoquadgu
zzg⁴ = bizogugu
z³g⁴ = trizoaquadgu
z⁴g⁴ = quadzogu
z⁴g³ = quadzoatrigu
z⁴gg = bizozogu (Breedsmic)
z⁴g = quadzoagu

g⁵ = quingu
zg⁵ = zoquingu
zzg⁵ = zozoquingu
z³g⁵ = trizoaquingu
z⁴g⁵ = quadzoaquingu
z⁵g⁵ = quinzogu
z⁵g⁴ = quinzoaquadgu
z⁵g³ = quinzoatrigu
z⁵gg = quinzoagugu
z⁵g = quinzoagu

g⁶ = tribigu
zg⁶ = zotribigu
zzg⁶ = bizotrigu
z³g⁶ = trizogugu
z⁴g⁶ = bizozotrigu
z⁵g⁶ = quinzoatribigu
z⁶g⁶ = tribizogu
z⁶g⁵ = tribizoaquingu
z⁶g⁴ = tribizoaquadgu*
z⁶g³ = trizozogu
z⁶gg = tribizoagugu*
z⁶g = tribizoagu

g⁷ = sepgu
zg⁷ = zosepgu
zzg⁷ = zozosepgu
z³g⁷ = trizoasepgu
z⁴g⁷ = quadzoasepgu
z⁵g⁷ = quinzoasepgu
z⁶g⁷ = tribizoasepgu
z⁷g⁷ = sepzogu
z⁷g⁶ = sepzoatribigu
z⁷g⁵ = sepzoaquingu
z⁷g⁴ = sepzoaquadgu
z⁷g³ = sepzoatrigu
z⁷gg = sepzoagugu
z⁷g = sepzoagu

g⁸ = quabigu
zg⁸ = zoquabigu
zzg⁸ = bizoquadgu
z³g⁸ = trizoaquabigu
z⁴g⁸ = quadzogugu
z⁵g⁸ = quinzoaquabigu
z⁶g⁸ = tribizoaquabigu*
z⁷g⁸ = sepzoaquabigu
z⁸g⁸ = quabizogu
z⁸g⁷ = quabizoasepgu
z⁸g⁶ = quabizoatribigu*
z⁸g⁵ = quabizoaquingu
z⁸g⁴ = quadzozogu
z⁸g³ = quabizoatrigu
z⁸gg = quabizoagugu*
z⁸g = quabizoagu

g⁹ = tritrigu
zg⁹ = zotritrigu
zzg⁹ = zozotritrigu
z³g⁹ = trizotrigu
z⁴g⁹ = quadzoatritrigu
z⁵g⁹ = quinzoatritrigu
z⁶g⁹ = trizozotrigu
z⁷g⁹ = sepzoatritrigu
z⁸g⁹ = quabizoatritrigu
z⁹g⁹ = tritrizogu
z⁹g⁸ = tritrizoaquabigu
z⁹g⁷ = tritrizoasepgu
z⁹g⁶ = tritrizoatribigu*
z⁹g⁵ = tritrizoaquingu
z⁹g⁴ = tritrizoaquadgu
z⁹g³ = tritrizoatrigu*
z⁹gg = tritrizoagugu
z⁹g = tritrizoagu

Possible solution to the GCD problem:

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

z⁶g⁴ = tribizoaquadgu* = biatrizoagugu?
z⁶gg = tribizoagugu* = biatrizoagu?
z⁶g⁸ = tribizoaquabigu* = biatrizoaquadgu?
z⁸g⁶ = quabizoatribigu* = biaquadzoatrigu?
z⁸gg = quabizoagugu* = biaquadzoagu?
z⁹g⁶ = tritrizoatribigu* = triatrizoagugu?
z⁹g³ = tritrizoatrigu* = triatrizoagu?

Tricolored examples

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

1ozgg = lozogugu
1ozzgg = lobizogu
1ozzg = lozozogu
1ozg³ = lozotrigu
etc.

1oozg = lolozogu
1oozgg = lolozogugu
1oozzzgg = bilozogu
1oozzg = bilozoagu

1oozg³ = lolozotrigu
1oozzg³ = bilozoatrigu
1ooz³g³ = lolotrizogu
1ooz³gg = lolotrizoagugu
1ooz³g = lolotrizoagu

1o³zg = triloazogu
1o³zgg = triloazogugu
1o³zzgg = triloabizogu
1o³zzg = triloazozogu

1o³zg³ = triloazotrigu
1o³zzg³ = triloazozotrigu
1o³z³g³ = trilozogu
1o³z³gg = trilozoagugu
1o³z³g = trilozoagu

Quadricolored examples

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

3o1uzg = tholuzogu
3o1uzgg = tholuzogugu
3o1uzzgg = tholubizogu
3o1uuzzg = thobiluzoagu
etc.

3oo1uzg = thotholuzogu
3oo1uzgg = thotholuzogugu
3oo1uzzg = thotholuzozogu
3oo1uzzgg = thotholubizogu
3oo1uuzg = bitholuazogu
3oo1uuzgg = bitholuazogugu
3oo1uuzzg = bitholuzoagu
3oo1uuzzgg = bitholuzogu
etc.