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

Note that sep- means 7-fold, while se- means 17-fold. Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: Trizogu = z³g³ and Trizo-agu = z³g. The "a" in la and sa acts as a delimiter: Trilayo = L³y, not L³y³, which would be Trila-triyo.   

Hyphens are used to make the name easier to parse. There are strict rules for hyphenation, to ensure uniformity.

Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. See below for further discussion.  

A non-wa comma can also imply an edo, but that edo isn't part of the temperament's name. Tempering out 128/125 from 2.3.5 makes Trigu, not 3-edo + wa. This avoids a non-wa comma having the same name as a wa comma.  

More examples of temperaments:

#The choice of commas must allow elimination of commas via downward inheritances.

#[[Odd limit|Double odd limit]] must be minimized.

Rule #1 makes a comma list that, if viewed as a matrix, has zeros in the upper right corner. Thus each comma's rightmost nonzero number is a pivot of the matrix. The mapping matrix always has zeros in the lower left, thus each row's leftmost nonzero number is a pivot. Every prime is either a comma pivot or a mapping pivot. The sign of the pivot is unimportant, so we'll define the pivot as the absolute value of the number in the matrix. As long as the comma matrix has no torsion (rule #2) and the mapping matrix isn't contorted, the product of the commas' pivots equals the product of the mappings' pivots. This number is called the temperament's '''pivot product'''. Torsion always causes the two products to differ, thus eliminating torsion means minimizing the commas' pivots.  

The pivot product indicates the amount of splitting in the [[pergen]]. 2 means something is split in half. 4 means either one thing is split into quarters, or two things are split into halves. Some double-split pergens have more splitting than the pivot product implies, thus a quad- comma can make an 8-fold split, e.g. Laquadlo = (P8/2, M2/4). But M2 = P5 + P5 - P8. Thus if M2 has a genspan of 4, P5 has a genspan of 2, and the pivot product is 2 x 2 = 4. For a pergen (P8/m, (a,b)/n), where (a,b) is the multigen, the pivot product is m·n/|b|. Pergens with an imperfect multigen (|b| > 1) are fairly rare, only about 3% of all rank-2 pergens. For a rank-3 pergen (P8/m, (a,b)/n, (a',b',c')/n'), the pivot product is m·n·n'/|b·c'|.

Because of rule #2, <u>the color name always indicates strong vs. weak upward extensions</u>. A strong extension always has the same pivot product, and a weak extension never does. Thus a strong upward extension always adds a comma with a pivot of 1, and a weak upward extension always adds a comma with a pivot > 1. (See "Issues" for downward extensions.) Gugu = 27/25, and Zozo = 49/48, and each one is (P8, P4/2). Combining both commas, Gugu & Zozo is a bad name, because it looks like a weak extension of Gugu (and of Zozo) when it is actually strong. This is because Gugu & Zozo has torsion. We can't change the ya comma, because rule #1 completely determines the 1st comma. Instead we change the 2nd one, and call it Gugu & Zogu. The Zogu comma is 21/20, so this name also has the advantage of using a lower odd-limit comma. However, often the effect of avoiding torsion is to raise the odd limit. For example, Pajara is Sagugu & Ru (2048/2025 & 64/63), not Sagugu & Biruyo, even though the Biruyo comma 50/49 has a lower odd limit.  

To apply rule #3, remove that comma's pivot color from all other commas on the list by adding/subtracting it from them. You may need to multiply the other comma first. If given Gu & Trizo-agugu and told that Gu should be excluded, eliminate gu by subtracting two Gu commas from Trizo-agugu, making Satrizo. The cents become negative, so invert to get Latriru.

=== Identifying vanishing commas ===

Yaza Lasepyo sounds too much like yazala Sepyo. Possible solutions:

* Put the subgroup last: Lasepyo yaza

* Add another word: Lasepyo on yaza, or yaza-ish Lasepyo

Eliding the final -a ("yaz' Lasepyo") won't always work, yazal' Lasepyo sounds too much like yaza Lasepyo.

SELECTING THE COMMA SET:

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 (7 and 5) and 2 implicit colors wa and clear (3 and 2). 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 pivot product, and thus 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 one 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 adding either 2 or 3 to the subgroup is a weak extension. For example, Gu & Biruyo must be a weak extension of Gu, and a strong extension of Biruyo. The commas in a multi-comma temperament name are chosen to indicate strong & weak extensions.

* Bibi- is never used, use quad- instead.

* Avoid using the -a- delimiter if possible: z⁴gg = bizozogu, not quadzo-agugu.

Bi- is not used with primary colors (zogugu not zobigu, and zozotrigu not bizo-atrigu), unless preceded by another multiplier (tribigu not trigugu). Bi- is always used with compound colors, to indicate the GCD: bizogugu not zozoquadgu.