Kite's color notation/Temperament names: Difference between revisions
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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. 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. | 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. 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. Tempering out the | 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. | |||
== Rules for naming remote colors | == 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 ([[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. <u>Subtract edos, but not plusses</u>. 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. However, it's not obvious that Sagugu & Biruyo is a strong extension, nor that Sagugu & Zozo isn't. | |||
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/ro adds a seven-axis step, etc. | |||
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 we 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. | 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. | * Colors are always listed highest primes first. | ||
* Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: trizogu = z<sup>3</sup>g<sup>3</sup>, but | * Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: trizogu = z<sup>3</sup>g<sup>3</sup>, but trizo-agu = z<sup>3</sup>g. | ||
* The "a" in la and sa acts as a delimiter: trilayo = L<sup>3</sup>y and | * The "a" in la and sa acts as a delimiter: trilayo = L<sup>3</sup>y and trila-triyo = L<sup>3</sup>y<sup>3</sup> | ||
* [''Is this rule needed?''] Multiplier syllables fuse into one multiplier word: tribizogu = (tribi) x (zogu) = z<sup>6</sup>g<sup>6</sup>, not (tri) x (bizogu). | * [''Is this rule needed?''] Multiplier syllables fuse into one multiplier word: tribizogu = (tribi) x (zogu) = z<sup>6</sup>g<sup>6</sup>, not (tri) x (bizogu). | ||
* Avoid using the -a- delimiter if possible: z<sup>4</sup>gg = bizozogu, not quadzo-agugu | * Avoid using the -a- delimiter if possible: z<sup>4</sup>gg = bizozogu, not quadzo-agugu | ||
Therefore if the name starts with a multiplier word, and there's no -a- delimiter, that first multiplier word usually indicates the | 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|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 | 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). 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. | 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. | There follows examples of remote colors, for illustration. These examples don't all correspond to actual temperaments. | ||
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''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>gg = tribizo-agugu* = biatrizo-agu?<br /> | ||
z<sup>6</sup>g<sup>8</sup> = tribizo-aquadbigu* = biatrizo-aquadgu?<br /> | z<sup>6</sup>g<sup>8</sup> = tribizo-aquadbigu* = biatrizo-aquadgu?<br /> | ||
z<sup>8</sup>g<sup>6</sup> = quadbizo-atribigu* = biaquadzo-atrigu?<br /> | z<sup>8</sup>g<sup>6</sup> = quadbizo-atribigu* = biaquadzo-atrigu?<br /> | ||
z<sup>8</sup>gg = quadbizo-agugu* = biaquadzo-agu?<br /> | 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>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 === | === Tricolored examples === | ||