Kite's color notation/Temperament names: Difference between revisions
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# [[Negri]] = Laquadyo, [[Tetracot]] = Saquadyo, [[Vulture]] = Sasa-quadyo, [[Diminished]] = Quadgu. | # [[Negri]] = Laquadyo, [[Tetracot]] = Saquadyo, [[Vulture]] = Sasa-quadyo, [[Diminished]] = Quadgu. | ||
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''' ("eh") for exponent: | 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''' ("eh") for exponent: | ||
* 11-fold = '''le-''' (" | * 11-fold = '''le-''' (as in "<u>le</u>git"), 13-fold = '''the-''' (as in "<u>the</u>saurus"). 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|Color Notation/Translations]]. Number words like bi or tri are always unaccented. Quad may optionally be spoken as "kwah", and quin as "kwih" or "kwee" or "keen". To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''gu'''gu, bi'''ru'''yo, bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''. | Sep- means 7-fold, while se- means 17-fold. | ||
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|Color Notation/Translations]]. Number words like bi or tri are always unaccented. Quad may optionally be spoken as "kwah", and quin as "kwih" or "kwee" or "keen". To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''gu'''gu, bi'''ru'''yo, bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''. | |||
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 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. | 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 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. | ||
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=== Choosing the commas === | === Choosing the commas === | ||
Rules for choosing the comma set, in order of priority: | Any multi-comma temperament tempers out infinitely many commas, but only a few are needed for the name. Rules for choosing the comma set, in order of priority: | ||
#The prime limit of each comma must be higher than the one before. | #The prime limit of each comma must be higher than the one before. | ||
#The color name must indicate strong vs. weak extensions, if possible. | #The color name must indicate strong vs. weak extensions, if possible. | ||
# | #Eliminate commas via upward and/or downward inheritances. | ||
#Double odd limit must be minimized. | #Double odd limit must be minimized. | ||
The first rule completely determines the first comma, except for the edo problem (see issues below). | The first rule completely determines the first comma, except for the edo problem (see issues below). | ||
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More importantly, the name can differentiate between strong extensions and weak 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. Instead we 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 implying the right kind of extension 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 is a lower odd limit. | More importantly, the name can differentiate between strong extensions and weak 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. Instead we 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 implying the right kind of extension 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 is a lower odd limit. | ||
=== Inheriting temperament names === | |||
Multi-comma temperament names can get quite long. To shorten them, certain extensions inherit the name of what they are extended from. The best (lowest badness) strong (same pergen) extension of a temperament inherits the name of the temperament. Thus every temperament implies certain other commas. Consider extensions of Gu. Gu & Ru is a strong extension, but not the best strong extension, so nothing is inherited and the name can't be shortened. The best extension of Gu is Gu & Zotrigu. This is usually called simply Gu, or perhaps yaza Gu or 7-limit Gu. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma set. In the table below, it's written as Gu (& Zotrigu). Any combination of the Gu and Zotrigu commas, e.g. Ruyoyo, makes the same extension, so Gu could be said to imply Ruyoyo as well. But such a comma will have a higher odd limit, and isn't part of the name, so the canonical best za extension for Gu is Zotrigu. | |||
Triyo implies Ru, and Triyo & Ru is called simply Triyo, or perhaps yaza Triyo. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is yazala Lasepyo, or simply Lasepyo. | |||
Extensions can be downward (adding lower primes) as well as upward. Every two-comma (i.e. codimension = 2) temperament can be viewed as a strong or weak extension in either direction. For example, Sayo & Ru is a strong extension of Sayo, and also of Ru. These both happen to be the best strong extensions, and Sayo & Ru could be called either yaza Sayo or yaza Ru. But the upward extension always takes priority, so Sayo & Ru is called Sayo. Often strong extensions are not possible in one or both directions, because each comma individually creates a different pergen. For example, Gu & Zozo is upwardly weak but downwardly strong, so it can't be called Gu, but it can be called Zozo. And Sagugu & Zozo is weak both ways, so it can't be shortened, it's always called Sagugu & Zozo. | |||
If the upward extension is weak and the downward extension is the best, the name must reflect that by excluding the lower prime. For example, za Liese is called Latriru, after its comma |-9 11 0 -3>. The best downward extension of Liese has commas 81/80, 686/675 (z<sup>3</sup>gg) and 1029/1000 (z<sup>3</sup>g<sup>3</sup>), all lower odd limit than the Latrilu comma. But Yaza Liese is called not Gu & Trizo-aquadgu or Gu & Trizogu, but Latriru. | |||
(This rule could possibly be extended to include when the upward extension is weak, and the downward extension is strong but not the best strong one. For example, Injera is Gu & Biruyo, but under this rule, it would become Gu & Laruru. However, this would greatly increase the odd limit.) | |||
Some rank-2 temperaments have 3-limit commas. These commas are written as edos, because "12-edo" is more informative than "Lalawa". Every edo implies other commas, which are simply the best strong extension of the 3-limit temperament to higher primes. 12-edo implies Gu and Ru. 5-edo implies Gubi and Zo (and also Ru, but Zo is the canonical comma). 7-edo implies Gu and Ru. 19-edo implies Gu and Lazo. 22-edo implies Triyo and Ru. Tweaks aka warts change the implied comma: 22c-edo implies Gu and Ru. | Some rank-2 temperaments have 3-limit commas. These commas are written as edos, because "12-edo" is more informative than "Lalawa". Every edo implies other commas, which are simply the best strong extension of the 3-limit temperament to higher primes. 12-edo implies Gu and Ru. 5-edo implies Gubi and Zo (and also Ru, but Zo is the canonical comma). 7-edo implies Gu and Ru. 19-edo implies Gu and Lazo. 22-edo implies Triyo and Ru. Tweaks aka warts change the implied comma: 22c-edo implies Gu and Ru. | ||
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=== Issues === | === Issues === | ||
COMPLEXITY: | |||
It would be simpler to select commas solely on prime limit and double odd limit. It's not always possible to indicate the correct amount of splitting in the pergen, e.g. Clyde is Tribiyo & Zozoyo. The name implies a 6-fold split, but the pergen is (P8, W<sup>4</sup>P4/12). | |||
DEFINITION OF BADNESS: | DEFINITION OF BADNESS: | ||
The definition should take into account both error and complexity. There are two main definitions, logflat badness and cangwu badness. The latter | The definition should take into account both error and complexity. There are two main definitions, logflat badness and cangwu badness. The latter uses various weighting parameters. So the definition will inevitably be somewhat arbitrary. The best extension will also not be obvious from merely examining the commas, but will require lengthy computations. This removes one of the main advantages of color names, that the comma set, and hence the mapping, the pergen, etc., can be derived from the name. | ||
The best metric for naming purposes is one that tends to give the same inheritances that have already been agreed on. This hasn't been determined yet. | The best metric for naming purposes is one that tends to give the same inheritances that have already been agreed on. This hasn't been determined yet. | ||
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QUARTER-SPLIT PROBLEM: | QUARTER-SPLIT PROBLEM: | ||
Sagugu & Quadru = (P8/4, P5), the name implies a different pergen, so make it | Sagugu & Quadru = (P8/4, P5), the name implies a different pergen, so make it Quadru & Rurugu? (Breaks rule #1) | ||
Yoyo & Quadlo = Yoyo & Quadluyo = (P8, P4/4), so make it | Yoyo & Quadlo = Yoyo & Quadluyo = (P8, P4/4), so make it Quadlo & Lologu? (Breaks rule #1) | ||
Clyde: Tribiyo & Sasa-quadtrizo = (P8, W<sup>4</sup>P4/12) | Clyde: Tribiyo & Sasa-quadtrizo = (P8, W<sup>4</sup>P4/12) | ||