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

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La means both large and 11-all, and sa means both small and 17-all. To avoid confusion, large and small should never be abbreviated unless part of a longer word. 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.

Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. two single-comma temperaments. The list minimizes the prime-limit for each comma, so the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. Within each prime limit, the list uses the comma of least [[Odd limit|double odd limit]], with occasional exceptions. See [[User:TallKite/Catalog of seven-limit rank two temperaments with Color names]] for further discussion.

Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. discussing two single-comma temperaments. The comma list minimizes the prime-limit for each comma, so the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. Within each prime limit, the list uses the comma of least [[Odd limit|double odd limit]], with occasional exceptions. 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, because "12-edo" is more informative than "Lalawa". Tempering out the pythagorean comma and 225/224 makes 12-edo & Ruyoyo.

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If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with a plus sign: [[Blackwood]] is 5-edo + ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is Gu + 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 3-edo + wa.

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 3-edo + wa. This avoids both 256/243 and Laquinzo being called 5-edo.

Temperaments can be abbreviated ~~just~~ as colors are: zzT = Zozo, y3T = Triyo, Ly#2T = Layobi, and g+z1aT = Gu ~~plus~~ zala.

Temperaments can be abbreviated much as colors are: zzT = Zozo, y3T = Triyo, g&rgT = Gu & Rugu, Ly#2T = Layobi, and g+z1aT = Gu + zala.

== Naming multi-comma temperaments ==

=== Choosing the commas ===

Rules for choosing the comma set, in order of priority:

#The prime limit of each comma must be higher than the one before.

#The color name must indicate strong vs. weak extensions, if possible.

#Double odd limit must be minimized.

The first rule completely determines the first comma, except for the edo problem (see below).

The name of a comma indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo something into 3 parts, Ruyoyo splits nothing, etc. 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. Likewise, Liese is Gu & Trizogu (81/80 & 1029/1000), not Gu & Trizo-aquadgu (81/80 & 686/675), because the latter would imply a strong extension of Gu, whereas Liese is a weak extension.

=== 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. Gu & Ru is a strong extension of Gu, but not the best strong extension, so Gu & Ru 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, if you want to indicate all the commas. 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.

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 can't be called Gu, but it can be called Zozo. And Sagugu & Zozo can't be shortened, it's always called Sagugu & Zozo.

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.

Edos become rank-2 in two ways. One way is by adding an untempered prime, as in Blackwood, which is 5-edo + Ya. The "+ Ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. 12-edo & Ruyoyo. Since Ruyoyo is yaza, the Gu & Ru commas are no longer implied.

Rules #1 and #2 conflict sometimes:

QUARTER-SPLIT PROBLEM:

Sagugu & Quadru = (P8/4, P5), the name implies a different pergen, so make it Rurugu & Quadru?

Yoyo & Quadlo = Yoyo & Quadluyo = (P8, P4/4), so make it Lologu & Quadlo?

Clyde: Tribiyo & Sasa-quadtrizo = (P8, W4P4/12)

Also a ninth-split problem, sixteenth-split problem, etc.

EDO PROBLEM:

4-edo & Yoyo = (P8/4) implies a different pergen. 9/8 halves the octave, Yoyo halves the 5th, so make it 2-edo & Yoyo?

Better example: Sawa (5-edo) and Yoyo = 10-edo, 10-edo & Yoyo implies more splitting, call it 5-edo & Yoyo?

== Advantages of color names ==

A temperament's color name is fairly concise. Assuming a reasonable prime-limit, 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.

@@ Line 21: / Line 21: @@
 La means both large and 11-all, and sa means both small and 17-all. To avoid confusion, large and small should never be abbreviated unless part of a longer word. 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.
-Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. two single-comma temperaments. The list minimizes the prime-limit for each comma, so the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. Within each prime limit, the list uses the comma of least [[Odd limit|double odd limit]], with occasional exceptions. See [[User:TallKite/Catalog of seven-limit rank two temperaments with Color names]] for further discussion.
+Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. discussing two single-comma temperaments. The comma list minimizes the prime-limit for each comma, so the 1st comma is ya, the 2nd yaza, the 3rd yazala, etc. Within each prime limit, the list uses the comma of least [[Odd limit|double odd limit]], with occasional exceptions. 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, because "12-edo" is more informative than "Lalawa". Tempering out the pythagorean comma and 225/224 makes 12-edo & Ruyoyo.
@@ Line 27: / Line 27: @@
 If the comma(s) don't include every prime in the subgroup, some primes are untempered. These primes are added with a plus sign: [[Blackwood]] is 5-edo + ya. The 2.3.5.7.11 subgroup with 81/80 tempered out is Gu + 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 3-edo + wa.
+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 3-edo + wa. This avoids both 256/243 and Laquinzo being called 5-edo.
-Temperaments can be abbreviated just as colors are: zzT = Zozo, y<sup>3</sup>T = Triyo, Ly#2T = Layobi, and g+z1aT = Gu plus zala.
+Temperaments can be abbreviated much as colors are: zzT = Zozo, y<sup>3</sup>T = Triyo, g&rgT = Gu & Rugu, Ly#2T = Layobi, and g+z1aT = Gu + zala.
+== Naming multi-comma temperaments ==
+=== Choosing the commas ===
+Rules for choosing the comma set, in order of priority:
+#The prime limit of each comma must be higher than the one before.
+#The color name must indicate strong vs. weak extensions, if possible.
+#Double odd limit must be minimized.
+The first rule completely determines the first comma, except for the edo problem (see below).
+The name of a comma indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo something into 3 parts, Ruyoyo splits nothing, etc. 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. Likewise, Liese is Gu & Trizogu (81/80 & 1029/1000), not Gu & Trizo-aquadgu (81/80 & 686/675), because the latter would imply a strong extension of Gu, whereas Liese is a weak extension.
+=== 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. Gu & Ru is a strong extension of Gu, but not the best strong extension, so Gu & Ru 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, if you want to indicate all the commas. 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.
+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 can't be called Gu, but it can be called Zozo. And Sagugu & Zozo can't be shortened, it's always called Sagugu & Zozo.
+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.
+Edos become rank-2 in two ways. One way is by adding an untempered prime, as in Blackwood, which is 5-edo + Ya. The "+ Ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. 12-edo & Ruyoyo. Since Ruyoyo is yaza, the Gu & Ru commas are no longer implied.
+Rules #1 and #2 conflict sometimes:
+QUARTER-SPLIT PROBLEM:
+Sagugu & Quadru = (P8/4, P5), the name implies a different pergen, so make it Rurugu & Quadru?
+Yoyo & Quadlo = Yoyo & Quadluyo = (P8, P4/4), so make it Lologu & Quadlo?
+Clyde: Tribiyo & Sasa-quadtrizo = (P8, W<sup>4</sup>P4/12)
+Also a ninth-split problem, sixteenth-split problem, etc.
+EDO PROBLEM:
+-edo & Yoyo = (P8/4) implies a different pergen. 9/8 halves the octave, Yoyo halves the 5th, so make it 2-edo & Yoyo?
+Better example: Sawa (5-edo) and Yoyo = 10-edo, 10-edo & Yoyo implies more splitting, call it 5-edo & Yoyo?
 == Advantages of color names ==
 A temperament's color name is fairly concise. Assuming a reasonable prime-limit, 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.