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

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== Explanation ==

[[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.). For example, Semaphore is the Zozo temperament. ~~Examples:~~

[[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.). For example, Semaphore is the Zozo temperament. The name of the temperament and the comma is always capitalized, to distinguish it from the color. Thus zozo refers to all zozo ratios, whereas Zozo refers to one specific zozo ratio, the zozo 2nd = zz2 = 49/48.

* [[User:TallKite/~~Catalog of seven-limit rank two temperaments with Color names]]~~

* [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]]

The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. A name without a magnitude, e.g. Zozo, refers to the central segment. Each segment contains 7 ratios. The comma that is tempered out is the smallest in cents of those 7.

* [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]]

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~~. Any possible color combined with any possible magnitude defines a comma, and thus a temperament, but not all of these will be musically useful~~.

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-'''

* -bi or -tri at the end of a name indicates that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Mavila = Layobi

* '''-bi''' or '''-tri''' at the end of a name indicates that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Mavila = Layobi

* 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 comma pages [[Comma|here]], [[Medium commas|here]], [[Large commas|here]] and [[Unnoticeable comma|here]].

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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-''' (as in "legit"), 13-fold = '''the-''' (as in "thesaurus"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc.

~~Sep~~- means 7-fold, while se- means 17-fold.

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 and trila-triyo = L3y3. More examples of temperaments:

* [[User:TallKite/Catalog of seven-limit rank two temperaments with Color names]]

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'''.

* [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]]

* [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]]

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 not 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 not be the smallest, and -bi or -tri 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.

Sometimes the smallest ratio in a segment is a multiple of another comma. For example, the smallest ratio in the central segment of the zozogugu row is 441/400. But since this is (21/20)2, tempering it out would simply result in the Zogu temperament. Thus there is no Bizogu temperament, although there is a Bizogubi one.

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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. 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 below for further discussion.

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

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". For example, tempering out the pythagorean comma and 225/224 makes 12-edo & Ruyoyo.

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.

If the commas 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. This avoids two commas having the same name, e.g. 256/243 is 5-edo and |-14 0 0 5> is Laquinzo.

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

It's fairly easy to find the color name for a temperament~~, except for multi-comma temperaments~~. 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 fairly easy to find the color name for a temperament. If the comma is < 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.

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== Naming multi-comma temperaments ==

Multi-comma temperaments are named as a list of commas, e.g. Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. discussing two single-comma temperaments.

=== Choosing the commas ===

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

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

#~~Eliminate~~ commas via upward and/or downward inheritances.

#The choice of commas must allow elimination of commas via upward and/or downward inheritances.

#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).

The color name of a comma indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo something into 3 parts, Ru and Ruyoyo split nothing, etc. Some double-splits are false doubles, which means a quad- comma can make an 8-fold split, e.g. Laquadlo = (P8/2, M2/4). The comma splits the 8ve "by accident".

The color name of a comma indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo splits something into 3 parts, Ru and Ruyoyo split nothing, etc. Some double-splits are false doubles, which means a quad- comma can make an 8-fold split, e.g. Laquadlo = (P8/2, M2/4). The comma splits the 8ve "by accident".

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 color name differentiates 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.

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 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.

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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 (z3gg) and 1029/1000 (z3g3), all lower odd limit than the Latrilu comma. But Yaza Liese is called ~~not~~ Gu & Trizo-~~aquadgu or~~ Gu & Trizogu, but Latriru.

Rule #3 says that 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 (z3gg) and 1029/1000 (z3g3), all lower odd limit than the Latrilu comma. But Yaza Liese is called neither Gu & Trizo-agugu nor Gu & Trizogu, but Latriru.

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, which are written as edos. 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. [''needs checking: The best extension sometimes creates tweaks, e.g. 12-edo's best 11-limit extension is 33/32, not 729/702, thus 12-edo becomes 12e-edo.'']

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.

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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?

Better example: 256/243 (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.

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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.

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 > 90¢ but somewhat far away, and is medium damage. Sasa-quadyo is < 204¢ and quite far away, and low damage.

Color names are easier than conventional temperament names for non-Anglophones to learn, spell, pronounce and remember. No need to learn obscure English words like porcupine and hedgehog. Color names don't rely on obscure facts to indicate extensions, such as that porcupines and hedgehogs both have quills.

One last advantage: Color names are very flowing, and fun to say out loud. :)

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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 trizo-agu = z3g.

* The "a" in la and sa acts as a delimiter: trilayo = L3y and trila-triyo = L3y3.

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

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. See below for a possible solution.

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 == Explanation ==
-[[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.). For example, Semaphore is the Zozo temperament. Examples:
+[[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.). For example, Semaphore is the Zozo temperament. The name of the temperament and the comma is always capitalized, to distinguish it from the color. Thus zozo refers to all zozo ratios, whereas Zozo refers to one specific zozo ratio, the zozo 2nd = zz2 = 49/48.
-* [[User:TallKite/Catalog of seven-limit rank two temperaments with Color names]]
-* [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]]
+The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. A name without a magnitude, e.g. Zozo, refers to the central segment. Each segment contains 7 ratios. The comma that is tempered out is the smallest in cents of those 7.
-* [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]]
-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. Any possible color combined with any possible magnitude defines a comma, and thus a temperament, but not all of these will be musically useful.
 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-'''
-* -bi or -tri at the end of a name indicates that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Mavila = Layobi
+* '''-bi''' or '''-tri''' at the end of a name indicates that the comma is the 2nd or 3rd largest ratio in that segment, e.g. Mavila = Layobi
 * 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 comma pages [[Comma|here]], [[Medium commas|here]], [[Large commas|here]] and [[Unnoticeable comma|here]].
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 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-''' (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.
-Sep- means 7-fold, while se- means 17-fold.
+Note that sep- means 7-fold, while se- means 17-fold. Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: trizogu = z<sup>3</sup>g<sup>3</sup> and trizo-agu = z<sup>3</sup>g. 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>. More examples of temperaments:
+* [[User:TallKite/Catalog of seven-limit rank two temperaments with Color names]]
-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'''.
+* [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]]
+* [[User:TallKite/Catalog of eleven-limit rank three temperaments with Color names]]
+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 or -tri 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.
 Sometimes the smallest ratio in a segment is a multiple of another comma. For example, the smallest ratio in the central segment of the zozogugu row is 441/400. But since this is (21/20)<sup>2</sup>, tempering it out would simply result in the Zogu temperament. Thus there is no Bizogu temperament, although there is a Bizogubi one.
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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. 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 below for further discussion.
+Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. See below 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.
+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". For example, tempering out the pythagorean comma and 225/224 makes 12-edo & Ruyoyo.
-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.
+If the commas 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. This avoids two commas having the same name, e.g. 256/243 is 5-edo and |-14 0 0 5> is Laquinzo.
@@ Line 37: / Line 37: @@
 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.
-It's fairly easy to find the color name for a temperament, except for multi-comma temperaments. 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 fairly easy to find the color name for a temperament. If the comma is < 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.
@@ Line 44: / Line 44: @@
 == Naming multi-comma temperaments ==
+Multi-comma temperaments are named as a list of commas, e.g. Triyo & Ru. Always use an ampersand, never the word "and", to distinguish between discussing a two-comma temperament vs. discussing two single-comma temperaments.
 === Choosing the commas ===
@@ Line 49: / Line 50: @@
 #The prime limit of each comma must be higher than the one before.
 #The color name must indicate strong vs. weak extensions, if possible.
-#Eliminate commas via upward and/or downward inheritances.
+#The choice of commas must allow elimination of commas via upward and/or downward inheritances.
 #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).
-The color name of a comma indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo something into 3 parts, Ru and Ruyoyo split nothing, etc. Some double-splits are false doubles, which means a quad- comma can make an 8-fold split, e.g. Laquadlo = (P8/2, M2/4). The comma splits the 8ve "by accident".
+The color name of a comma indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo splits something into 3 parts, Ru and Ruyoyo split nothing, etc. Some double-splits are false doubles, which means a quad- comma can make an 8-fold split, e.g. Laquadlo = (P8/2, M2/4). The comma splits the 8ve "by accident".
-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 color name differentiates 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.
+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 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.
@@ Line 64: / Line 65: @@
 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.
+Rule #3 says that 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 neither Gu & Trizo-agugu nor Gu & Trizogu, but Latriru.
-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, which are written as edos. 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. [''needs checking: The best extension sometimes creates tweaks, e.g. 12-edo's best 11-limit extension is 33/32, not 729/702, thus 12-edo becomes 12e-edo.'']
 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.
@@ Line 95: / Line 96: @@
 -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?
+Better example: 256/243 (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 111: / Line 112: @@
 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.
+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 > 90¢ but somewhat far away, and is medium damage. Sasa-quadyo is < 204¢ and quite far away, and low damage.
+Color names are easier than conventional temperament names for non-Anglophones to learn, spell, pronounce and remember. No need to learn obscure English words like porcupine and hedgehog. Color names don't rely on obscure facts to indicate extensions, such as that porcupines and hedgehogs both have quills.
 One last advantage: Color names are very flowing, and fun to say out loud. :)
@@ Line 118: / Line 121: @@
 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 = 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 trila-triyo = L<sup>3</sup>y<sup>3</sup>.
 * Avoid using the -a- delimiter if possible: z<sup>4</sup>gg = bizozogu, not quadzo-agugu.
 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. See below for a possible solution.