Kite's color notation/Temperament names: Difference between revisions
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[[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. | [[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. | ||
The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. A name without a magnitude, | The color defines a lattice row, and the magnitude (large, small, etc.) defines a '''segment''' of that row. A name without a magnitude, like 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. | ||
Words like large, small, double, etc. are abbreviated, to make the names a reasonable length. | Words like large, small, double, etc. are abbreviated, to make the names a reasonable length. | ||
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* [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]] | * [[User:TallKite/Catalog of eleven-limit rank two temperaments with Color names]] | ||
* [[User:TallKite/Catalog of eleven-limit rank three 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 | 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. Quad may optionally be spoken as "kwah", and quin as "kwee" or "keen". 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. 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 smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma bigger than 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. | 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. | ||
Any comma smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment. Any comma bigger than 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. | 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. | ||
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. | 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, only intervals are. | ||
Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ru. 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. | ||
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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. | 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 | 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 two commas having the same name, e.g. 256/243 is 5-edo and |-14 0 0 5> is Laquinzo. | ||
Temperaments can be abbreviated | Temperaments can be abbreviated using "T" like so: Zozo = zzT, Triyo = y<sup>3</sup>T, Gu & Rugu = g&rgT, Layobi = Ly#2T, and Gu + zala = g+z1aT. | ||
Every ratio can be named either as a standard interval or as a comma/temperament, e.g. 128/125 is both the trigu 2nd and the Trigu comma. The latter is awkward for low-odd-limit ratios: 5/4 would be Yobi and 6/5 would be Gutri. But the former is awkward for high odd-limit ratios, because there will be many 2nds and 3rds and even 4ths, and many of them will be negative. | Every ratio can be named either as a standard interval or as a comma/temperament, e.g. 128/125 is both the trigu 2nd and the Trigu comma. The latter is awkward for low-odd-limit ratios: 5/4 would be Yobi and 6/5 would be Gutri. But the former is awkward for high odd-limit ratios, because there will be many 2nds and 3rds and even 4ths, and many of them will be negative. | ||
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=== Choosing the commas === | === Choosing the commas === | ||
Any multi-comma temperament tempers out infinitely many commas, but only a few are needed for the name. Rules for choosing the comma | Any multi-comma temperament tempers out infinitely many commas, but only a few are needed for the name. Rules for choosing the comma list, 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 | #The comma list must be torsion-free. | ||
#The choice of commas must allow elimination of commas via | #The choice of commas must allow elimination of commas via downward inheritances. | ||
#Double odd limit must be minimized. | #[[Odd limit|Double odd limit]] must be minimized. | ||
Rule #1 ensures linear independence. It completely determines the first comma, except for the edo problem (see Issues below). | |||
Rules #1 and #2 make 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. 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 and the mapping matrix isn't contorted, <u>the product of the commas' pivots equals the product of the mappings' pivots</u>. This number is called the temperament's '''pivot product'''. | |||
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-splits are false doubles, which means 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 pergens. | |||
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 | A comma's pivot is the absolute value of the last number in the comma's monzo. The color name of a comma indicates its pivot directly: it's the multiplier of the first color: Sagugu has a pivot of 2, as does Biruyo. Both Rugu and Zotrigu have 1, and Trizo-agugu has 3. For multi-comma temperaments, the color name indicates the pivot product directly: it's the product of each comma's pivot. Sagugu & Latrizo = 2·3 = 6, Gu & Biruyo = 1·2 = 2, etc. Thus the color name indicates the amount of splitting in the pergen: Zozo splits something in half, Triyo splits something into 3 parts, as does Trizo-agugu. Ru and Ruyoyo split nothing. | ||
Because of rule #2, <u>the color name always indicates strong extensions vs. weak extensions</u>. A strong extension always has the same pivot product, and a weak extension never does. Thus a strong extension always adds a comma with a pivot of 1, and a weak extension always adds a comma with a pivot > 1. 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 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 has a lower odd limit. | |||
Rule #3 is justified in the next section. Rule #4 is needed to ensure a unique comma list. An alternative rule would require the comma list to be in Hermite normal form, but with negative pivots allowed to ensure that the comma's cents are positive. But this would result in more obscure commas. For example, Layo & Rugu would become Layo & Laru, and 36/35 would become 59049/57344. This is far less useful musically, thus rule #4 uses the double odd limit. | |||
=== Inheriting temperament names === | === 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 (i.e. lowest badness) strong (i.e. 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 | Multi-comma temperament names can get quite long. To shorten them, certain extensions inherit the name of what they are extended from. The best (i.e. lowest badness) strong (i.e. 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. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma list. In the giant table of temperaments, 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. | 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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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. | 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. | ||
=== Identifying vanishing commas === | |||
Rule #2 ensures that every vanishing comma is some combination of those in the list. This allows an easy way to check if a given comma is tempered out. Repeatedly reduce the prime limit of the comma in question by adding/subtracting the appropriate comma from the list. If the prime limit can be reduced to 1, the comma vanishes. The color name indicates what needs to be subtracted. | |||
For example, consider the Quadgu & Rugu temperament. Does the Zotrigu comma vanish? Remove zo by adding rugu to get quadgu. Remove gu by subtracting quadgu to get wa. Yes, it vanishes. Does the Biruyo comma vanish? Biruyo = ruruyoyo. Remove ru by subtracting rugu twice to get quadyo. Remove yo by adding quadgu to get wa. Yes, it vanishes. Does the Ruyoyo comma vanish? Remove ru by subtracting rugu to get triyo. Adding quadgu gives gu, so the comma can't be reduced to wa, and hence doesn't vanish. | |||
Sometimes this test returns false positives, because the prime limit is reduced to 3, but not necessarily to 1. In other words, the final wa interval may not be the wa unison. But the test never gives false negatives. If the comma's color can't be reduced to wa, the comma definitely does not vanish. | |||
Thus a 2nd test is needed... cents method. degree method. visual method = lattice vectors. | |||
The full set of commas tempered out by Quadgu & Rugu is easily found. Any ya comma must have a pivot that is a multiple of 4. The Rugu comma essentially equates gu with zo, and yo to ru. Thus any yaza comma must have the 3rd and 4th numbers of its monzo add up to a multiple of 4. | |||
=== Issues === | === Issues === | ||
SELECTING THE COMMA SET: | SELECTING THE COMMA SET: | ||
For some temperaments, it would be nice to change the order of higher primes: 2.3.5.7 becomes 2.3.7.5. This means the first comma is za and the second one is yaza. | |||
DEFINITION OF BADNESS: | DEFINITION OF BADNESS: | ||
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Better example: 256/243 (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 == | == 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. | 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 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>. 12edo&ryyT (4 colors minus 1 edo and 1 comma) is rank-2. 5edo+yT (3 colors minus 1 edo) is also 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 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>. 12edo&ryyT (4 colors minus 1 edo and 1 comma) is rank-2. 5edo+yT (3 colors minus 1 edo) is also 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 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. | 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. | ||
The color name also indicates splitting of colors other than wa. For example, Ruyoyo equates every zo ratio with a yoyo ratio. Every other yoyo ratio is some yo ratio doubled, so every other zo ratio is halved. The zo ratio may need to be widened by an 8ve, so actually every other voicing of every other zo ratio is halved. Likewise every other ru ratio equals two gu ratios. For example, two yo 3rds equals a zo 6th, and two gu 2nds equals a ru 2nd. | The color name also indicates splitting of colors other than wa. For example, Ruyoyo equates every zo ratio with a yoyo ratio. Every other yoyo ratio is some yo ratio doubled, so every other zo ratio is halved. The zo ratio may need to be widened by an 8ve, so actually every other voicing of every other zo ratio is halved. Likewise every other ru ratio equals two gu ratios. For example, two yo 3rds equals a zo 6th, and two gu 2nds equals a ru 2nd. | ||
The color name of a multi-comma temperament creates an easy test to see if any other comma vanishes, see above. | |||
The length of the color name is a rough indication of the comma's [[Commas by taxicab distance|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/ru adds a seven-axis step, etc. If [[Commas by taxicab distance|triangularized]] taxicab distance is desired, let over-colors (yo, zo, ilo, etc.) cancel under-colors of smaller primes (gu, ru, etc.), and let under-colors cancel smaller over-colors. | The length of the color name is a rough indication of the comma's [[Commas by taxicab distance|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/ru adds a seven-axis step, etc. If [[Commas by taxicab distance|triangularized]] taxicab distance is desired, let over-colors (yo, zo, ilo, etc.) cancel under-colors of smaller primes (gu, ru, etc.), and let under-colors cancel smaller over-colors. | ||
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* 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 color (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. | Therefore if the color (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. | ||
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. | 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. | ||