Kite's color notation/Temperament names: Difference between revisions
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== Definition== | == Definition== | ||
[[Color notation]] can name every regular temperament. The name is the same as that of the comma(s) tempered out, but using an alternate format designed for commas. This format omits the degree (unison, 2nd, etc.). For example, [[Semaphore]] tempers out the zozo 2nd and is called the Zozo temperament. The name of the comma | [[Color notation]] can name every regular temperament. The name is the same as that of the comma(s) tempered out, but using an alternate format designed for commas. This format omits the degree (unison, 2nd, etc.). For example, [[Semaphore]] tempers out the zozo 2nd and is called the Zozo temperament, written Zozoti or zzT, where "ti" and "T" mean temperament. The name of the comma or the temperament is always capitalized, to distinguish it from the color. Thus zozo refers to all zozo ratios, whereas Zozo refers to one specific zozo ratio, 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, like Zozo, refers to the central segment. Each segment contains 7 ratios. The comma that is tempered out is usually the smallest in cents of those 7. If not, '''-bi''' or '''-tri''' is added to the end of the name to indicate that the comma is the 2nd or 3rd largest ratio in that segment, e.g. [[Mavila]] | 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 usually the smallest in cents of those 7. If not, '''-bi''' or '''-tri''' is added to the end of the name to indicate that the comma is the 2nd or 3rd largest ratio in that segment, e.g. the [[Mavila]] comma is Layobi or Ly#2. The Mavila temperament is Layobiti or Ly#2T. Any comma smaller than 256/243 = 90¢ is guaranteed to be the smallest ratio in its segment, thus -bi and -tri are only used for very large commas. | ||
Some 5-limit examples, sorted by color depth. Many more examples can be found on the comma pages ([[Small comma]], [[Medium comma]], [[Large comma]] and [[Unnoticeable comma]]). | Some 5-limit examples, sorted by color depth. Many more examples can be found on the comma pages ([[Small comma]], [[Medium comma]], [[Large comma]] and [[Unnoticeable comma]]). | ||
#[[Schismatic]] = | #[[Schismatic]] = Layoti, [[Mavila]] = Layobiti, [[Superpyth]] = Sayoti, [[Meantone]] = Guti, [[Father]] = Gubiti. | ||
#[[Dicot]] = | #[[Dicot]] = Yoyoti, [[Immunity family|Immunity]] = Sasa-yoyoti, [[Bug]] = Guguti, [[Diaschismic]] = Saguguti, [[Beatles]] = Sasa-guguti. | ||
#[[Porcupine]] = | #[[Porcupine]] = Triyoti, [[Augmented]] = Triguti, [[Laconic family|Laconic]] = Latriguti, [[Misty comma|Misty]] = Sasa-triguti. | ||
#[[Negri]] = | #[[Negri]] = Laquadyoti, [[Tetracot]] = Saquadyoti, [[Vulture]] = Sasa-quadyoti, [[Diminished]] = Quadguti. | ||
Exponent syllables like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: Bi'''r<u>u</u>''' | Exponent syllables like bi or tri are always unaccented. The final "-ti" is too. To emphasize the prime limit, the first occurrence of the highest prime is always accented: Bi'''r<u>u</u>'''yoti, Bi'''<u>zo</u>'''zoguti. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''Sa'''sa-'''gu'''guti, '''Zo'''zotri'''gu'''ti. | ||
Sometimes the smallest ratio in a segment is some other comma raised to some power. 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 some other comma raised to some power. 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, only intervals are. | 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 & | Multi-comma temperaments are named as a list of commas. For example, 7-limit porcupine is Triyo & Ruti. See below for further discussion. | ||
If the commas don't include every prime in the subgroup, some primes are untempered. These primes are added with a plus sign: the 2.3.5.7.11 subgroup with 81/80 tempered out is | If the commas don't include every prime in the subgroup, some primes are untempered. These primes are added with a plus sign: the 2.3.5.7.11 subgroup with 81/80 tempered out is Guti + zala. Primes 2 and 3 are always assumed to be present in the subgroup, even if the commas don't contain them. They are never added, but are sometimes removed. Prime 3 is removed with the term "Nowa", and prime 2 with "Noca" (ca for clear). Thus 2.5.7 with 50/49 is named Biruyoti Nowa or rryyT-w. "Nowaca" removes both 2 and 3. | ||
If the comma is wa, an edo is implied. For the most common cases of 5-edo, 7-edo and 12-edo, the temperament is named after the wa comma. Thus [[Blackwood]] is | If the comma is wa, an edo is implied. For the most common cases of 5-edo, 7-edo and 12-edo, the temperament is named after the wa comma. Thus [[Blackwood]] is Sawati + ya, [[Whitewood]] is Lawati + ya, and [[Catler]] is Lalawati + za. | ||
Any other wa comma is named using the Wa-N format. Thus [[Countercomp family|Countercomp]] is Wa-41 + ya, not the difficult-to-decipher | Any other wa comma is named using the Wa-N format. Thus [[Countercomp family|Countercomp]] is Wa-41 + ya, not the difficult-to-decipher Tribisawati + ya. Note that multi-ring edos such as 10-edo can't be implied by a wa comma, and Wa-10 is not a valid comma name. However 10-edo can be created by a non-wa comma, or by a wa comma plus a non-wa comma, e.g. Sawa & Yoyoti. | ||
More examples of temperaments: | More examples of temperaments: | ||
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==Naming multi-comma temperaments== | ==Naming multi-comma temperaments== | ||
Multi-comma temperaments are named as a list of commas, e.g. Triyo & | Multi-comma temperaments are named as a list of commas, e.g. Triyo & Ruti. 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=== | ===Choosing the commas=== | ||
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Rule #1 makes 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. Every prime is either a comma pivot or a mapping 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 (rule #2) and the mapping matrix isn't contorted, the product of the commas' pivots equals the product of the mappings' pivots. This number is called the temperament's '''pivot product'''. Torsion always makes the first product bigger, and contorsion likewise increases the 2nd product. Thus if the products differ, one can identify the problem. In particular, one can identify torsion in the comma list and remove it. (But if the products are the same, it's possible that there is both torsion <u>and</u> contorsion, which is bad. So one can't rely on unequal pivot products to detect torsion.) | Rule #1 makes 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. Every prime is either a comma pivot or a mapping 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 (rule #2) and the mapping matrix isn't contorted, the product of the commas' pivots equals the product of the mappings' pivots. This number is called the temperament's '''pivot product'''. Torsion always makes the first product bigger, and contorsion likewise increases the 2nd product. Thus if the products differ, one can identify the problem. In particular, one can identify torsion in the comma list and remove it. (But if the products are the same, it's possible that there is both torsion <u>and</u> contorsion, which is bad. So one can't rely on unequal pivot products to detect torsion.) | ||
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 number of times the first color occurs. Sagugu has a pivot of 2, as does Biruyo. Both Rugu and Zotrigu have 1, and Trizo-agugu has 3. For wa commas, the pivot is the edo: Sawa has a pivot of 5. For multi-comma temperaments, the pivot product is the product of each comma's pivot. Sagugu & | 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 number of times the first color occurs. Sagugu has a pivot of 2, as does Biruyo. Both Rugu and Zotrigu have 1, and Trizo-agugu has 3. For wa commas, the pivot is the edo: Sawa has a pivot of 5. For multi-comma temperaments, the pivot product is the product of each comma's pivot. Sagugu & Latrizoti = 2·3 = 6, Gu & Biruyoti = 1·2 = 2, etc. Thus the color name directly indicates the pivot product. | ||
For a rank-2 temperament with primes 2 and 3 both being mapping pivots, the pivot product indicates how many chains of 5ths are in the temperament's lattice, i.e. the amount of splitting in the [[pergen]]. A pivot product of 2 means something is split in half, e.g. Yoyo is half-fifth and Sagugu is half-octave. Triyo splits something into 3 parts. Neither Ru nor Layobi split anything. 4 means either one thing is split into quarters (e.g. Quadgu), or two things are split into halves (e.g. Zozo & | For a rank-2 temperament with primes 2 and 3 both being mapping pivots, the pivot product indicates how many chains of 5ths are in the temperament's lattice, i.e. the amount of splitting in the [[pergen]]. A pivot product of 2 means something is split in half, e.g. Yoyo is half-fifth and Sagugu is half-octave. Triyo splits something into 3 parts. Neither Ru nor Layobi split anything. 4 means either one thing is split into quarters (e.g. Quadgu), or two things are split into halves (e.g. Zozo & Luluti). | ||
Some double-split pergens have more splitting than the pivot product implies, thus a "quad-" comma can make an 8-fold split, e.g. | Some double-split pergens have more splitting than the pivot product implies, thus a "quad-" comma can make an 8-fold split, e.g. Laquadloti = (P8/2, M2/4). But M2 = P5 + P5 - P8, and P5 = P8/2 + M2/2 = 1 period + 2 generators. Thus P5 has a genspan of 2, and the mapping's pivot product is 2 x 2 = 4. And indeed Laquadlo's lattice has 4 chains of 5ths. 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 the only pergens where the pergen's splitting is more than the pivot product implies. Fortunately imperfect pergens are fairly rare, only about 3% of all rank-2 pergens. For a rank-3 pergen (P8/m, (a,b)/n, (a',b',c')/n'), the pivot product is m·n·n'/|b·c'|. | ||
Eliminating torsion means minimizing the commas' pivots. For example, Quadgu & | Eliminating torsion means minimizing the commas' pivots. For example, Quadgu & Quadruti has a comma pivot product of 16, but the pergen is (P8/4, P5), which means the mapping's pivot product is only 4. Since the ya comma is fixed, the solution is to add/subtract some number of ya commas to the yaza comma to get a new yaza comma that can be simplified. Quadgu plus Quadru equals Quadrugu, which simplifies to Rugu. Quadgu & Ruguti has no torsion, and is a better name than Quadgu & Quadruti. | ||
Because of rule #2, <u>the color name always indicates strong vs. weak upward extensions</u>. A strong extension always has the same pivot product, and a weak extension never does. Thus a strong upward extension always adds a comma with a pivot of 1, and a weak upward extension always adds a comma with a pivot > 1. (See "Issues" for downward extensions.) Gugu = 27/25, and Zozo = 49/48, and each one is (P8, P4/2). Combining both commas, Gugu & | Because of rule #2, <u>the color name always indicates strong vs. weak upward extensions</u>. A strong extension always has the same pivot product, and a weak extension never does. Thus a strong upward extension always adds a comma with a pivot of 1, and a weak upward extension always adds a comma with a pivot > 1. (See "Issues" for downward extensions.) Gugu = 27/25, and Zozo = 49/48, and each one is (P8, P4/2). Combining both commas, Gugu & Zozoti 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 & Zoguti. 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 avoiding torsion is to raise the odd limit. For example, Pajara is Sagugu & Ruti (2048/2025 & 64/63), not Sagugu & Biruyoti, 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, Gu & | 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, Gu & Zotriguti would become Gu & Laruti, and 126/125 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 | 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 Guti. Gu & Ruti 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 Guti adds Zotrigu. This is called za Guti, or Guti-d. The "d" is analogous to '''tweaks''' aka edo warts and indicates prime 7. But unlike tweaks, "-d" is the best extension, and "-dd" is the 2nd best. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma list. | ||
Triyoti implies Ru, and Triyo & Ruti is called Triyo-d. Lasepyoti (Orson) implies Ruyoyo and Loruru (Orwell), which is zala Lasepyo, or Lasepyo-de. | |||
Extensions can be downward (adding lower primes) as well as upward. Every two-comma temperament (i.e. codimension = 2) can be viewed as an extension in either direction. For example, Sayo & | Extensions can be downward (adding lower primes) as well as upward. Every two-comma temperament (i.e. codimension = 2) can be viewed as an extension in either direction. For example, Sayo & Ruti is an upward extension of Sayoti, and also a downward extension of Ruti. These both happen to be not only strong extensions but also the best strong extensions, and this extension could be called either Sayo-d or Ru-c. But the smaller prime is preferred, so it's called Sayo-d. Often strong extensions are not possible in one or both directions, because each comma individually creates a different pergen. For example, Gu & Zozoti is upwardly weak but downwardly strong, so it can't be called Guti, but it can be (and is) called ya Zozoti. And Sagugu & Zozoti is weak both ways, so it can't be shortened. | ||
[''Possible refinement of this: given two commas that are each the strongest extension of the other, and having to choose just one to name the temperament, choose not the lower prime, but the prime with the simplest mapping. Simplest means fewest steps on the genchain from some 3-limit interval. For example, yazala Orwell has mapping [(1 0 3 1 3) (0 7 -3 8 2)]. We have a choice of Lasepyo, Sepru or Laseplo. The genchain mappings for 5, 7 and 11 are -3, 8 and 2. 5/4 is 3 steps away from P1, 7/6 is 1 step from P5, and 11/8 is 2 steps from P1. Thus 7/6 is closest, and Orwell is named Sepru-ce. Another example: yaza Superpyth has commas Sayo and Ru, and mapping [(1 1 -3 4) (0 1 9 -2)]. Here 5/4 and 7/4 both coincide with a 3-limit interval, so instead we use the numbers 9 and -2 and choose 7/4, and name the temperament Ru-c.''] | [''Possible refinement of this: given two commas that are each the strongest extension of the other, and having to choose just one to name the temperament, choose not the lower prime, but the prime with the simplest mapping. Simplest means fewest steps on the genchain from some 3-limit interval. For example, yazala Orwell has mapping [(1 0 3 1 3) (0 7 -3 8 2)]. We have a choice of Lasepyo, Sepru or Laseplo. The genchain mappings for 5, 7 and 11 are -3, 8 and 2. 5/4 is 3 steps away from P1, 7/6 is 1 step from P5, and 11/8 is 2 steps from P1. Thus 7/6 is closest, and Orwell is named Sepru-ce. Another example: yaza Superpyth has commas Sayo and Ru, and mapping [(1 1 -3 4) (0 1 9 -2)]. Here 5/4 and 7/4 both coincide with a 3-limit interval, so instead we use the numbers 9 and -2 and choose 7/4, and name the temperament Ru-c.''] | ||
Rule #3 says that if the upward extension is weak and the downward extension is not only strong but also the best, the name must reflect that by excluding the lower prime. Thus 2.3.5.7 in effect becomes 2.3.7.5. For example, za [[Liese]] is called | Rule #3 says that if the upward extension is weak and the downward extension is not only strong but also the best, the name must reflect that by excluding the lower prime. Thus 2.3.5.7 in effect becomes 2.3.7.5. For example, za [[Liese]] is called Latriruti, after its comma (-9 11 0 -3). The best downward extension of Liese has commas 81/80 and 686/675 (z<sup>3</sup>gg). Both are lower odd limit than the Latriru comma, thus without rule #3 7-limit Liese would be called Gu & Trizo-agugu. But then excluding the Gu comma would make Trizo-aguguti, which is rank-3, not rank-2. Thus the 2nd comma must be za, not yaza. | ||
To apply rule #3, remove that comma's pivot color from all other commas on the list by adding/subtracting it from them. You may need to multiply the other comma first. If given Gu & Trizo-agugu and told that Gu should be excluded, eliminate gu by subtracting two Gu commas from Trizo-agugu, making Satrizo. The cents become negative, so invert to get Latriru. Thus 7-limit Liese is called Latriru-c. | To apply rule #3, remove that comma's pivot color from all other commas on the list by adding/subtracting it from them. You may need to multiply the other comma first. If given Gu & Trizo-agugu and told that Gu should be excluded, eliminate gu by subtracting two Gu commas from Trizo-agugu, making Satrizo. The cents become negative, so invert to get Latriru. Thus 7-limit Liese is called Latriru-c. | ||
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Some rank-2 temperaments have wa commas, which imply edos. Every edo implies other commas, which are simply the best strong extension of the wa 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 by rule #4). 7-edo implies Gu and Ru. 19-edo implies Gu and Lazo. 22-edo implies Triyo and Ru. Tweaks 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 729/704, thus 12-edo becomes 12e-edo.''] | Some rank-2 temperaments have wa commas, which imply edos. Every edo implies other commas, which are simply the best strong extension of the wa 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 by rule #4). 7-edo implies Gu and Ru. 19-edo implies Gu and Lazo. 22-edo implies Triyo and Ru. Tweaks 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 729/704, 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 | Edos become rank-2 in two ways. One way is by adding an untempered prime, as in Blackwood, which is Sawati + ya. The "+ ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. Lalawa & Ruyoyoti. Since Ruyoyo is yaza, the Gu & Ru commas are no longer implied. | ||
===Identifying vanishing commas === | ===Identifying vanishing commas === | ||