Kite's color notation/Temperament names: Difference between revisions
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#The choice of commas must allow elimination of commas via downward inheritances. | #The choice of commas must allow elimination of commas via downward inheritances. | ||
#[[Odd limit|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. | Rule #1 ensures linear independence. It completely determines the first comma. Given two yaza commas, one can always derive the ya comma by combining the two commas such that the za component becomes zero. For example, take Ruyoyoo and Biruyo. Subtract Ruyoyo twice from Biruyo to get Sagugu. Next take Latrizo and Biruyo. The za-exponents are 3 and -2 respectively, so two Latrizos plus three Biruyos make a ya comma, Latribiyo. | ||
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 | 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. (But if the products are the same, it's possible that there is both torsion <u>and</u> contorsion.) | ||
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 & Latrizo = 2·3 = 6, Gu & Biruyo = 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 & Lulu). | |||
Some double-split pergens have more splitting than the pivot product implies, thus a "quad-" comma can make an 8-fold split, e.g. Laquadlo = (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 & Quadru 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 & Rugu has no torsion, and is a better name than Quadgu & Quadru. | |||
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 & 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 avoiding torsion 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. | 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 & 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 avoiding torsion 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. | ||
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=== 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 adds Zotrigu. This is called simply Gu, or Gu | 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 adds Zotrigu. This is called simply Gu, or Gu-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. | ||
Triyo implies Ru, and Triyo & Ru is called Triyo | Triyo implies Ru, and Triyo & Ru is called Triyo-d. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is 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 & Ru is an upward extension of Sayo, and also a downward extension of Ru. These both happen to be not only strong extensions but also the best strong extensions, and this extension could be called either 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 & Ru is an upward extension of Sayo, and also a downward extension of Ru. 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 & Zozo is upwardly weak but downwardly strong, so it can't be called Gu, but it can be (and is) called Zozo. And Sagugu & Zozo 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. 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.''] | [''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. For example, za [[Liese]] is called Latriru, 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-agugu, which is rank-3, not rank-2. Thus the 2nd comma must be za, not yaza. | 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 Latriru, 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-agugu, 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. | 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. | ||
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). 7-edo implies Gu and Ru. 19-edo implies Gu and Lazo. 22-edo implies Triyo and Ru. | 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 Sawa + ya. The "+ ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. Lalawa & 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 Sawa + ya. The "+ ya" means the Gu comma is no longer implied. The other way is to add a bicolored comma, e.g. Lalawa & Ruyoyo. Since Ruyoyo is yaza, the Gu & Ru commas are no longer implied. | ||
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=== Issues === | === Issues === | ||
<u>SELECTING THE COMMA SET</u>: | |||
SELECTING THE COMMA SET: | |||
For some temperaments, the commas' odd limits are much smaller if one changes 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. The 2nd comma's pivot is the ya-exponent. | For some temperaments, the commas' odd limits are much smaller if one changes 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. The 2nd comma's pivot is the ya-exponent. | ||
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weak up & weak down: Triyo + Zozo = Triyo & Zozo | weak up & weak down: Triyo + Zozo = Triyo & Zozo | ||
DEFINITION OF BADNESS: | <u>DEFINITION OF BADNESS</u>: | ||
The definition should take into account both error and complexity. There are two main definitions, logflat badness and cangwu badness. The | The definition should take into account both error and complexity. There are two main definitions, logflat badness and cangwu badness. The Cangwu badness is sqrt (k*complexity^2 + (complexity*error)^2) for a weighting parameter k. So the definition will inevitably be somewhat arbitrary. The best extension will also not be obvious from merely examining the commas, but will require lengthy computations. This removes one of the main advantages of color names, that the comma set, and hence the mapping, the pergen, etc., can be derived directly from the name. | ||
When two different extensions could both arguably be considered the best, depending on the exact metric, one way to resolve the matter is to not allow either one to inherit the name. | When two different extensions could both arguably be considered the best, depending on the exact metric, one way to resolve the matter is to not allow either one to inherit the name. | ||
The best metric for naming purposes is one that tends to give the same inheritances that have already been agreed on. This hasn't been determined yet. | The best metric for naming purposes is one that tends to give the same inheritances that have already been agreed on. This hasn't been determined yet. | ||
<u>AN ATTEMPT TO NAME MEANTONE STRONG EXTENSIONS (AKA MEANTONE'S IMMEDIATE FAMILY) WITH TWEAKS AKA WARTS</u>: | |||
2.3.5.7 | |||
The badness is from the xenwiki page on the meantone family. I just took the only 5 strong extensions listed and ranked them by badness. | |||
Meantone-d is septimal, 7/4 = A6, Badness: 0.0170 | |||
Meantone-dd is dominant, 7/4 = m7, Badness: 0.0207 | |||
Meantone-ddd is sharptone, 7/4 = M6, Badness: 0.0248 | |||
Meantone-dddd is flattone, 7/4 = d7, Badness: 0.0386 | |||
Meantone-ddddd is Plutus, 7/4 = M7, Badness: 0.0453 | |||
2.3.5.11 | |||
Unfortunately the page doesn't list any 2.3.5.11 strong extensions at all, so I don't know the badnesses. So I just guessed at the rankings. | |||
Meantone-e is unidecimal, 11/8 = AA3 | |||
Meantone-ee is meanpop, 11/8 = dd5 | |||
Meantone-eee is Meanenneadecal, 11/8 = A4 | |||
Meantone-eeee is Meanundeci or Meanertone, 11/8 = P4 | |||
11-limit dominant, 11/8 = d5 | |||
Domination, 11/8 = A3 | |||
== Advantages of color names == | == Advantages of color names == | ||
The color name can be derived from the comma list, and vice versa. The color name can be derived from the mapping matrix, and vice versa. However, inheritances have the same name. | The color name can be derived from the comma list, and vice versa. The color name can be derived from the mapping matrix, and vice versa. However, inheritances have the same name. | ||