Kite's color notation/Temperament names: Difference between revisions
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== | == Definition == | ||
[[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. | ||
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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. | 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 using "T" | Temperaments can be abbreviated using "T": 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. | == Finding the comma from the name and vice versa == | ||
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 the Yobi "comma" and 6/5 would be the Gutri "comma". 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. | |||
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 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. | ||
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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, except for the edo problem (see Issues below). | Rule #1 ensures linear independence. It completely determines the first comma, except possibly for the edo problem (see Issues below). | ||
Rule #1 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 (rule #2) 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'''. Torsion always causes the two products to differ, thus eliminating torsion means minimizing the commas' pivots. | |||
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- | 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-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. 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 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'|. | ||
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 | 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: 5-edo 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 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 | 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 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. | ||
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. | 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 Gu. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma list | 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. 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. | ||
Triyo implies Ru, and Triyo & Ru is called | Triyo implies Ru, and Triyo & Ru is called Triyo or yaza Triyo. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is yazala Lasepyo, or simply Lasepyo. | ||
Extensions can be downward (adding lower primes) as well as upward. Every two-comma (i.e. codimension = 2) | 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 Sayo & Ru could be called either yaza Sayo or yaza Ru. But the smaller prime is preferred, 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 (and is) called Zozo. And Sagugu & Zozo is weak both ways, so it can't be shortened, and is always called Sagugu & Zozo. | ||
''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. 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.'' | ||
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 | 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 Latrilu comma, thus without rule #3 yaza Liese would be called Gu & Trizo-agugu. But 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 is easy to apply. Any comma to be excluded will have a pivot of 1. Simply remove that comma's pivot color from any other commas on the list by adding/subtracting it from them. 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. | |||
Edos become rank-2 in two ways. One way is by adding an untempered prime, as in Blackwood, which is 5-edo + | Some rank-2 temperaments have wa commas, which are written as 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. '''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 729/704, 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 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 === | === Identifying vanishing commas === | ||
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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. | 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 | Sometimes removing colors returns a false positive, 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 this 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. | Thus once the color is reduced to wa, a 2nd test is needed. If you know the cents of each of the commas on the list as well as the one being tested, you can simply keep rough track of the cents as you add and subtract commas. If it's roughly zero, the comma vanishes. If you know each comma's 3-exponent, you can simply add and subtract those instead, and check that the end result is zero. (Presumably the commas won't add up to an entire octave.) | ||
=== Issues === | === Issues === | ||
SELECTING THE COMMA SET: | SELECTING THE COMMA SET: | ||
For some temperaments, | 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. | ||
Foe example, Octokaidecal is Sayoyo & Zo, but could be called Zo & Biruyo. | |||
EDO PROBLEM: | |||
Certain edos can't be created by a wa comma, such as 10-edo. However, they can be created by two commas, e.g. 256/243 & 25/24 make 10-edo. This temperament would logically be called 5-edo & Yoyo, but 10-edo seems a much better name. | |||
DEFINITION OF BADNESS: | |||
The definition should take into account both error and complexity. There are two main definitions, logflat badness and cangwu badness. The latter uses various weighting parameters. 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. | |||
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. | |||
== 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. | |||
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 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 | The color name of a multi-comma temperament creates an easy test to see if some 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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4thward commas sharpen the 5th and 5thward ones flatten it. This indicates where in the scale tree compatible edos are likely to be. Thus temperaments that start with sa-, e.g. Sayo, tend to be compatible with sharp-5th edos 15, 17, 22, etc. And la- temperaments, e.g. Laru, tend to be compatible with edos 19, 21, 26, etc. Several caveats: central comma names like Triyo don't indicate 4thwd vs. 5thwd. Also, it's possible for a la- comma to be 4thwd if the color depth is >= 5 and the color is over, e.g. Laquinyo = Magic = (-10 -1 5). Sasa- or lala- commas are safe up to color depth 11. Furthermore, Layo, while quite 5thwd, is compatible only with accurate-5th edos like 12, 24, etc. | 4thward commas sharpen the 5th and 5thward ones flatten it. This indicates where in the scale tree compatible edos are likely to be. Thus temperaments that start with sa-, e.g. Sayo, tend to be compatible with sharp-5th edos 15, 17, 22, etc. And la- temperaments, e.g. Laru, tend to be compatible with edos 19, 21, 26, etc. Several caveats: central comma names like Triyo don't indicate 4thwd vs. 5thwd. Also, it's possible for a la- comma to be 4thwd if the color depth is >= 5 and the color is over, e.g. Laquinyo = Magic = (-10 -1 5). Sasa- or lala- commas are safe up to color depth 11. Furthermore, Layo, while quite 5thwd, is compatible only with accurate-5th edos like 12, 24, etc. | ||
Color names are easier than [[Tour of Regular Temperaments| | === Advantages over current temperament names === | ||
Color names are easier than [[Tour of Regular Temperaments|current temperament names]] for non-Anglophones. No need to learn to spell and pronounce obscure English words like porcupine, hedgehog and opossum. Color names are based on only those words that a first-year student of English would know, and spelling and pronunciation are greatly simplified. | |||
Color names don't use mnemonics that rely on obscure facts, many with an implicit cultural bias, such as: | Color names don't use mnemonics that rely on obscure facts, many with an implicit cultural bias, such as: | ||
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* Sensis/Sensus | * Sensis/Sensus | ||
* Wurschmidt/Worschmidt/Whirrschmidt | * Wurschmidt/Worschmidt/Whirrschmidt | ||
Temperaments have the same name as commas, reducing memorization, unlike | Temperaments have the same name as commas, reducing memorization, unlike current names, in which: | ||
* The schisma creates Helmholtz | |||
* The diaschisma creates Srutal | * The diaschisma creates Srutal | ||
* The semicomma creates Orson | * The semicomma creates Orson | ||
* The gamelisma creates Slendric | * The gamelisma creates Slendric | ||
One last advantage: Color names are very flowing, and fun to say out loud. :) | One last advantage: Color names are very flowing, and fun to say out loud. :) | ||
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z<sup>9</sup>gg = tritrizo-agugu<br /> | z<sup>9</sup>gg = tritrizo-agugu<br /> | ||
z<sup>9</sup>g = tritrizo-agu | z<sup>9</sup>g = tritrizo-agu | ||
=== Tricolored examples === | === Tricolored examples === | ||