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

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=== 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 set, in order of priority:

Any multi-comma temperament tempers out infinitely many commas, but only a few are needed for the name. Rules for choosing the comma set, in order of priority (rules 2 & 3 are not mandatory):

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

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

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=== 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 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 (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 or 7-limit Gu. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma set. 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.

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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.

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.

''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, 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, 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 729/704, thus 12-edo becomes 12e-edo.'']

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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 from the name.

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.

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.

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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?

== Example ==

* Ya examples

** [[Tour of Regular Temperaments#Meantone family|3.1.1 Meantone family]] (P8, P5) Gu

** [[Tour of Regular Temperaments#Schismatic family|3.1.2 Schismatic family]] (P8, P5) Layo

** [[Tour of Regular Temperaments#Kleismic family|3.1.3 Kleismic family]] (P8, P12/6) Tribiyo

** [[Tour of Regular Temperaments#Magic family|3.1.4 Magic family]] (P8, P12/5) Quinyo

** [[Tour of Regular Temperaments#Diaschismic family|3.1.5 Diaschismic family]] (P8/2, P5) Sagugu

** [[Tour of Regular Temperaments#Pelogic family|3.1.6 Pelogic family]] (P8, P5) Layobi

** [[Tour of Regular Temperaments#Porcupine family|3.1.7 Porcupine family]] (P8, P4/3) Triyo

** [[Tour of Regular Temperaments#W.C3.BCrschmidt family|3.1.8 Würschmidt family]] (P8, WWP5/8) Saquadbigu

** [[Tour of Regular Temperaments#Augmented family|3.1.9 Augmented family]] (P8/3, P5) Trigu

** [[Tour of Regular Temperaments#Dimipent family|3.1.10 Dimipent family]] (P8/4, P5) Quadgu

** [[Tour of Regular Temperaments#Dicot family|3.1.11 Dicot family]] (P8, P5/2) Yoyo

** [[Tour of Regular Temperaments#Tetracot family|3.1.12 Tetracot family]] (P8, P5/4) Saquadyo

** [[Tour of Regular Temperaments#Sensipent family|3.1.13 Sensipent family]] (P8, WWP5/7) Sepgu

** [[Tour of Regular Temperaments#Orwell and the semicomma family|3.1.14 Orwell and the semicomma family]] (P8, P12/7) Lasepyo

** [[Tour of Regular Temperaments#Pythagorean family|3.1.15 Pythagorean family]] (P8/12, ^1) 12-edo + ya

** [[Tour of Regular Temperaments#Apotome family|3.1.16 Apotome family]] (P8/7, ^1) 7-edo + ya

** [[Tour of Regular Temperaments#Gammic family|3.1.17 Gammic family]] (P8, P5/20) Laquinquadyo

** [[Tour of Regular Temperaments#Minortonic family|3.1.18 Minortonic family]] (P8, WWP5/17) Trila-segu

** [[Tour of Regular Temperaments#Bug family|3.1.19 Bug family]] (P8, P4/2) Gugu

** [[Tour of Regular Temperaments#Father family|3.1.20 Father family]] (P8, P5) Gubi

** [[Tour of Regular Temperaments#Sycamore family|3.1.21 Sycamore family]] (P8, P5/11) Laleyo

** [[Tour of Regular Temperaments#Escapade family|3.1.22 Escapade family]] (P8, P4/9) Sasa-tritrigu

** [[Tour of Regular Temperaments#Amity family|3.1.23 Amity family]]

** [[Tour of Regular Temperaments#Vulture family|3.1.24 Vulture family]]

** [[Tour of Regular Temperaments#Vishnuzmic family|3.1.25 Vishnuzmic family]]

** [[Tour of Regular Temperaments#Luna family|3.1.26 Luna family]]

** [[Tour of Regular Temperaments#Laconic family|3.1.27 Laconic family]]

** [[Tour of Regular Temperaments#Immunity family|3.1.28 Immunity family]]

** [[Tour of Regular Temperaments#Ditonmic family|3.1.29 Ditonmic family]]

** [[Tour of Regular Temperaments#Shibboleth family|3.1.30 Shibboleth family]]

** [[Tour of Regular Temperaments#Comic family|3.1.31 Comic family]]

** [[Tour of Regular Temperaments#Wesley family|3.1.32 Wesley family]]

** [[Tour of Regular Temperaments#Fifive family|3.1.33 Fifive family]]

** [[Tour of Regular Temperaments#Maja family|3.1.34 Maja family]]

** [[Tour of Regular Temperaments#Pental family|3.1.35 Pental family]]

** [[Tour of Regular Temperaments#Qintosec family|3.1.36 Qintosec family]]

** [[Tour of Regular Temperaments#Trisedodge family|3.1.37 Trisedodge family]]

** [[Tour of Regular Temperaments#Maquila family|3.1.38 Maquila family]]

** [[Tour of Regular Temperaments#Mutt family|3.1.39 Mutt family]]

* [[Tour of Regular Temperaments#Clans|3.2 Za and yaza clans]]

** [[Tour of Regular Temperaments#Gamelismic clan|3.2.1 Gamelismic clan]] (P8, P5/3) Latrizo

** [[Tour of Regular Temperaments#Trienstonic clan|3.2.2 Trienstonic clan]] (P8, P5) Zo

** [[Tour of Regular Temperaments#Slendro clan|3.2.3 Slendro clan]] (P8, P4/2) Zozo

** [[Tour of Regular Temperaments#Jubilismic clan|3.2.4 Jubilismic clan]] (P8/2, P5, ^1) Biruyo (rank-3), or (P8/2, vM3) Biruyo nowa (rank-2)

** [[Tour of Regular Temperaments#Archytas clan|3.2.5 Archytas clan]] (P8, P5) Ru

** [[Tour of Regular Temperaments#Sensamagic clan|3.2.6 Sensamagic clan]] (P8, P5, ^1) Zozoyo (rank-3)

** [[Tour of Regular Temperaments#Hemimean clan|3.2.7 Hemimean clan]] (P8, P5, vM3/2) Zozoquingu (rank-3)

** [[Tour of Regular Temperaments#Mirkwai clan|3.2.8 Mirkwai clan]] (P8, P5) Quinru-aquadyo

** [[Tour of Regular Temperaments#Quince clan|3.2.9 Quince clan]] (P8, P5, ^m6/7) Lasepzo-agugu (rank-3)

* [[Tour of Regular Temperaments#Temperaments for a given comma|4 Temperaments for a given comma]]

** [[Tour of Regular Temperaments#Septisemi temperaments|4.1 Septisemi temperaments]] (P8, P5, ^1) Zogu (rank-3)

** [[Tour of Regular Temperaments#Mint temperaments|4.2 Mint temperaments]] (P8, P5, ^1) Rugu (rank-3)

** [[Tour of Regular Temperaments#Greenwoodmic temperaments|4.3 Greenwoodmic temperaments]] (P8, P5, ^1) Laruruyo (rank-3)

** [[Tour of Regular Temperaments#Avicennmic temperaments|4.4 Avicennmic temperaments]] (P8, P5, ^1) Lazozoyo (rank-3)

** [[Tour of Regular Temperaments#Keemic temperaments|4.5 Keemic temperaments]] (P8, P5, ^1) Zotriyo (rank-3)

** [[Tour of Regular Temperaments#Starling temperaments|4.6 Starling temperaments]] (P8, P5, ^1) Zotrigu (rank-3)

** [[Tour of Regular Temperaments#Marvel temperaments|4.7 Marvel temperaments]] (P8, P5, ^1) Ruyoyo (rank-3)

** [[Tour of Regular Temperaments#Orwellismic temperaments|4.8 Orwellismic temperaments]]

** [[Tour of Regular Temperaments#Octagar temperaments|4.9 Octagar temperaments]]

** [[Tour of Regular Temperaments#Hemifamity temperaments|4.10 Hemifamity temperaments]]

** [[Tour of Regular Temperaments#Porwell temperaments|4.11 Porwell temperaments]]

** [[Tour of Regular Temperaments#Stearnsmic temperaments|4.12 Stearnsmic temperaments]]

** [[Tour of Regular Temperaments#Hemimage temperaments|4.13 Hemimage temperaments]]

** [[Tour of Regular Temperaments#Cataharry temperaments|4.14 Cataharry temperaments]]

** [[Tour of Regular Temperaments#Horwell temperaments|4.15 Horwell temperaments]]

** [[Tour of Regular Temperaments#Breedsmic temperaments|4.16 Breedsmic temperaments]]

** [[Tour of Regular Temperaments#Ragismic microtemperaments|4.17 Ragismic microtemperaments]]

** [[Tour of Regular Temperaments#Landscape microtemperaments|4.18 Landscape microtemperaments]]

** [[Tour of Regular Temperaments#Wizmic microtemperaments|4.19 Wizmic microtemperaments]]

** [[Tour of Regular Temperaments#31 comma temperaments|4.20 31 comma temperaments]]

** [[Tour of Regular Temperaments#Turkish maqam music temperaments|4.21 Turkish maqam music temperaments]]

** [[Tour of Regular Temperaments#Very low accuracy temperaments|4.22 Very low accuracy temperaments]]

** [[Tour of Regular Temperaments#Very high accuracy temperaments|4.23 Very high accuracy temperaments]]

** [[Tour of Regular Temperaments#High badness temperaments|4.24 High badness temperaments]]

** [[Tour of Regular Temperaments#11-limit comma temperaments|4.25 11-limit comma temperaments]]

* [[Tour of Regular Temperaments#Rank-3 temperaments|5 Rank-3 temperaments]]

** [[Tour of Regular Temperaments#Marvel family|5.1 Marvel family]] (P8, P5, ^1) Ruyoyo

** [[Tour of Regular Temperaments#Starling family|5.2 Starling family]] (P8, P5, ^1) Zotrigu

** [[Tour of Regular Temperaments#Gamelismic family|5.3 Gamelismic family]] (P8, P5, ^1) Latrizo + ya

** [[Tour of Regular Temperaments#Breed family|5.4 Breed family]] (P8, P5/2, ^1) Bizozogu

** [[Tour of Regular Temperaments#Ragisma family|5.5 Ragisma family]] (P8, P5, ^1) Zoquadyo

** [[Tour of Regular Temperaments#Landscape family|5.6 Landscape family]] (P8/3, P5, ^1) Trizogugu

** [[Tour of Regular Temperaments#Hemifamity family|5.7 Hemifamity family]]

** [[Tour of Regular Temperaments#Porwell family|5.8 Porwell family]]

** [[Tour of Regular Temperaments#Horwell family|5.9 Horwell family]]

** [[Tour of Regular Temperaments#Hemimage family|5.10 Hemimage family]]

** [[Tour of Regular Temperaments#Sensamagic family|5.11 Sensamagic family]]

** [[Tour of Regular Temperaments#Keemic family|5.12 Keemic family]]

** [[Tour of Regular Temperaments#Sengic family|5.13 Sengic family]]

** [[Tour of Regular Temperaments#Orwellismic family|5.14 Orwellismic family]]

** [[Tour of Regular Temperaments#Nuwell family|5.15 Nuwell family]]

** [[Tour of Regular Temperaments#Octagar family|5.16 Octagar family]]

** [[Tour of Regular Temperaments#Mirkwai family|5.17 Mirkwai family]]

** [[Tour of Regular Temperaments#Hemimean family|5.18 Hemimean family]]

** [[Tour of Regular Temperaments#Mirwomo family|5.19 Mirwomo family]]

** [[Tour of Regular Temperaments#Dimcomp family|5.20 Dimcomp family]]

** [[Tour of Regular Temperaments#Tolermic family|5.21 Tolermic family]]

** [[Tour of Regular Temperaments#Kleismic rank three family|5.22 Kleismic rank three family]]

** [[Tour of Regular Temperaments#Diaschismic rank three family|5.23 Diaschismic rank three family]]

** [[Tour of Regular Temperaments#Didymus rank three family|5.24 Didymus rank three family]]

** [[Tour of Regular Temperaments#Porcupine rank three family|5.25 Porcupine rank three family]]

** [[Tour of Regular Temperaments#Archytas family|5.26 Archytas family]]

** [[Tour of Regular Temperaments#Jubilismic family|5.27 Jubilismic family]]

** [[Tour of Regular Temperaments#Semiphore family|5.28 Semiphore family]]

** [[Tour of Regular Temperaments#Mint family|5.29 Mint family]]

** [[Tour of Regular Temperaments#Valinorismic temperaments|5.30 Valinorismic temperaments]]

** [[Tour of Regular Temperaments#Rastmic temperaments|5.31 Rastmic temperaments]]

** [[Tour of Regular Temperaments#Werckismic temperaments|5.32 Werckismic temperaments]]

** [[Tour of Regular Temperaments#Swetismic temperaments|5.33 Swetismic temperaments]]

** [[Tour of Regular Temperaments#Lehmerismic temperaments|5.34 Lehmerismic temperaments]]

** [[Tour of Regular Temperaments#Kalismic temperaments|5.35 Kalismic temperaments]]

== 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 49: / Line 49: @@
 === 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 set, in order of priority:
+Any multi-comma temperament tempers out infinitely many commas, but only a few are needed for the name. Rules for choosing the comma set, in order of priority (rules 2 & 3 are not mandatory):
 #The prime limit of each comma must be higher than the one before.
 #The color name must indicate strong vs. weak extensions, if possible.
@@ Line 61: / Line 61: @@
 === 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 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 (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 or 7-limit Gu. It can also be called by its full name Gu & Zotrigu, to explicitly indicate the full comma set. 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.
@@ Line 67: / Line 67: @@
 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.
-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.
+''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, 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, 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 729/704, thus 12-edo becomes 12e-edo.'']
@@ Line 82: / Line 84: @@
 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 from the name.
+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.
 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.
@@ Line 101: / Line 103: @@
 Better example: 256/243 (5-edo) and Yoyo = 10-edo, 10-edo & Yoyo implies more splitting, call it 5-edo & Yoyo?
+== Example ==
+* Ya examples
+** [[Tour of Regular Temperaments#Meantone family|3.1.1 Meantone family]] (P8, P5) Gu
+** [[Tour of Regular Temperaments#Schismatic family|3.1.2 Schismatic family]] (P8, P5) Layo
+** [[Tour of Regular Temperaments#Kleismic family|3.1.3 Kleismic family]] (P8, P12/6) Tribiyo
+** [[Tour of Regular Temperaments#Magic family|3.1.4 Magic family]] (P8, P12/5) Quinyo
+** [[Tour of Regular Temperaments#Diaschismic family|3.1.5 Diaschismic family]] (P8/2, P5) Sagugu
+** [[Tour of Regular Temperaments#Pelogic family|3.1.6 Pelogic family]] (P8, P5) Layobi
+** [[Tour of Regular Temperaments#Porcupine family|3.1.7 Porcupine family]] (P8, P4/3) Triyo
+** [[Tour of Regular Temperaments#W.C3.BCrschmidt family|3.1.8 Würschmidt family]] (P8, WWP5/8) Saquadbigu
+** [[Tour of Regular Temperaments#Augmented family|3.1.9 Augmented family]] (P8/3, P5) Trigu
+** [[Tour of Regular Temperaments#Dimipent family|3.1.10 Dimipent family]] (P8/4, P5) Quadgu
+** [[Tour of Regular Temperaments#Dicot family|3.1.11 Dicot family]] (P8, P5/2) Yoyo
+** [[Tour of Regular Temperaments#Tetracot family|3.1.12 Tetracot family]] (P8, P5/4) Saquadyo
+** [[Tour of Regular Temperaments#Sensipent family|3.1.13 Sensipent family]] (P8, WWP5/7) Sepgu
+** [[Tour of Regular Temperaments#Orwell and the semicomma family|3.1.14 Orwell and the semicomma family]] (P8, P12/7) Lasepyo
+** [[Tour of Regular Temperaments#Pythagorean family|3.1.15 Pythagorean family]] (P8/12, ^1) 12-edo + ya
+** [[Tour of Regular Temperaments#Apotome family|3.1.16 Apotome family]] (P8/7, ^1) 7-edo + ya
+** [[Tour of Regular Temperaments#Gammic family|3.1.17 Gammic family]] (P8, P5/20) Laquinquadyo
+** [[Tour of Regular Temperaments#Minortonic family|3.1.18 Minortonic family]] (P8, WWP5/17) Trila-segu
+** [[Tour of Regular Temperaments#Bug family|3.1.19 Bug family]] (P8, P4/2) Gugu
+** [[Tour of Regular Temperaments#Father family|3.1.20 Father family]] (P8, P5) Gubi
+** [[Tour of Regular Temperaments#Sycamore family|3.1.21 Sycamore family]] (P8, P5/11) Laleyo
+** [[Tour of Regular Temperaments#Escapade family|3.1.22 Escapade family]] (P8, P4/9) Sasa-tritrigu
+** [[Tour of Regular Temperaments#Amity family|3.1.23 Amity family]]
+** [[Tour of Regular Temperaments#Vulture family|3.1.24 Vulture family]]
+** [[Tour of Regular Temperaments#Vishnuzmic family|3.1.25 Vishnuzmic family]]
+** [[Tour of Regular Temperaments#Luna family|3.1.26 Luna family]]
+** [[Tour of Regular Temperaments#Laconic family|3.1.27 Laconic family]]
+** [[Tour of Regular Temperaments#Immunity family|3.1.28 Immunity family]]
+** [[Tour of Regular Temperaments#Ditonmic family|3.1.29 Ditonmic family]]
+** [[Tour of Regular Temperaments#Shibboleth family|3.1.30 Shibboleth family]]
+** [[Tour of Regular Temperaments#Comic family|3.1.31 Comic family]]
+** [[Tour of Regular Temperaments#Wesley family|3.1.32 Wesley family]]
+** [[Tour of Regular Temperaments#Fifive family|3.1.33 Fifive family]]
+** [[Tour of Regular Temperaments#Maja family|3.1.34 Maja family]]
+** [[Tour of Regular Temperaments#Pental family|3.1.35 Pental family]]
+** [[Tour of Regular Temperaments#Qintosec family|3.1.36 Qintosec family]]
+** [[Tour of Regular Temperaments#Trisedodge family|3.1.37 Trisedodge family]]
+** [[Tour of Regular Temperaments#Maquila family|3.1.38 Maquila family]]
+** [[Tour of Regular Temperaments#Mutt family|3.1.39 Mutt family]]
+* [[Tour of Regular Temperaments#Clans|3.2 Za and yaza clans]]
+** [[Tour of Regular Temperaments#Gamelismic clan|3.2.1 Gamelismic clan]] (P8, P5/3) Latrizo
+** [[Tour of Regular Temperaments#Trienstonic clan|3.2.2 Trienstonic clan]] (P8, P5) Zo
+** [[Tour of Regular Temperaments#Slendro clan|3.2.3 Slendro clan]] (P8, P4/2) Zozo
+** [[Tour of Regular Temperaments#Jubilismic clan|3.2.4 Jubilismic clan]] (P8/2, P5, ^1) Biruyo (rank-3), or (P8/2, vM3) Biruyo nowa (rank-2)
+** [[Tour of Regular Temperaments#Archytas clan|3.2.5 Archytas clan]] (P8, P5) Ru
+** [[Tour of Regular Temperaments#Sensamagic clan|3.2.6 Sensamagic clan]] (P8, P5, ^1) Zozoyo (rank-3)
+** [[Tour of Regular Temperaments#Hemimean clan|3.2.7 Hemimean clan]] (P8, P5, vM3/2) Zozoquingu (rank-3)
+** [[Tour of Regular Temperaments#Mirkwai clan|3.2.8 Mirkwai clan]] (P8, P5) Quinru-aquadyo
+** [[Tour of Regular Temperaments#Quince clan|3.2.9 Quince clan]] (P8, P5, ^m6/7) Lasepzo-agugu (rank-3)
+* [[Tour of Regular Temperaments#Temperaments for a given comma|4 Temperaments for a given comma]]
+** [[Tour of Regular Temperaments#Septisemi temperaments|4.1 Septisemi temperaments]] (P8, P5, ^1) Zogu (rank-3)
+** [[Tour of Regular Temperaments#Mint temperaments|4.2 Mint temperaments]] (P8, P5, ^1) Rugu (rank-3)
+** [[Tour of Regular Temperaments#Greenwoodmic temperaments|4.3 Greenwoodmic temperaments]] (P8, P5, ^1) Laruruyo (rank-3)
+** [[Tour of Regular Temperaments#Avicennmic temperaments|4.4 Avicennmic temperaments]] (P8, P5, ^1) Lazozoyo (rank-3)
+** [[Tour of Regular Temperaments#Keemic temperaments|4.5 Keemic temperaments]] (P8, P5, ^1) Zotriyo (rank-3)
+** [[Tour of Regular Temperaments#Starling temperaments|4.6 Starling temperaments]] (P8, P5, ^1) Zotrigu (rank-3)
+** [[Tour of Regular Temperaments#Marvel temperaments|4.7 Marvel temperaments]] (P8, P5, ^1) Ruyoyo (rank-3)
+** [[Tour of Regular Temperaments#Orwellismic temperaments|4.8 Orwellismic temperaments]]
+** [[Tour of Regular Temperaments#Octagar temperaments|4.9 Octagar temperaments]]
+** [[Tour of Regular Temperaments#Hemifamity temperaments|4.10 Hemifamity temperaments]]
+** [[Tour of Regular Temperaments#Porwell temperaments|4.11 Porwell temperaments]]
+** [[Tour of Regular Temperaments#Stearnsmic temperaments|4.12 Stearnsmic temperaments]]
+** [[Tour of Regular Temperaments#Hemimage temperaments|4.13 Hemimage temperaments]]
+** [[Tour of Regular Temperaments#Cataharry temperaments|4.14 Cataharry temperaments]]
+** [[Tour of Regular Temperaments#Horwell temperaments|4.15 Horwell temperaments]]
+** [[Tour of Regular Temperaments#Breedsmic temperaments|4.16 Breedsmic temperaments]]
+** [[Tour of Regular Temperaments#Ragismic microtemperaments|4.17 Ragismic microtemperaments]]
+** [[Tour of Regular Temperaments#Landscape microtemperaments|4.18 Landscape microtemperaments]]
+** [[Tour of Regular Temperaments#Wizmic microtemperaments|4.19 Wizmic microtemperaments]]
+** [[Tour of Regular Temperaments#31 comma temperaments|4.20 31 comma temperaments]]
+** [[Tour of Regular Temperaments#Turkish maqam music temperaments|4.21 Turkish maqam music temperaments]]
+** [[Tour of Regular Temperaments#Very low accuracy temperaments|4.22 Very low accuracy temperaments]]
+** [[Tour of Regular Temperaments#Very high accuracy temperaments|4.23 Very high accuracy temperaments]]
+** [[Tour of Regular Temperaments#High badness temperaments|4.24 High badness temperaments]]
+** [[Tour of Regular Temperaments#11-limit comma temperaments|4.25 11-limit comma temperaments]]
+* [[Tour of Regular Temperaments#Rank-3 temperaments|5 Rank-3 temperaments]]
+** [[Tour of Regular Temperaments#Marvel family|5.1 Marvel family]] (P8, P5, ^1) Ruyoyo
+** [[Tour of Regular Temperaments#Starling family|5.2 Starling family]] (P8, P5, ^1) Zotrigu
+** [[Tour of Regular Temperaments#Gamelismic family|5.3 Gamelismic family]] (P8, P5, ^1) Latrizo + ya
+** [[Tour of Regular Temperaments#Breed family|5.4 Breed family]] (P8, P5/2, ^1) Bizozogu
+** [[Tour of Regular Temperaments#Ragisma family|5.5 Ragisma family]] (P8, P5, ^1) Zoquadyo
+** [[Tour of Regular Temperaments#Landscape family|5.6 Landscape family]] (P8/3, P5, ^1) Trizogugu
+** [[Tour of Regular Temperaments#Hemifamity family|5.7 Hemifamity family]]
+** [[Tour of Regular Temperaments#Porwell family|5.8 Porwell family]]
+** [[Tour of Regular Temperaments#Horwell family|5.9 Horwell family]]
+** [[Tour of Regular Temperaments#Hemimage family|5.10 Hemimage family]]
+** [[Tour of Regular Temperaments#Sensamagic family|5.11 Sensamagic family]]
+** [[Tour of Regular Temperaments#Keemic family|5.12 Keemic family]]
+** [[Tour of Regular Temperaments#Sengic family|5.13 Sengic family]]
+** [[Tour of Regular Temperaments#Orwellismic family|5.14 Orwellismic family]]
+** [[Tour of Regular Temperaments#Nuwell family|5.15 Nuwell family]]
+** [[Tour of Regular Temperaments#Octagar family|5.16 Octagar family]]
+** [[Tour of Regular Temperaments#Mirkwai family|5.17 Mirkwai family]]
+** [[Tour of Regular Temperaments#Hemimean family|5.18 Hemimean family]]
+** [[Tour of Regular Temperaments#Mirwomo family|5.19 Mirwomo family]]
+** [[Tour of Regular Temperaments#Dimcomp family|5.20 Dimcomp family]]
+** [[Tour of Regular Temperaments#Tolermic family|5.21 Tolermic family]]
+** [[Tour of Regular Temperaments#Kleismic rank three family|5.22 Kleismic rank three family]]
+** [[Tour of Regular Temperaments#Diaschismic rank three family|5.23 Diaschismic rank three family]]
+** [[Tour of Regular Temperaments#Didymus rank three family|5.24 Didymus rank three family]]
+** [[Tour of Regular Temperaments#Porcupine rank three family|5.25 Porcupine rank three family]]
+** [[Tour of Regular Temperaments#Archytas family|5.26 Archytas family]]
+** [[Tour of Regular Temperaments#Jubilismic family|5.27 Jubilismic family]]
+** [[Tour of Regular Temperaments#Semiphore family|5.28 Semiphore family]]
+** [[Tour of Regular Temperaments#Mint family|5.29 Mint family]]
+** [[Tour of Regular Temperaments#Valinorismic temperaments|5.30 Valinorismic temperaments]]
+** [[Tour of Regular Temperaments#Rastmic temperaments|5.31 Rastmic temperaments]]
+** [[Tour of Regular Temperaments#Werckismic temperaments|5.32 Werckismic temperaments]]
+** [[Tour of Regular Temperaments#Swetismic temperaments|5.33 Swetismic temperaments]]
+** [[Tour of Regular Temperaments#Lehmerismic temperaments|5.34 Lehmerismic temperaments]]
+** [[Tour of Regular Temperaments#Kalismic temperaments|5.35 Kalismic temperaments]]
 == 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.