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

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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.

To find the comma from the color name, first find the ratio of the midpoint of the segment, which has a 3-exponent that is a multiple of 7. Then find the cents of this ratio, and use this chart:

To find the comma from the color name, first find the ratio of the midpoint of the segment, which has a 3-exponent that is a multiple of 7. Then find the cents of this ratio, and use this chart, which is based on the 3-limit Dorian scale:

{| class="wikitable"

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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 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~~.

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 pivot product, and the amount of splitting in the pergen: Zozo splits something in half, Triyo splits something into 3 parts, as does Trizo-agugu. Neither Ru nor Ruyoyo split anything.

Because of rule #2, the color name always indicates strong vs. weak upward extensions. 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 ===

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 yaza. (The adjective yaza comes last, otherwise yazala Gu ~~would~~ be confused with yaza Lagu.) 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 adds Zotrigu. This is called simply Gu, or Gu yaza. (The adjective yaza comes last, otherwise yazala Gu might be confused with yaza Lagu.) 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 yaza. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is Lasepyo, or Lasepyo yazala.

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[''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 and 686/675 (z3gg). Both are lower odd limit than the Latrilu comma, thus without rule #3 7-limit 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 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 (z3gg). Both are lower odd limit than the Latrilu 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 ~~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.

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.

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.'']

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=== Issues ===

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 ya is a much better name.

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 example, Octokaidecal is Sayoyo & Zo, but could be called Zo & Biruyo.

For example, Octokaidecal is Sayoyo & Zo, but could be called Zo & Biruyo. Miracle is Lala-tribiyo & Ruyoyo, but could be Latrizo & Ruyoyo.

A strong downward extension always removes the original name if the new comma's pivot is > 1. A strong upward extension never removes it.

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Squares is Laquadru = (P8, P11/4). Sidi adds the Yoyo comma, (P8, P5/2) which is also (P8, P11/2). Sidi is a strong extension of Laquadru, but it's called Yoyo & Zozoyo, so it doesn't look like a strong extension, or even a weak one. Adding a lower prime with a similar pergen changes the higher prime's comma. za Orwell is Sepru, yaza Orwell is Lasepyo (& Ruyoyo).

There could be a rule that if two primes make the same pergen, choose the one who's IRREF comma has the lowest double odd limit to head up the subgroup. Thus Yoyo + Lulu = 2.3.5.11 = Yoyo (& Luyo), but Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). But Beep remains 2.3.5.7 = Gugu & Zogu.

Beep = Gugu + Zozo = Gugu (& Zogu). It's named after the badder of the two commas, so that the less bad comma can be the best extension. We use bad commas in order to get fewer commas.

To isolate each prime's effect on the temperament, put the comma list in [[Normal lists|IRREF]] form.

''There could be a rule that if two primes make the same pergen, choose the one who's IRREF comma has the lowest double odd limit to head up the subgroup. Thus Yoyo + Lulu = 2.3.5.11 = Yoyo (& Luyo) as before, but Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). Before, it was Trisa-yoyo (& Saluyo), so this new rule makes a shorter name. But Beep remains 2.3.5.7 = Gugu (& Zogu), which we don't want, because the Gugu comma is so high-error.''

~~Latrizo~~

''Or we could choose the IRREF comma that has the lowest badness. This makes Yoyo + Lulu = 2.3.11.5 = Lulu (& Luyo), Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). Gugu + Zogu = Zozo (& Zogu). But sometimes the names in parentheses are NOT the best extensions, and they can't be dropped.''

Old names: Hemififths = P5/2 = Sasa-zozo ==> Trisa-yoyo ==> Lulu ==> Thuthu. All commas have the same pergen. Lulu = 243/242, Thuthu = 512/507. Ordering the primes by odd limit of the commas makes a 2.3.11.13.7.5 temperament, called Lulu (& Thulu & Saluzo/Tholuluzo & Saluyo/Tritho-aquadlu-ayo/Luzozogu/Thuzozogu).

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Trirubi + Triyo = Triyo & Rugu, best ext of Trirubi but NOT best ext of Triyo. Trirubi + Triyo could be Trirubi (& Rugu) if viewed as 2.3.7.5.

''Best extension = IRREF comma makes same pergen, has least double odd limit? No, makes Gu (& Ru). Can't ignore error. Has least badness? No, Triyo + Ru = Triyo (& Ru), not the same pergen but still the best ext.''

''Best extension = IRREF comma makes same pergen, has least double odd limit? No, makes Gu (& Ru). Can't ignore error. Has least badness? No, Triyo + Ru = Triyo (& Ru), not the same pergen but still the best ext. Two low badness commas can make a high-badness temperament.''

best up & best down: Vulture Sasa-quadyo + Saquadru = Sasa-quadyo (& Saquadru)

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weak up & weak down: Triyo + Zozo = Triyo & Zozo

~~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 ya is a much better name.

DEFINITION OF BADNESS:

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 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.
-To find the comma from the color name, first find the ratio of the midpoint of the segment, which has a 3-exponent that is a multiple of 7. Then find the cents of this ratio, and use this chart:
+To find the comma from the color name, first find the ratio of the midpoint of the segment, which has a 3-exponent that is a multiple of 7. Then find the cents of this ratio, and use this chart, which is based on the 3-limit Dorian scale:
 {| class="wikitable"
 |+
@@ Line 87: / Line 87: @@
 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 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.
+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 pivot product, and the amount of splitting in the pergen: Zozo splits something in half, Triyo splits something into 3 parts, as does Trizo-agugu. Neither Ru nor Ruyoyo split anything.
 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 ===
-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 yaza. (The adjective yaza comes last, otherwise yazala Gu would be confused with yaza Lagu.) 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 adds Zotrigu. This is called simply Gu, or Gu yaza. (The adjective yaza comes last, otherwise yazala Gu might be confused with yaza Lagu.) 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 yaza. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is Lasepyo, or Lasepyo  yazala.
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 [''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 and 686/675 (z<sup>3</sup>gg). Both are lower odd limit than the Latrilu comma, thus without rule #3 7-limit 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 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 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 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.
+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.
 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.'']
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 === Issues ===
+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 ya is a much better name.
 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 example, Octokaidecal is Sayoyo & Zo, but could be called Zo & Biruyo.
+For example, Octokaidecal is Sayoyo & Zo, but could be called Zo & Biruyo. Miracle is Lala-tribiyo & Ruyoyo, but could be Latrizo & Ruyoyo.
 A strong downward extension always removes the original name if the new comma's pivot is > 1. A strong upward extension never removes it.
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 Squares is Laquadru = (P8, P11/4). Sidi adds the Yoyo comma, (P8, P5/2) which is also (P8, P11/2). Sidi is a strong extension of Laquadru, but it's called Yoyo & Zozoyo, so it doesn't look like a strong extension, or even a weak one. Adding a lower prime with a similar pergen changes the higher prime's comma. za Orwell is Sepru, yaza Orwell is Lasepyo (& Ruyoyo).
-There could be a rule that if two primes make the same pergen, choose the one who's IRREF comma has the lowest double odd limit to head up the subgroup. Thus Yoyo + Lulu = 2.3.5.11 = Yoyo (& Luyo), but Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). But Beep remains 2.3.5.7 = Gugu & Zogu.
+Beep = Gugu + Zozo = Gugu (& Zogu). It's named after the badder of the two commas, so that the less bad comma can be the best extension. We use bad commas in order to get fewer commas.
+To isolate each prime's effect on the temperament, put the comma list in [[Normal lists|IRREF]] form.
+''There could be a rule that if two primes make the same pergen, choose the one who's IRREF comma has the lowest double odd limit to head up the subgroup. Thus Yoyo + Lulu = 2.3.5.11 = Yoyo (& Luyo) as before, but Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). Before, it was Trisa-yoyo (& Saluyo), so this new rule makes a shorter name. But Beep remains 2.3.5.7 = Gugu (& Zogu), which we don't want, because the Gugu comma is so high-error.''
-Latrizo
+''Or we could choose the IRREF comma that has the lowest badness. This makes Yoyo + Lulu = 2.3.11.5 = Lulu (& Luyo), Trisa-yoyo + Lulu = 2.3.11.5 = Lulu (& Saluyo). Gugu + Zogu = Zozo (& Zogu). But sometimes the names in parentheses are NOT the best extensions, and they can't be dropped.''
 Old names: Hemififths = P5/2 = Sasa-zozo ==> Trisa-yoyo ==> Lulu ==> Thuthu. All commas have the same pergen. Lulu = 243/242, Thuthu = 512/507. Ordering the primes by odd limit of the commas makes a 2.3.11.13.7.5 temperament, called Lulu (& Thulu & Saluzo/Tholuluzo & Saluyo/Tritho-aquadlu-ayo/Luzozogu/Thuzozogu).
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 Trirubi + Triyo = Triyo & Rugu, best ext of Trirubi but NOT best ext of Triyo. Trirubi + Triyo could be Trirubi (& Rugu) if viewed as 2.3.7.5.
-''Best extension = IRREF comma makes same pergen, has least double odd limit? No, makes Gu (& Ru). Can't ignore error. Has least badness? No, Triyo + Ru = Triyo (& Ru), not the same pergen but still the best ext.''
+''Best extension = IRREF comma makes same pergen, has least double odd limit? No, makes Gu (& Ru). Can't ignore error. Has least badness? No, Triyo + Ru = Triyo (& Ru), not the same pergen but still the best ext. Two low badness commas can make a high-badness temperament.''
 best up & best down: Vulture  Sasa-quadyo + Saquadru = Sasa-quadyo (& Saquadru)
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 weak up & weak down: Triyo + Zozo = Triyo & Zozo
-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 ya is a much better name.
 DEFINITION OF BADNESS: