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

Line 59:

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, the product of the commas' pivots equals the product of the mappings' pivots. 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.

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 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-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'|.

Line 65:

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, the color name always indicates strong ~~extensions~~ vs. weak extensions. 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.

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.

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 ===

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.

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 or yaza Triyo. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is yazala Lasepyo, or simply Lasepyo.

Line 101:

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

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

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.

Latrizo

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).

Combining 2 commas: an upward ext must equal a downward extension: A + B must equal B + A

Yoyo + Lulu = Yoyo (& Loyo), because Lulu is the best ila extension of Yoyo, so the name Yoyo is inherited.

Lulu + Yoyo must also be Yoyo (& Loyo). Yoyo is a strong but not best downward extension of Lulu. it steals the name, removing Lulu from the list to avoid torsion. Lulu is "dis-inherited". The name reflects the worst commas, not the best ones.

If we treat it as 2.3.11.5, Lulu + Yoyo = Lulu & Loyo, strong but not best. Yoyo + Lulu is "downward", Lulu steals the name.

Triyo + Trirubi = Triyo & Rugu, strong but not best upwd ext

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

best up & strong down: Yoyo + Lulu = Yoyo (& Loyo)

best up & weak down: Triyo + Ru = Triyo (& Ru) ............. Sagugu + Ru = Sagugu (& Ru)

strong up & best down:

strong up & strong down: Gu + Ru = Gu & Rugu

strong up & weak down: Triyo + Rugu = Triyo & Rugu

weak up & best down: Liese, Gu + Latruru = (Gu &) Latriru .............. Gu + Laquadru = (Gu &) Laquadru

weak up & strong down: Gu + Zozo = (Gu &) Zozo

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 ~~seems~~ a much better name.

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 ya 10-edo is a much better name.

DEFINITION OF BADNESS:

@@ Line 59: / Line 59: @@
 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.
+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, <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-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'|.
@@ Line 65: / Line 65: @@
 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 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.
 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 ===
-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.
+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 or yaza Triyo. Lasepyo (Orson) implies Ruyoyo and Loruru (Orwell), which is yazala Lasepyo, or simply Lasepyo.
@@ Line 101: / Line 101: @@
 Foe example, Octokaidecal is Sayoyo & Zo, but could be called Zo & Biruyo.
+A strong downward extension always removes the original name if the new comma's pivot is > 1. A strong upward extension never removes it.
+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.
+Latrizo
+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).
+Combining 2 commas: an upward ext must equal a downward extension: A + B must equal B + A
+Yoyo + Lulu = Yoyo (& Loyo), because Lulu is the best ila extension of Yoyo, so the name Yoyo is inherited.
+Lulu + Yoyo must also be Yoyo (& Loyo). Yoyo is a strong but not best downward extension of Lulu. it steals the name, removing Lulu from the list to avoid torsion. Lulu is "dis-inherited". The name reflects the worst commas, not the best ones.
+If we treat it as 2.3.11.5, Lulu + Yoyo = Lulu & Loyo, strong but not best. Yoyo + Lulu is "downward", Lulu steals the name.
+Triyo + Trirubi = Triyo & Rugu, strong but not best upwd ext
+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 up & best down: Vulture  Sasa-quadyo + Saquadru = Sasa-quadyo (& Saquadru)
+best up & strong down: Yoyo + Lulu = Yoyo (& Loyo)
+best up & weak down: Triyo + Ru = Triyo (& Ru) ............. Sagugu + Ru = Sagugu (& Ru)
+strong up & best down:
+strong up & strong down: Gu + Ru = Gu & Rugu
+strong up & weak down: Triyo + Rugu = Triyo & Rugu
+weak up & best down: Liese, Gu + Latruru = (Gu &) Latriru .............. Gu + Laquadru = (Gu &) Laquadru
+weak up & strong down: Gu + Zozo = (Gu &) Zozo
+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 seems a much better name.
+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 ya 10-edo is a much better name.
 DEFINITION OF BADNESS: