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

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Multipliers like bi-, tri-, etc. can be combined: 6-fold = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. Higher primes use their color word, but with the suffix '''-e''' ("eh") for exponent:

* 11-fold = '''le-''' (as in "legitimate"), 13-fold = '''the-''' (as in "thesaurus"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc.

Note that sep- means 7-fold, while se- means 17-fold. Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: Trizogu = z3g3 and Trizo-agu = z3g. The "a" in la and sa acts as a delimiter: Trilayo = L3y, not L3y3, which would be Trila-triyo.

Note that sep- means 7-fold, while se- means 17-fold. Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: Trizogu = z3g3 and Trizo-agu = z3g. The "a" in la- and sa- acts as a delimiter: Trilayo = L3y, not L3y3, which would be Trila-triyo.

To make the names easier for non-Anglophones, and to make the names quicker to say, the 5 vowels are the basic vowels found in Spanish, ah-eh-ee-oh-oo. Quin is an exception. Quad may optionally be spoken as "kwah", and quin as "kwee~~" or "keen~~". In Spanish and many other languages, "th" would become "tr". See [[Color notation/Translations|Color Notation/Translations]]. Multiplier words like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''gu'''gu, bi'''ru'''yo, bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''.

It's theoretically possible for a comma to be larger than an octave and still be tempered out, especially a very complex comma. Such a comma is indicated by the prefix '''co-''', which stands for compound. For example, tempering out (4 -18 11) = 1214¢ makes the Cosa-leyo temperament. There's coco-, trico-, quadco-, etc. Multipliers of co- don't affect the magnitude or color.

To make the names easier for non-Anglophones, and to make the names quicker to say, the 5 vowels are the basic vowels found in Spanish, ah-eh-ee-oh-oo. Quin is an exception. Quad may optionally be spoken as "kwah", and quin as "kwee". In Spanish and many other languages, "th" would become "tr". See [[Color notation/Translations|Color Notation/Translations]]. Multiplier words like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''gu'''gu, bi'''ru'''yo, bi'''zo'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''.

Hyphens are used to make the name easier to parse. There are strict rules for hyphenation, to ensure uniformity.

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* Put a hyphen before and after every "twenty", "thirty", "forty", etc.

* Put a hyphen after the magnitude (after the final la- or sa-)

The hyphen after the magnitude is omitted if it would create a subunit of 1 syllable. Thus Layo, Lalagu and Sagugu are all unhyphenated. However, the 2nd rule always holds, e.g. 736/729 is Sa-twenty-tho.

* Put a hyphen after coco-, trico-, etc.

The hyphen after the magnitude is omitted if it would create a subunit of 1 syllable. Thus Layo, Lalagu and Sagugu are all unhyphenated. Likewise, there is no hyphen after co-. However, the 2nd rule always holds, e.g. 736/729 is Sa-twenty-tho.

Sometimes the smallest ratio in a segment is some other comma raised to some power. For example, the smallest ratio in the central segment of the zozogugu row is 441/400. But since this is (21/20)2, tempering it out would simply result in the Zogu temperament. Thus there is no Bizogu temperament, although there is a Bizogubi one.

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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'|.

For a rank-2 temperament with primes 2 and 3 both being mapping pivots, the pivot product indicates how many chains of 5ths are in the temperament's lattice. The pivot product also indicates the amount of splitting in the [[pergen]]. In general, 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: Sawa has a pivot of 5. For multi-comma temperaments, the pivot product is the product of each comma's pivot. Sagugu & Latrizo = 2·3 = 6, Gu & Biruyo = 1·2 = 2, etc. Thus the color name directly indicates the pivot product, 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.

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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 then excluding the Gu comma would make Trizo-agugu, which is rank-3, not rank-2. Thus the 2nd comma must be za, not yaza.

Rule #3 says that if the upward extension is weak and the downward extension is not only strong but also the best, the name must reflect that by excluding the lower prime. 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 Latriru comma, thus without rule #3 7-limit Liese would be called Gu & Trizo-agugu. But then excluding the Gu comma would make Trizo-agugu, which is rank-3, not rank-2. Thus the 2nd comma must be za, not yaza.

To apply rule #3, remove that comma's pivot color from all other commas on the list by adding/subtracting it from them. You may need to multiply the other comma first. If given Gu & Trizo-agugu and told that Gu should be excluded, eliminate gu by subtracting two Gu commas from Trizo-agugu, making Satrizo. The cents become negative, so invert to get Latriru.

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 Multipliers like bi-, tri-, etc. can be combined: 6-fold = tribi-, 8-fold = quadbi-, 9-fold = tritri-, 10-fold = quinbi-, 12-fold = quadtri-, 14-fold = sepbi-, 15-fold = quintri-, 16-fold = quadquad-, etc. Higher primes use their color word, but with the suffix '''-e''' ("eh") for exponent:
 * 11-fold = '''le-''' (as in "<u>le</u>gitimate"), 13-fold = '''the-''' (as in "<u>the</u>saurus"). 17 = '''se-''', 19 = '''ne-''', 23 = '''twenty-the-''', 29 = '''twenty-ne-''', etc.
-Note that sep- means 7-fold, while se- means 17-fold. Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: Trizogu = z<sup>3</sup>g<sup>3</sup> and Trizo-agu = z<sup>3</sup>g. The "a" in la and sa acts as a delimiter: Trilayo = L<sup>3</sup>y, not L<sup>3</sup>y<sup>3</sup>, which would be Trila-triyo.
+Note that sep- means 7-fold, while se- means 17-fold. Multipliers affect all subsequent syllables until the '''-a-''' delimiter occurs: Trizogu = z<sup>3</sup>g<sup>3</sup> and Trizo-agu = z<sup>3</sup>g. The "a" in la- and sa- acts as a delimiter: Trilayo = L<sup>3</sup>y, not L<sup>3</sup>y<sup>3</sup>, which would be Trila-triyo.
-To make the names easier for non-Anglophones, and to make the names quicker to say, the 5 vowels are the basic vowels found in Spanish, ah-eh-ee-oh-oo. Quin is an exception. Quad may optionally be spoken as "kwah", and quin as "kwee" or "keen". In Spanish and many other languages, "th" would become "tr". See [[Color notation/Translations|Color Notation/Translations]]. Multiplier words like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''<u>gu</u>'''gu, bi'''r<u>u</u>'''yo, bi'''<u>zo</u>'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''.
+It's theoretically possible for a comma to be larger than an octave and still be tempered out, especially a very complex comma. Such a comma is indicated by the prefix '''co-''', which stands for compound. For example, tempering out (4 -18 11) = 1214¢ makes the Cosa-leyo temperament. There's coco-, trico-, quadco-, etc. Multipliers of co- don't affect the magnitude or color.
+To make the names easier for non-Anglophones, and to make the names quicker to say, the 5 vowels are the basic vowels found in Spanish, ah-eh-ee-oh-oo. Quin is an exception. Quad may optionally be spoken as "kwah", and quin as "kwee". In Spanish and many other languages, "th" would become "tr". See [[Color notation/Translations|Color Notation/Translations]]. Multiplier words like bi or tri are always unaccented. To emphasize the prime limit, the first occurrence of the highest prime is always accented: sasa-'''<u>gu</u>'''gu, bi'''r<u>u</u>'''yo, bi'''<u>zo</u>'''zogu. In longer names, the 1st occurrence of sa/la and/or of lower primes may also be accented: '''sa'''sa-'''gu'''gu, '''zo'''zotri'''gu'''.
 Hyphens are used to make the name easier to parse. There are strict rules for hyphenation, to ensure uniformity.
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 * Put a hyphen before and after every "twenty", "thirty", "forty", etc.
 * Put a hyphen after the magnitude (after the final la- or sa-)
-The hyphen after the magnitude is omitted if it would create a subunit of 1 syllable. Thus Layo, Lalagu and Sagugu are all unhyphenated. However, the 2nd rule always holds, e.g. 736/729 is Sa-twenty-tho.
+* Put a hyphen after coco-, trico-, etc.
+The hyphen after the magnitude is omitted if it would create a subunit of 1 syllable. Thus Layo, Lalagu and Sagugu are all unhyphenated. Likewise, there is no hyphen after co-. However, the 2nd rule always holds, e.g. 736/729 is Sa-twenty-tho.
 Sometimes the smallest ratio in a segment is some other comma raised to some power. For example, the smallest ratio in the central segment of the zozogugu row is 441/400. But since this is (21/20)<sup>2</sup>, tempering it out would simply result in the Zogu temperament. Thus there is no Bizogu temperament, although there is a Bizogubi one.
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 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'|.
+For a rank-2 temperament with primes 2 and 3 both being mapping pivots, the pivot product indicates how many chains of 5ths are in the temperament's lattice. The pivot product also indicates the amount of splitting in the [[pergen]]. In general, 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: Sawa has a pivot of 5. For multi-comma temperaments, the pivot product is the product of each comma's pivot. Sagugu & Latrizo = 2·3 = 6, Gu & Biruyo = 1·2 = 2, etc. Thus the color name directly indicates the pivot product, 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.
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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 then excluding the Gu comma would make Trizo-agugu, which is rank-3, not rank-2. Thus the 2nd comma must be za, not yaza.
+Rule #3 says that if the upward extension is weak and the downward extension is not only strong but also the best, the name must reflect that by excluding the lower prime. For example, za [[Liese]] is called Latriru, after its comma (-9 11 0 -3). The best downward extension of Liese has commas 81/80 and 686/675 (z<sup>3</sup>gg). Both are lower odd limit than the Latriru comma, thus without rule #3 7-limit Liese would be called Gu & Trizo-agugu. But then excluding the Gu comma would make Trizo-agugu, which is rank-3, not rank-2. Thus the 2nd comma must be za, not yaza.
 To apply rule #3, remove that comma's pivot color from all other commas on the list by adding/subtracting it from them. You may need to multiply the other comma first. If given Gu & Trizo-agugu and told that Gu should be excluded, eliminate gu by subtracting two Gu commas from Trizo-agugu, making Satrizo. The cents become negative, so invert to get Latriru.