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

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Rule #1 ensures linear independence. It completely determines the first comma. Given two yaza commas, one can always derive the ya comma by combining the two commas such that the za component becomes zero. For example, take Ruyoyoo and Biruyo. Subtract Ruyoyo twice from Biruyo to get Sagugu. Next take Latrizo and Biruyo. The za-exponents are 3 and -2 respectively, so two Latrizos plus three Biruyos make a ya comma, Latribiyo.

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 makes the first product bigger, and contorsion likewise increases the 2nd product. Thus if the products differ, one can identify the problem. (But if the products are the same, it's possible that there is both torsion <u>and</u> contorsion.)

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 makes the first product bigger, and contorsion likewise increases the 2nd product. Thus if the products differ, one can identify the problem. In particular, one can identify torsion in the comma list and remove it. (But if the products are the same, it's possible that there is both torsion <u>and</u> contorsion, which is bad. So one can't rely on unequal pivot products to detect torsion.)

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.

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 Rule #1 ensures linear independence. It completely determines the first comma. Given two yaza commas, one can always derive the ya comma by combining the two commas such that the za component becomes zero. For example, take Ruyoyoo and Biruyo. Subtract Ruyoyo twice from Biruyo to get Sagugu. Next take Latrizo and Biruyo. The za-exponents are 3 and -2 respectively, so two Latrizos plus three Biruyos make a ya comma, Latribiyo.
-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 makes the first product bigger, and contorsion likewise increases the 2nd product. Thus if the products differ, one can identify the problem. (But if the products are the same, it's possible that there is both torsion <u>and</u> contorsion.)
+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 makes the first product bigger, and contorsion likewise increases the 2nd product. Thus if the products differ, one can identify the problem. In particular, one can identify torsion in the comma list and remove it. (But if the products are the same, it's possible that there is both torsion <u>and</u> contorsion, which is bad. So one can't rely on unequal pivot products to detect torsion.)
 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.