Interval matrix: Difference between revisions

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Working with a [[mos]] or any scale in a [[TAMNAMS|temperament-agnostic]] sense means that the cent values or JI ratios may not be known beforehand. However, it's still possible to generate an interval matrix as follows.

Consider the familiar diatonic scale (or [[5L 2s]]), represented as the string LLsLLLs, for example. (WWHWWWH also works, but to be general, L and s are used instead.) The intervals between the scale's root and any other scale degree can be considered as being a substring of "LLsLLLs" that starts at the first character (or step) and ends at any other character, including itself; for example, a 2nd is "L", a 3rd is "LL", a 4th is "LLs", and s on. The order of L's and s's is not important, rather the number of L's and s's. In other words, what makes a perfect 5th a perfect 5th is that it's reached by going up from the root by 3 large steps and 1 small step, no matter what the order of steps are. Note that a unison, or 1st, corresponds to a substring consisting of zero characters, or an empty string, and thus its sum of L's and s's is zero.

Consider the familiar diatonic scale (or [[5L 2s]]), represented as the string LLsLLLs, for example. (WWHWWWH also works, but to be general, L and s are used instead.) The intervals between the scale's root and any other scale degree can be considered as being the number of L's and s's from a substring of "LLsLLLs" that starts at the first character (or step) and ends at any other character, including itself; for example, a 2nd is "L", a 3rd is "LL" (or 2L), a 4th is "LLs" (or 2L + s), and s on. The order of L's and s's is not important, rather the number of L's and s's. In other words, what makes a perfect 5th a perfect 5th is that it's reached by going up from the root by 3 large steps and 1 small step (or 3L + 1s), no matter what the order of steps are. Note that a unison, or 1st, corresponds to a substring consisting of zero characters, or an empty string, and thus its sum of L's and s's is zero.

The first row of the matrix can then be populated as such:

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 Working with a [[mos]] or any scale in a [[TAMNAMS|temperament-agnostic]] sense means that the cent values or JI ratios may not be known beforehand. However, it's still possible to generate an interval matrix as follows.
-Consider the familiar diatonic scale (or [[5L 2s]]), represented as the string LLsLLLs, for example. (WWHWWWH also works, but to be general, L and s are used instead.) The intervals between the scale's root and any other scale degree can be considered as being a substring of "LLsLLLs" that starts at the first character (or step) and ends at any other character, including itself; for example, a 2nd is "L", a 3rd is "LL", a 4th is "LLs", and s on. The order of L's and s's is not important, rather the number of L's and s's. In other words, what makes a perfect 5th a perfect 5th is that it's reached by going up from the root by 3 large steps and 1 small step, no matter what the order of steps are. Note that a unison, or 1st, corresponds to a substring consisting of zero characters, or an empty string, and thus its sum of L's and s's is zero.
+Consider the familiar diatonic scale (or [[5L 2s]]), represented as the string LLsLLLs, for example. (WWHWWWH also works, but to be general, L and s are used instead.) The intervals between the scale's root and any other scale degree can be considered as being the number of L's and s's from a substring of "LLsLLLs" that starts at the first character (or step) and ends at any other character, including itself; for example, a 2nd is "L", a 3rd is "LL" (or 2L), a 4th is "LLs" (or 2L + s), and s on. The order of L's and s's is not important, rather the number of L's and s's. In other words, what makes a perfect 5th a perfect 5th is that it's reached by going up from the root by 3 large steps and 1 small step (or 3L + 1s), no matter what the order of steps are. Note that a unison, or 1st, corresponds to a substring consisting of zero characters, or an empty string, and thus its sum of L's and s's is zero.
 The first row of the matrix can then be populated as such: