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