Interval matrix: Difference between revisions

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=== Using step sizes ===

Working with a ~~[[mos]] or any~~ scale described as a sequence of steps 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 scale described as a sequence of steps, such as a [[MOS scale|mos]], 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 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.

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|LLsLLL

|5L + s

|}

|}To find the next row, we need to rotate the the scale (by moving the first step to the end of the sequence) and find the substrings that start at the first step of that rotated scale. Repeating this process finds the intervals sizes for each of the scale's modes. Since LLsLLLs represents the ionian mode, shifting this way produces the dorian mode (LsLLLsL), then the phrygian mode (sLLLsLL), and so on. The completed matrix is shown below:

To find the next row ~~means finding the substrings starting at the second L and ending at any other step after it. However~~, ~~this is equivalent~~ to ~~rotating~~ the the scale (~~specifically~~ moving the first L to the end) and ~~finding~~ the substrings that start at the first step of that rotated scale. Since LLsLLLs represents the ionian mode, shifting this way produces the dorian mode (LsLLLsL), ~~and the mode after that is~~ the phrygian mode (sLLLsLL), so ~~populating the rest of the interval matrix means finding the quantities of L's and s's for every substring for every mode~~. The completed matrix is shown below:

{| class="wikitable"

|+

@@ Line 131: / Line 131: @@
 === Using step sizes ===
-Working with a [[mos]] or any scale described as a sequence of steps 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 scale described as a sequence of steps, such as a [[MOS scale|mos]], 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 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.
@@ Line 177: / Line 177: @@
 |LLsLLL
 |5L + s
-|}
+|}To find the next row, we need to rotate the the scale (by moving the first step to the end of the sequence) and find the substrings that start at the first step of that rotated scale. Repeating this process finds the intervals sizes for each of the scale's modes. Since LLsLLLs represents the ionian mode, shifting this way produces the dorian mode (LsLLLsL), then the phrygian mode (sLLLsLL), and so on. The completed matrix is shown below:
-To find the next row means finding the substrings starting at the second L and ending at any other step after it. However, this is equivalent to rotating the the scale (specifically moving the first L to the end) and finding the substrings that start at the first step of that rotated scale. Since LLsLLLs represents the ionian mode, shifting this way produces the dorian mode (LsLLLsL), and the mode after that is the phrygian mode (sLLLsLL), so populating the rest of the interval matrix means finding the quantities of L's and s's for every substring for every mode. The completed matrix is shown below:
 {| class="wikitable"
 |+