Recursive structure of MOS scales: Difference between revisions

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=== General algorithm ===

This is the general algorithm for finding a moment of symmetry scale's generator and its [[octave complement]] for a mos xL ys as two, not-necessarily-equal halves of the mos pattern xL ys~~, or if~~ multi-period, two halves of the ~~mos's~~ period. This algorithm will produce ~~that scale's generators~~ without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.

This is the general algorithm for finding a moment of symmetry scale's generator and its [[octave complement]] for a mos xL ys as two, not-necessarily-equal halves of the mos pattern xL ys. If the mos is multi-period, its generator and period complement are returned as two halves of the period. This algorithm will produce these intervals without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.

# If either x or y is equal to 1 (base cases):

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# If neither x nor y is equal to 1 (recursive cases):

## Let k be the greatest common factor of x and y.

## If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the ~~generators~~ for (x/k)L (y/k)s; the ~~generators~~ returned this way will apply to the period rather than the octave.

## If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the generator and complement for (x/k)L (y/k)s; the intervals returned this way will apply to the period rather than the octave.

## If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:

### Let m1 = min(x, y) and m2 = max(x, y).

### Let z = m2 mod m1 and w = m1 - z.

### Let gen be the scale's generator and comp be the generator's octave complement. for the mos zL ws. Recursively call this algorithm to find ~~the generators~~ for zL ws; the final scale's ~~generators~~ will be based on this.

### Let gen be the scale's generator and comp be the generator's octave complement for the mos zL ws. Recursively call this algorithm to find these intervals for zL ws; the final scale's generator and complement will be based on this.

### If x < y, reverse the order of characters in gen and comp, then swap gen and comp. This is only needed if there are more L's than s's in the scale.

### To produce the scale's ~~generators~~, the L's and s's of ~~the two generators~~ must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).

### To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).

#### If x > y, every instance of an L in both ~~generators~~ is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale's ~~generators~~.

#### If x > y, every instance of an L in both intervals is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale's generator and complement.

#### If y > x, every instance of an L in both ~~generators~~ is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale's ~~generators~~.

#### If y > x, every instance of an L in both intervals is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale's generator and complement.

==== Notes on output ====

The algorithm described here returns what is known as the chroma-positive generator, or bright generator, and is usually what is meant by ''the'' generator of a scale. The octave-complement returned this way is the scale's chroma-negative generator, or dark generator. More specifically, the bright generator ~~returned~~ is in its perfect form, and the dark generator is in its augmented form. If the perfect dark generator is needed instead, a quick way to produce it is to replace one L in the dark generator with an s.

The algorithm described here returns what is known as the chroma-positive generator, or bright generator, and is usually what is meant by ''the'' generator of a scale. The octave-complement returned this way is the scale's chroma-negative generator, or dark generator. More specifically, the bright generator is in its perfect form, and the dark generator is in its augmented form. If the perfect dark generator is needed instead, a quick way to produce it is to replace one L in the dark generator with an s.

If the number of steps in the generators is needed, simply count the total number of L's and s's in each generator.

@@ Line 290: / Line 290: @@
 === General algorithm ===
-This is the general algorithm for finding a moment of symmetry scale's generator and its [[octave complement]] for a mos xL ys as two, not-necessarily-equal halves of the mos pattern xL ys, or if multi-period, two halves of the mos's period. This algorithm will produce that scale's generators without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.
+This is the general algorithm for finding a moment of symmetry scale's generator and its [[octave complement]] for a mos xL ys as two, not-necessarily-equal halves of the mos pattern xL ys. If the mos is multi-period, its generator and period complement are returned as two halves of the period. This algorithm will produce these intervals without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. Let x be the number of L's and y the number of s's.
 # If either x or y is equal to 1 (base cases):
@@ Line 298: / Line 298: @@
 # If neither x nor y is equal to 1 (recursive cases):
 ## Let k be the greatest common factor of x and y.
-## If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the generators for (x/k)L (y/k)s; the generators returned this way will apply to the period rather than the octave.
+## If x and y share a common factor k, where k is greater than 1, then recursively call this algorithm to find the generator and complement for (x/k)L (y/k)s; the intervals returned this way will apply to the period rather than the octave.
 ## If x and y don't share a common factor that is greater than 1 (if x and y are coprime), then:
 ### Let m1 = min(x, y) and m2 = max(x, y).
 ### Let z = m2 mod m1 and w = m1 - z.
-### Let gen be the scale's generator and comp be the generator's octave complement. for the mos zL ws. Recursively call this algorithm to find the generators for zL ws; the final scale's generators will be based on this.
+### Let gen be the scale's generator and comp be the generator's octave complement for the mos zL ws. Recursively call this algorithm to find these intervals for zL ws; the final scale's generator and complement will be based on this.
 ### If x < y, reverse the order of characters in gen and comp, then swap gen and comp. This is only needed if there are more L's than s's in the scale.
-### To produce the scale's generators, the L's and s's of the two generators must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).
+### To produce the scale's generator and complement, the L's and s's of both intervals must be replaced with substrings consisting of L's and s's. Let u = ceil(m2/m1) and v = floor(m2/m1).
-#### If x > y, every instance of an L in both generators is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale's generators.
+#### If x > y, every instance of an L in both intervals is replaced with one L and u s's, and every s replaced with one L and v s's. This produces the final scale's generator and complement.
-#### If y > x, every instance of an L in both generators is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale's generators.
+#### If y > x, every instance of an L in both intervals is replaced with u L's and one s, and every s replaced with v L's and one s. This produces the final scale's generator and complement.
 ==== Notes on output ====
-The algorithm described here returns what is known as the chroma-positive generator, or bright generator, and is usually what is meant by ''the'' generator of a scale. The octave-complement returned this way is the scale's chroma-negative generator, or dark generator. More specifically, the bright generator returned is in its perfect form, and the dark generator is in its augmented form. If the perfect dark generator is needed instead, a quick way to produce it is to replace one L in the dark generator with an s.
+The algorithm described here returns what is known as the chroma-positive generator, or bright generator, and is usually what is meant by ''the'' generator of a scale. The octave-complement returned this way is the scale's chroma-negative generator, or dark generator. More specifically, the bright generator is in its perfect form, and the dark generator is in its augmented form. If the perfect dark generator is needed instead, a quick way to produce it is to replace one L in the dark generator with an s.
 If the number of steps in the generators is needed, simply count the total number of L's and s's in each generator.