Recursive structure of MOS scales: Difference between revisions
→Recursive structure (chunking operation): I've gotten around to adding the pseudocode and did a bit of rewording. It may contain errors, so please feel free to edit if there are any. (Also please feel free to edit any of my other changes should it be too wordy.) |
m →Recursive structure (chunking operation): Formatted the sections so that they appear in the table of contents |
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By looking at the "tetrachords" L…s of an MOS scale in word form and giving them the names "L" and "s", we get out another MOS scale. The MOS thus obtained preserves a number of important properties, such as which interval is the generator. To find properties of complex MOS word patterns, we can then just compare them to the simpler ones, whose properties we know. | By looking at the "tetrachords" L…s of an MOS scale in word form and giving them the names "L" and "s", we get out another MOS scale. The MOS thus obtained preserves a number of important properties, such as which interval is the generator. To find properties of complex MOS word patterns, we can then just compare them to the simpler ones, whose properties we know. | ||
== Recursive | == Recursive Structure == | ||
Let w be an arbitrary moment-of-symmetry scale in either its brightest mode (where L's come before s's) or darkest mode (where s's come before L's). Let x be the number of L's in the scale and let y be the number of s's in the scale. Such a scale will fall under one of three general cases or one of three special cases | === Chunking operation === | ||
Let w be an arbitrary moment-of-symmetry scale in either its brightest mode (where L's come before s's) or darkest mode (where s's come before L's). Let x be the number of L's in the scale and let y be the number of s's in the scale. Such a scale will fall under one of three general cases or one of three special cases described below. | |||
==== General cases ==== | |||
* '''Case 1:''' The scale is a single-period moment-of-symmetry scale where there are more L's than s's, and there is more than one s (or x > y > 1). Scale w can broken up into y substrings (or chunks) where each substring contains at least one L and exactly one s. Among these chunks, there are two unique chunks whose sizes differ by exactly one L: the larger of the two contains ceil(x/y) L's and exactly one s, and the smaller of the two contains floor(x/y) L's and exactly one s. | * '''Case 1:''' The scale is a single-period moment-of-symmetry scale where there are more L's than s's, and there is more than one s (or x > y > 1). Scale w can broken up into y substrings (or chunks) where each substring contains at least one L and exactly one s. Among these chunks, there are two unique chunks whose sizes differ by exactly one L: the larger of the two contains ceil(x/y) L's and exactly one s, and the smaller of the two contains floor(x/y) L's and exactly one s. | ||
* '''Case 2:''' The scale is a single-period moment-of-symmetry scale where there are more s's than L's, and there is more than one L (or y > x > 1). Scale w can broken up into x substrings (or chunks) where each substring contains at least one s and exactly one L. Among these chunks, there are two unique chunks whose sizes differ by exactly one s: the larger of the two contains ceil(y/x) s's and exactly one L, and the smaller of the two contains floor(y/x) s's and exactly one L. | * '''Case 2:''' The scale is a single-period moment-of-symmetry scale where there are more s's than L's, and there is more than one L (or y > x > 1). Scale w can broken up into x substrings (or chunks) where each substring contains at least one s and exactly one L. Among these chunks, there are two unique chunks whose sizes differ by exactly one s: the larger of the two contains ceil(y/x) s's and exactly one L, and the smaller of the two contains floor(y/x) s's and exactly one L. | ||
* '''Case 3:''' The scale is a multi-period moment-of-symmetry scale. Because a multi-period scale consists of a single MOS-like chunk that's repeated throughout the scale, x and y necessarily share a common factor c, where c is the greatest common factor of x and y, and c is the number of times that chunk is repeated. Therefore, when considering a multi-period moment-of-symmetry scale, only the first chunk, the first (x+y)/c steps of the scale, needs to be considered. This chunk will fall under case 1 or case 2, or one of the special cases. | * '''Case 3:''' The scale is a multi-period moment-of-symmetry scale. Because a multi-period scale consists of a single MOS-like chunk that's repeated throughout the scale, x and y necessarily share a common factor c, where c is the greatest common factor of x and y, and c is the number of times that chunk is repeated. Therefore, when considering a multi-period moment-of-symmetry scale, only the first chunk, the first (x+y)/c steps of the scale, needs to be considered. This chunk will fall under case 1 or case 2, or one of the special cases. | ||
'''Special cases | ==== '''Special cases''' ==== | ||
* The special case for case 1 is when there is only one s in the scale. The scale is a moment-of-symmetry scale where the entire scale is considered to be one chunk. | * The special case for case 1 is when there is only one s in the scale. The scale is a moment-of-symmetry scale where the entire scale is considered to be one chunk. | ||
* The special case for case 2 is when there is only one L in the scale. The scale is a moment-of-symmetry scale where the entire scale is considered to be one chunk. | * The special case for case 2 is when there is only one L in the scale. The scale is a moment-of-symmetry scale where the entire scale is considered to be one chunk. | ||
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Note that in this reduction process, the order of L's and s's may reverse. This is normal. | Note that in this reduction process, the order of L's and s's may reverse. This is normal. | ||
=== Reversing the reduction rules into production rules === | |||
Just as reducing a moment-of-symmetry scale using the aforementioned process produces another moment-of-symmetry scale, reversing the reduction rules into production rules will produce another scale w' that is also a moment-of-symmetry scale. One of the following sets of production rules can be used: | Just as reducing a moment-of-symmetry scale using the aforementioned process produces another moment-of-symmetry scale, reversing the reduction rules into production rules will produce another scale w' that is also a moment-of-symmetry scale. One of the following sets of production rules can be used: | ||
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== Finding the MOS pattern from xL ys == | == Finding the MOS pattern from xL ys == | ||
=== General algorithm === | |||
This is the general algorithm for generating a maximally even scale consisting of x large steps and y small steps (a moment-of-symmetry scale). This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. | This is the general algorithm for generating a maximally even scale consisting of x large steps and y small steps (a moment-of-symmetry scale). This algorithm will produce that scale without any prior knowledge of how the scale's steps are ordered. This algorithm is recursive. | ||