Mathematics of MOS: Difference between revisions
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<math>\lfloor in/m \rfloor + \lfloor kn/m\rfloor - \lfloor in/m\rfloor = \lfloor kn/m\rfloor ≤ \lfloor(i+k)n/m \rfloor - \lfloor in/m \rfloor | <math>\lfloor in/m \rfloor + \lfloor kn/m\rfloor - \lfloor in/m\rfloor = \lfloor kn/m\rfloor ≤ \lfloor(i+k)n/m \rfloor - \lfloor in/m \rfloor | ||
≤ \ | ≤ \lfloor in/m \rfloor + \lfloor kn/m \rfloor - \lfloor in/m \rfloor + 1 = \lfloor kn/m \rfloor + 1.</math> | ||
As floor((''i'' + ''k'')''n''/''m'') − floor(''in''/''m'') is an integer, the above proves that there are at most two possible values for it and that if there are two sizes for k-steps, the two sizes must differ by 1. Steps of ME(''m'', ''n'') have exactly two sizes because if it were one size, we would have m | n, which is a contradiction. | As floor((''i'' + ''k'')''n''/''m'') − floor(''in''/''m'') is an integer, the above proves that there are at most two possible values for it and that if there are two sizes for k-steps, the two sizes must differ by 1. Steps of ME(''m'', ''n'') have exactly two sizes because if it were one size, we would have m | n, which is a contradiction. | ||
This maximally even MOS has n % m large steps and m − (n % m) small steps. | This maximally even MOS has n % m large steps and m − (n % m) small steps. | ||
=== Binary generated scales with #L coprime to #s within each period are MOS === | === Binary generated scales with #L coprime to #s within each period are MOS === | ||
This proof justifies the common description of "stack until binary" for MOS building and the original terminology "moment of symmetry" where such sizes are "moments" in time (when stacking) where this "symmetry" holds. | This proof justifies the common description of "stack until binary" for MOS building and the original terminology "moment of symmetry" where such sizes are "moments" in time (when stacking) where this "symmetry" holds. | ||