Mathematics of MOS: Difference between revisions

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=== MOS scales are generated ===

See [[Recursive structure of MOS scales#Proofs]].

=== Maximally even scales are MOS ===

An arbitrary k-step in ME(''m'', ''n'') has size floor((''i'' + ''k'')''n''/''m'') − floor(''in''/''m''), and

<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

≤ \floor 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.

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 ===

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.

By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''−g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.

@@ Line 235: / Line 235: @@
 === MOS scales are generated ===
 See [[Recursive structure of MOS scales#Proofs]].
+=== Maximally even scales are MOS ===
+An arbitrary k-step in ME(''m'', ''n'') has size floor((''i'' + ''k'')''n''/''m'') &minus; floor(''in''/''m''), and
+<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
+≤ \floor in/m \rfloor + \lfloor kn/m \rfloor - \lfloor in/m \rfloor + 1 = \lfloor kn/m \rfloor + 1.</math>
+As floor((''i'' + ''k'')''n''/''m'') &minus; 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 &minus; (n % m) small steps.
 === 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.
 By ''generatedness'', we mean that every interval in the scale is of the form ''jg'' + ''kp'' where ''g'' is a generator, ''p'' is the period, and ''j, k'' ∈ '''Z''', and that either ''g'' or ''&minus;g'' occurs on every note. We claim that any interval class not ''p''-equivalent to 0 has ''exactly'' 2 sizes in any scale satisfying the antecedent.