Harmonotonic tuning: Difference between revisions

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Monotonicity must also be restricted to within the interval of repetition to have much value. Otherwise, only scales with equal step sizes would qualify as monotonic, because any increase or decrease in step size would be countered with an opposite change once the scale repeated.

In mathematics, monotonicity is sometimes distinguished as strictly monotonic, when it is not permitted for a value to stay the same ~~if the values are increasing or decreasing~~. In this sense, all monotonic tunings classified here are strictly monotonic.

In mathematics, monotonicity is sometimes distinguished as strictly monotonic, when it is not permitted for a value to stay the same. In this sense, all tunings are strictly monotonic. But only some tunings are strictly step-monotonic. Of the tunings classified here, only those with increasing or decreasing step size are strictly step-monotonic.

~~Sometimes also absolutely~~ monotonic ~~sequences are distinguished~~, whose ~~derivatives~~ are ~~all also~~ monotonic~~. This~~ is ~~true for all tunings classified here~~. ~~So all~~ tunings here are absolutely ~~and strictly~~ monotonic ~~by step size~~.

The sequence of step-sizes is the "first difference" of the sequence of pitches. If you were to list the differences between the sizes of successive steps, that would be the second difference. A monotonic sequence, all of whose differences are monotonic, is called "absolutely monotonic". All tunings categorized here are absolutely monotonic.

== History ==

@@ Line 354: / Line 354: @@
 Monotonicity must also be restricted to within the interval of repetition to have much value. Otherwise, only scales with equal step sizes would qualify as monotonic, because any increase or decrease in step size would be countered with an opposite change once the scale repeated.
-In mathematics, monotonicity is sometimes distinguished as strictly monotonic, when it is not permitted for a value to stay the same if the values are increasing or decreasing. In this sense, all monotonic tunings classified here are strictly monotonic.
+In mathematics, monotonicity is sometimes distinguished as strictly monotonic, when it is not permitted for a value to stay the same. In this sense, all tunings are strictly monotonic. But only some tunings are strictly step-monotonic. Of the tunings classified here, only those with increasing or decreasing step size are strictly step-monotonic.
-Sometimes also absolutely monotonic sequences are distinguished, whose derivatives are all also monotonic. This is true for all tunings classified here. So all tunings here are absolutely and strictly monotonic by step size.
+The sequence of step-sizes is the "first difference" of the sequence of pitches. If you were to list the differences between the sizes of successive steps, that would be the second difference. A monotonic sequence, all of whose differences are monotonic, is called "absolutely monotonic". All tunings categorized here are absolutely monotonic.
 == History ==