MOS scale: Difference between revisions

Line 47:

== Properties ==

=== Basic properties ===

* For every MOS scale with an ~~arbitrary~~ period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (L) step and small (s) step are the same size), then by definition it is an ''x''L (''y''-''x'')s MOS scale, and the [[basic]] tuning where L = 2s is thus (''x''+''y'')-[[edo]]. ~~Because this is true regardless of the period (as long as the period can be expressed in both ''x''-[[edo]] and ''y''-[[edo]]), this~~ is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p''.

* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (L) step and small (s) step are the same size), then by definition it is an ''x''L (''y''-''x'')s MOS scale, and the [[basic]] tuning where L = 2s is thus (''x''+''y'')-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if ''p'' > 1) you are considering a "multiperiod" MOS scale.

* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[patent val]]s) while simultaneously also being used to define the ''px''L (''py'' - ''px'')s MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)

* For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the ''X'' &''Y'' rank 2 temperament, we can say that any [[Wikipedia:Natural number|natural]]-coefficient [[Wikipedia:Linear combination|linear combination]] of vals {{val| ''X'' ...}} and {{val| ''Y'' ...}} (where ''X'' < ''Y'') corresponds uniquely to a tuning of the ''X'' &''Y'' rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff gcd(''a'', ''b'') = 1, because if ''k'' = gcd(''a'', ''b'') > 1 then the val ''a''{{val| ''X'' ...}} + ''b''{{val| ''Y'' ...}} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the [[Wikipedia:Rational number|rational]] ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).

@@ Line 47: / Line 47: @@
 == Properties ==
 === Basic properties ===
-* For every MOS scale with an arbitrary period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (L) step and small (s) step are the same size), then by definition it is an ''x''L (''y''-''x'')s MOS scale, and the [[basic]] tuning where L = 2s is thus (''x''+''y'')-[[edo]]. Because this is true regardless of the period (as long as the period can be expressed in both ''x''-[[edo]] and ''y''-[[edo]]), this is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p''.
+* For every MOS scale with an [[octave]] period (which is usually the [[octave]]), if ''x''-[[edo]] is the [[collapsed]] tuning (where the small step vanishes) and ''y''-[[edo]] is the [[equalized]] tuning (where the large (L) step and small (s) step are the same size), then by definition it is an ''x''L (''y''-''x'')s MOS scale, and the [[basic]] tuning where L = 2s is thus (''x''+''y'')-[[edo]]. This is also true if the period is 1\''p'', that is, 1 step of ''p''-[[edo]], which implies that ''x'' and ''y'' are divisible by ''p'', though note that in that case (if ''p'' > 1) you are considering a "multiperiod" MOS scale.
 * More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[val]]s (usually but not necessarily through taking the [[patent val]]s) while simultaneously also being used to define the ''px''L (''py'' - ''px'')s MOS scale (where ''p'' is the number of periods per octave), then the ''px'' & ''py'' temperament corresponds to that MOS scale, and adding ''x'' and/or ''y'' corresponds to tuning closer to ''x''-[[edo]] and/or ''y''-[[edo]] respectively. (Optionally, see the below more precise statement for the mathematically-inclined.)
 * For the mathematically-inclined, we can say that whenever we consider a MOS with ''X''/''p'' notes per period in the [[collapsed]] tuning and ''Y''/''p'' notes per period in the [[equalized]] tuning and ''p'' periods per [[octave stretching|tempered octave]] (or more generally tempered [[equave]]), and whenever we want to associate that MOS with the ''X'' &''Y'' rank 2 temperament, we can say that any [[Wikipedia:Natural number|natural]]-coefficient [[Wikipedia:Linear combination|linear combination]] of vals {{val| ''X'' ...}} and {{val| ''Y'' ...}} (where ''X'' < ''Y'') corresponds uniquely to a tuning of the ''X'' &''Y'' rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff gcd(''a'', ''b'') = 1, because if ''k'' = gcd(''a'', ''b'') > 1 then the val ''a''{{val| ''X'' ...}} + ''b''{{val| ''Y'' ...}} has a common factor ''k'' in all of its terms, meaning it is guaranteed to be [[contorted]]. The tuning corresponding to the [[Wikipedia:Rational number|rational]] ''a''/''b'' is technically only unique up to (discarding of) [[octave stretching]] (or more generally [[equave]]-tempering).