MOS scale: Difference between revisions

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* 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).

* 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).

: The period of this temperament is 1\gcd(''X'', ''Y''), and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because 1{{val| ''X'' ...}} + 0{{val| ''Y'' ...}} is the L = 1, s = 0 tuning while 0{{val| ''X'' ...}} + 1{{val| ''Y'' ...}} is the L = 1, s = 1 tuning and 1{{val| ''X'' ...}} + 1{{val| ''Y'' ...}} is the L = 2, s = 1 tuning, so that L = ''a'' + ''b'' and s = ''b'' and therefore:

: 1/([[step ratio]]) = ''s''/''L'' = ''b''/(''a'' + ''b'') implying [[step ratio]] = (''a'' + ''b'')/''b'' >= 1 for [[Wikipedia:Natural number|natural]] ''a'' and ''b'', where if ''b'' = 0 then the step ratio is infinite, corresponding to the [[collapsed]] tuning.

: '''*''' It is '''important to note''' that the correspondence to the ''X'' &''Y'' rank 2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where ''X'' &''Y'' describe a contorted temperament on the subgroup given. An example is the 5 & 19 temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L 14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.

* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are ({{nowrap|''a'' + ''b''}})L ''a''s (generated by generators of soft-of-basic ''a''L ''b''s) and ''a''L ({{nowrap|''a'' + ''b''}})s (generated by generators of hard-of-basic ''a''L'' b''s).

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 * 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).
+* 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).
 : The period of this temperament is 1\gcd(''X'', ''Y''), and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because 1{{val| ''X'' ...}} + 0{{val| ''Y'' ...}} is the L = 1, s = 0 tuning while 0{{val| ''X'' ...}} + 1{{val| ''Y'' ...}} is the L = 1, s = 1 tuning and 1{{val| ''X'' ...}} + 1{{val| ''Y'' ...}} is the L = 2, s = 1 tuning, so that L = ''a'' + ''b'' and s = ''b'' and therefore:
 : 1/([[step ratio]]) = ''s''/''L'' = ''b''/(''a'' + ''b'') implying [[step ratio]] = (''a'' + ''b'')/''b'' >= 1 for [[Wikipedia:Natural number|natural]] ''a'' and ''b'', where if ''b'' = 0 then the step ratio is infinite, corresponding to the [[collapsed]] tuning.
+: '''*''' It is '''important to note''' that the correspondence to the ''X'' &''Y'' rank 2 temperament only works in all cases if we allow the temperament to be [[contorted]] on its [[subgroup]]; alternatively, it works if we exclude cases where ''X'' &''Y'' describe a contorted temperament on the subgroup given. An example is the 5 & 19 temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L 14s]]), so we either need to consider the temperament itself to be contorted (generated by something lacking an interpretation in the subgroup given, two of which yielding a meantone-tempered [[~]][[4/3]]) or we exclude it because of its contortion.
 * Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are ({{nowrap|''a'' + ''b''}})L ''a''s (generated by generators of soft-of-basic ''a''L ''b''s) and ''a''L ({{nowrap|''a'' + ''b''}})s (generated by generators of hard-of-basic ''a''L'' b''s).