MOS scale: Difference between revisions

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=== 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''.

* 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) on top of also being used to define the ''px''L ''py''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.

* For the mathematically-inclined, we can say any [[Wikipedia:Natural number|natural]]-coefficient linear combination of the vals {{val| ''X'' ...}} and {{val| ''Y'' ...}} (where ''X'' < ''Y''), ''a''{{val| ''X'' ...}} + ''b''{{val| ''Y'' ...}}, is only ''not'' [[contorted]] if gcd(''a'',''b'') = 1, meaning that every positive [[Wikipedia:Rational number|rational]] ''a''/''b'' corresponds uniquely to a tuning of the ''X'' & ''Y'' rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive), though technically this tuning is 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'' = (''a'' + ''b'')/''b'' implying [[step ratio]] = ''b''/(''a'' + ''b'') >= 1 for [[Wikipedia:Natural number|natural]] ''a'' and ''b''.

* 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).

* Every MOS scale (with a specified [[equave]] ''E''), excluding ''a''L ''a''s⟨''E''⟩, has a ''parent MOS''. If {{nowrap|''a'' > ''b''}}, the parent of ''a''L ''b''s is ''b''L ({{nowrap|''a'' − ''b''}})s; if {{nowrap|''a'' < ''b''}}, the parent of ''a''L ''b''s is ''a''L ({{nowrap|''b'' − ''a''}})s.

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 === 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''.
 * 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) on top of also being used to define the ''px''L ''py''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.
+* For the mathematically-inclined, we can say any [[Wikipedia:Natural number|natural]]-coefficient linear combination of the vals {{val| ''X'' ...}} and {{val| ''Y'' ...}} (where ''X'' < ''Y''), ''a''{{val| ''X'' ...}} + ''b''{{val| ''Y'' ...}}, is only ''not'' [[contorted]] if gcd(''a'',''b'') = 1, meaning that every positive [[Wikipedia:Rational number|rational]] ''a''/''b'' corresponds uniquely to a tuning of the ''X'' & ''Y'' rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive), though technically this tuning is 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'' = (''a'' + ''b'')/''b'' implying [[step ratio]] = ''b''/(''a'' + ''b'') >= 1 for [[Wikipedia:Natural number|natural]] ''a'' and ''b''.
 * 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).
 * Every MOS scale (with a specified [[equave]] ''E''), excluding ''a''L ''a''s⟨''E''⟩, has a ''parent MOS''. If {{nowrap|''a'' &gt; ''b''}}, the parent of ''a''L ''b''s is ''b''L ({{nowrap|''a'' &minus; ''b''}})s; if {{nowrap|''a'' &lt; ''b''}}, the parent of ''a''L ''b''s is ''a''L ({{nowrap|''b'' &minus; ''a''}})s.