MOS scale: Difference between revisions

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

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

: {{nowrap|1/([[step ratio]])}} = ''s''/''L'' = {{nowrap|''b''/(''a'' + ''b'')}} implying {{nowrap|[[step ratio]] {{=}} (''a'' + ''b'')/''b'' ≥ 1}} for [[Wikipedia:Natural number|natural]] ''a'' and ''b'', where if {{nowrap|''b'' {{=}} 0}} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is '''important to note''' that the correspondence to the {{nowrap|''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 {{nowrap|''X'' & ''Y''}} describe a contorted temperament on the subgroup given. An example is the {{nowrap|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.</ref>

~~: '''*'''~~ It is '''important to note''' that the correspondence to the {{nowrap|''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 {{nowrap|''X'' & ''Y''}} describe a contorted temperament on the subgroup given. An example is the {{nowrap|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 {{nowrap|''a''L (''a'' + ''b'')s}} (generated by generators of hard-of-basic ''a''L'' b''s).

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** [[User:Contribution/Gallery of MOS patterns (97 - 112 units)|97–112 units]]

** [[User:Contribution/Gallery of MOS patterns (113 - 128 units)|113–128 units]]

== Notes ==

[[Category:Math]]

@@ Line 53: / Line 53: @@
 : The period of this temperament is {{nowrap|1\gcd(''X'', ''Y'')}}, and the rational ''a''/''b'' is very closely related to the [[step ratio]] of the corresponding MOS scale, because {{nowrap|1{{val| ''X'' ...}} + 0{{val| ''Y'' ...}}}} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 0}} tuning while {{nowrap|0{{val| ''X'' ...}} + 1{{val| ''Y'' ...}}}} is the {{nowrap|''L'' {{=}} 1|''s'' {{=}} 1}} tuning and {{nowrap|1{{val| ''X'' ...}} + 1{{val| ''Y'' ...}}}} is the {{nowrap|''L'' {{=}} 2|''s'' {{=}} 1}} tuning, so that {{nowrap|''L'' {{=}} ''a'' + ''b''}} and {{nowrap|''s'' {{=}} ''b''}} and therefore:
-: {{nowrap|1/([[step ratio]])}} = ''s''/''L'' = {{nowrap|''b''/(''a'' + ''b'')}} implying {{nowrap|[[step ratio]] {{=}} (''a'' + ''b'')/''b'' &ge; 1}} for [[Wikipedia:Natural number|natural]] ''a'' and ''b'', where if {{nowrap|''b'' {{=}} 0}} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.
+: {{nowrap|1/([[step ratio]])}} = ''s''/''L'' = {{nowrap|''b''/(''a'' + ''b'')}} implying {{nowrap|[[step ratio]] {{=}} (''a'' + ''b'')/''b'' &ge; 1}} for [[Wikipedia:Natural number|natural]] ''a'' and ''b'', where if {{nowrap|''b'' {{=}} 0}} then the step ratio is infinite, corresponding to the [[collapsed]] tuning.<ref group="note">It is '''important to note''' that the correspondence to the {{nowrap|''X'' &amp; ''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 {{nowrap|''X'' &amp; ''Y''}} describe a contorted temperament on the subgroup given. An example is the {{nowrap|5 &amp; 19}} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L&nbsp;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.</ref>
-: '''*''' It is '''important to note''' that the correspondence to the {{nowrap|''X'' &amp; ''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 {{nowrap|''X'' &amp; ''Y''}} describe a contorted temperament on the subgroup given. An example is the {{nowrap|5 &amp; 19}} temperament is contorted in the [[5-limit]] (having a generator of a semifourth, corresponding to [[5L&nbsp;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&nbsp;''b''s are {{nowrap|(''a'' + ''b'')L ''a''s}} (generated by generators of soft-of-basic ''a''L&nbsp;''b''s) and {{nowrap|''a''L (''a'' + ''b'')s}} (generated by generators of hard-of-basic ''a''L''&nbsp;b''s).
@@ Line 99: / Line 97: @@
 ** [[User:Contribution/Gallery of MOS patterns (97 - 112 units)|97&ndash;112 units]]
 ** [[User:Contribution/Gallery of MOS patterns (113 - 128 units)|113&ndash;128 units]]
+== Notes ==
+<references group="note" />
 [[Category:Math]]