MOS scale - Revision history

ArrowHead294: /* Basic properties */

2026-07-18T16:30:39Z

Basic properties

← Older revision		Revision as of 16:30, 18 July 2026
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	* 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).		* 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).
	* Every MOS scale (with a specified [[equave]] ''Ɛ''), excluding {{nowrap\|''a''L ''a''s{{angbr\|''Ɛ''}} }}, has a ''parent MOS''. If {{nowrap\| ''a'' > ''b'' }}, the parent of ''a''L ''b''s is {{nowrap\| ''b''L (''a'' − ''b'')s }}; if {{nowrap\| ''a'' < ''b'' }}, the parent of ''a''L ''b''s is {{nowrap\| ''a''L (''b'' − ''a'')s }}.		* Every MOS scale (with a specified [[equave]] ''Ɛ''), excluding {{nowrap\|''a''L ''a''s{{angbr\|''Ɛ''\|-)}} }}, has a ''parent MOS''. If {{nowrap\| ''a'' > ''b'' }}, the parent of ''a''L ''b''s is {{nowrap\| ''b''L (''a'' − ''b'')s }}; if {{nowrap\| ''a'' < ''b'' }}, the parent of ''a''L ''b''s is {{nowrap\| ''a''L (''b'' − ''a'')s }}.

	=== Advanced discussion ===		=== Advanced discussion ===

Hkm: /* See also */

2026-06-12T05:59:41Z

Overthink: /* Individual pages for MOS scales */ the cluster temp ones feel more useful

2026-06-02T04:11:28Z

Individual pages for MOS scales: the cluster temp ones feel more useful

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Overthink: /* Larger MOS scales */ add the rest

2026-06-02T04:08:53Z

Larger MOS scales: add the rest

← Older revision		Revision as of 04:08, 2 June 2026
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	=== Larger MOS scales ===		=== Larger MOS scales ===
	[[12L 29s]], [[12L 41s]], [[13L 14s]], [[14L 13s]], [[17L 14s]], [[25L 6s]]		[[7L 34s]], [[9L 29s]], [[12L 29s]], [[12L 41s]], [[13L 14s]], [[14L 13s]], [[17L 14s]], [[25L 6s]], [[41L 12s]]

	== Variations ==		== Variations ==

ArrowHead294 at 13:22, 1 May 2026

2026-05-01T13:22:13Z

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Overthink: add pages section

2026-04-04T19:07:28Z

add pages section

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FloraC: - unnecessary HTML codes

2026-03-31T12:46:57Z

- unnecessary HTML codes

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ArrowHead294 at 02:56, 30 March 2026

2026-03-30T02:56:31Z

← Older revision		Revision as of 02:56, 30 March 2026
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	Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap\|C D E F G A C}} does not produces a MOS, because there are more than 2 sizes of each interval class.		Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap\|C D E F G A C}} does not produces a MOS, because there are more than 2 sizes of each interval class.

	The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53~~...~~ However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50~~...~~		The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53… However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50…

	== Step ratio spectrum ==		== Step ratio spectrum ==
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	== Properties ==		== Properties ==
	=== Basic properties ===		=== Basic properties ===

	* 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 {{nowrap\|''x''L (''y'' − ''x'')s}} MOS scale, and the [[basic]] tuning where {{nowrap\|''L'' {{=}} 2''s''}} is thus {{nowrap\|(''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 {{nowrap\|''p'' > 1}}) you are considering a "multiperiod" MOS scale.		* 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 {{nowrap\|''x''L (''y'' − ''x'')s}} MOS scale, and the [[basic]] tuning where {{nowrap\|''L'' {{=}} 2''s''}} is thus {{nowrap\|(''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 {{nowrap\|''p'' > 1}}) you are considering a "multiperiod" MOS scale.
	* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[Val\|vals]] (usually but not necessarily through taking the [[Patent val\|patent vals]]) while simultaneously also being used to define the {{nowrap\|''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.)		* More generally, whenever ''px''-[[edo]] and ''py''-[[edo]] are used to define two [[Val\|vals]] (usually but not necessarily through taking the [[Patent val\|patent vals]]) while simultaneously also being used to define the {{nowrap\|''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 {{nowrap\|''X'' & ''Y''}} rank 2 temperament'''*''', we can say that any {{w\|natural number\|natural}}-coefficient {{w\|linear combination}} of vals {{val\|''X'' ~~...~~}} and {{val\|''Y'' ~~...~~}} (where {{nowrap\|''X'' < ''Y''}}) corresponds uniquely to a tuning of the {{nowrap\|''X'' & ''Y''}} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap\|gcd(''a'', ''b'') {{=}} 1}}, because if {{nowrap\|''k'' {{=}} gcd(''a'', ''b'') > 1}} then the val {{nowrap\|''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 {{nowrap\|''X'' & ''Y''}} rank 2 temperament'''*''', we can say that any {{w\|natural number\|natural}}-coefficient {{w\|linear combination}} of vals {{val\|''X'' …}} and {{val\|''Y'' …}} (where {{nowrap\|''X'' < ''Y''}}) corresponds uniquely to a tuning of the {{nowrap\|''X'' & ''Y''}} rank 2 temperament between ''X''-[[ET]] and ''Y''-[[ET]] (inclusive) iff {{nowrap\|gcd(''a'', ''b'') {{=}} 1}}, because if {{nowrap\|''k'' {{=}} gcd(''a'', ''b'') > 1}} then the val {{nowrap\|''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 {{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:		: 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.<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>		: {{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>

ArrowHead294 at 01:13, 29 March 2026

2026-03-29T01:13:19Z

← Older revision		Revision as of 01:13, 29 March 2026
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	Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.		Note that the melodic minor scale (LsLLLLs) has only two step sizes, but it is not MOS since it has three different sizes of fifths: perfect, diminished, and augmented.

	Other MOS scales include [[2L 3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L 4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).		Other MOS scales include [[2L 3s]], where among its 5 modes are the major pentatonic scale (ssLsL) and the minor pentatonic scale (LssLs), and less commonly [[4L 4s]], also known as the octatonic scale in 12edo, which alternates between large and small steps and comes in two modes (LsLsLsLs and sLsLsLsL).

	See the [[catalog of MOS]] for other MOS scales.		See the [[catalog of MOS]] for other MOS scales.
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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:		: 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.<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>		: {{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>

	* 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).		* 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).

ArrowHead294 at 21:46, 18 January 2026

2026-01-18T21:46:47Z

← Older revision		Revision as of 21:46, 18 January 2026
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	== Examples ==		== Examples ==
	The most widely used MOS scale is the [[5L 2s\|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L 2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).		The most widely used MOS scale is the [[5L 2s\|diatonic scale]]. It has 7 steps: 5 large ones (major 2nds) and 2 small ones (minor 2nds), and thus is named 5L 2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).

	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| Interval classes in the 5L 2s MOS scale		\|+ style="font-size: 105%;" \| Interval classes in the 5L 2s MOS scale
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	== Periods and generators ==		== Periods and generators ==
	Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as C D E F G A C does not produces a MOS, because there are more than 2 sizes of each interval class.		Every MOS scale can be ''generated'' by stacking a certain interval called the [[generator]] and octave-reducing (or more generally, [[period]]-reducing). For example, the diatonic scale is generated by stacking 6 fifths (or equivalently, 6 fourths) and octave-reducing to get a 7 note scale. Another example, 2L 3s is generated by stacking 4 fifths to get 5 notes. However, stacking 5 fifths to get a hexatonic scale such as {{nowrap\|C D E F G A C}} does not produces a MOS, because there are more than 2 sizes of each interval class.

	The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50...		The amount of stacking that produces a MOS scale depends only on the size of the generator relative to the size to the period. For a just fifth and a just octave, the valid scale sizes are 2, 3, 5, 7, 12, 17, 29, 41, 53... However for a quarter-comma meantone fifth, the valid sizes are 2, 3, 5, 7, 12, 19, 31, 50...
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	When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L 2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.		When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L 2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.

	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| 5L 2s step ratios in various edos		\|+ style="font-size: 105%;" \| 5L 2s step ratios in various edos
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	Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.		Sometimes, scales are defined with respect to a period and an additional [[equivalence interval]], the interval at which pitch classes repeat. MOSes in which the equivalence interval is a multiple of the period, and in which there is more than one period per equivalence interval, are sometimes called '''Multi-MOSes'''. For example, a MOS with a half-octave period is called a '''2mos''', with a 1/3-octave period a '''3mos''', and so on. MOSes in which the equivalence interval is equal to the period are sometimes called '''Strict MOSes'''. MOSes in which the equivalence interval and period are simply disjunct, with no rational relationship between them, are simply MOS and have no additional distinguishing label.

	With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness\|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[Pentatonic\|pentatonics]] used in traditional [[Japanese music]] (e.g. A B C E F A), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.		With a few notable exceptions, Wilson generally focused his attention on MOS with period equal to the equivalence interval. Hence, some people prefer to use the term [[Distributional evenness\|distributionally even scale]], with acronym DE, for the more general class of scales which are MOS with respect to other intervals. MOS/DE scales are also sometimes known as ''well-formed scales'', the term used in the 1989 paper by Norman Carey and David Clampitt<ref>Norman Carey and David Clampitt. "Aspects of Well-Formed Scales", ''Music Theory Spectrum'', Vol. 11, No. 2 (Autumn, 1989), pp. 187-206.</ref>. A great deal of work has been done in academic circles extending these ideas. The idea of MOS also includes secondary or bi-level MOS scales which are actually the inspiration of Wilson's concept. They are in a sense the MOS of MOS patterns. This is used to explain the [[Pentatonic\|pentatonics]] used in traditional [[Japanese music]] (e.g. {{nowrap\|A B C E F A}}), where the 5-tone cycles are derived from a 7-tone MOS, which are not found in the concept of DE.

	== Equivalent definitions and generalizations ==		== Equivalent definitions and generalizations ==

← Older revision		Revision as of 05:59, 12 June 2026
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	== See also ==		== See also ==
			* Pailiaq's [https://lkorr.github.io/mos-explorer/ MOS explorer], an interactive tool for visualizing MOSses and the MOS spectrum.
	* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales		* [[Diamond-mos notation]], a microtonal [[notation]] system focused on MOS scales
	* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]		* [[Metallic MOS]], an article focusing on MOS scales based on metallic means, such as [[phi]]