MOS scale: Difference between revisions
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== Example: the diatonic scale == | == Example: the diatonic scale == | ||
The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L 2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) An important property of MOS scales is that all the intervals come in two sizes: major and minor seconds, major and minor thirds, perfect and augmented fourths, perfect and diminished fifths, etc. This is not true for something like the melodic minor scale (LsLLLLs), which has three kinds of fifths: perfect, diminished and augmented. Therefore, the melodic minor scale is not an MOS scale. | The [[5L 2s|diatonic scale]] is a classic example of an MOS scale. It has 7 steps: 5 large ones (whole tones) and 2 small ones (diatonic semitones). As a shorthand, the large step is denoted with 'L' and the small step with 's', so the diatonic scale may be abbreviated [[5L 2s]]. Writing out the pattern of the major mode, we get LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode.) An important property of MOS scales is that all the intervals come in two sizes: major and minor seconds, major and minor thirds, perfect and augmented fourths, perfect and diminished fifths, etc. This is not true for something like the melodic minor scale (LsLLLLs), which has three kinds of fifths: perfect, diminished, and augmented. Therefore, the melodic minor scale is not an MOS scale. | ||
== Definition == | == Definition == | ||
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# [[Binary]] and has a generator | # [[Binary]] and has a generator | ||
# Binary and [[distributionally even]] | # Binary and [[distributionally even]] | ||
# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap|L − s {{=}} c}}) | # Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or {{nowrap|''L'' − ''s'' {{=}} ''c''}}) | ||
# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''p'' | # Mode of a Christoffel word. (A ''Christoffel word with rational slope'' {{sfrac|''p''|''q''}} is the unique path from (0, 0) and (''p'', ''q'') in the 2-dimensional integer lattice graph above the ''x''-axis and below the line {{nowrap|''y'' {{=}} {{sfrac|''p''|''q''}} ''x''}} that stays as close to the line without crossing it.) | ||
While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]]. | While each characterization has a generalization to scale structures with more step sizes, the generalizations are not equivalent. For more information, see [[Mathematics of MOS]]. | ||
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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. A great deal of interesting work has been done on scales 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 pentatonics used in traditional Japanese music, 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. A great deal of interesting work has been done on scales 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 pentatonics used in traditional Japanese music, where the 5 tone cycles are derived from a 7 tone MOS, which are not found in the concept of DE. | ||
As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where {{nowrap|''s'' {{=}} 0}}. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. The [[TAMNAMS #Step ratio spectrum|TAMNAMS]] system has names for both specific ratios and ranges of ratios. | As for using MOS scales in practice for making music, the period and equivalence interval are often taken to be the octave, but an additional parameter is required for defining a scale: the ''step ratio'', which is the ratio of the small step (usually denoted ''s'') to the large step (usually denoted ''L''). This is usually written as ''L''/''s'', however, using ''s''/''L'' has the advantage of avoiding division by zero in the trivial case where {{nowrap|''s'' {{=}} 0}}. Different step ratios can produce very varied sounding scales (and very varied corresponding potential temperament interpretations) for a given MOS pattern and period, so it's useful to consider a spectrum of simple step ratios for tunings. The [[TAMNAMS#Step ratio spectrum|TAMNAMS]] system has names for both specific ratios and ranges of ratios. | ||
== Naming == | == Naming == | ||
Any MOS can be clearly specified by giving its [[signature]], i.e. the number of small and large steps, which is typically notated e.g. "5L 2s," and its equave. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a 2b" is used (which could refer to either [[5L 2s|diatonic]] or [[2L 5s|anti-diatonic]]). | Any MOS can be clearly specified by giving its [[signature]], i.e. the number of small and large steps, which is typically notated e.g. "5L 2s," and its equave. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a 2b" is used (which could refer to either [[5L 2s|diatonic]] or [[2L 5s|anti-diatonic]]). | ||
By default, the [[equave]] of a mos | By default, the [[equave]] of a mos ''a''L ''b''s is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. [[4L 5s (3/1-equivalent)|4L 5s{{angbr|3/1}}]]. Using angle brackets (<code>&#x27E8;</code> and <code>&#x27E9;</code>, or <code>&#10216;</code> and <code>&#10217;</code>) is recommended; using greater-than and less-than signs ("<nowiki><equave></nowiki>") can also be done, but this can conflict with HTML and other uses of these symbols. | ||
Several naming systems have also been proposed for MOS's, which can be seen at [[MOS naming]]. | Several naming systems have also been proposed for MOS's, which can be seen at [[MOS naming]]. | ||
== Step ratio spectrum == | == Step ratio spectrum == | ||
The melodic sound of a MOS is not just affected by the tuning of its intervals, but by the sizes of its steps. MOSes with L more similar to s sound smoother and more mellow. MOSes with L much larger than s sound jagged and dramatic. The ''step ratio'', the ratio between the sizes of L and s, is thus important to the sound of the scale. | The melodic sound of a MOS is not just affected by the tuning of its intervals, but by the sizes of its steps. MOSes with ''L'' more similar to ''s'' sound smoother and more mellow. MOSes with ''L'' much larger than ''s'' sound jagged and dramatic. The ''step ratio'', the ratio between the sizes of ''L'' and ''s'', is thus important to the sound of the scale. | ||
An in-depth analysis of this can be found at [[Step ratio]]. | An in-depth analysis of this can be found at [[Step ratio]]. | ||
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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 ''x''L (''y'' | * 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]]s (usually but not necessarily through taking the [[patent val]]s) while simultaneously also being used to define the ''px''L (''py'' | * 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 {{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 ''X'' &''Y'' rank 2 temperament'''*''', we can say that any | * 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 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 | : 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: | ||
: 1/([[step ratio]]) = ''s''/''L'' = ''b''/(''a'' + ''b'') implying [[step ratio]] = (''a'' + ''b'')/''b'' | : {{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. | ||
: '''*''' 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. | : '''*''' 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 | * 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]] '' | * 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 === | ||
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This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste. | This is an algorithmically generated recording of every MOS scale that has 14 or fewer notes for a total of 91 scales being showcased here. Each MOS scale played has its simplest step ratio (large step is 2 small step is 1) and therefore is inside the smallest EDO that can support it. Each MOS scale is also in its brightest mode. And rhythmically, each scale is being played with its respective MOS rhythm. Note that changing the mode or step ratio of any of these MOSes may dramatically alter the sound and therefore this recording is not thoroughly representative of each MOS but rather a small taste. | ||
[[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] | [[File:Every-MOS-Scale-With-14-Or-Fewer-Notes.mp3|left|800x800px]] {{clear}} | ||
{{clear}} | |||
== See also == | == See also == | ||
* [[Diamond-mos notation]], a microtonal notation system focussed on MOS scales | * [[Diamond-mos notation]], a microtonal notation system focussed on MOS scales | ||