MOS scale: Difference between revisions

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# Binary and [[distributionally even]]

# Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or L − s = c)

# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''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 ''y'' = ''p''/''q''*''x'' that stays as close to the line ''y'' = ''p''/''q''*''x'' without crossing it.)

# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''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'' {{=}} ''p''/''q'' * ''x''}} that stays as close to the line {{nowrap|''y'' = ''p''/''q'' * ''x''}} 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]].

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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.

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 ''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 ==

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 aL bs 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 ~~U+27E8 and U+27E9~~ angle brackets is recommended; using greater-than and less-than signs ("<equave>") is acceptable, but ~~they~~ can conflict with HTML and other uses of these symbols.

By default, the [[equave]] of a mos aL bs 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>⟩</code>, or <code>&#10216;</code> and <code>&#10217;</code>) is recommended; using greater-than and less-than signs ("{{^(}}equave{{)^}}") is acceptable, 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]].

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== Properties ==

=== Basic properties ===

* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are (''a'' + ''b'')L ''a''s (generated by generators of soft-of-basic ''a''L ''b''s) and ''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 ''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 ''a'' > ''b'', the parent of ''a''L ''b''s is ''b''L (''a'' − ''b'')s; if ''a'' < ''b'', the parent of ''a''L ''b''s is ''a''L (''b'' − ''a'')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.

=== Advanced discussion ===

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[[Category:Math]]

[[Category:MOS scale| ]]

[[Category:MOS scale| ]]

[[Category:Scale]]

[[Category:Erv Wilson]]

@@ Line 19: / Line 19: @@
 # Binary and [[distributionally even]]
 # Binary and balanced (for any ''k'', any two ''k''-steps ''u'' and ''v'' differ by either 0 or L &minus; s = c)
-# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''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 ''y'' = ''p''/''q''*''x'' that stays as close to the line ''y'' = ''p''/''q''*''x'' without crossing it.)
+# Mode of a Christoffel word. (A ''Christoffel word with rational slope'' ''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'' {{=}} ''p''/''q'' * ''x''}} that stays as close to the line {{nowrap|''y'' = ''p''/''q'' * ''x''}} 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]].
@@ Line 30: / Line 30: @@
 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 ''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 ==
 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 aL bs 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 U+27E8 and U+27E9 angle brackets is recommended; using greater-than and less-than signs ("<equave>") is acceptable, but they can conflict with HTML and other uses of these symbols.
+By default, the [[equave]] of a mos aL bs 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>&amp;#x27E8;</code> and <code>&#x27E9;</code>, or <code>&amp;#10216;</code> and <code>&amp;#10217;</code>) is recommended; using greater-than and less-than signs ("{{^(}}equave{{)^}}") is acceptable, 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]].
@@ Line 46: / Line 46: @@
 == Properties ==
 === Basic properties ===
-* Every MOS scale has two ''child MOS'' scales. The two children of the MOS scale ''a''L ''b''s are (''a'' + ''b'')L ''a''s (generated by generators of soft-of-basic ''a''L ''b''s) and ''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 ''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 ''a'' > ''b'', the parent of ''a''L ''b''s is ''b''L (''a'' &minus; ''b'')s; if ''a'' < ''b'', the parent of ''a''L ''b''s is ''a''L (''b'' &minus; ''a'')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.
 === Advanced discussion ===
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 [[Category:Math]]
-[[Category:MOS scale| ]] <!-- sort order in category: this page shows above A -->
+[[Category:MOS scale| ]] <!-- Sort order in category: this page shows above A -->
 [[Category:Scale]]
 [[Category:Erv Wilson]]