User:Ganaram inukshuk/MOS scale: Difference between revisions

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{{User:Ganaram inukshuk/Template:Rewrite draft|MOS scale|compare=https://en.xen.wiki/w/Special:ComparePages?page1=MOS+scale&rev1=&page2=User%3AGanaram+inukshuk%2FMOS+scale&rev2=&action=&diffonly=&unhide=

|changes=general rewrites; definition; wrangle different ways to say "mos"

}}A '''moment-of-symmetry scale''' (also called '''moment-of-symmetry''', commonly abbreviated as '''MOS scale''', '''MOSS''', or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]].

}}A '''moment-of-symmetry scale''' (also called '''moment-of-symmetry''', commonly abbreviated as '''MOS scale''', '''MOSS''', or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]] originally invented by [[Erv Wilson]].

== Definition ==

=== Erv Wilson's original definition ===

The concept of MOS scales were invented by [[Erv Wilson]] in 1975 in his paper ''Moments of Symmetry''. A moment-of-symmetry scale consists of:

The concept of MOS scales were invented by Erv Wilson in 1975 in his paper ''Moments of Symmetry''. A moment-of-symmetry scale consists of:

* A generator and an [[equivalence interval]], called the period, usually the octave.

* A generator and an [[equivalence interval]], called the period, which is usually the [[octave]].

** The generator is commonly denoted using a quantity of steps from an [[EDO|equal division of the octave]], where both the edo and generator are coprime.

** The generator is commonly denoted using a quantity of steps from an [[EDO|equal division of the octave]], where both the edo and generator are coprime, meaning they do not share any common factors greater than 1.

* Two unique step sizes, called ''large'' and ''small'', commonly denoted using the letters L and s.

** The quantities of these steps are coprime~~, meaning they do not share any common factors greater than 1~~.

** The quantities of these steps are also coprime.

The prototypical example of a moment-of-symmetry is the common diatonic scale of [[12edo]], which can be produced using a generator of 7 edosteps.

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|The common diatonic scale, denoted as '''5L 2s'''. This is the lydian mode, equivalent to WWWHWWH.

|}

=== Equivalent definitions ===

There are several equivalent definitions of MOS scales:

*

*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.

*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.

*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.

The term ''well-formed'', from Norman Carey and David Clampitt's paper ''Aspects of well-formed scales'', is sometimes used to equivalently describe the above definitions, and is used in academic research.

=== Single-period and multi-period MOS scales ===

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MOS scales in which the equivalence interval is a multiple of the period (or alternatively, the step pattern repeats multiple times within the equivalence interval), is commonly called a '''multi-MOS''' or '''multi-period MOS'''. This is to distinguish them from what Wilson had defined, called '''strict MOS''' or '''single-period MOS'''.

An alternate definition of a multi-period MOS scale is a MOS scale in which the quantities of large and small steps are ''not'' coprime.

== Notation ==

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== Properties ==

~~The definition of a moment-of-symmetry scale has several equivalences to other concepts:~~

*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.

*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.

*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.

== ~~Applications~~ ==

=== Step ratio and basic properties ===

=== Advanced discussion ===

== Non-tuning applications ==

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~~==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.

~~==Definition==~~

~~There are several equivalent definitions of MOS scales:~~

#

~~#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.)

~~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]].~~

==History and terminology==

The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [http://anaphoria.com/mos.PDF ''Moments of Symmetry'']. There is also an introduction by Kraig Grady here: [http://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].

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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.

~~==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.

~~An in-depth analysis of this can be found at [[Step ratio]].~~

==Properties==

===Basic properties===

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 {{User:Ganaram inukshuk/Template:Rewrite draft|MOS scale|compare=https://en.xen.wiki/w/Special:ComparePages?page1=MOS+scale&rev1=&page2=User%3AGanaram+inukshuk%2FMOS+scale&rev2=&action=&diffonly=&unhide=
 |changes=general rewrites; definition; wrangle different ways to say "mos"
-}}A '''moment-of-symmetry scale''' (also called '''moment-of-symmetry''', commonly abbreviated as '''MOS scale''', '''MOSS''', or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]].
+}}A '''moment-of-symmetry scale''' (also called '''moment-of-symmetry''', commonly abbreviated as '''MOS scale''', '''MOSS''', or '''MOS''', pronounced "em-oh-ess"; also spelled as '''mos''', pronounced "moss"; plural '''MOS scales''', '''MOSes''', or '''mosses''') is a type of [[binary]], [[Periods and generators|periodic scale constructed using a generator]] originally invented by [[Erv Wilson]].
 == Definition ==
 === Erv Wilson's original definition ===
-The concept of MOS scales were invented by [[Erv Wilson]] in 1975 in his paper ''Moments of Symmetry''. A moment-of-symmetry scale consists of:
+The concept of MOS scales were invented by Erv Wilson in 1975 in his paper ''Moments of Symmetry''. A moment-of-symmetry scale consists of:
-* A generator and an [[equivalence interval]], called the period, usually the octave.
+* A generator and an [[equivalence interval]], called the period, which is usually the [[octave]].
-** The generator is commonly denoted using a quantity of steps from an [[EDO|equal division of the octave]], where both the edo and generator are coprime.
+** The generator is commonly denoted using a quantity of steps from an [[EDO|equal division of the octave]], where both the edo and generator are coprime, meaning they do not share any common factors greater than 1.
 * Two unique step sizes, called ''large'' and ''small'', commonly denoted using the letters L and s.
-** The quantities of these steps are coprime, meaning they do not share any common factors greater than 1.
+** The quantities of these steps are also coprime.
 The prototypical example of a moment-of-symmetry is the common diatonic scale of [[12edo]], which can be produced using a generator of 7 edosteps.
@@ Line 58: / Line 58: @@
 |The common diatonic scale, denoted as '''5L 2s'''. This is the lydian mode, equivalent to WWWHWWH.
 |}
+=== Equivalent definitions ===
+There are several equivalent definitions of MOS scales:
+*
+*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.
+*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.
+*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.
+The term ''well-formed'', from Norman Carey and David Clampitt's paper ''Aspects of well-formed scales'', is sometimes used to equivalently describe the above definitions, and is used in academic research.
 === Single-period and multi-period MOS scales ===
@@ Line 63: / Line 71: @@
 MOS scales in which the equivalence interval is a multiple of the period (or alternatively, the step pattern repeats multiple times within the equivalence interval), is commonly called a '''multi-MOS''' or '''multi-period MOS'''. This is to distinguish them from what Wilson had defined, called '''strict MOS''' or '''single-period MOS'''.
+An alternate definition of a multi-period MOS scale is a MOS scale in which the quantities of large and small steps are ''not'' coprime.
 == Notation ==
@@ Line 74: / Line 84: @@
 == Properties ==
-The definition of a moment-of-symmetry scale has several equivalences to other concepts:
-*[[Maximum variety]] 2: every interval that spans the same number of steps has two distinct varieties.
-*Binary and [[distributionally even]]: there are two distinct step sizes that are distributed as evenly as possible. This is equivalent to maximum variety 2.
-*Binary and [[balanced]]: every interval that spans the same number of steps differs by having one large step being replaced with one small step.
-== Applications ==
+=== Step ratio and basic properties ===
+=== Advanced discussion ===
 == Non-tuning applications ==
@@ Line 87: / Line 97: @@
 <original stuff below here>
-==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.
-==Definition==
-There are several equivalent definitions of MOS scales:
-#
-#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.)
-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]].
 ==History and terminology==
 The term ''MOS'', and the method of scale construction it entails, were invented by [[Erv Wilson]] in 1975. His original paper is archived on Anaphoria.com here: [http://anaphoria.com/mos.PDF ''Moments of Symmetry'']. There is also an introduction by Kraig Grady here: [http://anaphoria.com/wilsonintroMOS.html ''Introduction to Erv Wilson's Moments of Symmetry''].
@@ Line 103: / Line 105: @@
 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.
-==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.
-An in-depth analysis of this can be found at [[Step ratio]].
 ==Properties==
 ===Basic properties===