MOS scale: Difference between revisions

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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"

|+~~interval~~ classes in the 5L 2s MOS scale

|+ style="font-size: 105%;" | Interval classes in the 5L 2s MOS scale

~~! rowspan="2" |interval class~~

~~! colspan="2" |small version~~

~~! colspan="2" |large version~~

|-

!~~quality~~

! rowspan="2" | Interval class

!~~size~~

! colspan="2" | Small version

!~~quality~~

! colspan="2" | Large version

~~!size~~

|-

!~~2nds (1 step)~~

! Quality

~~|minor~~

! Size

|s

! Quality

~~|major~~

! Size

|L

|-

!~~3rds~~ (~~2 steps~~)

! 2nds (1 step)

|minor

| minor

|~~1L + 1s~~

| s

|major

| major

|2L

| L

|-

!~~4ths~~ (3 steps)

! 3rds (2 steps)

|~~perfect~~

| minor

|2L + 1s

| {{nowrap|1L + 1s}}

|~~augmented~~

| major

|3L

| 2L

|-

!~~5ths~~ (4 steps)

! 4ths (3 steps)

|~~diminished~~

| perfect

|2L + 2s

| {{nowrap|2L + 1s}}

|~~perfect~~

| augmented

|3L ~~+ 1s~~

| 3L

|-

!~~6ths~~ (5 steps)

! 5ths (4 steps)

|~~minor~~

| diminished

|3L + 2s

| {{nowrap|2L + 2s}}

|~~major~~

| perfect

|4L + 1s

| {{nowrap|3L + 1s}}

|-

!~~7ths~~ (6 steps)

! 6ths (5 steps)

|minor

| minor

|4L + 2s

| {{nowrap|3L + 2s}}

|major

| major

|5L + 1s

| {{nowrap|4L + 1s}}

|-

!8ves (7 steps)

! 7ths (6 steps)

|perfect

| minor

|5L + 2s

| {{nowrap|4L + 2s}}

| colspan="2" |(only one version)

| major

| {{nowrap|5L + 1s}}

|-

! 8ves (7 steps)

| perfect

| {{nowrap|5L + 2s}}

| colspan="2" | (only one version)

|}

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.

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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.

{| class="wikitable"

|+5L 2s step ratios in various edos

|+ style="font-size: 105%;" | 5L 2s step ratios in various edos

~~!example~~

~~edo~~

~~!step~~

~~ratio~~

~~!TAMNAMS~~

~~name~~

~~!likely temperament~~

~~interpretations~~

|-

!12

! Example edo

~~|2:1~~

! Step ratio

~~|basic~~

! TAMNAMS name

~~|[[Meantone]] or [[Schismatic]]~~

! Likely temperament<br />interpretations

|-

!19

! 12

|3:2

| 2:1

|~~soft~~

| basic

|[[Meantone]]

| [[Meantone]] or [[Schismatic]]

|-

!22

! 19

|4:1

| 3:2

|superhard

| soft

|[[Archy]] or [[Superpyth]]

| [[Meantone]]

|-

! 22

| 4:1

| superhard

| [[Archy]] or [[Superpyth]]

|}

== Naming ==

Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L 2s,". Every possible signature corresponds to a valid MOS scale. 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 diatonic or [[2L 5s|anti-diatonic]] = 2L 5s).

Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L 2s,". Every possible signature corresponds to a valid MOS scale. 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 diatonic or {{nowrap|[[2L 5s|anti-diatonic]] {{=}} 2L 5s}}).

By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x27E8;</code> and <code>&#x27E9;</code>) is recommended; using greater-than and less-than signs ("<equave>") can also be done, but this can conflict with HTML and other uses of these symbols.

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=== Advanced discussion ===

See:

* [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.

** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.

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== Variations ==

* [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap|L − s}}, the "chroma".

* [[Muddle~~|Muddles~~]] are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.

* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.

* [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.

* [[Operations on MOSes]]

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== References ==

[[Category:Math]]

@@ Line 13: / Line 13: @@
 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&nbsp;2s. The major mode is LLsLLLs. The other modes are rotations of this pattern (e.g. LsLLsLL is the minor mode).
 {| class="wikitable"
-|+interval classes in the 5L 2s MOS scale
+|+ style="font-size: 105%;" | Interval classes in the 5L&nbsp;2s MOS scale
-! rowspan="2" |interval class
-! colspan="2" |small version
-! colspan="2" |large version
 |-
-!quality
+! rowspan="2" | Interval class
-!size
+! colspan="2" | Small version
-!quality
+! colspan="2" | Large version
-!size
 |-
-!2nds (1 step)
+! Quality
-|minor
+! Size
-|s
+! Quality
-|major
+! Size
-|L
 |-
-!3rds (2 steps)
+! 2nds (1 step)
-|minor
+| minor
-|1L + 1s
+| s
-|major
+| major
-|2L
+| L
 |-
-!4ths (3 steps)
+! 3rds (2 steps)
-|perfect
+| minor
-|2L + 1s
+| {{nowrap|1L + 1s}}
-|augmented
+| major
-|3L
+| 2L
 |-
-!5ths (4 steps)
+! 4ths (3 steps)
-|diminished
+| perfect
-|2L + 2s
+| {{nowrap|2L + 1s}}
-|perfect
+| augmented
-|3L + 1s
+| 3L
 |-
-!6ths (5 steps)
+! 5ths (4 steps)
-|minor
+| diminished
-|3L + 2s
+| {{nowrap|2L + 2s}}
-|major
+| perfect
-|4L + 1s
+| {{nowrap|3L + 1s}}
 |-
-!7ths (6 steps)
+! 6ths (5 steps)
-|minor
+| minor
-|4L + 2s
+| {{nowrap|3L + 2s}}
-|major
+| major
-|5L + 1s
+| {{nowrap|4L + 1s}}
 |-
-!8ves (7 steps)
+! 7ths (6 steps)
-|perfect
+| minor
-|5L + 2s
+| {{nowrap|4L + 2s}}
-| colspan="2" |(only one version)
+| major
+| {{nowrap|5L + 1s}}
+|-
+! 8ves (7 steps)
+| perfect
+| {{nowrap|5L + 2s}}
+| colspan="2" | (only one version)
 |}
 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.
@@ Line 80: / Line 82: @@
 When the step ratio is a rational number, the MOS is tuned to an edo. A counter-example is 5L&nbsp;2s tuned to quarter-comma meantone, which has a step ratio of about 1.649.
 {| class="wikitable"
-|+5L 2s step ratios in various edos
+|+ style="font-size: 105%;" | 5L&nbsp;2s step ratios in various edos
-!example
-edo
-!step
-ratio
-!TAMNAMS
-name
-!likely temperament
-interpretations
 |-
-!12
+! Example edo
-|2:1
+! Step ratio
-|basic
+! TAMNAMS name
-|[[Meantone]] or [[Schismatic]]
+! Likely temperament<br />interpretations
 |-
-!19
+! 12
-|3:2
+| 2:1
-|soft
+| basic
-|[[Meantone]]
+| [[Meantone]] or [[Schismatic]]
 |-
-!22
+! 19
-|4:1
+| 3:2
-|superhard
+| soft
-|[[Archy]] or [[Superpyth]]
+| [[Meantone]]
+|-
+! 22
+| 4:1
+| superhard
+| [[Archy]] or [[Superpyth]]
 |}
 == Naming ==
-Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L&nbsp;2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a&nbsp;2b" is used (which could refer to either diatonic or [[2L 5s|anti-diatonic]] = 2L 5s).
+Every MOS can be uniquely specified by giving its [[signature]], i.e. the number of large and small steps, which is typically notated e.g. "5L&nbsp;2s,". Every possible signature corresponds to a valid MOS scale. Sometimes, if one simply wants to talk about step sizes without specifying which is large and small, the notation "5a&nbsp;2b" is used (which could refer to either diatonic or {{nowrap|[[2L 5s|anti-diatonic]] {{=}} 2L 5s}}).
 By default, the [[equave]] of a MOS is assumed to be [[2/1]]. To specify a non-octave equave, "{{angbr|equave}}" is placed after the signature, e.g. {{mos scalesig|4L 5s<3/1>|link=1}}. Using angle brackets (<code>&#x26;#x27E8;</code> and <code>&#x26;#x27E9;</code>) is recommended; using greater-than and less-than signs ("&#x3C;equave&#x3E;") can also be done, but this can conflict with HTML and other uses of these symbols.
@@ Line 145: / Line 144: @@
 === Advanced discussion ===
 See:
 * [[Mathematics of MOS]], a more formal definition and a discussion of the mathematical properties.
 ** [[Recursive structure of MOS scales]], a description of how MOS scales are recursive and how one scale can be converted into a related scale.
@@ Line 154: / Line 152: @@
 == Variations ==
 * [[MODMOS scales]] are derived from chromatic alterations of one or more tones of an MOS scale, typically by the interval of {{nowrap|L &minus; s}}, the "chroma".
-* [[Muddle|Muddles]] are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.
+* [[Muddle]]s are subsets of MOS parent scales with the general shape of a smaller (and possibly unrelated) MOS scale.
 * [[MOS cradle]] is a technique of embedding MOS-like structures inside MOS scales and may or may not produce subsets of MOS scales.
 * [[Operations on MOSes]]
@@ Line 176: / Line 173: @@
 == References ==
-<references/>
+<references />
 [[Category:Math]]