SN scale: Difference between revisions

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==Examples ==

The diatonic scale can be generated by iterating a) twice, introducing first the octave, then the perfect fifth, and then iterating b) 3 times.~~{{clarify}}~~ It has step signature 5L 2s, and in the symmetric mode, it has step arrangement LsLLLsL. No other arrangement of 5 large and 2 small step sizes results in a SN scale.

The diatonic scale can be generated by iterating a) twice, introducing first the octave, then the perfect fifth, and then iterating b) 3 times, replacing '''a''' with '''sa''' for the current smallest step '''s''' and for every step '''a''' such that '''a''' > '''s''' at the current stage. It has step signature 5L 2s, and in the symmetric mode, it has step arrangement LsLLLsL. No other arrangement of 5 large and 2 small step sizes results in a SN scale.

{| class="wikitable"

|+ Producing the diatonic MOS via the SNS procedure

|-

! Stage !! Move !! Scale (Cumulative form) !! Smallest step !! Step signature !! Word !! Visualization

|-

! 1

|| a): add '''a'''

|| {1/1, 2/1} || 2/1 = '''a''' || 1'''a''' || '''a''' || {{step vis|53 }}┤

|-

! 2

|| a): add '''b'''

|| {1/1, '''3/2''', 2/1} || 4/3 = '''a''' || 1'''a'''1'''b''' || '''ba''' || {{step vis|31 22}}

|-

! 3

|| b): '''b''' → '''ab'''

|| {1/1, '''4/3''', 3/2, 2/1} || 9/8 = '''b''' || 2'''a'''1'''b''' || '''aba''' || {{step vis|22 9 22}}

|-

! 4

|| b): '''a''' → '''ba'''

|| {1/1, '''9/8''', 4/3, 3/2, '''27/16''', 2/1} || 9/8 = '''b''' || 2'''a'''3'''b''' || '''babba''' || {{Step vis|9 13 9 9 13}}

|-

! 5

|| b): '''a''' → '''ba'''

|| {1/1, 9/8, '''81/64''' 4/3, 3/2, 27/16, '''243/128''', 2/1} || 256/243 = '''a''' || 2'''a'''5'''b''' || '''bbabbba''' || {{Step vis|9 9 4 9 9 9 4}}

|}

MET-24 can be generated from the diatonic scale by iterating b) once more, and then applying a), introducing a quarter-tone type step. It has step signature 5L 12M 7s. A capital 'M' specifies that the size of the medium step is closer to the size of the large step than to the size of the small step. A lower case 'm' would specify the converse. We may write the signature alternatively as (5,12,7).

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For more examples of 3-SN scales, see [[Gallery of 3-SN scales]].

The simplest 4-SN scale is generated by iterating a) 4 times, leading to the scale abacabad. If we map the intervals introduced with a) as 2/1, 3/2, 7/6, and 15/14, we get the scale 15/14 7/6 5/4 3/2 45/28 7/4 15/8 2/1, with step signature (1,2,4,1), mapped to (6/5, 7/6, 15/14, 16/15).

== Denoting SN scales ==

@@ Line 14: / Line 14: @@
 ==Examples ==
-The diatonic scale can be generated by iterating a) twice, introducing first the octave, then the perfect fifth, and then iterating b) 3 times.{{clarify}} It has step signature 5L 2s, and in the symmetric mode, it has step arrangement LsLLLsL. No other arrangement of 5 large and 2 small step sizes results in a SN scale.
+The diatonic scale can be generated by iterating a) twice, introducing first the octave, then the perfect fifth, and then iterating b) 3 times, replacing '''a''' with '''sa''' for the current smallest step '''s''' and for every step '''a''' such that '''a''' > '''s''' at the current stage. It has step signature 5L 2s, and in the symmetric mode, it has step arrangement LsLLLsL. No other arrangement of 5 large and 2 small step sizes results in a SN scale.
-<!-- TODO: write the scale generating procedure explicitly. -->
+{| class="wikitable"
+|+ Producing the diatonic MOS via the SNS procedure
+|-
+! Stage !! Move !! Scale (Cumulative form) !! Smallest step !! Step signature !! Word !! Visualization
+|-
+! 1
+|| a): add '''a'''
+|| {1/1, 2/1} || 2/1 = '''a''' || 1'''a''' || '''a''' || {{step vis|53 }}┤
+|-
+! 2
+|| a): add '''b'''
+|| {1/1, '''3/2''', 2/1} || 4/3 = '''a''' || 1'''a'''1'''b''' || '''ba''' || {{step vis|31 22}}
+|-
+! 3
+|| b): '''b''' → '''ab'''
+|| {1/1, '''4/3''', 3/2, 2/1} || 9/8 = '''b''' || 2'''a'''1'''b''' || '''aba''' || {{step vis|22 9 22}}
+|-
+! 4
+|| b): '''a''' → '''ba'''
+|| {1/1, '''9/8''', 4/3, 3/2, '''27/16''', 2/1} || 9/8 = '''b''' || 2'''a'''3'''b''' || '''babba''' || {{Step vis|9 13 9 9 13}}
+|-
+! 5
+|| b): '''a''' → '''ba'''
+|| {1/1, 9/8, '''81/64''' 4/3, 3/2, 27/16, '''243/128''', 2/1} || 256/243 = '''a''' || 2'''a'''5'''b''' || '''bbabbba''' || {{Step vis|9 9 4 9 9 9 4}}
+|}
 MET-24 can be generated from the diatonic scale by iterating b) once more, and then applying a), introducing a quarter-tone type step. It has step signature 5L 12M 7s. A capital 'M' specifies that the size of the medium step is closer to the size of the large step than to the size of the small step. A lower case 'm' would specify the converse. We may write the signature alternatively as (5,12,7).
@@ Line 24: / Line 48: @@
 For more examples of 3-SN scales, see [[Gallery of 3-SN scales]].
 The simplest 4-SN scale is generated by iterating a) 4 times, leading to the scale abacabad. If we map the intervals introduced with a) as 2/1, 3/2, 7/6, and 15/14, we get the scale 15/14 7/6 5/4 3/2 45/28 7/4 15/8 2/1, with step signature (1,2,4,1), mapped to (6/5, 7/6, 15/14, 16/15).
 == Denoting SN scales ==