SN scale: Difference between revisions

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~~== Definition ==~~

A '''step-nested scale''', '''SN scale''', or '''SNS''' is a scale generated through iteratively performing the following two moves:

A ~~Step~~-nested scale, or SN scale (or SNS) is a scale generated through iteratively ~~placing an instance of~~

a) A new smaller step at the top or bottom of every existing step, or

a) Add a new smaller step at the top or bottom of every existing step, or

b) ~~The~~ existing smallest step at the top or bottom of every larger step

b) Add the existing smallest step at the top or bottom of every larger step: i.e. replacing '''x''' with '''xs''' or '''sx''' for every occurrence of any step '''x''' such that '''x''' > '''s''' at the current stage, where '''s''' is the current smallest step.

Each iteration of a) ~~increasing~~ the rank of the scale by 1.

Each iteration of a) increases the rank of the scale by 1.

An SN scale of rank 2, a 2-SN scale, is a [[MOS scale]]. Accordingly, SN scales are a generalization of MOS scales into arbitrary rank. [[ET]]~~<nowiki/>~~s ~~can be considered to be~~ 1-SN scales, which can be generated by applying a) once, introducing a step of a single degree of the ET.

An SN scale of rank 2, a 2-SN scale, is a [[MOS scale]]. Accordingly, SN scales are a generalization of MOS scales into arbitrary rank. [[Equal division]]s are rank-1 SN scales, which can be generated by applying a) once, introducing a step of a single degree of the ET.

SN scales are [[chirality|mirror-symmetric]], and may be uniquely defined by a ''step signature'' - a generalization of the MOS signature into arbitrary rank.

==Examples ==

== 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. It has step signature 5'''L'''2'''s''', 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).

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 5'''L'''12'''M'''7'''s'''. 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 opposite. We may write the signature alternatively as (5,12,7).

The double harmonic scale can be generated by iterating a) three times, introducing first the octave, then the fifth, then the major third, leading to a major seven tetrad, and then applying b) once. It has step signature ~~2L 1M 4s~~, and in the symmetric mode, it has step arrangement sLsMsLs.

The double harmonic scale can be generated by iterating a) three times, introducing first the octave, then the fifth, then the major third, leading to a major seven tetrad, and then applying b) once. It has step signature 2'''L'''1'''M'''4'''s''', and in the symmetric mode, it has step arrangement '''sLsMsLs'''.

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 ==

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If at any point in the application of T a negative number is reached, that combination of step incidences does not correspond to an SN scale. Accordingly, though for rank-2, any possible step signature corresponds to an SN scale, for higher ranks only a small portion of possible step signatures correspond to SN scales. The step signature (2,2,3), for example, does not correspond to an SN scale, as the iterative application of T leads to a negative number, i.e., (2,2,3)->(2,2,-1).

TODO: Prove that this algorithm yields the same result as the definition given ~~in the Definitions section~~.

TODO: Prove that this algorithm yields the same result as the first definition given.

== Step-nested differential scales ==

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SNDS ((2/1, 3/2)[5], ''x''))[10] - (2/1, 3/2)[5] = SNS (2/1, 3/2)[5] (dipentatonic SNS)

[[Category:~~Step-nested scales|~~ ]] ~~~~

[[Category:Scale]]

[[Category:MOS scale]]

@@ Line 1: / Line 1: @@
-== Definition ==
+A '''step-nested scale''', '''SN scale''', or '''SNS''' is a scale generated through iteratively performing the following two moves:
-A Step-nested scale, or SN scale (or SNS) is a scale generated through iteratively placing an instance of
-a) A new smaller step at the top or bottom of every existing step, or
+a) Add a new smaller step at the top or bottom of every existing step, or
-b) The existing smallest step at the top or bottom of every larger step
+b) Add the existing smallest step at the top or bottom of every larger step: i.e. replacing '''x''' with '''xs''' or '''sx''' for every occurrence of any step '''x''' such that '''x''' > '''s''' at the current stage, where '''s''' is the current smallest step.
-Each iteration of a) increasing the rank of the scale by 1.
+Each iteration of a) increases the rank of the scale by 1. <!-- In any of the steps, "bottom" may be replaced with "top", but the choice of "bottom" and "top" must be consistent. Todo: Prove this or find relevant literature on episturmian words to clarify this.-->
-An SN scale of rank 2, a 2-SN scale, is a [[MOS scale]]. Accordingly, SN scales are a generalization of MOS scales into arbitrary rank. [[ET]]<nowiki/>s can be considered to be 1-SN scales, which can be generated by applying a) once, introducing a step of a single degree of the ET.
+An SN scale of rank 2, a 2-SN scale, is a [[MOS scale]]. Accordingly, SN scales are a generalization of MOS scales into arbitrary rank. [[Equal division]]s are rank-1 SN scales, which can be generated by applying a) once, introducing a step of a single degree of the ET.
 SN scales are [[chirality|mirror-symmetric]], and may be uniquely defined by a ''step signature'' - a generalization of the MOS signature into arbitrary rank.
-==Examples ==
+== 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. It has step signature 5'''L'''2'''s''', 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).
+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 5'''L'''12'''M'''7'''s'''. 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 opposite. We may write the signature alternatively as (5,12,7).
-The double harmonic scale can be generated by iterating a) three times, introducing first the octave, then the fifth, then the major third, leading to a major seven tetrad, and then applying b) once. It has step signature 2L 1M 4s, and in the symmetric mode, it has step arrangement sLsMsLs.
+The double harmonic scale can be generated by iterating a) three times, introducing first the octave, then the fifth, then the major third, leading to a major seven tetrad, and then applying b) once. It has step signature 2'''L'''1'''M'''4'''s''', and in the symmetric mode, it has step arrangement '''sLsMsLs'''.
 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 58: / Line 81: @@
 If at any point in the application of T a negative number is reached, that combination of step incidences does not correspond to an SN scale. Accordingly, though for rank-2, any possible step signature corresponds to an SN scale, for higher ranks only a small portion of possible step signatures correspond to SN scales. The step signature (2,2,3), for example, does not correspond to an SN scale, as the iterative application of T leads to a negative number, i.e., (2,2,3)->(2,2,-1).
-TODO: Prove that this algorithm yields the same result as the definition given in the Definitions section.
+TODO: Prove that this algorithm yields the same result as the first definition given.
 == Step-nested differential scales ==
@@ Line 91: / Line 114: @@
 SNDS ((2/1, 3/2)[5], ''x''))[10] - (2/1, 3/2)[5] = SNS (2/1, 3/2)[5] (dipentatonic SNS)
-[[Category:Step-nested scales| ]] <!-- main article -->
+[[Category:Scale]]
 [[Category:MOS scale]]