SN scale: Difference between revisions

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== Definition ==

A Step-nested scale, or SN scale, is a scale generated through iteratively placing an incidence of

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SN scales are inversionally symmetric / invariant under reversal, and may be uniquely defined by a ''step signature'' - a generalisation of the MOS signature into arbitrary rank.

== 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. 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, for example,~~ 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 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~~.

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 7M 12s. 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, 7, 12).

~~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 7M 12s. 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~~.

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 2L 1M 4s~~, ~~and in~~ the ~~symmetric mode~~, ~~it has~~ step ~~arrangement sLsMsLs~~.

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).

== Labeling ==

Where the [[Meantone]] tempered diatonic scale can be labelled as Meantone[7], we may instead describe it through its derivation as a SN scale through labeling it (2/1, 3/2: 81/80)[7], which specifies that a) introduces the intervals 2/1 and 3/2, and then b) is applied until a 7-note scale is reached, and that 81/80 is tempered out in the scale.

The scale (2/1, 3/2, 5/4: 225/224)[7] describes a [[Marvel]] tempered double harmonic scale, with step signatures (2, 1, 4) mapped to (~7/6, ~9/8, 16/15~15/14).

MET-24, as a (2.3.7.11.13) Parapyth tempered scale can be labelled (2/1, 3/2)[12], 28/27~33/32: 352/351, 364/363)[24] (the simplest basis set for commas tempered out is chosen to specify the temperament), with step signatures (5, 7, 12) mapped to (~14/13, ~22/21, 28/27~33/32).

== Further definition ==

These scales all have a period of an octave - and therefore a) first introduces the interval of an octave, however, the period of a SN scale, and the mapping of any new smallest step introduced, is arbitrary.

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+== Definition ==
 A Step-nested scale, or SN scale, is a scale generated through iteratively placing an incidence of
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 SN scales are inversionally symmetric / invariant under reversal, and may be uniquely defined by a ''step signature'' - a generalisation of the MOS signature into arbitrary rank.
+== 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. 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, for example, 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 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.
+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 7M 12s. 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, 7, 12).
-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 7M 12s. 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.
+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 2L 1M 4s, and in the symmetric mode, it has step arrangement sLsMsLs.
+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).
+== Labeling ==
+Where the [[Meantone]] tempered diatonic scale can be labelled as Meantone[7], we may instead describe it through its derivation as a SN scale through labeling it (2/1, 3/2: 81/80)[7], which specifies that a) introduces the intervals 2/1 and 3/2, and then b) is applied until a 7-note scale is reached, and that 81/80 is tempered out in the scale.
+The scale (2/1, 3/2, 5/4: 225/224)[7] describes a [[Marvel]] tempered double harmonic scale, with step signatures (2, 1, 4) mapped to (~7/6, ~9/8, 16/15~15/14).
+MET-24, as a (2.3.7.11.13) Parapyth tempered scale can be labelled (2/1, 3/2)[12], 28/27~33/32: 352/351, 364/363)[24] (the simplest basis set for commas tempered out is chosen to specify the temperament), with step signatures (5, 7, 12) mapped to (~14/13, ~22/21, 28/27~33/32).
+== Further definition ==
 These scales all have a period of an octave - and therefore a) first introduces the interval of an octave, however, the period of a SN scale, and the mapping of any new smallest step introduced, is arbitrary.