SN scale: Difference between revisions

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Each iteration of a) increasing 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. ~~ETs~~ 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. [[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.

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

SN scales are [[Scale properties simplified|symmetric]], and may be uniquely defined by a ''step signature'' - a generalization 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.

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 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).

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).

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 an 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).

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, 12, 7) mapped to (~27/26, 28/27~33/32, ~64/63).

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We can then apply mappings to the step sizes to defined the word as a scale.

If at any point of the process 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 there the application of the generative algorithm leads to a negative number, i.e., (2,2,3)->(2,2,-1).

If at any point of the process 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 there the application of the generative algorithm leads to a negative number, i.e., (2,2,3)->(2,2,-1).

@@ Line 8: / Line 8: @@
 Each iteration of a) increasing 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. ETs 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. [[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.
-SN scales are inversionally symmetric / closed under reversal, and may be uniquely defined by a ''step signature'' - a generalization of the MOS signature into arbitrary rank.
+SN scales are [[Scale properties simplified|symmetric]], and may be uniquely defined by a ''step signature'' - a generalization 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.
-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 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).
 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).
+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 an 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).
+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, 12, 7) mapped to (~27/26, 28/27~33/32, ~64/63).
@@ Line 49: / Line 49: @@
 We can then apply mappings to the step sizes to defined the word as a scale.
-If at any point of the process 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 there the application of the generative algorithm leads to a negative number, i.e., (2,2,3)->(2,2,-1).
+If at any point of the process 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 there the application of the generative algorithm leads to a negative number, i.e., (2,2,3)->(2,2,-1).