SN scale: Difference between revisions

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== Step-nested differential scales ==

~~A step~~-nested different scales, or SNDS ~~is a scale~~ derived from the subtraction of a parent SNS from a child SNS.

Step-nested different scales, or SNDS are scales derived from the subtraction of a parent SNS from its child SNS.

For example, consider SNS (2/1, 3/2, 6/5)[7], which is 10/9 6/5 4/3 3/2 5/3 9/5 2/1 in mode 0, its symmetric mode.

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Take, instead, however, mode 3 of SNS (2/1, 3/2, 6/5)[12], for example, which is 27/25 10/9 6/5 162/125 4/3 36/25 3/2 81/50 5/3 9/5 243/125 2/1. In this case, the notes added to SNS (2/1, 3/2, 6/5)[7] mode 0, from 162/125 are 10/9 5/4 3/2 5/3 2/1, which is a mode of the inverse of 9/8 5/4 3/2 5/3 2/1.

3-~~SNDS are pairs~~ of ~~[[chiral]]~~ scales, ~~i.e., scales that are not mirror-symmetric~~.

SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1.

~~SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales~~ 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1.

Note that 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1 are both subsets of SNS (2/1, 3/2, 6/5)[7], and therefor of SNS (2/1, 3/2, 6/5)[12].

This can be generalised: Where both child and parent scales are rank-3, 3-SNDS are pairs of [[chiral]] scales, i.e., scales that are not mirror-symmetric; all first order step-nested differential scales 3-SNDS[''j - k]'' are subsets of 3-SNS[''k''], and therefore also subsets of 3-SNS[''j''] (due to the derivation of SN scales, since a child SNS can have no more than twice the cardinality of its parent SNS, ''j - k <= k'')''.''

When the parent scale is rank-''n'' and the child scale is rank-''n+1'' the parent scale is also the first order differential. The child scale comprises two instances of the parent scale, spaced apart by at an interval of the new smaller step introduced to the parent scale to derive the child scale.

2-SNDS are 2-SNS, which we know are mirror-symmetric.

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4-SNDS have not yet been explored.

== Examples of 3-SNDS (rank-3 step-nested differential scales) ==

SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1

SNDS (2/1, 3/2, 5/4: 225/224)[19-10] is the pair of scales ~ 16/15 7/6 5/4 4/3 3/2 8/5 7/4 15/8 2/1, and ~ 16/15 8/7 5/4 4/3 3/2 8/5 12/7 15/8 2/1

SNDS (2/1, 3/2, 6/5)[20-12] is the pair of scales

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:MOS]]

@@ Line 54: / Line 54: @@
 == Step-nested differential scales ==
-A step-nested different scales, or SNDS is a scale derived from the subtraction of a parent SNS from a child SNS.
+Step-nested different scales, or SNDS are scales derived from the subtraction of a parent SNS from its child SNS.
 For example, consider SNS (2/1, 3/2, 6/5)[7], which is 10/9 6/5 4/3 3/2 5/3 9/5 2/1 in mode 0, its symmetric mode.
@@ Line 64: / Line 64: @@
 Take, instead, however, mode 3 of SNS (2/1, 3/2, 6/5)[12], for example, which is 27/25 10/9 6/5 162/125 4/3 36/25 3/2 81/50 5/3 9/5 243/125 2/1. In this case, the notes added to SNS (2/1, 3/2, 6/5)[7] mode 0, from 162/125 are 10/9 5/4 3/2 5/3 2/1, which is a mode of the inverse of 9/8 5/4 3/2 5/3 2/1.
--SNDS are pairs of [[chiral]] scales, i.e., scales that are not mirror-symmetric.
+SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1.
-SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1.
+Note that 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1 are both subsets of SNS (2/1, 3/2, 6/5)[7], and therefor of SNS (2/1, 3/2, 6/5)[12].
+This can be generalised: Where both child and parent scales are rank-3, 3-SNDS are pairs of [[chiral]] scales, i.e., scales that are not mirror-symmetric; all first order step-nested differential scales 3-SNDS[''j - k]'' are subsets of 3-SNS[''k''], and therefore also subsets of 3-SNS[''j''] (due to the derivation of SN scales, since a child SNS can have no more than twice the cardinality of its parent SNS, ''j - k <= k'')''.''
+When the parent scale is rank-''n'' and the child scale is rank-''n+1'' the parent scale is also the first order differential. The child scale comprises two instances of the parent scale, spaced apart by at an interval of the new smaller step introduced to the parent scale to derive the child scale.
 -SNDS are 2-SNS, which we know are mirror-symmetric.
@@ Line 72: / Line 76: @@
 -SNDS have not yet been explored.
+== Examples of 3-SNDS (rank-3 step-nested differential scales) ==
+SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1
+SNDS (2/1, 3/2, 5/4: 225/224)[19-10] is the pair of scales ~ 16/15 7/6 5/4 4/3 3/2 8/5 7/4 15/8 2/1, and ~ 16/15 8/7 5/4 4/3 3/2 8/5 12/7 15/8 2/1
+SNDS (2/1, 3/2, 6/5)[20-12] is the pair of scales
+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:MOS]]