SN scale: Difference between revisions

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

a) A new ~~small(est)~~ step at the top or bottom of every existing step, or

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

b) The existing ~~small(est)~~ step at the top or bottom of every larger step

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

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

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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).

== 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.

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.

Consider then its child SNS, SNS (2/1, 3/2, 6/5)[12], which is 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1 is mode -3.

The notes added to SNS (2/1, 3/2, 6/5)[7] mode 0 to get to SNS (2/1, 3/2, 6/5)[12] mode -3 are 250/243, 100/81, 25/18, 125/81, and 50/27. Setting 100/81 as 1/1, this scale is 9/8 5/4 3/2 5/3 2/1, so we might think to say SNDS (2/1, 3/2, 6/5)[12-7] is 9/8 5/4 3/2 5/3 2/1.

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.

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

4-SNDS have not yet been explored.

[[Category:Step-nested scales| ]]

[[Category:MOS]]

@@ Line 2: / Line 2: @@
 A Step-nested scale, or SN scale (or SNS) is a scale generated through iteratively placing an instance of
-a) A new small(est) step at the top or bottom of every existing step, or
+a) A new smaller step at the top or bottom of every existing step, or
-b) The existing small(est) step at the top or bottom of every larger step
+b) The existing smallest step at the top or bottom of every larger step
 Each iteration of a) increasing the rank of the scale by 1.
@@ Line 52: / Line 52: @@
 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).
+== 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.
+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.
+Consider then its child SNS, SNS (2/1, 3/2, 6/5)[12], which is 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1 is mode -3.
+The notes added to SNS (2/1, 3/2, 6/5)[7] mode 0 to get to SNS (2/1, 3/2, 6/5)[12] mode -3 are 250/243, 100/81, 25/18, 125/81, and 50/27. Setting 100/81 as 1/1, this scale is 9/8 5/4 3/2 5/3 2/1, so we might think to say SNDS (2/1, 3/2, 6/5)[12-7] is 9/8 5/4 3/2 5/3 2/1.
+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 are 2-SNS, which we know are mirror-symmetric.
+-SNDS have not yet been explored.
 [[Category:Step-nested scales| ]] <!-- main article -->
 [[Category:MOS]]