SN scale: Difference between revisions

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SN scales are based on epi-Christoffel words form combinatorics, which generalize finite Sturmian words. Finite Sturmian words are equivalent to well-formed scales, and equivalently equivalent to MOS scales that are not ''Multi-MOS'' scales, wich are MOS scales of more than one period, typically with a period that divides the octave evenly. The algorithm for generating SN scales introduced above is equivalent to the two epi-Sturmian morphisms that generate epi-Christoffel words.

To find the step arrangement of an ''r''-SN scale for arbitrary step sizes treated as letters of alphabet size ''r'', we iteratively apply the epi-Sturmian moprhism M in which a particular letter from the alphabet is added before each incidence of a different letter. To uncover the order of letters associated the iterated application of the morphism we follow an algorithm in which, from incidences (''X''1, ''X''2, ..., ''Xr'' ) of arbitrary letters ''S''1, ''S''2, ..., and ''Sr,'' respectively, we subtract from the highest incidence value the sum of all other incidence values:

To find the step arrangement of an ''r''-SN scale for arbitrary step sizes treated as letters of alphabet size ''r'', we iteratively apply the epi-Sturmian moprhism M in which a particular letter from the alphabet is added before each incidence of a different letter. To uncover the order of letters associated with the iterated application of the morphism we follow an algorithm T in which, from incidences (''X''1, ''X''2, ..., ''Xr'' ) of arbitrary letters ''S''1, ''S''2, ..., and ''Sr,'' respectively, we subtract from the highest incidence value the sum of all other incidence values:

~~Using~~ (10,5,2) as an example,

Iteratively applying T to (10,5,2) as an example:

(10,5,2)->(3,5,2)->(3,0,2)->(1,0,2)->(1,0,1)->(0,0,1)

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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 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).

[[Category: Scale]]

[[Category: MOS]]

@@ Line 35: / Line 35: @@
 SN scales are based on epi-Christoffel words form combinatorics, which generalize finite Sturmian words. Finite Sturmian words are equivalent to well-formed scales, and equivalently equivalent to MOS scales that are not ''Multi-MOS'' scales, wich are MOS scales of more than one period, typically with a period that divides the octave evenly. The algorithm for generating SN scales introduced above is equivalent to the two epi-Sturmian morphisms that generate epi-Christoffel words.
-To find the step arrangement of an ''r''-SN scale for arbitrary step sizes treated as letters of alphabet size ''r'', we iteratively apply the epi-Sturmian moprhism M in which a particular letter from the alphabet is added before each incidence of a different letter. To uncover the order of letters associated the iterated application of the morphism we follow an algorithm in which, from incidences (''X''1, ''X''2, ..., ''Xr'' ) of arbitrary letters ''S''1, ''S''2, ..., and ''Sr,'' respectively, we subtract from the highest incidence value the sum of all other incidence values:
+To find the step arrangement of an ''r''-SN scale for arbitrary step sizes treated as letters of alphabet size ''r'', we iteratively apply the epi-Sturmian moprhism M in which a particular letter from the alphabet is added before each incidence of a different letter. To uncover the order of letters associated with the iterated application of the morphism we follow an algorithm T in which, from incidences (''X''1, ''X''2, ..., ''Xr'' ) of arbitrary letters ''S''1, ''S''2, ..., and ''Sr,'' respectively, we subtract from the highest incidence value the sum of all other incidence values:
-Using (10,5,2) as an example,
+Iteratively applying T to (10,5,2) as an example:
 (10,5,2)->(3,5,2)->(3,0,2)->(1,0,2)->(1,0,1)->(0,0,1)
@@ 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 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).
 [[Category: Scale]]
 [[Category: MOS]]