SN scale: Difference between revisions

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SN scales are a subset of [[MOS Cradle Scales]].

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 ~~episturmian~~ morphisms that generate epi-Christoffel words.

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.

~~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~~. For a step signature to correspond to an SN scale, there must be at least as many incidences of each step size as there are incidences of all less highly incident step sizes~~. The step signature (2, 2, 3), for example, does not correspond to an SN scale, as there ~~are more incidences of less highly incident step sizes than there are~~ of the ~~most highly incident step size~~ (2+2>3).

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:

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

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

We list in order the letter with the highest incidence in each step (relabeling ''S''1, ''S''2, and ''S''3 as ''a'', ''b'', and ''c'' respectively): ''abacac'' (we arbitrarily treated ''a'' as the largest step in (1,0,1) when we wrote (0,0,1) as the next step)

To generate the word, we apply M''abaca''(''c''), as in

M''abaca''(''c'') = M''abac''(''ac'') = M''aba''(''cac'') = M''ab''(''acaac'') = M''a''(''babcbababc'') = ''abaabacabaabaabac''.

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).

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 SN scales are a subset of [[MOS Cradle Scales]].
-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 episturmian morphisms that generate epi-Christoffel words.
+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.
-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. For a step signature to correspond to an SN scale, there must be at least as many incidences of each step size as there are incidences of all less highly incident step sizes. The step signature (2, 2, 3), for example, does not correspond to an SN scale, as there are more incidences of less highly incident step sizes than there are of the most highly incident step size (2+2>3).
+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:
+Using (10,5,2) as an example,
+(10,5,2)->(3,5,2)->(3,0,2)->(1,0,2)->(1,0,1)->(0,0,1)
+We list in order the letter with the highest incidence in each step (relabeling ''S''1, ''S''2, and ''S''3 as ''a'', ''b'', and ''c'' respectively): ''abacac'' (we arbitrarily treated ''a'' as the largest step in (1,0,1) when we wrote (0,0,1) as the next step)
+To generate the word, we apply M''abaca''(''c''), as in
+M''abaca''(''c'') = M''abac''(''ac'') = M''aba''(''cac'') = M''ab''(''acaac'') = M''a''(''babcbababc'') = ''abaabacabaabaabac''.
+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).