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. | 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 with the iterated application of the morphism we follow an algorithm T in which, from incidences (''X''1, ''X''2, ..., '' | 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''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>r</sub> ) of arbitrary letters '''S<sub>1</sub>''', '''S<sub>2</sub>''', ..., and '''S<sub>r</sub>''' respectively, we subtract from the highest incidence value the sum of all other incidence values: | ||
Iteratively applying T to (10,5,2) as an example: | Iteratively applying T to (10,5,2) as an example: | ||
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(10,5,2)->(3,5,2)->(3,0,2)->(1,0,2)->(1,0,1)->(0,0,1) | (10,5,2)->(3,5,2)->(3,0,2)->(1,0,2)->(1,0,1)->(0,0,1) | ||
In the last step, since both '''S<sub>1</sub>''' and '''S<sub>3</sub>''' have the same incidence value, we can pick either of them to subtract from (in this case, '''S<sub>1</sub>'''). | |||
M''abaca''(''c'') = M''abac''(''ac'') = M''aba''(''cac'') = M''ab''(''acaac'') = M''a''(''babcbababc'') = ''abaabacabaabaabac''. | We list in order the letter with the highest incidence in each step (relabeling '''S<sub>1</sub>''', '''S<sub>2</sub>''', and '''S<sub>3</sub>''' as '''a''', '''b''', and '''c''' respectively): '''abacac''' | ||
To generate the word, we apply ''M''<sub>'''abaca'''</sub>('''c'''). We proceed: | |||
''M''<sub>'''abaca'''</sub>('''c''') = ''M''<sub>'''abac'''</sub>('''ac''') = ''M''<sub>'''aba'''</sub>('''cac''') = ''M''<sub>'''ab'''</sub>('''acaac''') = ''M''<sub>'''a'''</sub>('''babcbababc''') = '''abaabacabaabaabac'''. | |||
We can then apply mappings to the step sizes to defined the word as a scale. | We can then apply mappings to the step sizes to defined the word as a scale. | ||
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TODO: Prove that this algorithm yields the same result as the definition given in the Definitions section. | TODO: Prove that this algorithm yields the same result as the definition given in the Definitions section. | ||
== Step-nested differential scales == | == Step-nested differential scales == | ||
Step-nested differential scales, or SNDS are scales derived from the subtraction of a parent SNS from its child SNS. | Step-nested differential scales, or SNDS are scales derived from the subtraction of a parent SNS from its child SNS. | ||