SN scale: Difference between revisions
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Each iteration of a) increases the rank of the scale by 1. <!-- In any of the steps, "bottom" may be replaced with "top", but the choice of "bottom" and "top" must be consistent. Todo: Prove this or find relevant literature on episturmian words to clarify this.--> | Each iteration of a) increases the rank of the scale by 1. <!-- In any of the steps, "bottom" may be replaced with "top", but the choice of "bottom" and "top" must be consistent. Todo: Prove this or find relevant literature on episturmian words to clarify this.--> | ||
An SN scale of rank 2, a 2-SN scale, is a [[MOS scale]]. Accordingly, SN scales are a generalization of MOS scales into arbitrary rank. [[Equal division]]s are 1 | An SN scale of rank 2, a 2-SN scale, is a [[MOS scale]]. Accordingly, SN scales are a generalization of MOS scales into arbitrary rank. [[Equal division]]s are rank-1 SN scales, which can be generated by applying a) once, introducing a step of a single degree of the ET. | ||
SN scales are [[chirality|mirror-symmetric]], and may be uniquely defined by a ''step signature'' - a generalization of the MOS signature into arbitrary rank. | SN scales are [[chirality|mirror-symmetric]], and may be uniquely defined by a ''step signature'' - a generalization of the MOS signature into arbitrary rank. | ||
==Examples == | == Examples == | ||
The diatonic scale can be generated by iterating a) twice, introducing first the octave, then the perfect fifth, and then iterating b) 3 times. It has step signature 5'''L'''2'''s''', and in the symmetric mode, it has step arrangement '''LsLLLsL'''. No other arrangement of 5 large and 2 small step sizes results in a SN scale. | The diatonic scale can be generated by iterating a) twice, introducing first the octave, then the perfect fifth, and then iterating b) 3 times. It has step signature 5'''L'''2'''s''', and in the symmetric mode, it has step arrangement '''LsLLLsL'''. No other arrangement of 5 large and 2 small step sizes results in a SN scale. | ||
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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). | 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). | ||
TODO: Prove that this algorithm yields the same result as the definition given | TODO: Prove that this algorithm yields the same result as the first definition given. | ||
== Step-nested differential scales == | == Step-nested differential scales == | ||
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SNDS ((2/1, 3/2)[5], ''x''))[10] - (2/1, 3/2)[5] = SNS (2/1, 3/2)[5] (dipentatonic SNS) | SNDS ((2/1, 3/2)[5], ''x''))[10] - (2/1, 3/2)[5] = SNS (2/1, 3/2)[5] (dipentatonic SNS) | ||
[[Category: | |||
[[Category:Scale]] | |||
[[Category:MOS scale]] | [[Category:MOS scale]] | ||