SN scale: Difference between revisions
→Further definition: You don't actually need to treat a as larger than c for the corresponding reverse operation c -> ac where a is a new step size. |
mNo edit summary |
||
| Line 23: | Line 23: | ||
The simplest 4-SN scale is generated by iterating a) 4 times, leading to the scale abacabad. If we map the intervals introduced with a) as 2/1, 3/2, 7/6, and 15/14, we get the scale 15/14 7/6 5/4 3/2 45/28 7/4 15/8 2/1, with step signature (1,2,4,1), mapped to (6/5, 7/6, 15/14, 16/15). | The simplest 4-SN scale is generated by iterating a) 4 times, leading to the scale abacabad. If we map the intervals introduced with a) as 2/1, 3/2, 7/6, and 15/14, we get the scale 15/14 7/6 5/4 3/2 45/28 7/4 15/8 2/1, with step signature (1,2,4,1), mapped to (6/5, 7/6, 15/14, 16/15). | ||
== | == Notation == | ||
Where the [[Meantone]] tempered diatonic scale can be labelled as Meantone[7], we may instead describe it through its derivation as an SN scale through labeling it (2/1, 3/2: 81/80)[7], which specifies that a) introduces the intervals 2/1 and 3/2, and then b) is applied until a 7-note scale is reached, and that 81/80 is tempered out in the scale. | Where the [[Meantone]] tempered diatonic scale can be labelled as Meantone[7], we may instead describe it through its derivation as an SN scale through labeling it (2/1, 3/2: 81/80)[7], which specifies that a) introduces the intervals 2/1 and 3/2, and then b) is applied until a 7-note scale is reached, and that 81/80 is tempered out in the scale. | ||
| Line 30: | Line 30: | ||
MET-24, as a (2.3.7.11.13) Parapyth tempered scale can be labelled ((2/1, 3/2)[12], 28/27~33/32: 352/351, 364/363))[24] (the simplest basis set for commas tempered out is chosen to specify the temperament), with step signatures (5, 12, 7) mapped to (~27/26, 28/27~33/32, ~64/63). | MET-24, as a (2.3.7.11.13) Parapyth tempered scale can be labelled ((2/1, 3/2)[12], 28/27~33/32: 352/351, 364/363))[24] (the simplest basis set for commas tempered out is chosen to specify the temperament), with step signatures (5, 12, 7) mapped to (~27/26, 28/27~33/32, ~64/63). | ||
== | == Algorithm == | ||
The above scales all have a period of an octave - and therefore a) first introduces the interval of an octave, however, the period of an SN scale, as with the mapping of any new smallest step introduced, is arbitrary. | The above scales all have a period of an octave - and therefore a) first introduces the interval of an octave, however, the period of an SN scale, as with the mapping of any new smallest step introduced, is arbitrary. | ||
| 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). | 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 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. | ||