SN scale
A step-nested scale, SN scale, or SNS is a scale generated through iteratively performing the following two moves:
a) Add a new smaller step at the top or bottom of every existing step, or
b) Add the existing smallest step at the top or bottom of every larger step: i.e. replacing x with xs or sx for every occurrence of any step x such that x > s at the current stage, where s is the current smallest step.
Each iteration of a) increases the rank of the scale by 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 divisions 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 mirror-symmetric, and may be uniquely defined by a step signature - a generalization of the MOS signature into arbitrary rank.
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 5L2s, 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.
Stage | Move | Scale (Cumulative form) | Smallest step | Step signature | Word | Visualization |
---|---|---|---|---|---|---|
1 | a): add a | {1/1, 2/1} | 2/1 = a | 1a | a | ├────────────────────────────────────────────────────┤ |
2 | a): add b | {1/1, 3/2, 2/1} | 4/3 = a | 1a1b | ba | ├──────────────────────────────┼─────────────────────┤ |
3 | b): b → ab | {1/1, 4/3, 3/2, 2/1} | 9/8 = b | 2a1b | aba | ├─────────────────────┼────────┼─────────────────────┤ |
4 | b): a → ba | {1/1, 9/8, 4/3, 3/2, 27/16, 2/1} | 9/8 = b | 2a3b | babba | ├────────┼────────────┼────────┼────────┼────────────┤ |
5 | b): a → ba | {1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1} | 256/243 = a | 2a5b | bbabbba | ├────────┼────────┼───┼────────┼────────┼────────┼───┤ |
MET-24 can be generated from the diatonic scale by iterating b) once more, and then applying a), introducing a quarter-tone type step. It has step signature 5L12M7s. A capital M specifies that the size of the medium step is closer to the size of the large step than to the size of the small step. A lower case m would specify the opposite. We may write the signature alternatively as (5,12,7).
The double harmonic scale can be generated by iterating a) three times, introducing first the octave, then the fifth, then the major third, leading to a major seven tetrad, and then applying b) once. It has step signature 2L1M4s, and in the symmetric mode, it has step arrangement sLsMsLs.
For more examples of 3-SN scales, see Gallery of 3-SN scales.
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).
Denoting SN scales
Where the Meantone tempered diatonic scale can be denoted Meantone[7], we may instead describe it through its derivation as an SN scale through denoting 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.
The scale SNS (2/1, 3/2, 5/4: 225/224)[7] describes a Marvel tempered double harmonic scale, with step signatures (2,1,4) mapped to (~7/6, ~9/8, 16/15~15/14).
MET-24, as a (2.3.7.11.13) Parapyth tempered scale can be denoted ((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.
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 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}, ..., X_{r} ) of arbitrary letters S_{1}, S_{2}, ..., and S_{r} 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:
(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_{1} and S_{3} have the same incidence value, we can pick either of them to subtract from (in this case, S_{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
To generate the word, we apply M_{abaca}(c). We proceed:
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 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 first definition given.
Step-nested differential scales
Step-nested differential scales, or SNDS are scales derived from the subtraction of a parent SNS from its child SNS.
For example, consider SNS (2/1, 3/2, 6/5)[7], which is 10/9 6/5 4/3 3/2 5/3 9/5 2/1 in mode 0, its symmetric mode.
Consider then its child SNS, SNS (2/1, 3/2, 6/5)[12], which is 250/243 10/9 6/5 100/81 4/3 25/18 3/2 125/81 5/3 9/5 50/27 2/1 is mode -3.
The notes added to SNS (2/1, 3/2, 6/5)[7] mode 0 to get to SNS (2/1, 3/2, 6/5)[12] mode -3 are 250/243, 100/81, 25/18, 125/81, and 50/27. Setting 100/81 as 1/1, this scale is 9/8 5/4 3/2 5/3 2/1, so we might think to say SNDS (2/1, 3/2, 6/5)[12-7] is 9/8 5/4 3/2 5/3 2/1.
Take, instead, however, mode 3 of SNS (2/1, 3/2, 6/5)[12], for example, which is 27/25 10/9 6/5 162/125 4/3 36/25 3/2 81/50 5/3 9/5 243/125 2/1. In this case, the notes added to SNS (2/1, 3/2, 6/5)[7] mode 0, from 162/125 are 10/9 5/4 3/2 5/3 2/1, which is a mode of the inverse of 9/8 5/4 3/2 5/3 2/1.
SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1.
Note that 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1 are both subsets of SNS (2/1, 3/2, 6/5)[7], and therefor of SNS (2/1, 3/2, 6/5)[12].
This can be generalised: Where both child and parent scales are rank-3, 3-SNDS are pairs of chiral scales, i.e., scales that are not mirror-symmetric; all first order step-nested differential scales 3-SNDS[j - k] are subsets of 3-SNS[k], and therefore also subsets of 3-SNS[j] (due to the derivation of SN scales, since a child SNS can have no more than twice the cardinality of its parent SNS, j - k <= k).
When the parent scale is rank-n and the child scale is rank-n+1 the parent scale is also the first order differential. The child scale comprises two instances of the parent scale, spaced apart by at an interval of the new smaller step introduced to the parent scale to derive the child scale.
2-SNDS are 2-SNS, which we know are mirror-symmetric.
4-SNDS have not yet been explored.
Examples of SNDS (step nested differential scales)
SNDS (2/1, 3/2, 6/5)[12-7] is the pair of scales 9/8 5/4 3/2 5/3 2/1, and 10/9 5/4 3/2 5/3 2/1
SNDS (2/1, 3/2, 5/4: 225/224)[19-10] is the pair of scales ~ 16/15 7/6 5/4 4/3 3/2 8/5 7/4 15/8 2/1, and ~ 16/15 8/7 5/4 4/3 3/2 8/5 12/7 15/8 2/1
SNDS (2/1, 3/2, 6/5)[20-12] is the pair of scales 10/9 6/5 4/3 25/18 3/2 5/3 9/5 2/1, 10/9 6/5 4/3 36/25 3/2 5/3 9/5 2/1
SNDS ((2/1, 3/2)[5], x))[10] - (2/1, 3/2)[5] = SNS (2/1, 3/2)[5] (dipentatonic SNS)