SN scale
Definition
A Step-nested scale, or SN scale, is a scale generated through iteratively placing an incidence of
a) A new small(est) step at the top or bottom of every existing step, or
b) The existing small(est) step at the top or bottom of every larger step
Each iteration of a) increasing 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. ETs can be considered to be 1-SN scales, which can be generated by applying a) once, introducing a step of a single degree of the ET.
SN scales are inversionally symmetric / closed under reversal, 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 5L 2s, 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.
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 5L 7M 12s. 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 converse. We may write the signature alternatively as (5, 7, 12).
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 2L 1M 4s, and in the symmetric mode, it has step arrangement sLsMsLs.
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).
Labeling
Where the Meantone tempered diatonic scale can be labelled as Meantone[7], we may instead describe it through its derivation as a 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.
The scale (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 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, 7, 12) mapped to (~14/13, ~22/21, 28/27~33/32).
Further definition
These scales all have a period of an octave - and therefore a) first introduces the interval of an octave, however, the period of a SN scale, and 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, which are equivalent to well-formed scales, and equivalently equivalent to MOS scales that are not Multi-MOS scales, MOS scales of more than one period, typically with a period that divides the octave evenly.