SN scale

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.

SN scales are inversionally symmetric / invariant under reversal, and may be uniquely defined by a step signature - a generalisation of the MOS signature into arbitrary rank.

The diatonic scale, for example, 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.

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.

These scales all have a period of an octave - and therefore a) first introduces the interval of an octave, but the period of a SN scale is arbitrary.