Re properties

The property of the diatonic scale in 12-tet mentioned by Bailey in TD
#717 was, as far as I can tell, first described by Terry Winograd.
Winograd called the scales that are generated by cycles of any interval
relatively prime to the octave (or any interval of equivalence) "Deep
Scales" if they have [C/2] or [C/2]+1 tones where C is the cardinality
of the Interval of Equivalence ( Tones per octave usually) and [C/2] is
the largest integer less than or equal to C/2/. Such scales have
interval vectors that contain all the interval classes of the chromatic
set (C) with unique multiplicity (meaning that an interval like as a
major third occurs a unique number of times compared to the number of
times other intervals appear).

The interval vector is a vector whose elements are the number of times
each interval of the chromatic scale (C) of which the scale is embedded
or is a subset, appears. Conventionally, the intervals are ordered from
1 degree to [C/2] For example, the interval vector of the diatonic
scale in 12-tet is [254361]. The interval vector of the chromatic
heptachord (0 1 2 3 4 5 6) is [654321] where the generator, g, is 1
degree. Two other Deep Scales exist in 12 tet: the chromatic hexachord (
0 1 2 3 4 5), whose interval vector is [543210] and the hexatonic scale
(0 2 4 5 7 9) with IV [143250].

Winograd described these relations originally in a hard to obtain term
paper at MIT. However, Carlton Gamer discussed them at length in "Some
Combinational Resources of Equal-Tempered Systems," J. 1967. Journal of
Music Theory 11(1): 32-59.

Needless to say, the 12 out of 19-tet MOS scale lacks this property, but
Wilson's 12-tone pseudo-pythagorean MOS scale in 22 does.. Mandelbaum's
9 and 10 tone "Quasi-equal-interval-symmetric" ( MOS where the two
interval sizes differ by one unit of the temperament) scales in 19-tet
are DS.

--John