Equilateral triangle scales

Of the dilation scales, if we exclude chains of generators such as
DE/MOS, the simplest examples will be planar--that is, the note
classes can be arranged in a planar lattice in terms of two
generators. The simplest example of this is 5-limit JI, and the
simplest structure to dilate is the major triad. If we do that, we get
equilateral triangle scales.

The nth equilateral triangle scale can be defined as the following, up
to transpositions and octave equivalence: the set {3^i 5^j}, where
0 <= i, j <= n, i+j <= n. If we call this Tn, then Tn has (n+1)(n+2)/2
notes, n(n+2)/2 major triads, and (n-1)n/2 minor triads, for a total
of n^2 triads.

As is ususally the case with geometrically regular scales, they aren't
particularly regular in terms of scale properties, but they do pack a
lot of chords efficiently, with an emphasis on major over minor. Good
ways of tempering them are not always what you might expect. For T3
and above, meantone is a good choice, with keemun another possibility.
For T5 and above, catakleismic becomes very nice; the fact that it is
complex in terms of chains of minor thirds should not obscure its
usefulness in other scales. Another temperament with a reputation for
uselessness, pontiac, is useful for T6 and above. Of course simply
using an appropriate equal temperament such as 53 or 72 makes a lot of
sense also, but I'm amused to note how powerful pontiac seems to be
for a range of these temperaments. Since this is getting into nano
territory, we can probably stop there.