Some 5-limit diamond scales

If s is a set of notes, then diamond(s) is the set of notes which are
octave reductions of ratios of elements of s--the diamond of s. By a
5-limit diamond scale I mean a 5-limit scale which is the diamond of
some set s. I give some examples which have turned up before below.
The "marvell" scales are things I came up with before, by a method
very similar to the "trab" scales I gave recently, but which
surprisingly led to quite different results. "Rosatim" is the marvel
projection back to the 5-limit of Dante Rosati's 21 note 7-limit
scale, which of course one really expects to be marvel tempered, like
the rest of the scales presented here. Dwarf9 and dwarf15 are
marvelous dwarves, and the 7-note transformed ball ("trab7") scale is
Helmholtz's gypsy diatonic.

A survey of 5-limit diamonds seems to be in order.

diff = ptolemy_diat = al_farabi_diat2 {3,5,9}

diff7b = helmholtz = tartini_7 = trab7 {1,3,15}

diff7c {3,5,25}

dwarf9_5 = efg3355 {1,3,5,15}

marvell13 {1,3,15,45,225}

dwarf15_5 {1,3,5,9,15,45}

trab19 {1,3,5,15,45,75,225}

marvell19 {1,3,15,45,225,675,3375}

marvell21 {1,3,5,15,225,3375}

rosatim {1,3,5,9,15,45,225}

marvell31 {1,3,5,15,45,75,225,3375,50625}

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> A survey of 5-limit diamonds seems to be in order.

A few observations:

If we use four generators for the diamond, the only way to get a
tetrad is from {1,3,5,7}--so we get the 7-limit tonality diamond. The
marvel version is from {1,3,5,225}. If we start with {1,3,5,225} and
add another 5-limit integer, we can get seven otonal tetrads from
{1,3,5,45,3375}, {1,3,5,75,3375}, or {1,3,5,225,3375}, each of which
leads to a scale of 21 notes. If we add two such integers, we get many
possibilites with 7 otonal tetrads or more, but the only one of 19
notes or less appears to be {1,3,5,45,75,225}, which is the "rather
remarkable scale" trab19. We can get an epimorphic scale of 31 notes from
{1,3,5,225,2025,30375} with 12 otonal tetrads, and we get two 29 note
scales with 13 otonal tetrads from {1,3,5,9,225,3375} and
{1,3,5,25,225,3375}.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

If we add two such integers, we get many
> possibilites with 7 otonal tetrads or more, but the only one of 19
> notes or less appears to be {1,3,5,45,75,225}, which is the "rather
> remarkable scale" trab19.

This doesn't make it completely unique, I should add. If we don't
assume we need to start from {1,3,5,225} we get many more
possibilities; if
we start from {1,3*{1,3,5,225}} = {1,3,9,15,675} we get another very
interesting scale, also of 19 notes and seven tetrads. Scala for some
reason does not show it as being or having this diamond, but that is
how I generated it.