Tranformed ball scales

The symmetrical lattice norm on 5-limit note classes is given by

|| |* a b> || = a^2 + ab + b^2

This produces the familiar lattice where triads are equilateral
triangles, and the 5-limit diamond is the ball of radius 1 around the
unison.

It occured to me that interesting scales from the point of view of
marvel tempering might follow from using

|| |* a b> || = a^2 - ab + b^2

intead. This looks just the same, but now triangles are not so
interesting, being 1-3/2-15/8 kind of things. However, the balls that
we get are interesting from the point of view of marvel tempering, we
already have two tetrads from the first ball, of just seven notes. For
a 19 note scale, we get an extremely impressive 7 each of major and
minor tetrads. The next scale is of 31 notes.

shell 1 radius 1

{16/15, 5/4, 4/3, 3/2 8/5 15/8}

ball 1 7 notes <7 11 16| epimorphic {1,3,15} diamond
[1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8]

1 otonal 1 utonal tetrads

shell 2 radius sqrt(3)
{75/64, 128/75, 64/45, 45/32, 5/3, 6/5}

ball 2 13 notes <13 23 30| permutation epimorphic {1,3,5,15} diamond

3 otonal 3 utonal tetrads

shell 3 radius 2
{9/8, 16/9, 25/16, 32/25, 256/225, 225/128}

ball 3 19 notes <19 33 41| permutation epimorphic {1,3,5,15,45,75,225}
diamond

7 otonal 7 utonal tetrads

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
...
> However, the balls that
> we get are interesting from the point of view of marvel tempering,
> wealready have two tetrads from the first ball, of just seven notes.
> For a 19 note scale, we get an extremely impressive 7 each of major
> and minor tetrads. The next scale is of 31 notes.
>
> shell 1 radius 1
>
> {16/15, 5/4, 4/3, 3/2 8/5 15/8}
>
> ball 1 7 notes <7 11 16| epimorphic {1,3,15} diamond
> [1, 16/15, 5/4, 4/3, 3/2, 8/5, 15/8]
>
> 1 otonal 1 utonal tetrads
>
> shell 2 radius sqrt(3)
> {75/64, 128/75, 64/45, 45/32, 5/3, 6/5}
>
> ball 2 13 notes <13 23 30| permutation epimorphic {1,3,5,15} diamond
>
> 3 otonal 3 utonal tetrads
>
> shell 3 radius 2
> {9/8, 16/9, 25/16, 32/25, 256/225, 225/128}
>
> ball 3 19 notes <19 33 41| permutation epimorphic {1,3,5,15,45,75,225}
> diamond
>
> 7 otonal 7 utonal tetrads

This differs from Rami Vitale's Byzantine superset in only two notes.

I think that this is clearly superior from the point of view of
numbers of available harmonies, but not from a melodic point of view.

Byzantine Chant is primarily a melodic thing and is purely vocal (a
capella). Why then should it have anything to do with small
whole-number ratios, you might ask. Because it is often sung against a
sung drone (which they call an "ison"). And the ison sometimes changes
during a piece.

We math types have often been accused, sometimes rightly, of being too
concerned with harmony and not enough with melody, when designing or
searching for "good" scales.

One way in which Rami Vitale's scale is superior is in containing a
number of distinctly different tetrachordal scales (as used in
Byzantine chant). These are proper diatonics, improper diatonics
(which they oddly call "enharmonics"), proper soft chromatics and hard
chromatics.

The 7-note ball you give above is the Byzantine soft chromatic, the
obvious white-note scale of Marvel, and so your 19-note ball certainly
contains lots of _them_. But it's a bit short on the others.

Take a look at this diagram
/tuning-math/files/Keenan/TetrachordChart.d\
oc

Rami Vitale's superset scale contains at least the following
tetrachords, given with their intervals in descending order of size
(in steps of 72-EDO), which is not necessarily a permutation that is
actually used.

12 11 7 diatonic (proper)
14 11 5 improper diatonic (Byzantine "enharmonic")
14 12 4 improper diatonic (Byzantine "enharmonic")
16 7 7 soft chromatic
19 7 4 hard chromatic

-- Dave Keenan