9 note scale

Paul Erlich [TD226.3]

>Ray Tomes wrote,
>>> C D Eb E F G A Bb B
>>>24 27 28 30 32 36 40 42 45

>Ken Moore wrote,
>>Very reasonable scale, but what leads you to name the new notes Eb and
>>Bb? A major third of 9/7 is larger even than the Pythagorean 81/64,
>>which is itself larger than ET12. They would be good as D# and A#,
>>though even then the major third from B to D# is flatter than just.

I figure that if a scale has these particular two black notes then they
are normally called flats. They are as you say very flat flats. In his
book on Indian music Clements has a lovely notation where the flat looks
like a vertical line with a seven written through it (which also
symbolizes the 7 ratio). Like this (magnified):

|
|
_|__
| /
|/
/

>If you're going to demand a logical notation here, start by noticing that
>the fifth from D to A is flatter than just.

OK, I am going to have to try and do a lattice.

Eb------Bb *
/|\ /|\
, | , , | ,
$ ' A-'---'-E-'-----B
/_/ \_\ /_/ \_\ _/ \_
/ \ / \ / \
F-------C-------G-------D

This shows lines for ratios of 3/2, 5/4, 6/5, 7/4, 7/5 and 7/6.
Instead of being three major triads (4:5:6) this scale is two 4:5:6:7
chords and one triad. The third triad does not have a note at * as this
would be very close to F (63/64) and Paul's comment indicates that the D
is not at $ (which would be 80/81 of the D shown), however it is just
with G and B.

If these intervals were attempted to be closed then it would be some
sort of meantone tuning and not a purely just scale based on exact
harmonics.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
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Ray Tomes wrote,

> Eb------Bb *
> /|\ /|\
> , | , , | ,
>$ ' A-'---'-E-'-----B
> /_/ \_\ /_/ \_\ _/ \_
> / \ / \ / \
> F-------C-------G-------D

There seem to be some "ghost" triangles there that shouldn't be there. This
is the way I do ASCII lattices:

$ A---------E---------B
/|\ /|\ / \
/ | \ / | \ / \
/ Eb \ / Bb \ / * \
/,' `.\ /,' `.\ / \
F---------C---------G---------D

That way you have the traditional triangular 5-limit lattice in clear view.

Ken Moore wrote,

>But any complete just scale of C major needs two As. This one is the
>major third above F. Two major tones (which is what four perfect fifths
>generate) is different.

Usually the just scale of C major is considered to have two Ds, not two As.
With two Ds you get all six consonant triads:

D---------A---------E---------B
\ / \ / \ / \
\ / \ / \ / \
\ / \ / \ / \
\ / \ / \ / \
F---------C---------G---------D

while with two As you get only five:

A---------E---------B
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
F---------C---------G---------D---------A

However, the latter diagram is applicable to Indian music, where the basic
scale (sa-grama) is recognized as the one with the rightmost A, and a
historical variant (ma-grama) is the one with the other A.

Paul H. Erlich [TD 228.6]

>There seem to be some "ghost" triangles there that shouldn't be there. This
>is the way I do ASCII lattices:

>$ A---------E---------B
> /|\ /|\ / \
> / | \ / | \ / \
> / Eb \ / Bb \ / * \
> /,' `.\ /,' `.\ / \
> F---------C---------G---------D

But you have missed out the line from Eb to Bb which should be there.
Otherwise more nicely drawn than mine.

This shape is interesting because it is the same as Kepler's sphere
stacking problem. It can be represented as a triangular lattice with
another one on top or a square lattice with another one on top. For
those that don't know this is a famous unproven conjecture about the
most efficient way of packing spheres.

-- Ray Tomes -- http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm --
Cycles email list -- http://www.kcbbs.gen.nz/users/af/cyc.htm
Alexandria eGroup list -- http://www.kcbbs.gen.nz/users/af/alex.htm
Boundaries of Science http://www.kcbbs.gen.nz/users/af/scienceb.htm

Ray Tomes wrote...

>This shape is interesting because it is the same as Kepler's sphere
>stacking problem. It can be represented as a triangular lattice with
>another one on top or a square lattice with another one on top. For
>those that don't know this is a famous unproven conjecture about the
>most efficient way of packing spheres.

FYI it was recently proven.

-C.