Reply to Willaim Annis

William Annis wrote,

> Yes, I did. That's what I'm using as the base scale, with a
>transposition available a 6:7 away. I've not yet tried incorporating
>the bottom system into the chords built using the top one. That is,
>using this shape:

> <14:9> <7:6> <7:4> <21:16> ; first three,
> <4:3> <1:1> <3:2> <9:8> ; all four
> <8:7> <12:7> <9:7> <27:14> ; "chromatic"

>I haven't yet tinkered with chords using the bottom row, although
>using the bottom two, you get everything you get in the top one, just
>down 7/6. I realize this is obvious to many on this list, but it's
>still a new way of thinking for me.

> By restricting myself to the 12 notes above I have no doubt
>I'm missing a lot of interesting possibilites, which I'm sure I'll be
>seeing in forthcoming email :), but as I said earlier, I have to start
>somewhere.

Actually, using 12 for the number of notes turns out not to be such a bad
idea in this case; it's just that the particular "chromatic" notes you've
chosen don't fall where you (I?) would want them to. For example, if try to
map your scale to a standard keyboard, with the base scale on the white
keys, you'd have

A - 1/1
B - 9/8
C - 7/6
D - 4/3
E - 3/2
F - 14/9
G - 7/4

which would place your chromatic notes

12/7 - F#
27/14 - G#
9/7 - C#
21/16 - C#
8/7 - between B and C

So you'd have a conflict between two of the notes and no good place to put
another. You'd get a more even "chromatic" scale, and thus a better mapping
to the keyboard, if you used 9:7 instead of 6:7 as the vertical interval in
your chart. You'd still have the same base "white note" scale:

> <28:27> <14:9> <7:6> <7:4> ; last three,
> <4:3> <1:1> <3:2> <9:8> ; all four
> <12:7> <9:7> <27:14> <81:56> ; "chromatic"

which would place your chromatic notes

28/27 - A#
12/7 - F#
9/7 - C#
27/14 - G#
81/56 - D#

Ah, much better!

Now, either of our 12-note schemes above allow for your "base" scale
(consisting of 6:7:9 triads on I, IV and V) to be played in only two
different keys. But using 12 notes, you can get a virtually identical "base"
scale to play in six different keys! This magic happens by constructing the
12-tone tuning as a single chain of slighly modified fifths. This is
analagous to the application of meantone temperament. Here it is the
septimal comma, rather than the syntonic comma, that we wish to distribute;
for example, if we increase the size of each fifth by 1/4 of a septimal
comma, the 7:9 intervals comes out just, and the 6:7 intervals will be 1/4
septimal comma wide; if we increase each fifth by 1/3 of a septimal comma,
the 6:7 intervals will be just, and the 7:9 intervals will 1/3 septimal
comma wide. My old friend 22-tone equal temperament is happily between these
two bounds of optimality. So, as I mentioned before, I'd tune your base
scale 4 1 4 4 1 4 4 in 22-tET, but this time I'd tune the 12-tone scale 1 3
1 1 3 1 3 1 1 3 1 3 in 22-tET, and now you have your "base" scale available
in six different keys! Try that and see how it strikes your ears.