Yahoo Tuning Groups Ultimate Backup tuning Re: quick this n that -- Bob Valentine's new scale

> You might be interested in or familiar with the 9-tone scale I
> mention in this post.

Hello, there, and thank you also for your kind words about
<http://www.medieval.org>; Todd McComb has done a most impressive job
as Early Music FAQ editor, and I'm delighted that it is helping to
make the music itself and some performances in period intonations more
accessible.

> Lars, that 9-of-17tet was a nice little piece. I'll be going back to
> check out more of your stuff. I had been working on the same scale
> form from another vantage point. I reversed the L and s and had for
> all practical purposes a 9 of 24

> L s L s L s L s L
> 0 204 249 453 498 702 747 951 996

This is a fascinating scale, and interestingly could be one
interpretation (or approximation for a fixed-pitch instrument) of the
vocal intonation system advocated by Marchettus of Padua in 1318. He
calls for pure 2:1 octaves, 3:2 fifths, 4:3 fourths, and 9:8
whole-tones, plus the use of cadential dieses equal to "one of the
five parts of a tone."

When I first saw your scale, I took it as a kind of 9 out of 29
(although 9 out of 24 is also quite close), but then realized that it
could exactly fit one late Gothic model deriving from Pythagorean
tuning -- if we take a rounded 204 cents, for example, to represent a
pure 9:8 (~203.91 cents), and so forth.

What this looks like to me, in rounded cents, is the precise dividing
of each Pythagorean diatonic semitone or limma (256:243, ~90.22 cents)
into two equal parts of around 45 cents (~45.11 cents). These parts
are sometimes known in late medieval theory as "diaschismas," with a
whole-tone equal to "four diaschismas plus a comma."[1]

Since a usual Pythagorean diatonic scale has five tones and two
diatonic semitones or limmas, dividing each semitone in half gives us
your nine-tone scale, or 5L + 4s.

What makes your scale especially radical is that the 45-cent steps are
used _in place of_ the usual Pythagorean semitones, rather than in
addition to them. It's maybe a bit of "Marchettus meets Dan Stearns,"
and I mean that as the deepest of compliments all around.

> I'm showing the symmetrical form, I don't really have a modal
> preference yet). I was playing around with dyadic resolutions, two
> of which seemed effective (at least I could wander off and come back
> to them).

> 453 -> 702 204 -> 249
> 249 -> 0 996 -> 951

Here the two left-hand resolutions, as you suggest in your reference
to my neo-Gothic posts, look like usual neo-Gothic: basically a small
minor third to unison (249-0) and a large major third to fifth
(453-702).

Am I correctly reading these two dyadic resolutions as together
defining the following three-voice progression, with degree numbers
from ^0 to ^9 used to identify the steps of the scale from unison to
octave inclusive, along with the position of each step in cents on
your diagram (e.g. ^5=702), an approach which at least avoids imposing
a usual diatonic worldview:

^2=1404 -- + 45 -- ^3=1449
(453) (702)
^7=951 -- -204 -- ^6=747
(702,249) (702,0)
^5=702 -- + 45 -- ^6=747

This 9-tone scale has all kinds of possibilities in terms of
specialized progressions available only on certain degrees, and the
like -- a new kind of "map" to explore!

The two right-hand resolutions suggest to me what I'm tempted to call
"neo-Renaissance suspensions," although there are 14th-century
precedents also: I might take 204-249 as going from a usual major
second to a small minor third (also possibly a large major second),
and 996-951 as from a usual minor seventh to a large major sixth (also
possibly a small minor seventh).

If these are suspensions or suspension-like resolutions in a
Renaissance manner, I'd take it that they involve oblique descending
motion. Is this the intended three-voice reading, with the outer
voices forming an octave, and the lower voice descending from ^9 to ^8
while the other two remain stationary?

^0=1200
^9=996 ^8=951
^0=0

This gives us a 996-951 resolution between the lower voices, and a
204-249 resolution between the upper voices, maybe suggesting
something like the Renaissance 7-6 and 2-3 suspensions (the latter, as
in this reading, involving stepwise descent of the _lower_ of the pair
of voices while the upper one remains stationary).

Please let me apologize if my three-voice interpretations are
mistaken -- but regardless, your mix of dyadic resolutions plus the
special qualities of this scale as a 9-note set could open the way to
some really new musics. Using an interval such as 249 cents or 951
cents as a point of repose and resolution seems to me a door to
beckoning universes -- a not inappropriate anticipation at the
beginning of a new century and millennium.

Above all, I'd say, congratulations!

----
Note
----

1. To see how an approximate ninefold division results, let us start
from the usual division of the 9:8 tone into a diatonic semitone or
limma (~90.22 cents) plus a chromatic semitone or apotome (2187:2048,
~113.69 cents). The apotome exceeds the limma by a Pythagorean comma
(531441:524288, ~23.46 cents), so we can divide the tone into two
limmas plus this comma. Each limma, in turn, divides into two
45.11-cent diaschismas, giving us in all four diaschismas plus a
comma. If we take a 45.11-cent diaschisma as roughly twice a
23.46-cent comma, then our two limmas or four diaschismas come to
something like eight commas -- which, together with the remaining
Pythagorean comma, gives approximately nine in all. As a diagram may
suggest, this approximation is quite close, although half a diaschisma
is actually slightly less than a Pythagorean comma:

~PC ~PC ~PC ~PC PC ~PC ~PC ~PC ~PC
~22.56 ~22.56 ~22.56 ~22.56 ~23.46 ~22.56 ~22.56 ~22.56 ~22.56
|------|------|------|------|-------|-------|------|------|------|
diaschisma diaschisma PC diaschisma diaschisma
~45.11c ~45.11c ~23.46c ~45.11c ~45.11c
|-------------|-------------|-------|--------------|-------------|
limma PC limma
~90.22c ~23.46c ~90.22c
|---------------------------|-------|----------------------------|
apotome -- 2187:2048 (~113.69c) limma -- 256:243 (~90.22c)
|-----------------------------------|----------------------------|
|----------------------------------------------------------------|
9:8 tone
~203.91c

Most appreciatively,

Margo Schulter
mschulter@value.net

Re: quick this n that -- Bob Valentine's new scale

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