Re: Sagittal Chord notation

Hi George

> > How would you notate Bohlen Pierce in
> > Sagittal? Or Wendy Carlos's
> > Gamma?

> A non-octave ET can be approximated by an edo if one goes high
> enough. When Dave Keenan looked at Bohlen-Pierce, he concluded that
> 41-ET notation would suffice.

Right. but the thing there is that the concept behind Bohlen Pierce
is that notes that are a 3/1 apart are treated as "the same", I
believe, and generally that's the non octave kind of philosophy
as I've been understanding it anyway. After all they do sound
"the same" if not quite as strongly as a 2/1, but if your
scale has no 2s in any of its ratios, then the 3/1 is the
nearest you have to an equivalence relation between the notes so the ear
may be expected to home in on that in its place possibly.
That's particularly clear with the just intonation Bohlen Pierce.

So you should have the same notation for e.g. 9/7 and 27/7.

If you take a chord like

1/1 25/21 5/3 7/3

it's inversions would be

1/1 25/21 5/3 7/3
25/21 5/3 7/3 3/1
5/3 7/3 3/1 25/7
7/3 3/1 25/7 5/1
3/1 25/7 5/1 7/1

to take an example.

Those should all be notated as the same chord with
notes shifted up by a 3/1 which would be the equivalent
of a 2/1 shift in a normal 2/1 repeating scale.

That's how I have used non octave scales in
what pieces I've tried out myself in those tunings.

Maybe I'll do a piece in j.i Bohlen Pierce as my next
composing project :-). Really bring out the 3/1
by using e.g. echo effects at 3/1 "octaves" between
two voices for instance, one a 3/1 below the other.

If that isn't the idea it is nevertheless a natural
way to compose in these non octave tunings
and I'm sure microtonalists will explore such things
more in due course as the field develops.

I'm sure a suitable notation can be developed
for this on the same philosophy as Sagittal.
Though I'm not sure what quite would take
the place of the Pythagorean scale with
no 2/1s. Maybe powers of 5 reduced
modulo 3? The j.i. Bohlen Pierce uses
powers of 5 ranging from 1/25 to 25 rather
analogously to the use of powers of 3
in ordinary octave scales.

Robert

--- In tuning-math@yahoogroups.com, "Robert Walker"
<robertwalker@n...> wrote:
> Right. but the thing there is that the concept behind Bohlen Pierce
> is that notes that are a 3/1 apart are treated as "the same", I
> believe, and generally that's the non octave kind of philosophy
> as I've been understanding it anyway.

That's right. This approach wouldn't require any accidentals other
than the existing Sagittals. What it needs is new nominals and a
corresponding new staff. This is akin to the extended system of
nominals for linear temperaments that Herman, Paul and I are working
on (extending some ideas of Erv Wilson). But we are only considering
octave-based linear temperaments so far. It seems difficult enough
to come up with a small set of (around 24) nominals whose subsets
can be used consistently across various octave-based linear
temperaments.

But it's nice that one _can_ notate these non-octave scales in
Sagittal with the standard diatonic nominals and a standard staff.
Shifting by a tritave isn't so bad. It just looks like modulation by
a fifth (in addition to the octave). Not too hard to follow. And
certainly this approach is going to be easier for a performer who
doesn't have time to learn a completely new notation. But not so
good for the composer or someone analysing it on its own terms.

...
> I'm sure a suitable notation can be developed
> for this on the same philosophy as Sagittal.
> Though I'm not sure what quite would take
> the place of the Pythagorean scale with
> no 2/1s. Maybe powers of 5 reduced
> modulo 3? The j.i. Bohlen Pierce uses
> powers of 5 ranging from 1/25 to 25 rather
> analogously to the use of powers of 3
> in ordinary octave scales.

I'd probably base a native BP notation on Dan Stearns' linear
version of BP with a generator which is an approximate 7:9 (and
tritave period) and has a proper "MOS" with 9 notes which could
provide the nominals. The simplest comma/diesis that corresponds to
9 of these generators would tell you what Sagittal accidental to use
with these nominals.

> Since the undecimal comma vanishes [in 14-ET] I assume that
> it will manage fine with an 11/9 and notate that as
> just an E. While the 9/7 will be fine too.

Yes that's correct, at least in the notation that uses the native
fifth.