More on 9-notes scales

More from an off-list exchange I had with John Chalmers.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
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From manynote@library.wustl.edu Thu Jun 17 10:46:00 1999
Date: Tue, 1 Sep 1998 11:24:31 -0500 (CDT)
From: Paul Hahn <manynote@library.wustl.edu>
To: John Chalmers <non12@deltanet.com>
Subject: Re: 9-toners again

Just figured out (don't know why it took me so long) that

3-3-4-3-5-3-4-3-3

actually has _four_ complete tetrads using only nine pitches--in JI you
would need at least ten pitches. This is analogous to the diatonic set
containing six complete triads using seven pitches, when in JI you would
need at least eight. Not only that, but it has no 2-step intervals,
making the stepsizes more even. I think I'm going to have to work some
more with this scale . . .

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
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From manynote@library.wustl.edu Thu Jun 17 10:46:00 1999
Date: Tue, 6 Oct 1998 12:37:09 -0500 (CDT)
From: Paul Hahn <manynote@library.wustl.edu>
To: John Chalmers <jhchalmers@UCSD.Edu>
Subject: Re: 9-toners

On Thu, 3 Sep 1998, John Chalmers wrote:
>> 3-3-4-3-5-3-4-3-3
>
> I see this symmetric arrangement as two similar perfect "pentachords"
> separated by a whole tone, an arrangement similar to the major scale in JI.
> I presume this is better than the form with identical pentachords a fifth
> apart.

(i.e. 3-4-3-3-5-3-4-3-3)

Well, better in some ways and worse in others. This one has only three
complete tetrads instead of four, but it has 22 7-limit consonances vs.
21 in the first, so it's a tradeoff. It also contains the lovely
7-limit pentatonic scale 1/1 7/6 4/3 3/2 7/4 (2/1) as a subset. I'm
going to have to study _this_ one some more, too.

--pH <manynote@lib-rary.wustl.edu> http://library.wustl.edu/~manynote
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

NOTE: dehyphenate node to remove spamblock. <*>

Thanks Paul (Hahn) for your archived posts... I'm enjoying checking them
out.

[On Thu, 3 Sep 1998, John Chalmers wrote:]
>3-3-4-3-5-3-4-3-3
>
>I see this symmetric arrangement as two similar perfect >"pentachords"
separated by a whole tone, an arrangement similar >to the major scale in
JI. I presume this is better than the >form with identical pentachords a
fifth apart.
>
>(i.e. 3-4-3-3-5-3-4-3-3)

I've also used methods of identical 'tetrachords' an interval apart to
generate scales. Here I would see this as:

1/1 (+3/2=3/2), 16/15 (+3/2=8/5), 7/6 (+3/2=7/4), 5/4 (+3/2=15/8), 4/3
(+3/2=2/1)

where n=31, and:

[(log1-log1)*(12/log2)]*n/12
[(log16-log15)*(12/log2)]*n/12
[(log7-log6)*(12/log2)]*n/12
[(log5-log4)*(12/log2)]*n/12
[(log4-log3)*(12/log2)]*n/12
[(log3-log2)*(12/log2)]*n/12
[(log8-log5)*(12/log2)]*n/12
[(log7-log4)*(12/log2)]*n/12
[(log15-log8)*(12/log2)]*n/12
[(log2-log1)*(12/log2)]*n/12

derives a 0, 3, 7, 10, 13, 18, 21, 25, 28, 31 ("3-4-3-3-5-3-4-3-3") in
31-tET where the 225/224 the 385/384 are 'ignored.' (And the 45/32 and the
64/45, are a 7/5 and a 10/7.):

1/1, 16/15, 7/6, 5/4, 4/3, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 12/11, 7/6, 5/4, 7/5, 3/2, 18/11, 7/4, 15/8, 2/1
1/1, 16/15, 8/7, 9/7, 11/8, 3/2, 8/5, 12/7, 11/6, 2/1
1/1, 16/15, 6/5, 9/7, 7/5, 3/2, 8/5, 12/7, 15/8, 2/1
1/1, 9/8, 6/5, 21/16, 7/5, 3/2, 8/5, 7/4, 15/8, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 10/7, 14/9, 5/3, 16/9, 2/1
1/1, 12/11, 7/6, 5/4, 4/3, 16/11, 14/9, 5/3, 15/8, 2/1
1/1, 16/15, 8/7, 11/9, 4/3, 10/7, 32/21, 12/7, 11/6, 2/1
1/1, 16/15, 8/7, 5/4, 4/3, 10/7, 8/5, 12/7, 15/8, 2/1
1/1, 16/15, 7/6, 5/4, 4/3, 3/2, 8/5, 7/4, 15/8, 2/1

Dan