RE: [tuning] A 9-tone system in 23-tET

>As far as theoretically obscure lower numbered equal temperaments go,
>I'm pretty sure that 23-tET must rank pretty high on the list; in fact
>I can't recall ever seeing anything about any possible tempered
>systems for it.

Tempered systems for it? Not sure what that means, but

(a) 23-tET contains a recognizable representation of pelog tuning (3 3 4 3 3
3 4) and the corresponding pelog scale (3 7 3 3 7), as well as a slendro
tuning (presumably some mode of 5 4 5 5 4), which is what Blackwood used
23-tET for.

(b) In my first days on the list ('96) I found 23-tET to be the worst of all
the ETs at approximating just ratios, given a fairly wide range of
parameters for odd limit and auditory resolution, relative to what one would
expect (i.e., relative to how well 23-tET approximates a random set of
intervals, on average).

(c) In my discussions here with Stephen Soderberg, I used 13-out-of-23-tET
as an example (in fact, the only one I could find) of a system that
satisfied his requirements for a hyper-diatonic system but didn't have a
chance of satisfying mine (due, of course, to the unfavorable acoustical
qualities of the scale).

Dear Paul H. Erlich:
We seem to remember Hornbostel, whose first name was you last. believing this was the
basis of Pelog and Slendro. He is quoted in Kunst's books for one. I would assume it is the
quoted parties source of the idea. We have always found this a bit of a stretch and
psychologically seeing a mountain while one has a whole range in front of oneself. Not
uncommon in tuning analysis.

"Paul H. Erlich" wrote:

> >As far as theoretically obscure lower numbered equal temperaments go,
> >I'm pretty sure that 23-tET must rank pretty high on the list; in fact
> >I can't recall ever seeing anything about any possible tempered
> >systems for it.

-- Banaphshu
North American Embassy of Anaphoria island
www.anaphoria.com

Paul Erlich wrote,

> Tempered systems for it? Not sure what that means,

Simply 23 as a tempered something or the other. In the case that I
posted that would be a tempering of a:

a-b
|
|
|
|
|
bi | a
\ | /
\ | /
\ | /
\ | /
\|/
(a+b)i----------t---------a+b
/|\
/ | \
/ | \
/ | \
/ | \
ai | b
|
|
|
|
|
(a-b)i

where "t" "a" and "b" are an 18:21:23, and the resulting scale is a
1/1 23/21 7/6 23/18 216/161 161/108 36/23 12/7 42/23 2/1.

> (a) 23-tET contains a recognizable representation of pelog tuning (3
3 4 3 3 3 4) and the corresponding pelog scale (3 7 3 3 7), as well as
a slendro tuning (presumably some mode of 5 4 5 5 4), which is what
Blackwood used 23-tET for.

Right, I've used these sorts of flip-flopped step structure scales in
a much more general way quite a bit before as well:

L s
L+s
2L+s L+2s
3L+s 3L+2s 2L+3s L+3s

(etc.)

So, here 23 (or 3 3 4 3 3 3 4) is one of the flip-flopped 5L & 2s
scales:

3/5 1/2
4/7
7/12 5/9
10/17 11/19 9/16 6/11
13/22 17/29 18/31 15/26 13/23 14/25 11/20 7/13

(etc.)

> (b) In my first days on the list ('96) I found 23-tET to be the
worst of all the ETs at approximating just ratios, given a fairly wide
range of parameters for odd limit and auditory resolution, relative to
what one would expect (i.e., relative to how well 23-tET approximates
a random set of intervals, on average).

Well, 23-tET works quite nicely here in my -- has little regard for
unfavorable acoustical qualities! -- opinion.

Dan

I wrote,

> where "t" "a" and "b" are an 18:21:23, and the resulting scale is a
1/1 23/21 7/6 23/18 216/161 161/108 36/23 12/7 42/23 2/1.

Though I had just posted it, I probably should have reiterated (as I
believe it's relevant amongst the talk of tempering) that 23-tET
recognizes the diaschisma-like 285768/279841 while hiding the syntonic
comma-like 7889/7776 in this scale.

Dan