a request

Could someone point me to the current consensus (you know what I mean,
the Dave K. Gene etc. consensus) note names and their corresponding
ascii glyphs for 31-tet again?

thanks,

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Could someone point me to the current consensus (you know what I
mean,
> the Dave K. Gene etc. consensus) note names and their corresponding
> ascii glyphs for 31-tet again?

hi dan.

i didn't know there was such a thing. 31-equal can of course be
notated conventionally, using double sharps and double flats and a
meantone genesis. if you meant the 72-equal consensus, gene was not
involved, but joseph, dave, and i have been using

] = 1/4-tone high
> = 1/6-tone high
^ = 1/12-tone high
v = 1/12-tone low
< = 1/6-tone low
[ = 1/4-tone low

so what did you mean?

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Could someone point me to the current consensus (you know what I
mean,
> the Dave K. Gene etc. consensus) note names and their corresponding
> ascii glyphs for 31-tet again?
>
>
> thanks,
>
> --Dan Stearns

I don't know if there is such a consensus. Here's a suggestion. I
understand it is pretty standard except for the choice of symbols for
half-sharp and half-flat, which are intended to be replaced with the
Sims "quarter-tone" symbols on a score.

C
C]
C#
Db
D[
D
D]
D#
Eb
E[
E
E] or Fb
F[ or E#
F
F]
F#
Gb
G[
G
G]
G#
Ab
A[
A
A]
A#
Bb
B[
B
B] or Cb
C[ or B#
C

Hmm, I thought that Dave Keenan's note naming scheme used Fokker's as
a starting or reference point--perhaps I'm confused, or perhaps it did
but as applied to 72-tet?

----- Original Message -----
From: "paulerlich" <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, February 11, 2002 7:09 PM
Subject: [tuning] Re: a request

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Could someone point me to the current consensus (you know what I
> mean,
> > the Dave K. Gene etc. consensus) note names and their
corresponding
> > ascii glyphs for 31-tet again?
>
> hi dan.
>
> i didn't know there was such a thing. 31-equal can of course be
> notated conventionally, using double sharps and double flats and a
> meantone genesis. if you meant the 72-equal consensus, gene was not
> involved, but joseph, dave, and i have been using
>
> ] = 1/4-tone high
> > = 1/6-tone high
> ^ = 1/12-tone high
> v = 1/12-tone low
> < = 1/6-tone low
> [ = 1/4-tone low
>
> so what did you mean?
>
>
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Okay, I guess what I had in mind was the 11-limit 72 notation, and 31
as a subset of this--in other words let the 7/72 generator be
commensurate with the 3/31.

This would be one way to use consistent ETs to spell other
inconsistent ETs--for example, 20-tet where 17/31 is commensurate with
11/20, etc.

See what I'm getting at?

----- Original Message -----
From: "dkeenanuqnetau" <d.keenan@uq.net.au>
To: <tuning@yahoogroups.com>
Sent: Monday, February 11, 2002 7:09 PM
Subject: [tuning] Re: a request

> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> > Could someone point me to the current consensus (you know what I
> mean,
> > the Dave K. Gene etc. consensus) note names and their
corresponding
> > ascii glyphs for 31-tet again?
> >
> >
> > thanks,
> >
> > --Dan Stearns
>
> I don't know if there is such a consensus. Here's a suggestion. I
> understand it is pretty standard except for the choice of symbols
for
> half-sharp and half-flat, which are intended to be replaced with the
> Sims "quarter-tone" symbols on a score.
>
> C
> C]
> C#
> Db
> D[
> D
> D]
> D#
> Eb
> E[
> E
> E] or Fb
> F[ or E#
> F
> F]
> F#
> Gb
> G[
> G
> G]
> G#
> Ab
> A[
> A
> A]
> A#
> Bb
> B[
> B
> B] or Cb
> C[ or B#
> C
>
>
> ------------------------ Yahoo! Groups
Sponsor ---------------------~-->
> Get your FREE credit report with a FREE CreditCheck
> Monitoring Service trial
> http://us.click.yahoo.com/ACHqaB/bQ8CAA/ySSFAA/RrLolB/TM
> --------------------------------------------------------------------
-~->
>
> You do not need web access to participate. You may subscribe
through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning
group.
> tuning-nomail@yahoogroups.com - put your email message delivery on
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> tuning-help@yahoogroups.com - receive general help information.
>
>
> Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
>
>

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> Okay, I guess what I had in mind was the 11-limit 72 notation, and 31
> as a subset of this--in other words let the 7/72 generator be
> commensurate with the 3/31.

That's incompatible with the system you want to adapt, it seems to me.
The "right" semitone to use should be h31(2187/2048) = 2, which goes nicely with h31(33/32) = h31(64/63) = 1. 16/15 or 15/14 would then
be a #].

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> C
> C]
> C#
> Db
> D[
> D
> D]

I suggested > and < instead of ] and [; this strikes me as better on the grounds that 7 is a more important prime than 11. Do you have a reason for preferring ] and [?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > C
> > C]
> > C#
> > Db
> > D[
> > D
> > D]
>
> I suggested > and < instead of ] and [; this strikes me as better on
the grounds that 7 is a more important prime than 11. Do you have a
reason for preferring ] and [?

Yes. Tradition and ... Melodically, the symbols beg to be read as
"half-sharp" and "half-flat" and 32:33 functions as such far more
often than 63:64. Neutral thirds get spelled the same as in 72-tET.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Yes. Tradition

Tradition???

and ... Melodically, the symbols beg to be read as
> "half-sharp" and "half-flat" and 32:33 functions as such far more
> often than 63:64. Neutral thirds get spelled the same as in 72-tET.

That means you are telling people coming from 72-et who are used to this having an 11-limit meaning that it now functions as 64/63. I think life would be a lot simpler for such people (and any theoretical people used to its use in JI, for that matter) if we kept on telling them that G-F< was a 7/4 interval, which is what they would be used to seeing. The idea as I see it is to get the notations to agree as much as possible.

hi Dan,

> From: D.Stearns <STEARNS@CAPECOD.NET>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 11, 2002 10:50 PM
> Subject: Re: [tuning] Re: a request
>
>
> Okay, I guess what I had in mind was the 11-limit 72 notation, and 31
> as a subset of this--in other words let the 7/72 generator be
> commensurate with the 3/31.
>
> This would be one way to use consistent ETs to spell other
> inconsistent ETs--for example, 20-tet where 17/31 is commensurate with
> 11/20, etc.
>
> See what I'm getting at?

well, now, aren't you clever?! :)

i see what you're getting at . . . sort of,
but don't really grasp the whole thing.

i did an analysis of these constructions you mentioned
in your post, giving edo degrees, cents, and notations.

(. . . and for once, i'm caving in and using the
72edo "standard" notation!)

(use "expand message" mode in Yahoo web interface)

legend:

lower raise

b # semitone = 100 cents
] [ 1/4-tone = 50 cents
< > 1/6-tone = 33&1/3 cents
v ^ 1/12-tone = 16&2/3 cents

on the far right, i also show the nearest semitone in
cents and the additional correction in cents, to help
make the 72edo notation clearer.

"let the 7/72 generator be commensurate with the 3/31"

7/72 generator 3/31 generator

gen 72edo "standard" 72edo 31edo Keenan 31edo
notation degree cents degree cents notation

16 G< 700 -33 1/3 40 666&2/3 17 658& 2/31 G[
15 F#] 600 -50 33 550 14 541&29/31 F]
14 E> 400 +33 1/3 26 433&1/3 11 425&25/31 E] or Fb
13 Eb^ 300 +16 2/3 19 316&2/3 8 309&21/31 Eb
12 D 200 0 12 200 5 193&17/31 D
11 C#v 100 -16 2/3 5 83&1/3 2 77&13/31 C#
10 C< 1200 -33 1/3 70 1166&2/3 30 1161& 9/31 C[ or B#
9 B] 1100 -50 63 1050 27 1045& 5/31 B[
8 A> 900 +33 1/3 56 933&1/3 24 929& 1/31 A]
7 G#^ 800 +16 2/3 49 816&2/3 21 812&28/31 Ab
6 G 700 0 42 700 18 696&24/31 G
5 F#v 600 -16 2/3 35 583&1/3 15 580&20/31 F#
4 F< 500 -33 1/3 28 466&2/3 12 464&16/31 F[ or E#
3 E] 400 -50 21 350 9 348&12/31 E[
2 D> 200 +33 1/3 14 233&1/3 6 232& 8/31 D]
1 C#^ 100 +16 2/3 7 116&2/3 3 116& 4/31 Db
0 C 0 0 0 0 0 0 C
-1 Bv 1100 -16 2/3 65 1083&1/3 28 1083&27/31 B
-2 Bb< 1000 -33 1/3 58 966&2/3 25 967&23/31 A#
-3 A] 900 -50 51 850 22 851&19/31 A[
-4 G> 700 +33 1/3 44 733&1/3 19 735&15/31 G]
-5 F#^ 600 +16 2/3 37 616&2/3 16 619&11/31 Gb
-6 F 500 0 30 500 13 503& 7/31 F
-7 Ev 400 -16 2/3 23 383&1/3 10 387& 3/31 E
-8 Eb< 300 -33 1/3 16 266&2/3 7 270&30/31 D#
-9 D] 200 -50 9 150 4 154&26/31 D[
-10 C> 0 +33 1/3 2 33&1/3 1 38&22/31 C]
-11 B^ 1100 +16 2/3 67 1116&2/3 29 122&18/31 B] or Cb
-12 Bb 1000 0 60 1000 26 1006&14/31 Bb
-13 Av 900 -16 2/3 53 883&1/3 23 890&10/31 A
-14 G#< 800 -33 1/3 46 766&2/3 20 774& 6/31 G#
-15 G] 700 -50 39 650 17 658& 2/31 G[

so, the +16th and -15th 31edo generators give the same 31edo note,
2^(17/31) = 658& 2/31 = G[. but this is mapped to two different
72edo pitches:

2^(39/72) = 650 cents = G] and
2^(40/72) = 666&2/3 cents = G< .

so, are you talking about choosing one of those to notate
2^(17/31) and thus 2^(11/20) ? or the other way around,
using 2^(17/31) to notate the two 72edo pitches and 2^(11/20) ?

"20-tet where 17/31 is commensurate with 11/20, etc"

Keenan 72edo
MT 20edo 31edo 31edo 72edo 72edo "standard"
gen deg cents deg cents notation deg cents degree cents notation

10 10 600 15 580&20/31 F# 30 500 40 666&2/3 G<
9 19 1140 29 1122&18/31 B] or Cb 63 1050 0 0 C
8 8 480 12 464&16/31 F[ or E# 24 400 32 533&1/3 F>
7 17 1020 26 1006&14/31 Bb 57 950 64 1066&2/3 B<
6 6 360 9 348&12/31 E[ 18 300 24 400 E
5 15 900 23 890&10/31 A 51 850 56 933&1/3 A>
4 4 240 6 232& 8/31 D] 12 200 16 266&2/3 Eb<
3 13 780 20 774& 6/31 G# 45 750 48 800 G#
2 2 120 3 116& 4/31 Db 6 100 8 133&1/3 C#>
1 11 660 17 658& 2/31 G[ 39 650 40 666&2/3 G<
0 0 0 0 0 C 0 0 0 0 C
-1 9 540 14 541&29/31 F] 33 550 32 533&1/3 F>
-2 18 1080 28 1083&27/31 B 66 1100 64 1066&2/3 B
-3 7 420 11 425&25/31 E] or Fb 27 450 24 400 E
-4 16 960 25 967&23/31 A# 60 1000 56 933&1/3 A>
-5 5 300 8 309&21/31 Eb 21 350 16 266&2/3 Eb<
-6 14 840 22 851&19/31 A[ 54 900 48 800 G#
-7 3 180 5 193&17/31 D 15 250 8 133&1/3 C#>
-8 12 720 19 735&15/31 G] 48 800 40 666&2/3 G<
-9 1 60 2 77&13/31 C# 9 150 0 0 C
-10 10 600 16 619&11/31 Gb 42 700 32 533&1/3 F>

the +10 and -10 generators in the 20edo "meantone-like"
chain are both 600 cents in 20edo, giving a closed
"circle of 5ths", altho the "5th" is 660 cents, not
exactly what i'd call a "recognizable 5th".

31edo also inconsistently to 20edo: +10 meantone generators
is 2^(15/31) = 580&20/31 cents = F# , and -10 meantone generators
is 2^(16/31) = 619&11/31 cents = Gb .

here i illustrate the better of the two 72edo notations
with note-names. notice that it is a 9-tone repeating
system in the notation. but both 72edo notations are
inconsistent with the 20edo "meantone" chain.

i see that this 20edo "circle of 5ths" has cent values
that are always between the 2^(17/31) mapping and the
2^(40/72) mapping. so you've gotten two close approximations
on either side of it.

but what are you trying to do with the notation?

-monz

_________________________________________________________
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Get your free @yahoo.com address at http://mail.yahoo.com

In-Reply-To: <011701c1b391$8d5d0d00$fe58d63f@stearns>
D.Stearns wrote:

> Okay, I guess what I had in mind was the 11-limit 72 notation, and 31
> as a subset of this--in other words let the 7/72 generator be
> commensurate with the 3/31.

That's how George's Sag{1,2}it{1,2}al notation works. As the 31-equal
steps are notated using the quartertone symbols, the equivalent from the
house-standard ASCII notation would be [] as originally suggested.

I use + and t for half sharps and flats, because they're similar to the
Tartini/Fokker, and nothing like any of the symbols for 72=.

Graham

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_34027.html#34081

> i did an analysis of these constructions you mentioned
> in your post, giving edo degrees, cents, and notations.
>
> (. . . and for once, i'm caving in and using the
> 72edo "standard" notation!)
>

****Monz, I really appreciate your doing this, *especially* since I
know you didn't want to. In a way, it was like my converting from my
beloved F-C-G Blackjack "key" and all my charts, diagrams, etc., etc.

But, I *do* believe, in the interest of communication, we need to
stick to the "standards" that we have developed together.

I note that some of the people on this list with *scientific*
backgrounds have no trouble adopting "community standards." I think
science may work a bit like that, but I may be wrong....

J. Pehrson

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> "let the 7/72 generator be commensurate with the 3/31"
>
>
> 7/72 generator 3/31 generator
>
> gen 72edo "standard" 72edo 31edo Keenan
31edo
> notation degree cents degree cents
notation
>
> 16 G< 700 -33 1/3 40 666&2/3 17 658& 2/31 G[
> 15 F#] 600 -50 33 550 14 541&29/31 F]
> 14 E> 400 +33 1/3 26 433&1/3 11 425&25/31 E] or
Fb
> 13 Eb^ 300 +16 2/3 19 316&2/3 8 309&21/31 Eb
> 12 D 200 0 12 200 5 193&17/31 D
> 11 C#v 100 -16 2/3 5 83&1/3 2 77&13/31 C#
> 10 C< 1200 -33 1/3 70 1166&2/3 30 1161& 9/31 C[ or
B#
> 9 B] 1100 -50 63 1050 27 1045& 5/31 B[
> 8 A> 900 +33 1/3 56 933&1/3 24 929& 1/31 A]
> 7 G#^ 800 +16 2/3 49 816&2/3 21 812&28/31 Ab
> 6 G 700 0 42 700 18 696&24/31 G
> 5 F#v 600 -16 2/3 35 583&1/3 15 580&20/31 F#
> 4 F< 500 -33 1/3 28 466&2/3 12 464&16/31 F[ or
E#
> 3 E] 400 -50 21 350 9 348&12/31 E[
> 2 D> 200 +33 1/3 14 233&1/3 6 232& 8/31 D]
> 1 C#^ 100 +16 2/3 7 116&2/3 3 116& 4/31 Db
> 0 C 0 0 0 0 0 0 C
> -1 Bv 1100 -16 2/3 65 1083&1/3 28 1083&27/31 B
> -2 Bb< 1000 -33 1/3 58 966&2/3 25 967&23/31 A#
> -3 A] 900 -50 51 850 22 851&19/31 A[
> -4 G> 700 +33 1/3 44 733&1/3 19 735&15/31 G]
> -5 F#^ 600 +16 2/3 37 616&2/3 16 619&11/31 Gb
> -6 F 500 0 30 500 13 503& 7/31 F
> -7 Ev 400 -16 2/3 23 383&1/3 10 387& 3/31 E
> -8 Eb< 300 -33 1/3 16 266&2/3 7 270&30/31 D#
> -9 D] 200 -50 9 150 4 154&26/31 D[
> -10 C> 0 +33 1/3 2 33&1/3 1 38&22/31 C]
> -11 B^ 1100 +16 2/3 67 1116&2/3 29 122&18/31 B] or
Cb
> -12 Bb 1000 0 60 1000 26 1006&14/31 Bb
> -13 Av 900 -16 2/3 53 883&1/3 23 890&10/31 A
> -14 G#< 800 -33 1/3 46 766&2/3 20 774& 6/31 G#
> -15 G] 700 -50 39 650 17 658& 2/31 G[

This is great work Monz!

Now try centering them both on D, and notice that [] and <> are the
same thing in 31-tET and v^ has no effect in 31-tET, and they should
agree a lot better than they seem to in the above table. In fact they
should agree totally.

Also check out

http://dkeenan.com/Music/NotatingEts.xls.zip

Hi Joe,

What I had in mind was using consistent temperaments to spell
inconsistent temperaments by looking at different adjacent fraction
series--for instance, here's the 20-tet example I gave:

4 1 5 6 11 17
--, --, --, --, --, --, ...
7 2 9 11 20 31

By using 31-tet as a template you could either spell say 11 or 20
traditionally--with multiple sharps and flats, or by using some other
scheme--say Fokker's or the 31 out of 72:

2 1 3 4 7
--, --, --, --, --, ...
21 10 31 41 72

See what I'm getting at now?

take care,

--Dan Stearns

----- Original Message -----
From: "monz" <joemonz@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: Tuesday, February 12, 2002 1:24 AM
Subject: Re: [tuning] Re: a request

>
> hi Dan,
>
>
> > From: D.Stearns <STEARNS@CAPECOD.NET>
> > To: <tuning@yahoogroups.com>
> > Sent: Monday, February 11, 2002 10:50 PM
> > Subject: Re: [tuning] Re: a request
> >
> >
> > Okay, I guess what I had in mind was the 11-limit 72 notation, and
31
> > as a subset of this--in other words let the 7/72 generator be
> > commensurate with the 3/31.
> >
> > This would be one way to use consistent ETs to spell other
> > inconsistent ETs--for example, 20-tet where 17/31 is commensurate
with
> > 11/20, etc.
> >
> > See what I'm getting at?
>
>
> well, now, aren't you clever?! :)
>
>
> i see what you're getting at . . . sort of,
> but don't really grasp the whole thing.
>
>
> i did an analysis of these constructions you mentioned
> in your post, giving edo degrees, cents, and notations.
>
> (. . . and for once, i'm caving in and using the
> 72edo "standard" notation!)
>
> (use "expand message" mode in Yahoo web interface)
>
>
> legend:
>
> lower raise
>
> b # semitone = 100 cents
> ] [ 1/4-tone = 50 cents
> < > 1/6-tone = 33&1/3 cents
> v ^ 1/12-tone = 16&2/3 cents
>
> on the far right, i also show the nearest semitone in
> cents and the additional correction in cents, to help
> make the 72edo notation clearer.
>
>
>
> "let the 7/72 generator be commensurate with the 3/31"
>
>
> 7/72 generator 3/31 generator
>
> gen 72edo "standard" 72edo 31edo Keenan
31edo
> notation degree cents degree cents
notation
>
> 16 G< 700 -33 1/3 40 666&2/3 17 658& 2/31 G[
> 15 F#] 600 -50 33 550 14 541&29/31 F]
> 14 E> 400 +33 1/3 26 433&1/3 11 425&25/31 E] or
Fb
> 13 Eb^ 300 +16 2/3 19 316&2/3 8 309&21/31 Eb
> 12 D 200 0 12 200 5 193&17/31 D
> 11 C#v 100 -16 2/3 5 83&1/3 2 77&13/31 C#
> 10 C< 1200 -33 1/3 70 1166&2/3 30 1161& 9/31 C[ or
B#
> 9 B] 1100 -50 63 1050 27 1045& 5/31 B[
> 8 A> 900 +33 1/3 56 933&1/3 24 929& 1/31 A]
> 7 G#^ 800 +16 2/3 49 816&2/3 21 812&28/31 Ab
> 6 G 700 0 42 700 18 696&24/31 G
> 5 F#v 600 -16 2/3 35 583&1/3 15 580&20/31 F#
> 4 F< 500 -33 1/3 28 466&2/3 12 464&16/31 F[ or
E#
> 3 E] 400 -50 21 350 9 348&12/31 E[
> 2 D> 200 +33 1/3 14 233&1/3 6 232& 8/31 D]
> 1 C#^ 100 +16 2/3 7 116&2/3 3 116& 4/31 Db
> 0 C 0 0 0 0 0 0 C
> -1 Bv 1100 -16 2/3 65 1083&1/3 28 1083&27/31 B
> -2 Bb< 1000 -33 1/3 58 966&2/3 25 967&23/31 A#
> -3 A] 900 -50 51 850 22 851&19/31 A[
> -4 G> 700 +33 1/3 44 733&1/3 19 735&15/31 G]
> -5 F#^ 600 +16 2/3 37 616&2/3 16 619&11/31 Gb
> -6 F 500 0 30 500 13 503& 7/31 F
> -7 Ev 400 -16 2/3 23 383&1/3 10 387& 3/31 E
> -8 Eb< 300 -33 1/3 16 266&2/3 7 270&30/31 D#
> -9 D] 200 -50 9 150 4 154&26/31 D[
> -10 C> 0 +33 1/3 2 33&1/3 1 38&22/31 C]
> -11 B^ 1100 +16 2/3 67 1116&2/3 29 122&18/31 B] or
Cb
> -12 Bb 1000 0 60 1000 26 1006&14/31 Bb
> -13 Av 900 -16 2/3 53 883&1/3 23 890&10/31 A
> -14 G#< 800 -33 1/3 46 766&2/3 20 774& 6/31 G#
> -15 G] 700 -50 39 650 17 658& 2/31 G[
>
>
>
>
> so, the +16th and -15th 31edo generators give the same 31edo note,
> 2^(17/31) = 658& 2/31 = G[. but this is mapped to two different
> 72edo pitches:
>
> 2^(39/72) = 650 cents = G] and
> 2^(40/72) = 666&2/3 cents = G< .
>
> so, are you talking about choosing one of those to notate
> 2^(17/31) and thus 2^(11/20) ? or the other way around,
> using 2^(17/31) to notate the two 72edo pitches and 2^(11/20) ?
>
>
>
> "20-tet where 17/31 is commensurate with 11/20, etc"
>
> Keenan
72edo
> MT 20edo 31edo 31edo 72edo 72edo
"standard"
> gen deg cents deg cents notation deg cents degree cents
notation
>
> 10 10 600 15 580&20/31 F# 30 500 40 666&2/3 G<
> 9 19 1140 29 1122&18/31 B] or Cb 63 1050 0 0 C
> 8 8 480 12 464&16/31 F[ or E# 24 400 32 533&1/3 F>
> 7 17 1020 26 1006&14/31 Bb 57 950 64 1066&2/3 B<
> 6 6 360 9 348&12/31 E[ 18 300 24 400 E
> 5 15 900 23 890&10/31 A 51 850 56 933&1/3 A>
> 4 4 240 6 232& 8/31 D] 12 200 16 266&2/3 Eb<
> 3 13 780 20 774& 6/31 G# 45 750 48 800 G#
> 2 2 120 3 116& 4/31 Db 6 100 8 133&1/3 C#>
> 1 11 660 17 658& 2/31 G[ 39 650 40 666&2/3 G<
> 0 0 0 0 0 C 0 0 0 0 C
> -1 9 540 14 541&29/31 F] 33 550 32 533&1/3 F>
> -2 18 1080 28 1083&27/31 B 66 1100 64 1066&2/3 B
> -3 7 420 11 425&25/31 E] or Fb 27 450 24 400 E
> -4 16 960 25 967&23/31 A# 60 1000 56 933&1/3 A>
> -5 5 300 8 309&21/31 Eb 21 350 16 266&2/3 Eb<
> -6 14 840 22 851&19/31 A[ 54 900 48 800 G#
> -7 3 180 5 193&17/31 D 15 250 8 133&1/3 C#>
> -8 12 720 19 735&15/31 G] 48 800 40 666&2/3 G<
> -9 1 60 2 77&13/31 C# 9 150 0 0 C
> -10 10 600 16 619&11/31 Gb 42 700 32 533&1/3 F>
>
>
> the +10 and -10 generators in the 20edo "meantone-like"
> chain are both 600 cents in 20edo, giving a closed
> "circle of 5ths", altho the "5th" is 660 cents, not
> exactly what i'd call a "recognizable 5th".
>
> 31edo also inconsistently to 20edo: +10 meantone generators
> is 2^(15/31) = 580&20/31 cents = F# , and -10 meantone generators
> is 2^(16/31) = 619&11/31 cents = Gb .
>
> here i illustrate the better of the two 72edo notations
> with note-names. notice that it is a 9-tone repeating
> system in the notation. but both 72edo notations are
> inconsistent with the 20edo "meantone" chain.
>
>
> i see that this 20edo "circle of 5ths" has cent values
> that are always between the 2^(17/31) mapping and the
> 2^(40/72) mapping. so you've gotten two close approximations
> on either side of it.
>
> but what are you trying to do with the notation?
>
>
>
> -monz
>
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