A classification of 31-et 4-tone chords up to the 11-limit

This is a complete list of 4-tone odd-limit-consonant chords up to the 11 limit, where "consonant" means every interval is consonant in the given odd limit. Only one version of each chord is given, and no attempt has been made to find the best root, or see if it is more otonal or utonal. The two chords in the 7-limit are therefore not otonal and utonal versions of the tetrad; one is the tetrad, and another is a chord in the 126/125 temperament, which means this interesting chord appears in some version in the 12, 15, 19, 27, 31, 34, 43, 46, 50, 58, 65, 74 and 77 ets.

7-limit chords

[0, 6, 13, 21]

[0, 7, 15, 23]

9-limit chords

[0, 5, 13, 21]

[0, 5, 13, 20]

[0, 5, 11, 18]

[0, 5, 15, 21]

[0, 5, 10, 15]

[0, 5, 11, 16]

[0, 5, 11, 21]

[0, 5, 10, 16]

[0, 11, 16, 26]

[0, 5, 10, 20]

[0, 5, 13, 23]

[0, 5, 10, 18]

[0, 13, 18, 26]

[0, 7, 13, 20]

11-limit chords

[0, 7, 16, 20]

[0, 4, 10, 18]

[0, 4, 11, 18]

[0, 4, 9, 15]

[0, 4, 9, 18]

[0, 7, 18, 27]

[0, 7, 14, 22]

[0, 4, 8, 14]

[0, 4, 9, 14]

[0, 4, 8, 15]

[0, 5, 13, 22]

[0, 6, 13, 22]

[0, 6, 14, 22]

[0, 7, 14, 21]

[0, 4, 13, 22]

[0, 14, 21, 25]

[0, 5, 16, 20]

[0, 11, 17, 25]

[0, 9, 17, 23]

[0, 9, 16, 25]

[0, 7, 13, 27]

[0, 4, 9, 20]

[0, 13, 17, 27]

[0, 11, 17, 21]

[0, 11, 15, 25]

[0, 6, 10, 27]

[0, 10, 15, 24]

[0, 10, 16, 24]

[0, 5, 22, 27]

[0, 15, 20, 24]

[0, 5, 11, 20]

[0, 9, 16, 24]

[0, 4, 10, 20]

[0, 18, 22, 27]

[0, 16, 21, 25]

[0, 4, 8, 13]

[0, 16, 20, 26]

[0, 13, 20, 24]

[0, 6, 13, 20]

[0, 9, 13, 27]

[0, 5, 11, 25]

[0, 9, 17, 27]

[0, 10, 16, 25]

[0, 10, 17, 27]

[0, 11, 17, 22]

[0, 4, 8, 18]

[0, 10, 17, 23]

[0, 11, 16, 25]

[0, 6, 10, 16]

[0, 4, 11, 21]

[0, 8, 17, 22]

[0, 9, 13, 26]

[0, 4, 11, 20]

[0, 9, 13, 18]

[0, 4, 10, 17]

[0, 4, 11, 15]

[0, 4, 9, 17]

[0, 4, 8, 17]

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

The two chords in the 7-limit are therefore not otonal and utonal versions of the tetrad; one is the tetrad, and another is a chord in the 126/125 temperament, which means this interesting chord appears in some version in the 12, 15, 19, 27, 31, 34, 43, 46, 50, 58, 65, 74 and 77 ets.

> [0, 7, 15, 23]

I should have added that in the 12-et this is a diminished 7th, and therefore there is a natural version of a diminished 7th in the 126/125 planar temperament, a fact of evident importance if you are making use of it or of some temperament which includes it.

> 7-limit chords
>
> [0, 6, 13, 21]
>
> [0, 7, 15, 23]

Awesome! Howabout about 37-tET?

-C.

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Awesome! Howabout about 37-tET?

I know why Paul likes the 22-et, but what is your reason for being fond of the 37-et? I had some other prospects in mind.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, January 14, 2002 12:59 AM
> Subject: [tuning] Re: A classification of 31-et 4-tone chords up to the
11-limit
>
>
> --- In tuning@y..., "clumma" <carl@l...> wrote:
>
> > Awesome! Howabout about 37-tET?
>
> I know why Paul likes the 22-et, but what is your reason
> for being fond of the 37-et? I had some other prospects in mind.

This is really great, Gene.

I'd like to see what results you get for 19, 55, 72,

... and 31-tone Canasta.
http://www.ixpres.com/dict/canasta.htm

-monz

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In-Reply-To: <a1u3j0+a959@eGroups.com>
gene wrote:

> > [0, 7, 15, 23]
>
> I should have added that in the 12-et this is a diminished 7th, and
> therefore there is a natural version of a diminished 7th in the 126/125
> planar temperament, a fact of evident importance if you are making use
> of it or of some temperament which includes it.

As intervals, that'll be 7-8-8-8. I'd write it as 8-7-8-8 so you have a
6:5 and 7:5 from the root. As well as containing tempered 7-limit
intervals, it's close to the JI diminished 7th (given by somebody like
Ellis) of 10:12:14:17:20, although 8-7-9-7 is closer.

In decimal notation, the latter could be [0 3v 5 8] which looks like an
11-limit pseudo-ass. In 41-equal, intervals are 11-9-12-9, so in 72
19-16-21-16. I think this is the best Miracle diminished 7th, at least in
theory. 8-7-8-8 would have to be [0 3v 5 8v] and 8v-0 isn't an 11-limit
interval. Still, you might get away with it.

Graham

>> Awesome! Howabout about 37-tET?
>
> I know why Paul likes the 22-et, but what is your reason for being
> fond of the 37-et? I had some other prospects in mind.

I know of no reason to like it more than any other "good" 7-limit
tuning. I'm interested in it in general because it's almost never
mentioned -- unfairly I think -- so it might could use some
attention. For the purpose at hand it seems ideal,

() It represents the worst positive fifth that's still acceptable,
to my ear.

() It's not very consistent, or Hahn-complete.

Thus, it is one of the "worst" "good" 7-limit temperaments. I
haven't taken the time to find out if it does anything that 22-tET
doesn't do, but I'd be surprised if it doesn't.

-Carl

--- In tuning@y..., "clumma" <carl@l...> wrote:

> Thus, it is one of the "worst" "good" 7-limit temperaments. I
> haven't taken the time to find out if it does anything that 22-tET
> doesn't do, but I'd be surprised if it doesn't.

Any equal temperament can always do something another cannot, in a sense I could make mathematically precise. In the 7-limit, 3136/3125
and 2401/2400 are commas of the 37-et, and this means the corresponding linear temperament works with it and not the 22-et. This is the [-16,-2,-5,-6,-37,34] temperament, with generator matrix

[ 0 1]
[15 -1]
[ 2 2]
[ 5 2]

which is an excellent temperament, particularly for concentrating on 5 and 7. The generator is 5.9789/37 = 15.9976/99, and a 37 tone MOS taken from the 99 et would be a logical system, though no more so than 31 out of 99. Of course from your point of view this would totally wreck your sharp fifths!

GWS:
>Any equal temperament can always do something another cannot, in
>a sense I could make mathematically precise. In the 7-limit,
>3136/3125 and 2401/2400 are commas of the 37-et, and this means
>the corresponding linear temperament works with it and not the
>22-et. This is the [-16,-2,-5,-6,-37,34] temperament
<snip>

Thanks, Gene! But what is [-16,-2,-5,-6,-37,34]?

-C.

--- In tuning@y..., "clumma" <carl@l...> wrote:
> GWS:
> >Any equal temperament can always do something another cannot, in
> >a sense I could make mathematically precise. In the 7-limit,
> >3136/3125 and 2401/2400 are commas of the 37-et, and this means
> >the corresponding linear temperament works with it and not the
> >22-et. This is the [-16,-2,-5,-6,-37,34] temperament
> <snip>
>
> Thanks, Gene! But what is [-16,-2,-5,-6,-37,34]?
>
> -C.

Wedge Invariant, or Wedgie!

(tuning-math would be a good place to find out about this)

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> > Thanks, Gene! But what is [-16,-2,-5,-6,-37,34]?

> Wedge Invariant, or Wedgie!

Sorry to drag it over here, but I didn't know another name for the temperament. Wedgies serve as names which define the temperament.