English sixth in JI?

The three most common types of augmented sixth chords are the Italian sixth (P1 M3 A6), French sixth (P1 M3 A4 A6) and German sixth (P1 M3 P5 A6). If the augmented sixth itself is tuned as 7/4, then in just intonation, the chords are voiced:

Italian: 1/1 5/4 7/4 > 4:5:7
German: 1/1 5/4 3/2 7/4 > 4:5:6:7
French: 1/1 5/4 7/5 7/4 > 20:25:28:35

There's another augmented sixth chord called an English sixth, which replaces the perfect fifth of the German sixth with an enharmonically-equivalent double augmented fourth. If the root is A-flat, that's Ab C D# F#. The fifth is lowered by a quarter-tone in 31-TET.

My question is: how should an English sixth be voiced in JI? In 11-limit, I'm thinking 1/1 5/4 16/11 7/4 > 44:55:64:77 in 11-limit, but are there better ideas and better prime limits?

Thanks,

~D.

--- In tuning@yahoogroups.com, "Danny Wier" <dawiertx@...> wrote:

> There's another augmented sixth chord called an English sixth,
which
> replaces the perfect fifth of the German sixth with an
> enharmonically-equivalent double augmented fourth. If the root is A-
flat,
> that's Ab C D# F#. The fifth is lowered by a quarter-tone in 31-TET.

Lowered by a diesis, rather.

> My question is: how should an English sixth be voiced in JI? In 11-
limit,
> I'm thinking 1/1 5/4 16/11 7/4 > 44:55:64:77 in 11-limit, but are
there
> better ideas and better prime limits?

Excellent question. Interpreting it mostly as 35/24 actually makes
more sense to me; (35/24)/(16/11) = 385/384, by the way. It is then a
7/6 above 5/4 and a 6/5 below 7/4, and if you equate it with 16/11,
you have a 385/384-planar magic chord. This is valid in 31-et, but
also in 41, 46, 72, 94, 118 and even 284. If I ever finish this 118-
et thing, you can find it in there.

From: "Gene Ward Smith"

[me]
>> My question is: how should an English sixth be voiced in JI? In 11-
> limit,
>> I'm thinking 1/1 5/4 16/11 7/4 > 44:55:64:77 in 11-limit, but are
> there
>> better ideas and better prime limits?
>
> Excellent question. Interpreting it mostly as 35/24 actually makes
> more sense to me; (35/24)/(16/11) = 385/384, by the way. It is then a
> 7/6 above 5/4 and a 6/5 below 7/4, and if you equate it with 16/11,
> you have a 385/384-planar magic chord. This is valid in 31-et, but
> also in 41, 46, 72, 94, 118 and even 284. If I ever finish this 118-
> et thing, you can find it in there.

And 53-tone (as imperfect as it is in 11-limit); both intervals map to 29 steps.

So 1/1 5/4 35/24 7/4 > 24:30:35:42, and that does make more sense. I completely overlooked 36/35 as a diesis, thanks.

I was also thinking of 16:18:23:28 as a no-limit version, but 23/16 seems kind of low.

~D.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> It is then a
> 7/6 above 5/4 and a 6/5 below 7/4, and if you equate it with 16/11,
> you have a 385/384-planar magic chord.

Spacial, sorry. If you look at the 7-limit lattice, you can see how
using 48/35 in place of 11/8 gives you a good 11 close to 1 in lattice
distance, which is convenient. In fact, 48/35 has a Hahn distance of
only 2, which is awfully short considering how small a comma 385/384
is. You get something similar from 100/99 or 99/98, but they are much
larger, and 56/55 equates 11/8 to 7/5, which is a Hahn distance of 1,
but this does not do justice to the 11-limit. So there is arguably
something fairly special about 385/384 tempering.