Diaschismic, Negri and Blackwood 10

Diaschismic[10] looks like an attractive alternative to meantone[7].

Diaschismic[10]
rms error 2.61 cents

2048/2025
[1, 16/15, 9/8, 5/4, 4/3, 45/32, 3/2, 8/5, 16/9, 15/8]
[[1, 3/2, 9/8, 16/9, 4/3], [45/32, 16/15, 8/5, 5/4, 15/8]]

Two identical circles of fifths
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 25/16]

Negri[10]
rms error 5.94 cents

16875/16384
[1, 16/15, 75/64, 5/4, 4/3, 64/45, 3/2, 8/5, 128/75, 15/8]
[3/2, 8/5, 128/75, 15/8, 1, 16/15, 75/64, 5/4, 4/3, 64/45]

Generator = 16/15
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 25/16]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 5/4, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]
[1, 6/5, 3/2]

Blackwood[10]
rms error 12.76 cents

256/243
[1, 10/9, 9/8, 5/4, 4/3, 40/27, 3/2, 5/3, 16/9, 15/8]
[[16/9, 4/3, 1, 3/2, 9/8], [15/8, 40/27, 10/9, 5/3, 5/4]]

Circles of fifths
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]
[1, 5/4, 3/2] [1, 6/5, 3/2]

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

> Diaschismic[10] looks like an attractive alternative to
>meantone[7].

especially if you look though 7-limit glasses :) :)

there is evidence that the original 10 pitches implied by the ancient
indian music treatises formed a 10-tone diaschsimic scale, though the
diaschsima was probably only used to "average out" 16/15 and 135/128,
the rest of the scale remaining just. see my paper.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Diaschismic[10] looks like an attractive alternative to
> >meantone[7].
>
> especially if you look though 7-limit glasses :) :)

Very true--did I put something up about diaschismic[10] in the 7-limit
yet?

> there is evidence that the original 10 pitches implied by the ancient
> indian music treatises formed a 10-tone diaschsimic scale, though the
> diaschsima was probably only used to "average out" 16/15 and 135/128,
> the rest of the scale remaining just. see my paper.

Whether they did so or not, it's a very nice system with a stamp of
approval from the Rule of 36, in case that helps.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > Diaschismic[10] looks like an attractive alternative to
> > >meantone[7].
> >
> > especially if you look though 7-limit glasses :) :)
>
> Very true--did I put something up about diaschismic[10] in the
7-limit
> yet?

i wrote an entire paper about it :)

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"

> > Very true--did I put something up about diaschismic[10] in the
> 7-limit
> > yet?
>
> i wrote an entire paper about it :)

I wasn't thinking of pajara, but of the 34&46 system I've been calling
"diaschismic". My endorsement was premature; I knew diaschismic[10]
was not only a 25/24 system, but also a 21/20 system, so I expected to
get asses in place of regular tetrads, and I did. Trouble is, these
are all 5-limit asses, so it's still 5-limit harmony. However, it's in
much better tune 5-limitwise than pajara.

Gene Ward Smith wrote:

> I wasn't thinking of pajara, but of the 34&46 system I've been calling
> "diaschismic". My endorsement was premature; I knew diaschismic[10]
> was not only a 25/24 system, but also a 21/20 system, so I expected to
> get asses in place of regular tetrads, and I did. Trouble is, these
> are all 5-limit asses, so it's still 5-limit harmony. However, it's in
> much better tune 5-limitwise than pajara.

Sorry, I've lost the original message. 34-equal isn't consistent in the 7-limit, so it won't define the system. Do you mean 46&58?

2 3 5 7
0 1 -2 -8

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:

> Sorry, I've lost the original message. 34-equal isn't consistent in
the
> 7-limit, so it won't define the system. Do you mean 46&58?
>
> 2 3 5 7
> 0 1 -2 -8

Yes; I'm afraid all I meant is that h34^h46 (standard vals) will give
the wedgie.