768 and the Hexe temperament

Here are some TM-reduced comma bases for the 768-et:

5-limit
[476837158203125/474989023199232,
1350851717672992089/1342177280000000000]

7-limit
[65625/65536, 250047/250000, 49589822592/49433168575]

11-limit
[9801/9800, 19712/19683, 703125/702464, 1296000/1294139]

If we take 65625/65536 and 240047/250000 we get a linear temperament
I'll name Hexe (which, despite being German, you aren't required to
capitalize unless you wish.)

Wedgie [12, 3, -36, -44, -116, -92]

Mapping [[3, 5, 7, 8], [0, -7, -1, 12]]

Generators (in terms of 768-et) [256, 9]

Hexe is really pretty much of a 171-et specialty, but 768 does it
also. Aside from 171, we get decent MOS for 84 and 87.

thanks, Gene!

i suddenly have a tremendous interest in
768edo, since i've found out that that's
what my microtonal music has effectively
been tuned in for years!

i'm looking forward to making lattices of
this data, but at this point any new lattices
will have to wait until my software can
produce them ... which it will soon.

-monz

----- Original Message -----
From: "Gene Ward Smith" <gwsmith@svpal.org>
To: <tuning-math@yahoogroups.com>
Sent: Tuesday, July 08, 2003 5:25 PM
Subject: [tuning-math] 768 and the Hexe temperament

> Here are some TM-reduced comma bases for the 768-et:
>
> 5-limit
> [476837158203125/474989023199232,
> 1350851717672992089/1342177280000000000]
>
> 7-limit
> [65625/65536, 250047/250000, 49589822592/49433168575]
>
> 11-limit
> [9801/9800, 19712/19683, 703125/702464, 1296000/1294139]
>
>
> If we take 65625/65536 and 240047/250000 we get a linear temperament
> I'll name Hexe (which, despite being German, you aren't required to
> capitalize unless you wish.)
>
> Wedgie [12, 3, -36, -44, -116, -92]
>
> Mapping [[3, 5, 7, 8], [0, -7, -1, 12]]
>
> Generators (in terms of 768-et) [256, 9]
>
> Hexe is really pretty much of a 171-et specialty, but 768 does it
> also. Aside from 171, we get decent MOS for 84 and 87.

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
> thanks, Gene!
>
> i suddenly have a tremendous interest in
> 768edo, since i've found out that that's
> what my microtonal music has effectively
> been tuned in for years!

Next we need to measure the pitch accuracy of the sound samples
people use if we are really going to pin this down.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Here are some TM-reduced comma bases for the 768-et:
>
> 5-limit
> [476837158203125/474989023199232,
> 1350851717672992089/1342177280000000000]
>
> 7-limit
> [65625/65536, 250047/250000, 49589822592/49433168575]
>
> 11-limit
> [9801/9800, 19712/19683, 703125/702464, 1296000/1294139]

iirc, 768-equal isn't consistent in the 9-limit, so there is probably
a different 11-limit mapping, besides the "standard" one, which gives
good results.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> iirc, 768-equal isn't consistent in the 9-limit, so there is
probably
> a different 11-limit mapping, besides the "standard" one, which
gives
> good results.

I don't think so, but I'll look again. It's occurred to be that the
gram or zeta tunings might be another way to define a "standard" val,
by the way. Would that be too far out? It would tend to home in on
the more interesting stuff.

Gene Ward Smith wrote:

> I don't think so, but I'll look again. It's occurred to be that the > gram or zeta tunings might be another way to define a "standard" val, > by the way. Would that be too far out? It would tend to home in on > the more interesting stuff.

The best 9-limit mapping is

[768, 1217, 1783, 2156]

which has a worst error of 0.502 scale steps.

Graham