A few full 11-limit 896/891 temperaments

I'm not sure that these haven't already been discovered (I'm sure one
of them has to be Margo's "Pepper" temperament), but I'm looking to
find some ideal temperaments in which 896/891 vanishes. It would be
nice to start with those in which the generator is 3/2 and expand out
from there.

My rationale here is
1) These temperaments are in an important middle ground in that the
generator is a bit too sharp for meantone, but too flat for a really
good superpyth.
2) These temperaments tend to be in the "sweet spot" for fifths, at
least if you're like me where you prefer fifths that are slightly
sharp.
3) These temperaments might lead to some interesting possibilities for
tonality, although I doubt we'll be able to squeeze anything out of
the diatonic MOS. This might apply more to temperaments that do
interesting things to the generator.

The first one that comes to mind is the 29&46 temperament, seen here:
http://x31eq.com/cgi-bin/rt.cgi?ets=29%2C46&limit=11

I can't figure out what commas this vanishes just from looking at it.
It's pretty accurate, 46-tet seems to be a good tuning. 4:5:6 is
pretty complex here, so that even in the 29-note MOS it doesn't pop up
that much.

Here's another one, which vanishes 896/891, 64/63, and 55/54, which is
pretty much cheating. It's just what happens if you take superpyth and
equate 14/11 with 9/7. It's cheating. Cheating!
http://x31eq.com/cgi-bin/rt.cgi?ets=22_5&error=11.488&limit=11&invariant=1_9_-2_-6_1_0_-12_6_13

Cheating. Now, here's one in which 896/891, 225/224, and 32805/32768
vanish, giving a beautiful 11-limit extension of schismatic
temperament apparently called "Cassandra" temperament, which seems to
be optimal for around 41-equal:
http://x31eq.com/cgi-bin/rt.cgi?ets=41_12&error=2.971&limit=11&invariant=1_-8_-14_-18_1_0_15_25_32

Does anyone have any other interesting options or ideas here? It would
be nice to somehow find a temperament that keeps the bright 17-tet
melodic "feel" but supports 5/4 in an interesting way.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I'm not sure that these haven't already been discovered (I'm sure one
> of them has to be Margo's "Pepper" temperament), but I'm looking to
> find some ideal temperaments in which 896/891 vanishes. It would be
> nice to start with those in which the generator is 3/2 and expand out
> from there.

Speaking of Pepper temperament, I tried looking it up a few days ago on what Carl claims is now a good search algorithm here, and didn't find the exposition I wanted.

> The first one that comes to mind is the 29&46 temperament, seen here:
> http://x31eq.com/cgi-bin/rt.cgi?ets=29%2C46&limit=11
>
> I can't figure out what commas this vanishes just from looking at it.
> It's pretty accurate, 46-tet seems to be a good tuning.

That one is leapday:

http://xenharmonic.wikispaces.com/Hemifamity+temperaments#Leapday

> Here's another one, which vanishes 896/891, 64/63, and 55/54, which is
> pretty much cheating. It's just what happens if you take superpyth and
> equate 14/11 with 9/7. It's cheating. Cheating!
> http://x31eq.com/cgi-bin/rt.cgi?ets=22_5&error=11.488&limit=11&invariant=1_9_-2_-6_1_0_-12_6_13

I have a different one down as 11-limit superpyth. Want to name this?

> Cheating. Now, here's one in which 896/891, 225/224, and 32805/32768
> vanish, giving a beautiful 11-limit extension of schismatic
> temperament apparently called "Cassandra" temperament, which seems to
> be optimal for around 41-equal:
> http://x31eq.com/cgi-bin/rt.cgi?ets=41_12&error=2.971&limit=11&invariant=1_-8_-14_-18_1_0_15_25_32

http://xenharmonic.wikispaces.com/Schismatic+family#Garibaldi

> Does anyone have any other interesting options or ideas here? It would
> be nice to somehow find a temperament that keeps the bright 17-tet
> melodic "feel" but supports 5/4 in an interesting way.

Well, of course there are other 896/891 temperaments such as diaschismic, rodan or mystery which don't use the fifth as a generator.

On 3/11/2011 8:45 PM, Mike Battaglia wrote:
> I'm not sure that these haven't already been discovered (I'm sure one
> of them has to be Margo's "Pepper" temperament), but I'm looking to
> find some ideal temperaments in which 896/891 vanishes. It would be
> nice to start with those in which the generator is 3/2 and expand out
> from there.
>
> My rationale here is
> 1) These temperaments are in an important middle ground in that the
> generator is a bit too sharp for meantone, but too flat for a really
> good superpyth.
> 2) These temperaments tend to be in the "sweet spot" for fifths, at
> least if you're like me where you prefer fifths that are slightly
> sharp.
> 3) These temperaments might lead to some interesting possibilities for
> tonality, although I doubt we'll be able to squeeze anything out of
> the diatonic MOS. This might apply more to temperaments that do
> interesting things to the generator.
>
> The first one that comes to mind is the 29&46 temperament, seen here:
> http://x31eq.com/cgi-bin/rt.cgi?ets=29%2C46&limit=11
>
> I can't figure out what commas this vanishes just from looking at it.
> It's pretty accurate, 46-tet seems to be a good tuning. 4:5:6 is
> pretty complex here, so that even in the 29-note MOS it doesn't pop up
> that much.

I've called that one "leapday" because of the 29-note MOS. It tempers out 121/120.

> Here's another one, which vanishes 896/891, 64/63, and 55/54, which is
> pretty much cheating. It's just what happens if you take superpyth and
> equate 14/11 with 9/7. It's cheating. Cheating!
> http://x31eq.com/cgi-bin/rt.cgi?ets=22_5&error=11.488&limit=11&invariant=1_9_-2_-6_1_0_-12_6_13
>
> Cheating. Now, here's one in which 896/891, 225/224, and 32805/32768
> vanish, giving a beautiful 11-limit extension of schismatic
> temperament apparently called "Cassandra" temperament, which seems to
> be optimal for around 41-equal:
> http://x31eq.com/cgi-bin/rt.cgi?ets=41_12&error=2.971&limit=11&invariant=1_-8_-14_-18_1_0_15_25_32
>
> Does anyone have any other interesting options or ideas here? It would
> be nice to somehow find a temperament that keeps the bright 17-tet
> melodic "feel" but supports 5/4 in an interesting way.

If you don't mind the lower accuracy there's a version of dominant (5&12).

[<1 2 4 2 1|, <0 -1 -4 2 6|] 1194.105, 494.306

Then there's an 11-limit version of "quasisuper" which may be more along the lines of what you're looking for. The 22-note MOS looks nice.

[<1 2 -3 2 1|, <0 -1 13 2 6|] 1197.587 490.806

Other octave-based temperaments with different generators include magic and rodan.

[<1 0 2 -1 6|, <0 5 1 12 -8|] 1200.143, 380.742 : magic 19&22
[<1 -1 -1 -2 9|, <0 7 9 13 -15|] 1200.043, 443.310 : sensi 19&27
[<1 1 -1 3 6|, <0 3 17 -1 -13|] 1200.057, 234.470 : rodan 5&41
[<1 1 -5 -1 2|, <0 2 25 13 5|] 1199.286, 351.311 : hemififths 41&58

Then there's a few temperaments with two periods to the octave, e.g.:

[<2 3 5 5 7|, <0 2 -4 7 -1|] 599.775 52.660 : shrutar 22&24
[<2 3 5 7 9|, <0 1 -2 -8 -12|] 599.449 103.619 : diaschismic 12&46

On Fri, Mar 11, 2011 at 10:14 PM, genewardsmith
<genewardsmith@...> wrote:
>
> Speaking of Pepper temperament, I tried looking it up a few days ago on what Carl claims is now a good search algorithm here, and didn't find the exposition I wanted.

I hope this doesn't turn out to be one of those brilliant but forever
lost type things.

> That one is leapday:
>
> http://xenharmonic.wikispaces.com/Hemifamity+temperaments#Leapday

She's a winner.

> > Here's another one, which vanishes 896/891, 64/63, and 55/54, which is
> > pretty much cheating. It's just what happens if you take superpyth and
> > equate 14/11 with 9/7. It's cheating. Cheating!
> > http://x31eq.com/cgi-bin/rt.cgi?ets=22_5&error=11.488&limit=11&invariant=1_9_-2_-6_1_0_-12_6_13
>
> I have a different one down as 11-limit superpyth. Want to name this?

What do you have down as 11-limit superpyth?

So this is the one where the diatonic diminished fifth becomes 11/8,
right? Isn't that what you just called "supra?"

> > Cheating. Now, here's one in which 896/891, 225/224, and 32805/32768
> > vanish, giving a beautiful 11-limit extension of schismatic
> > temperament apparently called "Cassandra" temperament, which seems to
> > be optimal for around 41-equal:
> > http://x31eq.com/cgi-bin/rt.cgi?ets=41_12&error=2.971&limit=11&invariant=1_-8_-14_-18_1_0_15_25_32
>
> http://xenharmonic.wikispaces.com/Schismatic+family#Garibaldi

This is the greatest temperament ever made.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > > Here's another one, which vanishes 896/891, 64/63, and 55/54, which is
> > > pretty much cheating. It's just what happens if you take superpyth and
> > > equate 14/11 with 9/7. It's cheating. Cheating!
> > > http://x31eq.com/cgi-bin/rt.cgi?ets=22_5&error=11.488&limit=11&invariant=1_9_-2_-6_1_0_-12_6_13
> >
> > I have a different one down as 11-limit superpyth. Want to name this?
>
> What do you have down as 11-limit superpyth?

http://xenharmonic.wikispaces.com/Archytas+clan#Superpyth

Commas: 64/63, 100/99, 245/243
Map: [<1 0 -12 6 -22|, <0 1 9 -2 16|]
22&49

> So this is the one where the diatonic diminished fifth becomes 11/8,
> right? Isn't that what you just called "supra?"

Supra is a no-fives temperament, tempering out 64/63 and 99/98. If you remove five from consideration in the "cheat" temperament, you get supra. And you shouldn't complain too hard about supra, considering how closely it's related to machine.

On Sat, Mar 12, 2011 at 12:31 AM, genewardsmith
<genewardsmith@...> wrote:
>
> Supra is a no-fives temperament, tempering out 64/63 and 99/98. If you remove five from consideration in the "cheat" temperament, you get supra. And you shouldn't complain too hard about supra, considering how closely it's related to machine.

Might as well call it "suprapyth" then. I can't see a better name for it.

-Mike

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

> Speaking of Pepper temperament, I tried looking it up a few days
> ago on what Carl claims is now a good search algorithm here, and
> didn't find the exposition I wanted.

I didn't say it was good, just that it's better than the search
that you get on facebook, reddit, quora, slashdot, google reader,
and so on.

-Carl