The Two Diadie Scales

Let me start out by saying that 81/80 with either the schisma or the
pythagorean commas (or those two taken together) give us the 12-note
Pythagorean scale, and that this completes the classification for
scales using 81/80, unless you want to go past 0.75 in epimericity.

The two scales using the DIAschisma and the DIEsis of 128/125 are
both known, and this seems like a Carl Lumma speciality. They don't
reduce to Meantone[12], but 22-et, pajara or orwell seem more to the
point. Reduction by 22-et or pajara leads to Pajara[12], but
reduction by orwell leads to two interesting new scales. Or at least
one is new, reducing the first diadie scale gives us something quite
close to lumma.scl, which Carl presented back in 1999.

! diadie1.scl
First Diadie 2048/2025 128/125 scale = lumma5r.scl
12
!
16/15
9/8
75/64
5/4
4/3
45/32
3/2
8/5
5/3
225/128
15/8
2

! diadie2.scl
Second Diadie 2048/2025 128/125 scale ~ pipedum_12a.scl
12
!
16/15
9/8
6/5
5/4
4/3
45/32
3/2
8/5
128/75
225/128
15/8
2

! diadieorw1.scl
84-et version of diadie1.scl (similar to lumma.scl)
12
!
114.285714
200.000000
271.428571
385.714286
500.000000
585.714286
700.000000
814.285714
885.714286
971.428571
1085.714286
1200.000000

! diadieorw2.scl
84-et version of diadie2.scl
12
!
114.285714
200.000000
314.285714
385.714286
500.000000
585.714286
700.000000
814.285714
928.571429
971.428571
1085.714286
1200.000000

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

I made a triad circle (around the circle of fifths) for the two
diadieorw scales, but neglected to include them. Here they are.

Diadieorw1 triad circle

-5 [385.714286, 314.285714, 700.0000000]
-4 [385.714286, 271.4285714, 657.1428571]
-3 [428.5714286, 271.4285714, 700.0000000]
-2 [428.5714286, 300.0000000, 728.5714286]
-1 [385.7142854, 314.2857143, 700.0000000]
0 [385.714286, 314.2857143, 700.0000000]
1 [385.714286, 314.2857143, 700.0000000]
2 [385.714286, 300.0000000, 685.7142857]
3 [428.5714286, 271.4285714, 700.0000000]
4 [428.5714286, 271.4285717, 700.0000000]
5 [385.714286, 314.2857143, 700.0000000]
6 [385.714286, 342.8571429, 728.5714286]

I count five excellent 84-et major triads, one weird flat one,
and three supermajor triads.

Diadieorw2 triad circle

-5 [385.714286, 314.285714, 700.000000]
-4 [385.714286, 314.2857143, 700.0000000]
-3 [385.714286, 271.4285714, 657.1428571]
-2 [428.5714286, 300.0000000, 728.5714286]
-1 [428.5714283, 271.4285714, 700.0000000]
0 [385.714286, 314.2857143, 700.0000000]
1 [385.714286, 314.2857143, 700.0000000]
2 [385.714286, 342.8571429, 728.5714286]
3 [385.714286, 271.4285714, 657.1428571]
4 [428.5714286, 271.4285717, 700.0000000]
5 [428.5714286, 271.4285714, 700.0000000]
6 [385.714286, 342.8571429, 728.5714286]

Four 84-et major triads, three supermajor triads.

>The two scales using the DIAschisma and the DIEsis of 128/125 are
>both known, and this seems like a Carl Lumma speciality. They don't
>reduce to Meantone[12], but 22-et, pajara or orwell seem more to the
>point. Reduction by 22-et or pajara leads to Pajara[12], but
>reduction by orwell leads to two interesting new scales. Or at least
>one is new, reducing the first diadie scale gives us something quite
>close to lumma.scl, which Carl presented back in 1999.

I did a non-thorough by-hand search for 12-tone 5- and 7-limit
'Fokker blocks' (before I knew the term, and before the subject had
been explored by the list -- I certainly wasn't checking for
epimorphism or monotonicity). Some of this was done before I joined
the list, on paper with the rectangular lattices I'd learned about
from Doty's JI Primer.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> Let me start out by saying that 81/80 with either the schisma or
the
> pythagorean commas (or those two taken together) give us the 12-
note
> Pythagorean scale, and that this completes the classification for
> scales using 81/80, unless you want to go past 0.75 in epimericity.
>
> The two scales using the DIAschisma and the DIEsis of 128/125 are
> both known, and this seems like a Carl Lumma speciality. They don't
> reduce to Meantone[12], but 22-et, pajara or orwell seem more to
the
> point. Reduction by 22-et or pajara leads to Pajara[12],

Gene -- you keep saying Pajara but don't you mean Diaschismic?

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

> > The two scales using the DIAschisma and the DIEsis of 128/125 are
> > both known, and this seems like a Carl Lumma speciality. They
don't
> > reduce to Meantone[12], but 22-et, pajara or orwell seem more to
> the
> > point. Reduction by 22-et or pajara leads to Pajara[12],
>
> Gene -- you keep saying Pajara but don't you mean Diaschismic?

I'm assuming that in 22-equal, it is more correctly called Pajara.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > The two scales using the DIAschisma and the DIEsis of 128/125
are
> > > both known, and this seems like a Carl Lumma speciality. They
> don't
> > > reduce to Meantone[12], but 22-et, pajara or orwell seem more
to
> > the
> > > point. Reduction by 22-et or pajara leads to Pajara[12],
> >
> > Gene -- you keep saying Pajara but don't you mean Diaschismic?
>
> I'm assuming that in 22-equal, it is more correctly called Pajara.

No, Pajara is simply the 7-limit extension of Diaschsimic that you do
get in 22-equal (and pretty much in no other ET):

http://www.cix.co.uk/~gbreed/diaschis.htm

As long as you're talking 5-limit though, there's no reason to bring
Pajara into it.

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

> No, Pajara is simply the 7-limit extension of Diaschsimic that you do
> get in 22-equal (and pretty much in no other ET):
>
> http://www.cix.co.uk/~gbreed/diaschis.htm
>
> As long as you're talking 5-limit though, there's no reason to bring
> Pajara into it.

We are tuning Diaschismic so that the 7-limit works well, whether we
want to acknowledge that fact or not. It's a Pajara tuning of
Diaschismic, in other words, and so correctly called Pajara.
Your method gives two precisely equal scales, one of which is called
Diaschismic[12] and the other Pajara[12].

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > No, Pajara is simply the 7-limit extension of Diaschsimic that
you do
> > get in 22-equal (and pretty much in no other ET):
> >
> > http://www.cix.co.uk/~gbreed/diaschis.htm
> >
> > As long as you're talking 5-limit though, there's no reason to
bring
> > Pajara into it.
>
> We are tuning Diaschismic so that the 7-limit works well, whether we
> want to acknowledge that fact or not.

You mean in the particular case of 22-equal?

> It's a Pajara tuning of
> Diaschismic, in other words, and so correctly called Pajara.

By "It" you mean 22-equal?

> Your method gives two precisely equal scales, one of which is called
> Diaschismic[12] and the other Pajara[12].

My method?

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

> > We are tuning Diaschismic so that the 7-limit works well, whether
we
> > want to acknowledge that fact or not.
>
> You mean in the particular case of 22-equal?

Or any other tuning where the fifth is quite sharp. You'd have us
call it "Diaschismic" even if the tuning was the pure 9/7's tuning!

> > Your method gives two precisely equal scales, one of which is
called
> > Diaschismic[12] and the other Pajara[12].
>
> My method?

Your scale naming method.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > We are tuning Diaschismic so that the 7-limit works well,
whether
> we
> > > want to acknowledge that fact or not.
> >
> > You mean in the particular case of 22-equal?
>
> Or any other tuning where the fifth is quite sharp. You'd have us
> call it "Diaschismic" even if the tuning was the pure 9/7's tuning!

But this is nonsense. Why would you use 22-equal or any good Pajara
tuning when there are far better Diaschismic tunings for doing what
you mention in connection with Pajara in all your recent posts? All
those posts concerned 5-limit only. What if we were to substitute
the "Dominant Sevenths" temperament. for "Meantone" in all those
posts? Would you be happy?

> > > Your method gives two precisely equal scales, one of which is
> called
> > > Diaschismic[12] and the other Pajara[12].
> >
> > My method?
>
> Your scale naming method.

Not any more than yours would give two precisely equal scales, one of
which is called DominantSevenths[12] and the other Meantone[12].

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

> > Or any other tuning where the fifth is quite sharp. You'd have us
> > call it "Diaschismic" even if the tuning was the pure 9/7's
tuning!
>
> But this is nonsense. Why would you use 22-equal or any good Pajara
> tuning when there are far better Diaschismic tunings for doing what
> you mention in connection with Pajara in all your recent posts?

Because I want higher than 5-limit harmonies, of course.

All
> those posts concerned 5-limit only.

Not if you read carefully. I mentioned the various approximate higher
limit consonances, after all.

> Not any more than yours would give two precisely equal scales, one
of
> which is called DominantSevenths[12] and the other Meantone[12].

I'd call 12-equal both of these, but I'd hardly term the 31-et
version of DominantSevenths[12] by that name.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > Or any other tuning where the fifth is quite sharp. You'd have
us
> > > call it "Diaschismic" even if the tuning was the pure 9/7's
> tuning!
> >
> > But this is nonsense. Why would you use 22-equal or any good
Pajara
> > tuning when there are far better Diaschismic tunings for doing
what
> > you mention in connection with Pajara in all your recent posts?
>
> Because I want higher than 5-limit harmonies, of course.
>
> All
> > those posts concerned 5-limit only.
>
> Not if you read carefully.

I do read carefully. Actually, my brain is beginning to deteriorate,
I should join a gym.

> I mentioned the various approximate higher
> limit consonances, after all.

I saw no mention of those in connection with the "Pajara" cases.

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

> > I mentioned the various approximate higher
> > limit consonances, after all.
>
> I saw no mention of those in connection with the "Pajara" cases.

It was in the same posting as the rest of them. What did you think I
meant by saying it was a good candidate for 22-et tempering?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > > I mentioned the various approximate higher
> > > limit consonances, after all.
> >
> > I saw no mention of those in connection with the "Pajara" cases.
>
> It was in the same posting as the rest of them.

I've searched and found tuning post #50759, and tuning-math post
#8332, and in both of them Pajara seems like a leap since you make no
mention of anything beyond 5-limit. You start with Meantone[12], so
if anything, the logical comparison would be to Diaschismic[12].

Another thing -- perhaps some of the non-Pajara 7-limit extensions of
Diaschsimic would in fact be useful for some of those scales?

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

> I've searched and found tuning post #50759, and tuning-math post
> #8332, and in both of them Pajara seems like a leap since you make
no
> mention of anything beyond 5-limit. You start with Meantone[12], so
> if anything, the logical comparison would be to Diaschismic[12].

I hope you don't assume that I always mean 5-limit when I
say "meantone". :)

> Another thing -- perhaps some of the non-Pajara 7-limit extensions
of
> Diaschsimic would in fact be useful for some of those scales?

I'm planning to see how many distinct scales one obtains in various
temperaments after completing the classification.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > I've searched and found tuning post #50759, and tuning-math post
> > #8332, and in both of them Pajara seems like a leap since you
make
> no
> > mention of anything beyond 5-limit. You start with Meantone[12],
so
> > if anything, the logical comparison would be to Diaschismic[12].
>
> I hope you don't assume that I always mean 5-limit when I
> say "meantone". :)

Since the PBs were 5-limit to begin with, it was by far the most
natural assumption in this case.