22 7-limit temperaments in the upper uv quadrant

I found 22 of these; this is probably all that exist. I list the
temperament and the product uv. The list can be used as is, but if
Dave wants rounded corners, we could fix a value for uv and only
accept temperaments above it. Paul may dislike the unboundedness of
the upper quadrant; we can fix that by transforming coordinates back
to complexity and error. We have

C = exp((8 + u - v)/3)

E = exp((4 - 4u - v)/3)

If we use the cuttoff hyperbola v = k/u, then we may plot that in the
C-E plane in parametric form as

C(u) = exp((8 + u - k/u))
E(u) = exp((4 - 4u - k/u))

This gives a curved line in an unbounded region, where zero error and
complexity (though not any temperaments exhibiting them) may be found.

Other objections might be that you don't find your favorite
temperament, or you do find one you can't stand. I'm not too impressed
by the second, but for either, or if you think you see a "moat"
somewhere, you have a 5-parameter family to play with--two for the
origin, two for the slopes of the coordinate axes, and one for the
constant k in the hyperbola. That should accomodate anyone's desire to
fiddle, or even cook the books.

Are there any remaining objections I have not answered above?

Ennealimmal
1 <18, 27, 18, 1, -22, -34| <3.629230331, .575465612| 2.088497

Meantone
2 <1, 4, 10, 4, 13, 12| <1.005097996, 1.609665600| 1.617872

Magic
3 <5, 1, 12, -10, 5, 25| <1.012524503, .7830372729| .792844

Pajara
4 <2, -4, -4, -11, -12, 2| <.524469574, 1.498444023| .785888

Dominant seventh
5 <1, 4, -2, 4, -6, -16| <.363511386, 2.141744135| .778548

Semisixths
6 <7, 9, 13, -2, 1, 5| <.8521893702, .8383344292| .714420

Tripletone
7 <3, 0, -6, -7, -18, -14| <.426316706, .940456192| .400932

Blackwood
8 <0, 5, 0, 8, 0, -14| <.152491081, 2.548674345| .388650

Miracle
9 <6, -7, -2, -25, -20, 15| <1.411065061, .2629785497| .371080

Diminished
10 <4, 4, 4, -3, -5, -2| <.160875338, 1.953852436| .314327

Negri
11 <4, -3, 2, -14, -8, 13| <.345586529, .859876933| .297162

Hemifourths
12 <2, 8, 1, 8, -4, -20| <.283811920, 1.034875923| .293710

Kleismic
13 <6, 5, 3, -6, -12, -7| <.322424097, .767227201| .247373

Superpythagorean
14 <1, 9, -2, 12, -6, -30| <.4535440254, .4454275914| .202021

Injera
15 <2, 8, 8, 8, 7, -4| <.245867248, .811824633| .199601

Augmented
16 <3, 0, 6, -7, 1, 14| <.113185821, 1.762756409| .199519

"Number 43" {50/49, 245/243} Supermajor?
17 <6, 10, 10, 2, -1, -5| <.287044794, .548744901| .157514

"Number 55" {81/80, 128/125} Duodecatonic?
18 <0, 0, 12, 0, 19, 28| <.178510293, .520800106| .092968

Orwell
19 <7, -3, 8, -21, -7, 27| <1.060730636, .7657743908e-1| .081228

Schismic
20 <1, -8, -14, -15, -25, -10| <1.080908962, .5026039173e-1| .054327

Flattone
21 <1, 4, -9, 4, -17, -32| <.3364294610, .1379787620| .046420

Porcupine
22 <3, 5, -6, 1, -18, -28| <.176167209, .93101647e-1| .016401

Gene posted this on Feb. 11th (click "up thread" or see below).

I have no idea what you did here, Gene, but this list is awfully
close to my Feb. 8th list of 23:

/tuning-math/message/9317

that led to my current list.

Only #21 (SupermajorSeconds) and #22 (Nonkleismic) of the 23 in my
list are omitted, and only Ennealimmal is added.

We were so busy arguing that we didn't notice how very close our
lists were.

So what exactly led to your results here? I can't understand how you
arrived at them.

Your list has four distinct domains. All temperaments in domain n+1
are both more accurate and more complex than any temperament in
domain n.

Domain 1: Blackwood, Diminished, Augmented, DominantSevenths
Domain 2: Everything on your list below that's not in other domains.
The five temperaments I've just added, including Gawel, can be put in
this domain too, without ruining the domains property above
Domain 3: Orwell, Schismic, Miracle
Domain 4: Ennealimmal

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I found 22 of these; this is probably all that exist. I list the
> temperament and the product uv. The list can be used as is, but if
> Dave wants rounded corners, we could fix a value for uv and only
> accept temperaments above it. Paul may dislike the unboundedness of
> the upper quadrant; we can fix that by transforming coordinates back
> to complexity and error. We have
>
> C = exp((8 + u - v)/3)
>
> E = exp((4 - 4u - v)/3)
>
> If we use the cuttoff hyperbola v = k/u, then we may plot that in
the
> C-E plane in parametric form as
>
> C(u) = exp((8 + u - k/u))
> E(u) = exp((4 - 4u - k/u))
>
> This gives a curved line in an unbounded region, where zero error
and
> complexity (though not any temperaments exhibiting them) may be
found.
>
> Other objections might be that you don't find your favorite
> temperament, or you do find one you can't stand. I'm not too
impressed
> by the second, but for either, or if you think you see a "moat"
> somewhere, you have a 5-parameter family to play with--two for the
> origin, two for the slopes of the coordinate axes, and one for the
> constant k in the hyperbola. That should accomodate anyone's desire
to
> fiddle, or even cook the books.
>
> Are there any remaining objections I have not answered above?
>
> Ennealimmal
> 1 <18, 27, 18, 1, -22, -34| <3.629230331, .575465612| 2.088497
>
> Meantone
> 2 <1, 4, 10, 4, 13, 12| <1.005097996, 1.609665600| 1.617872
>
> Magic
> 3 <5, 1, 12, -10, 5, 25| <1.012524503, .7830372729| .792844
>
> Pajara
> 4 <2, -4, -4, -11, -12, 2| <.524469574, 1.498444023| .785888
>
> Dominant seventh
> 5 <1, 4, -2, 4, -6, -16| <.363511386, 2.141744135| .778548
>
> Semisixths
> 6 <7, 9, 13, -2, 1, 5| <.8521893702, .8383344292| .714420
>
> Tripletone
> 7 <3, 0, -6, -7, -18, -14| <.426316706, .940456192| .400932
>
> Blackwood
> 8 <0, 5, 0, 8, 0, -14| <.152491081, 2.548674345| .388650
>
> Miracle
> 9 <6, -7, -2, -25, -20, 15| <1.411065061, .2629785497| .371080
>
> Diminished
> 10 <4, 4, 4, -3, -5, -2| <.160875338, 1.953852436| .314327
>
> Negri
> 11 <4, -3, 2, -14, -8, 13| <.345586529, .859876933| .297162
>
> Hemifourths
> 12 <2, 8, 1, 8, -4, -20| <.283811920, 1.034875923| .293710
>
> Kleismic
> 13 <6, 5, 3, -6, -12, -7| <.322424097, .767227201| .247373
>
> Superpythagorean
> 14 <1, 9, -2, 12, -6, -30| <.4535440254, .4454275914| .202021
>
> Injera
> 15 <2, 8, 8, 8, 7, -4| <.245867248, .811824633| .199601
>
> Augmented
> 16 <3, 0, 6, -7, 1, 14| <.113185821, 1.762756409| .199519
>
> "Number 43" {50/49, 245/243} Supermajor?
> 17 <6, 10, 10, 2, -1, -5| <.287044794, .548744901| .157514

This is "Biporky", not "Supermajor Seconds".

> "Number 55" {81/80, 128/125} Duodecatonic?
> 18 <0, 0, 12, 0, 19, 28| <.178510293, .520800106| .092968
>
> Orwell
> 19 <7, -3, 8, -21, -7, 27| <1.060730636, .7657743908e-1| .081228
>
> Schismic
> 20 <1, -8, -14, -15, -25, -10| <1.080908962, .5026039173e-1| .054327
>
> Flattone
> 21 <1, 4, -9, 4, -17, -32| <.3364294610, .1379787620| .046420
>
> Porcupine
> 22 <3, 5, -6, 1, -18, -28| <.176167209, .93101647e-1| .016401

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

> We were so busy arguing that we didn't notice how very close our
> lists were.

It's hardly the case that I didn't notice that. Do you began to
understand why I became so frustrated? I still can't figure out why
it all blew up the way it did; normally, we communicate better than
this.

> So what exactly led to your results here? I can't understand how
you
> arrived at them.

My idea was to get something closer to what people seemed to want,
two exponents could be used; this could be smoothed out if we used a
hyperbolic boundry in the log-log plane. In order to accomodate two
exponents, making them the vertical and horizontal axis seemed like a
good plan. This posting follows up a previous one which explains all
of that.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > We were so busy arguing that we didn't notice how very close our
> > lists were.
>
> It's hardly the case that I didn't notice that.

Well, that makes a lot more sense.

> Do you began to
> understand why I became so frustrated? I still can't figure out why
> it all blew up the way it did; normally, we communicate better than
> this.
>
> > So what exactly led to your results here? I can't understand how
> you
> > arrived at them.
>
> My idea was to get something closer to what people seemed to want,
> two exponents could be used; this could be smoothed out if we used
a
> hyperbolic boundry in the log-log plane. In order to accomodate two
> exponents, making them the vertical and horizontal axis seemed like
a
> good plan. This posting follows up a previous one which explains
all
> of that.

I still don't see the intuition behind it. Could you draw some
pictures to help?