Request for Gene

Can you (would you) provide a nice list of commas that vanish in each
of the following 7-limit wedgies?

<< 1 4 10 4 13 12 ]]
<< 2 -4 -4 -11 -12 2 ]]
<< 5 1 12 -10 5 25 ]]
<< 7 9 13 -2 1 5 ]]
<< 1 4 -2 4 -6 -16 ]]
<< 3 0 -6 -7 -18 -14 ]]
<< 4 -3 2 -14 -8 13 ]]
<< 2 8 1 8 -4 -20 ]]
<< 6 5 3 -6 -12 -7 ]]
<< 1 9 -2 12 -6 -30 ]]
<< 2 8 8 8 7 -4 ]]
<< 6 -7 -2 -25 -20 15 ]]
<< 6 10 10 2 -1 -5 ]]
<< 7 -3 8 -21 -7 27 ]]
<< 4 4 4 -3 -5 -2 ]]
<< 1 -8 -14 -15 -25 -10 ]]
<< 3 0 6 -7 1 14 ]]
<< 0 0 12 0 19 28 ]]
<< 1 4 -9 4 -17 -32 ]]
<< 0 5 0 8 0 -14 ]]
<< 3 12 -1 12 -10 -36 ]]
<< 10 9 7 -9 -17 -9 ]]
<< 3 5 -6 1 -18 -28 ]]

For example, for the second wedgie, I'd at least want to see 50:49,
64:63, and 225:224, if not several more.

Thanks,
Paul

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

> Can you (would you) provide a nice list of commas that vanish in each
> of the following 7-limit wedgies?

Below I give a list in descending size order of the subgroup commas
for each temperament, plus any on a 7-limit comma list which I either
cooked up for you or which you computed, of commas with relative error
less than 0.06 and epimericity less than 0.5.

[1, 4, 10, 4, 13, 12]
[59049/57344, 81/80, 126/125, 225/224, 3136/3125, 703125/702464]

[2, -4, -4, -11, -12, 2]
[50/49, 64/63, 2048/2025, 225/224]

[5, 1, 12, -10, 5, 25]
[3125/3072, 875/864, 245/243, 225/224, 10976/10935]

[7, 9, 13, -2, 1, 5]
[686/675, 245/243, 126/125, 78732/78125, 4375/4374]

[1, 4, -2, 4, -6, -16]
[256/245, 36/35, 64/63, 81/80, 5120/5103]

[3, 0, -6, -7, -18, -14]
[128/125, 64/63, 126/125, 4000/3969, 250047/250000]

[4, -3, 2, -14, -8, 13]
[16875/16384, 525/512, 49/48, 686/675, 225/224]

[2, 8, 1, 8, -4, -20]
[49/48, 81/80, 245/243, 19683/19600]

[6, 5, 3, -6, -12, -7]
[1029/1000, 49/48, 875/864, 126/125, 15625/15552]

[1, 9, -2, 12, -6, -30]
[20480/19683, 64/63, 245/243, 1728/1715, 420175/419904]

[2, 8, 8, 8, 7, -4]
[6561/6272, 405/392, 50/49, 81/80, 4000/3969]

[6, -7, -2, -25, -20, 15]
[34171875/33554432, 1063125/1048576, 1029/1024, 225/224, 16875/16807,
2401/2400]

[6, 10, 10, 2, -1, -5]
[250/243, 50/49, 2430/2401, 245/243]

[7, -3, 8, -21, -7, 27]
[2430/2401, 1728/1715, 2109375/2097152, 225/224, 6144/6125, 65625/65536]

[4, 4, 4, -3, -5, -2]
[360/343, 648/625, 36/35, 50/49, 3125/3087, 126/125]

[1, -8, -14, -15, -25, -10]
[3125/3087, 4000/3969, 225/224, 5120/5103, 33554432/33480783, 32805/32768]

[3, 0, 6, -7, 1, 14]
[405/392, 36/35, 128/125, 225/224]

[0, 0, 12, 0, 19, 28]
[648/625, 128/125, 531441/524288, 81/80, 2048/2025, 32805/32768]

[1, 4, -9, 4, -17, -32]
[137781/131072, 525/512, 875/864, 81/80, 4375/4374]

[0, 5, 0, 8, 0, -14]
[256/243, 28/27, 49/48, 64/63, 1029/1024]

[3, 12, -1, 12, -10, -36]
[81/80, 1728/1715, 1029/1024]

[10, 9, 7, -9, -17, -9]
[10077696/9765625, 559872/546875, 126/125, 1728/1715, 2401/2400]

[3, 5, -6, 1, -18, -28]
[250/243, 64/63, 875/864, 6144/6125]

Thanks for the quick work, Gene! I appreciate it. (this is for the
paper)

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Can you (would you) provide a nice list of commas that vanish in
each
> > of the following 7-limit wedgies?
>
> Below I give a list in descending size order of the subgroup commas
> for each temperament, plus any on a 7-limit comma list which I
either
> cooked up for you or which you computed, of commas with relative
error
> less than 0.06 and epimericity less than 0.5.
>
>
> [1, 4, 10, 4, 13, 12]
> [59049/57344, 81/80, 126/125, 225/224, 3136/3125, 703125/702464]
>
> [2, -4, -4, -11, -12, 2]
> [50/49, 64/63, 2048/2025, 225/224]
>
> [5, 1, 12, -10, 5, 25]
> [3125/3072, 875/864, 245/243, 225/224, 10976/10935]
>
> [7, 9, 13, -2, 1, 5]
> [686/675, 245/243, 126/125, 78732/78125, 4375/4374]
>
> [1, 4, -2, 4, -6, -16]
> [256/245, 36/35, 64/63, 81/80, 5120/5103]
>
> [3, 0, -6, -7, -18, -14]
> [128/125, 64/63, 126/125, 4000/3969, 250047/250000]
>
> [4, -3, 2, -14, -8, 13]
> [16875/16384, 525/512, 49/48, 686/675, 225/224]
>
> [2, 8, 1, 8, -4, -20]
> [49/48, 81/80, 245/243, 19683/19600]
>
> [6, 5, 3, -6, -12, -7]
> [1029/1000, 49/48, 875/864, 126/125, 15625/15552]
>
> [1, 9, -2, 12, -6, -30]
> [20480/19683, 64/63, 245/243, 1728/1715, 420175/419904]
>
> [2, 8, 8, 8, 7, -4]
> [6561/6272, 405/392, 50/49, 81/80, 4000/3969]
>
> [6, -7, -2, -25, -20, 15]
> [34171875/33554432, 1063125/1048576, 1029/1024, 225/224,
16875/16807,
> 2401/2400]
>
> [6, 10, 10, 2, -1, -5]
> [250/243, 50/49, 2430/2401, 245/243]
>
> [7, -3, 8, -21, -7, 27]
> [2430/2401, 1728/1715, 2109375/2097152, 225/224, 6144/6125,
65625/65536]
>
> [4, 4, 4, -3, -5, -2]
> [360/343, 648/625, 36/35, 50/49, 3125/3087, 126/125]
>
> [1, -8, -14, -15, -25, -10]
> [3125/3087, 4000/3969, 225/224, 5120/5103, 33554432/33480783,
32805/32768]
>
> [3, 0, 6, -7, 1, 14]
> [405/392, 36/35, 128/125, 225/224]
>
> [0, 0, 12, 0, 19, 28]
> [648/625, 128/125, 531441/524288, 81/80, 2048/2025, 32805/32768]
>
> [1, 4, -9, 4, -17, -32]
> [137781/131072, 525/512, 875/864, 81/80, 4375/4374]
>
> [0, 5, 0, 8, 0, -14]
> [256/243, 28/27, 49/48, 64/63, 1029/1024]
>
> [3, 12, -1, 12, -10, -36]
> [81/80, 1728/1715, 1029/1024]
>
> [10, 9, 7, -9, -17, -9]
> [10077696/9765625, 559872/546875, 126/125, 1728/1715, 2401/2400]
>
> [3, 5, -6, 1, -18, -28]
> [250/243, 64/63, 875/864, 6144/6125]

On <<0, 0, 12, 0, 19, 28]] . . .

When I said 'we discussed this years ago', it turns out I had the
wrong tuning.

I was thinking of a tuning where 5120/5103 vanished, but 81/80 didn't.

<<0, 0, 12, 0, 19, 28]], on the other hand, seems to be functionally
the same as Jon Catler's '12-tone plus' tuning -- except that the
offset is cleverly an eighthtone instead of a sixthtone, making for
better 7:5s . . . right?

So what's the wedgie, TOP primes, period, and generator for the
tuning where 531441/524288 and 5120/5103 vanish?

Waage is of course a *third* member of this 'family'.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Thanks for the quick work, Gene! I appreciate it. (this is for the
> paper)
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > Can you (would you) provide a nice list of commas that vanish
in
> each
> > > of the following 7-limit wedgies?
> >
> > Below I give a list in descending size order of the subgroup
commas
> > for each temperament, plus any on a 7-limit comma list which I
> either
> > cooked up for you or which you computed, of commas with relative
> error
> > less than 0.06 and epimericity less than 0.5.
> >
> >
> > [1, 4, 10, 4, 13, 12]
> > [59049/57344, 81/80, 126/125, 225/224, 3136/3125, 703125/702464]
> >
> > [2, -4, -4, -11, -12, 2]
> > [50/49, 64/63, 2048/2025, 225/224]
> >
> > [5, 1, 12, -10, 5, 25]
> > [3125/3072, 875/864, 245/243, 225/224, 10976/10935]
> >
> > [7, 9, 13, -2, 1, 5]
> > [686/675, 245/243, 126/125, 78732/78125, 4375/4374]
> >
> > [1, 4, -2, 4, -6, -16]
> > [256/245, 36/35, 64/63, 81/80, 5120/5103]
> >
> > [3, 0, -6, -7, -18, -14]
> > [128/125, 64/63, 126/125, 4000/3969, 250047/250000]
> >
> > [4, -3, 2, -14, -8, 13]
> > [16875/16384, 525/512, 49/48, 686/675, 225/224]
> >
> > [2, 8, 1, 8, -4, -20]
> > [49/48, 81/80, 245/243, 19683/19600]
> >
> > [6, 5, 3, -6, -12, -7]
> > [1029/1000, 49/48, 875/864, 126/125, 15625/15552]
> >
> > [1, 9, -2, 12, -6, -30]
> > [20480/19683, 64/63, 245/243, 1728/1715, 420175/419904]
> >
> > [2, 8, 8, 8, 7, -4]
> > [6561/6272, 405/392, 50/49, 81/80, 4000/3969]
> >
> > [6, -7, -2, -25, -20, 15]
> > [34171875/33554432, 1063125/1048576, 1029/1024, 225/224,
> 16875/16807,
> > 2401/2400]
> >
> > [6, 10, 10, 2, -1, -5]
> > [250/243, 50/49, 2430/2401, 245/243]
> >
> > [7, -3, 8, -21, -7, 27]
> > [2430/2401, 1728/1715, 2109375/2097152, 225/224, 6144/6125,
> 65625/65536]
> >
> > [4, 4, 4, -3, -5, -2]
> > [360/343, 648/625, 36/35, 50/49, 3125/3087, 126/125]
> >
> > [1, -8, -14, -15, -25, -10]
> > [3125/3087, 4000/3969, 225/224, 5120/5103, 33554432/33480783,
> 32805/32768]
> >
> > [3, 0, 6, -7, 1, 14]
> > [405/392, 36/35, 128/125, 225/224]
> >
> > [0, 0, 12, 0, 19, 28]
> > [648/625, 128/125, 531441/524288, 81/80, 2048/2025, 32805/32768]
> >
> > [1, 4, -9, 4, -17, -32]
> > [137781/131072, 525/512, 875/864, 81/80, 4375/4374]
> >
> > [0, 5, 0, 8, 0, -14]
> > [256/243, 28/27, 49/48, 64/63, 1029/1024]
> >
> > [3, 12, -1, 12, -10, -36]
> > [81/80, 1728/1715, 1029/1024]
> >
> > [10, 9, 7, -9, -17, -9]
> > [10077696/9765625, 559872/546875, 126/125, 1728/1715, 2401/2400]
> >
> > [3, 5, -6, 1, -18, -28]
> > [250/243, 64/63, 875/864, 6144/6125]

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

> So what's the wedgie, TOP primes, period, and generator for the
> tuning where 531441/524288 and 5120/5103 vanish?

wedgie <<0 12 12 19 19 -6||

mapping [<12 19 28 34|, <0 0 -1 -1|]

TM basis [50/49 3645/3584]

TOP tuning [1198.015473 1896.857833 2778.846497 3377.854234]

TOP generators [99.83462277 16.52294019]

Not the best of the "family", I guess.

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

> <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be functionally
> the same as Jon Catler's '12-tone plus' tuning -- except that the
> offset is cleverly an eighthtone instead of a sixthtone, making for
> better 7:5s . . . right?

One way to look at it is that it is 5-limit 12-equal with 7s tacked
on, which may as well be pure 7s (the TOP tuning.) TOP 12-et of course
has flat octaves, but this is hardly a requirement. "12 tone plus"
seems like a good description of this--is this Catler's idea? Is
"catler" a good name for this temperament?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be
functionally
> > the same as Jon Catler's '12-tone plus' tuning -- except that the
> > offset is cleverly an eighthtone instead of a sixthtone, making
for
> > better 7:5s . . . right?
>
> One way to look at it is that it is 5-limit 12-equal with 7s tacked
> on, which may as well be pure 7s (the TOP tuning.) TOP 12-et of
course
> has flat octaves, but this is hardly a requirement. "12 tone plus"
> seems like a good description of this--is this Catler's idea?

Apparently. See the microtones.com website.

> Is
> "catler" a good name for this temperament?

He might object, since his extended 13-limit JI systems are clearly
dearer to his heart.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > <<0, 0, 12, 0, 19, 28]], on the other hand, seems to be
functionally
> > the same as Jon Catler's '12-tone plus' tuning -- except that the
> > offset is cleverly an eighthtone instead of a sixthtone, making
for
> > better 7:5s . . . right?
>
> One way to look at it is that it is 5-limit 12-equal with 7s tacked
> on, which may as well be pure 7s (the TOP tuning.) TOP 12-et of
course
> has flat octaves, but this is hardly a requirement. "12 tone plus"
> seems like a good description of this--is this Catler's idea? Is
> "catler" a good name for this temperament?

I might use "catler" in my paper, but I'm having trouble with some of
the other names.

In 5-limit, <3 0 -7] is called "augmented" -- fine.

In 7-limit, we have
<3 0 6 -7 1 14]
and
<3 0 -6 -7 -18 -14]
Why is one of these "augmented" and the other "tripletone"?

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

> In 5-limit, <3 0 -7] is called "augmented" -- fine.
>
> In 7-limit, we have
> <3 0 6 -7 1 14]
> and
> <3 0 -6 -7 -18 -14]
> Why is one of these "augmented" and the other "tripletone"?

By my proposal to use TOP to guide us, the second one should be called
"augmented" also, and the name "tripletone" retired.

TOP generators:

128/125 [399.020, 93.145]

<3 0 6 -7 1 14| [399.992 107.311]

<3 0 -6 -7 -18 -14| [399.020 92.460]

That leaves us without a name for the other system, which is also an
important one.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > In 5-limit, <3 0 -7] is called "augmented" -- fine.
> >
> > In 7-limit, we have
> > <3 0 6 -7 1 14]
> > and
> > <3 0 -6 -7 -18 -14]
> > Why is one of these "augmented" and the other "tripletone"?
>
> By my proposal to use TOP to guide us, the second one should be
called
> "augmented" also, and the name "tripletone" retired.
>
> TOP generators:
>
> 128/125 [399.020, 93.145]
>
> <3 0 6 -7 1 14| [399.992 107.311]
>
> <3 0 -6 -7 -18 -14| [399.020 92.460]

Close, but no cigar. In my paper, I will only use the same name if
the tuning is exactly the same, because the names will be references
to horagrams.

> That leaves us without a name for the other system, which is also an
> important one.

I propose both names be some sort of variants of "augmented".

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

> Close, but no cigar. In my paper, I will only use the same name if
> the tuning is exactly the same, because the names will be
references
> to horagrams.

There's no reason not to use the same tuning for both, which argues
for the same name for both.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Close, but no cigar. In my paper, I will only use the same name
if
> > the tuning is exactly the same, because the names will be
> references
> > to horagrams.
>
> There's no reason not to use the same tuning for both,

I consoled John Chalmers by telling him I'd use only one tuning --
the TOP tuning -- for each temperament. Are you suggesting a
deviation from this strategy? Can you make it quantitative?

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

> I consoled John Chalmers by telling him I'd use only one tuning --
> the TOP tuning -- for each temperament. Are you suggesting a
> deviation from this strategy? Can you make it quantitative?

I was talking about in practice, not for a theory paper. If we wanted
to quantize it, perhaps a place to start would be to look at the ratio
between the difference between the two top tunings over the error, for
consonances which are not just. When the tuning difference is a lot
less than the error, in practical terms it doesn't mean much.

Thanks again for this, Gene.

Would it be too much trouble to also do

[8, 6, 6, -9, -13, -3]

and

[6, -2, -2, -17, -20, 1]

?

Those would be great.

Also add these to the tratio request . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Can you (would you) provide a nice list of commas that vanish in
each
> > of the following 7-limit wedgies?
>
> Below I give a list in descending size order of the subgroup commas
> for each temperament, plus any on a 7-limit comma list which I
either
> cooked up for you or which you computed, of commas with relative
error
> less than 0.06 and epimericity less than 0.5.
>
>
> [1, 4, 10, 4, 13, 12]
> [59049/57344, 81/80, 126/125, 225/224, 3136/3125, 703125/702464]
>
> [2, -4, -4, -11, -12, 2]
> [50/49, 64/63, 2048/2025, 225/224]
>
> [5, 1, 12, -10, 5, 25]
> [3125/3072, 875/864, 245/243, 225/224, 10976/10935]
>
> [7, 9, 13, -2, 1, 5]
> [686/675, 245/243, 126/125, 78732/78125, 4375/4374]
>
> [1, 4, -2, 4, -6, -16]
> [256/245, 36/35, 64/63, 81/80, 5120/5103]
>
> [3, 0, -6, -7, -18, -14]
> [128/125, 64/63, 126/125, 4000/3969, 250047/250000]
>
> [4, -3, 2, -14, -8, 13]
> [16875/16384, 525/512, 49/48, 686/675, 225/224]
>
> [2, 8, 1, 8, -4, -20]
> [49/48, 81/80, 245/243, 19683/19600]
>
> [6, 5, 3, -6, -12, -7]
> [1029/1000, 49/48, 875/864, 126/125, 15625/15552]
>
> [1, 9, -2, 12, -6, -30]
> [20480/19683, 64/63, 245/243, 1728/1715, 420175/419904]
>
> [2, 8, 8, 8, 7, -4]
> [6561/6272, 405/392, 50/49, 81/80, 4000/3969]
>
> [6, -7, -2, -25, -20, 15]
> [34171875/33554432, 1063125/1048576, 1029/1024, 225/224,
16875/16807,
> 2401/2400]
>
> [6, 10, 10, 2, -1, -5]
> [250/243, 50/49, 2430/2401, 245/243]
>
> [7, -3, 8, -21, -7, 27]
> [2430/2401, 1728/1715, 2109375/2097152, 225/224, 6144/6125,
65625/65536]
>
> [4, 4, 4, -3, -5, -2]
> [360/343, 648/625, 36/35, 50/49, 3125/3087, 126/125]
>
> [1, -8, -14, -15, -25, -10]
> [3125/3087, 4000/3969, 225/224, 5120/5103, 33554432/33480783,
32805/32768]
>
> [3, 0, 6, -7, 1, 14]
> [405/392, 36/35, 128/125, 225/224]
>
> [0, 0, 12, 0, 19, 28]
> [648/625, 128/125, 531441/524288, 81/80, 2048/2025, 32805/32768]
>
> [1, 4, -9, 4, -17, -32]
> [137781/131072, 525/512, 875/864, 81/80, 4375/4374]
>
> [0, 5, 0, 8, 0, -14]
> [256/243, 28/27, 49/48, 64/63, 1029/1024]
>
> [3, 12, -1, 12, -10, -36]
> [81/80, 1728/1715, 1029/1024]
>
> [10, 9, 7, -9, -17, -9]
> [10077696/9765625, 559872/546875, 126/125, 1728/1715, 2401/2400]
>
> [3, 5, -6, 1, -18, -28]
> [250/243, 64/63, 875/864, 6144/6125]

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> Thanks again for this, Gene.
>
>
> Would it be too much trouble to also do
>
> [8, 6, 6, -9, -13, -3]

390625/373248, 5971968/5764801, 50/49, 875/864, 1728/1715

> and
>
> [6, -2, -2, -17, -20, 1]

140625/131072, 525/512, 50/49, 1029/1024

> Also add these to the tratio request . . .

You'd better give me the whole request, because doing it piecemeal
adds to the work.

Are you giving names to all the temperaments you plan on tabulating,
and if so, which names?

The tratio request I referred to would involve finding the smallest
tratios, and/or the tratios with smallest LCM, for the 23 (now 25)
wedgies listed below.

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> Thanks again for this, Gene.
>
>
> Would it be too much trouble to also do
>
> [8, 6, 6, -9, -13, -3]
>
> and
>
> [6, -2, -2, -17, -20, 1]
>
> ?
>
> Those would be great.
>
>
> Also add these to the tratio request . . .
>
>
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> >
> > > Can you (would you) provide a nice list of commas that vanish
in
> each
> > > of the following 7-limit wedgies?
> >
> > Below I give a list in descending size order of the subgroup
commas
> > for each temperament, plus any on a 7-limit comma list which I
> either
> > cooked up for you or which you computed, of commas with relative
> error
> > less than 0.06 and epimericity less than 0.5.
> >
> >
> > [1, 4, 10, 4, 13, 12]
> > [59049/57344, 81/80, 126/125, 225/224, 3136/3125, 703125/702464]
> >
> > [2, -4, -4, -11, -12, 2]
> > [50/49, 64/63, 2048/2025, 225/224]
> >
> > [5, 1, 12, -10, 5, 25]
> > [3125/3072, 875/864, 245/243, 225/224, 10976/10935]
> >
> > [7, 9, 13, -2, 1, 5]
> > [686/675, 245/243, 126/125, 78732/78125, 4375/4374]
> >
> > [1, 4, -2, 4, -6, -16]
> > [256/245, 36/35, 64/63, 81/80, 5120/5103]
> >
> > [3, 0, -6, -7, -18, -14]
> > [128/125, 64/63, 126/125, 4000/3969, 250047/250000]
> >
> > [4, -3, 2, -14, -8, 13]
> > [16875/16384, 525/512, 49/48, 686/675, 225/224]
> >
> > [2, 8, 1, 8, -4, -20]
> > [49/48, 81/80, 245/243, 19683/19600]
> >
> > [6, 5, 3, -6, -12, -7]
> > [1029/1000, 49/48, 875/864, 126/125, 15625/15552]
> >
> > [1, 9, -2, 12, -6, -30]
> > [20480/19683, 64/63, 245/243, 1728/1715, 420175/419904]
> >
> > [2, 8, 8, 8, 7, -4]
> > [6561/6272, 405/392, 50/49, 81/80, 4000/3969]
> >
> > [6, -7, -2, -25, -20, 15]
> > [34171875/33554432, 1063125/1048576, 1029/1024, 225/224,
> 16875/16807,
> > 2401/2400]
> >
> > [6, 10, 10, 2, -1, -5]
> > [250/243, 50/49, 2430/2401, 245/243]
> >
> > [7, -3, 8, -21, -7, 27]
> > [2430/2401, 1728/1715, 2109375/2097152, 225/224, 6144/6125,
> 65625/65536]
> >
> > [4, 4, 4, -3, -5, -2]
> > [360/343, 648/625, 36/35, 50/49, 3125/3087, 126/125]
> >
> > [1, -8, -14, -15, -25, -10]
> > [3125/3087, 4000/3969, 225/224, 5120/5103, 33554432/33480783,
> 32805/32768]
> >
> > [3, 0, 6, -7, 1, 14]
> > [405/392, 36/35, 128/125, 225/224]
> >
> > [0, 0, 12, 0, 19, 28]
> > [648/625, 128/125, 531441/524288, 81/80, 2048/2025, 32805/32768]
> >
> > [1, 4, -9, 4, -17, -32]
> > [137781/131072, 525/512, 875/864, 81/80, 4375/4374]
> >
> > [0, 5, 0, 8, 0, -14]
> > [256/243, 28/27, 49/48, 64/63, 1029/1024]
> >
> > [3, 12, -1, 12, -10, -36]
> > [81/80, 1728/1715, 1029/1024]
> >
> > [10, 9, 7, -9, -17, -9]
> > [10077696/9765625, 559872/546875, 126/125, 1728/1715, 2401/2400]
> >
> > [3, 5, -6, 1, -18, -28]
> > [250/243, 64/63, 875/864, 6144/6125]

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

> Are you giving names to all the temperaments you plan on tabulating,
> and if so, which names?

I guess this is all up for grabs at the moment. I'm
using "semifourths" instead of "hemifourths" because generating
by "semifourths" was already mentioned in an XH article. The 5-limit
temperament where the Pythagorean comma vanishes will probably be
called Compton since, thanks to Carl, Compton's patent is the
earliest we know of . . . I'm planning to use Hanson only in the 5-
limit, to be true to Larry's intentions . . .

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
>
> > Are you giving names to all the temperaments you plan on
tabulating,
> > and if so, which names?
>
> I guess this is all up for grabs at the moment. I'm
> using "semifourths" instead of "hemifourths" because generating
> by "semifourths" was already mentioned in an XH article. The 5-
limit
> temperament where the Pythagorean comma vanishes will probably be
> called Compton since, thanks to Carl, Compton's patent is the
> earliest we know of . . . I'm planning to use Hanson only in the 5-
> limit, to be true to Larry's intentions . . .

The hanson thing doesn't matter since catakleismic/hanson7 isn't
being discussed anyway. I could give what I presently have down as
names for these 25 temperaments, including switching to compton if
you like.

What theory are you operating under regarding the point beyond which
increasing accuracy of tuning no longer makes a practical difference?
Another question: are you stopping at the 7-limit? It's a fact of
life that more and more temperaments are going to crop up as you go
uplimit.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> >
> > > Are you giving names to all the temperaments you plan on
> tabulating,
> > > and if so, which names?
> >
> > I guess this is all up for grabs at the moment. I'm
> > using "semifourths" instead of "hemifourths" because generating
> > by "semifourths" was already mentioned in an XH article. The 5-
> limit
> > temperament where the Pythagorean comma vanishes will probably be
> > called Compton since, thanks to Carl, Compton's patent is the
> > earliest we know of . . . I'm planning to use Hanson only in the
5-
> > limit, to be true to Larry's intentions . . .
>
> The hanson thing doesn't matter since catakleismic/hanson7 isn't
> being discussed anyway. I could give what I presently have down as
> names for these 25 temperaments, including switching to compton if
> you like.

Sure.

> What theory are you operating under regarding the point beyond
>which
> increasing accuracy of tuning no longer makes a practical
>difference?

None.

> Another question: are you stopping at the 7-limit?

Yes; you said you couldn't handle the 11-limit case, so I'll save
that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future
supplements. This paper is part 1.

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

> Yes; you said you couldn't handle the 11-limit case, so I'll save
> that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future
> supplements. This paper is part 1.

My computer keeps crashing now every 1000 temperaments or so, but I
could finish doing 11 limit if your graph was truly crucial; I have
in mind buying a new one sometime and seeing if that helps, for that
matter. I think however that discussing 5 and 7 limit temperaments
is enough for one paper, and that if you go to the 11 limit you
should probably continue on to 13. Are you putting much theory in
it?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > Yes; you said you couldn't handle the 11-limit case, so I'll
save
> > that, as well as {2,3,7}, {3,5,7}, etc., for part 2 and future
> > supplements. This paper is part 1.
>
> My computer keeps crashing now every 1000 temperaments or so, but I
> could finish doing 11 limit if your graph was truly crucial; I have
> in mind buying a new one sometime and seeing if that helps, for
that
> matter. I think however that discussing 5 and 7 limit temperaments
> is enough for one paper,

Yes.

> and that if you go to the 11 limit you
> should probably continue on to 13.

Yes, in the next paper or two.

Math content is minimal, at John's request.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> > Thanks again for this, Gene.
> >
> >
> > Would it be too much trouble to also do
> >
> > [8, 6, 6, -9, -13, -3]
>
> 390625/373248, 5971968/5764801, 50/49, 875/864, 1728/1715
>
> > and
> >
> > [6, -2, -2, -17, -20, 1]
>
> 140625/131072, 525/512, 50/49, 1029/1024

Very interesting. Now, how about:

[3, 12, 11, 12, 9, -8] (Gawel?)
[6, 10, 3, 2, -12, -21]
[2, -9, -4, -19, -12, 16]

The idea is that I'm moving my boundary out to

error/10 + complexity/24 < 1

This adds these three 7-limit temperaments, for a total of 28. The
number of 5-limit temperaments remains at 21. If I include
ennealimmal as a "bonus" temperament, that's 50 altogether -- "Fifty
Temperaments" will make a nice subtitle for the paper. 10, 24, and 50
are all nice, round numbers. Finality approaches.

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

> Very interesting. Now, how about:
>
> [3, 12, 11, 12, 9, -8] (Gawel?)

TM basis: {81/80, 686/675}
commas: [1029/1000, 686/675, 81/80, 177147/175616, 10976/10935]

> [6, 10, 3, 2, -12, -21]

TM basis: {49/48, 250/243}
commas: [282475249/262144000, 250/243, 49/48, 4000/3969]

> [2, -9, -4, -19, -12, 16]

TM basis: {64/63, 686/675}
commas: [524288/492075, 6272/6075, 686/675, 64/63, 2401/2400}

> The idea is that I'm moving my boundary out to

> error/10 + complexity/24 < 1

Moving to a fixed exponent after all!

> This adds these three 7-limit temperaments, for a total of 28. The
> number of 5-limit temperaments remains at 21. If I include
> ennealimmal as a "bonus" temperament, that's 50 altogether -- "Fifty
> Temperaments" will make a nice subtitle for the paper. 10, 24, and 50
> are all nice, round numbers. Finality approaches.

I think ennealimmal is so striking that discussing it anyway makes
sense--while your exponent favors low complexity over low error, there
is a school of thought which would favor low error, and even interest
itself in microtempering. From that point of view the Heavenly Hemis
(hemiwuerschmidt and hemififths) are also interesting, but ennealimmal
makes the point that we can get effective JI while making use of
useful approximations in scales of Partchian dimensions.

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

> [3, 12, 11, 12, 9, -8] (Gawel?)

Gawel is OK by me.

> [6, 10, 3, 2, -12, -21]

I don't have a name for this one.

> [2, -9, -4, -19, -12, 16]

This I called beatles, on the highly questionable grounds that 19/64
is a poptimal generator. Does that suit you?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
>
> > Very interesting. Now, how about:
> >
> > [3, 12, 11, 12, 9, -8] (Gawel?)
>
> TM basis: {81/80, 686/675}
> commas: [1029/1000, 686/675, 81/80, 177147/175616, 10976/10935]
>
> > [6, 10, 3, 2, -12, -21]
>
> TM basis: {49/48, 250/243}
> commas: [282475249/262144000, 250/243, 49/48, 4000/3969]
>
> > [2, -9, -4, -19, -12, 16]
>
> TM basis: {64/63, 686/675}
> commas: [524288/492075, 6272/6075, 686/675, 64/63, 2401/2400}
>
> > The idea is that I'm moving my boundary out to
>
> > error/10 + complexity/24 < 1
>
> Moving to a fixed exponent after all!

Huh? How does changing 23.5 to 24 make it a fixed exponent?

> > This adds these three 7-limit temperaments, for a total of 28.
The
> > number of 5-limit temperaments remains at 21. If I include
> > ennealimmal as a "bonus" temperament, that's 50 altogether --
"Fifty
> > Temperaments" will make a nice subtitle for the paper. 10, 24,
and 50
> > are all nice, round numbers. Finality approaches.
>
> I think ennealimmal is so striking that discussing it anyway makes
> sense--while your exponent favors low complexity over low error,

I have no idea what you're talking about. What do you mean?

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

> Huh? How does changing 23.5 to 24 make it a fixed exponent?

Sorry, I was thinking logs, but you mean the no logs.

How do you calculate if a given comma vanishes in the temperament
represented by a given wedgie?