7-limit comma Monzos and Ratios

Monzos:

/tuning/files/Erlich/herman2m.gif

Ratios:

/tuning/files/Erlich/herman2r.gif
/tuning/files/Erlich/herman2s.gif

The following all seem like excellent candidates for tempering out to
get a planar temperament, Herman:

numerator denominator e2 e3 e5 e7
28 27 2 -3 0 1
36 35 2 2 -1 -1
49 48 -4 -1 0 2
50 49 1 0 2 -2
81 80 -4 4 -1 0
126 125 1 2 -3 1
128 125 7 0 -3 0
135 128 -7 3 1 0
200 189 3 -3 2 -1
225 224 -5 2 2 -1
245 243 0 -5 1 2
250 243 1 -5 3 0
256 243 8 -5 0 0
256 245 8 0 -1 -2
360 343 3 2 1 -3
392 375 3 -1 -3 2
405 392 -3 4 1 -2
525 512 -9 1 2 1
648 625 3 4 -4 0
686 675 1 -3 -2 3
875 864 -5 -3 3 1
1029 1000 -3 1 -3 3
1029 1024 -10 1 0 3
1728 1715 6 3 -1 -3
2048 2025 11 -4 -2 0
2401 2400 -5 -1 -2 4
2430 2401 1 5 1 -4
3125 3072 -10 -1 5 0
3125 3087 0 -2 5 -3
3136 3125 6 0 -5 2
4000 3969 5 -4 3 -2
4375 4374 -1 -7 4 1
5120 5103 10 -6 1 -1
6144 6125 11 1 -3 -2
10976 10935 5 -7 -1 3
15625 15552 -6 -5 6 0
16875 16807 0 3 4 -5
19683 19600 -4 9 -2 -2
32805 32768 -15 8 1 0
65625 65536 -16 1 5 1
703125 702464 -11 2 7 -3
78125000 78121827 3 -13 10 -2

On Tue, 13 Jan 2004 11:26:01 -0000, "wallyesterpaulrus"
<paul@stretch-music.com> wrote:

>Monzos:
>
>/tuning/files/Erlich/herman2m.gif

No surprise that [-5 -1 -2 4] stands out so well -- it's one of the
miracle-72 commas. But the [-1 -7 4 1] is more unexpected. It works with a
handful of kleismic temperaments, such as 19, 53, 72, and 91-ET, but it
takes 22 steps of a kleismic generator according to the temperament finder
(plugging in 19, 53, 7), which doesn't seem as useful as [-4 -1 0 2], but
it might work better with some other 5-limit temperament. It certainly
looks potentially interesting for a planar temperament if nothing else.

>The following all seem like excellent candidates for tempering out to
>get a planar temperament, Herman:

These are definitely worth looking at; there's some I don't recall seeing
before, like 200;189 [3 -3 2 -1] and 256;245 [8 0 -1 -2]. I notice, though,
that you're including 5- and 3-limit commas on the list. Do you get
something other than meantone when you make a planar temperament from
81/80? I suppose you could pick any two generators that add up to a
meantone generator, as long as one of the generators is some fraction of a
7-limit interval; is that how it works?

--
see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> No surprise that [-5 -1 -2 4] stands out so well -- it's one of the
> miracle-72 commas. But the [-1 -7 4 1] is more unexpected.

Not to me. 2401/2400 and 4375/4374 are both high-voltage commas for
microtempering. If you put them together you get Ennealimmal, which
is more relevant to the discussion than the less-accurate
temperaments you name.

It certainly
> looks potentially interesting for a planar temperament if nothing
else.

Tell me about it. I got flamed to a cinder for analyzing this planar
temperament on this group.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> I notice, though,
> that you're including 5- and 3-limit commas on the list. Do you get
> something other than meantone when you make a planar temperament
from
> 81/80?

Of course you do. Remember, there were 3-limit commas on the 5-limit
linear temperament list; for example, 256:243 gives you blackwood,
and 531441:524288 gives you aristoxenean.

> I suppose you could pick any two generators that add up to a
> meantone generator, as long as one of the generators is some
fraction of a
> 7-limit interval; is that how it works?

No, what happens (in TOP) is that you simply construct a chain of
pure 7:1s above and below each note in the 5-limit linear
temperament. Note that this does *not* imply pure 7:4, because in
most cases, the octaves are tempered.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

/tuning/topicId_51719.html#51749

> --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> > I notice, though,
> > that you're including 5- and 3-limit commas on the list. Do you
get
> > something other than meantone when you make a planar temperament
> from
> > 81/80?
>
> Of course you do. Remember, there were 3-limit commas on the 5-
limit
> linear temperament list; for example, 256:243 gives you blackwood,
> and 531441:524288 gives you aristoxenean.
>
> > I suppose you could pick any two generators that add up to a
> > meantone generator, as long as one of the generators is some
> fraction of a
> > 7-limit interval; is that how it works?
>
> No, what happens (in TOP) is that you simply construct a chain of
> pure 7:1s above and below each note in the 5-limit linear
> temperament. Note that this does *not* imply pure 7:4, because in
> most cases, the octaves are tempered.

***So, in other words, the addition of this "dimension" is what makes
it *two dimensional* or *planar* rather than linear??

Didn't Ben Johnston also experiment with this kind of thing??

Thanks!

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_51719.html#51749
>
>
> > --- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >
> > > I notice, though,
> > > that you're including 5- and 3-limit commas on the list. Do you
> get
> > > something other than meantone when you make a planar
temperament
> > from
> > > 81/80?
> >
> > Of course you do. Remember, there were 3-limit commas on the 5-
> limit
> > linear temperament list; for example, 256:243 gives you
blackwood,
> > and 531441:524288 gives you aristoxenean.
> >
> > > I suppose you could pick any two generators that add up to a
> > > meantone generator, as long as one of the generators is some
> > fraction of a
> > > 7-limit interval; is that how it works?
> >
> > No, what happens (in TOP) is that you simply construct a chain of
> > pure 7:1s above and below each note in the 5-limit linear
> > temperament. Note that this does *not* imply pure 7:4, because in
> > most cases, the octaves are tempered.
>
>
> ***So, in other words, the addition of this "dimension" is what
makes
> it *two dimensional* or *planar* rather than linear??

Basically, yes.

> Didn't Ben Johnston also experiment with this kind of thing??

Not with temperament, but yes with adding dimensions to JI by adding
more primes. Not only Johnston but also Fokker, Doty, Monzo . . . who
didn't?

Gene,

Am I missing anything with epimericity < 0.5 and max. weighted error
< 0.006 in the list below?

-Paul

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> Monzos:
>
> /tuning/files/Erlich/herman2m.gif
>
> Ratios:
>
> /tuning/files/Erlich/herman2r.gif
> /tuning/files/Erlich/herman2s.gif
>
> The following all seem like excellent candidates for tempering out
to
> get a planar temperament, Herman:
>
> numerator denominator e2 e3 e5
e7
> 28 27 2 -3 0
1
> 36 35 2 2 -1 -
1
> 49 48 -4 -1 0
2
> 50 49 1 0 2 -
2
> 81 80 -4 4 -1
0
> 126 125 1 2 -3
1
> 128 125 7 0 -3
0
> 135 128 -7 3 1
0
> 200 189 3 -3 2 -
1
> 225 224 -5 2 2 -
1
> 245 243 0 -5 1
2
> 250 243 1 -5 3
0
> 256 243 8 -5 0
0
> 256 245 8 0 -1 -
2
> 360 343 3 2 1 -
3
> 392 375 3 -1 -3
2
> 405 392 -3 4 1 -
2
> 525 512 -9 1 2
1
> 648 625 3 4 -4
0
> 686 675 1 -3 -2
3
> 875 864 -5 -3 3
1
> 1029 1000 -3 1 -3
3
> 1029 1024 -10 1 0
3
> 1728 1715 6 3 -1 -
3
> 2048 2025 11 -4 -2
0
> 2401 2400 -5 -1 -2
4
> 2430 2401 1 5 1 -
4
> 3125 3072 -10 -1 5
0
> 3125 3087 0 -2 5 -
3
> 3136 3125 6 0 -5
2
> 4000 3969 5 -4 3 -
2
> 4375 4374 -1 -7 4
1
> 5120 5103 10 -6 1 -
1
> 6144 6125 11 1 -3 -
2
> 10976 10935 5 -7 -1
3
> 15625 15552 -6 -5 6
0
> 16875 16807 0 3 4 -
5
> 19683 19600 -4 9 -2 -
2
> 32805 32768 -15 8 1
0
> 65625 65536 -16 1 5
1
> 703125 702464 -11 2 7 -
3
> 78125000 78121827 3 -13 10 -
2

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> Gene,
>
> Am I missing anything with epimericity < 0.5 and max. weighted error
> < 0.006 in the list below?

In tuning theory, you might describe choice of base for logarithms as
being a choice of unit, analogous to choosing feet or meters in
physics. From that point of view, both relative error and epimericity,
being ratios of logs, are dimensionless constants, and I like the idea
of using them to define lists of commas.

I went way, way, way past the point of totally demolished returns, and
the list of 45 commas below should be complete.

[200/189, 135/128, 256/243, 360/343, 392/375, 256/245, 28/27,
648/625,405/392, 1029/1000, 250/243, 36/35, 525/512, 128/125,
49/48, 50/49, 3125/3072, 686/675, 64/63, 875/864, 81/80, 3125/3087,
2430/2401, 2048/2025, 245/243, 126/125, 4000/3969, 1728/1715,
1029/1024, 15625/15552, 225/224, 19683/19600, 16875/16807,
10976/10935, 3136/3125, 5120/5103, 6144/6125, 65625/65536,
32805/32768, 703125/702464, 420175/419904, 2401/2400, 4375/4374,
250047/250000, 78125000/78121827]

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

> 250047/250000

Obviously I did something wrong because this one didn't even show up.
My apologies to Herman, and I'll try to correct the graphs and list
as soon as possible.

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > 250047/250000
>
> Obviously I did something wrong because this one didn't even show
up.

It's a nice comma, too. Erlich's comma, in honor of your not
discovering it. :)

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> > 250047/250000
>
> Obviously I did something wrong because this one didn't even show
up.
> My apologies to Herman, and I'll try to correct the graphs and list
> as soon as possible.

The graphs are now corrected, so please download them again:

/tuning/files/Erlich/herman2r.gif
/tuning/files/Erlich/herman2s.gif
/tuning/files/Erlich/herman2m.gif

Next I'll produce 11-limit comma graphs, equal temperament graphs,
and what Gene and I are currently discussing on tuning-math, 7-limit
linear temperament graphs . . .

>The graphs are now corrected, so please download them again:
>
>/tuning/files/Erlich/herman2r.gif
>/tuning/files/Erlich/herman2s.gif

What's the difference between these graphs?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >The graphs are now corrected, so please download them again:
> >
>
>/tuning/files/Erlich/herman2r.gif
>
>/tuning/files/Erlich/herman2s.gif
>
> What's the difference between these graphs?

The angle at which the ratios are printed. If you can't read an
overlapping pair of ratios on one of them, you might be able to read
them on the other one.