Seven limit comma names from pairs of temperament names

Given two rank-two seven-limit temperaments, either the kernels
intersect trivially, or the intersection uniquely defines a comma. In
the latter case, we can use the temperament names to cook up a comma
name. We want the comma to be characteristic of the temperaments,
which means the error of the temperament should be small, close to
comma error; of course also we want the temperaments to have
reasonable badness figures. Here are my name proposals for some commas
based on this notion, along with the temperament names if someone
wants to complain about those also.

What do people think?

1029/1000: keemun, gawel: keega
686/675 sensi, gawel: senga
875/864 superkleismic, magic: supermagic
3125/3087 bohpier, garibaldi: gariboh
2430/2401: orwell, nusecond: nuwell
245/243: rodan, octacot: octarod
4000/3969: garibaldi, octacot: octagari
19683/19600: harry, catakleismic: cataharry
16875/16807: miracle, kwai: mirkwai
10976/10935: parakleismic, hemififths: parahemfi
3136/3125: parakleismic, hemiwuerschmidt: parahemwuer
5120/5103: hemififths, amity: hemifamity
6144/6125: hemiwuerschmidt, amity: hewuermity
65625/65536: tertiaseptal, pontiac: tertiapont
703125/702464: tertiaseptal, enneadecal: tertiaendec

keemun <<6 5 3 -6 -12 -7|| 15&19
gawel <<3 12 11 12 9 -8|| 19&36
sensi <<7 9 13 -2 1 5|| 19&27
superkleismic <<9 10 -3 -5 -30 -35|| 26&41
magic <<5 1 12 -10 5 25|| 19&22
bohpier <<13 19 23 0 0 0|| 41&49
garibaldi <<1 -8 -14 -15 -25 -10|| 41&53
orwell <<7 -3 8 -21 -7 27|| 22&31
nusecond <<11 13 17 -5 -4 3|| 31&70
rodan <<3 17 -1 20 -10 -50|| 41&46
octacot <<8 18 11 10 -5 -25|| 41&68
harry <<12 34 20 26 -2 -49|| 58&72
catakleismic <<6 5 22 -6 18 37|| 53&72
miracle <<6 -7 -2 -25 -20 15|| 31&41
kwai <<1 33 27 50 40 -30|| 41&152
parakleimsic <<13 14 35 -8 19 42|| 99&118
hemififths <<2 25 13 35 15 -40|| 99&140
hemiwuerschmidt <<16 2 5 -34 -37 6|| 99&130
amity <<5 13 -17 9 -41 -76|| 99&152
tertiaseptal <<22 -5 3 -59 -57 21|| 140&171
pontiac <<1 -8 39 -15 59 113|| 118&171
enneadecal <<19 19 57 -14 37 79|| 152&171

Looks good to me, though probably some will hate it.

-Carl

At 12:49 PM 10/15/2005, you wrote:
>Given two rank-two seven-limit temperaments, either the kernels
>intersect trivially, or the intersection uniquely defines a comma. In
>the latter case, we can use the temperament names to cook up a comma
>name. We want the comma to be characteristic of the temperaments,
>which means the error of the temperament should be small, close to
>comma error; of course also we want the temperaments to have
>reasonable badness figures. Here are my name proposals for some commas
>based on this notion, along with the temperament names if someone
>wants to complain about those also.
>
>What do people think?
>
>1029/1000: keemun, gawel: keega
>686/675 sensi, gawel: senga
>875/864 superkleismic, magic: supermagic
>3125/3087 bohpier, garibaldi: gariboh
>2430/2401: orwell, nusecond: nuwell
>245/243: rodan, octacot: octarod
>4000/3969: garibaldi, octacot: octagari
>19683/19600: harry, catakleismic: cataharry
>16875/16807: miracle, kwai: mirkwai
>10976/10935: parakleismic, hemififths: parahemfi
>3136/3125: parakleismic, hemiwuerschmidt: parahemwuer
>5120/5103: hemififths, amity: hemifamity
>6144/6125: hemiwuerschmidt, amity: hewuermity
>65625/65536: tertiaseptal, pontiac: tertiapont
>703125/702464: tertiaseptal, enneadecal: tertiaendec
>
>
>keemun <<6 5 3 -6 -12 -7|| 15&19
>gawel <<3 12 11 12 9 -8|| 19&36
>sensi <<7 9 13 -2 1 5|| 19&27
>superkleismic <<9 10 -3 -5 -30 -35|| 26&41
>magic <<5 1 12 -10 5 25|| 19&22
>bohpier <<13 19 23 0 0 0|| 41&49
>garibaldi <<1 -8 -14 -15 -25 -10|| 41&53
>orwell <<7 -3 8 -21 -7 27|| 22&31
>nusecond <<11 13 17 -5 -4 3|| 31&70
>rodan <<3 17 -1 20 -10 -50|| 41&46
>octacot <<8 18 11 10 -5 -25|| 41&68
>harry <<12 34 20 26 -2 -49|| 58&72
>catakleismic <<6 5 22 -6 18 37|| 53&72
>miracle <<6 -7 -2 -25 -20 15|| 31&41
>kwai <<1 33 27 50 40 -30|| 41&152
>parakleimsic <<13 14 35 -8 19 42|| 99&118
>hemififths <<2 25 13 35 15 -40|| 99&140
>hemiwuerschmidt <<16 2 5 -34 -37 6|| 99&130
>amity <<5 13 -17 9 -41 -76|| 99&152
>tertiaseptal <<22 -5 3 -59 -57 21|| 140&171
>pontiac <<1 -8 39 -15 59 113|| 118&171
>enneadecal <<19 19 57 -14 37 79|| 152&171
>
>
>
>
>
>
>
>
>Yahoo! Groups Links
>
>
>
>

I'll have to consider this later, but I repeat again my plea for
Gawel (the name of the messenger) to be ditched and replaced with
Liese (the name of the theorist); you'll find the latter name, not
the former, in the paper I snail-mailed around.

Meanwhile, I've been thinking that single syllables can stand for
commas, or . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Given two rank-two seven-limit temperaments, either the kernels
> intersect trivially, or the intersection uniquely defines a comma.
In
> the latter case, we can use the temperament names to cook up a comma
> name. We want the comma to be characteristic of the temperaments,
> which means the error of the temperament should be small, close to
> comma error; of course also we want the temperaments to have
> reasonable badness figures. Here are my name proposals for some
commas
> based on this notion, along with the temperament names if someone
> wants to complain about those also.
>
> What do people think?
>
> 1029/1000: keemun, gawel: keega
> 686/675 sensi, gawel: senga
> 875/864 superkleismic, magic: supermagic
> 3125/3087 bohpier, garibaldi: gariboh
> 2430/2401: orwell, nusecond: nuwell
> 245/243: rodan, octacot: octarod
> 4000/3969: garibaldi, octacot: octagari
> 19683/19600: harry, catakleismic: cataharry
> 16875/16807: miracle, kwai: mirkwai
> 10976/10935: parakleismic, hemififths: parahemfi
> 3136/3125: parakleismic, hemiwuerschmidt: parahemwuer
> 5120/5103: hemififths, amity: hemifamity
> 6144/6125: hemiwuerschmidt, amity: hewuermity
> 65625/65536: tertiaseptal, pontiac: tertiapont
> 703125/702464: tertiaseptal, enneadecal: tertiaendec
>
>
> keemun <<6 5 3 -6 -12 -7|| 15&19
> gawel <<3 12 11 12 9 -8|| 19&36
> sensi <<7 9 13 -2 1 5|| 19&27
> superkleismic <<9 10 -3 -5 -30 -35|| 26&41
> magic <<5 1 12 -10 5 25|| 19&22
> bohpier <<13 19 23 0 0 0|| 41&49
> garibaldi <<1 -8 -14 -15 -25 -10|| 41&53
> orwell <<7 -3 8 -21 -7 27|| 22&31
> nusecond <<11 13 17 -5 -4 3|| 31&70
> rodan <<3 17 -1 20 -10 -50|| 41&46
> octacot <<8 18 11 10 -5 -25|| 41&68
> harry <<12 34 20 26 -2 -49|| 58&72
> catakleismic <<6 5 22 -6 18 37|| 53&72
> miracle <<6 -7 -2 -25 -20 15|| 31&41
> kwai <<1 33 27 50 40 -30|| 41&152
> parakleimsic <<13 14 35 -8 19 42|| 99&118
> hemififths <<2 25 13 35 15 -40|| 99&140
> hemiwuerschmidt <<16 2 5 -34 -37 6|| 99&130
> amity <<5 13 -17 9 -41 -76|| 99&152
> tertiaseptal <<22 -5 3 -59 -57 21|| 140&171
> pontiac <<1 -8 39 -15 59 113|| 118&171
> enneadecal <<19 19 57 -14 37 79|| 152&171
>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Given two rank-two seven-limit temperaments, either the kernels
> intersect trivially, or the intersection uniquely defines a comma.

I'm just dying to know how you combine two kernels to get a single
comma. I know that in Octave, if I take two 7 limit temperaments,
I get an orthonormalized kernel with two values, but I have never
gotten the result to make any sense in this case.

Are you taking the kernel of each seven-limit temperament, based on
period + generator? Then what are you applying to find the intersection?

And of course, my other question: How to get two commas from two 7-
limit temperaments, short of TM-reduction, Hermite normal form etc.

If you could even give me clues - that would be great. (Rome wasn't
built in a day)

Thanks

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> I'm just dying to know how you combine two kernels to get a single
> comma.

Even Octave should be able to manage this. If you have two wedgies
u and v such that

u[1]*v[6]-u[2]*v[5]+u[3]*v[4]+u[6]*v[1]-u[5]*v[2]+u[4]*v[3] = 0

then the temperaments share a comma. If a1, a2 are a pair of vals for
the first temperament (period and generator, two et vals, etc.) and
b1, b2 for the second, take the kernel of [a1, a2, b1, b2]. Octave may
goof this up, but divide through to get rational values, and then uswe
those to clear denominators and get integer values, and hence a monzo.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > I'm just dying to know how you combine two kernels to get a
single
> > comma.
>
> Even Octave should be able to manage this. If you have two wedgies
> u and v such that
>
> u[1]*v[6]-u[2]*v[5]+u[3]*v[4]+u[6]*v[1]-u[5]*v[2]+u[4]*v[3] = 0
>
> then the temperaments share a comma. If a1, a2 are a pair of vals
for
> the first temperament (period and generator, two et vals, etc.) and
> b1, b2 for the second, take the kernel of [a1, a2, b1, b2]. Octave
may
> goof this up, but divide through to get rational values, and then
uswe
> those to clear denominators and get integer values, and hence a
monzo.
>
Yes, this works. I get (-4, 4, -1) from two pairs of meantone
temperaments. Is there any way to get it to generate two commas from
two vals?

Thanks

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> Yes, this works. I get (-4, 4, -1) from two pairs of meantone
> temperaments. Is there any way to get it to generate two commas from
> two vals?

I don't have Octave, so I don't know what I could get it to do. From two
vals, you could get the wedgie, and from the wedgie, you can get
commas; that is one approach.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > Yes, this works. I get (-4, 4, -1) from two pairs of meantone
> > temperaments. Is there any way to get it to generate two commas
from
> > two vals?
>
> I don't have Octave, so I don't know what I could get it to do. From
two
> vals, you could get the wedgie, and from the wedgie, you can get
> commas; that is one approach.
>
Yes that is a good method. I will see if Octave can do wedgies.
Any guesses as to what the function name would be? Maybe wedge(x),
exterior(x)? Sorry, I know you don't have Octave...

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> > I don't have Octave, so I don't know what I could get it to do.
From
> two
> > vals, you could get the wedgie, and from the wedgie, you can get
> > commas; that is one approach.

I know I have asked many times how to get commas from a wedge product,
so I am going to test Yahoo's search capabilities and find it again. I
think I will also load Python up so I am using the same software as
other people on this discussion group. I feel I have mastered the 5-
limit, so I am working methodically to understand the 7-limit, etc.
The double comma thing as two planes in 3-D space is quite interesting.
If holograms were more prevalent I would love to see a zoom diagram
in 3-D space with all the 7-limit temperaments mapped in it. Would it
be 3-5-7 as x-y-z, or perhaps something more complex? Perhaps 3/2, 5/4,
6/5, 7/6, 7/5, 8/7 axes in a 3-D arrangement?

> Yes that is a good method. I will see if Octave can do wedgies.
> Any guesses as to what the function name would be? Maybe wedge(x),
> exterior(x)? Sorry, I know you don't have Octave...

Darn. I don't think Octave does wedge products.
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> If holograms were more prevalent I would love to see a zoom diagram
> in 3-D space with all the 7-limit temperaments mapped in it. Would it
> be 3-5-7 as x-y-z, or perhaps something more complex? Perhaps 3/2, 5/4,
> 6/5, 7/6, 7/5, 8/7 axes in a 3-D arrangement?

Unfortunately, in general temperaments are represented by points on
something called a Grassmann variety. In the case of rank two 7-limit
temperaments, this is an algebraic variety (which is to say, an
algebraically defined geometric object; curve, surface, etc.) in 5
dimensional projective space, which does not make for easy visualization.

If x1 x2 x3 x4 x5 x6 are the homogenous coordiates of this 5D
projective space, then the variety is defined by the single condition
that x1*x6 - x2*x5 + x3*x4 = 0. This a 4D hypersurface (variety
defined by a single equation) in 5D projective space. In higher prime
limits the Grassmann varieties involve more such equations.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:

> If holograms were more prevalent I would love to see a zoom diagram
> in 3-D space with all the 7-limit temperaments mapped in it. Would it
> be 3-5-7 as x-y-z, or perhaps something more complex? Perhaps 3/2,
5/4,
> 6/5, 7/6, 7/5, 8/7 axes in a 3-D arrangement?

That (the latter) would be the symmetrical or Hahn approach -- the
six "axes" here each connect two opposite corners (through the center)
of a cuboctahedron (which is of course our familiar representation of
the Partch 7-limit Tonality Diamond) . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
>
> > If holograms were more prevalent I would love to see a zoom diagram
> > in 3-D space with all the 7-limit temperaments mapped in it. Would
it
> > be 3-5-7 as x-y-z, or perhaps something more complex? Perhaps 3/2,
5/4,
> > 6/5, 7/6, 7/5, 8/7 axes in a 3-D arrangement?
>
> Unfortunately, in general temperaments are represented by points on
> something called a Grassmann variety.

I don't think he's asking for them all to be represented as points,
Gene. I don't think you're following his analogy with the "zoom"
diagrams, since there, the rank 2 temperaments are represented as
lines, not points (the rank 1 temperaments are represented as points).

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

> > Unfortunately, in general temperaments are represented by points on
> > something called a Grassmann variety.
>
> I don't think he's asking for them all to be represented as points,
> Gene. I don't think you're following his analogy with the "zoom"
> diagrams, since there, the rank 2 temperaments are represented as
> lines, not points (the rank 1 temperaments are represented as points).

The rank 1 temperaments start off as points in 3-space, <a2 a3 a5|;
considered projectively, they are points in 2D projective space, which
you can coordinatize by dividing through, getting <1 a3/a2 a5/a2|. The
origin of the coordinates can be moved to <1 log2(3) log2(5)| and then
you have the points of your zoom space. The points of a projective
plane are dual to the lines of another projective plane, leading to
the dual zoom point of view.

If we do this for 7-limit temperaments, the projective points
corresponding to a projective val (provo) are now <1 a3/a2 a5/a2
a7/a2|. These determine points in 3D projective space, and lines
between these points correspond to rank-two temperaments. As before,
we can shift coordinates so the origin is <1 log2(3) log2(5) log2(7)|.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul_hjelmstad@a...> wrote:
> >
> > > I'm just dying to know how you combine two kernels to get a
> single
> > > comma.
> >
> > Even Octave should be able to manage this. If you have two wedgies
> > u and v such that
> >
> > u[1]*v[6]-u[2]*v[5]+u[3]*v[4]+u[6]*v[1]-u[5]*v[2]+u[4]*v[3] = 0
> >
> > then the temperaments share a comma. If a1, a2 are a pair of vals
> for
> > the first temperament (period and generator, two et vals, etc.)
and
> > b1, b2 for the second, take the kernel of [a1, a2, b1, b2].
Octave
> may
> > goof this up, but divide through to get rational values, and then
> uswe
> > those to clear denominators and get integer values, and hence a
> monzo.

Is there an easy way to get exactly one comma from two 7-limit
temperaments? (As opposed to two pairs of 7-limit temperaments shown
here). Thanks.

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@a...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@a...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <gwsmith@s...>
> > wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul_hjelmstad@a...> wrote:
> > >
> > > > I'm just dying to know how you combine two kernels to get a
> > single
> > > > comma.
> > >
> > > Even Octave should be able to manage this. If you have two
wedgies
> > > u and v such that
> > >
> > > u[1]*v[6]-u[2]*v[5]+u[3]*v[4]+u[6]*v[1]-u[5]*v[2]+u[4]*v[3] = 0
> > >
> > > then the temperaments share a comma. If a1, a2 are a pair of
vals
> > for
> > > the first temperament (period and generator, two et vals, etc.)
> and
> > > b1, b2 for the second, take the kernel of [a1, a2, b1, b2].
> Octave
> > may
> > > goof this up, but divide through to get rational values, and
then
> > uswe
> > > those to clear denominators and get integer values, and hence a
> > monzo.
>
> Is there an easy way to get exactly one comma from two 7-limit
> temperaments? (As opposed to two pairs of 7-limit temperaments
shown
> here). Thanks.

Sorry! Read the post incorrectly. Never mind. (Two rank two
temperaments is different from two simple ETs)
>