Rich Holmes temperaments

In this article

http://web.syr.edu/~rsholmes/music/xen/scale_mos.html

Rich Holmes sets himself the task of finding good 7-limit linear
temperaments with low Graham complexity, and for some reason doesn't
find any. One problem is that he didn't consider anything other than
strictly linear temperaments, so temperaments like pajara were out.

He does, however, find a temperament unfamiliar to me. His least
complex is a temperament with a generator of 919 cents, with major
thirds flat by 29 cents. This seems to be describing the <<6 3 5 -9 -9
3|| temperament, which is indeed pretty atrocious, as he notes. It has
a generator like orwell's evil twin, and like orwell, a 9-note MOS.
What it doesn't have is good 7-limit harmony. Another temperament he
mentions has a generator of 881 cents and an 11-note MOS. This is
keemun, tuned to a major sixth which gives pretty sharp fifths and thirds.

The Graham complexity of orwell, incidentially, is 11, so the
Orwell[13] MOS has two otonal and two utonal tetrads, which seems to
satisfy his requirements. Holmes doesn't much like Superpyth[12],
which has only one otonal and one utonal tetrad, but Orwell[13] seems
to fill his minimum requirements, as well as being in far better tune
than the tunings he considers. Magic[13] doesn't quite fit the bill,
alas, but perhaps the most obvious choice of all is Meantone[12]; I
have no idea why that wasn't considered.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> In this article
>
> http://web.syr.edu/~rsholmes/music/xen/scale_mos.html
>
> Rich Holmes sets himself the task of finding good 7-limit linear
> temperaments with low Graham complexity, and for some reason doesn't
> find any. One problem is that he didn't consider anything other than
> strictly linear temperaments, so temperaments like pajara were out.

Actually, on another page he does get to "nonlinear" temperaments and
pajara comes into its own.

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> Orwell[13] seems
> to fill his minimum requirements, as well as being in far better tune
> than the tunings he considers. Magic[13] doesn't quite fit the bill,
> alas, but perhaps the most obvious choice of all is Meantone[12]; I
> have no idea why that wasn't considered.

Where would I go to find definitions of these?

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Where would I go to find definitions of these?

Meantone[12] is 12 contiguous notes of septimal meantone:

http://en.wikipedia.org/wiki/Septimal_meantone_temperament

For example, Eb to G#.

Orwell is discussed here:

http://66.98.148.43/~xenharmo/orwell.html

Since good ets for it are 53 and 84, you can see the tuning is not
bad, especially for the 5-limit part.

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> > Where would I go to find definitions of these?
>
> Meantone[12] is 12 contiguous notes of septimal meantone:
>
> http://en.wikipedia.org/wiki/Septimal_meantone_temperament

OK, so 1200 cent period and a generator of 696.88 cents? Then you
need 1, 4, 10 generators to reach 3/2, 5/4, 7/4 (e.g. 7/4 = 969 cents
~= 10*696.88 mod 1200). (Sorry about not using the more compact
notation you are used to, but I'm not.) Therefore in Meantone[12] you
have just two otonal tetrads, and two utonal. It's been quite a while
since I did the stuff I wrote up on the page you mention and I'm not
sure why Meantone[12] seems to have been overlooked, but in any case
the smallish number of tetrads would've been regarded as a
shortcoming.

> Orwell is discussed here:
>
> http://66.98.148.43/~xenharmo/orwell.html

So 1200 cent period and about 271 cents generator? Here you need 7,
-3, 8 generators for 3/2, 5/4, 7/4. So you need 1+8-(-3) = 12 notes
for your first otonal tetrad and Orwell[13] will contain two of them.
Again, I'd been hoping for something better at the time. (I haven't
been thinking much about microtonal music lately, but when I was, I
was moving away from 7-limit scales and getting more interested in
things like scales based on factors 2, 3, and 11.)

- Rich Holmes

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> In this article
>
> http://web.syr.edu/~rsholmes/music/xen/scale_mos.html
>
> Rich Holmes sets himself the task of finding good 7-limit linear
> temperaments with low Graham complexity, and for some reason doesn't
> find any. One problem is that he didn't consider anything other than
> strictly linear temperaments, so temperaments like pajara were out.

I tried very hard to inform Rich of this limitation, but he seems not
to have taken my pleas into account so far. He mentions me in
connection with 22-equal but seems not to have looked at what my
paper on 22-equal is really about -- the decatonic scales.

> The Graham complexity of orwell, incidentially, is 11, so the
> Orwell[13] MOS has two otonal and two utonal tetrads, which seems to
> satisfy his requirements. Holmes doesn't much like Superpyth[12],
> which has only one otonal and one utonal tetrad, but Orwell[13]
seems
> to fill his minimum requirements, as well as being in far better
tune
> than the tunings he considers. Magic[13] doesn't quite fit the bill,
> alas, but perhaps the most obvious choice of all is Meantone[12]; I
> have no idea why that wasn't considered.

I thought Meantone[12] got highest props on his page. Wait -- it
looks like he re-wrote it. What happened?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
> >
> > In this article
> >
> > http://web.syr.edu/~rsholmes/music/xen/scale_mos.html
> >
> > Rich Holmes sets himself the task of finding good 7-limit linear
> > temperaments with low Graham complexity, and for some reason doesn't
> > find any. One problem is that he didn't consider anything other than
> > strictly linear temperaments, so temperaments like pajara were out.
>
> Actually, on another page he does get to "nonlinear" temperaments and
> pajara comes into its own.

But pajara *is* rank 2!

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> So 1200 cent period and about 271 cents generator? Here you need 7,
> -3, 8 generators for 3/2, 5/4, 7/4. So you need 1+8-(-3) = 12 notes
> for your first otonal tetrad and Orwell[13] will contain two of them.
> Again, I'd been hoping for something better at the time. (I haven't
> been thinking much about microtonal music lately, but when I was, I
> was moving away from 7-limit scales and getting more interested in
> things like scales based on factors 2, 3, and 11.)

Actually, Orwell[13] does pretty well for the 11-limit, if you find
the tuning acceptable. You can extend orwell to the 11-limit by
tempering out 99/98, in which case 11/8 is two generators. In 53-et,
the 11 is 7.9 cents flat, and if that is tolerable, you are in
business. The worst 11-limit interval is 11/7, which in 53-et is 12.7
cents flat.

"wallyesterpaulrus" <perlich@aya.yale.edu> writes:

> I tried very hard to inform Rich of this limitation, but he seems not
> to have taken my pleas into account so far. He mentions me in
> connection with 22-equal but seems not to have looked at what my
> paper on 22-equal is really about -- the decatonic scales.

Not so. The second page Gene referred to is almost entirely about the
decatonic scales.

> I thought Meantone[12] got highest props on his page. Wait -- it
> looks like he re-wrote it. What happened?

Yes... I've rewritten it several times (though not very recently.)

> But pajara *is* rank 2!

I probably chose a bad title for the first of those pages. There I
was limiting myself (initally through failure to recognize it as an
unnecessary assumption) to octave-period scales.

Is "linear" still the approved term for octave-period rank 2 scales?

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Is "linear" still the approved term for octave-period rank 2 scales?

I suppose. Paul likes it for historical reasons, and I like it because
the distinction is useful. I'd suggest that "cylindrical" would be a
much better term than "nonlinear" for rank two temperaments which are
not linear, however. Calling them "nonlinear" suggests they are going
to break out into chaotic dynamics. It's a cylindrical shape of
modulatory space:

http://en.wikipedia.org/wiki/Modulatory_space

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Not so. The second page Gene referred to is almost entirely about the
> decatonic scales.

Here's a listing in order of increasing rms error for various Graham
complexity cutoff figures. I used rms error because TOP, Kees or NOT
errors are likely to give two different temperaments the same value.
It looks to me that augene/tripletone is another one you might want to
look at. Also, if you go up to complexity 10, myna and meantone become
very attractive, with a much lower error than anything of less
complexity. Going up to complexity 11 adds orwell to the list, with a
lower error yet. Complexity 12 sees valentine come into the picture,
and complexity 13 sees miracle, which is an absolute killer; many
people would not *want* a temperament more accurate than this, and as
an extra trick, it does the 11-limit. The only problem is that for the
size of scales you wanted, you can't get both complete tetrads and a MOS.

I'd add some discussion of all of these; not just pajara but augene,
myna, meantone, orwell, valentine, and miracle. Otherwise it seems to
me a false picture has been presented of the problem.

Graham complexity < 8

Pajara [2, -4, -4, -11, -12, 2] 6 10.903177
Negri [4, -3, 2, -14, -8, 13] 7 12.188571
Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
Blackwood [0, 5, 0, 8, 0, -14] 5 15.815353
Augie [3, 0, 6, -7, 1, 14] 6 16.598678
Diminished [4, 4, 4, -3, -5, -2] 4 19.136992
Dominant seventh [1, 4, -2, 4, -6, -16] 6 20.163282
Decimal [4, 2, 2, -6, -8, -1] 4 23.945252

Graham complexity < 10

Augene [3, 0, -6, -7, -18, -14] 9 8.100679
Doublewide [8, 6, 6, -9, -13, -3] 8 10.132266
Pajara [2, -4, -4, -11, -12, 2] 6 10.903177
Injera [2, 8, 8, 8, 7, -4] 8 11.218941
Negri [4, -3, 2, -14, -8, 13] 7 12.188571
Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
Godzilla [2, 8, 1, 8, -4, -20] 8 12.690079
Blackwood [0, 5, 0, 8, 0, -14] 5 15.815353
Augie [3, 0, 6, -7, 1, 14] 6 16.598678
Mavila [1, -3, 5, -7, 5, 20] 8 18.584500

Graham complexity < 12

Orwell [7, -3, 8, -21, -7, 27] 11 2.589238
Myna [10, 9, 7, -9, -17, -9] 10 3.320167
Meantone [1, 4, 10, 4, 13, 12] 10 3.665035
Beatles [2, -9, -4, -19, -12, 16] 11 6.245316
Superpyth [1, 9, -2, 12, -6, -30] 11 6.410458
Porcupine [3, 5, -6, 1, -18, -28] 11 6.808962
Augene [3, 0, -6, -7, -18, -14] 9 8.100679
Doublewide [8, 6, 6, -9, -13, -3] 8 10.132266
Pajara [2, -4, -4, -11, -12, 2] 6 10.903177
Injera [2, 8, 8, 8, 7, -4] 8 11.218941

Graham complexity < 14

Miracle [6, -7, -2, -25, -20, 15] 13 1.637405
Orwell [7, -3, 8, -21, -7, 27] 11 2.589238
Valentine [9, 5, -3, -13, -30, -21] 12 3.065962
Myna [10, 9, 7, -9, -17, -9] 10 3.320167
Mothra [3, 12, -1, 12, -10, -36] 13 3.579262
Meantone [1, 4, 10, 4, 13, 12] 10 3.665035
Superkleismic [9, 10, -3, -5, -30, -35] 13 4.052704
Magic [5, 1, 12, -10, 5, 25] 12 4.139051
Sensi [7, 9, 13, -2, 1, 5] 13 5.052931
Beatles [2, -9, -4, -19, -12, 16] 11 6.245316

>Kemun [6, 5, 3, -6, -12, -7] 6 12.273810

Another list of numbers from you, Gene, without column
headings. :( I assume 12.27... is the rms error you
mentioned. I can't say I know what the other numbers
are.

Now let's try to identify this temperament. "Kemun"
doesn't occur on your page

http://66.98.148.43/~xenharmo/sevnames.htm

or in Herman's doc file. But "Keemun" does. It's a
kind of tea and perhaps a region in China. What's the
anecdote behind this name?

Ideally your page above would not be limited to the
7-limit, and ideally it would be in alpha order by
temperament name.

Herman -- what are the "catalog number"s in your Word
file?

-Carl

At 12:44 PM 1/4/2006, you wrote:
>>Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
>
>Another list of numbers from you, Gene, without column
>headings. :( I assume 12.27... is the rms error you
>mentioned. I can't say I know what the other numbers
>are.

I assume the thing between [] is the map, but why the
period and generator are apparently given on the same
line....

-C.

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> I'd add some discussion of all of these; not just pajara but augene,
> myna, meantone, orwell, valentine, and miracle. Otherwise it seems to
> me a false picture has been presented of the problem.

I never intended it to be a definitive and complete treatise. Just
some notes on some of the lines of thought I was pursuing.

Anyway, I don't know what most of the scales you list are; what the
numbers you show mean; or what "Graham complexity cutoff" is.

- Rich Holmes

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
>
> Another list of numbers from you, Gene, without column
> headings. :( I assume 12.27... is the rms error you
> mentioned. I can't say I know what the other numbers
> are.

Name of temperament (sometimes spelled correctly), then wedgie, then
Graham complexity, then rms error.

> or in Herman's doc file. But "Keemun" does. It's a
> kind of tea and perhaps a region in China. What's the
> anecdote behind this name?

It's Paul's name, so he'd better tell it.

> Ideally your page above would not be limited to the
> 7-limit, and ideally it would be in alpha order by
> temperament name.

Well, that's a thought.

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Anyway, I don't know what most of the scales you list are; what the
> numbers you show mean; or what "Graham complexity cutoff" is.

7-limit Graham complexity is what you want for your purpose; If you
take a generator chain scale of n notes, then the number of tetrads of
each kind will be n - Graham. Hence septimal meantone, which has a
Graham complexity of 10 (extending from 0 to 10 generators to get 3,
5, and 7) will have 2=12-10 tetrads of each kind for Meantone[12]. The
other numbers are the rms error of the 7-limit consonancs in cents,
and the wedgie, which identifies the temperament.

>> >Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
>>
>> Another list of numbers from you, Gene, without column
>> headings. :( I assume 12.27... is the rms error you
>> mentioned. I can't say I know what the other numbers
>> are.
>
>Name of temperament (sometimes spelled correctly), then wedgie,
>then Graham complexity, then rms error.

Aha. What threw me is, I thought wedgies supposed to be
in braket notation.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Aha. What threw me is, I thought wedgies supposed to be
> in braket notation.

They are, but after getting Maple to print the file I forgot to edit it.

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> "wallyesterpaulrus" <perlich@a...> writes:
>
> > I tried very hard to inform Rich of this limitation, but he seems
not
> > to have taken my pleas into account so far. He mentions me in
> > connection with 22-equal but seems not to have looked at what my
> > paper on 22-equal is really about -- the decatonic scales.
>
> Not so. The second page Gene referred to is almost entirely about
the
> decatonic scales.

I missed that, since there was no link from the first page stating
that it was continued on the second page.

> > I thought Meantone[12] got highest props on his page. Wait -- it
> > looks like he re-wrote it. What happened?
>
> Yes... I've rewritten it several times (though not very recently.)
>
> > But pajara *is* rank 2!
>
> I probably chose a bad title for the first of those pages.

Well, if you stated that it was continued on that second page, it
wouldn't be so bad.

> There I
> was limiting myself (initally through failure to recognize it as an
> unnecessary assumption) to octave-period scales.

> Is "linear" still the approved term for octave-period rank 2 scales?

Yes.

If you have your pages in something like a final form at some point,
I'd like an opportunity to proofread them. Also, it might be worth at
least mentioning the approach I took in my paper, that focuses on the
*kernel* of each of these temperaments, gleaning both the complexity
and the error (or damage) from the kernel, and doing a search over
all possible kernels within some bounds on complexity and error (and
then finally converting those that pass into period-generator-mapping
specifications of the corresponding temperaments). This seems like
the a more mathematically sophisticated approach than a search
stepping by fractions of a cent across generator sizes (and #s of
periods per octave) for "good" temperaments or scales, which is what
some of us used to do . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> > Is "linear" still the approved term for octave-period rank 2 scales?
>
> I suppose. Paul likes it for historical reasons, and I like it because
> the distinction is useful. I'd suggest that "cylindrical" would be a
> much better term than "nonlinear" for rank two temperaments which are
> not linear, however.

George Secor agrees with you on this, Gene.

> Calling them "nonlinear" suggests they are going
> to break out into chaotic dynamics.

:) I agree, "nonlinear" isn't really getting the regular or uniform
nature of these systems across.

> It's a cylindrical shape of
> modulatory space:
>
> http://en.wikipedia.org/wiki/Modulatory_space

Why would you call this modulatory space when absolutely no modulation
is required or need be involved? That seems to muddy the issue.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...>
wrote:
>
> > Not so. The second page Gene referred to is almost entirely
about the
> > decatonic scales.
>
> Here's a listing in order of increasing rms error for various Graham
> complexity cutoff figures. I used rms error because TOP, Kees or NOT
> errors are likely to give two different temperaments the same value.

I think Graham is clamoring for us to use an rms (that is, L_2)
version of TOP, and possibly of Kees as well. Unlike the L_inf
versions, these should give unique values for a given temperament and
different values to different temperaments, yes? BTW, will rms-TOP
and rms-Kees always be stretched versions of one another?

> It looks to me that augene/tripletone is another one you might want
to
> look at. Also, if you go up to complexity 10, myna and meantone
become
> very attractive, with a much lower error than anything of less
> complexity. Going up to complexity 11 adds orwell to the list, with
a
> lower error yet. Complexity 12 sees valentine come into the picture,
> and complexity 13 sees miracle, which is an absolute killer; many
> people would not *want* a temperament more accurate than this, and
as
> an extra trick, it does the 11-limit. The only problem is that for
the
> size of scales you wanted, you can't get both complete tetrads and
a MOS.
>
> I'd add some discussion of all of these; not just pajara but augene,
> myna, meantone, orwell, valentine, and miracle. Otherwise it seems
to
> me a false picture has been presented of the problem.
>
> Graham complexity < 8
>
> Pajara [2, -4, -4, -11, -12, 2] 6 10.903177
> Negri [4, -3, 2, -14, -8, 13] 7 12.188571
> Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
> Blackwood [0, 5, 0, 8, 0, -14] 5 15.815353
> Augie [3, 0, 6, -7, 1, 14] 6 16.598678
> Diminished [4, 4, 4, -3, -5, -2] 4 19.136992
> Dominant seventh [1, 4, -2, 4, -6, -16] 6 20.163282
> Decimal [4, 2, 2, -6, -8, -1] 4 23.945252
>
> Graham complexity < 10
>
> Augene [3, 0, -6, -7, -18, -14] 9 8.100679
> Doublewide [8, 6, 6, -9, -13, -3] 8 10.132266
> Pajara [2, -4, -4, -11, -12, 2] 6 10.903177
> Injera [2, 8, 8, 8, 7, -4] 8 11.218941
> Negri [4, -3, 2, -14, -8, 13] 7 12.188571
> Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
> Godzilla [2, 8, 1, 8, -4, -20] 8 12.690079
> Blackwood [0, 5, 0, 8, 0, -14] 5 15.815353
> Augie [3, 0, 6, -7, 1, 14] 6 16.598678
> Mavila [1, -3, 5, -7, 5, 20] 8 18.584500
>
> Graham complexity < 12
>
> Orwell [7, -3, 8, -21, -7, 27] 11 2.589238
> Myna [10, 9, 7, -9, -17, -9] 10 3.320167
> Meantone [1, 4, 10, 4, 13, 12] 10 3.665035
> Beatles [2, -9, -4, -19, -12, 16] 11 6.245316
> Superpyth [1, 9, -2, 12, -6, -30] 11 6.410458
> Porcupine [3, 5, -6, 1, -18, -28] 11 6.808962
> Augene [3, 0, -6, -7, -18, -14] 9 8.100679
> Doublewide [8, 6, 6, -9, -13, -3] 8 10.132266
> Pajara [2, -4, -4, -11, -12, 2] 6 10.903177
> Injera [2, 8, 8, 8, 7, -4] 8 11.218941
>
> Graham complexity < 14
>
> Miracle [6, -7, -2, -25, -20, 15] 13 1.637405
> Orwell [7, -3, 8, -21, -7, 27] 11 2.589238
> Valentine [9, 5, -3, -13, -30, -21] 12 3.065962
> Myna [10, 9, 7, -9, -17, -9] 10 3.320167
> Mothra [3, 12, -1, 12, -10, -36] 13 3.579262
> Meantone [1, 4, 10, 4, 13, 12] 10 3.665035
> Superkleismic [9, 10, -3, -5, -30, -35] 13 4.052704
> Magic [5, 1, 12, -10, 5, 25] 12 4.139051
> Sensi [7, 9, 13, -2, 1, 5] 13 5.052931
> Beatles [2, -9, -4, -19, -12, 16] 11 6.245316
>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
>
> Another list of numbers from you, Gene, without column
> headings. :( I assume 12.27... is the rms error you
> mentioned. I can't say I know what the other numbers
> are.
>
> Now let's try to identify this temperament. "Kemun"
> doesn't occur on your page
>
> http://66.98.148.43/~xenharmo/sevnames.htm
>
> or in Herman's doc file. But "Keemun" does. It's a
> kind of tea and perhaps a region in China. What's the
> anecdote behind this name?

It was called "Keenan" after Dave, whose page

http://dkeenan.com/Music/ChainOfMinor3rds.htm

implies this temperament. Dave didn't like the name, but I didn't
want to have to redo my horagrams which were alphabetical, so Herman
suggested "Keemun" so as to preserve the alphabetical ordering.

> Ideally your page above would not be limited to the
> 7-limit,

A lot of Gene's names would then have more than one entry.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> At 12:44 PM 1/4/2006, you wrote:
> >>Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
> >
> >Another list of numbers from you, Gene, without column
> >headings. :( I assume 12.27... is the rms error you
> >mentioned. I can't say I know what the other numbers
> >are.
>
> I assume the thing between [] is the map, but why the
> period and generator are apparently given on the same
> line....

They are? I don't see that. Where do you see period and generator?

Oh, the thing between [] is the wedgie, of course!

>> > Not so. The second page Gene referred to is almost entirely
>> > about the decatonic scales.
>>
>> Here's a listing in order of increasing rms error for various Graham
>> complexity cutoff figures. I used rms error because TOP, Kees or NOT
>> errors are likely to give two different temperaments the same value.
>
>I think Graham is clamoring for us to use an rms (that is, L_2)
>version of TOP, and possibly of Kees as well. Unlike the L_inf
>versions, these should give unique values for a given temperament and
>different values to different temperaments, yes? BTW, will rms-TOP
>and rms-Kees always be stretched versions of one another?

Hiya Paul - is there a name for this L_ notation that I could
look up somewhere?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >> > Not so. The second page Gene referred to is almost entirely
> >> > about the decatonic scales.
> >>
> >> Here's a listing in order of increasing rms error for various
Graham
> >> complexity cutoff figures. I used rms error because TOP, Kees or
NOT
> >> errors are likely to give two different temperaments the same
value.
> >
> >I think Graham is clamoring for us to use an rms (that is, L_2)
> >version of TOP, and possibly of Kees as well. Unlike the L_inf
> >versions, these should give unique values for a given temperament
and
> >different values to different temperaments, yes? BTW, will rms-TOP
> >and rms-Kees always be stretched versions of one another?
>
> Hiya Paul - is there a name for this L_ notation that I could
> look up somewhere?

It's similar to that in the mathematical definitions here:

http://mathworld.wolfram.com/L1-Norm.html
http://mathworld.wolfram.com/L2-Norm.html
http://mathworld.wolfram.com/L-Infinity-Norm.html

except that we don't assume that the elements in the "vector" of
errors have to be linearly independent. And we're specifically
talking about the tunings that *minimize* the given "norm" on
the "vector" of errors.

>> >Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
>>
>> Another list of numbers from you, Gene, without column
>> headings. :( I assume 12.27... is the rms error you
>> mentioned. I can't say I know what the other numbers
>> are.
>>
>> Now let's try to identify this temperament. "Kemun"
>> doesn't occur on your page
>>
>> http://66.98.148.43/~xenharmo/sevnames.htm
>>
>> or in Herman's doc file. But "Keemun" does. It's a
>> kind of tea and perhaps a region in China. What's the
>> anecdote behind this name?
>
>It was called "Keenan" after Dave, whose page
>
>http://dkeenan.com/Music/ChainOfMinor3rds.htm
>
>implies this temperament. Dave didn't like the name, but I didn't
>want to have to redo my horagrams which were alphabetical, so Herman
>suggested "Keemun" so as to preserve the alphabetical ordering.

Thanks.

>> Ideally your page above would not be limited to the
>> 7-limit,
>
>A lot of Gene's names would then have more than one entry.

That wouldn't bother me from a chart-organization point of view,
but perhaps you think it's inappropriate from an ontology point
of view. While in principle I would agree that different things
should have different names, I think some sort of family system
is desirable, and that names should reflect it (I realize all
the bugs with having the tunings be the same, etc, aren't worked
out yet)...

-Carl

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

> Why would you call this modulatory space when absolutely no modulation
> is required or need be involved? That seems to muddy the issue.

I called it a modulatory space because other people did. The term was
not invented on Wikipedia, as a google search shows, but I don't know
its provinance. I suppose the idea is that it is a space you can
modulate in, as just changing octaves doesn't count as modulation.

If you can find a more widely used term I could rewrite the page and
move it, leaving a redirect.

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

> BTW, will rms-TOP
> and rms-Kees always be stretched versions of one another?

I doubt it. I'd need to write some code to find examples.

Carl Lumma wrote:
>>Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
> > > Another list of numbers from you, Gene, without column
> headings. :( I assume 12.27... is the rms error you
> mentioned. I can't say I know what the other numbers
> are.

The numbers in brackets are the wedgie. That's usually the best way to identify temperaments; this sequence of numbers uniquely identifies this particular rank 2 temperament (one of the larger category of "kleismic" temperaments).

> Now let's try to identify this temperament. "Kemun"
> doesn't occur on your page
> > http://66.98.148.43/~xenharmo/sevnames.htm
> > or in Herman's doc file. But "Keemun" does. It's a
> kind of tea and perhaps a region in China. What's the
> anecdote behind this name?

It's just one of my favorite varieties of tea. Paul needed something that was alphabetized between "Injera" and "Lemba" for his paper, and that's the suggestion I came up with.

http://twiningsteashopusa.com/index.asp?PageAction=VIEWPROD&ProdID=89

> Ideally your page above would not be limited to the
> 7-limit, and ideally it would be in alpha order by
> temperament name.
> > Herman -- what are the "catalog number"s in your Word
> file?

Those are from Gene's old list of 114 7-limit "linear temperaments" (as we were still calling them in those days). Almost all of the useful 7-limit ones are on that list somewhere (with a tiny handful of exceptions like "superpelog"), so it's convenient for referring to temperaments that don't yet have names.

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> 7-limit Graham complexity is what you want for your purpose; If you
> take a generator chain scale of n notes, then the number of tetrads of
> each kind will be n - Graham.

Ah, I see. Thanks.

> and the wedgie, which identifies the temperament.

Well, to some people, it does. I took a look at
<http://66.98.148.43/~xenharmo/wedgie.html>. Now, I've studied enough
math to believe I could figure out what that definition means if I were
to work at it for a few to several hours, but on the whole I'd rather
not. I'm sure the group theory connections, the jargon, and so on are
useful to you and to some other people, but to me they're just
obstacles.

(Or maybe I wouldn't be able to figure it out. I remember trying to
puzzle out a definition of wedge product some time ago, and not
succeeding. Though given a clearer definition than the one I was
looking at I probably would have gotten it.)

- Rich Holmes

Once again, Yahoo sent spam to my account, which was bounced by SpamCop, causing Yahoo to stop sending mail. So I've probably missed some of this discussion. There's really no excuse for Yahoo to do this sort of thing. Here's the reply I originally wrote before Yahoo bounced it.

Carl Lumma wrote:
>>Kemun [6, 5, 3, -6, -12, -7] 6 12.273810
> > > Another list of numbers from you, Gene, without column
> headings. :( I assume 12.27... is the rms error you
> mentioned. I can't say I know what the other numbers
> are.

The numbers in brackets are the wedgie. That's usually the best way to
identify temperaments; this sequence of numbers uniquely identifies this
particular rank 2 temperament (one of the larger category of "kleismic"
temperaments).

> Now let's try to identify this temperament. "Kemun"
> doesn't occur on your page
> > http://66.98.148.43/~xenharmo/sevnames.htm
> > or in Herman's doc file. But "Keemun" does. It's a
> kind of tea and perhaps a region in China. What's the
> anecdote behind this name?

It's just one of my favorite varieties of tea. Paul needed something
that was alphabetized between "Injera" and "Lemba" for his paper, and
that's the suggestion I came up with.

http://twiningsteashopusa.com/index.asp?PageAction=VIEWPROD&ProdID=89

> Ideally your page above would not be limited to the
> 7-limit, and ideally it would be in alpha order by
> temperament name.
> > Herman -- what are the "catalog number"s in your Word
> file?

Those are from Gene's old list of 114 7-limit "linear temperaments" (as
we were still calling them in those days). Almost all of the useful
7-limit ones are on that list somewhere (with a tiny handful of
exceptions like "superpelog"), so it's convenient for referring to
temperaments that don't yet have names.

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> (Or maybe I wouldn't be able to figure it out. I remember trying to
> puzzle out a definition of wedge product some time ago, and not
> succeeding. Though given a clearer definition than the one I was
> looking at I probably would have gotten it.)

The history of multilinear algebra on this group has beem that no one
liked it at first, when I introduced it to eliminate "torsion"
problems, but after a while people came to find it convenient. The
hump was gotten over, not much help to me, since by trying to make
things simple for people, and presenting a kind of dumbed-down
version, I apparently made it impossible to understand (since it was
just waving a magic wand, I suppose.) So, if you really want to get
into this stuff, it's useful.

However, as you no doubt already know, multilinear algebra is good for
you if you are a physicist, so it's not as if it would be a waste of
your time anyway. Studying its application to music would probably be
a good way to get up to speed with differential forms, where things
are more complicated. It's often remarked how much simpler Maxwell's
equations become if the shift is made to vector analysis notation, but
a similar simplification and clarification comes when the change is
made to differential forms, exterior derivatives, and the Hodge-*
operator. But here in music land, we've laid the groundwork--for
instance the complement of a wedge product (Hodge-* for people who
don't have to worry about differential topology) is something you
become familiar working with music applications.

"Gene Ward Smith" <gwsmith@svpal.org> writes:

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> The history of multilinear algebra on this group has beem that no one
> liked it at first, when I introduced it to eliminate "torsion"
> problems, but after a while people came to find it convenient.

All people? Or just the ones who didn't give up and go away?

> However, as you no doubt already know, multilinear algebra is good for
> you if you are a physicist, so it's not as if it would be a waste of
> your time anyway. Studying its application to music would probably be
> a good way to get up to speed with differential forms, where things
> are more complicated.

True maybe for theoretical physicists; I'm an experimentalist.
Differential forms aren't really very useful in designing data
acquisition systems or simulating Cerenkov detectors. In fact, not
only do I not know multilinear algebra is good for me, I don't even
know what multilinear algebra is.

Maybe there needs to be a tuning-simple-math group.

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> True maybe for theoretical physicists; I'm an experimentalist.
> Differential forms aren't really very useful in designing data
> acquisition systems or simulating Cerenkov detectors. In fact, not
> only do I not know multilinear algebra is good for me, I don't even
> know what multilinear algebra is.

Multilinear algebra is where you start if you want to do determinants
and vector analysis in the right way. It's a pity it isn't more or
less a requirment for undergrad math and physics majors as a part of
the standard sophomore linear algebra course; in large measure that's
because of historical reasons. Grassmann confused the hell out of
people, whereas the approach of Gibbs seemed easier, and it *was* an
improvement over Hamilton's quaterion approach. But in the end, people
paid for it in lessened understanding and more complex notation.
Anyway, it isn't really non-simple mathematics, it's just that people
are not introduced to it as soon as they should be (I didn't learn
about it as an undergraduate, and as I say I think I should have.)

> Maybe there needs to be a tuning-simple-math group.

What is simple? To get rank two temperaments, stepping through
generator sizes sounds simple, but in practice it turns out to be
harder than the alternatives. Combining pairs of equal temperaments is
a good idea, and combining commas is a good idea, and working from
generators in an equal temperament is a good idea, but how do you
treat these all at once? Wedge products let you do so easily; anything
which saves effort is in some significant sense "simple".

We probably have to agree to disagree. By "simple" I mean simple
concepts, not ease of computation. As efficient and beautiful as you
think your approach may be, insisting on doing everything in terms of
the jargon and concepts you use can only frighten away a lot of people
who would like to, and could, learn some basic things about tuning
math but are not mathematically sophisticated and cannot be expected
to work at your conceptual level. As l said, I'm sure your approach
is useful to some here, and you're welcome to it; but to expect all
others to either use it or go away is unreasonable.

Actually I'm not even sure things like wedgies are not simple
concepts. It may be a matter of how they're explained. Can it be
done more clearly than <http://66.98.148.43/~xenharmo/wedgie.html>?
Paul Erlich has, I think, been pretty successful at making things like
periodicity blocks and TOP accessible to the novice. Maybe he or
someone else can do likewise (or already has) with wedgies.

- Rich Holmes

Obviously those pages weren't organized as well as they should have
been, and I've since added links between them and others. Perhaps now
it becomes more clear that they are not intended as any sort of
definitive and complete treatise; they're just notes on some of the
avenues of thought I've followed. If concepts are omitted it's
because they're concepts I haven't explored.

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> We probably have to agree to disagree. By "simple" I mean simple
> concepts, not ease of computation. As efficient and beautiful as you
> think your approach may be, insisting on doing everything in terms of
> the jargon and concepts you use can only frighten away a lot of people
> who would like to, and could, learn some basic things about tuning
> math but are not mathematically sophisticated and cannot be expected
> to work at your conceptual level.

Most people here who undersrand this stuff are not professional
mathematicians; they are like you.

As l said, I'm sure your approach
> is useful to some here, and you're welcome to it; but to expect all
> others to either use it or go away is unreasonable.

Tuning-math was created precisely to be a place where one could speak
math without hearing complaints.

> Actually I'm not even sure things like wedgies are not simple
> concepts.

They *are* simple concepts. If you want non-simple concepts used in
extreme math overkill mode, check out Mazzola's "The Topos of Music".
To understand that you'd need to study a whole lot of graduate-level
math, and if like me you *did* in fact learn it for some other reason,
you'd discover, alas, that it didn't matter. The math being used on
this group has a much more practical slant, though questions of purely
theoretical interest are asked and answered, no one introduces math
just for the sake of having more and more complicated math.

It may be a matter of how they're explained. Can it be
> done more clearly than <http://66.98.148.43/~xenharmo/wedgie.html>?

Everyone seems to think almost anything I say can be said more clearly.

> Paul Erlich has, I think, been pretty successful at making things like
> periodicity blocks and TOP accessible to the novice. Maybe he or
> someone else can do likewise (or already has) with wedgies.

Didn't Graham or Dave do something like that?

>> Now let's try to identify this temperament. "Kemun"
>> doesn't occur on your page
>>
>> http://66.98.148.43/~xenharmo/sevnames.htm
>>
>> or in Herman's doc file. But "Keemun" does. It's a
>> kind of tea and perhaps a region in China. What's the
>> anecdote behind this name?
>
>It's just one of my favorite varieties of tea.

I'm a tea nut -- I like it too. Also Yunnan Gold. And Pu-er.
But my favorite of all is Wenshan Baozhong oolong (from Taiwan).
Their first snow in 150 years ruined last year's crop of that,
though. :(

:)

-Carl

>> The history of multilinear algebra on this group has beem that no
>> one liked it at first, when I introduced it to eliminate "torsion"
>> problems, but after a while people came to find it convenient.
>
>All people? Or just the ones who didn't give up and go away?

I'm afraid I never got over it, but all my attempts to find simple
alternatives have failed (based on generator sizes, etc.). I have
collected copious notes if I should ever find time to study it.

>Maybe there needs to be a tuning-simple-math group.

Here's to that!

-Carl

...
>because of historical reasons.
>... But in the end, people
>paid for it in lessened understanding and more complex notation.
>Anyway, it isn't really non-simple mathematics, it's just that people
>are not introduced to it as soon as they should be

Lots of things in life like this...

-Carl

Herman Miller wrote:

> The numbers in brackets are the wedgie. That's usually the best way to > identify temperaments; this sequence of numbers uniquely identifies this > particular rank 2 temperament (one of the larger category of "kleismic" > temperaments).

In practical terms, the octave-equivalent wedgie is all you need to uniquely identify the temperament. That's the same as the generator mapping multiplied by the number of periods to an octave. I always prefer to give the full period and generator mapping to the wedgie and I don't see why anybody should have to read the linearized wedgie to identify a temperament.

Graham

Gene Ward Smith wrote:

> The history of multilinear algebra on this group has beem that no one
> liked it at first, when I introduced it to eliminate "torsion"
> problems, but after a while people came to find it convenient. The
> hump was gotten over, not much help to me, since by trying to make
> things simple for people, and presenting a kind of dumbed-down
> version, I apparently made it impossible to understand (since it was
> just waving a magic wand, I suppose.) So, if you really want to get
> into this stuff, it's useful.

Not true in my case. I liked it at first (at least as soon as I understood it) but I'm still waiting for a practical use. Everything I can do with wedge products now I could do with matrices before. That includes eliminating torsion. Because I wrote my own Python code for wedge products, I don't need an external library. But I could have written matrix routines instead (I did for C++). Which method's easier largely depends on what you're already familiar with and have libraries for.

The wedge products may be useful for quantifying the complexity of regular temperaments. The "Graham complexity" used here only works for linear temperaments with an odd limit. There is an equivalent weighted primes measure that I'm getting to grips with. I don't know what Paul's doing in his paper, but it looks like it could relate to weighted wedgies. I'll look into it more, anyway. I think it can still be done easily enough without wedge products, but it's easiest to think of in that way, and the wedgies may come into their own for higher rank temperaments.

Graham

Hiya Rich! I'll jump in here.

Rich Holmes wrote:
> We probably have to agree to disagree. By "simple" I mean simple
> concepts, not ease of computation. As efficient and beautiful as you
> think your approach may be, insisting on doing everything in terms of
> the jargon and concepts you use can only frighten away a lot of people
> who would like to, and could, learn some basic things about tuning
> math but are not mathematically sophisticated and cannot be expected
> to work at your conceptual level. As l said, I'm sure your approach
> is useful to some here, and you're welcome to it; but to expect all
> others to either use it or go away is unreasonable.

Wedgies are a simple concept, once you understand them. Gene's explanations aren't. They really seem to be pitched at professional mathematicians, to describe a branch of applied mathematics, which could prove very interesting but leaves the rest of us behind.

> Actually I'm not even sure things like wedgies are not simple
> concepts. It may be a matter of how they're explained. Can it be
> done more clearly than <http://66.98.148.43/~xenharmo/wedgie.html>?
> Paul Erlich has, I think, been pretty successful at making things like
> periodicity blocks and TOP accessible to the novice. Maybe he or
> someone else can do likewise (or already has) with wedgies.

I've typed out explanations on the list before, but they've probably been lost among the noise.

You'll have come across cross products somewhere? Wedge products are a generalized version of cross products, that work with any number of dimensions. Historically, wedge products were around first. That's what Gene was talking about with Grassman and Gibbs.

The basic property of wedge products is that they're antisymmetric, so

a^b = -b^a

most of the time. There are some objects it doesn't work for. I can't remember if the strict definition of exterior algebra excludes them, but for musical purposes assume that the relation holds.

It follows that the wedge product of a vector with itself is zero

a^a = -a^a = 0

So the wedge product is a test of linear independence. That's the simple concept behind it.

The reason multivectors come into it is that the wedge product a^b isn't the same thing as a and b. That's the simplification Gibbs brought in with the cross product. The cross product of two vectors in three dimensional space is another vector in three dimensional space. That means you can do your electromagnetism with only vectors and scalars, and not have to worry about multivectors.

The wedge product of two vectors in three dimensional space can be thought of in two different ways. One is that the result is the complement of a vector. You write the complement as a bar over the vector. For ASCII (and Python) simplification, I use a ~. Then the cross product can be defined in terms of the wedge product and complement:

aXb = ~(a^b)

In the special case of three dimensions, this means the cross product of two vectors is also a vector.

The other way of thinking of it is that you have two different spaces, a normal space and a dual space. The complement operation moves you between the two different spaces. So the wedge product of two vectors in three dimensional space is a vector in dual space. This way, you can leave the complements implicit but you have to remember which space you're in. Gene prefers dual spaces but I prefer explicit complements.

The complement is a bit like the transpose of a matrix. It doesn't alter the magnitude of the multivector. So we can define the magnitude of a vector as

|~(a^~a)|

then, substituting ~a for a gives

|~(~a^~~a)| = |~(a^~a)| = |~(~a^a)|

as reversing the order of the wedge product only changes the sign, and doing the complement twice gets you back where you started (give or take a change in sign). In general, this operation:

~(a^~b)

returns a scalar if a and b are the same kind of multivector. For simple vectors, this is the same as the dot product. So you can define a generalized dot product accordingly:

a*b = ~(a^~a)

And that's about all you need to know about the mathematics. You can dig an implementation out of my Python code at

http://x31eq.com/temper/

So the point of this for music theory is that any family of regular temperaments can be fully and uniquely defined by a wedge product. That wedge product could be arrived at as:

1) The wedge product of a set of commas that are tempered out.

2) The wedge product of a set of equal temperaments belonging to the family.

The two results are complements of each other. To see why this works, take an equal temperament mapping e and a comma c. If the temperament tempers out the comma, the dot product is zero:

e*c = 0

That's all we need for the 3-limit. It can be expanded as

~(e^~c) = 0

and simplified to

e^~c = 0

because the complement of 0 is still 0.

In the 5-limit, each equal temperament can temper out two linearly-independent commas:

e*c1 = 0 and e*c2 = 0

and, because the dot product is symmetric for simple vectors,

c1*e = 0 and c2*e = 0

it also happens that

(c1^c2)*e = 0

To show this, expand it as

~((c1^c2)^~e) = 0

And that's the same as

c1^(c2^~e) = 0

It isn't obvious that you can do that, but you can. Because c2^~e is zero, so is the whole thing. You can also write the equation as

c1^-(~e^c2) = 0

or

-(c1^~e)^c2 = 0

so it works if c1^~e is zero as well. As it happens,

(c1^c2)*e = 0

if e tempers out either c1 or c2, but

e*(c1^c2) = 0

requres e to temper out both c1 and c2. That's a bit more obscure, but the upshot is that

c1^c2 = +/- ~e or e = +/- ~(c1^c2)

if e tempers out both c1 and c2. It also works for any number of commas:

e = +/- ~(c1^c2^...^cm)

Now, take a lot of commas and a lot of equal temperament mappings. From similar reasoning as above,

e1^e2^...^en = +/- ~(c1^c2^...^cm)

if all the equal temperaments temper out all of the commas. So this demonstrates why the two wedgies are complements of each other. For linear temperaments, there are only two linearly independent equal temperament mappings:

e1^e2 = +/- ~(c1^c2^...^cm)

We're particularly interested in the octave-equivalent generator mapping. This is like defining an equal temperament with the octave as an additional comma, so it can be written as an equation with wedge products:

mapping = ~(octave^c1^c2^...^cm)
mapping = ~(octave^~(e1^e2))

That means the mapping can easily be derived from the wedgie, which is useful to know.

The lists of numbers Gene gives are one way of writing out the wedgie. To understand that you have to understand multivectors, which I'll leave for now. But I hope this gives you an idea of at least what the wedge products are for.

Graham

Graham Breed <gbreed@gmail.com> writes:

> Hiya Rich! I'll jump in here.

There's a lot more than I can instantly digest here, but I'll respond
to part.

> The reason multivectors come into it is that the wedge product a^b isn't

Definition of "multivector"?

> The wedge product of two vectors in three dimensional space can be
> thought of in two different ways. One is that the result is the
> complement of a vector....
> The other way of thinking of it is that you have two different spaces, a
> normal space and a dual space. The complement operation moves you
> between the two different spaces.

Neither one makes much sense to me so far.

> The complement is a bit like the transpose of a matrix.

In what way, and perhaps more importantly, in what way is it not?

You've talked about some properties of wedge products and vector
complements but I can't dig out from what you've said what the
complement represents, or how it differs from a vector, or the
definition of wedge product.

> And that's about all you need to know about the mathematics.

Sorry, but so far I'm still lost.

- Rich Holmes

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> In practical terms, the octave-equivalent wedgie is all you need to
> uniquely identify the temperament.

In theoretical terms, it isn't. There are many temperaments each of
which goes to the same octave-equivalent one, and forcing someone to
sort it out simply adds an unneeded layer of agony and potential error.

To reconstruct septimal meantone from <1 4 10|, you can first find the
two-parameter family of possiblities, which is <<1 4 10 a b 4b-10a||.
*Any* of these is a theoretical temperament; now find the good one.

That's the same as the generator
> mapping multiplied by the number of periods to an octave. I always
> prefer to give the full period and generator mapping to the wedgie
and I
> don't see why anybody should have to read the linearized wedgie to
> identify a temperament.

When you read the wedgie, the first numbers are exactly what you
talked about--the octave equivalent part. So wedgies are not hard to read.

The full period and generator mapping is a good method of identifying
the temperament, but it isn't unique unless you constrain the
generator, for instance by making it the smallest choice greater than
one. To do that, you need to calculate it, first choosing a definition
of optimal tuning, and then finding which generator, using this
definition, is minimal among those greater than one. After this
rigamarole, you may find the generator selected isn't necessarily the
most interesting one; for certain you will find it takes time compared
to computing the wedgie, so if you are doing a massive hunt through
13-limit temperaments, doing this step when getting an initial
temperament list will slow you down.

You can wait until after you've gotten the list, with a complexity
cutoff (complexity being easy from the wedgie) and *then* doing it to
compute error and badness. Trying to work directly with periods and
generators simply makes things harder. Moreover, you don't necessarily
want to compute error and badness this way anyway. You can compute,
say, TOP error and Graham complexity directly from the wedgie, use
that to compute badness, sort everything out, and then after you have
your list finally compute the mapping.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> The basic property of wedge products is that they're antisymmetric, so
>
> a^b = -b^a
>
> most of the time. There are some objects it doesn't work for. I can't
> remember if the strict definition of exterior algebra excludes them,
but
> for musical purposes assume that the relation holds.

The relationship holds for vectors, and everything else is a sum of
products of vectors, so the definition for vectors is all you need.

> The reason multivectors come into it is that the wedge product a^b
isn't
> the same thing as a and b. That's the simplification Gibbs brought in
> with the cross product. The cross product of two vectors in three
> dimensional space is another vector in three dimensional space.

I think historically Gibbs started from Hamilton and quaterions, not
Grassman. Mathematicians do have a use for antisymmetric products
which map vecors to vectors, in Lie algebras. But using the Lie
algebra o(3), which is what the cross-product is in mathematise, is
the wrong thing to do in most physics applications, leading to weird
concepts such as "pseudovectors". The "bivector", the wedge product of
two vectors, is a different animal than the vector, even though you
can relate it back to a vector. This appears to make things harder but
actually ends up removing some of the confusion in physics applications.

That
> means you can do your electromagnetism with only vectors and scalars,
> and not have to worry about multivectors.

And end up more confused and far less able to deal with it in terms of
relativity.

> The other way of thinking of it is that you have two different
spaces, a
> normal space and a dual space. The complement operation moves you
> between the two different spaces. So the wedge product of two vectors
> in three dimensional space is a vector in dual space.

Well, in three dimensions. But why only three?

Graham Breed <gbreed@gmail.com> writes:

> Hiya Rich! I'll jump in here.

OK, having looked at the Wikipedia entry for exterior algebra and
thought about it a bit, things seems a bit clearer.

> The basic property of wedge products is that they're antisymmetric, so
>
> a^b = -b^a
>
> most of the time. There are some objects it doesn't work for.

As I understand it, this is true for vectors, not for all elements of
the algebra defined by the vector space and the wedge product.

> It follows that the wedge product of a vector with itself is zero
>
> a^a = -a^a = 0
>
> So the wedge product is a test of linear independence.

Again paraphrasing from Wikipedia, the wedge product is bilinear and
associative, satisfies the antisymmetry requirement shown above, and
also a^b != 0 if a and b are linearly independent. Then it follows
that a^b^c..^z = 0 iff a, b, c, ... z are not linearly independent.

Then, I think, but this part's still hazy to me, the wedge product is
*defined* by the above requirements.

> The wedge product of two vectors in three dimensional space can be
> thought of in two different ways. One is that the result is the
> complement of a vector. ... The other way of thinking of it is that
> you have two different spaces, a normal space and a dual space. The
> complement operation moves you between the two different spaces. So
> the wedge product of two vectors in three dimensional space is a
> vector in dual space.

This is still murky but I think I'm getting used to it. The point is
that the wedge product of two vectors is a vector, but not a vector in
the same space as the two vectors. So if you have two vectors a and
b, you can't do, for example, the (ordinary vector) dot product of a
and a^b, because a^b is a vector in a different space from a.

Then the complements you speak of are a mapping from the vectors in
the dual space to vectors in the original space. So you *can* do the
dot product of a and ~(a^b), because both are vectors in the same
space. Yes?

In the case of 3 dimensional vectors with basis i, j, k:

a = a1i + a2j + a3k
b = b1i + b2j + b3k

a^b = (a1b2-a2b1)(i^j) + (a1b3-a3b1)(i^k) + (a2b3-a3b2)(j^k)

a vector in a space with basis i^j, i^k, j^k. (I was tempted to put a
minus sign in front of the second term, but Wikipedia says it's a plus
sign.) Presumably that formula is dictated by the requirements stated
above. The complement is

~(a^b) = (a2b3-a3b2)i - (a1b3-a3b1)j + (a1b2-a2b1)k
= a x b

the usual vector cross product. (Here there *is* a minus sign.)

> In general, this operation:
>
> ~(a^~b)
>
> returns a scalar if a and b are the same kind of multivector.

I still haven't seen a definition of multivector (well, I've seen the
one on Wikipedia but it doesn't help much). But I assume when you say
"same kind of multivector", vectors in the algebra's vector space are
the "same kind", wedge products of vectors are another "kind", and so
on. As for ~(a^~b) being a scalar if a and b are the same kind, I
hope that's not supposed to be obvious?

> So the point of this for music theory is that any family of regular
> temperaments can be fully and uniquely defined by a wedge product. That
> wedge product could be arrived at as:
>
> 1) The wedge product of a set of commas that are tempered out.

That is, each comma is represented by a vector and their wedge product
"fully and uniquely" defines a regular temperament? I presume the
vectors are the ones I've seen here before, with the nth component
being the exponent of the nth prime factor. By "fully and uniquely"
you mean the wedge product specifies exactly one regular temperament,
and two different wedge products specify two different temperaments?

> 2) The wedge product of a set of equal temperaments belonging to the family.

So each equal temperament is represented by a vector and their wedge
product "fully and uniquely" defines a regular temperament. And the
nth component of each vector is the number of ET steps corresponding
to the nth prime factor?

> The two results are complements of each other. To see why this works,
> take an equal temperament mapping e and a comma c. If the temperament
> tempers out the comma, the dot product is zero:
>
> e*c = 0
>
> That's all we need for the 3-limit. It can be expanded as
>
> ~(e^~c) = 0
>
> and simplified to
>
> e^~c = 0
>
> because the complement of 0 is still 0.

Let's see if I get the point here: e and c are vectors in two
different spaces -- comma space and ET space -- but these spaces are
dual to one another, so ~c is a vector in ET space, for instance; and
in particular, if e tempers out c, then ~c and e are not linearly
independent -- that is, ~c is equal to e (up to a constant factor).
Yes?

> In the 5-limit, each equal temperament can temper out two
> linearly-independent commas:
>
> e*c1 = 0 and e*c2 = 0
>
> and, because the dot product is symmetric for simple vectors,
>
> c1*e = 0 and c2*e = 0
>
> it also happens that
>
> (c1^c2)*e = 0
>
> To show this, expand it as
>
> ~((c1^c2)^~e) = 0
>
> And that's the same as
>
> c1^(c2^~e) = 0
>
> It isn't obvious that you can do that, but you can.

It's not? Isn't that just associativity?

> We're particularly interested in the octave-equivalent generator
> mapping. This is like defining an equal temperament with the octave as
> an additional comma, so it can be written as an equation with wedge
> products:
>
> mapping = ~(octave^c1^c2^...^cm)
> mapping = ~(octave^~(e1^e2))
>
> That means the mapping can easily be derived from the wedgie, which is
> useful to know.

You lost me there.

- Rich Holmes

Rich Holmes wrote:

> Definition of "multivector"?

A multivector's something you get by multiplying vectors. How about that?

The point is that it isn't indexed by a single number. If you think of a vector elementwize, it's a list of v[i]. A multivector needs more than one index. So a bivector's indexed as m[i,j]. A trivector as m[i,j,k] and so on.

For a bivector you get from a wedge product, you know that each v[i,i]=0 and v[i,j] = -v[j,i].

>>The wedge product of two vectors in three dimensional space can be >>thought of in two different ways. One is that the result is the >>complement of a vector....
>> The other way of thinking of it is that you have two different spaces, a >>normal space and a dual space. The complement operation moves you >>between the two different spaces. > > Neither one makes much sense to me so far.

Imagine a world populated by row and column vectors. The complement is the operation that turns a row into a column, and vice versa.

>>The complement is a bit like the transpose of a matrix. > > In what way, and perhaps more importantly, in what way is it not?

There similar in that the shape and numbers are the same, but the numbers move around. They're different in that the sign of some numbers changes

> You've talked about some properties of wedge products and vector
> complements but I can't dig out from what you've said what the
> complement represents, or how it differs from a vector, or the
> definition of wedge product.

Then perhaps you think it's more complicated than it really is. How do you like your algebras defined?

Graham

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Then, I think, but this part's still hazy to me, the wedge product is
> *defined* by the above requirements.

What more do you need? Mathmaticians have a way of "clarifying" that
which is probably guaranteed to confuse you much more if you are not a
mathematician, called a "universal property":

http://en.wikipedia.org/wiki/Universal_property

but basically, it's pretty clear. You have a certain set of rules, and
no others. Those rules give you the wedge product.

> This is still murky but I think I'm getting used to it. The point is
> that the wedge product of two vectors is a vector, but not a vector in
> the same space as the two vectors.

Right. Though there is something called the exterior algebra (or in
more general situations a Clifford algebra) if you want to toss
everything into the same space, it doesn't seem to be useful for
musical purposes.

http://en.wikipedia.org/wiki/Exterior_algebra

> Then the complements you speak of are a mapping from the vectors in
> the dual space to vectors in the original space. So you *can* do the
> dot product of a and ~(a^b), because both are vectors in the same
> space. Yes?

Right. The dot product is a way of making something in the space act
like something in the dual space. But the above formula is only valid
in three dimensions.

> In the case of 3 dimensional vectors with basis i, j, k:
>
>
> a = a1i + a2j + a3k
> b = b1i + b2j + b3k
>
> a^b = (a1b2-a2b1)(i^j) + (a1b3-a3b1)(i^k) + (a2b3-a3b2)(j^k)
>
> a vector in a space with basis i^j, i^k, j^k. (I was tempted to put a
> minus sign in front of the second term, but Wikipedia says it's a plus
> sign.)

It's helpful to standardize the basis in some way, and the obvious way
is to put it into alphabetical order. The basis you wanted is
j^k, k^i, i^k which corresponds to i, j, k.

> That is, each comma is represented by a vector and their wedge product
> "fully and uniquely" defines a regular temperament?

The wedge product, projectivized to a "wedgie", fully and uniquely
defines a regular temperament. To do that, take the GCD of the terms
and divide out any common factors, and then make the first nonzero
entry positive.

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

> The full period and generator mapping is a good method of identifying
> the temperament, but it isn't unique unless you constrain the
> generator, for instance by making it the smallest choice greater than
> one. To do that, you need to calculate it, first choosing a definition
> of optimal tuning, and then finding which generator, using this
> definition, is minimal among those greater than one.

There is a way around this, which is purely algebraic and fast, but it
never became accepted; that is to define the standard mapping
identifier for a temperament to be Hermite reduced.

Rich Holmes wrote:

> Actually I'm not even sure things like wedgies are not simple
> concepts. It may be a matter of how they're explained. Can it be
> done more clearly than <http://66.98.148.43/~xenharmo/wedgie.html>?
> Paul Erlich has, I think, been pretty successful at making things like
> periodicity blocks and TOP accessible to the novice. Maybe he or
> someone else can do likewise (or already has) with wedgies.

I have a brief description of wedgies at the bottom of my Zireen Music page. It doesn't go into much detail, and it's (deliberately) not technically precise, but it does describe some reasons why they can be useful for dealing with temperaments.

http://www.io.com/~hmiller/music/zireen-music.html

Calculating wedgies is actually not that difficult; here's C code for wedging two "vals" (a kind of vector) of the same size (valsize), placing the result in the "wedgie" array.

wedgiesize = 0;
for (i = 0; i < valsize - 1; i++)
{
for (j = i + 1; j < valsize; j++)
{
wedgie[wedgiesize] = val1[i] * val2[j] - val2[i] * val1[j];
wedgiesize++;
}
}

These so-called "vals" are usually used to describe a tuning map, which specifies how many periods and generators (or any other two intervals, like tones and semitones) are used for approximating each of the prime number intervals (2/1, 3/1, 5/1, etc.) In the case of meantone, for instance, you have these two vals:

<1, 2, 4, 7] (for the period, an octave in the case of meantone)
<0, -1, -4, -10] (for the generator, a perfect fourth).

(You get a different set of vals if you use a fifth as the generator, but the end result is the same wedgie.) What this means is that:

2/1 is approximated by one octave up.
3/1 is approximated by two octaves up and one fourth down.
5/1 is approximated by 4 octaves up and 4 fourths down.
7/1 is approximated by 7 octaves up and 10 fourths down.

Wedge these vals and you end up with a so-called "bival", which identifies meantone temperament.

<<1, 4, 10, 4, 13, 12]]

You can also wedge commas, represented in vector form as the prime factorization of the comma ratio:

[-4, 4, -1> 81/80 (2^-4 * 3^4 * 5^-1)
[-5, 2, 2, -1> 225/224 (2^-5 * 3^2 * 5^2 * 7^-1)

[[12, -13, 4, 10, -4, 1>>

which after a bit of rearranging and sign flipping ends up with the same result.

Herman Miller wrote:

> Calculating wedgies is actually not that difficult; here's C code for > wedging two "vals" (a kind of vector) of the same size (valsize), > placing the result in the "wedgie" array.
> > wedgiesize = 0;
> for (i = 0; i < valsize - 1; i++)
> {
> for (j = i + 1; j < valsize; j++)
> {
> wedgie[wedgiesize] = val1[i] * val2[j] - val2[i] * val1[j];
> wedgiesize++;
> }
> }

Oh, thanks! That is really simple. I can't currently do wedge products in C, and this is the most useful special case. So I'll borrow this code if that's okay.

Graham

Rich Holmes wrote:
> Graham Breed <gbreed@gmail.com> writes:

>>We're particularly interested in the octave-equivalent generator >>mapping. This is like defining an equal temperament with the octave as >>an additional comma, so it can be written as an equation with wedge >>products:
>>
>>mapping = ~(octave^c1^c2^...^cm)
>>mapping = ~(octave^~(e1^e2))
>>
>>That means the mapping can easily be derived from the wedgie, which is >>useful to know.
> > > You lost me there.

You're getting the hang if it pretty well, so I think the best thing's to ponder it until it makes more sense.

Graham

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > >>In practical terms, the octave-equivalent wedgie is all you need to >>uniquely identify the temperament. > > > In theoretical terms, it isn't. There are many temperaments each of
> which goes to the same octave-equivalent one, and forcing someone to
> sort it out simply adds an unneeded layer of agony and potential error.

It's a good thing I said "I always prefer to give the full period and generator mapping" below, isn't it?

Even theoretically, I don't know how two practically useful temperaments can have the same octave-equivalent definition. Nobody's produced any examples.

> When you read the wedgie, the first numbers are exactly what you
> talked about--the octave equivalent part. So wedgies are not hard to read.

Maybe not much harder than the full mapping, if you know what they are. Which most people don't.

> The full period and generator mapping is a good method of identifying
> the temperament, but it isn't unique unless you constrain the
> generator, for instance by making it the smallest choice greater than
> one. To do that, you need to calculate it, first choosing a definition
> of optimal tuning, and then finding which generator, using this
> definition, is minimal among those greater than one. After this
> rigamarole, you may find the generator selected isn't necessarily the
> most interesting one; for certain you will find it takes time compared
> to computing the wedgie, so if you are doing a massive hunt through
> 13-limit temperaments, doing this step when getting an initial
> temperament list will slow you down. For certain? Would you care to place money on that? This process is linear in the number of primes (with a suitable optimization) whereas the wedge product grows quadratically. There must be a prime limit in which the wedge product takes more time to calculate.

For simple cases like the 13-limit, yes, maybe it is faster. But neither is going to be very slow unless you use a slow optimization.

There's also the Hermite reduction, which you mentioned in another post. Interestingly you think that's fast, although I don't see it should be any faster than the things you say are slow. It's still a single pass over the primes.

> You can wait until after you've gotten the list, with a complexity
> cutoff (complexity being easy from the wedgie) and *then* doing it to
> compute error and badness. Trying to work directly with periods and
> generators simply makes things harder. Moreover, you don't necessarily
> want to compute error and badness this way anyway. You can compute,
> say, TOP error and Graham complexity directly from the wedgie, use
> that to compute badness, sort everything out, and then after you have
> your list finally compute the mapping.

A list??? Ordered by complexity I suppose? The Python code works a bit like this, but using a dictionary (implemented as a hash table). I reckoned when I moved to C that the hash lookups would be equivalent in complexity to the error calculation anyway. So I switched to keeping a small list ordered by badness. Currently I have to do a linear search to check for duplicates, because temperament families don't always have distinct badness the way I calculate it. Maybe that's avoidable.

It's pretty fast. Over a million linear temperaments in all odd limits from 5 to 21 in about thirty seconds. It gets a lot slower for higher limits, but I've shown that's mostly the time taken to find the seed equal temperaments. I haven't profiled it in the 13-limit yet. If the error calculation is a problem, it'll get faster when I switch to weighted primes.

I noticed you mentioned the TOP error calculation there, which is currently horribly slow by the method you suggested, and I don't plan on researching any further.

But yes, there's no need to calculate the octave-specific part of the mapping until you display the temperaments. The two seed ETs work fine as an octave specific mapping for the purposes of calculating the error. And if you don't think the octave-equivalent mapping is a practically unique key, adding the badness will make it so for any temperaments we'd care about distinguishing. So it all depends on which way is faster to calculate the octave-equivalent mapping. Or even easier, if you leave the display to a higher level language.

The wedge product's certainly worth considering. I think it would come into its own for rank 3 temperaments and beyond because there isn't a single octave-equivalent mapping to calculate the complexity from. Now I have a simple function from Herman I might give it a try.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > Why would you call this modulatory space when absolutely no
modulation
> > is required or need be involved? That seems to muddy the issue.
>
> I called it a modulatory space because other people did. The term was
> not invented on Wikipedia, as a google search shows, but I don't know
> its provinance. I suppose the idea is that it is a space you can
> modulate in, as just changing octaves doesn't count as modulation.
>
> If you can find a more widely used term I could rewrite the page and
> move it, leaving a redirect.

In the "modulatory space" that other people defined, is each point a
note, a chord, or a key? At least some of the modulatory spaces I have
seen (such as Krumhansl's) plot each *key* as a point (with C major, C
minor, and A minor all at different points), which makes sense because
modulation means a change of key.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > BTW, will rms-TOP
> > and rms-Kees always be stretched versions of one another?
>
> I doubt it. I'd need to write some code to find examples.

I'd absolutely love to know about those. I'm sure Graham would too.

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> "Gene Ward Smith" <gwsmith@s...> writes:
>
> > 7-limit Graham complexity is what you want for your purpose; If
you
> > take a generator chain scale of n notes, then the number of
tetrads of
> > each kind will be n - Graham.
>
> Ah, I see. Thanks.
>
> > and the wedgie, which identifies the temperament.
>
> Well, to some people, it does. I took a look at
> <http://66.98.148.43/~xenharmo/wedgie.html>. Now, I've studied
enough
> math to believe I could figure out what that definition means if I
were
> to work at it for a few to several hours, but on the whole I'd
rather
> not. I'm sure the group theory connections, the jargon, and so on
are
> useful to you and to some other people, but to me they're just
> obstacles.

Don't close your mind to something that may open it up greatly for
the future.

> (Or maybe I wouldn't be able to figure it out. I remember trying to
> puzzle out a definition of wedge product some time ago, and not
> succeeding. Though given a clearer definition than the one I was
> looking at I probably would have gotten it.)
>
> - Rich Holmes

I'm sure you can figure this all out with patience and our help. It
will be worth it.

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Maybe there needs to be a tuning-simple-math group.
>
> - Rich Holmes

It may take you a few years (as it took most of us), but after that,
I'm sure you'll think of wedgies as "simple". And I think you'll
correspondingly get a better sense of how to fully, economically and
mathematically tackle the kinds of scale searches you're interested in.
And probably teach us a few things as well!

Hi Rich,

You know about the "cross product" in physics, right? That's a kind
of wedge product . . .

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> We probably have to agree to disagree. By "simple" I mean simple
> concepts, not ease of computation. As efficient and beautiful as
you
> think your approach may be, insisting on doing everything in terms
of
> the jargon and concepts you use can only frighten away a lot of
people
> who would like to, and could, learn some basic things about tuning
> math but are not mathematically sophisticated and cannot be expected
> to work at your conceptual level. As l said, I'm sure your approach
> is useful to some here, and you're welcome to it; but to expect all
> others to either use it or go away is unreasonable.
>
> Actually I'm not even sure things like wedgies are not simple
> concepts. It may be a matter of how they're explained. Can it be
> done more clearly than <http://66.98.148.43/~xenharmo/wedgie.html>?

This page sure is confusing -- and it seems to contradict what Gene
himself has said; he told me that frequency ratios (or "monzos" or
vectors in the JI lattice) cannot be wedgies; that wedgies are always
vals or multivals.

> Paul Erlich has, I think, been pretty successful at making things
like
> periodicity blocks and TOP accessible to the novice. Maybe he or
> someone else can do likewise (or already has) with wedgies.
>
> - Rich Holmes

I'd like to (not this instant though). But even Gene has written much
more readably about "what the numbers in the wedgie mean" as well as
how to compute the wedgie.

Anyway, a simple example would be to show how the wedgie for a
particular temperament can be arrived at by some simple calculations
(like the calculation of the cross product), resulting in the same
answer whether one starts with vanishing commas, ET mappings, or
period-generator mappings. Do it for two different-rank temperaments,
12-equal and meantone (both in the 5-limit case), and the power of
the approach begins to become clear . . .

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Obviously those pages weren't organized as well as they should have
> been, and I've since added links between them and others. Perhaps now
> it becomes more clear that they are not intended as any sort of
> definitive and complete treatise; they're just notes on some of the
> avenues of thought I've followed. If concepts are omitted it's
> because they're concepts I haven't explored.
>
> - Rich Holmes

Please let me know if you'd like me to give you comments on them (in
private or public). I don't want to if it'll only annoy you; I just
want to be helpful and keep everyone up to speed with the latest
information . . .

"wallyesterpaulrus" <perlich@aya.yale.edu> writes:

> Anyway, a simple example would be to show how the wedgie for a
> particular temperament can be arrived at by some simple calculations
> (like the calculation of the cross product), resulting in the same
> answer whether one starts with vanishing commas, ET mappings, or
> period-generator mappings. Do it for two different-rank temperaments,
> 12-equal and meantone (both in the 5-limit case), and the power of
> the approach begins to become clear . . .

It would show you how different descriptions of the same temperament
lead to the same wedgie, while different temperaments lead to
different wedgies, I suppose. It wouldn't tell me how a wedgie can be
used, however.

It also is something I wouldn't be able to do until I learned just
what a wedgie *is*, which still is not clear, other than that it's
somehow related to a wedge product. I find it hard to understand why
clear definitions of these terms and examples of their use, which
people are insisting are required knowledge on this group, don't seem
to exist anywhere.

In particular: Way back a few days ago, I was complaining because Gene
Ward Smith was telling me "here, look at these tunings" and followed
that with a cryptic list of names and wedgies.

That sent me off on a tangent, but to return to the starting point:
Given a wedgie, how does one get e.g. a corresponding generator and
period for that tuning?

After all, I presume that's the sort of thing Gene expected me to do
with his list.

- Rich Holmes

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> The wedge products may be useful for quantifying the complexity of
> regular temperaments. The "Graham complexity" used here only works
for
> linear temperaments with an odd limit. There is an equivalent
weighted
> primes measure that I'm getting to grips with. I don't know what
Paul's
> doing in his paper,

Really? Why didn't you ask?

> but it looks like it could relate to weighted
> wedgies.

Yes, it's the weighted L_1 norm of the wedgie -- that's why it agrees
with (is proportional to) the Tenney Harmonic Distance of the
vanishing ratio in the codimension-1 case.

It's also extremely close to the "minimax Kees complexity", which is
defined similarly to Graham complexity, but uses a Kees-
expressibility or "log-of-odd-limit" weighting and in principle looks
over all intervals, not just some set of "consonances".

> I'll look into it more, anyway. I think it can still be done
> easily enough without wedge products, but it's easiest to think of
in
> that way, and the wedgies may come into their own for higher rank
> temperaments.

Sometimes, the wedgie has more elements than the full set of mapping
vectors, so it's inefficient, but certainly still valuable for a lot
of the calculation you might conceivably want to do.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> We're particularly interested in the octave-equivalent generator
> mapping. This is like defining an equal temperament with the
octave as
> an additional comma, so it can be written as an equation with wedge
> products:
>
> mapping = ~(octave^c1^c2^...^cm)
> mapping = ~(octave^~(e1^e2))
>
> That means the mapping can easily be derived from the wedgie, which
is
> useful to know.

But don't you have to also divide by the LCM (the LCM giving you the
number of periods per octave)?

> The lists of numbers Gene gives are one way of writing out the
wedgie.
> To understand that you have to understand multivectors, which I'll
leave
> for now. But I hope this gives you an idea of at least what the
wedge
> products are for.

I hope to provide some concrete examples later, if no one beats me to
it . . .

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> Even theoretically, I don't know how two practically useful
temperaments
> can have the same octave-equivalent definition. Nobody's produced any
> examples.

It's tough, since the same generator mapping numbers need to go to two
different period mapping numbers. Hence the tuning for at least one
must be either atrocious, or the temperaments must be complex. It can
happen in a more or less practical context, however. I have an old
list of 32201 seven-limit temperaments I used as a preliminary list
before eliminaing almost all of them, and somewhere on that list is a
pair of temperaments with the same octave-equivalent numbers. Of
course if I had been using octave-equivalent numbers the whole thing
would have taken too long anyway; it's not a practical system for that
sort of thing.

> For certain? Would you care to place money on that? This process is
> linear in the number of primes (with a suitable optimization) whereas
> the wedge product grows quadratically. There must be a prime limit in
> which the wedge product takes more time to calculate.

It's not even well-defined anyway. You need to use something like a
Hermite reduction step. I'm not too worried about what happens in the
101-limit, and for low limits the whole business slows things down.
Moreover, as far as I can see it's likely to be quadratic also;
certainly rms will be, because the tonality diamond grows quadratically.

> There's also the Hermite reduction, which you mentioned in another
post.
> Interestingly you think that's fast, although I don't see it
should be
> any faster than the things you say are slow.

It's a kind of souped up Gaussian elimination, applied to a problem
where most of the work has been done already. Anyway, it's
well-defined; in theory smallest generator depends on your optimization.

> A list??? Ordered by complexity I suppose?

You've seen my lists ordered by various things--complexity, error,
badness.

Currently I have to do a linear search
> to check for duplicates, because temperament families don't always have
> distinct badness the way I calculate it. Maybe that's avoidable.

It should be avoidable by using rms error.

> It's pretty fast. Over a million linear temperaments in all odd limits
> from 5 to 21 in about thirty seconds.

I'm using Maple, which is much slower. I suppose writing C code would
be an alternative.

The two seed ETs work fine
> as an octave specific mapping for the purposes of calculating the
error.

But not as a means of uniquely naming the temperaments.

> The wedge product's certainly worth considering. I think it would come
> into its own for rank 3 temperaments and beyond because there isn't a
> single octave-equivalent mapping to calculate the complexity from.

You need a different complexity measure than Graham complexity,
certainly. The rank 3 temperaments can be defined by a wedgie, but
also by a Hermite reduced mapping.

Hi Rich, I think you were doing OK up to here:

> Let's see if I get the point here: e and c are vectors in two
> different spaces -- comma space and ET space -- but these spaces are
> dual to one another, so ~c is a vector in ET space, for instance; and
> in particular, if e tempers out c, then ~c and e are not linearly
> independent -- that is, ~c is equal to e (up to a constant factor).
> Yes?

This is only true in the 3-limit -- I don't know if you followed Graham
on that. More generally, if e tempers out c, it is true as Graham said
that e*c = 0 (wedge product of e and ~c equals zero), but in general
they're not equal up to a constant factor. Example: 12-equal tempers
out the syntonic comma:

<12 19 28|-4 4 -1> = 12*(-4) + 19*4 + 28*(-1) = 0

but clearly, <12 19 28] and [-4 4 -1> are not equal up to a constant
factor, even if you take the complement of one of them . . .

Personally, I'd use the concept of 2 or 4 different spaces to allow
all these operations to be understood a bit less abstractly. You have
tone-space, which is like the JI lattice -- each point is a note (or
interval). You have tuning-space, where each point is a "regular"
tuning system. In both cases, the axes are the primes. But in the
first case, the coordinate on a given axis tells you the number of
times you have to multiply by that prime to get to the note (or
interval) in question. In the second case, the coordinate on a given
axis tells you how that prime is tuned in the tuning system in
question. Then you have projective versions of both spaces, where
vectors from the original spaces that were scalar multiples of one
another are identified (thus the projective spaces have one fewer
dimension than the original spaces from which they were defined). If
you adopt these pictures, it's much more "tangible" what the various
vectors, wedge products, and complements "represent" . . .

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> Rich Holmes wrote:
>
> > Definition of "multivector"?
>
> A multivector's something you get by multiplying vectors. How
about that?
>
> The point is that it isn't indexed by a single number. If you
think of
> a vector elementwize, it's a list of v[i]. A multivector needs
more
> than one index. So a bivector's indexed as m[i,j]. A trivector as
> m[i,j,k] and so on.
>
> For a bivector you get from a wedge product, you know that each v
[i,i]=0
> and v[i,j] = -v[j,i].
>
> >>The wedge product of two vectors in three dimensional space can
be
> >>thought of in two different ways. One is that the result is the
> >>complement of a vector....
> >> The other way of thinking of it is that you have two different
spaces, a
> >>normal space and a dual space. The complement operation moves
you
> >>between the two different spaces.
> >
> > Neither one makes much sense to me so far.
>
> Imagine a world populated by row and column vectors. The
complement is
> the operation that turns a row into a column, and vice versa.
>
> >>The complement is a bit like the transpose of a matrix.
> >
> > In what way, and perhaps more importantly, in what way is it not?
>
> There similar in that the shape and numbers are the same, but the
> numbers move around. They're different in that the sign of some
numbers
> changes
>
> > You've talked about some properties of wedge products and vector
> > complements but I can't dig out from what you've said what the
> > complement represents, or how it differs from a vector, or the
> > definition of wedge product.
>
> Then perhaps you think it's more complicated than it really is.
How do
> you like your algebras defined?
>
>
> Graham
>

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> Gene Ward Smith wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
> >
> >
> >>In practical terms, the octave-equivalent wedgie is all you need
to
> >>uniquely identify the temperament.
> >
> >
> > In theoretical terms, it isn't. There are many temperaments each
of
> > which goes to the same octave-equivalent one, and forcing someone
to
> > sort it out simply adds an unneeded layer of agony and potential
error.
>
> It's a good thing I said "I always prefer to give the full period
and
> generator mapping" below, isn't it?
>
> Even theoretically, I don't know how two practically useful
temperaments
> can have the same octave-equivalent definition. Nobody's produced
any
> examples.

The 7-limit wedgies, treated as vectors in a six-dimensional space,
all lie in a four-dimensional subspace of that six-dimensional space.
There's also a three-dimensional subspace within that that
corresponds to zero error, so all the low-error temperaments lie
quite close to that three-dimensional subspace. This is why three
numbers is usually enough in the 7-limit case. But certainly there
are examples of high-error temperaments where the octave-equivalent
part of the two wedgies is the same.

> And if you don't think the octave-equivalent mapping is a
practically
> unique key, adding the badness will make it so for any temperaments
we'd
> care about distinguishing.

What sort of 'badness' do you propose to use for this, and why?
Badness is a function of both error and complexity, both of which can
be defined in several ways, as can the badness function itself. (For
those who didn't know . . .)

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> "wallyesterpaulrus" <perlich@a...> writes:
>
> > Anyway, a simple example would be to show how the wedgie for a
> > particular temperament can be arrived at by some simple
calculations
> > (like the calculation of the cross product), resulting in the
same
> > answer whether one starts with vanishing commas, ET mappings, or
> > period-generator mappings. Do it for two different-rank
temperaments,
> > 12-equal and meantone (both in the 5-limit case), and the power
of
> > the approach begins to become clear . . .
>
> It would show you how different descriptions of the same temperament
> lead to the same wedgie, while different temperaments lead to
> different wedgies, I suppose. It wouldn't tell me how a wedgie can
be
> used, however.

It can be used, for example, to find out which ET results when you
temper a certain set of commas out, or to find out which comma is
tempered out by a certain set of ETs . . . Also it's very direct to
get a measure of a temperament's complexity from it.

> It also is something I wouldn't be able to do until I learned just
> what a wedgie *is*, which still is not clear, other than that it's
> somehow related to a wedge product.

It took a while for this to become clear to me too, but the wedgie as
I thought Gene had defined it can always be seen as an object in
projective tuning-space. For an ET, this object is simply a point.
For example, <12 19 28] tells you that the mappings/approximations
(in cents or any other logarithmic measure) of primes 2, 3, and 5 are
in the ratio 12:19:28, which means (a possibly stretched or
compressed) 12-equal, and this is a single point in the (in this case
2-dimensional) projective tuning space. For a 2D temperament, this
object is a line (and hence requires more numbers to parameterize it
if there are more than 3 primes involved). For a 3D temperament, this
object is a plane. These geometrical figures (or subspaces, to be
more precise) show us which particular "regular" tuning systems (up
to an overall stretching, since the space is projective) belong to
the temperament "family" that the wedgie represents.

More later . . .

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
>
> > Even theoretically, I don't know how two practically useful
> temperaments
> > can have the same octave-equivalent definition. Nobody's
produced any
> > examples.
>
> It's tough, since the same generator mapping numbers need to go to
two
> different period mapping numbers. Hence the tuning for at least one
> must be either atrocious, or the temperaments must be complex. It
can
> happen in a more or less practical context, however. I have an old
> list of 32201 seven-limit temperaments I used as a preliminary list
> before eliminaing almost all of them, and somewhere on that list is
a
> pair of temperaments with the same octave-equivalent numbers.

I was going to mention that.

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

> > The wedge product's certainly worth considering. I think it would
come
> > into its own for rank 3 temperaments and beyond because there isn't
a
> > single octave-equivalent mapping to calculate the complexity from.
>
> You need a different complexity measure than Graham complexity,
> certainly. The rank 3 temperaments can be defined by a wedgie, but
> also by a Hermite reduced mapping.

I didn't follow what you guys were saying here . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...> wrote:
>
> Hi Rich, I think you were doing OK up to here:
>
> > Let's see if I get the point here: e and c are vectors in two
> > different spaces -- comma space and ET space -- but these spaces
are
> > dual to one another, so ~c is a vector in ET space, for instance;
and
> > in particular, if e tempers out c, then ~c and e are not linearly
> > independent -- that is, ~c is equal to e (up to a constant
factor).
> > Yes?
>
> This is only true in the 3-limit -- I don't know if you followed
Graham
> on that. More generally, if e tempers out c, it is true as Graham
said
> that e*c = 0 (wedge product

Sorry, that's not a wedge product. It's been called a "dot product"
here, but Gene wouldn't let me get away with that in my paper --
technically it's a "bracket product".

> of e and ~c equals zero), but in general
> they're not equal up to a constant factor. Example: 12-equal
>tempers
> out the syntonic comma:
>
> <12 19 28|-4 4 -1> = 12*(-4) + 19*4 + 28*(-1) = 0
>
> but clearly, <12 19 28] and [-4 4 -1> are not equal up to a
constant
> factor, even if you take the complement of one of them . . .
>

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

> In the "modulatory space" that other people defined, is each point a
> note, a chord, or a key? At least some of the modulatory spaces I have
> seen (such as Krumhansl's) plot each *key* as a point (with C major, C
> minor, and A minor all at different points), which makes sense because
> modulation means a change of key.

Chords, but apparently I should add a section on keys. In effect,
there is not much difference, as the tonic chord defines the key.

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

> This page sure is confusing -- and it seems to contradict what Gene
> himself has said; he told me that frequency ratios (or "monzos" or
> vectors in the JI lattice) cannot be wedgies; that wedgies are always
> vals or multivals.

This page seems to have been written for mathematicians; at one time I
was mostly trying to document the ideas. What I maybe should do is
write something easier, by taking what Herman wrote and adding more or
something.

But, nothing on the page says anything other than that wedgies are
multivals that I can see.

> I'd like to (not this instant though). But even Gene has written much
> more readably about "what the numbers in the wedgie mean" as well as
> how to compute the wedgie.

I could mess around with that and make it a web page. Dp you have
article numbers by any chance?

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

> This is only true in the 3-limit -- I don't know if you followed Graham
> on that. More generally, if e tempers out c, it is true as Graham said
> that e*c = 0 (wedge product of e and ~c equals zero)...

Inner product: <e|c> = 0.

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

> The 7-limit wedgies, treated as vectors in a six-dimensional space,
> all lie in a four-dimensional subspace of that six-dimensional space.

Unfortunately, it's not a subspace, it's an algebraic variety--the
same sort of thing as for instance an ellipsoid in 3-space. If

<<x1 x2 x3 x4 x5 x6||

is a wedge product, then

x1*x6 - x2*x5 + x3*x4 = 0.

This defines a hypersurface in 6-space. But wedgies are projective;
you go through a step of taking out the GCD and picking the right
sign. Hence, you want to look at the above as a 4-D projective variety
in 5-D real projective space.

> There's also a three-dimensional subspace within that that
> corresponds to zero error, so all the low-error temperaments lie
> quite close to that three-dimensional subspace.

Once again, it's more complicated than that. I think you are thinking
of something else: the lattice of intervals, living inside something
like TOP space.

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

> It took a while for this to become clear to me too, but the wedgie as
> I thought Gene had defined it can always be seen as an object in
> projective tuning-space.

These objects in projective tuning space are in 1-1 correspondence
with wedgies, but except for vals are not wedgies themselves. The
k-dimensional subspaces of an n-dimensional space are called the
Grassmanians, Grassmann varities, or Grassmann manifolds G_k(n); in
case k is 1, these are the lines through the origin, or in other words
projective space. G_k(n) is a generalization of projective space. The
Plucker map maps the elements of G_k(n) to another space; it is simply
what you get by taking k-fold wedge products of a basis for the
subspaces in G_k(n) and projectivizing. The image under the Plucker
map is identical as an abstract algebraic variety with the
Grassmannian and so is also called the Grassmannian;
"homogenous coordinates" or "Plucker coordinates" or "Grassmann
coordinates" for the Grassmannian.

Fortunately, we don't actually need to know this...

wallyesterpaulrus wrote:

> Really? Why didn't you ask?

I only started looking at this a few days ago, and I haven't checked if my hunch was correct yet. Anyway, here I am with a new subject line.

>>but it looks like it could relate to weighted >>wedgies.
> > Yes, it's the weighted L_1 norm of the wedgie -- that's why it agrees > with (is proportional to) the Tenney Harmonic Distance of the > vanishing ratio in the codimension-1 case.

That's the sum-abs? And weighted? That's what I guessed you were doing, anyway. Being told doesn't make it that much clearer :P

> It's also extremely close to the "minimax Kees complexity", which is > defined similarly to Graham complexity, but uses a Kees-
> expressibility or "log-of-odd-limit" weighting and in principle looks > over all intervals, not just some set of "consonances".

What I'm doing now is taking the max-min of the weighted generator mapping (times the number of periods per octave if that's already been divided out). That's the nearest thing to odd-limit complexity without actually using odd-limits. Taking the straight weighted mean or minimax doesn't work, because it makes things like miracle look far too good.

So it happens that my definition of the weighted-prime complexity is very similar to that of the TOP error of an equal temperament.

> Sometimes, the wedgie has more elements than the full set of mapping > vectors, so it's inefficient, but certainly still valuable for a lot > of the calculation you might conceivably want to do.

The wedgie gets dramatically larger the further you go beyond the 11-limit for rank 2 temperaments. But it doesn't look unmanageably large for rank 3 temperaments with sensible prime limits. And it might mean the code can be made general for any rank.

Graham

Gene Ward Smith wrote:

> It's tough, since the same generator mapping numbers need to go to two
> different period mapping numbers. Hence the tuning for at least one
> must be either atrocious, or the temperaments must be complex. It can
> happen in a more or less practical context, however. I have an old
> list of 32201 seven-limit temperaments I used as a preliminary list
> before eliminaing almost all of them, and somewhere on that list is a
> pair of temperaments with the same octave-equivalent numbers. Of
> course if I had been using octave-equivalent numbers the whole thing
> would have taken too long anyway; it's not a practical system for that
> sort of thing.

I've added an assertion to the C code, and it shows that all apparent duplicates have the same badness to four figure accuracy. If they are different, they're not significantly so. These are only being compared with the current best 100 though.

>>For certain? Would you care to place money on that? This process is >>linear in the number of primes (with a suitable optimization) whereas >>the wedge product grows quadratically. There must be a prime limit in >>which the wedge product takes more time to calculate.
> > It's not even well-defined anyway. You need to use something like a
> Hermite reduction step. I'm not too worried about what happens in the
> 101-limit, and for low limits the whole business slows things down.
> Moreover, as far as I can see it's likely to be quadratic also;
> certainly rms will be, because the tonality diamond grows quadratically. What's not defined?

You don't need the Hermite reduction if you can get the optimum error. It'd make sense to put off the optimization if you don't need it 90% of the time. That sounds unlikely to me.

No, RMS of a tonality diamond is cubic. The tonality diamond itself grows quadratically, and you also have to loop over the primes for each interval. The RMS of the primes themselves however is linear. So when choosing my suitable optimization I obviously wouldn't go for one that involves tonality diamonds.

We're talking about:

- find the generator mapping (linear time plus the inverse modulus)
- find the period mapping (linear time)
- find the complexity (linear time with floating point)
- check the hcf of the generator mapping (linear time)
- optimize (linear time with floating point)
- find the error (linear time with floating point)
- adjust the period mapping (linear time)
- use the full mapping as a unique identifier (linear time for each comparison)

as compared to

- find the wedgie (quadratic time)
- check the hcf of the wedgie (quadratic time)
- extract the generator mapping (linear time)
- maybe check the hcf of the generator mapping anyway (linear time)
- find the complexity (linear time with floating point)
- use the wedgie as a unique identifier (quadratic time for each comparison)
- optimize at some later stage (same as above when you get round to it)

So yes, there may be fewer steps if you use wedgies, but three of them are quadratic time. That hardly makes it obvious that the process will be faster, even for sensible prime limits. You also have to consider the extra memory to store all the candidates, which may exhuast the cache.

If you use a wedgie-based complexity measure, you lose two linear steps for another quadratic one.

> It's a kind of souped up Gaussian elimination, applied to a problem
> where most of the work has been done already. Anyway, it's
> well-defined; in theory smallest generator depends on your optimization.

For temperaments from commas the mapping depends only on the chromatic unison vectors(s) and so will be consistent within a given search. For linear from equal temperaments, the initial generator is usually fairly sensible. You don't need to get the smallest generator, only one of the pair within the period. Then you constrain the generator mapping to start with a positive number.

>>A list??? Ordered by complexity I suppose? > > You've seen my lists ordered by various things--complexity, error,
> badness.

The point is that searching a list will be harder than searching a set, and all you need is a set, but you nonchalantly threw this inefficiency in. You don't have error or badness yet. Complexity ordering will give a lot of collisions, that you will have to resolve by checking the wedgie.

> Currently I have to do a linear search > >>to check for duplicates, because temperament families don't always have >>distinct badness the way I calculate it. Maybe that's avoidable.
> > It should be avoidable by using rms error.

Yes. The thing is I need to be consistent between different implementations, with some weird results. But I'll look into it one day.

> The two seed ETs work fine >>as an octave specific mapping for the purposes of calculating the
> error. > > But not as a means of uniquely naming the temperaments.

Indeed.

>>The wedge product's certainly worth considering. I think it would come >>into its own for rank 3 temperaments and beyond because there isn't a >>single octave-equivalent mapping to calculate the complexity from.
> > You need a different complexity measure than Graham complexity,
> certainly. The rank 3 temperaments can be defined by a wedgie, but
> also by a Hermite reduced mapping.

Yes. This is in a different thread now.

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > In the "modulatory space" that other people defined, is each point
a
> > note, a chord, or a key? At least some of the modulatory spaces I
have
> > seen (such as Krumhansl's) plot each *key* as a point (with C
major, C
> > minor, and A minor all at different points), which makes sense
because
> > modulation means a change of key.
>
> Chords,

So why not call it "chord space"? Or better yet, "triad space", "tetrad
space", or whatever particular chords you're using.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > This page sure is confusing -- and it seems to contradict what
Gene
> > himself has said; he told me that frequency ratios (or "monzos"
or
> > vectors in the JI lattice) cannot be wedgies; that wedgies are
always
> > vals or multivals.
>
> This page seems to have been written for mathematicians; at one
time I
> was mostly trying to document the ideas. What I maybe should do is
> write something easier, by taking what Herman wrote and adding more
or
> something.
>
> But, nothing on the page says anything other than that wedgies are
> multivals that I can see.

You talk about 16/15 being a wedgie.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> What's not defined?

If you try to uniquely define the mapping in terms of the smallest
generator, that is not well-defined unless you also give your
opitimization method for determing the optimal tuning and hence the
exact size of the generator. In practice it would be.

> > You've seen my lists ordered by various things--complexity, error,
> > badness.
>
> The point is that searching a list will be harder than searching a set,
> and all you need is a set, but you nonchalantly threw this inefficiency
> in.

Well, I never search lists, unless it's a list which came from ordering
a set.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > The 7-limit wedgies, treated as vectors in a six-dimensional
space,
> > all lie in a four-dimensional subspace of that six-dimensional
space.
>
> Unfortunately, it's not a subspace, it's an algebraic variety--the
> same sort of thing as for instance an ellipsoid in 3-space.

Sorry -- I should have said "subset" instead of "subspace", right?

> If
>
> <<x1 x2 x3 x4 x5 x6||
>
> is a wedge product, then
>
> x1*x6 - x2*x5 + x3*x4 = 0.
>
> This defines a hypersurface in 6-space. But wedgies are projective;
> you go through a step of taking out the GCD and picking the right
> sign. Hence, you want to look at the above as a 4-D projective
variety
> in 5-D real projective space.

OK, so I should have said a 4-D subset of the 5-D projective space
that comes from the full 6-D vector space . . . yes?

> > There's also a three-dimensional subspace within that that
> > corresponds to zero error, so all the low-error temperaments lie
> > quite close to that three-dimensional subspace.
>
> Once again, it's more complicated than that.

I see I said something wrong. Could you correct it then?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > It took a while for this to become clear to me too, but the
wedgie as
> > I thought Gene had defined it can always be seen as an object in
> > projective tuning-space.
>
> These objects in projective tuning space are in 1-1 correspondence
> with wedgies, but except for vals are not wedgies themselves.

Is it just that the wedgies have all integers and GCD = 1? But then
you wouldn't have made an exception for vals. So I don't understand
why you say this.

> The
> k-dimensional subspaces of an n-dimensional space are called the
> Grassmanians, Grassmann varities, or Grassmann manifolds G_k(n); in
> case k is 1, these are the lines through the origin, or in other
words
> projective space. G_k(n) is a generalization of projective space.
The
> Plucker map maps the elements of G_k(n) to another space; it is
simply
> what you get by taking k-fold wedge products of a basis for the
> subspaces in G_k(n) and projectivizing. The image under the Plucker
> map is identical as an abstract algebraic variety with the
> Grassmannian and so is also called the Grassmannian;
> "homogenous coordinates" or "Plucker coordinates" or "Grassmann
> coordinates" for the Grassmannian.
>
> Fortunately, we don't actually need to know this...

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> wallyesterpaulrus wrote:
>
> > Really? Why didn't you ask?
>
> I only started looking at this a few days ago, and I haven't
checked if
> my hunch was correct yet. Anyway, here I am with a new subject
line.
>
> >>but it looks like it could relate to weighted
> >>wedgies.
> >
> > Yes, it's the weighted L_1 norm of the wedgie -- that's why it
agrees
> > with (is proportional to) the Tenney Harmonic Distance of the
> > vanishing ratio in the codimension-1 case.
>
> That's the sum-abs? And weighted? That's what I guessed you were
> doing, anyway.

Good!

> Being told doesn't make it that much clearer :P

OK, let me know what would make it clearer, then . . .

> > It's also extremely close to the "minimax Kees complexity", which
is
> > defined similarly to Graham complexity, but uses a Kees-
> > expressibility or "log-of-odd-limit" weighting and in principle
looks
> > over all intervals, not just some set of "consonances".
>
> What I'm doing now is taking the max-min of the weighted generator
> mapping (times the number of periods per octave if that's already
been
> divided out).

Isn't the min always zero (because the octave takes zero generators)?

> That's the nearest thing to odd-limit complexity without
> actually using odd-limits. Taking the straight weighted mean or
>minimax

You mean maximum?

> doesn't work, because it makes things like miracle look far too
>good.

Can you elaborate? I don't see that my numbers, which are virtually
identical to max Kees-weighted complexity according to Gene, make
miracle look far too good . . .

> So it happens that my definition of the weighted-prime complexity
is
> very similar to that of the TOP error of an equal temperament.

I've noticed some things along these lines, and posted them in '04,
but never got an explanation for them . . .

> > Sometimes, the wedgie has more elements than the full set of
mapping
> > vectors, so it's inefficient, but certainly still valuable for a
lot
> > of the calculation you might conceivably want to do.
>
> The wedgie gets dramatically larger the further you go beyond the
> 11-limit for rank 2 temperaments. But it doesn't look unmanageably
> large for rank 3 temperaments with sensible prime limits. And it
might
> mean the code can be made general for any rank.

Go for it!

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> Complexity ordering will give
> a lot of collisions, that you will have to resolve by checking the >
wedgie.

Why do you say that? I don't see how complexity collisions would be
possible if you use Tenney-weighted L_1 (or even L_inf) wedgie
complexity.

>> Hiya Paul - is there a name for this L_ notation that I could
>> look up somewhere?
>
>It's similar to that in the mathematical definitions here:
>
>http://mathworld.wolfram.com/L1-Norm.html
>http://mathworld.wolfram.com/L2-Norm.html
>http://mathworld.wolfram.com/L-Infinity-Norm.html
>
>except that we don't assume that the elements in the "vector" of
>errors have to be linearly independent. And we're specifically
>talking about the tunings that *minimize* the given "norm" on
>the "vector" of errors.

I'm trying to remember a post in which you stated there's a
difference between this sort of choice and choosing between...
something else. Bah, it was recent, but I have no idea how
to find it.

-Carl

This is great, Herman. Maybe even the best explanation I've
seen. I wish you'd flesh it out more!

-Carl

At 06:55 PM 1/5/2006, you wrote:
>Rich Holmes wrote:
>
>> Actually I'm not even sure things like wedgies are not simple
>> concepts. It may be a matter of how they're explained. Can it be
>> done more clearly than <http://66.98.148.43/~xenharmo/wedgie.html>?
>> Paul Erlich has, I think, been pretty successful at making things like
>> periodicity blocks and TOP accessible to the novice. Maybe he or
>> someone else can do likewise (or already has) with wedgies.
>
>I have a brief description of wedgies at the bottom of my Zireen Music
>page. It doesn't go into much detail, and it's (deliberately) not
>technically precise, but it does describe some reasons why they can be
>useful for dealing with temperaments.
>
>http://www.io.com/~hmiller/music/zireen-music.html
>
>Calculating wedgies is actually not that difficult; here's C code for
>wedging two "vals" (a kind of vector) of the same size (valsize),
>placing the result in the "wedgie" array.
>
>wedgiesize = 0;
>for (i = 0; i < valsize - 1; i++)
>{
> for (j = i + 1; j < valsize; j++)
> {
> wedgie[wedgiesize] = val1[i] * val2[j] - val2[i] * val1[j];
> wedgiesize++;
> }
>}
>
>These so-called "vals" are usually used to describe a tuning map, which
>specifies how many periods and generators (or any other two intervals,
>like tones and semitones) are used for approximating each of the prime
>number intervals (2/1, 3/1, 5/1, etc.) In the case of meantone, for
>instance, you have these two vals:
>
><1, 2, 4, 7] (for the period, an octave in the case of meantone)
><0, -1, -4, -10] (for the generator, a perfect fourth).
>
>(You get a different set of vals if you use a fifth as the generator,
>but the end result is the same wedgie.) What this means is that:
>
>2/1 is approximated by one octave up.
>3/1 is approximated by two octaves up and one fourth down.
>5/1 is approximated by 4 octaves up and 4 fourths down.
>7/1 is approximated by 7 octaves up and 10 fourths down.
>
>Wedge these vals and you end up with a so-called "bival", which
>identifies meantone temperament.
>
><<1, 4, 10, 4, 13, 12]]
>
>You can also wedge commas, represented in vector form as the prime
>factorization of the comma ratio:
>
>[-4, 4, -1> 81/80 (2^-4 * 3^4 * 5^-1)
>[-5, 2, 2, -1> 225/224 (2^-5 * 3^2 * 5^2 * 7^-1)
>
>[[12, -13, 4, 10, -4, 1>>
>
>which after a bit of rearranging and sign flipping ends up with the same
>result.

"wallyesterpaulrus" <perlich@aya.yale.edu> writes:

> It can be used, for example, to find out which ET results when you
> temper a certain set of commas out, or to find out which comma is
> tempered out by a certain set of ETs . . . Also it's very direct to
> get a measure of a temperament's complexity from it.

Fine, but how?

> > It also is something I wouldn't be able to do until I learned just
> > what a wedgie *is*, which still is not clear, other than that it's
> > somehow related to a wedge product.
>
> It took a while for this to become clear to me too, but the wedgie as
> I thought Gene had defined it can always be seen as an object in
> projective tuning-space.

I may not have been clear; I wasn't talking about what the wedgie's
definition *implies* or *means*, but what its definition *is*. I
still haven't seen a definition -- none that I can make any sense of,
anyway.

> More later . . .

I look forward to it.

- Rich Holmes

Graham Breed wrote:

> Oh, thanks! That is really simple. I can't currently do wedge products > in C, and this is the most useful special case. So I'll borrow this > code if that's okay.

Sure, no problem; feel free to use this. I didn't really have any need for the general case, so I just did the easy version.

Carl Lumma wrote:
> This is great, Herman. Maybe even the best explanation I've
> seen. I wish you'd flesh it out more!

I probably ought to write up a general page on regular temperaments. Maybe if I have some time over the weekend....

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

> OK, so I should have said a 4-D subset of the 5-D projective space
> that comes from the full 6-D vector space . . . yes?

Right.

> > > There's also a three-dimensional subspace within that that
> > > corresponds to zero error, so all the low-error temperaments lie
> > > quite close to that three-dimensional subspace.
> >
> > Once again, it's more complicated than that.
>
> I see I said something wrong. Could you correct it then?

Two wedgies can be right next to each other in projective space and
have competely different error properties, so this isn't a good point
of view. You can use wedge products, of course. If you wedge a 7-limit
val with <1 log2(3) log2(5) log2(7)| then you get a 6-D object which
denotes relative error; multiplying it by 1200/n, where n is the equal
division, gives cents vales. The first three are the errors of the primes.

wallyesterpaulrus wrote:

> The 7-limit wedgies, treated as vectors in a six-dimensional space, > all lie in a four-dimensional subspace of that six-dimensional space. > There's also a three-dimensional subspace within that that > corresponds to zero error, so all the low-error temperaments lie > quite close to that three-dimensional subspace. This is why three > numbers is usually enough in the 7-limit case. But certainly there > are examples of high-error temperaments where the octave-equivalent > part of the two wedgies is the same.

If there are examples, why can't anybody find one?

But this is interesting to know, anyway. What are the effective numbers of dimensions in higher limits?

I can't really visualize what distinct, practical temperaments with the same octave-equivalent mapping(s) (including octave division) but different period mappings would look like. Maybe there are some borderline cases where an interval approximates close to the period, like with inconsistent ETs. But why should we care about them?

>> And if you don't think the octave-equivalent mapping is a > practically >>unique key, adding the badness will make it so for any temperaments > we'd >>care about distinguishing. > > What sort of 'badness' do you propose to use for this, and why? > Badness is a function of both error and complexity, both of which can > be defined in several ways, as can the badness function itself. (For > those who didn't know . . .)

I'm using error*complexity for equal temperaments, and error*error*complexity for rank 2 temperaments. Along with fudges to differentiate temperaments with zero complexity, or identical error and complexity (which may never happen if the prime intervals are linearly independent). I'd have to use less fudge for them to be one-one with a temperament.

Error and complexity are what we're talking about elsewhere. If the complexity only depends on the generator mapping, naturally two temperaments with the same generator mapping will have the same complexity.

But with any badness, you can always substitute the better temperament when you have an apparent duplicate. If the mappings are the same, why would you care about the poorer one?

With contorsion, two temperaments have the same octave-equivalent mapping and optimal error. But they also have the same wedgie, so this doesn't help.

Graham

Gene Ward Smith wrote:

> If you try to uniquely define the mapping in terms of the smallest
> generator, that is not well-defined unless you also give your
> opitimization method for determing the optimal tuning and hence the
> exact size of the generator. In practice it would be.

You're arguing on a false predicate there. You don't have to use the smallest generator, as I've already said. You could use the one within a period that makes the first non-zero term of the generator mapping positive.

It's also not true that you need to optimize the tuning to get a standard generator size. I've already said this as well. If you're generating the temperament from commas, you have to supply a chromatic unison vector (CUV). The generator size is determined by the CUV. As long as you always use the same CUVs, you'll always get the same generators. And all this works with matrices -- no need for wedgies.

It's very unlikely that temperaments generated from equal temperaments won't get the generator right to the nearest period. With sufficiently inconsistent ETs, it is possible, but you could always do the optimization for those rare occasions where the generator-mapping matches but the period-mapping doesn't. It'll have a negligible effect on the running speed.

> Well, I never search lists, unless it's a list which came from ordering > a set.

Unfortunately, my C library does use a list to hold the temperaments, on the assumption that it never gets too big. That means it takes a long time to search for duplicates if I do want to remember them. So far, I can only confirm that there are none amongst the 10,536 7-limit temperaments returned by searching for a maximum complexity of 100 and a maximum error of 120 cents, derived from 100 equal temperaments with a worst error of 0.6 scale steps.

Graham

wallyesterpaulrus wrote:

> Isn't the min always zero (because the octave takes zero generators)?

No, it's usually negative because some intervals approximate different ways.

>>That's the nearest thing to odd-limit complexity without >>actually using odd-limits. Taking the straight weighted mean or >>minimax > > You mean maximum?

Er, yes.

> Can you elaborate? I don't see that my numbers, which are virtually > identical to max Kees-weighted complexity according to Gene, make > miracle look far too good . . .

Miracle has the mapping (6, -7, -2, 15, -3). Using 13-limit weighting, that becomes (12, -7, -2, 15, -3). A max-abs of those numbers gives the complexity as 15. The true complexity is 22. So you have to do max-min, not max(abs). Other kinds of average and weighting have the same problem. There's a 13-limit meantone with mapping (1,4,10,18,15) so ther the max(abs) is the same as the max(min). Under max-abs, it's no simpler than miracle although it clearly is. (As it happens, this is currently the best 13-limit temperament in my general weighted-prime RMS search. The results are substantially different to the worst case odd-limit search.)

>>So it happens that my definition of the weighted-prime complexity > is >>very similar to that of the TOP error of an equal temperament.
> > I've noticed some things along these lines, and posted them in '04, > but never got an explanation for them . . .

Oh, I can't have been following it then. I remember looking a lot at geometric complexity and not being able to understand it.

They're both octave-equivalent prime-based measures. So you have to correct for 15:8 and 5:3 having different octave-specific complexities. Either tempering the octaves or considering the full wedgie gives you a proper octave-specific measure.

Graham

wallyesterpaulrus wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > >>Complexity ordering will give >>a lot of collisions, that you will have to resolve by checking the > > > wedgie.
> > Why do you say that? I don't see how complexity collisions would be > possible if you use Tenney-weighted L_1 (or even L_inf) wedgie > complexity.

Probably, yes. So I said that because I hadn't thought of t'other.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> If there are examples, why can't anybody find one?

I gave you infinitely many a few days ago. You can find your own;
simply take the same generator map with a different octaves map, and
check it doesn't lead to the same wedgie.

> I can't really visualize what distinct, practical temperaments with the
> same octave-equivalent mapping(s) (including octave division) but
> different period mappings would look like.

Who said there are any? I only found one pair out of 32201 wedgies
which I picked to be reasonable starting points; that's a long way
from saying they were all practical.

You *could* therefore use the octave equivalent map to name
temperaments, but the question is if we want to, since then we have to
go through a calculation to get the whole wedgie and/or full mapping
back again. If, however, this is acceptable, it saves a lot of space
as it grows linearly, not quadratically.

Here's an algorithm for going from an oe map to the full wedgie, which
should be much faster than a brute force search. Take the oe map, and
use it to get the generator map and an octave map with indeterminates
in the odd primes places to solve for. Wedge these two together, and
then wedge the result with <1 log2(3) log2(5) ... log2(p)|. Now take
the coefficients of this, and solve for the least squares minimum
value; that is, take the sum of the squares, take derivatives, and
solve the resulting linear equation, subsituting the result back into
the wedge product of the generator map and the generic octave map. The
result should give coefficients close to integers, particularly if the
temperament is a good one. Round these off, reduce to the standard
form if needed, and you have your wedgie. I'll implement the code
myself, so at least I can do it. I don't imagine Graham will have any
problem.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> It's also not true that you need to optimize the tuning to get a
> standard generator size. I've already said this as well. If you're
> generating the temperament from commas, you have to supply a chromatic
> unison vector (CUV). The generator size is determined by the CUV.

(1) No you don't

(2) How do you select the CUV?

As
> long as you always use the same CUVs, you'll always get the same
> generators.

What in the world does "the same CUVs" mean?

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

> Here's an algorithm for going from an oe map to the full wedgie, which
> should be much faster than a brute force search.

I tried this out on a list of 7-limit temperaments, and the only
problem I had is that using floating point values, rather than
rational approximations, to the logarithms base two can lead to a
sitation where the computer thinks there isn't a solution but there
is, since it's set the cutoff too finely. Substiting rational number
approximations seems to cure the problem, which arose for diminised
([4, 4, 4] oe.)

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> >>If there are examples, why can't anybody find one?
> > I gave you infinitely many a few days ago. You can find your own;
> simply take the same generator map with a different octaves map, and
> check it doesn't lead to the same wedgie.

That must have bounced, as I can only receive finite email.

>>I can't really visualize what distinct, practical temperaments with the >>same octave-equivalent mapping(s) (including octave division) but >>different period mappings would look like. > > Who said there are any? I only found one pair out of 32201 wedgies
> which I picked to be reasonable starting points; that's a long way
> from saying they were all practical.

You did, that's where this whole discussion started.

> You *could* therefore use the octave equivalent map to name
> temperaments, but the question is if we want to, since then we have to
> go through a calculation to get the whole wedgie and/or full mapping
> back again. If, however, this is acceptable, it saves a lot of space
> as it grows linearly, not quadratically.

What does "name" mean? They're useful as a tag, but it's useful to remember what you were tagging.

> Here's an algorithm for going from an oe map to the full wedgie, which
> should be much faster than a brute force search. Take the oe map, and
> use it to get the generator map and an octave map with indeterminates
> in the odd primes places to solve for. Wedge these two together, and
> then wedge the result with <1 log2(3) log2(5) ... log2(p)|. Now take
> the coefficients of this, and solve for the least squares minimum
> value; that is, take the sum of the squares, take derivatives, and
> solve the resulting linear equation, subsituting the result back into
> the wedge product of the generator map and the generic octave map. The
> result should give coefficients close to integers, particularly if the
> temperament is a good one. Round these off, reduce to the standard
> form if needed, and you have your wedgie. I'll implement the code
> myself, so at least I can do it. I don't imagine Graham will have any
> problem.

Sounds good, yes.

Anyway, my big C search did get some results. Here's a fairly simple duplicate pair:

3/7

401.6 cents period
152.6 cents generator

mapping by period and generator:
[(3, 0), (5, -1), (7, 0), (10, -4)]

mapping by steps:
[(15, 6), (23, 9), (35, 14), (42, 16)]

complexity measure: 4.274
RMS weighted error: 0.014266
max weighted error: 0.024524
TOP damage 20.743

7/8

397.3 cents period
340.1 cents generator

mapping by period and generator:
[(3, 0), (4, 1), (7, 0), (5, 4)]

mapping by steps:
[(21, 3), (34, 5), (49, 7), (59, 9)]

complexity measure: 4.274
RMS weighted error: 0.008632
max weighted error: 0.014490
TOP damage 12.565

and here's a relatively undamaged one:

4/9

133.7 cents period
56.0 cents generator

mapping by period and generator:
[(9, 0), (15, -2), (23, -5), (26, -2)]

mapping by steps:
[(63, 18), (99, 28), (146, 41), (176, 50)]

complexity measure: 19.380
RMS weighted error: 0.003128
max weighted error: 0.004483
TOP damage 4.464

5/6

133.0 cents period
109.3 cents generator

mapping by period and generator:
[(9, 0), (16, -2), (25, -5), (27, -2)]

mapping by steps:
[(45, 9), (72, 14), (105, 20), (127, 25)]

complexity measure: 19.380
RMS weighted error: 0.002763
max weighted error: 0.003993
TOP damage 3.953

There are lots more, too many to list here. None of them are any good.

Graham

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> >>It's also not true that you need to optimize the tuning to get a >>standard generator size. I've already said this as well. If you're >>generating the temperament from commas, you have to supply a chromatic >>unison vector (CUV). The generator size is determined by the CUV. > > (1) No you don't

Don't I? How not?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> > Who said there are any? I only found one pair out of 32201 wedgies
> > which I picked to be reasonable starting points; that's a long way
> > from saying they were all practical.
>
> You did, that's where this whole discussion started.

Did not.

> > You *could* therefore use the octave equivalent map to name
> > temperaments, but the question is if we want to, since then we have to
> > go through a calculation to get the whole wedgie and/or full mapping
> > back again. If, however, this is acceptable, it saves a lot of space
> > as it grows linearly, not quadratically.
>
> What does "name" mean? They're useful as a tag, but it's useful to
> remember what you were tagging.

A "name" is a unique identifier. A mapping to primes isn't a "name"
because more than one mapping denotes the same temperament. Same with
N&M, where N and M are vals or numbers denoting divisions of the octave.

> Anyway, my big C search did get some results. Here's a fairly simple
> duplicate pair:

> mapping by period and generator:
> [(3, 0), (5, -1), (7, 0), (10, -4)]
>
> mapping by steps:
> [(15, 6), (23, 9), (35, 14), (42, 16)]

> mapping by period and generator:
> [(3, 0), (4, 1), (7, 0), (5, 4)]
>
> mapping by steps:
> [(21, 3), (34, 5), (49, 7), (59, 9)]

Using the algorithm I suggested you get

<<3 0 12 -6.97 10.60 27.86||

as the thing to round off. Rounding 10.6 to 11 gives the first system,
rounding it to 10 the second.

> and here's a relatively undamaged one:

> mapping by period and generator:
> [(9, 0), (15, -2), (23, -5), (26, -2)]
>
> mapping by steps:
> [(63, 18), (99, 28), (146, 41), (176, 50)]

> mapping by period and generator:
> [(9, 0), (16, -2), (25, -5), (27, -2)]
>
> mapping by steps:
> [(45, 9), (72, 14), (105, 20), (127, 25)]

Doing the same thing, the pre-wedgie looks like

<<18 45 18 29.53 -22.00 -84.54||

The wedgie for the first warped ennealimmal is

<<18 45 18 29 -22 -84||

and for the second is

<<18 45 18 30 -22 -85||

> There are lots more, too many to list here. None of them are any good.

There's a limit on how good they can be; in fact the difference
between the pre-wedgie and the nearest integer values serves as a (non
logflat) badness measure.

Incidentially, in the algorithm I gave I suppose you should check when
done that you actually have a wedgie. If you tried to round off to
either <<18 45 18 29 -22 85|| or <<18 45 18 30 -22 84|| it's *not* a
wedgie.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> Gene Ward Smith wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> >
> >>It's also not true that you need to optimize the tuning to get a
> >>standard generator size. I've already said this as well. If you're
> >>generating the temperament from commas, you have to supply a
chromatic
> >>unison vector (CUV). The generator size is determined by the CUV.
> >
> > (1) No you don't
>
> Don't I? How not?

You don't need to pick a CUV, since you can choose a generator on
other grounds, such as Hermite generators or smallest generators.

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

> Two wedgies can be right next to each other in projective space and
> have competely different error properties, so this isn't a good point
> of view. You can use wedge products, of course. If you wedge a 7-limit
> val with <1 log2(3) log2(5) log2(7)| then you get a 6-D object which
> denotes relative error; multiplying it by 1200/n, where n is the equal
> division, gives cents vales. The first three are the errors of the
primes.

I screwed up my answer to this; since by "error properties" I had
error relative to complexity on the brain, in other words, badness.
You can get arbitrarily close to a good temperament and be arbitrarily
bad, because of complexity. But even there it's not as bad as I was
thinking.

The second part of what I said shows how to approach what Paul wants;
do the same thing with the bival as I said to do with the val, namely
wedge it with <1 log2(3) log2(5) log2(7)|. The result will be a 4-D
object, and squaring the coordinates will give a quadric hypersurface,
where wedgies on the wedgie hypersurface which get close to this
hypersurface will do what Paul wants, namely be more or less good.
Appropriately weighting things can even improve the metric here.

Gene Ward Smith wrote:

> You don't need to pick a CUV, since you can choose a generator on
> other grounds, such as Hermite generators or smallest generators.

Is this "Hermite generators" like Hermite normal form? I think I was thrown before because "Hermite reduction" has a different meaning. Hermite normal form, anyway, is the same as using the smallest linearly independent odd prime as the CUV.

You may be able to find the smallest generators without using a CUV, but I can't, and won't likely be able to until you tell me how.

Graham

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > >>>Who said there are any? I only found one pair out of 32201 wedgies
>>>which I picked to be reasonable starting points; that's a long way
>>>from saying they were all practical.
>>
>>You did, that's where this whole discussion started.
> > > Did not.

6th Jan 2006
"""
It's tough, since the same generator mapping numbers need to go to two
different period mapping numbers. Hence the tuning for at least one
must be either atrocious, or the temperaments must be complex. It can
happen in a more or less practical context, however. I have an old
list of 32201 seven-limit temperaments I used as a preliminary list
before eliminaing almost all of them, and somewhere on that list is a
pair of temperaments with the same octave-equivalent numbers. Of
course if I had been using octave-equivalent numbers the whole thing
would have taken too long anyway; it's not a practical system for that
sort of thing.
"""

Okay, "more or less practical"

> A "name" is a unique identifier. A mapping to primes isn't a "name"
> because more than one mapping denotes the same temperament. Same with
> N&M, where N and M are vals or numbers denoting divisions of the octave.

So your "name" isn't Gene?

The generator mapping (including hcf) uniquely identifies all temperaments of practical interest.

> There's a limit on how good they can be; in fact the difference
> between the pre-wedgie and the nearest integer values serves as a (non
> logflat) badness measure. Ah, now we're getting somewhere! So it's possible to theoretically that ambiguous temperaments aren't practically interesting?

> Incidentially, in the algorithm I gave I suppose you should check when
> done that you actually have a wedgie. If you tried to round off to
> either <<18 45 18 29 -22 85|| or <<18 45 18 30 -22 84|| it's *not* a
> wedgie.

It's good that it works, anyway. Is a good temperament more likely to round off to a true wedgie?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> You may be able to find the smallest generators without using a CUV,
but
> I can't, and won't likely be able to until you tell me how.

Eh? You know how--once you have the period and any generator, then
(speaking additively) make the generator positive, subtract off as
many periods as divide the generator with remainder, then take the
smallest between g and p-g, where g is your penultimate generator.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> Okay, "more or less practical"

I said the *context* was more or less practical, not the temperaments
in question.

> Ah, now we're getting somewhere! So it's possible to theoretically
that
> ambiguous temperaments aren't practically interesting?

The only problem might be in Herman Miller type low complexity
temperaments. Anything at all high with a decent badness cutoff will
be too good to be ambiguous.

On the other hand, start with tempering out 9/8 and 7/6, which gives
you wedgie <<0 2 0 3 0 -5||. The pre-wedgie for <0 2 0| is
<<0 2 0 2*log2(3) 0 2*log2(7)||, which is about <<0 2 0 3.17 0 -5.61||
Rounding off to the nearest integer gives <<0 2 0 3 0 -6||, which
checks out as being a wedgie. In fact, it is the wedgie for the
temperament tempering out 8/7 and 9/7. These kinds of examples are all
over the place in extremely low complexity.

> It's good that it works, anyway. Is a good temperament more likely to
> round off to a true wedgie?

Any decent temperament must round off to a true wedgie, by most
notions of "decent". If you define "decent" to simply mean the
pre-wedgie rounds off to the correct wedgie, that would be a
tautology, and then the question would be what can you show about
"decent" temperaments. I don't think we would lose anything of value
by defining rank one temperaments to simply be those in which the
pre-wedgie rounds off to the wedgie; this would at least take away the
murky boundry. It is analogous to defining an equal temperament in a
given p-limit as by definition a standard val.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> >>You may be able to find the smallest generators without using a CUV,
> but >>I can't, and won't likely be able to until you tell me how.
> > Eh? You know how--once you have the period and any generator, then
> (speaking additively) make the generator positive, subtract off as
> many periods as divide the generator with remainder, then take the
> smallest between g and p-g, where g is your penultimate generator.

Whoa! Where did this period and generator come from? I use the CUV to find the period and generator. I'm asking you how to find the period and generator without a CUV, and you tell me I've already got them???

Graham

Gene Ward Smith wrote:

> The only problem might be in Herman Miller type low complexity
> temperaments. Anything at all high with a decent badness cutoff will
> be too good to be ambiguous.
> > On the other hand, start with tempering out 9/8 and 7/6, which gives
> you wedgie <<0 2 0 3 0 -5||. The pre-wedgie for <0 2 0| is > <<0 2 0 2*log2(3) 0 2*log2(7)||, which is about <<0 2 0 3.17 0 -5.61||
> Rounding off to the nearest integer gives <<0 2 0 3 0 -6||, which
> checks out as being a wedgie. In fact, it is the wedgie for the
> temperament tempering out 8/7 and 9/7. These kinds of examples are all
> over the place in extremely low complexity. They're in the realm where a scale large enough to play the chords you're optimizing for is going to be less accurate than the same sized equal temperament. That's the kind of thing I'd call "not a practical temperament".

> Any decent temperament must round off to a true wedgie, by most
> notions of "decent". If you define "decent" to simply mean the
> pre-wedgie rounds off to the correct wedgie, that would be a
> tautology, and then the question would be what can you show about
> "decent" temperaments. I don't think we would lose anything of value
> by defining rank one temperaments to simply be those in which the
> pre-wedgie rounds off to the wedgie; this would at least take away the
> murky boundry. It is analogous to defining an equal temperament in a
> given p-limit as by definition a standard val.

How standard does a standard val have to be? Insisting on rounding to a correct wedgie is like insisting on consistency. There's plenty of life in inconsistent temperaments. Having only one temperament per octave-equivalent mapping is the same as having only one equal temperament for a given octave division. Which is fine most of the time, different inconsistent ET mappings are useful in my linear temperament search. Perhaps I'd find indecent linear temperaments also become important when searching for planar temperaments. 24 and 34 in the 7-limit as well.

What do you mean by "rank one temperaments"? I thought "rank 2" was the new name for linear temperaments, so wouldn't rank one temperaments be equal temperaments?

Anyway, I switched my wedgie search to using the full wedgie as a the unique key, and I found one ambiguous pair got into a top 100. This one:

(0, 0, 0, 0, 0, 0, 0, 0, 46, 0)

It's the 76th best inconsistently generated 31-limit temperament. But it looks exactly the same as number 33 in the same list. I don't know what the mappings are, but I have reason to believe the TOP error of the poor one is the same as that for 92-equal. The moral is that good 31-limit temperaments are hard to find, and you soon end up amongst the noise. Quite possibly the whole idea of 31-limit linear temperament's absurd, but here's one that stands out:

261/286

1200.0 cents period
1095.1 cents generator

mapping by period and generator:
[(1, 0), (39, -41), (37, -38), (53, -55), (50, -51), (11, -8), (5, -1), (59, -60), (10, -6), (66, -67), (88, -91)]

mapping by steps:
[(183, 103), (290, 163), (425, 239), (514, 289), (633, 356), (677, 381), (748, 421), (777, 437), (828, 466), (889, 500), (907, 510)]

complexity measure: 25.868
RMS weighted error: 0.000274
max weighted error: 0.000517

And with odd-limit values:

25/286, 104.9 cent generator

basis:
(1.0, 0.087419227697998383)

mapping by period and generator:
[(1, 0), (-2, 41), (-1, 38), (-2, 55), (-1, 51), (3, 8), (4, 1), (-1, 60), (4, 6), (-1, 67), (-3, 91)]

mapping by steps:
[(183, 103), (290, 163), (425, 239), (514, 289), (633, 356), (677, 381), (748, 421), (777, 437), (828, 466), (889, 500), (907, 510)]

highest interval width: 123
complexity measure: 123 (183 for smallest MOS)
highest error: 0.003727 (4.473 cents)

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> 261/286
>
> 1200.0 cents period
> 1095.1 cents generator

Cool. It has a nonstandard 286 val:

<286 453 664 803 989 1058 1169 1214 1294 1389 1417|

The mapping for 19 differs, and is worse; however it leads to lower
complexity. The generator is 17/16, so it doesn't make sense unless
you go at least that high, and up to the 17-limit we are in standard
val territory; it's 80&103 in terms of standard vals, 286 is standard
also.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >> Hiya Paul - is there a name for this L_ notation that I could
> >> look up somewhere?
> >
> >It's similar to that in the mathematical definitions here:
> >
> >http://mathworld.wolfram.com/L1-Norm.html
> >http://mathworld.wolfram.com/L2-Norm.html
> >http://mathworld.wolfram.com/L-Infinity-Norm.html
> >
> >except that we don't assume that the elements in the "vector" of
> >errors have to be linearly independent. And we're specifically
> >talking about the tunings that *minimize* the given "norm" on
> >the "vector" of errors.
>
> I'm trying to remember a post in which you stated there's a
> difference between this sort of choice and choosing between...
> something else.

Different choices for how to weight the various intervals?

> Bah, it was recent, but I have no idea how
> to find it.
>
> -Carl

I think it was on MMM.

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> "wallyesterpaulrus" <perlich@a...> writes:
>
> > It can be used, for example, to find out which ET results when
you
> > temper a certain set of commas out, or to find out which comma is
> > tempered out by a certain set of ETs . . . Also it's very direct
to
> > get a measure of a temperament's complexity from it.
>
> Fine, but how?

For the former, just wedge . . . For the latter, what I used in my
paper was a weighted sum of the absolute values of the elements in
the wedgie. The weight on each component of the wedgie is inversely
proportional to the product of the logs of the associated primes. We
can discuss this in more detail if you wish . . .

> > > It also is something I wouldn't be able to do until I learned
just
> > > what a wedgie *is*, which still is not clear, other than that
it's
> > > somehow related to a wedge product.
> >
> > It took a while for this to become clear to me too, but the
wedgie as
> > I thought Gene had defined it can always be seen as an object in
> > projective tuning-space.
>
> I may not have been clear; I wasn't talking about what the wedgie's
> definition *implies* or *means*, but what its definition *is*. I
> still haven't seen a definition -- none that I can make any sense
of,
> anyway.

Be patient -- we'll get there!

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > OK, so I should have said a 4-D subset of the 5-D projective
space
> > that comes from the full 6-D vector space . . . yes?
>
> Right.
>
> > > > There's also a three-dimensional subspace within that that
> > > > corresponds to zero error, so all the low-error temperaments
lie
> > > > quite close to that three-dimensional subspace.
> > >
> > > Once again, it's more complicated than that.
> >
> > I see I said something wrong. Could you correct it then?
>
> Two wedgies can be right next to each other in projective space and
> have competely different error properties, so this isn't a good
point
> of view.

But I wasn't talking about a projective space there . . .

> You can use wedge products, of course. If you wedge a 7-limit
> val with <1 log2(3) log2(5) log2(7)| then you get a 6-D object which
> denotes relative error;

Defined how? I'm wondering if there's something here I didn't know
before and need to learn . . .

> multiplying it by 1200/n, where n is the equal
> division,

What equal division? Which equal division?

> gives cents vales. The first three are the errors of the primes.

What does this have to do with low-error wedgies being or not being
near a particular hypersurface?

P.S. NONE OF MY QUESTIONS ARE RHETORICAL.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> wallyesterpaulrus wrote:
>
> > Isn't the min always zero (because the octave takes zero
generators)?
>
> No, it's usually negative because some intervals approximate
different ways.

Oh, you're looking at signed generator mappings . . .

> >>That's the nearest thing to odd-limit complexity without
> >>actually using odd-limits. Taking the straight weighted mean or
> >>minimax
> >
> > You mean maximum?
>
> Er, yes.
>
> > Can you elaborate? I don't see that my numbers, which are
virtually
> > identical to max Kees-weighted complexity according to Gene, make
> > miracle look far too good . . .
>
> Miracle has the mapping (6, -7, -2, 15, -3). Using 13-limit
weighting,
> that becomes (12, -7, -2, 15, -3). A max-abs of those numbers
gives the
> complexity as 15. The true complexity is 22. So you have to do
> max-min, not max(abs).

I don't know what you're comparing here or why. This bears no
similarity to my complexity calculation . . . I think you must be
talking to someone else here :)

> Other kinds of average and weighting have the
> same problem. There's a 13-limit meantone with mapping
(1,4,10,18,15)
> so ther the max(abs) is the same as the max(min). Under max-abs,
it's
> no simpler than miracle although it clearly is.

Again, I see no resemblance with my calculations . . .

> (As it happens, this is
> currently the best 13-limit temperament in my general weighted-
prime RMS
> search. The results are substantially different to the worst case
> odd-limit search.)

Perhaps you could offer some 7-limit examples so we can use my paper
to compare?

> They're both octave-equivalent prime-based measures. So you have
to
> correct for 15:8 and 5:3 having different octave-specific
complexities.

I don't know what context I'm supposed to take the above in. My
complexity numbers are virtually identical to the max-Kees complexity
(according to Gene), which is an octave-equivalent measure despite my
measure being octave-specific.

> Either tempering the octaves or considering the full wedgie gives
you
> a proper octave-specific measure.

?

BTW, a chromatic unison (vector) is supposed to be a small interval
that functions as a sharp or flat in a diatonic-like scale. Not a
fifth, not a fourth, not a third, not a second, but a *unison* --
thus the name. How this term came to mean something completely
different (which has no recognizable relationship with the term
itself) which can include fifths or even octaves is baffling to me;
hence I continue to rail against this random act of ridiculousness.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...>
wrote:
>
> > It's also not true that you need to optimize the tuning to get a
> > standard generator size. I've already said this as well. If
you're
> > generating the temperament from commas, you have to supply a
chromatic
> > unison vector (CUV). The generator size is determined by the
CUV.
>
> (1) No you don't
>
> (2) How do you select the CUV?
>
> As
> > long as you always use the same CUVs, you'll always get the same
> > generators.
>
> What in the world does "the same CUVs" mean?
>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > Two wedgies can be right next to each other in projective space
and
> > have competely different error properties, so this isn't a good
point
> > of view. You can use wedge products, of course. If you wedge a 7-
limit
> > val with <1 log2(3) log2(5) log2(7)| then you get a 6-D object
which
> > denotes relative error; multiplying it by 1200/n, where n is the
equal
> > division, gives cents vales. The first three are the errors of the
> primes.
>
> I screwed up my answer to this; since by "error properties" I had
> error relative to complexity on the brain, in other words, badness.
> You can get arbitrarily close to a good temperament and be
arbitrarily
> bad, because of complexity. But even there it's not as bad as I was
> thinking.
>
> The second part of what I said shows how to approach what Paul
wants;
> do the same thing with the bival as I said to do with the val,
namely
> wedge it with <1 log2(3) log2(5) log2(7)|. The result will be a 4-D
> object, and squaring the coordinates will give a quadric
hypersurface,
> where wedgies on the wedgie hypersurface which get close to this
> hypersurface will do what Paul wants, namely be more or less good.
> Appropriately weighting things can even improve the metric here.

So how can we alter my original statement so that it becomes correct?

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> Gene Ward Smith wrote:
>
> > You don't need to pick a CUV, since you can choose a generator on
> > other grounds, such as Hermite generators or smallest generators.
>
> Is this "Hermite generators" like Hermite normal form? I think I was
> thrown before because "Hermite reduction" has a different meaning.
> Hermite normal form, anyway, is the same as using the smallest
linearly
> independent odd prime as the CUV.

How can a small odd prime be a chromatic unison vector? Chromatic
unison vectors are how things like sharps and flats are represented in
the lattice. This is crazy!

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

> > You can use wedge products, of course. If you wedge a 7-limit
> > val with <1 log2(3) log2(5) log2(7)| then you get a 6-D object which
> > denotes relative error;
>
> Defined how? I'm wondering if there's something here I didn't know
> before and need to learn . . .

The above *is* a defintion, isn't it?

Let's say I do this with <12 19 28 34|; I get

<cents(2) cents(3) cents(5) cents(7)|^<12 19 28 34| =
<<-23.46 164.24 374.09 314.78 658.78 407.54||

These are what I was calling "relative cents" until you objected from
one point of view, though they are in fact the absolute size in cents
of two-prime commas of 12-et. Dividing through by 12, we get absolute
cents:

<<-1.955 13.686 31.714 26.232 54.898 33.962||

The first three coefficients are the errors in cents of 3, 5, and 7 in
terms of an equal division of the octave into 12 parts.

> What equal division? Which equal division?

The one the val is for.

> What does this have to do with low-error wedgies being or not being
> near a particular hypersurface?

The point is you can do something analogous for wedgies.

"wallyesterpaulrus" <perlich@aya.yale.edu> writes:

> --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
> >
> > I may not have been clear; I wasn't talking about what the wedgie's
> > definition *implies* or *means*, but what its definition *is*. I
> > still haven't seen a definition -- none that I can make any sense
> of,
> > anyway.
>
> Be patient -- we'll get there!

Patience, my ass... I've got that part figured out, finally.

But when Gene Ward Smith says something like

> Negri <<4, -3, 2, -14, -8, 13|| 7 12.188571

(I fixed the brackets) it's still not clear to me what I'm supposed to
do with that information.

Or more generally: if all I know about a temperament is its wedgie,
what can I extract from it?

- Rich Holmes

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > > You can use wedge products, of course. If you wedge a 7-limit
> > > val with <1 log2(3) log2(5) log2(7)| then you get a 6-D object
which
> > > denotes relative error;
> >
> > Defined how? I'm wondering if there's something here I didn't
know
> > before and need to learn . . .
>
> The above *is* a defintion, isn't it?

See below . . .

> Let's say I do this with <12 19 28 34|; I get
>
> <cents(2) cents(3) cents(5) cents(7)|^<12 19 28 34| =
> <<-23.46 164.24 374.09 314.78 658.78 407.54||
>
> These are what I was calling "relative cents" until you objected
from
> one point of view, though they are in fact the absolute size in
cents
> of two-prime commas of 12-et. Dividing through by 12, we get
absolute
> cents:
>
> <<-1.955 13.686 31.714 26.232 54.898 33.962||
>
> The first three coefficients are the errors in cents of 3, 5, and 7
in
> terms of an equal division of the octave into 12 parts.

Now see, when you said "Relative error", and I said "Defined how?",
you could have given this definition of relative error instead of
saying "The above *is* a defintion, isn't it?" There's a long, long
distance between the ease of understanding a definition that's stated
in terms of the errors of primes in an ET vs. a definition that's
stated in terms of a 6-dimensional wedge product. For most purposes,
I would think the former is preferable, and if the latter happens to
be equal to that, that should be taken as a mathematical *result*,
not a definition . . .

So, what about the other three coefficients?

>> What equal division? Which equal division?

>The one the val is for.

Oh, OK. I goofed. But in this case, aren't you interpreting that
object more as a "tuning map" and less as a "val"? Since it seems
you're specifically identifying it with pure-octaves 12-equal . . .

> > What does this have to do with low-error wedgies being or not
being
> > near a particular hypersurface?
>
> The point is you can do something analogous for wedgies.

I wish I could say I was enlightened. Graham responded to my
hypersurface description which you said was wrong, and I still don't
know how to respond to his response, since I don't know how to
correct my description.

At 03:49 PM 1/10/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>
>> >> Hiya Paul - is there a name for this L_ notation that I could
>> >> look up somewhere?
//
>> I'm trying to remember a post in which you stated there's a
>> difference between this sort of choice and choosing between...
>> something else.
>
>Different choices for how to weight the various intervals?

I don't think so...

>> Bah, it was recent, but I have no idea how
>> to find it.
>>
>> -Carl
>
>I think it was on MMM.

Oh, thanks! Maybe it was this (though I thought what I'm
remembering was addressed to me, perhaps back when we were
talking about hexagons vs. circles on tuning, whereas this
was addressed to Gene):

>>Because different people have different tastes, this is a question
>>with no answer, I think. The minimax tuning for porcupine has a
>>stronger theoretical claim, but I think porcupine actually sounds
>>better in 22-et.
//
>Minimax vs. RMS vs. Sum-abs is another dimension along which one can
>vary the definition of "optimal", in addition to the dimension I was
>referring to above, and others. So one example is not enough to
>determine what exactly it is about our "optimality" criteria that we
>wish to change. Anyway, let's continue on tuning-math or tuning.

How is minimax in the 1st paragraph different from that in the 2nd?

-Carl

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> "wallyesterpaulrus" <perlich@a...> writes:
>
> > --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...>
wrote:
> > >
> > > I may not have been clear; I wasn't talking about what the
wedgie's
> > > definition *implies* or *means*, but what its definition *is*.
I
> > > still haven't seen a definition -- none that I can make any
sense
> > of,
> > > anyway.
> >
> > Be patient -- we'll get there!
>
> Patience, my ass... I've got that part figured out, finally.
>
> But when Gene Ward Smith says something like
>
> > Negri <<4, -3, 2, -14, -8, 13|| 7 12.188571
>
> (I fixed the brackets) it's still not clear to me what I'm supposed
to
> do with that information.

Especially if he doesn't label the columns :)

> Or more generally: if all I know about a temperament is its wedgie,
> what can I extract from it?

Everything! But in many cases, it would indeed be more convenient to
give the mapping from generator and period to the primes instead;
it's a bit of work to derive this from the wedgie.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> At 03:49 PM 1/10/2006, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>
> >> >> Hiya Paul - is there a name for this L_ notation that I could
> >> >> look up somewhere?
> //
> >> I'm trying to remember a post in which you stated there's a
> >> difference between this sort of choice and choosing between...
> >> something else.
> >
> >Different choices for how to weight the various intervals?
>
> I don't think so...
>
> >> Bah, it was recent, but I have no idea how
> >> to find it.
> >>
> >> -Carl
> >
> >I think it was on MMM.
>
> Oh, thanks! Maybe it was this (though I thought what I'm
> remembering was addressed to me, perhaps back when we were
> talking about hexagons vs. circles on tuning, whereas this
> was addressed to Gene):
>
> >>Because different people have different tastes, this is a question
> >>with no answer, I think. The minimax tuning for porcupine has a
> >>stronger theoretical claim, but I think porcupine actually sounds
> >>better in 22-et.
> //
> >Minimax vs. RMS vs. Sum-abs is another dimension along which one
can
> >vary the definition of "optimal", in addition to the dimension I
was
> >referring to above, and others. So one example is not enough to
> >determine what exactly it is about our "optimality" criteria that
we
> >wish to change. Anyway, let's continue on tuning-math or tuning.
>
> How is minimax in the 1st paragraph different from that in the 2nd?
>
> -Carl

Why do you think it's different?

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...> wrote:
>
[snip]
> > Patience, my ass... I've got that part figured out, finally.
> >
> > But when Gene Ward Smith says something like
> >
> > > Negri <<4, -3, 2, -14, -8, 13|| 7 12.188571
> >
> > (I fixed the brackets) it's still not clear to me what I'm supposed to
> > do with that information.
>
> Especially if he doesn't label the columns :)
>
> > Or more generally: if all I know about a temperament is its wedgie,
> > what can I extract from it?
>
> Everything! But in many cases, it would indeed be more convenient to
> give the mapping from generator and period to the primes instead;
> it's a bit of work to derive this from the wedgie.

Delurking briefly to say that so far I've seen a lot of postings containing wedgies for
various temperaments, and defining what a wedgie is, but so far haven't seen what I'd
consider the most useful bit of information: how to take a wedgie and extract enough info
from it to form a usable scale. Any takers?

bfowol wrote:
> Delurking briefly to say that so far I've seen a lot of postings containing wedgies for > various temperaments, and defining what a wedgie is, but so far haven't seen what I'd > consider the most useful bit of information: how to take a wedgie and extract enough info > from it to form a usable scale. Any takers?

See the thread "Understanding wedgies". Gene gave a pretty good explanation of this. I'd be happy to clarify anything, now that I understand it.

Keenan

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

> > <<-1.955 13.686 31.714 26.232 54.898 33.962||
> >
> > The first three coefficients are the errors in cents of 3, 5, and 7
> in
> > terms of an equal division of the octave into 12 parts.

> So, what about the other three coefficients?

The first three can be written 100*(19*log2(2)-12*log2(3)),
100*(28*log2(2)-12*log2(5)), 100*(34*log2(2)-12*log2(7)), giving the
error in cents. The next three are 100*(28*log2(3)-19*log2(5)),
100*(34*log2(3)-19*log2(7)), 100*(34*log2(5)-28*log2(7)); more or less
they are weighted errors.

> Oh, OK. I goofed. But in this case, aren't you interpreting that
> object more as a "tuning map" and less as a "val"? Since it seems
> you're specifically identifying it with pure-octaves 12-equal . . .

Yes, you may certainly view this as wedging the tuning map with the
JIP and getting the error.

At 06:02 PM 1/10/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>>
>> At 03:49 PM 1/10/2006, you wrote:
>> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>> >>
>> >> >> Hiya Paul - is there a name for this L_ notation that I could
>> >> >> look up somewhere?
>> //
>> >> I'm trying to remember a post in which you stated there's a
>> >> difference between this sort of choice and choosing between...
>> >> something else.
>> >
>> >Different choices for how to weight the various intervals?
>>
>> I don't think so...
>>
>> >> Bah, it was recent, but I have no idea how
>> >> to find it.
>> >>
>> >> -Carl
>> >
>> >I think it was on MMM.
>>
>> Oh, thanks! Maybe it was this (though I thought what I'm
>> remembering was addressed to me, perhaps back when we were
>> talking about hexagons vs. circles on tuning, whereas this
>> was addressed to Gene):
>>
>> >>Because different people have different tastes, this is a question
>> >>with no answer, I think. The minimax tuning for porcupine has a
>> >>stronger theoretical claim, but I think porcupine actually sounds
>> >>better in 22-et.
>> //
>> >Minimax vs. RMS vs. Sum-abs is another dimension along which one
>> >can vary the definition of "optimal", in addition to the dimension
>> >I was referring to above, and others. So one example is not enough
>> >to determine what exactly it is about our "optimality" criteria
>> >that we wish to change. Anyway, let's continue on tuning-math or
>> >tuning.
>>
>> How is minimax in the 1st paragraph different from that in the 2nd?
>>
>> -Carl
>
>Why do you think it's different?

"Minimax vs. RMS vs. Sum-abs is another dimension along which one
can vary the definition of "optimal", in addition to the dimension
I was referring to above, and others."

-Carl

Keenan Pepper <keenanpepper@gmail.com> writes:

> bfowol wrote:
> > Delurking briefly to say that so far I've seen a lot of postings containing wedgies for
> > various temperaments, and defining what a wedgie is, but so far haven't seen what I'd
> > consider the most useful bit of information: how to take a wedgie and extract enough info
> > from it to form a usable scale. Any takers?
>
> See the thread "Understanding wedgies". Gene gave a pretty good explanation of
> this. I'd be happy to clarify anything, now that I understand it.

I don't think I understood every bit of that post, but it looks like
it did not tell you how to do anything with a wedgie if you don't
already know the period and generator.

If it did, then I need clarification.

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> I don't think I understood every bit of that post, but it looks like
> it did not tell you how to do anything with a wedgie if you don't
> already know the period and generator.

What things do you want to do?

wallyesterpaulrus wrote:
> BTW, a chromatic unison (vector) is supposed to be a small interval > that functions as a sharp or flat in a diatonic-like scale. Not a > fifth, not a fourth, not a third, not a second, but a *unison* -- > thus the name. How this term came to mean something completely > different (which has no recognizable relationship with the term > itself) which can include fifths or even octaves is baffling to me; > hence I continue to rail against this random act of ridiculousness.

We had this discussion before, and I don't think it got anywhere. Certainly nobody convinced me that there was a better term than "unison vector" for unison vectors (and I note you didn't propose one there).

The earliest reference I have in English is this from Fokker:

http://www.xs4all.nl/~huygensf/doc/fokkerpb.html

"All pairs of notes differing by octaves only are considered unisons. There are other pairs of notes which in musical practice are considered to be unisonous...The vectors, which in the harmonic note lattice connect such notes taken for unisons will be called unison vectors."

Nothing about them being small, and almost includes octaves (not quite because they aren't vectors in Fokker's scheme). So that's your answer about how they "came to mean something completely different" -- it's people making the entirely reasonable assumption that chromatic unison vectors are unison vectors.

At least, saying "you don't need a chromatic unison vector because you can use a vector too large to count as a unison" is evading the question.

Now, the "chromatic" part of "chromatic unison vector" is wrong. They're considered to be unisonous in a diatonic scale, and so should be "diatonic unison vectors". Although in this context it doesn't matter if they lead to a diatonic or chromatic scale. But if you want to claim ownership of "chromatic unison vector" I'll switch to "diatonic unison vector" and reclaim Fokker's original meaning.

It also follows that commas and octaves can be classed as "enharmonic unison vectors".

Graham

Rich Holmes wrote:
> I don't think I understood every bit of that post, but it looks like
> it did not tell you how to do anything with a wedgie if you don't
> already know the period and generator.
> > If it did, then I need clarification.
> > - Rich Holmes

But that's the point, you can use any period and generator (really just two independent generators) you want. For example, meantone is generated by the octave and the fifth, or the octave and the fourth, or the fifth and the fourth, or the fourth and the whole tone. The only intervals that can't be generators are multiples of other intervals, for example the minor seventh can't be a generator because it's twice a fourth.

The wedgie is so useful because it describes a temperament independantly of its generators.

Keenan

--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:

> The wedgie is so useful because it describes a temperament
independantly of its
> generators.

You can also compute the tuning map from the wedgie independently of
generators, and it's the tuning map which tells you how to tune
things. It's also extremely easy directly from the wedgie to find the
ets which support the temperament in question, and easy to find commas
for it.

Keenan Pepper <keenanpepper@gmail.com> writes:

> But that's the point, you can use any period and generator (really just two
> independent generators) you want.

Either I'm completely misunderstanding something, or this is
incorrect.

> For example, meantone is generated by the
> octave and the fifth, or the octave and the fourth, or the fifth and the fourth,
> or the fourth and the whole tone.

But not the octave and the whole tone. Nor the octave and the semitone.

> The only intervals that can't be generators
> are multiples of other intervals, for example the minor seventh can't be a
> generator because it's twice a fourth.

Since every interval is twice some other interval, I suspect this
needs to be stated more carefully.

Obviously I'm not understanding something, either because you've
misstated it or because I've misunderstood what you're saying.
Clarification welcome.

- Rich Holmes

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> You can also compute the tuning map from the wedgie independently of
> generators, and it's the tuning map which tells you how to tune
> things. It's also extremely easy directly from the wedgie to find the
> ets which support the temperament in question, and easy to find commas
> for it.

Which answers your question about what I want to know how to do with a
wedgie. I think I understand how to find commas. The rest I'd like
to see an explanation of.

- Rich Holmes

Rich Holmes wrote:
> Keenan Pepper <keenanpepper@gmail.com> writes:
> > >>But that's the point, you can use any period and generator (really just two >>independent generators) you want. > > > Either I'm completely misunderstanding something, or this is
> incorrect.

What I meant was: Pick any meantone interval that's not a multiple of another meantone interval. Then you can find some other interval that generates all of meantone together with that one.

>>For example, meantone is generated by the >>octave and the fifth, or the octave and the fourth, or the fifth and the fourth, >>or the fourth and the whole tone. > > > But not the octave and the whole tone. Nor the octave and the semitone.

Right, you can't pick any pair of intervals, but you can pick one and then there will be many (I think infinitely many?) choices for the other one. I think the determinant of the matrix has to be 1 or something? I haven't even taken linear algebra yet, give me a break!

>>The only intervals that can't be generators >>are multiples of other intervals, for example the minor seventh can't be a >>generator because it's twice a fourth.
> > > Since every interval is twice some other interval, I suspect this
> needs to be stated more carefully.

Not twice some _meantone_ interval.

> Obviously I'm not understanding something, either because you've
> misstated it or because I've misunderstood what you're saying.
> Clarification welcome.
> > - Rich Holmes

My point was just that there are a heck of a lot of generators, so it doesn't make sense to ask for a way to get "the generator" from a wedgie.

Keenan

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> Which answers your question about what I want to know how to do with a
> wedgie. I think I understand how to find commas. The rest I'd like
> to see an explanation of.

Suppose T is a wedgie, and v is an equal temperament val. Then
v supports T if and only if T^v = 0.

What an optimal tuning map is depends on how you define "optimal", so
various algorithms are required if we want to consider these various
notions of optimality. However, suppose s is a set of intervals and
you wish to find the least squares optimum with respect to s. Then you
may take a generic tuning map--in the 7-limit limit, m = <1200 a b c|.
Them we must have T^m = 0; the coefficients therefore give us linear
equations, which we can solve and substitute back to get M, with one
indeterminate. Now take the sum of squares of the error for each
element of s using this M, differentiate and solve; you now have a
least squares tuning map. This actually works for any rank of
temperament. It's actually not hard to find exact solutions, in fact.

bfowol wrote:

> Delurking briefly to say that so far I've seen a lot of postings containing wedgies for > various temperaments, and defining what a wedgie is, but so far haven't seen what I'd > consider the most useful bit of information: how to take a wedgie and extract enough info > from it to form a usable scale. Any takers?

You can solve the wedgie equations to find a tuning map, if you specify certain conditions for the mapping (e.g., an integer number of periods and no generators to produce an octave). We had a discussion of this a couple years ago, but I don't remember the details. From the tuning map you can determine the size of the period and generator. So you could use the wedgie as an intermediate step between commas and tuning maps. Take 81/80 and 225/224, wedge to get [[12, -13, 4, 10, -4, 1>>, take the complement <<1, 4, 10, 4, 13, 12]], and the first 3 numbers are part of the tuning map if you define the period as an octave. So the tuning map is something like:

<1, x, y, z] (periods)
<0, 1, 4, 10] (generators)

Because zero generators are used in the approximation of the octave, the first three numbers of the wedgie come from the generator map multiplied by the number of periods in an octave.

(If the first three numbers of a 7-limit wedgie have a common factor, factor it out and use it for the period mapping.) Then solve the wedgie equations for x, y, and z.

4x - y = 4
10x - z = 13
10y - 4z = 12

Possible solutions include:

x = 0, y = -4, z = -13
x = 1, y = 0, z = -3
x = 2, y = 4, z = 7

etc. You'll have to try a few of them to find one with an appropriate size generator. Take the last one: 3/1 is approximated by two periods (i.e., two octaves) and one generator. This results in a fourth downward as the size of the generator (you can negate the sign of the generator map to make it a fourth upward).

<1, 2, 4, 7] (octaves)
<0, -1, -4, -10] (fourths)

This is cheating a bit, since I started with a pair of commas that I knew would turn out to give meantone temperament. But it's an example of the sort of thing you can do with wedgies.

Keenan Pepper <keenanpepper@gmail.com> writes:

> What I meant was: Pick any meantone interval that's not a multiple of another
> meantone interval. Then you can find some other interval that generates all of
> meantone together with that one.

OK, that makes more sense to me. But... it requires knowledge of the
meantone intervals. If I knew nothing about meantone other than its
wedgie, how then would I find a period/generator? Or equivalently (I
guess), how would I construct a meantone scale knowing only the
wedgie?

> My point was just that there are a heck of a lot of generators, so it doesn't
> make sense to ask for a way to get "the generator" from a wedgie.

Sure, sorry, I knew that and should've said "a period/generator pair".

- Rich Holmes

Rich Holmes<rsholmes@mailbox.syr.edu> writes:

> OK, that makes more sense to me. But... it requires knowledge of the
> meantone intervals. If I knew nothing about meantone other than its
> wedgie, how then would I find a period/generator? Or equivalently (I
> guess), how would I construct a meantone scale knowing only the
> wedgie?

Looks like Herman's answered this. Thanks!

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> OK, that makes more sense to me. But... it requires knowledge of the
> meantone intervals. If I knew nothing about meantone other than its
> wedgie, how then would I find a period/generator?

Herman already gave an answer to this, so I'll give a completely
different one. If T is the wedgie, then take the "v" product with all
the primes in the p-limit; recall that Tvc, where c is a "monzo"
(prime exponents vector) representing an interval, then Tvc = ~(~T^c).
These give a set of vals; you can take out the GCD at this point and
make them reduced, though it isn't required. Now reduce this set of
vals, written as a row matrix of vals, to Hermite normal form, which
works a lot like Gaussian elimination. Take the two rows which aren't
all zeros, and those are your period and generator.

> Or equivalently (I
> guess), how would I construct a meantone scale knowing only the
> wedgie?

Compute a tuning map, take a JI scale, and reduce to meantone using
this scale. If you want to use Fokker blocks with 81/80 used in
contructing the block, that would make sense. If the number of steps
in the scale is n and the standard n-val v supports T, as shown by
T^v = 0, then you'll likely get a Fokker block. Even if you don't get
a Fokker block, you *will* get a meantone scale.

On the other hand, if you want a MOS you should proably go ahead and
compute a period-generator pair.

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

> If the number of steps
> in the scale is n and the standard n-val v supports T, as shown by
> T^v = 0, then you'll likely get a Fokker block.

Sorry, I meant DE scale.

wallyesterpaulrus wrote:

> Oh, you're looking at signed generator mappings . . .

Yes.

> I don't know what you're comparing here or why. This bears no > similarity to my complexity calculation . . . I think you must be > talking to someone else here :)

Perhaps when you look back throught the thread, you'll see I was always talking about max-min of the weighted generator mapping, and not whatever calculations you were doing.

> Perhaps you could offer some 7-limit examples so we can use my paper > to compare?

Here are some. The "OE Complexity" column is the octave-equivalent complexity, where you do the max-min of the weighted generator mapping. Then there are two different values for the wedgie complexity. The first one is divided through by the number of primes. That means it scales up roughly proportional to the octave equivalent complexity, and the complexity of an equal temperament is roughly the number of notes to the octave. The other wedige complexity is the raw sum-abs which agrees with your paper. Then there are the optimal TOP and weighted-primes RMS errors in cents/octave.

Some temperaments may be wrong. I originally found representative ETs with the odd-limit program. Now, I notice that a different value for 7-equal is optimal in the prime limit so I had to stop it being used. There may be others I missed.

Temperament OE Complexity Wedgie Complex. Errors
Blacksmith 2.153 1.619 6.475 7.240 5.393
Dimisept 2.524 1.979 7.917 5.872 4.918
Dominant 2.435 1.989 7.956 6.144 4.715
August 2.137 2.076 8.305 5.871 4.733
Pajara 2.985 2.601 10.402 3.107 2.572
Semaphore 3.445 2.801 11.204 3.669 2.755
Meantone 3.562 2.941 11.765 1.699 1.382
Injera 3.445 2.979 11.918 3.583 3.138
Negrisept 3.816 3.031 12.125 3.187 2.575
Augene 4.030 3.031 12.125 2.940 2.228
Keemun 3.786 3.102 12.409 3.834 2.583
Catler 4.274 3.210 12.840 3.557 2.690
Hedgehog 4.307 3.297 13.190 3.107 2.780
Superpyth 4.589 3.608 14.431 2.404 1.917
Sensisept 4.631 3.615 14.459 1.610 1.323
Lemba 4.647 3.657 14.627 3.762 3.200
Porcupine 4.291 3.699 14.796 3.278 2.531
Flattone 4.929 3.844 15.376 2.536 2.117
Magic 4.274 3.884 15.536 1.277 1.074
Doublewide 5.047 3.899 15.596 3.278 2.483
Nautilus 4.307 3.906 15.623 3.480 3.248
Beatles 5.138 4.219 16.877 2.896 2.301
Liese 5.168 4.372 17.490 2.625 2.219
Cynder 5.524 4.612 18.448 1.805 1.425
Orwell 5.709 4.995 19.980 0.946 0.748
Garibaldi 5.618 5.073 20.292 0.913 0.726
Myna 6.309 5.081 20.326 1.460 0.952
Miracle 6.800 5.275 21.102 0.631 0.515
Ennealimmal 11.628 9.957 39.829 0.036 0.030

But I don't know why you want 7-limit tables to investigate a 13-limit phenomenon. Here are some more relevant results:

Temperament OE Complexity Wedgie Complex. Errors
Mystery 12.490 15.671 94.027 0.652 0.512
Diaschismic 9.369 11.950 71.702 1.268 0.826
Cassandra1 11.635 14.225 85.348 0.969 0.696
Cassandra2 6.306 8.103 48.617 2.032 1.517

Mystery is 29&58, with mapping
[(29, 0), (46, 0), (67, 1), (81, 1), (100, 1), (107, 1)]
and a 13 or 15-limit complexity of 29.

Diaschismic is the 46&58 temperament with mapping
[(2, 0), (3, 1), (5, -2), (7, -8), (9, -12), (10, -15)]
and a 13 or 15-limit complexity of 34.

Cassandra1 is 41%53 with mapping
[(1, 0), (2, -1), (-1, 8), (-3, 14), (13, -23), (12, -20)]
and a 13 or 15-limit complexity of 37.

So in either the 13 or 15 limits, Mystery is the simplest of the three. But in the weighted 13-limit, Mystery is the most complex.

>>They're both octave-equivalent prime-based measures. So you have > to >>correct for 15:8 and 5:3 having different octave-specific > complexities. > > I don't know what context I'm supposed to take the above in. My > complexity numbers are virtually identical to the max-Kees complexity > (according to Gene), which is an octave-equivalent measure despite my > measure being octave-specific.

So what's the formula for max-Kees complexity?

>> Either tempering the octaves or considering the full wedgie gives > you >>a proper octave-specific measure.
> > ?

The worst weighted error won't be the TOP error if you don't optimize your octaves properly, will it? And the sum-abs of the weighted octave-equivalent wedgie won't give equivalent results to the sum-abs of the full wedgie either. It happens that the formula for the TOP error of unweighted octaves is similar to the octave-equivalent complexity formula.

5-limit
Temperament TOP basis RMS basis Errors
Father 1185.9 1181.3 447.4 448.9 14.131 13.194
Bug 1185.8 1200.0 246.1 260.4 14.176 11.574
Dicot 1207.7 1206.4 353.2 350.5 7.658 7.095
Meantone 1201.7 1201.4 504.1 504.3 1.700 1.582
Augmented 399.0 399.0 88.5 93.1 2.940 2.400
Mavila 1206.6 1208.4 521.5 523.8 6.549 6.065
Porcupine 1196.9 1199.6 162.3 163.9 3.094 2.678
Blackwood 238.9 238.9 66.9 80.0 5.666 4.626
Dimipent 299.2 299.7 101.7 99.4 3.359 3.104
Srutal 599.6 599.4 104.7 104.8 0.889 0.835
Magic 1201.3 1201.2 380.8 380.5 1.279 1.110
Ripple 1203.3 1200.3 102.0 100.9 3.324 2.819
Hanson 1200.3 1200.2 317.1 317.1 0.291 0.274
Negripent 1201.8 1202.3 126.1 126.0 1.823 1.690
Tetracot 1201.7 1201.4 348.8 348.5 1.699 1.582
Superpyth 1197.6 1197.7 489.4 489.0 2.404 2.112
Helmholtz 1200.1 1200.1 498.3 498.3 0.066 0.057
Sensipent 1199.6 1199.9 443.0 443.0 0.413 0.356
Passion 1198.3 1197.8 98.4 98.5 1.686 1.567
Wuerschmidt 1199.7 1199.7 387.6 387.7 0.309 0.262
Compton 100.1 100.1 13.7 15.1 0.617 0.504
Amity 1199.9 1199.9 339.5 339.5 0.150 0.140
Orson 1200.2 1200.3 271.7 271.7 0.240 0.215
Vishnu 600.0 600.0 71.1 71.1 0.051 0.047
Luna 1200.0 1200.0 193.2 193.2 0.018 0.015

Temperament OE Complexity Wedgie Complex. Errors
Father 1.062 0.716 2.149 14.967 13.194
Bug 1.292 0.851 2.554 14.176 11.574
Dicot 1.262 0.836 2.508 7.658 7.095
Meantone 1.723 1.147 3.441 1.699 1.582
Augmented 1.893 1.265 3.795 3.557 2.400
Mavila 1.923 1.275 3.825 6.548 6.065
Porcupine 2.153 1.439 4.318 3.216 2.678
Blackwood 2.153 1.442 4.327 5.666 4.626
Dimipent 2.524 1.687 5.062 3.557 3.104
Srutal 2.985 1.991 5.974 0.889 0.835
Magic 3.155 2.101 6.303 1.277 1.110
Ripple 3.445 2.291 6.872 3.324 2.819
Hanson 3.786 2.523 7.569 0.291 0.274
Negripent 3.816 2.540 7.620 1.823 1.690
Tetracot 3.445 2.294 6.881 1.699 1.582
Superpyth 3.876 2.589 7.768 2.404 2.112
Helmholtz 4.076 2.717 8.152 0.065 0.057
Sensipent 4.417 2.945 8.836 0.412 0.356
Passion 4.877 3.256 9.768 1.686 1.567
Wuerschmidt 5.047 3.366 10.097 0.428 0.262
Compton 5.168 3.444 10.331 0.617 0.504
Amity 5.599 3.733 11.199 0.303 0.140
Orson 5.709 3.805 11.415 0.240 0.215
Vishnu 8.833 5.889 17.667 0.051 0.047
Luna 10.325 6.884 20.651 0.109 0.015

7-limit
Temperament TOP basis RMS basis Errors
Blacksmith 239.2 239.4 67.0 87.0 7.240 5.393
Dimisept 298.5 299.1 101.5 99.2 5.872 4.918
Dominant 1195.2 1195.4 495.9 496.5 4.771 4.715
August 400.0 399.1 107.3 103.8 5.871 4.733
Pajara 598.4 598.9 106.6 106.8 3.107 2.572
Semaphore 1203.7 1203.9 252.5 253.4 3.670 2.755
Meantone 1201.7 1201.2 504.1 504.0 1.700 1.382
Injera 600.9 600.7 93.6 94.5 3.583 3.138
Negrisept 1203.2 1203.5 124.8 126.0 3.188 2.575
Augene 399.0 398.8 88.5 90.5 2.940 2.228
Keemun 1203.2 1202.6 317.8 317.2 3.188 2.583
Catler 99.8 99.9 14.6 26.8 3.557 2.690
Hedgehog 598.4 599.6 162.3 164.2 3.107 2.780
Superpyth 1197.6 1197.1 489.4 488.5 2.404 1.917
Sensisept 1198.4 1199.7 443.2 443.3 1.611 1.323
Lemba 601.7 601.5 230.9 232.7 3.741 3.200
Porcupine 1196.9 1197.8 162.3 162.6 3.094 2.531
Flattone 1202.5 1203.6 507.1 507.8 2.537 2.117
Magic 1201.3 1201.1 380.8 380.7 1.279 1.074
Doublewide 599.3 600.0 327.0 325.7 3.269 2.483
Nautilus 1202.7 1202.2 83.0 82.7 3.481 3.248
Beatles 1197.1 1196.6 354.7 354.9 2.897 2.301
Liese 1202.6 1201.6 569.0 568.3 2.625 2.219
Cynder 1201.7 1200.9 232.5 232.4 1.699 1.425
Orwell 1199.5 1200.0 271.5 271.5 0.947 0.748
Garibaldi 1200.8 1200.1 498.1 498.0 0.914 0.726
Myna 1198.8 1199.3 309.9 310.0 1.172 0.952
Miracle 1200.6 1200.8 116.7 116.8 0.632 0.515
Ennealimmal 133.3 133.3 49.0 49.0 0.036 0.030

Temperament OE Complexity Wedgie Complex. Errors
Blacksmith 2.153 1.619 6.475 7.240 5.393
Dimisept 2.524 1.979 7.917 5.872 4.918
Dominant 2.435 1.989 7.956 6.144 4.715
August 2.137 2.076 8.305 5.871 4.733
Pajara 2.985 2.601 10.402 3.107 2.572
Semaphore 3.445 2.801 11.204 3.669 2.755
Meantone 3.562 2.941 11.765 1.699 1.382
Injera 3.445 2.979 11.918 3.583 3.138
Negrisept 3.816 3.031 12.125 3.187 2.575
Augene 4.030 3.031 12.125 2.940 2.228
Keemun 3.786 3.102 12.409 3.834 2.583
Catler 4.274 3.210 12.840 3.557 2.690
Hedgehog 4.307 3.297 13.190 3.107 2.780
Superpyth 4.589 3.608 14.431 2.404 1.917
Sensisept 4.631 3.615 14.459 1.610 1.323
Lemba 4.647 3.657 14.627 3.762 3.200
Porcupine 4.291 3.699 14.796 3.278 2.531
Flattone 4.929 3.844 15.376 2.536 2.117
Magic 4.274 3.884 15.536 1.277 1.074
Doublewide 5.047 3.899 15.596 3.278 2.483
Nautilus 4.307 3.906 15.623 3.480 3.248
Beatles 5.138 4.219 16.877 2.896 2.301
Liese 5.168 4.372 17.490 2.625 2.219
Cynder 5.524 4.612 18.448 1.805 1.425
Orwell 5.709 4.995 19.980 0.946 0.748
Garibaldi 5.618 5.073 20.292 0.913 0.726
Myna 6.309 5.081 20.326 1.460 0.952
Miracle 6.800 5.275 21.102 0.631 0.515
Ennealimmal 11.628 9.957 39.829 0.036 0.030

11-limit
Temperament TOP basis RMS basis Errors
Miracle 1200.6 1200.8 116.7 116.7 0.632 0.484
Diaschismic 599.4 599.4 103.8 103.6 1.268 0.905
Orwell 1201.2 1200.6 271.4 271.6 1.364 1.151
Shrutar 599.8 599.8 52.4 52.7 1.425 1.146
Schismic 1201.4 1200.2 497.9 497.7 1.790 1.310
Microschismic 1200.8 1200.3 498.1 498.0 0.914 0.673
Magic 1200.7 1200.1 380.9 380.7 1.678 1.226
Meantone 1201.6 1200.8 504.0 503.4 1.740 1.439
Vicentino1 1201.7 1201.2 348.8 348.8 1.701 1.471
Vicentino2 1201.7 1201.8 348.8 348.7 1.699 1.396
Mystery 41.4 41.4 16.1 16.0 0.652 0.539
Hemiennialimmal 66.7 66.7 17.6 17.6 0.049 0.038

Temperament OE Complexity Wedgie Complex. Errors
Miracle 7.351 7.675 38.375 0.631 0.484
Diaschismic 8.199 8.847 44.237 1.268 0.905
Orwell 5.709 5.704 28.520 1.363 1.151
Shrutar 8.432 7.989 39.946 1.425 1.146
Schismic 5.834 6.656 33.280 1.785 1.310
Microschismic 11.635 10.936 54.678 0.913 0.673
Magic 6.587 6.530 32.649 1.676 1.226
Meantone 5.203 5.332 26.661 1.805 1.439
Vicentino1 7.364 6.461 32.306 1.699 1.471
Vicentino2 7.124 6.563 32.813 1.699 1.396
Mystery 12.490 14.128 70.642 0.652 0.539
Hemiennialimmal 23.257 25.608 128.040 0.049 0.038

13-limit
Temperament TOP basis RMS basis Errors
Mystery 41.4 41.4 16.1 15.9 0.652 0.512
Diaschismic 599.4 599.4 103.8 103.6 1.268 0.826
Cassandra1 1200.4 1200.2 498.1 498.0 0.971 0.696
Cassandra2 1201.5 1200.3 497.9 497.6 2.035 1.517

Temperament OE Complexity Wedgie Complex. Errors
Mystery 12.490 15.671 94.027 0.652 0.512
Diaschismic 9.369 11.950 71.702 1.268 0.826
Cassandra1 11.635 14.225 85.348 0.969 0.696
Cassandra2 6.306 8.103 48.617 2.032 1.517

17-limit
Temperament TOP basis RMS basis Errors
Mystery 41.4 41.4 16.1 16.2 1.219 0.852
Diaschismic 599.4 599.6 103.9 103.7 1.351 0.890

Temperament OE Complexity Wedgie Complex. Errors
Mystery 12.490 16.974 118.820 1.219 0.852
Diaschismic 9.369 13.912 97.383 1.351 0.890

Graham

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...>
> wrote:
>
> > > <<-1.955 13.686 31.714 26.232 54.898 33.962||
> > >
> > > The first three coefficients are the errors in cents of 3, 5,
and 7
> > in
> > > terms of an equal division of the octave into 12 parts.
>
> > So, what about the other three coefficients?
>
> The first three can be written 100*(19*log2(2)-12*log2(3)),
> 100*(28*log2(2)-12*log2(5)), 100*(34*log2(2)-12*log2(7)), giving the
> error in cents. The next three are 100*(28*log2(3)-19*log2(5)),
> 100*(34*log2(3)-19*log2(7)), 100*(34*log2(5)-28*log2(7)); more or
less
> they are weighted errors.

I don't understand the weighting or why it arises. Can you provide
some insight?

> > Oh, OK. I goofed. But in this case, aren't you interpreting that
> > object more as a "tuning map" and less as a "val"? Since it seems
> > you're specifically identifying it with pure-octaves 12-
equal . . .
>
> Yes, you may certainly view this as wedging the tuning map with the
> JIP and getting the error.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> At 06:02 PM 1/10/2006, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>
> >> At 03:49 PM 1/10/2006, you wrote:
> >> >--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...>
wrote:
> >> >>
> >> >> >> Hiya Paul - is there a name for this L_ notation that I
could
> >> >> >> look up somewhere?
> >> //
> >> >> I'm trying to remember a post in which you stated there's a
> >> >> difference between this sort of choice and choosing between...
> >> >> something else.
> >> >
> >> >Different choices for how to weight the various intervals?
> >>
> >> I don't think so...
> >>
> >> >> Bah, it was recent, but I have no idea how
> >> >> to find it.
> >> >>
> >> >> -Carl
> >> >
> >> >I think it was on MMM.
> >>
> >> Oh, thanks! Maybe it was this (though I thought what I'm
> >> remembering was addressed to me, perhaps back when we were
> >> talking about hexagons vs. circles on tuning, whereas this
> >> was addressed to Gene):
> >>
> >> >>Because different people have different tastes, this is a
question
> >> >>with no answer, I think. The minimax tuning for porcupine has a
> >> >>stronger theoretical claim, but I think porcupine actually
sounds
> >> >>better in 22-et.
> >> //
> >> >Minimax vs. RMS vs. Sum-abs is another dimension along which one
> >> >can vary the definition of "optimal", in addition to the
dimension
> >> >I was referring to above, and others. So one example is not
enough
> >> >to determine what exactly it is about our "optimality" criteria
> >> >that we wish to change. Anyway, let's continue on tuning-math or
> >> >tuning.
> >>
> >> How is minimax in the 1st paragraph different from that in the
2nd?
> >>
> >> -Carl
> >
> >Why do you think it's different?
>
> "Minimax vs. RMS vs. Sum-abs is another dimension along which one
> can vary the definition of "optimal", in addition to the dimension
> I was referring to above, and others."
>
> -Carl

So? Seems like the same sense of Minimax (L_inf) is used there, as
it's being contrasted against RMS (L_2) and Sum-abs (L_1) . . .

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> wallyesterpaulrus wrote:
> > BTW, a chromatic unison (vector) is supposed to be a small
interval
> > that functions as a sharp or flat in a diatonic-like scale. Not a
> > fifth, not a fourth, not a third, not a second, but a *unison* --
> > thus the name. How this term came to mean something completely
> > different (which has no recognizable relationship with the term
> > itself) which can include fifths or even octaves is baffling to
me;
> > hence I continue to rail against this random act of
ridiculousness.
>
> We had this discussion before, and I don't think it got anywhere.
> Certainly nobody convinced me that there was a better term
than "unison
> vector" for unison vectors (and I note you didn't propose one
there).

A perfect fifth is no kind of unison vector.

> The earliest reference I have in English is this from Fokker:
>
> http://www.xs4all.nl/~huygensf/doc/fokkerpb.html
>
> "All pairs of notes differing by octaves only are considered
unisons.
> There are other pairs of notes which in musical practice are
considered
> to be unisonous...The vectors, which in the harmonic note lattice
> connect such notes taken for unisons will be called unison vectors."
>
> Nothing about them being small,

What do you think "unisonious" means??

> and almost includes octaves (not quite
> because they aren't vectors in Fokker's scheme). So that's your
answer
> about how they "came to mean something completely different" --
it's
> people making the entirely reasonable assumption that chromatic
unison
> vectors are unison vectors.

Again, it's entirely unreasonable to assume a fifth is a unison.

> At least, saying "you don't need a chromatic unison vector because
you
> can use a vector too large to count as a unison" is evading the
question.

It is?

> Now, the "chromatic" part of "chromatic unison vector" is wrong.
> They're considered to be unisonous in a diatonic scale, and so
should be
> "diatonic unison vectors".

Relative to the diatonic scale, they create chromatic alterations.
The term musicians use is "augmented unison", and musicians also call
any augmented or diminished intervals "chromatic". "Diatonic unison
vector" seems like, at best, it could refer to either a chromatic
unison vector or a commatic unison vector.

> Although in this context it doesn't matter
> if they lead to a diatonic or chromatic scale. But if you want to
claim
> ownership of "chromatic unison vector" I'll switch to "diatonic
unison
> vector" and reclaim Fokker's original meaning.

It still won't be "unisonious" contrary to Fokker's description.

> It also follows that commas and octaves can be classed
>as "enharmonic
> unison vectors".

Commas? Sure.

--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:
>
> Rich Holmes wrote:
> > I don't think I understood every bit of that post, but it looks
like
> > it did not tell you how to do anything with a wedgie if you don't
> > already know the period and generator.
> >
> > If it did, then I need clarification.
> >
> > - Rich Holmes
>
> But that's the point, you can use any period and generator (really
just two
> independent generators) you want.

Not really.

> For example, meantone is generated by the
> octave and the fifth, or the octave and the fourth, or the fifth
and the fourth,
> or the fourth and the whole tone.

Yes, but meantone is *not* generated by the octave and whole tone. So
you can't just use any two independent intervals.

> The only intervals that can't be generators
> are multiples of other intervals, for example the minor seventh
can't be a
> generator because it's twice a fourth.

This is not so. The whole tone in meantone is not a multiple of any
other interval. Yet the octave and whole tone don't work as a basis
for meantone.

> The wedgie is so useful because it describes a temperament
independantly of its
> generators.

Yes, but only certain pairs of intervals can generate a given
temperament.

--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@g...>
wrote:

> My point was just that there are a heck of a lot of generators, so it
doesn't
> make sense to ask for a way to get "the generator" from a wedgie.

It does if you insist that "the other generator", or "the period",
generates the octave all by itself. Then there are only two possible
choices for "the generator" which are smaller than the period, and in
fact these two add up to the period. If you have your optimization
criteria fixed, you can then use a convention such as always choosing
the smaller value for "the generator".

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...>
wrote:
>
> > OK, that makes more sense to me. But... it requires knowledge of
the
> > meantone intervals. If I knew nothing about meantone other than
its
> > wedgie, how then would I find a period/generator?
>
> Herman already gave an answer to this, so I'll give a completely
> different one. If T is the wedgie, then take the "v" product with
all
> the primes in the p-limit; recall that Tvc, where c is a "monzo"
> (prime exponents vector) representing an interval, then Tvc = ~
(~T^c).

Is this the interior or regressive product? I'd write this as T\/c to
avoid confusion.

> These give a set of vals; you can take out the GCD at this point and
> make them reduced, though it isn't required. Now reduce this set of
> vals, written as a row matrix of vals, to Hermite normal form, which
> works a lot like Gaussian elimination. Take the two rows which
aren't
> all zeros, and those are your period and generator.
>
> > Or equivalently (I
> > guess), how would I construct a meantone scale knowing only the
> > wedgie?
>
> Compute a tuning map, take a JI scale, and reduce to meantone using
> this scale.

This may give funny results, especially for systems other than
meantone. And JI scales are poor relatives of tempered scales. I
would think it makes more sense to start with one note per period,
and then keep building more notes on each of these using the
generator, iteratively. Useful stopping points for a scale would be
those where a distributionally even scale (or equivalently, one with
two step sizes) results.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > If the number of steps
> > in the scale is n and the standard n-val v supports T, as shown by
> > T^v = 0, then you'll likely get a Fokker block.
>
> Sorry, I meant DE scale.

Aha. This is a good thing to point out to people, and I missed it
entirely when it read "Fokker block". But that could be seen as correct
too, if you add "tempered" in front of "Fokker", because of my
Hypothesis . . .

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>
> wallyesterpaulrus wrote:
>
> > Oh, you're looking at signed generator mappings . . .
>
> Yes.
>
> > I don't know what you're comparing here or why. This bears no
> > similarity to my complexity calculation . . . I think you must be
> > talking to someone else here :)
>
> Perhaps when you look back throught the thread, you'll see I was
always
> talking about max-min of the weighted generator mapping, and not
> whatever calculations you were doing.

OK. But Gene said that max Kees-weighted complexity was nearly
identical to my calculations. Is something not checking up right here?

> > Perhaps you could offer some 7-limit examples so we can use my
paper
> > to compare?
>
> Here are some. The "OE Complexity" column is the octave-equivalent
> complexity, where you do the max-min of the weighted generator
mapping.
> Then there are two different values for the wedgie complexity.
The
> first one is divided through by the number of primes. That means
it
> scales up roughly proportional to the octave equivalent complexity,
and
> the complexity of an equal temperament is roughly the number of
notes to
> the octave. The other wedige complexity is the raw sum-abs which
agrees
> with your paper. Then there are the optimal TOP and weighted-
primes RMS
> errors in cents/octave.
>
> Some temperaments may be wrong. I originally found representative
ETs
> with the odd-limit program. Now, I notice that a different value
for
> 7-equal is optimal in the prime limit so I had to stop it being
used.
> There may be others I missed.
>
> Temperament OE Complexity Wedgie Complex. Errors
> Blacksmith 2.153 1.619 6.475 7.240 5.393
> Dimisept 2.524 1.979 7.917 5.872 4.918
> Dominant 2.435 1.989 7.956 6.144 4.715
> August 2.137 2.076 8.305 5.871 4.733
> Pajara 2.985 2.601 10.402 3.107 2.572
> Semaphore 3.445 2.801 11.204 3.669 2.755
> Meantone 3.562 2.941 11.765 1.699 1.382
> Injera 3.445 2.979 11.918 3.583 3.138
> Negrisept 3.816 3.031 12.125 3.187 2.575
> Augene 4.030 3.031 12.125 2.940 2.228
> Keemun 3.786 3.102 12.409 3.834 2.583
> Catler 4.274 3.210 12.840 3.557 2.690
> Hedgehog 4.307 3.297 13.190 3.107 2.780
> Superpyth 4.589 3.608 14.431 2.404 1.917
> Sensisept 4.631 3.615 14.459 1.610 1.323
> Lemba 4.647 3.657 14.627 3.762 3.200
> Porcupine 4.291 3.699 14.796 3.278 2.531
> Flattone 4.929 3.844 15.376 2.536 2.117
> Magic 4.274 3.884 15.536 1.277 1.074
> Doublewide 5.047 3.899 15.596 3.278 2.483
> Nautilus 4.307 3.906 15.623 3.480 3.248
> Beatles 5.138 4.219 16.877 2.896 2.301
> Liese 5.168 4.372 17.490 2.625 2.219
> Cynder 5.524 4.612 18.448 1.805 1.425
> Orwell 5.709 4.995 19.980 0.946 0.748
> Garibaldi 5.618 5.073 20.292 0.913 0.726
> Myna 6.309 5.081 20.326 1.460 0.952
> Miracle 6.800 5.275 21.102 0.631 0.515
> Ennealimmal 11.628 9.957 39.829 0.036 0.030
>
>
> But I don't know why you want 7-limit tables to investigate a 13-
limit
> phenomenon. Here are some more relevant results:
>
> Temperament OE Complexity Wedgie Complex. Errors
> Mystery 12.490 15.671 94.027 0.652 0.512
> Diaschismic 9.369 11.950 71.702 1.268 0.826
> Cassandra1 11.635 14.225 85.348 0.969 0.696
> Cassandra2 6.306 8.103 48.617 2.032 1.517

I don't see any disagreement in the complexity rankings . . .

> Mystery is 29&58, with mapping
> [(29, 0), (46, 0), (67, 1), (81, 1), (100, 1), (107, 1)]
> and a 13 or 15-limit complexity of 29.
>
> Diaschismic is the 46&58 temperament with mapping
> [(2, 0), (3, 1), (5, -2), (7, -8), (9, -12), (10, -15)]
> and a 13 or 15-limit complexity of 34.
>
> Cassandra1 is 41%53 with mapping
> [(1, 0), (2, -1), (-1, 8), (-3, 14), (13, -23), (12, -20)]
> and a 13 or 15-limit complexity of 37.
>
> So in either the 13 or 15 limits, Mystery is the simplest of the
three.
> But in the weighted 13-limit, Mystery is the most complex.

Weighted vs. unweighted? I think it's fair to weight the simpler
intervals more heavily in the complexity calculation, but that's a
different issue. I thought the issue here was weighted vs. weighted
(under different definitions of complexity).

> >>They're both octave-equivalent prime-based measures. So you have
> > to
> >>correct for 15:8 and 5:3 having different octave-specific
> > complexities.
> >
> > I don't know what context I'm supposed to take the above in. My
> > complexity numbers are virtually identical to the max-Kees
complexity
> > (according to Gene), which is an octave-equivalent measure
despite my
> > measure being octave-specific.
>
> So what's the formula for max-Kees complexity?

I think it's just the maximum, over all ratios, of the number of
generators needed to reach the ratio (times the number of periods per
octave), divided by the expressibility (log of the "odd limit") of
the ratio. Gene?

> >> Either tempering the octaves or considering the full wedgie
gives
> > you
> >>a proper octave-specific measure.
> >
> > ?
>
> The worst weighted error won't be the TOP error if you don't
optimize
> your octaves properly, will it?

How did we get from complexity to error?

> And the sum-abs of the weighted
> octave-equivalent wedgie won't give equivalent results to the sum-
abs of
> the full wedgie either.

Right, but there appears to be a close relationship when you use the
max of the weighted octave-equivalent wedgie instead -- Gene?

> It happens that the formula for the TOP error
> of unweighted octaves is similar to the octave-equivalent
complexity
> formula.

OK, but is this demostrated in the tables below? What do they
demonstrate? (I think you switched the second and third columns or
something, BTW . . .)

> 5-limit
> Temperament TOP basis RMS basis Errors
> Father 1185.9 1181.3 447.4 448.9 14.131 13.194
> Bug 1185.8 1200.0 246.1 260.4 14.176 11.574
> Dicot 1207.7 1206.4 353.2 350.5 7.658 7.095
> Meantone 1201.7 1201.4 504.1 504.3 1.700 1.582
> Augmented 399.0 399.0 88.5 93.1 2.940 2.400
> Mavila 1206.6 1208.4 521.5 523.8 6.549 6.065
> Porcupine 1196.9 1199.6 162.3 163.9 3.094 2.678
> Blackwood 238.9 238.9 66.9 80.0 5.666 4.626
> Dimipent 299.2 299.7 101.7 99.4 3.359 3.104
> Srutal 599.6 599.4 104.7 104.8 0.889 0.835
> Magic 1201.3 1201.2 380.8 380.5 1.279 1.110
> Ripple 1203.3 1200.3 102.0 100.9 3.324 2.819
> Hanson 1200.3 1200.2 317.1 317.1 0.291 0.274
> Negripent 1201.8 1202.3 126.1 126.0 1.823 1.690
> Tetracot 1201.7 1201.4 348.8 348.5 1.699 1.582
> Superpyth 1197.6 1197.7 489.4 489.0 2.404 2.112
> Helmholtz 1200.1 1200.1 498.3 498.3 0.066 0.057
> Sensipent 1199.6 1199.9 443.0 443.0 0.413 0.356
> Passion 1198.3 1197.8 98.4 98.5 1.686 1.567
> Wuerschmidt 1199.7 1199.7 387.6 387.7 0.309 0.262
> Compton 100.1 100.1 13.7 15.1 0.617 0.504
> Amity 1199.9 1199.9 339.5 339.5 0.150 0.140
> Orson 1200.2 1200.3 271.7 271.7 0.240 0.215
> Vishnu 600.0 600.0 71.1 71.1 0.051 0.047
> Luna 1200.0 1200.0 193.2 193.2 0.018 0.015
>
> Temperament OE Complexity Wedgie Complex. Errors
> Father 1.062 0.716 2.149 14.967 13.194
> Bug 1.292 0.851 2.554 14.176 11.574
> Dicot 1.262 0.836 2.508 7.658 7.095
> Meantone 1.723 1.147 3.441 1.699 1.582
> Augmented 1.893 1.265 3.795 3.557 2.400
> Mavila 1.923 1.275 3.825 6.548 6.065
> Porcupine 2.153 1.439 4.318 3.216 2.678
> Blackwood 2.153 1.442 4.327 5.666 4.626
> Dimipent 2.524 1.687 5.062 3.557 3.104
> Srutal 2.985 1.991 5.974 0.889 0.835
> Magic 3.155 2.101 6.303 1.277 1.110
> Ripple 3.445 2.291 6.872 3.324 2.819
> Hanson 3.786 2.523 7.569 0.291 0.274
> Negripent 3.816 2.540 7.620 1.823 1.690
> Tetracot 3.445 2.294 6.881 1.699 1.582
> Superpyth 3.876 2.589 7.768 2.404 2.112
> Helmholtz 4.076 2.717 8.152 0.065 0.057
> Sensipent 4.417 2.945 8.836 0.412 0.356
> Passion 4.877 3.256 9.768 1.686 1.567
> Wuerschmidt 5.047 3.366 10.097 0.428 0.262
> Compton 5.168 3.444 10.331 0.617 0.504
> Amity 5.599 3.733 11.199 0.303 0.140
> Orson 5.709 3.805 11.415 0.240 0.215
> Vishnu 8.833 5.889 17.667 0.051 0.047
> Luna 10.325 6.884 20.651 0.109 0.015
>
> 7-limit
> Temperament TOP basis RMS basis Errors
> Blacksmith 239.2 239.4 67.0 87.0 7.240 5.393
> Dimisept 298.5 299.1 101.5 99.2 5.872 4.918
> Dominant 1195.2 1195.4 495.9 496.5 4.771 4.715
> August 400.0 399.1 107.3 103.8 5.871 4.733
> Pajara 598.4 598.9 106.6 106.8 3.107 2.572
> Semaphore 1203.7 1203.9 252.5 253.4 3.670 2.755
> Meantone 1201.7 1201.2 504.1 504.0 1.700 1.382
> Injera 600.9 600.7 93.6 94.5 3.583 3.138
> Negrisept 1203.2 1203.5 124.8 126.0 3.188 2.575
> Augene 399.0 398.8 88.5 90.5 2.940 2.228
> Keemun 1203.2 1202.6 317.8 317.2 3.188 2.583
> Catler 99.8 99.9 14.6 26.8 3.557 2.690
> Hedgehog 598.4 599.6 162.3 164.2 3.107 2.780
> Superpyth 1197.6 1197.1 489.4 488.5 2.404 1.917
> Sensisept 1198.4 1199.7 443.2 443.3 1.611 1.323
> Lemba 601.7 601.5 230.9 232.7 3.741 3.200
> Porcupine 1196.9 1197.8 162.3 162.6 3.094 2.531
> Flattone 1202.5 1203.6 507.1 507.8 2.537 2.117
> Magic 1201.3 1201.1 380.8 380.7 1.279 1.074
> Doublewide 599.3 600.0 327.0 325.7 3.269 2.483
> Nautilus 1202.7 1202.2 83.0 82.7 3.481 3.248
> Beatles 1197.1 1196.6 354.7 354.9 2.897 2.301
> Liese 1202.6 1201.6 569.0 568.3 2.625 2.219
> Cynder 1201.7 1200.9 232.5 232.4 1.699 1.425
> Orwell 1199.5 1200.0 271.5 271.5 0.947 0.748
> Garibaldi 1200.8 1200.1 498.1 498.0 0.914 0.726
> Myna 1198.8 1199.3 309.9 310.0 1.172 0.952
> Miracle 1200.6 1200.8 116.7 116.8 0.632 0.515
> Ennealimmal 133.3 133.3 49.0 49.0 0.036 0.030
>
> Temperament OE Complexity Wedgie Complex. Errors
> Blacksmith 2.153 1.619 6.475 7.240 5.393
> Dimisept 2.524 1.979 7.917 5.872 4.918
> Dominant 2.435 1.989 7.956 6.144 4.715
> August 2.137 2.076 8.305 5.871 4.733
> Pajara 2.985 2.601 10.402 3.107 2.572
> Semaphore 3.445 2.801 11.204 3.669 2.755
> Meantone 3.562 2.941 11.765 1.699 1.382
> Injera 3.445 2.979 11.918 3.583 3.138
> Negrisept 3.816 3.031 12.125 3.187 2.575
> Augene 4.030 3.031 12.125 2.940 2.228
> Keemun 3.786 3.102 12.409 3.834 2.583
> Catler 4.274 3.210 12.840 3.557 2.690
> Hedgehog 4.307 3.297 13.190 3.107 2.780
> Superpyth 4.589 3.608 14.431 2.404 1.917
> Sensisept 4.631 3.615 14.459 1.610 1.323
> Lemba 4.647 3.657 14.627 3.762 3.200
> Porcupine 4.291 3.699 14.796 3.278 2.531
> Flattone 4.929 3.844 15.376 2.536 2.117
> Magic 4.274 3.884 15.536 1.277 1.074
> Doublewide 5.047 3.899 15.596 3.278 2.483
> Nautilus 4.307 3.906 15.623 3.480 3.248
> Beatles 5.138 4.219 16.877 2.896 2.301
> Liese 5.168 4.372 17.490 2.625 2.219
> Cynder 5.524 4.612 18.448 1.805 1.425
> Orwell 5.709 4.995 19.980 0.946 0.748
> Garibaldi 5.618 5.073 20.292 0.913 0.726
> Myna 6.309 5.081 20.326 1.460 0.952
> Miracle 6.800 5.275 21.102 0.631 0.515
> Ennealimmal 11.628 9.957 39.829 0.036 0.030
>
> 11-limit
> Temperament TOP basis RMS basis Errors
> Miracle 1200.6 1200.8 116.7 116.7 0.632 0.484
> Diaschismic 599.4 599.4 103.8 103.6 1.268 0.905
> Orwell 1201.2 1200.6 271.4 271.6 1.364 1.151
> Shrutar 599.8 599.8 52.4 52.7 1.425 1.146
> Schismic 1201.4 1200.2 497.9 497.7 1.790 1.310
> Microschismic 1200.8 1200.3 498.1 498.0 0.914 0.673
> Magic 1200.7 1200.1 380.9 380.7 1.678 1.226
> Meantone 1201.6 1200.8 504.0 503.4 1.740 1.439
> Vicentino1 1201.7 1201.2 348.8 348.8 1.701 1.471
> Vicentino2 1201.7 1201.8 348.8 348.7 1.699 1.396
> Mystery 41.4 41.4 16.1 16.0 0.652 0.539
> Hemiennialimmal 66.7 66.7 17.6 17.6 0.049 0.038
>
> Temperament OE Complexity Wedgie Complex. Errors
> Miracle 7.351 7.675 38.375 0.631 0.484
> Diaschismic 8.199 8.847 44.237 1.268 0.905
> Orwell 5.709 5.704 28.520 1.363 1.151
> Shrutar 8.432 7.989 39.946 1.425 1.146
> Schismic 5.834 6.656 33.280 1.785 1.310
> Microschismic 11.635 10.936 54.678 0.913 0.673
> Magic 6.587 6.530 32.649 1.676 1.226
> Meantone 5.203 5.332 26.661 1.805 1.439
> Vicentino1 7.364 6.461 32.306 1.699 1.471
> Vicentino2 7.124 6.563 32.813 1.699 1.396
> Mystery 12.490 14.128 70.642 0.652 0.539
> Hemiennialimmal 23.257 25.608 128.040 0.049 0.038
>
> 13-limit
> Temperament TOP basis RMS basis Errors
> Mystery 41.4 41.4 16.1 15.9 0.652 0.512
> Diaschismic 599.4 599.4 103.8 103.6 1.268 0.826
> Cassandra1 1200.4 1200.2 498.1 498.0 0.971 0.696
> Cassandra2 1201.5 1200.3 497.9 497.6 2.035 1.517
>
> Temperament OE Complexity Wedgie Complex. Errors
> Mystery 12.490 15.671 94.027 0.652 0.512
> Diaschismic 9.369 11.950 71.702 1.268 0.826
> Cassandra1 11.635 14.225 85.348 0.969 0.696
> Cassandra2 6.306 8.103 48.617 2.032 1.517
>
> 17-limit
> Temperament TOP basis RMS basis Errors
> Mystery 41.4 41.4 16.1 16.2 1.219 0.852
> Diaschismic 599.4 599.6 103.9 103.7 1.351 0.890
>
> Temperament OE Complexity Wedgie Complex. Errors
> Mystery 12.490 16.974 118.820 1.219 0.852
> Diaschismic 9.369 13.912 97.383 1.351 0.890
>
>
>
>
> Graham
>

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

> > The only intervals that can't be generators
> > are multiples of other intervals, for example the minor seventh
> can't be a
> > generator because it's twice a fourth.
>
> This is not so. The whole tone in meantone is not a multiple of any
> other interval. Yet the octave and whole tone don't work as a basis
> for meantone.

Yet 9/8 does work as a generator: for instance, with 16/15.

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

> > Herman already gave an answer to this, so I'll give a completely
> > different one. If T is the wedgie, then take the "v" product with
> all
> > the primes in the p-limit; recall that Tvc, where c is a "monzo"
> > (prime exponents vector) representing an interval, then Tvc = ~
> (~T^c).
>
> Is this the interior or regressive product? I'd write this as T\/c to
> avoid confusion.

Interior product. I don't think your notation is very clear on Yahoo.

> > Compute a tuning map, take a JI scale, and reduce to meantone using
> > this scale.
>
> This may give funny results, especially for systems other than
> meantone.

Eh? The whole point of your "hypothesis" is that you can get DE scales
in exactly this way.

And JI scales are poor relatives of tempered scales. I
> would think it makes more sense to start with one note per period,
> and then keep building more notes on each of these using the
> generator, iteratively. Useful stopping points for a scale would be
> those where a distributionally even scale (or equivalently, one with
> two step sizes) results.

The question was, do we need to first compute the period and generator
to get to the scales, and the answer is, no we don't. That doesn't
mean computing them isn't a good idea anyway.

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

> I think it's just the maximum, over all ratios, of the number of
> generators needed to reach the ratio (times the number of periods per
> octave), divided by the expressibility (log of the "odd limit") of
> the ratio. Gene?

Right, but it's just a finite computation. If T[1] through T[n] is the
OE part of the wedgie, then take the maximum of |T[i]/p_i| and
|T[i]/p_j - T[j]/p_j|, where p_i is the ith odd prime.

> Right, but there appears to be a close relationship when you use the
> max of the weighted octave-equivalent wedgie instead -- Gene?

That's the Kees complexity.

>> My point was just that there are a heck of a lot of generators, so
>> it doesn't make sense to ask for a way to get "the generator" from
>> a wedgie.
>
>It does if you insist that "the other generator", or "the period",
>generates the octave all by itself. Then there are only two possible
>choices for "the generator" which are smaller than the period, and in
>fact these two add up to the period. If you have your optimization
>criteria fixed, you can then use a convention such as always choosing
>the smaller value for "the generator".

Are there any named temperaments which require both "period" and
"generator" to map the octave? I'd always thought such cases would
make distinguishing one of the generators as a "period" inappropriate.

-Carl

Carl Lumma wrote:

> Are there any named temperaments which require both "period" and
> "generator" to map the octave? I'd always thought such cases would
> make distinguishing one of the generators as a "period" inappropriate.

I don't think that makes sense. If both generators divide the octave, you can divide the octave into more parts and get an equal temperament. So it means you don't have a rank 2 temperament at all.

Graham

>> Are there any named temperaments which require both "period" and
>> "generator" to map the octave? I'd always thought such cases would
>> make distinguishing one of the generators as a "period" inappropriate.
>
>I don't think that makes sense. If both generators divide the octave,
>you can divide the octave into more parts and get an equal temperament.
> So it means you don't have a rank 2 temperament at all.
>
>
> Graham

Why is the octave special in this regard?

-Carl

Carl Lumma wrote:

>>I don't think that makes sense. If both generators divide the octave, >>you can divide the octave into more parts and get an equal temperament. >> So it means you don't have a rank 2 temperament at all.
> > > Why is the octave special in this regard?

Because we made it the equivalence interval, and so defined the period such that it always divides the octave.

Graham

>Carl Lumma wrote:
>
>>>I don't think that makes sense. If both generators divide the octave,
>>>you can divide the octave into more parts and get an equal temperament.
>>> So it means you don't have a rank 2 temperament at all.
>>
>> Why is the octave special in this regard?
>
>Because we made it the equivalence interval, and so defined the period
>such that it always divides the octave.
>
> Graham

This sounds circular to me.

-Carl

wallyesterpaulrus wrote:

> A perfect fifth is no kind of unison vector.

If an octave, why not a fifth?

>>http://www.xs4all.nl/~huygensf/doc/fokkerpb.html
>>
>>"All pairs of notes differing by octaves only are considered > unisons. >>There are other pairs of notes which in musical practice are > considered >>to be unisonous...The vectors, which in the harmonic note lattice >>connect such notes taken for unisons will be called unison vectors."
>>Nothing about them being small,
> > What do you think "unisonious" means??

As he doesn't define it or use it anywhere else we can both take it to mean whatever we want it to mean. The next part of the ... is

"Three major thirds, from the origin 0,0,0 lead to a note 0,3,0. On the ordinary key-instruments this note is taken to be identical with the octave 0,0,0. It should be 125/128 flat. As a rule no attention is paid to the difference of c (-2,0,0) and /c (2,-1,0)."

So that implies that "unisonous" means "identical on ordinary key-instruments" for "musical practice" and that "as a rule no attention is paid to the difference".

> Again, it's entirely unreasonable to assume a fifth is a unison.

Why are you still arguing this? I've given you a quote from the originator of the term that shows that unison vectors are like octaves but also vectors in octave-equivalent space. You haven't even acknowledged this let alone redefined your term to acknowledge that unison vectors don't have to be small.

>>At least, saying "you don't need a chromatic unison vector because > you >>can use a vector too large to count as a unison" is evading the > question.
> > It is?

The point is that Gene claimed that you need to define a optimize the generator to get a unique octave-equivalent mapping, or do a reduction. I said that's not the case. If you start from equal temperaments it's extremely unlikely that you'll get an unreasonable generator from them (I couldn't find any examples with a huge search). A reasonable equal temperament will have its primes in the right order.

If you start from commmas then the mapping depends on the additional unison vector you use to get a representative ET. That mapping (or even the original representative ET mapping) will do fine as a supplement to the generator mapping as a unique key to distinguish temperaments produced by a search through commas, given that you'll always use the same additional unison vectors within that search.

It also happens that I don't know how to go from commas to mapping without an additional unison vector. If Gene has a way of doing this it would be useful to know. All he's said, in another thread, is a vague "solve for x and y" without detailing the method. If there is a faster method than using an additional unison vector it would mean that a result which depends on that vector isn't of practical importance. But insisting on a strict definition of "unison vector" so that excludes some useful vectors outside it isn't at all helpful.

It may be that the other methods also give unique mappings. It should be obvious that any deterministic wedgie-to-mapping algorithm will always give the same mapping for the same wedgie. But that's still somewhat irrelevant as the idea is to get rid of wedgies altogether. Hence the big point is that unless you go beyond linear temperaments (which nobody is interested in doing in a systematic way) there's nothing you can do with wedgies that you can't do as easily without them.

> Relative to the diatonic scale, they create chromatic alterations. > The term musicians use is "augmented unison", and musicians also call > any augmented or diminished intervals "chromatic". "Diatonic unison > vector" seems like, at best, it could refer to either a chromatic > unison vector or a commatic unison vector.

Yes, it could be either. But if it were a commatic unison vector there'd be no point in specifying it as a unison vector of the diatonic scale.

I agree that 25:24 is both a chromatic interval and a unison. But it isn't both at the same time. And if we're going to be strict about the terminology it isn't a "chromatic unison vector" of the usual musical system. I can understand where the name comes from but it doesn't make sense when I think it through.

Musicians also talk about "the chromatic scale". And they talk about instruments that can play all and only 12 notes as chromatic. On such instruments, two notes differing by 25:24 don't share a note. They do on a diatonic instrument (or would if you ever tried to play both).

The term "augmented unison" follows diatonic naming. The "unison" has the same status as "second", "third", etc. as defining an interval on a diatonic scale. It doesn't carry any connotations of chromaticism.

So what about these "chromatic intervals"? I'm not sure if they relate to unison vectors at all. The best I can think of is that a chromatic interval must be the same as either a chromatic or diatonic interval on a chromatic scale. Also, a chromatic interval always differs from a diatonic interval by a chromatic semitone. So if the chromatic semitone were a unison vector, there wouldn't be any need to talk about "chromatic intervals" because they wouldn't be any different from diatonic ones. But maybe you can make the opposite argument. I think it's a red herring. Perhaps it shows that musical usages aren't consistent on this but they weren't mathematically based in the first place.

I'm not a musician, but I think this is why I was originally confused by the term "chromatic unison vector". I was always thinking of it in terms of a chromatic scale when it really generates a diatonic scale.

>>Although in this context it doesn't matter >>if they lead to a diatonic or chromatic scale. But if you want to > claim >>ownership of "chromatic unison vector" I'll switch to "diatonic > unison >>vector" and reclaim Fokker's original meaning.
> > It still won't be "unisonious" contrary to Fokker's description.

Why not?

Graham

wallyesterpaulrus wrote:

> OK. But Gene said that max Kees-weighted complexity was nearly > identical to my calculations. Is something not checking up right here?

I don't see anything wrong. We have three different complexity measures:

1) sum-abs of the weighted wedgie
2) max-min of the weighted generator mapping
3) max Kees-weighted complexity

They may all relate to each other but the definitions, at least, aren't the same. I'd hope that (2) and (3) are identical but I don't know how to prove it.

It also that I don't have an equivalent of (2) using means instead of max and min. But as there's nothing to optimize for it probably doesn't matter. I also don't have an equivalent of (2) for higher rank temperaments. In that case it doesn't matter because nobody seems to be interested in them.

> Weighted vs. unweighted? I think it's fair to weight the simpler > intervals more heavily in the complexity calculation, but that's a > different issue. I thought the issue here was weighted vs. weighted > (under different definitions of complexity).

Yes, the weighting's doing what we asked it to. The issue is that the odd-limit complexity I was happy with won't work without an odd-limit. The max-min of the weighted mapping is the nearest weighted equivalent to the odd limit complexity. But in practice it seems to give more divergent results than changing to weighted errors. I'm not saying it's wrong, or even that I don't like the results, but I'm pointing out that the 13-limit R2 landscale looks a lot different as a result. A temperament that used to be the clear number 1 now doesn't even make the top 10 (although that may be partly because the cutoffs are different).

In other limits the weighting doesn't make much difference to the ordering. Schismic is the best 5-limit R2T, then Hanson I think and meantone. In the 7-limit, it's meantone, orwell, schismic, then Orwell I think. Ennealimmal beats them if you allow it through. In the 11-limit, miracle still comes first, and meantone and schismic variants are in the top 10. Also, my 7-limit neutral thirds mapping is the number 1 11-limit planar temperament (strictly defined -- there's an R3 temperament that beats it, but it uses half-octaves, and so wouldn't work as a lattice).

In case it isn't obvious, the odd-limit complexity is a max-min as well. You need to do harmonics rather than primes. For the 9-limit, you take the maximum steps for 1:1, 3:1, 5:1, 7:1 and 9:1 and add the the maximum steps for 1:1, 1:3, 5:1, 7:1 and 9:1 to get the maximum steps for the full tonality diamond. As the maximum of one is minus the minumum of the other, that's the same as a maximum-minumum of the same list. And it's the same as a max-min of the primes if you give 3:1 a weight of 2 (or 0.5 depending on how you define the weighting) because 9:1 has twice the error of 3:1. That breaks down in the 15-limit, but still the log prime weighted max-min is an approximation to arbitrary prime-limited tonality diamond.

>>So what's the formula for max-Kees complexity?
> > I think it's just the maximum, over all ratios, of the number of > generators needed to reach the ratio (times the number of periods per > octave), divided by the expressibility (log of the "odd limit") of > the ratio. Gene?

That's a definition, not a formula. I can't calculate a maximum over all ratios.

>>The worst weighted error won't be the TOP error if you don't > optimize >>your octaves properly, will it?
> > How did we get from complexity to error?

I'm just saying the formulae are similar. And conjecturing that this is because there are deep similarities between the two problems.

>>And the sum-abs of the weighted >>octave-equivalent wedgie won't give equivalent results to the sum-
> abs of >>the full wedgie either.
> > Right, but there appears to be a close relationship when you use the > max of the weighted octave-equivalent wedgie instead -- Gene?

No. The max of the weighted octave-equivalent wedgie doesn't relate to anything. Neither does the max-abs. But the max-abs of the octave-specific wedgie probably has something to do with the max-min of the generator mapping (all properly weighted).

>>It happens that the formula for the TOP error >>of unweighted octaves is similar to the octave-equivalent > complexity >>formula.
> > OK, but is this demostrated in the tables below? What do they > demonstrate? (I think you switched the second and third columns or > something, BTW . . .)

Er, yes, the basis labels are wrong. What you have is:

TOP period, RMS period, TOP generator, RMS generator

But no, the tables don't demonstrate that the formulae are similar. Only the formulae do:

Octave specific TOP error = max(w, -w) - 1 = max(abs(e))
Octave equivalent TOP error = (max(w) - min(w))/(max(w)+min(w))
Approximate TOP error = (max(e) - min(e))/2
Wedgie complexity = max(W, -W) = max(abs(W))
Octave-equivalent complexity = max(M) - min(M)

Where:

w is the set of weighted primes
e is the set of signed, weighted errors where e = w-1
W are the elements of the weighted wedgie in vector form
M are the elements of the weighted generator mapping

Note that M is not the same as the weighted octave-equivalent wedgie contrary to loose remarks I may have made in other places. The generator mapping has to include a zero to show that there are zero generators to an octave. You have to add this zero to the octave-equivalent wedgie for the formula to work.

It's just struck me that any mapping might do in place of M. I wonder...

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> The point is that Gene claimed that you need to define a optimize the
> generator to get a unique octave-equivalent mapping, or do a reduction.

What I said was that you needed to define an optimization method to
produce a precise definition.

> I said that's not the case. If you start from equal temperaments
it's
> extremely unlikely that you'll get an unreasonable generator from them
> (I couldn't find any examples with a huge search).

You were looking for temperaments where two different optization
methods led to different minimal generators?

A reasonable equal
> temperament will have its primes in the right order.

Don't know what this means.

> If you start from commmas then the mapping depends on the additional
> unison vector you use to get a representative ET.

If you start from commas and then produce a wedgie, the result
obviously is going to work in just the same way as starting from ets.
If you are not getting this, then you've just discovered the use for
wedgies you were looking for.

In fact, always starting from ets doesn't seem to me to be a good
idea; you are guaranteed commas, but not standard vals. Examples are
father and jamesbond, with only one standard val each.

That mapping (or even
> the original representative ET mapping) will do fine as a supplement to
> the generator mapping as a unique key to distinguish temperaments
> produced by a search through commas, given that you'll always use the
> same additional unison vectors within that search.

By all means, let's make things far more complicated than necessary.
What in the world would be the point of doing this?

> It also happens that I don't know how to go from commas to mapping
> without an additional unison vector.

Compute the wedgie first springs to mind as the obvious method.

If Gene has a way of doing this it
> would be useful to know. All he's said, in another thread, is a vague
> "solve for x and y" without detailing the method.

You are asking for what, exactly? Starting from a wedgie, you can get
a list of commas from the triprime commas, which can be reduced to a
generating pair if you want one.

> It may be that the other methods also give unique mappings. It should
> be obvious that any deterministic wedgie-to-mapping algorithm will
> always give the same mapping for the same wedgie. But that's still
> somewhat irrelevant as the idea is to get rid of wedgies altogether.

And why would you want to do that?

> Hence the big point is that unless you go beyond linear temperaments
> (which nobody is interested in doing in a systematic way) there's
> nothing you can do with wedgies that you can't do as easily without
them.

We've got people on this list interested in rank three, you included.
I've not only written about it a lot, I've written music in such
temperaments. I was just pointing out on the main tuning list how
tempering a 5-limit scale using {100/99, 225/224} seems to lead to
interesting results. So I think there is interest.

> I'm not a musician, but I think this is why I was originally
confused by
> the term "chromatic unison vector". I was always thinking of it in
> terms of a chromatic scale when it really generates a diatonic scale.

The term made no sense to me when I first heard it, and still doesn't.
My preference would be that people quit using it rather than argue
about what it means.

Gene Ward Smith wrote:

> What I said was that you needed to define an optimization method to
> produce a precise definition. Right.

>> I said that's not the case. If you start from equal temperaments
> it's >>extremely unlikely that you'll get an unreasonable generator from them >>(I couldn't find any examples with a huge search). > > You were looking for temperaments where two different optization
> methods led to different minimal generators?

I don't think so. What's a minimal generator?

> A reasonable equal >>temperament will have its primes in the right order.
> > Don't know what this means.

If the initial choice of generator isn't within a period, it means the original equal temperaments must have been so bad that they didn't get the prime intervals correct to within the nearest period. If you avoid such ETs there's no need to optimize to define the mapping.

>>If you start from commmas then the mapping depends on the additional >>unison vector you use to get a representative ET.
> > If you start from commas and then produce a wedgie, the result
> obviously is going to work in just the same way as starting from ets.
> If you are not getting this, then you've just discovered the use for
> wedgies you were looking for.

Not only is this not obvious but it isn't true. The commas won't give me the same equal temperaments. Why is throwing away information a "use"?

> In fact, always starting from ets doesn't seem to me to be a good
> idea; you are guaranteed commas, but not standard vals. Examples are
> father and jamesbond, with only one standard val each.

What's a "standard val"? What's "jamesbond"? Why do I care? I get exactly the same "father" as is in Paul's paper, from the best weighted-RMS mappings of 3 and 8. It isn't in the results because it's too inaccurate.

Starting with ETs is a good, practical idea as I have explained before. It's an O(n2) search with n equal temperaments, regardless of the prime limit. It's also fairly easy to get the initial list -- and I still don't know an efficient way to get the list of commas.

> That mapping (or even >>the original representative ET mapping) will do fine as a supplement to >>the generator mapping as a unique key to distinguish temperaments >>produced by a search through commas, given that you'll always use the >>same additional unison vectors within that search.
> > By all means, let's make things far more complicated than necessary.
> What in the world would be the point of doing this?

The point of doing this as compared to what other method?

>>It also happens that I don't know how to go from commas to mapping >>without an additional unison vector.
> > Compute the wedgie first springs to mind as the obvious method.

Which gives exactly the same result as using matrices, and still requires that additional unison vector.

> You are asking for what, exactly? Starting from a wedgie, you can get
> a list of commas from the triprime commas, which can be reduced to a
> generating pair if you want one.

Starting from a list of commas (or a wedgie, I can do that bit) get an octave-specific mapping.

>>It may be that the other methods also give unique mappings. It should >>be obvious that any deterministic wedgie-to-mapping algorithm will >>always give the same mapping for the same wedgie. But that's still >>somewhat irrelevant as the idea is to get rid of wedgies altogether.
> > And why would you want to do that?

Because most people don't know anything about wedgies, and find them difficult to understand. That's where this whole thread started.

>>Hence the big point is that unless you go beyond linear temperaments >>(which nobody is interested in doing in a systematic way) there's >>nothing you can do with wedgies that you can't do as easily without
> > them.
> > We've got people on this list interested in rank three, you included.
> I've not only written about it a lot, I've written music in such
> temperaments. I was just pointing out on the main tuning list how
> tempering a 5-limit scale using {100/99, 225/224} seems to lead to
> interesting results. So I think there is interest.

I posted here about my sytematic rank 3 search, and got no replies. I take that as meaning no interest. Your example funnily enough doesn't make the list, which goes to show that the search isn't very interesting. It's easy enough to produce:

[ 4.53113367 22.85170121 32.33068401] cent steps

wedgie:
(1, 2, 2, -2, 2, 8, -5, 2, 14, 6)

mapping by steps:
[[12 19 28 34 42]
[19 30 44 53 66]
[22 35 51 62 76]]

wedgie based complexity: 0.856
RMS weighted error: 0.000997
max weighted error: 0.001831

Those results are genearlly biased towards more accurate temperament, but this one still beats yours in both error and complexity:

[ 26.58663049 11.15023396 36.07450767] cent steps

wedgie:
(2, -3, 1, -1, -1, 2, 11, 3, -10, 4)

mapping by steps:
[[ 9 14 21 25 31]
[15 24 35 42 52]
[22 35 51 62 76]]

wedgie based complexity: 0.802
RMS weighted error: 0.000713
max weighted error: 0.001363

It also works with the best mappings of 31 and 46.

You also said 99:98 instead of 100:99 in a message on the tuning list, and that would give

[ 32.53442332 25.8490932 14.48034347] cent steps

wedgie:
(1, 2, 4, -2, -2, 4, -5, -9, 2, 8)

mapping by steps:
[[12 19 28 34 42]
[19 30 44 53 65]
[22 35 51 62 76]]

wedgie based complexity: 0.819
RMS weighted error: 0.000895
max weighted error: 0.001522

Note the alternative mapping of 19. If you want a best-mappings definition, use 9 instead.

>>I'm not a musician, but I think this is why I was originally
> confused by >>the term "chromatic unison vector". I was always thinking of it in >>terms of a chromatic scale when it really generates a diatonic scale.
> > The term made no sense to me when I first heard it, and still doesn't.
> My preference would be that people quit using it rather than argue
> about what it means.

Yes, but you were confused by the "unison vector" part, which seemed to follow from your not knowing what "vector" means.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> > You were looking for temperaments where two different optization
> > methods led to different minimal generators?
>
> I don't think so. What's a minimal generator?

A generator > 1 of smallest size.

> If the initial choice of generator isn't within a period, it means the
> original equal temperaments must have been so bad that they didn't get
> the prime intervals correct to within the nearest period. If you avoid
> such ETs there's no need to optimize to define the mapping.

This makes no sense to me whatever. The generator can always be larger
than the period. Where are these initial choices coming from?

> > If you start from commas and then produce a wedgie, the result
> > obviously is going to work in just the same way as starting from ets.
> > If you are not getting this, then you've just discovered the use for
> > wedgies you were looking for.
>
> Not only is this not obvious but it isn't true. The commas won't give
> me the same equal temperaments.

Of course they will; this must be the case, since the equal
temperaments can be defined in terms of commas. Or are you on about
torsion?

> > In fact, always starting from ets doesn't seem to me to be a good
> > idea; you are guaranteed commas, but not standard vals. Examples are
> > father and jamesbond, with only one standard val each.
>
> What's a "standard val"?

A standard val is what you get by rounding the number of steps to a
prime off to the nearest integer.

> What's "jamesbond"? Why do I care?

Jamesbond has an OE part of 007. The reason to care is that I was
using it as an example.

I get
> exactly the same "father" as is in Paul's paper, from the best
> weighted-RMS mappings of 3 and 8. It isn't in the results because it's
> too inaccurate.

You seem to be using a different choice of val standarization, I'd
give counterexamples there but I don't know what it is.

> Starting with ETs is a good, practical idea as I have explained before.

Duh. This is obvious. The question is, might you not miss something?

> > That mapping (or even
> >>the original representative ET mapping) will do fine as a
supplement to
> >>the generator mapping as a unique key to distinguish temperaments
> >>produced by a search through commas, given that you'll always use the
> >>same additional unison vectors within that search.
> >
> > By all means, let's make things far more complicated than necessary.
> > What in the world would be the point of doing this?
>
> The point of doing this as compared to what other method?

As compared to *not* distinguishing between temperaments produced from
commas and those produced from vals, which strikes me as an absurd idea.

> >>It also happens that I don't know how to go from commas to mapping
> >>without an additional unison vector.
> >
> > Compute the wedgie first springs to mind as the obvious method.
>
> Which gives exactly the same result as using matrices, and still
> requires that additional unison vector.

I have no idea wha you mean by "that additional unison vector", but I
do know it isn't required, whatever it is.

> > You are asking for what, exactly? Starting from a wedgie, you can get
> > a list of commas from the triprime commas, which can be reduced to a
> > generating pair if you want one.
>
> Starting from a list of commas (or a wedgie, I can do that bit) get an
> octave-specific mapping.

I explained this once again quite recently. By taking the interior
product with each of the primes, you get something which reduces to
such a mapping. If you don't have LLL or Hermite reduction, then use
Gaussian elimination and adjust to get integers if necessary.
Alternatively, find the period mapping from the wedgie, and solve the
equations for the generator mapping, by requiring that when wedged
with the period mapping you get the wedgie.

That's 2.5 methods.

> > And why would you want to do that?
>
> Because most people don't know anything about wedgies, and find them
> difficult to understand. That's where this whole thread started.

But you know what they are, so you don't need to get rid of them. All
this amounts to is an argument that some other naming method should be
used.

> I posted here about my sytematic rank 3 search, and got no replies. I
> take that as meaning no interest.

I've responded to your postings about rank 3. Which article did you
mean? I may not have seen anything new in it, and hence didn't
respond, or I may have just missed responding, which often happens for
one reason or another.

Your example funnily enough doesn't
> make the list, which goes to show that the search isn't very
> interesting.

You mean, isn't yet very complete.

It's easy enough to produce:
>
> [ 4.53113367 22.85170121 32.33068401] cent steps

How are these defined--in terms of your three originating vals? That
doesn't strike me as such a good system.

> wedgie:
> (1, 2, 2, -2, 2, 8, -5, 2, 14, 6)
>
> mapping by steps:
> [[12 19 28 34 42]
> [19 30 44 53 66]
> [22 35 51 62 76]]

> wedgie:
> (1, 2, 4, -2, -2, 4, -5, -9, 2, 8)
>
> mapping by steps:
> [[12 19 28 34 42]
> [19 30 44 53 65]
> [22 35 51 62 76]]

> Note the alternative mapping of 19. If you want a best-mappings
> definition, use 9 instead.

I think commas are more illuminating. The first is {100/99, 225/224},
and the second {99/98, 225/224}, or equivalently {99/98, 176/175}.
The first is a relative of magic, and this is a relative of orwell.
"Tictactoe diagrams" for both, the first under the name apollo, and
the second which I called minerva, can be found here:

/tuning-math/message/12298

Obviously both by my reckoning are significant temperaments. Both fit
the tictactoe diagram into a 3x5 rectangle, and I don't think there is
much to choose between them in terms of complexity.

The point of these tictactoe diagrams, incidentally, emerges if you
use them with Tonescape.

> > The term made no sense to me when I first heard it, and still doesn't.
> > My preference would be that people quit using it rather than argue
> > about what it means.
>
> Yes, but you were confused by the "unison vector" part, which seemed to
> follow from your not knowing what "vector" means.

No, I *objected* to the term vector; I thought that implied that
Monzo's rational valued vectors should be allowed, and as far as I
could tell they weren't allowed. What confused me was that there was
all this murky business of "commatic" versus "chromatic" unison
vectors. Now you've made it worse by suggesting the octave can be a
unison vector. I think it's degenerated into babbling, where nobody
knows what anyone else is saying.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
>>What's a "standard val"? > > > A standard val is what you get by rounding the number of steps to a
> prime off to the nearest integer.

I should add this to my page; this seems to come up a lot....

>>What's "jamesbond"? Why do I care? > > > Jamesbond has an OE part of 007. The reason to care is that I was
> using it as an example.

It's a temperament with TOP tuning really close to 14-ET (so named from the "0, 0, 7" in the wedgie).

wedgie:
<<0, 0, 7, 0, 11, 16]]

map:
<7, 11, 16, 20]
<0, 0, 0, -1]

Oh, I missed this before

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
> > >>I think it's just the maximum, over all ratios, of the number of >>generators needed to reach the ratio (times the number of periods per >>octave), divided by the expressibility (log of the "odd limit") of >>the ratio. Gene?
> > > Right, but it's just a finite computation. If T[1] through T[n] is the
> OE part of the wedgie, then take the maximum of |T[i]/p_i| and
> |T[i]/p_j - T[j]/p_j|, where p_i is the ith odd prime.

I take it your T[i]/p_i is the same as my W[i], the ith element of the weighted octave-equivalent mapping (multiplied by the number of periods to the octave, I keep forgetting that), and you meant to put a "log of" in there. In that case, max(|W[i]|) for W > 0 is the same as max(W,-W) for all W, or max(max(W, 0), -min(W, 0)). Given that W[0]=0 always (as this is the octave equivalent mapping) that's also max(max(W), -min(W)). If it were T[i]/p_i instead of T[i]/p_j, that part would be max(|W[i] - W[j]|) for i, j > 0. The || is redundant because i and j are interchangeable, so that's max(W[i] - W[j]). It'll be the same as max(W[i]) - min(W[i]) still for i > 0. So the formula is max(max(W), -min(W), max(W[i], i>0) - min(W[i], i>0)). As W[i]=0 always, that's in turn the same as max(W) - min(W) isn't it? So your formula would be the same as mine without two weirdnesses.

>>Right, but there appears to be a close relationship when you use the >>max of the weighted octave-equivalent wedgie instead -- Gene?
> > That's the Kees complexity.

It is? I thought the above formula was for Kees complexity. They clearly aren't the same.

Graham

> > > > > > > > Yahoo! Groups Links
> > > > > > >

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > >>>You were looking for temperaments where two different optization
>>>methods led to different minimal generators?
>>
>>I don't think so. What's a minimal generator?
> > A generator > 1 of smallest size.

Well, I certainly wasn't looking for them.

>>If the initial choice of generator isn't within a period, it means the >>original equal temperaments must have been so bad that they didn't get >>the prime intervals correct to within the nearest period. If you avoid >>such ETs there's no need to optimize to define the mapping.
> > This makes no sense to me whatever. The generator can always be larger
> than the period. Where are these initial choices coming from?

They come from the extended Euclidian algorithm. They can be less than a unison, so I change them so they aren't. If I cared about a unique mapping I'd also change them so that the first entry in the generator mapping was positive. That's more reliable than using the minimal generator when the temperament isn't optimized.

Incidentally, it's also possible to write a function to compare two full mappings for equality without them having to be the same numbers. That's why we don't need to specify a universal standard.

>>>If you start from commas and then produce a wedgie, the result
>>>obviously is going to work in just the same way as starting from ets.
>>>If you are not getting this, then you've just discovered the use for
>>>wedgies you were looking for.
>>
>>Not only is this not obvious but it isn't true. The commas won't give >>me the same equal temperaments. > > Of course they will; this must be the case, since the equal
> temperaments can be defined in terms of commas. Or are you on about
> torsion?

There are an infinite number of equal temperaments, and the wedgie doesn't tell me which two I want. In that case it might find a different period mapping. The two methods don't give the same results but they are both internally consistent.

>>>In fact, always starting from ets doesn't seem to me to be a good
>>>idea; you are guaranteed commas, but not standard vals. Examples are
>>>father and jamesbond, with only one standard val each.
>>
>>What's a "standard val"? > > A standard val is what you get by rounding the number of steps to a
> prime off to the nearest integer.

I don't care about that.

>>What's "jamesbond"? Why do I care? > > Jamesbond has an OE part of 007. The reason to care is that I was
> using it as an example.

Is it an example of something I'm likely to care about?

> I get >>exactly the same "father" as is in Paul's paper, from the best >>weighted-RMS mappings of 3 and 8. It isn't in the results because it's >>too inaccurate.
> > You seem to be using a different choice of val standarization, I'd
> give counterexamples there but I don't know what it is.

Please, do not accuse me of standardizing ambiguous vals. I've argued against that before and I haven't changed my position. That's why I always say explicitly how I arrive at my mappings -- as I did in the bit you quoted. That way nobody has to remember my jargon from one message to another. Which part of "best weighted-RMS mapping" are you having difficulty weith?

As to your eagerness to produce counterexamples, you still haven't told my why I'd care.

>>Starting with ETs is a good, practical idea as I have explained before. > > Duh. This is obvious. The question is, might you not miss something?

As compared to what method that guarantees you won't miss anything? You're less likely to miss something the more ETs you work with, which means the program has to be fast. It's fastest when it does prime-based RMS calculations with no wedgies.

>>> That mapping (or even >>>
>>>>the original representative ET mapping) will do fine as a
> > supplement to > >>>>the generator mapping as a unique key to distinguish temperaments >>>>produced by a search through commas, given that you'll always use the >>>>same additional unison vectors within that search.
>>>
>>>By all means, let's make things far more complicated than necessary.
>>>What in the world would be the point of doing this?
>>
>>The point of doing this as compared to what other method?
> > As compared to *not* distinguishing between temperaments produced from
> commas and those produced from vals, which strikes me as an absurd idea.

Are you suggesting this idea came from me? Does it have anything to do with the quote above?

>>>>It also happens that I don't know how to go from commas to mapping >>>>without an additional unison vector.
>>>
>>>Compute the wedgie first springs to mind as the obvious method.
>>
>>Which gives exactly the same result as using matrices, and still >>requires that additional unison vector.
> > I have no idea wha you mean by "that additional unison vector", but I
> do know it isn't required, whatever it is.

The thing Paul objects to calling a "chromatic unison vector" in the general case. To get an ET mapping (or "val" as you call them, for anybody following along) you need n-1 unison vectors for n prime dimensions. For a rank 2 temperament, you need n-2 unison vectors. So one additional unison vector gives you the mapping for an equal temperament that belongs to the rank 2 family (maybe not a sensible temperament, but a mapping nonetheless).

>>>You are asking for what, exactly? Starting from a wedgie, you can get
>>>a list of commas from the triprime commas, which can be reduced to a
>>>generating pair if you want one.
>>
>>Starting from a list of commas (or a wedgie, I can do that bit) get an >>octave-specific mapping.
> > I explained this once again quite recently. By taking the interior
> product with each of the primes, you get something which reduces to
> such a mapping. If you don't have LLL or Hermite reduction, then use
> Gaussian elimination and adjust to get integers if necessary.
> Alternatively, find the period mapping from the wedgie, and solve the
> equations for the generator mapping, by requiring that when wedged
> with the period mapping you get the wedgie.
> > That's 2.5 methods.

No, you didn't explain it, you sketched it.

When you say "each of the primes" I think you mean using "each of the primes" as additional unison vectors (which Paul says we aren't allowed to call "chromatic unison vectors" because they're too big). I'll have to look up "interior product" to be sure. It probably depends on which wedgie you started with, which you neglected to mention. But it sounds like additional unison vectors to me.

"Find something" and "solve the problem" isn't a method at all. It's an aspiration.

>>>And why would you want to do that?
>>
>>Because most people don't know anything about wedgies, and find them >>difficult to understand. That's where this whole thread started.
> > But you know what they are, so you don't need to get rid of them. All
> this amounts to is an argument that some other naming method should be > used.

I do need to get rid of them because I'm trying to communicate with people who won't know about them. Yes, that is largely what this argument's about. I place a high value on ease of communication.

>>I posted here about my sytematic rank 3 search, and got no replies. I >>take that as meaning no interest.
> > I've responded to your postings about rank 3. Which article did you
> mean? I may not have seen anything new in it, and hence didn't
> respond, or I may have just missed responding, which often happens for
> one reason or another.

The one about the search.

> Your example funnily enough doesn't > >>make the list, which goes to show that the search isn't very >>interesting.
> > > You mean, isn't yet very complete.

Only 10 results for each limit, plus 10 more microtemperaments.

> It's easy enough to produce:
> >>[ 4.53113367 22.85170121 32.33068401] cent steps
> > > How are these defined--in terms of your three originating vals? That
> doesn't strike me as such a good system.

Yes. It's the easiest way to do it, and accurate enough with the right function from Numeric.

>>wedgie:
>>(1, 2, 2, -2, 2, 8, -5, 2, 14, 6)
>>
>>mapping by steps:
>>[[12 19 28 34 42]
>> [19 30 44 53 66]
>> [22 35 51 62 76]]
> > >>wedgie:
>>(1, 2, 4, -2, -2, 4, -5, -9, 2, 8)
>>
>>mapping by steps:
>>[[12 19 28 34 42]
>> [19 30 44 53 65]
>> [22 35 51 62 76]]
> > >>Note the alternative mapping of 19. If you want a best-mappings >>definition, use 9 instead.
> > > I think commas are more illuminating. The first is {100/99, 225/224},
> and the second {99/98, 225/224}, or equivalently {99/98, 176/175}.
> The first is a relative of magic, and this is a relative of orwell.
> "Tictactoe diagrams" for both, the first under the name apollo, and
> the second which I called minerva, can be found here:
> > /tuning-math/message/12298

They're both related to orwell and magic because they both contain 19, 22 and 31 (=12+19). They differ in the 11-mapping. This can be deduced from the ET mappings, no need for commas.

> Obviously both by my reckoning are significant temperaments. Both fit
> the tictactoe diagram into a 3x5 rectangle, and I don't think there is
> much to choose between them in terms of complexity.
> > The point of these tictactoe diagrams, incidentally, emerges if you
> use them with Tonescape.

Why are they significant? You haven't given any reasons they stand out.

Anyway, as you're using this as an excuse to push your results, here are mine. They're limited by a wedgie complexity of 0.9 and an optimal RMS weighted error of 0.001.

[ 26.58663049 11.15023396 36.07450767] cent steps

wedgie:
(2, -3, 1, -1, -1, 2, 11, 3, -10, 4)

mapping by steps:
[[ 9 14 21 25 31]
[15 24 35 42 52]
[22 35 51 62 76]]

wedgie based complexity: 0.802
RMS weighted error: 0.000713
max weighted error: 0.001363

[ 37.40033296 16.5838239 26.41660661] cent steps

wedgie:
(1, 3, 5, 2, 2, -4, -1, -5, -10, -8)

mapping by steps:
[[12 19 28 34 42]
[15 24 35 42 52]
[19 30 44 53 65]]

wedgie based complexity: 0.864
RMS weighted error: 0.000665
max weighted error: 0.001314

[ 28.53863771 0.4963866 42.58473932] cent steps

wedgie:
(4, 1, 2, -5, -2, 2, 1, 6, 1, -7)

mapping by steps:
[[ 9 14 21 25 31]
[14 22 32 39 48]
[22 35 51 62 76]]

wedgie based complexity: 0.768
RMS weighted error: 0.000959
max weighted error: 0.001530

[-14.48034347 47.01476679 40.32943667] cent steps

wedgie:
(1, 2, 4, -2, -2, 4, -5, -9, 2, 8)

mapping by steps:
[[ 9 14 21 25 31]
[12 19 28 34 42]
[19 30 44 53 65]]

wedgie based complexity: 0.819
RMS weighted error: 0.000895
max weighted error: 0.001522

[ 22.66876973 9.61561008 26.71269058] cent steps

wedgie:
(0, 0, 5, 0, 1, 12, 0, -10, 5, 25)

mapping by steps:
[[19 30 44 53 66]
[19 30 44 53 65]
[22 35 51 62 76]]

wedgie based complexity: 0.898
RMS weighted error: 0.000801
max weighted error: 0.001240

[ 3.8832202 25.26699935 10.66869258] cent steps

wedgie:
(4, 0, 0, -9, -10, 0, -3, -2, 0, -3)

mapping by steps:
[[ 14 22 32 39 48]
[ 31 49 72 87 107]
[ 34 54 79 96 118]]

wedgie based complexity: 0.859
RMS weighted error: 0.000916
max weighted error: 0.001497

[ 7.44209837 21.21783287 24.26574802] cent steps

wedgie:
(0, 0, 7, 0, 9, 13, 0, -2, 1, 5)

mapping by steps:
[[19 30 44 53 66]
[19 30 44 53 65]
[27 43 63 76 94]]

wedgie based complexity: 0.836
RMS weighted error: 0.000986
max weighted error: 0.001678

[ 21.41009811 26.31523098 5.56119297] cent steps

wedgie:
(0, 4, 0, 9, 0, -10, 5, 0, -2, 8)

mapping by steps:
[[ 14 22 32 39 48]
[ 27 43 63 76 94]
[ 34 54 79 96 118]]

wedgie based complexity: 0.856
RMS weighted error: 0.000985
max weighted error: 0.001498

[ 4.53113367 22.85170121 32.33068401] cent steps

wedgie:
(1, 2, 2, -2, 2, 8, -5, 2, 14, 6)

mapping by steps:
[[12 19 28 34 42]
[19 30 44 53 66]
[22 35 51 62 76]]

wedgie based complexity: 0.856
RMS weighted error: 0.000997
max weighted error: 0.001831

[ -7.61340071 26.92124317 36.10794037] cent steps

wedgie:
(2, -1, 4, -5, 4, 8, 0, 4, -2, -10)

mapping by steps:
[[14 22 32 39 48]
[19 30 44 53 66]
[22 35 51 62 76]]

wedgie based complexity: 0.895
RMS weighted error: 0.000941
max weighted error: 0.001401

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> > Right, but it's just a finite computation. If T[1] through T[n] is the
> > OE part of the wedgie, then take the maximum of |T[i]/p_i| and
> > |T[i]/p_j - T[j]/p_j|, where p_i is the ith odd prime.
>
> I take it your T[i]/p_i is the same as my W[i], the ith element of the
> weighted octave-equivalent mapping (multiplied by the number of periods
> to the octave, I keep forgetting that), and you meant to put a "log of"
> in there.

Sorry about that. I meant max(|W[i]|, |W[i]-W[j]|) in your terminology.

In that case, max(|W[i]|) for W > 0 is the same as max(W,-W)

So your formula would be the
> same as mine without two weirdnesses.

Apparently.

> >>Right, but there appears to be a close relationship when you use the
> >>max of the weighted octave-equivalent wedgie instead -- Gene?
> >
> > That's the Kees complexity.
>
> It is? I thought the above formula was for Kees complexity.

That's what I was trying to say.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> Incidentally, it's also possible to write a function to compare two
full
> mappings for equality without them having to be the same numbers.

Obviously; the simplest way to do it I think is to take the wedgie.

> That's why we don't need to specify a universal standard.

How do you avoid duplicate entries in your list without a unique
naming scheme?

> There are an infinite number of equal temperaments, and the wedgie
> doesn't tell me which two I want.

You have a way of defining a unique pair of equal temperament vals for
a given rank two temperament? If so, what is it?

In that case it might find a
> different period mapping. The two methods don't give the same results
> but they are both internally consistent.

I don't know what methods you are referring to.

> > Jamesbond has an OE part of 007. The reason to care is that I was
> > using it as an example.
>
> Is it an example of something I'm likely to care about?

If you want your list to cover higher error temperaments of the kind
Herman cares about, yes.

> > You seem to be using a different choice of val standarization, I'd
> > give counterexamples there but I don't know what it is.
>
> Please, do not accuse me of standardizing ambiguous vals. I've argued
> against that before and I haven't changed my position. That's why I
> always say explicitly how I arrive at my mappings -- as I did in the
bit
> you quoted.

"Best weighted-RMS mappings" isn't explicit.

That way nobody has to remember my jargon from one message
> to another. Which part of "best weighted-RMS mapping" are you having
> difficulty weith?

Are the weightings logs of primes? Do we do the least squares
calculation with respect to the primes, a tonality diamond, or what?

> > Duh. This is obvious. The question is, might you not miss something?
>
> As compared to what method that guarantees you won't miss anything?

I've tried to avoid missing things by using three separate methods:
sets of et vals, sets of commas, and (in the case of rank two) periods
and generators. The problem of using vals only becomes a concern as we
go to higher prime limits, where more and more choices are possible,
but where the sets of primes for the low ranks which are of the most
interest become unpleasantly large and therefore numerous. Either way
you seem to need to look at a lot of possibilities.

> You're less likely to miss something the more ETs you work with, which
> means the program has to be fast. It's fastest when it does
prime-based
> RMS calculations with no wedgies.

I'll take your word for it, but what are these calculations? Where
does the list of starting ets come from?

> > As compared to *not* distinguishing between temperaments produced from
> > commas and those produced from vals, which strikes me as an absurd
idea.
>
> Are you suggesting this idea came from me? Does it have anything to do
> with the quote above?

As best I could tell, that is what you said. What did you mean?

> > I have no idea wha you mean by "that additional unison vector", but I
> > do know it isn't required, whatever it is.
>
> The thing Paul objects to calling a "chromatic unison vector" in the
> general case. To get an ET mapping (or "val" as you call them, for
> anybody following along) you need n-1 unison vectors for n prime
> dimensions. For a rank 2 temperament, you need n-2 unison vectors. So
> one additional unison vector gives you the mapping for an equal
> temperament that belongs to the rank 2 family (maybe not a sensible
> temperament, but a mapping nonetheless).

Right. So what you are talking about in other terms is an interval
which, when you take the interior product with a wedgie, gives you a
val. But why call that a "chromatic unison vector"? It's neither
chromatic nor a unison; in fact, it can't temper to the unison if it
is to be of any use. If two of these so-called "chromatic unison
vectors" are a choice of representatives for the period and generator,
the result will be a standard period-generator mapping. Why not just
call it a "generator", or "generating interval" if you need to draw a
distinction?

> When you say "each of the primes" I think you mean using "each of the
> primes" as additional unison vectors (which Paul says we aren't allowed
> to call "chromatic unison vectors" because they're too big).

Well, I object to gibberish about "unison vectors" in application to
primes, but I think this is right. Using your method of taking n-2
commas, you add to the list each of the primes successively, calculate
the resulting val using determinants, and you get a list of n vals,
which are linearally dependent. Reducing to remove the linear
dependence leaves you with a basis of two of them.

I'll have
> to look up "interior product" to be sure. It probably depends on which
> wedgie you started with, which you neglected to mention.

Eh? There is *only one* wedgie for a given temperament. That one,
obviously, is the one you start with. There are scads of n-2 sets of
primes for a rank two temperament, but not scads of wedgies. Just one.

But it sounds
> like additional unison vectors to me.
>
> "Find something" and "solve the problem" isn't a method at all.
It's an
> aspiration.

It's a solution which works in every case, and which I've defined.
That's not an aspiration, it's a method.

> I do need to get rid of them because I'm trying to communicate with
> people who won't know about them. Yes, that is largely what this
> argument's about. I place a high value on ease of communication.

Your non-confusing language confuses the hell out of me, and I wonder
if anyone else knows what you mean with all of this talk of chromatic
unison vectors. I just now figured it out (and found it is really,
really, bad terminology, incidentally.)

> The one about the search.

Which is? Title? Message number?

> > How are these defined--in terms of your three originating vals? That
> > doesn't strike me as such a good system.
>
> Yes. It's the easiest way to do it, and accurate enough with the right
> function from Numeric.

More useful would be to give a tuning map.

> They're both related to orwell and magic because they both contain 19,
> 22 and 31 (=12+19). They differ in the 11-mapping. This can be
deduced
> from the ET mappings, no need for commas.

It's a more delicate question than that in my meaning of "related"
here, having to do with what commas the tuning shrinks without
reducing it entirely to a unison. A temperament which reduces
3125/3072 to two cents is in an obvious sense, different from what you
are using, related to magic.

> Why are they significant? You haven't given any reasons they stand out.

Because they have relatively low error, given their complexity, from
most of the alternatives. If you have a best-of 11-limit planar
temperaments list they should be on it unless it is awfully short, for
basically the same reason that magic and orwell pop up on lists.

> Anyway, as you're using this as an excuse to push your results, here
are
> mine.

Eh? You were the one who mentioned these temperaments, not me. I've
posted on more than one occasion about planar temperaments, but the
cited article was about tictactoe diagrams, not "results", whatever
you mean by that.

They're limited by a wedgie complexity of 0.9 and an optimal RMS
> weighted error of 0.001.

Is this everything under those constraints?

Gene Ward Smith wrote:

> How do you avoid duplicate entries in your list without a unique
> naming scheme?

Only use the octave-equivalent part of the mapping.

>>There are an infinite number of equal temperaments, and the wedgie >>doesn't tell me which two I want. > > You have a way of defining a unique pair of equal temperament vals for
> a given rank two temperament? If so, what is it?

No, I don't, not as such. That's why I get different results for different methods.

> In that case it might find a >>different period mapping. The two methods don't give the same results >>but they are both internally consistent.
> > I don't know what methods you are referring to.

1) start with commas

2) start with equal temperaments

>>>Jamesbond has an OE part of 007. The reason to care is that I was
>>>using it as an example.
>>
>>Is it an example of something I'm likely to care about?
> > > If you want your list to cover higher error temperaments of the kind
> Herman cares about, yes.

I won't even bother to check if it's covered unless you give me a reason why it wouldn't be.

> "Best weighted-RMS mappings" isn't explicit.

It's a good indicator, at least.

> That way nobody has to remember my jargon from one message >>to another. Which part of "best weighted-RMS mapping" are you having >>difficulty weith?
> > Are the weightings logs of primes? Do we do the least squares
> calculation with respect to the primes, a tonality diamond, or what?

Yes, logs of primes. The mapping. I said the mapping. That means the mapping. If I wanted you to use a tonality diamond I would have said that.

> I've tried to avoid missing things by using three separate methods:
> sets of et vals, sets of commas, and (in the case of rank two) periods
> and generators. The problem of using vals only becomes a concern as we
> go to higher prime limits, where more and more choices are possible,
> but where the sets of primes for the low ranks which are of the most
> interest become unpleasantly large and therefore numerous. Either way
> you seem to need to look at a lot of possibilities. Wow! You've been doing temperament searches! How far do you go? Do you have results? Code?

>>You're less likely to miss something the more ETs you work with, which >>means the program has to be fast. It's fastest when it does
> prime-based >>RMS calculations with no wedgies.
> > I'll take your word for it, but what are these calculations? Where
> does the list of starting ets come from?

http://x31eq.com/temper/regular.py
http://x31eq.com/temper/regular_wedgie.py

>>>As compared to *not* distinguishing between temperaments produced from
>>>commas and those produced from vals, which strikes me as an absurd
> idea.
>>Are you suggesting this idea came from me? Does it have anything to do >>with the quote above?
> > As best I could tell, that is what you said. What did you mean?

I have a method for getting the full mapping from the commas. It isn't pretty, but the code's at

http://x31eq.com/temper.py

The generator is uniquely defined by the choice of what we still don't have a better term for than "chromatic unison vector". As such there's no need to optimize the temperament to get a unique key, if you're sufficiently obsessive that the octave equivalent part isn't good enough.

The method for getting the mapping from equal temperaments is much more stable, so we can assume that always gives the correct mapping.

These two methods are entirely different, and there's no need to compare the results without optimizing.

>>>I have no idea wha you mean by "that additional unison vector", but I
>>>do know it isn't required, whatever it is.
>>
>>The thing Paul objects to calling a "chromatic unison vector" in the >>general case. To get an ET mapping (or "val" as you call them, for >>anybody following along) you need n-1 unison vectors for n prime >>dimensions. For a rank 2 temperament, you need n-2 unison vectors. So >>one additional unison vector gives you the mapping for an equal >>temperament that belongs to the rank 2 family (maybe not a sensible >>temperament, but a mapping nonetheless).
> > > Right. So what you are talking about in other terms is an interval
> which, when you take the interior product with a wedgie, gives you a
> val. But why call that a "chromatic unison vector"? It's neither
> chromatic nor a unison; in fact, it can't temper to the unison if it
> is to be of any use. If two of these so-called "chromatic unison
> vectors" are a choice of representatives for the period and generator,
> the result will be a standard period-generator mapping. Why not just
> call it a "generator", or "generating interval" if you need to draw a
> distinction?

Probably, yes. I call it a chromatic unison vector because somebody, maybe Paul, called it that. It is a unison vector. It tempers to a unison in the equal temperament whose mapping you get by doing whatever operation it is with wedge products and complements, or by taking a column of the inverse of the matrix multiplied by the determinant as an extension of Fokker's work.

We already have a meaning for "generator" and it's different.

>>When you say "each of the primes" I think you mean using "each of the >>primes" as additional unison vectors (which Paul says we aren't allowed >>to call "chromatic unison vectors" because they're too big).
> > Well, I object to gibberish about "unison vectors" in application to
> primes, but I think this is right. Using your method of taking n-2
> commas, you add to the list each of the primes successively, calculate
> the resulting val using determinants, and you get a list of n vals,
> which are linearally dependent. Reducing to remove the linear
> dependence leaves you with a basis of two of them.

Yes, that happens not to be what I do, but it certainly uses additional unison vectors.

> Eh? There is *only one* wedgie for a given temperament. That one,
> obviously, is the one you start with. There are scads of n-2 sets of
> primes for a rank two temperament, but not scads of wedgies. Just one.

No, there are two important ones, related by the complement operation. It really does matter which one you use, but never mind.

>>"Find something" and "solve the problem" isn't a method at all. > It's an >>aspiration.
> > It's a solution which works in every case, and which I've defined.
> That's not an aspiration, it's a method.

Where have you defined it? Show me the code!

>>I do need to get rid of them because I'm trying to communicate with >>people who won't know about them. Yes, that is largely what this >>argument's about. I place a high value on ease of communication.
> > Your non-confusing language confuses the hell out of me, and I wonder
> if anyone else knows what you mean with all of this talk of chromatic
> unison vectors. I just now figured it out (and found it is really,
> really, bad terminology, incidentally.)

Well, Lordy, if you didn't now what I meant why didn't you say so instead of confidently asserting that I was wrong?

For unison vectors, see the online Fokker paper:

http://www.xs4all.nl/~huygensf/doc/fokkerpb.html

We use octave-specific vectors but otherwise it's the same. The chromatic unison vector is the one that you don't temper out. The Fokker determinant tells you how many notes there are in the MOS.

>>The one about the search.
> > Which is? Title? Message number?

"Rank 3 temperament search" 13976

>>>How are these defined--in terms of your three originating vals? That
>>>doesn't strike me as such a good system.
>>
>>Yes. It's the easiest way to do it, and accurate enough with the right >>function from Numeric.
> > More useful would be to give a tuning map.

What's a tuning map?

>>Why are they significant? You haven't given any reasons they stand out.
> > Because they have relatively low error, given their complexity, from
> most of the alternatives. If you have a best-of 11-limit planar
> temperaments list they should be on it unless it is awfully short, for
> basically the same reason that magic and orwell pop up on lists. But they don't! Well, we have different ideas of what counts as "short". I'd expect magic and orwell to both make their relevant top tens. For some criteria magic is optimal. These ones don't make it.

>>Anyway, as you're using this as an excuse to push your results, here
> > are > >>mine. > > > Eh? You were the one who mentioned these temperaments, not me. I've
> posted on more than one occasion about planar temperaments, but the
> cited article was about tictactoe diagrams, not "results", whatever
> you mean by that.

So do you think there's anything special about them, or was I wasting my time looking at them?

> They're limited by a wedgie complexity of 0.9 and an optimal RMS > >>weighted error of 0.001.
> > > Is this everything under those constraints? Of course not.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> Wow! You've been doing temperament searches! How far do you go? Do
> you have results? Code?

Is this sarcasm, and if so, about what? I've done a lot of searching
using Maple, which is very slow but still fast enough to get the job done.

/tuning-math/message/2791

/tuning-math/message/4556

/tuning-math/message/4557

/tuning-math/message/5620

/tuning-math/message/5648

/tuning-math/message/8809

/tuning-math/message/10783

/tuning-math/message/10142

/tuning-math/message/11104

> > I'll take your word for it, but what are these calculations? Where
> > does the list of starting ets come from?
>
> http://x31eq.com/temper/regular.py
> http://x31eq.com/temper/regular_wedgie.py

Python script isn't much help.

> I have a method for getting the full mapping from the commas. It isn't
> pretty, but the code's at
>
> http://x31eq.com/temper.py

Why is it hard? You can use an n-fold wedge product, for which the
code isn't too hard, since you already have a parity function.

> The generator is uniquely defined by the choice of what we still don't
> have a better term for than "chromatic unison vector".

I gave a better term, "generating interval".

> The method for getting the mapping from equal temperaments is much more
> stable, so we can assume that always gives the correct mapping.

Why is there any instability? Are you using floating point operations
for some reason?

> These two methods are entirely different, and there's no need to
compare
> the results without optimizing.

Seems to me to be a bad plan. Why not combine both approaches?

> Probably, yes. I call it a chromatic unison vector because somebody,
> maybe Paul, called it that. It is a unison vector. It tempers to a
> unison in the equal temperament whose mapping you get by doing whatever
> operation it is with wedge products and complements, or by taking a
> column of the inverse of the matrix multiplied by the determinant as an
> extension of Fokker's work.

That produces a val. It may or may not be a val for an equal
temperament, and in the cases we are discussing, probably won't be.

> No, there are two important ones, related by the complement operation.
> It really does matter which one you use, but never mind.

Not if you follow my definition, which is that a wedgie is *always* a
multival. I take no responsibility for confusions introduced by Paul.

> >>"Find something" and "solve the problem" isn't a method at all.
> > It's an
> >>aspiration.
> >
> > It's a solution which works in every case, and which I've defined.
> > That's not an aspiration, it's a method.
>
> Where have you defined it? Show me the code!

Code for what? I've got 194 kb of Maple code, mostly not very elegant,
but effective. The way I've written interior products follows my usual
inelegant prodceedure of doing a separate program in each prime
limits. For example, however, the interior product ("down") of a
7-limit bival by a monzo I compute this way:

a7down := proc(l, w)
# product down to val of wedgie with monzo
[l[3]*w[4]+l[2]*w[3]+w[2]*l[1], l[5]*w[4]+l[4]*w[3]-w[1]*l[1],
w[4]*l[6]-l[4]*w[2]-w[1]*l[2], -w[3]*l[6]-l[5]*w[2]-w[1]*l[3]] end:

Then I have something else for the 11-limit,

a11down := proc(x, y)
# linear wedgie x times monzo y down
[x[1]*y[2]+x[2]*y[3]+x[3]*y[4]+x[4]*y[5],
-x[1]*y[1]+x[5]*y[3]+x[6]*y[4]+x[7]*y[5],
-x[2]*y[1]-x[5]*y[2]+x[8]*y[4]+x[9]*y[5],
-x[3]*y[1]-x[6]*y[2]-x[8]*y[3]+x[10]*y[5],
-x[4]*y[1]-x[7]*y[2]-x[9]*y[3]-x[10]*y[4]] end:

and so forth. Nothing to write home about, and I've been trying to
write some more general code lately.

> Well, Lordy, if you didn't now what I meant why didn't you say so
> instead of confidently asserting that I was wrong?

Because I didn't need to know what the hell one was to know you can do
your calculations with commas and not bother with them.

> >>The one about the search.
> >
> > Which is? Title? Message number?
>
> "Rank 3 temperament search" 13976

The reason I didn't comment on it is that it was just a comma list,
followed by only one rank three, thirteen limit temperament.

> > More useful would be to give a tuning map.
>
> What's a tuning map?

A mapping primes-->cents. For instance, <1200 1896.578 2786.314| for
1/4-comma 5-limit meantone.

> But they don't! Well, we have different ideas of what counts as
> "short". I'd expect magic and orwell to both make their relevant top
> tens. For some criteria magic is optimal. These ones don't make it.

And the ten better temperaments of comparable complexity are which?

> So do you think there's anything special about them, or was I
wasting my
> time looking at them?

They are strictly planar, of reasonably low complexity, and fairly
accurate. Where again are those ten better temperaments of comparable
complexity?

Graham Breed wrote:
> I have a method for getting the full mapping from the commas. It isn't > pretty, but the code's at
> > http://x31eq.com/temper.py
> > The generator is uniquely defined by the choice of what we still don't > have a better term for than "chromatic unison vector". As such there's > no need to optimize the temperament to get a unique key, if you're > sufficiently obsessive that the octave equivalent part isn't good enough.

I was kind of thinking that "comma" was our substitute for "unison vector" in cases like this. I don't have any particular objections to "unison vector", since I'm familiar with various meanings of the word "vector" in different fields that are clear from context, but I've been avoiding it recently since it doesn't seem to have caught on. Still, it's useful to have "comma" for a specific size of small interval, and the possible ambiguity could be avoided by using "unison vector" or something like it.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > >>Wow! You've been doing temperament searches! How far do you go? Do >>you have results? Code?
> > > Is this sarcasm, and if so, about what? I've done a lot of searching
> using Maple, which is very slow but still fast enough to get the job done.

No, I really don't know what you're doing unless you tell me. None of these links say "I've been searching for temperaments". Most of them say "here's a list of temperaments" blah blah blah. Some of them came when I either had a poor internet connection, or was bedridden, so sorry for not studying them in detail. This one:

/tuning-math/message/11104

says you "used standard vals", but not what for and in the middle of such a boring paragraph I probably skipped it.

Really, I thought you were only doing searches by commas and in low limits. Even these links don't go above the 13-limit. But I'll certainly look at them.

>>>I'll take your word for it, but what are these calculations? Where
>>>does the list of starting ets come from?
>>
>>http://x31eq.com/temper/regular.py
>>http://x31eq.com/temper/regular_wedgie.py
> > > Python script isn't much help.

Whyever not? You can try these, but they're out of date:

http://x31eq.com/temper.ml
http://x31eq.com/write_temper.ml

http://x31eq.com/temper/temper.c
http://x31eq.com/temper/temper.h

>>I have a method for getting the full mapping from the commas. It isn't >>pretty, but the code's at
>>
>>http://x31eq.com/temper.py
> > Why is it hard? You can use an n-fold wedge product, for which the
> code isn't too hard, since you already have a parity function. What's an n-fold wedge product? I can't really remember how I did it. It uses a few loops to find numbers that work.

>>The generator is uniquely defined by the choice of what we still don't >>have a better term for than "chromatic unison vector".
> > I gave a better term, "generating interval". Am I supposed to comment on this? You think calling two different things by the same name is "better" and without getting anybody to agree with you think the language is changed?

>>The method for getting the mapping from equal temperaments is much more >>stable, so we can assume that always gives the correct mapping.
> > Why is there any instability? Are you using floating point operations
> for some reason?

It was unstable because I used primes in place of chromatic unison vectors. Now I use more sensible intervals it's probably giving sensible generators but I haven't looked at it much since then.

>>These two methods are entirely different, and there's no need to
> compare >>the results without optimizing.
> > Seems to me to be a bad plan. Why not combine both approaches?

Why do it? I can see value in cross-checking the methods with each other. If that does mean doing the search twice, checking that the right generator mappings are in the right badness order is good enough. But I haven't started any research like that.

>>Probably, yes. I call it a chromatic unison vector because somebody, >>maybe Paul, called it that. It is a unison vector. It tempers to a >>unison in the equal temperament whose mapping you get by doing whatever >>operation it is with wedge products and complements, or by taking a >>column of the inverse of the matrix multiplied by the determinant as an >>extension of Fokker's work.
> > That produces a val. It may or may not be a val for an equal
> temperament, and in the cases we are discussing, probably won't be.

I don't care if it's a sensible equal temperament any more than if it's a sensible unison vector.

>>No, there are two important ones, related by the complement operation. >>It really does matter which one you use, but never mind.
> > Not if you follow my definition, which is that a wedgie is *always* a
> multival. I take no responsibility for confusions introduced by Paul.

Please, Gene, try to stop assuming we can all remember all your definitions. I can't. I really can't. I take "wedgie" as being a slack term for "the things you can do wedge products with".

>>>>"Find something" and "solve the problem" isn't a method at all. >>>
>>>It's an >>>
>>>>aspiration.
>>>
>>>It's a solution which works in every case, and which I've defined.
>>>That's not an aspiration, it's a method.
>>
>>Where have you defined it? Show me the code!
> > Code for what? I've got 194 kb of Maple code, mostly not very elegant,
> but effective. The way I've written interior products follows my usual
> inelegant prodceedure of doing a separate program in each prime
> limits. For example, however, the interior product ("down") of a
> 7-limit bival by a monzo I compute this way:

Code for finding the period mapping by some method that doesn't use a chromatic unison vector. Simplified or converted to pseudocode if it's too complicated in the raw.

>>"Rank 3 temperament search" 13976
> > The reason I didn't comment on it is that it was just a comma list,
> followed by only one rank three, thirteen limit temperament. No it wasn't, it was also an announcement that I'd done something. If you've done something similar you could have offered help. And if you were interested in what I did you could have asked for more details.

>>>More useful would be to give a tuning map.
>>
>>What's a tuning map?
> > A mapping primes-->cents. For instance, <1200 1896.578 2786.314| for
> 1/4-comma 5-limit meantone.

Okay.

>>But they don't! Well, we have different ideas of what counts as >>"short". I'd expect magic and orwell to both make their relevant top >>tens. For some criteria magic is optimal. These ones don't make it.
> > And the ten better temperaments of comparable complexity are which?

I already gave you a list. Magic an orwell don't sit at the bottom of a list of comparable complexity.

>>So do you think there's anything special about them, or was I
> wasting my >>time looking at them?
> > > They are strictly planar, of reasonably low complexity, and fairly
> accurate. Where again are those ten better temperaments of comparable
> complexity?

So you do think there's something special about them! Well, some of them are good, some are bad. You have the best ones.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> No, I really don't know what you're doing unless you tell me. None of
> these links say "I've been searching for temperaments". Most of them
> say "here's a list of temperaments" blah blah blah.

The blah blah blah being where the cutoffs are and what methods I used
to generate candidates.

> Really, I thought you were only doing searches by commas and in low
> limits. Even these links don't go above the 13-limit.

You mean 17-limit. But my own belief is that there is little utility
in going beyond the 19-limit.

> Whyever not? You can try these, but they're out of date:

It's hard to read.

> > Why is it hard? You can use an n-fold wedge product, for which the
> > code isn't too hard, since you already have a parity function.
>
> What's an n-fold wedge product?

You start off with v1, v2 ... vn and compute v1^v2^...^vn from that.
It's not too hard.

> > I gave a better term, "generating interval".
>
> Am I supposed to comment on this? You think calling two different
> things by the same name is "better" and without getting anybody to
agree
> with you think the language is changed?

What in the world are you saying? Anyway, almost anything would be
better than calling it a "chromatic unison vector", which is actively
pernicious. The point of sticking the word "generating" in there is
that it is relevant; if you take an interior product of a bival with
an interval, you get a val which tempers out the interval, and hence
could be used as a part of a generator map if you have another val
sending the interval to +-1.

However, for any rank two temperament, if you take an interval you get
a unique val in this way, which you could call an "associated val", or
use some other word since you hate "val", just so long as it doesn't
contain the word "unison". Representing val? It represents the
interval in the temperament in question.

> >>The method for getting the mapping from equal temperaments is much
more
> >>stable, so we can assume that always gives the correct mapping.
> >
> > Why is there any instability? Are you using floating point operations
> > for some reason?
>
> It was unstable because I used primes in place of chromatic unison
> vectors.

If you were using integers and not floats, how can that be unstable?
Is this from an optimization step? Anyway, using primes always works
and should not be giving you any instability problems if you compute
generators from the primes.

> > Seems to me to be a bad plan. Why not combine both approaches?
>
> Why do it? I can see value in cross-checking the methods with each
> other.

That's a point, so long as you do finally end up naming them uniquely.

> > That produces a val. It may or may not be a val for an equal
> > temperament, and in the cases we are discussing, probably won't be.
>
> I don't care if it's a sensible equal temperament any more than if it's
> a sensible unison vector.

Same problem as with "unison vector". If it isn't a unison, don't call
it one. If it isn't an equal temperament, don't call it one. Do you
think <0 1 4 10| is a 7-limit equal temperament?

> Please, Gene, try to stop assuming we can all remember all your
> definitions. I can't. I really can't. I take "wedgie" as being a
> slack term for "the things you can do wedge products with".

Well, it isn't and never was. It means a multival, reduced in a
standardized way so that it uniquely represents a given regular
temperament. Since you yourself took part in and influenced the
discussion of how to do the reduction, that shouldn't have been as
difficult as it appears to have been.

Unlike "chromaticm unison vector", it has a precise definition which
does not degenerate into seemingly contradictory babbling.

> Code for finding the period mapping by some method that doesn't use a
> chromatic unison vector. Simplified or converted to pseudocode if it's
> too complicated in the raw.

Here's Maple code; see if that will do:

lval7 := proc(l)
# subgroup vals of 7-limit wedgie
-[[0, -l[1], -l[2], -l[3]], [l[1], 0, -l[4], -l[5]],
[l[2], l[4], 0, -l[6]], [l[3], l[5], l[6], 0]] end:

mherm7 := proc(l)
# Hermite generators for 7-limit wedgie
local a;
a := convert(ihermite(lval7(l)), listlist);
[a[1], a[2]] end:

"ihermite" is a Maple function call. Here's the Maple help file, sans
examples:

Calling Sequence
ihermite(A)
ihermite(A, U)
Parameters
A - rectangular matrix of integers
U - name

Description

The function ihermite computes the Hermite Normal Form (reduced row
echelon form) of a rectangular matrix of integers.

The Hermite normal form of A is an upper triangular matrix H with
rank(A) = the number of nonzero rows of H. If A is an n by n matrix of
full rank then |det(A)| = product(H[i, i], i=1..n).

This is not an efficient method for computing the rank or determinant
except that this may yield a partial factorization of det(A) without
doing any explicit factorizations.

The Hermite normal form is obtained by doing elementary row
operations. This includes interchanging rows, multiplying through a
row by -1, and adding an integral multiple of one row to another.
One can use transposes to obtain column form of the Hermite Normal Form.

In the case of two arguments, the second argument U will be assigned
the transformation matrix on output, such that the following holds:
ihermite(A) = U A.

The command with(linalg,ihermite) allows the use of the abbreviated
form of this command.

> No it wasn't, it was also an announcement that I'd done something. If
> you've done something similar you could have offered help. And if you
> were interested in what I did you could have asked for more details.

I assumed you knew I'd been posting about this stuff myself, and that
if you had had a list of temperaments you wanted to give, you would
have done so. I completely misunderstood the purpose of your post.

> > And the ten better temperaments of comparable complexity are which?
>
> I already gave you a list.

Title? Message number?

Magic an orwell don't sit at the bottom of a
> list of comparable complexity.

They usually don't sit at the top either. They should be on it.

> > They are strictly planar, of reasonably low complexity, and fairly
> > accurate. Where again are those ten better temperaments of comparable
> > complexity?
>
> So you do think there's something special about them! Well, some of
> them are good, some are bad. You have the best ones.

Except you seem to be saying you have some a lot better hidden away
somewhere. It's certainly easy to beat them in terms of error by going
up a tad in complexity, or in terms of complexity by going down a tad,
but what about sticking close to this region?

> Not if you follow my definition, which is that a wedgie is *always* a
> multival. I take no responsibility for confusions introduced by Paul.

I thought Herman had something somewhere that it could also be the
wedge product of two commas. Yes, his language here, reviewed by
Gene, discusses ""bicomma" wedgie"s and the like.

http://www.io.com/~hmiller/music/regular-temperaments.html

-Carl

>You mean 17-limit. But my own belief is that there is little utility
>in going beyond the 19-limit.

Agreed. Though it might be nice to do away with limits entirely.
I seem to remember seeing hints of methods around here that might
do this.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > Not if you follow my definition, which is that a wedgie is *always* a
> > multival. I take no responsibility for confusions introduced by Paul.
>
> I thought Herman had something somewhere that it could also be the
> wedge product of two commas. Yes, his language here, reviewed by
> Gene, discusses ""bicomma" wedgie"s and the like.
>
> http://www.io.com/~hmiller/music/regular-temperaments.html

I don't see anything really wrong with using the phrase "bicomma
wedgie", if you make it clear it is not the same as a multival wedgie;
though I think there is a better way to say it. However in this
instance I've already remarked I didn't like the reference to the
wedge product of two commas as a "wedgie". It's not even clearly
defined--is it reduced, or is it just the pure wedge product? If it is
reduced, what is the canonical reduction? I'd suggest, if we want a
standard definition for a multimonzo wedgie, we define it as the
complement of the corresponding multival wedgie. Of course then
"complement of a wedgie" is perfectly adequate, and avoids any
possibility of confusion. Hence I suggest using "complement of a
wedgie" for these.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >>>Not if you follow my definition, which is that a wedgie is *always* a
>>>multival. I take no responsibility for confusions introduced by Paul.
>>
>>I thought Herman had something somewhere that it could also be the
>>wedge product of two commas. Yes, his language here, reviewed by
>>Gene, discusses ""bicomma" wedgie"s and the like.
>>
>>http://www.io.com/~hmiller/music/regular-temperaments.html
> > > I don't see anything really wrong with using the phrase "bicomma
> wedgie", if you make it clear it is not the same as a multival wedgie;
> though I think there is a better way to say it. However in this
> instance I've already remarked I didn't like the reference to the
> wedge product of two commas as a "wedgie". It's not even clearly
> defined--is it reduced, or is it just the pure wedge product? If it is
> reduced, what is the canonical reduction? I'd suggest, if we want a
> standard definition for a multimonzo wedgie, we define it as the
> complement of the corresponding multival wedgie. Of course then
> "complement of a wedgie" is perfectly adequate, and avoids any
> possibility of confusion. Hence I suggest using "complement of a
> wedgie" for these.

Well, now's the time to get the terminology right before I go too far with something that isn't commonly accepted or at least understood. I've seen these just called "bicommas", I believe; how about just skipping the "wedgie" and just using "bicomma"? Or should it be a "bimonzo"? I still have a ways to go before I get all the things into the page that I've been planning, so you can take your time looking over the page.

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

I've
> seen these just called "bicommas", I believe; how about just skipping
> the "wedgie" and just using "bicomma"?

I would say that is fine if the two things you wedge together can
reasonably be called commas.

> Or should it be a "bimonzo"?

That would be any wedge product of two monzos.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Agreed. Though it might be nice to do away with limits entirely.
> I seem to remember seeing hints of methods around here that might
> do this.

This can be done, but I don't see the utility of it.

The positive rational numbers under multiplication, Q*, is a "free
abelian group of infinite rank". It can be represented by infinite
monzos |e2 e3 e5 ...> where all but a finite number of the
coefficients, all of which must be integers, are zero. The group of
homomorphisms from Q* to Z, called in mathematise Hom(Q*, Z), is
another group, *not* free, but "torsion free" or "flat" (these mean
the same for abelian groups) called the Baer-Specker group, or Z^N,
whose properties are trickier than you might think when I tell you it
is just the infinite vals <v2 v3 v5 ...| where it is *not* assumed all
but a finite number of the coefficients, which again must be integers,
are zero. Any finite number of such infinitary vals, members of the
Baer-Specker group, can be wedged together, and produce an infinitary
n-val. These can be reduced to wedgies in the usual way, producing
infinitary temperaments which temper out an infinitude of independent
commas, for which a basis for the rank two temperaments is still the
triprime commas.

Similarly, any finite number of commas can be wedged and give an
infinitary multimonzo. You can't relate this to a multival by taking
the complement, which doesn't exist. One kind of wedgie tempers out
commas giving a kernel which has a countably infinite basis, and the
other kind has a kernel with a finite basis. They are therefore
different things.

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

>These can be reduced to wedgies in the usual way, producing
> infinitary temperaments which temper out an infinitude of independent
> commas, for which a basis for the rank two temperaments is still the
> triprime commas.

I meant these span or generate the kernal, sorry.

>> These can be reduced to wedgies in the usual way, producing
>> infinitary temperaments which temper out an infinitude of independent
>> commas, for which a basis for the rank two temperaments is still the
>> triprime commas.
>
>I meant these span or generate the kernal, sorry.

I'm afraid this doesn't help me with this statement. And additionally,
it raises the question: What's the difference between having commas of a
certain type be a basis, and having a kernal containing commas only of
that type?

-Carl

>> Agreed. Though it might be nice to do away with limits entirely.
>> I seem to remember seeing hints of methods around here that might
>> do this.
>
>This can be done, but I don't see the utility of it.
//
>These can be reduced to wedgies in the usual way, producing
>infinitary temperaments which temper out an infinitude of independent
>commas, for which a basis for the rank two temperaments is still the
>triprime commas.

I'm not sure what you mean by triprime commas here -- monzos with
only three nonzero coefficients, or?

I don't think I'd seen hints of this approach, but there's your
zeta stuff, and convergent weightings. I don't know how what goes
into the zeta computation -- can it work for rank 2 temperaments
as well as equal temperaments? By convergent weightings, I refer
to an idea I had (which I thought I detected in one of Graham's
messages perhaps) whereby steep enough error and complexity
weightings are used that they always converge as the harmonic limit
goes to infinity. I have no idea how such a thing might actually
work.

Oh, Paul often has phrases like 'without need for limits'... I wish
I understood this better.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >I meant these span or generate the kernal, sorry.
>
> I'm afraid this doesn't help me with this statement. And additionally,
> it raises the question: What's the difference between having commas of a
> certain type be a basis, and having a kernal containing commas only of
> that type?

If you start multiplying together commas with three primes in their
factorization, you can end up with commas with any number of primes in
the factorization.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I'm not sure what you mean by triprime commas here -- monzos with
> only three nonzero coefficients, or?

Right, or in multiplicative terms, with at most three primes in their
factorization.

> I don't think I'd seen hints of this approach, but there's your
> zeta stuff, and convergent weightings. I don't know how what goes
> into the zeta computation -- can it work for rank 2 temperaments
> as well as equal temperaments?

I don't see how, but there may be something which could be done along
those lines.

>> >I meant these span or generate the kernal, sorry.
>>
>> I'm afraid this doesn't help me with this statement. And additionally,
>> it raises the question: What's the difference between having commas of a
>> certain type be a basis, and having a kernal containing commas only of
>> that type?
>
>If you start multiplying together commas with three primes in their
>factorization, you can end up with commas with any number of primes in
>the factorization.

It will be at most three times the number of commas you've multiplied.
I'm not sure what the significance of this is.

I admit I don't have a clue what your proposal is, here, but it sounds
like you're not doing away with limits at all... just allowing
infinitely-long monzos and vals.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I admit I don't have a clue what your proposal is, here, but it sounds
> like you're not doing away with limits at all... just allowing
> infinitely-long monzos and vals.

Why isn't infinite-limit the same as doing away with limits?

>>> These can be reduced to wedgies in the usual way, producing
>>> infinitary temperaments which temper out an infinitude of independent
>>> commas, for which a [kernel] for the rank two temperaments is still
>>> the triprime commas.

>> I'm not sure what you mean by triprime commas here -- monzos with
>> only three nonzero coefficients, or?
>
>Right, or in multiplicative terms, with at most three primes in their
>factorization.

O-K.

>>It will be at most three times the number of commas you've multiplied.
>>I'm not sure what the significance of this is.
>>
>>I admit I don't have a clue what your proposal is, here, but it sounds
>>like you're not doing away with limits at all... just allowing
>>infinitely-long monzos and vals.

I guess I need to understand why it's important that they only have
three primes each. Does this somehow make computations with
infinitely-long monzos and vals possible?

-Carl

>> I admit I don't have a clue what your proposal is, here, but it sounds
>> like you're not doing away with limits at all... just allowing
>> infinitely-long monzos and vals.
>
>Why isn't infinite-limit the same as doing away with limits?

Is it computable?

Even if you stick to finite monzos but with no restrictions on
the highest prime (which is what I did a few months back), you
need heavy weighting to punish large primes like 1663.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I guess I need to understand why it's important that they only have
> three primes each. Does this somehow make computations with
> infinitely-long monzos and vals possible?

For a rank n temperament, where everything is generated by n
generators, n+1 primes can cancel out, giving a comma. But this has
nothing specifically to do with the infinitely long vals and monzos.
Computation there is possibly when only finite sums and products of
non-zero integers arise. This will be the case in evaluating <val |
monzo>, since the monzo has only finitely many nonzero coefficients.
It also works for va11 ^ val2, since for each coefficient, you take a
difference of products of two integers, which is an integer.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >Why isn't infinite-limit the same as doing away with limits?
>
> Is it computable?

You can't compute a lot of things, such as unwieghted optimizations.
So, you weight.

> Even if you stick to finite monzos but with no restrictions on
> the highest prime (which is what I did a few months back), you
> need heavy weighting to punish large primes like 1663.

Absolute convergence will do it.

Gene:
>>Why isn't infinite-limit the same as doing away with limits?

Carl:
> Is it computable?

A lot of calculations can be done without a prime-limit bound. You need to define you equal temperament mappings as lazy lists. As a Schemer this should be familiar to you. You could write a function to compute Gene's standard val (the number of notes to an octave multiplied by the size in octaves of a given prime number, rounded to the nearest integer). Generation of prime numbers is a standard example of lazy evaluation ;) If the laziness is working, you can look at 5-limit intervals with your infinite-limit mapping and all will be well. Start looking at the 7-limit and the lazy lists pop off some more elements and all remains well. You can define higher rank temperaments based on those lazy lists no problem. You can't be sure a temperament is contorted with a finite number of calculations, although you may be able to tell that it is.

The second issue is about the errors converging. There's a popular branch of mathematics involving infinite limits. It's well known that the sum of reciprocals of squares of integers 1/n2 has a stable value. (n2 is going to mean "n squared" for the rest of this paragraph.) The 1/log(p) weighted prime errors look arbitrary to me. So with a weighting of 1/p2log(p) you can guarantee convergence. I think Gene said 1/p is good enough. It may be that 1/p converges where p is prime, but not for all integers, or the RMS calculation means you're really summing 1/p2. I'm sure Gene's right, anyway. So the weightings have to be something like that.

If you set the weightings such that the total error converges quickly, then you can do calculations to within a certain accuracy with a limited number of primes. You could probably get the first few entries of the optimal RMS-error mapping for a given weighting. But you'd have to think about what you were doing -- lazy evaluation won't do it for you.

You can also determing the weighted minimax for nearest-prime mappings. The worst error in any prime is half a scale step. So that's 1/2n for n-EDO. If the weighting's 1/p, you know that the highest weighted error for prime p is 1/2np. Once the weighted minimax already exceeds that for the prime you're looking at you can stop calculating.

> Even if you stick to finite monzos but with no restrictions on
> the highest prime (which is what I did a few months back), you
> need heavy weighting to punish large primes like 1663.

Yes. My guess is that either the weighting is so light that stupidly high primes end up dominating the calculation, or it's so heavy that only the first few primes matter. But I'd rather know than guess, so see what you find.

Graham

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> I've
> > seen these just called "bicommas", I believe; how about just skipping
> > the "wedgie" and just using "bicomma"?
>
> I would say that is fine if the two things you wedge together can
> reasonably be called commas.

The wedge product of two commas is not necessarily (the complement of)
a wedgie, is it?

- Rich Holmes

--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:

> The wedge product of two commas is not necessarily (the complement of)
> a wedgie, is it?

If one comma is not the multiple of another, then by taking out any
common factors and changing the sign if necessary (or equivalently,
changing the order in which you do the wedge product) then it will be
the complement of a wedgie for a codimension 2 temperament.

At 04:17 AM 1/18/2006, you wrote:
>Gene:
>>>Why isn't infinite-limit the same as doing away with limits?
>
>Carl:
>> Is it computable?
>
>A lot of calculations can be done without a prime-limit bound. You need
>to define you equal temperament mappings as lazy lists. As a Schemer
>this should be familiar to you.

I could be wrong, but I think this requires a true lazy language
like Haskell. I'd heard that Scheme has lazy evaluation, but in
fact it's pretty limited.

>You could write a function to compute
>Gene's standard val (the number of notes to an octave multiplied by the
>size in octaves of a given prime number, rounded to the nearest
>integer).

I do have that.

>Generation of prime numbers is a standard example of lazy
>evaluation ;)

My prime number generator takes an upper bound as an input. :(

>If the laziness is working, you can look at 5-limit
>intervals with your infinite-limit mapping and all will be well. Start
>looking at the 7-limit and the lazy lists pop off some more elements and
>all remains well.

Still sounds like this is using the concept of limits.

>The second issue is about the errors converging. There's a popular
>branch of mathematics involving infinite limits. It's well known that
>the sum of reciprocals of squares of integers 1/n2 has a stable value.
>(n2 is going to mean "n squared" for the rest of this paragraph.)

If you used /\ for wedge you could reserve ^ for powers.

>The 1/log(p) weighted prime errors look arbitrary to me.

That's the TOP weighting, yes?

What about Gene's suggestion of 1/sqrt(p), "apeing the Zeta function
on the critical line"?

>So with a weighting of 1/p2log(p) you can guarantee convergence. I
>think Gene said 1/p is good enough. It may be that 1/p converges
>where p is prime, but not for all integers, or the RMS calculation means
>you're really summing 1/p2. I'm sure Gene's right, anyway. So the
>weightings have to be something like that.

Hm...

>If you set the weightings such that the total error converges quickly,
>then you can do calculations to within a certain accuracy with a limited
>number of primes.

Right.

>> Even if you stick to finite monzos but with no restrictions on
>> the highest prime (which is what I did a few months back), you
>> need heavy weighting to punish large primes like 1663.
>
>Yes. My guess is that either the weighting is so light that stupidly
>high primes end up dominating the calculation, or it's so heavy that
>only the first few primes matter. But I'd rather know than guess, so
>see what you find.

No time to revise my search code right now... maybe in a few weeks.

-Carl

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> >>No, I really don't know what you're doing unless you tell me. None of >>these links say "I've been searching for temperaments". Most of them >>say "here's a list of temperaments" blah blah blah. > > The blah blah blah being where the cutoffs are and what methods I used
> to generate candidates.

Yes, you say where the candidates come from, but not why you want us to look at them.

>>Really, I thought you were only doing searches by commas and in low >>limits. Even these links don't go above the 13-limit. > > You mean 17-limit. But my own belief is that there is little utility
> in going beyond the 19-limit.

Sorry, yes, 17-limit. Except you say that wasn't really a search.

19-limit rank 2 temperaments are pretty well covered by equal temperament searches. You may have to wait a few minutes, but you'll get the results. It may be worth going beyond the 19-limit to learn how the searches behave, in order to optimize the higher rank searches. I don't know if the rank 3 search is good enough or not, or if we're ever going to want higher ranks.

If you can get reasonable 19-limit searches working with commas, that's news. I think it's difficult.

>>Whyever not? You can try these, but they're out of date:
> > It's hard to read.

Why didn't you say so before? I posted the links before and nobody said they had trouble. Which bits aren't clear?

>>>Why is it hard? You can use an n-fold wedge product, for which the
>>>code isn't too hard, since you already have a parity function. >>
>>What's an n-fold wedge product? > > You start off with v1, v2 ... vn and compute v1^v2^...^vn from that.
> It's not too hard.

Right, the wedgie. And what was I supposed to do with it again?

>>>I gave a better term, "generating interval". >>
>>Am I supposed to comment on this? You think calling two different >>things by the same name is "better" and without getting anybody to
> agree >>with you think the language is changed?
> > What in the world are you saying? Anyway, almost anything would be
> better than calling it a "chromatic unison vector", which is actively
> pernicious. The point of sticking the word "generating" in there is
> that it is relevant; if you take an interior product of a bival with
> an interval, you get a val which tempers out the interval, and hence
> could be used as a part of a generator map if you have another val
> sending the interval to +-1. But the generator's already the generating interval.

Yuk, "interior product", "bival", "val", "generator", "val". I don't really feel like parsing that.

A unison vector is an interval from just intonation which is tempered to be the same as a unison. How complicated is that?

The best term in any situation is usually the one people already understand. Last time we had a consensus on this list, it was about "chromatic unison vector".

> However, for any rank two temperament, if you take an interval you get
> a unique val in this way, which you could call an "associated val", or
> use some other word since you hate "val", just so long as it doesn't
> contain the word "unison". Representing val? It represents the
> interval in the temperament in question.

I don't hate the word "val". I'm avoiding using it without defining it. And I've yet to find it easier to give the definition than avoid the word.

But yes, you get one of them.

>>>>The method for getting the mapping from equal temperaments is much
> more >>>>stable, so we can assume that always gives the correct mapping.
>>>
>>>Why is there any instability? Are you using floating point operations
>>>for some reason?
>>
>>It was unstable because I used primes in place of chromatic unison >>vectors. > > If you were using integers and not floats, how can that be unstable?
> Is this from an optimization step? Anyway, using primes always works
> and should not be giving you any instability problems if you compute
> generators from the primes.

It's unstable in that it's less likely to give a generator that lies between a unison and a period for an optimal (or any sensible) tuning. The reason is that the mapping you produce may not be for a sensible equal temperament.

>>>Seems to me to be a bad plan. Why not combine both approaches?
>>
>>Why do it? I can see value in cross-checking the methods with each >>other.
> > That's a point, so long as you do finally end up naming them uniquely.

Either that or I write an "__eq__" method to determine if the two temperament objects are the same.

Actually, no, you caught me out there. You don't need to give unique names to cross-check the methods. If you generated the temperaments from equal temperaments, any temperament family that the same equal temperaments belong to is the same. If you generated the temperaments from commas, any temperament with the same unison vectors is the same.

>>>That produces a val. It may or may not be a val for an equal
>>>temperament, and in the cases we are discussing, probably won't be.
>>
>>I don't care if it's a sensible equal temperament any more than if it's >>a sensible unison vector.
> > Same problem as with "unison vector". If it isn't a unison, don't call
> it one. If it isn't an equal temperament, don't call it one. Do you
> think <0 1 4 10| is a 7-limit equal temperament?

A unison vector the same as a unison once it's tempered. Was that your only problem? It was easily disposed of.

<0 1 4 10| could be the mapping for an equal temperament. That is, it maps primes to tempered intervals. I can't see any point in calling the thing it maps to anything other than an equal temperament. There are cases when I'd call it an "equal mapping" to show we don't care about the putative temperament.

In general I don't want to define two different terms when a simpler definition of a single term says anything I need it to.

>>Please, Gene, try to stop assuming we can all remember all your >>definitions. I can't. I really can't. I take "wedgie" as being a >>slack term for "the things you can do wedge products with".
> > Well, it isn't and never was. It means a multival, reduced in a
> standardized way so that it uniquely represents a given regular
> temperament. Since you yourself took part in and influenced the
> discussion of how to do the reduction, that shouldn't have been as
> difficult as it appears to have been.

Can I guess what a multival is? At least I made the same mistake as Herman, and that you didn't spot.

> Unlike "chromaticm unison vector", it has a precise definition which
> does not degenerate into seemingly contradictory babbling.

I've defined "unison vector" now. A chromatic unison vector is a unison vector of an equal temperament which belongs to a family of regular temperaments. However, it is only a unison vector of that one equal temperament.

An equal temperament is an object with a list of integers mapping primes to integers representing scale steps. Or, in group theory terms, a homomorphism from ratios to a free abelian group of rank one. Did I get that right?

Where's the contradiction?

>>Code for finding the period mapping by some method that doesn't use a >>chromatic unison vector. Simplified or converted to pseudocode if it's >>too complicated in the raw.
> > > Here's Maple code; see if that will do:
> > lval7 := proc(l)
> # subgroup vals of 7-limit wedgie
> -[[0, -l[1], -l[2], -l[3]], [l[1], 0, -l[4], -l[5]], > [l[2], l[4], 0, -l[6]], [l[3], l[5], l[6], 0]] end:

Aren't these the chromatic unison vectors you said you weren't using.

> mherm7 := proc(l)
> # Hermite generators for 7-limit wedgie
> local a;
> a := convert(ihermite(lval7(l)), listlist);
> [a[1], a[2]] end:
> > "ihermite" is a Maple function call. Here's the Maple help file, sans
> examples:
<snip>

That's useful, because you haven't defined that before.

Now, how about the method that doesn't use a chromatic unison vector?

>>>And the ten better temperaments of comparable complexity are which?
>>
>>I already gave you a list. > > > Title? Message number?

Same title as this, follow it back.

Anyway, I now know the example I singled out is your "Zeus". It beats both Apollo and Minerva for both complexity and optimal error. Minerva is much better than Apollo. I don't know why you singled out Apollo.

If you're sticking to strictly planar temperaments, that makes them more significant.

> Magic an orwell don't sit at the bottom of a >>list of comparable complexity.
> > They usually don't sit at the top either. They should be on it.

Magic does usually sit at the top of 9-limit lists of equivalent complexity. If you disagree, give a counter-example. Orwell might be 2 or 3.

>>>They are strictly planar, of reasonably low complexity, and fairly
>>>accurate. Where again are those ten better temperaments of comparable
>>>complexity?
>>
>>So you do think there's something special about them! Well, some of >>them are good, some are bad. You have the best ones.
> > Except you seem to be saying you have some a lot better hidden away
> somewhere. It's certainly easy to beat them in terms of error by going
> up a tad in complexity, or in terms of complexity by going down a tad,
> but what about sticking close to this region?

No, you can't get better than best. You have 7 of my top 10 whic is too many for it to be an accident. The other three are of relatively high complexity. Then you have white elephants like Skadi and Apollo that don't look special. Probably it means you're using a different badness measure to me. But unless you say what it is I can't comment on it.

My top 10 is (the numbers are arbitrary tags, there are two kinds of 19):

1. Wonder (11-limit neutral-thirds lattice)
2. 7&12&14
3. 7&8&14
4. Portent
5. 7&12&19
6. Freya
7. Marvel
8. Zeus
9. Indra
10. Thrush

My current badness measure is complexity cubed times error. So it looks like we agree more with low errors. My number 2 may be outside your error cutoff:

[ 20.99907587 53.56660308 29.45074257] cent steps

wedgie:
(0, 1, 0, 4, 0, -6, 4, 0, -6, 0)

mapping by steps:
[[ 7 11 16 19 24]
[12 19 28 34 42]
[14 22 32 39 48]]

tuning map (cents):
[ 1202.10316403 1896.67162962 2778.27386239 3368.82590647 4167.41079
359]

wedgie based complexity: 0.447
RMS weighted error: 3.136 cents/octave
max weighted error: 4.652 cents/octave

but this one has a lower error than Vulcan:

[ 23.54749922 51.471315 44.55341478] cent steps

wedgie:
(0, 3, 0, 5, 0, -4, 1, 0, 4, 8)

mapping by steps:
[[ 7 11 16 19 24]
[ 8 13 19 23 28]
[14 22 32 39 48]]

tuning map (cents):
[ 1200.35082141 1908.32471151 2780.42424537 3368.82590647 4144.90071
056]

wedgie based complexity: 0.513
RMS weighted error: 2.287 cents/octave
max weighted error: 4.019 cents/octave

This one is number 5:

[ 10.12758683 43.56517558 31.97720745] cent steps

wedgie:
(0, 0, 1, 0, 4, 10, 0, 4, 13, 12)

mapping by steps:
[[ 7 11 16 19 24]
[12 19 28 34 42]
[19 30 44 53 65]]

tuning map (cents):
[ 1201.24215627 1898.45801457 2788.86343321 3368.43211421 4151.31794
236]

wedgie based complexity: 0.680
RMS weighted error: 1.236 cents/octave
max weighted error: 2.206 cents/octave

even lower error, and much less complex then Freya and the like. It tempers out 81:80 and 126:125. You gave it in this list:

/tuning-math/message/4557

where you said it had

comp 15.10180563 rms 5.640679273 bad 4999.235518

You don't say what that means. Presumably, "comp" is complexity, "rms" is an error and "bad" is a function of the other two.

Of Vulcan, you said:

comp 9.571162783 rms 11.60092948 bad 3287.793582

I make it slightly more complex (0.632 compared to 0.680) but your difference is more dramatic. I use a the sum-abs of the Tenney-weighted wedgie divided by the number of primes. In the "Complexity" thread you didn't chip in to say you had an alternative.

We agree about the RMS errors but they must have a low weight in your badness. You say badness < 5000 at the top so this one only just made it in.

You give Apollo as

comp 14.53451469 rms 4.698274549 bad 3783.900689

So it's slightly less complex and the bigger difference in the errors swings it.

Graham

"Gene Ward Smith" <gwsmith@svpal.org> writes:

> --- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>
> If one comma is not the multiple of another, then by taking out any
> common factors and changing the sign if necessary (or equivalently,
> changing the order in which you do the wedge product) then it will be
> the complement of a wedgie for a codimension 2 temperament.

Exactly -- a bicomma is a wedge product of two commas, not necessarily
without common factors and not necessarily with the right sign to be
the complement of a wedgie. So if Herman means the complement of a
wedgie he should say so; if he means the wedge product of commas, he
should say that. They're related, but they're not interchangeable.

- Rich Holmes

Carl Lumma wrote:
> At 04:17 AM 1/18/2006, you wrote:
>>
>>A lot of calculations can be done without a prime-limit bound. You need >>to define you equal temperament mappings as lazy lists. As a Schemer >>this should be familiar to you.
> > > I could be wrong, but I think this requires a true lazy language
> like Haskell. I'd heard that Scheme has lazy evaluation, but in
> fact it's pretty limited.

Probably, yes.

>>You could write a function to compute >>Gene's standard val (the number of notes to an octave multiplied by the >>size in octaves of a given prime number, rounded to the nearest >>integer).
> > I do have that.

Right ho.

>>Generation of prime numbers is a standard example of lazy >>evaluation ;)
> > My prime number generator takes an upper bound as an input. :(

I think I saw it in a Haskell tutorial. It's probably less efficient because it can't use a sieve. Haskell in practice tends to be like that.

>>If the laziness is working, you can look at 5-limit >>intervals with your infinite-limit mapping and all will be well. Start >>looking at the 7-limit and the lazy lists pop off some more elements and >>all remains well.
> > Still sounds like this is using the concept of limits.

It isn't limited by limits. You can have a function to factorize a ratio and return a vector only as large as it needs to be. No need for the user to ever specify a limit. You can even skip primes if they never happen to crop up. My wedge product things already work like this ... until you need the complement.

>>The second issue is about the errors converging. There's a popular >>branch of mathematics involving infinite limits. It's well known that >>the sum of reciprocals of squares of integers 1/n2 has a stable value. >>(n2 is going to mean "n squared" for the rest of this paragraph.)
> > If you used /\ for wedge you could reserve ^ for powers.

Or I could use ** in the tradition of mathematical ASCIIfication going back to Fortran. But it's easier to see precedence by using nothing.

>>The 1/log(p) weighted prime errors look arbitrary to me.
> > That's the TOP weighting, yes?

Yes.

> What about Gene's suggestion of 1/sqrt(p), "apeing the Zeta function
> on the critical line"?

Gene knows more about it than I do. But I'll humbly suggest that it might be slow to converge.

> No time to revise my search code right now... maybe in a few weeks.

Keep us posted!

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> Gene Ward Smith wrote:

> > The blah blah blah being where the cutoffs are and what methods I used
> > to generate candidates.
>
> Yes, you say where the candidates come from, but not why you want us to
> look at them.

I didn't want people to look at them, so I didn't give them. Instead,
I gave the search results.

> Sorry, yes, 17-limit. Except you say that wasn't really a search.

Not much of one, but it beats saying how well your program could do a
search, and then not giving any results.

> > It's hard to read.
>
> Why didn't you say so before? I posted the links before and nobody
said
> they had trouble. Which bits aren't clear?

You misunderstand me. It's Python code, whereas what is needed is a
precise explanation in English of what your algorithm is.

> >>>Why is it hard? You can use an n-fold wedge product, for which the
> >>>code isn't too hard, since you already have a parity function.
> >>
> >>What's an n-fold wedge product?
> >
> > You start off with v1, v2 ... vn and compute v1^v2^...^vn from that.
> > It's not too hard.
>
> Right, the wedgie. And what was I supposed to do with it again?

You wanted to know how you can start from commas, and this would be
how. As to what you do with it, exactly the same as what you'd do with
a wedgie obtained in any other way. I do a quick complexity check, and
if it passes add it to a set to be evaluated later. Exactly the same
thing which I do if I start from et vals, or periods and generators.

> Yuk, "interior product", "bival", "val", "generator", "val". I don't
> really feel like parsing that.

At least it isn't gibberish.

> A unison vector is an interval from just intonation which is
tempered to
> be the same as a unison. How complicated is that?

But you don't use this definition. Instead, you use "chromatic unison
vectors". Moreover, even without the "chromatic", the term is not used
consistently to mean this.

And "comma" is shorter.

> The best term in any situation is usually the one people already
> understand. Last time we had a consensus on this list, it was about
> "chromatic unison vector".

We have no consensus whatever on that, as a check of recent postings
here by you, me and Paul will show. Not only is there is no agreed on
meaning, we are not even close to one. People do NOT understand it.
How could they?

> I don't hate the word "val". I'm avoiding using it without defining
it.

It's a mapping denoted by <v2 v3 ... vp|, and given by integers vq
which map the prime q to the integer vq, from p-limit intervals to the
integers.

> And I've yet to find it easier to give the definition than avoid the
word.

So instead you call it an "equal temperament", which is outright
gibberish.

> It's unstable in that it's less likely to give a generator that lies
> between a unison and a period for an optimal (or any sensible) tuning.

Why call that stabiility? And what does it matter?

> The reason is that the mapping you produce may not be for a sensible
> equal temperament.

Therefore, don't call it an equal temperament.

> A unison vector the same as a unison once it's tempered. Was that your
> only problem? It was easily disposed of.

Wrong, because it doesn't cover your "chromatic" unison vectors, which
are not unisons when tempered. Nor does it cover the octave-equivalent
thingie people insisted on using when I first arrived here. It also
introduces unnecessary technobabble, which you object to. "Comma" is
better.

> <0 1 4 10| could be the mapping for an equal temperament. That is, it
> maps primes to tempered intervals.

It maps primes to integers, not to intervals. Moreover, it maps 2 to
0. Is this the 0-equal temperament? In fact, of course, it isn't an
equal temperament at all; it is the generator mapping for meantone.

I can't see any point in calling the
> thing it maps to anything other than an equal temperament.

An equal temperament involves an equal division of the octave, and
some kind of actual tempering. If you divide the coefficiets of an
equal temperament val <v2 v3 ... vp| by vq/(log2(q)v2) for the vq
term, the result should be reasonably close to <1 1 1 ... 1|. In
particular, v2 can't be zero, and the terms should be non-decreasing.

There are
> cases when I'd call it an "equal mapping" to show we don't care about
> the putative temperament.

I see--"equal mapping" is better than "val" because it is longer,
somewhat confusing (equal to what? What is equal?) and doesn't have a
precise definition.

> I've defined "unison vector" now. A chromatic unison vector is a
unison
> vector of an equal temperament which belongs to a family of regular
> temperaments.

You've completely changed your definition. Definitions have to stick
with meaning one thing. You haven't defined "family of regular
temperaments". Moreover, if I don't know what you mean by "an equal
temperament", and of course I don't, this makes no sense whatever.

However, it is only a unison vector of that one equal
> temperament.

It's a unison vector for one equal temperament, the definition of
which we don't know. It is *not* a unison vector for something else,
the nature of which you have given us no clue about.

> An equal temperament is an object with a list of integers mapping
primes
> to integers representing scale steps.

Ie, a val. Except your definition is just wrong, because that's not
what people mean by an equal temperament. Hence, both you and the rest
of us on this list would be far better off if you simply said "val".

Or, in group theory terms, a
> homomorphism from ratios to a free abelian group of rank one. Did I
get
> that right?

More or less. Make it "p-limit ratios".

> > Here's Maple code; see if that will do:
> >
> > lval7 := proc(l)
> > # subgroup vals of 7-limit wedgie
> > -[[0, -l[1], -l[2], -l[3]], [l[1], 0, -l[4], -l[5]],
> > [l[2], l[4], 0, -l[6]], [l[3], l[5], l[6], 0]] end:
>
> Aren't these the chromatic unison vectors you said you weren't using.

Nope, because they are vals belonging to the temperament defined by l.

> That's useful, because you haven't defined that before.

I've given cites on a number of occasions.

> Now, how about the method that doesn't use a chromatic unison vector?

And this chromatic unison vector you think you saw is what?

> >>I already gave you a list.
> >
> >
> > Title? Message number?
>
> Same title as this, follow it back.

I did that. I found zero (0) temperaments.

> Anyway, I now know the example I singled out is your "Zeus". It beats
> both Apollo and Minerva for both complexity and optimal error.

By your measure of complexity. However, a look at the tictactoe
diagram here:

/tuning-math/message/12298

shows it is in a 6x4 rectangle, whereas apollo is 3x5 and minerva is
5x3. So one complexity measure which arises in a practical situation,
using the temperament with Tonalsoft, rates it as more complex. In any
case a list which restricts itself only to temperaments which are
better in either complexity or error is pretty restrictive. It's an
interesting idea, but I don't recall it being used before.

Minerva
> is much better than Apollo. I don't know why you singled out Apollo.

Because apollo arose as the relevant to one particular issue, whereas
minerva wasn't. And much better is baloney; if you think that I think
you should examine your premises.

Here is the least squares optimal tuning map for apollo:

<1200 1903.6 2781.1 3369.5 4155.0|

Here is the difference between that and JI:

<0 1.67 -5.20 0.66 3.66|

Here is the same thing for minerva:

<1200 1900.7 2786.7 3374.7 4148.0|

<0 -1.29 0.36 5.85 -3.29|

Minerva beats apollo in the 5-limit, and that might be a big
consideration for some. In the 7 and 11 limits they are pretty
similar. But minerva, with its flat fifth, was completely different
than what I needed for the case I was discussing.

Minerva, by the way, shrinks 78732/78125 to nearly nothing (-0.7
cents) in the tuning I gave, so adding it is a freebie. However, the
resulting temperament is very complex past the 5-limit, so minerva has
what you might call an independent existence. Apollo does the same, a
little less extremely, for 3125/3072, and adding that gives magic.
Hence apollo is not much distinguished from 11-limit magic.

> If you're sticking to strictly planar temperaments, that makes them
more
> significant.

I'd say so, but it wasn't relevant at the time.

> Magic does usually sit at the top of 9-limit lists of equivalent
> complexity. If you disagree, give a counter-example. Orwell might
be 2
> or 3.

Well, I may check on that. It gets competition from meantone, which is
clearly the one for magic to beat. I'm pretty sure you could cook the
books so that whichever one you wanted came out on top. If you add a
little more complexity, garibaldi comes into the picture.

> My top 10 is (the numbers are arbitrary tags, there are two kinds of
19):

Then for gosh sakes say which 19! How many of these use non-standard vals?

> 1. Wonder (11-limit neutral-thirds lattice)
> 2. 7&12&14
> 3. 7&8&14
> 4. Portent
> 5. 7&12&19
> 6. Freya
> 7. Marvel
> 8. Zeus
> 9. Indra
> 10. Thrush

> [ 20.99907587 53.56660308 29.45074257] cent steps

Ouch!

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> > What about Gene's suggestion of 1/sqrt(p), "apeing the Zeta function
> > on the critical line"?
>
> Gene knows more about it than I do. But I'll humbly suggest that it
> might be slow to converge.

I wouldn't even guarantee it does converge, but it's about as far as you
can push the issue.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> My top 10 is (the numbers are arbitrary tags, there are two kinds of
19):
>
> 1. Wonder (11-limit neutral-thirds lattice)

This has TM basis {243/242, 441/440}. With little damage one can add
4000/3993 to the commas, and get 11-limit "harry" temperament, with
a wedgie of <<12 24 20 30 26 -2 6 -49 -48 15||. This is the 58&72
(standard vals, as always) temperament. The planar temperament could
be then be written 41&58&72.

> 2. 7&12&14

With standard vals, this would mean {36/35, 729/704}, but the vals are
not defined so it isn't of much use to say this. Below we find the TM
basis is actually {45/44, 81/80}. Hence in terms of standard vals this
could have been written 7&12&19 or 12&19&26, for example. Name?

> 3. 7&8&14

Once again the notation is obscure; it turns out this is nothing you
could get from superparticulars, the TM basis being {250/243,
1331/1296}. Both commas are 7-free, and the corresponding no-sevens
temperament has a no-sevens bival of <<9 15 12 3 -12 -24||. Given the
rather high error, the complexity seems to be nothing to write home
about, and I wonder why this comes so high on Graham's list, and also
why he says it has a lower complexity than vulcan, the {56/55, 64/63}
temperament. A comparison between wedgies does not seem to support
that view, so I wonder if we are really talking about another
temperament. In terms of standard vals this can be written 7&8&29. Name?

> 4. Portent

Portent is rated lower by Graham than I rated it. It tempers out
385/384 and 441/440, and is on my list of things I should try out. In
terms of standard vals, you can write it 31&41&46.

> 5. 7&12&19

As you can tell by looking at the wedgie, which Graham gave below,
this is just septimal meantone with 11 tacked on, so the comma basis
is {81/80, 126/125}. Name?

> 6. Freya

{2401/2400, 3025/3024}

Note the badness measure is not log-flat, so the higher complexity
temperaments come in low.

> 7. Marvel

{225/224, 385/384}

I'm surprised this comes in lower than Freya.

> 8. Zeus

{121/120, 176/175}

> 9. Indra

{540/539, 1375/1372}

> 10. Thrush

{126/125, 176/175}

These are not, of course, ten temperaments of complexity better than
minerva or apollo and lower error.

> I make it slightly more complex (0.632 compared to 0.680) but your
> difference is more dramatic. I use a the sum-abs of the
Tenney-weighted
> wedgie divided by the number of primes.

In the "Complexity" thread you
> didn't chip in to say you had an alternative.

Max of the Tenney-weighted (I presume this means by 1/(log(p)
log(q))?) absolute error is an obvious alternative. RMS will give
something in between. I wonder if this is why you rate {250/243,
1331/1296} so high. Your method may be unduly weighted in favor of
things with a lot of zeros in the wedgie, resulting from adding a
prime to a temperament with commas lacking it.

> We agree about the RMS errors but they must have a low weight in your
> badness.

As usual, I use logflat badness.

Rich Holmes wrote:
> "Gene Ward Smith" <gwsmith@svpal.org> writes:
> > >>--- In tuning-math@yahoogroups.com, Rich Holmes<rsholmes@m...> wrote:
>>
>>If one comma is not the multiple of another, then by taking out any
>>common factors and changing the sign if necessary (or equivalently,
>>changing the order in which you do the wedge product) then it will be
>>the complement of a wedgie for a codimension 2 temperament.
> > > Exactly -- a bicomma is a wedge product of two commas, not necessarily
> without common factors and not necessarily with the right sign to be
> the complement of a wedgie. So if Herman means the complement of a
> wedgie he should say so; if he means the wedge product of commas, he
> should say that. They're related, but they're not interchangeable.

Which is why I preferred not using "complement of a wedgie".

But I see that in the example I changed the signs. Wedging [-4, 4, -1> with [7, 0, -3> results in [[-28, 19, -12>>, not the result I have on the page, which is [[28, -19, 12>>. I guess I was assuming that bicommas would be normalized like bivals, but this is easily corrected.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
that in the example I changed the signs. Wedging [-4, 4, -1>
> with [7, 0, -3> results in [[-28, 19, -12>>, not the result I have on
> the page, which is [[28, -19, 12>>. I guess I was assuming that
bicommas
> would be normalized like bivals, but this is easily corrected.

Bivals aren't normalized, wedgies are normalized.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

>>Sorry, yes, 17-limit. Except you say that wasn't really a search.
> > Not much of one, but it beats saying how well your program could do a
> search, and then not giving any results.

No, because anybody can take my code and do the search they want.

> You misunderstand me. It's Python code, whereas what is needed is a
> precise explanation in English of what your algorithm is.

That's difficult. "Precise" and "English" don't really go together. The clearest way I know to precisely state the algorithm is to give the Python code. There's this explanation:

http://riters.com/microtonal/index.cgi/FindingLinearTemperaments

which doesn't say how I do the ET search, or even how I did it back then. Is that what you want? The idea is to take each mapping of each prime that doesn't already give a sum larger than what we're looking for.

>>A unison vector is an interval from just intonation which is
> tempered to >>be the same as a unison. How complicated is that?
> > But you don't use this definition. Instead, you use "chromatic unison
> vectors". Moreover, even without the "chromatic", the term is not used
> consistently to mean this.

How don't I follow the definition? How isn't it used consistently?

> And "comma" is shorter.

Also more ambiguous because it has other meanings. But "comma" is good. I prefer "unison vector" because it's the first term in the literature with this specific meaning. I'm thinking of a "comma" as being something which might be used as a unison vector.

>>The best term in any situation is usually the one people already >>understand. Last time we had a consensus on this list, it was about >>"chromatic unison vector".
> > We have no consensus whatever on that, as a check of recent postings
> here by you, me and Paul will show. Not only is there is no agreed on
> meaning, we are not even close to one. People do NOT understand it.
> How could they?

No, we don't have a consensus, that's why I said "last time". I don't know where Paul is. Probably he has more important things to do than argue is. When he comes back I'll try and sort it out.

If people want to understand "unison vector" they could start by reading the Fokker paper.

>>I don't hate the word "val". I'm avoiding using it without defining
> it. > > It's a mapping denoted by <v2 v3 ... vp|, and given by integers vq
> which map the prime q to the integer vq, from p-limit intervals to the
> integers.

So it's the notation, not the thing being notated?

>>And I've yet to find it easier to give the definition than avoid the
> word.
> > So instead you call it an "equal temperament", which is outright
> gibberish.

Why?

>>It's unstable in that it's less likely to give a generator that lies >>between a unison and a period for an optimal (or any sensible) tuning. > > Why call that stabiility? And what does it matter?

I called it "stability" because it was the first word that came into my head when I was writing it. It doesn't really matter much. I didn't expect you to worry about it.

>>The reason is that the mapping you produce may not be for a sensible >>equal temperament.
> > Therefore, don't call it an equal temperament.

Whether I choose to call it an equal temperament is distinct from whether it is an equal temperament by my definition.

>>A unison vector the same as a unison once it's tempered. Was that your >>only problem? It was easily disposed of.
> > Wrong, because it doesn't cover your "chromatic" unison vectors, which
> are not unisons when tempered. Nor does it cover the octave-equivalent
> thingie people insisted on using when I first arrived here. It also
> introduces unnecessary technobabble, which you object to. "Comma" is
> better.

It does, as I define it below. My definition does cover the octave-equivalent case but in a subtle way that I didn't expect you to notice. It looks like I succeeded because you falsely made "are not" the opposite of "the same as". But that's not important and I don't insist on it.

It doesn't introduce anything. It was introduced into the literature by Fokker before I was born. I don't have any control over that. It doesn't have any shady existing meanings like "comma".

>><0 1 4 10| could be the mapping for an equal temperament. That is, it >>maps primes to tempered intervals.
> > It maps primes to integers, not to intervals. Moreover, it maps 2 to
> 0. Is this the 0-equal temperament? In fact, of course, it isn't an
> equal temperament at all; it is the generator mapping for meantone.

Yes, it maps primes to integers. If those integers represented intervals it would be the mapping for an equal temperament. That equal temperament would indeed have 0 steps to the octave. You can happily call it a "generator mapping" and never worry your reader about whether it also refers to an equal temperament.

> I can't see any point in calling the >>thing it maps to anything other than an equal temperament. > > An equal temperament involves an equal division of the octave, and
> some kind of actual tempering. If you divide the coefficiets of an
> equal temperament val <v2 v3 ... vp| by vq/(log2(q)v2) for the vq
> term, the result should be reasonably close to <1 1 1 ... 1|. In
> particular, v2 can't be zero, and the terms should be non-decreasing.

Yes, that's a definition, and I don't see the point of it. Mine is simpler.

Do you include 88CET and the like? Division of the octave implies that octaves have to be present.

> There are >>cases when I'd call it an "equal mapping" to show we don't care about >>the putative temperament.
> > I see--"equal mapping" is better than "val" because it is longer,
> somewhat confusing (equal to what? What is equal?) and doesn't have a
> precise definition.

"Equal mapping" is better than "val" because it relates to the terms "equal temperament" and "mapping" which have to be introduced first. As such it's much easier for the reader to remember what it refers to than a word that doesn't sound like anything. Either term has a precise definition if given a precise definition.

I'd prefer "linear mapping" to "equal mapping" if that didn't conflict with "linear temperament". Maybe "rank one mapping", or "one-map" when you want to be terse. Usually "mapping" is fine in context.

Short words are useful if you want to use them a lot. I find I don't need a word for "val" often enough in a text to justify the expense of a new term. Of course, as you keep obstinately using it against all objections it is becoming part of the community's language and perhaps I'll have to give in one day. Thankfully I don't think it's appeared in print yet.

>>I've defined "unison vector" now. A chromatic unison vector is a
> unison >>vector of an equal temperament which belongs to a family of regular >>temperaments. > > You've completely changed your definition. Definitions have to stick
> with meaning one thing. You haven't defined "family of regular
> temperaments". Moreover, if I don't know what you mean by "an equal
> temperament", and of course I don't, this makes no sense whatever.

What have I changed?

No, I haven't defined "family of regular temperaments" but that's what we're talking about. So:

A family of rank-r regular temperaments consists of different tunings that all share the same rank-r mapping. For example, meantone is a family of regular temperaments of rank 2. 5, 7, 12 and 19 are examples of rank 1 temperaments that belong to this rank 2 family.

Now I need to define "rank", "temperament", "regular temperament", "tuning" and "mapping". But not being a mathematician I won't bother.

> However, it is only a unison vector of that one equal >>temperament.
> > It's a unison vector for one equal temperament, the definition of
> which we don't know. It is *not* a unison vector for something else,
> the nature of which you have given us no clue about.

The definiton of what? It's a unison vector for an equal temperament, and that equal temperament belongs to the family we're interested in.

>>An equal temperament is an object with a list of integers mapping
> primes >>to integers representing scale steps.
> > Ie, a val. Except your definition is just wrong, because that's not
> what people mean by an equal temperament. Hence, both you and the rest
> of us on this list would be far better off if you simply said "val".

Your definition of "val" above doesn't say anything about scale steps so it isn't the same.

Why is my definition wrong?

Thank you for speaking for the entire list. I didn't realize I was away at the meeting when you were appointed dictator. Do you also speak for people not on the list who might want to understand what we're talking about?

> Or, in group theory terms, a >>homomorphism from ratios to a free abelian group of rank one. Did I
> get >>that right?
> > More or less. Make it "p-limit ratios".

Why the limit?

> I've given cites on a number of occasions.

Which usually refer to square matrices.

>>Now, how about the method that doesn't use a chromatic unison vector?
> > And this chromatic unison vector you think you saw is what?

I give up. You win.

>>>>I already gave you a list. >>>
>>>
>>>Title? Message number?
>>
>>Same title as this, follow it back.
> > I did that. I found zero (0) temperaments.

It doesn't matter.

>>Anyway, I now know the example I singled out is your "Zeus". It beats >>both Apollo and Minerva for both complexity and optimal error. > > By your measure of complexity. However, a look at the tictactoe
> diagram here:
> > /tuning-math/message/12298
> > shows it is in a 6x4 rectangle, whereas apollo is 3x5 and minerva is
> 5x3. So one complexity measure which arises in a practical situation,
> using the temperament with Tonalsoft, rates it as more complex. In any
> case a list which restricts itself only to temperaments which are
> better in either complexity or error is pretty restrictive. It's an
> interesting idea, but I don't recall it being used before.

So what do those diagrams mean?

> Minerva >>is much better than Apollo. I don't know why you singled out Apollo.
> > Because apollo arose as the relevant to one particular issue, whereas
> minerva wasn't. And much better is baloney; if you think that I think
> you should examine your premises.

Ah, there you go. I think most practical uses of rank 3 temperaments depend strongly on the context they're required. That's why I don't place much value on these general searches. They leave out everything important.

Maybe my premises are wrong, yes. I haven't been able to do these searches for long.

> Here is the least squares optimal tuning map for apollo:
> > <1200 1903.6 2781.1 3369.5 4155.0|

That's not at all what I get :-S

> Here is the difference between that and JI:
> > <0 1.67 -5.20 0.66 3.66|
> > Here is the same thing for minerva:
> > <1200 1900.7 2786.7 3374.7 4148.0|

Still different :-S

> <0 -1.29 0.36 5.85 -3.29|
> > Minerva beats apollo in the 5-limit, and that might be a big
> consideration for some. In the 7 and 11 limits they are pretty
> similar. But minerva, with its flat fifth, was completely different
> than what I needed for the case I was discussing. The 5-limit? We were talking about the 11-limit.

> Minerva, by the way, shrinks 78732/78125 to nearly nothing (-0.7
> cents) in the tuning I gave, so adding it is a freebie. However, the
> resulting temperament is very complex past the 5-limit, so minerva has
> what you might call an independent existence. Apollo does the same, a
> little less extremely, for 3125/3072, and adding that gives magic.
> Hence apollo is not much distinguished from 11-limit magic.

Yes, it looks like we're getting into the noise here and there's no advantage on simple linear temperaments.

>>Magic does usually sit at the top of 9-limit lists of equivalent >>complexity. If you disagree, give a counter-example. Orwell might
> be 2 >>or 3.
> > Well, I may check on that. It gets competition from meantone, which is
> clearly the one for magic to beat. I'm pretty sure you could cook the
> books so that whichever one you wanted came out on top. If you add a
> little more complexity, garibaldi comes into the picture.

I was taking meantone(12&19), magic(19&22) and garibaldi(12&29) as not being of equivalent complexity. Meantone can be ahead of magic when they're compared. It partly depends on the difference between an unweighted 7-limit, weighted 7-limit and unweighted 9-limit error because the fifths suffer in meantone.

>>My top 10 is (the numbers are arbitrary tags, there are two kinds of
> 19):
> Then for gosh sakes say which 19! How many of these use non-standard vals?

I defined them lower down, and you make a lot of fuss about using your scrollbar. I don't give a toss about standard vals.

Graham

Gene Ward Smith wrote:

> With standard vals, this would mean {36/35, 729/704}, but the vals are
> not defined so it isn't of much use to say this. Below we find the TM
> basis is actually {45/44, 81/80}. Hence in terms of standard vals this
> could have been written 7&12&19 or 12&19&26, for example. Name?

Why does it need a name? I don't know if it's accurate enough to be useful but it happened to float to the top.

> Once again the notation is obscure; it turns out this is nothing you
> could get from superparticulars, the TM basis being {250/243,
> 1331/1296}. Both commas are 7-free, and the corresponding no-sevens
> temperament has a no-sevens bival of <<9 15 12 3 -12 -24||. Given the
> rather high error, the complexity seems to be nothing to write home
> about, and I wonder why this comes so high on Graham's list, and also
> why he says it has a lower complexity than vulcan, the {56/55, 64/63}
> temperament. A comparison between wedgies does not seem to support
> that view, so I wonder if we are really talking about another
> temperament. In terms of standard vals this can be written 7&8&29. Name?

This wedgie

(0, 3, 0, 5, 0, -4, 1, 0, 4, 8)

Vulcan's

(1, 0, -1, 2, 2, 2, 6, 9, 6, 6)

I think I agree with my program, but it probably depends on how you measure it.

>>4. Portent
> > Portent is rated lower by Graham than I rated it. It tempers out
> 385/384 and 441/440, and is on my list of things I should try out. In
> terms of standard vals, you can write it 31&41&46.

It's still the second from your list.

>>5. 7&12&19
> > As you can tell by looking at the wedgie, which Graham gave below,
> this is just septimal meantone with 11 tacked on, so the comma basis
> is {81/80, 126/125}. Name?

As you can tell by looking at the equal temperaments.

How about calling it "septimal meantone with 11 tacked on"?

>>6. Freya
> > {2401/2400, 3025/3024}
> > Note the badness measure is not log-flat, so the higher complexity
> temperaments come in low.

Yes, I changed the badness measure in the middle of doing this and I think it makes more sense now. Before it was all these nanotemperaments at the top.

> These are not, of course, ten temperaments of complexity better than
> minerva or apollo and lower error.

No, they aren't.

>>I make it slightly more complex (0.632 compared to 0.680) but your >>difference is more dramatic. I use a the sum-abs of the
> > Tenney-weighted > >>wedgie divided by the number of primes. > > > In the "Complexity" thread you > >>didn't chip in to say you had an alternative.
> > Max of the Tenney-weighted (I presume this means by 1/(log(p)
> log(q))?) absolute error is an obvious alternative. RMS will give
> something in between. I wonder if this is why you rate {250/243,
> 1331/1296} so high. Your method may be unduly weighted in favor of
> things with a lot of zeros in the wedgie, resulting from adding a
> prime to a temperament with commas lacking it.

So is that the alternative you used? It's the difference between a mean and a maximum then. Yes, means will tend to favor zeros. If it's so important to the ordering it suggests they're all mediocre and there aren't many good 11-limit planar temperaments with this kind of complexity.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> No, because anybody can take my code and do the search they want.

Sure. All they need to do is install Python, and become a Python
programmer.

> >>A unison vector is an interval from just intonation which is
> > tempered to
> >>be the same as a unison. How complicated is that?
> >
> > But you don't use this definition. Instead, you use "chromatic unison
> > vectors". Moreover, even without the "chromatic", the term is not used
> > consistently to mean this.
>
> How don't I follow the definition? How isn't it used consistently?

If you are counting octaves and fifths as unison vectors, that is not
consistent with your definition.

> I prefer "unison vector" because it's the first term in the
literature
> with this specific meaning.

But it doesn't have this meaning in Fokker, so that's not correct. At
least, I understand Fokker used it for pitch classes, not pitches.

> > It's a mapping denoted by <v2 v3 ... vp|, and given by integers vq
> > which map the prime q to the integer vq, from p-limit intervals to the
> > integers.
>
> So it's the notation, not the thing being notated?

I said "mapping". It's a mapping. "Denoted" means notation.

> > So instead you call it an "equal temperament", which is outright
> > gibberish.
>
> Why?

Because you are giving the name "equal temperament" to things which
cannot remotely be considered equal temperaments, such as
<0 6 -7 -2 15|. That cannot be used as an equal temperament.

> Yes, it maps primes to integers. If those integers represented
> intervals it would be the mapping for an equal temperament.

No it wouldn't. You don't even know how big the steps are.

> Yes, that's a definition, and I don't see the point of it. Mine is
simpler.

Yours is simply wrong, like calling an elephant a kind of dog, and
therefore confusing.

> "Equal mapping" is better than "val" because it relates to the terms
> "equal temperament" and "mapping" which have to be introduced first.

"Dog" is better than "elephant" for anyone familiar with dogs, but who
only read about elephants in history class when Hannibal came up.
Hence, we are going to tell the new arrivals that the huge animals
with the huger noses are dogs. It will be less confusing, and they do
have four legs.

> No, I haven't defined "family of regular temperaments" but that's what
> we're talking about. So:
>
> A family of rank-r regular temperaments consists of different tunings
> that all share the same rank-r mapping. For example, meantone is a
> family of regular temperaments of rank 2.

No one else is using this definition of temperament, and contradictory
notions of what families of temperaments are are already in use.

5, 7, 12 and 19 are examples
> of rank 1 temperaments that belong to this rank 2 family.

Why is this better than saying they are meantone tunings, or support
meantone?

> Now I need to define "rank", "temperament", "regular temperament",
> "tuning" and "mapping". But not being a mathematician I won't bother.
>
> > However, it is only a unison vector of that one equal
> >>temperament.
> >
> > It's a unison vector for one equal temperament, the definition of
> > which we don't know. It is *not* a unison vector for something else,
> > the nature of which you have given us no clue about.
>
> The definiton of what? It's a unison vector for an equal temperament,
> and that equal temperament belongs to the family we're interested in.
>
> >>An equal temperament is an object with a list of integers mapping
> > primes
> >>to integers representing scale steps.

> Your definition of "val" above doesn't say anything about scale
steps so
> it isn't the same.
>
> Why is my definition wrong?

Sorry, I overlooked the "scale steps" part.

This could certainly be an equal temperament, but it could represent a
far stranger sounding transformation of a JI original than that. For
instance, the mapping <-12 -19 -28| applied to 5-limit JI and using a
scale step of size 2^(1/12) inverts things. A mapping <0 19 28| using
the same scale step, or 3^(1/19) if you prefer, will produce highly
exotic results, collapsing all octaves to unisons. <12 21 29| and a
scale step of 2^(1/12) will send 5-limit JI to something which
approximates 5-limit JI, but it will be an approximation of a
different JI original.

> Thank you for speaking for the entire list. I didn't realize I was
away
> at the meeting when you were appointed dictator.

I doubt you will find much support on the list for the idea that
<0 1 4 10| should be called an "equal temperament", a fifth a "unison
vector", or that "meantone temperament" should be replaced with
"family of meantone temperaments" but if someone does think that they
could chime in now.

> > Or, in group theory terms, a
> >>homomorphism from ratios to a free abelian group of rank one. Did I
> > get
> >>that right?
> >
> > More or less. Make it "p-limit ratios".
>
> Why the limit?

I orginally defined it without the limit, but in practice we always
seem to assume the limit, and write it <v2 v3 ... vp|.

> > Here is the least squares optimal tuning map for apollo:
> >
> > <1200 1903.6 2781.1 3369.5 4155.0|
>
> That's not at all what I get :-S

Mine is an unweighted least-squares on the tonality diamond. What's yours?

> I defined them lower down, and you make a lot of fuss about using your
> scrollbar. I don't give a toss about standard vals.

But unless other people have an easy way to determine what you mean by
8&14&15 they get little from your notation, and unless they have some
way to compute it they get next to nothing. When I use something like
that, it is clearly defined and extremely easy to compute.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@g...> wrote:

> This wedgie
>
> (0, 3, 0, 5, 0, -4, 1, 0, 4, 8)

This is the wedgie of a different but related, and more interesting,
temperament. It has TM basis {55/54, 100/99}, so of course it *is*
obtainable from superparticular commas. In terms of standard vals, you
could call it 22&29&37. Probably I screwed up.

The Hermite mapping is

[<1 2 3 0 4|, <0 3 5 0 4|, <0 0 0 1 0|]

In the 5-limit this is just porcupine, and if you add 55/54 to 5-limit
porcupine you get this. It didn't quite make the cut on this old list:

/tuning-math/message/4557

Of course with such small lists it gets more arbitary, in that
reasonable systems get left off.

> Vulcan's
>
> (1, 0, -1, 2, 2, 2, 6, 9, 6, 6)
>
> I think I agree with my program, but it probably depends on how you
> measure it.

If you take a weighted version of the wedgie, you get that the maximum
weighted absolute value for {55/54, 100/99} is 0.767, and for vulcan,
0.707. Not much of a difference, whereas your temperament is more
accurate, so I don't know why it didn't make my cut. 225/224 can be
introduced into it with essentially no tuning damage, at least if the
least-squares tuning is your ideal. This gives you a planar
temperament <<3 5 16 4 1 -18 -4 -28 -8 32||.