Temperament calculations online

<http://63.99.108.42/temper/>

Early days yet, but it is working.

Graham

--- In tuning-math@y..., graham@m... wrote:
> <http://63.99.108.42/temper/>
>
> Early days yet, but it is working.
>
>
> Graham

That's awesome!

--- In tuning-math@y..., graham@m... wrote:
> <http://63.99.108.42/temper/>
>
> Early days yet, but it is working.
>
>
> Graham

Nice work, Graham!

Hey Dave,

Continuing our conversation from the tuning list, I plugged in the
unison vectors 243:245 and 224:225 into Graham's temperament finder,
and got Graham's MAGIC temperament. Graham gives a generator of
380.39 cents. The 19-tone MOS would have 7 otonal and 7 utonal
tetrads, with a maximum error of 5+ cents.

How many tetrads did your MIRACLE Vitale 19 have, Dave? (by which I
mean Rami Vitale's scale, without 21/16, 63/32, 8/7, 12/7, and
Miraclized.)

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Hey Dave,
>
> Continuing our conversation from the tuning list, I plugged in the
> unison vectors 243:245 and 224:225 into Graham's temperament finder,
> and got Graham's MAGIC temperament. Graham gives a generator of
> 380.39 cents. The 19-tone MOS would have 7 otonal and 7 utonal
> tetrads, with a maximum error of 5+ cents.
>
> How many tetrads did your MIRACLE Vitale 19 have, Dave? (by which I
> mean Rami Vitale's scale, without 21/16, 63/32, 8/7, 12/7, and
> Miraclized.)

It has 5 otonal and 5 utonal 7-limit tetrads with max error of 2.7 c.

It's like this on a chain of secors.
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Canasta
+-+-+-+-----+-+-+-+-------+-+-+-------+-+-+-+-----+-+-+-+ MV19
5---------7---1-----------3-----------9----11 11-limit ratios

Dave, didn't you once show that the number of o- and
u-tonal chords must be the same in any linear temp.,
of any number of notes?

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> Dave, didn't you once show that the number of o- and
> u-tonal chords must be the same in any linear temp.,
> of any number of notes?

Hmm. I don't think I showed it. I just claimed it was obvious when you
look at a linear tempered tuning as chains of generators, at least in
the single-chain case.

The utonal chord pattern on the chains must always be the mirror image
of the otonal. If the tuning has a point of reflective symmetry (not
necessarily at a note) then there will be the same number of otonal as
utonal.

With multiple chains they must be considered to be arranged uniformly
around the surface of a cylinder (with the chains parallel to the axis
of the cylinder). If the tuning has a point of reflective symmetry in
this geometry, then there will be the same number of otonal as utonal.

Miracle-tempered Vitale 19 is not a linear temperament since it is not
contiguous on the chain. But it is symmetrical on the chain so it has
equal o and u.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > Hey Dave,
> >
> > Continuing our conversation from the tuning list, I plugged in
the
> > unison vectors 243:245 and 224:225 into Graham's temperament
finder,
> > and got Graham's MAGIC temperament. Graham gives a generator of
> > 380.39 cents. The 19-tone MOS would have 7 otonal and 7 utonal
> > tetrads, with a maximum error of 5+ cents.
> >
> > How many tetrads did your MIRACLE Vitale 19 have, Dave? (by which
I
> > mean Rami Vitale's scale, without 21/16, 63/32, 8/7, 12/7, and
> > Miraclized.)
>
> It has 5 otonal and 5 utonal 7-limit tetrads with max error of 2.7
c.

Now -- if you think of this as a linear temperament where _only_
224:225 is tempered out, I bet you can reduce that error even further.

But: maybe you want those extra tetrads that MAGIC gives you. I
would. (Clearly I'm using a _subjective_ goodness measure, but I
would want to impose that _after_ I had a nice _flat_ survey.)

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> Dave, didn't you once show that the number of o- and
> u-tonal chords must be the same in any linear temp.,
> of any number of notes?
>
> -Carl

Huh? Compare 7-meantone with 7-chain-of-minor thirds.

I wrote,

> Now -- if you think of this as a linear temperament where _only_
> 224:225 is tempered out

"linear" should read "planar"

>> Dave, didn't you once show that the number of o- and
>> u-tonal chords must be the same in any linear temp.,
>> of any number of notes?
>>
>> -Carl
>
> Huh? Compare 7-meantone with 7-chain-of-minor thirds.

I realized this ambiguity after I posted. I meant,
any given linear temp. taken to any given number of
notes.

-C.

> If the tuning has a point of reflective symmetry (not necessarily
> at a note) then there will be the same number of otonal as utonal.

Symmetry with respect to what? If it doesn't have to be a note,
any continuous single-generator chain will have it.

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> > If the tuning has a point of reflective symmetry (not necessarily
> > at a note) then there will be the same number of otonal as utonal.
>
> Symmetry with respect to what? If it doesn't have to be a note,
> any continuous single-generator chain will have it.
>
> -Carl

You bet! But if it isn't a continuous single-generator chain,
then . . .

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >> Dave, didn't you once show that the number of o- and
> >> u-tonal chords must be the same in any linear temp.,
> >> of any number of notes?
> >>
> >> -Carl
> >
> > Huh? Compare 7-meantone with 7-chain-of-minor thirds.
>
> I realized this ambiguity after I posted. I meant,
> any given linear temp. taken to any given number of
> notes.

That doesn't quite do it. You meant:

.. that the number of o-tonal chords is the same as the number
of u-tonal chords in any linear temp. taken to any number of notes.

>> I realized this ambiguity after I posted. I meant,
>> any given linear temp. taken to any given number of
>> notes.
>
> That doesn't quite do it. You meant:
>
> .. that the number of o-tonal chords is the same as the number
> of u-tonal chords in any linear temp. taken to any number of notes.

You took out the "given"??

-Carl

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >> I realized this ambiguity after I posted. I meant,
> >> any given linear temp. taken to any given number of
> >> notes.
> >
> > That doesn't quite do it. You meant:
> >
> > .. that the number of o-tonal chords is the same as the number
> > of u-tonal chords in any linear temp. taken to any number of
notes.
>
> You took out the "given"??

Put 'em back if you like, but there doesn't seem to be any ambiguity
now without them, or am I missing something?

>>>> I realized this ambiguity after I posted. I meant,
>>>> any given linear temp. taken to any given number of
>>>> notes.
>>>
>>> That doesn't quite do it. You meant:
>>>
>>> .. that the number of o-tonal chords is the same as the number
>>> of u-tonal chords in any linear temp. taken to any number of
>>> notes.
>>
>> You took out the "given"??
>
> Put 'em back if you like, but there doesn't seem to be any
> ambiguity now without them, or am I missing something?

You said, "that doesn't quite do it"....

Anyway, without the givens, one could read... "all linear
temperaments have the same number of o- and u-tonal chords",
as Paul seems to have done.

-Carl

Dave Keenan:
> > It has 5 otonal and 5 utonal 7-limit tetrads with max error of 2.7
> c.

Paul Erlich
> Now -- if you think of this as a planar temperament where _only_
> 224:225 is tempered out, I bet you can reduce that error even further.

224:225 comes from 14:15 and 15:16 being equivalent. These are both
second-order 7-limit intervals. So, the error has to be shared amongst 4
7-limit intervals. 224:225 is 7.7 cents, so any scale tempering it out
can't be closer than 7.7/4=1.9 cents to 7-limit JI. So that's what the
minimax for the planar temperament will be.

This is a useful thing to know for a temperament finder. When considering
unison vectors, you can check which ones can never produce a temperament
as accurate as the one you want. Do people have other rules of thumb for
filtering unison vectors, ETs or wedgies according to the simplest or most
accurate temperaments they can give rise to? It would make the search
less arbitrary.

Graham

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> You said, "that doesn't quite do it"....
>
> Anyway, without the givens, one could read... "all linear
> temperaments have the same number of o- and u-tonal chords",
> as Paul seems to have done.

You seem to have missed the important change I made. I put "same" in
between "o" and "u", instead of after them.

Instead of

the number of o and u are the same in any ...

I made it

the number of o is the same as the number of u in any ...

--- In tuning-math@y..., graham@m... wrote:
> This is a useful thing to know for a temperament finder. When
considering
> unison vectors, you can check which ones can never produce a
temperament
> as accurate as the one you want. Do people have other rules of
thumb for
> filtering unison vectors, ETs or wedgies according to the simplest
or most
> accurate temperaments they can give rise to? It would make the
search
> less arbitrary.

See http://dkeenan.com/Music/DistributingCommas.htm

--- In tuning-math@y..., graham@m... wrote:

> This is a useful thing to know for a temperament finder. When
considering
> unison vectors, you can check which ones can never produce a
temperament
> as accurate as the one you want. Do people have other rules of
thumb for
> filtering unison vectors, ETs or wedgies according to the simplest
or most
> accurate temperaments they can give rise to?

Yes, I've talked about this before, but my version does not
correspond to the minimax view of things.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> See http://dkeenan.com/Music/DistributingCommas.htm

Seems like a dead end. Time to redo this page with linear programming?

--- In tuning-math@y..., graham@m... wrote:

> 224:225 comes from 14:15 and 15:16 being equivalent. These are
both
> second-order 7-limit intervals. So, the error has to be shared
amongst 4
> 7-limit intervals.

Or, the Hahn length of 224:225 in the 7-limit is 4. (Scala is
supposedly able to compute this)

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > See http://dkeenan.com/Music/DistributingCommas.htm
>
> Seems like a dead end. Time to redo this page with linear
programming?

Be my guest. It's not something I'm interested in doing, and it's a
very long time since I learnt about linear programming, and I've never
used it for anything real since.

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > See http://dkeenan.com/Music/DistributingCommas.htm
> >
> > Seems like a dead end. Time to redo this page with linear
> programming?
>
> Be my guest. It's not something I'm interested in doing, and it's a
> very long time since I learnt about linear programming, and I've
never
> used it for anything real since.

Actually, what would be the point. The point of my attempt on that
page, is that you can do it with nothing more than pen and paper and
you can follow what and why.

If you just want an algorithm for computer, then numerical methods
(sucessive approximations) work just fine.

In-Reply-To: <9vlfh8+9o62@eGroups.com>
Paul wrote:

> Yes, I've talked about this before, but my version does not
> correspond to the minimax view of things.

About what? When? Where?

Graham

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Actually, what would be the point. The point of my attempt on that
> page, is that you can do it with nothing more than pen and paper
and
> you can follow what and why.

But it doesn't work right -- though of course if you could find a
general fix, I'd be all for it . . . Linear programming can usually
be done with pen and paper too.

> If you just want an algorithm for computer, then numerical methods
> (sucessive approximations) work just fine.

You'd be surprised what a black-box minimization program can do with
absolute value functions.

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9vlfh8+9o62@e...>
> Paul wrote:
>
> > Yes, I've talked about this before, but my version does not
> > correspond to the minimax view of things.
>
> About what? When? Where?
>
>
> Graham

/tuning-math/message/1437

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Linear programming can usually
> be done with pen and paper too.

Why so it can. I'd love to see such a method explained.

> You'd be surprised what a black-box minimization program can do with
> absolute value functions.

I don't think it is hard to make it "absolute-value aware".

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > Linear programming can usually
> > be done with pen and paper too.
>
> Why so it can. I'd love to see such a method explained.

Well, above the 5-limit, you might need a 3-dimensional "paper" :)

> > You'd be surprised what a black-box minimization program can do
with
> > absolute value functions.
>
> I don't think it is hard to make it "absolute-value aware".

Well, I wish the Matlab people had thought of that!