--- 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.

--- 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

--- 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!