In-Reply-To: <9g2ab8+cpvi@eGroups.com>

Paul wrote:

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

>

> > FWIW, this gives the Miracle unison vector 540:539.

>

> Hmm . . . how do you get this from 224:225 and 2400:2401? I know

> 1024:1029 was one, but . . . I haven't slept so I'm not going to

> attempt to prime-factorize in my head . . . you can reply to

> tuning-math . . . BTW tuning-math lives on . . .

You can't, it's 11-prime limit. I haven't checked back, but I think it's

one of the unison vectors I originally gave. I'm not sure offhand how

you get it from the 385:384, 243:232 and 225:224 but I checked it with

the defining matrix and it works.

(2 3 1 -2 -1)(10 1) = (0 0)

(16 1)

(23 3)

(28 3)

(35 2)

It's interesting that there are so many superparticular Miracle unison

vectors. Is that unexpected?

Graham

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

> In-Reply-To: <9g2ab8+cpvi@e...>

> Paul wrote:

>

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

> >

> > > FWIW, this gives the Miracle unison vector 540:539.

> >

> > Hmm . . . how do you get this from 224:225 and 2400:2401? I know

> > 1024:1029 was one, but . . . I haven't slept so I'm not going to

> > attempt to prime-factorize in my head . . . you can reply to

> > tuning-math . . . BTW tuning-math lives on . . .

>

> You can't, it's 11-prime limit.

Duh!

> I haven't checked back, but I think it's

> one of the unison vectors I originally gave.

As I recall, those didn't seem to give Fokker periodicity blocks that quite agreed with the

MIRACLE MOSs. I wonder why that is?

> I'm not sure offhand how

> you get it from the 385:384, 243:232 and 225:224

I'll try that triplet along with 35:36 and see what kinds of blackjacks my FPB program gives.

> It's interesting that there are so many superparticular Miracle unison

> vectors. Is that unexpected?

I think superparticulars are the smallest unison vectors for a given taxicab distance in the triangular

lattice, if the lattice is constructed Kees' way. I was conjecturing to Dave Keenan that tempering

out superparticular unison vectors, with number-size proportional to N, generally cause the

constituent consonant intervals to be tempered by an amount proportional to 1/N^2. If a unison

vector is not superparticular, but is instead K-particular, where K is the difference between

numerator and denominator, then the amount of tempering of the consonances will be more like

K/(N^2) -- i.e., more tempering.

I wrote:

> I think superparticulars are the smallest unison vectors for a given taxicab distance in the

triangular

> lattice, if the lattice is constructed Kees' way.

This seems to be true until you run out of superparticulars for the given prime limit. This happens

at 81:80 for the 5-prime-limit. The first smaller unison vector obtained by searching slightly larger

regions of the lattice is 2025:2048. 2048 - 2025 = 23, so it's not too surprising that the numbers

in this ratio are on the order of 23 times the numbers in 80:81.

In the 7-prime-limit, this happens at 4374:4375. The first smaller unison vector obtained by

searching slightly larger regions of the lattice is 250000:250047. 250047 - 250000 = 47, so it's

not too surprising that the numbers in this ratio are on the order of 47 times the numbers in

4374:4375. Make sense?

In-Reply-To: <9g2dms+eaf5@eGroups.com>

Paul wrote:

> > I haven't checked back, but I think it's

> > one of the unison vectors I originally gave.

>

> As I recall, those didn't seem to give Fokker periodicity blocks that

> quite agreed with the MIRACLE MOSs. I wonder why that is?

You mean the hyperparallelopiped doesn't agree? I don't see why it

should.

> I think superparticulars are the smallest unison vectors for a given

> taxicab distance in the triangular lattice, if the lattice is

> constructed Kees' way. I was conjecturing to Dave Keenan that tempering

> out superparticular unison vectors, with number-size proportional to N,

> generally cause the constituent consonant intervals to be tempered by

> an amount proportional to 1/N^2. If a unison vector is not

> superparticular, but is instead K-particular, where K is the difference

> between numerator and denominator, then the amount of tempering of the

> consonances will be more like K/(N^2) -- i.e., more tempering.

Don't know about this, it's getting complex. But a superparticular

ratio will tend to be the simplest way of expressing a relationship.

The thing is, are all temperaments expressible in terms of

superparticulars? I think septimal schismic is 5120:5103 and 225:224, so

am I missing a superparticular?

Miracle temperament has so many of them: 225:224, 243:242, 385:384,

441:440, 540:539, 2401:2400, and I'm sure there are more. Does the

11-limit make it easier to get them?

Graham

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

> In-Reply-To: <9g2dms+eaf5@e...>

> Paul wrote:

>

> > > I haven't checked back, but I think it's

> > > one of the unison vectors I originally gave.

> >

> > As I recall, those didn't seem to give Fokker periodicity blocks that

> > quite agreed with the MIRACLE MOSs. I wonder why that is?

>

> You mean the hyperparallelopiped doesn't agree? I don't see why it

> should.

It did when I used 224:225 and 2400:2401! Did you miss that?

>

> The thing is, are all temperaments expressible in terms of

> superparticulars?

No -- Pythagorean pentatonic has a unison vector of 256:243; tempered out, that leads to

5-equal. Also, very large temperaments will lie beyond the reach of the superparticulars in a

given prime limit (they go up to 81:80 in 5-prime-limit and 4374:4375 in 7-prime limit, so very

large temperaments may involve 2025:2048, 250000:250047, etc.).

>

> Miracle temperament has so many of them: 225:224, 243:242, 385:384,

> 441:440, 540:539, 2401:2400, and I'm sure there are more. Does the

> 11-limit make it easier to get them?

Each higher limit will make it easier to get them, of course -- they'll be denser, and they'll extend

farther out, in the lattice.