hi Gene,

in an old post, you wrote:

> tuning-math Message 1168

> From: genewardsmith@j...

> Date: Fri Oct 5, 2001 6:18 am

> Subject: Re: 3rd-best 11-limit temperament

>

>

> . . . I've been meaning to suggest that Manuel

> consider putting into Scala a routine to calculate

> Gen(m, n, p) and Mos(n,m,p) for two ets m and n and

> a prime limit p; in case m and n are not relatively

> prime this needs to be adjusted by working inside of

> the interval of repetition. Of course one can also

> think of this in terms of the ets generated by linear

> combinations of hm and hn, as for instance

> h53 = h22 + h31 and h72 = h31 + h41.

by "h72 = h31 + h41", do you mean the following?

72-udo (unequal division of the octave),

the combination of 31edo and 41edo

~cents

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0

and can you explain a little more fully what

you wrote there?

-monz

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In-Reply-To: <004101c1b2d7$1f67e5a0$af48620c@dsl.att.net>

monz wrote:

> by "h72 = h31 + h41", do you mean the following?

>

>

> 72-udo (unequal division of the octave),

> the combination of 31edo and 41edo

No, that's "the nearest-prime mapping of 72edo which is consistent with

the nearest-prime mappings of 31edo and 41edo." If you know how many

steps an interval approximates to in 31edo and 41edo, you can add them

together to get the approximation to 72edo where this equation holds.

Graham

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> by "h72 = h31 + h41", do you mean the following?

>

>

> 72-udo (unequal division of the octave),

> the combination of 31edo and 41edo

Not at all. I start by assuming the temperaments I am looking at are both regular and consistent, and regard them as defined by mappings from JI to abstract "notes" consisting of generator steps, and the tuning of the temperament by a mapping in turn of the "notes" to "tones" which are real numbers. In the case of the 11-limit,

"h31" is the mapping defined b sending 2 to 31, 3 to 49, 5 to 72,

7 to 87 and 11 to 107. BY definition h31(a*b) = h31(a) + h31(b), so this defines a mapping from any 11-limit interval to an integer. This one-dimensional mapping I call a "val"; it is in a sense the dual concept to interval. There is likewise a [41,65,95,115,142] mapping I call h41, and a [72,114,167,202,249] mapping I call h72. Denoting by

"g+h" the mapping which sends a to g(a)+h(a), we have h72=h31+h41; in terms of the mappings above regarded as column vectors, this is vector addition.

hi Graham and Gene,

> From: <graham@microtonal.co.uk>

> To: <tuning-math@yahoogroups.com>

> Sent: Monday, February 11, 2002 1:44 AM

> Subject: [tuning-math] Re: h72 = h31 + h41

>

>

> In-Reply-To: <004101c1b2d7$1f67e5a0$af48620c@dsl.att.net>

> monz wrote:

>

> > by "h72 = h31 + h41", do you mean the following?

> >

> >

> > 72-udo (unequal division of the octave),

> > the combination of 31edo and 41edo

>

> No, that's "the nearest-prime mapping of 72edo which is consistent with

> the nearest-prime mappings of 31edo and 41edo." If you know how many

> steps an interval approximates to in 31edo and 41edo, you can add them

> together to get the approximation to 72edo where this equation holds.

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Monday, February 11, 2002 2:06 AM

> Subject: [tuning-math] Re: h72 = h31 + h41

>

>

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

>

> > by "h72 = h31 + h41", do you mean the following?

> >

> >

> > 72-udo (unequal division of the octave),

> > the combination of 31edo and 41edo

>

> Not at all. I start by assuming the temperaments I am looking

> at are both regular and consistent, and regard them as defined

> by mappings from JI to abstract "notes" consisting of generator

> steps, and the tuning of the temperament by a mapping in turn

> of the "notes" to "tones" which are real numbers. In the

> case of the 11-limit, "h31" is the mapping defined b sending

> 2 to 31, 3 to 49, 5 to 72, 7 to 87 and 11 to 107. BY definition

> h31(a*b) = h31(a) + h31(b), so this defines a mapping from any

> 11-limit interval to an integer. This one-dimensional mapping

> I call a "val"; it is in a sense the dual concept to interval.

> There is likewise a [41,65,95,115,142] mapping I call h41, and

> a [72,114,167,202,249] mapping I call h72. Denoting by "g+h"

> the mapping which sends a to g(a)+h(a), we have h72=h31+h41;

> in terms of the mappings above regarded as column vectors,

> this is vector addition.

so, then does this express what you're saying? :

[ 31 41 72] [2 3 5 7 11]

[ 49 65 114]

[ 72 95 167]

[ 87 115 202]

[107 142 249]

if it does, then i think i understand what you're doing.

and can you explain what you mean by 'a "val" ... is in

a sense the dual concept to interval' ? i don't get it.

-monz

_________________________________________________________

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In-Reply-To: <00cb01c1b332$dcdd1e60$af48620c@dsl.att.net>

monz wrote:

> so, then does this express what you're saying? :

>

> [ 31 41 72] [2 3 5 7 11]

> [ 49 65 114]

> [ 72 95 167]

> [ 87 115 202]

> [107 142 249]

Not really. The multiplication's the wrong way round. It should be

[2 3 5 7 11][ 31 41 72]

[ 49 65 114]

[ 72 95 167]

[ 87 115 202]

[107 142 249]

And then, you shouldn't be multiplying frequency integers like that, so

change it to

[log(2) log(3) log(5) log(7) log(11)][ 31 41 72]

[ 49 65 114]

[ 72 95 167]

[ 87 115 202]

[107 142 249]

And you still aren't saying anything about nearest approximations or how

they add up.

> and can you explain what you mean by 'a "val" ... is in

> a sense the dual concept to interval' ? i don't get it.

It helps if you understand wedge products. But yes, it's something like

that. I think the definition is that the wedge product of a val and an

integer will always be a scalar.

Graham

> From: <graham@microtonal.co.uk>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, February 12, 2002 4:49 AM

> Subject: [tuning-math] Re: h72 = h31 + h41

>

>

> In-Reply-To: <00cb01c1b332$dcdd1e60$af48620c@dsl.att.net>

> monz wrote:

>

> > so, then does this express what you're saying? :

> >

> > [ 31 41 72] [2 3 5 7 11]

> > [ 49 65 114]

> > [ 72 95 167]

> > [ 87 115 202]

> > [107 142 249]

>

> Not really. The multiplication's the wrong way round. It should be

>

> [2 3 5 7 11][ 31 41 72]

> [ 49 65 114]

> [ 72 95 167]

> [ 87 115 202]

> [107 142 249]

hmmm . . . i had a hunch that that was the case.

i only wrote it that way so that the Yahoo interface

would put the bigger matrix in proper columns.

but why does it make a difference?

> And then, you shouldn't be multiplying frequency integers like that, so

> change it to

>

> [log(2) log(3) log(5) log(7) log(11)][ 31 41 72]

> [ 49 65 114]

> [ 72 95 167]

> [ 87 115 202]

> [107 142 249]

i had a hunch about that too. can't that be written

more easily as:

log([2 3 5 7 11]) [ 31 41 72]

[ 49 65 114]

[ 72 95 167]

[ 87 115 202]

[107 142 249]

> And you still aren't saying anything about nearest approximations or how

> they add up.

why not? please clarify, because that's exactly what i'm

trying to understand here.

> > and can you explain what you mean by 'a "val" ... is in

> > a sense the dual concept to interval' ? i don't get it.

>

> It helps if you understand wedge products. But yes, it's something like

> that. I think the definition is that the wedge product of a val and an

> integer will always be a scalar.

ok, thanks . . . i still haven't learned what a wedgie is.

still studying.

-monz

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Me:

> > Not really. The multiplication's the wrong way round. It should be

> >

> > [2 3 5 7 11][ 31 41 72]

> > [ 49 65 114]

> > [ 72 95 167]

> > [ 87 115 202]

> > [107 142 249]

>

Monz:

> hmmm . . . i had a hunch that that was the case.

> i only wrote it that way so that the Yahoo interface

> would put the bigger matrix in proper columns.

> but why does it make a difference?

Try multiplying it out in Excel, using MMULT. You'll get different

results.

> i had a hunch about that too. can't that be written

> more easily as:

>

> log([2 3 5 7 11]) [ 31 41 72]

> [ 49 65 114]

> [ 72 95 167]

> [ 87 115 202]

> [107 142 249]

There is a definition of the logarithm of a matrix, but I don't think this

is it.

Me:

> > And you still aren't saying anything about nearest approximations or

> > how they add up.

Monz:

> why not? please clarify, because that's exactly what i'm

> trying to understand here.

There's nothing in the formula to say "take the nearest approximation" or

"this column plus this other one makes that column".

You can expend h31+h41=h72 to

[ 31] [ 41] [ 72]

[ 49] [ 65] [114]

[ 72] + [ 95] = [167]

[ 87] [115] [202]

[107] [142] [249]

If you want to understand it, try setting up a spreadsheet that calculates

the last column from the first two. Then get it to generate those columns

from their top entries, and the list of prime numbers.

> > It helps if you understand wedge products. But yes, it's something

> > like that. I think the definition is that the wedge product of a val

> > and an integer will always be a scalar.

>

>

> ok, thanks . . . i still haven't learned what a wedgie is.

> still studying.

The thing you have here as h72 is a val:

[ 72]

[114]

[167]

[202]

[249]

for an interval, take the comma 81:80. That's 2**-4 * 3**4 * 5**-1 or [-4

4 -1 0 0]. You get the representation of comma in h72 from the wedge

product comma^h72. That expands to give the matrix product

`[-4 4 1 0 0][ 72] = [1]

` [114]

` [167]

` [202]

` [249]

If you want to play with wedge products, you'll have to get my Python

library from <http://x31eq.com/temper.html>. Then you can do

things like

>>> import temper

>>> comma = temper.WedgableRatio(81,80)

>>> h72 = temper.PrimeET(72, temper.primes[:4])

>>> comma^temper.Wedgable(h72).complement()

{(0, 1, 2, 3, 4): 1}

I think my complement() method is doing what Gene calls the "dual" so I'll

rename it sometime.

Graham

hi Graham,

> From: <graham@microtonal.co.uk>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, February 12, 2002 1:20 PM

> Subject: [tuning-math] Re: h72 = h31 + h41

>

>

> The thing you have here as h72 is a val:

>

> [ 72]

> [114]

> [167]

> [202]

> [249]

>

> for an interval, take the comma 81:80. That's 2**-4 * 3**4 * 5**-1 or [-4

> 4 -1 0 0]. You get the representation of comma in h72 from the wedge

> product comma^h72.

ahh . . . so now we have to use ** instead of ^ for

"raise to the power of", because now ^ is the wedge product.

yes?

> That expands to give the matrix product

>

> `[-4 4 1 0 0][ 72] = [1]

> ` [114]

> ` [167]

> ` [202]

> ` [249]

there's a typo in the row matrix on the left. the exponent of 5

should be -1 not 1, so that matrix should read `[-4 4 -1 0 0].

so then if ^ is the wedge product, is (81/80)^h72 = 1

the proper way to notate this?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> so then if ^ is the wedge product, is (81/80)^h72 = 1

> the proper way to notate this?

It's certainly *possible* to interpret this in a way which makes sense, and notate it thusly, but I think it would be much preferable to write this as h72(81/80) = 1.

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, February 13, 2002 2:46 PM

> Subject: [tuning-math] Re: h72 = h31 + h41

>

>

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:

>

> > so then if ^ is the wedge product, is (81/80)^h72 = 1

> > the proper way to notate this?

>

> It's certainly *possible* to interpret this in a way which

> makes sense, and notate it thusly, but I think it would be

> much preferable to write this as h72(81/80) = 1.

well, ok . . . now t h a t notation looks familiar, and

i understand it.

but i've seen ^ used in connection with wedgies, so my

question is still not really answered: do we have to use **

now to represent "raise to the power of"? apparently,

whatever ^ is being used for, it's something else other

than that.

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> but i've seen ^ used in connection with wedgies, so my

> question is still not really answered: do we have to use **

> now to represent "raise to the power of"? apparently,

> whatever ^ is being used for, it's something else other

> than that.

The "^" symbol is well-established as a notation both for exponentiation and wedge product; I would use it for either myself so long as there seemed no potential for confusion. Fortran gave us

"**" for exponentiation also, which is fine, and not used for anything else to my knowledge.