In-Reply-To: <002c01c0ff33$9751a860$4448620c@att.com>

I'm replying to this here, as we've had complaints about over-precise

cents values. Perhaps Monz could post back to TBL (The Big List) when he

understands it.

> > > > mapping by period and generator:

> > > > ([1, 0], ([0, 2, -1], [5, 1, 12]))

> >

> > The first two-element list shows the mapping of the octave. The

> > second

> > element is always zero for both my scripts, as the period is always a

> > fraction of an octave.

>

> So in other words you always use the nearest integer here?

> I'm still confused about that "0".

It isn't the nearest integer to anything: these lists are the definition

of the temperament.

All the temperaments I'm currently considering use a period that's an

equal division of the octave. So you never need the generator to get the

octave, and that parameter's always zero.

> > So the first number tells you how many equal

> > parts the octave is being divided into. Here it's 1 which is the

> > simplest case.

>

> Confused about this too... I thought this example divided the

> octave into 41 parts? Again, is this pair of numbers expressing

> the nearest integer fraction of an octave?

The octave is divided into 41 *unequal* parts. There are no equal

divisions of the octave. Compare with this 13-limit temperament I gave

before:

9/52, 103.897 cent generator

basis:

(0.5, 0.086580634742799478)

mapping by period and generator:

([2, 0], ([3, 5, 7, 9, 10], [1, -2, -8, -12, -15]))

mapping by steps:

[(58, 46), (92, 73), (135, 107), (163, 129), (201, 159), (215, 170)]

unison vectors:

[[1, 2, -3, 1, 0, 0], [-4, 0, 2, 1, -1, 0], [1, -3, 2, 1, 0, -1], [2, -1,

0, 1, -2, 1]]

highest interval width: 17

complexity measure: 34 (46 for smallest MOS)

highest error: 0.004911 (5.893 cents)

unique

and diaschismic temperament:

2/11, 105.214 cent generator

basis:

(0.5, 0.087678135277931377)

mapping by period and generator:

([2, 0], ([3, 5], [1, -2]))

mapping by steps:

[(12, 10), (19, 16), (28, 23)]

unison vectors:

[[11, -4, -2]]

highest interval width: 3

complexity measure: 6 (8 for smallest MOS)

highest error: 0.002716 (3.259 cents)

unique

Both of them divide the octave in 2 equal parts all the time.

> Also, I think it's confusing the way you give the "octave correction"

> first and the "number of generators" second in this line, but

> it's reversed in all the following lines, generator first and

> octave second.

Octave is always first in the printouts. I explained the generator

mapping first because it makes more sense that way round. As another

thread shows, you can ignore octaves completely and define the scale

using the generator mapping and number of equal divisions of the octave.

You don't know how many notes the MOS will contain, or whether it's

unique assuming octave invariance, but you can still work out this

information.

> > In more familiar terms, the generator is a 5:4 major third. 5 major

> > thirds are a 3:1 perfect twelfth.

>

> (2^(380.391/1200))^5 does indeed equal exactly 3.

>

> Following you so far...

Oh good!

> So I follow this too. Now comes the tricky part...

>

> > (2*(5,0) - (12,-1) = (-2, 1))

>

> OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 .

> The "2*" means that we square that, and so the first group

> stands for 3^2 = 9 .

Oh, that is putting the generator first, isn't it? That's wrong, you're

right to pull me up on it. So it should be

(2*(0, 5) - (12, -1)) = (1, -2))

I think you're complicating it by bringing ratios and exact pitches back

into it. A 9:1 is two 3:1 steps, hence 2*(0, 5). Any ratio can be prime

factorized, and worked out in terms of octaves and generators using the

conversion matrix.

> And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 ,

> which agrees with your definition above as ~7.

>

> The minus sign means we divide the terms, and...

> Voil�! ... ~9/7 .

7:1 is written as (2,-1). You read that straight off. So 9:7 or (0 2 0

-1)H is 2*(0,5)-(-1,12). You could write that

(0 2 0 -1)( 1 0)

( 0 5)

( 2 1)

(-1 12)

> And checking the answer:

> ((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7.

>

> So you're putting an equivalence relationship in here.

That's a check, yes, but a simpler one would be to use the relationship

at the top of the printout, that the generator is 13 steps from 41.

41-2*13=41-26=15. And 9:7 does approximate to 15 steps from 41.

> That was confusing... I had a hard time understanding how

> ~9/7 = "An octave less two major thirds". Now it's clear.

It means you can construct an augmented triad with two 5-limit and one

9-identity thirds.

> > > > mapping by steps:

> > > > [(22, 19), (35, 30), (51, 44), (62, 53)]

> >

> > Each pair shows the size of a prime interval in terms of scale steps.

> > Call the steps x and y. An octave is 22x+19y. For the case where

> > x=y,

> > you have 41-equal. Where x=0, you have 19-equal. Where y=0, you

have

> > 22-equal. So 19, 22 and 41-equal are all members of this temperament

> > family.

> >

> > 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any

> > 7-prime limit interval in terms of x and y by combining these.

>

> OK, I understand all the math here, but I'm not quite following the

> logic which deterimines that they are "all members of this temperament

> family". How does your program find the 19, 22 and 41 in this example?

My program *starts* with 19 and 22, along with the prime intervals and

odd limit, and works everything out from them. It gets 41 by adding 19

and 22.

My other program starts with the unison vectors, but that's more complex.

A third program could go from unison vectors to a mapping in terms of

generators within a period that's a fraction of an octave. In that case,

it'd have to get the equal temperaments by optimizing for the best

generator/period ratio, and walking the scale tree.

> > and use simpler coordinates. Here, q=x+y and p=x

> >

> > 3:2 is 11q + 2p

> > 5:4 is 6q + p

> > 8:7 is 4q

>

> Oops... now you lost me.

Substitute in for p and q:

11q+2p = 11(x+y) + 2x = 13x + 11y

6q+p = 6(x+y) + x = 7x + 6y

4q = 4(x+y) = 4x + 4y

> > So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal

> > and p is 1 step in 41-or 22-equal, and no steps in 19-equal.

>

> Getting foggier...

| 19= | 22= | 41=

---------------------------------

x | 0 | 1 | 1

y | 1 | 0 | 1

p | 0 | 1 | 1

q | 1 | 1 | 2

> > > > unison vectors:

> > > > [[-10, -1, 5, 0], [5, -12, 0, 5]]

>

>

> So these are the ratios 3125/3072 and 537824/531441 ?

The might well be, you're as capable as me of working them out.

I missed off the bottom of the file before:

>>highest interval width: 12

This is the maximum number of generators needed for an interval within

the consonance limit.

>>complexity measure: 12 (13 for smallest MOS)

The complexity measure is the previous number multiplied by the number of

equal divisions of the octave. The number of otonal or utonal chords is

the number of notes in the scale minus this measure. Roughly. In this

case there's a 13 note MOS that can hold that complete otonality.

>>highest error: 0.004936 (5.923 cents)

This comes from the minimax optimisation. It's a bit over-precise so it

can be checked against another program, and because it's possible for the

program to throw out *very* accurate temperaments.

The temperaments are sorted assuming the fewer notes needed and the

smaller the error the better.

>>unique

This is either there or not. I added it to be able to assess keyboard

mappings as well as temperaments. I have lists of keyboard mappings, not

uploaded yet, that rate a mapping twice as badly if it isn't unique, and

ignore the accuracy.

Graham

Hi Graham,

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

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

> Sent: Thursday, June 28, 2001 4:00 AM

> Subject: [tuning-math] Re: questions about Graham's matrices (was:

13-limit mappin

>

>

> I'm replying to this here, as we've had complaints about over-precise

> cents values. Perhaps Monz could post back to TBL (The Big List) when he

> understands it.

Yup - I should have posted the other one here only, and not also on TBL.

(Thanks for that convenient abbreviation).

> > > > > [Graham]

> > > > > mapping by period and generator:

> > > > > ([1, 0], ([0, 2, -1], [5, 1, 12]))

> > >

> > > [Graham]

> > > The first two-element list shows the mapping of the octave.

> > > The second element is always zero for both my scripts, as

> > > the period is always a fraction of an octave.

> >

> > [me, monz]

> > So in other words you always use the nearest integer here?

> > I'm still confused about that "0".

>

> [Graham]

> It isn't the nearest integer to anything: these lists are the definition

> of the temperament.

>

> All the temperaments I'm currently considering use a period that's an

> equal division of the octave. So you never need the generator to get the

> octave, and that parameter's always zero.

OK, I think I understand that now. A counter-illustration

using a period that's an unequal division, which would "need the

generator to get the octave", would help.

> > > So the first number tells you how many equal

> > > parts the octave is being divided into. Here it's 1 which is the

> > > simplest case.

> >

> > Confused about this too... I thought this example divided the

> > octave into 41 parts? Again, is this pair of numbers expressing

> > the nearest integer fraction of an octave?

>

> The octave is divided into 41 *unequal* parts. There are no equal

> divisions of the octave.

This was a bit confusing, because I got it tangled with what you

said above about "a period that's an equal division of the octave".

Now I see the difference.

The scale under consideration is an temperament which is an

*unequal* division of the octave (the period of equivalence),

because it's a result of multiples of the generator and not of

an equal division of anything.

So all thru the rest of the explanation when you refer to

"steps of 19=, 22=, 41="... they're all approximations to

the generated scale. Right?

I think what's been confusing me is that you refer to both ratios

and EDOs as approximations of the scale resulting from your generator,

and perhaps I've been taking them more literally than I should

have been. I realize now that every interval is to be understood

in terms of this temperament's approximations to the basic prime

intervals 2, 3, 5, 7. So your matrices are presenting a set of

transformations.

> > > (2*(5,0) - (12,-1) = (-2, 1))

> >

> > OK, so as I said above, ((2^(380.391/1200))^5) * (2^0) = 3 .

> > The "2*" means that we square that, and so the first group

> > stands for 3^2 = 9 .

>

> Oh, that is putting the generator first, isn't it? That's wrong, you're

> right to pull me up on it. So it should be

>

> (2*(0, 5) - (12, -1)) = (1, -2))

Oops!... your bad. You didn't reverse (12, -1) into (-1, 12)

as you meant to do.

I really think it's much more intuitive to have it the other way around

(your mistake here shows the persistence of that way of thinking).

Put the generator first and the octave second, consistently.

I agree with you that the number of generators is the more important

figure, and to me it makes sense to *see* that number first.

(I think your unconscious switch in the original post shows that.)

So reverse the "correction" you made here and put it back like

it was, and reverse the *other* lines to agree with these.

So your illustrated calcuation (call it "two fifths less a seventh")

translates into approximate ratios as ~(3:2)^2 / ~7:4 = ~9:7 ,

and would look like:

(2*(5, 0) - (12, -1)) = (-2, 1)) .

And the "octave less two major thirds" translates into

approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 .

When I did the matrix calculation I got

(0, 1) - 2*(1, 2) = (-2, -3) .

Hmmm... the important number, the generator, works out to be

the same -2, which is correct. But why is the period calculation

not working out when the octave is included? Is is because

there is no zero period?

So your opening lines, with extra labels, would look like:

basis (generator, period) as fraction of octave:

(0.31699250014423125, 1.0)

mapping by [generator], [period] (~2:1, (~3:2, ~5:4, ~7:4)) :

([0, 1], ([5, 1, 12], [0, 2, -1]))

Actually, I think that second line would be better rearranged

to agree with the octave notation:

mapping by (generator, period) [~2:1, ~3:2, ~5:4, ~7:4]:

[(0, 1), (5, 0), (1, 2), (12, -1)]

To me, that's as plain as day. The octave can still be seen

as set apart by virtue of being first/leftmost on the list.

>

> I think you're complicating it by bringing ratios and exact pitches back

> into it.

Agreed... but using the ratios allowed *me* to do the math in

an Excel spreadsheet so that I could follow your reasoning.

I went thru it step by step, looking at the cents values all

along the way.

If I understood better how to manipulate the matrices, I certainly

would have done it that way too. I can see that it's *much* more

elegant that way, even tho I've been having trouble understanding it.

This is along the lines of what I was trying to get Paul to

understand a couple of different times in the past. It's not

necessary to always use prime-factors as the basis for lattice

metrics... any numbers that give even, consistent divisions

of the pitch-space *in SOME way* will do. The different ways

of dividing (and multiplying) produce different kinds of lattices.

(Of course Paul already understands all this, *and* the math to

manipulate it, as do you. But I don't think he was following

my reasoning when I was trying to make that point... probably

because *I don't* understand the math! I'm not speaking the

same language you guys are... altho I'm trying hard...)

> A 9:1 is two 3:1 steps, hence 2*(0, 5).

Yes, now that's *very* clear. I caught it, but certainly

didn't explain it as elegantly as this.

> Any ratio can be prime factorized, and worked out in terms of

> octaves and generators using the conversion matrix.

*That's* what's been giving me such trouble!

Relating the calculations given in terms of this temperament

to the approximate ratios really did confuse me, even tho in

hindsight now I think it helped in the process of understanding.

>

>

> > And ((2^(380.391/1200))^12) * (2^-1) = ~6.983305074 ,

> > which agrees with your definition above as ~7.

> >

> > The minus sign means we divide the terms, and...

> > Voilï¿½! ... ~9/7 .

>

> 7:1 is written as (2,-1).

Oops! Your bad again... you meant (12, -1).

... well, actually you meant (-1, 12).

(Boy, this interval sure keeps giving you the slip!)

> You read that straight off.

> So 9:7 or (0 2 0 -1)H is 2*(0,5)-(-1,12). You could write that

>

> (0 2 0 -1)( 1 0)

> ( 0 5)

> ( 2 1)

> (-1 12)

>

Thanks, Graham. Seeing the matrix conversion broken down like

this helps me a lot.

So keeping to my (generator, period) reversal of your notation,

(0 2 0 -1)H = 2*(5,0)-(12,-1) because ~3 = (5,0) and ~7 = (12,-1).

Of course, in the case of an octave-invariant scale

like this it's much simpler to just omit the period. So

ignoring the first column in "H" because it's powers of 2,

ratio prime vector 380.391-cent generators

2:1 = ( 1 0 0 0)H = ~ 0

3:2 = (-1 1 0 0)H = ~ 5

5:4 = (-2 0 1 0)H = ~ 1

7:4 = (-2 0 0 1)H = ~12

9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2.

> > And checking the answer:

> > ((2^(380.391/1200))^-2) * (2^1) does indeed equal the ~9/7.

> >

> > So you're putting an equivalence relationship in here.

>

> That's a check, yes, but a simpler one would be to use the relationship

> at the top of the printout, that the generator is 13 steps from 41.

> 41-2*13=41-26=15. And 9:7 does approximate to 15 steps from 41.

This helps a lot too. Thanks.

So here you're back again to "an octave less two major thirds".

~2:1 / ~(5/4)^2 = ~32/25 or (0, 1) - 2*(1, 2) = (-2, -3) .

> > That was confusing... I had a hard time understanding how

> > ~9/7 = "An octave less two major thirds". Now it's clear.

>

> It means you can construct an augmented triad with two 5-limit and one

> 9-identity thirds.

Er... this is a little confusing, because a triad is constructed of

only two intervals.

You mean that if one measured all the intervals in an augmented triad

*and its inversions*, the result would be two ~5:4s and one ~9:7.

> > > > > mapping by steps:

> > > > > [(22, 19), (35, 30), (51, 44), (62, 53)]

> > >

> > > Each pair shows the size of a prime interval in terms of scale steps.

> > > Call the steps x and y. An octave is 22x+19y. For the case where

> > > x=y, you have 41-equal. Where x=0, you have 19-equal. Where y=0,

> > > you have 22-equal. So 19, 22 and 41-equal are all members of this

> > > temperament family.

> > >

> > > 3:1 is 35x+30y, 5:1 is 51x+44y and 7:1 is 62x+53y. You can get any

> > > 7-prime limit interval in terms of x and y by combining these.

> >

> > OK, I understand all the math here, but I'm not quite following the

> > logic which deterimines that they are "all members of this temperament

> > family". How does your program find the 19, 22 and 41 in this example?

>

> My program *starts* with 19 and 22, along with the prime intervals and

> odd limit, and works everything out from them. It gets 41 by adding 19

> and 22.

Ahhh!... that certainly explains *that*!

> My other program starts with the unison vectors, but that's more complex.

> A third program could go from unison vectors to a mapping in terms of

> generators within a period that's a fraction of an octave. In that case,

> it'd have to get the equal temperaments by optimizing for the best

> generator/period ratio, and walking the scale tree.

Hmmm.... that last algorithm sounds like a good one! Exactly the

kind of thing I always wanted to include in my JustMusic software,

applicable to rational systems as well as irrational.

> > > and use simpler coordinates. Here, q=x+y and p=x

> > >

> > > 3:2 is 11q + 2p

> > > 5:4 is 6q + p

> > > 8:7 is 4q

> >

> > Oops... now you lost me.

>

> Substitute in for p and q:

>

> 11q+2p = 11(x+y) + 2x = 13x + 11y

> 6q+p = 6(x+y) + x = 7x + 6y

> 4q = 4(x+y) = 4x + 4y

Ah... if only I had been paying attention in algebra class...

OK, now it's perfectly clear.

> > > So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal

> > > and p is 1 step in 41-or 22-equal, and no steps in 19-equal.

> >

> > Getting foggier...

>

> | 19= | 22= | 41=

> ---------------------------------

> x | 0 | 1 | 1

> y | 1 | 0 | 1

> p | 0 | 1 | 1

> q | 1 | 1 | 2

>

Uh-oh... still not getting this part. Please elaborate.

For some reason I'm not seeing the connection between

p and q and the steps of the EDOs.

Too many layers of abstraction for me to follow ...

> > > > > unison vectors:

> > > > > [[-10, -1, 5, 0], [5, -12, 0, 5]]

> >

> >

> > So these are the ratios 3125/3072 and 537824/531441 ?

>

> The[y] might well be, you're as capable as me of working them out.

OK... I was simply double-checking with you that the numbers

stand for exponents of the prime-factors 2, 3, 5, and 7.

So I guess they do.

So your parenthetical lists are elegant, but IMO could use

a little bit more of a legend explaining what those lists

represent. Otherwise one has to learn the sequences beforehand

and keep them in mind. I suggest adding a label giving the

parameter list before each line.

-monz

http://www.monz.org

"All roads lead to n^0"

_________________________________________________________

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

>

> Er... this is a little confusing, because a triad is constructed of

> only two intervals.

Last time I checked, a triad had three intervals, a tetrad six, a

pentad ten, and a hexad fifteen.

monz wrote:

> > All the temperaments I'm currently considering use a period that's an

> > equal division of the octave. So you never need the generator to get

> > the

> > octave, and that parameter's always zero.

>

> OK, I think I understand that now. A counter-illustration

> using a period that's an unequal division, which would "need the

> generator to get the octave", would help.

As you never need it, it's difficult to find a counter-example...

My schismic fourth keyboard mapping would be an example, where the period

is a fourth. So the octave would be two periods plus a generator I think.

But I've never worked that out with matrices.

> > The octave is divided into 41 *unequal* parts. There are no equal

> > divisions of the octave.

>

> This was a bit confusing, because I got it tangled with what you

> said above about "a period that's an equal division of the octave".

> Now I see the difference.

>

> The scale under consideration is an temperament which is an

> *unequal* division of the octave (the period of equivalence),

> because it's a result of multiples of the generator and not of

> an equal division of anything.

It's an unequal division of the period, which is an equal division of the

interval of equivalence (in this case the trivial division of one) which

for these example is always an octave.

> So all thru the rest of the explanation when you refer to

> "steps of 19=, 22=, 41="... they're all approximations to

> the generated scale. Right?

There isn't "a generated scale". The generator can be used for a whole

family of scales. 19, 22 and 41-equal are merely special cases of the

scales it can generate.

> I think what's been confusing me is that you refer to both ratios

> and EDOs as approximations of the scale resulting from your generator,

> and perhaps I've been taking them more literally than I should

> have been. I realize now that every interval is to be understood

> in terms of this temperament's approximations to the basic prime

> intervals 2, 3, 5, 7. So your matrices are presenting a set of

> transformations.

Yes, matrices are all about transformations. In this case between the

harmonic and melodic ways of looking at things. It's a one-way

transformation because you loose information in going from the just to

tempered mapping.

> > (2*(0, 5) - (12, -1)) = (1, -2))

>

>

> Oops!... your bad. You didn't reverse (12, -1) into (-1, 12)

> as you meant to do.

>

> I really think it's much more intuitive to have it the other way around

> (your mistake here shows the persistence of that way of thinking).

>

> Put the generator first and the octave second, consistently.

> I agree with you that the number of generators is the more important

> figure, and to me it makes sense to *see* that number first.

> (I think your unconscious switch in the original post shows that.)

No, because that would contradict the usual way of writing vectors from

low to high primes.

> And the "octave less two major thirds" translates into

> approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 .

> When I did the matrix calculation I got

>

> (0, 1) - 2*(1, 2) = (-2, -3) .

>

> Hmmm... the important number, the generator, works out to be

> the same -2, which is correct. But why is the period calculation

> not working out when the octave is included? Is is because

> there is no zero period?

You're using 5:1 instead of 5:4.

> > I think you're complicating it by bringing ratios and exact pitches

> > back

> > into it.

>

> Agreed... but using the ratios allowed *me* to do the math in

> an Excel spreadsheet so that I could follow your reasoning.

> I went thru it step by step, looking at the cents values all

> along the way.

>

> If I understood better how to manipulate the matrices, I certainly

> would have done it that way too. I can see that it's *much* more

> elegant that way, even tho I've been having trouble understanding it.

If you've got Excel, you can do that! I explain it on my website. You

use MINVERSE, MDETERM and MMULT, pressing CTRL-SHIFT-RETURN to enter the

formulae.

> This is along the lines of what I was trying to get Paul to

> understand a couple of different times in the past. It's not

> necessary to always use prime-factors as the basis for lattice

> metrics... any numbers that give even, consistent divisions

> of the pitch-space *in SOME way* will do. The different ways

> of dividing (and multiplying) produce different kinds of lattices.

Yes, I think this is what Pierre Lamothe was trying to get across before

he left the list as well.

> Of course, in the case of an octave-invariant scale

> like this it's much simpler to just omit the period. So

> ignoring the first column in "H" because it's powers of 2,

>

> ratio prime vector 380.391-cent generators

>

> 2:1 = ( 1 0 0 0)H = ~ 0

> 3:2 = (-1 1 0 0)H = ~ 5

> 5:4 = (-2 0 1 0)H = ~ 1

> 7:4 = (-2 0 0 1)H = ~12

> 9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2.

If you're thinking octave invariantly, you can simplify it further.

ratio prime vector 380.391-cent generators

3:2 = (1 0 0)H = ~ 5

5:4 = (0 1 0)H = ~ 1

7:4 = (0 0 1)H = ~12

9:7 = (2 0 -1)H = ~(2*5)-12 = ~-2.

> > > That was confusing... I had a hard time understanding how

> > > ~9/7 = "An octave less two major thirds". Now it's clear.

> >

> > It means you can construct an augmented triad with two 5-limit and one

> > 9-identity thirds.

>

> Er... this is a little confusing, because a triad is constructed of

> only two intervals.

>

> You mean that if one measured all the intervals in an augmented triad

> *and its inversions*, the result would be two ~5:4s and one ~9:7.

Oh, however you count it, I was thinking of the octave as part of the

chord.

> > My other program starts with the unison vectors, but that's more

> > complex.

> > A third program could go from unison vectors to a mapping in terms of

> > generators within a period that's a fraction of an octave. In that

> > case,

> > it'd have to get the equal temperaments by optimizing for the best

> > generator/period ratio, and walking the scale tree.

>

> Hmmm.... that last algorithm sounds like a good one! Exactly the

> kind of thing I always wanted to include in my JustMusic software,

> applicable to rational systems as well as irrational.

Try it. Type the octave-invariant unison vectors into the spreadsheet as

an array, with the chromatic one at the top. Then select an equal sized

square, type "=minverse(?:?)*mdeterm(?:?)" where ?:? is that original

array, and the left hand column will be the mapping by generators.

> > > > So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal

> > > > and p is 1 step in 41-or 22-equal, and no steps in 19-equal.

> > >

> > > Getting foggier...

> >

> > | 19= | 22= | 41=

> > ---------------------------------

> > x | 0 | 1 | 1

> > y | 1 | 0 | 1

> > p | 0 | 1 | 1

> > q | 1 | 1 | 2

> >

>

>

> Uh-oh... still not getting this part. Please elaborate.

> For some reason I'm not seeing the connection between

> p and q and the steps of the EDOs.

>

> Too many layers of abstraction for me to follow ...

If you tune to a given ET, the table shows you how many steps will be in

each interval.

> So your parenthetical lists are elegant, but IMO could use

> a little bit more of a legend explaining what those lists

> represent. Otherwise one has to learn the sequences beforehand

> and keep them in mind. I suggest adding a label giving the

> parameter list before each line.

The "parenthetical lists" are the stringifications of the objects the

program uses. I could get it to write out an HTML file for each

temperament, but for the moment it's cutting-edge data.

Graham

> From: Paul Erlich <paul@stretch-music.com>

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

> Sent: Thursday, June 28, 2001 12:31 PM

> Subject: [tuning-math] Re: questions about Graham's matrices (was:

13-limit mappin

>

>

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

> >

> > Er... this is a little confusing, because a triad is constructed of

> > only two intervals.

>

> Last time I checked, a triad had three intervals, a tetrad six, a

> pentad ten, and a hexad fifteen.

Oops... my bad this time! I compounded my correction of Graham's

not-entirely-correct statement by introducing an actual error.

Yes, Paul, of course a triad *does* have three intervals.

I was speaking specifically of the types of "3rds". Thanks.

-monz

http://www.monz.org

"All roads lead to n^0"

_________________________________________________________

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

> monz wrote:

> > This is along the lines of what I was trying to get Paul to

> > understand a couple of different times in the past. It's not

> > necessary to always use prime-factors as the basis for lattice

> > metrics... any numbers that give even, consistent divisions

> > of the pitch-space *in SOME way* will do. The different ways

> > of dividing (and multiplying) produce different kinds of lattices.

Funny -- I would have said that this is something I was trying to get

Monz to understand, rather than something Monz was trying to get me

to understand . . . Maybe Monz could restate the context in which he

was trying to get me to understand this?

From: <graham@microtonal.co.uk>

To: <tuning-math@yahoogroups.com>

Cc: <graham@microtonal.co.uk>

Sent: Thursday, June 28, 2001 2:18 PM

Subject: [tuning-math] Re: questions about Graham's matrices

> > So all thru the rest of the explanation when you refer to

> > "steps of 19=, 22=, 41="... they're all approximations to

> > the generated scale. Right?

>

> There isn't "a generated scale". The generator can be used for a whole

> family of scales. 19, 22 and 41-equal are merely special cases of the

> scales it can generate.

OK... how about "the *potential* generated scales" instead?

But my point is... they're *approximately* 19, 22, and 41-equal, right?

> > I really think it's much more intuitive to have it the other way around

> > (your mistake here shows the persistence of that way of thinking).

> >

> > Put the generator first and the octave second, consistently.

> > I agree with you that the number of generators is the more important

> > figure, and to me it makes sense to *see* that number first.

> > (I think your unconscious switch in the original post shows that.)

>

> No, because that would contradict the usual way of writing vectors from

> low to high primes.

Huh? In a general sense, generators can be any interval, so

what does prime ordering have to do with it? Clarification on

this would be appreciated.

>

> > And the "octave less two major thirds" translates into

> > approximate ratios as ~2:1 / ~(5/4)^2 = ~32/25 .

> > When I did the matrix calculation I got

> >

> > (0, 1) - 2*(1, 2) = (-2, -3) .

> >

> > Hmmm... the important number, the generator, works out to be

> > the same -2, which is correct. But why is the period calculation

> > not working out when the octave is included? Is is because

> > there is no zero period?

>

> You're using 5:1 instead of 5:4.

Thanks, Graham... I had figured that out by the time I sent this,

but neglected to delete that bit before sending. (I *did* delete

a lot of it as I learned-by-figuring-out...)

> > > I think you're complicating it by bringing ratios and exact pitches

> > > back

> > > into it.

> >

> > Agreed... but using the ratios allowed *me* to do the math in

> > an Excel spreadsheet so that I could follow your reasoning.

> > I went thru it step by step, looking at the cents values all

> > along the way.

> >

> > If I understood better how to manipulate the matrices, I certainly

> > would have done it that way too. I can see that it's *much* more

> > elegant that way, even tho I've been having trouble understanding it.

>

> If you've got Excel, you can do that! I explain it on my website. You

> use MINVERSE, MDETERM and MMULT, pressing CTRL-SHIFT-RETURN to enter the

> formulae.

Thanks for that. Since buying my new system, and no longer having

the CD of Excel, I can only run it with the minimal set of formulae.

I recall having to add some extra files to Excel on my old machine

to make it do some more esoteric math, and have avoided even trying

on the new PC. I'll try it and see if these will work.

>

> > This is along the lines of what I was trying to get Paul to

> > understand a couple of different times in the past. It's not

> > necessary to always use prime-factors as the basis for lattice

> > metrics... any numbers that give even, consistent divisions

> > of the pitch-space *in SOME way* will do. The different ways

> > of dividing (and multiplying) produce different kinds of lattices.

>

> Yes, I think this is what Pierre Lamothe was trying to get across before

> he left the list as well.

Wow... now *that's* a revelation, because I've been struggling to

understand Pierre's work. It looks to me to be very closely related

to some of my own, and I'm anxious to cut thru the comprehension

barrier with respect to his work, as I've finally done with yours.

>

> > Of course, in the case of an octave-invariant scale

> > like this it's much simpler to just omit the period. So

> > ignoring the first column in "H" because it's powers of 2,

> >

> > ratio prime vector 380.391-cent generators

> >

> > 2:1 = ( 1 0 0 0)H = ~ 0

> > 3:2 = (-1 1 0 0)H = ~ 5

> > 5:4 = (-2 0 1 0)H = ~ 1

> > 7:4 = (-2 0 0 1)H = ~12

> > 9:7 = ( 0 2 0 -1)H = ~(2*5)-12 = ~-2.

>

> If you're thinking octave invariantly, you can simplify it further.

>

> ratio prime vector 380.391-cent generators

>

> 3:2 = (1 0 0)H = ~ 5

> 5:4 = (0 1 0)H = ~ 1

> 7:4 = (0 0 1)H = ~12

> 9:7 = (2 0 -1)H = ~(2*5)-12 = ~-2.

Right, I knew that and just didn't bother to omit the "2" column.

> > > My other program starts with the unison vectors, but that's more

> > > complex.

> > > A third program could go from unison vectors to a mapping in terms of

> > > generators within a period that's a fraction of an octave. In that

> > > case,

> > > it'd have to get the equal temperaments by optimizing for the best

> > > generator/period ratio, and walking the scale tree.

> >

> > Hmmm.... that last algorithm sounds like a good one! Exactly the

> > kind of thing I always wanted to include in my JustMusic software,

> > applicable to rational systems as well as irrational.

>

> Try it. Type the octave-invariant unison vectors into the spreadsheet as

> an array, with the chromatic one at the top. Then select an equal sized

> square, type "=minverse(?:?)*mdeterm(?:?)" where ?:? is that original

> array, and the left hand column will be the mapping by generators.

Cool... I can't wait to check this out. Hope it works.

> > > > > So q is 2 steps in 41-equal, or 1 step in 22- or 19-equal

> > > > > and p is 1 step in 41-or 22-equal, and no steps in 19-equal.

> > > >

> > > > Getting foggier...

> > >

> > > | 19= | 22= | 41=

> > > ---------------------------------

> > > x | 0 | 1 | 1

> > > y | 1 | 0 | 1

> > > p | 0 | 1 | 1

> > > q | 1 | 1 | 2

> > >

> >

> >

> > Uh-oh... still not getting this part. Please elaborate.

> > For some reason I'm not seeing the connection between

> > p and q and the steps of the EDOs.

> >

> > Too many layers of abstraction for me to follow ...

I was hoping for another remedial-algebra lesson on how this

table could be calculated from the other data you gave. :)

> If you tune to a given ET, the table shows you how many steps will be in

> each interval.

>

> > So your parenthetical lists are elegant, but IMO could use

> > a little bit more of a legend explaining what those lists

> > represent. Otherwise one has to learn the sequences beforehand

> > and keep them in mind. I suggest adding a label giving the

> > parameter list before each line.

>

> The "parenthetical lists" are the stringifications of the objects the

> program uses. I could get it to write out an HTML file for each

> temperament, but for the moment it's cutting-edge data.

Keep up the good work, Graham! Please come on over to JustMusic

when you're interested in helping us graph this stuff.

(And that goes for you two too, Dave and Paul.)

-monz

http://www.monz.org

"All roads lead to n^0"

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

From: Paul Erlich <paul@stretch-music.com>

To: <tuning-math@yahoogroups.com>

Sent: Thursday, June 28, 2001 2:47 PM

Subject: [tuning-math] Re: questions about Graham's matrices

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

> > monz wrote:

>

> > > This is along the lines of what I was trying to get Paul to

> > > understand a couple of different times in the past. It's not

> > > necessary to always use prime-factors as the basis for lattice

> > > metrics... any numbers that give even, consistent divisions

> > > of the pitch-space *in SOME way* will do. The different ways

> > > of dividing (and multiplying) produce different kinds of lattices.

>

> Funny -- I would have said that this is something I was trying to get

> Monz to understand, rather than something Monz was trying to get me

> to understand . . . Maybe Monz could restate the context in which he

> was trying to get me to understand this?

Oh... one time was back around when I asked you to show me how

to prime-factorize my meantone formulae. As I said, I don't have

the mathematical understanding to even imagine accurately, let alone

describe, some of the vague latticing ideas I have. I'm trying.

But I'm glad to see that somehow we managed to agree on this,

without understanding each other! Cool.

-monz

http://www.monz.org

"All roads lead to n^0"

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

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

> But I'm glad to see that somehow we managed to agree on this,

> without understanding each other! Cool.

>

Yes, and I think it's a very important point. Rather than the basis being 3, 5, 7, it could just as

easily be, say, 5/4, 6/5, 7/6, or any other basis that spans the 3D lattice (though consonant

intervals are somewhat preferable here).

In-Reply-To: <003101c1005d$50edba00$4448620c@att.com>

monz wrote:

> OK... how about "the *potential* generated scales" instead?

>

> But my point is... they're *approximately* 19, 22, and 41-equal, right?

They all contain MOS subsets with these numbers of notes.

> Huh? In a general sense, generators can be any interval, so

> what does prime ordering have to do with it? Clarification on

> this would be appreciated.

The period is a stand-in for the octave, and the generator is a stand-in

for the twelfth (or fifth or whatever but not always). So it makes sense

to put them in the same positions as the octave and fifth. In some

common cases they really are an octave and fifth. Although you can

define them the other way round, it's simpler not to.

> > > > | 19= | 22= | 41=

> > > > ---------------------------------

> > > > x | 0 | 1 | 1

> > > > y | 1 | 0 | 1

> > > > p | 0 | 1 | 1

> > > > q | 1 | 1 | 2

> I was hoping for another remedial-algebra lesson on how this

> table could be calculated from the other data you gave. :)

An octave is 22x+19y steps. So when x=0 you have 19-equal and when y=0

you have 22-equal. When x=y you have 41-equal. As p is the same as x,

it must have the same size in those temperaments. As q is the sum of x

and y it'll either be 1 (from 1+0 or 0+1) or 2 (from 1+1).

In algebraic terms

(x) = (1)oct/22 or (0)oct/19 or (1)oct/41

(y) (0) (1) (1)

(q) = (1 1)(x) = (1 1)(1)oct/22 = (1)oct/22

(p) (1 0)(y) (1 0)(0) (1)

or (q) = (1 1)(0)oct/19 = (1)oct/19

(p) (1 0)(1) (0)

or (q) = (1 1)(1)oct/41 = (2)oct/41

(p) (1 0)(1) (1)

Graham