Yahoo Tuning Groups Ultimate Backup tuning-math Re: questions about Graham's matrices (was: 13-limit mappin

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

🔗monz <joemonz@yahoo.com>

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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🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

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

🔗monz <joemonz@yahoo.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

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"

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