Tuning-Math messages 725 - 749

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Message: 725

Date: Mon, 20 Aug 2001 20:31:55

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:
> 
> > But see the message I just posted about why MOSs appear to be
> > _harmonically_ special for the class of scales with given step
> > sizes and number of notes.
> 
> I didn't catch the why, but I am of course familiar with the
> example you gave.
> 
Roughly, the reasoning is that slicing the lattice with parallel, 
hyperplanar slices is likely to minimize the number of "wolves" or 
broken consonances relative to using "bumpy" slices.


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Message: 726

Date: Mon, 20 Aug 2001 20:33:30

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:

> I mean, I caught that they are non-parallelpiped PBs, but not
> why this should translate into fewer harmonic structures

See the last post.

> (do
> you mean only complete chords? total consonant dyads?).

I'm thinking both, but I suppose the latter might do if we're trying 
to mathematize this.


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Message: 727

Date: Mon, 20 Aug 2001 20:35:34

Subject: Re: Mea culpa

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Carl,
> 
> It's a 12-tet scale with a 5/12 generator.

I'm not seeing the 12-tET-ness or the 5/12-ness of this at all:

> > > This is not quite true -- for example, LssssLssss is MOS but not
> WF
> > > and doesn't have Myhill's property.


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Message: 728

Date: Mon, 20 Aug 2001 20:51:12

Subject: Re: Mea culpa

From: carl@l...

>>>> MOS, WF, and Myhill's property are all equivalent.
>>> 
>>> This is not quite true -- for example, LssssLssss is MOS but not 
>>> WF and doesn't have Myhill's property.
>> 
>> What single generator produces the scale?
>> 
>> -Carl
> 
> One possibility is s -- here the interval of repetition is the half-
> octave.

Then my reply is that the scale is MOS/WF at the half-octave.

-Carl


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Message: 729

Date: Mon, 20 Aug 2001 20:55:57

Subject: Re: Mea culpa

From: Paul Erlich

--- In tuning-math@y..., carl@l... wrote:

> Then my reply is that the scale is MOS/WF at the half-octave.

Well there's no point in going into a big debate on terminology here, 
but note that Clampitt' list of WFs in 12-tET is sorely incomplete if 
you allow this kind of construction.

Let's just say "MOS" and forget about it.


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Message: 731

Date: Mon, 20 Aug 2001 21:53:18

Subject: Re: Mea culpa

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul Erlich wrote,
> 
> <<I'm not seeing the 12-tET-ness or the 5/12-ness of this at all>>
> 
> Hmm, why not?

Sorry, I hadn't seen your subsequent message where you said you were 
interpreting this scale as a 12-tET scale. As I'm sure you know, the 
discussion of this scale started with them in 22-tET or something 
close to it. There one could say the generator is 1/11 of an octave.


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Message: 732

Date: Mon, 20 Aug 2001 23:18:53

Subject: Re: The hypothesis

From: genewardsmith@j...

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

> 2 is certainly prime, but most of the time we consider octave-
invariant 
> scales.

Considering scales is another level of generality altogether--first 
we have approximations (kernels, unison vectors, and so forth) then 
we have tuning, and finally we select a subset and have scales.

Of course, unless you have an infinite number of notes in your scale, 
which you may have conceptually but not in practice, you don't have 
octave invariance anyway.

> So the kernel has dimension 1 because it contains 1 unison vector? 

Because it is generated by one unison vector. I'm not clear yet if a 
unison vector is supposed to be an element of the kernel or a 
generator of the kernel, as I mentioned.

> So in octave-invariant terms, 5-limit is rank 2, but cyclic about 
the 
> octave. 

If we consider equivalence classes modulo octaves, the 5-limit is 
free of rank two, but I don't know what you mean by "cyclic around 
the octave".

>An ET would be rank 0 I suppose, but you've already given the 
> real name for that case.

An ET would be free of rank 1, or "cyclic of infinite order". If we 
mod out by octaves, it would no longer be free but would (still) be 
cyclic, which implies one generator.

> In the octave-invariant case the octave lies outside the system, so 
you 
> can't say anything about it.

Whether to tune the octave exactly or not is a question which lies at 
a more specific, less abstract level than that created by defining 
certain things to be unison vectors. As a general rule, you only 
confuse things by insisting on concrete particulars when they are not 
required. About all one can say for certain is that you can't toss 2 
out of a discussion involving unison vectors, because without 2 we 
can't tell what is a small interval and what is not.

> For the octave invariant case, the fourth or fifth is the 
generator, which 
> I think agrees with both meanings of "generator".

You can generate by fifths and octaves if you want, but you don't 
need to.

You have octaves on the brain, which is the usual situation in music 
theory; however when discussing tuning and temperment it really is 
just another interval.

Suppose I decide to have a mean-tone system, so that 81/80 is a 
unison vector. I could tune things so that octaves were pure 2's, but 
I don't have to. Suppose instead I decide that I want the major sixth 
to be exact. Now I can look at the circle of octaves, and notice that 
it approximately returns after 14 octaves--14 octaves is almost the 
same as 19 major sixths; 2^14 = (5/3)^18.9968... Suppose I decide to 
tune octaves so that I represent 2 by (5/3)^(19/14); this is equal to
2.000232... and is sharp by about 1/5 of a cent. Since I have fixed 
two values and I am making 81/80 a unison vector, major thirds are 
now determined also. Since 2 and 5/3 are not now incommensurable, I 
actually have a rank 1 group. It is the 14 equal division of the 
major sixth, with a very slightly sharp octave; it is in practice 
more or less indistinguishable from the 19 equal division of the 
octave, with very slightly flat major sixths.

However, there is nothing in the nature of the problem to suggest I 
need to make any interval exact. One obvious way to decide would be 
to pick a set of intervals {t1, ... , tn} which I want to be well 
approximated, and a corresponding set of weights {w1, ... , wn} 
defining how important I think it is to have that interval 
approximate nicely. Perhaps I could do this using harmonic entropy? 
In any case, having done this I now have an optimization problem 
which I can decide using the method of least squares. If I have two 
generators, which I have in the case of the 5-limit with 81/80 a 
unison vector, then solving this will give me tunings for the 
generators and hence tunings for the entire system. There is no 
special treatment given to the octave in this method, but I see no 
reason in terms of psychological acoustics why there needs to be.


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Message: 733

Date: Tue, 21 Aug 2001 00:00:39

Subject: Re: Mea culpa

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., carl@l... wrote:

> > MOS, WF, and Myhill's property are all equivalent.

> This is not quite true -- for example, LssssLssss is MOS but not WF 
> and doesn't have Myhill's property.

If "L" is a large scale step and "s" is a small scale step, then this 
has two sizes of steps. If that is Myhill's property then it should 
have it, so why doesn't it?


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Message: 734

Date: Tue, 21 Aug 2001 00:08:59

Subject: Re: Microtemperament and scale structure

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Are you saying that both keyboards are tuned identically, or that 
> there may be an offset?

It certainly would be more interesting musically with an offset, but 
it doesn't matter in the sense that this is a tuning question, not a 
structure question. Either way, a comma interval is represented by 
jumping from the green keyboard to the red keyboard or vice-versa--
therefore, there is no distinction between a comma up and a comma 
down, and two commas are a unison.

> I was just asking what it stood for. "Just Tuning"?

Sorry, just trying to be one of the boys in this alphabet soup of 
acronyms around here.


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Message: 735

Date: Tue, 21 Aug 2001 00:08:57

Subject: Re: Microtemperament and scale structure

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Are you saying that both keyboards are tuned identically, or that 
> there may be an offset?

It certainly would be more interesting musically with an offset, but 
it doesn't matter in the sense that this is a tuning question, not a 
structure question. Either way, a comma interval is represented by 
jumping from the green keyboard to the red keyboard or vice-versa--
therefore, there is no distinction between a comma up and a comma 
down, and two commas are a unison.

> I was just asking what it stood for. "Just Tuning"?

Sorry, just trying to be one of the boys in this alphabet soup of 
acronyms around here.


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Message: 736

Date: Tue, 21 Aug 2001 00:13:34

Subject: Re: The hypothesis

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:

> In any case, having done this I now have an optimization problem 
> which I can decide using the method of least squares.

We've done these sorts of things many times.

> If I have two 
> generators, which I have in the case of the 5-limit with 81/80 a 
> unison vector, then solving this will give me tunings for the 
> generators and hence tunings for the entire system. There is no 
> special treatment given to the octave in this method, but I see no 
> reason in terms of psychological acoustics why there needs to be.

Right -- so mathematically, why don't we just call the octave (or in 
some cases, like the BP scale, another simple interval) the 
equivalence interval, and deal with ETs as cyclic groups, etc., 
ignoring the question of whether the octaves are slightly tempered or 
not?


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Message: 737

Date: Tue, 21 Aug 2001 00:15:47

Subject: Re: Mea culpa

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning-math@y..., carl@l... wrote:
> 
> > > MOS, WF, and Myhill's property are all equivalent.
> 
> > This is not quite true -- for example, LssssLssss is MOS but not 
WF 
> > and doesn't have Myhill's property.
> 
> If "L" is a large scale step and "s" is a small scale step, then 
this 
> has two sizes of steps. If that is Myhill's property then it should 
> have it, so why doesn't it?

Myhill's property isn't just about the step sizes. Recall the melodic 
minor scale, which has two step sizes but isn't WF. Myhill's property 
says it has two sizes of _every_ generic interval size. But in the 
case of LssssLssss, all "sixths" are the same size: L+4*s. There's 
only one size of "sixth" -- so Myhill fails.


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Message: 738

Date: Tue, 21 Aug 2001 00:32:56

Subject: Re: Hypothesis

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> Strict, fixed-pitch just intonation has almost never been used in 
> actual music with these scales. This is because of the so-
> called "comma problem". Don't let the JI advocates fool you: 
> Pythagorean tuning and various meantone-like temperaments have been 
> far more important than fixed-pitch 5-limit just intonation for the 
> actual performance of these scales -- even in China!

It seems to me the comma problem is less of a problem if you are only 
interested in melody, and this whole business is justified in terms 
of melody. Is it really true that a pentatonic or diatonic melody 
sounds better in a meantone tuning than it does in just tuning? 
Moreover, the smaller the scale steps the harder it becomes to tell 
the difference between them. If hearing the difference between 9/8 
and 10/9 is hard, hearing the difference between 16/15 and 15/14 will 
certainly be harder.


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Message: 739

Date: Tue, 21 Aug 2001 01:09:57

Subject: Re: Hypothesis

From: Paul Erlich

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> > Strict, fixed-pitch just intonation has almost never been used in 
> > actual music with these scales. This is because of the so-
> > called "comma problem". Don't let the JI advocates fool you: 
> > Pythagorean tuning and various meantone-like temperaments have 
been 
> > far more important than fixed-pitch 5-limit just intonation for 
the 
> > actual performance of these scales -- even in China!
> 
> It seems to me the comma problem is less of a problem if you are 
only 
> interested in melody, and this whole business is justified in terms 
> of melody.

In terms of harmony?

> Is it really true that a pentatonic or diatonic melody 
> sounds better in a meantone tuning than it does in just tuning? 

Probably Pythagorean is everyone's favorite melodic tuning. And yes, 
I do dislike the melodic jaggedness of just scales . . . but why 
don't we just assume harmony _is_ important for the purposes of the 
Hypothesis. Let's assume that the only reason for tempering is to 
tame those nasty wolves.

> Moreover, the smaller the scale steps the harder it becomes to tell 
> the difference between them. If hearing the difference between 9/8 
> and 10/9 is hard, hearing the difference between 16/15 and 15/14 
will 
> certainly be harder.

Probably . . . let's just say that tempering out the 225:224 is more 
of a harmonic, than a melodic, consideration.


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Message: 740

Date: Tue, 21 Aug 2001 03:18:23

Subject: Re: The hypothesis

From: Dave Keenan

--- In tuning-math@y..., genewardsmith@j... wrote:
> Sounds like we may be getting there, but there seems to be some 
> confusion as to whether 2 counts as a prime, and so whether for 
> instance the 5-limit is 2D or 3D. ...

There's no confusion over whether 2 is a prime. We understand quite 
well, all that you wrote. Each of us has probably railed against it at 
some time. But it would be too confusing to change it now. You'd best 
just learn to accept it.

Rank 1 = equal temperament
Rank 2 = linear temperament
Rank 3 = planar temperament

Regards,
-- Dave Keenan


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Message: 741

Date: Tue, 21 Aug 2001 03:38:23

Subject: Re: Microtemperament and scale structure

From: Dave Keenan

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> The diatonic scale (LsssLss) is MOS: the IoR is an octave, and the 
> generator is L+s+s.
> 
> The melodic minor scale (LssssLs) is not MOS: there is no generator 
> that produces all the notes and no others.

Shouldn't all those "L"s be "s"s and vice versa?


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Message: 742

Date: Tue, 21 Aug 2001 03:50:10

Subject: Tetrachordal alterations (was: Hi gang.)

From: Dave Keenan

> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> What we need is a really user-friendly, _practical_ guide to a 
bunch 
> of the new temperaments and their MOSs (and ideally, tetrachordal 
> alterations of those MOSs in cases like 10-of-22 and 22-of-46).

Are "tetrachordal alterations" only possible when the interval of 
repetition is some whole-number fraction of octave?

How do you do them, in general?

What would be a "tetrachordal alteration" of Blackjack?

-- Dave Keenan


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Message: 744

Date: Tue, 21 Aug 2001 04:16:36

Subject: Re: Hypothesis

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > Is it really true that a pentatonic or diatonic melody 
> > sounds better in a meantone tuning than it does in just tuning? 

> Probably Pythagorean is everyone's favorite melodic tuning.

I don't know--to my ears, melodically Pythagorean is brighter and 
more aggressive, (and actually not too much different from 12 ET), 
but JI diatonic melody is smooth and refined, so to speak. Maybe my 
ears are no good. :)

 And yes, 
> I do dislike the melodic jaggedness of just scales . . . but why 
> don't we just assume harmony _is_ important for the purposes of the 
> Hypothesis. Let's assume that the only reason for tempering is to 
> tame those nasty wolves.

As you can see, "jagged" is not how JI diatonic melodies strike me at 
all. If you are tempering merely to tame wolves, why does this WF 
stuff concern you, however?


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Message: 745

Date: Tue, 21 Aug 2001 04:22:49

Subject: Re: The hypothesis

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> > There is no 
> > special treatment given to the octave in this method, but I see 
no 
> > reason in terms of psychological acoustics why there needs to be.

> Right -- so mathematically, why don't we just call the octave (or 
in 
> some cases, like the BP scale, another simple interval) the 
> equivalence interval, and deal with ETs as cyclic groups, etc., 
> ignoring the question of whether the octaves are slightly tempered 
or 
> not?

There are two distinct questions involved--tuning, and scale 
construction. If you are discussing tuning, the octave is an interval 
and needs to be tuned--even leaving it a 2 is after all a choice of 
tuning. If you are constructing scales which repeat a particular 
pattern of steps, the psycoacoustic properties of the octave make it 
by far the most interesting choice.


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Message: 747

Date: Tue, 21 Aug 2001 07:49:27

Subject: Chromatic = commatic?

From: genewardsmith@j...

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> 1) The number of notes in the scale should be (normally) the 
> determinant of the matrix of unison vectors. One has to include 
both 
> the chromatic and the commatic unison vectors in this calculation.

What you are calling the determinant is just the determinant of the 
minor you get by setting 2 aside--and there is 2 on the brain again. 
From the point of view of approximations and real life, your comment 
is true. From the point of view of pure algebra, it isn't. From an 
algebraic point of view, the 7-et might be [7, 11, 17] and not 
[7, 11, 16]--they are different homomorphisms. To recover the whole 
homomorphism, and not just the number of steps in an octave, we need 
all three minor determinants.

> 2) In the "prototypical" case, the commatic unison vector is "the 
> comma", 81:80; and the chromatic unison vector is "the chromatic 
> unison" or "augmented unison", 25:24. These define a 7-tone 
> periodicity block: the diatonic scale. You see how the terminology 
is 
> just a generalization of this case.

Both of these are elements of the kernel of the 7-homomorphism 
[7, 11, 16] and together they generate it. There really is no 
distinction to be drawn beyond the obvious fact that 25/24 is bigger 
than 81/80. You could always call the biggest element in your 
generating set the chromatic unison and the rest commatic unison 
vectors, but I don't see the point. Anyway, what is chromatic for one 
set will end up being commatic for another!

> > > The weak form of the hypothesis simply says that 
> > > if there is 1 chromatic unison vector, and n-1 
> > > commatic unison vectors, then what you have is a 
> > > linear temperament, with some generator and 
> > > interval of repetition (which is usually equal to the 
> > > interval of equivalence, but sometimes turns out to 
> > > be half, a third, a quarter . . . of it).

Now I translate this to saying that if the rank of the kernel is n, 
then we get a linear temperament. Since the rank of the set of notes 
is n+1, this means the codimension is 1 and hence the rank of the 
homomorphic image is 1, meaning we have an et--which is precisely 
what we did get in the case where we had the 7-et. Why do you say 
linear temperament, which we've just determined means rank 2?

> > At last we are making progress! I don't see much role for 
> > the "chromatic" element here, though.
> 
> You're right . . . it plays no role here.

Aha! So perhaps what you are saying is if the codimension is 2, then 
the rank of the homomorphic image is 2, and we have a linear 
temperament.

> No -- you did that with n unison vectors -- I'm not counting the 2 
> axis as a "dimension" here.

Not a good idea in this context--you should.

> MOS means that there is an interval of repetition 

What do the letters of the acronym stand for?


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Message: 748

Date: Tue, 21 Aug 2001 08:10:08

Subject: Re: Microtemperament and scale structure

From: genewardsmith@j...

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

> It's what <Unison vector to MOS script *> is all about.  
> <Unison vectors *> is a list of examples.

It was pretty hard to figure out what they were examples of.

Let me give an example matrix computation, and see if it looks 
familiar. Let's take three et's in the 5-limit, for 12, 19, and 34. 
If we make a matrix out of them, we have

     [12 19 34]
S =  [19 30 54]
     [28 44 79]

Since this consists of three column vectors pointing in more or less 
the same direction, the determinant is likely to be small; however 
none of these three is a linear combination of the other two (as 
often will happen--ets tend to be sums of other ets) the determinant 
is nonzero--in this case, 1. If we invert it, we get

          [-6 -5  6]
S^(-1) =  [11 -4 -2]
          [-4  4 -1]

The row vectors of S^(-1) are now 15625/15552, 2048/2025, and 81/80. 
Taken in pairs, these give generators for the kernel of each of the 
above systems, and hence good unison vectors for a PB. Each is a step 
vector in one system, and a unison vector in the other two, in the 
obvious way (given how matrix multiplication works.)

In the same way, we could start with three linearly independent 
unison vector candidates, and get a matrix of three ets by inverting.

The single vectors generate the intersection of the kernels of a pair 
of ets, and so define a linear temperament which factors through to 
each of the ets. That is, 81/80 generates the intersection of the 
kernel of the 12-system and the 19-system, and produces the mean tone 
temperaments. Both 12 and 19 belong to this system--we can send it to 
first the mean tone, then to either 12 or 19 (then to tuning as the 
last step!) Similarly, 2048/2025 defines a temperament which is 
common to both the 12 and the 34 system. It essentially defines what 
they have in common.

There are other types of matrix computations we could make, but I'm 
wondering if this seems familiar?


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Message: 749

Date: Tue, 21 Aug 2001 15:38 +0

Subject: Re: Microtemperament and scale structure

From: graham@m...

In-Reply-To: <9lt510+dcmg@e...>
In article <9lt510+dcmg@e...>, genewardsmith@j... () wrote:

> --- In tuning-math@y..., graham@m... wrote:
> 
> > It's what <Unison vector to MOS script *> is all about.  
> > <Unison vectors *> is a list of examples.
> 
> It was pretty hard to figure out what they were examples of.

Um, yes, it would be.  You'll find the discussion in the archives.

> Let me give an example matrix computation, and see if it looks 
> familiar. Let's take three et's in the 5-limit, for 12, 19, and 34. 
> If we make a matrix out of them, we have
> 
>      [12 19 34]
> S =  [19 30 54]
>      [28 44 79]
> 
> Since this consists of three column vectors pointing in more or less 
> the same direction, the determinant is likely to be small; however 
> none of these three is a linear combination of the other two (as 
> often will happen--ets tend to be sums of other ets) the determinant 
> is nonzero--in this case, 1. If we invert it, we get
> 
>           [-6 -5  6]
> S^(-1) =  [11 -4 -2]
>           [-4  4 -1]
> 
> The row vectors of S^(-1) are now 15625/15552, 2048/2025, and 81/80. 
> Taken in pairs, these give generators for the kernel of each of the 
> above systems, and hence good unison vectors for a PB. Each is a step 
> vector in one system, and a unison vector in the other two, in the 
> obvious way (given how matrix multiplication works.)

Aaaaaah!  So they are!  I hadn't thought of doing that.  Suddenly 
everything is a lot clearer.

The main difference with what I do is that I consider two ETs and perfect 
octaves instead of three ETs.  Presumably, chromatizing a unison vector is 
the same as junking one of the three ETs you get from the inverse.

> In the same way, we could start with three linearly independent 
> unison vector candidates, and get a matrix of three ets by inverting.

So is that what I'm doing?  Hmm.  Actually, I'd take two UVs along with 
the octave.  Hmm.

> The single vectors generate the intersection of the kernels of a pair 
> of ets, and so define a linear temperament which factors through to 
> each of the ets. That is, 81/80 generates the intersection of the 
> kernel of the 12-system and the 19-system, and produces the mean tone 
> temperaments. Both 12 and 19 belong to this system--we can send it to 
> first the mean tone, then to either 12 or 19 (then to tuning as the 
> last step!) Similarly, 2048/2025 defines a temperament which is 
> common to both the 12 and the 34 system. It essentially defines what 
> they have in common.

Yes, that figures.  Let's go back to the example.

>      [12 19 34]
> S =  [19 30 54]
>      [28 44 79]

>           [-6 -5  6]
> S^(-1) =  [11 -4 -2]
>           [-4  4 -1]

Rows and columns are interchanged when you take the inverse.  So the [-4 4 
-1] corresponds to 34.  This is the unison vector that results from taking 
34 *out* of the system.  Take out the bottom two, and you should end up 
with 12=

|[ 1  0  0]|
|[11 -4 -2]| = 12
|[-4  4 -1]|

so that works.

Generalizing to more dimensions, presumably considering n unison vectors, 
and taking out n-2 of them, will give you the linear temperament.

So, as I already have a program for generating consistent ETs, I could use 
it to generate a list of candidate unison vectors.  And then use them to 
go back to ETs.  All of it without assuming octave equivalence anywhere.

> There are other types of matrix computations we could make, but I'm 
> wondering if this seems familiar? 

Okay, I think I see the connection.  What you're doing with octave 
*specific* matrices is analogous to what I did with octave *invariant* 
matrices.  So you invert the matrix of unison vectors to get sets of 
generators, where the generator is a step size in an ET.  I invert a 
matrix of one less unison vector to get sets of generators, where the 
generator is the interval so called in WF or MOS theory.  Algebraically, 
it's exactly the same operation.

What I do with octave specific matrices is a bit more complicated.  
Because I'm considering one fewer ET, and setting the octave just, that 
means the inverse contains a mixture of step-size and MOS generators.  I 
really should be *reducing* the number of unison vectors, but it is very 
interesting to see your method that works with one more.


                      Graham


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