Yahoo Tuning Groups Ultimate Backup tuning-math Hypothesis

Paul wrote:
> No one responded to my Hypothesis on the tuning list. Search
for "hypothesis".

If you accidentally hit Ctrl-W in IE, the window you're typing in
disappears. What a crock!

So, I'll have to start this again.

Hypotheses got mentioned a lot, so this is the relevant post:

"Hypothesis: If you temper out all but one of the unison vectors in a
periodicity block, you get a distributionally even scale."

Cross-reference with the tuning dictionary

<http://www.ixpres.com/interval/dict/disteven.htm>

"""
distributional evenness

The scale has no more than two sizes of interval in each interval
class.
"""

Does this mean the hyperparallelopiped has to become the
distributionally even scale? Or only that the relevant linear
temperament with that number of notes can be distributionally even?

The example of the 24 note periodicity block from the schisma and
diesis might be a counterexample.

|-8 -1|
| 0 -3|

It won't be an MOS anyway, and I think Carey and Clampitt showed that
an MOS is always distributionally even.

Graham

🔗paul@stretch-music.com

--- In tuning-math@y..., "Graham Breed" <graham@m...> wrote:

> Does this mean the hyperparallelopiped has to become the
> distributionally even scale?

Not necessarily -- I was thinking more along the lines of, the form
of the periodicity block with the most consonances.

> Or only that the relevant linear
> temperament with that number of notes can be distributionally even?

It might not be a linear temperament!

>
> The example of the 24 note periodicity block from the schisma and
> diesis might be a counterexample.
>
> |-8 -1|
> | 0 -3|
>
> It won't be an MOS anyway,

Uhh . . . which unison vector are you tempering out?

> and I think Carey and Clampitt showed that
> an MOS is always distributionally even.

But not all distributionally even scales are MOS!

🔗graham@microtonal.co.uk

In-Reply-To: <9ebk3d+5dqs@eGroups.com>
Paul wrote:

> --- In tuning-math@y..., "Graham Breed" <graham@m...> wrote:
>
> > Does this mean the hyperparallelopiped has to become the
> > distributionally even scale?
>
> Not necessarily -- I was thinking more along the lines of, the form
> of the periodicity block with the most consonances.

How would that relate to the unison vectors?

> > Or only that the relevant linear
> > temperament with that number of notes can be distributionally even?
>
> It might not be a linear temperament!

One fewer unison vectors than you need for an ET will always give a linear
temperament of some kind. That follows from my matrix definitions.

> > The example of the 24 note periodicity block from the schisma and
> > diesis might be a counterexample.
> >
> > |-8 -1|
> > | 0 -3|
> >
> > It won't be an MOS anyway,
>
> Uhh . . . which unison vector are you tempering out?

I don't know, I haven't tried. How are you defining "interval class"?

> > and I think Carey and Clampitt showed that
> > an MOS is always distributionally even.
>
> But not all distributionally even scales are MOS!

But if we could prove that all linear temperaments give something like an
MOS, that would prove the hypothesis.

Graham

🔗paul@stretch-music.com

--- In tuning-math@y..., graham@m... wrote:
> >
> > Not necessarily -- I was thinking more along the lines of, the
form
> > of the periodicity block with the most consonances.
>
> How would that relate to the unison vectors?

For example, the melodic minor scale and the major scale are both
periodicity blocks of the unison vectors 25:24 and 81:80, with the
81:80 tempered out. But the melodic minor scale is not
distributionally even. The major scale has more consonances . . .

>
> > > Or only that the relevant linear
> > > temperament with that number of notes can be distributionally
even?
> >
> > It might not be a linear temperament!
>
> One fewer unison vectors than you need for an ET will always give a
linear
> temperament of some kind. That follows from my matrix definitions.

Something must be wrong with your definitions then. For example, my
decatonic system comes from the unison vectors 64:63, 50:49, and
49:48, with 64:63 and 50:49 tempered out. But it's not represented by
any linear temperament. However, the decatonic with the most
consonances is distributionally even.

> > > The example of the 24 note periodicity block from the schisma
and
> > > diesis might be a counterexample.
> > >
> > > |-8 -1|
> > > | 0 -3|
> > >
> > > It won't be an MOS anyway,
> >
> > Uhh . . . which unison vector are you tempering out?
>
> I don't know, I haven't tried. How are you defining "interval
class"?

Where did I use that term?
>
> > > and I think Carey and Clampitt showed that
> > > an MOS is always distributionally even.
> >
> > But not all distributionally even scales are MOS!
>
> But if we could prove that all linear temperaments give something
like an
> MOS, that would prove the hypothesis.

Something _like_ an MOS, yes.

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗jpehrson@rcn.com

🔗graham@microtonal.co.uk

In-Reply-To: <9ebn2i+9ksu@eGroups.com>
Paul wrote:

> --- In tuning-math@y..., graham@m... wrote:
> > >
> > > Not necessarily -- I was thinking more along the lines of, the
> form
> > > of the periodicity block with the most consonances.
> >
> > How would that relate to the unison vectors?
>
> For example, the melodic minor scale and the major scale are both
> periodicity blocks of the unison vectors 25:24 and 81:80, with the
> 81:80 tempered out. But the melodic minor scale is not
> distributionally even. The major scale has more consonances . . .

Does it matter which major scale we take? Or are we contracting the
lattice to one dimension?

> > One fewer unison vectors than you need for an ET will always give a
> linear
> > temperament of some kind. That follows from my matrix definitions.
>
> Something must be wrong with your definitions then. For example, my
> decatonic system comes from the unison vectors 64:63, 50:49, and
> 49:48, with 64:63 and 50:49 tempered out. But it's not represented by
> any linear temperament. However, the decatonic with the most
> consonances is distributionally even.

It is, the equivalence interval's a half-octave and the generating
interval's a fourth.

> > > > |-8 -1|
> > > > | 0 -3|
> > > >
> > > > It won't be an MOS anyway,
> > >
> > > Uhh . . . which unison vector are you tempering out?
> >
> > I don't know, I haven't tried. How are you defining "interval
> class"?
>
> Where did I use that term?

It's part of Monz's definition of "distributionally equal": "The scale
has no more than two sizes of interval in each interval
class."

> > But if we could prove that all linear temperaments give something
> like an
> > MOS, that would prove the hypothesis.
>
> Something _like_ an MOS, yes.

Usually that comes out fine. The unison vectors define a linear
temperament, which forms an MOS with the right number of notes.
Hopefully this will maximise the consonances (another hypothesis?).

The diaschismic temperaments give an MOS with the half-octave as a
generator. In general, for octave-equivalent unison vectors, the
equivalence interval will always be some fraction of an octave.

So the only outstanding problems are those temperaments where the
determinant comes out as a multiple of the number of notes in the
relevant ET. In that case, you get overcounting. The linear temperament
can still be calculated, but not in its lowest terms. And the
periodicity block contains twice as many notes as it needs to.

It's not clear to me what's going on here, especially after one vector
gets tempered out, but I'm sure the MOS concept can be expanded to cover
it.

Graham

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ebn2i+9ksu@e...>
> Paul wrote:
>
> > --- In tuning-math@y..., graham@m... wrote:
> > > >
> > > > Not necessarily -- I was thinking more along the lines of,
the
> > form
> > > > of the periodicity block with the most consonances.
> > >
> > > How would that relate to the unison vectors?
> >
> > For example, the melodic minor scale and the major scale are both
> > periodicity blocks of the unison vectors 25:24 and 81:80, with
the
> > 81:80 tempered out. But the melodic minor scale is not
> > distributionally even. The major scale has more consonances . . .
>
> Does it matter which major scale we take? Or are we contracting
the
> lattice to one dimension?

When the 81:80 is tempered out, the lattice curls into a cylinder.
Then all major scales are identical.
>
> > > One fewer unison vectors than you need for an ET will always
give a
> > linear
> > > temperament of some kind. That follows from my matrix
definitions.
> >
> > Something must be wrong with your definitions then. For example,
my
> > decatonic system comes from the unison vectors 64:63, 50:49, and
> > 49:48, with 64:63 and 50:49 tempered out. But it's not
represented by
> > any linear temperament. However, the decatonic with the most
> > consonances is distributionally even.
>
> It is, the equivalence interval's a half-octave and the generating
> interval's a fourth.

OK! If that falls out of your matrix formalism, then let's go with it!
>
> > > > > |-8 -1|
> > > > > | 0 -3|
> > > > >
> > > > > It won't be an MOS anyway,
> > > >
> > > > Uhh . . . which unison vector are you tempering out?
> > >
> > > I don't know, I haven't tried. How are you defining "interval
> > class"?
> >
> > Where did I use that term?
>
> It's part of Monz's definition of "distributionally equal": "The
scale
> has no more than two sizes of interval in each interval
> class."

In this case, an interval class is the set of all intervals subtended
by n consecutive scale degrees in a given scale, for some whole
number n.
>
> > > But if we could prove that all linear temperaments give
something
> > like an
> > > MOS, that would prove the hypothesis.
> >
> > Something _like_ an MOS, yes.
>
> Usually that comes out fine. The unison vectors define a linear
> temperament, which forms an MOS with the right number of notes.

Let's prove this.

> Hopefully this will maximise the consonances (another hypothesis?).

It would be good to prove that too, but I'm afraid it won't always
work. It works if all the consonances come out to simple powers of
the generator, though.
>
> The diaschismic temperaments give an MOS with the half-octave as a
> generator.

You mean, the half-octave as an equivalence interval?

> In general, for octave-equivalent unison vectors, the
> equivalence interval will always be some fraction of an octave.

Great. But you can't take "equivalence interval" too far here -- the
strongest consonances become dissonances when altered by the half-
octave.
>
> So the only outstanding problems are those temperaments where the
> determinant comes out as a multiple of the number of notes in the
> relevant ET. In that case, you get overcounting. The linear
temperament
> can still be calculated, but not in its lowest terms. And the
> periodicity block contains twice as many notes as it needs to.
>
> It's not clear to me what's going on here, especially after one
vector
> gets tempered out, but I'm sure the MOS concept can be expanded to
cover
> it.

We'll figure it out!

🔗graham@microtonal.co.uk

In-Reply-To: <9eecso+amc3@eGroups.com>
Paul wrote:

> > It is, the equivalence interval's a half-octave and the generating
> > interval's a fourth.
>
> OK! If that falls out of your matrix formalism, then let's go with it!

You should always be able to define a linear temperament using two
intervals. Finding the right two can be tricky.

> >
> > > > > > |-8 -1|
> > > > > > | 0 -3|
> > > > > >
> > > > > > It won't be an MOS anyway,
> > > > >
> > > > > Uhh . . . which unison vector are you tempering out?
> > > >
> > > > I don't know, I haven't tried. How are you defining "interval
> > > class"?
> > >
> > > Where did I use that term?
> >
> > It's part of Monz's definition of "distributionally equal": "The
> scale
> > has no more than two sizes of interval in each interval
> > class."
>
> In this case, an interval class is the set of all intervals subtended
> by n consecutive scale degrees in a given scale, for some whole
> number n.

Temper out the schisma from the periodicity block above. You end up with
a 24-note schismic scale. No way can that have two step sizes!

That looks like a refutation with the definitions I have. How about a
weaker hypothesis using propriety instead? Schismic-24 is still proper,
but not strictly proper. I'm sure some even hairier examples would break
this. Remember unison vectors don't even have to be small intervals.

> > Usually that comes out fine. The unison vectors define a linear
> > temperament, which forms an MOS with the right number of notes.
>
> Let's prove this.

I'm sure you can always get the linear temperament. You can describe it
with fractions of the octave and chromatic unison vector if needs be.
Getting to the MOS is more difficult, if you have a formula for that it
would be useful anyway.

Seeing as this is the mathematical list, I'll give the matrix equation:

(R1) (R2)
(R2) (R2)
(M1) (00)
(. )H' = (. )H'
(. ) (. )
(. ) (. )
(Mn) (00)

Where R1 and R2 are the chromatic unison vectors (one of which will
usually be the octave) as row vectors. M1 to Mn are the commatic unison
vectors. 00 is a row of zeros. So the things that look like column
matrices are actually square. H' is the tempered equivalent of the list
of prime axes, including 2.

Multiply on the left by the inverse of the matrix with the unison vectors
in, and you have an equation defining H' in terms of itself. You can
then get your chromatic unison vectors in terms of H', and you have a
two-dimensional system.

Usually the chromatic vectors are an octave and a twelfth. For meantone
and schismic this works fine. For diaschismic, they both have to be
divided by two, but only the octave actually needs to be divided. For
Miracle, the octave and twelfth both have to be divided by 6, so you have
to re-define it with a fifth as a unison vector. For other scales it'll
be hard to find the chromatic UV that leads nicely to the MOS.

> > Hopefully this will maximise the consonances (another hypothesis?).
>
> It would be good to prove that too, but I'm afraid it won't always
> work. It works if all the consonances come out to simple powers of
> the generator, though.

It probably would work for a sufficiently large MOS. But we could frame
the hypothesis so that the MOS is used as the default PB. I think that
would make sense.

> > The diaschismic temperaments give an MOS with the half-octave as a
> > generator.
>
> You mean, the half-octave as an equivalence interval?

Yes, sorry.

> > In general, for octave-equivalent unison vectors, the
> > equivalence interval will always be some fraction of an octave.
>
> Great. But you can't take "equivalence interval" too far here -- the
> strongest consonances become dissonances when altered by the half-
> octave.

I think this is in line with what Wilson intended by using "equivalence
interval" instead of "octave".

> > So the only outstanding problems are those temperaments where the
> > determinant comes out as a multiple of the number of notes in the
> > relevant ET. In that case, you get overcounting. The linear
> temperament
> > can still be calculated, but not in its lowest terms. And the
> > periodicity block contains twice as many notes as it needs to.
> >
> > It's not clear to me what's going on here, especially after one
> vector
> > gets tempered out, but I'm sure the MOS concept can be expanded to
> cover
> > it.
>
> We'll figure it out!

You still get a chain of generators, but not closed to give only two step
sizes.

Graham

🔗Paul Erlich <paul@stretch-music.com>

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

> Temper out the schisma from the periodicity block above. You end
up with
> a 24-note schismic scale. No way can that have two step sizes!
>
> That looks like a refutation with the definitions I have.

I think the problem is that, as you said before, the scale really has
12 pitch classes, not 24, due to the syntonic comma squared vanishing.

> How about a
> weaker hypothesis using propriety instead?

Blackjack it not proper.

> Schismic-24 is still proper,
> but not strictly proper. I'm sure some even hairier examples would
break
> this. Remember unison vectors don't even have to be small
intervals.

Exactly.
>
>
> > > Usually that comes out fine. The unison vectors define a
linear
> > > temperament, which forms an MOS with the right number of notes.
> >
> > Let's prove this.
>
> I'm sure you can always get the linear temperament. You can
describe it
> with fractions of the octave and chromatic unison vector if needs
be.
> Getting to the MOS is more difficult, if you have a formula for
that it
> would be useful anyway.
>
> Seeing as this is the mathematical list, I'll give the matrix
equation:
>
> (R1) (R2)
> (R2) (R2)
> (M1) (00)
> (. )H' = (. )H'
> (. ) (. )
> (. ) (. )
> (Mn) (00)
>
> Where R1 and R2 are the chromatic unison vectors (one of which will
> usually be the octave) as row vectors. M1 to Mn are the commatic
unison
> vectors. 00 is a row of zeros. So the things that look like
column
> matrices are actually square. H' is the tempered equivalent of the
list
> of prime axes, including 2.
>
> Multiply on the left by the inverse of the matrix with the unison
vectors
> in, and you have an equation defining H' in terms of itself. You
can
> then get your chromatic unison vectors in terms of H', and you have
a
> two-dimensional system.

OK, good so far.
>
> Usually the chromatic vectors are an octave and a twelfth.

Lost me there.

? For
> Miracle, the octave and twelfth both have to be divided by 6, so
you have
> to re-define it with a fifth as a unison vector.

??
> It probably would work for a sufficiently large MOS. But we could
frame
> the hypothesis so that the MOS is used as the default PB. I think
that
> would make sense.

All right, as long as we don't get circular.

🔗monz <joemonz@yahoo.com>

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

/tuning-math/message/14

> --- In tuning-math@y..., graham@m... wrote:
>
> > Temper out the schisma from the periodicity block above.
> > You end up with a 24-note schismic scale. No way can that
> > have two step sizes!
> >
> > That looks like a refutation with the definitions I have.
>
> I think the problem is that, as you said before, the scale
> really has 12 pitch classes, not 24, due to the syntonic
> comma squared vanishing.
>
> <etc.>

Can you guys please illustrate all this with lattices and
other tables and diagrams?

My math abilities are far below a lot of you others on
*this* list, and it's all I can do to keep up with the
other tuning lists now, to say nothing of the difficulties
I have understanding what's written here.

I know it would slow down the rate of discourse when those
of you who speak the lingo start rapid-fire exchange, as
happened over the past month with the MIRACLEs, but I for one
would greatly appreciate help by seeing lots of visuals.

Thanks.

> > Seeing as this is the mathematical list, I'll give the matrix
> > equation:
> >
> > (R1) (R2)
> > (R2) (R2)
> > (M1) (00)
> > (. )H' = (. )H'
> > (. ) (. )
> > (. ) (. )
> > (Mn) (00)
> >
> > Where R1 and R2 are the chromatic unison vectors (one of
> > which will usually be the octave) as row vectors. M1 to Mn
> > are the commatic unison vectors. 00 is a row of zeros.
> > So the things that look like column matrices are actually
> > square. H' is the tempered equivalent of the list
> > of prime axes, including 2.

Graham, again, could you please use lattices etc. to show this?

I use vector addition for the prime factors in my work,
and I thought at first that your matrix notation used here
was similar, but I still don't understand it after having
read your webpages. Could you (or someone else?) help
guide me thru it? And explain how it's different from
what I use, if it indeed is? My version of this "matrix
addition" is here:
http://www.ixpres.com/interval/monzo/article/article.htm#calculate

I'd like detailed contributions like these I'm requesting
to be made with a view toward incorporation into my Tuning
Dictionary. I'm willing to make audio examples for as much
of what we discuss as I can, but I have to understand what
it is.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning-math/message/14
>
> > --- In tuning-math@y..., graham@m... wrote:
> >
> > > Temper out the schisma from the periodicity block above.
> > > You end up with a 24-note schismic scale. No way can that
> > > have two step sizes!
> > >
> > > That looks like a refutation with the definitions I have.
> >
> > I think the problem is that, as you said before, the scale
> > really has 12 pitch classes, not 24, due to the syntonic
> > comma squared vanishing.
> >
> > <etc.>
>
> Can you guys please illustrate all this with lattices and
> other tables and diagrams?

Hi Monz.

What we're discussing here is the 24-tone periodicity block you came up with to derive the
22-shruti system of Indian music.

The unison vectors of that periodicity block are the schisma and the diesis.

As you can see, half the notes in that periodicity block differ from the other half by a syntonic
comma. You can see that either in the lattice diagram or in the list of ratios.

But here's the rub. If the schisma is a unison vector, and the diesis is a unison vector, then the
schisma+diesis (multiply the ratios) is a unison vector. But you can verify that the ratio for the
schisma times the ratio for the diesis is the square of the ratio of the syntonic comma. In other
words, it represents _two_ syntonic commas.

Now, if _two_ of anything is a unison vector, then the thing itself must be either a unison or a
half-octave. But in your scale, the syntonic comma separates pairs of adjacent pitches, so it's
clearly not acting as a half-octave. So it must be a unison. In a sense, it's logically contradictory to
say that the schisma and diesis are both unison vectors while maintaining syntonic comma
differences in the scale. The scale is "degenerate", or perhaps more accurately, it's a "double
exposure" -- it seems to have twice as many pitch classes than it really has.

As for Graham's matrix methods, I'd suggest that, rather than remain fairly confused indefinitely,
you take a linear algebra course, or get a linear algebra book with exercises and solutions and
work through it. Then, perhaps you might be able to make tremendous contributions of your
own! Linear algebra is pretty abstract, so lattices, tables, and diagrams might never be able to
get across some of the wisdom that Graham is making use of.

🔗graham@microtonal.co.uk

In-Reply-To: <9ehtrn+iaeg@eGroups.com>
monz wrote:

> > I think the problem is that, as you said before, the scale
> > really has 12 pitch classes, not 24, due to the syntonic
> > comma squared vanishing.
> >
> > <etc.>
>
> Can you guys please illustrate all this with lattices and
> other tables and diagrams?

Hey, this *is* tuning-math, you know!

> My math abilities are far below a lot of you others on
> *this* list, and it's all I can do to keep up with the
> other tuning lists now, to say nothing of the difficulties
> I have understanding what's written here.

Oh, alright then. The periodicity block could be

F\--C\--G\--D\
/ \ / \ / \ /
A---E---B---F#--C#--GA--Eb--Bb
/ \ / \ / \ / \ / \ / \ / \ /
F---C---G---D---A/--E/--B/--F#/
/ \ / \ / \ /
C#/-GA/-Eb/-Bb/

That's using my schismic notation. Which is convenient, because the
schisma is one of the unison vectors. So we can temper it out to get

C\ C C# C#/ D\ ...

I've got the whole thing in my PDA and might copy it over to be cut and
pasted one day. But we already have 3 intervals involved. C\ to C is a
comma. C to C# is a limma. But C#/ to D\ is a limma less a comma. So
we've already got 3 intervals involved for the same interval class, and
the definition only allows 2.

Paul suggested using 12 rather then 24 interval classes. It may be
possible to make such a scale distributionally even, but I don't see how.

> > > Seeing as this is the mathematical list, I'll give the matrix
> > > equation:
> > >
> > > (R1) (R2)
> > > (R2) (R2)
> > > (M1) (00)
> > > (. )H' = (. )H'
> > > (. ) (. )
> > > (. ) (. )
> > > (Mn) (00)
> > >
> > > Where R1 and R2 are the chromatic unison vectors (one of
> > > which will usually be the octave) as row vectors. M1 to Mn
> > > are the commatic unison vectors. 00 is a row of zeros.
> > > So the things that look like column matrices are actually
> > > square. H' is the tempered equivalent of the list
> > > of prime axes, including 2.
>
>
> Graham, again, could you please use lattices etc. to show this?

No, that's not latticeable. It's essentially a generalisation of Fokker's
periodicity blocks to

1) remove implicit octave equivalence

2) express linear as well as equal temperaments

If you have some 4-dimensional graph paper to hand, you could try drawing
some periodicity blocks. The won't work too well in ASCII.

> I use vector addition for the prime factors in my work,
> and I thought at first that your matrix notation used here
> was similar, but I still don't understand it after having
> read your webpages. Could you (or someone else?) help
> guide me thru it? And explain how it's different from
> what I use, if it indeed is? My version of this "matrix
> addition" is here:
> http://www.ixpres.com/interval/monzo/article/article.htm#calculate

How far did you get? I added an introduction to the math. If that isn't
enough, you may have to follow Paul's suggestion and get a book on linear
algebra.

Your addition is the same as matrix addition, but matrices can be
multiplied as well which is the clever bit.

Looks like my lunch hour's definitely over, so no time for guiding now.
But I can't explain it better than I do on my website anyway. If I could,
I'd alter the website!

Graham

🔗graham@microtonal.co.uk

In-Reply-To: <9ehp0r+cb7i@eGroups.com>
Paul Erlich wrote:

> --- In tuning-math@y..., graham@m... wrote:
>
> > Temper out the schisma from the periodicity block above. You end
> up with
> > a 24-note schismic scale. No way can that have two step sizes!
> >
> > That looks like a refutation with the definitions I have.
>
> I think the problem is that, as you said before, the scale really has
> 12 pitch classes, not 24, due to the syntonic comma squared vanishing.

It defines 12-equal, but must contain 24 notes.

> > How about a
> > weaker hypothesis using propriety instead?
>
> Blackjack it not proper.

That's that one disposed of ...

> > Usually the chromatic vectors are an octave and a twelfth.
>
> Lost me there.

The top of the equation will look like

(1 0 0 ...) (1 0 0 ...)
(0 1 0 ...)H' = (0 1 0 ...)H'

So the prime axes 2 and 3 aren't being tempered out.

> ? For
> > Miracle, the octave and twelfth both have to be divided by 6, so
> you have
> > to re-define it with a fifth as a unison vector.
>
> ??

I don't have Excel running on this machine, so I can't show the example.
But you can define Miracle temperament as:

( 1 0 0 0 0) ( 1 0 0 0 0)
( 0 1 0 0 0) ( 0 1 0 0 0)
( 5 -2 -2 1 0)H' = ( 0 0 0 0 0)H'
(-1 5 0 0 -2) ( 0 0 0 0 0)
(-7 -1 1 1 1) ( 0 0 0 0 0)

So to get H' in terms of the octave and twelfth, you invert the matrix on
the left and take the first two columns. I don't know what that is, but
it's of the form

(6 0)
1(0 6)
-(? ?)
6(? ?)
(? ?)

To define it in terms of octaves and fifths, you add one column to the
other:

(6 0)
1(6 6)
-(? ?)
6(? ?)
(? ?)

If you work it out, you should find that every number in the left hand
column is a multiple of 6. So the octave can be taken as the equivalence
interval without being divided, with the fifth divided into 6 parts to be
the generator.

Graham

🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <9ejj0r+mrao@e...>
> Paul wrote:
>
> > > > > Usually the chromatic vectors are an octave and a twelfth.
> > > >
> > > > Lost me there.
> > >
> > > The top of the equation will look like
> > >
> > > (1 0 0 ...) (1 0 0 ...)
> > > (0 1 0 ...)H' = (0 1 0 ...)H'
> > >
> > > So the prime axes 2 and 3 aren't being tempered out.
> >
> > Still confused. Since when are octaves and twelfths "chromatic"?
>
> They're intervals used to specify the temperament that aren't being
> tempered out. I thought this was the definition of "chromatic
unison
> vector".

A unison vector defines an equivalence relation in the lattice. Hence
the name "unison". A periodicity block results from N independent
unison vectors in an N-dimensional lattice. A twelfth or fifth as a
unison vector would be really crazy, as far as I can tell.

In the diatonic scale, the commatic unison vector is 81:80, and the
chromatic unison vector is 25:24 (or 135:128).

In the decatonic scale, the commatic unison vectors are any two of
{225:224, 64:63, 50:49}; and the chromatic unison vector is any one
of {49:49, 28:27, 25:24}.

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗John Starrett <jstarret@carbon.cudenver.edu>

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., graham@m... wrote:
>
> > But if we could prove that all linear temperaments give something
> like an
> > MOS, that would prove the hypothesis.
>
> I might try, but first I need some definitions. :)

A sketch of the proof of the Hypothesis is
provided in post #591 on this list. Hopefully, all the
terminology in that post should be self-explanatory,
or explained by context. If not,

Fokker periodicity blocks are explained in

www.ixpres.com/interval/td/erlich/intropblock1.htm

and the pages that follow.

MOS is almost synonymous with WF (well-
formed) and that concept is explained in many
papers, such as

http://depts.washington.edu/~pnm/CLAMPITT.pdf

except that in an MOS, the interval of repetition
(which Clampitt calls interval of periodicity) can be
a half, third, quarter, etc. of the interval of
equivalence, and not necessarily equal to it.

See what you can make of it all . . . I've ignored
pathological cases, so hopefully you can come up
with a mathematical framework that covers it all!

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > MOS is almost synonymous with WF (well-
> > formed) and that concept is explained in many
> > papers, such as
> >
> > http://depts.washington.edu/~pnm/CLAMPITT.pdf
> >
> > except that in an MOS, the interval of repetition
> > (which Clampitt calls interval of periodicity) can be
> > a half, third, quarter, etc. of the interval of
> > equivalence, and not necessarily equal to it.
>
> I looked at CLAMPITT.pdf, and it seems to me the argument that there
> is something interesting about WF scales is extremely unconvincing.
> Can anyone actually *hear* this? I notice that when you talk about
> periodiciy blocks, you ignore this stuff yourself, as well you might
> so far as I can see.
>
> What gives? Am I missing something?

There are a tremendous number of arguments as
to why there is something interesting about WF or
MOS scales in the literature. Personally, I buy very
few of them, if any. But there are some very
powerful WF/MOS scales around, especially, of
course, the usual diatonic scale, and the usual
pentatonic scale. The whole point of my
Hypothesis is to show that these scales, and
perhaps ultimately the entire interest of WF/MOS
scales, in fact has a deeper basis in just intonation
and periodicity blocks.

🔗genewardsmith@juno.com

🔗carl@lumma.org

> I looked at CLAMPITT.pdf, and it seems to me the argument that
> there is something interesting about WF scales is extremely
> unconvincing. Can anyone actually *hear* this? I notice that
> when you talk about periodiciy blocks, you ignore this stuff
> yourself, as well you might so far as I can see.
>
> What gives? Am I missing something?

Howdy, Gene!

I doubt the "synechdochic property" (the "self-similarity" at the
center of the Carey and Clampitt article) is significant, except
maybe in very special kinds of musical examples and with a lot of
training. In my opinion the Carey and Clampitt article amounts to
some interesting ideas for algorithmic composition.

I don't think MOS itself means much for the perception of melody.
Rather, I think it works together, or is often confounded with
other properties:

() Symmetry at the 3:2. The idea is that the 3:2 is a special
interval, a sort of 2nd-order octave. When a scale's generator
is 3:2, MOS means that a given pattern can more often be repeated
a 3:2 away. Chains of 5, 7, and 12 "fifths" are historically
favored, but where are all the MOS chains of 5:4, 7:4, etc.? In
my experience, MOS chains of non-fifth generators can be special
too, but we should be careful not to give MOS credit for symmetry
at the 3:2.

() Myhill's property -- every scale interval comes in exactly two
acoustic sizes. This may make it easier for listeners to track
scale intervals. Consider a musical phrase that is transposed to
a different mode of the diatonic scale -- it is changed with
respect to acoustic intervals but unchanged with respect to scalar
intervals. I think this is an important musical device that is
only possible with certain kinds of scales. Myhill's property
may make it easier for the listener to access such a device, but
probably doesn't mean much if the scale can't support the device
in the first place. Here, I believe a property called "stability"
comes into play.[1] Fortunately, we can test this by listening
to un-stable MOS scales. I've done some of this listening
informally.

-Carl

[1]
Rothenberg, David. "A Model for Pattern Perception with Musical
Applications. Part I: Pitch Structures as Order-Preserving Maps",
Mathematical Systems Theory vol. 11, 1978, pp. 199-234.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part II: The Information Content of Pitch structures",
Mathematical Systems Theory vol. 11, 1978, pp. 353-372.

Rothenberg, David. "A Model for Pattern Perception with Musical
Applications Part III: The Graph Embedding of Pitch Structures",
Mathematical Systems Theory vol. 12, 1978, pp. 73-101.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > There are a tremendous number of arguments as
> > to why there is something interesting about WF or
> > MOS scales in the literature. Personally, I buy very
> > few of them, if any. But there are some very
> > powerful WF/MOS scales around, especially, of
> > course, the usual diatonic scale, and the usual
> > pentatonic scale.
>
> Unless I am missing something (highly likely at this point!) the
> pentatonic and diatonic scales are WF in mean tone intonation but
not
> in just intonation. Is that right? If it is right, doesn't that
serve
> to make the whole idea seem fishy?

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!

One thing I forgot to mention about the hypothesis: if you don't use
the parallelepiped, you might end up with a scale that is not MOS,
but has the same number of notes as the MOS that comes from the
parallelepiped. I conjecture that in some precise sense, the MOS has
more consonant structures (intervals and/or chords) than the
corresponding non-MOS. This is seen, for example, in the decatonic
case, where the chromatic unison vector is one member of the set
{25:24, 28:27, 49:48}, and the commatic unison vectors are two
members of the set {50:49, 64:63, 225:224}. The MOS is LssssLssss,
which has 8 consonant 7-limit tetrads (4 4:5:6:7s and 4
1/7:1/6:1/5:1/4s); while a melodically superior non-MOS scale,
LsssssLsss, has only 6.

🔗Paul Erlich <paul@stretch-music.com>

🔗carl@lumma.org

>> () Symmetry at the 3:2. The idea is that the 3:2 is a special
>> interval, a sort of 2nd-order octave. When a scale's generator
>> is 3:2, MOS means that a given pattern can more often be repeated
>> a 3:2 away. Chains of 5, 7, and 12 "fifths" are historically
>> favored, but where are all the MOS chains of 5:4, 7:4, etc.? In
>> my experience, MOS chains of non-fifth generators can be special
>> too, but we should be careful not to give MOS credit for symmetry
>> at the 3:2.
>
> Did you get this from me? 'Cause you know I agree.

Absolutely -- I've long credited you with it, even in a pre-send
version of that post.

> 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.

-Carl

🔗carl@lumma.org

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

--- 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.

🔗genewardsmith@juno.com

--- 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?

🔗Paul Erlich <paul@stretch-music.com>

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

You like JI diatonic melody even when there is a direct leap of 40:27?
>
> 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.

Even when there is a direct leap of 40:27?

P.S. The only culture where the major scale is tuned in JI is in
Indian music. But they tune it 1/1 9/8 5/4 4/3 3/2 27/16 15/8 2/1,
which has two identical tetrachords, and avoid direct leaps between
5/4 and 27/16.

> If you are tempering merely to tame wolves, why does this WF
> stuff concern you, however?

I don't understand this question. If we prove the hypothesis, we'll
essentially be saying that in a sense, the most harmonically
interesting fixed-pitch scales are all MOS scales. Think of MOS as a
nice, abstract property, and please disregard all the recent music-
theory literature! I think it's extremely interesting if we can
determine a nice, simple property that will have to hold for any
scale that comes out of the PB-tempering process.

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

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

> > > 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. :)

> You like JI diatonic melody even when there is a direct leap of
40:27?

I see what you are saying--I was assuming the JI melody was something
appropriate to the scale. Of course if you translate something to JI
it's likely not to be.

> Think of MOS as a
> nice, abstract property, and please disregard all the recent music-
> theory literature! I think it's extremely interesting if we can
> determine a nice, simple property that will have to hold for any
> scale that comes out of the PB-tempering process.

That sounds like an excellent plan.

Hypothesis

🔗paul@stretch-music.com

🔗Graham Breed <graham@microtonal.co.uk>

🔗paul@stretch-music.com

🔗graham@microtonal.co.uk

🔗paul@stretch-music.com

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗jpehrson@rcn.com

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

🔗graham@microtonal.co.uk

🔗Paul Erlich <paul@stretch-music.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗John Starrett <jstarret@carbon.cudenver.edu>

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗carl@lumma.org

🔗Paul Erlich <paul@stretch-music.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗carl@lumma.org

🔗carl@lumma.org

🔗Paul Erlich <paul@stretch-music.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com