No one responded to my Hypothesis on the tuning list. Search for "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:

</tuning/topicId_22135.html#22135>

"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

--- 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!

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

--- 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.

Thanks, Paul, for inviting me to participate in your new group!

Good luck with it!

Joseph

>From: paul@stretch-music.com

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

>To: tuning-math@yahoogroups.com

>Subject: [tuning-math] Hypothesis

>Date: Mon, 21 May 2001 04:54:45 -0000

>

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

>

>

>To unsubscribe from this group, send an email to:

>tuning-math-unsubscribe@yahoogroups.com

>

>

>

>Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

>

>

_________________________________________________________________

Get your FREE download of MSN Explorer at http://explorer.msn.com

--- In tuning-math@y..., paul@s... wrote:

> No one responded to my Hypothesis on the tuning list. Search

for "hypothesis".

I remember this interesting hypothesis mentioned on the Tuning List...

So, I can take the "hint..." Somehow it can be "proven"

mathematically...

Where's Keenan or Walker??

______ _____ ______

Joseph Pehrson

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

--- 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!

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

--- 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.

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

>

> <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"

--- 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.

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

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

--- In tuning-math@y..., graham@m... 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"?

In-Reply-To: <9ejj0r+mrao@eGroups.com>

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".

Graham

--- 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}.

I wrote,

> and the chromatic unison vector is any one

> of {49:49

Oops -- should be 49:48, of course!

--- 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. :)

--- 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. :)

Joe Monzo's dictionary is chock full of definitions:

http://www.ixpres.com/interval/dict/index.htm

--- 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!

--- 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?

--- 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.

--- 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?

> 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.

--- 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.

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

>

> 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.

Did you get this from me? 'Cause you know I agree. 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.

>> () 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

I 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.

I mean, I caught that they are non-parallelpiped PBs, but not

why this should translate into fewer harmonic structures (do

you mean only complete chords? total consonant dyads?).

-Carl

--- 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.

--- 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.

--- 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.

--- 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.

--- 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?

--- 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.

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

> but JI diatonic melody is smooth and refined, so to speak. Maybe my

> ears are no good. :)

Do you have an example of an actual diatonic melody that sounds good

to you in JI? Most classical melodies, say Mozart for example, sound

bad to me in JI -- the motivic unity between statements is disturbed

by the variation in the size of the whole tone.

--- 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.