I was thinking about what n "unison vectors" in n-dimensional space could

mean. In fact, it's what I've already called a "basis". See

<http://x31eq.com/matrix.htm>. I assume that's analogous to

Gene's "kernel". Let's revisit the example there:

(t) ( 1 -2 1)

(s) = ( 4 -1 -1)H

(p) (-4 4 -1)

Which can be used to define this scale

C D E F G A B C

t+p t s t+p t p+t s

The inverse of the matrix is

( 5 2 3)

( 8 3 5)

(12 4 7)

It defines H in terms of t, s and p. That I already knew. Now, the

interesting thing is, taking p out of the scale

C D E F G A B C

t t s t t t s

is the same as making p a commatic unison vector. And we can define the

approximate H in terms of t and s

( 5 2)(t)

H' = ( 8 3)(s)

(12 4)

I already knew this, but I didn't think of it as "designating a commatic

unison vector" or whatever. Before we've been thinking about starting

with commatic unison vectors and making them chromatic. It actually makes

more sense the other way around.

Making (4 -1 -1)H or 16:15 another commatic unison vector gives us

5-equal:

( 5)(t)

H' = ( 8)

(12)

Gene's already explained this in algebraic terms, but I think the example

makes it clearer. It means I can now interpret

( 1 -2 1)

( 4 -1 -1)

(-4 4 -1)

as the octave-specific equivalent of a periodicity block. Unfortunately

it only has 1 note, so the analogy breaks down. But whatever, I'll call

all the intervals unison vectors anyway.

Now, we can also define meantone as

( 1 0 0) (1 0)

( 0 1 0)C = (0 1)

(-4 4 -1) (0 0)

Here, the octave and twelfth are taking the place of the chromatic unison

vectors. So, can we call them unison vectors? I think not, because C

evaluates to

( 1 0)

( 0 1)

(-4 4)

That has negative numbers in it, which means the generators (for so they

are) don't add up to give all the primary consonances. So now we have a

definition for what interval qualifies as a "unison vector".

Interestingly, a fifth would work as a generator here, but not for a

diatonic scale.

An alternative to "unison vector" would be "melodic generator".

Usually we call the octave the period and a fifth (which is therefore

equivalent to the twelfth) the generator. It isn't much of a stretch to

think of two generators.

In octave-equivalent terms, a unison is:

v v v v v v v v v v v w

where v is s fifth and w is a wolf. The same algebra applies, but it's

harder to visualise. I think the fifth here does count as a unison

vector.

Defining an MOS in octave-equivalent terms is easy, once you realize that

the generators that pop out needn't be unison vectors in octave-specific

space. So is defining temperaments by chromatic unison vectors with

octave-specific matrices. The difficult part is when you make the octave

a generator in those octave-specific matrices. Usually the octave won't

be a unison vector. Obviously not if you want to look at intervals

smaller than an octave! We end up with generators that aren't unison

vectors, and it all gets complicated.

The hard part of my original MOS finding script was getting the second

scale step from the one chromatic unison vector. It's because I was

getting mixed results: one column showing scale steps, the other the

generator mapping. Keep to only unison vectors (in the relevant space)

and it's all nice and simple.

That's the way I see it now. I've also realized that I've been saying

"octave invariant" when "octave equivalent" would probably be more to the

point.

Graham

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

> I was thinking about what n "unison vectors" in n-dimensional space

could

> mean. In fact, it's what I've already called a "basis". See

> <http://x31eq.com/matrix.htm>. I assume that's

analogous to

> Gene's "kernel". Let's revisit the example there:

>

> (t) ( 1 -2 1)

> (s) = ( 4 -1 -1)H

> (p) (-4 4 -1)

>

> Which can be used to define this scale

>

> C D E F G A B C

> t+p t s t+p t p+t s

How did you get this, without appealing to a parallelogram or some

such construction?

>

> The inverse of the matrix is

>

> ( 5 2 3)

> ( 8 3 5)

> (12 4 7)

>

> It defines H in terms of t, s and p. That I already knew. Now,

the

> interesting thing is, taking p out of the scale

>

> C D E F G A B C

> t t s t t t s

>

> is the same as making p a commatic unison vector.

By definition.

> And we can define the

> approximate H in terms of t and s

>

> ( 5 2)(t)

> H' = ( 8 3)(s)

> (12 4)

>

> I already knew this, but I didn't think of it as "designating a

commatic

> unison vector" or whatever. Before we've been thinking about

starting

> with commatic unison vectors and making them chromatic. It

actually makes

> more sense the other way around.

If you mean, start with the unison vectors as non-zero intervals and

then selectively bring some of them to zero, yes, that's how I think

of it in the lattice/PB regime.

>

> Making (4 -1 -1)H or 16:15 another commatic unison vector gives us

> 5-equal:

>

> ( 5)(t)

> H' = ( 8)

> (12)

>

> Gene's already explained this in algebraic terms, but I think the

example

> makes it clearer. It means I can now interpret

>

> ( 1 -2 1)

> ( 4 -1 -1)

> (-4 4 -1)

>

> as the octave-specific equivalent of a periodicity block.

Unfortunately

> it only has 1 note,

How so?

> so the analogy breaks down. But whatever, I'll call

> all the intervals unison vectors anyway.

_All_ the intervals?

> Now, we can also define meantone as

>

> ( 1 0 0) (1 0)

> ( 0 1 0)C = (0 1)

> (-4 4 -1) (0 0)

>

> Here, the octave and twelfth are taking the place of the chromatic

unison

> vectors. So, can we call them unison vectors?

A unison vector has to come out to a musical _unison_! An augmented

unison maybe, but still a unison.

> I think not, because C

> evaluates to

>

> ( 1 0)

> ( 0 1)

> (-4 4)

>

> That has negative numbers in it, which means the generators (for so

they

> are) don't add up to give all the primary consonances.

Don't follow.

> So now we have a

> definition for what interval qualifies as a "unison vector".

> Interestingly, a fifth would work as a generator here, but not for

a

> diatonic scale.

You've completely lost me.

>

> An alternative to "unison vector" would be "melodic generator".

Ouch! Are you sure? The steps in the scale can't be unison vectors,

nor can the generator of the scale be a unison vector.

>

> Usually we call the octave the period and a fifth (which is

therefore

> equivalent to the twelfth) the generator. It isn't much of a

stretch to

> think of two generators.

Gene has referred to that concept, I believe.

>

> In octave-equivalent terms, a unison is:

>

> v v v v v v v v v v v w

>

> where v is s fifth and w is a wolf. The same algebra applies, but

it's

> harder to visualise. I think the fifth here does count as a unison

> vector.

Please, please, please don't refer to a fifth as a unison vector.

Musical intervals go: unison, second, third, fourth, fifth. There's a

long way between a unison and a fifth.

> Defining an MOS in octave-equivalent terms is easy, once you

realize that

> the generators that pop out needn't be unison vectors in octave-

specific

> space.

They can't be unison vectors in any space! If the generator is a

unison vector, you never get beyond the first note of the scale and

its chromatic alterations.

>

> That's the way I see it now. I've also realized that I've been

saying

> "octave invariant" when "octave equivalent" would probably be more

to the

> point.

I suppose -- but you got me doing it too :)

In-Reply-To: <9lug7f+8e65@eGroups.com>

Paul wrote:

> > C D E F G A B C

> > t+p t s t+p t p+t s

>

> How did you get this, without appealing to a parallelogram or some

> such construction?

It's an arbitrary scale, not generated from anything more fundamental.

> > Making (4 -1 -1)H or 16:15 another commatic unison vector gives us

> > 5-equal:

> >

> > ( 5)(t)

> > H' = ( 8)

> > (12)

> >

> > Gene's already explained this in algebraic terms, but I think the

> example

> > makes it clearer. It means I can now interpret

> >

> > ( 1 -2 1)

> > ( 4 -1 -1)

> > (-4 4 -1)

> >

> > as the octave-specific equivalent of a periodicity block.

> Unfortunately

> > it only has 1 note,

>

> How so?

Everything's a unison!

> > so the analogy breaks down. But whatever, I'll call

> > all the intervals unison vectors anyway.

>

> _All_ the intervals?

Yep!

> > Now, we can also define meantone as

> >

> > ( 1 0 0) (1 0)

> > ( 0 1 0)C = (0 1)

> > (-4 4 -1) (0 0)

> >

> > Here, the octave and twelfth are taking the place of the chromatic

> unison

> > vectors. So, can we call them unison vectors?

>

> A unison vector has to come out to a musical _unison_! An augmented

> unison maybe, but still a unison.

Fokker says, in "Unison Vectors and Periodicity Blocks"

<http://www.xs4all.nl/~huygensf/doc/fokkerpb.html> "All pairs of notes

differing by octaves only are considered unisons." That's not far from

"an octave is a unison vector".

> > I think not, because C

> > evaluates to

> >

> > ( 1 0)

> > ( 0 1)

> > (-4 4)

> >

> > That has negative numbers in it, which means the generators (for so

> they

> > are) don't add up to give all the primary consonances.

>

> Don't follow.

You can't define the 5:4 approximation by adding octaves and twelfths.

This gives us a way of defining "small" intervals without needing interval

sizes. We need to construct a scale somehow or other, and say that scale

has to be constructed by adding together a set of linearly independent

small intervals. The only problem is that they can be smaller than a

unison. So adding a major and minor tone can be replaced by a major third

and a descending minor tone. So the next constraint is that the scale has

to have monotonically increasing pitch, which you need the metric for.

> > So now we have a

> > definition for what interval qualifies as a "unison vector".

> > Interestingly, a fifth would work as a generator here, but not for

> a

> > diatonic scale.

>

> You've completely lost me.

Four fifths add up to a 5:1 approximation. But you can't define a

diatonic scale by adding fifths and octaves.

> > An alternative to "unison vector" would be "melodic generator".

>

> Ouch! Are you sure? The steps in the scale can't be unison vectors,

> nor can the generator of the scale be a unison vector.

I don't know what to call them.

> > Usually we call the octave the period and a fifth (which is

> therefore

> > equivalent to the twelfth) the generator. It isn't much of a

> stretch to

> > think of two generators.

>

> Gene has referred to that concept, I believe.

Yes.

> > In octave-equivalent terms, a unison is:

> >

> > v v v v v v v v v v v w

> >

> > where v is s fifth and w is a wolf. The same algebra applies, but

> it's

> > harder to visualise. I think the fifth here does count as a unison

> > vector.

>

> Please, please, please don't refer to a fifth as a unison vector.

> Musical intervals go: unison, second, third, fourth, fifth. There's a

> long way between a unison and a fifth.

Probably it'd be better written as

v v v v v v v v v v v v p

where v is a fifth and p is a Pythagorean comma. That means p is the

(chromatic) unison vector. That leaves v as "the thing you have left when

you take out all the unison vectors". That's what we need a name for.

> > Defining an MOS in octave-equivalent terms is easy, once you

> realize that

> > the generators that pop out needn't be unison vectors in octave-

> specific

> > space.

>

> They can't be unison vectors in any space! If the generator is a

> unison vector, you never get beyond the first note of the scale and

> its chromatic alterations.

But the generator does have a close relationship to the chromatic unison

vector, in an octave equivalent system. You're taking out everything

except the chromatic unison vector, and describing what you have left in

terms of the generator.

Graham

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

> > > Here, the octave and twelfth are taking the place of the

chromatic

> > unison

> > > vectors. So, can we call them unison vectors?

> >

> > A unison vector has to come out to a musical _unison_! An

augmented

> > unison maybe, but still a unison.

>

> Fokker says, in "Unison Vectors and Periodicity Blocks"

> <http://www.xs4all.nl/~huygensf/doc/fokkerpb.html> "All pairs of

notes

> differing by octaves only are considered unisons." That's not far

from

> "an octave is a unison vector".

The octave, OK, but the twelfth?

>

> You can't define the 5:4 approximation by adding octaves and

twelfths.

Why not? We've been doing so all along. The 5:4 approximation is four

twelfths minus six octaves.

>

> Four fifths add up to a 5:1 approximation. But you can't define a

> diatonic scale by adding fifths and octaves.

Why not? We've been doing so all along, right?

> > > An alternative to "unison vector" would be "melodic generator".

> >

> > Ouch! Are you sure? The steps in the scale can't be unison

vectors,

> > nor can the generator of the scale be a unison vector.

>

> I don't know what to call them.

Something else, I beseech you!

>

> Probably it'd be better written as

>

> v v v v v v v v v v v v p

>

> where v is a fifth and p is a Pythagorean comma. That means p is

the

> (chromatic) unison vector. That leaves v as "the thing you have

left when

> you take out all the unison vectors". That's what we need a name

for.

The generator?

>

> > > Defining an MOS in octave-equivalent terms is easy, once you

> > realize that

> > > the generators that pop out needn't be unison vectors in octave-

> > specific

> > > space.

> >

> > They can't be unison vectors in any space! If the generator is a

> > unison vector, you never get beyond the first note of the scale

and

> > its chromatic alterations.

>

> But the generator does have a close relationship to the chromatic

unison

> vector, in an octave equivalent system. You're taking out

everything

> except the chromatic unison vector, and describing what you have

left in

> terms of the generator.

Great! That's a valuable insight! But that doesn't make the generator

a unison!

Look, the unison vectors together define N equivalence classes in the

infinite lattice. All that unison vectors can do is to move you

around within a single equivalence class. You'll never get to the

other N-1 equivalence classes using unison vectors.

In-Reply-To: <9m12f6+qhkq@eGroups.com>

Paul wrote:

> > You can't define the 5:4 approximation by adding octaves and

> twelfths.

>

> Why not? We've been doing so all along. The 5:4 approximation is four

> twelfths minus six octaves.

Do you remember, back when you were at school, a distinction being made

between "adding" and "subtracting"?

> > Probably it'd be better written as

> >

> > v v v v v v v v v v v v p

> >

> > where v is a fifth and p is a Pythagorean comma. That means p is

> the

> > (chromatic) unison vector. That leaves v as "the thing you have

> left when

> > you take out all the unison vectors". That's what we need a name

> for.

>

> The generator?

Yes, isn't that where I started? In octave-specific terms, when you take

out 25:24 and 81:80 you're left with a generator equivalent to 16:15. In

octave-invariant terms, when you take out 81:80 you're left with a

generator of a fifth. In octave-specific terms, when you take out only

81:80 you have a choice of generators. They could be 2:1 and 3:2 or 16:15

and 25:24. Taking 2:1 and 25:24 as a pair is different, because the

generators and unison vectors don't form a unitary matrix. So perhaps

that's why we call one an "interval of equivalence" and the other a

"chromatic unison vector".

Graham

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

> In-Reply-To: <9m12f6+qhkq@e...>

> Paul wrote:

>

> > > You can't define the 5:4 approximation by adding octaves and

> > twelfths.

> >

> > Why not? We've been doing so all along. The 5:4 approximation is

four

> > twelfths minus six octaves.

>

> Do you remember, back when you were at school, a distinction being

made

> between "adding" and "subtracting"?

Ha ha. I thought we weren't scared of negative numbers anymore.

>

> > > Probably it'd be better written as

> > >

> > > v v v v v v v v v v v v p

> > >

> > > where v is a fifth and p is a Pythagorean comma. That means p

is

> > the

> > > (chromatic) unison vector. That leaves v as "the thing you

have

> > left when

> > > you take out all the unison vectors". That's what we need a

name

> > for.

> >

> > The generator?

>

> Yes, isn't that where I started? In octave-specific terms, when

you take

> out 25:24 and 81:80 you're left with a generator equivalent to

16:15. In

> octave-invariant terms, when you take out 81:80 you're left with a

> generator of a fifth. In octave-specific terms, when you take out

only

> 81:80 you have a choice of generators. They could be 2:1 and 3:2

or 16:15

> and 25:24. Taking 2:1 and 25:24 as a pair is different, because

the

> generators and unison vectors don't form a unitary matrix. So

perhaps

> that's why we call one an "interval of equivalence" and the other a

> "chromatic unison vector".

You lost me. Gene, can you shed any light?