Generators and unison vectors

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?