Unison Vectors

I notice the previous discussion on the meaning of "unison vector" has dwindled out for the second time. We still don't agree on the meaning, or on a term to use when "unison vector" is too constricting. This isn't good enough.

Unison vectors are a fundamental concept to the whole temperaments-from-commas approach we looked at a few years ago. They lie behind some of the more advanced concepts. They're also directly related to musical objects and so should be the first concepts we use to describe algorithms where they could be used.

I don't particularly care about the corner cases. If you want to say that some weird interval isn't really a unison vector then you can do that. But if it works in a situation where a unison vector could also do the job you should still say that it's working like a unison vector.

For clarity, the best thing is only to choose sensible unison vectors for examples. Then you can mention that silly vectors can be used instead as an aside or a footnote.

There are four different things I'd like to be able to talk about and be understood. With my suggested terms they are:

1) Periodicity blocks

2) Unison vectors

3) Chromatic/commatic unison vectors

4) Equivalence vectors

So, to take them in turn

Periodicity Blocks
------------------

I don't know if anybody's noticed a problem about this but I don't expect we all agree on the meaning. Questions are:

a) Is an equal temperament a periodicity block?

b) Is an MOS of a linear temperament a periodicity block?

c) Does the period have to be the equivalence interval?

d) What about torsion?

e) Do we need an equivalence interval at all?

For this message I'll assume yes to (a) and (b) and no to (c). If anybody disagrees we need a new term.

I'll also note that I think Fokker only used the term for three dimensions.

Unison Vectors
--------------

The definition of unison vectors should closely follow that of periodicity blocks. A unison vector's the interval between the same note in different periodicity blocks that fill harmonic space without overlapping. Or it's a unison in the equal temperament you get by making all intervals in the periodicity block equal.

I don't believe that unison vectors have to be small intervals, although you can usually assume they will be. If you want to talk about small intervals use the term "comma" which does carry this association.

If a vector matches a period rather than a unison then I'm happy not to call it a unison vector. I don't think we need a new term. Mention in a footnote that the specific algorithm also works with such vectors, but explain it in terms of unison vectors.

Chromatic/Commatic Unison Vectors
---------------------------------

The basic concept is that you start with an equal temperament, and the scale becomes progressively lumpy until you get a JI periodicity block. The chromatic-unison vectors are the ones that make it lumpy. A chromatic-unison vector isn't actually tuned as a unison.

Somebody said in the previous discussion that a JI periodicity block is defined by all commatic unison vectors. That doesn't make sense. Equal temperaments are all commatic, JI is all chromatic.

I still don't like the term "chromatic-unison vector" and I'm not convinced that "chromatic unison" is an existing musical term. But I'd rather use a bad term that we agree on than keep arguing about it.

With higher rank temperaments it is possible to cast aside the periodicity block and talk only about commatic unison vectors. In practice it may be worth leaving the chromatic-unison vectors in so that we can talk about a concrete scale. It's also still the case that all the algorithms I can implement to find a temperament from unison vectors do require the chromatic-unison vectors, or something like them, to be there.

Equivalence Vectors
-------------------

Usually the octave is our equivalence vector. Fokker used octave-equivalent vectors so he didn't need to talk about the octave itself. Because our vectors are octave-specific we need this extra concept. I suggest "equivalence vector".

The equivalence vector works pretty much like a unison vector, and it's worth drawing that analogy. But actually calling it a unison vector seems to cause great distress so let's avoid that.

We may also want a term for things like unison vectors that approximate equivalence intervals rather than unisons. But it would probably be easier to avoid confusion by not talking about such things.

So those are my positions, and if anybody disagrees can we please sort it out?

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Periodicity Blocks
> ------------------
> c) Does the period have to be the equivalence interval?

I don't see how this is a question about periodicity blocks at all.
Rather, it seems to be a question about R2 tunings. Certainly, if you
start with any periodicity block and temper out all but one of the
unison vectors, you have to be prepared for the possibility that the
period might be a fraction of the equivalence interval.

> d) What about torsion?

I prefer to say "torsional block" when the defining unison vectors
lead to torsion (a multiple of the number of notes that would would
get if you tempered all the unison vectors out and obtained an ET.

> e) Do we need an equivalence interval at all?

Yes, although an alternative concept of periodicity block might deal
with scales that go infinitely high and low in pitch and would not
need an equivalence interval.

> For this message I'll assume yes to (a) and (b) and no to (c). If
> anybody disagrees we need a new term.
>
> I'll also note that I think Fokker only used the term for three
>dimensions.

What the ??? He wrote one paper on them in the 5-limit, and one paper
on them in the 7-limit -- these can't both be three-dimensional!

> Unison Vectors
> --------------
>
> The definition of unison vectors should closely follow that of
> periodicity blocks. A unison vector's the interval between the
same
> note in different periodicity blocks that fill harmonic space
without
> overlapping. Or it's a unison in the equal temperament you get by
> making all intervals in the periodicity block equal.
>
> I don't believe that unison vectors have to be small intervals,
although
> you can usually assume they will be. If you want to talk about
small
> intervals use the term "comma" which does carry this association.

So does "chroma" in some cases.

> If a vector matches a period

So we're talking about 2D tunings here?

> rather than a unison

Will you admit no other possibility?

> then I'm happy not to
> call it a unison vector. I don't think we need a new term.
>Mention in
> a footnote that the specific algorithm also works with such
>vectors, but
> explain it in terms of unison vectors.

In order to maximize pain and confusion?

> Chromatic/Commatic Unison Vectors
> ---------------------------------
>
> The basic concept is that you start with an equal temperament, and
the
> scale becomes progressively lumpy until you get a JI periodicity
block.

:) It just becomes lumpy on its own?

> The chromatic-unison vectors are the ones that make it lumpy. A
> chromatic-unison vector isn't actually tuned as a unison.

Also, it's distinguished in the notation.

> With higher rank temperaments it is possible to cast aside the
> periodicity block and talk only about commatic unison vectors. In
> practice it may be worth leaving the chromatic-unison vectors in so
that
> we can talk about a concrete scale.

Sometimes that is what we want to do.

> We may also want a term for things like unison vectors that
approximate
> equivalence intervals rather than unisons. But it would probably
be
> easier to avoid confusion by not talking about such things.

What's the big deal? Call them vectors, don't call them unisons, and
everything will be fine.

wallyesterpaulrus wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> > >>Periodicity Blocks
>>------------------
>>c) Does the period have to be the equivalence interval?
> > > I don't see how this is a question about periodicity blocks at all. > Rather, it seems to be a question about R2 tunings. Certainly, if you > start with any periodicity block and temper out all but one of the > unison vectors, you have to be prepared for the possibility that the > period might be a fraction of the equivalence interval. It depends if periodicity blocks have to be JI or not. I said I was assuming they didn't.

>>d) What about torsion?
> > I prefer to say "torsional block" when the defining unison vectors > lead to torsion (a multiple of the number of notes that would would > get if you tempered all the unison vectors out and obtained an ET.

Okay.

>>e) Do we need an equivalence interval at all?
> > Yes, although an alternative concept of periodicity block might deal > with scales that go infinitely high and low in pitch and would not > need an equivalence interval.

Right.

>>For this message I'll assume yes to (a) and (b) and no to (c). If >>anybody disagrees we need a new term.
>>
>>I'll also note that I think Fokker only used the term for three >>dimensions.
> > What the ??? He wrote one paper on them in the 5-limit, and one paper > on them in the 7-limit -- these can't both be three-dimensional!

I don't have any papers on 5-limit periodicities. But I do have a 7-limit paper which says "In a previous paper I dealt with the periodicity meshes, formed by unison vectors in the two-dimensional plane lattice of perfect fifths and major thirds." So I take it the 5-limit things are "periodicity meshes" and the 7-limit things "periodicity blocks".

>>I don't believe that unison vectors have to be small intervals, > although >>you can usually assume they will be. If you want to talk about > small >>intervals use the term "comma" which does carry this association.
> > So does "chroma" in some cases.

Does it? I don't know what the usual meaning of "chroma" is.

>>If a vector matches a period
> > So we're talking about 2D tunings here?

More than 1D.

>>rather than a unison
> > Will you admit no other possibility?

It could be any number of periods. If not, it isn't like a unison vector.

>>then I'm happy not to >>call it a unison vector. I don't think we need a new term. >>Mention in >>a footnote that the specific algorithm also works with such >>vectors, but >>explain it in terms of unison vectors.
> > In order to maximize pain and confusion?

In comparison to what other way of doing it?

>> The chromatic-unison vectors are the ones that make it lumpy. A >>chromatic-unison vector isn't actually tuned as a unison.
> > Also, it's distinguished in the notation.

Yes.

>>With higher rank temperaments it is possible to cast aside the >>periodicity block and talk only about commatic unison vectors. In >>practice it may be worth leaving the chromatic-unison vectors in so > that >>we can talk about a concrete scale.
> > Sometimes that is what we want to do.

Both? Yes.

>>We may also want a term for things like unison vectors that > approximate >>equivalence intervals rather than unisons. But it would probably > be >>easier to avoid confusion by not talking about such things.
> > What's the big deal? Call them vectors, don't call them unisons, and > everything will be fine.

Because a vector is the line between any two points on the lattice. It could be any interval. There's nothing to say it's treated as a unison.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> wallyesterpaulrus wrote:
> > --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> >
> >>Periodicity Blocks
> >>------------------
> >>c) Does the period have to be the equivalence interval?
> >
> >
> > I don't see how this is a question about periodicity blocks at all.
> > Rather, it seems to be a question about R2 tunings. Certainly, if you
> > start with any periodicity block and temper out all but one of the
> > unison vectors, you have to be prepared for the possibility that the
> > period might be a fraction of the equivalence interval.
>
> It depends if periodicity blocks have to be JI or not. I said I was
> assuming they didn't.
>
> >>d) What about torsion?
> >
> > I prefer to say "torsional block" when the defining unison vectors
> > lead to torsion (a multiple of the number of notes that would would
> > get if you tempered all the unison vectors out and obtained an ET.
>
> Okay.
>
> >>e) Do we need an equivalence interval at all?
> >
> > Yes, although an alternative concept of periodicity block might deal
> > with scales that go infinitely high and low in pitch and would not
> > need an equivalence interval.
>
> Right.
>
> >>For this message I'll assume yes to (a) and (b) and no to (c). If
> >>anybody disagrees we need a new term.
> >>
> >>I'll also note that I think Fokker only used the term for three
> >>dimensions.
> >
> > What the ??? He wrote one paper on them in the 5-limit, and one paper
> > on them in the 7-limit -- these can't both be three-dimensional!
>
> I don't have any papers on 5-limit periodicities. But I do have a
> 7-limit paper which says "In a previous paper I dealt with the
> periodicity meshes, formed by unison vectors in the two-dimensional
> plane lattice of perfect fifths and major thirds." So I take it the
> 5-limit things are "periodicity meshes" and the 7-limit things
> "periodicity blocks".

Don't take it that way. The "mesh" formed by unison vectors extends over the whole lattice, and divides it up into congruent periodicity blocks. This is true in 2D and in 3D as well.

> >>I don't believe that unison vectors have to be small intervals,
> > although
> >>you can usually assume they will be. If you want to talk about
> > small
> >>intervals use the term "comma" which does carry this association.
> >
> > So does "chroma" in some cases.
>
> Does it? I don't know what the usual meaning of "chroma" is.

You'll see both "major chroma" and "minor chroma" in lists of JI intervals. Wanna guess their ratios?

> >>If a vector matches a period
> >
> > So we're talking about 2D tunings here?
>
> More than 1D.
>
> >>rather than a unison
> >
> > Will you admit no other possibility?
>
> It could be any number of periods. If not, it isn't like a unison vector.
> >>then I'm happy not to
> >>call it a unison vector. I don't think we need a new term.
> >>Mention in
> >>a footnote that the specific algorithm also works with such
> >>vectors, but
> >>explain it in terms of unison vectors.
> >
> > In order to maximize pain and confusion?
>
> In comparison to what other way of doing it?

Understanding what the calculations mean in terms of the entities you're plugging into them.

> >> The chromatic-unison vectors are the ones that make it lumpy. A
> >>chromatic-unison vector isn't actually tuned as a unison.
> >
> > Also, it's distinguished in the notation.
>
> Yes.
>
> >>With higher rank temperaments it is possible to cast aside the
> >>periodicity block and talk only about commatic unison vectors. In
> >>practice it may be worth leaving the chromatic-unison vectors in so
> > that
> >>we can talk about a concrete scale.
> >
> > Sometimes that is what we want to do.
>
> Both? Yes.
>
> >>We may also want a term for things like unison vectors that
> > approximate
> >>equivalence intervals rather than unisons. But it would probably
> > be
> >>easier to avoid confusion by not talking about such things.
> >
> > What's the big deal? Call them vectors, don't call them unisons, and
> > everything will be fine.
>
> Because a vector is the line between any two points on the lattice. It
> could be any interval. There's nothing to say it's treated as a unison.

If it's a period or a number of periods, then treating it as a unison makes no sense. Maybe that's what the math looks like on the face of it, but there has to be a way of meaninfully interpreting the calculations in their own right for what they really mean.

wallyesterpaulrus wrote:
> --- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> >>wallyesterpaulrus wrote:
>>
>>>--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@> wrote:
>>>

>>>>I'll also note that I think Fokker only used the term for three >>>>dimensions.
>>>
>>>What the ??? He wrote one paper on them in the 5-limit, and one paper >>>on them in the 7-limit -- these can't both be three-dimensional!
>>
>>I don't have any papers on 5-limit periodicities. But I do have a >>7-limit paper which says "In a previous paper I dealt with the >>periodicity meshes, formed by unison vectors in the two-dimensional >>plane lattice of perfect fifths and major thirds." So I take it the >>5-limit things are "periodicity meshes" and the 7-limit things >>"periodicity blocks".
> > Don't take it that way. The "mesh" formed by unison vectors extends over the whole lattice, and divides it up into congruent periodicity blocks. This is true in 2D and in 3D as well.

Maybe, but there are still two instances in that paper where it would have been natuaral to talk about 2-D periodicity blocks, but he doesn't. He also doesn't use the term "base de p�riodicit�" for two dimensions in the other online paper.

>>>>I don't believe that unison vectors have to be small intervals, >>>although >>>>you can usually assume they will be. If you want to talk about >>>small >>>>intervals use the term "comma" which does carry this association.
>>>
>>>So does "chroma" in some cases.
>>
>>Does it? I don't know what the usual meaning of "chroma" is.
> > You'll see both "major chroma" and "minor chroma" in lists of JI intervals. Wanna guess their ratios?

Dunno, but I take it "chroma" is an abbreviation of "chromatic semitone".

Incidentally, in that online English paper, Fokker uses the word "comma" for steps of the 53 note scale, so using "comma" as a replacement for "unison vector" directly contradicts Fokker's usage.

>>>>If a vector matches a period
<snip>
>>>>rather than a unison
<snip>
>>>>then I'm happy not to >>>>call it a unison vector. I don't think we need a new term. >>>>Mention in >>>>a footnote that the specific algorithm also works with such >>>>vectors, but >>>>explain it in terms of unison vectors.
>>>
>>>In order to maximize pain and confusion?
>>
>>In comparison to what other way of doing it?
> > Understanding what the calculations mean in terms of the entities you're plugging into them.

How am I not doing that? How could you do it better?

>>>>We may also want a term for things like unison vectors that >>>approximate >>>>equivalence intervals rather than unisons. But it would probably >>>be >>>>easier to avoid confusion by not talking about such things.
>>>
>>>What's the big deal? Call them vectors, don't call them unisons, and >>>everything will be fine.
>>
>>Because a vector is the line between any two points on the lattice. It >>could be any interval. There's nothing to say it's treated as a unison.
> > If it's a period or a number of periods, then treating it as a unison makes no sense. Maybe that's what the math looks like on the face of it, but there has to be a way of meaninfully interpreting the calculations in their own right for what they really mean.

Yes, so they're not unison vectors, like I said. You're losing the focus. The question is, do we need a name for them?

Graham

Hi Graham,

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Incidentally, in that online English paper, Fokker uses
> the word "comma" for steps of the 53 note scale, so using
> "comma" as a replacement for "unison vector" directly
> contradicts Fokker's usage.

I'm so glad you noticed that and wrote about it here.
I've hated the use of "comma" as a replacement for
"unison-vector" ever since folks around here started
using it as such -- and this is a good example of why.

A 53-edo step is indeed of a size which i think should
typically be called a "comma", but a unison-vector can
cover anything from ~120 cents all the way down to
near zero cents: it's much more general.

> Yes, so they're not unison vectors, like I said.
> You're losing the focus. The question is, do we
> need a name for them?

I came up with "promo" and "vapro" as terms related
to, or possibly instead of, "unison-vector".

http://tonalsoft.com/enc/p/promo.aspx

http://tonalsoft.com/enc/v/vapro.aspx

-monz
http://tonalsoft.com
Tonescape microtonal music software