why use "vector"?

There has been a lot of recent discussion of this term "unison vector". Since learning about
this term only five years ago, I've never used it, because it seemed to me semantically bad.
I realize it has a historical context, but it is my opinion that the term should be left in its
historical context, and should not be used except when discussion Fokker's original idea.

With all due respect to Fokker, this thing is not a vector. On a list like "tuning math" I
would think that a term like this would simply not be used because it is mathematically
incorrect usage of the term "vector".

If you want to be taken seriously by mathematicians or by anyone who knows what a
vector is, I think you should find a term based on what this thing is mathematically, that
means what it is supposed to mean. For example, when you are talking about something
that is a convergent, then you call it a convergent.

In my own work, I needed a term to describe two pitches which are indistinguishable from
one another by human ears. Because this concerns perception, the term I use for this
comes not from mathematics, but from psychophysics. It is "metamer". You probably know
that the term "metamer" has been used for years in visual psychophisics to describe two
colors of light which appear the same to the human eye, when in fact each color is made
up of differing intensities of wavelength components. Nobody that I know of in
psychoacoustics has picked up on this term to describe pitches within the boundary of a
JND, but the term is useful because it means exactly what it is supposed to mean; that is
to say, someone who knows what a "conventional" metamer is will understand what this
other new metamer is - and isn't that the point? To make things clear?

I realize you are looking for a strictly mathematical term, and not a perceptual term. I
think you can find something appropriate.

Aaron Hunt

Hi Aaron,

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@...> wrote:
>
> There has been a lot of recent discussion of this term
> "unison vector". Since learning about this term only
> five years ago, I've never used it, because it seemed
> to me semantically bad. I realize it has a historical
> context, but it is my opinion that the term should be
> left in its historical context, and should not be used
> except when discussion Fokker's original idea.
>
> With all due respect to Fokker, this thing is not a vector.
> On a list like "tuning math" I would think that a term
> like this would simply not be used because it is
> mathematically incorrect usage of the term "vector".

I've heard this complaint before but still don't
understand why anyone says it. I can see why there
would be trouble with the "unison" part, and perhaps
"convergent" should replace that, but what's wrong
with the "vector" part?

A lattice diagram is an example of a vector space,
and so any interval which is an element of that lattice
can be described as a vector.

If one models 5-limit just-intonation as a 3-D lattice
whose axes represent prime-factors 2, 3, and 5, and
then discusses the syntonic-comma as a "unison-vector",
the syntonic-comma of ratio 81/80 appears on the lattice
as the 2,3,5-monzo (i.e., vector) [-4 4, -1> .

Anyway, i came up with the terms "promo" and "vapro"
a few years ago because of the arguments people had
like yours. But alas, no-one seems interested in using them.

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

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

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

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@...> wrote:

> With all due respect to Fokker, this thing is not a vector.

I've objected to this usage on those grounds. It's really an abelian
group (Z-module) element. I call those monzos, myself, when expressed
in vector notation. However, if you tensor with the rationals, you
embed these into a vector space. Joe Monzo likes those sorts of
things, and they are nice for various purposes, such as temperament
tunings.

> If you want to be taken seriously by mathematicians or by anyone who
knows what a
> vector is, I think you should find a term based on what this thing
is mathematically, that
> means what it is supposed to mean.

I don't think this is that big of a problem. I make snarky comments
from time to time, but after all physicists use the term "vector"
also, as Paul Erlich likes to point out, and Fokker comes from that
background.

The problem, such as it is, is this: to a mathematician, a "vector" is
an element of a vector space, and we haven't defined one. We are
representing elements of finite rank subgroups of Q*, the positive
rational numbers under multiplication. These are abelian groups, but
*not* vector spaces. You can embed them in a vector space, however. If
you embed into a vector over the rationals, you still have a unique
correspondence between vector space elements and positive real numbers
they represent. If you embed it into the reals, that unique
correspondence goes away.

On 2/7/06, gooseplex <cfaah@eiu.edu> wrote:
[snip]
> In my own work, I needed a term to describe two pitches which are indistinguishable from
> one another by human ears. Because this concerns perception, the term I use for this
> comes not from mathematics, but from psychophysics. It is "metamer". You probably know
> that the term "metamer" has been used for years in visual psychophisics to describe two
> colors of light which appear the same to the human eye, when in fact each color is made
> up of differing intensities of wavelength components. Nobody that I know of in
> psychoacoustics has picked up on this term to describe pitches within the boundary of a
> JND, but the term is useful because it means exactly what it is supposed to mean; that is
> to say, someone who knows what a "conventional" metamer is will understand what this
> other new metamer is - and isn't that the point? To make things clear?
>
> I realize you are looking for a strictly mathematical term, and not a perceptual term. I
> think you can find something appropriate.
>
> Aaron Hunt

In my opinion, "metamer" is a much worse name than "unison vector"
because the analogy to vision is not very good. The space of possible
optical spectra is infinite-dimensional, and a normal human eye
(humans being trichromats) projects it onto a three-dimensional
subspace. There is no such projection with hearing, so there is no
direct analogy to visual metamers.

Also, a unison vector does not have to be a small interval in JI. For
example, mavila (one of my favorite temperaments) has a unison vector
of 135/128. At 92 cents, this is an undeniably noticeable interval,
but in mavila it is tempered away to a unison.

Keenan

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

> I've heard this complaint before but still don't
> understand why anyone says it. I can see why there
> would be trouble with the "unison" part, and perhaps
> "convergent" should replace that, but what's wrong
> with the "vector" part?

Because you have to make it into a vector before you can call it one.
Or at least, that's how mathematicians think about these things.

> A lattice diagram is an example of a vector space,
> and so any interval which is an element of that lattice
> can be described as a vector.

Not exactly. If you embed the alleged vectors into the vector space
Rn, they now are honest vectors. Since Rn has a metric structure, you
now also have a lattice (discrete subgroup of Rn.) If you draw a
diagram of that, then you have a lattice diagram.

> Anyway, i came up with the terms "promo" and "vapro"
> a few years ago because of the arguments people had
> like yours. But alas, no-one seems interested in using them.

Those are for projective elements, I thought. Embedding into
projective space is a whole other story, and mathematicians would
certainly not be happy with calling those vectors.

--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@...> wrote:

>
> In my opinion, "metamer" is a much worse name than "unison vector"
> because the analogy to vision is not very good. The space of possible
> optical spectra is infinite-dimensional, and a normal human eye
> (humans being trichromats) projects it onto a three-dimensional
> subspace. There is no such projection with hearing, so there is no
> direct analogy to visual metamers.
>
> Also, a unison vector does not have to be a small interval in JI. For
> example, mavila (one of my favorite temperaments) has a unison vector
> of 135/128. At 92 cents, this is an undeniably noticeable interval,
> but in mavila it is tempered away to a unison.
>
> Keenan
>

If you are familiar with the plethora of models which exist to describe both vision and
hearing, you may realize that your comment above is quite easy to refute. Anyway, it is
irrelevant. The idea is simply this: that there is an effective perceptual stand-in
relationship between two things which are physically (or mathematically, or quantitaively)
different. This is all that needs to be taken in analogy between vision and hearing in order
to understand the concept of metamer as I am using it. It is not a mathematical construct,
either. If you'll reread my original post, I never suggested that "metamer" replace "unison
vector", as these concepts exist in two completely different domains. I only gave
"metamer" as my example of borrowed jargon which is actually meaningful. As it is not a
mathematical concept, so one should not expect mathematical rigor to be applied to it, as
perception cannot be absolutely quantified anyway.

Aaron Hunt

On 2/7/06, gooseplex <cfaah@eiu.edu> wrote:
[snip]
> If you are familiar with the plethora of models which exist to describe both vision and
> hearing, you may realize that your comment above is quite easy to refute.

Which comment?

> Anyway, it is
> irrelevant. The idea is simply this: that there is an effective perceptual stand-in
> relationship between two things which are physically (or mathematically, or quantitaively)
> different.

Temperament makes pitches separated by a unison vector physically the same.

> This is all that needs to be taken in analogy between vision and hearing in order
> to understand the concept of metamer as I am using it. It is not a mathematical construct,
> either. If you'll reread my original post, I never suggested that "metamer" replace "unison
> vector", as these concepts exist in two completely different domains. I only gave
> "metamer" as my example of borrowed jargon which is actually meaningful. As it is not a
> mathematical concept, so one should not expect mathematical rigor to be applied to it, as
> perception cannot be absolutely quantified anyway.

So, what do you suggest we call unison vectors?

> Aaron Hunt

Keenan

--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@...>
wrote:

> So, what do you suggest we call unison vectors?

"Commas" springs to mind.

Hi Gene, Aaron, and Keenan,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > I've heard this complaint before but still don't
> > understand why anyone says it. I can see why there
> > would be trouble with the "unison" part, and perhaps
> > "convergent" should replace that, but what's wrong
> > with the "vector" part?
>
> Because you have to make it into a vector before you can
> call it one. Or at least, that's how mathematicians think
> about these things.
>
> > A lattice diagram is an example of a vector space,
> > and so any interval which is an element of that lattice
> > can be described as a vector.
>
> Not exactly. If you embed the alleged vectors into the
> vector space Rn, they now are honest vectors. Since Rn
> has a metric structure, you now also have a lattice
> (discrete subgroup of Rn.) If you draw a diagram of that,
> then you have a lattice diagram.
>
> > Anyway, i came up with the terms "promo" and "vapro"
> > a few years ago because of the arguments people had
> > like yours. But alas, no-one seems interested in using them.
>
> Those are for projective elements, I thought.

Yes, that's correct.

> Embedding into projective space is a whole other story,
> and mathematicians would certainly not be happy with
> calling those vectors.

But it's my understanding that if a "unison-vector"
vanishes (i.e., is tempered out), then we are indeed
dealing with a projective space.

Anyway, i thought i'd toss my 2 cents in, but i can't
digest any of the follow-up, because this is all way over
my head.

I really enjoyed taking that algebra course in 2004,
and i ended it at the top of my class, but unfortunately
the duties of working on Tonescape prevented me from
continuing my math education, despite the fact that
i really wanted to.

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

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@>
> wrote:
>
> > So, what do you suggest we call unison vectors?
>
> "Commas" springs to mind.

Yuck.

I really wish you guys had left "comma" to refer only
to an interval in the ~20-30 cent range.

I put "anomaly" into the Encyclopedia way back in 1998,
hoping that people would use that for the generic term.
By my interpretation, "anomaly" can refer to anything
from just over zero cents to ~100 cents, or possibly
even more.

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

At 08:31 PM 2/7/2006, you wrote:
>--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
><genewardsmith@...> wrote:
>>
>> --- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@>
>> wrote:
>>
>> > So, what do you suggest we call unison vectors?
>>
>> "Commas" springs to mind.
>
>Yuck.
>
>I really wish you guys had left "comma" to refer only
>to an interval in the ~20-30 cent range.

I was just going to say, "comma" typically means any
small interval (can't it be less than 20 cents, though?).

It still might make sense to talk about 'the commas
of the temperament' or the 'commas in the kernel'. Even
though I don't have a problem with the "uv" term.

>I put "anomaly" into the Encyclopedia way back in 1998,
>hoping that people would use that for the generic term.
>By my interpretation, "anomaly" can refer to anything
>from just over zero cents to ~100 cents, or possibly
>even more.

I'm not sure I like "anomaly".

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> I was just going to say, "comma" typically means any
> small interval (can't it be less than 20 cents, though?).

Commas typically can be, or are being, tempered out. However, if you
want another term, try "kernel element".

> I'm not sure I like "anomaly".

Sounds anomalous to me.

AHD:

-l)
n. pl. a·nom·a·lies

1. Deviation or departure from the normal or common order, form, or
rule.
2. One that is peculiar, irregular, abnormal, or difficult to
classify: "Both men are anomalies: they have... likable personalities
but each has made his reputation as a heavy" (David Pauly).

At 12:30 AM 2/8/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
>> I was just going to say, "comma" typically means any
>> small interval (can't it be less than 20 cents, though?).
>
>Commas typically can be, or are being, tempered out.

What does "can be" mean? In many circles, the basic steps
of the scale are referred to as commas!

>However, if you want another term, try "kernel element".

Perfect!

-Carl

monz wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@...> wrote:
> >>--- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@>
>>wrote:
>>
>>
>>>So, what do you suggest we call unison vectors?
>>
>>"Commas" springs to mind.
> > > > Yuck.
> > I really wish you guys had left "comma" to refer only > to an interval in the ~20-30 cent range.
> > I put "anomaly" into the Encyclopedia way back in 1998,
> hoping that people would use that for the generic term.
> By my interpretation, "anomaly" can refer to anything
> from just over zero cents to ~100 cents, or possibly
> even more.

I don't know about "anomaly"; it suggests something out of place or irregular. I agree that the ambiguity of "comma" as it's been used is unfortunate, and think it might have been better to use it for intervals of the size of the syntonic or Pythagorean commas. You could say that a periodicity block has "anomalies" at the edges, but the edges can be arbitrarily moved, and if you're tempering it out in a regular temperament, there isn't any single point in the chain that's anomalous.

Personally I don't have a problem with "unison vector". Different specialized fields (computer science, biology, etc.) have different meanings of "vector" that don't always agree with the usage of mathematicians. (Programmers even have to deal with different meanings of the word "vector" in computer terminology!) Context should make it clear. But how about "univec" or "UV" as a shorthand? In any case where there's a need to define this term with mathematical precision (which I'll leave to the mathematicians), it can be defined in a way that doesn't refer to "vectors".

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@...> wrote:
>
> On a list like "tuning math" I
> would think that a term like this would simply not be used because it
is mathematically
> incorrect usage of the term "vector".

Why do you say that?

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@>
wrote:
>
> >
> > In my opinion, "metamer" is a much worse name than "unison vector"
> > because the analogy to vision is not very good. The space of
possible
> > optical spectra is infinite-dimensional, and a normal human eye
> > (humans being trichromats) projects it onto a three-dimensional
> > subspace. There is no such projection with hearing, so there is no
> > direct analogy to visual metamers.
> >
> > Also, a unison vector does not have to be a small interval in JI.
For
> > example, mavila (one of my favorite temperaments) has a unison
vector
> > of 135/128. At 92 cents, this is an undeniably noticeable
interval,
> > but in mavila it is tempered away to a unison.
> >
> > Keenan
> >
>
>
> If you are familiar with the plethora of models which exist to
describe both vision and
> hearing, you may realize that your comment above is quite easy to
refute. Anyway, it is
> irrelevant.

I'd like to see this refutation, however irrelevant.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

> Anyway, i came up with the terms "promo" and "vapro"
> a few years ago because of the arguments people had
> like yours.

Really? I thought the reasons were completely different; in fact, my
recollection is that we came up with these terms together when
discussing temperament theory on Messenger.

> But alas, no-one seems interested in using them.
>
> http://tonalsoft.com/enc/p/promo.aspx
>
> http://tonalsoft.com/enc/v/vapro.aspx

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@...>
wrote:

> Really? I thought the reasons were completely different; in fact, my
> recollection is that we came up with these terms together when
> discussing temperament theory on Messenger.

What's Messenger, and are there any good discussions there?

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

> > Embedding into projective space is a whole other story,
> > and mathematicians would certainly not be happy with
> > calling those vectors.
>
>
> But it's my understanding that if a "unison-vector"
> vanishes (i.e., is tempered out), then we are indeed
> dealing with a projective space.

No, it's just that for many situations in which tempering is going
on, the projective space is a more convenient picture for
understanding the various calculations we do than the original, non-
projective space. But in general, it's best to always keep all four
spaces in mind -- tone-space, tuning-space, and the projective
versions of both.

> I really enjoyed taking that algebra course in 2004,
> and i ended it at the top of my class,

Congratulations!

> but unfortunately
> the duties of working on Tonescape prevented me from
> continuing my math education, despite the fact that
> i really wanted to.

Hope you can get back to it someday!

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Keenan Pepper <keenanpepper@>
> > wrote:
> >
> > > So, what do you suggest we call unison vectors?
> >
> > "Commas" springs to mind.
>
>
> Yuck.
>
> I really wish you guys had left "comma" to refer only
> to an interval in the ~20-30 cent range.

Who is "you guys?"

> I put "anomaly" into the Encyclopedia way back in 1998,
> hoping that people would use that for the generic term.
> By my interpretation, "anomaly" can refer to anything
> from just over zero cents to ~100 cents, or possibly
> even more.

If a unison vector is tempered out, it's 0 cents by definition.

Really, our "vectors" are a whole heck of a lot closer to being,
mathematically, vectors, than what the music theory world out there
calls "vectors". The "interval-class vector" you see discussed in
music theory textbooks and journals is, for the diatonic scale for
example, [2 5 4 3 6 1]. It counts the number of times each 12-equal
interval class (in order of size) occurs in the diatonic scale. Now,
is this a vector? Can we, in any sense, "transform" this vector so
that we could see what it looks like in a different "basis"? For true
vectors, you can; here, this doesn't even seem to make sense.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@> wrote:
>
> > With all due respect to Fokker, this thing is not a vector.
>
> I've objected to this usage on those grounds. It's really an abelian
> group (Z-module) element. I call those monzos, myself, when
expressed
> in vector notation. However, if you tensor with the rationals, you
> embed these into a vector space. Joe Monzo likes those sorts of
> things, and they are nice for various purposes, such as temperament
> tunings.
>
> > If you want to be taken seriously by mathematicians or by anyone
who
> knows what a
> > vector is, I think you should find a term based on what this thing
> is mathematically, that
> > means what it is supposed to mean.
>
> I don't think this is that big of a problem. I make snarky comments
> from time to time, but after all physicists use the term "vector"
> also, as Paul Erlich likes to point out, and Fokker comes from that
> background.
>
> The problem, such as it is, is this: to a mathematician, a "vector"
is
> an element of a vector space, and we haven't defined one. We are
> representing elements of finite rank subgroups of Q*, the positive
> rational numbers under multiplication. These are abelian groups, but
> *not* vector spaces. You can embed them in a vector space, however.
If
> you embed into a vector over the rationals, you still have a unique
> correspondence between vector space elements and positive real
numbers
> they represent. If you embed it into the reals, that unique
> correspondence goes away.
>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@>
> wrote:
>
> > Really? I thought the reasons were completely different; in fact,
my
> > recollection is that we came up with these terms together when
> > discussing temperament theory on Messenger.
>
> What's Messenger, and are there any good discussions there?

It was either MSN Messenger or Yahoo Messenger, and I used it to teach
Monz a whole lot of stuff when I took a hiatus.