A few notes on notation and terminology

I've started writing a series of posts over on XA about how I've come
to understand regular temperament theory, deliberately geared to the
kind of person who already sort of knows what vals are, kind of, and
also what monzos are, sort of, and doesn't get what the "big picture"
is. A page in progress with all of this is on the wiki here:
http://xenharmonic.wikispaces.com/Mike%27s+Lectures+On+Regular+Temperament+Theory

While my goal with this was basically to innovate as little as
possible and just say things exactly as they're described, there were
a few things that I've consistently seen trip up EVERYONE who's new to
this stuff, to the point where I thought a slight alteration would
really help people understand much better. All of these things usually
have to do with contorsion and so on. In these few cases I took the
liberty of proposing some subtle extensions to bra-ket notation to
make it clear exactly what idea I'm talking about. These extensions
are:

<a b c| - val
|a b c> - monzo

(a b c| - projective val
|a b c) - projective monzo

{a b c| - tuning map

The one that's most important is the projective one. Get any
reasonably intelligent person who you've explained the bare basics to,
and then tell them the only 5-limit temperament tempering out 81/80
and 128/125 is <12 19 28|, and watch the look of confusion on their
faces. At this juncture, which is inevitable, I believe it's
suboptimal to attempt explanations of the nature that <24 38 56| is
"not a temperament," or that <24 38 56| tempers out (81/80)^2 but not
81/80 itself or what have you, or that <24 38 56| and <12 19 28| "are
the same thing," or that there's a difference between 24-EDO and
24-TET, etc. I've tried all of these things at different points and
I've found they're just great ways to confuse the hell out of anyone.

There's no need to ever get into that, because we all know exactly
what's going on. Rank-1 temperaments aren't vals, but are instead
these magical things that correspond to lines through the origin. We
can talk about <12 19 28| in a projective sense, but to suddenly
conflate that with the other sort of val without being very clear is,
I've found, a recipe for disaster. And sweeping this distinction under
the rug by defining the word "temperament" cleverly enough also is
something I think is a bad idea. In this case, the set of all vals
tempering out both 81/80 and 128/125 forms a line through the origin,
and is obviously not just the single val <12 19 28|, and any new
person who thinks enough about that will get it.

I think it's best to just be as explicit as possible about it from the
beginning. And a good way to be explicit is to just alter the notation
slightly so that it's always clear when you're talking about a
projective vector or a subspace being spanned by a vector.

To notate a projective vector or a line through the origin and a
vector, something like (12 29 28| fits the bill, with parentheses as a
logical choice since they're used in homogenous coordinates anyway and
look like a semicircle. The actual rank-1 temperament eliminating
81/80 and 128/125 is thus (12 29 28|, which is a fundamental
mathematical object that's independent from any one val. Then <12 19
28|, <24 38 56|, etc, are all vals supporting this temperament, just
like a val can support any temperament.

I also suggested {a b c| for tuning maps. This is because we just got
in a big discussion over whether or not tuning maps are in "the same
space" as vals and just interpreted differently, or if they're in "a
different space" as vals or whatever. Clearly it doesn't matter
because those two options are basically the same.

Whether or not you think tuning maps are in the same space as vals or
not, they definitely imply a clear conceptual difference in how one is
interpreting the musical relevance of the space of covectors. Since
they're different enough to talk about as being different in English,
there's no reason we can't just use shorthand for it and write {a b c|
for them. This is especially useful, I think, because I don't know
anyone who hasn't gotten tripped up at some point on the difference
between vals and tuning maps, thinking that there's some mystical
connection between them when there isn't, or confusing subtle aspects
of the two.

This also leaves open the possibility of there being something like |a
b c}, but I have no idea what that would be. If anyone has any ideas,
I'm all ears.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> Whether or not you think tuning maps are in the same space as vals or
> not, they definitely imply a clear conceptual difference in how one is
> interpreting the musical relevance of the space of covectors. Since
> they're different enough to talk about as being different in
> This also leaves open the possibility of there being something like |a
> b c}, but I have no idea what that would be. If anyone has any ideas,
> I'm all ears.

Since you are proposing {a b c| for what the Xenwiki calls an element of tuning space and you are calling a tuning map, clearly |a b c} should be an element of interval spacee. I wouldn't call an element of tuning space a "tuning map" unless it makes some sort of sense as one myself.

On Sat, Apr 28, 2012 at 2:56 PM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Whether or not you think tuning maps are in the same space as vals or
> > not, they definitely imply a clear conceptual difference in how one is
> > interpreting the musical relevance of the space of covectors. Since
> > they're different enough to talk about as being different in
> > This also leaves open the possibility of there being something like |a
> > b c}, but I have no idea what that would be. If anyone has any ideas,
> > I'm all ears.
>
> Since you are proposing {a b c| for what the Xenwiki calls an element of
> tuning space and you are calling a tuning map, clearly |a b c} should be an
> element of interval spacee. I wouldn't call an element of tuning space a
> "tuning map" unless it makes some sort of sense as one myself.

As you know, there was some disagreement over all of the different
ways to look at the same exact thing. Paul wanted tuning maps to be in
a separate vector space with units of "cents" or some such, along with
a canonical isomorphism from <a b c| in the first space to <a b c| in
the other space. You thought that was silly, and thought they should
just be two different ways to interpret the same space. I think we
left off that it was all the same.

All -I'm- saying is, regardless of how you want to model things, that
<a b c| is a val, and {a b c| is a tuning map. Since there are
obviously different mathematical ways to model the two, which are
trivially isomorphic to one another then the mathematical
interpretation of these elements will also change.

If you want to say that tuning maps lie in a separate vector space
with units, then that's what the {a b c| is denoting - that it's in
this space which has units of cents or millioctaves or whatever, as
per Paul's use. Then <a b c| refers to the elements in the normal dual
space to interval space.

If you want to say that tuning maps are just another interpretation of
the same elements in the same space, then {a b c| serves instead to
denote the intended interpretation explicitly, rather than leaving it
implicit, and assuming that a new student is going to be smart enough
to figure things out from context.

It seemed above like you were suggesting that {a b c| is just the
default notation for covectors. If that's how you want to do it, then
instead <a b c| becomes the "special" thing, this time denoting those
special covectors which lie on the lattice of vals which has been
embedded into the space. And then |a b c} can be the default for
vectors, and then |a b c> denotes the special vectors which lie on the
embedded lattice of monzos.

It's all the same. I'll leave the philosophizing out of it and just
suggest it's a useful notation for tuning maps vs vals.

For instance, you said you wouldn't call an element of tuning space a
tuning map unless it makes some sort of sense as one. If your paradigm
includes some notion of things "making sense" as tuning maps and using
the English-language phrase "tuning map" to describe a bra like <a b
c| when it makes sense, then I simply suggest that it may be useful to
shorten things like "prepare! this is a tuning map: <a b c|" to just
"{a b c|."

-Mike