undefined

Gene writes:

> This is why I'm complaining. If "pitch" is an integer, a "pitch-
> class" a mod 12 equivalence class of integerss, then a set of such
> classes is a set of pitch classes, and so should be a set-pitch-class.

No, it should be a pitch-class set. A room for the debriefing of agents is
an agent-debriefing room, not a room-debriefing-agent. A scheme for the
identification of widgets is a widget-identification scheme, not a
scheme-widget-identification A contingency plan for aviation disasters is
an aviation-disaster contingency plan, not a
contingency-plan-aviation-disaster. A list of rodent species is a
rodent-species list, not a list-rodent-species. Etc.! Hyphenate the two
nouns that make a compound modifier.

So a set of pitch classes is definitely a pitch-class set. Can we move on?

> Isn't that what you are calling by the hideous name "set-class"?

No, a set class is a class of sets; specifically it's an equivalence class
of pc sets (sets of pcs). Is this really so confusing?

> It would be even better with less confusing name behind the acronyms,
> but a good beginning would be clear definitions, which I'm not sure
> we have here.

See "Basic Atonal Theory" by John Rahn, or "Generalised Musical Intervals
and Transformations" by David Lewin, or "Composition with Pitch-classes"
by Robert Morris. I'm perhaps not doing a good job of making any of this
utterly clear--I wasn't trying very hard to provide total watertightness,
which you can get from any of those books.

> We seem to be conflating pitches with inegers
> representing them

No, we're modelling pitches with integers.

> {... -20, -8, 4, 16, 30...} isn't a residue mod 12, it is an
> equivalence class mod 12;

sorry, I always screw that up. I think of the residue "4" as representing
the equivalence class, I know it's not quite right.

> Are you sure music theorists don't know we are talking about the
> dihedral group of degree 12? There's a certain amount of group
> awareness percolating through academia--e.g., Lewin.

Lewin and many others certainly know it! Do they carry music theory
journals at your library? Or the three books I mentioned above? Rahn's is
for beginners but is solid, Lewin's has some quite advanced stuff in the
appendices. But the most advanced treatment I know of, which I'm sure
you'd appreciate a lot, is Guerino Mazzola's new book "The Topos of
Music", which recasts everything in category theory. Don't ask me to
explain it, it's way over my head. But it's a beautiful book. Or have a
look at some of the papers at the MaMuX website, which is part of IRCAM in
Paris. Or some of the abstracts at the upcoming special session of
January's national AMS meeting on music and mathematics.

By the way, I don't know if anyone else here knew Lewin, but you might not
have heard the very sad news that he died in May at the age of 69. He was
a really lovely guy and certainly the giant of music theory in the second
half of the 20th-century.

> Is the definition of "pitch" an integer associated to standard
> musical notation in such a way that C corresponds to 0, sharp to the
> addition of 1, and flat to the subtraction of 1? If so, at some point
> it ought to be said; if not, whatever else it means ought to be given.
> I presume someone actually has done this?

Yes. It's the first chapter in Rahn's book "Basic Atonal Theory". The
chapter is called "The Integer Model of Pitch". Sorry, I thought you knew
about that.

Best --Jon

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
> Gene writes:

> No, a set class is a class of sets; specifically it's an
equivalence class
> of pc sets (sets of pcs). Is this really so confusing?

Mathematicians are always using sets of sets of sets, etc, but no one
calls them set-set-sets--how would you tell one sort of set of sets
of sets of sets from another? The definitions involved here are way,
way easier than trying to define schemes or vertex operator algebras
or suchlike complicated things, but the nomenclature is a mess. If
you tried to define rings in terms of sets of sets, its spectrum in
terms of sets of sets of sets of sets, its locally ringed space in
terms of sets of sets of sets of sets of sets, and schemes in terms
of sets of sets of sets of sets of sets of sets, and called schemes
set-set-set-set-set-sets you would drive people even more nuts than
they already are trying to understand it all now.

> > {... -20, -8, 4, 16, 30...} isn't a residue mod 12, it is an
> > equivalence class mod 12;
>
> sorry, I always screw that up. I think of the residue "4" as
representing
> the equivalence class, I know it's not quite right.

"4" is a class representive and can be and often is used to represent
the equivalence class in question.

> > Are you sure music theorists don't know we are talking about the
> > dihedral group of degree 12? There's a certain amount of group
> > awareness percolating through academia--e.g., Lewin.
>
> Lewin and many others certainly know it! Do they carry music theory
> journals at your library? Or the three books I mentioned above?
Rahn's is
> for beginners but is solid, Lewin's has some quite advanced stuff
in the
> appendices. But the most advanced treatment I know of, which I'm
sure
> you'd appreciate a lot, is Guerino Mazzola's new book "The Topos of
> Music", which recasts everything in category theory.

Years ago I read something by someone trying to apply category theory
to music. I felt like taking a gun and shooting him, because the
categories he was so pround of were abelian groups--not the
*category* of abelian groups, the groups themselves. I hope Mazzola
does better. I've looked at Lewin's book, and the man is sound, but
he is using groups, not categories, which makes a great deal of sense.

> By the way, I don't know if anyone else here knew Lewin, but you
might not
> have heard the very sad news that he died in May at the age of 69.
He was
> a really lovely guy and certainly the giant of music theory in the
second
> half of the 20th-century.

Sorry to hear that. He made some definitions that music theory was in
sore need of.

> > Is the definition of "pitch" an integer associated to standard
> > musical notation in such a way that C corresponds to 0, sharp to
the
> > addition of 1, and flat to the subtraction of 1? If so, at some
point
> > it ought to be said; if not, whatever else it means ought to be
given.
> > I presume someone actually has done this?
>
> Yes. It's the first chapter in Rahn's book "Basic Atonal Theory".
The
> chapter is called "The Integer Model of Pitch". Sorry, I thought
you knew
> about that.

Why would I know a definition in a book I havn't read? Defining
things by means of a map from the integers to a system of notation is
not the most obvious approach, at least to me. "Integer model of
pitch", by the way, doesn't exactly sound like this is what Rahn does.