Yahoo Tuning Groups Ultimate Backup tuning-math undefined

Gene wrote:

>> Yeah funnily enough even stochastic music with the right parameters
>> will sound like serial atonal music with just a casual listen.
>
> But completely atonal stochastic music with triadic harmony will not
> sound at all like what people think atonal music ought to sound like.

Which is why I said "with the right parameters". Point being that what the
surface of the music sounds like might not tell you about its method of
composition, or about its structural features.

>> > It sounds ugly either way, but shouldn't that be "class-set",
>> > not "set-class"? And what is a "set-class correspondence"?
>>
>> A pitch is, for example, {16}. It belongs to the pitch-class {4}.
>>
>> A pitch-set is, e.g. {3,6,7}. It belongs to the set-class [014]. [014]
>> is an equivalence class of pitch sets, so its a pitch-set equivalence
>> class, or set-class for short.
>
> This is getting totally out of hand. A pitch denoted by an integer. A
> pitch-class is denoted by an element of Z/12Z, as an integer reduced
> to the range 0-11 mod 12. A set-class, short for set-pitch-class, is
> a element of an equivalence class of sets of pitch-classes under the
> dihedreal group acting as a permutation group on Z/12Z. What next?

I'll tell you what's next: Klumpenhouwer networks, which describe
complexes of set-class relationships abstracted from any particular
set-class. There's a David Lewin tutorial in the Journal of Music Theory
called "Klumpenhouwer networks and some isographies that involve them" (or
"that they involve", I can't remember).

"Set-class" isn't short for "set-pitch-class". It's a class of sets of
pitches==>a class of pitch sets==>a "pitch-set class", if anything.
Alternatively you could think of it as a class of sets of pcs, i.e. a
class of pc-sets==>"pc-set class" or "pitch-class-set class". Sure it
sounds ridiculous if you use full names like that, but in general it's
pretty easy to say something like "such-and-such a sc has 3 pcs from one
wholetone collection and 2 pcs from the other"

I'm just reporting on what general accepted usage is. I don't much like
the name "pitch" either, really, but what else do you want to call it that
makes sense to musicians? Yes of course it's a mod 12 residue, but it's
easier to say "pc 4" (you're right, I shouldn't have written "pc {4}")
than "the mod 12 residue {... -20, -8, 4, 16, 30...}". It's a useful
musical concept--it's nice to have a simple name to refer to any or all
E's and Fb's in any or every octave--so it gets a simple name for the
benefit of musicians. Most musicians don't care that transposition and
inversion form a dihedral group acting as a permutation group on Z/12Z,
though presumably tuning-math readers do. Tuning-math writers, then, can
use the terminology that everyone who's *not* a tuning-math writer uses,
or they can invent their own. At the very least though it's good to know
what musicians call things so if you meet one you can explain your theory
to them.

You can't call pitches "notes" because of the ambiguity in the question
"what's the second note of Beethoven's 5th?" Is it the second G? that's
what most people would answer. But they'd also say the two G's are the
same note. How can the first note be the same as the second note? They're
two notes, but one pitch. You can't call them "tones" because that's
similarly ambiguous, plus it has the meaning "wholetone" in some contexts,
plus a non-technical meaning of "timbre". Maybe "chroma" would have been a
good name, but pc is it now, for better or for worse, as far as
Anglo-American theory goes.

Calling them "pitches" doesn't imply anything about the tuning. In this
sense of "pitches", a score contains the same pitches whether or not the
piece is performed at A-440 or A-445 or A-415.

--Jon

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
> > This is getting totally out of hand. A pitch denoted by an
integer. A
> > pitch-class is denoted by an element of Z/12Z, as an integer
reduced
> > to the range 0-11 mod 12. A set-class, short for set-pitch-class,
is
> > a element of an equivalence class of sets of pitch-classes under
the
> > dihedreal group acting as a permutation group on Z/12Z. What next?

> I'll tell you what's next: Klumpenhouwer networks, which describe
> complexes of set-class relationships abstracted from any particular
> set-class.

I've seen the buzzword when googling; is there something online which
gives the definition?

> "Set-class" isn't short for "set-pitch-class". It's a class of sets
of
> pitches==>a class of pitch sets==>a "pitch-set class", if
anything.

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

> Alternatively you could think of it as a class of sets of pcs, i.e.
a
> class of pc-sets==>"pc-set class" or "pitch-class-set class".

Now you've got a set of sets of equivalence classes of sets, right?
Is that a so-called "set-class"?

Sure it
> sounds ridiculous if you use full names like that, but in general
it's
> pretty easy to say something like "such-and-such a sc has 3 pcs
from one
> wholetone collection and 2 pcs from the other"

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. We seem to be conflating pitches with inegers
representing them, and equivalence classes mod 12 with mod 12
residues, sets of such classes with sets of residues, and then we try
to sort out what, exactly, we are defining when we look at orbits of
such sets of classes under a G-action, and take a representive of the
orbit. This has gotten to the point where you need to get
mathematical about it, and give definitions which you really mean.

> I'm just reporting on what general accepted usage is. I don't much
like
> the name "pitch" either, really, but what else do you want to call
it that
> makes sense to musicians? Yes of course it's a mod 12 residue, but
it's
> easier to say "pc 4" (you're right, I shouldn't have written "pc
{4}")
> than "the mod 12 residue {... -20, -8, 4, 16, 30...}".

{... -20, -8, 4, 16, 30...} isn't a residue mod 12, it is an
equivalence class mod 12; it can also be called an element of the
group Z/12Z, whose 12 elements are precisely these classes. The
corresponding residue mod 12 would be 4.

It's a useful
> musical concept--it's nice to have a simple name to refer to any or
all
> E's and Fb's in any or every octave--so it gets a simple name for
the
> benefit of musicians.

This is not simple, this is a crazy-quilt of needless complexity.

Most musicians don't care that transposition and
> inversion form a dihedral group acting as a permutation group on
Z/12Z,
> though presumably tuning-math readers do.

Most musicians probably don't give much of a hoot for any of this,
but what do music theorists call it, if not the usual?

Tuning-math writers, then, can
> use the terminology that everyone who's *not* a tuning-math writer
uses,
> or they can invent their own.

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.

> Calling them "pitches" doesn't imply anything about the tuning. In
this
> sense of "pitches", a score contains the same pitches whether or
not the
> piece is performed at A-440 or A-445 or A-415.

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?

undefined

🔗jon wild <wild@fas.harvard.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>