Interval Vectors - The "Set theory" kind...

First, I should say that what some musicians call "Set theory", is
not really "Set Theory" as described by mathematicians, that is a
field in itself. The problem is, it's hard to know what else to call
it.

One can analyze all the subsets of a given ET, such as 12-tet, and
look at the "interval vectors" which are derived from them. For
example, take (0,1,2,3,4,5) which is a hexachord subset of 12-tet.
It has an "interval vector" of <5,4,3,2,1,0> if one counts all the
intervals in the set: 5 1-steps (or semitones), 4 2-steps (wholetones)
3 3-steps (minor thirds) 2 4-steps (major thirds) 1 5-steps (perfect
fourths and 0 6-steps (tritones). Going higher, you just mirror image
the vector (so it is not neccessary to give 7- through 11-steps)

Well, it turns out that subsets that are the mirror-image of a set
have the same interval vector. Also, sets related by the "Z-relation"
have the same interval vector (This is when two or more sets of
different Tn/TnI type have the same interval vector). I have been
analyzing subsets in various ETs to see if there is a pattern to
the unique interval-vector counts they have. This may or may not
be calculable, because the "Z-relation" is so elusive. For example
in 12-tet there is 1 Z-relation among Tetrachords, 3 among
Pentachords and 15 among hexachords (This is AFTER reducing for
Tn/TnI, that is, transposition and transposition of inverses)

I'd like to stimulate discussion on this issue, I have been
corresponding with John Wild out at Harvard who has compiled a large
database of sets with interval vectors, all the way up to 30-tet.
Anyone else out there interested in this line of attack? Thought I
would start this up again on Tuning, and if neccessary, we can go
over to Tuning-math with any discussion that is more mathematical!

Paul Hjelmstad

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> First, I should say that what some musicians call "Set theory", is
> not really "Set Theory" as described by mathematicians, that is a
> field in itself. The problem is, it's hard to know what else to call
> it.

Wikipedia calls it "musical set theory", which seems to me to be a big
improvement over calling it, very misleadingly, "set theory":

http://en.wikipedia.org/wiki/Musical_set_theory

> Anyone else out there interested in this line of attack? Thought I
> would start this up again on Tuning, and if neccessary, we can go
> over to Tuning-math with any discussion that is more mathematical!

Neither group is overloaded with postings at the moment.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > First, I should say that what some musicians call "Set theory",
is
> > not really "Set Theory" as described by mathematicians, that is a
> > field in itself. The problem is, it's hard to know what else to
call
> > it.
>
> Wikipedia calls it "musical set theory", which seems to me to be a
big
> improvement over calling it, very misleadingly, "set theory":
>
> http://en.wikipedia.org/wiki/Musical_set_theory
>

Musical Set Theory it is, then!

> > Anyone else out there interested in this line of attack? Thought
I
> > would start this up again on Tuning, and if neccessary, we can go
> > over to Tuning-math with any discussion that is more mathematical!
>
> Neither group is overloaded with postings at the moment.

Well, I need to analyze more data. We have been having trouble
tranferring data between Jon's server at Harvard and my webpage. (The
files are huge). What I have found out on my own is that there really
is no pattern to the behavior of Z-relations. However, I am still
hopeful that there may be a pattern to sets based on interval vectors:
(Intvec(24,12)) for example, would be the count of interval vectors
of sets reduced from C{24,12}. It equals 24,220, which is 4 * 5 * 13
* 97. However, I have found that sets reduced from C{n,9} have been
prime numbers. I was hoping to find a pattern in the factorization of
the resultant set-count which relates somehow to the initial C{n,m}
function count. (Can I call it a function?) For sets reduced from
C{12,m} You obtain (Intvec(12,m)) for m=0_to_12:
1,1,6,12,28,35,35,35,28,12,6,1,1. None are prime numbers, and 5 are
based on 7, and 4 are based on 6, etc. Unfortunately there is nothing
(that I know of) like Polya's algorithm to find this "polynomial"
series. (If you put it in the form of ax^12+bx^11+cx^10...). Jon has
directed me to the work of David Lewin, and others, which I plan on
looking at currently.

Paul Hj

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"

/tuning/topicId_52920.html#52923
>
> Well, I need to analyze more data. We have been having trouble
> tranferring data between Jon's server at Harvard and my webpage.
(The
> files are huge). What I have found out on my own is that there
really
> is no pattern to the behavior of Z-relations. However, I am still
> hopeful that there may be a pattern to sets based on interval
vectors:
> (Intvec(24,12)) for example, would be the count of interval vectors
> of sets reduced from C{24,12}. It equals 24,220, which is 4 * 5 *
13
> * 97. However, I have found that sets reduced from C{n,9} have been
> prime numbers. I was hoping to find a pattern in the factorization
of
> the resultant set-count which relates somehow to the initial C{n,m}
> function count. (Can I call it a function?) For sets reduced from
> C{12,m} You obtain (Intvec(12,m)) for m=0_to_12:
> 1,1,6,12,28,35,35,35,28,12,6,1,1. None are prime numbers, and 5 are
> based on 7, and 4 are based on 6, etc. Unfortunately there is
nothing
> (that I know of) like Polya's algorithm to find this "polynomial"
> series. (If you put it in the form of ax^12+bx^11+cx^10...). Jon has
> directed me to the work of David Lewin, and others, which I plan on
> looking at currently.
>
> Paul Hj

***It seems a "hot item" these days in Academia... well, OK, at least
at Columbia, cf. Chris Bailey, to examine microtonality through the
lens of "musical set theory..." I think it's a good development,
since at least it gets microtonality studied at such places.
However, there is still an even *more* exciting intellectual study
just going after the ratios and such like of just intonation theory
and temperaments and harmonic entropy... as we have been doing on
these lists. Personally, I think these studies are even more
pertinent and practical than set theory for "real" composing...

JP

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "Paul G Hjelmstad"
>
> /tuning/topicId_52920.html#52923
> >
> >>
> ***It seems a "hot item" these days in Academia... well, OK, at
least
> at Columbia, cf. Chris Bailey, to examine microtonality through the
> lens of "musical set theory..." I think it's a good development,
> since at least it gets microtonality studied at such places.
> However, there is still an even *more* exciting intellectual study
> just going after the ratios and such like of just intonation theory
> and temperaments and harmonic entropy... as we have been doing on
> these lists. Personally, I think these studies are even more
> pertinent and practical than set theory for "real" composing...
>
> JP

...But wouldn't it be exciting if one could join these two sides of
music theory (musical set theory and tuning theory) into some
unifing principle? Group Theory may hold the key...

Paul Hj

--- In tuning@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:

> ...But wouldn't it be exciting if one could join these two sides of
> music theory (musical set theory and tuning theory) into some
> unifing principle? Group Theory may hold the key...

To some extent they are joined already, since transposition and
inversion taken together on sets in n-et produce a group (the dihedral
group Dn) which preserves tuning properties. However, a tuning focus
will tend to prune the number of such sets we consider worth
characterizing.

There is nothing about musical set theory which confines it to be a
language only for serial music or only for 12-equal, so we are at
liberty to make use of it elsewhere. It is also entirely possible to
extend many of the ideas to linear temperaments, and to a lesser
extent even beyond, where now we would find ourselves dealing with
infinite groups instead of finite ones. I don't know if anyone has
done this, but I suspect not.

I have to say also that this whole topic seems more in the nature of a
tuning-math issue.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>

/tuning/topicId_52920.html#52927

wrote:
> --- In tuning@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > ...But wouldn't it be exciting if one could join these two sides
of
> > music theory (musical set theory and tuning theory) into some
> > unifing principle? Group Theory may hold the key...
>
> To some extent they are joined already, since transposition and
> inversion taken together on sets in n-et produce a group (the
dihedral
> group Dn) which preserves tuning properties. However, a tuning
focus
> will tend to prune the number of such sets we consider worth
> characterizing.
>
> There is nothing about musical set theory which confines it to be a
> language only for serial music or only for 12-equal, so we are at
> liberty to make use of it elsewhere. It is also entirely possible
to
> extend many of the ideas to linear temperaments, and to a lesser
> extent even beyond, where now we would find ourselves dealing with
> infinite groups instead of finite ones. I don't know if anyone has
> done this, but I suspect not.
>
> I have to say also that this whole topic seems more in the nature
of a
> tuning-math issue.

***Well, knowledge is power... (since when? :) but the danger, I
think lies in the generally-accepted compositional procedures used
with these kind of methodologies... ie., people tend to manipulate
sets rather than to actually *listen* to the material.

If something *audible* comes out of this kind of study, so much the
better. I'm just saying that, generally speaking, that's *not* the
direction composition has taken with these approaches in the past...
maybe because it's just too easy to compose this way without
listening to the material... :)

JP

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:

>
> ***Well, knowledge is power... (since when? :) but the danger, I
> think lies in the generally-accepted compositional procedures used
> with these kind of methodologies... ie., people tend to manipulate
> sets rather than to actually *listen* to the material.
>
> If something *audible* comes out of this kind of study, so much the
> better. I'm just saying that, generally speaking, that's *not* the
> direction composition has taken with these approaches in the past...
> maybe because it's just too easy to compose this way without
> listening to the material... :)
>

Errm... But you cannot blame the methodologies for this, can you?
Composing without listening is definitely a thing NOT to be done. If a
composer does so, all the worse for him - but that's hardly the
methodology's fault; i'd rather call it a wrong understanding what is
appropriate or even lacking mastery of the method.

I occasionally hear arguments like this in the context of questions
like whether you should study counterpoint or Schenkerian analysis or
music theory at all, and there they are not more convincing to me...

The other side of the coin is if somebody claims that this and this
methododogy (couterpoint, set theory or whatever) is absolutely
necessary for becoming a composer - this is, of course, equally wrong,
for the same reasons.
--
Hans Straub

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

/tuning/topicId_52920.html#52937

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
>
> >
> > ***Well, knowledge is power... (since when? :) but the danger, I
> > think lies in the generally-accepted compositional procedures
used
> > with these kind of methodologies... ie., people tend to
manipulate
> > sets rather than to actually *listen* to the material.
> >
> > If something *audible* comes out of this kind of study, so much
the
> > better. I'm just saying that, generally speaking, that's *not*
the
> > direction composition has taken with these approaches in the
past...
> > maybe because it's just too easy to compose this way without
> > listening to the material... :)
> >
>
> Errm... But you cannot blame the methodologies for this, can you?
> Composing without listening is definitely a thing NOT to be done.
If a
> composer does so, all the worse for him - but that's hardly the
> methodology's fault; i'd rather call it a wrong understanding what
is
> appropriate or even lacking mastery of the method.
>
> I occasionally hear arguments like this in the context of questions
> like whether you should study counterpoint or Schenkerian analysis
or
> music theory at all, and there they are not more convincing to
me...
>
> The other side of the coin is if somebody claims that this and this
> methododogy (couterpoint, set theory or whatever) is absolutely
> necessary for becoming a composer - this is, of course, equally
wrong,
> for the same reasons.
> --
> Hans Straub

***Thanks, Hans, for this amplification. Your view makes sense...

JP

--- In tuning@yahoogroups.com, "hstraub64" <hstraub64@t...> wrote:

> Errm... But you cannot blame the methodologies for this, can you?
> Composing without listening is definitely a thing NOT to be done.

It worked for Beethoven.:)