Interval Vectors - The Musical Set Theory Kind

All,

Jon Wild has been so kind as to let me analyze his database of
interval vectors (The Musical Set Theory Kind). Define Intvec(n,m)
as being a count of a musical subset based on unique interval vectors.

An interval vector gives counts of all the intervals in a set. For
example, the set (0,1,2,3,4,5) has interval vector of <5,4,3,2,1,0>
because it has 5 semitones, 4 wholetones, 3 minor thirds, 2 major
thirds and 1 perfect fifth and zero tritones. It is not neccessary
to count interval above floor[[n/2]] because they are just a mirror-
image.(1 perfect fifth, 2 minor sixths etc.

Continuing on: one can reduce C(12,6) which has 954 members to Intvec
(12,6) which is 35. Jon has supplied me with sets which have been
reduced for transposition and inversion (related to the Dihedral
Group) with counts for Z-relations. I merely have counted up these
numbers and divided by them to get the unique interval-vector count.

For example, for Intvec(24,12) you have 984 singletons (non-Z related)
41676 Z-related pairs, 138 Z-related triples, 12512 Z-related
quadruples, 876 Z-related sextuples, 600 Z-related octuples, and 36
Z-related 12-tuples. Now:

984+41676/2+138/3+12512/4+876/6+600/8+36/12=25220 unique interval
vectors.

For those who need a review on Z-relations: This is when 2 or more
sets (in the Dihedral Group) have the same interval vector.

So far, I have analyzed Intvec(n,m) where n=2m. This gives the
series for Intvec(2,1), Intvec(4,2),...Intvec(24,12) as follows:

1,2,3,7,13,35,85,254,701,2376,7944,25220. These are the "master
subsets" for even sets. (hexachords are the "master subset" for the
12 tone set because all other subsets (or supersets) can be derived
from them, where the superset is merely the complement of the subset
of this set).

Anyone see a pattern to the above series? I think I will graph it
and post it to my Files section. Once again, thanks Jon Wild for
sharing this data with us.

Paul Hjelmstad

set

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> All,
>
> Jon Wild has been so kind as to let me analyze his database of
> interval vectors (The Musical Set Theory Kind). Define Intvec(n,m)
> as being a count of a musical subset based on unique interval
vectors.
>
> An interval vector gives counts of all the intervals in a set. For
> example, the set (0,1,2,3,4,5) has interval vector of <5,4,3,2,1,0>
> because it has 5 semitones, 4 wholetones, 3 minor thirds, 2 major
> thirds and 1 perfect fifth and zero tritones. It is not neccessary
> to count interval above floor[[n/2]] because they are just a mirror-
> image.(1 perfect fifth, 2 minor sixths etc.
>
> Continuing on: one can reduce C(12,6) which has 954 members to
Intvec
> (12,6) which is 35. Jon has supplied me with sets which have been
> reduced for transposition and inversion (related to the Dihedral
> Group) with counts for Z-relations. I merely have counted up these
> numbers and divided by them to get the unique interval-vector count.
>
> For example, for Intvec(24,12) you have 984 singletons (non-Z
related)
> 41676 Z-related pairs, 138 Z-related triples, 12512 Z-related
> quadruples, 876 Z-related sextuples, 600 Z-related octuples, and 36
> Z-related 12-tuples. Now:
>
> 984+41676/2+138/3+12512/4+876/6+600/8+36/12=25220 unique interval
> vectors.
>
> For those who need a review on Z-relations: This is when 2 or more
> sets (in the Dihedral Group) have the same interval vector.
>
> So far, I have analyzed Intvec(n,m) where n=2m. This gives the
> series for Intvec(2,1), Intvec(4,2),...Intvec(24,12) as follows:
>
> 1,2,3,7,13,35,85,254,701,2376,7944,25220. These are the "master
> subsets" for even sets. (hexachords are the "master subset" for the
> 12 tone set because all other subsets (or supersets) can be derived
> from them, where the superset is merely the complement of the
subset
> >
> Anyone see a pattern to the above series? I think I will graph it
> and post it to my Files section. Once again, thanks Jon Wild for
> sharing this data with us.
>
Okay, I've run all the "oddballs": Intvec(3,1), Intvec(5,2)...Intvec
(21,10). They come to 1,2,4,10,26,74,222,698,2338,7866. This makes
a series of:
1,2,2,3,4,7,10,13,26,35,74,85,222,254,698,701,2338,2376,7866,7944,****
*,25220 (with Intvec(23,11) missing for now. One thing to note
is that the oddballs cling close to their upper evenball neighbors.

PHj

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

> For those who need a review on Z-relations: This is when 2 or more
> sets (in the Dihedral Group) have the same interval vector.

Don't you mean two or more sets in the orbit of an action by the
dihedral group? In other words, s1 Z s2 iff Intervalvector(s1) =
Intervalvalvector(g(s2)) for some g in the dihedral group Dn?

> So far, I have analyzed Intvec(n,m) where n=2m. This gives the
> series for Intvec(2,1), Intvec(4,2),...Intvec(24,12) as follows:
>
> 1,2,3,7,13,35,85,254,701,2376,7944,25220. These are the "master
> subsets" for even sets. (hexachords are the "master subset" for the
> 12 tone set because all other subsets (or supersets) can be derived
> from them, where the superset is merely the complement of the subset
> of this set).
>
> Anyone see a pattern to the above series?

No, but see

http://www.research.att.com/projects/OEIS?Anum=A045611

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > For those who need a review on Z-relations: This is when 2 or more
> > sets (in the Dihedral Group) have the same interval vector.
>
> Don't you mean two or more sets in the orbit of an action by the
> dihedral group? In other words, s1 Z s2 iff Intervalvector(s1) =
> Intervalvalvector(g(s2)) for some g in the dihedral group Dn?

Yes I think you are correct
>
> > So far, I have analyzed Intvec(n,m) where n=2m. This gives the
> > series for Intvec(2,1), Intvec(4,2),...Intvec(24,12) as follows:
> >
> > 1,2,3,7,13,35,85,254,701,2376,7944,25220. These are the "master
> > subsets" for even sets. (hexachords are the "master subset" for
the
> > 12 tone set because all other subsets (or supersets) can be
derived
> > from them, where the superset is merely the complement of the
subset
> > of this set).
> >
> > Anyone see a pattern to the above series?
>
> No, but see
>
> http://www.research.att.com/projects/OEIS?Anum=A045611

Thanks

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > For those who need a review on Z-relations: This is when 2 or more
> > sets (in the Dihedral Group) have the same interval vector.
>
> Don't you mean two or more sets in the orbit of an action by the
> dihedral group? In other words, s1 Z s2 iff Intervalvector(s1) =
> Intervalvalvector(g(s2)) for some g in the dihedral group Dn?

Actually, I take that back. I think you're reading too much into
this. There is no functional relationship between two sets that are
z-related, they just happen to have the same interval vector, by
coincidence. (But I might be wrong...) What I mean is Intvec(s1)
=Intvec(s2), there is no g, that I can find...

>
> > So far, I have analyzed Intvec(n,m) where n=2m. This gives the
> > series for Intvec(2,1), Intvec(4,2),...Intvec(24,12) as follows:
> >
> > 1,2,3,7,13,35,85,254,701,2376,7944,25220. These are the "master
> > subsets" for even sets. (hexachords are the "master subset" for
the
> > 12 tone set because all other subsets (or supersets) can be
derived
> > from them, where the superset is merely the complement of the
subset
> > of this set).
> >
> > Anyone see a pattern to the above series?
>
> No, but see
>
> http://www.research.att.com/projects/OEIS?Anum=A045611

I thought of looking there. Wow! This is the sort of thing I was
looking for. Great news. Obviously I have an error with Intvec(20,10)
which I will correct (2377 not 2376) I also looked up the whole
sequence (with the oddballs) but did not get anything. Thanks Gene

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > So far, I have analyzed Intvec(n,m) where n=2m. This gives the
> > > series for Intvec(2,1), Intvec(4,2),...Intvec(24,12) as follows:
> > >
> > > 1,2,3,7,13,35,85,254,701,2376,7944,25220. These are the "master
> > > subsets" for even sets.

> > > Anyone see a pattern to the above series?
> >
> > No, but see
> >
> > http://www.research.att.com/projects/OEIS?Anum=A045611

Also, A089404 is the oddball sequence - Paul

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > Anyone see a pattern to the above series?
> >
> > No, but see
> >
> > http://www.research.att.com/projects/OEIS?Anum=A045611
>
The part I don't get is why adding up all the multiplicities for 2377
{Intvec(20,10)} equals C{19,9} instead of C{20,10}. They are off by a
factor of two.

This gives the sum 1+1+3+22+362+1855+1+130+2 multiplied by
1,2,5,10,20,40,60,80,120. What I think they are doing is collapsing a
set and its complement in C{20,10} - and of course every set has
exactly 1 complement. So that must be why there are C{19,9}
configurations.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > Anyone see a pattern to the above series?
>
> No, but see
>
> http://www.research.att.com/projects/OEIS?Anum=A045611

Hmm - might this be a case for Polya's enumeration theory?

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@t...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > Anyone see a pattern to the above series?
> >
> > No, but see
> >
> > http://www.research.att.com/projects/OEIS?Anum=A045611
>
> Hmm - might this be a case for Polya's enumeration theory?

If anyone could show how Polya's method could used to find interval
vector counts, I would be elated beyond measure. But, I would tend
to doubt it. Polya's method would then have to account for the
elusive z-relation, something that (as far as I know) cannot be
calculated. However, if there is a way to count interval vectors
independent of knowing z-related counts, then perhaps they can
be counted/enumerated.

Does anyone know what practical application the "number of different
energy states of n positive and n negative charges on a necklace..."
would have? What else could this tie into?

Paul Hj