Combinatorics and Tuning Systems?

Thought you all might be interested in another area of music theory, that I
have been working on for years - the study of combinatoric sets. I am
trying to see if there are any connections between the math of tuning
systems and sets. (It's possible there is no connection at all). For
example, there are 50 hexachord sets after reducing for symmetry (80 -> 50,
there are 20 symmetrical sets and 30 asymmetrical sets). You can take this
one step further, and reduce for the "Z-relation" (a la Allen Forte; this
is when 2 or more sets of different normal set-type have the same interval
vector) and count sets based on their "interval vectors" to obtain 35
hexachord sets (in the 12-ET system, of course). Could 35 here be related
to the 35/36 comma of a 7-limit model of Z12? Maybe not, but maybe so. One
neat thing about all the Z-related sets in Z12 is that there are 15
interlocking, Z-related hexachords....

I have come up with a method for reducing sets for transpositional symmetry
to obtain the count of normal set types. For example, C{12,6} reduces to 80
sets after reducing for transposition. Since some sets have extra symmetry
(for example 0,2,4,6,8,10 has hexagonal radial symmetry) one must consider
all the symmetrical possibilities. Here is my technique for C{12,6}:

C{12,6}, C{6,3}, C{4,2}, C{2,1}. Start on the furthest right and reduce
that set first. C{2,1}/2=1. (C{4,2}-C{2,1})/4=1. (C{6,3}-C{2,1})/6=3.
(C{12,6}-C{6,3}-C{4,2}+C{2,1}*)/12=75. Add all these up to obtain 80 sets
types. *C{2,1} has to be added back in once due to the interaction of
C{4,2} and C{6,3} with C{2,1}, Sorry, I don't know a better way to say
it...

Anyway, using this technique I found that C{24,12} reduces to 112720.
Reducing for symmetry gives 61822. I have predicted that there are 31373
sets after reducing for Z-related sets, but I have not proven this.

The programming I have been working on with my brother is a method for
counting sets based on their interval vectors. We have run many sets,
however, a programming problem has prevented him from running any sets with
more than 9 in the "denominator" (Like C{20,9}). Have found some very
lovely patterns with Z-relations, etc. but I won't know if any of this
relates to tuning systems until I can run C{24,12}...

Pitch class set theory is a study of relationships between
classes of pitches and intervals, and therefore assumes
enharmonic equivalence and octave equivalence, though not
necessarily 12tet. I would consider it applicable to most
musics in some form at least.

If you're interested in combinatoriality, don't waste your
time on hexachords. That area of research is pretty well
played out at this point.

Somehow, even the great serialists failed to much
exploit combinatoriality between sets of 5 and 7.
the 5-12/7-12 aggregate is particularly interesting
in that 7-12 does not actually include any forms of 5-12.
It's such an obvious candidate for serialist treatment...
...ok, I'll drop that.

I have a special interest in 5-28 (and 7-28) because
of the inclusion of the all-interval tetrachords in 5-28
into a set with a minimum prime form right entry of 8.
I consider 5-28 to be the maximally cognitively transparent
scale set, and I'm intending to eventually build a small
gamelan...blah...blah...

OK... combinatoriality...

I think the basic principle as it applies to actual
music perception is that whenever you have an implied
scale in which there are appreciable differences in
interval size, the mind may be primed in some way
to anticipate the introduction of more tones sooner
in the larger gaps.

The extension of the quintal pentatonic (5-35)into
it's inclusion-transposed complement can take any
of 3 forms before one of the original 5 tones is displaced;
all three forms fall in the larger gaps.

That the quintal pentatonic is a ubiquitous reference
point across cultures and tuning systems, I don't think
is open to dispute. I consider, though, that the advantages
of 5-35 over 5-28 are mechanical and technical, rather
than cognitive. 5-28 also expands into its gap as it
becomes 7-28, though. With pre-tuned instruments and
a more controlled repertoire and performance practice...
(uh...).

We're a bit off topic, but this is my thing, even
more than intonation.
I think it's relevant, though, so we can discuss it
further if people think it's appropriate.

--- In tuning-math@y..., <Josh@o...> wrote:

> Somehow, even the great serialists failed to much
> exploit combinatoriality between sets of 5 and 7.
> the 5-12/7-12 aggregate is particularly interesting
> in that 7-12 does not actually include any forms of 5-12.
> It's such an obvious candidate for serialist treatment...
> ...ok, I'll drop that.

A t-(v,k,n) design is a set of subsets called "blocks" of a set of v elements, each of size k, and such that any t-tuple is contained in exactly n blocks; for example a projective plane of order q is
a 2-(q^2+q+1,q+1,1) design. There are some very interesting designs related to the (sporadic, simple) Mathieu groups; in connection with the 12-et, the 5-(12,6,1) design related to the Mathieu group M12, a subgroup of the permutation group S12 of twelve elements, is especially interesting; it consists of 132 hexads, a set invariant under M12, such that if we know five elements of the hexad this determines the sixth.

Some web stuff:

http://mathworld.wolfram.com/SteinerSystem.html

http://mathworld.wolfram.com/SteinerTripleSystem.html

http://web.usna.navy.mil/~wdj/m_12.htm#Steiner Hexad

I am interested by your response-

Could you provide me a reference of a published paper or web site where what
you call "Pitch class set theory" state of the art or latest status is
described?

Thank you very much.

Carlos

----- Original Message -----
From: <Josh@orangeboxman.com>
To: <tuning-math@yahoogroups.com>
Sent: Wednesday, September 11, 2002 9:59 PM
Subject: Re: [tuning-math] Combinatorics and Tuning Systems?

> Pitch class set theory is a study of relationships between
> classes of pitches and intervals, and therefore assumes
> enharmonic equivalence and octave equivalence, though not
> necessarily 12tet. I would consider it applicable to most
> musics in some form at least.
>
> If you're interested in combinatoriality, don't waste your
> time on hexachords. That area of research is pretty well
> played out at this point.
>
> Somehow, even the great serialists failed to much
> exploit combinatoriality between sets of 5 and 7.
> the 5-12/7-12 aggregate is particularly interesting
> in that 7-12 does not actually include any forms of 5-12.
> It's such an obvious candidate for serialist treatment...
> ...ok, I'll drop that.
>
> I have a special interest in 5-28 (and 7-28) because
> of the inclusion of the all-interval tetrachords in 5-28
> into a set with a minimum prime form right entry of 8.
> I consider 5-28 to be the maximally cognitively transparent
> scale set, and I'm intending to eventually build a small
> gamelan...blah...blah...
>
> OK... combinatoriality...
>
> I think the basic principle as it applies to actual
> music perception is that whenever you have an implied
> scale in which there are appreciable differences in
> interval size, the mind may be primed in some way
> to anticipate the introduction of more tones sooner
> in the larger gaps.
>
> The extension of the quintal pentatonic (5-35)into
> it's inclusion-transposed complement can take any
> of 3 forms before one of the original 5 tones is displaced;
> all three forms fall in the larger gaps.
>
> That the quintal pentatonic is a ubiquitous reference
> point across cultures and tuning systems, I don't think
> is open to dispute. I consider, though, that the advantages
> of 5-35 over 5-28 are mechanical and technical, rather
> than cognitive. 5-28 also expands into its gap as it
> becomes 7-28, though. With pre-tuned instruments and
> a more controlled repertoire and performance practice...
> (uh...).
>
> We're a bit off topic, but this is my thing, even
> more than intonation.
> I think it's relevant, though, so we can discuss it
> further if people think it's appropriate.
>
>
> To unsubscribe from this group, send an email to:
> tuning-math-unsubscribe@yahoogroups.com
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>

Interesting. But I disagree with part of this: Actually, most pentachords
in 12tET fit into septachords, through regular hexachords. There are 2 that
fit in by means of Z-related hexachords, and one (0,1,3,5,6) does not fit
in to its septachord complement by means of ANY hexachord. This is Allen
Forte's "weakly related 7-5 set complexes (complices?)" Unless you were
talking about something else when you stated "7-12 does not include any
forms of 5-12"

It is true that the study of hexachords in 12tET is pretty exhausted, but I
am also interested in C{24,6} for example.Created a program (my brother
wrote it actually) to count sets based on their interval vectors. Counting
interval vectors in 12tET for diads through hexachords gives 6, 12, 28, 35,
35. (Before reducing for Forte's Z-relation you get 6,12,29,38,50 Tn/TnI
types). So I feel something is going on here, which I have extended to
16tET, 19tET, and (am working on) 24tET. If anything, there are some
amazing patterns in the behaviour of the Z-relations (in 16tET and 19tET
for example.) Once I run sets in 24tET for diads through dodecads(?) I
will post the results. Hope to find a beautiful pattern!

Josh@orangeboxman
.com To: tuning-math@yahoogroups.com
cc:
09/11/2002 02:59 Subject: Re: [tuning-math] Combinatorics and Tuning Systems?
PM
Please respond to
tuning-math

Pitch class set theory is a study of relationships between
classes of pitches and intervals, and therefore assumes
enharmonic equivalence and octave equivalence, though not
necessarily 12tet. I would consider it applicable to most
musics in some form at least.

If you're interested in combinatoriality, don't waste your
time on hexachords. That area of research is pretty well
played out at this point.

Somehow, even the great serialists failed to much
exploit combinatoriality between sets of 5 and 7.
the 5-12/7-12 aggregate is particularly interesting
in that 7-12 does not actually include any forms of 5-12.
It's such an obvious candidate for serialist treatment...
...ok, I'll drop that.

I have a special interest in 5-28 (and 7-28) because
of the inclusion of the all-interval tetrachords in 5-28
into a set with a minimum prime form right entry of 8.
I consider 5-28 to be the maximally cognitively transparent
scale set, and I'm intending to eventually build a small
gamelan...blah...blah...

OK... combinatoriality...

I think the basic principle as it applies to actual
music perception is that whenever you have an implied
scale in which there are appreciable differences in
interval size, the mind may be primed in some way
to anticipate the introduction of more tones sooner
in the larger gaps.

The extension of the quintal pentatonic (5-35)into
it's inclusion-transposed complement can take any
of 3 forms before one of the original 5 tones is displaced;
all three forms fall in the larger gaps.

That the quintal pentatonic is a ubiquitous reference
point across cultures and tuning systems, I don't think
is open to dispute. I consider, though, that the advantages
of 5-35 over 5-28 are mechanical and technical, rather
than cognitive. 5-28 also expands into its gap as it
becomes 7-28, though. With pre-tuned instruments and
a more controlled repertoire and performance practice...
(uh...).

We're a bit off topic, but this is my thing, even
more than intonation.
I think it's relevant, though, so we can discuss it
further if people think it's appropriate.

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I see how to calculate the Steiner system S(5,6,12) to obtain 132 hexads
from r=66. And I see how that is connected with M12 Matthieu group. But
what I don't see is how M12 has order 2^6*3^3*5*11, the number of elements
in the group. How does this relate to the Steiner system? Thanks

"Gene Ward Smith"
<genewardsmith@ju To: tuning-math@yahoogroups.com
no.com> cc:
Subject: [tuning-math] Re: Combinatorics and Tuning Systems?
09/11/2002 05:13
PM
Please respond to
tuning-math

--- In tuning-math@y..., <Josh@o...> wrote:

> Somehow, even the great serialists failed to much
> exploit combinatoriality between sets of 5 and 7.
> the 5-12/7-12 aggregate is particularly interesting
> in that 7-12 does not actually include any forms of 5-12.
> It's such an obvious candidate for serialist treatment...
> ...ok, I'll drop that.

A t-(v,k,n) design is a set of subsets called "blocks" of a set of v
elements, each of size k, and such that any t-tuple is contained in exactly
n blocks; for example a projective plane of order q is
a 2-(q^2+q+1,q+1,1) design. There are some very interesting designs related
to the (sporadic, simple) Mathieu groups; in connection with the 12-et, the
5-(12,6,1) design related to the Mathieu group M12, a subgroup of the
permutation group S12 of twelve elements, is especially interesting; it
consists of 132 hexads, a set invariant under M12, such that if we know
five elements of the hexad this determines the sixth.

Some web stuff:

http://mathworld.wolfram.com/SteinerSystem.html

http://mathworld.wolfram.com/SteinerTripleSystem.html

http://web.usna.navy.mil/~wdj/m_12.htm#Steiner Hexad

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--- In tuning-math@y..., paul.hjelmstad@u... wrote:
>
> I see how to calculate the Steiner system S(5,6,12) to obtain 132 hexads
> from r=66. And I see how that is connected with M12 Matthieu group. But
> what I don't see is how M12 has order 2^6*3^3*5*11, the number of elements
> in the group. How does this relate to the Steiner system? Thanks

|M12| = 132*6! = 132 |S6|. M12 is 5-transitive, so all n-tuples aside from 6-tuples are permutable. For 6-tuples, M12 sends a hexad of the Steiner system to another hexad of the Steiner system; since it is a simple group this gives a permutation representation of M12 on the 132 hexands.

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> |M12| = 132*6! = 132 |S6|. M12 is 5-transitive, so all n-tuples aside from 6-tuples are permutable. For 6-tuples, M12 sends a hexad of the Steiner system to another hexad of the Steiner system; since it is a simple group this gives a ^ permutation representation of M12 on the 132 hexands. faithful

Thanks for the explanation. I'm still working to understand why knowing 5
elements determines the 6th one, though. Looks like I just need to study
this more...

"Gene Ward Smith"
<genewardsmith@ju To: tuning-math@yahoogroups.com
no.com> cc:
Subject: [tuning-math] Re: Combinatorics and Tuning Systems?
09/21/2002 06:32
AM
Please respond to
tuning-math

--- In tuning-math@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> |M12| = 132*6! = 132 |S6|. M12 is 5-transitive, so all n-tuples aside
from 6-tuples are permutable. For 6-tuples, M12 sends a hexad of the
Steiner system to another hexad of the Steiner system; since it is a simple
group this gives a ^ permutation representation of M12 on the 132 hexands.
faithful

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back around September 10,

> --- In tuning-math@y..., <Josh@o...> wrote:
>
> Somehow, even the great serialists failed to much
> exploit combinatoriality between sets of 5 and 7.
> the 5-12/7-12 aggregate is particularly interesting
> in that 7-12 does not actually include any forms of 5-12.
> It's such an obvious candidate for serialist treatment...
> ...ok, I'll drop that.

i wasn't following this thread, and only remembered seeing
the word "combinatorics" in the subject line. but i just
stumbled across this:

"Some Combinational Resources of Equal-Tempered Systems"
by Carlton Gamer
_Journal of Music Theory_ 11:1, Spring 1967

in which the opening paragraph gives the following abstract:

>> "The purpose of this article is to reveal and discuss
>> certain resources available to the composer who wishes to
>> employ equal-tempered systems containing either more or
>> less than twelve tones per octave, with particular emphasis
>> upon the former, the so-called "microtonal" systems."

i'm going to completely skirt the issue of Gamer's specific
definition of "microtonal": see the Tuning Dictionary and
the list archives for those arguments.

anyway, i wasn't following the thread, but those who were
would find this paper relevant.

-monz
"all roads lead to n^0"