C24 Request

This is a request for information on all the group theoretical
symmetries of C24 (Would that be C8 X C3?)

Particularly, the Polya Polynomials for every sort of decomposition,
D8 X C3, etc. I could calculate by hand, but that would take
forever.

Are there online resources to do this? Gene, what language did
you use for your C12 printouts? (with both Polynomials and Generators)
Was that MAPLE?

Thanks,

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:
>
> This is a request for information on all the group theoretical
> symmetries of C24 (Would that be C8 X C3?)
>
> Particularly, the Polya Polynomials for every sort of decomposition,
> D8 X C3, etc. I could calculate by hand, but that would take
> forever.
>
> Are there online resources to do this? Gene, what language did
> you use for your C12 printouts? (with both Polynomials and Generators)
> Was that MAPLE?
>
> Thanks,
>
> PGH

Here is what I get in GAP, for two of them: C8 X C3 and D8 X S3:

C8 X C3

Group([
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) ])

Cycle Index:

1/24*x_1^24+1/24*x_2^12+1/12*x_3^8+1/12*x_4^6+1/12*x_6^4+1/6*x_8^3+1/6*x
_12^2+1/3*x_24

Expanded:

2/3*x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+30
667*x^16+54484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+81752
*x^10+54484*x^9+30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+12*
x^2+x+1/3*x_24+2/3

D8 X S3

Group([ (1,2,3,4), (2,4), (5,6,7), (5,6) ])

Cycle Index:

1/48*x_1^7+5/48*x_1^5*x_2+1/24*x_1^4*x_3+3/16*x_1^3*x_2^2+1/24*x_1^3*x_4
+1/12*x_1^2*x_2*x_3+3/16*x_1*x_2^3+1/8*x_1*x_2*x_4+1/8*x_2^2*x_3+1/12*x_
3*x_4

Expanded

5/24*x^24+x^23+23/2*x^22+85*x^21+1773/4*x^20+1771*x^19+11221/2*x^18+1442
1*x^17+245167/8*x^16+54484*x^15+81719*x^14+104006*x^13+1/24*1^12+338042/
3*x^12+104006*x^11+81719*x^10+54484*x^9+245167/8*x^8+14421*x^7+11221/2*x
^6+1771*x^5+1/12*6^4+1773/4*x^4+1/6*x^3+85*x^3+1/6*4^2+23/2*x^2+x+1/3*x_
24+5/24

Does this look right? I struggled a little with converting x_1 -> x+1,
x_2 -> x^2+1

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
> >
> > This is a request for information on all the group theoretical
> > symmetries of C24 (Would that be C8 X C3?)
> >
> > Particularly, the Polya Polynomials for every sort of
decomposition,
> > D8 X C3, etc. I could calculate by hand, but that would take
> > forever.
> >
> > Are there online resources to do this? Gene, what language did
> > you use for your C12 printouts? (with both Polynomials and
Generators)
> > Was that MAPLE?
> >
> > Thanks,
> >
> > PGH
>
> Here is what I get in GAP, for two of them: C8 X C3 and D8 X S3:
>
> C8 X C3
>
> Group([
> (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) ])
>
> Cycle Index:
>
>
1/24*x_1^24+1/24*x_2^12+1/12*x_3^8+1/12*x_4^6+1/12*x_6^4+1/6*x_8^3+1/6
*x
> _12^2+1/3*x_24
>
>
> Expanded:
>
>
>
2/3*x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+
30
>
667*x^16+54484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+817
52
>
*x^10+54484*x^9+30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+1
2*
> x^2+x+1/3*x_24+2/3
>
>
> D8 X S3
>
> Group([ (1,2,3,4), (2,4), (5,6,7), (5,6) ])
>
> Cycle Index:
>
>
1/48*x_1^7+5/48*x_1^5*x_2+1/24*x_1^4*x_3+3/16*x_1^3*x_2^2+1/24*x_1^3*x
_4
>
+1/12*x_1^2*x_2*x_3+3/16*x_1*x_2^3+1/8*x_1*x_2*x_4+1/8*x_2^2*x_3+1/12*
x_
> 3*x_4
>
> Expanded
>
>
5/24*x^24+x^23+23/2*x^22+85*x^21+1773/4*x^20+1771*x^19+11221/2*x^18+14
42
>
1*x^17+245167/8*x^16+54484*x^15+81719*x^14+104006*x^13+1/24*1^12+33804
2/
>
3*x^12+104006*x^11+81719*x^10+54484*x^9+245167/8*x^8+14421*x^7+11221/2
*x
>
^6+1771*x^5+1/12*6^4+1773/4*x^4+1/6*x^3+85*x^3+1/6*4^2+23/2*x^2+x+1/3*
x_
> 24+5/24
>
> Does this look right? I struggled a little with converting x_1 ->
x+1,
> x_2 -> x^2+1
>
> PGH

I forgot about x_24:

Here is C8 X C3 Expanded correctly:

C8 X C3

x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+3066
7*x^16+54484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+81752
*x^10+54484*x^9+30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+1
2*x^2+x+1

I cannot get D8 X S3 to cooperate, to fix the same problem

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@>
> > wrote:
> > >
> > > This is a request for information on all the group theoretical
> > > symmetries of C24 (Would that be C8 X C3?)
> > >
> > > Particularly, the Polya Polynomials for every sort of
> decomposition,
> > > D8 X C3, etc. I could calculate by hand, but that would take
> > > forever.
> > >
> > > Are there online resources to do this? Gene, what language did
> > > you use for your C12 printouts? (with both Polynomials and
> Generators)
> > > Was that MAPLE?
> > >
> > > Thanks,
> > >
> > > PGH
> >
> > Here is what I get in GAP, for two of them: C8 X C3 and D8 X S3:
> >
> > C8 X C3
> >
> > Group([
> >
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24) ])
> >
> > Cycle Index:
> >
> >
>
1/24*x_1^24+1/24*x_2^12+1/12*x_3^8+1/12*x_4^6+1/12*x_6^4+1/6*x_8^3+1/6
> *x
> > _12^2+1/3*x_24
> >
> >
> > Expanded:
> >
> >
> >
>
2/3*x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+
> 30
> >
>
667*x^16+54484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+817
> 52
> >
>
*x^10+54484*x^9+30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+1
> 2*
> > x^2+x+1/3*x_24+2/3
> >
> >
> > D8 X S3
> >
> > Group([ (1,2,3,4), (2,4), (5,6,7), (5,6) ])
> >
> > Cycle Index:
> >
> >
>
1/48*x_1^7+5/48*x_1^5*x_2+1/24*x_1^4*x_3+3/16*x_1^3*x_2^2+1/24*x_1^3*x
> _4
> >
>
+1/12*x_1^2*x_2*x_3+3/16*x_1*x_2^3+1/8*x_1*x_2*x_4+1/8*x_2^2*x_3+1/12*
> x_
> > 3*x_4
> >
> > Expanded
> >
> >
>
5/24*x^24+x^23+23/2*x^22+85*x^21+1773/4*x^20+1771*x^19+11221/2*x^18+14
> 42
> >
>
1*x^17+245167/8*x^16+54484*x^15+81719*x^14+104006*x^13+1/24*1^12+33804
> 2/
> >
>
3*x^12+104006*x^11+81719*x^10+54484*x^9+245167/8*x^8+14421*x^7+11221/2
> *x
> >
>
^6+1771*x^5+1/12*6^4+1773/4*x^4+1/6*x^3+85*x^3+1/6*4^2+23/2*x^2+x+1/3*
> x_
> > 24+5/24
> >
> > Does this look right? I struggled a little with converting x_1 ->
> x+1,
> > x_2 -> x^2+1
> >
> > PGH
>
> I forgot about x_24:
>
> Here is C8 X C3 Expanded correctly:
>
> C8 X C3
>
>
x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+3066
>
7*x^16+54484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+81752
>
*x^10+54484*x^9+30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+1
> 2*x^2+x+1
>
> I cannot get D8 X S3 to cooperate, to fix the same problem

D8 X S3

3/16*x^7+5/16*x^6+5/8*x^5+15/16*x^4+13/12*x^3+43/48*x^2+9/16*x+5/24

Okay sorry for multiple posts.

PGH

Paul G Hjelmstad wrote:

> x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+3066
> 7*x^16+54484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+81752
> *x^10+54484*x^9+30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+1
>> 2*x^2+x+1
>>
>> I cannot get D8 X S3 to cooperate, to fix the same problem
> > D8 X S3
> > 3/16*x^7+5/16*x^6+5/8*x^5+15/16*x^4+13/12*x^3+43/48*x^2+9/16*x+5/24
> > Okay sorry for multiple posts. > > PGH

All those superscripts look pretty, but does any of this really have anything to do with tuning? You know, the topic of this mailing list? Otherwise we might as well just change the name to "math"....

Maybe I've missed some deep significance of group theoretical symmetry to the selection of pitches for actual music (you know, sound waves in the air, that kind of stuff). But all I see is a bunch of polynomials involving huge numbers without a single connection to tuning theory, expressed or implied. I don't mind big lists of huge numbers if they can be of use in tuning theory, but this kind of stuff looks like it belongs in a general math-related mailing list or discussion group.

I know this list hasn't been getting much discussion lately, but if it starts turning into "random off-topic math" instead of "tuning-math", I may start losing interest. Not everyone here is in it for the math, you know....

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...>
wrote:
>
> Paul G Hjelmstad wrote:
>
> >
x^24+x^23+12*x^22+85*x^21+446*x^20+1771*x^19+5620*x^18+14421*x^17+306
6
> >
7*x^16+54484*x^15+81752*x^14+104006*x^13+112720*x^12+104006*x^11+8175
2
> >
*x^10+54484*x^9+30667*x^8+14421*x^7+5620*x^6+1771*x^5+446*x^4+85*x^3+
1
> >> 2*x^2+x+1
> >>
> >> I cannot get D8 X S3 to cooperate, to fix the same problem
> >
> > D8 X S3
> >
> >
3/16*x^7+5/16*x^6+5/8*x^5+15/16*x^4+13/12*x^3+43/48*x^2+9/16*x+5/24
> >
> > Okay sorry for multiple posts.
> >
> > PGH
>
> All those superscripts look pretty, but does any of this really
have
> anything to do with tuning? You know, the topic of this mailing
list?
> Otherwise we might as well just change the name to "math"....
>
> Maybe I've missed some deep significance of group theoretical
symmetry
> to the selection of pitches for actual music (you know, sound
waves in
> the air, that kind of stuff). But all I see is a bunch of
polynomials
> involving huge numbers without a single connection to tuning
theory,
> expressed or implied. I don't mind big lists of huge numbers if
they can
> be of use in tuning theory, but this kind of stuff looks like it
belongs
> in a general math-related mailing list or discussion group.
>
> I know this list hasn't been getting much discussion lately, but
if it
> starts turning into "random off-topic math" instead of "tuning-
math", I
> may start losing interest. Not everyone here is in it for the
math, you
> know....

Thanks for your message. Actually, D8 X S3 is wrong, I need to
figure out how to expand this out correctly from the Cycle Index.

But in answer to your question, yes, it is a little off-topic, but
not completely, I believe it does tie into tuning theory.

For example, we have discussed how, for example, the 19 different
triads of a 12 note system (not neccessary 12t-ET) map under
different tuning systems, and the patterns that ensue from such
mappings.

Another aspect is for example, how 1,5,7,11 are based on the Klein
quartic in 12t-ET, are relatively prime to 12, and how they generate
the affine group. (And the multiplicative modulo group).

I know everyone's sick of my hexachord theory stuff, so I am trying
to expand out a little. One application that relates to tuning a
little more closely is an examination of the interval vector content
of different sets, instead of just the sets themselves. Of course
one has to be careful to use a term like "interval vector" because
the different meanings and contexts it has on this discussion group.

Another poster has written paper(s) on this topic, which tie in
Steiner systems, difference sets, and Hadamard, Singer and planar
sets.

No, I haven't found the strong connections with tuning that I have
been looking for now for five years.

We've discussed some number theory things recently, in conjunction
with Euler's seminal paper, maybe if I got that and read it through
interlibrary loan or something I'd have a better sense of direction
with my music-math aspirations. Apparently he does discuss lattices..

I know I am on the fringe, and it would make more sense if perhaps
if my messages on the topic were only, say 1% of all the messages
that are posted - I am learning lattices which, if I discussed with
some sense of enlightenment would probably be a lot more welcome
here than my musical-set theory posts. I'm just fascinated by the
math, Polya theory and so forth, the 35 hexachords etc.

Others have questioned the relevance of some of my posts, that they
are a little off the wall so I have been reprimanded so to speak.

But as one of my best teachers said in a master-class:

"Harmony, harmony, harmony! None of us here understands all there is
to know about harmony" (Pianists of course!)

So, tuning without harmony? tuning without melody? Do we only
consider the most ordinary scales in a given tuning system?

I guess I like Bartok more than Brittany Spears but I guess that is
just me.

But I see your point and know it is representative. I'm looking for
a silver bullet, and there probably isn't one. It's almost like the
different maths are at loggerheads! But that just makes me all the
more captivated by it. (Musical Set Theory<->Tuning Theory)

So for example, there are 30667 different octads in C24. This is
337*13*7. Which is (256+81)*13*7=(2^16+3^4)*13*7. Seems to have
absolutely nothing to do with the primes in a 24t-ET system right?

Do logarithmic divisions of the octave have anything to do with the
primes in their respective systems? I think they do. If someone
(with the authority) would say, "They definitely aren't related"
(about musical sets, and primes in tuning) I would have to take that
seriously (I wouldn't like it, but I would have to admit I might be
wrong) Of course, primes are big in groups, rings and finite
fields...

So math can be used to analyze, but art is used to synthesize, so I
am just saying "why not"? Right now my best bet is something like
coverings of a space, enumerative combinatorics - I'm sorry I'm not
smarter, I really wish I was.

Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> <snip> ... I'm sorry I'm not smarter, I really wish I was.
>
> Paul Hj

I know how you feel, Paul. I've been struggling to learn
math beyond algebra so that i can speak more clearly to
my tuning-math colleagues about my tuning-theory ideas.
I've had to take a break after trig and precalculus, so
that's where i'm stuck right now.

I also never really got very deeply into your ideas
about hexachords, but i am interested and would like
to learn about it when i can.

My personal breakthru was in finding a way to lattice
harmonic spaces larger than 3 dimensions, in 1998. See:

http://tonalsoft.com/monzo/lattices/lattices.htm

I'm convinced that the hearing apparatus is better
suited than the sight apparatus to modeling
multi-dimensional space.

To me, this is one of the fascinations of music.
A great composer can elaborate processes on many
different levels simultaneously. Music that is
this deep rewards a lifetime of repeated listening.
Anyway, i digress ...

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > <snip> ... I'm sorry I'm not smarter, I really wish I was.
> >
> > Paul Hj
>
>
> I know how you feel, Paul. I've been struggling to learn
> math beyond algebra so that i can speak more clearly to
> my tuning-math colleagues about my tuning-theory ideas.
> I've had to take a break after trig and precalculus, so
> that's where i'm stuck right now.
>
> I also never really got very deeply into your ideas
> about hexachords, but i am interested and would like
> to learn about it when i can.
>
>
> My personal breakthru was in finding a way to lattice
> harmonic spaces larger than 3 dimensions, in 1998. See:
>
> http://tonalsoft.com/monzo/lattices/lattices.htm
>
>
> I'm convinced that the hearing apparatus is better
> suited than the sight apparatus to modeling
> multi-dimensional space.
>
> To me, this is one of the fascinations of music.
> A great composer can elaborate processes on many
> different levels simultaneously. Music that is
> this deep rewards a lifetime of repeated listening.
> Anyway, i digress ...

Thanks monz I'll look at that. Don't underestimate yourself, just
remember what a small percentage of people even know this stuff at
all, for better or for worse, anyway the breakthroughs will happen,
I'm convinced of that. The mind does a lot on it own, doesn't it?
Whatever that means.

My stuff is basically just Polya theory. I've developed kind of a
grid format for doing it, including complementability of sets,
I've worked as high as 5 colors, I can imagine some interesting
applications to composition where notes are like colored beads (a la
The Glass Bead Game)

Now this is so silly someone should flame me but here goes:

1,6,19,43,66,80 are the subsets (1-6) in 12t-ET. (C12=C4 X C3)
1,6,12,29,38,50 are reduced for mirror-image (D12)

Of these 1,12,19,38,43,50 and 66 are also meantone temperaments
(doubling 19,33->38,66)

29 is schismic

So a representative from each column is a meantone temperament,
except for 6. Coincidence? 6 and 80 are no good, they are dicot and
blackwood.

Of course with meantone you have 5x+7y for every temperament, so for
example, 43 is 5(3)+ 7(4), 3 and 4 are contiguous, with Polya theory
you also run into the 5x + 7y thing too in 12-tET, symmetrical sets
are 5x and the assymmetrical ones are 7y, so that explains the
coincidence in part.

It would be fun to run other meantone temperaments and see if their
subsets have this property

Now it is I who digress!

Paul Hj

>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> This is a request for information on all the group theoretical
> symmetries of C24 (Would that be C8 X C3?)
>
> Particularly, the Polya Polynomials for every sort of decomposition,
> D8 X C3, etc. I could calculate by hand, but that would take
> forever.

*** With help from the GAP Forum, here finally, is D8 X S3, Expanded:

x^24+x^23+9*x^22+31*x^21+151*x^20+496*x^19+1553*x^18+3777*x^17+8028*x^
16+13981*x^15+21042*x^14+26516*x^13+28898*x^12+26516*x^11+21042*x^10+1
3981*x^9+8028*x^8+3777*x^7+1553*x^6+496*x^5+151*x^4+31*x^3+9*x^2+x+1

Now, working with multiplication modulo groups (M24, but not the
Mathieu group!), one gets a decomposition of C2 X C2 X C2

This is based on C2 X C2 = 1,3,5,7 for 2^3 part, and C2 = 1,2 for
the 3 part, scaled up one obtains 3,9,15,21 and 8,16 respectively,
added together (8 combinations), one obtains 1,5,7,11,13,17,23.

Now the generators for D8 X S3 are just 1,7,17,23, the remaining
are exactly 12 higher, I am not sure what group theoretical structure
would use all 8. I know where the other four come from, this is
where C2 X C2 can be broken down into (0,4)+(3,5) to obtain 1,3,5,7
(Only 1,7 are used in D8 itself, and only 3,5 -> 9,15 are needed in
D8 X S3, in conjunction with 8,16 from S3)

So what group structure would generate the full affine group, having
24 x 8 = 192 group order? Would that have any bearing on calculations
in a quarter-tone system? Maybe in terms of (musical-set) interval-
vectors used?

(I know, A little knowledge is a dangerous thing)

Happy Epiphany

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > This is a request for information on all the group theoretical
> > symmetries of C24 (Would that be C8 X C3?)
> >
> > Particularly, the Polya Polynomials for every sort of
decomposition,
> > D8 X C3, etc. I could calculate by hand, but that would take
> > forever.
>
> *** With help from the GAP Forum, here finally, is D8 X S3,
Expanded:
>
>
x^24+x^23+9*x^22+31*x^21+151*x^20+496*x^19+1553*x^18+3777*x^17+8028*x^
>
16+13981*x^15+21042*x^14+26516*x^13+28898*x^12+26516*x^11+21042*x^10+1
> 3981*x^9+8028*x^8+3777*x^7+1553*x^6+496*x^5+151*x^4+31*x^3+9*x^2+x+1
>
> Now, working with multiplication modulo groups (M24, but not the
> Mathieu group!), one gets a decomposition of C2 X C2 X C2
>
> This is based on C2 X C2 = 1,3,5,7 for 2^3 part, and C2 = 1,2 for
> the 3 part, scaled up one obtains 3,9,15,21 and 8,16 respectively,
> added together (8 combinations), one obtains 1,5,7,11,13,17,23.
>
> Now the generators for D8 X S3 are just 1,7,17,23, the remaining
> are exactly 12 higher, I am not sure what group theoretical
structure
> would use all 8. I know where the other four come from, this is
> where C2 X C2 can be broken down into (0,4)+(3,5) to obtain 1,3,5,7
> (Only 1,7 are used in D8 itself, and only 3,5 -> 9,15 are needed in
> D8 X S3, in conjunction with 8,16 from S3)
>
> So what group structure would generate the full affine group, having
> 24 x 8 = 192 group order? Would that have any bearing on
calculations
> in a quarter-tone system? Maybe in terms of (musical-set) interval-
> vectors used?
>
> (I know, A little knowledge is a dangerous thing)
>
> Happy Epiphany
>
> PGH

Well, I found it. But I don't know the group theoretical name. (It's
probably one of those weird ones I never understood the name of.

Here's my GAP code, I've left out all the boring steps. I have to
switch from intransitive to transitive group direct product
(Cartesian). Hope you don't hate it. I found that is is D8 X S3, with
an extra rotation of odd points of the octagon one half turn. based
on x + 12x (evens cancel). This will use all primes totient to 24, as
generators (1 gen), (2 2*gen) etc.

1. First, my Group (Intransitive, it's the octogon and the triangle,
this will make 8 x 3 = 24).

Group( [ (1,2,3,4,5,6,7,8), (1,3), (2,6), (5,7), (1,5), (3,7), (1,7),
(3,5), ( 9,10), ( 9,10,11) ] )

2. Second, the cartesian product thing

gap> cart:=Cartesian([1..8],[9..11]);
[ [ 1, 9 ], [ 1, 10 ], [ 1, 11 ], [ 2, 9 ], [ 2, 10 ], [ 2, 11 ], [
3, 9 ],
[ 3, 10 ], [ 3, 11 ], [ 4, 9 ], [ 4, 10 ], [ 4, 11 ], [ 5, 9 ], [
5, 10 ],
[ 5, 11 ], [ 6, 9 ], [ 6, 10 ], [ 6, 11 ], [ 7, 9 ], [ 7, 10 ], [
7, 11 ],
[ 8, 9 ], [ 8, 10 ], [ 8, 11 ] ]

3. Third, the Group, as Transitive Group Direct Product, up to
relabelling:

Group(
[ ( 1, 4, 7,10,13,16,19,22)( 2, 5, 8,11,14,17,20,23)( 3, 6,
9,12,15,18,21,24)
, ( 1, 7)( 2, 8)( 3, 9), ( 4,16)( 5,17)( 6,18), (13,19)(14,20)
(15,21),
( 1,13)( 2,14)( 3,15), ( 7,19)( 8,20)( 9,21), ( 1,19)( 2,20)( 3,21),
( 7,13)( 8,14)( 9,15), ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)
(19,20)
(22,23), ( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)
(16,17,18)
(19,20,21)(22,23,24) ] )

4. Fourth, the CycleIndex:

1/6912*x_1^24+1/576*x_1^18*x_2^3+1/432*x_1^15*x_3^3+7/1152*x_1^12*x_2^
6+1/230\
4*x_1^8*x_2^8+1/576*x_1^12*x_4^3+1/72*x_1^9*x_2^3*x_3^3+1/96*x_1^6*x_2
^9+7/38\
4*x_1^4*x_2^10+1/64*x_1^2*x_2^11+1/96*x_1^6*x_2^3*x_4^3+1/108*x_1^6*x_
3^6+1/1\
44*x_1^5*x_2^5*x_3*x_6+1/144*x_1^3*x_2^6*x_3^3+11/576*x_2^12+1/192*x_1
^4*x_2^\
4*x_4^3+1/24*x_1^3*x_2^6*x_3*x_6+1/32*x_1^2*x_2^5*x_4^3+1/48*x_1*x_2^7
*x_3*x_\
6+1/72*x_1^3*x_3^3*x_4^3+5/48*x_2^6*x_4^3+1/36*x_1^2*x_2^2*x_3^2*x_6^2
+3/128*\
x_3^8+1/24*x_1*x_2*x_3*x_4^3*x_6+1/32*x_3^6*x_6+1/9*x_2^3*x_6^3+5/192*
x_3^4*x\
_6^2+1/16*x_4^6+1/32*x_3^4*x_12+1/96*x_3^2*x_6^3+1/48*x_3^2*x_6*x_12+2
5/384*x\
_6^4+5/96*x_6^2*x_12+1/12*x_8^3+1/32*x_12^2+1/24*x_24

And finally, the Expanded Polya Polynomial. There are 538 octads.
Each power represents a cardinality, so the coefficient of x^8 for
example is octads. Tomorrow I will run S8 X S3.

x_1^24+x_1^23+5*x_1^22+12*x_1^21+35*x_1^20+74*x_1^19+169*x_1^18+303*x_
1^17+53\
8*x_1^16+796*x_1^15+1098*x_1^14+1288*x_1^13+1401*x_1^12+1288*x_1^11+10
98*x_1^\
10+796*x_1^9+538*x_1^8+303*x_1^7+169*x_1^6+74*x_1^5+35*x_1^4+12*x_1^3+
5*x_1^2\
+x_1+1

The most helpful command from the GAP Forum was a function to convert
to a transitive product using the Cartesian thing. That held me up
for a while. Now, in the interval vector for sets in 24t-ET, you have
(1,2,3,4,5,6,7,8,9,10,11,12) up to the "tritone" or half-octave. So
the top 4 multipliers just make a set the inverse of the lower ones,
not a big deal. (M13, M17, M19, M23) Now M1 is trivial. (Mere
transposition.) The remaining then are just [(1,5)(2,10)(3,15), etc],
[(1,7),(2,14),(3,21) etc], and [(1,11),(2,22),(3,9) etc], now these
reduce to absolute value of the interval vector, so 15->9, 22->2 etc.

So the swapping of these values is trickier than with 12tET. I need
to look up my Z-relation stuff too, and see which Z-related sets
might be invariant under these swaps, and perhaps there are some non-
Z_related sets that are also...

Thanks Sorry for any overkill

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > This is a request for information on all the group theoretical
> > symmetries of C24 (Would that be C8 X C3?)
> >
> > Particularly, the Polya Polynomials for every sort of
decomposition,
> > D8 X C3, etc. I could calculate by hand, but that would take
> > forever.
>
> *** With help from the GAP Forum, here finally, is D8 X S3,
Expanded:
>
>
x^24+x^23+9*x^22+31*x^21+151*x^20+496*x^19+1553*x^18+3777*x^17+8028*x^
>
16+13981*x^15+21042*x^14+26516*x^13+28898*x^12+26516*x^11+21042*x^10+1
> 3981*x^9+8028*x^8+3777*x^7+1553*x^6+496*x^5+151*x^4+31*x^3+9*x^2+x+1
>
> Now, working with multiplication modulo groups (M24, but not the
> Mathieu group!), one gets a decomposition of C2 X C2 X C2
>
> This is based on C2 X C2 = 1,3,5,7 for 2^3 part, and C2 = 1,2 for
> the 3 part, scaled up one obtains 3,9,15,21 and 8,16 respectively,
> added together (8 combinations), one obtains 1,5,7,11,13,17,23.
>
> Now the generators for D8 X S3 are just 1,7,17,23, the remaining
> are exactly 12 higher, I am not sure what group theoretical
structure
> would use all 8. I know where the other four come from, this is
> where C2 X C2 can be broken down into (0,4)+(3,5) to obtain 1,3,5,7
> (Only 1,7 are used in D8 itself, and only 3,5 -> 9,15 are needed in
> D8 X S3, in conjunction with 8,16 from S3)
>
> So what group structure would generate the full affine group, having
> 24 x 8 = 192 group order? Would that have any bearing on
calculations
> in a quarter-tone system? Maybe in terms of (musical-set) interval-
> vectors used?
>
> (I know, A little knowledge is a dangerous thing)
>
> Happy Epiphany
>
> PGH

Well, I found it. But I don't know the group theoretical name. (It's
probably one of those weird ones I never understood the name of.

Here's my GAP code, I've left out all the boring steps. I have to
switch from intransitive to transitive group direct product
(Cartesian). Hope you don't hate it. I found that is is D8 X S3, with
an extra rotation of odd points of the octagon one half turn. based
on x + 12x (evens cancel). This will use all primes totient to 24, as
generators (1 gen), (2 2*gen) etc.

1. First, my Group (Intransitive, it's the octogon and the triangle,
this will make 8 x 3 = 24).

Group( [ (1,2,3,4,5,6,7,8), (1,3), (2,6), (5,7), (1,5), (3,7), (1,7),
(3,5), ( 9,10), ( 9,10,11) ] )

2. Second, the cartesian product thing

gap> cart:=Cartesian([1..8],[9..11]);
[ [ 1, 9 ], [ 1, 10 ], [ 1, 11 ], [ 2, 9 ], [ 2, 10 ], [ 2, 11 ], [
3, 9 ],
[ 3, 10 ], [ 3, 11 ], [ 4, 9 ], [ 4, 10 ], [ 4, 11 ], [ 5, 9 ], [
5, 10 ],
[ 5, 11 ], [ 6, 9 ], [ 6, 10 ], [ 6, 11 ], [ 7, 9 ], [ 7, 10 ], [
7, 11 ],
[ 8, 9 ], [ 8, 10 ], [ 8, 11 ] ]

3. Third, the Group, as Transitive Group Direct Product, up to
relabelling:

Group(
[ ( 1, 4, 7,10,13,16,19,22)( 2, 5, 8,11,14,17,20,23)( 3, 6,
9,12,15,18,21,24)
, ( 1, 7)( 2, 8)( 3, 9), ( 4,16)( 5,17)( 6,18), (13,19)(14,20)
(15,21),
( 1,13)( 2,14)( 3,15), ( 7,19)( 8,20)( 9,21), ( 1,19)( 2,20)( 3,21),
( 7,13)( 8,14)( 9,15), ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)
(19,20)
(22,23), ( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)
(16,17,18)
(19,20,21)(22,23,24) ] )

4. Fourth, the CycleIndex:

1/6912*x_1^24+1/576*x_1^18*x_2^3+1/432*x_1^15*x_3^3+7/1152*x_1^12*x_2^
6+1/230\
4*x_1^8*x_2^8+1/576*x_1^12*x_4^3+1/72*x_1^9*x_2^3*x_3^3+1/96*x_1^6*x_2
^9+7/38\
4*x_1^4*x_2^10+1/64*x_1^2*x_2^11+1/96*x_1^6*x_2^3*x_4^3+1/108*x_1^6*x_
3^6+1/1\
44*x_1^5*x_2^5*x_3*x_6+1/144*x_1^3*x_2^6*x_3^3+11/576*x_2^12+1/192*x_1
^4*x_2^\
4*x_4^3+1/24*x_1^3*x_2^6*x_3*x_6+1/32*x_1^2*x_2^5*x_4^3+1/48*x_1*x_2^7
*x_3*x_\
6+1/72*x_1^3*x_3^3*x_4^3+5/48*x_2^6*x_4^3+1/36*x_1^2*x_2^2*x_3^2*x_6^2
+3/128*\
x_3^8+1/24*x_1*x_2*x_3*x_4^3*x_6+1/32*x_3^6*x_6+1/9*x_2^3*x_6^3+5/192*
x_3^4*x\
_6^2+1/16*x_4^6+1/32*x_3^4*x_12+1/96*x_3^2*x_6^3+1/48*x_3^2*x_6*x_12+2
5/384*x\
_6^4+5/96*x_6^2*x_12+1/12*x_8^3+1/32*x_12^2+1/24*x_24

And finally, the Expanded Polya Polynomial. There are 538 octads.
Each power represents a cardinality, so the coefficient of x^8 for
example is octads. Tomorrow I will run S8 X S3.

x_1^24+x_1^23+5*x_1^22+12*x_1^21+35*x_1^20+74*x_1^19+169*x_1^18+303*x_
1^17+53\
8*x_1^16+796*x_1^15+1098*x_1^14+1288*x_1^13+1401*x_1^12+1288*x_1^11+10
98*x_1^\
10+796*x_1^9+538*x_1^8+303*x_1^7+169*x_1^6+74*x_1^5+35*x_1^4+12*x_1^3+
5*x_1^2\
+x_1+1

The most helpful command from the GAP Forum was a function to convert
to a transitive product using the Cartesian thing. That held me up
for a while. Now, in the interval vector for sets in 24t-ET, you have
(1,2,3,4,5,6,7,8,9,10,11,12) up to the "tritone" or half-octave. So
the top 4 multipliers just make a set the inverse of the lower ones,
not a big deal. (M13, M17, M19, M23) Now M1 is trivial. (Mere
transposition.) The remaining then are just [(1,5)(2,10)(3,15), etc],
[(1,7),(2,14),(3,21) etc], and [(1,11),(2,22),(3,9) etc], now these
reduce to absolute value of the interval vector, so 15->9, 22->2 etc.

So the swapping of these values is trickier than with 12tET. I need
to look up my Z-relation stuff too, and see which Z-related sets
might be invariant under these swaps, and perhaps there are some non-
Z_related sets that are also...

Thanks Sorry for any overkill

PGH

All,

My apologies for an error in this posting thread. Misplacement of the
commas in the Group completely gave the wrong Cycle Index and Polya
Polynomial.

I have dubbed this group structure D8(X2)X S3 for now, until I
determine the correct name. It uses all of the primes totient to
24 as generators.

I know this isn't that central to tuning, but I feel I should at
least present correct data if I post anything. (Measure twice! Cut
once!)

G:= Group( [ (1,2,3,4,5,6,7,8), (1,3)(2,6)(5,7), (1,5)(3,7), (1,7)
(2,6)(3,5),( 9,10), ( 9,10,11) ] );

Converted to Cartesian format:

Group([ ( 1, 4, 7,10,13,16,19,22)( 2, 5, 8,11,14,17,20,23)( 3, 6,
9,12,15,18,21,24),
( 1, 7)( 2, 8)( 3, 9)( 4,16)( 5,17)( 6,18)(13,19)(14,20)(15,21),
( 1,13)( 2,14)( 3,15)( 7,19)( 8,20)( 9,21),
( 1,19)( 2,20)( 3,21)( 4,16)( 5,17)( 6,18)( 7,13)( 8,14)( 9,15),
( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23),
( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)
(19,20,21)(22,23,24) ] )

Correct Cycle Index:

1/192*x_1^24+1/96*x_1^12*x_2^6+1/64*x_1^8*x_2^8+1/24*x_1^6*x_2^9+1/32*
x_1^4*x_2^10+1/8*x_1^2*x_2^11+5/48*x_2^12+1/96*x_3^8+1/48*x_3^4*x_6^2+
1/6*x_4^6+1/12*x_3^2*x_6^3+5/96*x_6^4+1/6*x_8^3+1/12*x_12^2+1/12*x_24

Correct Polya Polynomial:

x_1^24+x_1^23+7*x_1^22+23*x_1^21+97*x_1^20+294*x_1^19+870*x_1^18+2051*
x_1^17+4272*x_1^16+7352*x_1^15+10980*x_1^14+13790*x_1^13+15008*x_1^12+
13790*x_1^11+10980*x_1^10+7352*x_1^9+4272*x_1^8+2051*x_1^7+870*x_1^6+2
94*x_1^5+97*x_1^4+23*x_1^3+7*x_1^2+x_1+1

There are 4,272 octads....

Also, the notation M1, M5, M7, etc is confusing, same as both Mathieu
Groups and Multiplication Modulo Groups. They stand for the
generators, such as 1,5,7,11,13,17,19,23 in this case.

Thanks - I will not "cry wolf" again!

PGH

(Garbage from before deleted!)