Interesting numerical coincidences (Combinatorics etc)

Just wanted to post an interesting discovery I just made. This
concerns Interval Vectors of Hexachords in 12-ET and 24-ET. If one
counts all the Hexachords in 12-ET, you get 80. Reducing for
asymmetry, you get 50, which is the Tn/TnI set count. (John Rahn)
Reducing again for Z-relations (a la Allen Forte... this is when two
or more sets have the same interval vector) you get 35. Factorizing
35, you get 7 * 5. Note that these numbers lie along the meantone
line in zoomq.gif Taxicab diagram (for example).

Okay, here's the good part. In 24-ET, there are 5620 hexachords.
Reducing for asymmetry gives 2920 Tn/TnI set types. Reducing for the
Z-relation gives 2635 sets. Factorizing gives 5 * 17 * 31. These sets
ALL LINE ALONG THE MEANTONE LINE in zoomr.gif (for example).

I need to run C{18,6} and C{30,6} to see if this pattern holds up.
For example for C{24,8}, it doesn't work, there are 12,844 octochords
which is 4 * 169 *19.

In 12-ET, it works for all chords (see below)

chord: interval vector count comma-line
trichord 12 = 4 * 3 dicot
tetrachord 28 = 4 * 7 dicot
pentachord 35 = 7 * 5 meantone
hexachord 35 = 7 * 5 meantone

Just a coincidence? Also, C{24,4} has 253 interval vectors, that's
13 * 11, which also lies along dicot.

I plan to run this for as many hexachords as I can. In 12-ET, you can
arrange the hexachords in a nice 7 X 5 grid based on frequencies of
the various intervals (1,2,3,4,5.. tritone excluded). Is C{24,6} based
on a 3-dimensional grid of 5 X 17 X 31???? Also note that these
numbers add up to 53, 5 + 17 + 31 = 22 + 31 = 53. Another coincidence?
Comments, anyone?

Paul (A music B.M. and math B.A.)

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:
> Just wanted to post an interesting discovery I just made. This
> concerns Interval Vectors of Hexachords in 12-ET and 24-ET. If one
> counts all the Hexachords in 12-ET, you get 80. Reducing for
> asymmetry, you get 50, which is the Tn/TnI set count. (John Rahn)
> Reducing again for Z-relations (a la Allen Forte... this is when
two
> or more sets have the same interval vector) you get 35. Factorizing
> 35, you get 7 * 5. Note that these numbers lie along the meantone
> line in zoomq.gif Taxicab diagram (for example).
>
> Okay, here's the good part. In 24-ET, there are 5620 hexachords.
> Reducing for asymmetry gives 2920 Tn/TnI set types. Reducing for
the
> Z-relation gives 2635 sets. Factorizing gives 5 * 17 * 31. These
sets
> ALL LINE ALONG THE MEANTONE LINE in zoomr.gif (for example).
>
> I need to run C{18,6} and C{30,6} to see if this pattern holds up.
> For example for C{24,8}, it doesn't work, there are 12,844
octochords
> which is 4 * 169 *19.
>
> In 12-ET, it works for all chords (see below)
>
> chord: interval vector count comma-line
> trichord 12 = 4 * 3 dicot
> tetrachord 28 = 4 * 7 dicot
> pentachord 35 = 7 * 5 meantone
> hexachord 35 = 7 * 5 meantone
>
> Just a coincidence? Also, C{24,4} has 253 interval vectors, that's
> 13 * 11, which also lies along dicot.
>
> I plan to run this for as many hexachords as I can. In 12-ET, you
can
> arrange the hexachords in a nice 7 X 5 grid based on frequencies of
> the various intervals (1,2,3,4,5.. tritone excluded). Is C{24,6}
based
> on a 3-dimensional grid of 5 X 17 X 31???? Also note that these
> numbers add up to 53, 5 + 17 + 31 = 22 + 31 = 53. Another
coincidence?
> Comments, anyone?
>
> Paul (A music B.M. and math B.A.)

A couple quick addendums/corrections: When I said "count all the
hexachords in 12-ET, I meant all the hexachords after being reduced
for transposition, this is 80" When I said "all the sets that have
the same interval vector, I really meant "all the set types" Lastly,
when I said these sets ALL LINE ALONG THE MEANTONE LINE, I meant the
factors of 2635 (5 * 17 * 31). not sets, but factors of 2635, the
count based on UNIQUE interval vectors of hexachords in the
quartertone system. Thanks

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:
> Just wanted to post an interesting discovery I just made. This
> concerns Interval Vectors of Hexachords in 12-ET and 24-ET. If one
> counts all the Hexachords in 12-ET, you get 80. Reducing for
> asymmetry, you get 50, which is the Tn/TnI set count. (John Rahn)
> Reducing again for Z-relations (a la Allen Forte... this is when
two
> or more sets have the same interval vector) you get 35. Factorizing
> 35, you get 7 * 5. Note that these numbers lie along the meantone
> line in zoomq.gif Taxicab diagram (for example).
>
> Okay, here's the good part. In 24-ET, there are 5620 hexachords.
> Reducing for asymmetry gives 2920 Tn/TnI set types. Reducing for
the
> Z-relation gives 2635 sets. Factorizing gives 5 * 17 * 31. These
sets
> ALL LINE ALONG THE MEANTONE LINE in zoomr.gif (for example).
>
> I need to run C{18,6} and C{30,6} to see if this pattern holds up.
> For example for C{24,8}, it doesn't work, there are 12,844
octochords
> which is 4 * 169 *19.
>
> In 12-ET, it works for all chords (see below)
>
> chord: interval vector count comma-line
> trichord 12 = 4 * 3 dicot
> tetrachord 28 = 4 * 7 dicot
> pentachord 35 = 7 * 5 meantone
> hexachord 35 = 7 * 5 meantone
>
> Just a coincidence? Also, C{24,4} has 253 interval vectors, that's
> 13 * 11, which also lies along dicot.
>
> I plan to run this for as many hexachords as I can. In 12-ET, you
can
> arrange the hexachords in a nice 7 X 5 grid based on frequencies of
> the various intervals (1,2,3,4,5.. tritone excluded). Is C{24,6}
based
> on a 3-dimensional grid of 5 X 17 X 31???? Also note that these
> numbers add up to 53, 5 + 17 + 31 = 22 + 31 = 53. Another
coincidence?
> Comments, anyone?
>
> Paul (A music B.M. and math B.A.)
New finding: C{18,6} Reduces to 493 = 29 X 17 (in terms of unique
interval vectors). 17 is on meantone line, but 29 is on schismic. So,
not along the same line. However, there may be a different pattern:
C{6n,6} may all be composed of factors equalling (5x + 7y). C{4n,4}
might also have a pattern...Isn't this lovely?

Paul

--- In tuning-math@yahoogroups.com, "paulhjelmstad"

> New finding: C{18,6} Reduces to 493 = 29 X 17 (in terms of unique
> interval vectors). 17 is on meantone line, but 29 is on schismic.
So,
> not along the same line.

what do you mean? look again. the schismic line has a 17 on it as
close to the ji center as the 17 on the meantone line!

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
>
> > New finding: C{18,6} Reduces to 493 = 29 X 17 (in terms of unique
> > interval vectors). 17 is on meantone line, but 29 is on schismic.
> So,
> > not along the same line.
>
> what do you mean? look again. the schismic line has a 17 on it as
> close to the ji center as the 17 on the meantone line!

Hooray! Thanks for pointing the out. What a totally weird coincidence.
My brother is crunching C{30,6} as we speak. There are 19,811 sets
reduced to 10,133 Tn/TnI set types... Maybe C{4n,6} have factors on
meantone, and the remaining C{2n,6} are along schismic.

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

> Hooray! Thanks for pointing the out. What a totally weird coincidence.
> My brother is crunching C{30,6} as we speak. There are 19,811 sets
> reduced to 10,133 Tn/TnI set types... Maybe C{4n,6} have factors on
> meantone, and the remaining C{2n,6} are along schismic.

I don't see any weird coincidences; however, here is C(n, 6) for n
from 7 to 60:

7 7
8 2^2 * 7
9 2^2 * 3 * 7
10 2 * 3 * 5 * 7
11 2 * 3 * 7 * 11
12 2^2 * 3 * 7 * 11
13 2^2 * 3 * 11 * 13
14 3 * 7 * 11 * 13
15 5 * 7 * 11 * 13
16 2^3 * 7 * 11 * 13
17 2^3 * 7 * 13 * 17
18 2^2 * 3 * 7 * 13 * 17
19 2^2 * 3 * 7 * 17 * 19
20 2^3 * 3 * 5 * 17 * 19
21 2^3 * 3 * 7 * 17 * 19
22 3 * 7 * 11 * 17 * 19
23 3 * 7 * 11 * 19 * 23
24 2^2 * 7 * 11 * 19 * 23
25 2^2 * 5^2 * 7 * 11 * 23
26 2 * 5 * 7 * 11 * 13 * 23
27 2 * 3^2 * 5 * 11 * 13 * 23
28 2^2 * 3^2 * 5 * 7 * 13 * 23
29 2^2 * 3^2 * 5 * 7 * 13 * 29
30 3^2 * 5^2 * 7 * 13 * 29
31 3^2 * 7 * 13 * 29 * 31
32 2^4 * 3^2 * 7 * 29 * 31
33 2^4 * 7 * 11 * 29 * 31
34 2^3 * 11 * 17 * 29 * 31
35 2^3 * 5 * 7 * 11 * 17 * 31
36 2^4 * 3 * 7 * 11 * 17 * 31
37 2^4 * 3 * 7 * 11 * 17 * 37
38 3 * 7 * 11 * 17 * 19 * 37
39 3 * 7 * 13 * 17 * 19 * 37
40 2^2 * 3 * 5 * 7 * 13 * 19 * 37
41 2^2 * 3 * 13 * 19 * 37 * 41
42 2 * 7 * 13 * 19 * 37 * 41
43 2 * 7 * 13 * 19 * 41 * 43
44 2^2 * 7 * 11 * 13 * 41 * 43
45 2^2 * 3 * 5 * 7 * 11 * 41 * 43
46 3 * 7 * 11 * 23 * 41 * 43
47 3 * 7 * 11 * 23 * 43 * 47
48 2^3 * 3 * 11 * 23 * 43 * 47
49 2^3 * 3 * 7^2 * 11 * 23 * 47
50 2^2 * 3 * 5^2 * 7^2 * 23 * 47
51 2^2 * 5 * 7^2 * 17 * 23 * 47
52 2^3 * 5 * 7^2 * 13 * 17 * 47
53 2^3 * 5 * 7^2 * 13 * 17 * 53
54 3^2 * 5 * 7^2 * 13 * 17 * 53
55 3^2 * 5^2 * 11 * 13 * 17 * 53
56 2^2 * 3^2 * 7 * 11 * 13 * 17 * 53
57 2^2 * 3^2 * 7 * 11 * 13 * 19 * 53
58 2 * 3^2 * 7 * 11 * 19 * 29 * 53
59 2 * 3^2 * 7 * 11 * 19 * 29 * 59
60 2^2 * 5 * 7 * 11 * 19 * 29 * 59

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> >
> > > New finding: C{18,6} Reduces to 493 = 29 X 17 (in terms of
unique
> > > interval vectors). 17 is on meantone line, but 29 is on
schismic.
> > So,
> > > not along the same line.
> >
> > what do you mean? look again. the schismic line has a 17 on it as
> > close to the ji center as the 17 on the meantone line!
>
> Hooray! Thanks for pointing the out. What a totally weird
coincidence.
> My brother is crunching C{30,6} as we speak. There are 19,811 sets
> reduced to 10,133 Tn/TnI set types... Maybe C{4n,6} have factors on
> meantone, and the remaining C{2n,6} are along schismic.

Well good news. C{30,6} reduces to 9,701 sets (based on unique
interval vector count). This is 89 * 109. I found 89 along "Schismic"
in Zoomr.gif (hard to read, its really small, and doesn't show up in
Zooms.gif. I can't find 109. Help! Can someone tell me where 109
would show up in a taxicab diagram. Thanks!

Paul
in

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > Hooray! Thanks for pointing the out. What a totally weird
coincidence.
> > My brother is crunching C{30,6} as we speak. There are 19,811
sets
> > reduced to 10,133 Tn/TnI set types... Maybe C{4n,6} have factors
on
> > meantone, and the remaining C{2n,6} are along schismic.
>
> I don't see any weird coincidences; however, here is C(n, 6) for n
> from 7 to 60:
>
> 7 7
> 8 2^2 * 7
> 9 2^2 * 3 * 7
> 10 2 * 3 * 5 * 7
> 11 2 * 3 * 7 * 11
> 12 2^2 * 3 * 7 * 11
> 13 2^2 * 3 * 11 * 13
> 14 3 * 7 * 11 * 13
> 15 5 * 7 * 11 * 13
> 16 2^3 * 7 * 11 * 13
> 17 2^3 * 7 * 13 * 17
> 18 2^2 * 3 * 7 * 13 * 17
> 19 2^2 * 3 * 7 * 17 * 19
> 20 2^3 * 3 * 5 * 17 * 19
> 21 2^3 * 3 * 7 * 17 * 19
> 22 3 * 7 * 11 * 17 * 19
> 23 3 * 7 * 11 * 19 * 23
> 24 2^2 * 7 * 11 * 19 * 23
> 25 2^2 * 5^2 * 7 * 11 * 23
> 26 2 * 5 * 7 * 11 * 13 * 23
> 27 2 * 3^2 * 5 * 11 * 13 * 23
> 28 2^2 * 3^2 * 5 * 7 * 13 * 23
> 29 2^2 * 3^2 * 5 * 7 * 13 * 29
> 30 3^2 * 5^2 * 7 * 13 * 29
> 31 3^2 * 7 * 13 * 29 * 31
> 32 2^4 * 3^2 * 7 * 29 * 31
> 33 2^4 * 7 * 11 * 29 * 31
> 34 2^3 * 11 * 17 * 29 * 31
> 35 2^3 * 5 * 7 * 11 * 17 * 31
> 36 2^4 * 3 * 7 * 11 * 17 * 31
> 37 2^4 * 3 * 7 * 11 * 17 * 37
> 38 3 * 7 * 11 * 17 * 19 * 37
> 39 3 * 7 * 13 * 17 * 19 * 37
> 40 2^2 * 3 * 5 * 7 * 13 * 19 * 37
> 41 2^2 * 3 * 13 * 19 * 37 * 41
> 42 2 * 7 * 13 * 19 * 37 * 41
> 43 2 * 7 * 13 * 19 * 41 * 43
> 44 2^2 * 7 * 11 * 13 * 41 * 43
> 45 2^2 * 3 * 5 * 7 * 11 * 41 * 43
> 46 3 * 7 * 11 * 23 * 41 * 43
> 47 3 * 7 * 11 * 23 * 43 * 47
> 48 2^3 * 3 * 11 * 23 * 43 * 47
> 49 2^3 * 3 * 7^2 * 11 * 23 * 47
> 50 2^2 * 3 * 5^2 * 7^2 * 23 * 47
> 51 2^2 * 5 * 7^2 * 17 * 23 * 47
> 52 2^3 * 5 * 7^2 * 13 * 17 * 47
> 53 2^3 * 5 * 7^2 * 13 * 17 * 53
> 54 3^2 * 5 * 7^2 * 13 * 17 * 53
> 55 3^2 * 5^2 * 11 * 13 * 17 * 53
> 56 2^2 * 3^2 * 7 * 11 * 13 * 17 * 53
> 57 2^2 * 3^2 * 7 * 11 * 13 * 19 * 53
> 58 2 * 3^2 * 7 * 11 * 19 * 29 * 53
> 59 2 * 3^2 * 7 * 11 * 19 * 29 * 59
> 60 2^2 * 5 * 7 * 11 * 19 * 29 * 59

Thanks, Gene, but actually look at my first post. I am not just
calculating C{6n,6} I am looking at sets based on unique interval
vectors. I just called the sets C{18,6} and C{30,6} as a kind of
shorthand. Should have said "sets reduced from ..."

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> > wrote:
> > > --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> > >
> > > > New finding: C{18,6} Reduces to 493 = 29 X 17 (in terms of
> unique
> > > > interval vectors). 17 is on meantone line, but 29 is on
> schismic.
> > > So,
> > > > not along the same line.
> > >
> > > what do you mean? look again. the schismic line has a 17 on it
as
> > > close to the ji center as the 17 on the meantone line!
> >
> > Hooray! Thanks for pointing the out. What a totally weird
> coincidence.
> > My brother is crunching C{30,6} as we speak. There are 19,811
sets
> > reduced to 10,133 Tn/TnI set types... Maybe C{4n,6} have factors
on
> > meantone, and the remaining C{2n,6} are along schismic.
>
> Well good news. C{30,6} reduces to 9,701 sets (based on unique
> interval vector count). This is 89 * 109. I found 89
along "Schismic"
> in Zoomr.gif (hard to read, its really small, and doesn't show up
in
> Zooms.gif. I can't find 109. Help! Can someone tell me where 109
> would show up in a taxicab diagram. Thanks!
>
> Paul

I found 109 at the intersection of Escapade and Tetracot. Not on
Schismic (Definitely). Well I hope you are not all sick of this
subject. The question is, does 89 appear along Escapade or Tetracot?
That would be cool. I will quit posting for today. I will present my
results on all this tomorrow. Thanks and bye for now. Paul

> This is 89 * 109. I found 89 along "Schismic" in
> Zoomr.gif (hard to read, its really small, and
> doesn't show up in Zooms.gif. I can't find 109. Help!
> Can someone tell me where 109 would show up in a
> taxicab diagram. Thanks!

The diagram appears at different scales here:

http://sonic-arts.org/dict/eqtemp.htm

These are not "taxicab" diagrams. I'm don't know of
any cute name for them. Maybe Paul E. would care to
coin one.

-Carl

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

> Thanks, Gene, but actually look at my first post. I am not just
> calculating C{6n,6} I am looking at sets based on unique interval
> vectors. I just called the sets C{18,6} and C{30,6} as a kind of
> shorthand. Should have said "sets reduced from ..."

I've taught this stuff too many times to be able to see C(n, m) in
more than one way. :)

Evidently, you have another function you want computed.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> <paul.hjelmstad@u...> wrote:
>
> > Thanks, Gene, but actually look at my first post. I am not just
> > calculating C{6n,6} I am looking at sets based on unique interval
> > vectors. I just called the sets C{18,6} and C{30,6} as a kind of
> > shorthand. Should have said "sets reduced from ..."
>
> I've taught this stuff too many times to be able to see C(n, m) in
> more than one way. :)
>
> Evidently, you have another function you want computed.

Sorry, I get excited and rush things. My brother has been computing
these sets, by indexing uniquely on the interval vectors. For example,
a set in C{24,6} would be (1,2,3,4,5,6) and have interval vector
<5,4,3,2,1,0,0,0,0,0,0,0> a la John Rahn or Allen Forte. I really
thought I was onto something, but I fear its just coincidences...:)

--- In tuning-math@yahoogroups.com, "paulhjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...>
> > > wrote:
> > > > --- In tuning-math@yahoogroups.com, "paulhjelmstad"
> > > >
> > > > > New finding: C{18,6} Reduces to 493 = 29 X 17 (in terms of
> > unique
> > > > > interval vectors). 17 is on meantone line, but 29 is on
> > schismic.
> > > > So,
> > > > > not along the same line.
> > > >
> > > > what do you mean? look again. the schismic line has a 17 on
it
> as
> > > > close to the ji center as the 17 on the meantone line!
> > >
> > > Hooray! Thanks for pointing the out. What a totally weird
> > coincidence.
> > > My brother is crunching C{30,6} as we speak. There are 19,811
> sets
> > > reduced to 10,133 Tn/TnI set types... Maybe C{4n,6} have
factors
> on
> > > meantone, and the remaining C{2n,6} are along schismic.
> >
> > Well good news. C{30,6} reduces to 9,701 sets (based on unique
> > interval vector count). This is 89 * 109. I found 89
> along "Schismic"
> > in Zoomr.gif (hard to read, its really small, and doesn't show up
> in
> > Zooms.gif. I can't find 109. Help! Can someone tell me where 109
> > would show up in a taxicab diagram. Thanks!
> >
> > Paul
>
> I found 109 at the intersection of Escapade and Tetracot. Not on
> Schismic (Definitely). Well I hope you are not all sick of this
> subject. The question is, does 89 appear along Escapade or Tetracot?
> That would be cool. I will quit posting for today. I will present
my
> results on all this tomorrow. Thanks and bye for now. Paul

Another cool coincidence. C{19,6} reduces to a count of 735 unique
interval vectors. That's 49 X 15. 49 and 15 appear along "Kleismic"
in the zoom diagram.

--- In tuning-math@yahoogroups.com, "paulhjelmstad"

> Another cool coincidence. C{19,6} reduces to a count of 735 unique
> interval vectors. That's 49 X 15. 49 and 15 appear along "Kleismic"
> in the zoom diagram.

what's the coincidence? any two ets will be collinear along some
line, whether the particular line connecting them is special enough
to merit inclusion on the chart or not -- and quite a few did. any
two ets define a 5-limit linear temperament (represented by a line in
the diagram).

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> --- In tuning-math@yahoogroups.com, "paulhjelmstad"
>
> > Another cool coincidence. C{19,6} reduces to a count of 735
unique
> > interval vectors. That's 49 X 15. 49 and 15 appear
along "Kleismic"
> > in the zoom diagram.
>
> what's the coincidence? any two ets will be collinear along some
> line, whether the particular line connecting them is special enough
> to merit inclusion on the chart or not -- and quite a few did. any
> two ets define a 5-limit linear temperament (represented by a line
in
> the diagram).
You got me. I'll cease and desist for now