Doublestacked Necklaces

I'm still working my way up to hexachords in 12-et. Here is trichords
in 6-et, which could also be whole tone trichords in 12-et. (Starting
very simple):

Comparing (0,1,2),(0,1,3),(0,2,3), (0,2,4) with themselves, with
6 different offsets (0,1,2,3,4,5)

Union: Multiplicity: Unique:
0 1+2+3=6 3
1 6+8+4+18+4+2=42 6
2 2+4+18+4+8+6=42 6
3 3+2+1=6 3

Totals 96 18

So right off there is a nice symmetry. I will be calculating double-
trichords in 12-et which can be defined with hexachords, pentachords
and the like. Stay tuned for tetrachords in 8-et, etc.

PH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> I'm still working my way up to hexachords in 12-et. Here is
trichords
> in 6-et, which could also be whole tone trichords in 12-et.
(Starting
> very simple):
>
> Comparing (0,1,2),(0,1,3),(0,2,3), (0,2,4) with themselves, with
> 6 different offsets (0,1,2,3,4,5)
>
> Union: Multiplicity: Unique:
> 0 1+2+3=6 3
> 1 6+8+4+18+4+2=42 6
> 2 2+4+18+4+8+6=42 6
> 3 3+2+1=6 3
>
> Totals 96 18
>
> So right off there is a nice symmetry. I will be calculating double-
> trichords in 12-et which can be defined with hexachords, pentachords
> and the like. Stay tuned for tetrachords in 8-et, etc.
>
> PH

Also I should add that all the vectors are completely symmetrical,
that is, for example, the interval vectors with multiplicity
18 are 2,2,4,1 for union=2 and 1,4,2,2 for union=1. I will have
to see if this pattern carries through for higher sets

I've run the calculations for doublestacked necklaces of 8 elements
with 4 positive and 4 negative charges in each one. I have found
71 unique energy states (interval vectors) in agreement with EIS
A045612. I matched all 10 types with themselves, and checked
T0,T1,T2,T3,T4 transpositions, making 500 vectors.

A couple cool patterns. Every interval vector adds to 16, as expected.
Taking column totals, you obtain 1033,1984,1966,1984,1033 for
0,1,2,3,4 interval vector values. This totals to 8000. (500*16) Also,
grouping vectors by "0" values (union) you obtain 7,14,29,14,7 unique
vectors based on unions of 0,1,2,3 and 4. This equals 71.

I'm still analyzing whether there is any significance to reversing
the interval vector (0<->4, 1<->3 and 2<->2) like there is for
Intvec(6,3)

Eventually, I'll get to calculating double-hexachords in 12-et, which
has more importance for tuning systems, obviously.

PH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> I've run the calculations for doublestacked necklaces of 8 elements
> with 4 positive and 4 negative charges in each one. I have found
> 71 unique energy states (interval vectors) in agreement with EIS
> A045612. I matched all 10 types with themselves, and checked
> T0,T1,T2,T3,T4 transpositions, making 500 vectors.
>
> A couple cool patterns. Every interval vector adds to 16, as
expected.
> Taking column totals, you obtain 1033,1984,1966,1984,1033 for
> 0,1,2,3,4 interval vector values. This totals to 8000. (500*16)
Also,
> grouping vectors by "0" values (union) you obtain 7,14,29,14,7
unique
> vectors based on unions of 0,1,2,3 and 4. This equals 71.
>
> I'm still analyzing whether there is any significance to reversing
> the interval vector (0<->4, 1<->3 and 2<->2) like there is for
> Intvec(6,3)

(Kind of interesting: The mirror image of the vector has the same
multiplicity. There are 23 (2,5,4,3,2) and also 23 (2,3,4,5,2) out
of the 500 vectors checked.)
>
> Eventually, I'll get to calculating double-hexachords in 12-et,
which
> has more importance for tuning systems, obviously.
>
> PH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > I've run the calculations for doublestacked necklaces of 8
elements
> > with 4 positive and 4 negative charges in each one. I have found
> > 71 unique energy states (interval vectors) in agreement with EIS
> > A045612. I matched all 10 types with themselves, and checked
> > T0,T1,T2,T3,T4 transpositions, making 500 vectors.
> >
> > A couple cool patterns. Every interval vector adds to 16, as
> expected.
> > Taking column totals, you obtain 1033,1984,1966,1984,1033 for
> > 0,1,2,3,4 interval vector values. This totals to 8000. (500*16)
> Also,
> > grouping vectors by "0" values (union) you obtain 7,14,29,14,7
> unique
> > vectors based on unions of 0,1,2,3 and 4. This equals 71.
> >
> > I'm still analyzing whether there is any significance to reversing
> > the interval vector (0<->4, 1<->3 and 2<->2) like there is for
> > Intvec(6,3)
>
> (Kind of interesting: The mirror image of the vector has the same
> multiplicity. There are 23 (2,5,4,3,2) and also 23 (2,3,4,5,2) out
> of the 500 vectors checked.)
> >
> > Eventually, I'll get to calculating double-hexachords in 12-et,
> which
> > has more importance for tuning systems, obviously.
> >
> > PH

Decided to analyze all 800 vectors (10*10*8) for T0 thru T7. The
patterns here are even nicer. Column totals are
1600,3200,3200,3200,1600 for interval vector values 0,1,2,3,4.
Totals for union=0,1,2,3,4 are 14,180,412,180,14. This reduces
to 7,14,29,14,7 unique vectors.

Here are the multiplicities for "0" from 0 to 4 (union of
doublestacked necklaces)

0: 1,2,2,2,1,2,4
1: 6,16,8,4,12,16,36,8,16,8,8,32,4,6
2: 2,2,4,4,8,40,8,8,8,2,4,16,4,2,188,2,4,16,4,2,8,8,8,40,8,4,4,2,2
3: 6,4,32,8,8,16,8,36,16,12,4,8,16,6
4: 4,2,1,2,2,2,1

Note the mirror image between 0 and 4, 1 and 3 and 2 alone.
Also, the mirror image of a particular vector will have the same
multiplicity as the "preimage" vector

PH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > I've run the calculations for doublestacked necklaces of 8
> elements
> > > with 4 positive and 4 negative charges in each one. I have found
> > > 71 unique energy states (interval vectors) in agreement with
EIS
> > > A045612. I matched all 10 types with themselves, and checked
> > > T0,T1,T2,T3,T4 transpositions, making 500 vectors.
> > >
> > > A couple cool patterns. Every interval vector adds to 16, as
> > expected.
> > > Taking column totals, you obtain 1033,1984,1966,1984,1033 for
> > > 0,1,2,3,4 interval vector values. This totals to 8000. (500*16)
> > Also,
> > > grouping vectors by "0" values (union) you obtain 7,14,29,14,7
> > unique
> > > vectors based on unions of 0,1,2,3 and 4. This equals 71.
> > >
> > > I'm still analyzing whether there is any significance to
reversing
> > > the interval vector (0<->4, 1<->3 and 2<->2) like there is for
> > > Intvec(6,3)
> >
> > (Kind of interesting: The mirror image of the vector has the same
> > multiplicity. There are 23 (2,5,4,3,2) and also 23 (2,3,4,5,2) out
> > of the 500 vectors checked.)
> > >
> > > Eventually, I'll get to calculating double-hexachords in 12-et,
> > which
> > > has more importance for tuning systems, obviously.
> > >
> > > PH
>
> Decided to analyze all 800 vectors (10*10*8) for T0 thru T7. The
> patterns here are even nicer. Column totals are
> 1600,3200,3200,3200,1600 for interval vector values 0,1,2,3,4.
> Totals for union=0,1,2,3,4 are 14,180,412,180,14. This reduces
> to 7,14,29,14,7 unique vectors.
>
> Here are the multiplicities for "0" from 0 to 4 (union of
> doublestacked necklaces)
>
> 0: 1,2,2,2,1,2,4
> 1: 6,16,8,4,12,16,36,8,16,8,8,32,4,6
> 2: 2,2,4,4,8,40,8,8,8,2,4,16,4,2,188,2,4,16,4,2,8,8,8,40,8,4,4,2,2
> 3: 6,4,32,8,8,16,8,36,16,12,4,8,16,6
> 4: 4,2,1,2,2,2,1
>
> Note the mirror image between 0 and 4, 1 and 3 and 2 alone.
> Also, the mirror image of a particular vector will have the same
> multiplicity as the "preimage" vector
>
> PH

Also, totals after reducing to 71 unique interval vectors is
142,284,284,284,142 for interval vector values 0,1,2,3,4.
These are all multiples of 71. Also, reflects the other totals
1600,3200,3200,3200,1600 which are all multiples of 800.

I realize I am straying from tuning-math subject matter, until I
do double-hexachords in 12-et...Not much use for 8 tones per octave.

PH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > I've run the calculations for doublestacked necklaces of 8
> > elements
> > > > with 4 positive and 4 negative charges in each one. I have
found
> > > > 71 unique energy states (interval vectors) in agreement with
> EIS
> > > > A045612. I matched all 10 types with themselves, and checked
> > > > T0,T1,T2,T3,T4 transpositions, making 500 vectors.
> > > >
> > > > A couple cool patterns. Every interval vector adds to 16, as
> > > expected.
> > > > Taking column totals, you obtain 1033,1984,1966,1984,1033 for
> > > > 0,1,2,3,4 interval vector values. This totals to 8000.
(500*16)
> > > Also,
> > > > grouping vectors by "0" values (union) you obtain
7,14,29,14,7
> > > unique
> > > > vectors based on unions of 0,1,2,3 and 4. This equals 71.
> > > >
> > > > I'm still analyzing whether there is any significance to
> reversing
> > > > the interval vector (0<->4, 1<->3 and 2<->2) like there is for
> > > > Intvec(6,3)
> > >
> > > (Kind of interesting: The mirror image of the vector has the
same
> > > multiplicity. There are 23 (2,5,4,3,2) and also 23 (2,3,4,5,2)
out
> > > of the 500 vectors checked.)
> > > >
> > > > Eventually, I'll get to calculating double-hexachords in 12-
et,
> > > which
> > > > has more importance for tuning systems, obviously.
> > > >
> > > > PH
> >
> > Decided to analyze all 800 vectors (10*10*8) for T0 thru T7. The
> > patterns here are even nicer. Column totals are
> > 1600,3200,3200,3200,1600 for interval vector values 0,1,2,3,4.
> > Totals for union=0,1,2,3,4 are 14,180,412,180,14. This reduces
> > to 7,14,29,14,7 unique vectors.
> >
> > Here are the multiplicities for "0" from 0 to 4 (union of
> > doublestacked necklaces)
> >
> > 0: 1,2,2,2,1,2,4
> > 1: 6,16,8,4,12,16,36,8,16,8,8,32,4,6
> > 2: 2,2,4,4,8,40,8,8,8,2,4,16,4,2,188,2,4,16,4,2,8,8,8,40,8,4,4,2,2
> > 3: 6,4,32,8,8,16,8,36,16,12,4,8,16,6
> > 4: 4,2,1,2,2,2,1
> >
> > Note the mirror image between 0 and 4, 1 and 3 and 2 alone.
> > Also, the mirror image of a particular vector will have the same
> > multiplicity as the "preimage" vector
> >
> > PH
>
> Also, totals after reducing to 71 unique interval vectors is
> 142,284,284,284,142 for interval vector values 0,1,2,3,4.
> These are all multiples of 71. Also, reflects the other totals
> 1600,3200,3200,3200,1600 which are all multiples of 800.
>
> I realize I am straying from tuning-math subject matter, until I
> do double-hexachords in 12-et...Not much use for 8 tones per octave.
>
> PH

One last comment: If you sort based on interval vector value 4,
you get the same amounts for v4=0,1,2,3,4 as for v0. Also, v1 and v3
mirror each other. Also, there is symmetry for each value, as you can
see. Here are the values:

v0,v4: 14,180,412,180,14 (v0,v4=0,1,2,3,4)
v1,v3: 4,0,64,136,392,136,64,0,4 (v1,v3=0,1,2,3,4,5,6,7,8)
v2: 6,0,88,96,420,96,88,0,6 (v2=0,1,2,3,4,5,6,7,8)

PH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul.hjelmstad@u...> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > <paul.hjelmstad@u...> wrote:
> > > > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > > > <paul.hjelmstad@u...> wrote:
> > >
> > > Decided to analyze all 800 vectors (10*10*8) for T0 thru T7.
The
> > > patterns here are even nicer. Column totals are
> > > 1600,3200,3200,3200,1600 for interval vector values 0,1,2,3,4.
> > > Totals for union=0,1,2,3,4 are 14,180,412,180,14. This reduces
> > > to 7,14,29,14,7 unique vectors.
> > >
> > > Here are the multiplicities for "0" from 0 to 4 (union of
> > > doublestacked necklaces)
> > >
> > > 0: 1,2,2,2,1,2,4
> > > 1: 6,16,8,4,12,16,36,8,16,8,8,32,4,6
> > > 2:
2,2,4,4,8,40,8,8,8,2,4,16,4,2,188,2,4,16,4,2,8,8,8,40,8,4,4,2,2
> > > 3: 6,4,32,8,8,16,8,36,16,12,4,8,16,6
> > > 4: 4,2,1,2,2,2,1
> > >
> > > Note the mirror image between 0 and 4, 1 and 3 and 2 alone.
> > > Also, the mirror image of a particular vector will have the same
> > > multiplicity as the "preimage" vector
> > >
> > > PH
> >
> > Also, totals after reducing to 71 unique interval vectors is
> > 142,284,284,284,142 for interval vector values 0,1,2,3,4.
> > These are all multiples of 71. Also, reflects the other totals
> > 1600,3200,3200,3200,1600 which are all multiples of 800.
> >
> > I realize I am straying from tuning-math subject matter, until I
> > do double-hexachords in 12-et...Not much use for 8 tones per
octave.
> >
> > PH
>
> One last comment: If you sort based on interval vector value 4,
> you get the same amounts for v4=0,1,2,3,4 as for v0. Also, v1 and v3
> mirror each other. Also, there is symmetry for each value, as you
can
> see. Here are the values:
>
> v0,v4: 14,180,412,180,14 (v0,v4=0,1,2,3,4)
> v1,v3: 4,0,64,136,392,136,64,0,4 (v1,v3=0,1,2,3,4,5,6,7,8)
> v2: 6,0,88,96,420,96,88,0,6 (v2=0,1,2,3,4,5,6,7,8)
>
> PH

I lied. This really is the last thing for this post:
Reduced (unique interval vector counts) for the following:

v0,v4: 7,14,29,14,7 (v0,v4=0,1,2,3,4)
v1,v3: 1,0,11,11,25,11,11,0,1 (v1,v3=0,1,2,3,4,5,6,7,8)
v2: 2,0,7,12,29,12,7,0,2 (v2=0,1,2,3,4,5,6,7,8)

Best, tomorrow is my last day at this company. I will be changing my
email to phjelmstad@msn.com, my home email:)