Steiner Systems

I've been analyzing the Steiner System (5,6,12) using "R.T. Curtis'
Kitten" on a webpage I found. (David Joyner). David supplied the
two hexads that were missing previously. So now I have solid results:

Of the 924 hexachords, there are 132 Steiner hexads, which is exactly
1/7th.

Of the 80 hexachords, 64 get used. This is 4/5ths. These of course,
expand out to 132. (Some are used more than once).

Of the 44 hexachord-partitions, 35 get used, and they expand to
66, which is of course 1/2 or 132, and this makes sense, because
the 132 Steiner hexads are 66 + 66. (Sets with complements). Actually,
72 are used, total, but 6 are complements of themselves, so it
depends on how you look at the representation...

Of the 35 hexachords, reduced for mirror-image and complementability,
28 get used. Interesting that this is also 4/5ths. U (0,2,4,6,8,10) P
(0,1,2,6,7,8) and I (0,1,3,6,7,9) are not used at all, but A is used,
which has transpositional symmetry. (0,1,4,5,8,9). These expand out
to 60. Now if you further reduce these 60 to include only one side of
a hexachord-partition you get 55. 55/66 is 5/6ths. Going the other
way: 35/60 is 7/12, 44/66 is 2/3. Etc.

The website shows weights of these hexads, I am working to
analyze interval vector frequencies, value totals and the like.

Anyone think this is leading anywhere? And what does it have to do
with tuning? I am hoping to develop this into a compositional tool,
at least, another way this relates to this newsgroup is the fact
that Steiner systems relate to projective planes, etc.

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

> Anyone think this is leading anywhere?

It might if you want to use this as a basis for music.

> And what does it have to do
> with tuning?

Given that it's basically a 12-equal thing, nothing.

I am hoping to develop this into a compositional tool,
> at least, another way this relates to this newsgroup is the fact
> that Steiner systems relate to projective planes, etc.

And projective planes relate to?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > Anyone think this is leading anywhere?
>
> It might if you want to use this as a basis for music.
>
> > And what does it have to do
> > with tuning?
>
> Given that it's basically a 12-equal thing, nothing.

I feared as much. It is interesting that Steiner systems do
include 24, 23, 22, 12, 11, and 24, 22, and 12 are popular ETs.
I'm also studying MOG, Miracle Octad Generator, which is S(5,8,24)
(Aren't all these Steiner systems subsets of this one?)
>
> I am hoping to develop this into a compositional tool,
> > at least, another way this relates to this newsgroup is the fact
> > that Steiner systems relate to projective planes, etc.
>
> And projective planes relate to?

Your lattices, I thought, of course this is a different application
of projective planes. I'm probably trying to find relationships
where there are none - but, for example, logarithms and primes are
related, so why not musical set theory and tuning theory?

>

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

> > Given that it's basically a 12-equal thing, nothing.
>
> I feared as much. It is interesting that Steiner systems do
> include 24, 23, 22, 12, 11, and 24, 22, and 12 are popular ETs.

It would be interesting to hear M22 applied to 22 notes to the octave.
I don't say it would be good, but interesting. :)

> > I am hoping to develop this into a compositional tool,
> > > at least, another way this relates to this newsgroup is the fact
> > > that Steiner systems relate to projective planes, etc.
> >
> > And projective planes relate to?
>
> Your lattices, I thought, of course this is a different application
> of projective planes.

I don't see how that connects. There is a relation to the projective
plane over the field of four elements, which has 21 points and 21
lines. That, I suppose, someone could inflict on the Blackjack scale.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
>
> > > Given that it's basically a 12-equal thing, nothing.
> >
> > I feared as much. It is interesting that Steiner systems do
> > include 24, 23, 22, 12, 11, and 24, 22, and 12 are popular ETs.
>
> It would be interesting to hear M22 applied to 22 notes to the
octave.
> I don't say it would be good, but interesting. :)

Yes, there are 77 hexads here, and all C{22,3} occur each in exactly
one hexad (Just reviewing for the general public...) How do you
envision using the M22 automorphism group over 22t-ET?

> > > I am hoping to develop this into a compositional tool,
> > > > at least, another way this relates to this newsgroup is the
fact
> > > > that Steiner systems relate to projective planes, etc.
> > >
> > > And projective planes relate to?
> >
> > Your lattices, I thought, of course this is a different
application
> > of projective planes.
>
> I don't see how that connects. There is a relation to the projective
> plane over the field of four elements, which has 21 points and 21
> lines. That, I suppose, someone could inflict on the Blackjack
scale.

Actually that's what I meant! (At least in terms of Steiner(3,6,22))
I'll review the Blackjack scale to see what you mean. Isn't that the
Steiner system (2,5,21)?

>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:

> > It would be interesting to hear M22 applied to 22 notes to the
> octave.
> > I don't say it would be good, but interesting. :)
>
> Yes, there are 77 hexads here, and all C{22,3} occur each in exactly
> one hexad (Just reviewing for the general public...) How do you
> envision using the M22 automorphism group over 22t-ET?

If you write a tune using one hexad as a scale, then applying M22 will
convert it to a tune in one of the 77 hexads. Or, you could prefer
working with pentatonic scales belong to a hexad. If you were really
brave, or foolish, you could use the hexads as chords and transform
those. The hexads containing interesting triads might be especially
nice to look at.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <paul_hjelmstad@> wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> > <genewardsmith@> wrote:
>
> > > It would be interesting to hear M22 applied to 22 notes to the
> > octave.
> > > I don't say it would be good, but interesting. :)
> >
> > Yes, there are 77 hexads here, and all C{22,3} occur each in
exactly
> > one hexad (Just reviewing for the general public...) How do you
> > envision using the M22 automorphism group over 22t-ET?
>
> If you write a tune using one hexad as a scale, then applying M22
will
> convert it to a tune in one of the 77 hexads. Or, you could prefer
> working with pentatonic scales belong to a hexad. If you were really
> brave, or foolish, you could use the hexads as chords and transform
> those. The hexads containing interesting triads might be especially
> nice to look at.

I see. Just to review, the automorphism group M22 is P{22,3} * 48,
where 48 is the three-point stabilizer in M22. When you say tune,
I take it you mean a collection of all the notes of the hexad in any
order? Of course the index (G/H), as I understand it, maps triads to
(any other unordered) triad. How do hexads get mapped to hexads?

Going back to S(5,6,12), do you think its a total waste of time
to analyze the interval vectors of these hexads? (I use Excel
and 'DCOUNT').

Thanx

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:

> > If you write a tune using one hexad as a scale, then applying M22
> will
> > convert it to a tune in one of the 77 hexads.

> I see. Just to review, the automorphism group M22 is P{22,3} * 48,
> where 48 is the three-point stabilizer in M22. When you say tune,
> I take it you mean a collection of all the notes of the hexad in any
> order?

No, I mean an actual tune, using the notes of a hexad as a scale.

Why don't you list 77 hexads here, and we can see what kinds of things
we get?

Of course the index (G/H), as I understand it, maps triads to
> (any other unordered) triad. How do hexads get mapped to hexads?

Apply a permutation group element.

> Going back to S(5,6,12), do you think its a total waste of time
> to analyze the interval vectors of these hexads? (I use Excel
> and 'DCOUNT').

Depends on whether you end up using it to compose something!

> As soon as I have a little program written, I should be able to find
> a modulo-11 labelled Steiner System. Alas, it does not look like it
> will be 22 sets at even tranpositions (I already have more than 22
> different hexads, so it's not that kind of SSS)

Hi Paul - do you know about the 11-tET set {0,1,2,4,7}? It generates a "biplane" geometry shown here:

http://www.maths.monash.edu.au/~bpolster/bi.html

The chord shares exactly two pitches with any of its transpositions in 11-tET. Equivalently, you could say it's "doubly all-interval", meaning that it contains every interval class of 11-tone equal temperament exactly twice. So it doesn't favour any particular interval from 11-tET; its distribution of intervals is the same as the parent universe. If you wanted to sum up the sound of 11-tET in one chord, this would be it. I'll leave you to find attractive voicings for it...

(ah I just noticed you mentioned the quadratic residues - that's what this is. You can get such a set in any n-tet where n is prime and congruent to 3 mod 4. Here is a nice one in 19-tET (voiced from bottom up): F# C# E B Eb G Bb D# E#. It has every interval-class 4 times. The condition n=prime can be relaxed, but then you don't construct them with quadratic residues. Other sets with the same properties can be found whenever n is a product of twin primes, or n is one less than a power of two.)

By the way another interesting feature of the 11-tET set above, which doesn't automatically follow from its construction as the set of quadratic residues, is that it contains every trichordal set-class exactly once. This is a very rare property; I only know of one other collection that has this (it's {0,1,2,4} mod 7).

Best --JW

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
>
> > As soon as I have a little program written, I should be
able to find
> > a modulo-11 labelled Steiner System. Alas, it does not look
like it
> > will be 22 sets at even tranpositions (I already have more
than 22
> > different hexads, so it's not that kind of SSS)
>
> Hi Paul - do you know about the 11-tET set {0,1,2,4,7}? It
generates a
> "biplane" geometry shown here:
>
> http://www.maths.monash.edu.au/~bpolster/bi.html
>
> The chord shares exactly two pitches with any of its transpositions
in
> 11-tET. Equivalently, you could say it's "doubly all-interval",
meaning
> that it contains every interval class of 11-tone equal temperament
exactly
> twice. So it doesn't favour any particular interval from 11-tET;
its
> distribution of intervals is the same as the parent universe. If
you
> wanted to sum up the sound of 11-tET in one chord, this would be
it. I'll
> leave you to find attractive voicings for it...
>
> (ah I just noticed you mentioned the quadratic residues - that's
what this
> is. You can get such a set in any n-tet where n is prime and
congruent to
> 3 mod 4. Here is a nice one in 19-tET (voiced from bottom up): F#
C# E B
> Eb G Bb D# E#. It has every interval-class 4 times. The condition
n=prime
> can be relaxed, but then you don't construct them with quadratic
residues.
> Other sets with the same properties can be found whenever n is a
product
> of twin primes, or n is one less than a power of two.)
>
> By the way another interesting feature of the 11-tET set above,
which
> doesn't automatically follow from its construction as the set of
quadratic
> residues, is that it contains every trichordal set-class exactly
once.
> This is a very rare property; I only know of one other collection
that has
> this (it's {0,1,2,4} mod 7).
>
> Best --JW
>

Thanks Jon, all very interesting. Too bad 11-tET doesn't sound better.
I'm interested very much in 19-tET, did you generate your Steiner
System from this seed (quadratic residues?) Did the SSS have any
nice properties?

PGH

> Thanks Jon, all very interesting. Too bad 11-tET doesn't sound better.
> I'm interested very much in 19-tET, did you generate your Steiner
> System from this seed (quadratic residues?) Did the SSS have any
> nice properties?

I used the 19-tet set of quadratic residues, yes (one voicing is given in my previous email). What I did with it wasn't exactly a Steiner system... I have to rerender the soundfiles I made originally, and I'll send it along. For one of my examples I found a very curious voice-leading structure involving that chord.

Composer Tom Johnson was at the math & music conference in Berlin too, and he presented lots of Steiner systems that he had worked into his compositions--they were all based on scales within 12-tet. There was also a display of some of his pre-compositional graphic sketches, with commentary on the designs by a mathematician from the university of Vermont.

Best - Jon

--- In tuning-math@yahoogroups.com, Jon Wild <wild@...> wrote:
>
>
> > As soon as I have a little program written, I should be
able to find
> > a modulo-11 labelled Steiner System. Alas, it does not look
like it
> > will be 22 sets at even tranpositions (I already have more
than 22
> > different hexads, so it's not that kind of SSS)
>
> Hi Paul - do you know about the 11-tET set {0,1,2,4,7}? It
generates a
> "biplane" geometry shown here:
>
> http://www.maths.monash.edu.au/~bpolster/bi.html
>
> The chord shares exactly two pitches with any of its transpositions
in
> 11-tET. Equivalently, you could say it's "doubly all-interval",
meaning
> that it contains every interval class of 11-tone equal temperament
exactly
> twice. So it doesn't favour any particular interval from 11-tET;
its
> distribution of intervals is the same as the parent universe. If
you
> wanted to sum up the sound of 11-tET in one chord, this would be
it. I'll
> leave you to find attractive voicings for it...
>
> (ah I just noticed you mentioned the quadratic residues - that's
what this
> is. You can get such a set in any n-tet where n is prime and
congruent to
> 3 mod 4. Here is a nice one in 19-tET (voiced from bottom up): F#
C# E B
> Eb G Bb D# E#. It has every interval-class 4 times. The condition
n=prime
> can be relaxed, but then you don't construct them with quadratic
residues.
> Other sets with the same properties can be found whenever n is a
product
> of twin primes, or n is one less than a power of two.)
>
> By the way another interesting feature of the 11-tET set above,
which
> doesn't automatically follow from its construction as the set of
quadratic
> residues, is that it contains every trichordal set-class exactly
once.
> This is a very rare property; I only know of one other collection
that has
> this (it's {0,1,2,4} mod 7).
>
> Best --JW
>
Projective Geometry and Steiner Systems go together of course.
Say, does anyone know of a free knock-off of MATLAB I can download
so I can write my little PSL(2,11)-> S(5,6,12) program?

Thanx

PGH