question for my more learned friends

Probably more of a set theory question, I'll try to phrase what
I was looking for such that a mathematician might be able to come
up with an answer.

Take the notes C C# E and F# in 12tet (lets call them 0, 1, 4, 6).
These four notes considered pairwise and octave transposed contain
all the unique intervals for 12tet. 0, 2, 5, 6 does this as well.

This makes perfectly good sense since there are 6 unique intervals
and "four choose two" is also equal to 6.

Repeating this experiment for 7tet gets a similar pair (0, 1, 3) and
(0,2,3), which makes sense since "three choose two" = 3 which is the
number of unique intervals in 7tet.

For those wondering, the solution for 7 is good for 6 and that for
12 is good for 13. Other ETs will be over-determined (which is fine,
I just happen to be interested in exact fits).

The question is this, there should be a bunch of five-note sets that
do this for 20 and 21tet, and (what I was looking for) a six-note set
that covers 31 (and 30). I can't produce them.

So either an example of a set or a "no you can't do that..." with a
brief explanation will suffice. Thanks,

Bob Valentine

Hi Bob,

Of course ( 0, 1, 3, 7 ) and its inversion ( 0, 1, 6, 10 ) also work in 12
tet.

For 31 tet I found the following pairs:

( 0, 1, 3, 8, 12, 18 )
( 0, 1, 14, 20, 24, 29 )

( 0, 1, 3, 10, 14, 26 )
( 0, 1, 6, 18, 22, 29 )

( 0, 1, 4, 6, 13, 21 )
( 0, 1, 11, 19, 26, 28 )

( 0, 1, 4, 10, 12, 17 )
( 0, 1, 15, 20, 22, 28 )

( 0, 1, 8, 11, 13, 17 )
( 0, 1, 15, 19, 21, 24 )

Let me know if you also want the 20, 21 and 30.

Kees

----- Original Message -----
From: "Robert C Valentine" <BVAL@IIL.INTEL.COM>
To: <tuning-math@yahoogroups.com>
Sent: Sunday, April 07, 2002 3:37 AM
Subject: [tuning-math] question for my more learned friends

Probably more of a set theory question, I'll try to phrase what
I was looking for such that a mathematician might be able to come
up with an answer.

Take the notes C C# E and F# in 12tet (lets call them 0, 1, 4, 6).
These four notes considered pairwise and octave transposed contain
all the unique intervals for 12tet. 0, 2, 5, 6 does this as well.

This makes perfectly good sense since there are 6 unique intervals
and "four choose two" is also equal to 6.

Repeating this experiment for 7tet gets a similar pair (0, 1, 3) and
(0,2,3), which makes sense since "three choose two" = 3 which is the
number of unique intervals in 7tet.

For those wondering, the solution for 7 is good for 6 and that for
12 is good for 13. Other ETs will be over-determined (which is fine,
I just happen to be interested in exact fits).

The question is this, there should be a bunch of five-note sets that
do this for 20 and 21tet, and (what I was looking for) a six-note set
that covers 31 (and 30). I can't produce them.

So either an example of a set or a "no you can't do that..." with a
brief explanation will suffice. Thanks,

Bob Valentine

To unsubscribe from this group, send an email to:
tuning-math-unsubscribe@yahoogroups.com

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For 21 I only found one pair:
( 0, 1, 4, 14, 16 )
( 0, 1, 6, 8, 18 )

And nothing for 30 and 20. But I just hurriedly written a simple program to
find this stuff and might very well have made mistakes, since I don't really
consider myself learned enough to approach this in an exact manner.
These constructions are of course totally in the line of, for instance,
Allen Forte's 'The Structure of Atonal Music', and maybe a warning in the
subject line:
'considered harmful to serialism allergies' would be appropriate:-)

Kees

----- Original Message -----
From: "Kees van Prooijen" <kees@dnai.com>
To: <tuning-math@yahoogroups.com>
Sent: Sunday, April 07, 2002 1:49 PM
Subject: Re: [tuning-math] question for my more learned friends

Hi Bob,

Of course ( 0, 1, 3, 7 ) and its inversion ( 0, 1, 6, 10 ) also work in 12
tet.

For 31 tet I found the following pairs:

( 0, 1, 3, 8, 12, 18 )
( 0, 1, 14, 20, 24, 29 )

( 0, 1, 3, 10, 14, 26 )
( 0, 1, 6, 18, 22, 29 )

( 0, 1, 4, 6, 13, 21 )
( 0, 1, 11, 19, 26, 28 )

( 0, 1, 4, 10, 12, 17 )
( 0, 1, 15, 20, 22, 28 )

( 0, 1, 8, 11, 13, 17 )
( 0, 1, 15, 19, 21, 24 )

Let me know if you also want the 20, 21 and 30.

Kees

----- Original Message -----
From: "Robert C Valentine" <BVAL@IIL.INTEL.COM>
To: <tuning-math@yahoogroups.com>
Sent: Sunday, April 07, 2002 3:37 AM
Subject: [tuning-math] question for my more learned friends

Probably more of a set theory question, I'll try to phrase what
I was looking for such that a mathematician might be able to come
up with an answer.

Take the notes C C# E and F# in 12tet (lets call them 0, 1, 4, 6).
These four notes considered pairwise and octave transposed contain
all the unique intervals for 12tet. 0, 2, 5, 6 does this as well.

This makes perfectly good sense since there are 6 unique intervals
and "four choose two" is also equal to 6.

Repeating this experiment for 7tet gets a similar pair (0, 1, 3) and
(0,2,3), which makes sense since "three choose two" = 3 which is the
number of unique intervals in 7tet.

For those wondering, the solution for 7 is good for 6 and that for
12 is good for 13. Other ETs will be over-determined (which is fine,
I just happen to be interested in exact fits).

The question is this, there should be a bunch of five-note sets that
do this for 20 and 21tet, and (what I was looking for) a six-note set
that covers 31 (and 30). I can't produce them.

So either an example of a set or a "no you can't do that..." with a
brief explanation will suffice. Thanks,

Bob Valentine

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/

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/

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:
>
> Probably more of a set theory question, I'll try to phrase what
> I was looking for such that a mathematician might be able to come
> up with an answer.

It's a combinatorics question, and has a vague relationship to the theory of cyclic difference sets. Mod 7, {1,2,4} will give you all the non-zero elements exactly once: 2-1=1, 4-2=2, 4-1=3, and then the negatives of those. This makes {1,2,4} a (7,3,1)-difference set. That's not what you were looking for, of course, but it seems to have the same theme.

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:
>
> Probably more of a set theory question, I'll try to phrase what
> I was looking for such that a mathematician might be able to come
> up with an answer.

By the way, I find Math World has a page for perfect difference sets:

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

If q is a prime power, then there is such a beast for q^2+q+1, corresponding to the finite projective plane over Fq with
q^2+q+1 points and lines. Now that I've actually read your question, I see that in fact this whole business *is* closely connected to cyclic difference sets and modular Golomb rulers. The 7-et solution you found is from q=2, and the 13-et is from q=3. The projective plane corresponding to q=4 gives a 21-et solution, and q=5 a 31-et solution, which is what you were looking for.