Math paper even more relevant to our interests

http://dx.doi.org/10.1023/A:1006513923148

This paper cites some other papers that go a long way toward proving my max-variety-3 (N=3 DE) conjecture, and furthermore it proves that, for any set S of affine dimension 3 (for example a nondegenerate scale in an octave-equivalent rank-4 lattice),

|S-S| >= 4.5 |S| - 9

where S-S is the set of differences of S. This has several immediate implications:

* The minimum mean variety for asymptotically large rank-4 scales is 4.5 (compared to 2 for rank 2 and 3 for rank 3). As in rank 3, such minimum-mean-variety scales will be quasi-one-dimensional, with a strictly limited width in 2 of the 3 lattice directions.

* The set of rank-4 max-variety-4 scale imprints is finite, with a maximum number of notes per period given by the solution to

4.5 x - 9 = 4 x - 3

that is, 12 notes per period. Therefore this is an *exhaustive* list of rank-4 max-variety-4 scale imprints:

1 1 1 1 ['abcd']

2 1 1 1 ['abacd']

2 2 1 1 ['abacdc']

2 2 2 1 ['aabcdcb']
3 2 1 1 ['abacabd', 'abacadc']

3 2 2 1 ['abacbdbc']
4 2 1 1 ['aabaacdc']

4 2 2 2 ['abacdabadc']
4 3 2 1 ['abacbadabc']
6 2 1 1 ['aabaacadac']

Keenan

On Tue, Sep 20, 2011 at 3:57 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> * The minimum mean variety for asymptotically large rank-4 scales is 4.5 (compared to 2 for rank 2 and 3 for rank 3). As in rank 3, such minimum-mean-variety scales will be quasi-one-dimensional, with a strictly limited width in 2 of the 3 lattice directions.
>
> * The set of rank-4 max-variety-4 scale imprints is finite, with a maximum number of notes per period given by the solution to

If the minimum mean variety is 4.5, then how can there be any rank-4
scales with max variety 4 at all?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> If the minimum mean variety is 4.5, then how can there be any rank-4
> scales with max variety 4 at all?

The equation is |S-S| >= 4.5 |S| - 9. It's the minus 9 that does it.

I said the mean variety for "asymptotically large" rank-4 scales is 4.5, so that means if you want a scale with 1000 notes in it you can't do much better than 4.49. But if you want a scale with only 10 notes in it, you can make the mean variety as low as 4.

If the coefficient were 4 instead of 4.5 there might be some hope that arbitrarily large rank-4 max-variety-4 scales exist. But it's actually 4.5, which means there's an upper limit to the number of notes per period.

Keenan

On Tue, Sep 20, 2011 at 9:08 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > If the minimum mean variety is 4.5, then how can there be any rank-4
> > scales with max variety 4 at all?
>
> The equation is |S-S| >= 4.5 |S| - 9. It's the minus 9 that does it.
>
> I said the mean variety for "asymptotically large" rank-4 scales is 4.5, so that means if you want a scale with 1000 notes in it you can't do much better than 4.49. But if you want a scale with only 10 notes in it, you can make the mean variety as low as 4.
>
> If the coefficient were 4 instead of 4.5 there might be some hope that arbitrarily large rank-4 max-variety-4 scales exist. But it's actually 4.5, which means there's an upper limit to the number of notes per period.

Ah, I see. OK, that makes sense.

One more thing I'm not sure I get about difference sets is - are they
going to yield just the set of chromata? For example, the difference
set for the JI major scale should have things like 15/8 / 4/3 = 45/32
in it, right?

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Ah, I see. OK, that makes sense.
>
> One more thing I'm not sure I get about difference sets is - are they
> going to yield just the set of chromata? For example, the difference
> set for the JI major scale should have things like 15/8 / 4/3 = 45/32
> in it, right?

That's right. The set of differences is simply the set of all intervals between notes of the scale.

The set of chromata is one level beyond: its elements aren't differences between scale pitches, they're differences between differences between scale pitches.

Specifically, the set of chromata is the union of the sets of differences of each intersection of the set of differences of the scale with the preimage of an integer under the val corresponding to scale steps. Whew, that's a mouthful!

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
Therefore this is an *exhaustive* list of rank-4 max-variety-4 scale imprints:
>
> 1 1 1 1 ['abcd']
>
> 2 1 1 1 ['abacd']
>
> 2 2 1 1 ['abacdc']
>
> 2 2 2 1 ['aabcdcb']
> 3 2 1 1 ['abacabd', 'abacadc']
>
> 3 2 2 1 ['abacbdbc']
> 4 2 1 1 ['aabaacdc']
>
> 4 2 2 2 ['abacdabadc']
> 4 3 2 1 ['abacbadabc']
> 6 2 1 1 ['aabaacadac']

Makes me wonder what some actual 7-limit scales might be.

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:

> 4 3 2 1 ['abacbadabc']

Here's an example scale. I found it by taking the patent vals for 1, 2, 3, and 4, which form a unimodular matrix. Inverting it and transposing gives monzos for 21/20, 35/32, 15/14 and 16/15 in that order, and substituting 16/15 for "a", etc, gives this scale:

! abacbadabc.scl
7-limit scale with mean variety four
10
!
16/15
8/7
128/105
4/3
10/7
32/21
8/5
128/75
64/35
2/1

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

Since this scale has both 16/15 and 15/14 as steps, it's no surprise that marvel tempering it gives a scale with mean variety three. Below is a 5-limit transversal, which tempering will turn into marvel; put it into Scala and look at the lattice and you'll see it's a nice one. I've got it listed as "ying.scl" and it's marvel tempering as "yingmarv.scl", so at some point I presumably knew something about this scale, but I seem to have forgotten what.

! abacbadabc-marvtrans.scl
!
Transversal of marvel tempering of 7-limit scale with mean variety four
10
!
16/15
256/225
4096/3375
4/3
64/45
1024/675
8/5
128/75
2048/1125
2/1

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Since this scale has both 16/15 and 15/14 as steps, it's no surprise that marvel tempering it gives a scale with mean variety three. Below is a 5-limit transversal, which tempering will turn into marvel; put it into Scala and look at the lattice and you'll see it's a nice one. I've got it listed as "ying.scl" and it's marvel tempering as "yingmarv.scl", so at some point I presumably knew something about this scale, but I seem to have forgotten what.
>
>
> ! abacbadabc-marvtrans.scl
> !
> Transversal of marvel tempering of 7-limit scale with mean variety four
> 10
> !
> 16/15
> 256/225
> 4096/3375
> 4/3
> 64/45
> 1024/675
> 8/5
> 128/75
> 2048/1125
> 2/1

This scale does not have mean variety 3. It has mean variety 28/9, because there are 4 different intervals in the 5-step class (512/375, 45/32, 64/45, and 375/256).

Lately I've been thinking about exactly this sort of "exception" or "failure" to temper down as you might expect, but I have no hints about a general theory yet.

Keenan

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> That's right. The set of differences is simply the set of all intervals between notes of the scale.
>
> The set of chromata is one level beyond: its elements aren't differences between scale pitches, they're differences between differences between scale pitches.
>
> Specifically, the set of chromata is the union of the sets of differences of each intersection of the set of differences of the scale with the preimage of an integer under the val corresponding to scale steps. Whew, that's a mouthful!

I just figured out that an equivalent and simpler way to say this is that a chroma is any element of (the set of differences of the set of differences of the scale) that maps to 0 steps under the val.

Also, if S - S is the set of x - y for x and y in S, and S + S is the set of x + y for x and y in S, we have the following equalities:

(S - S) - (S - S) = (S - S) + (S - S) = (S + S) - (S + S)

Looks pretty goofy when written that way, but it actually makes sense and says something nontrivial.

Keenan

Fascinating work, Keenan. I'm just now sitting down to try and
catch up on my recent backlog here. -Carl

At 12:57 PM 9/20/2011, you wrote:
> http://dx.doi.org/10.1023/A:1006513923148
>
>This paper cites some other papers that go a long way toward proving
>my max-variety-3 (N=3 DE) conjecture, and furthermore it proves that,
>for any set S of affine dimension 3 (for example a nondegenerate scale
>in an octave-equivalent rank-4 lattice),
>
>|S-S| >= 4.5 |S| - 9
>
>where S-S is the set of differences of S. This has several immediate
>implications:
>
>* The minimum mean variety for asymptotically large rank-4 scales is
>4.5 (compared to 2 for rank 2 and 3 for rank 3). As in rank 3, such
>minimum-mean-variety scales will be quasi-one-dimensional, with a
>strictly limited width in 2 of the 3 lattice directions.
>
>* The set of rank-4 max-variety-4 scale imprints is finite, with a
>maximum number of notes per period given by the solution to
>
>4.5 x - 9 = 4 x - 3
>
>that is, 12 notes per period. Therefore this is an *exhaustive* list
>of rank-4 max-variety-4 scale imprints:
>
>1 1 1 1 ['abcd']
>
>2 1 1 1 ['abacd']
>
>2 2 1 1 ['abacdc']
>
>2 2 2 1 ['aabcdcb']
>3 2 1 1 ['abacabd', 'abacadc']
>
>3 2 2 1 ['abacbdbc']
>4 2 1 1 ['aabaacdc']
>
>4 2 2 2 ['abacdabadc']
>4 3 2 1 ['abacbadabc']
>6 2 1 1 ['aabaacadac']
>
>Keenan
>
>
>
>------------------------------------
>
>Yahoo! Groups Links
>
>
>

Keenan wrote:

>>Specifically, the set of chromata is the union of the sets of
>>differences of each intersection of the set of differences of the
>>scale with the preimage of an integer under the val corresponding to
>>scale steps. Whew, that's a mouthful!
>
>I just figured out that an equivalent and simpler way to say this is
>that a chroma is any element of (the set of differences of the set of
>differences of the scale) that maps to 0 steps under the val.

Yes and yes!

>Also, if S - S is the set of x - y for x and y in S, and S + S is the
>set of x + y for x and y in S, we have the following equalities:
>
>(S - S) - (S - S) = (S - S) + (S - S) = (S + S) - (S + S)

Lost me with this expression.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Fascinating work, Keenan. I'm just now sitting down to try and
> catch up on my recent backlog here. -Carl
>
> At 12:57 PM 9/20/2011, you wrote:
> > http://dx.doi.org/10.1023/A:1006513923148
> >
> >This paper cites some other papers that go a long way toward proving
> >my max-variety-3 (N=3 DE) conjecture, and furthermore it proves that,
> >for any set S of affine dimension 3 (for example a nondegenerate scale
> >in an octave-equivalent rank-4 lattice),
> >
> >|S-S| >= 4.5 |S| - 9
> >
> >where S-S is the set of differences of S. This has several immediate
> >implications:
> >
> >* The minimum mean variety for asymptotically large rank-4 scales is
> >4.5 (compared to 2 for rank 2 and 3 for rank 3). As in rank 3, such
> >minimum-mean-variety scales will be quasi-one-dimensional, with a
> >strictly limited width in 2 of the 3 lattice directions.
> >
> >* The set of rank-4 max-variety-4 scale imprints is finite, with a
> >maximum number of notes per period given by the solution to
> >
> >4.5 x - 9 = 4 x - 3
> >
> >that is, 12 notes per period. Therefore this is an *exhaustive* list
> >of rank-4 max-variety-4 scale imprints:
> >
> >1 1 1 1 ['abcd']
> >
> >2 1 1 1 ['abacd']
> >
> >2 2 1 1 ['abacdc']
> >
> >2 2 2 1 ['aabcdcb']
> >3 2 1 1 ['abacabd', 'abacadc']
> >
> >3 2 2 1 ['abacbdbc']
> >4 2 1 1 ['aabaacdc']
> >
> >4 2 2 2 ['abacdabadc']
> >4 3 2 1 ['abacbadabc']
> >6 2 1 1 ['aabaacadac']
> >
> >Keenan
> >
> >
> >
> >------------------------------------
> >
> >Yahoo! Groups Links
> >
> >
> >
>

Is there a way I could get this wihout having to pay $$$ thanks - pgh

--- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
> Is there a way I could get this wihout having to pay $$$ thanks - pgh

I emailed you a copy of this one.

Anyone who wants one, please email me if you want a copy of a paper I mention.

Keenan

On Thu, Oct 6, 2011 at 3:25 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@...> wrote:
> > Is there a way I could get this wihout having to pay $$$ thanks - pgh
>
> I emailed you a copy of this one.
>
> Anyone who wants one, please email me if you want a copy of a paper I mention.
>
> Keenan

I'm all for this, but might we be opening ourselves up to hairy legal
territory by advertising this publically? Not sure.

-Mike

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Oct 6, 2011 at 3:25 PM, Keenan Pepper <keenanpepper@...> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul" <phjelmstad@> wrote:
> > > Is there a way I could get this wihout having to pay $$$ thanks - pgh
> >
> > I emailed you a copy of this one.
> >
> > Anyone who wants one, please email me if you want a copy of a paper I mention.
> >
> > Keenan
>
> I'm all for this, but might we be opening ourselves up to hairy legal
> territory by advertising this publically? Not sure.
>
> -Mike
>
There is always that concern --- but it's funny how many papers I can get free,
even when they are for sale (at up to $50 bucks a paper) on a legitimate website, often
the author's own --- site.