Maximally even scales

Suppose L is a multiset of intervals, the product of which is 2. Then
the various permutations of L lead to various scales whose scale steps
are the intervals of L. If we take all the permutations, there will be
a minimum value for the maximum bumber of intervals in an interval class.
A scale among the permutations of L which achieves this minimum value
can be called *maximally even*.

One example of a maximally even scale would be a DE scale, or more
generally an even scale of rank n. But given a multiset with n
distinct elements generating a group of rank n, it is not the case
that an even scale of rank n exists; however maximally even scales
will certainly exist.

For example, consider the multiset defined by

(10/9)^2 (15/14)^2 (16/15)^2 (21/20)^3 = 2

Trying all the permutations, we find that the minimum maximum number of
elements in an interval class is 5, and hence no even scales of rank 4
can be constructed from this multiset. Examples of maximally even
scales can be found, however (60 in number.) The most interesting are
the scales defined in this posting from a few years back:

/tuning-math/message/4563

At 06:59 PM 10/28/2005, you wrote:
>Suppose L is a multiset of intervals, the product of which is 2. Then
>the various permutations of L lead to various scales whose scale steps
>are the intervals of L. If we take all the permutations, there will be
>a minimum value for the maximum bumber of intervals in an interval class.
>A scale among the permutations of L which achieves this minimum value
>can be called *maximally even*.

I've done this in scheme.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I've done this in scheme.

Are you going to give us some results? What are the maximally even
scales for (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2, for example?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > I've done this in scheme.
>
> Are you going to give us some results? What are the maximally even
> scales for (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2, for example?

If you marvel project the above multiset to the 5-limit, you get
(16/15)^7 (135/128)^3 (25/24)^2 = 2. The maximally even scales from
this have a maximum of four interval sizes per interval class, and I
count 31 of them. The most interesting seems to be a scale already
familiar, and known variously as lumma5, diadie1, and the marvel
projection of prism. Lumma7 = prism itself uses the original set with
a maximum of six intervals per class.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> Suppose L is a multiset of intervals,

What's the difference between a set and a multiset?

> the product of which is 2. Then
> the various permutations of L lead to various scales whose scale
steps
> are the intervals of L. If we take all the permutations, there will
be
> a minimum value for the maximum bumber of intervals in an interval
class.
> A scale among the permutations of L which achieves this minimum
value
> can be called *maximally even*.

Have you read any of the academic papers on this term ("maximally
even")? They don't consider the diatonic scale in 31-equal to be
maximally even. Must we get all readers "maximally confused"?

> One example of a maximally even scale would be a DE scale, or more
> generally an even scale of rank n. But given a multiset with n
> distinct elements generating a group of rank n, it is not the case
> that an even scale of rank n exists; however maximally even scales
> will certainly exist.
>
> For example, consider the multiset defined by
>
> (10/9)^2 (15/14)^2 (16/15)^2 (21/20)^3 = 2
>
> Trying all the permutations, we find that the minimum maximum
number of
> elements in an interval class is 5, and hence no even scales of
rank 4
> can be constructed from this multiset.

Yes, I remember finding this before . . . I was thinking that 2^(n-1)
would be an upper bound on the number of elements in an interval
class . . .

> Examples of maximally even
> scales can be found, however (60 in number.) The most interesting
are
> the scales defined in this posting from a few years back:
>
> /tuning-math/message/4563

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:

> > Suppose L is a multiset of intervals,
>
> What's the difference between a set and a multiset?

Multisets have multiplicities. Multisets of intervals can more or less
be identified with monomial expressions in the intervals; as for
instance (16/15)^2 (10/9)^2 (9/8)^3.

> Have you read any of the academic papers on this term ("maximally
> even")? They don't consider the diatonic scale in 31-equal to be
> maximally even. Must we get all readers "maximally confused"?

So what name would you suggest?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
>
> > > Suppose L is a multiset of intervals,
> >
> > What's the difference between a set and a multiset?
>
> Multisets have multiplicities. Multisets of intervals can more or
less
> be identified with monomial expressions in the intervals; as for
> instance (16/15)^2 (10/9)^2 (9/8)^3.
>
> > Have you read any of the academic papers on this term ("maximally
> > even")? They don't consider the diatonic scale in 31-equal to be
> > maximally even. Must we get all readers "maximally confused"?
>
> So what name would you suggest?

Sounds like you're talking about higher-order versions of
distributional evenness. An Nth-order distributionally even scale
would have N+1 different step sizes arranged as evenly as
possible . . . You're familiar with distributional evenness vs.
maximal evenness, yes?

>Suppose L is a multiset of intervals, the product of which is 2. Then
>the various permutations of L lead to various scales whose scale steps
>are the intervals of L. If we take all the permutations, there will be
>a minimum value for the maximum bumber of intervals in an interval class.
>A scale among the permutations of L which achieves this minimum value
>can be called *maximally even*.

One n difference between this and Rothenberg's mean variety is that
it doesn't give credit for intervals appearing in more than one
interval class, such as the tritone in the diatonic scale in 12-ET.
Since Rothenberg seemed to consider such "ambiguous intervals" a bad
thing, he might have liked this proposal.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> One n difference between this and Rothenberg's mean variety is that
> it doesn't give credit for intervals appearing in more than one
> interval class, such as the tritone in the diatonic scale in 12-ET.
> Since Rothenberg seemed to consider such "ambiguous intervals" a bad
> thing, he might have liked this proposal.

So what's a good name for it?

>> One n difference between this and Rothenberg's mean variety is that
>> it doesn't give credit for intervals appearing in more than one
>> interval class, such as the tritone in the diatonic scale in 12-ET.
>> Since Rothenberg seemed to consider such "ambiguous intervals" a bad
>> thing, he might have liked this proposal.
>
>So what's a good name for it?

I dunno...

-Carlg