Questions for Graham Breed

1. Where is schismic temperament in your rankings of 5-limit
temperaments? Isn't it "better" than #1 by your measure?

2. Wouldn't you find an infinite number of temperaments that
are "better" than your #1 temperaments by your measure, if you
carried your search further and further out?

3. If the answer to question 2 is "yes", do you think that maybe one
should put more weight on having a low "complexity measure", and come
up with a ranking that, while not too subjective, is comprehensive in
that no, let along an infinite number, of possibilities are left out?

Paul asked:

> 1. Where is schismic temperament in your rankings of 5-limit
> temperaments? Isn't it "better" than #1 by your measure?

I'm not sure. It probably gets vetoed for being too complex. I don't pay
much attention to the 5-limit results.

> 2. Wouldn't you find an infinite number of temperaments that
> are "better" than your #1 temperaments by your measure, if you
> carried your search further and further out?

Probably, certainly enough that they would swamp the useful results. If
you recall the initial data, this is what happened with the 5-limit.

> 3. If the answer to question 2 is "yes", do you think that maybe one
> should put more weight on having a low "complexity measure", and come
> up with a ranking that, while not too subjective, is comprehensive in
> that no, let along an infinite number, of possibilities are left out?

If you can find one that makes sense. For now, it works well enough by
restricting the seed ETs and excluding anything beyond a given level of
complexity. You can then apply a truly subjective measure to the
temperaments that make the list.

Graham

--- In tuning@y..., graham@m... wrote:
> Paul asked:

> > do you think that maybe one
> > should put more weight on having a low "complexity measure", and
come
> > up with a ranking that, while not too subjective, is
comprehensive in
> > that no, let along an infinite number, of possibilities are left
out?
>
> If you can find one that makes sense.

I bet Gene can do that . . . also, I think a "weighted complexity"
measure, with ratios of 3 somewhat more important than ratios of 5,
etc., would be better. I propose weights of 1/log(3), 1/log(5), etc.

> For now, it works well enough by
> restricting the seed ETs and excluding anything beyond a given
level of
> complexity.

Well, you're cutting things off at a very arbitrary place, as I think
the absence of schismic (with a complexity measure of 9 and max.
error of around 1/4 of 1 cent) in the 5-limit results, and the
considerable complexity of the #1 7-limit result, make clear . . .
that is, the result of which tuning happens to come up as #1 is
highly sensitive to your cutoff point.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> I bet Gene can do that . . . also, I think a "weighted complexity"
> measure, with ratios of 3 somewhat more important than ratios of 5,
> etc., would be better. I propose weights of 1/log(3), 1/log(5), etc.

I've been sitting here trying to figure out what complexity it is you
want to measure. What is a complexity measure supposed to measure?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > I bet Gene can do that . . . also, I think a "weighted
complexity"
> > measure, with ratios of 3 somewhat more important than ratios of
5,
> > etc., would be better. I propose weights of 1/log(3), 1/log(5),
etc.
>
> I've been sitting here trying to figure out what complexity it is
you
> want to measure. What is a complexity measure supposed to measure?

Graham's complexity measure is currently defined as the maximum
number of generators necessary to generate any of the consonant
intervals of the system (times the number of intervals of repetition
per octave, usually 1). But I think one should favor temperaments
where the ratios of 3 are generated by fewer generators than ratios
of higher odd numbers, over those where the opposite is the case, all
else being equal.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Graham's complexity measure is currently defined as the maximum
> number of generators necessary to generate any of the consonant
> intervals of the system (times the number of intervals of
repetition
> per octave, usually 1). But I think one should favor temperaments
> where the ratios of 3 are generated by fewer generators than ratios
> of higher odd numbers, over those where the opposite is the case,
all
> else being equal.

Do you mean by "number of generators" the required span of the powers
of a generator--that is, if our generator is a meantone fifth m, we
get m ~ 3 mod 2, m^4 ~ 5 mod 2, so the complexity of the 5-limit
would be 4?

--- In tuning@y..., genewardsmith@j... wrote:

> if our generator is a meantone fifth m, we
> get m ~ 3 mod 2, m^4 ~ 5 mod 2, so the complexity of the 5-limit
> would be 4?

Yes, Graham's current complexity measure gives meantone a complexity
of 4 in the 5-limit.

Paul Erlich wrote:

> Graham's complexity measure is currently defined as the maximum
> number of generators necessary to generate any of the consonant
> intervals of the system (times the number of intervals of repetition
> per octave, usually 1). But I think one should favor temperaments
> where the ratios of 3 are generated by fewer generators than ratios
> of higher odd numbers, over those where the opposite is the case, all
> else being equal.

That'd give temperaments like meantone and schismic, that we already knew
about. What excited me about Miracle is that the intervals of 11 aren't
in general any more complex than the 9-limit. So it removes the
temptation to use a core 5-limit harmony with higher limit elaborations.
The aim of the program was to find more temperaments like Miracle, in
their respective limits.

Finding meantone-like temperaments should be easier without the program.
You can start with the 5-limit approximation, and then find the closest
ones to the higher limits. I suppose there may be some value in doing
this for alternative timbres. Getting the existing program to work with
such a complexity formula wouldn't be trivial, but you can probably work
it out if you poke around for long enough. A similar metric for
calculating the errors would probably be appropriate.

Graham