Is meantone inherently septimal?

Speaking of Milne, Sethares and Plamondon, they suggest defining a tuning range for a given regular temperament and list of target intervals as the range of tunings which maintains the size ordering of the intervals. So, for instance, taking 5-limit meantone and the 5-limit diamond, they get the range all the way from 7et to 5et, or 4/7 to 3/5 in terms of generators. Since this obviously isn't the correct meantone range, they end up calling it "syntonic", but this doesn't address the underlying problem, which is that this seems to often give pretty screwed up answers for the tuning range. For instance, for schismatic temperament this gives the range from 7et to 17et, which is completely crazy.

Using a 7-limit temperament with the 7 or 9 limit diamond often seems to work much better. As they point out, meantone really lives roughly from 19et to 12et, and this is exactly the range you get for septimal meantone with either the 7 or the 9 limit diamond as a target. Using the 11-limit diamond, meanpop extends from 19 to 31, and unidecimal meantone (huyghens) from 31 to 12, and this seems right also. Meanwhile, garibaldi temperament using either the 7 or 9 limit diamond extends from 12 to 29, and this seems reasonable.

Gene wrote:

> Speaking of Milne, Sethares and Plamondon, they suggest defining
> a tuning range for a given regular temperament and list of target
> intervals as the range of tunings which maintains the size
> ordering of the intervals.

What do you think of Graham's/my suggestion of using badness
instead?

/tuning/topicId_92040.html#92203

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Gene wrote:
>
> > Speaking of Milne, Sethares and Plamondon, they suggest defining
> > a tuning range for a given regular temperament and list of target
> > intervals as the range of tunings which maintains the size
> > ordering of the intervals.
>
> What do you think of Graham's/my suggestion of using badness
> instead?
>
> /tuning/topicId_92040.html#92203

Thst should work, but you'd need to pick a limiting value and I don't see how to do it canonically.

Gene wrote:

> That should work, but you'd need to pick a limiting value
> and I don't see how to do it canonically.

Howabout this:

Given a temperament t, make temperament ranking lists
(using logflat badness) at progressively higher complexity
cutoffs until t appears as the 2nd worst temperament in
a list. Then vary the tuning for t so its badness,
so tuned, is bounded to keep t's position in that list.

-Carl