Re: well-temperment wazoo

Paul,

I'm very interested in your method of relaxing tunings to achieve maximum
consonance, but I'm puzzled by these latest results. Correct me if I'm
wrong: theoretically we could graph the total harmonic entropy measure
against note values for a scale of cardinality n in n + 1 dimensions and
have a smooth "surface." Then any value your algorithm arrives at must be
a local minimum with repect to entropy. Now my question would be: can we
generalize about the positions of local minima or the general shape of the
surface for cardinality n? Your last set of imputs to the minimum-finding
algorithm showed a large number of local minima in the region of 12-tET.
If the graphs are this bumpy everywhere, then looking for minima tells us
very little. But maybe the graphs drop off in other regions, and we might
find that, while in the region of 12t-ET, there are a number of minima,
most other regions contain few minima, for instance. Perhaps there is
something to be seen by looking at the problem with a coarser resolution
which would smooth out the smaller lumps in the graph. I'm thinking of
something like having the algorithm consider the results of a step of 5
cents in some direction, rather than of .01 (or whatever) cents. Then you
could see if any other regions of the graph are worth considering. Also:
can you find an absolute minimum given some number of tones?

jason

Paul's total diadic harmonic entropy minimizer is reminiscent of other
methods in that it tries to maximise the number of consonant intervals
in a scale:
- Dave Keenan's comma distribution method. This method calculates
precisely how big a part of a given comma needs to be added to which
intervals in order to minimise the maximum deviation from a predefined
set of consonant intervals.
- Some time ago I wrote a new Scala routine which minimises the least
squares deviation of scale intervals from a predefined set of consonant
intervals. This routine first marks the intervals which are within a
given distance from one of the consonant intervals and then calculates
the optimal tempering for them. This has the advantage over Keenan's
method that one doesn't need to tell which comma should be distributed
(it can distribute more than one at the same time), but the result can
be less precise and slightly less optimal than Keenan's method.
The name of this routine is EGALIZE/MODEL, present in the next release.
- Paul's entropy minimizer has the advantage that one doesn't need to
specify the consonant intervals, since they are implied by the entropy
curve. The disadvantage may be that it's likely to get stuck in local
minima of what might be a very bumpy landscape.

By the way, when I feeded 12-tone equal temperament with one
randomly changed tone to my least-squares method, it also produced a
well-temperament.

Perhaps a combination of Paul's and my method would be useful. First
run the entropy minimizer to get the basic scale structure, and after
that run my method to fine-tune it.

Paul, the well-temperament that you found is close to Schlick (1511)
and Lambert (1774).

Manuel Op de Coul coul@ezh.nl