Vals vs commas

To find a linear temperament wedgie we can use two vals, pi(n)-2 commas. As n increases, the number of combinations of a list of N commas taken pi(n)-2 at a time increases, whereas the number of standard vals and the number taken in pairs is steady, so one question is, do we need to worry about commas?

We run into a problem with the strictly val approach if fewer than two standard vals cover the temperament in question; this can happen because it is of relatively low complexity, or because it has a small period. One way out of this is to include non-standard vals--that is, look at various second best choices for mappings to primes; this however, increases the computational burden also.

Graham reports that he didn't find he needed the comma list approach.
I can't quite see how this happened, since the Tricontaheximal temperament, with a period of 33 1/3 cents, has only one standard val, namely 72. Other temperaments are skating close to the wire by having only two standard vals--Hemififths with 41 and 58, and Diaschismic with 46 and 58 among the top four, which is all I have checked.

Gene Ward Smith wrote:

> We run into a problem with the strictly val approach if fewer than two standard vals cover the temperament in question; this can happen because it is of relatively low complexity, or because it has a small period. One way out of this is to include non-standard vals--that is, look at various second best choices for mappings to primes; this however, increases the computational burden also.

It doesn't add much to the computational burden. You can still search with the same number of ETs, because the new, simpler ones are as likely to give a hit as the old, accurate ones. Even using twice as many ETs only makes the search 4 times as hard. The vector search is orders of magnitude harder, and gets worse the more primes you use. It may be possible to tame it by pruning branches, but that'll also make it much more complex.

> Graham reports that he didn't find he needed the comma list approach.
> I can't quite see how this happened, since the Tricontaheximal temperament, with a period of 33 1/3 cents, has only one standard val, namely 72. Other temperaments are skating close to the wire by having only two standard vals--Hemififths with 41 and 58, and Diaschismic with 46 and 58 among the top four, which is all I have checked.

Because I'm using non-standard vals. And this Tricontaheximal works anyway. Inconsistent ETs are more important the higher the odd limit, as simple temperaments can be missed. Probably non-standard vals will be important at the same time. I wasn't confident the ET search would compete with the vectors until I implemented them.

There are other ways of doing the search that should complete with the pairs of ETs. One is to go through the list of ETs and choose each number of steps as the generator for a linear temperament. The number of LTs you look at is roughly proportional to the number of ETs, so it's equivalent to the pairs search in complexity. The list of ETs should be smaller. You also have to look at some alternative mappings with the same ET and generator.

The other way is to start with a huge list of 5-limit linear temperaments (which is easy) and keep adding primes. Reject any LTs that are too complex or inaccurate as you go along.

Graham