Harmonic circulating temperaments with no harmonic waste

I'm revisiting this topic:

/tuning/topicId_75816.html#75816

That 7-limit 22-tone tuning mentioned in the message should be

124:128:132:136:141:145:150:155:160:165:170:
176:181:187:193:199:206:212:219:226:233:241

or that is what an old program of mine gives as the lowest
22-tone tuning with no 7-limit harmonic waste. The problem
is that that program is a mess and I now find it hard to
understand what it is doing. :) I suppose it limits the search
space by incorporating the restriction that fifths must be
pure or sharp and similar constraints for some other intervals
and then doing some kind of tree search.

Another program gives these 3 tunings as the lowest 22-tone
tunings with no 9-limit harmonic waste:

321:332:342:353:364:376:388:401:413:427:440:
454:468:484:499:516:532:549:566:584:602:623

321:332:342:353:364:376:388:401:413:427:440:
454:468:484:499:516:532:549:566:584:603:623

321:332:342:353:364:376:388:401:413:427:440:
454:469:484:499:516:532:549:566:584:603:623

Can anybody better in math than me confirm these results?
More elegant and general methods for doing these searches
would also be great.

P.S. If you are wondering what's the point of these try that
12-tone

116:123:130:138:146:155:164:174:184:195:207:219:232

with a relatively steady timbre.

Thanks,
Kalle

--- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> I'm revisiting this topic:
>
> /tuning/topicId_75816.html#75816

What's "combined absolute error"? Do you mean the sum of the absolute values of the errors?

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning-math@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
> >
> > I'm revisiting this topic:
> >
> > /tuning/topicId_75816.html#75816
>
> What's "combined absolute error"? Do you mean the sum of the absolute values of the errors?

Yes, but you don't combine the errors of all n-limit intervals, you
look at them separately. For example, in 12-tone 5-limit case you sum
the absolute errors of fifths, minor thirds and major thirds
separately. If the fifths error is 23.46 cents, major thirds error
164.2354 cents and minor thirds error is 187.6954 then you have a
well temperament (well, sort of: you have a something that Scala
recognizes as a well temperament). Equivalently you must have all
fifths flat, major thirds sharp and minor thirds flat. If there are
both positive and negative errors for the same interval, you are
tempering more than what is necessary. The concept of harmonic waste
comes from Owen Jorgensen. Perhaps someone else can give a more
rigorous definition of harmonic waste.

Kalle