proposal

I would like to see the results of...

for all equal temperaments with 5 <= G <= 100 steps/octave
for all K such that 3 <= K <=6
for all K-out-of {2 3 5 7 11 13 17}
the val with the lowest TOP damage
sort (96 * 4 = 384) ET-val pairs by K, then 1 / (G * TOP damage)

-C.

A version of this weighted by chordal Tenney height would be
interesting also. -C.

At 12:35 AM 10/2/2006, you wrote:
>I would like to see the results of...
>
>for all equal temperaments with 5 <= G <= 100 steps/octave
> for all K such that 3 <= K <=6
> for all K-out-of {2 3 5 7 11 13 17}
> the val with the lowest TOP damage
>sort (96 * 4 = 384) ET-val pairs by K, then 1 / (G * TOP damage)
>
>-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> A version of this weighted by chordal Tenney height would be
> interesting also. -C.

What exactly are you looking for? Lowest errors? That seems a little
artificial, since errors get lower as divisions get larger anyway.

>> A version of this weighted by chordal Tenney height would be
>> interesting also. -C.
>
>What exactly are you looking for? Lowest errors? That seems a little
>artificial, since errors get lower as divisions get larger anyway.

Doesn't the original text make it clear that this is badness (* G)?

The idea is, instead of finding rank 2 temperaments and then embedding
them in a suitable ET, rather to take what we've learned and go back
to ETs-as-atomic. My hope is to better address issues formerly tackled
with consistency and (the aptly-named) harmonic "limits".

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Doesn't the original text make it clear that this is badness (* G)?

It seems to me that logflat badness figures have been posted a zillion
times. If that wasn't enough, there are related topics, including some
on OEIS:

http://www.research.att.com/~njas/sequences/A117536
http://www.research.att.com/~njas/sequences/A117537
http://www.research.att.com/~njas/sequences/A117538
http://www.research.att.com/~njas/sequences/A117539

http://www.research.att.com/~njas/sequences/A054540

http://www.research.att.com/~njas/sequences/A117554
http://www.research.att.com/~njas/sequences/A117555
http://www.research.att.com/~njas/sequences/A117556
http://www.research.att.com/~njas/sequences/A117557
http://www.research.att.com/~njas/sequences/A117558
http://www.research.att.com/~njas/sequences/A117559

Now, of course, if you want to do this for subgroups you open up a can
of worms in terms of the large numbers of subgroups. However,
restricting it to just prime generators makes sense, as that usually
seems to be what people want anyway. The logflat computations can be
done, and in the past I've presented some. The Keenan ambiguity thing
could be done also.

> The idea is, instead of finding rank 2 temperaments and then embedding
> them in a suitable ET, rather to take what we've learned and go back
> to ETs-as-atomic. My hope is to better address issues formerly tackled
> with consistency and (the aptly-named) harmonic "limits".

I'm still not clear on what it is you really want. The Zeta function
selected ets are free of constraints imposed by limits, however.

>> Doesn't the original text make it clear that this is badness (* G)?
>
>It seems to me that logflat badness figures have been posted a zillion
>times.

None of them return the result that I'm suggesting (characteristic
harmonic basis for each ET < 100).

If that wasn't enough, there are related topics, including some
>on OEIS:
>
>http://www.research.att.com/~njas/sequences/A117536
>http://www.research.att.com/~njas/sequences/A117537
>http://www.research.att.com/~njas/sequences/A117538
>http://www.research.att.com/~njas/sequences/A117539
>
>http://www.research.att.com/~njas/sequences/A054540
>
>http://www.research.att.com/~njas/sequences/A117554
>http://www.research.att.com/~njas/sequences/A117555
>http://www.research.att.com/~njas/sequences/A117556
>http://www.research.att.com/~njas/sequences/A117557
>http://www.research.att.com/~njas/sequences/A117558
>http://www.research.att.com/~njas/sequences/A117559
>
>Now, of course, if you want to do this for subgroups you open up a can
>of worms in terms of the large numbers of subgroups. However,
>restricting it to just prime generators makes sense, as that usually
>seems to be what people want anyway.

I list only primes in my pseudocode.

>> The idea is, instead of finding rank 2 temperaments and then embedding
>> them in a suitable ET, rather to take what we've learned and go back
>> to ETs-as-atomic. My hope is to better address issues formerly tackled
>> with consistency and (the aptly-named) harmonic "limits".
>
>I'm still not clear on what it is you really want. The Zeta function
>selected ets are free of constraints imposed by limits, however.

It selects them for limit infinity, is my understanding. My proposal
doesn't select ETs, however. It selects mappings for ETs.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> Doesn't the original text make it clear that this is badness (* G)?
> >
> >It seems to me that logflat badness figures have been posted a zillion
> >times.
>
> None of them return the result that I'm suggesting (characteristic
> harmonic basis for each ET < 100).

What's a characteristic harmonic basis? This doesn't seem to have much
to do that I can see with what you were asking before.

>> >> Doesn't the original text make it clear that this is badness (* G)?
>> >
>> >It seems to me that logflat badness figures have been posted a zillion
>> >times.
>>
>> None of them return the result that I'm suggesting (characteristic
>> harmonic basis for each ET < 100).
>
>What's a characteristic harmonic basis?

A set of n primes <= 17 which the ET is better at approximating than
any other.

-Carl

At 02:11 AM 10/4/2006, you wrote:
>>> >> Doesn't the original text make it clear that this is badness (* G)?
>>> >
>>> >It seems to me that logflat badness figures have been posted a zillion
>>> >times.
>>>
>>> None of them return the result that I'm suggesting (characteristic
>>> harmonic basis for each ET < 100).
>>
>>What's a characteristic harmonic basis?
>
>A set of n primes <= 17 which the ET is better at approximating than
>any other.

...Any other set of n primes <=17, not any other ET. -C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> ...Any other set of n primes <=17, not any other ET. -C.
>

That still isn't making sense--pick the best approximated prime, and
that singleton set is your answer.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> >What's a characteristic harmonic basis?
>
> A set of n primes <= 17 which the ET is better at approximating than
> any other.

That doesn't make sense to me--ets are never better than all other
ets. Do you mean all smaller ones?

>> ...Any other set of n primes <=17, not any other ET. -C.
>
>That still isn't making sense--pick the best approximated prime, and
>that singleton set is your answer.

The number of primes I'm considering starts at 3 and goes to 6.

-Carl

Carl Lumma wrote:
> I would like to see the results of...
> > for all equal temperaments with 5 <= G <= 100 steps/octave
> for all K such that 3 <= K <=6
> for all K-out-of {2 3 5 7 11 13 17}
> the val with the lowest TOP damage
> sort (96 * 4 = 384) ET-val pairs by K, then 1 / (G * TOP damage)

Taking your sorting instructions literally (so the list is mostly sorted by badness), here's my Python code:

import regular, temper

primeLookup = dict(zip(regular.primeNumbers, regular.primes))

allPrimes = 2, 3, 5, 7, 11, 13, 17
errorFunction = regular.topError

decorated = []

for K in range(3, 7):
for intBasis in temper.combinations(K, allPrimes):
basis = [primeLookup[p] for p in intBasis]
for G in range(5, 101):
et = regular.BestET(G, basis, errorFunction)
badness = errorFunction(et.weightedPrimes())*G
decorated.append((badness, K, intBasis, et))

decorated.sort()

for dec1, dec2, intBasis, et in decorated:
print '%40s%s'%(str(intBasis), str(et))

The decorated.sort() is unindented, so you can reconstruct the indentation if Yahoo still screws it up. It takes about 14 seconds to run on my laptop.

The required libraries are at

http://microtonal.co.uk/temper/regular.zip

the results along with all code are at

http://microtonal.co.uk/carl.zip

That includes results for sorting the other way round because I don't know if Carl meant what he said, which technically I'm not doing anyway.

Caveat: trivial changes in the code can lead to different results, so perhaps I'm doing something wrong

Graham