Re:Canright's post on algorithm

CF convergents are the best for given sizes of terms, but sometimes CF's
converge too fast and yield solutions with too many tones/octave (in a
music theory context). What one wants is are scales with reasonable
approximations to the just intervals in a reasonable number of tones.
For that reason, Barbour's mixed ternary CF expansion or Brun's (or
Selmer's or Pipping's variants) are useful.

--John

[John Chalmers]
> For that reason, Barbour's mixed ternary CF expansion or Brun's (or
Selmer's or Pipping's variants) are useful.

I was wondering if anyone could offer some examples of these, or point
me in a direction where I could find out more about them.

Thanks,
Dan

John Chalmers wrote,

>CF convergents are the best for given sizes of terms, but sometimes CF's
>converge too fast and yield solutions with too many tones/octave (in a
>music theory context). What one wants is are scales with reasonable
>approximations to the just intervals in a reasonable number of tones.
>For that reason, Barbour's mixed ternary CF expansion or Brun's (or
>Selmer's or Pipping's variants) are useful.

But John, we were not talking about deriving multi-tone systems -- we were
talking about approximating an irrational interval with a rational one. In
such a context, would anything other that CF's make sense? And again, isn't
the outer loop Canright was referring to equivalent to CF's?

In the context of deriving multi-tone systems, you must be talking about
approximating two or more just intervals at a time with a common smallest
step size. In such a context, it makes sense that one might want to trade
off an increase in the number of tones and a small worsening in one interval
for a considerable improvement in the other intervals. That's where mixed
expansions or Brun's algorithms can help. My guess is that in the standard
multi-term CF approach or whatever, which must be what you're referring to
above (since for the usual, single-term CF, "convergents are the best for
given sizes of terms" is synonymous with "scales with reasonable
approximations to the just intervals in a reasonable number of tones"), an
ET with a higher number of notes comes in if and only if _all_ the
approximations to the just intervals are improved over those of the last ET.

But with modern computer power, all these algorithms are a mere mathematical
curiousity, because you can easily specify what kind of trade-off you want
and then write a computer program to find the solution by brute force.