Musical sequence

[Jon Wild]

On Wed, 18 Jan 2006, David Wilson wrote:

> Is there a measure of goodness of r that would rate r = 2^(1/12) high? 
> Presumably this rating would be based on the closeness of tone ratios to 
> simple rationals which represent pleasing harmonies.  Given this rating, 
> what would be the best r?  If we return to octave-based scales, could be 
> construct a sequence of t with increasingly better ratings of 2^(1/t), 
> which would represent increasingly good choices for number of tones in 
> an octave?

There are several candidates for such a sequence; they generally include 
(5,7,12),19,31,53... (the closeness of 3^53 to a power of 2, which causes 
53-tone equal temperament to have an excellent perfect fifth, was 
discovered by the Chinese more than 5000 years ago)

As a couple of examples out of hundreds, you could look into Joseph 
Yasser's book "A Theory of Evolving Tonality" for precisely what you're 
talking about; Norman Carey & David Clampitt's article "Well-formed 
Scales" in _Music Theory Spectrum_ for a more mathematical treatment (and 
a slightly different series), or see the yahoo discussion groups "tuning" 
and "tuning-math" for many, many more details than you'd care to know 
about.

Jon

-- 
Jon Wild
Assistant Professor
Schulich School of Music
McGill University