Musical sequence

[Jud McCranie]

At 01:40 PM 1/18/2006, David Wilson wrote:
>In general music theory, we hold to the concept of an octave, 
>specifically, that if a musical scale includes a tone of frequency f, it 
>also includes a tone of frequency 2f.
>
>What if we abandon the notion of octave, but keep the idea of equally 
>spaced tones, so that our scale can be described by base tone frequency b 
>and a ratio r between adjacent tones on the scale.
>
>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?

I looked at this sometime ago.  The important thing about 12 equally-spaced 
tones is that r^4 is close to 5/4, r^5 is close to 4/3, and r^7 is close to 
3/2, r is the 12-th root of 2, as above.  The next values that come close 
to these important values are when there are 43 and 53 equally-spaced tones 
between frequencies f and 2f, IIRC.