Musical sequence

[Richard Guy]

See page 257 of the BoN, where the comma
of Pythagoras is incorrectly defined.  It
shouldbe `the difference between 12 fifths
and 7 octaves' ( not 7 and 4).   R.


On Wed, 18 Jan 2006, Jud McCranie wrote:

> 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.
>
>