Musical sequence

[franktaw at netscape.net]

It's not just those values.  You want all ratios up to a certain "size" to be approximated well - increasing this size is more important that getting a better approximation.  There is no obvious definition for size.  In general, divisibility only by small primes is important; 9/8 is clearly superior to 7/6, for example.
 
31 works pretty well.
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
 
 
-----Original Message-----
From: Jud McCranie <j.mccranie at adelphia.net>
To: David Wilson <davidwwilson at comcast.net>
Cc: Sequence Fans <seqfan at ext.jussieu.fr>
Sent: Wed, 18 Jan 2006 13:46:40 -0500
Subject: Re: Musical sequence


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. 
 
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