Kite's thoughts on Stern-Brocot ancestors and rank 2 temperaments
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- This revision was by author TallKite and made on 2014-02-20 04:26:56 UTC.
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==Rank 2 tunings== Let's say you're making a rank 2 tuning, with a period of P cents and a generator of G cents. You make an N-note scale with a chain of N-1 generators. The classic example of all this is a diatonic scale with an octave as period and a fifth as generator, with P = 1200¢, G = 702¢, and N = 7. Or perhaps the fifth is tempered down to 700¢ for 12-EDO, or 696¢ for quarter-comma meantone. Or perhaps N = 5 because you live in a pentatonic culture. Or perhaps you're using the tritave 3/1 as your period. Whatever, this discussion applies to all these cases. You stop at N notes because that's what you conceive as the __**stepspan**__ of the period. The stepspan is one less than the degree, so a third has stepspan 2, a fourth has 3, etc. The process of filling in the period with notes can be visualized as drawing a star of sorts in a circle. The generator has a stepspan too, determined by how many star-points fall between the tonic and the first generator. Let's call this stepspan A. For the octave/fifth/diatonic situation, A is 4. For the pentatonic situation, A is 3, because there are only 2 notes between the tonic and the fifth. The generator is A/N of a period. As long as the generator
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<html><head><title>Stern-Brocot ancestors and rank 2 temperaments</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Rank 2 tunings"></a><!-- ws:end:WikiTextHeadingRule:0 -->Rank 2 tunings</h2> Let's say you're making a rank 2 tuning, with a period of P cents and a generator of G cents. You make an N-note scale with a chain of N-1 generators. The classic example of all this is a diatonic scale with an octave as period and a fifth as generator, with P = 1200¢, G = 702¢, and N = 7. Or perhaps the fifth is tempered down to 700¢ for 12-EDO, or 696¢ for quarter-comma meantone. Or perhaps N = 5 because you live in a pentatonic culture. Or perhaps you're using the tritave 3/1 as your period. Whatever, this discussion applies to all these cases.<br /> <br /> You stop at N notes because that's what you conceive as the <u><strong>stepspan</strong></u> of the period. The stepspan is one less than the degree, so a third has stepspan 2, a fourth has 3, etc. The process of filling in the period with notes can be visualized as drawing a star of sorts in a circle. The generator has a stepspan too, determined by how many star-points fall between the tonic and the first generator. Let's call this stepspan A. For the octave/fifth/diatonic situation, A is 4. For the pentatonic situation, A is 3, because there are only 2 notes between the tonic and the fifth.<br /> <br /> The generator is A/N of a period. As long as the generator</body></html>