EDT: Difference between revisions

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== Rank two temperaments ==

If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [{{val|1 1 2}}, {{val|0 2 -1}}] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit.

If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [{{val|1 1 2}}, {{val|0 2 -1}}] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit. An alternative approach is to preserve 2s but use a sixth-based 3:4:5 triad rather than the fifth-based 4:5:6, which favors MOS of the form 2L7s with the generator as 5/3 or an approximation of it, and period as the tritave.

At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [{{val|1 3 3}}, {{val|0 -5 -4}}] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.

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 == Rank two temperaments ==
-If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [{{val|1 1 2}}, {{val|0 2 -1}}] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit.
+If factors of two are eliminated, the simplest possible triad is (1):(3):5:7:(9), with 1, 3 and 9 in parentheses as they're all tritave-equivalent to 1. Hence, 3:5:7 can be viewed as the fundamental consonant triad of no-twos music. The linear temperament that best approximates these chords in the moderate complexity range is the Bohlen-Pierce linear temperament eliminating 245/243, which has a no-twos mapping of [{{val|1 1 2}}, {{val|0 2 -1}}] and a pure-tritaves TE generator which is a sharp 9/7 of 440.488 cents. It possesses [[MOSScales|MOS]] of the forms 4L1s (pentatonic) and 4L5s (nonatonic), and larger MOS of size 13, 17, 30, 43, 56, 69 and 82. This temperament serves a function analogous to meantone in the 5-limit. An alternative approach is to preserve 2s but use a sixth-based 3:4:5 triad rather than the fifth-based 4:5:6, which favors MOS of the form 2L7s with the generator as 5/3 or an approximation of it, and period as the tritave.
 At higher complexities, the rank two 3.5.7 temperament tempering out 16875/16807 called [[Canopus]] begins to predominate. This has a mapping [{{val|1 3 3}}, {{val|0 -5 -4}}] and a pure-tritaves TE generator a slightly flat 7/5 at 581.512 cents. This has MOS of size 3, 4, 7, 10, 13, 23, 36, etc, with the 36 note MOS being particularly even.