Halftone: Difference between revisions
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[[File:halftone6.wav|Halftone[6] example in 16edf and 4<nowiki>|</nowiki>1 mode]] | [[File:halftone6.wav|thumb|Halftone[6] example in 16edf and 4<nowiki>|</nowiki>1 mode]] | ||
'''Halftone''' is a [[nonoctave]] (fifth-repeating) [[regular temperament]] in the 3/2.5/2.7/2 fractional subgroup that tempers out 9604/9375. It could be used as a harmonic basis for "1/2 prime" (3/2.5/2.7/2.11/2.13/2 etc.) systems with the equivalence as [[3/2]], similar to [[meantone]] for full prime-limit systems with the equivalence as [[2/1]] and [[BPS]] for no-twos systems with the equivalence as [[3/1]]. Halftone temperament can be extended to the 11-limit (3/2.5/2.7/2.11/2) by additionally tempering out 1232/1215, the difference between [[15/14]] and [[88/81]] (the fifth-reduction of 11/2). Small [[EDF]]s that [[support]] halftone with relatively low error include [[11edf]], [[16edf]], [[17edf]] (not in the patent val), and [[23edf]] (not in the patent val). 11edf particularly is an interesting case because it is also an approximation of [[19edo]], which allows for playing both meantone and halftone music. | '''Halftone''' is a [[nonoctave]] (fifth-repeating) [[regular temperament]] in the 3/2.5/2.7/2 fractional subgroup that tempers out 9604/9375. It could be used as a harmonic basis for "1/2 prime" (3/2.5/2.7/2.11/2.13/2 etc.) systems with the equivalence as [[3/2]], similar to [[meantone]] for full prime-limit systems with the equivalence as [[2/1]] and [[BPS]] for no-twos systems with the equivalence as [[3/1]]. Halftone temperament can be extended to the 11-limit (3/2.5/2.7/2.11/2) by additionally tempering out 1232/1215, the difference between [[15/14]] and [[88/81]] (the fifth-reduction of 11/2). Small [[EDF]]s that [[support]] halftone with relatively low error include [[11edf]], [[16edf]], [[17edf]] (not in the patent val), and [[23edf]] (not in the patent val). 11edf particularly is an interesting case because it is also an approximation of [[19edo]], which allows for playing both meantone and halftone music. | ||
Revision as of 23:41, 9 June 2023
Halftone is a nonoctave (fifth-repeating) regular temperament in the 3/2.5/2.7/2 fractional subgroup that tempers out 9604/9375. It could be used as a harmonic basis for "1/2 prime" (3/2.5/2.7/2.11/2.13/2 etc.) systems with the equivalence as 3/2, similar to meantone for full prime-limit systems with the equivalence as 2/1 and BPS for no-twos systems with the equivalence as 3/1. Halftone temperament can be extended to the 11-limit (3/2.5/2.7/2.11/2) by additionally tempering out 1232/1215, the difference between 15/14 and 88/81 (the fifth-reduction of 11/2). Small EDFs that support halftone with relatively low error include 11edf, 16edf, 17edf (not in the patent val), and 23edf (not in the patent val). 11edf particularly is an interesting case because it is also an approximation of 19edo, which allows for playing both meantone and halftone music.
If tone clusters with intervals of supraminor seconds or less are ignored, the most fundamental 3/2.5/2.7/2 chord that is narrower than a perfect fifth is 45:50:63 (1-10/9-7/5), essentially a diminished triad with a major second instead of a minor third. There is also a more "major-sounding" counterpart of it 50:63:70 (1-63/50-7/5), a diminished triad with a major third instead of a minor third. These chords generally sound more consonant than a standard diminished triad but not as much as a standard major or minor triad. Both of these are well approximated in halftone because it equates 4 7/5 with 10/9.