Halftone: Difference between revisions

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'''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 ~~Bohlen-Pierce-Stearns~~ for no-twos systems with the equivalence as 3/1. 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 is an interesting case because it is also an approximation of [[19edo]], which allows for playing both meantone and halftone music. Halftone temperament can be extended to the 11-limit (3/2.5/2.7/2.11/2) by additionally tempering out 1232/1515.

'''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]]. 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 is an interesting case because it is also an approximation of [[19edo]], which allows for playing both meantone and halftone music. Halftone temperament can be extended to the 11-limit (3/2.5/2.7/2.11/2) by additionally tempering out 1232/1515.

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.

Revision as of 23:37, 9 June 2023 view source CompactStar (talk \| contribs) 5,020 edits No edit summary ← Older edit		Revision as of 23:37, 9 June 2023 view source CompactStar (talk \| contribs) 5,020 edits No edit summary Newer edit →
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	'''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 ~~Bohlen-Pierce-Stearns~~ for no-twos systems with the equivalence as 3/1. 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 is an interesting case because it is also an approximation of [[19edo]], which allows for playing both meantone and halftone music. Halftone temperament can be extended to the 11-limit (3/2.5/2.7/2.11/2) by additionally tempering out 1232/1515.		'''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]]. 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 is an interesting case because it is also an approximation of [[19edo]], which allows for playing both meantone and halftone music. Halftone temperament can be extended to the 11-limit (3/2.5/2.7/2.11/2) by additionally tempering out 1232/1515.

	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.		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.