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 include ~~[[5edf]], [[6edf]], [[7edf]] (not in the patent val),~~ [[11edf]], [[16edf]], [[17edf]] (not in the patent val), and [[23edf]] (not in the patent val).

'''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.

If tone clusters with intervals of supraminor seconds or less are ignored, the most fundamental 3/2.5/2.7/2 chord that can fit inside 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 major or minor triad. Both 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 can fit inside 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 major or minor triad. Both of these are well approximated in halftone because it equates 4 7/5 with 10/9.

~~[[16edf]] is ay~~

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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 include [[5edf]], [[6edf]], [[7edf]] (not in the patent val), [[11edf]], [[16edf]], [[17edf]] (not in the patent val), and [[23edf]] (not in the patent val).
+'''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.
-If tone clusters with intervals of supraminor seconds or less are ignored, the most fundamental 3/2.5/2.7/2 chord that can fit inside 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 major or minor triad. Both 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 can fit inside 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 major or minor triad. Both of these are well approximated in halftone because it equates 4 7/5 with 10/9.
-[[16edf]] is ay