Halftone: Difference between revisions

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[[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 and has a generator of a flat 7/5 of around 570-580 cents. 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 and has a generator of a flat [[7/5]] of around 570-580 cents. 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.

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 far less than a standard major or minor triad. Both of these are well approximated in halftone because it equates 4 7/5 generators 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 far less than a standard major or minor triad. Both of these are well approximated in halftone because it equates 4 [[7/5]] generators with [[10/9]].

== Interval chain ==

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== ~~Scales~~ ==

== MOS scales ==

Halftone possesses MOS scales with 4 ([[1L 3s (3/2-equivalent)|1L 3s⟨3/2⟩]] or "neptunian"), 5 ([[1L 4s (3/2-equivalent)|1L 4s⟨3/2⟩]]), 6 ([[5L 1s (3/2-equivalent)|5L 1s⟨3/2⟩]]) and 11 ( ([[5L 6s (3/2-equivalent)|5L 6s⟨3/2⟩]]) ). The tetratonic scales ix usable, but the tempered 10/9 is not present in it, so really the pentatonic and hexatonic scales are the smallest options.

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 [[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 and has a generator of a flat 7/5 of around 570-580 cents. 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 and has a generator of a flat [[7/5]] of around 570-580 cents. 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.
-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 far less than a standard major or minor triad. Both of these are well approximated in halftone because it equates 4 7/5 generators 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-[[
+/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 far less than a standard major or minor triad. Both of these are well approximated in halftone because it equates 4 [[7/5]] generators with [[10/9]].
 == Interval chain ==
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 |}
-== Scales ==
+== MOS scales ==
+Halftone possesses MOS scales with 4 ([[1L 3s (3/2-equivalent)|1L 3s⟨3/2⟩]] or "neptunian"), 5 ([[1L 4s (3/2-equivalent)|1L 4s⟨3/2⟩]]), 6 ([[5L 1s (3/2-equivalent)|5L 1s⟨3/2⟩]]) and 11 ( ([[5L 6s (3/2-equivalent)|5L 6s⟨3/2⟩]]) ). The tetratonic scales ix usable, but the tempered 10/9 is not present in it, so really the pentatonic and hexatonic scales are the smallest options.