Halftone: Difference between revisions
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[[File:halftone6.wav|thumb|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 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 | '''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 include [[5edf]], [[6edf]], [[11edf]], [[16edf]], [[21edf]], and [[27edf]]. | ||
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]]. | ||
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{|class="wikitable" | {|class="wikitable" | ||
|- | |- | ||
|53.040 | | 53.040 | ||
|182.823 | | 182.823 | ||
|312.606 | | 312.606 | ||
|442.389 | | 442.389 | ||
|572.172 | | 572.172 | ||
|0 | | 0 | ||
|129.783 | | 129.783 | ||
|259.566 | | 259.566 | ||
|389.349 | | 389.349 | ||
|519.132 | | 519.132 | ||
|648.915 | | 648.915 | ||
|- | |- | ||
|[[28/27]] | | [[28/27]] | ||
|[[10/9]] | | [[10/9]] | ||
|[[25/21]] | | [[25/21]] | ||
|125/98~98/75 | | 125/98~98/75 | ||
|[[7/5]] | | [[7/5]] | ||
|[[1/1]] | | [[1/1]] | ||
|[[15/14]] | | [[15/14]] | ||
|225/196~147/125 | | 225/196~147/125 | ||
|[[63/50]] | | [[63/50]] | ||
|[[27/20]] | | [[27/20]] | ||
|81/56 | | 81/56 | ||
|} | |} | ||
== MOS 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 | 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⟩]]/[[6L 5s (3/2-equivalent)|6L 5s⟨3/2⟩]]) notes. The tetratonic scale is usable, but the tempered 10/9 is not present in it, so the pentatonic and hexatonic scales are the smallest options for halftone. | ||
[[Category:Halftone| ]] <!-- main article --> | |||
[[Category:Rank-2 temperaments]] | |||
[[Category:Non-octave temperaments]] | |||
Latest revision as of 09:23, 29 April 2025
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 EDFs that support halftone include 5edf, 6edf, 11edf, 16edf, 21edf, and 27edf.
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.
For technical data, see Subgroup temperaments#Halftone.
Interval chain
| 53.040 | 182.823 | 312.606 | 442.389 | 572.172 | 0 | 129.783 | 259.566 | 389.349 | 519.132 | 648.915 |
| 28/27 | 10/9 | 25/21 | 125/98~98/75 | 7/5 | 1/1 | 15/14 | 225/196~147/125 | 63/50 | 27/20 | 81/56 |
MOS scales
Halftone possesses MOS scales with 4 (1L 3s⟨3/2⟩ or "neptunian"), 5 (1L 4s⟨3/2⟩), 6 (5L 1s⟨3/2⟩) and 11 (5L 6s⟨3/2⟩/6L 5s⟨3/2⟩) notes. The tetratonic scale is usable, but the tempered 10/9 is not present in it, so the pentatonic and hexatonic scales are the smallest options for halftone.