Halftone

Halftone[6] example in 16edf and 4|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 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.