Carlos harmonic scale
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The harmonic scale is a scale of twelve just intonation pitches that repeats at the octave. Its pitches are derived from the harmonic series of a single frequency up to the 21st harmonic, meaning they go up to the 19-limit. The harmonic scale can also be described as an arbitrary subset of 16-ODO where the 23rd, 25th, 29th, and 31st harmonics are removed.
| Harmonic | Ratio | Decimal | Cents | Deviation from 12-TET |
|---|---|---|---|---|
| 16 | 1/1 | 1.0000 | 0.000 | 0\12 ± 0.000 |
| 17 | 17/16 | 1.0625 | 104.955 | 1\12 + 4.955 |
| 18 | 9/8 | 1.1250 | 203.910 | 2\12 + 3.910 |
| 19 | 19/16 | 1.1875 | 297.513 | 3\12 - 2.487 |
| 20 | 5/4 | 1.2500 | 386.314 | 4\12 - 13.686 |
| 21 | 21/16 | 1.3125 | 470.781 | 5\12 - 29.219 |
| 22 | 11/8 | 1.3750 | 551.318 | 6\12 - 48.682 |
| 24 | 3/2 | 1.5000 | 701.955 | 7\12 + 1.955 |
| 26 | 13/8 | 1.6250 | 840.528 | 8\12 + 40.528 |
| 27 | 27/16 | 1.6875 | 905.865 | 9\12 + 5.865 |
| 28 | 7/4 | 1.7500 | 968.826 | 10\12 - 31.174 |
| 30 | 15/8 | 1.8750 | 1088.269 | 11\12 - 11.731 |
| 32 | 2/1 | 2.0000 | 1200.000 | 12\12 ± 0.000 |
As a NEJI
The harmonic scale can be viewed as an intentionally inaccurate 12-NEJI. From 12-TET, the harmonic scale has a total error of 194.193 cents and an average error of 16.183 cents.
Usage and History
The harmonic scale is typically used as an alternative tuning for regular twelve-tone pianos to play spectral or otonal music. Versions of the scale are known to have been used by composers Ezra Sims, Franz Richter Herf, and especially Wendy Carlos in her Beauty and the Beast (1986) and Ben Johnston in his works for retuned piano.
Scala file
! carlos_harm.scl ! Carlos Harmonic & Ben Johnston's scale of 'Blues' from Suite f.micr.piano (1977) & David Beardsley's scale of 'Science Friction' 12 ! 17/16 9/8 19/16 5/4 21/16 11/8 3/2 13/8 27/16 7/4 15/8 2/1