Carlos harmonic scale
The harmonic scale is a twelve note scale in just intonation 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 be described as a subset of mode 16 of the harmonic series where harmonics 23, 25, 29, and 31 are removed, so that it's CS. It's also an example of including every harmonic as far out as possible while maintaining this property, so that it's an example of a Ringer scale.
Interval table
| 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, Wendy Carlos in her Beauty and the Beast (1986)[1] and Ben Johnston in Suite for Microtonal Piano (1978).
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
Music
See also
References
- ↑ Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.