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 21-odd-limit, or 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, producing a constant structure. It is also an example of including every harmonic as far out as possible while maintaining constant structure, which means it is 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 |
Perspectives
As a NEJI
The harmonic scale can be viewed as an intentionally inaccurate 12-NEJI. From 12edo, the harmonic scale has a total error of 194.193 cents and an average error of 16.183 cents.
As a Ringer Scale
The harmonic scale can be interpreted as a ringer scale detempered from 12edo devised for spectralist purposes; specifically, it is devised such that the root is a [math]\displaystyle{ 2^n }[/math]th harmonic, allowing it to act as the "fundamental frequency" pitch class (in an octave-repeating scale). Typical Ringer 12 scales, however, do not have this particular focus.
The harmonic scale can be derived as such: a ringer scale that specifically starts on a [math]\displaystyle{ 2^n }[/math]th harmonic. To fit twelve pitches while fitting this requirement, the scale must therefore start on the 16th harmonic and end on the 32nd; in other words, it must be a subset of 16::32.
Consider the 12edo patent val up to the 31-limit: ⟨12 19 28 34 42 44 49 51 54 58 59] Based on this patent val, we can deduce that 12edo tempers out the superparticular ratios 23/22, 26/25, 29/28, and 31/30. This means that we can only use one of the harmonics listed in each ratio in the scale; otherwise, "retempering" the scale will lead to two notes with the same pitch. Even numbers can be prioritized, since they reduce to simpler ratios when put over 16.
Thus, we can we can remove the 23, 25, 29, and 31 from the 16::32 scale to arrive at the 16:17:18:19:20:21:22:24:26:27:28:30:32 scale—the harmonic scale.
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
- from Microtones & Garden Gnomes (2017)
See also
References
- ↑ Milano, Dominic (November 1986). "A Many-Colored Jungle of Exotic Tunings", Keyboard.