Carlos harmonic scale

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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 21^st harmonic, meaning they go up to the 19-limit. The harmonic scale can be described as an arbitrary subset of the mode 16 of the harmonic series where harmonics 23, 25, 29, and 31 are removed. It is also a constant structure.

Interval table

Intervals of the harmonic scale

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) 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

Xotla

• from Microtones & Garden Gnomes (2017)
  • "Flying Dutchman's Folly" Bandcamp | YouTube
  • "Bonus - Flying Dutchman's Folly Pixelized" Bandcamp

See also

• Carlos Alpha
• Carlos Beta
• Carlos Gamma
• Carlos Super Just