Shirt a Phobia

Sample Page

Wendy Carlos devised several musical scales. She showcased them on her 1986 album Beauty in the Beast. Several are non-octave repeating scales, which Carlos named alpha, beta, and gamma. Each approximates just intervals using multiples of a single interval. She also used the upper partials of the harmonic series to tune a chromatic scale.

Background

Wendy Carlos‘ 1984 album, Digital Moonscapes, showcased the virtual orchestra she dubbed the “LSI Philharmonic” in reference to large-scale integration computer chips. For her next album, Carlos developed alternate tunings in order to generate new timbres. Beauty in the Beast features music written in these tuning systems.[1]: 156 

A 1980 study devised a method of comparing alternate tunings to consonant intervals.[2][3] Carlos plugged asymmetric ratios into the model and noticed several distinct peaks of consonance which she labeled alpha, beta, and gamma.[4]: 42 

Alpha scale

The alpha scale divides the octave into 15.385 steps, which precludes the traditional 2:1 octave ratio. The resulting scale yields a minor third with 4 steps, a major third with 5 steps, and a perfect fifth with 9 steps.[5][6][7][8][9]

Though it does not have a perfect octave, the alpha scale produces “wonderful triads,” and the beta scale has similar properties but the sevenths are more in tune.[6] Initially, Carlos overlooked inversions of the alpha scale. She discovered that they yield “excellent harmonic seventh chords“, which is part of why the alpha scale was one of Carlos’ favorite tunings.[5]

interval name size
(steps) 		size
(cents) 		just ratio just
(cents) 		error
septimal major second 3 233.89 8:7 231.17 +2.72
minor third 4 311.86 6:5 315.64 −3.78
major third 5 389.82 5:4 386.31 +3.51
perfect fifth 9 701.68 3:2 701.96 −0.27
harmonic seventh octave−3 966.11 7:4 968.83 −2.72
octave 15 1169.47 2:1 1200.00 −30.53
octave 16 1247.44 2:1 1200.00 +47.44

Beta scale

Perfect fourth (just: 498.04 cents, 12-tet: 500 cents, Beta scale: 512 cents)

The β (beta) scale may be approximated by splitting the perfect fifth (3:2) into eleven equal parts [(3:2)1⁄11 ≈ 63.8 cents], or by splitting the perfect fourth (4:3) into two equal parts [(4:3)1⁄2],[6] or eight equal parts [(4:3)1⁄8 = 64 cents],[7] totaling approximately 18.8 steps per octave.

The size of this scale step may also be precisely derived by putting the perfect fifth and the major third in an 11:6 ratio (not to be confused with the interval 11/6).

In order to make the approximation as good as possible we minimize the mean square deviation. … We choose a value of the scale degree so that eleven of them approximate a 3:2 perfect fifth, six of them approximate a 5:4 major third, and five of them approximate a 6:5 minor third.[8]

and

Although neither has an octave, one advantage to the beta scale over the alpha scale is that 15 steps, 957.494 cents, is a reasonable approximation to the seventh harmonic (7:4, 968.826 cents)[8][9] though both have nice triads[6]. “According to Carlos, beta has almost the same properties as the alpha scale, except that the sevenths are slightly more in tune.”[6]

interval name size
(steps) 		size
(cents) 		just ratio just
(cents) 		error
major second 3 191.50 9:8 203.91 −12.41
minor third 5 319.16 6:5 315.64 +3.52
major third 6 383.00 5:4 386.31 −3.32
perfect fifth 11 702.16 3:2 701.96 +0.21
harmonic seventh 15 957.49 7:4 968.83 −11.33
octave 18 1148.99 2:1 1200.00 −51.01
octave 19 1212.83 2:1 1200.00 +12.83

Gamma scale

The γ (gamma) scale may be approximated by splitting the perfect fifth (3:2) into 20 equal parts (3:21⁄20≈35.1 cents),[citation needed] of approximately 35.1 cents each for 34.188 steps per octave.[5]

As 20 is even, this scale contains true neutral thirds, which are not found in alpha or beta.

The size of this scale step may also be precisely derived by putting the perfect fifth and the major third in a 20:11 ratio (not to be confused with the interval 20/11). Thus the step is approximately 35.099 cents and there are 34.1895 per octave.[8]

and

“It produces nearly perfect triads.”[6] “A ‘third flavor’, sort of intermediate to ‘alpha’ and ‘beta’, although a melodic diatonic scale is easily available.”[5]

interval name size
(steps) 		size
(cents) 		just
ratio 		just
(cents) 		error
minor third 9 315.89 6:5 315.64 +0.25
major third 11 386.09 5:4 386.31 −0.22
perfect fifth 20 701.98 3:2 701.96 +0.02

Harmonic scale

Harmonic series above C.[10]

Carlos derived a chromatic scale from the fifth octave of the harmonic series. The scale begins on the 16th partial and runs to the 32nd, omitting numbers 23, 25, 29, and 31. Carlos called this a harmonic scale.[4]: 37f 

Using 100-cent semitones, Carlos transposed the harmonic scale on all 12 chromatic pitches to generate a theoretical division of the octave into 144 steps. In practice, she would retune the scale by triggering different fundamentals on a keyboard controller. Her use of the harmonic scale is showcased in “Just Beginnings” on Beauty in the Beast.[4]: 37f 

Audio examples

Alpha scale
Beta scale
Gamma scale

See also

References

  1. ^ Sewell, Amanda. Wendy Carlos: A Biography. Oxford University Press, 2020.
  2. ^ Yunik, M. and G. W. Swift. “A Microprocessor Based Keyboard Instrument for Microtonal Music“, Proceedings of the International Computer Music Conference. San Francisco: Computer Music Association, 1982. 589.
  3. ^ Yunik, M. and G. W. Swift. “Tempered Music Scales for Sound Synthesis“, Computer Music Journal 4, no. 4, 1980. 60–65.
  4. ^ a b c Carlos, Wendy. “Tuning: At the Crossroads“, Computer Music Journal, vol. 11, no. 1, 1987. 29–43.
  5. ^ a b c d Carlos, Wendy. “Three asymmetric divisions of the octave“, wendycarlos.com. Accessed April 8, 2026.
  6. ^ a b c d e f Milano, Dominic. “A many-colored jungle of exotic tunings“, Keyboard. November, 1986. 64f.
  7. ^ a b Carlos, Wendy. Beauty in the Beast. Liner notes. East Side Digital Records, 2000.
  8. ^ a b c d Benson, David J. Music: a Mathematical Offering. Cambridge University Press, 2007. 221–4.
  9. ^ a b Sethares, William. Tuning, Timbre, Spectrum, Scale New York: Springer., 2005. 60.
  10. ^ Sauveur, Joseph. Principes d’acoustique et de musique, ou système général des intervalles des sons. Paris: Academie Royal des Sciences, 1701. 52.