Wendy Carlos scales

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. A 1980 study devised a method of comparing alternate tunings to consonant intervals. Carlos plugged asymmetric ratios into the model and noticed several distinct peaks of consonance which she labeled alpha, beta, and gamma. In 1987, Carlos wrote about her tuning experiments in Computer Music Journal (CMJ). The issue included a vinyl flexi disc with several recordings that demonstrated the unique timbres she created. The soundsheet also included excerpts from Beauty in the Beast and her 1987 album Secrets of Synthesis. In 2006, CMJ reissued the contents of its vinyl inserts on DVD. ==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. 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. 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. ==Beta scale==

The β (beta) scale may be approximated by splitting the perfect fifth (3:2) into eleven equal parts [(3:2) ≈ 63.8 cents], or by splitting the perfect fourth (4:3) into two equal parts [(4:3)], or eight equal parts [(4:3) = 64 cents], 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). \frac{11\log_2{(3/2)}+6\log_2{(5/4)}+5\log_2{(6/5)}}{11^2+6^2+5^2}=0.05319411048 and 0.05319411048\times1200=63.832932576 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) though both have nice triads. "According to Carlos, beta has almost the same properties as the alpha scale, except that the sevenths are slightly more in tune." ==Gamma scale==

The γ (gamma) scale may be approximated by splitting the perfect fifth (3:2) into 20 equal parts (3:2≈35.1 cents), of approximately 35.1 cents each for 34.188 steps per octave. 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. \frac{20\log_2{(3/2)}+11\log_2{(5/4)}+9\log_2{(6/5)}}{20^2+11^2+9^2}=0.0292487852 and 0.0292487852\times1200=35.0985422804 "It produces nearly perfect triads." "A 'third flavor', sort of intermediate to 'alpha' and 'beta', although a melodic diatonic scale is easily available." ==Harmonic scale==

above C. 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. 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. ==Audio examples==

;Alpha scale • • • • • ;Beta scale • • • • • • ;Gamma scale • • ==See also==