Harmonic scale It was invented by Wendy Carlos and used on three pieces on her album "Beauty in the Beast" (1986): "Just Imaginings", "That's Just It", and "Yusae-Aisae". Versions of the scale have also been used by Ezra Sims and Frans Richter Herf. Though described by Carlos as containing "144 [= 12] distinct pitches to the octave", the twelve scales include 78 (= ) notes per octave. Technically there should then be duplicates and thus 57 (= 78 − 21) pitches (21 = ). For example, a perfect fifth above G (D) is the major tone above C. Harmonic scale The Harmonic
Scale (music) notes. A more mathematically flexible schema for binary representation, espoused by theorist Ian Ring in his study of musical scales, has bits representing ascending tones, with the smaller bits representing lower pitches and higher bits representing higher pitches. This causes the bits to appear reversed compared with Duncan's system. Under this schema, a major scale is 101010110101 = (2741), a single root tone is 1, and a chromatic scale is still 111111111111 (4095). All the tone possibilities that begin with a root tone, and are thus scale-like, are odd-numbered. Scales may also be shown as semitones from the tonic. For