Chromatic scale common conception of the chromatic scale before the 13th century was the Pythagorean chromatic scale (). Due to a different tuning technique, the twelve semitones in this scale have two slightly different sizes. Thus, the scale is not perfectly symmetric. Many other tuning systems, developed in the ensuing centuries, share a similar asymmetry. In Pythagorean tuning (3-limit just intonation) the chromatic scale is tuned as follows, with sharps higher than their enharmonic flats: These are 17-EDO Pythagorean tuning approximations. In 5-limit just intonation the chromatic scale, Ptolemy's intense chromatic scale, is as follows, with flats higher than their enharmonic sharps,
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