Imagining Musical Scales
Musical cultures adopt particular arrangements of tones as the alphabet in which musical works are written. Upon a second look, musical scales are not as obvious as they may seem. For example, naive listeners will hear the familiar do-re-mi “major” scale as comprising equally-spaced steps, like the rungs of a ladder (scala = staircase, ladder). The steps are, in fact, not all the same size. Until very recently, the origin of the major scale and other musical scales has been explained upon the basis of the overtone series. In particular, the overtone series offers a reasonable basis for the choice of our fundamental musical intervals, the octave and the fifth. Although overtone series can account for the material of the scale, it does not explain the arrangement of that material. A delicate balance of symmetry and asymmetry lies at the heart of the major scale and others. When generating scales with octaves and fifths, the balance is found in scales with 1, 2, 3, 5, 7, or 12 notes. The 5-note scale corresponds to the black keys on the piano, the 7-note scale with the white keys, and the 12 note scale with black and white keys consecutively. The series (1, 2, 3, 5, 7, 12…) can be infinitely extended. Puzzler for the talk: how many notes would be in the next scale, the one that follows the 12-note chromatic scale? (Hint: No, this isn’t the Fibonacci series, but you’re close.) Recently, these developments in musical scale theory have been found to have a strong connection with the mathematics of “word theory,” where a word is a string, either finite or infinite, of symbols based on a (normally) finite alphabet.
Optional Reading: Aspects of well-formed scales. Norman Carey and David Clampitt. Music Theory Spectrum, Vol. 11, No. 2 (Autumn, 1989), pp. 187-206
