| Just Intonation |
|
INTRODUCTION Each musical tone consists of a fundamental resonance and a series of overtones. The qualities and relative strengths of the fundamental and of these overtones affect our perception of the pitch itself to some extent and especially our perception of the timbre of the pitch. The overtones arrange themselves in a series of pure intervals, many of which constitute the building blocks of Western scale systems. The modern tuning system of “equal temperament” (= the tuning system of the modern piano) modifies all of these intervals to produce a scale of twelve equal half steps. In piano tuning, all intervals derive as multiples of the stable, unchanging half step. In the pure tuning of the overtone series, on the other hand, the intervals do not derive from a common half step; indeed, as will be shown below, the intervals of pure tuning are incommensurate with one another. JUST INTONATION The system of tuning (to the extent possible) by pure intervals is known as “just intonation.” Theoreticians sometimes dismiss this system as a hypothesis with no application, because just intonation cannot be used on fixed-pitch instruments such as normal (12-key-to-the-octave) keyboards, and it can be used in only a limited way on fretted strings and other instruments that have one or more fixed pitches. However, any consort of voices or instruments with the ability to adjust pitches will often want to strive toward just intonation, especially at cadences, because the pure intervals are more consonant (and resonant) than impure ones. (The increased consonance/resonance derives from the coincidence of many of the overtones of different pitches tuned in pure intervals.) This effect of pure intervals in a consort of voices or instruments is most telling when the notes sound for a prolonged time against one another. DEMONSTRATION OF THE RATIOS OF PURE INTERVALS For ease of demonstration, let the note C two octaves below middle C be assigned the numerical value of 1; this fundamental pitch is produced by a string or pipe vibrating at its whole length. The series of overtones of this note produced by the string or pipe vibrating at half its length, a third of its length, a quarter of its length, a fifth of its length, etc., will correspond to the “pure intervals,” where each overtone is calculated with respect to the previous one:
The pure intervals translated into decimals and cents (where 100 cents = one equal-tempered half step):
DEMONSTRATION OF THE INCOMMENSURABILITY OF PURE FIFTHS AND OCTAVES Two octaves = 1 x 2:1 x 2:1 = 4.0000 Seven octaves = 1 x 2:1 x 2:1 x 2:1...= 128.0000 [On the piano: C - C - C - C - C - C - C - C] One fifth = 1 x 3:2 = 1.5000 Two fifths = 1 x 3:2 x 3:2 = 2.25 Twelve fifths = 1 x 3:2 x 3:2 x 3:2...= 129.7463 [C - G - D - A - E - B - F# - C# - G# - D# -A# - E# - B#] * * * * * [On the piano: C - C] One major third = 1 x 5:4 = 1.2500 Two major thirds = 1 x 5:4 x 5:4 = 1.5625 Three major thirds = 1 x 5:4 x 5:4 x 5:4 = 1.9531 [C - E - G# - B#] [On the piano: C - C] Two minor thirds = 1 x 6:5 x 6:5 = 1.44 Three minor thirds = 1 x 6:5 x 6:5...= 1.728 Four minor thirds = 1 x 6:5 x 6:5...= 2.0736 [C - Eb - Gb - Bbb - Dbb] **As in all tuning systems, the increments by which pure and tempered intervals differ from one another are quite small.** |