Tuning Systems

When two tones sound together, they form a harmonic interval. When three or more tones sound together, they form a chord. Assuming that the tones are periodic so that their spectra are exactly harmonic, the subjective sound quality of an interval or chord depends on the ratios between their fundamental frequencies.

Musicians or professional tuners tune musical instruments prior to performances, and in many cases musicians also tune individual tones during performances, in real time. Tones are tuned by adjusting their fundamental frequencies, for example, by varying the tension in a string or the sounding length of a wind instrument. Chords such as major triads (e.g., the tones C, E, and G) and minor triads (e.g., C, Eb, and G) are tuned according to different subjective criteria. Well-tuned intervals and chords may be considered maximally “pleasing” or “stable,” but the exact meaning of these terms is unclear from a psychological perspective. First, it is unclear what musicians are aiming for exactly, and second, it is unclear which physical or other parameters are influencing their choices and behavior. Good tuning is often equated with (equally uncertain) notions of consonance and dissonance.

One principle that influences perceived consonance, and hence tuning, is beating or roughness between partials that almost coincide. From a purely physical viewpoint, two pure tones of almost the same frequency combine to form an amplitude-modulated waveform whose carrier frequency (which corresponds to the pitch that one hears, if it is in the audible range) is the mean of the original two frequencies and whose modulation or beat frequency corresponds to the difference between the two original frequencies. If the beat frequency is slower than about 20 cycles per second, one may hear individual beats; if it is faster than about 20, one may hear a sensation of roughness. In a typical musical chord, many pairs of partials almost coincide in this way, and may therefore produce audible beats or roughness. One approach to tuning is to try to minimize these disturbances. The result, if the original sounds are period (harmonic), corresponds to simple ratios of whole numbers. For example, one might tune the fundamental frequencies of the tones of a major triad in the ratio 4:5:6.

It is not possible to tune a keyboard to just tuning because of inherent conflicts within the diatonic scale. If the interval between scale degrees 1 and 2 in the major scale (a major second) is 8:9 and between 1 and 6 (a major sixth) is 3:5, then the interval between 2 and 6 (a perfect fifth) is not 2:3. Even if one were to create the harmonic foundation for the key of C major by tuning the chords C major, F major, and G major to just ratios (4:5:6), and the intervals between their roots to just fifths (3:2) and fourths (4:3), one could still not speak of a “just scale,” because the D-minor chord would be mistuned. The chord of D minor [Page 1152]is very common in the key of C major; in many styles, from Wolfgang Amadeus Mozart to bebop jazz, it is even more common than the F-major chord in progressions leading to cadences in C major (i.e., ii V I is more common than IV V I).

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