In tuning keyboard instruments, tempering of some or all of the intervals is inevitable. Different tuning systems have different values and advantages depending on how the intervals are tempered. The decision of which tuning system to employ in a musical setting depends (in addition to many other factors) upon the music to be played.
This study is of the perceived differences between equal-temperament and classic just-intonation within the context of tonal music. The sounds which are compared are diatonic chords built within the two different tunings. The tones used to build these chords are synthesized from 20 partials with the relative strength of the nth partial being 1/n . The different types of chords which are used in classical western tonal music are compared, as well as combinations of them within short musical examples.
In equal-temperament each occurrence of a minor triad is equivalent to any other minor triad in terms of its frequency ratios. Mediant and Supertonic triads share the same frequency ratio. In just-intonation, on the other hand, these chords represent distinct harmonies:
Cents Just from E.T. Intonation MAJOR (I, IV, or V): -14, +2 SUPERTONIC (ii): -6, -20 MEDIANT (iii or vi): +16, +2 DIMINISHED (vii): +16, +10Vos's experimentally derived threshold for discrimination of pitch deviation suggests that the members of each of these chord pairs would be distinguishable from each other. Experiments by Hall and Hess (1984); Vos and Von Vianen (1985) and Elliott, Platt and Racine (1987) support this, with the added point that tolerance for mistuning increased when going from more consonant to less consonant intervals.
Within a musical context equal-tempered chords are not heard against just-tuned chords, they are heard against equal-tempered chords. As a musical selection modulates, the basic equal-tempered frequency ratios of major, minor and diminished triads remain constant. That may not be said of just-intonation. Tables 3 and 4 show the frequency ratios and deviation from equal-temperament of diatonic triads in the key of the tonic and dominant, the most common destination of modulation.
1 2 3 4 5 6 7 Tonic Dominant Table 3. 1 2 3 4 5 6 7 Tonic -14,+2 -6,-20 +16,+2 -14,+2 -14, +2 +16,+2 +16,+10 Dominant -14,+2 +16,+2 +16,+2 -14,+2 +14,-20 +16,+2 -6,+10 Table 4.The further a musical selection modulates from the reference pitch of the tuning system, the less consonant its intervals are. Here is the diatonic subset based on F# in a C-reference just-tuned system:
This difference between just-intonation and equal-temperament has long been considered the great advantage of equal-temperament, which can modulate to F# or any other key and retain the same frequency ratios.
Rasch's study (1985) of large sequences of simultaneous tones found that mistuning of the intervals of the melody was more disturbing than mistuning of simultaneous intervals. This suggests that listeners compare melodic intervals to an abstract interval standard. Since Vos only used two-voice polyphonic settings, it is likely that his attempts to isolate preferences of harmony were partially undermined by melodic mistunings.
One of the difficulties with these studies is that they don't take into account the variability of dissonance inherent in just-intonation systems. Consider this within a tonal music setting. If one looks at the ratios of diatonic triads within tonic and dominant scales (within a tuning based on the tonic element) one sees that the greatest dissonances occur at different places:
1 2 3 4 5 6 7 Tonic Dominant Table 3.Instead of testing for listener preference in combined settings of melodic and harmonic material, Rasch and Vos could have begun their work with testing of distinguishability in isolated settings. While a listener's preference certainly implies distinction, it brings many unnecessary factors into consideration. Sound example 3 suggests that there are perceptual differences between different key areas in just intonation. Different cadential patterns than the classical ii-V-I cadence formula also produce different sonorities.