Distinctions Between Just-tuned Key Areas Within Musical Contexts

by Ken Overton

It is a widely accepted point that consonance is best accounted for in terms of beating or roughness. Vos and Von Vianen (1984) present one of many convincing studies showing that perceived dissonance increases monotonically with the integers needed to express an interval's frequency ratio. That is, small-integer ratios of fundamental frequencies are more consonant than large-interval ratios that are of close magnitude. They go on to show that the threshold for discriminating between pure and mistuned intervals is in the range of 10-30 cents. Tempered intervals, then, are fine deviations (usually below 15-20 cents) from the "pure" consonance of small integer ratios.

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.

Tuning Theory

In equal temperament the octave is divided precisely into 12 equal intervals. Thus, each half step in equal temperament is expressed as where f0 is the lower of the two pitches. Just intonation, on the other hand, is generated from a collection of small integer ratios (5-limit). The tuning is shown below.
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, +10
Vos'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.

Studies Concerning Subjective Preference

Vos (1988) conducted an experiment in which various tunings of diatonic scales were rated by listeners in terms of their subjective acceptability. The scales were used to synthesize performances of two-voice settings from Michael Praetorius' Musae Sioniae, Part VI (1609). In this study he found that individual intervals not more than 5 cents removed from the "pure" intervals (ratios of 5:4 for major thirds, 6:5 for minor thirds, 3:2 for perfect fifths) were all found to be close to equal "acceptability"; but deviations were found to dramatically decrease in "acceptability" as they increased above 5 cents.

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.

Conclusions

The evidence given above suggests that the harmonic relationships of different key areas may be audibly distinguishable in just intonation. This is accomplished only at key points, in chords that occur most often in cadences, while the majority of the chords with identical functions are identical in sonority as well. Future testing of this hypothesis may include attempts to train listeners to hear these differences within tonal musical contexts and link them to a change in the key area. This would be particularly powerful in classical era tonality because of the structural importance of modulation. Being able to distinguish between keys would help listeners without perfect pitch to infer structure in the work.

If you have comments/suggestions, e-mail me at: kov@onyx.dartmouth.edu