Historically, a number of musical theorists have commented regarding the best spacing of tones within chords. It is widely thought that it is better to have larger intervals between the lower voices in a chord. Some theorists suggested that the optimum spacing approximates the spacing of the harmonic series. That is, an octave would occur between the lowest voices, a perfect fifth would separate the next two voices, and so on.
Theorist Walter Piston noted that this idea does not correspond with actual musical practice. Wide spacing seems to be register-dependent. That is, when the bass voice is relatively high, it is less desireable to have a large interval between the bass and the tenor.
Since 1960, a better understanding of chord-tone spacing has arisen from work in psychoacoustics. In order to understand this view, one must consider what is going on in the human hearing organ. In particular, one must look at the behavior of the basilar membrane in the cochlea.
The following figure shows the relationship between three different scales: (1) frequency, (2) log frequency (pitch), and (3) cochlear coordinates (millimeters from the apex of the cochlea). The lines join equivalent points along each scale.
Plomp & Levelt (1965) supposed that in the arrangements of chords in music, composers would tend to space chords so that the spectral components would be relatively evenly spaced along the basilar membrane. Consider a four-note chord consisting of the pitches C3, G3, C4 and E4. Of course musicians generate complex tones rather than pure tones, so the four notes of this chord, generate a much larger number of harmonics. Consider the first five harmonics for each note. This will produce 20 components in total. Notice that several partials are duplicated leaving 14 unique frequencies.
In the illustration below, these 14 partials are plotted on all three scales. On a linear frequency scale, all 14 partials clump together in a tiny region. For both the pitch scale and cochlear coordinate scale the partials are more spread out.
In Huron & Sellmer (1992) it was shown that, when common timbres are taken into account, the majority of chords are consistent with relatively even spacing across the basilar membrane. This practice reduces masking of partials within the chord.
Consider the following spacing for C major chords. In each chord, the pitch-class C has been doubled. Five different voicings have been created with three different transpositions. The upper-most system shows the five chords in the lowest octave with the bass held constant at C2. The lowest system shows the five chords in the highest octave with the bass held constant at middle C (C4).
Under each chord is a numerical value. The value represents the average distance separating successive partials along the basilar membrane when each note in the chord is played with six equally-weighted harmonics. For example, the first chord has an average inter-partial distance of 0.57 millimeters.
Three chords are identified as having very similar inter-partial distances. The three chords have average distances of 0.73, 0.74 and 0.75 millimeters. From a tonotopic perspective, these chords are the most similar. Try playing these chords on a piano. The three chords have a similar "density" or "thickness" to them. This perception occurs even though the third chord is arranged in closed position (spanning one octave), while the lowest chord spans two octaves.
Note that the above numerical values were calculated using six equally-weighted harmonics. It turns out that the same qualitative results are found between 1 and 13 harmonics. That is, the three chords identified above remain the most tonotopically similar, even when the number of harmonics considered varies widely.
Aarden, B. (2001). An empirical study of chord-tone doubling in common era music. Masters Thesis, School of Music, Ohio State university.
Greenwood, D. (1961). Critical bandwidth and the frequency coordinates of the basilar membrane. Journal of the Acoustical Society of America, Vol. 33, No. 4, pp. 1344-1356.
Huron, D. (1993). Chordal-tone doubling and the enhancement of key perception. Psychomusicology, Vol. 12, No. 1, pp. 73-83.
Huron, D., & Sellmer, P. (1992). Critical bands and the spelling of vertical sonorities. Music Perception, Vol. 10, No. 2, pp. 129-149.
Plomp, R., & Levelt, W.J.M. (1965). Tonal consonance and critical bandwidth. Journal of the Acoustical Society of America, Vol. 37, pp. 548-560.
This document is available at http://dactyl.som.ohio-state.edu/Music829E/Huron.Sellmer.html