Volume 10, Number 2, June 2004
Copyright © 2004 Society for Music Theory
Bret Aarden and Paul T. von Hippel
Rules for Chord Doubling (and Spacing): Which Ones Do We Need?
 By definition, each voice in a complete four-voice triad has a 50% chance of being doubled. This is the case because doubling involves two voices, so any voice has two chances in four to be doubled. (By the same logic, if a PC is doubled in a two-voice texture, each voice has a 100% chance of being doubled.)
 As a result, specifying what inversion a triad is in constrains which doublings will occur. At first glance it might appear that each of the 3 PCs would have one chance in three of being doubled. Specifying an inversion, however, fixes one PC in the bottom voice. If the bottom voice has a 50% chance of being doubled, then so does its PC.
 Consider the specific example of a complete 4-voice root-position A-major triad. This triad has twelve possible configurations of PCs and voices, as shown in Figure 4.2.1a. Of those, there are six configurations that double the PC in the bass, but only three that double each of the other two PCs.
Figure 4.2.1a. The 12 PC/Voice configurations of complete four-note
root-position A-major triads. Dots indicate the 6 triads that double the bass.
 To complicate matters, not all triad inversions are equally likely. Root
position triads are the most common, and second-inversion triads are relatively
rare. If triads are more likely to occur in root position, it follows that roots
are correspondingly more likely to be doubled purely by chance.
 In order to compensate for this bias we paired the inversion of each random triad to that of its composed twin. By selecting a random triad only from the population of triads that have the same inversion as the composed triad, the random population has the same inversion bias as the composed triads.
Brent Yorgason, Managing Editor
Updated 03 June 2004