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?
KEYWORDS: pedagogy, doubling, spacing
ABSTRACT: Music theory instructors teach a number of rules for chord tone doubling. These rules fall under two general headings: rules for doubling certain triad members (e.g., double the root of a major root position triad) and rules for doubling certain scale degrees (e.g., avoid doubling the leading tone). The present study tests these doubling rules against large samples of triads by Bach, Mozart, and Haydn. Unlike previous analyses, the study controls for the confounding effects of inversion, range, and spacing. The results suggest that the rules governing triad members are almost entirely redundant with the rules governing scale degrees; that is, the two sets of rules are different ways of describing the same musical practices. It is clearly unnecessary to teach both sets of rules. We prefer the scale-degree rules since they are more concise, and easier to justify on theoretical grounds.
Author's Note: This article is meant to be "self-skimming." The basic thread is given here on the home page. Technical details, theoretical background, and supporting evidence are available through links.
Received January 2004
 Music theorists may teach more rules than anyone else in the academy. Whereas a semester of physics reinforces a consistent system of perhaps a dozen rules,(1) a semester of harmony and counterpoint uses several dozen rules, most of which admit exceptions and contradictions. It should be no surprise that students' grasp of the material is often weaker than we might hope. Videotaped homework sessions reveal students struggling to remember and apply the rules, taking perhaps half an hour to harmonize a 3- or 4-bar diatonic melody.(2)
 In light of students' difficulties assimilating new rules, it seems wise to keep the number of rules taught to a minimum. Toward that end, this study tests and winnows a set of rules taught in introductory harmony.
 The rules we have chosen to analyze are rules for chord tone doubling. As commonly taught, doubling rules prescribe which pitch-class (PC) to double when writing a 3-PC chord (triad) in a 4-voice texture. Doubling rules can be extended to 3- and 4-PC chords in 5- and 6-part textures. Such extensions are rarely taught, however, and it seems likely that the underlying principles would be the same.
 There are two reasons why doubling rules seem ripe for simplification:
 Although theorists disagree on particulars (see §2.1), there are two basic ways of stating doubling rules. Some theorists state rules in terms of triad members; others state rules in terms of scale degrees.
 In addition to scale-degree and triad-member rules, doubling is subject to the influence of chord spacing:
This turns out to be important because the chord spacing rule makes certain doublings more convenient than others. (See §2.2.)
 At least two previous studies have tested doubling rules. McHose(70) tested certain triad-member rules, and Huron(20) tested one version of the scale-degree rules. (See §3.1.) Our study builds on these precedents in three ways.
 To examine doubling practice rigorously, we assembled two large samples of four-part triads. (See §4.1.) In particular, we sampled:
 Each triad sampled from these composed repertoires was paired with a triad that we generated in a random fashion. (See §4.2.) The random triad was in the the same inversion as the composed triad, and followed similar conventions for part-crossing and pitch range.
 The difference between composed and random triads was that the random triads were innocent of rules for doubling or spacing. Rules governing these practices, therefore, can be tested for their ability to discriminate composed from random triads.
 Our task, then, was to use doubling and spacing features to decide which triad in each pair is composed, and which is random. An illustration of the task is given in the sidebar (see "Can you beat the computer?").
 Toward this end, we encoded each triad's doubling and spacing features. (See §5.1.) We then employed a statistical model to estimate which features were more common in the composed triads, and which were more common in the random triads. (See §5.2.)
 In general, the triad member results and scale degrees results were consistent with the most commonly taught rules for doubling. Compared to the random triads, the composed triads were more likely to double the root of a major root-position triad. (See §6.1.) They were also less likely to double the leading tone or a chromatic pitch. (See §6.2.) Not surprisingly, composed triads were also more likely to put the largest space between the two lowest parts. This suggests that the traditional rules are fairly accurate at describing which doubling and spacing practices were favored and avoided, at least in these touchstone repertoires.
 This is not to say that the results merely tell us what theorists already knew. It is not entirely clear what theorists knew, since theorists disagreed on a number of points. Moreover, the results show not only which doubling practices are preferred, but how strong those preferences are.
 Finally, the results confirm that many rules are redundant. Whether the model discriminates composed from random triads according to
the results are about the same, a respectable 70% accuracy. (See §6.3.) Haphazard guessing would be correct 50% of the time, so 70% accuracy may not seem very high. On the other hand, much higher accuracy may be unattainable, since the random triads are often quite plausible. To check your own level of accuracy, try the test in the sidebar (click "Can you beat the computer?").
 The parity between triad-member and scale-degree rules suggests that these groups of rules are redundant. That is, scale-degree and triad-member rules are different ways of describing the same musical practices.
 Since the empirical results suggest that the different groups of rules are
redundant with one another, the decision between triad-member rules and
scale-degree rules must be made on theoretical and pedagogical grounds.
 On either ground, it is difficult to justify teaching the triad-member rules. (See §6.1.) They consist of a large number of mild preferences which are tenuously related to any undergirding principle.
 By contrast, the scale degree rules boil down to a single, strong, and sweeping proscription: avoid doubling unstable "tendency tones". (See §6.2.) This rule has been justified in two different ways.
Our results do not decide between these perspectives. However, we think the latter perspective works better in the classroom.
 If we were to give a summary of doubling rules appropriate for students, we
would say this: Worry about the voice-leading, not the voice-doubling. Never
mind the triad-member rules, and simply avoid doubling tendency tones. Teachers
can justify this by pointing out that doubled tendency tones lead to parallel
octaves--a practice that students are already trying to avoid.
 Rather than burdening students with more to memorize, this well-motivated rule should reinforce what students already know. Moreover, our results suggest that following this rule and observing good spacing will bring students close to the doubling practices of Bach, Haydn, and Mozart.
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