4.2 “Random” Triads

[1] Behind every doubling rule is an implied hypothesis that composers purposefully choose to double particular notes and not to double others. If this hypothesis were false, then notes would be doubled apparently at random. But even in randomly doubled triads, some notes would be doubled more than others, since doubling can be affected by spacing (see §2.2), inversion (see §4.2.1), and range constraints (see §4.2.2). All these confounding factors must be controlled if doubling rules are to be properly tested.

[2] Our general strategy was to take a large sample of triads composed by Bach, Mozart, and Haydn (see §4.1), and pair each of these “composed” triads with a “random” triad generated by a computer program that was innocent of rules for triad tone doubling. If a doubling rule is valid, it should be able to tell, most of the time, which of two paired triads is composed and which is random.

[3] In order to control confounding factors, the random and composed triads were subject to similar constraints on inversion and range. (Spacing will be discussed separately.)

• To control the influence of inversion, we required that the random triad be in the same inversion as the composed triad. We also required that the random and composed triads have the same pitch-classes—and perforce the same quality and function.
• To control the influence of range, we required the random triads to use notes that were in the characteristic range of each part, and to avoid part-crossing. (Composed triads with part-crossing were also discarded.)

[4] For example, if we sampled a root-position C Major triad from a chorale or quartet, we would pair that triad with a randomly generated root-position C Major triad that was subject to similar constraints on range and part-crossing. Some illustrative triads are depicted in Figure 4.2a.

[5] It seemed problematic to build a spacing rule into the generation of random triads. (See §2.2.) Since there were many composed triads that did not have their widest space between their lowest voices, it appeared inappropriate to enforce this rule on the random triads. A better way to control the effect of spacing is to use a spacing rule, alongside the doubling rules, to distinguish composed from random triads.

[6] In generating random triads, we sometimes found that the random triad was identical to the composed triad with which it was paired. More subtly, we sometimes found that the composed and random triad could not be distinguished by any feature that was relevant to theories of doubling or spacing. (See §5.1.) For example, the random and composed triads might be identical except that the inner voices were exchanged. Since indistinguishable triads are useless for testing the theory, we continued to sample random triads until we arrived at one that was distinct from the composed triad with which it was paired.

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