Volume 10, Number 2, June 2004

## 6.3 Comparative Results

[1] In the pair of triads shown in Figure 6.3a, one is sampled from a chorale harmonization (see §4.1) by J.S. Bach (BWV 253, “Ach bleib’ bei uns, Herr Jesu Christ”). The other triad is randomly generated without regard to rules for doubling or spacing. (See §4.2.)

 Figure 6.3a. One triad sampled from BWV 253, and a random twin. In terms of the analysis, it is unknown a priori which is composed.

[2] A priori, we don’t know which triad is which. So each triad has an equal probability of being composed, or of being random. Our statistical models refine these probabilities by using features of the doubled tones, along with chord spacing. (See §5.2.)

1. One version of the model uses the triad member of the doubled tone. (See §6.1.) In the example above, the left triad doubles the root, and the right triad doubles the third. Since the root is a bit more likely to be doubled in a first-inversion major triad, this difference increases the probability that the left triad is composed. In addition, the left triad has its largest space between bass and tenor, while the right does not. Considering both doubling and spacing, this model assigns the left triad a higher probability of being composed (75%).
2. Another version of the model uses the scale degree of the doubled tone. (See §6.2.) In the example above, the left triad doubles the fifth degree, and the right doubles the leading tone. Combining this with the left triad’s better spacing, this model assigns the left triad a much higher probability of being composed (92%).
3. A final version of the model uses both the scale degree and the triad member of the doubled tone. Combining this information with the left triad’s better spacing, this model assigns the left triad a higher probability of being composed. The estimated probability (84%) is about midway between the probabilities assigned by the previous models, which use triad members or scale degrees alone.(84)

[3] In this case, all three models are accurate. The left triad is in fact composed. In other cases, however, one model or the other may be inaccurate, assigning the random triad a higher probability of being composed.

[4] Overall, all three models are about equally accurate (see Figure 6.3b), identifying the composed triad in about 70% of all pairs.

 Figure 6.3b. The accuracy of four models is shown: the spacing rule alone, the triad-member model, the scale-degree model, and both models together. All three doubling models include the chord-spacing predictor. Results are broken down by genre: quartets (squares) and chorales (diamonds).

[4] Since the model based on scale degrees is about as accurate as the model based on triad members, and since there is little improvement from using scale degrees and triad members together, it seems that the scale-degree and triad-member doubling rules are redundant with one another. They are merely different ways of representing the same musical practices.

[5] As remarked in section 1, some redundancies are obvious. In major dominant chords, for instance, doubling the leading tone is indistinguishable from doubling the third. Other redundancies are subtler, but probably inevitable when a large number of features are used to describe a fairly basic musical practice.

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