Volume 10, Number 2, June 2004

6.2 Scale-Degree Results

[1] Based on a statistical model (see §5.2) fit to triads by Haydn, Mozart, and J.S. Bach, the figures below estimate the probability that a triad is composed (see §4.1) rather than random (see §4.2). For each scale degree, the figures show an estimated probability and a 95% confidence interval for that estimate. Scale degrees that composers preferred to double have probabilities above 50%; scale degrees that composers avoided doubling have probabilities below 50%.

[2] In major mode, for example, a poorly-spaced triad that doubles the leading tone has a 15-24% probability of being composed. If the triad is well spaced, the probability of being composed is higher (25-35%), but still less than 50%. Either way, it seems that Bach, Haydn, and Mozart avoided doubling the leading tone.

 Figure 6.2a. Probability that a triad is composed if it doubles a specific triad member. For example, in major mode (top), a triad that doubles the 7th scale degree has a 25-35% chance of being composed rather than random--provided the chord is well-spaced (black). If the triad is poorly spaced (red), this probability drops to 15-24%. In either case, the results fit the idea that these composers avoided doubling the leading tone.

[3] On the whole, the graphs are consistent with the most commonly taught rules for doubling scale degrees: avoid doubling the leading tone and any chromatics. In minor mode, the second scale degree may also be avoided to some extent, perhaps because it can act as a leading tone to the relative major.(82)

[4] These results may be interpreted in at least two ways:

1. One interpretation is that doubling reinforces the key by emphasizing "strong" scale degrees--for example, the degrees with high values in the Krumhansl-Kessler key profiles. (See §6.2.1.)
2. A second interpretation is that doubling aids voice-leading by avoiding unstable "tendency tones." When tendency tones are doubled, it becomes impossible to use conjunct motion while avoiding parallel octaves. The avoidance of tendency tones makes it much easier to avoid parallels gracefully.

[5] Our results do not decide between these interpretations. Either could explain why the leading tone and chromatics are least likely to be doubled. But neither interpretation can explain the differences among the other scale degrees. According to the tendency tone interpretation, doubling should favor all non-tendency tones equally--but Figure 6.2a does not support this, at least not in major mode. According to the Krumhansl-Kessler profiles, doubling should favor the notes of the tonic triad over other diatonic tones (see §6.2.1)--but Figure 6.2a does not support this, either. On balance, it seems that either voice-leading or tonality concerns could explain doubling practice--but neither explains it perfectly.

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