String-based methods for tonal harmony: A corpus study of Haydn’s string quartets
Table 3: Contingency table for the bigram <7,2,5,11> → <0,4,⊥,⊥> (V7–I). Counts reflect tokens with up to five skips between chord events.
['[EMPTY]', 'I', 'Not I', 'Totals']
To resolve this issue, corpus linguists have developed a large family of association (or attraction) measures that quantify the statistical association between co-occurring words \autociteEvert2008. The majority of these measures use contingency tables. <7,2,5,11> → <0,4,⊥,⊥>. The counts reflect tokens with up to five skips between bigram members. The table has four cells for tokens containing both chord1 and chord2 ( a), tokens containing chord1 but not chord2 (i.e., any other chord) (b), tokens containing chord2 but not chord1 (c), and tokens containing neither chord (d). The marginal frequencies, so called because they appear at the margins of the table, represent the sum of each row and column. Thus, the co-occurence frequency for V7–I is 581. Asymmetric (or directional) association measures typically compute the conditional probabilities between the members of each bigram \autociteMichelbacher2007. In this case, P(I|V7) is aa+b, with a representing all of the instances in which I follows V7, and b representing all of the instances in which some other harmony follows V7. Thus, 5813873=.15, which tells us that I follows V7 roughly 15% of the time. This estimate does not represent the probability that V7 precedes I, however. Here, P(V7|I) is aa+c, or 5816216=.09. Thus, for this particular variant of the V7–I progression, V7 is a better predictor of I than the other way around. Or put another way, I exerts the greater attractional force. I have formalized this statistical inference in the following way: \textscasym=P(chord2|chord1)−P(chord1|% chord2)=aa+b−aa+c (3) In this equation, asym is simply the arithmetic difference between the two conditional probabilities. The estimates of asym fall in a range between −1 and 1, where positive values indicate that chord2 is the attractor, negative values indicate that chord1 is the attractor, and 0 indicates bidirectionality, since both harmonies exert equivalent attractional force.