David R. W. Sears Texas Tech University david.sears@ttu.edu Similarity, Prototypicality, and the Classical Cadence Typology: Classification based on Family Resemblance SMT 2017 Arlington, irginia November 3, 2017 Idea ˆ6 ˆ7 Resolution Arrival ii 6 4 5 3 Progression Category Example 1: Haydn, String Quartet in F, Op. 76/2, i, mm. 15 20. Types Categories Essential Characteristics Perfect Authentic (PAC) I ˆ1 I Imperfect Authentic (IAC) Deceptive (DC) Evaded (E) Half () I ˆ3 or ˆ5?, Typically vi ˆ1 or ˆ3??, Typically ˆ5? ˆ5, ˆ7, or ˆ2 Example 2: The cadential types and categories in Caplin (1998), along with the harmonic and melodic characteristics for each category. 1
DC 19 E 11 84 122 PAC 9 IAC Example 3: Pie chart of the cadences in the Haydn corpus. δ n (a, b) = δ(a, b) = f (A B) f (A B) + f (A \ B) + f (B \ A) f = I DF (τ, C ) = log C s : τ s I DFC (τ An B n ) I DFC (τ An B n ) + I DF C (τ An \B n ) + I DF C (τ Bn \A n ) (1) (2) (3) Example 4: (1) Tversky s ratio model. A and B denote the sets of features for items a and b. The function f measures the salience of the features that are either shared by a and b ( A B), distinct to a ( A \ B), or distinct to b (B \ A), where salience refers to the intensity, frequency, familiarity, good form, or informational content of a given feature within the larger set. (2) The Inverted Document Frequency (IDF) of n-gram τ from A or B in sequence s. C denotes the total number of sequences in the collection. (3) The full model estimated for each value of n. Each similarity estimate, δ n (a, b), is then combined into one composite similarity estimate, δ(a, b), which represents the arithmetic average of the similarity estimates calculated for each value of n, weighted by the entropies of the corresponding n-gram distributions from the larger collection. 2
a b ˆ3 ˆ4 ˆ5 ˆ5 ˆ1 ˆ3 ˆ4 ˆ2 ˆ5 ˆ5 ˆ1 n = 2 n = 3 n = 4 n = 2 n = 3 n = 4 ˆ5-ˆ1 ˆ5-ˆ5-ˆ1 ˆ4-ˆ5-ˆ5-ˆ1 ˆ5-ˆ1 ˆ5-ˆ5-ˆ1 ˆ2-ˆ5-ˆ5-ˆ1 ˆ4-ˆ1 ˆ4-ˆ5-ˆ1 ˆ3-ˆ4-ˆ5-ˆ1 ˆ2-ˆ1 ˆ2-ˆ5-ˆ1 ˆ4-ˆ5-ˆ5-ˆ1 ˆ3-ˆ1 ˆ3-ˆ5-ˆ1 ˆ3-ˆ5-ˆ5-ˆ1 ˆ4-ˆ1 ˆ4-ˆ5-ˆ1 ˆ4-ˆ2-ˆ5-ˆ1 ˆ3-ˆ4-ˆ1 ˆ3-ˆ1 ˆ4-ˆ2-ˆ1 ˆ3-ˆ5-ˆ5-ˆ1 ˆ3-ˆ5-ˆ1 ˆ3-ˆ2-ˆ5-ˆ1 ˆ3-ˆ2-ˆ1 ˆ3-ˆ4-ˆ5-ˆ1 ˆ3-ˆ4-ˆ1 ˆ3-ˆ4-ˆ2-ˆ1 Example 5: Top: a) Haydn, String Quartet in B-flat, Op. 50/1, iv, mm. 68 70. b) String Quartet in B-flat, Op. 64/3, i, mm. 3 5. Bottom: The distinct n-gram types in each cadential bass line. ˆ5-ˆ1.02.12.01 ˆ4-ˆ5-ˆ1.1.14.08 ˆ4-ˆ2-ˆ5-ˆ5-ˆ1.08 ˆ4-ˆ2-ˆ5-ˆ1 ˆ4-ˆ3-ˆ2-ˆ5-ˆ1.08.07.05.04.12 ˆ3-ˆ4-ˆ5-ˆ1.13 ˆ3-ˆ4-ˆ5-ˆ5-ˆ1 ˆ3-ˆ3-ˆ4-ˆ5-ˆ1 Example 6: Square dendrogram calculated with the neighbor-joining (NJ) algorithm for the sequence of chromatic scale degrees from eight authentic cadential bass lines in the Haydn corpus. 3
a) NJ Algorithm b) Cadence Typology 4 PAC IAC DC E Example 7: Equal-angle tree calculated with the NJ algorithm for the cadences from the Haydn corpus. a) Partitioned into the five clusters that minimized the maximum dissimilarity between cadences in each cluster. b) Partitioned into the five cadence categories from Caplin s typology.
Reinterpreted Converging Expanding Example 8: Equal-angle tree calculated with the NJ algorithm for the half cadences from the Haydn corpus. Expanding Exemplar ˆ1 ˆ4 ˆ5 4 3 a) b) ˆ1 ˆ4 ˆ5 ˆ1 ˆ4 ˆ5 vi I 6 I 7 I Ger +6 4 3 7 8 4 3 Example 9: Top: Expanding Exemplar from the lower-right branch of the half cadence subtree, String Quartet in F, Op 17/2, i, mm. 19 20. Bottom: ariants of the Expanding Do-Fi-Sol. a) String Quartet in G minor, Op. 20/3, iii, mm. 26 27; b) String Quartet in D minor, Op. 76/2, i, mm. 3 4. 5
Converging Exemplar ˆ1 ˆ7 standing on the dominant... 6 5 a) b) ˆ4 ˆ3 ˆ2 ˆ4 ˆ3 ˆ2 ˆ1 ˆ7 vi ii 6 6 5 I ii 6 6 5 4 3 Example 10: Top: Converging Exemplar from the second branch of the half cadence sub-tree, String Quartet in B-flat, Op. 64, No. 3, iv, mm. 52 53. Bottom: ariants of the Converging Half Cadence. a) String Quartet in E, Op. 54, No. 3, i, mm. 4 5; b) String Quartet in B-flat, Op. 55, No. 3, i, mm. 5 8. 6 5 Reinterpreted Exemplar b. i. % ˆ4 ˆ5 standing on the dominant I 7 vi ii 7 Example 11: Reinterpreted Exemplar from the third branch of the half cadence sub-tree, String Quartet in G, Op. 54, No. 1, i, mm. 17 23. 6
Selected References Caplin, William E. Classical Form: A Theory of Formal Functions for the Instrumental Music of Haydn, Mozart, and Beethoven. New York: Oxford University Press, 1998.. The Classical Cadence: Conceptions and Misconceptions. Journal of the American Musicological Society 57, no. 1 (2004): 51 118. Conklin, Darrell, and Ian H. Witten. Multiple iewpoint Systems for Music Prediction. Journal of New Music Research 24, no. 1 (1995): 51 73. Gjerdingen, Robert O. A Classic Turn of Phrase: Music and the Psychology of Convention. Philadelphia, PA: University of Pennsylvania Press, 1988.. Music in the Galant Style: Being an Essay on arious Schemata Characteristic of Eighteenth- Century Music. New York: Oxford University Press, 2007. Martin, Nathan John, and Julie Pedneault-Deslauriers. The Mozartean Half Cadence. In What is a cadence? Theoretical and Analytical Perspectives on Cadences in the Classical Repertoire, edited by Markus Neuwirth and Pieter Bergé, 185 214. Leuven: Leuven University Press, 2015. Posner, Michael I. Empirical Studies of Prototypes. In Noun Classes and Categorization, edited by Colette Craig, 53 61. Amsterdam: John Benjamin, 1986. Quinn, Ian. Are Pitch-Class Profiles Really Key for Key. Zeitschrift der Gesellschaft der Musiktheorie 7 (2010): 151 163. Rosch, Eleanor. Principles of Categorization. In Cognition and Categorization, edited by E. Rosch and B. B. Lloyd, 28 48. Hillsdale, NJ: Erlbaum, 1978. Rosch, Eleanor H. Natural Categories. Cognitive Psychology 4 (1973): 328 350. Rosch, Eleanor, and Carolyn B. Mervis. Family Resemblances: Studies in the Internal Structure of Categories. Cognitive Psychology 7 (1975): 573 605. Saitou, Naruya, and Masatoshi Nei. The Neighbor-Joining Method: A New Method for Reconstructing Phylogenetic Trees. Molecular Biology and Evolution 4, no. 4 (1987): 406 425. Tversky, Amos. Features of Similarity. Psychological Review 84, no. 4 (1977): 327 352. Tversky, Amos, and J. Wesley Hutchinson. Nearest Neighbor Analysis of Psychological Spaces. Psychological Review 93, no. 1 (1986): 3 22. 7