An Encoding Algorithm and Tables for the Digital Analysis of Harmony (II)

Abstract

1HE PROBLEM OF organizing a large universe of objects is common to several research areas. Especially in musical theory, it is important to be able to identify classes (or sets) in such a manner that objects within a class are similar to each other, but are dissimilar from objects outside the class. Musicians have found that much can be said about a rather bewildering variety of musical objects called chords by lumping them together into sets called harmonies. Even harmonies can be reduced to a more manageable number by grouping those of like character into classes called sonorities.1 In attempting to analyze the musical works of others, the student of traditional harmony is confronted repeatedly with different tonal assortments. Although he may have little trouble classifying a given tone combination as a melody, arpeggio, scale, or chord, he might require additional instruction to identify every underlying harmony correctly and to state its proper name, e.g., major seventh, if indeed such a name exists. The average student's diiculty in mastering the principles of elementary harmony is increased by vagaries and ambiguities stemming from the use of literal notation and the definition of cllord roots. A more pronounced difficulty awaits the musical analyst when he undertakes the study of musical products of the twentieth century. Polytonality has given rise to awkward signature conventions in modern scoring, and by condoning an indiscriniinate mingling of enharmonic values, atonality has led to an unnecessary proliferation of variant chord spellings. Unfortunately, certain natural difficulties also arise when it becomes necessary to consider complicated harmonies. First, it is not easy to list correctly all of the harmonies of the type wanted. Secondly, it is not easy to shift some harmonies to new pitch levels accurately. Thirdly, it is not easy to recognize some harmonies after their transposition to new keys. It is customary to say that two harmonies are identical if and only if they contain the same tones, regardless of spelling. Two harmonies may