The Invocation of Balzano

The words Uniqueness, Coherence, and Simplicity come from a paper by Gerald Balzano published in MUSIC, MIND AND BRAIN, a collection of articles on the neuropsychology of music. It was edited by Manfred Clynes and published by Plenum Press (New York) in 1982.

Balzano proposed the study of pitch sets as an alternative to the study of timbre or harmonicity for psychoacousticians. The bulk of his analysis was devoted to the 12-note collection: " ... depending on the tuning system (e.g. just, pythagorean, equal temperament), differences among these integers translate either approximately or exactly into log frequency differences." (p. 322) His analysis and discussion was devoted to various subsets of this 12-note collection, with extended discussion of the diatonic subsets.

Below are the general definitions of each word. If you would like to read a more complete discussion of a word, click on that word.

• Uniqueness: Uniqueness contributes to the emergence of a 'tonic' element. It refers to the ability to individuate the elements of a set by virtue of their relations with one another. If more than one element in a set have the same intervallic relationship to all the other members of that set, it is NOT unique.

• Scalestep-Semitone Coherence: This requires that the same distance is not traversed in one part of the scale by one step and in another part of the scale by more than one step.

• Simplicity of Scale Family: Simplicity is a criterion which guarantees transpositional equivalence by comparing patterns of overlap among scale family members and their vectors of relations.

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Tuning and Temperament