Diatonicity in a nutshell rev. 13 Feb. 2003. _______________________________________________________________ _ __ ___ Define... pitch - a musical sound characterized by a log-frequency value. scale - an ordered list of pitches. interval - a distance between two pitches, given as their difference. generic interval - a distance between two scale members, given as the number of scale members subtended, inclusive of the outer members. interval class - a list of intervals produced by moving a given generic interval sequentially over all positions in a scale. melody - an ordered list of signed generic intervals and a starting position in a scale, which generates a series of pitches when applied iteratively in the scale. Then define a scale's diatonicity with four quantities... (1) Pitch Tracking Generalized octave equivalence; Miller limit The scale is periodic with respect to a strong consonance, and addressing the pitches in a period stretches but does not exhaust human working memory. k, where k is the number of pitches in a period of the scale and 4 < k < 11. (2) Mode Autonomy Rothenberg Efficiency (higher values are better) A relatively long melody must be heard before the listener can unambiguously determine its starting position. Rothenberg, David. "A Model for Pattern Perception with Musical Applications." Parts I & II. Mathematical Systems Theory vol. 11, 1978. Note that for equal temperaments, Rothenberg considers a single element sufficient, and assigns minimal efficiency 1/k. These scales should have maximal mode autonomy however, and for our purposes efficiency 1. (3) Modal Transposition Stability (higher values are better) The relative sizes of the intervals produced by a melody may change when the melody's starting position is changed. A stable scale is one for which these changes are slight or nonexistent. The portion, in log-frequency space, of the scale's period that is not covered by the spans of its interval classes is the raw stability. We divide this by a factor based on the portion that is redundantly covered. s/(1+i), where s is the "Lumma stability" and i the "impropriety factor" as given by Scala's "show data". (4) Diatonic harmony Diatonic Property (higher values are better) There is at least one generic interval at which a melody may be harmonized without the voices becoming timbre-fused. Such a harmony is parallel with respect to generic intervals but not with respect to actual intervals. (k+1-c)(d^2)/(k^3) for a given interval class, where d is the number of times it produces a consonant triad with the interval of equivalence and c is the length of the longest consecutive run of any single interval through it. ~ Carl Lumma ___ __ _ _______________________________________________________________