Harmonic radius
is a theoretical tool for evaluating the relative harmonicity of any set of pitches expressed in terms of rational intonation. It is related to concepts like Leonhard Euler’s gradus suavitas, James Tenney’s harmonic distance and intersection, Clarence Barlow’s indigestibility, and Paul Ehrlich’s harmonic entropy. Its advantage over these other approaches is that it can be estimated directly, in real time, without computation, simply by knowing the odd partial pitch classes making up a pitch set. At any given time, it is possible to consider collections of tones that happen to be musically relevant (intervals, chords, melodies, gamuts). You can read my article, or view my presentation on youtube, and look at the slides as a pdf document.
Any set of tones may be expressed, within some margin of error or tolerance, in one or several ways as a proportion of odd whole numbers. Each such proportion represents harmonic partials over a single fundamental and expresses the most consonant voicing of a particular JI tuning of the chord in question. Harmonic radius takes the proportional average or geometric mean of the partial numbers. Smaller values represent tunings that, on average, lie closer to their common fundamental, and therefore have greater potential to be perceived harmonically. In this context, the most consonant harmonic sound combining three different partials is 1°-3°-5°, a 5-limit JI major triad in root position, while the complementary 5-limit JI minor triad is first found as 3°-5°-15°, with its minor third in the lowest voice and somewhat greater harmonic radius.
Odd partial pitch class sets
are a useful way of implementing the harmonic radius concept. Odd partials mark first occurrences of new notes above a given fundamental, and are the smallest numbers identifying those pitch classes. So collections of odd partials represent pitch class voicings with lowest harmonic radius. Depending on the musical unfolding of collections as melodies, counterpoints, or chords, it can be useful to look at harmonic radius statistically across the relevantly sized subsets (2-note, 3-note, etc.). In this way, it is possible to quantifiably describe the relative harmonicity of rational intonation pitch spaces for different musical textures. By implementing a tolerance for real-world inharmonicity or intentional detuning (i.e., temperaments), this approach may be extended to the analysis of arbitrary frequency collections.