Sean Gurd
This paper contextualizes one of the projects of a shadowy set of later-fifth-century musical theorists identified by Aristoxenus as “harmonikoi:” the attempt to identify the smallest audible musical interval. According to Aristoxenus, this smallest audible interval provided a basic measure for constructing a diagram of harmonic space: from this diagram, one could plot and interrelate the many different musical tunings or scales in use in Greek art music. But the work of the harmonikoi seems remarkable for one deficiency: having created a theoretical structure capable of accounting for all types of tunings, according to Aristoxenus they theorized only the enharmonic, ignoring diatonic and chromatic tunings altogether -- an omission Aristoxenus roundly criticized.
But if the harmonikoi’s purpose truly was to create an intervallic measure useful for diagraming all musical space, it strains credulity that they failed to explain chromatic and diatonic tunings. In fact, I argue, their purpose in identifying the smallest measure was not to account for the whole range of tonal systems used by 5th Century musicians. Rather, their aim was to perfect and advocate for the enharmonic, the only genus of tunings on which they are said to have worked.
The “enharmonic” was not called the enharmonic in Greek but simply the “harmonic” or just “the harmonia.” This linguistic fact allows the “harmonikoi’s” investigations of the “harmonic” genus to be considered in the context of broader currents of presocratic physical inquiry, where harmonia was a major topic of theorization (as Bundrick, amongst others, has pointed out; Barker identified the “harmonikoi” as a loosely affiliated set of musicians who not only made music but also offered theoretical demonstrations (apodeixeis), operating within the broad intellectual context of presocratic theory). Drawing on crucial texts from Heracleitus, Empedocles, the Hippocratic corpus and the Pythagorean tradition, I argue that despite major points of disagreement a general consensus can be identified: harmonia designated the coherence or coexistence of maximal differences. A musical harmonia, by extension, would be one in which the greatest differences could be included. A natural way to discover or design musical harmonia would be to find the smallest audible interval: since, as seems to have been a basic assumption in Greek music from an early period (see Franklin, Hagel 2005 and 2010, West), one covered the interval of a perfect fourth (or tetrachord) in four steps, passing through the smallest intervals at the bottom of a fourth would necessitate passing through a correspondingly large interval to complete the tetrachord. This would maximize intervallic differences within the tetrachord and create a musical scale that conformed to accepted theoretical assumptions about harmonia.