Harmony In Ancient Greek Writings On Music
In ancient Greek writings on the subject of music, harmony (also known as "harmonics") was the study of the formation of melody. This study began with the elements of melody—the individual notes—and continued with the specification of appropriate ways in which pairs of notes, a higher and a lower, could be combined successively into melodic intervals. (The simultaneous combination of notes was not a part of classical Greek musical practice.) These melodic intervals were in turn combined into a variety of complex scalar systems, the defining structures of complete melodies. In general terms, classical Greek harmonics falls into two traditions: the Aristoxenian and the Pythagorean.
To Aristoxenus (c. 375–300 B.C.E.), a prolific writer on a variety of philosophical and historical subjects and the son of a musician, harmonics was the study of music as we hear it. Its task was to arrive at an understanding of the musical sounds that the human ear hears as pleasing or melodic through a systematic analysis of the perceived phenomena. The definition of "melodic" must concern itself only with the sounded elements of music—the notes, described exactly as they are heard by the ear, namely, as different pitches on a melodic continuum; furthermore, the general rules that govern melodic structure must not be derived from any abstract principles. However, Aristoxenus, to support his phenomenalist argument for the existence of certain melodic combinations of notes that have an "affinity" with one another, makes an analogy to a related property of speech: "And yet the order which relates to the melodic and unmelodic is similar to that concerned with the combination of letters in speech: from a given set of letters a syllable is not generated in just any way, but in some ways and not in others" (Barker, p. 153).
By contrast, Pythagorean harmonics, the set of beliefs about music attributed to the contemporary followers of Pythagoras of Samos (c. 580–c. 500 B.C.E.) and his intellectual heirs in the fourth and third centuries B.C.E., was centered on the discovery that the fundamental melodic intervals of the octave, the fifth, and the fourth could be produced by the lengths of the two sections of a stretched string in the simple and elegant mathematical ratios of 2:1, 3:2, and 4:3, respectively. The Pythagoreans took this musical discovery as an affirmation of their belief in the mathematical nature of reality and argued that certain musical intervals are pleasing to the ear because of their underlying structure, not for any reason having to do with musical sound considered only as an audible phenomenon. Within Pythagorean harmonics, all subsequent combinations of tones into scalar systems were generated from these basic ratios, including 9:8, the mathematical ratio associated with a musical whole tone. To the Pythagoreans, these scalar constructions—the Pythagorean system of intonation—were musical embodiments of a cosmic scalar relationship among the planets governed by their distances from the earth and their revolutionary speed: the harmony of the spheres. (This idea may have had Mesopotamian roots; see Kilmer, "Mesopotamia.")