(fl late 1st – early 2nd century ce). Greek mathematician and music theorist. His Introduction to Arithmetic (Arithmētikē eisagōgē) won him high praise and instant fame in antiquity. This work and the Manual of Harmonics (Harmonikon encheiridion) have survived in their entirety; ten extracts (Excerpta ex Nicomacho, ed. Jan, 266–82) remain from a longer treatise on music, and portions of another work, Theology of Arithmetic (Theologoumena arithmetikēs), are preserved in an anonymous treatise of the same title.
The Manual of Harmonics is the only work on Greek music to have come down from the long period between the appearance of the Harmonic Elements of Aristoxenus and the Euclidean Division of the Canon in the 4th century bce, and that of the Harmonics of Nicomachus’s celebrated younger contemporary Claudius Ptolemy in the 2nd century ce. The Manual is important for its influence on numerous later writers, and especially for the canonical material that it alone preserves. It is the first work in the literature to transmit the time-honoured story of Pythagoras’s momentous discovery that musical pitch is ruled by number. As Nicomachus tells it (chap.vi), Pythagoras’s revelation was inspired by the ringing sounds he chanced to hear issuing from a blacksmith’s anvil. Recognizing them to be the very concords – octave, 5th and 4th – that he could produce on the strings of his lyre, Pythagoras performed a series of experiments and found the elegantly simple truth about musical sound: the pitch from a plucked string depends on the length of the string, and the concords are produced by strings whose lengths are to each other as the ratios of the whole numbers: 6:8:9:12.
No less significant is Nicomachus’s detailed account of the first unified theory of the cosmos. It contains what may be the most ancient version of the distinctly Pythagorean-Platonic concept that the harmonic properties of music, discoverable in the ratios of the concords, are implicit in the orderly distribution of the heavenly planets (see Music of the spheres). Nicomachus’s discussion of this theory, along with Plato’s in the Timaeus, influenced astronomical thought for centuries, converging eventually with cosmic reality in the celestial physics of Johannes Kepler.
Nicomachus was the first writer on music to attribute the invention of the octachord (eight-string lyre) to Pythagoras (Manual, v); he is also the source (ibid., ix) of one of the oldest pieces of evidence on musical scales, a fragment of Philolaus’s On Nature, the first written document on the teachings of Pythagoras.
In his discussion of the inverse proportion that obtains between a sounding body and musical pitch (the higher the pitch, the smaller the body, and conversely) and the reciprocal relation between pitch and tension (the greater the tension, the higher the pitch, and conversely), Nicomachus provides (Manual, iv and x) valuable information on diverse musical instruments. The most exotic and obscure of the strings mentioned by him is the spadix, apparently a lyre-type instrument shaped like a palm frond. Equally interesting is his evidence on the pandoura, a lute of remote antiquity which he likens to the Pythagorean research instrument, the monochord.
Perhaps the most curious feature of the Manual is the incorporation (in chap.ii) into an otherwise strictly Pythagorean programme of a decidedly non-Pythagorean concept imported from the theory of Aristoxenus. For Aristoxenus’s theory is based on the notion of a tonal continuum (topos) whose division by the placement of pitches and intervals is under the sole governance of the human voice and ear. In the conventional Pythagorean approach, however, the division of musical space is determined solely by the mathematical laws of harmonic proportion. Thus, without citing him, Nicomachus spoke the language of Aristoxenus, and in his effort to credit the Pythagoreans with the invention of all things musical, he attributed to them the very doctrine that is contravened by their mathematically based harmonic principles.
See also Greece, §I, 6(i).
C.E. Ruelle, trans.: Nicomaque de Gérase: Manuel d’harmonique et autres textes relatifs à la musique (Paris, 1881)
C. von Jan, ed.: Musici scriptores graeci (Leipzig, 1895/R), 209–82
A. Barker, ed.: Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge, 1989), 245–69
F. Levin, ed.: The Manual of Harmonics of Nicomachus the Pythagorean (Grand Rapids, MI, 1994)
FLORA R. LEVIN