(b ?Ptolemais, after 83 ce; d 161 ce). Greek mathematician, geographer, astronomer and music theorist. He probably had access to an observatory at Alexandria, where he spent his working life. His major work, on mathematics and astronomy, was the Almagest (Arabic, al-majistī; Ptolemy's original title was Mathēmatikē suntaxis); a compendium of the work of earlier Greek astronomers, it describes a geocentric universe.
His three-volume Harmonics (Harmonika), written during the mid-2nd century, constitutes the most learned and lucid exposition of music theory in antiquity, its logical, systematic comprehensiveness making it a ‘worthy counterpart’ to the Almagest (Düring). In it Ptolemy discusses the principles and purposes of the theory of harmonics (i.1–2); the principles of acoustics (i.3–4); the theory of intervals, with a critique of the theory of the Pythagorean and Aristoxenian schools (i.5–11); the theory of the genera, with a critique of various divisions of the tetrachord (i.12–ii.1); a description of the helicon (a geometrical instrument devised, like the monochord, for measuring interval ratios using stretched strings; ii.2); the theory of 4th, 5th and octave species (ii.3); the Perfect System (systēma teleion) and the derivation of modes by transposition or modulation, with a critique of the Aristoxenian theory of tonoi (ii.4–11); a description of the monochord (ii.12–13); tables of the genera and the ‘mixtures’ (migmata) of genera usual in practice (ii.14–15); the use of the 15-string ‘monochord’ (iii.1–2); comparisons of the relationships between notes and the parts of the human soul (iii.3–7) and between the heavenly bodies, with tables (iii.8–16). An inscription from Canopus (Heiberg, ii, 149ff) draws a more detailed comparison between the planets and elements on the one hand, and the outer notes of tetrachords and their related numbers on the other.
Ptolemy's basic postulate was that the two criteria of judgment or reason and empirical observation should not contradict each other. Believing sense perception to be fallible, he discovered in the monochord, which enables acoustic phenomena to be expressed in visual, geometric terms, a precise scientific instrument by which to measure the numerical ratios of consonances. He thus conceived of music theory in terms of precise mathematical calculation.
Citing such authorities as Didymus on the difference between the Pythagorean and Aristoxenian theories of music, he criticized the Pythagoreans for frequently postulating theoretical relationships that do not correspond with reality; on the other hand he criticized the Aristoxenians for designating intervals by diastēma (‘distance apart’, i.e. by summation; see Greece, §I, 6(iii)) rather than in terms of precise mathematical ratios.
The Pythagoreans are attacked for defining in different ways intervals smaller and greater than the octave. For example, Ptolemy claims that they wrongly excluded the 11th (8:3) from the consonances while admitting the 4th (4:3), because the 11th was a ratio neither multiple (i.e. in the form nx : x) nor superparticular (i.e. in the form (x + 1) : x, where x is a positive integer): these were the mathematical formulae favoured by the Pythagoreans to define consonances.
Ptolemy arranged intervals between notes of definite but unequal pitch (anisotonoi) according to the simplicity of their ratios, as homophōnoi, or multiple ratios (e.g. the octave, 2:1, or the double octave, 4:1); sumphōnoi, or the first two non-multiple superparticular ratios (the 4th, 4:3, and the 5th, 3:2) and their octave extensions (the 11th, 2:1 × 4:3 = 8:3, and the 12th, 2:1 × 3:2 = 3:1); and finally emmeleis (‘in the melos’, i.e. melodically useful), or the superparticular ratios according to the formula above where x is greater than 3, for example, the whole tone (9:8). He also proved mathematically and acoustically that the Aristoxenians wrongly defined the 4th as two and a half whole tones and the octave as six whole tones.
Similarly, in examining the classifications of the tetrachord by his predecessors, he showed that the results they obtained had not been confirmed by empirical observation. After citing the classifications of Archytas, Eratosthenes, Didymus and Aristoxenus, he calculated his own tetrachords, using superparticular ratios, as follows:
One enharmonic tetrachord: 5:4 × 24:23 × 46:45
Two chromatic tetrachords: the ‘soft’ (chrōma malakon) – 6:5 × 15:14 × 28:27 ; and the ‘high’ or ‘tense’ (chrōma syntonon) – 7:6 × 12:11 × 22:21
Three diatonic tetrachords: the ‘tense’ (diatonon syntonon) – 10:9 × 9:8 × 16:15; the ‘soft’ (diatonon malakon) – 8:7 × 10:9 × 21:20; and the ‘even’ (diatonon homalon) – 10:9 × 11:10 × 12:11
The enharmonic tetrachord entails the pure 3rd (5:4) of Archytas and Didymus. The divisions of the ‘soft’ and ‘tense’ chromatic tetrachords are probably very close to the corresponding divisions in Aristoxenus's genera of the same name. Of the diatonic tetrachords, the ‘tense’ adapts the ratios of Didymus by exchanging the order of the two upper intervals, and the ‘soft’ is a counterpart, though differing in some respects, to Aristoxenus's ‘soft’ diatonic tetrachord; the ‘even’ includes a harmonic division by three into intervals of three-quarters of a tone, as in the archaic spondeion scale.
Ptolemy also discusses tuning in instrumental practice, using different genera in the two tetrachords of one octave. Whereas the lyre has two tunings, sterea (‘hard’, ?diatonic) and malaka (‘soft’, ?chromatic), the kithara has six different tunings, called tropoi (in the Hypodorian mode), iasti-aiolia (Hypophrygian), hypertropa (Phrygian), tritai (Hypodorian), parhypatai (Dorian) and lydia (Dorian). Ptolemy's association of them with the ethnic names of the modes suggests that they were commonly used in the imperial era.
For the derivation of the octave species, Ptolemy used a ‘monochord’ with one string for each of the 15 notes of the double octave making up the Perfect System. The notes can be identified by thesis, that is, their absolute position on the strings, or by dunamis, their function relative to other notes within the mode in question.
In discussing metabolē (‘change’, including transposition and modulation) Ptolemy distinguishes between a simple transposition of a whole melody to another pitch while retaining the same intervals within it, and a more basic ‘modulation’ of part of the melody, involving a new sequence of intervals and, thus, a change of genus (see Metabolē). Only the second of these is true modulation, and it alters the ethos.
Instead of the 13 (or 15) transpositional notes, a semitone apart, of the Aristoxenian school, Ptolemy's system consists of seven tonoi, each spanning a complete octave. These are derived from the three oldest modes (Dorian, Phrygian and Lydian) with the aid of a cycle of 4ths, the result of which is a sequence of modes pitched either a whole tone (1) or a semitone (½) apart, in the following order: Mixolydian – (½) – Lydian – (1) – Phrygian – (1) – Dorian – (½) – Hypolydian – (1) – Hypophrygian – (1) – Hypodorian. Since the notes mesē in each are tuned in the middle octave e–e', Ptolemy's transpositional notes correspond to an octave species.
The Harmonics culminates in a metaphysical discussion, largely on Pythagorean and Platonic lines, of the music of the spheres. Ptolemy draws various analogies between the relationships of the elements of music and those of the human soul (the microcosm) and the movements of the planets (the macrocosm). For instance, he compares the intervals named by him symphōnoi to the parts of the soul, the genera to the virtues and the Perfect System to the ecliptic. His last chapter, which is fragmentary, contains astrological speculations about the characteristics of planets and musical notes.
A century after the composition of the Harmonics, the Neoplatonic philosopher Porphyry wrote an informative commentary on it, referring to Ptolemy's sources and giving his own evaluation of many earlier authors. He took particular pleasure in discussing such distinctions as the quantities and qualities of notes. The Harmonics was translated into Arabic in the 9th century, and two important Byzantine (Greek) editions were made by Nicephorus Gregoras (c1335) and his pupil Isaac Argyrus, the former including Gregoras's own emendations and original material, interlinear glosses and scholia (annotations).
The various theories put forward by Ptolemy greatly influenced the Byzantine theorists Georgios Pachymeres and Manuel Bryennius, and his influence also extended to Western Europe, where he was known in the Middle Ages chiefly through Boethius's De institutione musica. His musical and astronomical systems were still current in Kepler's writing. The edition of A. Gogavinus's Latin translation (Venice, 1562) by Wallis was exemplary in its day; the standard modern edition is that of Düring, and is based on 84 manuscripts.
See also Greece, §I, 6, (iii)(e).
J. Wallis, ed.: Klaudiou Ptolemaiou Harmonikōn biblia g (Oxford, 1682/R); also pubd in J. Wallis: Operum mathematicorum, iii (Oxford, 1699), pp.i ff, 1–152
J.L. Heiberg and others, eds.: Claudii Ptolemaei opera quae exstant omnia (Leipzig, 1898–1961)
K. Manitius, trans.: Handbuch der Astronomie (Leipzig, 1912, rev. 2/1963 by O. Neugebauer)
I. Düring, ed.: Die Harmonielehre des Klaudios Ptolemaios (Göteborg, 1930/R)
I. Düring, ed.: Porphyrios Kommentar zur Harmonielehre des Ptolemaios (Göteborg, 1932/R)
I. Düring, ed.: Ptolemaios und Porphyrios über die Musik (Göteborg, 1934/R)
F.E. Robbins, ed.: Tetrabiblos (London and Cambridge, MA, 1940/R)
J. Tommer, trans.: Ptolemy's Almagest (London, 1984)
A. Barker, ed.: Greek Musical Writings, ii: Harmonic and Acoustic Theory (Cambridge, 1989), 270–391 [Ptolemy's Harmonics]
K. von Jan: ‘Die Harmonie der Sphären’, Philologus, lii (1893), 13–39
F. Boll: Studien über Claudius Ptolemäus: ein Beitrag zur Geschichte der griechischen Philosophie und Astrologie (Leipzig, 1894); also pubd in Jahrbücher für classische Philologie, suppl.xxi (1894), 66–254
R. Issberner: ‘Dynamis und Thesis’, Philologus, lv (1896), 541–60
C. Stumpf: ‘Geschichte des Konsonanzbegriffes, I: Die Definition der Konsonanz im Altertum’, Abhandlungen der Bayerischen Akademie der Wissenschaften, Philosophisch-philologische Klasse, xxi (1897), 1–78, esp. 56ff; pubd separately (Munich, 1901)
S. Wantzloeben: Das Monochord als Instrument und als System, entwicklungsgeschichtlich dargestellt (Halle, 1911)
L. Schönberger: Studien zum 1. Buch der Harmonik des Claudius Ptolemäus (Augsburg, 1914)
J.F. Mountford: ‘The Harmonics of Ptolemy and the Lacuna in II, 14’, Transactions of the American Philological Association, lvii (1926), 71–80
R.P. Winnington-Ingram: ‘Aristoxenus and the Intervals of Greek Music’, Classical Quarterly, xxvi (1932), 195–208
R.P. Winnington-Ingram: Mode in Ancient Greek Music (Cambridge, 1936/R), esp. 62ff
O.J. Gombosi: Die Tonarten und Stimmungen der antiken Musik (Copenhagen, 1939/R), esp. 98ff
B.L. van der Waerden: ‘Die Harmonielehre der Pythagoreer’, Hermes, lxxviii (1943), 163–99
J. Handschin: Der Toncharakter: eine Einführung in die Tonpsychologie (Zürich, 1948/R)
J.M. Barbour: Tuning and Temperament: a Historical Survey (East Lansing, MI, 1951/R, 2/1953), 15ff
M. Shirlaw: ‘Claudius Ptolemy as Musical Theorist’, MR, xvi (1955), 181–90
K. Ziegler and others: ‘Ptolemaios’, Paulys Real-Encyclopädie der classischen Altertumswissenschaft, xlvi (Stuttgart, 1959), 1788–1859
H. Husmann: Grundlagen der antiken und orientalischen Musikkultur (Berlin, 1961), esp. 34ff, 48–9, 82ff
L. Richter: Zur Wissenschaftslehre von der Musik bei Platon und Aristoteles (Berlin, 1961), esp. 181ff
M. Vogel: Die Enharmonik der Griechen (Düsseldorf, 1963), esp. i, 33ff, 48ff
A. Machabey: ‘De Ptolémée aux Carolingiens’, Quadrivium, vi (1964), 37–48
B. Alexanderson: Textual Remarks on Ptolemy's Harmonica and Porphyry's Commentary (Göteborg, 1969)
O. Neugebauer: A History of Ancient Mathematical Astronomy (New York, 1975), i, 1–261; ii, 917–41
A. Barbera: ‘Arithmetic and Geometric Divisions of the Tetrachord’, JMT, xxi (1977), 294–323
M. Markovits: Das Tonsystem der abendländischen Musik im frühen Mittelalter (Bern, 1977)
F.L. Levin: ‘Plēgē and tasiz in the Harmonics of Klaudios Ptolemaios’, Hermes, cviii (1980), 205–29
W.J. Tucker: L'astrologie de Ptolémée (Paris, 1981)
A. Barbera: ‘Octave Species’, JM, iii (1984), 229–41
B.L. van der Waerden: Die Astronomie des Griechen (Darmstadt, 1988), esp. 252–302
J. Solomon: ‘A Preliminary Analysis of the Organization of Ptolemy's Harmonics’, Music Theory and its Sources, ed. A. Barbera (Notre Dane, IN, 1990), 68–84
LUKAS RICHTER