Waart, Edo de. 56 Wachmann, Eduard 56

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Wallis, John

(b Ashford, Kent, 23 Nov 1616; d Oxford, 28 Oct 1703). English mathematician, experimental philosopher and music theorist. He entered Emmanuel College, Cambridge, in 1632, graduated BA in 1637 and MA in 1640, the same year he was ordained. In recognition of his services to parliament during the Civil War (e.g. decoding ciphers used by the royalist army), Cromwell appointed Wallis to the Savilian Chair of Geometry at Oxford in 1649. He lectured on harmonics as a branch of mathematics and over the next half century established himself as the foremost English authority on musical science. He was also one of the experimental philosophers active in London and Oxford during the Interregnum who became founder members of the early Royal Society of London. His dual interest in mathematics and experimental philosophy is reflected in his music theory, which was written entirely from the perspective of a scholar rather than a practising musician.

Wallis's outstanding contribution to music theory was his Greek edition with Latin translation (1682) of Ptolemy's Harmonika (mid-2nd century) and the commentaries on it by Porphyry and Bryennius (1699). His edition of Ptolemy, one of the most important sources on ancient Greek music theory, remained a standard authority until the 20th century. He had been appointed as Keeper of the University Archives in 1658 and had had access to Greek manuscripts collated and studied by Peter Turner (a former Savilian Professor of Geometry) and Edmund Chilmead before the latter's ejection from Christ Church in the late 1640s. In an appendix to Ptolemy, Wallis identified the chief differences between ancient and modern music, and an English summary of his conclusions was published in the Philosophical Transactions of the Royal Society (1698).

It was also via the Philosophical Transactions in 1677 that he alerted readers to the discovery of the existence of nodes, made by two Oxford scholars, William Noble and Thomas Pigot, in about 1674. This curious property of vibrating strings had already been described in an essay by Narcissus Marsh in Robert Plot's Natural History of Oxfordshire (Oxford, 1677), and the physics of musical strings (studied by Mersenne and Galileo Galilei in the 1620s and 30s) had been investigated by Wallis and other Royal Society members in 1664–5. Wallis was also interested in the organs of speech and hearing, and wrote extensively on the structure and origins of language, as well as participating with William Holder in a debate on teaching the deaf and dumb.

Wallis's knowledge of ancient harmonic doctrine clearly informed his contribution to modern debates on tuning and temperament. It was in response to John Birchensha's theories presented to the Royal Society in 1664 that he first began to articulate his theory of tonality based on Ptolemy's authority in a series of letters to the society's secretary, Henry Oldenburg. Only by the third letter did Wallis acknowledge the contribution of more recent authors such as Kepler and Mersenne to this subject. In the 1670s and 80s he gave public support to Thomas Salmon's scheme for improving the tuning of fretted instruments described in the latter's A Proposal to Perform Musick; Perfect and Mathematical Proportions (London, 1688). By means of an adjustable fingerboard it would be possible to play the intervals of Ptolemy's syntonic diatonic scale (i.e. just intonation). The same Ptolemaic ideal underpins Wallis's 1698 article in which he describes the system of equal temperament used by the organ builder Renatus Harris. While accepting the need for some system of temperament in instruments, he seems to have shared Salmon's conviction that if music could be played using just intonation this might produce the kind of ethical effects described by ancient authors. In his letter to Andrew Fletcher ‘Concerning the Strange Effects reported of Musick in Former Times’ (1698), Wallis attributed most of these effects either to exaggeration by the authors themselves, or to the simplicity of ancient people. Yet having noted that the aim of modern music is to please the ear, he observed that musicians probably would still be able to excite the passions of their listeners if this was their intention.

The brief articles Wallis published in the Philosophical Transactions seem to have had a far greater influence on later readers than his more original Latin writings. Like other areas of his work, however, his musical scholarship relied considerably on other people's ideas, and he tended to pass them off as his own. As Anthony Wood observed, he could ‘at any time, make black white and white black, for his own ends, and hath a ready knack of sophistical evasion’.


only those on music

Correspondence with Henry Oldenburg on John Birchensha's tuning, 7, 14, 25 May 1664 (London, Royal Society Library, W1, nos.7–9); ed. A.R. Hall and M.B. Hall in The Correspondence of Henry Oldenburg (Madison, WI, 1965–77), ii

‘Of the Trembling of Consonant Strings: Dr Wallis's Letter to the Publisher, concerning a New Musical Discovery’, 14 March 1677, Philosophical Transactions, no.134 (23 April 1677), 12, 839–44

Claudii Ptolemaei Harmonicorum libri tres (Oxford, 1682, R/1699 in Opera mathematica, iii), incl. appx, ‘De veterum harmonica ad hodiernam comparata’

‘A Question in Musick lately proposed to Dr Wallis, concerning the Division of the Monochord, Or Section of the Musical Canon: With his Answer to it’, 5 March 1698, Philosophical Transactions, no.238 (March 1698), 20, 80–88

‘A Letter of Dr John Wallis to Samuel Pepys Esquire, relating to some supposed Imperfections in an Organ’, 27 June 1698, Philosophical Transactions, no.242 (July 1698), 20, 249–56

‘A Letter of Dr John Wallis to Mr Andrew Fletcher: Concerning the Strange Effects reported of Musick in Former Times, beyond what is to be found in Later Ages’, 18 Aug 1698, Philosophical Transactions, no.243 (Aug 1698), 20, 297–303

Porphyrii commentarius in librum primum Harmonicorum Claudii Ptolemaei: atque Manuelis Bryennii commentarius in tres libros Harmonicos ejusdem Ptolemaei … graece ac latine (Oxford, 1699, as pt of Opera mathematica), iii

Correspondence with Thomas Salmon on tuning systems (Ob Add. D.105, ff.92–3, 124–7; English Letters c.130)

Letter concerning an ancient Greek MS found at Budapest (GB-Lbl Lansdowne 763, f.124)



A. Wood: Athenae oxonienses … to which are added the Fasti or Annals (London, 1691–2, rev. and enlarged 3/1813–20/R by P. Bliss)

P.M. Gouk: ‘Speculative and Practical Music in Seventeenth-Century England: Oxford University as a Case Study’, IMSCR XIV: Bologna 1987, 199–205

L. Miller and A. Cohen: Music in the Royal Society of London 1660–1806 (Detroit, 1987)

M. Reeve: ‘John Wallis, Editor of Greek Mathematical Texts’, Aporemata: Kritische Studien zur Philologiegeschichte, ii: Editing Texts/Texte edieren, ed. G.W. Most (Göttingen, 1998), 77–93

P. Gouk: Music, Science and Natural Magic in Seventeenth Century England (forthcoming)


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