The standard printed edition currently available is Tinctoris 1975: the Liber imperfectionum notarum musicalium is at i. 143–67; the Tractatus alterationum at i. 173–9; the Scriptum … super punctis musicalibus at i. 185–98. New texts and translations of these treatises are, or will shortly become, available online as part of the present Tinctoris project.

One qualification to this statement of which I am aware is the French translation in manuscript of the bulk of Tinctoris’s theoretical writings, drafted by François-Joseph Fétis in the mid-nineteenth century from the primary Brussels source (Bibliothèque royale, MS II 4147) while Fétis owned this manuscript: see Woodley 2011: 138. Fétis’s translation, which includes some commentary notes, survives as Brussels, Bibliothèque royale, MS 5482–3, but it has not yet been studied at all.

For a recent edition of De Muris, see De Muris 1999; also De Muris 1972.

Throughout this essay I use the medieval -e form, rather than the classical or neo-classical –æ diphthong/ligature, except where quoting directly from a source which employs the latter.

A notable exception to this is Blackburn 1981: esp. 68–71.

It is worth mentioning here that the present essay does not attempt to discuss non-standard usages by, for instance, Machaut, who occasionally adapts the strict principle of imperfection to include notes which are not themselves partes propinque of perfect entities, such as minims in minor prolation, as agents of imperfection: see, for instance, Apel 1953: 112 and commentary ibid., 440. This adaptation of the system on Machaut’s part is cited by a number of theorists in the fourteenth and early fifteenth centuries, following De Muris’s basic precepts but categorizing Machaut’s usage as erroneous; and it is clear that even at the end of the fifteenth century Gafori, for example, was aware of this theoretical tradition: see Gaforus 1496: fol. bb iiij (Book 2, Chapter 11); also briefly discussed in Gaforus 1969: 98–9. Young also mentions Gafori’s citation of Machaut’s usage in his own hand-written version of the De Muris ‘Practica musice’, following his Glossemata super nonnullis partibus prime partis theorice musice Ioannis de muris, surviving as Milan, Biblioteca Ambrosiana, MS H165 inf., fols. 18v–22, at fol. 19v. I am most grateful to Bonnie J. Blackburn for enabling me to consult a microfilm of this work. See also, most recently, Lerch-Kalavrytinos 2007: 241. I am also grateful to Margaret Bent for pointing out to me that the English composer Leonel practises this extended use of the ‘imperfection’ of already imperfect values at the beginning of the five-voice Amen sections of his Glora and Credo settings surviving as Nos. 21 and 77 of the Old Hall manuscript; in both of these cases Leonel imperfects an already fully imperfect maxima with a single minim.

In addition to the use of the term in various of his mensural treatises, Tinctoris defines reductio in his Diffinitorium as ‘unius aut plurium notarum cum maioribus quas imperficiunt aut cum sociis annumeratio’ (‘the reckoning together of one or more notes with larger notes which they imperfect, or with [other] partner notes.’ (Tinctoris 2004: 40). On the use of the term [nota] socia, see below in the present essay.

See Tinctoris 1975: i. 187; as printed as Ex. 4 of Seay's edition, however, many of the dots are actually misplaced.           

See 17 below.

The reference to duple proportion here refers primarily to Tinctoris’s assertion, especially in his Proportionale but also the Tractatus de notis et pausis, that (within a void base notation) black ‘semiminims’ are actually minims coloured under proportio dupla. For the relevant passages of the Proportionale, see Tinctoris 2008: 36–48; Tinctoris 1975: iib. 17 and 19–25.

For instance, in Woodley 2006a: esp. 47–8.

For a brief recent discussion, however, of the unusual use in late fourteenth- and early fifteenth-century English sources of the cauda hirundinis (swallow’s tail) to indicate non-standard quasi-alteration, sometimes incorporating displacement and even indicating ‘alteration’ of the first, rather than second, of the two notes, see Lefferts 2007: 274–5; also Bent forthcoming.

See the legacy edition of the Tractatus alterationum embedded within the present Tinctoris project; also Tinctoris 1975: i. 173–9.

The best manuscript sources of this treatise (Brussels 4147, Valencia 835 and Bologna 2573) unanimously read ‘Quiquidem puncti vulgariter dicuntur asinei …’ in the relevant passage, leading to the slightly unusual form punctus asineus, while Ghent 70, fol. 176 reads ‘puncti … azinini’, from the more classical adjective asininus.

In his letter Del Lago also explains Tinctoris’s notion of reductio, as discussed in the present essay.

Elsewhere in the Liber imperfectionum Tinctoris does indulge in his characteristic ad hominem criticisms of contemporary composers, including De Domarto, Barbingant and Busnoys (see text of present project, esp. I.iii towards the end of the Twelfth General Rule; Tinctoris 1975: i. 153–4), though these examples do not specifically impinge on the discussions here. For a preliminary paper on the wider, more critically orientated issues of textuality and auctoritas which emerge from such citations, see Woodley 2005.

Tinctoris’s scheme of things seems not to extend explicitly to using coloration by itself to indicate note socie for purposes of alteration, though we have already seen how he accepts coloration as a valid means of indicating the correct reductio of displaced imperfection groupings. In one example of the Tractatus alterationum, however, Tinctoris uses the coloration of a syncopated pairing of blackened (but still perfect) breve plus (imperfected) long in order to indicate a mensural grouping which prevents the expected alteration of the second of a pair of uncoloured breves which stand immediately before that long. See the legacy edition of the treatise within the present project, Example 3, and commentary.

For discussions of the musical relations between England and continental Europe, at both technical and institutional levels, see for example Dumitrescu 2006 and Dumitrescu 2007). In the context of the longer fifteenth-century history of complex notation in England, there are also clear linkages to be explored from Bent forthcoming.