Tractatus alterationum
Commentary notes on text and music examples

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Sources:
Br, fols. 44v–47; V, fols. 67–70; Bu, fols. 75–79; G, fols. 169–171v.
Guide to manuscript sigla
Outline description of principal sources
Introduction to Tinctoris's writings

Related paper:
'At the Limits of Mensural Theory: Tinctoris on Imperfection and Alteration' (2006)


Abbreviations:
Mx maxima; longa; B brevis; S semibrevis; M minima; Sm semiminima (or, in Tinctoris's construction, flagged/black minim under dupla proportion).
p.a. punctus augmentationis; p.d. punctus divisionis; p.p. punctus perfectionis.

The location of individual notes in each music example (original notation) is referenced here by simple, linear note-count in bold, including components of ligatures. To make orientation easier for the reader, the note-count of each example is given every ten notes above the staff, with downward arrow for additional precision, e.g. with ligatures.

In the transcriptions of the examples embedded in the English translation of the text, units of minor modus (i.e. at the level of L) and major modus (i.e. at the level of Mx) are indicated respectively with double and treble barlines. Close study of the examples in treatises such as this, and the Liber imperfectionum notarum musicalium, shows how careful Tinctoris was to observe these larger-scale mensural patternings, so that even in examples where complex, multiple syncopations take place at various mensural levels, the units of minor and major modus almost invariably resolve correctly at the end. In the rare instances where a degree of ambiguity remains, a commentary note is added here.

For technical detail of terminology with regard to the various processes of imperfection described here, see the text and translation of Tinctoris's Liber imperfectionum notarum musicalium in the present online edition.


lines 4–9: The dedicatee of the treatise is given with unanimity in the three main MSS as Guillelmus Guinandi, though from Tinctoris's description of him as first chaplain of the Duke of Milan, as well as priest and lawyer, this must presumably refer to the abbot Antonio Guinati, who directed the cantori di cappella in Galeazzo Maria Sforza's chapel at the time of Tinctoris's writing in the (probably) mid-1470s. It is difficult to reconstruct the nature of Tinctoris's error here, which raises wider questions concerning the nature of dedications such as this, and the extent to which one can take them at face value as indications of personal contact, knowledge and communication. (See also Woodley 1988: 191.) The individual singer at Milan whose open criticism of Tinctoris's notation ostensibly prompted the writing of the treatise is not named, though Tinctoris's wording here implies that personal contact with him, too, had been lacking, and that the criticism had reached the theorist via a third party rather than through direct correspondence.

lines 9–10: Tinctoris's presumably early mass setting based on 'Nos amis' is lost: for the complexities of its attempted identification, its relationship to the song 'Nos amys vous vous abusés' by Adrien Basin and the German song 'Wünschlichen schön ist ir gestalt', and surviving mass settings in Trent 89 and other MSS, see especially Strohm 1979; Strohm 1989; Strohm 1993: 430, 470, 531; Fallows 1999: 295–6, 496–7). Note the emphatically pluperfect tense of 9. habuisset and 11. reperisset: this may imply that the unnamed critic had had the mass in his possession at some time before his arrival at the Sforza chapel, which may in turn imply (especially if the composition of the mass predated Tinctoris's own arrival in Italy in the early 1470s) that the critic himself was of northern European origin: possibly Compère or Martini, for example?

Example 1
There is no 'deficiency in number' here, in Tinctoris's view, since we should consider the S-equivalent group of two M to imperfect the B at 1, and the notated S to be counted with the initial two S rests, in syncopation (cf. Tinctoris's First General Rule below). The complaint by Tinctoris's unnamed Milanese accuser that the two S units in between two B in perfect tempus ought to demand alteration of the second is essentially undermined by his misreading of the initial two rests, since his criticism would only be valid if the second rest were alterable, making up a full, perfect B's worth of rests, which is manifestly impossible (cf. Sixth General Rule below), or that the unit of tempus preceding the notated B at 1 was in some other way rendered already complete. These two initial S rest, then, are crucial to Tinctoris's case, and to the correct interpretation of the theory underlying the notation.

line 30 cessante ... effectus: a long-established principle in canon law, which Tinctoris would have picked up during his legal studies in Orléans, if not before as part of his general education.

Example 2
3 Second of two L altered before Mx (perfect major modus); 2 and 3 cannot be regarded as counted together, through syncopation, with third L at 5, since the latter does not precede the former two 'either in continuous juxtaposition or through syncopation'.

5 L imperfects preceding Mx (perfect major modus).

7 Second of two B altered before L (perfect minor modus); cannot be regarded as counted together, through syncopation, with third B at 9, since the latter does not precede the former two (i.e. analogous situation to 3 at next lower mensural level).

15 p.d. attached to first of three M ensures both imperfection of preceding S and alteration of third M at 17 before following S, rather than the three M being counted together.

22 p.d. attached to S, together with p.p. attached to S at 24, ensures alteration of last of three M at 26 before following S.

line 35 reduci: the verb reducere, with its substantive reductio, is used here, as elsewhere in Tinctoris's writings (e.g. the Liber imperfectionum notarum musicalium, and the Proportionale musices) in this specific, technical sense of gathering together the constituent elements of a displaced grouping.

Example 3
49 This unit of perfect major modus (i.e. worth 3 L) is constituted as follows: (1) normal perfect L at 4; (2) one perfect L comprising the black B at 5 plus the black L at 8, in which the coloration indicates the displaced grouping (reductio) rather than imperfection, even though the L is indeed imperfected at a distance naturally by the black B; and (3) the third perfect L comprising two ligated B at 67, plus single B at 9, separated from the following S by p.d. in order to prevent its imperfection by the latter. Tinctoris's point here is that the second of the two ligated B at 7 would normally be altered before the following L (with the previous L at 4 imperfected by its following B), were it not for the black coloration which indicates the alternative mensural patterning. Note also that 49 could be renotated with metrical synonymy without coloration (all void, with p.p. added to 4), but the nature of the displaced groupings through reductio, which Tinctoris clearly wishes to demonstrate, would thus be lost.

1017 The initial S at 10 is separated by p.d. from the following syncopated, ligated group 1113 in order to ensure that it is counted with the remaining two S-worth (S plus three M at 1417), to complete the unit of perfect tempus. This also ensures the alteration of the second ligated S at 12, rather than counting these three S together as a B unit. The last note of the ligated group (B at 13) is also separated by p.p. from the following S to prevent its imperfection by the latter.

Example 4
This example contains, in addition to two 'normal' instances of alteration, two cases of the highly unusual syncopated alteration allowed for in Tinctoris's Second General Rule.

3 Second of two L altered before Mx (perfect major modus)

4 Mx identified as perfect by addition of p.p., to prevent imperfection by following L.

5 This perfect L forms a pair with 7, separated by the perfect Mx of 6. This second L must be altered in syncopation before the Mx at 8. The p.p. applied to 6 makes it clear that this Mx is not to be imperfected by 7. The unusual syncopated alteration here is the only way to achieve the desired note-value of two L-worth for 7, since there is no way to employ an imperfect Mx in this context.

8 Mx imperfected by according to normal rules by following L at 9.

916 This sequence applies similar thinking to that of the first half of the example, at lower mensural levels. Note especially, again, the unusual syncopated alteration of the S at 15, by analogy with the L at 7.

line 48: This technical sense of remotus, remotior and remotissimus is explained in more detail in Tinctoris's Liber imperfectionum notarum musicalium, Book 1, Ch. 1: see the present online edition, lines 40 ff.

Example 5
1 Major modus is indicated as imperfect (with perfect minor modus) by the single perfect L rest after the mensuration sign: Tinctoris, De regulari valore notarum, Ch. 8–9 (see Seay 1975, i. 129–30). In Ch. 10 of this treatise Tinctoris advocates a distinction between the placement of the signum modi before or after the tempus sign, in the latter case counting as a real rest, to be factored into the measure-count at the level of major modus. In the present example it seems unlikely that the calculation of major modus structuring is intended to incorporate the rest, unless one is to read the dot attached to the Mx at 4 as functioning simultaneously as p.d. and p.a. (cf. Tinctoris, Scriptum super punctis musicalibus, Ch. 11: Seay 1975, i. 192; new online edition forthcoming here 2008). Also, there are other examples in the present treatise in which Tinctoris employs the single L rest as sign of imperfect major modus, where such a calculation clearly cannot be intended (e.g. Example 15 below), though in others there is certainly a degree of ambiguity (cf. Example 14 below). It may be that at the time of writing the Tractatus alterationum this particular precept had not been fully formed in Tinctoris's mind, and it may therefore be an indication that the treatise predates De regulari valore notarum.

4 The dot following this Mx is primarily a p.d. (though see previous paragraph), preventing imperfection quantum ad partes propinquas by the following B.

6 First of two ligated L imperfected by preceding B.

9 Second of two ligated B altered before following L (perfect minor modus).

13 Second of two ligated S altered before following B (perfect tempus).

21 Second of two M altered before following S (major prolation).

25 Second of two ligated B not altered, since before following Mx, which, although intrinsically imperfect, is imperfected quantum ad partes propinquas (perfect minor modus) by these two B, giving it a value of four B.

29 Second of two ligated S not altered, since before following L, which is imperfected quantum ad partes propinquas (perfect tempus) by these two S, giving it a value of seven S.

33 Second of two M not altered, since before following B, which is imperfected quantum ad partes propinquas (major prolation) by these two M, giving it a value of seven M.

39 Second of two ligated S not altered, since before following Mx, which, although intrinsically imperfect, is imperfected quantum ad partes remotas (perfect tempus) by these two S, giving it a value of 16 S.

41 The dot attached to the L is a p.p., preventing imperfection quantum ad partes remotas by the following two M.

43 Second of two M not altered, since before following L, which is imperfected quantum ad partes remotas (major prolation) by these two M, giving it a value of 25 M.

47 The dot attached to the B is a p.p., preventing imperfection quantum ad partes propinquas by the following two M.

49 Second of two M not altered, since before following Mx, which, although intrinsically imperfect, is imperfected quantum ad partes remotiores (major prolation) by these two M, giving it a value of 52 M.

Example 6
3 Second of two B altered before following 'maior propinqua' L (perfect minor modus), without imperfection of preceding 'maior remota' Mx.

4 Second of two L altered before following Mx (perfect major modus). Note that the first of these L is divided into its component parts (two B at 23), although Tinctoris has yet to describe and authorize this practice in his Fifth General Rule immediately following this example. Also noteworthy is the use here of 'compound alteration' across 24.

69 The second of two S at 7 is altered before following B; Tinctoris's point here is that this B-equivalent unit does not imperfect the preceding Mx at 5; rather, the whole L-equivalent ligature 69 imperfects the following Mx at 10.

12 Second of two M altered before following S (major prolation), this alteration again being deemed by Tinctoris's precept to take priority over the (further) imperfection of the preceding Mx.

15 L imperfected both a parte ante by 1114 (quantum ad totum) and a parte post by the single S at 16 (quantum ad partes propinquas). The p.d. following this S prevents this and the following S from being counted together, with the second altered before the following B at 18, illustrating Tinctoris's precept.

18 B imperfected both a parte ante by the preceding S (quantum ad totum) and a parte post by the following two M (quantum ad partes propinquas). The p.d. following these two M prevents the second of them from being altered before the following S at 21, and forcing imperfection of the preceding B, according to Tinctoris's precept.

22 Second of two S altered before following B.

23 Note that the second of two B (of which the first is divided at 212) is not altered before following L, since the perfection at minor modus level is complete in any case (i.e. there is no 'deficiency of number').

29 Note that this is a rare instance of an example ending correctly with a L rather than Mx, Tinctoris having carefully calculated that the final unit of major modus is nevertheless duly completed.

Example 7
On the use of the single L rest as signum modi, and its potential implications, see Example 5 above.

3 Second of two ligated B imperfected by following S (perfect tempus).

13 Second of two ligated L imperfected by following B (perfect minor modus).

1517 B-equivalent group of three S followed by second (intact) B at 18, this last being altered before following L. This illustrates Tinctoris's precept that the 'nota veniens alteranda' is indivisible, whilst its 'socia' or partner note may be divided and represented by its constituent parts (here 1517).

19 L imperfected by following B (perfect minor modus).

213 A further illustration of Tinctoris's Fifth General Rule, combined with compound alteration: the second of two M at 22 is altered before following S (major prolation), and these two M themselves constitute the first of two S units as 'socia', the second of which (undivided) at 23 is altered before following B (perfect tempus).

24 B imperfected quantum ad totum by following perfect S-equivalent group of imperfected S plus M. The p.d. is added to the M to prevent it from being counted together with the following group of two Sm plus M, which would in turn undermine the illustration of alteration coming up at 30.

2730 A further illustration of Tinctoris's Fifth General Rule: the two Sm form a M-equivalent group ('socia') followed by a single, intact M which is altered before the following S. The p.p. is added to the S to prevent its imperfection by the following M, which instead imperfects the subsequent S at 32 a parte ante.

33 Note that this is a further instance of an example ending correctly on a L rather than Mx, with the final unit of major modus calculated by Tinctoris as complete.

Example 8
1 Alteration of L induced by preceding L rest (perfect major modus), illustrating Tinctoris's Sixth General Rule.

4 Alteration of B induced by preceding B rest (perfect minor modus).

6 p.p. added to Mx to prevent imperfection a parte post by following note or group of notes.

8 Alteration of S induced by preceding S rest (perfect tempus).

12 Alteration of M induced by preceding M rest (major prolation).

19 Perfect L rest following 19 not able to be altered following first notated L of the new mensuration, before following Mx at 20, again illustrating Tinctoris's Sixth General Rule.

20 The necessary imperfection of this Mx should probably be deemed to operate a parte post by the L-equivalent group of 212, rather than a parte ante by the previous L rest. In this case, the unit of perfect major modus commencing at 19 should be regarded as completed, through syncopation, by the L-equivalent group of 23 plus preceding B rest.

21 L imperfected by following B (perfect minor modus).

23 L imperfected by preceding B rest. This rest also functions to exemplify further Tinctoris's precept that a rest cannot itself be altered, since, if it were a actual B, it would require alteration as second of two B before following L.

24 A case could be made for adding a p.p. to this Mx, by analogy with 6, though no source transmits this. It may be that a degree of subjectivity enters the argument in deciding to what extent the subsequent groupings of notes form an obviously complete unit of major modus, in the eyes of an informed reader, before the subsequent change of mensuration after 32.

25 B imperfected by following S; i.e. by analogy with the group 213 at a lower mensural level the S at 26 cannot be counted together with its following S rest, with altered rest before following B, since this would again violate the Sixth General Rule. Instead, the B at 26 is imperfected by its preceding S rest.

368 The same argument is again presented at the next mensural level down: the second ligated S at 36 is imperfected by following M, and the S at 38 imperfected a parte ante by the M rest, rather than the M and M rest being taken together, with the rest falsely altered before following B at 39.

40 Second ligated B imperfected by following S, the p.d. attached to 41 confirming that 412 are not to be taken together, with second altered before following B at 43. Instead, the B at 43 is imperfected by its preceding S.

Example 9
5 Second of two ligated B altered before following perfect L rest, as per the second part of Tinctoris's Sixth General Rule. These two ligated B then obviously form the second complete unit of major modus with the two L rests taken together, all following the clear first unit of 13. All sources transmit these two juxtaposed L rests at this point, rather than the single one at least implicitly suggested by Tinctoris's text. There is, however, a small difficulty of interpretation for the working-out of the rest of this example, since, if these are regarded as rests proper, within the mensural context of the perfect major modus set up at the beginning, then it is impossible to resolve the example so that the final Mx remains perfect, as is the case in the vast majority of such examples in the treatises. The alternatives, then, are: (a) to regard the MSS as correct, and imperfect the final Mx a parte ante quantum ad totum, by the previous L-equivalent group at 1219; or (b) to regard the MSS as correct, but read the doubled L rest as functioning simultaneously to complete the second unit of major modus and also to indicate a change to imperfect major modus, in which case the remainder of the example can resolve correctly into units of major modus; or (c) to regard the unanimous MS reading as corrupt (the correct reading being a final L), the error being perhaps caused at some level by the synonymy of single and doubled L rests as indicators of imperfect major modus (cf. De regulari valore notarum, Ch. 8–10: Seay 1975, 129–30). Of these three alternatives (a) seems the least problematic, and the unit division of the transcription offered here follows this assumption.

6 L imperfected by following B.

10 Second of two ligated S altered before following B rest, as per Tinctoris's precept.

15 Second of two M altered before following S rest.

20 See discussion above under 5.

Example 10
34 Tinctoris's Seventh General Rule indicates that, in circumstances such as this, the ligation of these two L does not lead to the alteration of the second L before the following Mx, nor does the non-ligation of 2 imply association with, and imperfection of, the Mx at 1, but rather that the group of three L 24 should be counted together, just as if they were separate notes, as an independent Mx-equivalent group. The reading of V here, which ligates 23 instead of 34, clearly negates Tinctoris's very point.

9 An illustration of Tinctoris's further point that the same applies even if the second note (here, the second B at 9) is ligated not only to its preceding partner note, but also to the following next-larger note (here, the L at 10).

15 The same precept as 4 is illustrated at the level of tempus, the second ligated S not altered before following B, but counted with 13 and 14 as a complete unit of perfect tempus.

19 An illustration of the final statement of the Seventh General Rule, that the previously stated precept can be circumvented by the introduction of a p.d. Here, the p.d. attached to the S at 19 ensures that the first of the group of three S is counted with, and imperfects, the B at 18, and the second of the following two ligated S at 201 is altered before the following B at 22.

line 78 nisi fiat propter dicta supposita: a rare instance of Tinctoris referring to text underlay as a major factor in regulating ligature configuration.

Example 11
This example reproduces Example 10 with the ligature patterns modified according to Tinctoris's recommendations, appended as an adjunct to his Seventh General Rule. Specifically:

24 All three L are now unligated.

79 The three B become ligated, followed by separate L, rather than the first separate and last two ligated with the following L.

1315 The first two of three S become ligated, followed by one separate S.

1618 Three B become unligated; this seems to be the least necessary of Tinctoris's emendations.

Example 12
3
Second of two L altered before following Mx (perfect major modus).

4 The p.p. is added to the Mx to prevent its imperfection by following L.

6 The p.p. (NB not p.d.) is added to this Mx to indicate that the L at 5 and 7 should be counted together as a pair in syncopation, with the second altered before the following Mx at 8 in accordance with Tinctoris's Second General Rule. Although it may not be immediately apparent that the dot at 6 is not a p.d., an informed reader would presumably be expected to see readily that the groupings following 8 form complete units of perfect major modus, so that the syncopated alteration of 7 is the only feasible interpretation.

13 Second of two ligated L altered before following Mx.

Example 13
2 Second of two ligated B altered before following L (perfect minor modus).

5 Second of two ligated B altered before following L rest: see Tinctoris's Sixth General Rule.

7 p.d. added to B to ensure imperfection of preceding L, and to prevent this B from being counted together with 8 and 9, which would, in turn, prevent eventual alteration of 12.

12 Second of two ligated B (last of chain of five after previous p.d.) altered before following L.

14 p.d. added to B to ensure imperfection of preceding L, and to prevent this B from being counted together with 15 and 16, which would, in turn, prevent eventual alteration of 19.

19 Second of two ligated B (last of chain of five after previous p.d.) altered before following L rest: cf. 5 above.

213 Three ligated B between two L: this final group functions as a straightforward reminder that these are to be counted together as a complete unit of minor modus, rather than the first imperfecting the preceding L and the last being altered before the final L.

Example 14
3
Second of two S altered before following B (perfect tempus).

6 Second of two S altered before following B rest (cf. Tinctoris's Sixth General Rule above).

7 p.p. added to B to prevent its imperfection by following S.

810 As in Example 12 at 6, the p.p. (not p.d.) is attached to the B at 9 in order to prevent its imperfection either a parte ante by 8 or a parte post by 10. As a consequence, 8 and 10 should be counted together in syncopation, with 10 altered before the following B (cf. Tinctoris's Second General Rule). Again, as in Example 12, a musically literate reader of Tinctoris's time would presumably be expected to sight-read or plan ahead that the groupings starting at 11 form complete units of tempus without difficulty, illustrating that the syncopated alteration is the only possible interpretation for 810.

17 Second of two S (last in chain of five) altered before following B (extension of usage illustrated at 3).

23 Second of two S (last in chain of five) altered before following B rest (extension of usage illustrated at 6).

26 All the best MSS (Br, V, Bu) record the final note as Mx, which may indicate that this example is one in which Tinctoris is intending the single L rest placed after the tempus sign, functioning as sign of imperfect major modus (here also with imperfect minor modus), to be counted as a quasi-real rest in the calculation of major modus units: cf. commentary to Examples 5 above and 15 below. In the transcription of this example given here, the unit divisions have been indicated accordingly, with the first unit being completed at 3.

Example 15
3
Second of two M altered before following S (major prolation).

7 Second of two M altered before following S rest (cf. Tinctoris's Sixth General Rule above).

9 p.d. added to ensure imperfection of preceding S, and to prevent this M from being counted together with the following two as part of an incorrect chain of six.

14 Second of two M (last of chain of five) altered before following S.

20 Second of two M (last of chain of five) altered before following S rest.

25 p.d. added to ensure imperfection of preceding S, and to prevent this M from being counted together with the following two.

27 Second of two discrete M altered before following S.

29 p.d. added to ensure imperfection of preceding S, and to prevent alteration of 30 before following S, which would render the measure hypermetric.

31 imperfected by preceding M.

32 Note that the unanimity of the MSS that the final note is L rather than Mx suggests that the initial single L rest as sign of imperfect major (and minor) modus cannot this time be included in the unit calculation: cf. commentary notes to Examples 5 and 14 above.