1 Dr [John Wall] Callcott, , A Musical Grammar, in Four Parts, second edition (London: B. McMillan, 1809), section 417, 219Google Scholar; Weber, Gottfried, Versuch einer geordneten Theorie der Tonsezkunst zum Selbstunterricht (Mainz: Schott, 1818), volume 2, 273Google Scholar; Reicha, Antoine, Cours de composition musicale ou traité complet et raisonné d’harmonie pratique (Paris: Gambaro, no date), 151Google Scholar.

2 Callcott links this cadence explicitly to endings in church music; Reicha similarly restricts it to church music, and adds that it occurs after an ordinary perfect cadence; Weber makes no such restrictions.

3 Per contra, L. Poundie Burstein offers an illuminating reflection on the first of these dichotomies in his article ‘The Half Cadence and Other Such Slippery Events’, Music Theory Spectrum 36/2 (2014), 203–227.

4 The term plagius, and its correlate autenticus, are both Greek in origin, which doubtless means that the terms and concepts were borrowed from the Eastern Church's octoechos tradition. For a thorough list of treatments of the octoechos system see Cohen, David E., ‘Notes, Scales, and Modes in the Earlier Middle Ages’, in The Cambridge History of Western Music Theory, ed. Christensen, Thomas (Cambridge: Cambridge University Press, 2002), 310Google Scholar.

5 One such example may be found in Guido d’Arezzo, Micrologus, ed. Smits van Waesberghe, Joseph as Corpus scriptorum de musica, volume 4 (Rome: American Institute of Musicology, 1955), chapter 13, 155–156Google Scholar. On the earliest medieval conceptions of these terms see Cohen, ‘Notes, Scales, and Modes’, 311.

6 Early medieval discussions of ambitus were often rather ad hoc; a systematic theory of modal ambitus as conjunctions of fourths and fifths was developed in the eleventh century by Berno of Reichenau and his followers. Cohen, ‘Notes, Scales, and Modes’, 351–354.

7 Tinctoris, Johannes, Liber de natura et proprietate tonorum, ed. Seay, Albert in Corpus scriptorum de musica, volume 22–1 (Rome: American Institute of Musicology, 1975–1978), chapter 14, 85Google Scholar; Glareanus, Henricus, Dodecachordon (Basel: Henrichus Petrus, 1547), book 2, chapters 6–7, 75–83Google Scholar.

8 The earliest witness to this association is a text not mentioned by Zarlino, the anonymous theoretical compilation of the ninth and tenth centuries known as Alia musica; Cohen, ‘Notes, Scales, and Modes’, 337–338.

9 And directly analogous to this, of course, is the association of authentic modes’ fourth atop a fifth and the harmonic mean of the duple ratio. These mappings of plagal mode to arithmetic mean and authentic mode to harmonic mean assume that quantities in question refer to string lengths, as was usually the case in the canonist tradition stretching from antiquity to the Renaissance. Were the pitches’ frequencies to be considered instead, the contrary mappings would obtain, because of a reciprocal relationship between string length and pitch level already observed by the first century A. D. (Thrasyllus, according to Theon of Smyrna, De utilitate mathematicae, ed. Hiller, Eduard as Theonis Smyrnaei philosophi Platonici expositio (Leipzig: Teubner, 1878), 87Google Scholar; trans. Barker, Andrew in Greek Musical Writings, volume 2, Harmonic and Acoustic Theory (Cambridge: Cambridge University Press, 1989), 226)Google Scholar.

10 Caplin, William E. compellingly argues in favour of this distinction in ‘The Classical Cadence: Conceptions and Misconceptions’, Journal of the American Musicological Society 57/1 (2004), 81–83CrossRefGoogle Scholar.

11 For example, in Callcott's definition of the plagal cadence quoted above, he calls it simply a chord progression from subdominant to tonic, and only at the very beginning of his chapter on cadences does he acknowledge that ‘a cadence in harmony. . .is used to terminate the Sections and Periods of Musical Rhythm [that is, form]’ (Callcott, A Musical Grammar, 216).

12 For one of the few accounts of Blainville's theoretical contributions see Martin, Nathan John, ‘Rameau's Changing Views on Supposition and Suspension’, Journal of Music Theory 56/2 (2012), 121–167CrossRefGoogle Scholar.

13 The simphonie is scored for five instruments, and consists of three movements comprising 344 bars in total. Blainville occasionally writes two melodic lines in the lowest staff. In these cases he indicates that the upper should be played by the bassoon; there are no other instrumental indications. The uppermost two staves are notated in treble clef, the third in alto clef, and the bottom two switch between bass and tenor clefs.

14 de Blainville, Charles-Henri, Essay sur un troisieme mode presenté et aprouvé par M^rs. de l’Academie des Sciences, joint la simphonie executée au concert du Chateau des Thuilleries 30. May 1751 (Paris: Ballard, 1751)Google Scholar; facsimile of Essay in Basse continue: France 1600–1800, ed. Jean Saint-Arroman, volume 4 (Courlay: Fuzeau, 2006), 22–25 (page references are to original); simphonie ed. as Simphonie dans un troisième mode in Cahiers de musique 122 (Versailles: Centre de Musique Baroque de Versailles, 2005). Barry S. Brook states that the publication of the symphony and essay occurred in the month of October (La symphonie française dans la seconde moitié du XVIII^e siècle, volume 1 (Paris: Institute de Musicologie de l’Université de Paris, 1962), 130).

15 The most informative testimony about that concert we have is a terse summary published the following month in the Mercure de France, stating that the performance ‘began with a symphony by M. Blainville in a new style [genre] of modulation, to attempt a third mode’; ‘Concerts spirituels’, Mercure de France, June 1751, 173.

16 Blainville, Essay, 1.

17 Blainville, Essay, 2, 4–5.

18 Blainville, Essay, 2.

19 Blainville, Essay, 7. In positing such reciprocity for the melody and the fundamental bass, Blainville is, of course, departing from Rameau's conception that the basse fondamentale gives rise to the melody, and he may be confounding the basse fondamentale with the basse continuë.

20 Nivers, Guillaume Gabriel, Traité de la composition de musique (Paris: Ballard, 1667), 18–19Google Scholar; de Brossard, Sébastien, Dictionaire de musique (Paris: Ballard, 1703), entry for ‘tuono’Google Scholar.

21 The pioneering study of the ‘church keys’, therein called ‘“pitch-key” modes’, was Atcherson, Walter's ‘Key and Mode in Seventeenth-Century Music Theory Books’, Journal of Music Theory 17/2 (1973), 204–232CrossRefGoogle Scholar. See also Lester, Joel, Between Modes and Keys: German Theory 1592–1802 (Stuyvesant: Pendragon, 1989)Google Scholar; Powers, Harold S., ‘From Psalmody to Tonality’, in Tonal Structures in Early Music, ed. Judd, Cristle Collins (New York: Garland, 1998), 275–339Google Scholar; Barnett, Gregory, ‘Modal Theory, Church Keys, and the Sonata at the End of the Seventeenth Century’, Journal of the American Musicological Society 51/2 (1998), 245–281CrossRefGoogle Scholar; and Michael Robert Dodds, ‘The Baroque Church Tones in Theory and Practice’ (PhD dissertation, University of Rochester, 1998).

22 Nivers, for example, provides musical examples of each tone showing its key notes, and the fourth tone's notes are only E, A and C; Nivers, Traité de la composition, 18–19.

23 Blainville, Essay, 2.

24 Jean-Jacques Rousseau, ‘Lettre de M. Rousseau de Genève, à M. l’Abbé Raynal’, Mercure de France, June 1751, 175.

25 Rousseau, ‘Lettre de M. Rousseau’, 175.

26 Rousseau, ‘Lettre de M. Rousseau’, 176.

27 Blainville's discussion of this issue curiously addresses the conclusion of ‘the octave of his scale’, a musical entity that is at best monophonic, rather than of compositions in this third mode; Blainville, Essay, 4.

28 Blainville, Essay, 4.

29 See Harrison, Daniel, Harmonic Function in Chromatic Music: A Renewed Dualist Theory and an Account of Its Precedents (Chicago: University of Chicago Press, 1994), 26–27Google Scholar.

30 Blainville, Essay, 5.

31 Blainville himself had proposed a cadence with this name in his earlier treatise, but in that work he defined it as the motion from the tonic to the dominant; de Blainville, Charles-Henri, Harmonie theorico-pratique, divisee en six parties, second edition (Paris: Ballard, 1752), 5Google Scholar.

32 Blainville, Essay, 5.

33 Blainville, ‘Observation de M. de Blainville, sur la Lettre de M***, inserée dans le Mercure de mois de Septembre, pag.’, Mercure de France, November 1751, 120–124.

34 Blainville, Essay, 6.

35 Wollick, Nicolaus, Opus aureum musicae (Cologne: Henricus Quentel, 1501), f. H4vGoogle Scholar; Printz, Wolfgang Caspar, Phrynis oder satyrischer Componist (Quedlinburg: Christian Okels, 1676), ffGoogle Scholar. C2r, D1v. For other examples one can also consult two useful articles on the history of the cadence by Siegfried Schmalzriedt, with the proviso that the treatises he cites as examples are rarely the earliest or most pertinent sources (Schmalzriedt, ‘Clausula’ and ‘Kadenz’, in Handwörterbuch der musikalischen Terminologie, ed. Hans Heinrich Eggebrecht (Wiesbaden: Steiner, 1972–)).

36 Nivers, Traité de la composition, 24; Masson, Charles, Nouveau traité des regles pour la composition de la musique (Paris: Ballard, 1705), 54Google Scholar.

37 Rameau, Jean-Philippe, Traité de l’harmonie reduite à ses principes naturels (Paris: Ballard, 1722), book 2, chapter 7, 64–67Google Scholar; Rameau, , Code de musique pratique (Paris: Imprimerie Royale, 1760), 85Google Scholar.

38 Rameau, Jean-Philippe, Génération harmonique (Paris: Prault fils, 1737), chapter 6, 72Google Scholar; Rameau, , Observations sur notre instinct pour la musique (Paris: Prault fils, 1754), 48–49nGoogle Scholar; Christensen, Thomas, Rameau and Musical Thought in the Enlightenment (Cambridge: Cambridge University Press, 1993), 185–190Google Scholar.

39 The elegy delivered for Fouchy makes no mention of music (Condorcet, Nicolas de, ‘Eloge de M. de Fouchy’, in Éloges des académiciens de l’Académie royale des sciences, volume 4 (Brunswick and Paris: Vieweg and Fuchs, 1799), 336–364Google Scholar), and his signature may be on the report ex officio. Concerning Dortous de Mairan's musical predilections see Christensen, Rameau and Musical Thought, 139–141.

40 Blainville, Essay, 8.

41 In their discussion of the mode mixte, these authors also cite the works of Locatelli as an example of the mode's characteristic minor semitone (Blainville, Essay, 8). This intriguing connection most probably refers to a minor-mode idiom Locatelli frequently employs, involving a progression in which, after the final perfect cadence on the tonic, the bass descends by step to the fifth scale degree. Once the bass arrives on that note, there is a pause indicated by a fermata, and then the next movement begins. Thus, such movements conclude with a descending semitone in the lowest voice, a characteristic that evokes the scale of the troisiéme mode.

42 Blainville, Essay, 5.

43 It is, of course, possible that performances of these motets in Blainville's day differed from the manuscript scores, and that musicians often added ‘Amen’ cadences at the end of the motets. Yet in 1743 the theorist Charles Levens claimed that the cadence imparfaite (from tonic to dominant or subdominant to tonic) had formerly been used by ‘the ancients’ to end pieces, but was no longer used in this way (Abregé des regles de l’harmonie, pour aprendre la composition (Bourdeaux: J. Chappuis, 1743), 51–52).

44 This movement of Lalande's motet is written for five vocal parts in open score. No information is provided about how these vocal lines should be assigned to voice types.

45 Michel Richard de Lalande, ‘Dixit Dominus’, in Bibliothèque municipal de Versailles, Ms mus. 11, 94 <http://conquest.imslp.info/files/imglnks/usimg/8/8b/IMSLP58076-PMLP117031-Delalande_-_Motets_-_Tome_4.pdf> (3 November 2014). In this case, the major mode of the key means that the plagal cadence features only one descending semitone, rather than the two found in the cadence plagale proper to Blainville's third mode.

46 As Lionel Sawkins has shown, the motets of Lalande were a mainstay of the Parisian concerts spirituels throughout the middle of the eighteenth century, and the Mercure de France records that Lalande's motet ‘Cantate Domino’ was performed in the concert at which Blainville's simphonie made its debut. Sawkins, ‘Lalande and the Concert Spirituel’, The Musical Times 116 (April 1975), 334; ‘Concerts spirituels’, Mercure de France, June 1751, 173.

47 Choron, Alexandre, Principes de composition des ecoles d’Italie, volume 1 (Paris: Auguste Le Duc, 1808), 101Google Scholar.

48 de Blainville, Charles-Henri, Histoire générale, critique et philologique de la musique (Paris: Pissot, 1767), 75Google Scholar. My italics in the translation.

49 As Giovanni Guanti has pointed out, Tartini was educated within the neoplatonist Franciscan circles of the Veneto region, and his final treatise, the Scienza platonica fondata nel cerchio, engages extensively, and idiosyncratically, with the Platonist tradition. Guanti, , ‘La natura nel sogno platonizzante di Giuseppe Tartini’, in Tartini ‘maestro’ narodov in kulturno življenje v obalnih mestih današnje Slovenije med 16. in 18. stoletjem, ed. Kokole, Metoda (Ljubljana: Zal. ZRC, 2002), 60–67Google Scholar; Guanti, , ‘Giuseppe Tartini lettore di Platone’, in Florilegium musicae: studi in onore di Carolyn Gianturco, volume 2, ed. Radicchi, Patrizia and Burden, Michael (Pisa: ETS, 2004), 603–619Google Scholar. Concerning the musica speculativa tradition see, for example, Barker, Andrew, The Science of Harmonics in Classical Greece (Cambridge: Cambridge University Press, 2007); 263–411CrossRefGoogle Scholar; Meyer, Christian, Mensura monochordi: la division du monocorde, IXe–XVe siècles (Paris: Klincksieck, 1996)Google Scholar; and Gouk, Penelope, ‘The Role of Harmonics in the Scientific Revolution’, in The Cambridge History of Western Music Theory, ed. Christensen, Thomas (Cambridge: Cambridge University Press, 2002), 223–245CrossRefGoogle Scholar. One of the last important instances of a musica speculativa-influenced treatise before Tartini was Leonhard Euler's Tentamen novae theoriae musicae (St Petersburg: Typographia Academiae Scientiarum, 1639).

50 Tartini may be adopting these two cadential terms from a now obscure Italian tradition, since they had previously been employed in a seventeenth-century manuscript treatise by Christoph Bernhard, who, as Walter Hilse points out, made several trips to Italy in his lifetime. Hilse, Preface to ‘The Treatises of Christoph Bernhard’, Music Forum, ed. William J. Mitchell and Felix Salzer, volume 3 (New York: Columbia University Press, 1973), 2; for Bernhard's use of the terms see Tractatus compositionis augmentatus, trans. Walter Hilse in ‘The Treatises of Christoph Bernhard’, 67–68. In the mid-seventeenth century Jean Denis had also drawn a suggestive connection between an arithmetically divided modal octave on E and the impossibility of a perfect cadence in that situation; Denis, , Traité de l’accord de l’espinette (Paris: Ballard, 1650), 27–33Google Scholar.

51 Tartini, Giuseppi, Trattato di musica secondo la vera scienza dell’ armonia (Padua: Stamperia del Seminario, Giovanni Manfrè, 1754), 102–103Google Scholar; translation adapted from Fredric Johnson, ‘Tartini's Trattato di musica seconda la vera scienza dell’ armonia: An Annotated Translation with Commentary’ (PhD dissertation, Indiana University, 1985), 263–264.

52 In addition to the passage quoted above, Tartini also glosses cadenza aritmetica this way one other time; see the Trattato di musica, 137. In a letter to Padre Martini, however, Tartini reserves the arithmetic/harmonic terms for octave divisions alone, and refers to the cadences solely as ‘plagale’ and ‘autentica’; Giuseppe Tartini, Padua, to P. G. B. Martini, Bologna, 14 April 1752, ed. in Martini, Giovanni Battista, Carteggio inedito del P. Giambattista Martini, volume 1 (Bologna: Forni, 1969), 351–352Google Scholar.

53 Pierluigi Petrobelli, ‘Tartini, Giuseppe’, in Grove Music Online <www.oxfordmusiconline.com> (27 April 2014).

54 Petrobelli, Pierluigi, Giuseppe Tartini: le fonti biografiche (Florence: Universal Edition, 1968), 112Google Scholar.

55 Jean-Adam Serre, a Swiss theorist who was based in Paris from 1751 to 1756 and engaged extensively with Tartini's work, claimed not to have encountered Tartini's work until a trip to London in 1756, when the Trattato was given to him by an Englishman who had recently arrived from Italy; Serre, Jean-Adam, ‘Lettre aux auteurs de ce journal’, Journal encyclopédie 3/1 (April 1769), 132Google Scholar. Rousseau did not begin to engage with Tartini's work until about 1755, and Diderot's discussion of the Trattato was first published in 1757; see Boccadoro, Brenno, ‘Tartini, Rousseau et les lumières’, in Jean-Jacques Rousseau, Oeuvres complètes, ed. Gagnebin, Bernard and Raymond, Marcel, volume 5 (Paris: Gallimard, 1995), 1695, 1698Google Scholar. Additionally, in his Histoire of 1767 Blainville summarized the key points of an extract from the Trattato that he had received, which demonstrates that he thought that Tartini's ideas could still be unfamiliar to his readers; Histoire générale, 180–185.

56 Serre, Jean-Adam, Observations sur les principes de l’harmonie (Geneva: Henri-Albert Gosse et Jean Gosse, 1763), 144Google Scholar; Rousseau, Jean-Jacques, Dictionnaire de musique (Paris: Duchesne, 1768), 490, entry for ‘systeme’Google Scholar.

57 These descriptions are limited to Rousseau's commentary in his Dictionnaire (291, entry for ‘mode’), and a brief cribbing of Dortous de Mairan and Fouchy's extrait found in the Encyclopédie's volume of musical plates; Recueil de planches, volume 7 (Paris, Briasson, 1769), 18, entry for ‘musique’.

58 While it is not impossible that Tartini adopted the association of plagality and the cadence from Blainville, his independent justification and the obscure nature of Blainville's work make this highly unlikely.

59 One cannot disprove the possibility that Blainville adopted the term cadence plagale from French oral tradition, or some other yet more obscure source. Yet if this were the case, one could reasonably expect other French authors in the following years to use the term in contexts other than discussions of Blainville's third mode, which, as we have just seen, was not the case.

60 Rameau, Jean-Philippe, Nouveau systême de musique theorique (Paris: Ballard, 1726), 41Google Scholar.

61 [Odo,] Dialogus, ed. Gerbert, M. in Scriptores ecclesiastici de musica sacra potissimum, volume 1 (St Blaise: Typis San-Blasianis, 1784), 260–262Google Scholar.

62 Wollick, Opus aureum musicae, f. E1v.

63 de Menehou, Michel, Nouvelle Instruction Familière (Paris: Nicolas du Chemin, 1558), chapter 15, f. B3rGoogle Scholar. In his letter in the Mercure, Rousseau wrote of the mixed mode that its ‘fourth [degree] will be called, if one likes, dominant’, a description that is more historically apt than he probably knew (Rousseau, ‘Lettre de M. Rousseau’, 175).

64 Blainville, Histoire générale, 70.

65 Rousseau, Dictionnaire de musique, 518, entry for ‘tons de l’église’.

66 For an early instance of the species-based theory of modes see Berno of Reichenau, Bernonis Augiensis Abbatis De arte musica disputationes traditae: De mensurando monochordo, ed. Joseph Smits van Waesberghe (Buren: Knuf, 1978).

67 My transcription amends two typographical errors in Blainville's original: the authentic Phrygian progression (third mode) showed a mediating A instead of the proper B, and the notes in the first row (‘Two Hypodorian’) were all a third too high.

68 Blainville, Histoire générale, 126–127.

69 Jean Lebeuf's influential treatise on the history and practice of plainchant, for example, largely propagated traditional modal theory (Lebeuf, Traité historique et pratique sur le chant ecclesiastique (Paris: Herissant, 1741)).

70 Yolande de Brossard claims that it is the first French music dictionary in ‘Brossard, Sébastien de’, in Grove Music Online <www.oxfordmusiconline.com> (28 April 2014). For its publication history see Damschroder, David and Williams, David Russell, Music Theory from Zarlino to Schenker: A Bibliography and Guide (Stuyvesant: Pendragon, 1990), 38Google Scholar.

71 Concerning Rousseau's use of Brossard see de Brossard, Yolande, Sebastien de Brossard: theoricien et compositeur 1655–1730 (Paris: Éditions Picard, 1987), 55–56Google Scholar. Blainville cites Brossard in his ‘Dessertation [sic] où l’on examine les droits de la mélodie & de l’harmonie’, Mercure de France, May 1752, 146.

72 Brossard, Dictionaire de musique, entry for ‘tuono’, section 2.

73 Tinctoris, Liber de natura, chapter 44, 97–98; Zarlino, Gioseffo, Le istitutioni harmoniche (Venice, 1558), book 3, chapter 59, 243Google Scholar.

74 Brossard, Dictionaire de musique, entry for ‘finale’.

75 Brossard's assertion that authentic compositions end with one cadence type and plagal compositions with another is not true of musical practice; see, for instance, Ritter, A. G., Zur Geschichte des Orgelspiels, volume 2 (Leipzig: Max Hesse, 1884)Google Scholar, which contains a useful collection of modally identified organ compositions from a range of countries and centuries. Brossard may have come up with this claim by drawing upon the work of Nivers, whose treatise Brossard described as his composition primer; de Brossard, Sébastien, Catalogue des livres de musique, théorique et prattique, vocalle et instrumentalle, ed. de Brossard, Yolande as La Collection Sébastien de Brossard, 1655–1730: catalogue [Département de la musique, Rés. Vm.⁸ 20] (Paris: Bibliothèque Nationale de France, 1994), 69Google Scholar. Nivers had set forth two cadences: the perfect, which ends with a descending fifth (or a major sixth expanding to an octave), and the imperfect, which concludes with a descending fourth (Nivers, Traité de musique, 23–24). Brossard may have been drawing upon these cadential types to make his assertion, a possibility that is suggested by his definition of ‘finale’, quoted above. Therein Brossard takes the highly unconventional step of overlaying the distinction between authentic and plagal modes with the (perhaps cadence-borrowed) binary of perfect versus imperfect precisely when he describes the cadential endings of authentic and plagal pieces (Brossard, Dictionaire de musique, entry for ‘finale’).

76 Brossard, Dictionaire de musique, entry for ‘tuono’, section 2.

77 Brossard, Dictionaire de musique, entry for ‘modo’.

78 Rousseau, ‘Lettre de M. Rousseau de Genève’, 176; Blainville, Essay, 2.

79 Rousseau, ‘Lettre de M. Rousseau de Genève’, 175–176; Philaetius [Jean-Adam Serre], ‘A l’auteur du Mercure, sur la nature d’un mode en e-si-mi naturel’, Mercure de France, September 1751, 167–169.

80 Rousseau, Dictionnaire, 292.

81 Bouvier, Xavier, ‘Réponse à la critique de Rameau des basses de Corelli: un manuscrit inédit de Charles-Henri Blainville (1711–ca 1777)’, Schweizer Jahrbuch für Musikwissenschaft 18 (1998), 185Google Scholar.

82 In 1809 Callcott, one of the earliest writers to mention an ‘authentic cadence’, dismissed the term as a synonym for the perfect cadence, noting that ‘[it] is only so termed in contradistinction to the Plagal’ (A Musical Grammar, section 418, 220). The cause or causes of the nineteenth-century resurgence of the term ‘plagal cadence’ and the rise of its conceptual mate ‘authentic cadence’ are beyond the scope of this article, and could merit a separate study.