The author thanks Peter Westergaard and Elaine Sisman for their comments on earlier drafts of this articleGoogle Scholar

1 See Charles-Simon Catel, Traité d'harmonie par Catel, membre du Conservatoire de Musique, adopté par le Conservatoire pour servir à l'étude dans cet établissement (Paris, 1801), Préface, ‘Arrêtés relatifs à l'adoption du Traité d'harmonie’Google Scholar

2 Jean Mongrédien, La musique en France des lumières au romantisme (1789–1830) (Paris, 1986), 23 ‘Les principes définis par Catel dans son Traité sont encore, sur ce point, ceux sur lesquels repose l'essentiel de l'enseignement de l'harmonie aujourd'hui.’ Fétis viewed the adoption of Catel's system as a veritable overthrow of Rameau's system See François-Joseph Fétis, Henri-Montan Berton, Pierre-Marie-François-de-Sales Baillot, Hommage à la mémoire de Catel, Membre de l'Institut, Chevalier de la Légion d'Honneur, Ancien Inspecteur des Etudes au Conservatoire de Musique, décédé à Paris le 29 novembre 1830 Extrait de la Revue musicale, tome 10 (Paris, [1830]), 13Google Scholar

3 This reference to Rameau appeared in a eulogy published in the Mercure de France (October 1764), 194–5.Google Scholar

4 This view is articulated, for example, by William J Mitchell in ‘Chord and Context in Eighteenth-Century Theory’, Journal of the American Musicological Society, 16 (1963), 221–39.Google Scholar

5 Vandermonde is better remembered for his contributions in the fields of mathematics and engineering See Hecht, Jacqueline, ‘Un exemple de multidisciplínarité Alexandre Vandermonde (1735–1796)’, Population, 26 (1971), 641–75, Phillip S Jones, ‘Vandermonde, Alexandre-Théophile’, Dictionary of Scientific Biography (New York, 1976), xiii, 571–2, Henri Lebesque, ‘L'oeuvre mathématique de Vandermonde’, Enseignement mathématique, 2nd ser, 1 (1955), 203–23, and Arthur Birembaut, ‘Précisions sur la biographie du mathématicien Vandermonde et de sa famille’, 72^e Congrès de l'Association Française pour l'Avancement des Sciences (Paris, 1953), 530–3CrossRefGoogle Scholar

6 ‘Système d'harmonie applicable à l'état actuel de la musique’, Journal des savants, Paris edn (December 1778), 855–62 The paper was also printed independently as a brochure (Paris, Bibliothèque Nationale, Mus recueil 217(5)) Further citations are abbreviated as ‘Système’, page numbers refer to the brochureGoogle Scholar

7 Jean-Benjamin de Laborde, Essai sur la musique ancienne et moderne, 4 vols (Paris, 1780, repr New York, 1978)Google Scholar

8 ‘Second mémoire sur un nouveau système d'harmonie applicable à l'état actuel de la musique’, Journal des savants, Paris edn (January 1781), 32–41, (February 1781), 93–101 The two sections of the ‘Mémoire’ were also printed independently as a single brochure (Paris, Bibliothèque Nationale, Imp 4°B, pièce 305) This brochure, like the brochure of the ‘Système’, contains the same text as the Journal des savants articles. Further citations are abbreviated as ‘Mémoire’, page numbers refer to the brochureGoogle Scholar

9 The Complete Theoretical Works of Jean-Philippe Rameau (1683–1764), ed Erwin R Jacobi (Rome, 1967–72), vi, p. xxxi, n. 11.Google Scholar

10 It is Balthazar, Romeo's manservant, who delivers the false news of Juliet's death to his master and who is entrusted with Romeo's suicide noteGoogle Scholar

11 There is no biographical information on Vandermonde that describes the extent of his musical training Lebesque offers that Vandermonde's father encouraged him to study music as a child because of his poor health Hecht refers to Birembaut, who mentions the inventory made upon Vandermonde's death The inventory includes two violins, and leads Hecht to the conclusion that Vandermonde was a proficient violinist He could sightread music, as indicated by his participation in a compositional contest that took place in the salon of Pierre-Joseph Roussier on 16 July 1781, reported in Journal de Paris, 199 (18 July 1781), 801–2Google Scholar

12 Cohen, Albert, Music in the French Royal Academy of Sciences (Princeton, 1981) The extent of this reorientation is supported by the 1795 reorganization of the Académie Royale des Sciences into the Institut National des Sciences et des Arts In the restructured Institut, music was assigned to the division on literature and fine arts, rather than to the division on physical and mathematical sciences See esp pp 97–8 and 112Google Scholar

13 Engraved plates did occasionally appear with articles on various non-musical topics In the second half of the eighteenth century, Vandermonde's system was the only article on music that included engraved plates Plates of musical examples or diagrams required entirely different techniques from text-typesetting methods and added expense to the printing and compilation of a volume See George J Buelow, ‘Printing and Publishing of Music’, The New Grove Dictionary of Music and Musicians (London, 1980), xv, 232–74, esp p 253Google Scholar

14 For general studies of the Journal, see Birn, Raymond, ‘Le Journal des savants sous l'Ancien régime’, Journal des savants (January-March 1965), 15–35, Cyril B O'Keefe, Contemporary Reactions to the Enlightenment (1728–1762) A Study of Three Critical Journals The Jesuit ‘Journal de Trévoux’, the Jansenist ‘Nouvelles ecclésiastiques’, and the Secular Journal des Savants’ (Geneva, 1974), and Jean Ehrard and Jacques Roger, ‘Deux périodiques français du 18^e siècle Le Journal des savants et les Mémoires de Trévoux’, Livre et société dans la France du XVIII^e stècle, ed François Furet (Paris, 1965), 33–59 Ehrard and Roger provide a quantitative study of the overall contents of the Journal des savants in which they categorize articles by subject-matter theology, law, history, sciences and arts, and belles-lettres Unfortunately, it is impossible to discern where they place music-related articlesGoogle Scholar

15 In addition to reviews of these theoretical systems, pedagogical music methods and works on music history and aesthetics were occasionally reviewed Generally, methods reviewed by the Journal were those that had attracted some public attention, such as those of Antoine-Joseph Dumas, Joseph Lacassagne and Anton Bemetzrieder These methods are all somewhat unusual in comparison with the myriad of music methods circulating in the French language For example, Dumas's method was an exposition of his novel system for children's education applied to the teaching of the rudiments of music As for Lacassagne and Bemetzrieder, they had both presented their works to the Académie for approval, in 1766 and 1781 respectively (Cohen, Music, 101 and 105) Furthermore, these methods received considerable attention in other contemporary periodicals Historical studies reviewed by the Journal included Gerbert's De cantu et musica sacra (St Blasien, 1774) and Laborde's Essai. Rousseau's Dictionnaire de musique (Paris, 1768) commanded a two-part article Reviews of aesthetic writings included Chabanon's Observations sur la musique et principalement sur la métaphysique de l'art (Paris, 1779) and La Cépède's La poétique de la musique (Paris, 1785)Google Scholar

16 The Académie's official account of Vandermonde's papers appears in the Histoire de l'Académie Royale des Sciences, année 1778, avec les mémoires de mathématique et de physique tirés des registres de cette Académie (Paris, 1781), 51–5 The report for the year 1778 appeared in 1781, and thus Vandermonde's two presentations are conflated into a single accountGoogle Scholar

17 The Journal’s editorial boards had consisted almost exclusively of academicians since 1702 See Hahn, Roger, The Anatomy of a Scientific Institution The Paris Academy of Sciences, 1666–1803 (Berkeley, 1971), 64, and Ehrard and Roger, ‘Deux périodiques français’, 37Google Scholar

18 Vandermonde sat on 35 committees between 1773 and 1792 Only Jean-Paul Grandjean de Fouchy (49 committees) and Jean-Jacques Dortous de Mairan (38 committees) were assigned more frequently than Vandermonde to committees reviewing music proposals (Cohen, Music, 117–19)Google Scholar

19 Vandermonde presented his system initially on 8 August 1778 and again, after the public assembly of 14 November 1778, on 19 December 1778 (Cohen, Music, 97, notes 87 and 88)Google Scholar

20 Hahn, referring to the semi-annual assemblies, states ‘In Old Regime France, power and prestige mutually supported each other to a point where they were often inseparable Both were underscored by public ceremonial events, the symbolic meaning of which could hardly escape the public eye The Academy, no less than other corps d'état, developed its share of luster and pomp One occasion for display was the semiannual public meetings held at the Louvre’ (The Anatomy of a Scientific Institution, 73)Google Scholar

21 Vandermonde's ‘Système’ probably had a particular appeal to fellow members of the Académie In the first paper, Vandermonde proposed that the harmonica – an invention of another Academician, Benjamin Franklin – was the ideal instrument for teaching his system Franklin became an associate member of the Académie in 1772 and a translation of his works was presented to the Académie in 1773 (Cohen, Music, 65). Vandermonde, admitted to the Académie in 1771, would have had access to these works.Google Scholar

22 Mercure de France (15 February 1779), 135–44Google Scholar

23 Ehrard and Roger, ‘Deux périodiques français’, 35–6Google Scholar

24 One of the duties of the Secretary of the Académie was to provide a report of the public assemblies, but these reports were not generally printed in the Mercure Furthermore, d'Alembert did not hold this post See Bertrand, Joseph, L'Académie des Sciences et les Académiciens de 1666 à 1793 (Paris, 1869; repr Amsterdam, 1969), 207Google Scholar

25 Mercure de France (15 February 1779), 142–3 ‘Les Philosophes et les Gens de l'Art ne pourront bien juger du nouveau système, que lorsque cet Académicien l'aura donné au Public avec le détail nécessaire pour le développer et l'appuyer dans toutes ses parties M de Vandermonde est d'ailleurs trop éclairé, et se connaît trop bien en démonstration, pour croire qu'on puisse jamais parvenir à rien démontrer sur cette matière ’ French spelling has been modernized throughout this articleGoogle Scholar

26 D'Alembert's criticism appears in the preliminary comments to the 1762 edition of his Eléments de musique théorique et pratique, suivant les principes de M Rameau, éclaircis, développés et simplifiés Nouvelle édition, revue, corrigée et considérablement augmentée (Lyon, 1762, repr of 1779 edn Paris, 1984). Thomas Christensen details the ideological rupture between Rameau and d'Alembert in ‘Science and Music Theory in the Enlightenment D'Alembert's Critique of Rameau’ (Ph D dissertation, Yale University, 1985), 41–6.Google Scholar

27 Vandermonde's association with d'Alembert had begun even before his admittance to the Académie. He began frequenting the circle of the Encyclopedists, notably d'Alembert and Denis Diderot, as early as 1762 (Hecht, ‘Un exemple de multidisciphnarité’, 642)Google Scholar

28 ‘Système’, I ‘Cet homme célèbre [d'Alembert] qui n'a pas dédaigné d'associer ses idées à celles de Rameau, et qui, en exposant celles-ci leur a donné de l'ordre et de la suite, s'est toujours refusé hautement à l'emploi du mot de démonstration dont Rameau décorait mal-à-propos des raisonnements ingénieux sans doute, mais peu solides. Aussi personne n'est-il plus disposé que M d'Alembert à tolérer toutes les opinions en Musique‘Google Scholar

29 Laborde acknowledged Roussier's aid in two instances (1, 128, iii, 678)Google Scholar

30 Fétis, given his inveterate disapproval of Rameau's theories, was predictably outraged by the Essai‘s underlying homage to Rameau ‘Ce livre est un chef-d'oeuvre d'ignorance, de désordre et d'incurie’ (Biographie universelle des musiciens, 2nd edn, Paris, 1868–70, s v ‘Laborde‘)Google Scholar

31 The article on the theorist Jean-François Demoz also elicited two engraved plates (iii, 611–12) Diagrams were necessary to depict Demoz's unique system of musical notation published under the title Méthode de musique selon un nouveau système (Paris, 1728)Google Scholar

32 Laborde, Essai, iii, 690 ‘Vandermonde (M) de l'Académie Royale des Sciences a fait paraître dans le Journal des Savants, un Mémoire touchant un Système d'harmonie applicable à l'état actuel de la Musique ‘Google Scholar

33 A privilège was the legal permission to print a work in the Old Regime Dates for the privilèges of musical works are given by Michel Brenet, ‘La librairie musicale en France de 1653 à 1790 d'après les registres de privilèges’, Sammelbande der Internationalen Musikgesellschaft (1906–7), 401–66Google Scholar

34 Journal des savants (September 1780), 586–93 Emil Haraszti suggested that the Essai was already in print in the autumn of 1778 and was definitely in circulation in 1779. He was attempting, however, to prove a rather tenuous point about Laborde's sources (‘Jean-Benjamin de Laborde et la musique hongroise’, Revue de musicologie, 54 (1935), 100–7; 55 (1935), 168–78) The Essai may have been circulating in some form by late 1779, but could not have been entirely completed until after Vandermonde's presentation at the Académie on 14 November 1778 and the subsequent article in the Journal des savants in December 1778, because Laborde specifically referred to Vandermonde's paper as it appeared in the Journal Announcements of the Essai's availability did not appear until April and June 1780 (Mercure de France, 15 April 1780, 149, Journal des savants, June 1780, 445). Reviews of the Essai began surfacing in journals other than the Journal des savants after April 1780 (see Journal de Paris, 103 (12 April 1780), 425–7, 105 (14 April 1780), 433–5, Mercure de France, 6 May 1780, 14–36, Journal encyclopédique, 4 (June 1780), 276–95, 450–68; 5 (July 1780), 78–91, 285–303).Google Scholar

35 ‘Mémoire’, 2. Vandermonde stated that work on his system had been interrupted by his ‘proper occupations’, but the commentary that appeared in Laborde's Essai had reawakened his musical interestsGoogle Scholar

36 Our present-day conception of ‘nature’ does not have the same all-encompassing philosophical basis as did the eighteenth-century conception which implied that natural phenomena provided a systematic and unifying foundation for scientific inquiry See, for example, the article ‘Nature’ in the Encyclopédie ou dictionnaire raisonné des sciences, des arts et des métiers, ed. Denis Diderot and Jean Le Rond d'Alembert (Paris, 1751–72). One also might consult Jean Ehrard, L'idée de nature en France à l'aube des lumières (Paris, 1970), and Michel Baridon, ‘Le concept de nature dans l'esthétique de Rameau’, Jean-Philippe Rameau Colloque international organisé par la Société Rameau, Dijon, 21–24 septembre 1983, ed Jérôme de la Gorce (Paris and Geneva, 1987), 445–59Google Scholar

37 ‘Système’, 2. ‘On met en principe que notre Musique doit être puisée dans la Nature Elle l'est sans doute, si on prend cette expression dans un sens très-général; mais dans ce même sens notre Langue Française l'est aussi et il n'est guère moins impracticable, selon moi, de déduire de ce principe une bonne théorie de notre Musique que d'en déduire une bonne Grammaire de notre Langue Aussi les Systèmes de Rameau et de Tartini sont-lis inconciliables, quoique dérivés d'une même source.‘Google Scholar

38 Laborde, Essai, iii, 691, n (a)Google Scholar

39 Rameau's insistence on the corps sonore as the scientific basis for his system does, in fact, bias many of his speculative and practical propositions See Christensen, Thomas, ‘Rameau's “L'art de la basse fondamentale”’, Music Theory Spectrum, 9 (1987), 18–41; 12 (1990), 276–7Google Scholar

40 ‘Mémoire’, 1Google Scholar

41 The judgment of the ear was crucial to Rameau's theory, but Vandermonde must have found it gratifying that Rameau admitted the ear's inability to hear the difference between altered and pure fifths Rameau deferred to the ear in his ultimate support for equal temperament despite his system's basis on the triple progression of perfect fifths See Rameau's Génération harmonique (Paris, 1737), 75–104, esp pp 92 and 95Google Scholar

42 In the Code de musique pratique (Paris, 1760), which Rameau titled as a practical work, he did not enter into the physical properties relating to the generation of chords He began the treatise with the presentation of three scales that he posited as the ‘root of all practical music’ In the ‘plan’ of the work that precedes the main text, however, Rameau referred the reader to the appended Nouvelles réflexions sur le principe sonore, a lengthy diatribe on the validity of his mathematical propositions For this reason, it would be incorrect to say that Rameau based this practical treatise on the diatonic scale as Vandermonde did Instead, Rameau reinterpreted the hierarchy of his system, which he had established in his earlier works, to three scales – the diatonic scale, a scale in thirds and a scale in fifths Thus, the thirds and fifths of the overtone series are set forth in more accessible terms for readers of the CodeGoogle Scholar

43 Laborde, Essai, iii, 691 Laborde continued his argument on the assumption that, in the ‘science’ of harmony, no semitones could ever be equal ratios simply because they would support an ‘unnatural’ temperament In fact, the dispute between Laborde and Vandermonde on the issue of equal temperament continued at the Académie in following years Laborde gave a nascent description of a tuning system, based upon the overtone series and using pure fifths, in the Essai (i, 343–6) Laborde's system appeared formally in 1781 as the Mémoires sur les proportions musicales, le genre enharmonique des Grecs et celui des modernes, avec les observations de M Vandermonde, de l'Académie des Sciences, et des remarques de M l'Abbé Roussier Supplément à l'Essai sur la musique (Paris, 1781) This version of Laborde's temperament system was never presented at the Académie In a letter preceding the Mémoires, Laborde reported that he had sent his proposal to d'Alembert who had returned it with the written opinion of Vandermonde Laborde then called upon Roussier to rebut Vandermonde's comments and this exchange appeared as the addendum to the Mémoires, i e Vandermonde's ‘Observations’ and Roussier's ‘Remarques’ Then, in 1782, Roussier published a Mémoire sur le nouveau clavecin chromatique de M de Laborde Suite de supplément à l'Essai sur la musique (Paris, 1782, repr Geneva, 1972). When this work was submitted to the Académie, Vandermonde was one of the reviewers of the Roussier/Laborde proposal (Cohen, Music, 53)Google Scholar

44 Laborde, Essai, iii, 690Google Scholar

45 Shirlaw, Matthew, The Theory and Nature of Harmony (London, 1917, repr Sarasota, 1970), provides a detailed, though not unbiased, review of rationalizations given for the natural origin of the minor scaleGoogle Scholar

46 ‘Système’, 4Google Scholar

47 Examples of purely practical methods that begin with fundamentals of music abound in the eighteenth century One such example that is readily available in reprint is Dubugarre's Méthode plus courte et plus facile que l'ancienne pour l'accompagnement du clavecin (Paris. [1754], repr Geneva, 1972)Google Scholar

48 ‘Système’, 4 It is important to note that Vandermonde's nineteenth-century critics took no notice of his premiss of equal temperament, but instead rejected his species of the minor scale Choron and Fayolle mentioned Vandermonde's system under the article ‘Blainville’ in their Dictionnaire historique des musiciens, artistes et amateurs, morts ou vivants (Paris, 1810, repr. Hildesheim, 1971) Fétis applauded Vandermonde's system for what he believed was its basis in tonalité, but found fault with the four minor modes (Biographie, s v ‘Vandermonde‘)Google Scholar

49 ‘Système’, 2Google Scholar

50 Ibid., 5Google Scholar

51 In his mathematical writings, Vandermonde addressed questions of combinatorial analysis (see Lebesque, ‘L'oeuvre mathématique de Vandermonde’) Leonard G. Ratner discusses combinatorial procedures in mathematics and music in ‘Ars Combinatoria Chance and Choice in Eighteenth-Century Music’, Studies in Eighteenth Century Music A Tribute to Karl Geiringer on his Seventieth Birthday, ed. H. C Robbins Landon (New York, 1970), 343–63Google Scholar

52 Rameau did give a table in which chords and figures are grouped by intervals in his Dissertation sur les différentes méthodes d'accompagnement (Paris, 1732). Dissonant chords are arranged by sevenths, sixths/thirds, fifths, fourths/tritones, seconds and ninths in the table. Rameau, however, explained in the text how his system of fundamental bass related all chords to the two essential categories of the consonant tonic triad and of the dissonant dominant seventhGoogle Scholar

53 If one used all the pitches that are allowed in Vandermonde's ten major- and minor-scale types on C and G, the pitches C, C#, D, E♭, E, F, F#, G, A♭, A, B♭ and B would be available to form tertian triads. (The C# enters as the raised fourth in the fourth type of G minor scale.) The result would be 34 different possible tertian triads; eight of these are excluded because they fall outside the limits of Vandermonde's scale definitions The triads C#-E-G, F-A♭-C#, F#-A♭-C#, F-A-C#, A-C#-E, A♭-C#-E, A♭-C#-Eb are excluded because they include combinations of notes not in the G minor scale with raised fourth (i.e. G-A-Bb -C# -D-E♭ -F#). The chord A♭ -C-E is excluded because none of the scale types allows that a major third can appear with a minor sixth Chords that arise from the minor scale with augmented fourth (D-F#-A♭, F#-A♭-C, A-C#-Eb and C#-E♭-G) include the interval of an augmented sixth when inverted This fortuitous consequence of the scale species was also noted in the Histoire de l'Académie Royale des Sciences, année 1778 (p 53) In the ‘Mémoire’ (p. 5), Vandermonde came to the conclusion that because both the major and minor sixth appear in the ascending and descending minor scales, a major sixth could then also appear in the raised fourth minor scale, which usually has the minor sixth Later in the ‘Mémoire’ (p 12), he stated that a second major mode could be admitted in which the altered fourth appears providing for a major scale species with a raised fourth and major sixthGoogle Scholar

54 ‘Système’, 3; ‘Mémoire’, 3Google Scholar

55 I have purposely avoided translating the term base d'harmonie in order to emphasize its difference from basse fondamentale and basse continueGoogle Scholar

56 ‘Système’, 6Google Scholar

57 ‘Mémoire’, 6 ‘St la Base d'Harmonie des deux Accords est la même, tout est permis, si ce n'est d'altérer à-la-fois deux notes, dont l'une ne serait pas ou tierce ou quinte juste de l'autre Si la Base d'Harmonie est différente, toutes les notes du premier des deux Accords, supposé complet, doivent appartenir à la gamme où le second se trouve placé, et aucune note dissonante de ce premier ne doit faire partie d'un Accord parfait sur la Base d'Harmonie du second‘Google Scholar

58 Vandermonde had to rephrase the condition that regulated the progression of two chords which have the same base d'harmonie In 1778, he gave the General Law of Harmony as ‘When two consecutive chords have the same note for their base d'harmonie and two notes of the first chord form a dissonance between them, they cannot be changed simultaneously When the base d'harmonie is not the same and the two chords are made up only from the notes of a single scale, a dissonant note of the first (chord) cannot be consonant in the second ’ (‘QUE DEUX ACCORDS CONSÉCUTIFS AIENT LA MÊME NOTE POUR BASE D'HARMONIE, et que deux notes du premier accord, entre lesquelles il y aurait dissonance, ne soient pas altérées à-la-fois, OU, si la base d'harmonie n'est pas la même, QUE LES DEUX ACCORDS NE SOIENT COMPOSÉS QUE DES NOTES D'UNE MÊME GAMME, et qu'une note dissonante du premier ne soit pas consonante dans le second “Système”, 6) The condition ‘when two notes of the first chord form a dissonance between them’ was changed in 1781 to ‘all is allowed if one does not simultaneously alter two notes that do not form a third or perfect fifth between them’ This restatement was necessary because Vandermonde defined all fourths and sixths as dissonances. Therefore, in a progression such as the chord d-a-a‘-d“-f” passing to the chord d-a-g’ -c#“ -e” (as in bars 1–2 of Example 4 below), the intervals a‘-d“ and a’ -f” would be dissonant and their constituent pitches could not simultaneously change (as is obviously permissible)Google Scholar

59 ‘Mémoire’, 3Google Scholar

60 Ibid., 10Google Scholar

61 Ibid., 5–7, passim Vandermonde even stated that he felt that dissonance preparation was unnecessary when the key or key area was clear to a performerGoogle Scholar

62 Ibid, 7 ‘Alors toute note du premier Accord qui se trouvera dissonante sur cette Base d'Harmonie, doit, ou tenir, ou passer chromatiquement sur une note de même nom, ou aller diatoniquement sur une note consonante, en observant, lorsqu'il y a deux routes, que le repos absolu exige qu'on descende‘Google Scholar

63 The chord in bar 2 of Example 1 would be considered an ‘augmented seventh’ chord in conventional continuo practice The augmented seventh was usually completed by the intervals of the sixth, fourth and second According to Rameau, this continuo chord only occurred on the tonic and its fundamental bass was the dominant (see Rameau, Traité d'harmonie, Paris, 1722, 281–2)Google Scholar

64 ‘Mémoire’, 14Google Scholar

65 ‘Système’, 5.Google Scholar

66 Rameau's use of the term ‘accords par supposition’ was reserved for chords such as ninths, elevenths and thirteenths, which had roots subposed beneath the fundamental bass For example, a ninth chord (such as F-A-C#-E-G) has a root (F) added a third below the fundamental bass (A) of a seventh chord For further discussion, see Cohen, Albert, ’ “La supposition” and the Changing Concept of Dissonance in Baroque Theory’, Journal of the American Musicological Society, 24 (1971), 63–84, and Nancy K Baker, ‘Der Urstoff der Musik Implications for Harmony and Melody in the Theory of Heinrich Koch’, Music Analysis, 7 (1988), 3–30Google Scholar

67 Rameau, Traité, 41–2, 282–5 and 404–6Google Scholar

68 A number of Rameau's justifications for chord classifications and progressions lose their validity when related directly to the overtone series – as is the case with Rameau's borrowed chords Likewise, Rameau's ‘accords par supposition’ no longer have a physical basis in the overtone series because their roots are not harmonically generated See Ferns, Joan, ‘The Evolution of Rameau's Harmonic Theories’, Journal of Music Theory, 3 (1959), 231–56, and Emile Leipp, ‘Critique des fondements de la théorie de Jean-Philippe Rameau’, Revue musicale, 260 (1964), 97–111Google Scholar

69 ‘Système’, 4 ‘tout accord ne soit formé que des notes d'un certain mode, et ce n'est qu'entre les notes de ce mode qu'on peut chercher sa base d'harmonie‘Google Scholar

70 ‘Mémoire’, 5.Google Scholar

71 Ibid., 14Google Scholar

72 Vandermonde intimated that he preferred a more direct relationship between continuo figures and harmonic function. He felt that the use of figures and the standard practice of ‘completion’ of chords by filling in intervals not indicated by the figure did not necessarily aid in defining a chord's function (‘Système’, 5, ‘Mémoire’, 14) An example of the logical path that must be followed to find the relationship between a continuo chord and its fundamental bass chord is the ‘augmented second’ In Rameau's system, the chord was ultimately related to a dominant seventh chord, because he classified it as an ‘inversion’ of a diminished seventh chord. The diminished seventh chord, in turn, resulted from the ‘borrowing’ of a bass note, which was a semitone above the bass note of a dominant seventh chord (see n. 67 above)Google Scholar

73 See Mitchell, William J, ‘Modulation in C P. E Bach's Versuch’, Studies in Eighteenth Century Music, ed Robbins Landon, 533–42; and E Cynthia Verba, ‘Rameau's Views on Modulation and their Background in French Theory’, Journal of the American Musicological Society, 31 (1978), 467–79. Both Mitchell and Verba discuss the eighteenth-century use of the term ‘modulation’ in its twofold connotation Rameau used both connotations in his works (see also Philip Gossett, Treatise on Harmony Jean-Philippe Rameau, New York, 1971, 46, n 56)Google Scholar

74 ‘Système’, 4Google Scholar

75 ‘Mémoire’, 12 Some confusion may anse in the mind of the cautious reader, because the General Law of Part-Writing would authorize either G or D as the base d'harmonie According to this law, a note that is dissonant to the base d'harmonie should descend diatonically to a consonant note (as the f’ would descend to e’, relative to the suggested base d'harmonie of G) However, the passage does obey Vandermonde's law when the base d'harmonie is D, which he defined as the base d'harmonie In bar 3, the g, which is dissonant to the base d'harmonie, is held, and the b’, which is dissonant by Vandermonde's definition of dissonance, moves chromatically to b♭’Google Scholar

76 Rameau, Traité, 288Google Scholar

77 See Verba, , ‘Rameau's Views on Modulation’, and her article “The Development of Rameau's Thoughts on Modulation and Chromatics', Journal of the American Musicological Society, 26 (1973), 69–91.CrossRefGoogle Scholar

78 ‘Mémoire’, 12.Google Scholar

79 Vandermonde intended to prepare tables that would enumerate all the possible chord progressions, but this project was apparently never realized However, what appears to be a preliminary draft of such a table is included in the Dossier Vandermonde at the Archives of the Académie Royale des Sciences Vandermonde's scale types, his method for arriving at his class of elementary chords, and his views on modulation are better elucidated by the table, entitled ‘Table of All Admissible Modes in Music’ Unfortunately there is no accompanying textual explanation of how this table would provide rules for chord progressions The table is transcribed in my dissertation, ‘The Institutionalization of Music Theory in France 1764–1802’ (Ph D dissertation, Princeton University, 1989)Google Scholar

The table must have been drawn up after the ‘Système’ because it includes a fifth type of minor mode, the ‘ascending minor mode with the leading note of the fifth’, which Vandermonde defined in the ‘Mémoire’ (see n 53 above) This mode includes the minor third, the raised fourth, the major sixth and the major seventh The table shows the chromatic pitches numbered from 0 to 11 and set into a grid, the number 0 equals the tonic pitch. Two rows for the tonic and fifth of the key as the base d'harmonie appear vertically and intersect six columns for each of the major and minor scale types Under each scale type the pitches from 0 to 11 are given vertically Honzontally, each scale degree has seven pitches arranged by thirds (i e line 1 = 0 4 7 11 2 5 9, line 2 = 1 5 8 0 3 6 10, etc)Google Scholar

Examination of the table does shed some light on Vandermonde's ideas on ruling key areas and modulation. In the lower portion of the table, when the fifth as the base d'harmonie rules a progression, it can be inferred that modulation would occur only if the tonic were permanently altered by repeated intrusion of another scale's pitches Temporary cadences on the fifth of the key using diminished or augmented chords would not push a musical phrase beyond the control of the fifth of the tonic key By the little evidence given in this table, modulation (in the sense of change of key) in Vandermonde's system would require a protracted use of a new scale's pitches and would not necessarily be indicated by temporary cadences involving leading-note relationshipsGoogle Scholar

80 ‘Système’, 5 ‘Ici je commence à m'écarter des idées reçues dans la théorie, et cela doit être, puisque c'est le point d'où je déduis cette simplification des lois de l'harmonie que j'ai annoncée’Google Scholar

81 Laborde, Essai, iii, 693, n (a)Google Scholar

82 Ibid., iii, 694Google Scholar

83 Système', 2 Les Théoriciens se sont ménagé une porte pour expliquer tout, que je me suis fermée. Ce sont les notes de Passage et les notes de Goût Les premières servent à rendre la Mélodie naturelle, les secondes, à lui donner un agrément recherché Ces notes ne sont point comptées dans l'harmonie, quoiqu‘égales en durée à d'autres notes comptées dans le même morceau, quoique beaucoup plus longues quelquefois. Je trouverais plus naturel de supposer dans certains cas que l'Auteur a supprimé à dessein une note de liaison, que d'omettre des notes qu'il a eu soin d‘écrire, et je ne connais point de réponse à ce dilemme le note est de bon goût ou de mauvais goût, si elle est de mauvais goût, il faut la corriger, si elle est de bon goût, c'est se moquer que de l'exclure de l'harmonieGoogle Scholar

84 Tratte des accords et de leur succession, selon le système de la basse fondamentale, pour servir de principes d'harmonie à ceux qui étudient la composition ou l'accompagnement du clavecin, avec une méthode d'accompagnement (Paris, 1764, repr Geneva, 1972), Observations sur différents points d'harmonie (Geneva, 1765), and L'harmonie pratique, ou exemples pour le Traité des accords (Paris, 1775) The content and logic of Laborde's argument leads the reader to believe that Roussier undoubtedly had a part in this rebuttal. Fétis insisted that Roussier himself had written the article on Vandermonde in the Essai (Biographie, s v ‘Laborde‘)Google Scholar

85 Roussier insisted upon the difference of the subdominant chord's passage to the tonic in all of his treatises. See, for example, his Traité des accords, 134Google Scholar

86 Laborde, Essai, in, 699Google Scholar

87 Ibid., iii, 694–5, n (a) ‘lui faisait dieser une sous-dominante, lorsqu'elle forme ce qu'on appelle une cadence imparfaite sur sa tonique C'est, par example, fa-dièse ut, pour représenter la cadence imparfaite fa ut. Nous ne croyons pourtant pas que ce maître pratique lui-même le passage direct de fa-dièse à ut, l'oreille l'en empêcherait sans doute, mais ce qui est synonym à cette absurde harmonie, il fait pratiquer fa-dièse portant tierce, quinte et sixte, et passant à sol, accompagné de quarte et de sixte.‘Google Scholar

88 ‘Mémoire’, 13.Google Scholar

89 Indeed, Vandermonde remarked that the ideas of a number of theorists on double emploi were, to him, ‘very strange’ (‘Mémoire’, 15).Google Scholar

90 ‘Système’, 7: ‘de chaleur et d'altercations sur la Musique, qui a répandu dans les Sociétés plus de connaisseurs que de connaissances’.Google Scholar

91 Mercure de France (15 February 1779), 142 ‘L'objet de M Vandermonde n'est pas de décider nos grandes ou petites querelles sur cet objet, mais de proposer un nouveau système de Musique qui satisfait, selon lui, mieux que tous les précédents, aux lois observées dans l'harmonie par les plus habiles Compositeurs, à celles même qu'on pourrait suivre sans que l'oreille s'en offensât, quoique les Artistes ne s'en soient pas encore avisés ‘Google Scholar

92 Laborde, Essai, iii, 46 ‘La basse fondamentale est la preuve de la composition, comme en Arithmétique, l'addition est la preuve de la soustraction Toute harmonie ne peut être bonne, quand elle n'est pas soumise à la basse fondamentale ‘Google Scholar

93 ‘Système’, 5 ‘suffit pour expliquer, sans rien omettre, toute composition propre à satisfaire l'oreille des connaisseurs‘Google Scholar

94 Laborde, Essai, iii, 695 ‘La Musique Italienne, et toutes celles qui sont faites à l'imitation de l'Italienne, sont remplies de ces sortes de cinquièmes notes élevées d'un demi-ton, uniquement par goût de chant‘Google Scholar

95 Ibid., iii, 699 Laborde took Example 8(a) from Rousseau's Dictionnaire de musique (Amsterdam and Paris, 1768, repr Hildesheim, 1969), s v ‘Quinte’ and Planche K, Figure 5 He did not, however, tell the reader that Rousseau had also explained and given an example of the French usage of the augmented fifth chord (Planche K, Figure 3) Rousseau defined both an ‘Italian usage’, in which the augmented fifth chord appears only on the tonic of a major mode, and a ‘French usage’, in which the chord appears only on the mediant of a minor mode This follows Rameau who, when he first set forth his rule for the augmented fifth chord, limited its usage to the mediant only of the minor mode (Traité, Book 3, Chapter 29) Laborde's example, it should be noted, is in a major key The example of the French usage given by Rousseau, which Laborde did not cite, does show its usage in a minor key Laborde's interpretation can be attributed to the fact that he was in no way an admirer of Rousseau, whom he saw as Rameau's adversary. He referred to Rousseau's writings as being full of ‘deadly poisons’ in the preface to his Mémoires sur les proportions musicales (Paris, 1781). Laborde, therefore, had no motivation to quote Rousseau correctly nor to give him credit for understanding Rameau's views on the augmented fifth chordGoogle Scholar

96 ‘Mémoire’, 14–15Google Scholar

97 ‘Système’, 6 ‘la manière d’analyser un morceau de musique consiste à trouver une suite d'accords complets sous leur forme élémentaire dans laquelle soient insérés par ordre tous les accords employés par l'Auteur, et où l'application de la loi générale de l'harmonie s'observe sans discontinuité de proche en proche’Google Scholar

98 Ibid ‘Cette loi offre un champ si vaste que l'on pourrait, en s'y conformant rigoureusement, ne composer que de l'harmonie bizarre et du mauvais effet, si l'on négligeait entièrement l'observation des conseils. Ces conseils se réduisent à faire en sorte que l'harmonie soit claire et sans équivoque sur la note principale; et que les transitions où plusieurs notes sont altérées à-la-fois n'y soient employées que sobrement, à éviter rarement les repos qu'on annonce, enfin, à se conformer aux règles générales de tous les Beaux-Arts.‘Google Scholar

99 Ibid., 1 ‘Le choix entre les Systèmes parce que toutes les méthodes ont leur manière de se prêter aux licences que l'Autorité peut introduir ou que Génie peut suggérer‘Google Scholar

100 ‘Mémoire’, 2 ‘Il y a deux écueils opposés à éviter en proposant un système d'harmonie, l'un, de désapprouver des choses qui se pratiquent constamment et avec succès, sous toutes sortes de formes, l'autre, d'étendre les règles jusqu’ à des cas impraticables, ou qui ne peuvent se pratiquer qu'avec des précautions particulières 'Google Scholar

101 Jonathan Dunsby, in the editorial to the first issue of Mustc Analysis, summarizes how these same difficulties are presented to the present-day analyst (Mustc Analysts, 1 (1982), 3–8)Google Scholar

102 Laborde, Essai, iii, 680 ‘Nous n'avons laissé échapper aucune occasion de faire connaître dans notre ouvrage la vénération que nous avons pour lui [Roussier], et les voeux que nous formons pour qu'on profite des années qui lui restent encore, à pouvoir former des élèves dignes de lui L'établissement d'une chaire de Musique, remplie par l'Abbé Roussier, nous dédommagerait de la privation des conservatoires, et alors l'École Française serait fondée sur des principes invariables, avantage dont ne jouissent pas les écoles de l'Italie, où tout s'enseigne par la routine.’Google Scholar

103 ‘Mémoire’, 2. ‘Il [Piccinni] a trouvé que les deux règles, qui renferment le système dont il est ici question, étaient déduites de principes très-vrais, et qu'elles seraient d'un usage plus général, et incomparablement plus facile que toutes celles qu'on a enseignées jusqu'à présent dans les Écoles.’ It is ironic that Vandermonde calls upon the aid of the music teacher of Laborde's children – Piccinni (noted by Constant Pierre, Le Conservatoire National de Musique et de Déclamation Documents historiques et administratifs, Paris, 1900, 12–13)Google Scholar

104 Principes élémentaires de musique arrêtés par les membres du Conservatoire de musique pour servir à l'étude dans cet établissement suivis par des solfèges (Paris, 1799–1800), ‘Dispositions relatifs à la confection des principes élémentaires de musique et à leur emploi dans le Conservatoire, 12 fructidor, an II [29 August 1794]’Google Scholar

105 Catel, Traité d'harmonie, Avant-propos ‘Cette méthode .. évite aux élèves la difficulté presqu’ insurmontable de classer dans leur mémoire cette série nombreuse d'accords isolés qui, présentés comme autant d'accords différents, rendent l'harmonie difficile à concevoir et encore plus à bien pratiquer.'Google Scholar

106 Principes, ‘Dispositions‘Google Scholar

107 Shakespeare, William, Romeo and Juliet, Act 3, scene iii, lines 132–4Google Scholar