¹ These labels (‘2/7 comma’ and the like) say how much the 5ths and 4ths are tempered. They refer to some fraction of the syntonic comma (the amount by which the 3rds and 6ths will automatically be rendered impure if the 5ths and 4ths are tuned perfectly pure). If we call this fraction ‘t’, then the amounts by which the major 3rds and 6ths are tempered will be 1–4t and 1–3t respectively (see Figures 5a and c below). If the major 3rd were tempered 7/11 comma (as in equal temperament), the diminished 4th — the difference between two major 3rds and an octave — would be, as it happens, the same size. (This is because three pure major 3rds fall short of an octave by an amount practically indistinguishable from 21/11 of the syntonic comma.) Whatever amount ‘m’ the major 3rd is tempered less than 7/11 comma in any particular shade of meantone temperament, the diminished 4th will be 3m larger than the major 3rd, and will accordingly sound more or less inappropriate if used in its stead.

² Barnes, J., ‘The Specious Uniformity of Italian Harpsichords’, Keyboard Instruments: Studies in Keyboard Organology, ed. Ripin, E. M. (Edinburgh, 1971)Google Scholar, shows that extant sixteenthcentury instruments are liable to have been altered during the seventeenth and eighteenth centuries to eliminate the split keys. Ferdinando Tagliavini has assured me that the same is true of extant sixteenth-century Italian organs, for example, in Santa Maria Maggiore, Rome, and San Petronio. Bologna.

³ From his Ricercari per ogni sorte di stromenti da tasti (Venice, 1595)Google Scholar; ed. in Andrea Gabrieli, Ricercari für Orgel, i, ed. Pidoux, P. (Kassel, 1941)Google Scholar.

⁴ The locus classicus is later: the Tocata cromatica for the Elevation in Frescobaidi's Messa delta domenica from Fiori musicali (Venice, 1635)Google Scholar.

⁵ Numerous books and articles could be cited apropos; the following are taken more or less at random: Kaufmann, W., The Ragas of North India (Bloomington, 1968), pp. 8–10Google Scholar; Sonis, E., Classical Persian Music, an Introduction (Cambridge, Mass., 1973), pp. 52–3Google Scholar; Tamba, Akira, La structure musicale du Nô (Paris, 1974), pp. 49–65Google Scholar; Morton, D., The Traditional Music of Thailand (Berkeley, California, 1976), p. 28Google Scholar. See also Powers, H. S., ‘Mode’, The New Grove Dictionary of Music and Musicians, ed. Sadie, S., 20 vols. (London, 1980), xii, pp. 376–450, and its bibliographyGoogle Scholar.

⁶ P. 24. The mixture of tenses is due to the inversion: ‘was’ refers to p. 22, not to Vincentino.

⁷ An account of Vicentino's tuning prescriptions is given below.

⁸ Benedetti, G. B., Diversarum speculationem mathematicarum & physicorum liber (Turin, 1585), pp. 177–83Google Scholar. A good account of Benedetti's criticism, which goes awry only in attributing to Benedetti a wish to ‘show the composer why a system of equal temperament was a necessity for modern music’, is available in Palisca, C., ‘Scientific Empiricism in Musical Thought’, Seventeenth Century Science and the Arts, ed. Rhys, H. H. (Provincetown, 1958), pp. 113–18Google Scholar. For an analysis of Benedetti's tuning prescription, see Lindley, M., ‘Stimmung und Temperatur’, Geschichte der Musiktheorie, vi, ed. Zaminer, F. (Berlin, forthcoming)Google Scholar.

⁹ Dialogo di Vicentio Galilei nobilefiorentino della musica antica et della moderna (Florence, 1581), p. 31Google Scholar: ‘la Terza maggiore sia contenuta da vna proporzione irrationale assai vicina alia Sesquiquarta’ (‘The major 3rd is contained in an irrational proportion quite close to 5:4’), but the 5ths and 4ths ‘vengano accostarsi al Diatono Ditonieo’ (‘come to resemble those of [Pythagorean intonation]’). L'artvsi ovvero delle imperfettioni della moderna musica (Venice, 1600), fol. 31Google Scholar: ‘queste cose mi concludono, che la spetie Diatonica, che oggi si tiene per quella di Tolomeo, sia quella di Aristosseno; ò almeno vna che sia mista dall'vna e l'altra’.

¹⁰ Gerbert, Martin, Scriptores ecclesiastici de musica sacra potissimum, 3 vols. (St Blasien, 1784), in, pp. 75, 77, 80Google Scholar.

¹¹ An American academic composer, Easley Blackwood, has compiled a list of presumed mistakes in Vicentino's compositions; see his ‘The 31 Note Equal Tuning of Nicola Vicentino’, Proceedings of the 50th Anniversary Meeting of the National Association of Schools of Music (Houston, 1974)Google Scholar.

¹² This point strengthens our freedom to perform sixteenth-century music at whichever pitch level best suits the circumstances. See Brown, H. M., ‘Notes (and Transposing Notes) on the Viol in the Early Sixteenth Century’, Music in Medieval and Modern Europe: Patronage, Sources and Texts, ed. Fenlon, I. (Cambridge, 1981), pp. 61–78Google Scholar; and Mendel, A., ‘Pitch in Western Music since 1500: a Re-examination’, Ada Musicologica, 1 (1978), p. 91Google Scholar.

¹³ See for instance Boethius, De institutione musica, Book 4, Chap. 13.

¹⁴ This is, of course, Barbour, J. M., Tuning and Temperament, a Historical Survey (East Lansing, 1951)Google Scholar, by far the best secondary source available to Berger at the time.

¹⁵ The statement, as cited from Barbour, says that most shades of meantone temperament, when extended to twelve pitch classes with eleven uniform 5ths, will not yield as good an approximation to the 100-cent scale as they will contain when extended to some manageable number larger than eighteen. This has very little to do with the character of the non-uniform 5th which is bound to remain in all shades of meantone temperament (except 1/2.98-comma) when these are extended to nineteen pitch classes.

¹⁶ I have undertaken a fresh account of the role of logarithms in the history of tuning theory in Chapters 4 and 10 of ‘Stimmung und Temperatur’.

¹⁷ L'antica musica ridotta alia moderna prattica (Rome, 1555), fols. 103^v–104^vGoogle Scholar.

¹⁸ Not necessarily ¼-comma: see Lindley, M., ‘Early 16th-Century Keyboard Temperaments’, Musica Disciplina, 28 (1974), p. 150Google Scholar.

¹⁹ I assume that ‘più perfetto’ means not only ‘better’ (and ‘larger’) but also ‘more nearly pure’. If not, the statement indicates merely that Vicentino preferred to use major 3rds somewhat larger than they were on either of his keyboards.

²⁰ Inasmuch as 2/7 = 0.286, which is close to 0.3. Le istitutioni harmoniche (Venice, 1558), pp. 126 ffGoogle Scholar.

²¹ De musica libri vii (Salamanca, 1577), p. 139Google Scholar.

²² L'artvsi, fol. 20^v. I have abridged Dr Berger's translation (p. 90).

²³ Lowinsky, E., ‘Adrian Willaert's Chromatic “Duo” Re-examined’, Tijdschrift voor Muziekwetemchap, 18 (1956–1959), pp. 17, 19Google Scholar. Professor Lowinsky surmised, correctly, ‘What he must have meant is that they are placed in intervals of equal semitones.’

²⁴ For a list, see Lindley, M., Lutes, Viols and Temperaments (Cambridge, forthcoming)Google Scholar. In Book 1, Chapter 16 of De institutione musica Boethius argued: ‘16 ac 18 collati, sesquioctavam retinent proportionem, atque idcirdo tonum. Sed hanc proportionem 17 numeris medius non in aequalia partitur… Est enim minor pars septimadecima, maior sextadecima … inter haec unum maius semitonium nuncupatur, aliud minus.’ (‘16 and 18 together make a 9:8 proportion, and thus a whole-tone. But this proportion is not divided equally by the middle number 17 … For 1/17 is the lesser part, and 1/16 the greater … Of these one is called the major semitone, the other minor.’)

²⁵ For a modern calculation, see Jahnel, F., Die Gitarre undihrBau: Technologic von Gitarre, Laute, Mandoline, Sister, Tanbur und Saite (Frankfurt-am-Main, 1962), pp. 150Google Scholar

²⁶ Dialogo, p. 49.

²⁷ Musica instrumental deudsch (4th edn, Wittemberg, 1545)Google Scholar, fols. 53v–54: ‘fast das groste part der Lautnisten und Geiger art Alle bund machen gleich von ein … ein bund Der Semiton minus thut kind’.

²⁸ See Hieronymus Cardanus, Writings on Music, trans. Miller, C. A. (n.p., 1973), pp. 17–19Google Scholar

²⁹ Chap. 2, paragraph 26. Operum tomus decimus (London, 1663), p. 107Google Scholar: ‘sciendum, quod tonus consistit in sexquioctava vt demonstrabo & dimidium quodquod consistit in proportions 18. ad 17. & vocatur semitonium minus’.

³⁰ Opvs novvm deproportionibvs nvmerorvm (Basle, 1570), p. 170Google Scholar: ‘Hie subit admiratio quomodo semitonum minus aptent tarn gratem in symphonijs, maius autem nequaquam.’

³¹ L'artvsi, fol. 27^r–27^v. Dr Berger's translation (p. 92).

³² See Palisca, C., ‘The Artusi-Monteverdi Controversy’, The Monteverdi Companion, ed. Arnold, D. and Fortune, N. (London, 1968), pp. 134–5Google Scholar. Notice that in the passage cited on p. 392 above, Artusi did not disapprove of ‘li pratici più moderni’ (‘the most modern practitioners’ — Willaert, Porta, Rore et al.) even though he suggested that ‘non conoscano qual sia quella spetia d'Harmonia, che si Canti, e Suoni’ (‘they do not know which is [really] the type of harmony that is sung and played’); yet now he attributes the defects of ‘certi moderni compositori’ (Monteverdi) to capriciousness even though ‘they’ have at least offered a theory to justify their simultaneous use of a sharp and a flat.

³³ Discorso secondo mvsicale di Antonio Braccino da Todi (Venice, 1608), p. 6Google Scholar: ‘Sig. Claudio hebbe all'hora il torto lui; perche quando l'Artusi gli scrisse quelle lettere, gli le scrisse piene d'amoreuolezza e ciuiltà; & egli in vece di rispondergli nella istessa maniera, gli fece rispondere per vna terza persona, & lettere senza nome proprio.’ On the same page Artusi's fictitious spokesman ‘Braccino’ threatens ‘on a better occasion’ to publish ‘le lettere, le copie di cui sono nelle mani mie, & debbono ancora essere nelle mani del Sig. Monteverde’. However the letters of ‘Ottuso’ were actually drafted, there is no evidence that the views they expressed were distinct from those of the composer they defended.

³⁴ Artusi says ‘quinto’, but the lines he gives are from Book 3 (216–18).

³⁵ In addition to the apparently balletto-like rhythm, I suppose Artusi disapproved of placing the weak syllables ‘la’ and ‘-za’ on strong beats. For a modern discussion see Razzi, F., ‘Polyphony of the seconda prattica: Performance Practice in Italian Vocal Music of the Mannerist Era’, Early Music, 8 (1980), pp. 306–9Google Scholar.

³⁶ De institutione musica, Book 1, Chap. 34 ‘ratione perpensa canendi scientiam non servitio operis sed imperio speculationis adsumsit’.

³⁷ Tuning and Temperament, a Historical Survey (East Lansing, 1951), pp. 146–8Google Scholar

³⁸ (log 81/80 – ½ log 128/125) ÷ log 2/1200 = 0.98

³⁹ For a more complete discussion, see Lindley, Lutes, Viols and Temperaments.

⁴⁰ The Harmonics of Aristoxenus, trans. Macran, H. S. (Oxford, 1902), p. 189Google Scholar.

⁴¹ L'artvsi, fol. 31.

⁴² When one divides two integers the result if not itself an integer will be a fraction or else an integer plus a fraction. But roots of integers are never fractional. One can easily understand why by taking the opposite perspective: all the powers of, say, 2 1/n (where ‘n’ is an integer) are bound to be fractional! Numbers that are neither integer nor fractional have been called ‘irrational’ (that is, divorced from ratios) since the ancient Greeks. A synonym is ‘surd’. Of course there is nothing unreasonable or uncertain, pace Artusi, about these ‘irrational’ answers to root-extraction problems, just as there is nothing unwholesome or incomplete about fractional answers to division problems. It is just that the two categories are mutually exclusive.

⁴³ De arithmetics. Book 1, Chap. 1. In this passage the quadrivium is introduced and divided as follows: stable magnitudes are to be studied in geometry; rotating magnitudes in astronomy; multitudes per se (like 3 or 4) in arithmetic; ratios between multitudes in the science of music. That Boethius saw this as more than just an academic distinction might be inferred from Book 3, Chapter 1 of De institutione musica, where he criticised ‘Aristoxenus musicus’ for saying that the whole-tone comprised two equal semitones, and repeated in this connection the argument cited above in note 24. One infers that Aristoxenus was wrong to consider intervals as magnitudes.

⁴⁴ A weak argument, as the theory does not imply that any rational numbers should be associated with a ½-comma or with the semitones in question.