1. Diversarum speculationum mathematicarum & physicorum liber (Turin, 1585), pp. 277-83.

2. Istitutioni Harmoniche (Venice, 1558), especially the seconda parte; Dimostrationi Harmoniche (Venice, 1571), especially ragionamento secondo and quarto; Sopplimenti Musicali (Venice, 1588), libro quarto.

3. See his Humanism in Italian Renaissance Musical Thought (New Haven, 1986), pp. 257-65; and Studies in the History of Italian Music and Music Theory (Oxford, 1994), pp. 213-23.

4. The comment about the thirds is from the insert to Musurgia A-3 (1958), p. 31, and the sanity comment is from Tuning and Temperament, p. 105.

5. Both of these quotations are from the insert to Musurgia A-3, the first on p. 3, and the second on p. 31.

6. Musurgia A-3, p. 31.

7. Musurgia A-3 (1958), p. 31. The studies Barbour cited include those from the Psychological Laboratory at the University of Iowa, such as Carl E. Seashore, ed, Objective Analysis of Musical Performance, University of Iowa Studies in the Psychology of Music 4 (1936) (see in particular, Paul C. Greene, "Violin Performance with Reference to Tempered, Natural, and Pythagorean Intonation," University of Iowa Studies in the Psychology of Music 4 (1936), 232-51); studies from the Harvard Psychological Laboratory by Austin M. Brues (1927); studies from the University of Washington by E. R. Guthrie and H. Morill (1928); James F. Nickerson, "Intonation of Solo and Ensemble Performance of the Same Melody," Journal of the Acoustical Society of America 21 (1949), 593-95.

8. See Harold Seashore, "An Objective Analysis of Artistic Singing," in Carl E. Seashore, ed, Objective Analysis of Musical Performance, University of Iowa Studies in the Psychology of Music 4 (1936), 12-171, especially "The Accuracy of Intonation and Intervals," pp. 48-57. More recent studies from the Department of Speech Communication and Music Acoustics at the Royal Institute of Technology in Stockholm, include Sten Ternström and Johan Sundberg, "Acoustics of choir Singing," in Acoustics for Choir and Orchestra (Stockholm: Kungliga Musikaliska Akadamien, 1986, 12-22; "Intonation Precision of Choir Singers," in Journal of the Acoustical Society of America 84 (1988), 59-69. Most recently, Ternström and Duane Richard Karna opine that "there should be no inherent advantage of using just intonation in choir music," although this does seem to imply that it is possible. See their chapter, "Choir," in The Science and Psychology of Music Performance: Creative Strategies for Teaching and Learning, ed. Richard Parncutt and Gary E. McPherson (Oxford, 2002), 269-84.

9. Seashore and Barbour maintain that scooping and pitch wandering make it impossible for singers to come reliably closer to a certain frequency than about a fifth of a semitone. See Harold Seashore, "An Objective Analysis of Artistic Singing," pp. 25-77, and Barbour Tuning and Temperament, pp. 197-98. But the Hilliard Ensemble recording excerpt given below, in a sampling of major chords at eleven points throughout, gives an average major 10th (the most common voicing for the third) with a ratio of 2.507 above the root. This compares with the pure ratio of 2.500 and the ET ratio of 2.520, and represents a predilection for thirds that are twice as close to pure as they are to ET, as well as a tuning accuracy six times closer to the intended frequency than that predicted above. The excerpt also ends with a chord sustained with astonishingly pure tuning over a full eight seconds duration. It takes remarkable precision and extraordinary vocal control to maintain such a beautifully tuned chord for so long, but this recording shows that it is possible, and that our expectations for ensemble singing should be adjusted accordingly. Indeed, even Ternström and Karna acknowledge that "vocal groups that perform close harmony with one voice to a part ... strive to achieve harmonies that are so precisely tuned and so straight in pitch that the voices fuse together and we hear one instrumentlike chord rather than several part singers." See "Choir," p. 280.

10. Easley Blackwood, The Structure of Recognizable Diatonic Tunings (Princeton, c.1985), p. 153.

11. Musica Practica (Bologna, 1482). See commentary and translation by Clement A Miller, Musicological Studies and Documents 44 (1993), and by Luanne Eris Fose (PhD diss., University of North Texas, 1992).

12. De harmonia musicorum instrumentorum opus (Milan, 1518), Book II, ch. 34, 35.

13. See Anicius Manlius Severinus Boethius, Fundamentals of Music, trans. Calvin M. Bower, ed. Claude V. Palisca (New Haven & London, 1989), p. 26.

14. See his Structure of Recognizable Diatonic Tunings, pp. 67-68.

15. Vicente Lusitano, Introduttione facilissima, (Rome, 1553), p. 4. The practice of referring to the diesis as a comma, in fact, persisted into the 19th century.

16. See his Dialogo della Musica Antica et della Moderna (Florence, 1581); translation and commentary by Claude V. Palisca (New Haven, 2003), section 1 [The Tuning Question], pp. 10-134.

17. Christian Huygens, Cosmotheoros (1698), pp. 88-90. As his preferred tuning method, Huygens, in fact, recommended 31-note Equal Temperament, a system which is almost identical to extended quarter-comma meantone. Quarter-comma and other meantone temperaments are discussed below.

18. Robert Smith, Harmonics, or the Philosophy of Musical Sounds, second edition (1759), pp. 229, citing "Méthode genérale pour former les Systèmes temperés de musique," Memoires de l'Académie des Sciences (1707), p. 263.

19. Palisca, Humanism, p. 264.

20. Mark Lindley, "Early 16th-Century Keyboard Temperaments," Musica Disciplina 28 (1974), 149-50.

21. Arnolt Schlick, Spiegel der Orgelmacher und Organisten (Speyer, 1511), ch. 8.

22. Pietro Aaron, Toscanello (Venice, 1523), Book II, ch. 41.

23. Francisco de Salinas, De Musica Libri Septem (Salamanca, 1577), book 3.

24. See his L'Antica Musica Ridotta alla Moderna Prattica (Rome, 1555), Book V; translated by Rika Maniates as Ancient Music Adapted to Modern Practice, ed. Claude V. Palisca (New Haven & London, 1996), Book V, 315-443. This is a long and complicated discussion, but on pp. 432-433, for example, he describes his whole tone as made up of five "dieses" or commas, of which three belong to the major semitone and two to the minor semitone. Thus, an octave consists of seven major semitones (7 x 3 = 21) plus five minor semitones (5 x 2 = 10), or of five whole tones (5 x 5 = 25) plus two major semitones (2 x 3 = 6), both of which methods give a total of thirty-one equal parts.

25. It is not sufficiently recognized, I believe, that each of these meantone temperaments possesses one or more Just intervals. Pythagorean tuning is known to contain pure 5ths and 4ths, of course, but each of the regular meantone temperaments given in Example 1b has some purity as well.

Regular Meantone version
Pure interval
1/3 Comma
minor 3rd
Major 6th
2/7 Comma
minor semitone
1/4 Comma
Major 3rd
minor 6th
1/5 Comma
Major semitone
Major 7th
1/6 Comma
diminished 5th

The fact that each of these systems found adherents in the Renaissance may have been due, in part, to the presence of these pure intervals and the consequent enhancement to certain modes and progressions.

26. See my forthcoming article and practice resource, "Baroque Ensemble Tuning in Extended 1/6 Syntonic Comma Meantone," Digital Case (2006).

27. Benedetti, p. 281.

28. See his Structure, especially ch. 7.

29. Diversarum speculationum mathematicarum & physicorum liber (Turin, 1585), p. 278.

30. So characterized in Palisca, Studies, p. 224.

31. "... realmente le voci cosifatte cantano gl'intervalli musici nella vera et perfetta forma loro, dove la più parte degl'artifiziali strumenti gli suonano chi più et chi meno da essa lontana." Vincenzo Galilei, Discorso particolare intorno all'unisono (Florence, 1589), ed. and trans. by Claude Palisca (New Haven, 1989), 205-07. In saying this, Galilei was echoing a statement he had made in his Dialogo della Musica Antica et della Moderna (Florence, 1581), but whereas in the 1581 publication Galilei was attempting to refute the statement, in 1589 he was merely observing while stating his personal preference for the temperament of instruments--perhaps not surprising for a lutenist. For the 1581 discussion, see Vincenzo Galilei, Dialogo, pp. 54-56; Dialogue on Ancient and Modern Music, trans. Claude V. Palisca (New Haven, 2003), pp. 131-34.

32. Palisca, Studies, p. 222.

33. Blackwood, p. 153.

34. New Grove 2, "Just intonation."

35. For a detailed explanation of how Just intonation works in late-medieval polyphony, see my chapter, "Tuning," in A Performer's Guide to Medieval Music, ed. Ross W. Duffin (Bloomington, 2000), 545-62.

End of footnotes