By Ross W. Duffin

“The idea … that one can understand the ratios of musical consonances without experiencing them with the senses is wrong. Nor can one know the theory of music without being versed in its practice.”


     So begins the first of two letters sent by the mathematician Giovanni Battista Benedetti to the composer Cipriano de Rore in 1563. Subsequently publishing the letters in 1585, Fn1 Benedetti was attempting to demonstrate that adhering to principles of Just intonation, as championed most famously by Gioseffo Zarlino,Fn2would, in certain cases, cause the pitch of the ensemble to migrate. This, of course, raises questions about whether Just intonation can work in practice. His letters were discussed at length by the late Claude Palisca,Fn3 and I am greatly indebted to his work.  However, there remain many misconceptions about the subject of Just intonation, then and now, so Benedetti’s evidence will form one crucial part of this re-examination of its theory and practice in Renaissance music.

     In modern times, the use of acoustically pure intervals in musical performance—Just intonation—has not had a very positive reputation. J. Murray Barbour, whose book, Tuning and Temperament (East Lansing, 1951) remains a standard reference more than half a century after its publication, and whose 1958 recording,The Theory and Practice of Just Intonation (Musurgia A-3), has been one of the few resources available to anyone interested in the subject, referred to pure major and minor thirds as boring and insipid, and to anyone advocating a certain form of Just intonation as insane.Fn4 It is difficult to imagine an opinion on tuning more powerfully expressed, and it makes the consignment of Just intonation to the dustbin of history seem, as it were, all the more justified. But there was a fundamental flaw in Barbour’s conception of Just intonation, as may be seen in these two statements:

     The characteristic features of Just intonation in its practical application, are demonstrated best and most precisely on keyboard instruments. [emphasis original]

     Of course, it is the severe limitation of just intonation to twelve notes in the octave, with enharmonically equivalent notes forbidden, that has caused its outright rejection by musicians.Fn5

     These things caused Barbour, in his words, to “present just intonation more because of its significance for the history of musical theory than as a tuning system that was ever of importance in the world of practical Western music.”Fn6 It is a grievous misunderstanding of Just intonation to think that it is meant to work in a twelve-note octave as found on modern keyboard instruments, however. Nor does the theory of Just intonation as described by many writers in the Renaissance ever suggest that a choice must be made between enharmonics. Barbour’s error caused him to offer recordings of supposedly Just chords that were nothing of the kind. To give one example, his version of the triad on the second degree of the “Just” major scale, presented a 9:8 whole tone, a 4:3 fourth, and a 5:3 major sixth above the “tonic.” In such a chord, the “third” is not a pure minor third and the “fifth” is not a pure fifth above the root. There are ways to create a pure minor triad on the second degree of the scale—using a 10:9 rather than a 9:8 whole tone, for instance—but such solutions are not explored because they violate Barbour’s arbitrary twelve-note octave rule. Unfortunately, generations of students and scholars have depended on his flawed exposition as the last word on the issue.

     Another burden that Just intonation has had to bear are the studies from the 1930s by Carl Seashore and others, showing the supposed preference of musicians and listeners for “unjust” intervals, including wide major thirds and narrow diatonic semitones. As Barbour summarized:

     All of these experiments prove conclusively that … singers and instrumentalists have no predilection for the so-called natural intervals, and that the same is valid for the majority of listeners.Fn7

     This has been “gospel” in the psychology of music field for decades. I suggest, however, that much of a person’s preference for intervals depends not on biological predilection but on aural conditioning, and in the 1930s through the 1950s, singers and violinists (the usual subjects of such experiments) had probably never heard Just intervals nor knew any sort of context where their use would make musical sense. This is not to say that most musicians and listeners today would react any differently: the stranglehold of equal temperament in our musical culture has seen to that. But it does not diminish the value of Just intervals in general, and particularly not in the culture of the Renaissance.

     Some writers have also cast doubt on whether singers, especially, could sound frequencies to the kind of tolerances necessary to achieve intentional tuning choices.Fn8 Performances and recordings by modern early music groups, such as the Hilliard Ensemble recording used below, refute that supposition utterly in my opinion, presenting highly trained voices with minimal vibrato and a level of accomplishment in tuning accuracy that Seashore, Barbour and their colleagues probably never imagined.Fn9

     It is one thing, however, for a group of singers to have talent and good ears, and another for the theory behind a complicated tuning system to be understood, assimilated, and realized in performance. Refuting the notion that Just intonation is merely a theoretical system without any practical value will take a precise, note-by-note analysis of several relevant passages—the kind of rigorous examination that, to my knowledge, only theorist and composer Easley Blackwood has attempted before. But even Blackwood concluded, echoing Barbour, that “just tuning is of no practical use with regard to the existing Western repertoire.”Fn10

     The aim of this article is to show that view to be erroneous. To do so, it will be necessary to establish the remarkable level of agreement among Renaissance theorists on the subject of intervals, to deal with misgivings about Just intonation expressed during the Renaissance, and finally, to demonstrate how singers could have accomplished—and may today accomplish—Just intonation in the context of Renaissance polyphony.



Theoretical Background

Benedetti’s Puzzles

Problematic Passages

Is Just Tuning Possible?

A New Approach

The Exercises

    Problem Spots

    Rehearsal Usage



Footnotes   (Back to top of page)

1. Diversarum speculationum mathematicarum & physicorum liber (Turin, 1585), pp. 277–83.
2. Istitutioni Harmoniche (Venice, 1558), especially theseconda 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.