Dr. Ross W. Duffin

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Is Just Tuning Possible?


     Benedetti’s puzzles have reinforced the modern view that Just tuning is impossible in practice—that the theory of pure intervals was not transferable to the real world of musical sound even though the ratios were set forth so unanimously by theorists of the Renaissance. It is true that Zarlino, perhaps the most conspicuous champion of Just tuning, had to endure criticism from colleagues at the time. But even Vincenzo Galilei, Zarlino’s “most outspoken and severest critic,”Fn30 in 1589 had to admit that, in practice, singers do use pure intervals rather than tempered ones:

… truly, voices sing musical intervals in their true and perfect ratio, whereas most artificial musical instruments play them more or less distantly from their true form.Fn31

     Among modern critics, Palisca and others implicitly or explicitly call for some form of temperament as the only solution. As Palisca says:

     In [Benedetti’s] examples, the performance will continue on an even keel only if the true intervals of the senario recommended by Zarlino are abandoned through judicious adjustments by the singers. A system of temperament thus becomes a necessity.

     The solution to the problem is obviously equal semitones and tones, or what is called equal temperament.Fn32

     Without endorsing temperament, other critics simply discount the utility of Just intonation, as in Blackwood’s comments cited in the Introduction above,Fn33and Mark Lindley’s characterization of Zarlino’s flexible Just system based on the senario as “metaphysically inspired nonsense.”Fn34

     The fact is, the evidence suggests that Renaissance singers did try to follow the precepts of Just intonation, using simple ratios for both melodic and harmonic intervals wherever possible, and since they did not seem to migrate in pitch as would have been inevitable with strict adherence to Just ratios, it behooves us, I believe, to try to figure out what they did. As I hope I have shown, it is possible in problematic passages, and even in Benedetti’s puzzles, to devise solutions that follow principles of Just tuning but that modify only one or two pure intervals, rather than the entire system, as does temperament. This is a compromise, certainly, but I don’t believe it invalidates the whole concept of Just intonation nor necessitates the total replacement of Just tuning with some form of temperament.  Moreover, most Renaissance choral works do not contain many tuning problems, if any at all, and thus allow the singers to pursue a goal of Just intonation without the kind of severe difficulty encountered in the Benedetti examples.

     As a demonstration of this, I have chosen part of a piece apparently composed around the time Benedetti was writing to Rore—the end of the first part of the Tallis Lamentations (see Ex. 6)—and have annotated an untexted score with the notes as they appear, and changes as necessary. The sound examples include an electronic performance to demonstrate the Just ratios under controlled circumstances, a recording by the Hilliard Ensemble (Thomas Tallis: Lamentations of Jeremiah, ECM 1341 833 308-2 (1987), trk 1 excerpt), used with the kind permission of ECM Records, and lastly, for comparison, an electronic version in equal temperament. While there are one or two notes in the human performance that may not be absolutely perfect (I’m thinking in particular of the top part in mm. 132-34), it still illustrates, overall, that the ideals of Just intonation can be achieved in practice with stunning effect.

Example 6. Thomas Tallis, Lamentations of Jeremiah (end of the 1st part)

Electronic Just


Hilliard Ensemble


Electronic ET



    The members of The Hilliard Ensemble managed to solve the tuning problems successfully and create a version that approaches Just intonation, and my annotated score—buttressed by my electronic re-creation of a Just version—shows how I think that was accomplished. There are actually no tuning problem spots at all, which is why on the annotated score I’ve basically shown only where notes change from their previous positions. The entire passage is mostly in the A mode although it moves briefly to the G mode at m. 129. At m. 136, overlap with G+1 in the lower voices requires D to be at +1 in the top voice, but the excursion is brief and there is no awkwardness in getting back to D0. The essential point, of course, is how astonishing the live performance sounds with this kind of tuning. The final chord in the Hilliard Ensemble performance is one of the most extraordinary examples of perfect tuning I have ever heard, and once when I played it for a performance practice class, one student wrote afterwards that Just tuning was “shocking.” Indeed, it is shocking that the effect of Just intonation in performance is so striking, and shocking also that we don’t get to hear it more often!

      Remembering Benedetti’s epigrammatic comment about experiencing the ratios with the senses, I don’t think anyone would argue that such a performance would be improved by the use of tempered fifths and thirds, and I trust the electronic ET version makes that amply clear. In fact, such vivid practical demonstrations help us understand that music students in the Renaissance had good reason to reconcile their teachers’ basic theory of Just intonation with their own practice of it, and rather than abandoning it all for the compromise of temperament, they must have found a way to make it work. The answer to the question “Is it possible?” must therefore be a resounding “Yes!” The next question must be: “How can we begin to reconstruct such a practice in our own time?”



Theoretical Background

Benedetti’s Puzzles

Problematic Passages

Is Just Tuning Possible?

A New Approach

The Exercises

    Problem Spots

    Rehearsal Usage



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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.”

Page last modified: February 2, 2017