Dr. Ross W. Duffin

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Problematic Passages


     Benedetti’s puzzles were designed expressly to show potential difficulties in using Just intonation. Perhaps more relevant to Renaissance music in general is whether the principles of Just intonation can be made to work on “real” music from the period—especially passages that have been identified as problematic from a tuning standpoint.

     The first example for this purpose actually comes from a reference by Benedetti himself,Fn29 although he did not explain how the tuning was problematic, as he did for the puzzles. It is the end of Rore’s own chanson Hellas, comment which is the secunda pars of En vos adieux. It is not hard to see what Benedetti was referring to (see Ex. 3a). A succession of pure chords will require the Bb in the alto at the end of the second measure to be one comma higher than the first note of the piece, and an immediate reiteration of the imitative entries causes the texture to rise yet another comma. The piece is stable from there to the end but the damage is done.

Example 3a. Rore’s Hellas, comment (ending) in Benedetti’s presumed version


     Furthermore, this passage is a petite reprise, so its wholesale repetition would mean an ascent of four commas if Just ratios were maintained throughout. If, on the other hand, the singers were required to stay stable in pitch and find a quasi-Just solution, what could they do? One possibility is shown in Ex. 3b.

Example 3b. Rore’s Hellas, comment (ending) in Duffin version


     The hinge-points here occur at the beginning of bar 3 and the beginning of bar 5. In both cases, the Bb and F are allowed to make impure intervals against the D0, which remains pure to the G0 tied from the previous bar. (To be pure, Bb and F should be at +1, as in Benedetti’s version.) In the case of bar 3, this is a short, unaccented moment with the D in the bass. The clash is worse in bar 5, but is still not very long, and the overall effect of the stable pitch and otherwise pure harmonies is very favorable. This seems to me a reasonable solution consistent with tolerating short imperfections in order to maintain a consistent pitch.

     Another source for Renaissance musical passages that are problematic from a Just tuning standpoint is The Structure of Recognizable Diatonic Tunings, where theorist and composer Easley Blackwood introduces two examples (see chapter 7, pp. 129–53), both from a single motet: Orlando di Lasso’s four-voice Ave regina caelorum. The first passage begins with a problematic circle of fifths, with suspensions from chord to chord. Ex. 4a shows Blackwood’s interpretation of what the passage must have been like if strict Just ratios were applied.

Example 4a. Lasso’s Ave regina caelorum excerpt in Blackwood’s presumed “Just” version


     Describing the apparently inevitable downward “modulation” by two commas in five measures as “a distortion of the intent,” however, he prefers the following version as an alternative.

Example 4b. Lasso’s Ave regina caelorum excerpt in Blackwood’s own “Just” version


     While Blackwood avoids microtonal modulation with this version, I personally find both of these solutions unsatisfactory. They both include wrenching shifts from the third to the fourth measure that simply cannot be practical, even for musicians whose main aim is to perform in Just intonation. In the Renaissance, of course, performers did not have musical scores for calculating possible comma shifts in the middle of a note, so a wholesale modulation like that at mm. 3–4 seems impossible. If these solutions are what Easley Blackwood regards as plausible realizations of this passage, it is no wonder people have little faith in the practice of Just tuning.

     My own solution, shown in Ex. 4c, begins the same as Blackwood’s preferred version, but avoids microtonal modulation by allowing two impure intervals. In bar 3, the alto’s G-1 is impure against the C0 in the tenor and bass, and in bar 4, the A-1 in the top voice is impure to the D0 in the alto and bass. In both cases, these impure fifths occur as suspensions against notes a major second or minor seventh away. Thus, the impurity is masked to a large degree by the dissonant interval that occurs simultaneously, and there is no other microtonal awkwardness.Fn29a

Example 4c. Blackwood’s first Lasso example in Duffin version



     Blackwood’s second example is the last nine measures of Lasso’s same Ave regina caelorum motet (see Ex. 5a).

Example 5a. Blackwood’s second Lasso example in his own version


     Blackwood essentially starts a comma lower than he ends (with D at –1, for example) but keeps a smoothly consistent pitch throughout until he has to “pay the piper” and raise the pitch two measures from the end in order to finish at 0. One highly unusual aspect of his version—really a side-issue here—is his choice of tuning for the second note in the sixth measure of the Alto part. I have represented it as –x since it is outside the usual comma units of the Just system. What Blackwood says in his discussion of this passage is that, since the chord is like a dominant seventh, he wants a note at the 7th harmonic over G. This note is lower by about 27 cents (1.25 comma) than the note we expect, which is F0. Here, I personally think Blackwood’s choice is vertical priority run amok, especially since the note doesn’t even resolve like it should in a dominant seventh progression. Placing that note in a melodic context is something no performer of Renaissance music would find intuitive or gratifying.

     My own solution, shown in Ex. 5b, relies on what I perceive to be the most likely placement of the notes in certain modes. Since the passage starts by firmly outlining the A mode, the positions of the notes correspond most closely to the tendencies in that mode, including F and C at +1. The beginning of the third measure begins a transition to G mode. The A mode flickers again briefly in the second measure of the second line before G mode settles in once and for all.

Example 5b. Blackwood’s second Lasso example in Duffin version


     Whereas Blackwood delays the comma ascent to the final until the second last measure, my version manages to maintain a more stable pitch throughout by fudging the transition from A mode to G mode in the middle of bar 3. The problem there is that the C wants to be pure to the G in the middle of the measure, but also wants to be pure to the A at the beginning. What I have done is to make the C pure to the A at the beginning of the measure and allow the C to be impure against the G in the middle. This choice is made because the latter C is a dissonant suspension, so its impurity is more easily tolerated. The second awkward moment in my version occurs at the beginning of bar 6, where there is a shift to G at +1, and the tenor part even has to raise a consecutive B. The problem here is that the modal orientation is unstable and the temporary focus on A requires C at +1, necessitating G+1 when the two overlap. The B and E at the beginning of bar 6 could be kept at -1, as in Blackwood’s solution, but sooner or later the texture must rise by a comma, whether at the beginning of bar 7, or bar 8 (as in Blackwood’s version). Ex. 5b seems to me the best solution.



Theoretical Background

Benedetti’s Puzzles

Problematic Passages

Is Just Tuning Possible?

A New Approach

The Exercises

    Problem Spots

    Rehearsal Usage




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Back to fn29 mian text

29. Diversarum speculationum mathematicarum & physicorum liber (Turin, 1585), p. 278.
29a. A different approach to “justifying” the tuning in this passage is offered by by Joe Monzo with Paul Ehrlich at the Tonalsoft encyclopedia of microtonal music, and called Adaptive JI (Just Intonation). It uses parallel extended quarter-comma meantone systems a pure fifth apart, and provides both pure fifths and pure thirds by requiring adjustments of 1/4 of a syntonic comma to purify the fifths in the chords. Note that the annotations are deviations from 1/4-comma meantone, not from Just intonation, so that the series of fourths and fifths at the beginning of the bass part, for example, are of meantone intervals, not Just ones. It is therefore, essentially, a kind of super-extended 1/4-comma meantone.
Clicking on the score will cause a MIDI file of the version to play. There is a fair amount of vibrato which slightly obscures the tuning.
As in my solution, this version calls for imperfection between the G and the C in the middle of m. 3, and between the A and the D in the middle of m. 4. In addition, however, there are imperfections between the F and C at the beginning of m. 4 and between G and D at the beginning of m. 5. The lower neighbor eighths in mm. 4 & 5 are also impure but of short duration.
Perhaps more difficult from a performance standpoint are the numerous quarter-comma shifts in the middle of notes, such as the bass C tied from m. 3 to m. 4. This seems to place this solution squarely in the realm of computer realization, rather than as something that could be sung. The justification of extended meantone systems seems to me a fruitful approach, however, for consort instruments tuned by their makers to play basically in meantone.

Page last modified: February 2, 2017