Dr. Ross W. Duffin

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Theoretical Background


     Beginning with Ramos de Pareja in 1482,Fn11 the simple ratios of Just intonation began to be cited by theorists and used in their divisions of the monochord. Before that, the 3:2 ratio of the pure fifth and its resulting musical system—a whole tone derived from two pure fifths and a major third derived from four pure fifths (in other words, Pythagorean tuning)—comprised the almost universal theoretical model. The ratios of the Pythagorean system, after the 2:1 octave and the 3:2 fifth, were mathematically complex, however, so Ramos’s identification of intervals with simple harmonic ratios was greeted with near-unanimous acceptance. Just intonation was easy to support from both a scientific and practical standpoint because the intervals not only had simple mathematical ratios, they sounded demonstrably better than those derived from the Pythagorean system.

     Table 1 shows the interval ratios promulgated by a selection of prominent Renaissance musical theorists. The one famous “hold-out” who continued to favor the old Pythagorean system was Gafurius, although he did acknowledge that the simple ratios were used by practical musicians.Fn12 As may be seen, there is not much disagreement as to what constituted the correct ratios for intervals: for each theorist I have given those ratios that are either mentioned explicitly in the text, or can be deduced by the writer’s recommended combination of ratios to create other intervals.

     Many of the basic interval ratios are superparticular, where the first number exceeds the second by one. Thus, an octave is 2:1, a fifth is 3:2, a fourth is 4:3, a pure major third is 5:4, a pure minor third is 6:5. These “six-limit” ratios—the senarius, or senario, as Zarlino calls them—are the basic harmonic intervals. Zarlino attributed almost mystical significance to the senario in the same way that Pythagoras apparently did to the tetractys—the first four numbers. But the intervals represented in the senario cannot be the only ones in the system because they do not fully account for stepwise motion. The 9:8 whole tone is in common with the Pythagorean system, but Just intonation also possesses a 10:9 “minor tone” which is the difference between a 5:4 (that is, 10:8) major third and a 9:8 major tone. Like the common harmonic intervals, those smaller intervals are also part of the harmonic series (the series of overtones of any naturally produced sound), so the 10:9 ratio, for example, is simply the distance from the 9th to the 10th harmonic in the overtone series and thus, in itself, a natural acoustical phenomenon.

     Other intervals can be deduced by inversion: the 5:4 pure major third inverts to an 8:5 pure minor sixth. The 6:5 pure minor third similarly inverts to a 5:3 pure major sixth, the 9:8 major tone to the 16:9 lesser minor seventh, the 10:9 minor tone to the 9:5 greater minor seventh.

     It is perhaps surprising to see ratios for the tritone and the diminished fifth given by Renaissance theorists. Some of these are given explicitly as numerical ratios; others were derived by specified combinations of intervals, such as the tritone as a pure major third plus a major tone (5:4 + 9:8 = 45:32), or a pure fifth minus a major semitone (3:2 – 15:16 = 45:32). The diminished fifth (or semi-diapente) is the inversion of that at 64:45, but was also derived as a pure fourth plus a major semitone (4:3 + 16:15).

     The semitone with the lowest ratio in the system, 16:15, can be derived as the difference between the 4:3 fourth and the 5:4 major third, but of course, it also exists as that interval in the harmonic series, as the distance between the ascending overtones gets smaller and smaller. Regarding the other semitone sizes, two of them consist of differences between the 16:15 major semitone and the two different sizes of whole tone: Subtract a 16:15 major semitone from a 10:9 minor tone and the 25:24 semitone remains (10:9 – 15:16 = 25:24); subtract it from a 9:8 major tone and the 135:128 semitone remains (9:8 – 15:16 = 135:128).

     That accounts for most of the ratios in Table 1, with the exception of the semitones given by Vicentino, and the 10:7 ratio given for the tritone by Cardanus. Vicentino’s semitones seem a simple arithmetical way to divide the 9:8 (18:16) major tone, and the 18:17 ratio, in particular, found some use, at least, in lute fretting schemes in the Renaissance. The idea of dividing the whole tone in this way stems from Boethius’s early sixth-century De Institutione Musica I.16.Fn13 So-called septimal ratios—those using the seventh harmonic, like 10:7—are often considered by modern theorists as part of the Just scale, but with the exception of Cardan’s citation, they were clearly avoided by Renaissance theorists and, indeed, I believe they were not normally part of the Renaissance harmonic and melodic vocabulary.

     Table 2 gives the ratios and cents values for Just intervals, along with a selection of other intervals for comparison. One characteristic of the differences between Pythagorean-derived notes and notes in Just intonation is that they are separated by a syntonic comma. The syntonic comma is a microtonal interval which is the difference between the Pythagorean major third, or ditone, at 81:64—that is, two consecutive 9:8 whole tones—and the pure major third at 5:4, or 80:64. Thus, the syntonic comma has the ratio 81:80. This small interval can thus be recognized as the difference between a major tone and a minor tone, and between Just intervals and their Pythagorean equivalents. Modern theorist and composer Easley Blackwood’s comma-variant notation of C0, C+1, and so on, is extremely useful, both for discussions and charts of intervals, as in Tables 1 & 2, as well as in annotating musical scores.Fn14 Briefly, octaves, fifths, fourths, and major tones that are pure to one another have the same annotation; major thirds, major sixths, and minor tones are all narrow by one comma; minor thirds and minor sixths are wide by one comma. I will use Blackwood’s system for analyzing Benedetti’s problemmatic progressions, as well as the difficult passages he himself culled from the works of Orlando di Lasso.

     First, I need to mention some other very small intervals in Table 2. One is the Pythagorean comma, which is the difference between 12 pure fifths and 7 pure octaves. It is larger than the syntonic comma by a miniscule interval called a schisma. This is also the amount by which a pure fifth exceeds an equal-tempered fifth, as well as the amount by which a syntonic comma exceeds the diesis in the Just system—the diesis being the discrepancy between diatonic versions of C# and Db, for example. There are thus three closely related microtonal intervals of about the same size—the syntonic comma and intervals one schisma on either side of it—and these sometimes all get loosely referred to as commas without further definition. Vicente Lusitano, for example, refers to the discrepancy between the major and minor semitone—the diesis—as a comma, suggesting that for practical purposes in Renaissance performance, they were probably indistinguishable.Fn15 These three microtonal intervals are given in Table 3, with buttons for hearing their sound produced by electronic means. As with all of the sound examples in this article, these respresentations are correct to within one cent (1/100 of an equal-tempered semitone)—typically much closer than that.

Table 3. Common Microtonal intervals of Just intonation







C#-1 – Db+1



syntonic comma

C0 – C+1



Pythagorean comma

C0 – B#0




     These kinds of discrepancies and, indeed, the whole system of Just intervals may seem arcane to modern musicians (and even to musicologists), but, as shown by Table 1, these ratios constituted the essence of intervallic relationships in the Renaissance. The principal intervals of Just intonation are given in Table 4, along with sound files with consecutive and simultaneous demonstrations of the intervals.

Table 4. Just intervals with their sounds







minimal semitone

C0 – C#-2



minor semitone

C0 – C#-1



major semitone

C0 – Db+1



maximum semitone

C0 – Db+2



minor tone

C0 – D-1



major tone

C0 – D0



minor third

C0 – Eb+1



major third

C0 – E-1



perfect fourth

C0 – F0



augmented fourth

C0 – F#-1



diminished fifth

C0 – Gb+1



perfect fifth

C0 – G0



minor sixth

C0 – Ab+1



major sixth

C0 – A-1



minor seventh (lesser)

C0 – Bb0



minor seventh (greater)

C0 – Bb+1



major seventh

C0 – B-1




C0 – C0




     The remarkable degree of agreement among Renaissance theorists, particularly with regard to the most common harmonic intervals, leads inexorably to the conclusion that Just intervals were preferred in both theory and—as even the theoretically objecting Gafurius confirms—in practice. The “sounding number” of these pure ratios, furthermore, would be experienced by singers—reinforced, in a live acoustic, by various combination tones—as sonorous, beatless, tranquil chords which, I believe, would have been the constant goal of performers. At the very least, from the historical evidence, it may reasonably be inferred that Just intonation represents an aesthetic preference over other systems on the part of Renaissance musicians generally, and that attempts should be made to explore Just intonation as a way of understanding that preference. However, Benedetti’s shrewdly conceived progressions, discussed below, as well as the complaints of Zarlino’s erstwhile pupil Vincenzo Galilei,Fn16 have led modern scholars to believe that the use of these Just intervals as recommended by the theorists was unsustainable, and that temperament—even equal temperament—was the necessary result. I hope to show that such a conclusion is not inevitable.



Theoretical Background

Benedetti’s Puzzles

Problematic Passages

Is Just Tuning Possible?

A New Approach

The Exercises

    Problem Spots

    Rehearsal Usage



Footnotes –   (Back to the top of page)

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. Back to main text Fn14
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.

Page last modified: February 2, 2017