In Part Three of Gioseffo Zarlino's "Le Istitutioni Harmoniche", "The Art of Counterpoint", Zarlino argues that the diatessaron (or perfect fourth) is not a dissonance, as the musicians of his time had relegated it to, but a consonance. He defends this assertion (only in part, although I won't be discussing the other arguments here) by stating:

An interval that has a rational proportion between high and low pitch is consonant, as is shown in the definition of the philosopher [Aristotle] given in Chapter 12 of the second book [Posterior Analytics]. Therefore the fourth, having such a proportion, is consonant. A minor proposition is demonstrated by Philoponus, who, adding to the philospher's definition in the Second Book of the Poteriora, calls the sesquitertia (which is the true ration of the fourth) a rational number.

Zarlino verifies his second premise (that the fourth has a rational proportion) with Philoponus's statement that 4:3 (the ratio of the fourth), or the sesquitertia, is a rational number.

My first question is, is it safe to assume that by an interval with a "rational proportion", Zarlino means an interval that is a rational number, or is this a conflation on his part? If it is the case that he does mean this, what is his definition of a rational proportion/number? It cannot be synonymous with the modern definition (a number that can be made by dividing two integers), as dissonant intervals of every conventional tuning system in Europe at the time (as far as I know) would also be consonant because they are rational (albeit ugly) numbers. Can someone clear this up for me?

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    "Philoponus's statement that the fourth (or sesquitertia, as he calls it) is a rational number": Philoponus is not writing about musical intervals. "Sesquitertia" denotes the ratio 4:3, just as "sesquialtera" denotes the ratio 3:2. – phoog Jan 26 at 22:54
  • Thanks, noted and edited. – Massimo Asteriti Jan 26 at 22:57

What Zarlino (by way of the Greeks) is talking about is proportions of small integers. The octave, for example, has a ratio of 2:1; the fifth 3:2. Zarlino is arguing that since the fourth has a ratio of 4:3, it, too, like the fifth and octave, should be considered a consonance.

So even though other intervals are "rational" in the modern mathematical sense, the concern for Zarlino (i.e., the Greeks) was ratios of small integers.

Unlike the Greeks, who put primacy on the numbers 1 - 4, Zarlino centered his harmonic theory around the numbers 1 - 6, the senario. Those digits are capable of producing all of the just-intoned consonances:

  • 2:1 (octave)
  • 3:2 (fifth)
  • 4:3 (fourth)
  • 5:4 (major third)
  • 6:5 (minor third)

Other intervals (seconds and sevenths) could be calculated as derivations from the above.

Zarlino's whole theory of consonance, then, is related to a series of six numbers, from one to six, or the arithmetical series 1:2:3:4:5:6.... This is really an extension of the Pythagorean system.... Zarlino calls his series the Senario. Therefore,

Delle proprieta del numero Senario et delle sue parti et come tra loro si ritroua la forma d'ogni consonanze musicale.

TRANSLATION: From the propositions of the number Six and from its parts and the relation between them is found the form of every consonance.1

1 Robert W. Wienpahl, "Zarlino, the Senario, and Tonality", Journal of the American Musicological Society Vol. 12, No. 1 (Spring, 1959), p. 30. (Accessed 26 Jan 2021.) See also Benito V. Rivera, "Theory Ruled by Practice: Zarlino's Reversal of the Classical System of Proportions", Indiana Theory Review 16 (), pp. 145 - 170, which includes some discussion of the contemporaneous controversy, specifically with regard to "his former pupil Vincenzo Galilei"; Ross W. Duffin, "Theoretical Background" discusses the derivations of the various intervals.

  • It's not part of the OP, but why did Zarlino's contemporaries disagree with him? – Michael Curtis Jan 26 at 16:54
  • I understand that Zarlino is talking about the proportions of small integers; I'm still confused about his definition of "rational proportion". What makes 2:1, 3:2, and 4:2 rational proportions, but not 9:8, for instance? – Massimo Asteriti Jan 26 at 17:10
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    It may be helpful to include the concept of senario here. – Richard Jan 26 at 20:40
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    @MichaelCurtis Updated the post. In particular, see the footnote, which links to a paper with some discussion of the controversy. – Aaron Jan 26 at 21:28
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    @MassimoAsteriti Please see the current update, which includes, at Richard's suggestion, discussion of the "Senario", which explains why Zarlino didn't include 9:8. – Aaron Jan 26 at 21:29

The fourth gets a funny treatment in music theory. Acoustically, as the inversion of the fifth, it acts like a consonance. However, for some reasons that I'm not sure of, the fourth is treated as a dissonance against the bass note but a consonance between upper voices.

One explanation that I have read is that a fourth (in two-part harmony) tends to indicate a 6-4 chord. The interval of a fourth existed before the 6-4 chords were thought of as chords so that explanation seems anachronistic. More numerology suggests that the fourth obtained from the harmonic series on C is the G-C in higher harmonics. The idea is that the C (upper note of the fourth) is the root of the interval.

I prefer just to observe the treatment; a fourth against the bass is treated as a dissonance; a fourth in upper voices is treated as a consonance. This may be a culturally (or at least stylistically) determined preference.

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    It's mainly because the root is in the overtone series of the fourth rather than the other way around. – phoog Jan 26 at 22:41
  • The minor third doesn't occur either. Riemann had an "inverse" or "negative" version of these intervals through using undertones. The fourth is still the oddball. It's the only interval which has a different treatment than its inversion does. – ttw Jan 27 at 1:55
  • Acoustically, as the inversion of the fifth, it acts like a consonance. However, for some reasons that I'm not sure of, the fourth is treated as a dissonance against the bass note but a consonance between upper voices. Maybe this is the explanation: In the development of polyphony history we had first the organum the parallels of fih^fths or fourths, In this setting fourths have been considered as consonants. But when the harmony was developed from triads the fourth was conceived as a suspended third and thus as dissonant. – Albrecht Hügli Jan 27 at 10:02

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