What Zarlino (or, more specifically,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.