In theory, each justly intoned interval is represented by a numerical ratio. The larger number in the ratio represents the greater string length on the traditional Monochord and hence the lower pitch; in terms of wave frequencies it represents the higher pitch. The ratio for the octave is 2:1; for the 5th 3:2; for the 4th 4:3. Pythagorean intonation shares these pure intervals with just intonation, but excludes from its ratios any multiples of 5 or any higher prime number, whereas just-intonation theory admits multiples of 5 in order to provide for pure 3rds and 6ths.
To find the ratio for the sum of two intervals their ratios are multiplied; the ratio for the difference between two intervals is found by dividing their ratios. In Pythagorean intonation the whole tone normally has the ratio 9:8 (obtained by dividing the ratio of the 5th by that of the 4th), and so the major 3rd has the ratio 81:64 (obtained by squaring 9:8). But a pure major 3rd has the ratio 5:4, which is the same as 80:64 and thus smaller than 81:64. (The discrepancy between the two (81:80) is called the syntonic comma and amounts to about one ninth of a whole tone.) Since 5:4 divided by 9:8 equals 40:36, or rather 10:9 (a comma less than 9:8), just intonation has two different sizes of whole tone – a feature that tends to go against the grain of musical common sense and gives rise to various practical as well as theoretical complications. Some 18th-century advocates of just intonation and others since have admitted ratios with multiples of 7 (such as 7:5 for the diminished 5th in a dominant 7th chord; see Septimal system).
Two medieval British theorists, Theinred of Dover and Walter Odington, suggested that the proper ratio for a major 3rd might be 5:4 rather than 81:64, and some 15th-century manuscript treatises on clavichord making include quintal and, in one instance, septimal ratios (see Lindley, 1980). Quintal ratios were introduced into the mainstream of Renaissance musical thought by Ramis de Pareia, whose famous theoretical monochord (1482) provided just intonation for the notes of traditional plainchant, but with G–D, B–G and D–B implicitly left a comma impure (fig.1a). Thence Ramis derived the 12-note scale by adding two 5ths on the flat side (A and E) and two on the sharp (F and C); in this scheme (fig.1b), C–A would make a good 5th, hardly 2 cents smaller than pure. Ramis did not intend or expect this tuning to be used in any musical performances, however, for in his last chapter (giving advice to ‘cantors’ and describing what he called ‘instrumenta perfecta’) he said that G–D was a good 5th but C–A must be avoided (see Temperaments, §2).
Gioseffo Zarlino (1558) argued that although voices accompanied by artificial instruments would match their tempered intonation, good singers when unaccompanied would adhere to the pure intervals of the ‘diatonic syntonic’ tetrachord which he had selected (following the example of Ramis's disciple, Giovanni Spataro) from Ptolemy's various models of the tetrachord (fig.2a). Zarlino eventually became aware that this would entail a sour 5th in any diatonic scale consisting of seven rigidly fixed pitch classes (see fig.2b, where D–A is labelled ‘dissonant’); but he held that the singers' capacity to intone in a flexible manner would enable them to avoid such problems without recourse to a tempered scale – and that they must do so because otherwise the ‘natural’ intervals (those with simple ratios) ‘would never be put into action’, and ‘sonorous number … would be altogether vain and superfluous in Nature’. This metaphysically inspired nonsense was to prove a stimulating irritant in the early development of experimental physics, and during the next three centuries a number of distinguished scientists paid a remarkable amount of attention to the conundrum of just intonation (as well as to various attempts to explain the nature of consonance by something more real than sonorous numbers).
In the 1650s Giovanni Battista Benedetti, a mathematician and physicist, pointed out in two letters to the distinguished composer Cipriano de Rore (who had been Zarlino's predecessor as maestro di cappella at S Marco, Venice) that if progressions such as that shown in ex.1 were sung repeatedly in just intonation, the pitch level would change quite appreciably, going up or down a comma each time. In 1581 Vincenzo Galilei, a former pupil of Zarlino, denied that just intonation was used in vocal music, and asserted that the singers' major 3rd ‘is contained in an irrational proportion rather close to 5:4’ and that their whole tones made ‘two equal parts of the said 3rd’. In the ensuing quarrels, Vincenzo Galilei's search for evidence against Zarlino's mystical doctrine of the ‘senario’ (the doctrine that the numbers 1–6 are the essence of music) led him to discover by experiment that for any interval the ratio of thicknesses between two strings of equal length is the square root of the ratio of lengths between two strings of equal thickness. This undermined the theoretical status of the traditional ratios of just intonation as far as the eminent Dutch scientist Simon Stevin was concerned; it might have had further consequences had not Galilei retracted in 1589 his 1581 account of vocal intonation, and had not his son Galileo's generation devised the ‘pulse’ theory of consonance, according to which the eardrum is struck simultaneously by the wave pulses of the notes in any consonant interval or chord (thus mistakenly assuming that the waves are always in phase with one another). Such a theory tended rather to undermine the concept of tempered consonances, where the wave frequencies are theoretically incommensurate.
Descartes found Stevin's dismissal of simple ratios ‘so absurd that I hardly know any more how to reply’, but Marin Mersenne advanced the real argument that the superiority of justly intoned intervals is shown by the fact that they do not beat (1636–7). (He probably gained this argument from Isaac Beeckman, who seems to have invented the ‘pulse’ theory of consonance.) 50 years later, however, Wolfgang Caspar Printz wrote that a 5th tempered by 1/4-comma remains concordant because ‘Nature … transforms the confusion into a pleasant beating [which] should be taken not as a defect but rather as a perfection and gracing of the 5th’. Andreas Werckmeister agreed (Musicalische Temperatur, 2/1691/R).
About this time Christiaan Huygens developed Benedetti's point (although he did not associate it with Benedetti) in his assertion that if one sings the notes shown in ex.2 slowly, the pitch will fall (just as in ex.1); ‘but if one sings quickly, I find that the memory of the first C keeps the voice on pitch, and thus makes it state the consonant intervals a little falsely’. Rameau stated (Génération harmonique, 1737) that an accompanied singer is guided by the ‘temperament of the instruments’ only for the ‘fundamental sounds’ (the roots of the triads), and automatically modifies, in the course of singing the less fundamental notes, ‘everything contrary to the just rapport of the fundamental sounds’. While this represents a musicianly departure from the common error that there is something natural about the scheme shown in fig.2b, it does rather overlook the fact that the tuning of the ‘fundamental sounds’ was normally tempered on keyboard instruments and lutes.
The most eminent scientist among 18th-century music theorists, Leonhard Euler, developed an elaborate and remarkably broad mathematical theory of tonal structure (scales, modulations, chord progressions and gradations of consonance and dissonance) based exclusively upon just-intonation ratios. He failed to observe that a 5th tuned a comma smaller than pure sounds sour, and so allowed himself to be misled by an inept passage in Johann Mattheson's Grosse General-Bass-Schule (1731) into supposing that keyboard instruments of his day were actually tuned in just intonation. Euler at first rejected septimal intervals, saying in 1739 that ‘they sound too harsh and disturb the harmony’, but declared in 1760 that if they were introduced, ‘music would be carried to a higher degree’ (an idea previously voiced by Mersenne and Christiaan Huygens). He published two articles in 1764 to demonstrate that ‘music has now learnt to count to seven’ (Leibnitz had said that music could only ‘count to five’).
Another extreme of theoretical elaboration was reached in the early 19th century by John Farey, a geologist, who reckoned intervals by a combination of three mutually incommensurate units of measurement derived from just-intonation ratios. Farey's largest unit was the ‘schisma’, which was the difference between the syntonic and Pythagorean commas. (The Pythagorean comma is the amount by which six Pythagorean whole tones exceed an octave; the schisma is some 195 cents and has the ratio 32805:32768.) His smallest unit was the amount by which the syntonic comma theoretically exceeds 11 schismas (or by which 11 octaves theoretically exceed the sum of 42 Pythagorean whole tones and 12 pure major 3rds; this is some 1/65-cent, and its ratio would require 49 digits to write out). His intermediate unit (some 0·3 cent) was the amount by which each of the three most common types of just-intonation semitone (16:15, 25:24 and 135:128) theoretically exceeds some combination of the other two units (see fig.3) or the amount by which 21 octaves theoretically exceed the difference between 37 5ths and two major 3rds.