The Just Intonation System of Nicola Vicentino

This article originally appeared in 1/1: Journal of the Just Intonation Network, 5, No. 2 (Spring 1989), 8-13.

© 1989 by Bill Alves

Humanism began having a profound effect on music theory and aesthetics in the sixteenth century, when musicians began rediscovering or reevaluating the writings of ancient theorists such as Boethius and Ptolemy. The ancient theories of modes and tetrachordal divisions, though often misunderstood, were widely considered prerequisite knowledge to the art of composition. Vicentino presented his own theories and interpretations of ancient theory in his treatise L'antica musica ridotta alla moderna prattica (Ancient Music Restored to Modern Practice), first published in Rome in 1555.

The reevaluation of the sources to which Vicentino refers, especially Ptolemy, Aristoxenus, and Boethius, had become very controversial in the first half of the sixteenth century. Boethius, who advocated Pythagorean tuning, was the primary theoretical source throughout the Middle Ages. However, important discrepancies surfaced between theory and practice with the evolution of polyphony. Specifically, thirds and sixths were considered dissonances by Boethius because of their complex ratios (major third, or ditone, = 81/64 and minor third, or semiditone = 32/27 or 19/16), and yet they were used as "imperfect" consonances in polyphony. Likewise the fourth, considered by Boethius a consonance, was treated as a dissonance. The first Renaissance theorists who suggested that thirds and sixths were consonant because of their proximity to the more simple ratios found corroboration in Ptolemy, who had presented several just systems in his Harmonics, including those of Archytas, Didymus, and Eratosthenes. Ptolemy wrote that the best systems are those in which sense and mathematics agree, and such agreements to him were primarily superparticular just ratios.

Aristoxenus, though, presented an even more radical viewpoint, that the musician's ear should be the ultimate arbiter. Because pitch was a continuum, it could be divided into equal intervals, even if the whole number mathematics of the Pythagoreans could not express them as string lengths. Thus he was considered by sixteenth-century theorists to be the first writer to describe equal temperament, in theory though not in practice. Such use of irrational divisions of a continuum ultimately challenged the whole mystical relationship of music and number which was so much a part of medieval music theory. Vicentino put himself on the middle road of Ptolemy and just intonation, between the strict rationality of the Pythagoras/Boethius school, and the subjectivism of the Aristoxenians. Still, as Kaufmann points out, Vicentino's language, if not his tuning, comes much closer to the Aristoxenian viewpoint in his constant invocation of the musician's aural judgement and instinct in final musical decision [2]. While Vicentino's first-hand knowledge of ancient sources was probably not very extensive, he may have been inspired by humanist scholars around Ferrara [3].

Vicentino's defence of Ptolemy and just ratios had an immediate predecessor in Lodovico Fogliano, whose Musica Theoretica of 1529 was the first thoroughly logical justification of imperfect consonances as just ratios. He described a just tuning system which applied Ptolemy's syntonic diatonic tetrachord (though with the intervals in reverse order) to the modern keyboard. Fogliano was not unaware of the price of these pure intervals, and suggested that what one really needs are two keys for D (one to be a 5/3 from F and the other a 3/2 from G) and also Bb. Though split-keys were known at the time, Fogliano admits that they are an impractical option. However, the idea of extending a tuning system through multiple division of the octave was clearly a logical extrapolation, and one which would also bring into the system the long-rejected enharmonic and chromatic genera of the Greeks. This was Vicentino's innovation.

One concern of many composers of the emerging sixteenth-century "avant garde," which included Vicentino, was the expression of text and the moving of affectations, or passions, within the listener. The traditional and more conservative composers were primarily concerned with such compositional aspects as balance, unity, and harmony, but all of these, Vicentino says, could be sacrificed to the goal of better expressing the text. Ancient music theory seemed to provide clues to the achievement of this goal from a time when music had such powerful behavior-modifying properties that Pythagoras could calm a mad man, Orpheus could tame wild beasts, and Plato was very concerned about society's control over music. Among the most important passages in the ancient writings which became known around this time was the following statement from Plutarch's De musica, which was translated by Carlo Valgulio and published in 1507: "The musicians of our times, though, disdained completely the most beautiful genus of all [the enharmonic] and the most fitting, which the ancients cherished for its majesty and severity" [4].

Vicentino believed that it was primarily the chromatic and enharmonic genera which were the most "sweet" and expressive, but that most modern composers mixed the genera indiscriminately, did not use them to reflect the text, and concentrated on the diatonic genus. His theories and advocacy of chromatic and enharmonic music were considered quite radical and widely criticized by the conservatives. Even many later musicians of the progressive camp would agree that the diatonic was the only practical genus. Vicentino's most famous contemporary critic, Ghiselin Danckerts, said, "The harmony of the enharmonic or chromatic order does not produce greater miracles than that of the diatonic, even though the Don Nicola [Vicentino] will willingly suggest coarse people that with his enharmonic and chromatic music he stops rivers and indomitable wild beasts better than with the diatonic . . . if would find a fool to believe it"[5].

Vicentino defines the genera with the traditional tetrachords: the diatonic consisting of some combination of two whole tones and a semitone; the chromatic with two semitones, one major and one minor, and a minor third; and the enharmonic with two dieses (intervals smaller than a semitone) and a major third. Like the chromatic semitone, the enharmonic diesis may be either major, which would be the same size as the minor semitone, or minor, which would be one-half of that interval. Vicentino used a dot above the note to indicate that it was raised by a diesis. The way these three intervals are arranged within a perfect fourth defines the "species of the fourth," and four intervals may be analogously arranged within a perfect fifth to derive the "species of the fifth." When conjunctly joined, species of the fourth and fifth form the octave species, which include the traditional church modes.

The derivation of these species within the genera can be considered transformational; that is, the types and order of the intervals in the enharmonic species are analogous to the same in the chromatic, which is in turn a transformation of the diatonic [6]. Of course, these transformations and the church modes themselves are not a part of ancient theory, and Vicentino repeats the common confusion of the modern modes, with an implication of a tonic, with the Greek tonoi, which had no such center [7]. Vicentino also consciously strays from Boethius, his principal model for the definitions of the genera, in stating that the major diesis of the enharmonic genus may be split into two minor dieses, thus forming an extra interval within the tetrachord. This means that an octave species, or mode, constructed in the enharmonic genus may have from nine to thirteen tones per octave. Similarly, the extra whole tone in the diatonic species of the fifth becomes two semitones when transformed into the chromatic genus, forming a mode of eight notes per octave. Such concepts as the variable number of tones in an octave species as well as a gamut containing all intervals were also foreign the ancient theorists.

Vicentino's just ratios are not always defined explicitly but often in terms of other intervals. His gamut of intervals, with some holes filled in, is given in table 1. The whole tone (or "natural" tone) is exceptional. In order for two whole tones to add up to a pure major third (5/4), two different whole number ratios are needed: 9/8 between ut and re in the hexachord and 10/9 between re and mi. This is normal in just tuning, but the discrepancy apparently disturbed Vicentino, who finally justified it by saying that the difference was insignificant, especially in singing [8]. As can be seen in the table, this difference of 21.5 cents is in fact the syntonic comma and is no more insignificant than Vicentino's comma or many of the steps between the other adjacent intervals. Nevertheless, it does lead to some confusion for those other intervals which are defined in terms of the whole tone. The tritone, for example, is by definition made up of three whole tones, which gives four possibilities. The ratios chosen in this table, when there is a choice, were made in favor of Ptolemy's tuning systems.

Table 1: Vicentino's Gamut of Intervals and Their Ratios

Interval Name	Ratio	Cents	Remarks
Minor diesis	40/39	43.8	Difference between major and minor semitones
Major diesis	21/20	84.5	Same as minor semitone
Minor semitone	21/20	84.5	Same as major diesis
Major semitone	14/13	128.3
Minor tone	13/12	138.6
Whole tone	10/9 or 9/8	182.4 or 203.9	Two different sizes are needed for correct derivation of scales (see text)
Major tone	8/7	231.2
Minimal third	7/6	266.9	Whole tone (10/9) plus minor semitone (189/160 if 9/8 whole tone is used)
Minor third	6/5	315.6
Less than major third	39/32	342.5	Major third minus minor diesis
Major third	5/4	386.3
Greater than major third	50/39	430.1	Major third plus minor diesis
Less than perfect fourth	13/10	454.2	Perfect fourth minus minor diesis
Perfect fourth	4/3	498.0
Greater than fourth	160/117	541.9	Perfect fourth plus minor diesis
Tritone	45/32	590.2	Major third plus 9/8 whole tone (25/18 if 10/9 whole tone is used)
Diminished fifth	75/52	634.1	Tritone plus minor diesis
Less than fifth	117/80	658.1	Perfect fifth minus minor diesis
Perfect fifth	3/2	702.0
Greater than fifth	20/13	745.8	Perfect fifth plus minor diesis
Less than minor sixth	39/25	769.9	Minor sixth minus minor diesis
Minor sixth	8/5	813.7	Inversion of major third
Greater than minor sixth	64/39	857.5	Minor sixth plus minor diesis
Major sixth	5/3	884.4	Inversion of minor third
Greater than major sixth	12/7	933.1	Inversion of minimal third
Less than minor seventh	7/4	968.8	Inversion of major tone; not mentioned explicitly by Vicentino
Minor seventh	9/5 or 16/9	1017.6 or 996.1	Inversion of whole tone
Greater than minor seventh	24/13	1061.4	Inversion of minor tone
Major seventh	13/7	1071.7	Inversion of major semitone
Greater than major seventh	40/21	1115.5	Inversion of minor semitone/major diesis
Less than octave	39/20	1156.2	Inversion of minor diesis; not mentioned explicitly by Vicentino
Octave	2/1	1200

Table 2 compares Ptolemy's various just tetrachord tunings with the intervals within the tetrachord of Vicentino's system. It is clear that Vicentino did, true to his word, follow Ptolemy's example. Some notable exceptions are those ratios containing 13 either in the numerator or denominator, a number which does not appear in any of the tetrachords of Ptolemy, Didymus, or Eratosthenes, nor Boethius or Aristoxenus, for that matter. This choice is particularly puzzling for the interval of the minor tone, which would have been closer to the midpoint between the adjacent steps if Ptolemy's simpler ratios 12/11 or 11/10 had been chosen. Apparently, the 14/13 major semitone and 13/12 minor tone are derived from an arithmetic mean of the minor tone (14/13 x 13/12 = 7/6, see below). Ptolemy's 15/14 or 16/15 would have been much closer to an equal tempered semitone, if that indeed concerned Vicentino. In fact, Vicentino's language leads one to believe that he considers the ratios in his gamut to define more or less equal steps, when they, in fact, do not.

Table 2: Ptolemy's Tetrachord Tunings Compared with Vicentino's Intervals

Ptolemy							Vicentino
Enharmonic ratio/cents	Syntonon Chromatic ratio/cents	Malakon Chromatic ratio/cents	Malakon Diatonic ratio/cents	Tonaion Diatonic ratio/cents	Homalon Diatonic ratio/cents	Syntonon Diatonic ratio/cents
							13/10 454.2
							50/39 430.1
5/4 386.3							5/4 386.3
							39/32 342.5
		6/5 315.6					6/5 315.6
	7/6 266.9						7/6 266.9
			8/7 231.2	8/7 231.2			8/7 231.2
				9/8 203.9		9/8 203.9	9/8 203.9
			10/9 182.4		10/9 182.4	10/9 182.4	10/9 182.4
					11/10 165.0
	12/11 150.6				12/11 150.6
							13/10 138.6
							14/13 128.3
		15/14 119.4
						16/15 111.7
			21/20 84.5				21/20 84.5
	22/21 80.5
24/23 73.7
		28/27 63.0		28/27 63.0
						40/39 43.8
46/45 38.1

Vicentino either implicitly or explicitly defines the intervals within his system up to three different ways: as a just ratio, as a division of another interval, or through the tempered fifth tuning system of the archicembalo. Unfortunately these three rarely coincide. For example, he initially defines his "comma" as the difference between the tempered and pure fifth. If one accepts that he is using common meantone temperament for his practical tuning, this would be one-fourth of the syntonic comma, or 5.4 cents. However, he also defines it as one-half of his minor diesis, for which he gives a ratio of 40/39. Using the geometric mean, or square root, of this ratio, it would come out to 21.9 cents. In his archicembalo tuning the minor diesis varies in size from 13 to 65 cents, so that the comma would not be a constant interval. Perhaps it is best to just to accept the informal definition of it as the smallest audible interval [9]. The comma is not included in the gamut because it is smaller than the smallest step necessary to realize the enharmonic genus; however, it does play a role in the tuning of the archicembalo.

Vicentino's additive definitions of his intervals are given in table 3, and, if these definitions are to reconciled with the ratios in table 1, the same problem would result as with two whole tones adding up to a major third. However, these discrepancies are not mentioned by Vicentino, and one is left to assume that, like the discrepancy of temperament, these differences were, to Vicentino, insignificant enough to be ignored (at least in so far as it suited his purposes). Exactly what is meant by dividing these intervals is not clear either. Three different methods of dividing a ratio were known at this time. In the arithmetic proportion, used most by Ptolemy, ratios are calculated from adjacent numbers in a linear series. Thus Vicentino's minor tone (7/6), as mentioned above, could be represented as 14:12, and a mean then linearly interpolated: 14:13:12, giving the two dividing ratios, the major semitone (14:13) and the minor tone (13:12). Vicentino also gives an example of dividing the 9/8 whole tone arithmetically into a 17/16 major and an 18/17 minor semitone, which contradicts other ratios given.

Table 3: Vicentino's Interval Equivalencies

Minor diesis	= 2 commas
Major diesis	= 2 minor dieses
Minor semitone	= Major diesis
Major semitone	= Minor diesis + major diesis	= 3 minor dieses
Minor tone	= 2 major dieses	= 4 minor dieses
Whole tone	= Minor semitone + major semitone	= 5 minor dieses
Major tone	= Major semitone + major semitone	= 6 minor dieses
Minimal third	= Whole tone + major diesis	= 7 minor dieses
Minor third	= Whole tone + major semitone	= 8 minor dieses
Less than major third	= Major third + minor diesis	= 9 minor dieses
Major third	= 2 whole tones	= 10 minor dieses
Greater than major third	= Major third + minor diesis	= 11 minor dieses
Less than fourth	= Perfect fourth - minor diesis	= 12 minor dieses
Perfect fourth	= Major third + major semitone	= 13 minor dieses
Greater than fourth	= Perfect fourth + minor diesis	= 14 minor dieses
Tritone	= 3 whole tones	= 15 minor dieses

Boethius shows that, since this method of dividing an interval produces these component intervals of unequal sizes, that the whole tone is not divisible into equal parts, as the Aristoxenians held [10]. The concept of an irrational square root and thus the Aristoxenian viewpoint were still controversial at this time. However, the rediscovery of Euclid's Elements by the humanists had unearthed another method of dividing an interval, called the geometric mean, which produced an approximation of the square root [11]. Fogliano used this method to temper the D and Bb between the just ratios of his system. The third method for finding a mean is called the harmonic, and Vicentino gives a long algorithm for finding it, algebraically equivalent to 2ab/(a+b) [12].

After presenting all of this theory, Vicentino finally says that the archicembalo is tuned not tuned justly at all, but "according to the use of the other [keyboard] instruments with the fifths and fourths somewhat shortened, as the good masters do . . ."[13]. This is the statement Barbour interpreted as indicating 1/4-syntonic comma temperament, and he showed how his division of the octave into thirty-one tones very cleverly extends this temperament into practically a closed system closely approximating 31-tone equal temperament [14]. Is the elaborate justly tuned gamut simply a numerological justification for his practical tempered system, or did Vicentino really recognize the important differences between tempered and just systems? It is impossible to say for sure, and Vicentino offers evidence both ways. It is true that he does speak of the interval definitions as if they are equivalencies and calls the syntonic comma aurally insignificant. On the other hand, he does offer methods for achieving pure fifths on the archicembalo.

While Vicentino's gamut of intervals contains 31 notes, his archicembalo contains 36 keys per octave. The extra five keys, on the sixth "order" or keyboard rank,[15] are tuned a comma higher than the tempered D, E, G, A, and B. If the definition of a comma as the difference between the tempered and pure fifth is used, one could maintain just triads by playing the fifth in this order and the root and third in the appropriate tempered order. Furthermore, Vicentino gives an optional tuning in which all of the notes in orders four, five, and six are tuned likewise a comma sharp. Presumably, this would limit the number of available intervals from the gamut, but would provide more opportunity for consonant fifths, if one could reach the keys, of course. Both Kaufmann [16] and Barbour [17] have pertinent questions about the interpretation of this tuning system, but Vicentino does say that the main purpose of the comma is "to aid a consonance" [18].

As reflected in the carefully worded title of his treatise, Ancient Music Restored to Modern Practice, Vicentino did not attempt to resurrect ancient composition, but rather to use the theories of the ancients as the proper starting point for a modern theory of composition.Vicentino's view of music history is essentially as an evolution towards a more perfect art. Hence, while he often used ancient sources (somewhat selectively) to justify his own positions, he was not at all afraid to modify them for the sake of modern progress. The tempering of the fifth is an example of one of his "improvements" as well as his desire for practical system that is "downwardly compatible" with contemporary practice.

Vicentino's system was a unique solution to the problems of tuning faced by musicians of this period, and he made a noble effort to reconcile the wisdom of the ancients the needs of polyphony and modulation within one consistent system. His radical advocation of microtonality seems very modern to us but was still very much born of the spirit of his time, the same way Partch's system was in our own century. Vicentino's studies and applications of ancient music theory, though flawed, were very influential and controversial throughout the second half of the sixteenth century, and he was an important part of the late renaissance "avant garde" that ultimately led to the baroque era.

Notes

1 Barbour 117-118.

2 Kaufmann 1966, 105.

3 Palisca, 253.

4 Translated in Palisca, 109.

5 Danckerts, Part II, Ch. 9, fol 401v., translated in Berger, 41.

6 For a detailed reconstruction of these derivations, see Berger, 8-18.

7 Gombosi, 21.

8 Vicentino, fol. 143. Kaufmann 1966, 119.

9 Vicentino, fol. 143.

10 Bower-Boethius 171-178.

11 Palisca, 241-243.

12 Kaufmann 1966, 109-110.

13 Vicentino, fol. 103 v.

14 Barbour, 117-118.

15 Kaufmann 1970, 84.

16 Kaufmann 1966, 171.

17 Barbour, 118.

18 Vicentino, fol. 18.

Works Cited

Aristoxenus, The Harmonics, ed. with translation, introduction, and notes by Henry S. Macran. Oxford: Clarendon Press, 1902.

Barbour, J. Murray. Tuning and Temperament: A Historical Survey, 2nd ed. East Lansing: Michigan State College Press, 1953.

Berger, Karol. Theories of Chromatic and Enharmonic Music in Late Sixteenth Century Italy. Ann Arbor, Michigan: UMI Research Press, 1980.

Bower, Calvin M. Boethius' The Principles of Music, An Introduction, Translation, and Commentary. Ann Arbor: University Microfilms, Inc., 1967.

Danckerts, Ghiselin. Trattato sopra una differentia musicale. Rome: Biblioteca Vallicelliana, Ms. R. 56. Excerpts translated in Berger.

Gombosi, Otto. "Key, Mode, Species," Journal of the American Musicological Society, IV (1951).

Kaufmann, Henry. The Life and Works of Nicola Vicentino. American Institute of Musicology, 1966.

Kaufmann, Henry. "More on the Tuning of the Archicembalo", Journal of the American Musicological Society 23 (1970).

Palisca, Claude V. Humanism in Italian Renaissance Musical Thought. New Haven: Yale University Press, 1985.

Ptolemy, Claudius. Harmonicorum libri tres, Latin translation by John Wallis. Oxford: Oxford Press, 1682, facsimile in Monuments of Music and Music Literature in Facsimile, 2nd series, v. 60. New York: Boude Brothers, 1977.

Vicentino, Nicola, L'antica musica ridotta alla moderna prattica Rome, 1555. All translations, unless otherwise noted, are from Kaufmann 1966.