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

Nicola Vicentino (1511-c.1576) was a remarkable theorist and composer whose fame today rests chiefly withhis advocacy of chromatic and even microtonal music. Unlike the traditional theorist of the Middle Ages, hewas not content to just confine himself to abstract mathematical theory; he showed how his theories could beapplied to practical composition and tuning. In fact, he designed and built at least two keyboard instrumentsdesigned to play in all of the Greek genera: a harpsichord with thirty-six keys per octave which he called thearchicembalo and a comparable portative organ, the arciorgano. His tuning system included thirty-onepitches within an octave, which, as Barbour reasonably interprets it, was most probably applied to thearchicembalo through a cycle of fifths tempered by 1/4-syntonic comma, the same interval commonly used inmeantone temperament at the time [1]. However, it is less well known that he also defined the intervals of hissystem either implicitly or explicitly at least two other ways: as just ratios derived from ancient theorists, andas divisions of other intervals.

Humanism began having a profound effect on music theory and aesthetics in the sixteenth century, whenmusicians 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 widelyconsidered prerequisite knowledge to the art of composition. Vicentino presented his own theories andinterpretations of ancient theory in his treatise L'antica musica ridotta alla moderna prattica (Ancient MusicRestored to Modern Practice), first published in Rome in 1555.

The reevaluation of the sources to which Vicentino refers, especially Ptolemy, Aristoxenus, and Boethius, hadbecome very controversial in the first half of the sixteenth century. Boethius, who advocated Pythagoreantuning, was the primary theoretical source throughout the Middle Ages. However, important discrepanciessurfaced between theory and practice with the evolution of polyphony. Specifically, thirds and sixths wereconsidered dissonances by Boethius because of their complex ratios (major third, or ditone, = 81/64 and minorthird, 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 Renaissancetheorists who suggested that thirds and sixths were consonant because of their proximity to the more simpleratios found corroboration in Ptolemy, who had presented several just systems in his Harmonics, includingthose of Archytas, Didymus, and Eratosthenes. Ptolemy wrote that the best systems are those in whichsense 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 ultimatearbiter. Because pitch was a continuum, it could be divided into equal intervals, even if the whole numbermathematics of the Pythagoreans could not express them as string lengths. Thus he was considered bysixteenth-century theorists to be the first writer to describe equal temperament, in theory though not inpractice. Such use of irrational divisions of a continuum ultimately challenged the whole mystical relationshipof music and number which was so much a part of medieval music theory. Vicentino put himself on themiddle 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 auraljudgement and instinct in final musical decision [2]. While Vicentino's first-hand knowledge of ancient sourceswas 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, whoseMusica Theoretica of 1529 was the first thoroughly logical justification of imperfect consonances as justratios. He described a just tuning system which applied Ptolemy's syntonic diatonic tetrachord (though withthe intervals in reverse order) to the modern keyboard. Fogliano was not unaware of the price of these pureintervals, and suggested that what one really needs are two keys for D (one to be a 5/3 from F and the other a3/2 from G) and also Bb. Though split-keys were known at the time, Fogliano admits that they are animpractical option. However, the idea of extending a tuning system through multiple division of the octavewas clearly a logical extrapolation, and one which would also bring into the system the long-rejectedenharmonic 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 andmore 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 hadsuch powerful behavior-modifying properties that Pythagoras could calm a mad man, Orpheus could tamewild beasts, and Plato was very concerned about society's control over music. Among the most importantpassages in the ancient writings which became known around this time was the following statement fromPlutarch's De musica, which was translated by Carlo Valgulio and published in 1507: "The musicians of ourtimes, 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 toreflect the text, and concentrated on the diatonic genus. His theories and advocacy of chromatic andenharmonic music were considered quite radical and widely criticized by the conservatives. Even many latermusicians of the progressive camp would agree that the diatonic was the only practical genus. Vicentino'smost famous contemporary critic, Ghiselin Danckerts, said, "The harmony of the enharmonic or chromaticorder does not produce greater miracles than that of the diatonic, even though the Don Nicola [Vicentino] willwillingly suggest coarse people that with his enharmonic and chromatic music he stops rivers and indomitablewild 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 oftwo whole tones and a semitone; the chromatic with two semitones, one major and one minor, and a minorthird; and the enharmonic with two dieses (intervals smaller than a semitone) and a major third. Like thechromatic semitone, the enharmonic diesis may be either major, which would be the same size as the minorsemitone, or minor, which would be one-half of that interval. Vicentino used a dot above the note to indicatethat 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, whichinclude the traditional church modes.

The derivation of these species within the genera can be considered transformational; that is, the types andorder of the intervals in the enharmonic species are analogous to the same in the chromatic, which is in turn atransformation of the diatonic [6]. Of course, these transformations and the church modes themselves are nota part of ancient theory, and Vicentino repeats the common confusion of the modern modes, with animplication of a tonic, with the Greek tonoi, which had no such center [7]. Vicentino also consciously straysfrom Boethius, his principal model for the definitions of the genera, in stating that the major diesis of theenharmonic 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 tothirteen tones per octave. Similarly, the extra whole tone in the diatonic species of the fifth becomes twosemitones when transformed into the chromatic genus, forming a mode of eight notes per octave. Suchconcepts as the variable number of tones in an octave species as well as a gamut containing all intervals werealso foreign the ancient theorists.

Vicentino's just ratios are not always defined explicitly but often in terms of other intervals. His gamut ofintervals, with some holes filled in, is given in table 1. The whole tone (or "natural" tone) is exceptional. Inorder 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 thediscrepancy apparently disturbed Vicentino, who finally justified it by saying that the difference wasinsignificant, especially in singing [8]. As can be seen in the table, this difference of 21.5 cents is in fact thesyntonic comma and is no more insignificant than Vicentino's comma or many of the steps between the otheradjacent intervals. Nevertheless, it does lead to some confusion for those other intervals which are defined interms of the whole tone. The tritone, for example, is by definition made up of three whole tones, which givesfour possibilities. The ratios chosen in this table, when there is a choice, were made in favor of Ptolemy'stuning 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 tone10/9 or

9/8182.4 or

203.9Two different sizes are needed for correct derivation ofscales (see text)

Major tone 8/7 231.2

Minimal third 7/6 266.9Whole tone (10/9) plus minor semitone (189/160 if 9/8whole 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 majorthird

50/39 430.1 Major third plus minor diesis

Less than perfectfourth

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 45/32 590.2Major third plus 9/8 whole tone (25/18 if 10/9 whole toneis 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 minorsixth

64/39 857.5 Minor sixth plus minor diesis

Major sixth 5/3 884.4 Inversion of minor third

Greater than majorsixth

12/7 933.1 Inversion of minimal third

Less than minorseventh

7/4 968.8Inversion of major tone; not mentioned explicitly byVicentino

Minor seventh9/5 or16/9

1017.6 or996.1

Inversion of whole tone

Greater than minorseventh

24/13 1061.4 Inversion of minor tone

Major seventh 13/7 1071.7 Inversion of major semitone

Greater than majorseventh

40/21 1115.5 Inversion of minor semitone/major diesis

Less than octave 39/20 1156.2Inversion of minor diesis; not mentioned explicitly byVicentino

Octave 2/1 1200

Table 2 compares Ptolemy's various just tetrachord tunings with the intervals within the tetrachord ofVicentino's system. It is clear that Vicentino did, true to his word, follow Ptolemy's example. Some notableexceptions are those ratios containing 13 either in the numerator or denominator, a number which does notappear in any of the tetrachords of Ptolemy, Didymus, or Eratosthenes, nor Boethius or Aristoxenus, for thatmatter. This choice is particularly puzzling for the interval of the minor tone, which would have been closerto 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 equaltempered semitone, if that indeed concerned Vicentino. In fact, Vicentino's language leads one to believe thathe 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

Enharmonicratio/cents

SyntononChromaticratio/cents

MalakonChromaticratio/cents

MalakonDiatonicratio/cents

TonaionDiatonicratio/cents

HomalonDiatonicratio/cents

SyntononDiatonicratio/cents

13/10454.2

50/39430.1

5/4 386.3 5/4 386.3

39/32342.5

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

13/10138.6

14/13128.3

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 ajust 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 differencebetween the tempered and pure fifth. If one accepts that he is using common meantone temperament for hispractical tuning, this would be one-fourth of the syntonic comma, or 5.4 cents. However, he also defines it asone-half of his minor diesis, for which he gives a ratio of 40/39. Using the geometric mean, or square root, ofthis ratio, it would come out to 21.9 cents. In his archicembalo tuning the minor diesis varies in size from 13to 65 cents, so that the comma would not be a constant interval. Perhaps it is best to just to accept theinformal definition of it as the smallest audible interval [9]. The comma is not included in the gamut because itis smaller than the smallest step necessary to realize the enharmonic genus; however, it does play a role in thetuning of the archicembalo.

Vicentino's additive definitions of his intervals are given in table 3, and, if these definitions are to reconciledwith 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 thediscrepancy of temperament, these differences were, to Vicentino, insignificant enough to be ignored (at leastin so far as it suited his purposes). Exactly what is meant by dividing these intervals is not clear either. Threedifferent methods of dividing a ratio were known at this time. In the arithmetic proportion, used most byPtolemy, ratios are calculated from adjacent numbers in a linear series. Thus Vicentino's minor tone (7/6), asmentioned above, could be represented as 14:12, and a mean then linearly interpolated: 14:13:12, giving thetwo dividing ratios, the major semitone (14:13) and the minor tone (13:12). Vicentino also gives an example ofdividing the 9/8 whole tone arithmetically into a 17/16 major and an 18/17 minor semitone, which contradictsother 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 anirrational square root and thus the Aristoxenian viewpoint were still controversial at this time. However, therediscovery 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 thismethod to temper the D and Bb between the just ratios of his system. The third method for finding a mean iscalled 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 somewhatshortened, as the good masters do . . ."[13]. This is the statement Barbour interpreted as indicating1/4-syntonic comma temperament, and he showed how his division of the octave into thirty-one tones verycleverly extends this temperament into practically a closed system closely approximating 31-tone equaltemperament [14]. Is the elaborate justly tuned gamut simply a numerological justification for his practicaltempered system, or did Vicentino really recognize the important differences between tempered and justsystems? It is impossible to say for sure, and Vicentino offers evidence both ways. It is true that he doesspeak 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. Theextra 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, onecould maintain just triads by playing the fifth in this order and the root and third in the appropriate temperedorder. Furthermore, Vicentino gives an optional tuning in which all of the notes in orders four, five, and six aretuned 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. BothKaufmann [16] and Barbour [17] have pertinent questions about the interpretation of this tuning system, butVicentino 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, Vicentinodid not attempt to resurrect ancient composition, but rather to use the theories of the ancients as the properstarting point for a modern theory of composition.Vicentino's view of music history is essentially as anevolution towards a more perfect art. Hence, while he often used ancient sources (somewhat selectively) tojustify his own positions, he was not at all afraid to modify them for the sake of modern progress. Thetempering of the fifth is an example of one of his "improvements" as well as his desire for practical systemthat 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 hemade a noble effort to reconcile the wisdom of the ancients the needs of polyphony and modulation withinone consistent system. His radical advocation of microtonality seems very modern to us but was still verymuch born of the spirit of his time, the same way Partch's system was in our own century. Vicentino's studiesand applications of ancient music theory, though flawed, were very influential and controversial throughoutthe 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 StateCollege 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. AnnArbor: 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 Society23 (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: BoudeBrothers, 1977.

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

Back to my Home Page

Updated on August 1, 1996 by Bill Alves (alves@hmc.edu).