Although the entry for "tritone" in the New Grove Dictionary of Music and Musicians[1] traces the tradition of this interval as the "devil in music" (diabolus in musica) to "medieval" times, I'm not sure just how far back this usage goes. J. J. Fux (1725), in his famous counterpoint treatise Gradus ad Parnassum, gives the traditional rule against a direct melodic leap of a tritone under the more general axiom Mi contra fa diabolus est diabolus in musica[2] -- this general maxim showing a disinclination not only for the augmented fourth or dimnished fifth, but for other "mi contra fa" combinations such as the diminished fourth and the augmented or diminished octave.
As modern writers have suggested, in a context of early organum around the 9th-10th centuries with parallel fifths and fourths predominating, an augmented fourth or diminished fifth might be heard as a kind of a "Wolf" -- that is, an interval which seems an "out-of-tune" variant on an expected concord.
I experienced this directly and vividly around the end of 1966, after taking a music survey course in high school (happily in effect a music history course) which led me to fall in love first with Renaissance, and soon thereafter also with medieval.
I was just learning a bit of music theory and sight-singing as a result of this excitement, when I discovered that I could play the two notes of a fourth simultaneously on a piano, and produce one of the most delightful sounds I could imagine. Sitting down and playing parallel fourths or fifths at a keyboard became my great pastime.
Then I noticed that one such interval sounded horribly out of tune, and asked if the instrument might need retuning. At this point, someone obligingly explained to me what a tritone was.
From this experience, I can see why this interval might not have been received so fondly in early organum -- one solution was a scale with accidentals making it impossible to sing other than perfect fifths (although diminished or augmented octaves might result in some circumstances if octave doubling were applied), while another was the variation of organum at the fourth with oblique or contrary motion at some points to keep the voices within a "tritone-free" range. The latter solution might also reflect trends in popular practice; we cannot be sure.[3]
As has also been pointed out by modern writers, the melodic tritone can be difficult to sing -- a point also made, for example, by Jacobus of Liege around 1325.[4]
However, at least by the epoch of Perotin and his successors, while the tritone was typically classified in the 13th century as a "perfect discord" (along with m2 and M7), it nevertheless occurs, as do these intervals, quite frequently and prominently in practice.
Writing his retrospective exposition and defense of 13th-century style (by now the Ars Antiqua or "Ancient Art"), Jacobus includes the tritone (i.e. augmented fourth equal to precisely three 9:8 whole-tones, or 729:512, e.g. f-b) as one of the 13 basic intervals, and also proposes as a distinct 14th interval the "semitritonus" or diminished fifth of 1024:729, which he finds somewhat less discordant. Note, by the way, that these are not equal intervals in Pythagorean tuning, and that the diminished fifth (about 588 cents, 588/1200 octave) is smaller than the augmented fourth (about 612 cents -- as opposed to the even 600 cents for both intervals in the 12-tone equally tempered scale).
He says that although rare, these intervals do occur in the ecclesiastical chants; and granted that they are discordant and difficult to sing, nevertheless their theory is interesting and beautiful. He views these intervals as difficult in practice, but neither unknown nor apparently as "diabolical."[5]
In 1357, Johannes Boen -- thanks to Jason Stoessel for leading me to this source! -- classifies the tritone as a consonantia per accidens, that is, as a "consonance by circumstance." Specifically, he finds either the tritone or the diminished fourth acceptable when it is accompanied by a lower minor third, e.g. e-g-c#, and likewise for the diminished fourth d-f#-bb. The perfect fourth is likewise in this category of "situational consonance" (to use a modern expression) when accompanied by an octave and lower fifth, e.g. d-a-d'.[6]
Again, the tritone, far from being viewed as "diabolic," is treated as an interval which can be pleasing and even "consonant" in the right context.
Indeed, there are many 13th-century cadences where the tritone serves basically as a "counterfeit fourth or fifth," and Boen suggests that similar progressions may sometimes have occurred in the 14th century as well. In 13th-century practice, the tritonic fourth or fifth typically behaves much like a concordant fourth or fifth, often moving to a (nontritonic) fourth or fifth by parallel motion while other intervals resolve by directed contrary motion:
e'-f'
f'-g' d'-c'
d'-c' bb-c'
b -c' g -f
(m3-1 + m3-5) (M6-8 + m3-5 + m3-1 + M2-4)
Substituing Bb in the first example, or B-natural in the second, would change the flavor of the progression, but not the directed cadential logic.
With the shift from medieval to Renaissance style during the 15th-century, and the advent of an approach where the formerly active thirds and sixths are treated as increasingly stable and even conclusive, the tritone indeed takes on a new vertical role.
In a medieval context, where fifths and fourths are the most complex stable intervals, the tritone is unique among the usual intervals in neither being itself stable, nor in being to resolve to any stable interval by conjunct contrary motion: compare 2-4, 3-1 or 3-5, 6-8 or 6-4, and 7-5.
However, with the establishment of a third (or sixth) as a possible point of repose and arrival, the tritone becomes the one unstable interval which can resolve to this new standard of rich stability by such stepwise contrary motion (d5-m3, A4-m6). Thus around 1500, we may see three-voice progressions like this:
f'-e'
b -c'
d -c'
(M6-8 + d5-M3?)
Here I use a question mark to show that it may be problematic if at first the tritone resolution between the two upper voices was regarded as a distinct two-voice progression in its own right, or merely as a side-effect when the traditional M6-8 cadence between the two lower voices (common in the 13th century and an especially favored 14th-century ingredient of cadences) has the lower part doubled at the tenth by the upper voice, a texture characteristic of this epoch.
In any case, by 1558, Zarlino notes that while the diminished fifth is itself a "nonharmonic relation," it is pleasing as a simultaneous interval if resolved to the ditone (M3), and indeed that this dissonance may be sounded "in a single percussion" -- that is, in note-against-note, unlike the second and seventh which require a more caution treatment as ornamental tones or suspension dissonances. Likewise the augmented fourth is admitted if it resolves to the minor sixth.[7]
Zarlino takes such progressions as characteristic of both "moderns," and "older" composers, which might be translated to refer both to the Josquin generation and thereabouts, and to this theorist's model Willaert and others of his own epoch.[8]
In fact, tritone resolutions take place in one of the most characteristic internal cadences of this period, which may also occasionally occur as a final cadence, e.g.
c'' b' f#' g' c' d' a g m10-M10 M6 - 8 m3 - 5 (M6-8 + m3-5 + d5-M3)
Note that the lower three voices alone, interestingly, would make a typical 13th-century or early 15th-century cadence moving to a stable fifth and octave; but the new triadic harmony of the Renaissance makes the new d5-M3 resolution approved by Zarlino a primary cadential event in its own right, leading to what this theorist calls harmonia perfetta, with a third-plus-fifth-or-sixth above the bass, and what Johannes Lippius (1610, 1612) will soon call a complete "triad."
Thus the tritone comes into play as an essential cadential ingredient and is recognized as such by Zarlino well before this interval is combined with the bold use of the seventh (espoused by Vicenzo Galilei in the 1580's, and implemented by Monteverdi and Gesualdo, among others, in the following decade or so).
In fact, when Giovanni Maria Artusi condemns Monteverdi's new treatment of the seventh in his famous polemic of 1600 on The Imperfections of Modern Music, he faults this composer not for using the tritone, but for using it in a way other than that taught by good authority -- including Artusi himself, in his handbook on counterpoint![9]
While orthodox 16th-century practice and theory included a place for the tritone, Zarlino's more radical contemporary Nicola Vicentino (1511-c. 1572) was ready to go further, advocating the use of even the direct melodic leap of a tritone in his treatise on Ancient Music Adapted to Modern Practice (1555). "Although troublesome to sing, this interval is indispensable whenever the words require a marvelous effect, for by nature it is vivacious and shows great force when ascending, and when descending it makes a very funereal and sad effect.... Some singers do not hesitate to practice it. After all, with continual usage every difficult thing becomes easy in all professions..."[10]
Thus whatever may be the case with the diabolus in musica tradition, it didn't prevent composers from using this interval, nor theorists from endorsing this practice.
The above discussion has focused mainly on vertical tritones, but two related aspects of medieval and Renaissance styles concern tritones "outlined" by a series of melodic intervals in the same direction, and what might be called "diagonal" tritones occurring as "nonharmonic relations" between the consecutive rather than simultaneous tones of two separate parts.
In much medieval plainsong and in monophonic repertories such as the 13th-century Cantigas collection of Alfonso El Sabio, melodic figures outlining a tritone are quite common:
f-a-b-c' b-a-g-f-g ----- -------
The first ascending figure is characteristic of Gregorian chant, while the second is a very popular cadential figure in the Cantigas. While some modern editors have attempted to "repair" these uses by proposing accidental inflections, others have wisely observed that these tritonic figures are a vital part of the melodic style which need not be "sanitized" to conform to more cautious later standards.
A kind of "diagonal" tritone relationship evidently taken quite for granted by 13th-14th century musicians, but of concern to Renaissance theorists, occurs for example when a major third expands to a fifth, one of the most favored Gothic two-voice resolutions in theory and practice:
b - c' a - b
. .
. .
. .
g - f f - e
M3 - 5 M3 - 5
As shown by the diagonal dots, these progressions involve a "nonharmonic relation" (sometimes known in the modern literature as a "cross-relation") of a tritone (b-f or f-b) between the first tone of the upper voice and the following tone of the lower voice -- or vice versa.
Interestingly, it would seem that even as 15th-century musicians were finding new uses for the vertical tritone as discussed above, they may have been taking a more cautious attitude to such "diagonal" tritones.
Zarlino, in contrast to his endorsement of the vertical tritone when duly resolved (see above), urges composers to avoid this kind of tritonic "nonharmonic relation" in two-voice writing. He is ready to accept it in pieces for three or more voices, where the richness of the harmony tends to conceal such "faults" and may even make them pleasing.[11] Passages such as the following from a liturgical setting by Jacob Handl suggest that 16th-century practice agreed with Zarlino's liberty on the latter point:
e' - f#'- g' b - d' - e' g# - a - c' e - d - c
In moving from the first to the second sonority, the M3-5 progression of the lowest voices involves the tritonic relation of g#-d; in moving from the second to the third sonority, the M10-12 progression of the outer voices involves a similar relation of f#'-c.
Thus with the melodically outlined and diagonal tritone, as with other manifestations of this interval, we have a picture of stylistic variation in theory and practice rather than an absolute prohibition.
New Grove Dictionary of Music and Musicians 15:485-487, ed. Stanley Sadie (Washington, DC: Grove's Dictionaries of Music, 1980), ISBN 0333231112, "Tritone."
Excerpted in Oliver Strunk, ed. and annotator, Source Readings in Music History: The Baroque Era (New York: W. W. Norton, 1965), pp. 175-203, at 182: "I do not doubt that you have often heard the trite proverb, `Mi contra fa est diabolus in musica.'"
For an example of Chinese polyphony in a rather similar style, see Richard Hoppin, Medieval Music (New York: W. W. Norton, 1978), at p. 193.
Jacobus of Liege, Jacobus Leodiens Speculum Musicae, ed. Roger Bragard, Corpus Scriptorum de Musica 3 (7 vols.), (Rome: American Institute of Musicology, 1955-1973), Book 2, Chapter 85, p. 201: "... et non facilis pronuntiationis."
Of the semitritonus or diminished fifth, Jacobus observes: et ipsam vidi, etsi raro, in aliquibus ecclesiasticis planis cantibus ("and I have seen it, although rarely, in certain ecclestiastical plainsongs"), Book II, Chapter 80, p. 192. In Chapter 83, at 197, he concludes that since the semitritone concords "poorly" (male) as a simultaneous interval and is difficult to sing melodically, "non est mirus si rarus est eius usus qui praxim respicit. Ipsius tamen theorica subtilis est et pulchra." That is, "It is no wonder if its use is rare in practice, but nevertheless its theory is subtle and beautiful." In Chapter 85, at p. 201, Jacobus similarly finds the tritonus or augmented fourth rare in use (raro est usus).
Johannes Boen, Johannes Boens Musica und Seine Konsonanzenlehre, Wolf Frobenius, ed., Freiburger Schriften zur Musikwissenschaft (Band 2), (Stuttgart: Musikwissenschaftliche Verlags-Gesellschaft, 1971), ISBN 3-920670-30-2; see pp. 74-76 on the tritone and other consonances per accidens.
Gioseffo Zarlino, The Art of Counterpoint: Part Three of Le Istitutioni harmoniche, 1558, trans. Guy A. Marco and Claude Palisca (W. W. Norton, 1976), ISBN 0-393-00833-9. For the endorsement of the vertical diminished fifth or semidiapente when resolved to the major third, see Chapter 30, at pp. 67-68. On the augmented fourth or tritone, see Chapter 61, pp. 197-198: "... the small bit of dissonance heard in the tritone or semidiapente passes quickly and adds a sweetness to the following consonance it would not possess alone."
See Chapter 30, at p. 68: "This is practiced by the best modern musicians as it was in the past by some of the older ones."
Thus see excerpts from Artusi's L'Artusi, ovvero, Delle imperfezioni della moderna musica, in Strunk, n. 2 above, pp. 33-44 at 43, where the author objects to a "semidiapente" or diminished fifth in the works of the tactfully unnamed Monteverdi "which leaves the singer in doubt as to whether he is making an error or singing correctly." Condemning this use of the interval after a rest, he adds, "All composers have employed this interval, but in a different way.... As Artusi demonstrates in his Art of Counterpoint [1598], a sixth or some other consonance precedes it."
Nicola Vicentino, Ancient Music Adapted to Modern Practice, tr. Maria Rika Maniates, ed. Claude V. Palisca (New Haven: Yale University Press, 1996), Book I on Music Practice, Chapter 35, p. 77. Vicentino adds that "if the tritone is sung composite [passing through the steps of the interval, e.g. f-g-a-b], why cannot one -- with practice -- sing it incomposite [as a direct leap]?" This advocacy of the direct melodic tritone ties in with his interest in the expressive possibilities of an assortment of chromatic and microtonal intervals which he derives from ancient Greek theory, and also with his endorsement of other "awkward" melodic leaps such as major sixths, sevenths, and ninths when used to convey the emotions of a text. See, e.g., Book IV on Music Practice, Chapter 11, pp. 242-243, which suggests a technique for singing such intervals. Although rare in conventional 16th-century style, leaps of this kind were indeed used by Giaches de Wert in his madrigals, and later by Gesualdo and Monteverdi.
Zarlino, see n. 7 above, Chapter 31, pp. 68-71. He argues that just as there are in medicine "deadly ingredients which in combination with other substances are healthful," so in music, "there are intervals and relations that give little pleasure in themselves, but have wonderful effect when combined with others."
To some preliminary remarks on medieval chord structure
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Margo Schulter