Richard L. Crocker - Discant, Counterpoint, And Harmony

7/27/2019 Richard L. Crocker - Discant, Counterpoint, And Harmony

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Discant, Counterpoint, and Harmony

Richard L. Crocker

Journal of the American Musicological Society, Vol. 15, No. 1. (Spring, 1962), pp. 1-21.

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Discant, C ounterpo int, and H armonyBY R I C H A R D L. C R O C K E R

HOWFTEN ONE READS,n discussions of medieval music, remarks likethis: " H er e the voices sound a m ajor triad-but, of course, th e com-poser did not think of it that way." A comm endable reservation; but onethat raises the u rgen t question: ho w (didhe think of it?M any feel tha t th e medieval composer did no t think of vertical sonor-ity at all; or, if h e did, only in abstract, mathematical terms. T h is viewholds tha t medieval po lyp ho ny is "linear," tha t vertical sonorities are th eproduct of intersecting melodic lines, and that these sonorities are for-tuitous. If th e medieval composer did pay atten tion t o vertical sono rity,it was only to ensure the use of perfect consonances, that is, unison,fourth, fifth, and octave. This was obviously due (the argument con-tinues) to a mystical trust in num ber ra ther than to a musical trust in thejudgment of th e ear, since these "perfect" intervals sound bad, o r at bestdisembodied. In any case, neither the composer nor the listener is sup-posed to have listened to t h e vertical so nority.T his is a hard doctrine t o swallow. It seems to have arisen w he nmodern ears were first confronted with medieval sounds; accustomed to"traditional harmony," the ear found the sound of medieval music mean-ingless o r intolerable. B ut w he n view ed as the result of sim ultaneous melo-dies, the crudity of the progressions became acceptable, even interesting.In this way medieval music was made accessible to the modern mind,which was willing to attribute philosophic brilliance but not commonsense perception to the musical contemporaries of St. Thomas Aquinas.

Is suc h a d rastic, m erely cereb ral solution as this really necessary-oreven tenable-any longe r? Is it really necessary to den y th e evidence ofour senses (and theirs) that three melodic lines sung simultaneously doin fact strike the ear with a progression of three-note chords? Must weden y the logic of history, tha t to a monopho nic age the m ost striking factof polyphony must have been the presence of three pitches where thereshould be on ly one? Finally, must w e den y the facts of a polypho nic styleth at com pressed th e three "independe nt" melodies into a single octaveand then fused them together with modal rhythm, the most uniformrhythm known to the history of music? Reasonable observers have forsome time suggested a m ore reasonable interpretation. I t requires, I think,only a summing up of these suggestions in order to present an accountof the theory of medieval polyphony more in harmony, so to speak,with the facts.There is one reasonable observer who, it seems to me, must be cited


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2 JOURNAL OF TH E AMERICAN MUSICOLOGICAL SOCIETYb y more than a footnote. In 1937, a quarter of a cent ury ago, Prof.Thrasybulos Georgiades presented the whole matter quite clearly in histhesis, "Englische Diskanttraktate aus der ersten Halfte des 15. Jahrhund-erts." H e showed that the procedures of medieval polypho ny w ere to beexplained as progressions of intervals, a vo -n ote entities. T h e logic of suc hprogressions is distinct o n th e on e hand f ro m melodic, "linear," logic,and on the other from the logic of triads. Prof. Georgiades's interpreta-tion was made in conn ection w ith t he problem of discant in Englan d, andthe v er y fe w w ho have bothered to pursue his remarks have done so largelyin the same connection.] But a recent writer, Sylvia Kenney, has shownthat discant in England is in all essentials the same as discant anywhereelse;2 this makes it easy to ap pl y th e idea of interval progression to all ofmedieval polyp honic the ory .Fo r i t is the medieval view that w e want to understand. W e kno who w w e conce ive it ; wha t w e need to k now is how they conceived it . T odo this, we m ust take hold of their th eo ry book s w ith bo th hands and read.If this reading is done in the light o f Pr of . Georgiades's remarks, on e findsthat the discant authors from the ~ ~ t h ao th e 16th centuries provideclear, consistent, and pe rtin en t acc ou nt of medieval-and Renaissance-polyphony.

F or o u r purpo ses "discant" means a system of teaching tw o-p artcomposit ion, in use f rom the 13th to the 16th c e n tu r ie ~ . ~hi s is the mostcom prehensive definition of th e ter m ; it ca n also refer t o a specific musicalstyle or t o th e upper voice of a composition. W e will no t be concernedwith these more restricted meanings. Discant, so defined, shows how to1 E. Apfel, Studien z w Satztechnik der nzit te lal terlichen englischen Mzlsik , 2 vols.

(Abhandlungen der Heidelberger Akademie der Wissenschaften. Philosophisch-historische Klasse. Jahrg. 1959,j. Abhandlung); also "Der klangliche Satz und der freieDiskantsatz im I j. Jahrhundert," Archiv fur Musikwissenschaft XI1 (I~s!), pp. 297ff.See also G. Schmidt, "Zur Frage des Cantus firmus im 14. und beginnenden 15.Jahrhundert," Archiv fur Musikwissenschaft X V (19 j8) , pp. 23off.

2 " 'English Discant' and Discant in England," Musical Quarterly XLV (19~9) ,p.26ff. Most writers on medieval music are forced, in spite of any convictions to thecontrary, to acknowledge in some degree the existence of a vertical component; tolist all such references would be futile. As special studies one should mention E.Lowinsky, "The Function of Conflicting Signatures in Early Polyphonic hlusic,"Musical Quarterly XXXI (rglj), p. 227; H. E. Bush, "The Recognition of ChordalFormation by Early A4usic Theorists," Musical Quarterly XXXII (19+6), p. 227; G.Reaney, "Fourteenth Century Harmony and the Ballades, Rondeaux, and Virelais ofGuillaume de Machaut," A4usica disciplim VII (1953)~P. 129; H. Tischler, "TheEvolution of the Harmonic S q le in the Notre-Dame hlotet," Acta musicologicaXXVIII (1956), p. 87; K. v. Fischer, "On the Technique, Origin, and Evolution ofItalian Trecento Music," Musical Quarterly XLVII (rg61), pp. 41ff (too late to beconsidered in the present article).3Most of the discant treatises are published by C. E. H. de Coussemaker inScrip torum de musica medi i cevi nova series, q. vols. (Paris, 1864-76) (hereafter ab-breviated as CS), and in Histoke de l 'hamzonze au moyen 6ge (Paris, 1852). Theseversions are not, of course, completely reliable and will one day have to be replaced;for the present survey, however, they are adequate. Other texts will be cited a s needed.


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3ISCANT, COUNTERPOINT, AND HARMONYcombine one (and only one) note wi th each note of a given melodicprogression b y th e application of t w o basic principles. T h e first principledeals with the kinds of sono rity to b e used, th e second wit h the o rder inw hic h sonorities m ay appear.'The first principle requires discant to consist essentially of concordsand o nly accidentally of discords. W i t h slight adjustments in th e definitionof concord, this principle governs not only medieval discant and Renais-sance counterpoint, but also Baroque thorough-bass and traditional har-mony.T h e second principle requires con trary mot ion between the tw o parts .This principle is absolutely binding, but has many, many exceptions. Alarge par t of th e typical discant treatise is devoted t o circu mv ention of thisprinciple, laying down conditions under which similar or even parallelmotion may be used. Here again, reflecting on the nature of thorough-bass or traditional harmony, we can observe: "Plus $a change, plus c'estla m tm e chose."In applying the first principle the critical point is, clearly, the defini-tion of concord. Systematic treatment of the concords of discant appearsfirst in th e the orist Jo hn of G ~ l a n d , ~t a time (mid-I 3th cen tury ) whenthe new, international style of Leonin and Perotin had firmly establishedthe use of these concords. John's formulation, destined to become classic,is itself a clarification of the previous "common doctrine of discant," orDiscantus positio uzdgaris, a short exposition found immediately beforeJohn's in the compendium of Jerome of M ~ r a v i a . ~h e Positio vulgarissays that some intervals, namely unison, fifth, and octave, are better thanothers, and some are more dissonant, but "according to greater or lesserdegree." Ev en in this mode st treatise the re is no hint of an absolute dichot-omy between consonance and dissonance; indeed, the notion of a con-t inuum stretching from consonance to dissonance prevails throughoutthe Middle Ages and Renaissance.John of Garland arranges intervals on the continuum as follows:Perfect Middle Imperfect lmp erfec t Middle Perfectunison fifth major third major sixth major second major seventhoctave fourth minor third minor seventh minor sixth minor second

tritoneThese are rephrasings of Miss Kenney's second and third principles of discant.M y friend and colleague gives as a first principle the requirement that discant shouldconsist of only one note against another. I take this to be not a law like the other two,but rather a part of the definition of discant, like the provision that discant concernsonly two voices.

5 Or perhaps Anon. VII (CS I, 38z), if the passage in question really antedatesTohn's treatise.6 S. M. Oerba, Hieronymrs de Moravia O.P. Tractntus de musica (Regensburg,1935). Th e Discantus positio wuIgaris is on pp. 189-194; John's D e musica m m r a b i l ipositio on pp. 194-230, his discussion of consonance on pp. z g f f .


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4 J O U RN A L O F THE A M E R I C A N M U S I CO L O G IC A L S O C IE T YBeing a theorist as well as a practical musician, John goes on to draw theinference: "The more the string ratio of an interval approaches equality,the more concordant i t sounds." Certainly a justified inference, and onethat brings up the question of the judgment of concord and discord.Medieval writers, from John of Garland on, consistently invoke thejudgment of the ear in discussing the degree of concord and discord.Since this observation is in flat contradiction to the opinion commonlyheld about medieval musicians, it seems prudent to exhibit a quantityof texts sufficient to convince the most sceptical.John of Garland: "A concord is said to exist when two pitches are joined to-gether at the same time in such a way that the sense of hearing tolerates themone with the other. Discord is described c~ntrariwise."~Anonymous I: "A concord is the harmony (harmonia) of two or more dif-ferent sounds produced a t the same time, blending together and reaching theear in sweet uniformity (unifomziter suaviterque veniens ad a~di tum)."~Lambert: "Concord is said to exist when two pitches sounding a t the same timeblend together so that they render sweet melody (suavem melodiam) in theear. . . ."9Anonymous 11: "Discant is composed principally of consonances and onlyincidentally of dissonances, in order that the discant per se may be more beauti-ful, and that we may be more delighted by the consonances. Consonance ismade of diverse sounds mixed together. Dissonance is a rough collision (duracollisio) . " l oATScontrapunctus secundum Philippunz de Vitriaco: "These concords (unison,fifth, octave) are called perfect because they bring a perfect, pure (integrum)sound to the ears of the listeners."llJacob of Li6ge: "Discant is said to be a consonance of different songs: just asconsonance requires distinct pitches mixed together a t the same time, so discantrequires distinct songs (cantus) sounding simultaneously. Not all sounds, how-ever, can be combined into a mixture that will present itself smoothly andsweetly to the listener; similarly, not all songs when mixed together makediscant, but only those that harmonize with one another so that through theirconcord they make as it were one song. . . ."12Johannes Tinctoris: "A concord, therefore, is a mixture of two pitches renderedsweetly agreeable to the ear by a natural power (naturali virtute)."lSClearly, from these statements, it is false to believe that the Middle Agesrelied solely on mathematics and excluded the judgment of the ear in de-termining the nature of consonance. These authors say, in sum, that theear takes pleasure in consonance, and the greater the consonance thegreater the pleasure; and that for this reason one should use chiefly con-sonances in composing discant.14

7 Cserba, Hieronymus de Moravia,p. 207.8CSI,z97. QCSI,260. loCS1,311. 11CSIII,t7.lz Cs 11, 387. l3 CS IV, 78.14The formulas used by the discant writers to describe consonance go back, ofcourse, all the way to Boethius, De insrimtione mzcsica I, viii, ed. G. Friedlein (Leipzig,

1867)~p. 195.


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6 JOURNAL OF THE AMERICAN MUSICOLOGICAL SOCIETYand discant treatises described it as such. In th e 14th cen tury , however,th e discant teachers characterized the fo ur th as discordant, on the gro un dsthat it was now treated like a discord even if it did not sound like one.But this reason did no t satisfy the speculative theorist, wh o k ne w that th eratio of th e fou rth (4: 3) o ccurre d within the tetrad, the first fo ur num -bers, and th at it came imm ediately after the ratio f o r th e perfect fifth( 3 : ~ ) . ur thermore , the fo ur th was a basic e lement in the Pythagoreanharmony:

OCTAVELast bu t n ot least, the f ou rth sounded l ike a consonance. Co nfron ted b yall these arguments, a theorist could hardly assign the fourth a statusof discord for merely stylistic reasons. In fact the anomaly of the fourthis so deep-seated th at acco rdin g to latest re po rts th e issue is still in d ou bt.W e should n ot be upset, therefore, if the m edieval theorist is less tha nconclusive on this point.But th e need fo r forma l explanation did no t touch the practical teacherof discant. After all, he had already described the major and minor thirdsas concords despite their complex ratios ( 8 I :64 and 32 : 27 respectively15).I t cost him lit tle to place the fourth among th e discords. Th is was the onlytime that a concord was demoted to a discord; the oth er changes consistedin raising th e m ajor and mino r sixths to the status of concords-or bette r,in moving the dividing line between concord and discord further downthe continuum to include more complex intervals as concords. FollowingPro f. Georgiades's suggestion,16 w e can c ons truct a tentative genealog y ofthe anonymous discant authors on the basis of their treatment of theseconcords. In the 13th century the anonymous authors I, IV , and VI Ifrom Coussemaker's first volume maintain with Franco the classificationof John of Garland.17 Anonymous II,lB however, includes the major sixthalong with the thirds as an imperfect concord. This change reappears in

15 Their proper names in this tuning are "ditone" and "semiditone" respectively;they are so named by the early discant authors.

16 Englische Diskanttraktate aus der ersten Halfte des 15. Jabrbunderts (Schriften-reihe des Musikwissenschaftlichen Seminars der Universitat hlunchen, Vol. 111,Wurzburg, 1 9 3 7 ) ~p. 61.

17 CS I , 298, 3j8, 382; Cserba. Hiero~aymusde Moravia, pp. 207, rjo; 0 . Strunk,Source Readings i n Mzrsic Histo ry (Ne w York, 1 9 5 0 ) ~pp. 152f.CS 1, 312.


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7ISCANT, COUNTERPOINT, AND HARMONYWolf ' s C ~ m p e n d i m ~ ~I 336), which furtherm ore drops the fou rth fromthe l ist of concords, thus reduc ing their num ber back t o six. T h e num beris increased again to seven in the Ar s contrapunctus secund um Phi l ippumde Vitriaco20 (late 14th ce nt ur y? ) by th e addit ion of th e minor sixth; thisarrangement is reproduced in the Ars discantus secundum loannem deM ~ t r i s . ~ lAll this is better shown in the following table:

' 2 ~ 0 1700 (?) '336 a ft er 1 3 ~ 0?)unison unison unison unisonoctave octave octave octavefifth fifth fifth fifthfourth fourth - -major third major third major third major thirdminor third minor third minor third minor third

major sixth major sixth major sixthminor sixth

W h y th e fo urth should be dropped and the s ixths added to th e lis t ofconcords are questions that would detain us too long here. There are, Iam certain, very clear stylistic answers, towards which Prof. Georgiadespoints the w ay:22 he says that th e major sixth was adopted before th eminor one because the major was part of a progression from fifth tooctave by contrary motion, whereas the minor sixth could proceed onlyto a fifth and that with one voice stationary. Since this latter progres-sion was less congenial to the style as a whole, the minor sixth could notattain concordant status as easily.Concurrent with these changes in the classification of concords, therewas a change in the treatment of intervals larger than the octave. 13th-century discant viewed these as compounds of those smaller than the oc-tave, hence su bject t o similar treatm ent: a fifth and a twelfth, fo r example,we re handled in the same way. T h e authors speak of th e num ber of inter-vals as "infinite," envisaging an endless dup lication of intervals up wardsb y octaves, b u t all subject to th e rules gov erning those below th e octave.23Lamber t (ca. 1260 ), however, gives the six con cords as follows: octav eand double octave (perfect); f ifth and twelfth (middle); fourth andeleventh (imperfect) .24 In other words, h e insists on the Pythagorean con-sonances (fourth, fifth, and octave) as the only true concords, being al-most the only author to d o so; bu t he adapts this doctrine to John's sys-tem of six concords by including the respective octave compounds. Thisanticipates the later treatises Ars perfecta in ??zusica nzagistri Philippoti

1 9 J. Wolf, "Ein Beitrag zur Diskantlehre des 14. Jahrhunderts," Smmzelbande derInternationalen Musikgesellschaft X V (1914) ,pp. SO&2 0 C S 111, 27.cs 111, 70.22 Engliscbe Diskanttraktate, pp. 64f.23 For example, John of Garland: Cserba, Hieronynrus de Morm ia, p. 208.24 C S I, 260; the passage is corrupt but its meaning evident.


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8 JOURNAL OF THE AMERICAN MUSICOLOGICAT, SOCIETYde VitriacoZ5 nd the Libw ms ica l i m Philippi de Vitriac0,2~ hich in-clude the tenth and twelfth among the concords. Since these writersennumerate thirds and sixths witho ut specifying m ajor o r minor, th enumber of concords stays at seven. Both old and new listings are foundin the Ars contrapuncti secundum Ioannem de M~ris ,2~he older on e first.T h e newe r listing: unison, third, fifth, sixth, octave, tenth, and twe lfth,remains standard through the Renaissance, altered only by upward ex-tension.T h e specification of intervals larger than the octave also needs m orediscussion than w e can h ere afford. In passing it is interesting to observewhat the Compendiunz discantus,28 (ascribed to Fra nco bu t possibly later)says abo ut these larger intervals:And note that when you wish to ascend above the diapason, you will imag-ine yourself to be in unison with the tenor, and you will discant in the sameway you would over the tenor in the lower register, because there is really nodifference except that you are higher in pitch. Such elevation [of the discantlcan be repeated indefinitely (multiplex est in infinitum).The phrase "in infinitum" connects these remarks solidly with the doc-trine of John of Garland , bu t th e w or d "imaginabis," and the proceduredesignated thereby, is identical with the 15th-century English principleof "sights."29 Perhaps this modest Compendium is the link that connectsth e doc trine of sights direc tly to traditional discant.

T h e second principle of discant requires con trar y motion between thetwo parts, using the concords already described. This, of course, is thecatch: i t wou ld be simple to write t w o parts in co ntr ary motion if all in-tervals w ere permitted; i t w ould be equally simple to use th e concords ifcontra ry motion were n ot required. We ster n part-music, fro m then unti lnow, depends upon a delicate balance between the demands of verticalsonority and those of voice-leading. Sometimes the balance is threatenedby too much attention to the vert ical or the l inear dimension, but equi-librium is soon restored with the realization that each dimension is mean-ingless with ou t th e other.T h e linear interpretation of m edieval music depends fo r m uch of itsevidence on th e instructions f or com posing fo un d in the discant treatises.Th ese instructions are intended to produce the desired concords throug hcontrary, or at least oblique motion. They are apt to take the followingform:When one part ascends a step, the other, beginning at the octave above, maydescend two steps and be a t the fifth.I t is argued th at th e stress on co ntra ry mo tion in such instructions reflectsan emphasis o n th e linear dimension. I t is fu rth er argu ed th at such instruc-

25 CS 111, 28. 26 CS HI, 36. 27 CS 111, 59f. CS 1 156. 2s See Kenney, "English Discant," pp. 33ff .


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9ISCANT, COUNTERPOINT, AND HARMONYtion implies two simultaneous but independent melodies in the mind ofth e composer and also in the ear of th e listener.The discant teacher does, indeed, stress contrary motion, but so doesth e teacher of traditional fou r-part h armon y, and to t he same degree; onlyhere the result is not so apparent because in four-part writing there isnecessarily more parallel and similar motion, there being still only twodirections in which to go.As a furth er rebuttal , let me point ou t that th e discant treatise does notdescribe wh at th e listener hears, an y m ore th an does th e treatise o n tradi-t ional harmony. In both cases the teacher tells the student how to pro-ceed; he does no t analyze the result as it strikes the ear. T h e typical discanttreatise is a collection of practical precepts on how to make music, nota the or y of aesthetics. T h e instructions of discant, therefore, d o no t im-ply that th e listener hears t w o separate melodies; at most, these instructionsimply only that the composer proceeds by combining two melodies.Do the instructions imply even that? I think not, for i t seems to methat th e assumption that discant taugh t h ow t o write a second tune ov era first is open to question. Consider the typical instruction again: notethat while i t may be taken to regu late the leading of one voice co ntingentupon the leading of the other, the same instruction also regulates theprogression from the vertical sonority of an octave to one of a fifth.N o w this amb iguity arises on ly in describing two -part progressions; intriadic ones the vertical sonority is identified as something distinct (a"triad") fro m th e intervals (a "fifth" and tw o "thirds") th at describe thelocation of its constituent tones. In other words, terminology does notperm it a distinction betw een th e location of one note an octave away fro manother, and the interval of an octave that these two notes form.H en ce w e can speak of a progression of tw o tr iads, one on g and oneo n c, o r over a bass that moves do wn a fifth, or a dominan t triad followedby its tonic, and no one suspects us of describing linear counterpoint.But w hen w e speak of an octave followed b y a f ifth, wi th the lower pa rtascending one step, then w e m ay possibly be describing tw o melodic pro-gressions; and th en again w e ma y not. Assume fo r th e sake of argum entthat the medieval teacher does mean to describe a progression of verticalsonorities, each consisting of two notes: how else could he describe itbut the wa y he does?Just as 13th-century discant lays down the basic doctrine of concordand discord, subject o nly t o slight modification in th e cen turies following,so does it present rules of voice-leading th at go vern bo th th e later MiddleAges and the Renaissance. These rules are discussed and illustrated by13th-century wri ters in a bewildering variety of ways, yet as rules theyare broad and simple. T h e first rule is, of course, th e basic principle of con-tra ry m otion. T h e second (the o rder is mine, no t theirs) is tha t one shouldbegin w ith a concord and end w ith a perfect concord. Th is permits thirds


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I 0 JOURNAL OF THE AMERICAN MUSICOLOGICAL SOCIETYat the beginning b ut n ot at the end. John of G arland extends this rule t oinclude the beginning of modal feet,3O while Franco, carrying out theimplications of his mensural theory, applies the rule to the beginning of a" p e r f e ~ t i o n ; " ~ ~oth authors are speaking of imperfect as well as perfec tconcords.A third rule tolerates similar motion, without, however, precise dis-tinction b etwe en similar and parallel motion, and w itho ut spe cifying per-fect or imperfect concords. Franco says that one can use parallels forbeauty, "propter pulchri tudinem," which should not cause our eyebrowsto rise even if consecutive fifths are in question. Parallel motion, prop-erly handled, can indeed be beautiful, as will be apparent to all but themost academic observers.

It should also be apparent that music cannot consist entirely of con-cords, but must also include discords. A fourth rule of 13th-century dis-can t says that discords should be mixed in with the concords a t the properplaces;32 these places are sometimes described as being befo re o r betw eenconcords, but are not otherwise located. Here, more than ever, i t is im-porta nt t o rem embe r wh at discant aims to do: i t gives systematic instruc-t ion in w rit ing tw o parts. If a technique can not be presented wi th a t leasta semblance of system, discant does no t treat th at technique. C onco rds arethe substance of two-part writing and can be treated systematically; dis-cords are the accidents, to speak in Aristotelian terms, and 13th-centurydiscant foun d i t hard to treat them systematically. Fo r this reason, no t be-cause the discant teacher disapproves of them, discords are passed overin the typical treatise. T he re is no ga p here between th eo ry and practice,save that imposed by the needs of rational discourse.Du ring th e 14th ce ntu ry the name "discant" was gradually changedt o " c o ~ n t e r p o i n t. ~ ~ 3 3he re w as, however, no change in the basic princi-ples; the y w ere me rely applied in a more specific and refined w ay. T h is isbest illustrated b y th e 14th-c entury treatm ent of parallel motion, of w hic hthe first example is in the little-known Compendium of Petrus "dictuspalma ociosa" ( I 3 3 6 ) .z4 A n interesting and well-written w ork , it setsforth near the beginning an informative discussion of discant.

Song (cmtus) is an inflection from one pitch to another. Discant is sweetmelody made up of different songs, with two or more pitches reaching the earin combinations governed by modus and tempus. It is called "discant" as if i twere diverse songs, because the songs out of which discant is made ought todiffer so that when one goes up the other goes down, and conversely. But bothcan ascend or descend together for the sake of the song's beauty, or because of

30 Cserba, Hieronymus de Moravia , p. 2 1 I .31 Cserba, Hier onymts de Mor av ia , p. 254; Strunk, Source Readhgs , p. 155.32 LO C. i t.33 See Kenney, "English Discant," p. 43.34 Edited by J. Wolf; see note 19. The passage translated starts on p. 507. Ascent"in the same way" means by the same interval; "division" means diminution, as indiminished or florid counterpoint.


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D IS CA N T, C O U N T E RP O I N T, A N D H A R M O N Y I Ilimitation of range o r some oth er necessity. Such ascent or descent should notbe made in the same way; rather it should be as elegant and as graceful as pos-sible. I allow, nevertheless, ascents and descents in the same way either bymeans of some division of the intervals or by imperfect intervals, such as minorthird, major third, and major sixth. I do not advise using two or more conso-nances repeated in the same lines o r spaces, either in perfect species of m usicalintervals or imperfect or middle ones.T ha t finished, let us see briefly of which musical species discant should becomposed. Concerning this you should know that all simple discant (which isnothing but punctum against punctum or one note produced b y natural instru-ments placed against ano ther) can be composed and orde red sim ply with unison,minor and major thirds, fifth, major sixth, and octave.After stating the principle of contrary motion, the author allows similarmotion fo r the sake of beauty, as did Franco. Th en the author goes on toadm it parallel mo tion in thirds and m ajor sixths, and this, it seems, is new.Of greatest interest is the way in which the instruction is phrased: i tsnovelty lies not in the banishment of consecutive fifths and octaves, butin the tolerance of consecutive thirds and sixths. Consecutive fifths andoctaves had been banned categorically from the moment when contrarymotion became obligatory; consecutives of all kinds were tolerated onlyas exceptions. Now, in the 14th century, the musician seems to reason:"Some kinds of parallel m otion a re acceptable, oth er kinds are no t. Th os eintervals which are not perfec t concords and yet not discordant seemt o permit parallel m otion w itho ut upse tting the delicate balance ofpolypho ny, whereas consecutive perfec t concords are too str iking in theireffect." Incidentally, it is interes ting tha t discant auth ors freq ue ntly ban-no t parallel motion-but consecutive perfe ct concords, a clear indicationof th e medieval con ce rn fo r th e progression of vertical sonorities.T h e real impor tance , however, of th e imperfec t concords a t th is t imehas to do with contrary, not with para l le l motion. Tempting as i t i s toseize upon consecutive thirds and sixths in older music as evidence ofprogressive tendencies, these consecutives have but little significance forthe future development of musical style. Parallel motion does not producethe basic structures of part-music, as any authority on tr iadic harmonywill testify. By an accident of history, consecutive thirds and sixths re-mind us of traditional harmony, which causes us to apply the labels "har-monic" o r "functional;" b u t in medieval times as we ll as m od ern , paral-lelism is t he an tithes is of f ~ n c t i o n a l i t y . ~ ~O n the o the r ha nd , t he of thirds and sixths within c on tra rymo tion leads us to th e ce nter of 14th-century discant, and ult imately toth e foundations of tr iadic harm ony. A t this t ime the progressions m ajorsixth to octave, major third to f if th, and minor third to unison take onmore and m ore impo rtance as the building blocks of counterpoint. The se

35 C f. H. Besseler, "Tonalharmonik und Vollklang," Acta mussicologica XXIV( '952)t pp. ' 3 5 f .


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I 2 JOURNAL OF THE AMERICAN MUSICOLOGICAL SOCIETYprogressions acquire the force of necessity: their conclusion becomes ob-ligatory. "The sixth seeks out the octave, and this rule always holds,"says t he au thor of t he Ars con trapunctus s ecmd um P h i l i g p de Vitri-a ~ o . ~ ~iew ed fr om this angle, a succession of consecutive sixths or third sis th e inte rru ptio n of expected resolution--of "function," if y ou will.14th-century writers specifically allow several ascending or descendingconsecutive sixths (o r thirds) on condition that the y a re followed by anoctav e (o r fifth o r u nison).37 T h e parallelism allowed here is less signifi-can t than th e resolution required.T h e importance of these progressions is great enough to demand al-teration of the written pitches through musica ficta. If a sixth that pro-ceeds to an octa ve is w ritte n as a minor one because of its position in thescale (f or example, a-f proceeding to g -g) , hen according to 14th-centurywr iters this sixth is to be m ade m ajor b y raising th e upp er note.38 In t heI 3th ce nt ur y musica ficta (falsa) was used chiefly to avoid u-itones and im-perfect octaves, that is, to ensure that the intervals of discant would beconcords. This use of musica ficta, declared by Lambert and Philippe deVi t ry t o be no t false bu t nec es~ ary ,3 ~s another clear indication of themedieval concern for vertical sonority. But the alteration of sixths andthirds reveals the equally impo rtant con cern fo r progression that emergesduring the 14th century.Perhaps the strongest argument advanced on behalf of the linear inter-pretation has been based on the technique of "successive composition."After two-part writing, some discant and counterpoint treatises go on todescribe the addition of a third and even a fo ur th voice.40 I t is arguedthat since the medieval composer added his voices onto the tenor "suc-cessively," rather than conceiving of his vertical sonorities all at once,he is w riting linear coun terpoin t.But if, as I tried to show, the two-part framework was not linearcounterpoint in the first place, then the third voice may not be either.Here we must avoid the false dichotomy between linear counterpointand (triadic) harmony; we must think in terms of those two-note sonori-ties called con cord s o r accords-terms n o t fa r fr om "chord." If the firststep is the composition of a progression of two-note chords, then thethird voice is added not as a third melody but as enrichment of thosechords. T h e medieval writer says: " W hen adding the third voice, pro-ceed as in discant," meaning that the third voice will proceed throughthe proper concords in co ntrary mot ion with one of the other two. Forwhile the discant teach er can think in term s of tw o-no te entities, his com-

sa CS 111, 27.37 For example, CS 111, 40.38For example, UTolf,"Ein Beitrag zur Diskantlehre," pp. 513ff , esp. p. 515.39 CS I, 258; G. Reaney et al., "The 'Ars Nova' of Philippe de Vitry," Musicadisciplim X ( 1956 ) , p. 22 .40For example, John of G arland : Cserba, Hieronymus de Moravia, p. 225.


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D I SC A N T, C O U N T E R P O I N T , A N D H A R M O N Y I 3prehension seems to stop there; he can explain a three-note progressionon ly as th e summation of two-no te ones-much as traditional harmo nyexplains poly cho rds as comp lex or m odified triads. B ut if this is true, th enmedieval composition is no t m ore successive than ou r ow n. T h e really im-portant difference is that the medieval system uses a basic unit consistingof two notes, whereas we use a unit of three notes. And successivenessin both cases is a feature of teaching rather than of listening.Particularly in the 13th century the authors f ind l i t t le new to sayabout three-part writing, having said it all in connection with normaldiscant. They sometimes add, with the brevity it deserves, the maximthat t he third voice ma y ascend o r descend n ow w ith th e first voice, no ww ith the second, but not w ith both at once.41 But the 14th cen tury onceagain offers more specific instruction: the Quatuor pincipalia ( I 3 5 I )refers to the principle that the upper voices must concord with thelowest.42 T h e author describes ho w t o discant below the tenor, usingthe same concords and procedures as in discanting above, then addsthe proviso that while improvising below, no one else should discantabove unless he knows what tones are being sung below, "because allthe upper parts must be in concord with the lowest voice in order tomake good consonance." The Ars discantus secundum I o m e m de Muriscontains a passage on the composition of two counterpoints over onetenor;43 the auth or advises against t w o similar co ncords over t he samenote at the same time (that is, a fifth and a twelfth, an octave and a fif-teenth, a third and a tenth ) because in these there is no diversity. H e alsocautions against a fifth and a sixth at the same time, but recommends afifth and a tenth, for if the tenor were to rest, these two counterpointswo uld still be in concord with each o ther.Immediately following this passage in the Ars discantus (bu t no tnecessarily b y th e same au th or ), is a description of three-part w riting f o rtenor, c a m e n , and contratenor. For each concord of tenor and c m e n(unison, third, fifth, sixth, octave, tenth , and tw elfth) th e auth or gives allpossible conc ords fo r the co ntrate no r below. Some of these are describedas sweeter, some not so sweet. It should be noted that the procedure iscompletely vertical: there is no mention of progression from one sonorityto the next, progression being treated in the exposition of discant proper.T h e third voice is understood b y the au thor to be an expansion of thevertical sonority.

Th is helps t o explain th e curious m atter of th e Renaissance bass. I t isfrequently pointed out, with solid textual support, that the early Renais-sance teacher reckoned his bass notes down from the tenor, rather thanthe o ther wa y around. I t is concluded from this that the early Renaissance41For example, Franco: Cserba, Hieronymus de Moravia , p. 254; Strunk,SourceReadings, p. I55.42 CS IV, 292, 294. 43 C S 111, 92f.


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'4 J O U R N A L O F T H E A M E R I C A N M U S IC O LO G IC A L S OC IE TYbass was "non-functional," w hic h seems reasonable. But, th e argum entcontinues, if the bass is non-functional, then everything over it must belinear, and this is where I think the argument runs off the track. Themedieval sense of function resides, as we saw, in the progression of con-cords, especially in the progression sixth-to-octave. T he se progressions areinterrupted, obscured, all bu t obliterated in th e Renaissance; y e t som ehowthey continue to function. Example I shows ho w the sixth-to-octave (inwho le notes) can be enriched b y a third voice (in quarte r notes), which isbelow the tenor in the first chord, and either above or below it in thesecond.

Ex. I

In the first case this third voice is clea rly non-functional in its progression:i t merely enriches the first chord, then the second. It cannot be said toprogress fro m one t o the other, either harm onically o r melodically. But thesame is tru e in the second case: here, too , the thir d voice m erely enrichesth e sonority. I t can no t be said to have an y function-save fr om an 18th-century point of view. The functional parts of the second case are stillthe sixth and octave, even tho ugh masked b y th e bass. Su ch masking of aprogression, however, is very different from complete independence ofvoices.I t is also argued t ha t since discant was reckoned u p o r d o w n f rom thetenor, the tenor was never a foundation in the same sense that the 17th-ce nt ur y bass wras. T h e "fundam ental bass" is described as being invented ,or discovered, by the late Renaissance and early Baroque. But we alreadysaw in the Qzratuor principalin ( m i d - ~ ~ t he n t ur y ) t h at c on co rd s w e r ereckoned in some sense from the lowest sounding part. Indeed, 14th-century discant describes primarily the construction of intervals over th etenor. If w e w ere to survey 14th-century music we would find that in mo-tets n j the tenor is usually the lowest part , hence the foundation in everyconceivable sense. In mo tets n 4 (w ith a c ontraten or) i t sometimes seemsas though the contratenor, when below the tenor, is the lower part of astandard discant progression. Perhaps this is what Anonymous XI meanswhen he says: 44

Another general mle: contratenor can well descend with the tenor in im-perfect species, ending in a perfect species; and similarly the tenor with thecontratenor. And you should know that the contratenor is said to be the tenorwhen it is lower than the teno r.In other words: in the 14th ce ntu ry the lowest part is the foundation; to-gether with one of the uppe r parts i t form s the basic two -part framework.

44 CS 111, 466.


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DISCANT, COUNTERPOINT, AND HARMONY '5In th e e arly Renaissance this fram ew ork is masked b y othe r voices, aboveand below; th e lowest part is no longer th e foundation. Th en in the laterRenaissance the lowest part is once again described as the foundation,first of indiv idua l sonorities, and-much later-of progressions.Tw o-p art w riting remained the basis fo r instruction straight throu ghthe Renaissance. But in response to contemporary practice, teachers be-gan to offer additional instructions for composing in three parts, that is,they described certain stereotyped formulas for masking discant. Thesesolutions to the three-part problem were, in the early Renaissance, neces-sarily crude and pragmatic: they involved a good deal of parallel motionand sequence, and lacked the economy of classic ~~th-centuryrogres-sions. But th ey did manage to prod uce th at full, rich so nority so m uch indemand at the time. The treatise of William the Monk provides us witha catalogue of such instructions, summarized as follows:45I . T h e English "modes" (formulas) : Fauxbourdon, a 3 , Gymel , a 2.2. Ano ther formula a 3, "non mutatis."3 . Discant (here is the logical beginning of the treatment of composition).4. Table for f inding concords to notes in the c and g hexachords.5. Fauxbourdon and Gy m el (al ternate rules) .6. T h e low Cont ra t enor fo r No. 5 (hence a 4 ) .7. More rules, and two exceptions.8. Ano ther formula a 3 .9. And another .T h e only formula of real importance for the futu re is contained in N o. 7(Ex. 2 ) .

Ex. 2

*C n CS.T h e streng th of this form ula seems to lie in its avoidance of parallel mo-tion while producing a series of imperfect concords. Like the "clausulae"of the 16th ~ e n t u r y , 4 ~his formula taps the resources of traditional dis-cant.

T h e counterpoint treatises of th e 14th and early 15th centuries, wh entaken seriously and in order, provide a wealth of material and a fascinat-4 5 CS 111, 273; ~ 8 8 ff ; xample 2 f rom p. 296. The treatise had been ably describedand analyzed by Brian Trowell, "Faburden and Fauxbourdon," Mzcsica disciplina

XI11 ( 1 9 5 9 )~pp. 64ff.46 Se e B. Meler, "Die Ha rm onik im c antus firmus-halti en Satz des 15. Jahr-hundem," Archiv f i ir Musikwissenschaft IX ( ~ p j z ) , p . 27 f See also A. Schmitz,Ob erita lim isch e Figuralpassionen des 16 . Jahrhzmderts (Mainz, 1955), Vo1. I, p. IT*.


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I 6 JOURNAL OF THE AMERICAN MUSICOLOGICAL SOCIETYing va riety of detail. I t is remarkable tha t so little attention has been paidt o these treatises since the indefatigable H u g o Riemann described them inhis Geschichte der Musiktheorie i n 1898. T h e condition of th e sourcesbeing wha t it was (and still is), Riemann did not g et them entirely in or-der, bu t he did take the m seriously. H is seriousness, however, was on e thatlooked forward to the Messianic appearance of the Dual Nature of Har-mony in the Major and Minor TRIADS, the glory of whose comingblinded him to the actual meaning of the medieval authors. Nevertheless,many who are scandalized by his speculations could benefit from hisknowledge of the sources.T h e mo re one becomes acquainted w ith these authors of th e 14th andearly 15th centuries, the more one sees how dependent the Renaissanceauthors are upon them. Tinctoris's rules, for example, reveal no basicnovelty w hen compared t o earlier sources.47 T h e most imp ortan t differ-rence is the insistence on variety, with urgent prohibitions against repeti-tion. Th is seems to be related to a greater num ber of im perfect concords,and a relaxation of th e procedures g overn ing their use. N o lon ger does acomposer resolve thirds and sixths, but leads them in unending chains ofsuspended functions. It must be this th at gives Renaissance discant its ne wsound, since, as we saw, the sound of imperfect concords is as old as dis-cant, and a mere increase in their fre qu en cy seems a weak basis f or a ne wstyle.But this variety, being an avoidance of the obvious, can find no ex-pression in general principles. T he refo re th e old principles are surrou ndedin Renaissance treatises by an endless number of provisions against theobvious, and an even greate r num be r of examples showing borderline casesof similar motion and ways of exploiting discords. Composition becomesthe skill of producing continuous variety while avoiding on t he one handthe barbarous and on th e o ther th e too familiar.

I t is frequently said that in th e three-part formulas of 15th -cen turycoun terpoin t one can hear "a real feeling fo r functio nal harmony." W hi lena'ive, this observa tion is no t w ith ou t foundation-only w e must disen-tangle t he meanings of the terms involved. W e use the term "functionalharmon y" so often tha t w e say "functionalhannony"-one w ord with theaccent on th e four th syllable . W e forget that there are tw o words w ithtwo different meanings; that there might be "non-functional harmony,"o r even "function" in th e absence of "harmony." hT ow t seems clear that"function," since Riem ann, refe rs to relationships between triadic chords,relationships that may be actual or implied. Armed with a more compre-hensive view of history, we can proceed cautiously to speak of functionsbetween two-note entities instead of between triads; I have tried, througha discussion of discant, to s how h ow this might be done. W e m ight even47 CS IV, 147ff. G. Reese, illusic in th e Renaissance (New York, 1954)~p. 144.


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DISCANT, COUNTERPOINT, AND HARMONY I 7speak of functions in monophony, if we could find the appropriate terms-but tha t is anoth er matter.T h e formulas of the 15th century, then, are indeed functional: th eydepend upon the two-note progressions of discant. They also sound likethe familiar progressions of "functionalhannony," which simply meansthat triadic functions and progressions develop in unbroken continuityou t of discant. T h e difference between discant and "functionalharm ony~'has to do n ot wit h "function" (although th e specific functions are slightlydifferent in th e tw o systems) b ut w ith "harmony." T h e search fo r themeaning of this term takes us into quite another part of the forest, farfrom the counterpo int teacher and his practical precepts. W e must at longlast take o n th e speculative theorist an d his intricate calculations. I n com-pensation for the thorny mathematics the theorist offers us explanationsabout th e very nature of musical sound.T h e term "harmony" is not unk now n in the Middle Ages, whosewriters go t i t from the Greeks. In ord er to understand i ts use w e must re-mind ourselves that it is an every day wo rd fo r the marvellous quality thatcharacterizes a great painting, a successful piece of architecture, a happyfamily, and that for which the peoples of the world yearn. As musicianswe tend to forget this more basic meaning: our books on harmony donot usually tell us why their subject should be so named.T h e Middle Ages as well as th e Renaissance ap proac hed this marvel-lous quality by paradox, explaining it as the "concord of discords." Thisinscription finally turns up on the title page of Gaffurio's De hamzoniamsicorunz inst rumentorum ( I 5 I 8). N o t that "harmony" meant "poly-phonyv-far from it-but in poly pho ny the y saw ye t another manifesta-tion of th at quality that ran throu gh the whole creation. Indeed, polyph onybecomes the most tangible manifestation of harmony: in the Renaissance,the Prom ethean musicus speculator seizes upon th e Idea of harm ony andfixes it in t he m atter of counterpoint.N o t long after th e cou nterpoint teachers tackled t he three-part p rob-lem-around th e end of th e 15th century--certain theorists we re pon der-ing th e same problem fro m a d ifferent point of view. These theorists wer enot so much concerned with how to produce three-note chords, butrather why some chords sounded better, more harmonious than others.T he ir attention w as focussed up on th e ch ord itself as a vertical sonority;the y sought some tool for explaining its nature.

T h e tool had been at hand fo r some t ime. Every theorist in the We sthad presumably read Boethius, and thus knew about the several kinds of"mean" or division of a prop ortion. Boethius described thre e means:arithmetic, geometric, and harmonic, as follows:4*48 De institutione musica IT , 1 2 (Friedle in ed., p. 241).Eleven kinds of mean wereknown to antiquity; see P. H. Michel, De Pytl2agore d Euclide (Paris, ~gj-o),pp. 365,

369s.


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I 8 JOURNAL OF THE AMERICAN MUSICOLOGICAL SOCIETYArithmetic z:3:4 Geometric 2:4:8 Harmonic 3:4:6

The arithmetic mean divides the distance between the extremes intotwo equal parts; the resulting ratios, however, are not equal (2: 3 # 3 :4 ) .In the case of the geometric mean, the ratios are equal, but the middleterm does not divide the distance into two equal parts. In the harmonicdivision neither the parts nor the ratios are equal, but the ratio of theparts is equal to the ratios of the extremes. This curious affinity of thedivision to the w hole is the special pro pe rty of the harmonic mean.Its ver y name, "medietas harmonica," must have ca ug ht th e fanc y ofthe R enaissance theorist. T h e application to vertical son ority is striking,and the theorist must have pondered the result with excited s a t i s f a ~ t i o n ~ ~

Ex. 3

In this example the numbers to the right of the notes represent stringratios: thus strings in the arithmetic propo rtion (2: 3:4) sound a fifth witha fo ur th below; those in the geometric propo rtion (2:4:8 ) sound an oc-tave with another octave below; those in harmonic proportion (3 :4 :6 )sound a fo ur th wit h a fifth below. I t is characteristic of the a rithmeticmean that the larger ratio occurs between the smaller numbers, henceth e larger interval (th e fifth) comes at the top , between th e shorter strings;the harmonic m ean, on th e o ther hand, has the special pro pe rty of placingthe larger interval between the longer strings, hence below the fourth.W h e n judged b y a 15th-c entury ear-perhaps b y an y ear-the thre epitches produced by the harmonic division of the octave have a muchmore balanced, euphonious sonority than the others. Having reached thisconclusion, th e theorists reserved th e term "harmony" fo r a cho rd of th reepitches; chords of tw o pitches w ere concords o r discords. An d it was theharmonic mean-0 hap py coincidence-that produced the tru ly har-monious division of th e octave, that cho rd w hich fo r a long time had beenmost wo rth y to end a song.F o r reasons that th e subtle reader m ay fe rre t ou t fo r himself, the har-mon ic mean canno t be applied within the "Pythagorean" tuning beyondthe division of the octave. But at the m ome nt w he n theorists were m akingthis application, they were also busily engaged in modifying the system

49 All this and the following is available in Riemann's Gerchichte der Muriktheorie,Ch. XII: "Die Revision der mathematischen Akustik," p . 318ff; in addition to clarify-ing the material, my aim is to put it in a slightly di2'rent light. The first theoristactually to apply the harmonic mean to sonority seems to be Gaffurio (Riemann,p. 324). Walter Odington does not say exactly what Riemann suggests.


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DISCANT, COUNTERPOINT, AND HARMONY I9of tuning .60 T hi s dislocation of traditional con cepts resulted in consider-able controversy; w he n th e dus t had settled, the issue seemed decided, atleast temporarily, in fav or of those advocating a "pure" third. O ne conse-quence of this was an endless series of discussions about the "least" inter-vals, the commas and their kind, which are apparent to the reason butn ot the ear-this in an age w hic h allegedly trusted in th e sensible.Another consequence more germane to our topic was this: the purethirds 5: 4 and 6: 5, b y replacin g th e o ld ratios 8 I : 64 and 32: 27, could nowtake their rightful positions alongside the perfect concords of octave(2: I ) , fifth (3: 2), and fo ur th (4: 3). Of course the discant teachers hadgroup ed the thirds with the concords ever since John of Garland in theI jth cen tury , bu t the Renaissance theorist provided the mathematical jus-tification. And now, reasoned Zarlino, the principal concords of counter-point can be derived f ro m th e ratios of th e first six numbers, the ena aria.^^Surely a remarkable demonstration of the rational nature of music! Ademonstration as well that the moderns had surpassed the ancients, whohad used only th e tetrad, o r first fo ur numbers.If the major and minor thirds are expressed by these ratios, anotherapplication of the harmonic mean is possible. Gaffurio had perhaps al-ready made this application, but in passing; Zarlino takes it u p w ith m oredecision (Ex. 4) .5 2 Ex. 4

H er e are tw o m ore sonorities that are formed fro m concords, have threepitches, and fill th e ear w ith sweet harmo ny. I n th e first a fifth is dividedarithmetically, placing the larger interval at the top; in the second thefifth is divided harmonically, placing the larger interval at the bottom.On ce again reason coincides w ith t he judgment of th e ear in declaring thelatte r t o be mo re harmonious. Because i t consists of t hr ee d ifferent pitchesit is called a "triad;" because it uses th e ha rm onic m ean i t is the "har-m onic triad." F o r Zarlino and th e 16 th cen tur y this triad, of all sonori-ties, manifests most clearly that marvellous quality, harmony.It is important to observe that such analyses in no way conflict withdiscant. In the third part of the Istitutioni barnoniche, Zarlino discussesdiscant in the traditional fashion, and quite consciously so, for he refersfrequ ently to the ancients, meaning the earlier discant authors, and shows

6oAlthough treated by Riemann and others, this chapter in the history of theoryalso needs rewriting. J. Murray Barbour's analysis, (Tuning and Temperament [EastLansing, hlich., 19511) while authoritative, wants sympathetic insight; according toBarbour, "just intonation" is something that can and should be "confuted."5 1 lstitutioni hamzoniche (Venice, 1 5 7 3 ) , I, Cap. xiiiff.52 Ibid., I, Cap. xxxix, xi; 111, Cap . xxxi; S tru nk , Source Readings, p. 242.


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2 0 J O U R N A L O F T H E A M E R I C A N M U S IC O L OG IC A L S O C IE T Ylittle inclination to depart from their principles. But one should not con-clude from this that Zarlino is a dual mentality, half medieval and halfmodern, unaware of a contradiction between linear counterpoint andfunctional harmony. As I have tried to show, he discusses neither of thesethings; the two subjects he does discuss, counterpoint and harmony, arein no way contradictory. His precepts of counterpoint, like those of hispredecessors, teach how to get from one concord to the next, and howto expand this progression of concords into three or more parts. His the-o ry of harm ony analyzes the nature of three-part sonorities. Th is the oryof harmony does not treat, in principle, the progression from one har-mony to the next: the harmonic triads have no systematic relation, andtherefore no function, one to another. H is theor y is about harmony, butnot about functional harmony.A word might be said comparing Zarlino's theory of harmony toRameau's. Zarlino demonstrated that the principal concords could be de-rived from the ratios of the first six numbers, and that the principalharmon y w as formed fro m these concords arranged according to th e har-monic proportion. Rameau showed, in effect, that the principal con-cords could be derived fro m t he intervals in th e natural (th at is, physical)series of partials or overtones, and tha t th e principal ha rmo ny was fo rme dby taking certain concords in the order in which they actually appearedin this natural series. T h e one demon stration seems just as valid, and nomore so, than the other.53 T h e 16th -cen tury theorist believed th at if hecould find the form of music in the realm of number, he somehow mademusic m ore real or m ore tru e; th e 18th-century theorist believed thesame, on ly he looked f o r his proof in th e realm of physical phenomena.And the Creator in his Wisdom made the universe big enough so thatperhaps both are right.

G ran ting tha t Zarlino's argum ents are cogen t, it is still difficult to rec-oncile oneself t o his allotment of space am ong various topics. T h e senariaby no means dominates the scene, while the harmonic division, consider-in g its implications, seems actually s lighted. T h e "least intervals," here asin other Renaissance theorists, get the largest share of space, which in-clines one to reject the whole tedious discussion of commas as hopeless5 8 T h e frequency ratios of th e ascending partial series of course produc e--or areprod uce d by-the series of wh ole num bers: 1,2,3,4,5,6 . . . Strings arranged so as t oyield these sounds will have lengths correspon ding to t he series: I, 1/2,. 1/ 3, I/.+, / 5 ,1/6 . . .Any three consecutive terms of this latter series yield a harmonlc proportion;in fact this series (the reciprocals of the whole numbers) is the only continuousharmonic proportion, since all cases of the harmonic proportion involving wholenumbers (e.g. 2:3:6) come to an end. See P. H. Michel, De Pythagore d Euclide,

pp. 394f.If string ratios are used, therefore, the harmonic triad must be explained by theharmo nic prop ortion ; if freq uen cy ratios, by th e whole-num ber series, that is, arith-metic proportion. Not without logic is the whole-number series called the "harmonicseries" when it refers to frequencies.


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DISCANT, COUNTERPOINT, AND HARMONY 2 1pedantry, remote from musical art. Yet perhaps when an art is drawingto an end, w hen all the w ays o u t have b een explored, all limits reached-perhaps in this moment of closure, the investigation of the whole tonalsystem, all its cracks and crevices, becomes a matter of great importance.Perhaps w e will come t o understand this in o ur ow n time.

Yale University

Documents

Richard L. Crocker - Discant, Counterpoint, And Harmony