Mersenne on keyboard tuning

MERSENNE ON KEYBOARD TUNING MarkLindley Mersenneis an interestingfigure in the history of keyboardtemperaments not only for what he said but also for what he stimulated others, includingTitelouze, Descartesand Rousseau,to say.1 It is true that he tended to combine sophisticationand naivety as perhapsonly a very kindly and cloistered savant could do, and that his first-hand grasp of the craftsmanly and even mathematicalaspects of temperament was less secure than that of a Zarlinoor a Sauveur.(In the Latin counterpart of Harmonie Universelle, the Harmonicorum libri of 1636, he gave a prescription for tuning the organ which is, as far as I can tell, inept by any standard.') Nevertheless,he is one of the outstandingseventeenth-centurywriters on tuningsas well as other aspects of acoustics and organology; his treatment of the subject over the years reflects the growth of an intellectual personality of considerable importance in the history of science as well as musicology; and his instructions in French for meantone temperamentmay have exercised a particularly curious influence in the stylistic development of the Frenchharpsicordrepertoire. Perhapsthe fairest way to summarizeMersenne'sviews on keyboard tuning is to discuss in turn his attitude toward the three main varieties that he dealt with: just intonation, meantone temperament,and equal 167 temperament.This approachwill enable us to describecertainconflicting statementswithout forcingan unwarrantedinterpretativeresolution upon them. It seems clear that he regardedthese as philosophicalor scientific matters to be debated and researched,ratherthan theological ones requiringdoctrinalconsistency. Mersenne'sfirst major work, the monumental discussion of Questiones celeberrimae in Genesim (Paris 1623), includes a 100-page commentary on Genesis, Ch. IV, v. 21 ("And his brother'sname was Jubal: he was the father of all such as handle the harp and organ"). Here we find a good deal about just intonation: Salinas'selaborate diagram of a justly intoned monochord; several pages explaining the ratios, nomenclature, and mutual relationshipsof myriad pure intervals; and the copper engravingof a harp in just intonation ("Cithara nova et antiqua") that later appearedin the thirdvolume of Harmonie universelle.3 While none of the material is presented in the specific context of keyboardinstruments,it all shows that Mersenne'sapproach to temperament was based on a long-standingfamiliarity with the calculationsof just intonation. Considerable space is given in the Harmonicorumlibri and Harmonie universelle to various pure tuning schemes for keyboard, including some with the ordinary twelve-per-octavepattern of seven naturalsand five accidentals.Mersenneacknowledgedclearly,however, that temperament was a practical necessity for normal keyboard instruments. At the same time-and this shows the strength of his liking for just intonation-he urgedthe adoption of elaboratekeyboards with enough notes per octave to provide a complete set of justly intoned concordsfor each of the seven naturals(Plate 1): I wish to add here a keyboard with the keys necessaryto make all the consonances in their justness, for although the nineteen keys of its octave may be, it seems, more difficult to play than the thirteen of other keyboards, nevertheless the perfection of the harmony and the facility there is in tuning organswhich use this ... keyboard abundantlyrepays the difficulty of playing, which organists will be able to surmount in the space of one week, or in very little time. . . Augmented keyboards should not be deemed extraordinary,for they do nothing other than that whichvoices do; they simply place harmonyin [that state of] perfection which the mind and the ear desire... It is of no importancethat the difficulty of playing them is greater;one need not feel pity for the pains, nor evade the work, which lead to perfection. To this I add that they will be played as easily as the others, when the hands become accustomedto them...4. This keyboard design was refined by Joan Albert Ban for a harpsi168 Mes ene 7n? A* **&* o t at to # ~F Fllp 0 44 *D stoo032 C o2avo -D 2700o24-0021 E F G t A r o 1 co c m Plate 1 169 chord built in Haarlemin 1639.s The scale it accommodatesmay be derivedtheoretically from the seven diatonic naturalsas follows: first, the most economical way to provide all possible pure concordsamong the naturalsthemselvesis to have two D's, one pure with F and A, and the other, a comma higher, pure with G and B (see Fig. 1). Then these eight diatonic notes may be "surrounded"with chromaticnotes giving each natural all six of its possible concords (see Fig. 2). The result comprises eighteen pitch classes (i.e., nineteen "marches"or steps per octave), the same as those specified by Mersenneand used by Ban. Mersenne's interest in a justly intoned clavier was bolstered by philosophical considerations.The fact that a concord with a "pure," simple ratio sounds "pure"(whereasthose with theoreticallyirrational ratios, such as the fifth in any regulartemperament,producebatemans, i.e. beating) was presumablywelcomed by him as evidence againstthe kind of philosophical skepticism that he had vigorously opposed as a writer in the early 1620's and, no doubt, as a teacher of philosophy and theology at Nevers before moving to Parisin 1619.6 The skeptic's view, as Mersenne portrayed it in his book defending La Verite des sciences (1625), would be that: Music is nothing but appearance,since that which I find pleasant someone else will find discordant according to his whimsy, and if one cannot give a reason why the octave, fifth, and fourth are consonances, rather than a seventh or a second, perhpasthe latter are the true concords,and the others are discords,for if this number pleased one [person] that one will please someone else.7 In a book on Questions harmoniques published in 1634, Mersenne associated the advocacy of tempered tuning with skepticismby saying that one of the strongest argumentsagainst the status of music as a science with certain and evident principles lies precisely in the fact that no proof has been establishedthat concords are based on simple mathematicalratios; on the contrary, he reports, some geometricians, adept at music, give a higher theoretical status to the intervals of equal temperament,as practiced on lutes and viols, than to pure intervals, even though the former are "surd and irrational,"that is, are defined mathematicallyby quantities that cannot be expressedas the quotient of two integers: Question II: Is Musica science? Does it have evident, certain principles? Many believe that Musicis not a science but only a mechanical art, the rules of which are based on the senses, particularlythe sense of hearing.In support of this there are severalarguments,the most powerful of which is found in the uncertainty of its rules and its intervals, for it has not yet been demonstratedthat the ratio of 170 A DF E --C -- - B D -G--- G Figure 1 F B B E A D DO GO C$ FO G C FO D i Eb D , Figure2 171 A DF E --C -- - B D -G--- G Figure 1 F B B E A D DO GO C$ FO G C FO D i Eb D , Figure2 171 the fifth is 3:2, and one encounters excellent geometricians, adept at musical composition, who deny all the ratios of the concords and discords which the Pythagoreans, Euclid, Ptolemy, Boethius, Zarlino, Salinas, et al. have given, and who believe that the ratios of all the notes and intervals are "inexplicable," or surd and irrational. For they maintain that all the tones and semitones are equal, that three major thirds make a pure octave, that the augmented fifth is no different from a minor sixth, that the diminished fifth and augmented fourth are the same, that the performance and composition of music is much better, or easier, with equal tones and semitones than when one follows the theory which posits their inequality, and finally, that the concords and notes produced by lutes, viols, and other [such] instruments attest to this equalitylikewise the ear. I omit several other details pertaining to this opinion of Aristoxenus and of those who follow his views, because I discourse particularly about this in the treatise on the lute, and I respond in several other places to all that one can object against the hypotheses of those [like me] who perpetually join the senses to reason and who prefer the spirit to the body, and intelligence to sensation. Therefore I give here only [a sceptical discourse on music by my anonymous friend La Mothe Le Vayer], so that everyone can read it with the same enjoyment that I have derived from it, and so that the musician may consider that which can be said against his art-and consequently his obligation to study and become learned enough to respond to it.8 The source of the doctrine denying the traditional ratios and favoring equal semitones was the Flemish mathematician and engineer, Simon Stevin (1548-1620).9 Apparently unaware of the existence of meantone temperaments and unaware or heedless of the phenomenon of beating in imperfect concords, Stevin had calculated the intervals of equal temperament in 1585 and advocated equal temperament in a manuscript of the first decade of the 17th century entitled "Vande Spiegheling der Singconst." He incisively rebutted the argument that small-whole-number ratios are in some metaphysical sense better than ratios involving roots of 2; acoustically his argument refers to the "sweet sound of the fifth" but omits any characterization of thirds and sixths: Now someone might wonder according to the ancient view, how the sweet sound of the fifth [can] reside in so inexplicable, irrational, [and] inappropriate a number. To this we could answer at length, but since it is not our intention-in [connection with] the unspeakable irrationality and inappropriateness of such a misunderstanding-to teach here the articulateness, rationality, appropriate- 172 ness, and natural, wonderful perfection of these numbers, since we have provedit elsewhere,we shallleave it at that.10 The relative aesthetic value attributed by Mersennein 1634 to just intonation, meantone temperament(as practiced on keyboardinstruments of his day) and, by implication,equal temperament,is indicated in the following passage from the Prdludes de l'harmonie universelle, where meantone is praised for sounding rather like just intonation. Here we see Mersenne"joiningthe senses to reason"by expressingan acoustical as well as a philosophical preference for purely intoned intervals.He says that enharmonicdistinctions: are necessary to provide pure concords in several places in the [Guidonian] hand, or the musical scale, and in the keyboard of organs and virginals.Yet while the temperamentof organsand other instruments approachesso closely the purity of the concords that the ear is not offended, and endureswith ease diminishedfifths and augmentedfourths on instruments,still one does not deriveas much satisfactionas if all the concordswere perfect.11 From all this we may infer that since the "simplest" ratios (pace Stevin) produce intervalsof a qualitativelypotent character,Mersenne would have considered it perverseto assign a negativeaesthetic value to them by sayingthat temperedintervals,intervalstinged with beating, are more beautiful. Again and again he implicitly rejected the latter view. And in this he was implicitly supportedby his close friend, Rene Descartes, to whom he apparentlysent an advancecopy of the Questions harmoniques. In February 1634, Descartes wrote Mersenne a letter which, as it happens,recordsone of the most painfulmoments of their life-long friendship. In the first part of the letter Descartes explained at some length that the recent actions of the Inquisitionagainst Galileo "m'empechoit de vous envoyer mon traitte" (Le monde, ou Traitede la lumidre).Suddenly, however, the letter turns to music and not only upholds the status of the traditionalintervallic ratios, but also invokes the opinion of Jacques Mauduit (1557-1627), the composer, humanist, and friend of Mersenne,against equal temperament in particular: As for the reasons given by your musicianswho deny the proportions of the concords, I find them so absurdthat I hardlyknow any more how to reply. .. And indeed if M. M[auduit] were still alive, he could well testify that the difference between major and minor semitones is quite perceptible;for after I drewhis attention to them one time, he said he could not stand harmony in which it was not observed.12 173 A source of persistent encouragementfor Mersenne'sadvocacy of just intonation and his belief that an expanded keyboard would be practicable was Giovanni Battista Doni (1595-1647), erudite scholar of ancient Greek music, secretary(from 1629) of the Collegeof Cardinals, and inventor of the justly tuned "Lyra Barberina."13 Doni became acquaintedwith Mersennein 1621-22 while accompanyingArchbishop Ottavio Corsinito Paris on a mission of papal diplomacy. Throughout their correspondenceover the years Doni never relaxed his advocacy of pure intonation. It seems that in at least one instance he may have influenced for the worse Mersenne'spresentationof temperament.In the preliminary Traite'de 1lharmonieuniverselleof 1627 Mersenne included, in a discussion"Du temperamentdes Instrumensde Musique, and leur imperfection," an account of the three kinds of meantone described by Salinas (and Zarlino) in the 16th century, namely, 1/3comma (in which fifths are tempered 1/3 comma, minor thirds are pure, and major thirds are 1/3-comma smallerthan pure), 2/7-comma (in which fifths are tempered 2/7 comma and both major and minor thirds are 1/7 comma smaller than pure), and 1/4-comma (in which fifths are tempered 1/4 comma, major thirds are pure, and minor thirdsare 1/4 comma smallerthan pure): One can enlarge the minor whole-tone and diminish the major in several ways by distributingthe comma: First, one can divide the comma into three parts, [etc.] . . . Secondly, one can divide the comma in half, i.e. enlargingthe minor whole-toneby one part, and diminishing the major by the other part. The first method is the most imperfect, as I shall show in the Books of the Instruments, where I shall speak at length about the second method, and about several others, which can serve on the virginal,on the lute, and for the constructionof organs.14 The "liures de instruments" in Harmonie universelle, published ten years later, do not furnishthe additionalinformationabout 1/3-comma and 2/7-comma meantone that is promised here. Part of the reason might have been that in May 1636, Doni advised Mersennenot to bother with "toutes ces divisions espaisses et subtiles de Salinas et d'autres avec tant de commas majeurset mineurs, etc."15sDoni's persistence is illustrated by his reaction in a letter of August 7, 1638, to Mersenne'saccount in Harmonieuniverselleof Titelouze'sspecialharpsichorddividingthe octave into 19 equal intervals: If the way that harpsichordsounded was good, believe me, it did not have the [whole] tones dividedinto three equal parts,but ratheraccording to the ordinarytuning, wherein the middleintervalof these three: C- C sharp- D flat - D is much smallerthan the others.16 174 It would have been informativefor Doni to point out that the tuning he disapproved of was indistinguishablefrom Salinas' 1/3-comma meantone." My point is that this disdain might well have discouraged Mersenne from delving further into meantone temperaments. (Certainly there were plenty of other subjects for his avid curiosity.) His human capacity to slip is shown by the following remarkin Harmonie universelleon the tuning of the dpinette,which he said was similarto that of the organ: it is necessary to divide the fifths into major and minor thirds, so that the major [thirds] are a little small[er] and the minor [thirds] a little largerthan their justness demands.. .18 Minor thirdsare neverlargerthan purein any regularmeantone temperament; and while the major thirds in 2/7-comma meantone are slightly smaller than pure, there is no corroboratingevidence to suggest that Mersenneadvocatedthat form of meantone. Of much greater interest than this momentary and rather trivial lapse is the confusion befalling Mersenne'sattempts to give step-bystep instructionsfor setting meantone. He tripped over a simplemathematical point, but in the process unknowingly (I believe) prepared the way for a fruitful innovation. A few pages after the sentence quoted just above from Harmonie universelle, tuning instructionsare given for the virginal,accompanied by the illustration shown in Plate 2. Choosing to start from F, Mersenne explainedthat: il faut fairela Quintefoible en haut, c'est a direque la 2. note qui est en C sol ne monte pas iusquesa la quinteiuste du monochorde. One must weakenthe fifth above; i.e. the secondnote, whichis low C, shouldnot rise quite to the purefifth of the monochord. Cette diminutionest signifiee This diminutionis indicated parla lettre d, qui enseigneque by the letter d, whichmeansthat toutes les notes, all those notes surlesquellesil se rencontre, on whichit is found doiuentestreaffoiblies& diminuees. must be weakenedand diminished.19 Notice that the note itself is said to be weakenedor diminished,rather than the intervallic quantity between two notes. This metaphysical fine point presentlybecomes the crux of a realinterpretivedifficulty. According to the proceduredescribedby Mersenne-and by several of his predecessorswho give step-by-stepinstructions for some form of meantone-one tunes an ascendingchain of fifths from F to G sharp and a descendingpair of fifths from F to B flat to E flat. Instructions of this type tend to become awkwardat the point where it is explained 175 •- t a9 a, v~ 2 f 3 --4---~ b6 4 / 4 f /a cJ Plate 2 176 ;-- ? 4 to b a tt b1 . that B flat must be tuned higher than pure in relation to F in orderto produce the proper kind of tempered fifth (i.e. smallerthan pure). Of course it is the same with E flat in relationto B flat: you must raisethe note (E flat) in orderto diminishthe interval. Praetoriusin 1619 devoted half a page to clarifyingthis point." But Mersenneslippedinto ambiguity: L'on vient apres a l'accord des Feintes en commencant a I'F superieur, avec lequel il faut accorder le b fa a la quinte: mais cette quinte doit estre augmentee au lieu que les precedentes ont este diminuees: c'est pourquoi i'ay mis la lettre f dessouz pour signifier qu'elle doit estre forte, de meme que celle qui suit. Quant aux trois autres elles sont marquees par d, parce qu'elles doiuent estre diminuees. Le 7 . ou dernier rang de notes signifie le defaut de l'accord ... Next comes the tuningofthe black notes starting from the higher F, with which one must tune B-flat at the fifth: but this fifth must be increased, whereas the preceding ones were diminished; that is why I put the letter f underneath to indicate that it must be high, likewise the one which follows. As for the other three, they are marked by a d because they must be lowered. The 7th or last pair of notes indicate the fault in the temperament21 To determinewhether it is the intervalB flat - F or the note B flat that Mersennewanted set higherthan pure, one might consult the directions he gave some 200 pageslaterto "Expliquerla maniereet la methode d'accorder les Orguestant iustes que temper6es."Actually only one kind of tuning is discussed here, 1/4-comma meantone: "les Octaues doiuent estre iustes, et les Quintes foibles . . . it les faut affoiblir d'un quartde comma. . . "22 Mersenne'sillustrationin this case uses the poetic scansion signs%and - ,instead of little letters:(See Example1). The accompanying step-by-stepinstructionstell us, in referenceto the little dash under B flat and E flat: "le signe de dessouzmonstrequ'il faut tenir la note de dessouzvn peu forte.'"3 Thusthe lowernote wasto be tempered higher than pure, and the resultingintervalmade smallerthan pure.24 The Harmonicorumlibri contain nio equivalentto these instructions for the organ, giving instead the completely inept method mentioned on page 167 above. The instructionsfor virginals,however,areparalleled in the Latin book, and there the root of the confusion is plain: .. B molli, seu B fa... cum superiore F ut fa Diapente auctum efficit, hinc f forte significat, ut & sequens Diapente a B fa ad b fa dictionis E la mi... seu D fictum. ... B flat with the F above makes an increased fifth; hence "f" indicates "strong;" likewise afterwards the fifth from B-flat to E-flat.2s 177 desOrgues. notes Arcorddes pri•ples d Clauier S_ ~a - - --_ _4u--. - -4- . . accorddesFetes. --4Example 1 u o,1AIu"-u• p I * -jubi .- 4,A . _.. ~ ~ ~-+,~ •__~ •? , - i. - ~ i ~L., ~~~~~~~~~~j L -i .. A l . Example2 178 - ''-.•'f F. I . J.'a.qa'. * . . desOrgues. notes Arcorddes pri•ples d Clauier S_ ~a - - --_ _4u--. - -4- . . accorddesFetes. --4Example 1 u o,1AIu"-u• p I * -jubi .- 4,A . _.. ~ ~ ~-+,~ •__~ •? , - i. - ~ i ~L., ~~~~~~~~~~j L -i .. A l . Example2 178 - ''-.•'f F. I . J.'a.qa'. * . . How a readermight interpretthis sentence would depend on his understanding of the traditionalterminology of medievalmusic-a music so predominantlymelodic that the word "diapente"in theoreticalwritings very often meant "fifth melodic note." That some seventeenth-century readers innocent of this fact interpreted the passage-or indeed the equivalent passage from Harmonieuniverselle-as a prescriptionfor irregulartemperamentis suggestedby part of the "Methoded'accorderle Clauessin"publishedin Parisin 1695 by LambertChaumentat the end of his book of Pieces d'orgue sur le huit tons. The last four bars in Chaumont'stuning chartindicate that B flat and E flat might be treated in either of the ways that can be inferredfrom Mersenne(see Example 2). Did Mersenneconfuse two kind of tuning (regularand irregular)that he had heard or heard about? I doubt it. No descriptionof this kind of irregulartemperamenthas been found in his survivingcorrespondence, or in the writingsof any previousauthorexcept Arnolt Schlick(1511), with whom Mersenne was unfamiliar;and the compositions of Jean Titelouze and Jacques de Chambonnieres,the outstandingFrench organ and harpsichordcomposers of Mersenne'sday, sound better in a regularmeantone.26 Moreover,in the chapteron the harpsichordin his Cogitataphysico-mathematicaof 1644 Mersenneundertook to mend the "errorespaginis 364. and 365. obuios" from Harmonieuniverselle by giving a straightforward,step-by-step prescriptionfor 1/4-comma meantone without any special comment on the intervalsE flat-B flat-F, comment which certainly would have been in order had they not been intended to follow the regularpatternof the rest of the temperament.27 And finally, another explanation for his confusion or carelessnessof 1637 is available. In 1643 Jean Denis, organist at Saint-Barthelemyin Paris and builder of keyboard instruments,publisheda set of tuninginstructions. It is quite clearthat Denis used some form of regularmeantone,though his prescriptiondoes not narrowthis to 1/4-commaeither explicitly or by specifyingjust major thirds. In a later edition (1650) he includeda tuning chart wherein B flat and E flat are labeled "forte" (the tuning having started on F) while the other notes in the chain of fifths from E flat to G sharpare labeled "foible"28(see Example3). It seems likely that Denis was the source of Mersenne'sinstructions. He lived in Paris;he evidently was not shy or inclined to keep his own views secret; he is mentioned in Harmonieuniverselleas one of three excellent contemporaryFrench makersof the virginals;29and he is referred to, in connection with tuning, in the chapter cited above from the Cogitataphysico-mathematica.The first sentence of that chapter discussesplectra, as occasioned by the fact that Mersenne'sLatinterm for the harpsichord was "polyplectra harmonica." Then we read: 179 ilfiuraccorder Comme c~ lePreftan I'Ejinette desOrtues. foible foible_...-Ifoible julIe ufIc julte foible foible foible ..A jul ,0"-Freuc -71? foible I--- - )uttCe rrcut pafuaice :.prcuus parfaite -- julte foible foibc rtuue jufte derniere preuuc Example3 180 forte preute juflte forte'preauedes defiautde b mols. I'accord. Moreoverour practitionerslegitimately apply their own method of tuning; but this temperamentis not the equal division (with equal semitones) which M. Gall6wished for, [and] which-although [those] of us [with] more delicatehearing[auresdelicatiores] such as the very skilled harpsichordmaker, Denis, can hardly stand that temperamentwith equal semitones-satisfied other very experienced musiciansnonetheless.30 The reference to "delicatehearing"so apt for that blend of sensitivity, expertise, and arroganceexuded by Denis's book, was made by Mersenne on previous occasions-for example in the course of expressing his enthusiasmfor algebrain 1625: Thereare certainpeople who pretendto haveearsso delicate,or a mind so pure and refined, that they havea horrorof the termsof Algebra,as if its difficulties were so great that they could not be understood by a man who has other occupations;but their fear is puerile . . .3 The cryptic bond between delicate ears and a horror of algebramight perhapsbe explainedas follows: Algebradealswith a world of relations between objects rather than with objects themselves.32 One obvious musical application of this mode of thought-i.e. to quantify a diapente as an interval between two notes (an Aristoxenian diastema), rather than "locating" it as a note markingone of the boundaries(in Greek monochord theory, horoi) of such an interval33-would be resisted by the professionaltuner, the man of delicate ears, not because he thinks melodically but because what he must have clearly in mind when his hand is on the tuning wrench is whetherto raiseor lower the note produced by the stringhe is tuning. His m6tier virtually obliges him, in the last stage of the procedureoutlined above for setting regularmeantone, to think of B flat and E flat as "stronger"than pure. It seems that in this instance Mersenne,the "grandnegociantdes Lettres,"34passedon to innocent consumers,without sufficient examination,the raw material suppliedby his informant. In fairness,it could not have been easy for the savantto investigate such technicalities of the workshop. The social obstacles alone would be considerable.CertainlyMersenne'scorrespondentfrom Sens, Christophe Villiers, found it difficult to obtain information and services from the local organ technician in 1635. In FebruaryVilliers had to look over the man's shoulder to gain even a rathervagueaccount of his tuning procedure, because "L'organistequi est en ce pays (ignorant dansla theorie, passableen la pratique)m'a toujourscach6 la methode de l'accord des orgues qu'il tient pour secrette."35sDuring the next months, other difficulties with this gentleman are reported to Mersenne, culminatingin December: 181 He has gone to Paristo retrievehis wife, who fled there from him because of his excessive cruelty. But I think he will gain nothing from it. The wretch . .. doesn't know at all how to tune an organperfectly, i.e., without evident discordanceto the earsof those who understand intervals [les accords]. All the organs he has worked on are out of tune, and now [I] believe that he cannot recoverhis [professional] honor unless through the work of someone that he can bringback from Parisalong with his wife.36 Such vicissitudesmust have favored the influence of articulatenonprofessionals-humanists,astonomers,even lawyerslike ClaudeBredeau of Nevers;Bredeauwrote to Mersennein November1627, and againin March 1628, that on the virginalhis ear found the diatonicsemitone to be smallerthan the chromaticsemitone. Hereis a nice piece of evidence for the use of Pythagoreanintonation in a dilletantishand provincial context. (Bredeau asked Mersenne to "levez ... mon doubte de cette partie. Car je vous auray plus de foy qu'a mes sens.")37 Apparently Mersennereported Bredeau'scomments to Titelouze, who confirmed the prevalenceof meantone by responding: Those who say that [from] mi to fa is only a minorsemitone arenot at all based on harmonic numbers, nor on the practice of instruments, for there they will see just the opposite.38 So far as I know, Mersennenowhere refers clearly to any form of meantone with major thirds largerthan pure. In this respect he differs from his predecessorsSchlick, Lanfranco, Verheyen, and Cerone.39 When we add this sin of omissionto the confusion discussedabove, we are obliged to conclude that he did not treat meantone temperaments very masterfully,even though in 1637 he considered1/4-commameantone "la maniere d'accorder parfaictment les Orgues ordinaires".40 On the whole he did ratherbetter by equal temperament.Standard for fretted instruments such as lutes and viols, it would of course be discussed by him because of the prominenceof the lute in France at that time. (By comparison the French art of the clavecin was just emergingfrom infancy, underthe leadershipof Chambonnieres.) This circumstance,however, does not account for all of Mersenne'sstatements about equal temperament.He was liable to mention it at any appropriatemoment in Harmonieuniverselle,for instance apropos the archiviole in one of the prefacesto the first volume.41 The main discussions of equal temperament,however, are in the third volume, and then not only as a monochord scheme (in Book I) and as the normal scheme for fretted instruments (in Book II), but also aproposthe organ (in Book VI), and the harpsichord as well (in the Nouuelles 182 observationsphysiques et mathematiques,which form one of the appendices after Book VII). We shall see that he evidently favoredthe use of equal temperamenton keyboard instrumentsbut was wary of advocating the idea very explicitly because he was impressedby the fact that professionalkeyboardmusiciansopposed it. Proposition 14 of Book I explains "vn ... monochordequi sert pour diuiserle manche de Luth, de la Voile ... et pour fairele Diapasondes Orgues."The context shows that Mersennewas not prescribingequal temperament for organs to the exclusion of other tunings;but he did say at the outset that: Equal semitones and tones serve to avoid the embarrassmentof [that] great multitude of intervalswhich arisesfrom the discrepancies [in size] of the concordstaken in their usual termsand ratios.42 And in a Corollaryto Proposition 15 he added that "compositionwill thereforebe much easierand more agreeable,and a thousandthingswill be permitted that several [people] believe to be forbidden."43Proposition 15 is about whether the difference between pure intervals and those of equal temperament"can offend the delicate earsof our instrument makers and players,"44 but actually considers only fifths and fourths, where the impurity is found insensiblesince it is even less than in meantone.45 Thereis no considerationof the other intervalsin equal temperament "parce qu'il est tres-ayse de trouuer leurs differences d'auec les iustes interualles."The resultis a presentationbiasedin favor of equal temperament. Mersenne'sstrongest remarkin behalf of equal temperamenton keyboard instrumentsoccurs in Proposition 19 of the "Libertertius de instrumentis harmonicis"in the HarmonicorumLibri (1636), where he said, "quod ttperamentum omnium facillimft esse fatebiitur Organarij,cum illum ad praxim redegerint"("This temperamentthe organists will acknowledgeto be the easiest of all when they put it into practice").46But that remarkwas omitted from the parallelpassagein Harmonieuniversellethe following year, and no step-by-stepkeyboard tuning instructionswere ever elaboratedto match those for meantone. Equal temperamentis not mentioned in the book on harpsichordsin Harmonie universelle; but we do find, in the book on organs, two propositions devoted to its mathematics,the titles of which encourage the readerto consider it for keyboard instruments.Proposition38 begins: An explanation of a universalmethod for the octave on instruments and for dividingthe monochord and the neck of stringinstruments, wherein one sees a new theory of music. Mr. Boulliau, one of the finest astronomersof our time, has given me an HarmonicTable, 183 which deserves to be inserted here because it contains a complete theory of music and ought to be consideredby all those who love harmony.47 Boulliau'stheory of music seeks to explain consonance on the basis of equal-temperedsemitones by means of elaboratebut trivialnumerical operationssuch as the following: The minor thirdhas 3 semitones. The majorthirdhas 4. The octave has 12. 33= 27. 43= 64. 123= 1728. 27 x 64 = 1728, and thereforemajorand minor thirdsare consonant. (The reason for cubing is that musical tones are produced by three-dimensionalobjects.) Proposition45, added on at the end of the Book of the Organ,is entitled "Entre deux lignes droites inesgales donnees, trouuer deux moyennes continuellementproportionelles,pour diviserle Diapasondes Orgues en douze demitone esgaux" ("Between two given unequal straightlines, find two mean proportionalsfor dividingthe fundamental octave of the organ into twelve equal semitones").Apartfrom the title it is cast entirely as a geometricalratherthan a musicalproblem,and is surroundedby Avertissementsin the following vein: Since I have dealt at such length with all the difficulties of the organ and have traced its fundamental octave in so many ways, among which that based on the 11 proportionalmeansis one of the principal ones, I wish to add here a way of findingthem geometrically... If there be some ... artisanswho scorn this way of dividingthe neck of lutes, viols, etc., or the fundamentaloctave, and who think [it] better to rely upon practice and their good ear, then upon all the methods that I have prescribedabove, I do not impede them from following whatever [method] they wish; but I can assurethem they will never go awry by following the methods explained in several places in this book. For the most reliable methods, one should consult the best builders. .. .But if one is willingto use keyboardscontaining the three genera of music [diatonic, chromatic and enharmonic] in their perfection, of which I have spoken at length in several places, ... the purity of the intervals-both concords and discords-will lend new graceto music.48 Thus Mersenne encouraged organists or organ builders to try out 184 equal temperament,but inhibited them from anticipatinga favorable verdict by remindingthem of the beauty of other tunings. The same attitude is evident in Proposition 16, where a friendly reference to equal temperamentis lodged between a thorough descriptionof 1/4comma meantone and a wistful suggestionfor combiningthe virtuesof both. Shortly before this proposition there appearsa table, the second and third columns of which refer to 1/4-comma meantone temperament. Proposition 16 describesthis as the "perfect temperament"but implies that to obtain a complete scale for the diatonic genrewould require all the flats and sharpsprovidedby the 18-note system of just intonation. Apparently Mersennedid not realize that it could be done with 15 notes in meantone (by splitting three black keys: cf. Fig. 3.), since Proposition 16 refersto what appearsto be a non-existentchapter of Salinas ("the 33rd chapter of the third book") for authority that 19 "degrees"would be requiredto make meantone practicable.49 It is in this context that equal temperamentis now mentioned: Since the diatonicgenrein its perfectionrequiresat least 19 notes in each octave, and [since] ordinarykeyboardshave only 13, it follows that they cannot be pure. The major whole tone is diminished 1/2 comma and the minor whole tone augmented likewise; thus the major thirds remainperfect, the fifth is too weak by 1/4 comma... and the fourth is too strong by the same 1/4.... Finally, the major sixth is too strong by this same 1/4, since it makes an octave with the minor third, which is too weak by this 1/4. ... I omit the discords . . . since it matters little that they are altered because one does not perceive it as easily as in the concords. Now if you understand what I have just explained, it will be easy to describethe octave of the perfect temperament,as one can see in the second and third column of the precedingtable. . ... But since one cannot use this temperamentunless one has 20 keys or notes in each octave of the keyboard-as one can see in the 33rd chapter, Book III, of Salinas, who explains it-one must use another plan, for example the one which I have shown in the treatise on the lute, by means of which all the semitones of the octave are equal, though it would be better to leave the major thirds pure and divide [them] into four equal semitones. ..so The system suggestedat the end of this excerpt amountsto a keyboard equivalentto a lute method attributedby Barbourto Artusi(1603).s1 In effect the diatonic C-major scale would be tuned in 1/4-comma meantone, and then each of the five accidentalswould be tuned precisely half way between its neighbouringnaturals.Consequently,thirds or sixths between a naturaland an accidentalwould be about the same as in Pythagoreanintonation, and B flat-F and B-F sharp would be 185 00 CN GO C# - F D - A -- Ai B E C - G - B E- Figure3 rather sour (see Fig. 4). I think Mersenne did not have these details in mind but was merely listing alternatives schematically; the text goes on to mention some other possibilities, namely dividing the pure fourth, fifth, and minor and major sixths into equal semitones. Thus the entire passage displays a wish to combine enharmonic flexibility with good thirds and sixths in a 12-note keyboard tuning-impossible on instruments of normal timbre. The kind of argument in favor of equal temperament that eventually became decisive, in the 19th century, is what might be called an "argument for standardization," such as first advanced by a friend of Zarlino, Girolamo Roselli, sometime before 1588. According to Zarlino: This way of dividing the diapason or octave into 12 equal semitones has [been] singled out for praise, [by] the Rev. Don Girolamo Roselli, . .. as that which could alleviate all the difficulties of singer, players and composers, by enabling [them] to sing or play.. . DORE-MI-FA-SOL-LA upon whichever of the 12 notes they wish, touring through all the notes, making (as he says) a circular music; ... hence all instruments will be able to keep their tuning and be in unison, and organs (as he says) will be neither too high nor too low in pitch.52 Arguments similar to the one at the end of this passage figure prominently in 18th-century discussions of the relative merits of equal and unequal temperaments. The following passages from Neidhardt (1732) and Marpurg (1776) illustrate the importance of such arguments in fostering the adoption of equal temperament: Most people do not find in this tuning that which they seek. It lacks, they say, variety in the beating of its major thirds, and, consequently, a heightening of emotion. In a triad everything sounds tolerable; but if the major thirds alone, or minor thirds alone, are played, the former [sound] much too high, [and] the latter sound much too low. . . . [But] now if oboes, flutes, and the like, and also violins, lutes, gambas, and the rest, were all arranged in this same tuning, then the church and chamber pitch would inevitably blend together throughout in the purest [way]; and thereby this tuning can ingratiate itself in this respect as well.... Thus equal temperament brings with it its comfort and discomfort, like blessed matrimony.53 It is better, in music produced by various instruments [in concert], to use an equal rather than an unequal temperament; diversity in the character of the keys will serve only to increase a 'diversity' of bad sounds in the performance. . ... Let us now consider the clavier by itself, unaccompanied, outside the realm of concert[ized music], and imagine a composer who, in keeping with his own imaginative 187 00 00 1? 1 ? 8 1 1/ 0 1 1? ? 1? 0 1 r j 3 1 1/4 1? ? 1? 1 E-3 Figure4 1 0 1 1? 1 bent, creates a temperament, and for that same [temperament] writes a keyboard piece. . . . As soon as the piece contrived for this or any other particularkind of temperamentis played on a clavier tuned in some other way, the quality [that was] to have been achieved by the method of tuning is lost. How fortuitous therefore, are the benefits that unequaltuning bestows.54 Mersenne'sinterest in a "universaldiapason"suggestsa very similar frame of mind. His attitude emerges explicitly in the "Nouelles Observations"appended to Harmonie universelle in 1637. The eighth of these Observations deals numerically with "les 11 moyens proportionelles" of equal temperament. In the ninth Observation,which describes"le sentiment, et les observationsde nos Praticienssur ce qui appartient a l'accord de l'Epinette et de l'Orgue,"Mersenneimplies that an argument for standardizationshould not be applied against equal temperamenton fretted instruments. One can call those blacknotes whichmakea minorsemitone"sharps", in order to distinguish them from those that make a major [semitone]. Now although musiciansfind equal temperamentvery crude because of its large major thirds and small semitones, which lessen the solidity and goodness of cadences;. . and althoughthey deem it more appropriateto modify the frets of the lute and viol (by means of the monochord)so that they will be perfectly in tune with the virginal, than to corrupt or destroy the tuning [of the latter], the harmonies of which they find sweeter, . . . nevertheless it would not be [a] bad [idea] to retain equal temperamentin orderto promote differences [of intonation] in harmony, and in the same concert to play first one piece of music in this tuning, with lutes and viols joining in, and then another piece in the ordinary virginal tuning.ss But in the seventh Observationhe actually applied the argumentfor standardizationin favor of equal temperament for keyboard instruments ("ensembles . . . would seem more in tune . . ."), and also in- cluded some novel but rathercrude tuning advice, part of which "'the fifth beats once every second") many tuners today still follow: I know of no one except Mr. Gall6who has adapted this tuning to the organ and harpsichord.... The fifths are diminishedso little that it is almost impossible to distinguishthem from just [fifths]; only the major thirds are too large by a little more than half a comma, which hurts the ear of our practitioners. ... The equal semitones have also been judged too small to make agreeablethose cadences which one makes by going from mi to fa.... Some people believe that they can achieve equal temperamentby going up DO, 189 RE, MI, FA, etc. from each key of the harpsichord,or else by [measuring] the number of beats made by the fifth and the other tempered concords: for example, the fifth beats once every second when it is tempered to specification, on the organ as well as the harpsichord,whereas when it is just, it doesn't beat. Now I am astonished by the contention of some excellent lute players who do not believe that the best-soundingfifth is 3:2, nor that the other concordsare [defined] by the ratios ordinarilyascribedto them, and who think that the concords on their lutes are sweeter and more charming.... Accordingto the common sayingof musiciansthe lute is the charlatanof music, because it passes off as good that which, on good instruments,is bad. Now the beating . . . [of] tempered concords makes it evident that they are sweeter when they arejustly [intoned], for then there is no beating. . ... Now it is certain that were the organ and harpsichordtempered the same way lutes and viols are, ensembles [les concerts] in which they are combined would seem more in tune, because their tuning would be in agreement. But our musiciansare not willing to changethe tuning of the harpsichord,to make it agree with the lute tuning, for fear of losing the perfection of their thirds and their semitones, which constitute one of the greatest [sources of] beauty and variety in music.56 How, then, should Mersenne'sview be summarized?I consider Barbour's assessment that "Mersennewas a zealous advocate of equal temperament in practice""7to be less discriminatingthan Frederick Hyde's statements: "In spite of [his] diapason, Mersenneexhibited likewise an equal interest in the very opposite aim, to simplify, through temperament, all instruments to a universalthirteen-note diapason," and "Mersenneseems to us to promote equal temperament,or at least to present it in a favorablelight."ss Hyde does justice to the facts and to Mersenne'sessentially non-doctrinaireapproach. Only after recognizing this attitude in Mersennecan one quite make sense of his last statement on keyboard tuning, the one from the Cogitataphysicomathematicathat is quoted on page 181 above. Believing that Mersenne"zealously advocated"equal temperament, Barbour concluded that "the widespread influence of Mersenne's greatest work, Harmonie universelle, (Paris 1636-37), undoubtedly helped greatly to popularize a tuning [i. e. equal temperament] that was then still considered as suitable for lutes and viols only.""9 It is true that Ismael Boulliau, the erudite astronomer(and noble defender of Copernicanism)whose calculations for an equal division of the octave Mersenne had published in Proposition 38, in 1644 cited Mersenne putatively in support of his own zeal for equal temperament: Moreover,experience has shown that the division of the tones into 190 equal parts makes the harmony more precise and sweet; and if the whole octave be divided into 12 equal semitones systematicallyon the organ, the notes blend better with one another.Concerningthis one should read Father MarinMersennein his treatise on instruments [de organis] in his book, Harmonieuniverselle.60 But it is also true, and more demonstrablyof practicalimport, that the first part of the passagefrom the seventh of the Nouvelles observations quoted above was used successfully by Doni to put down a fad for equal temperamentin Rome in the late 1630's. (This at a time when, as one friend told Mersenne in 1634, fretted instruments were "quasi hors d'usagea Rome.")6' Doni wrote to Mersenneon February29, 1640: An old man has been here, who, after havinglived most of his life in Calabriaand Sicily, having [now] retired in Rome, has tried to introduce here, as [if it were] a fine, new invention, equal temperament on the harpsichord,and [he] found some among our musicians (ignorant as they are) who gave him credence. But at last, recognizingthe imperfection of this tuning, and [that] good singers did not want to sing with such instruments(as I had predicted),they abandoned it, and everything is back to normal. To this end your book in French also contributed, for I showed Monsgr.Cardinal [Francesco] Barbarinithat which you say of a certain Mr. Gall6, who had tried to introducethe same thing, but to no avail.62 On August 10, a group of more than sixty dignitariesof the Church inaugurated,by attending mass, Bernini'srenovatedapse in the Basilica of S. Lorenzo-in-Damaso,containing a magnificentorgan63which (as will be shown presently) would have been tuned in equal temperament had Frescobaldi'sendorsementbeen heeded, but instead was tuned in the normal fashion (i.e. meantone) thanksto the influence of Doni and of Mersenne'sHarmonieuniverselle. Frescobaldi'sadvocacy, however, was not lacking in consequences: some of the keyboard works of Froberger,a student of Frescobaldiin Rome at that time, appearto have been conceivedfor an instrumentin equal temperament.They do not fit meantone very well, and Froberger's reluctanceto publishhis music indicatesthat they werenot designed to be performedby musiciansat large(who indeed would not have used equal temperament).Although Frobergerwas later stronglyinfluenced by Frenchlute music-subsequently by Frenchharpsichordmusic-Mersenne is obviously less likely to have convertedhim to equal temperament than Frescobaldi, whose own greatest single composition, the "Cento partite sopra gli passacagli"of 1637, is bound to involve sour notes on an ordinarykeyboardinstrumentin meantonetemperament.64 It was in April of 1638 that Doni had sought to dampenMersenne's 191 admirationof Frescobaldiby, among other arguments,describinghim as: so unlearnedthat he doesn't know what a majoror minor semitone is and hardly every plays on the chromatickeys [i.e. black keys]; and when he does not understandsome word unusual in colloquial language, he asks advice about it from his wife, who knows more [Latin] than he.65 But if Doni was so biasedin his perceptionsas to concludethat Frescobaldi hardly every played chromatically, then the reference to not knowing about majorand minor semitonesshould be taken with a grain of salt to mean that probably Frescobaldidid not care about them, having perhapshad his fill of that sort of thing while yet a student of Luzzaschi.66Thereis some "broadertruth" to the accusation,however, in that duringthese sameyears the use of equal temperamentwas being advocatedby musicallyinexpert or provincialindividualsin three major cities. Accordingto an account publishedby Doni in 1647, Frescobaldiin Rome had been led into error, againstthe evidenceof his own ears,by frequent and gratuitousbeveragesconsumedin the companyof the old Sicilian to whom Doni had referredin his letter to Mersenneof February 1640: You [may] remember,not long ago, a certainold man in ragscame into our city; he knew nothing except how to play the harpsichord, and he thrust forth as a new and very useful inventionthat equality of semitones which is commonly but erroneouslyattribed to Aristoxenus. And the noble organist,Frescobaldi[Psychogaurum],who was then in chargeof music at the [Barbarini]palace,he reducedto such a state, by [givinghim] frequentandfree drinks,thathe [Frescobaldi] was not ashamed,againstthe proof of his own ears,to extol that invention to the heavens to the good Prince [CardinalFrancesco Barberini] : . . . he [the old man] carried the thing so far that this same Prince, who at the time was refashioningone of the foremost and oldest basilicas of the city [S. Lorenzo in Damaso], especially the very fine apse and auditorium,ordered that the noble organ of this church be reduced to that dissonantkind of tuning; and had not our Doni with the most certain reasons explained to him the vainness of the work, the waste of money, [and] the bad name of improprietywhich was being reflected upon Roman musicians, he would have had it done.67 In 1643 Jean Denis publishedhis "Trait6de l'Accordde l'Espinette" in order to counter the efforts of a certain man, very learnedin mathematics, who had arrivedin Parisand apparentlyhad convincedat least 192 one circle of "fort honnestes gens" that equal temperamentwas better than meantone, though it might requiresome getting used to. The man who had tuned the harpsichord[espinette] told me that it [equal temperament] was good to play in [on the harpsichord],and to transposeby half-steps, and that all chords turn out good everywhere, and that it fits, better than ours does, the tuning of the lute and the viola da gamba. I told him that it was poor thinking to be willing to ruin a good and perfect tuning in orderto accommodateit to imperfect instruments,and that he should instead seek to perfect the lute and the gamba, by finding a way to make the semitones major and minor the way we have them on the harpsichord,which cannot be done with the strings' frets with which one fingersthe lute, becausethey would have to be staggered,which can be done by means of ivory frets.68 Meanwhile in London in 1636 CharlesButler, deeming the scale of twelve equal semitones to be a form of just intonation, commented: [I] n a Lute . .. from fret to fret is but half a Tone. . . . Des Halftones weider dey bee Eqal or Uneqal, it is a Question .... Philo- laus . . . devides de Tone into 2 unequal Parts. ... Dis opinion of Philolaus, concerning de unequal partes of a Tone, Boethius takes much pains, by his qeint ArithematicalConclusions, to maintein. But dat it is indeede a meere fanci, forged onely by Melankolik imaginations,dere is noe Musicianso simple, dat knowes not: and dat . . . de Diatessaron consistes of 2 Tones, and de Diapente of 3, with one eqal Semitone: wich if it bee raisedor deprestfrom its just sound, de quantiti of a Diesis . . . or a Comma or Schisma, or les, if les may bee; it is out of Tune: and noe good Musik,or true Concord can bee made with it, til it bee rectifyed, and browgt to de perfect Hemitonium.69 The invective tone of these little discourses,together with the fact that it was a master craftsman(Denis) and an erudite scholarof high station (Doni) who opposed equal temperament,might well have prompted Mersenneto forego any zealous advocacy of its adoption on keyboard instruments.His scholarlyor scientificcomprehensiveness-letus not call it equivocation-enabled future participantsin this debate to extract passagesfavorableto their own partisanviews. Jean-JacquesRousseau, for instance,writingfor the Encyclopedieabout ten years after Rameau had retracted (in 1737) his favorable comments on unequal temperament, attacked "Rameau's tuning," as equal temperamentwas now being called in France, and cited part of Mersenne'sseventh "Observation" of 1637: 193 Father Mersennereports that it was said in his day that those who first used semitoneson keyboardinstruments... originallytuned all the fifths in something like the equal temperamentproposed by M. Rameaubut that [since] their earscould not bear the discordanceof the major thirds obliged to be too strong,they temperedthe tuning by weakeningthe first fifths in orderto lower the majorthirds.Thus it would seem that to become accustomedto this kind of tuning is not, for a trainedand sensitiveear, an easy habit to form.70 Actually Mersennedid not say that the first musiciansto use the chromatic keyboardhad used or tried out an approximationof equal temperament;he merely said that some people imaginedso ("quelques-vns s'imaginent").71In any case that speculation remindsone of the myth later attachedto Ramos(namely, that he prescribedequal temperament in 1482), and indeed might have inspiredBarbour'sattempt to find evidence of equal temperamentin Gaffurioand Lanfranco.72 I have not discussed to what extent Mersenne'sown views evolved over the years. The principalchange that I can detect is that in about 1635 he apparentlyovercamehis philosophicalobjections to temperament in general and equal temperamentin particular.Thereafterhis evaluations were based exclusively on mundane considerationslike practicabilityand aural quality. This change is very much in keeping with his intellectual development at large, as portrayed by Robert Lenoble,73 whereinthe concern for demonstrabletruthsgrew while the enthusiasmfor metaphysicalspeculationand humanisticelegancefaded. On a rathersubtle level one might detect subsequentlyanothershift in favor of equal temperament:whereas in Harmonie universellethe question, as far as aural quality is concerned, seems to have been whether equal temperament could be tolerated on keyboard instruments in spite of objections voiced by "les praticiens,"in the passage cited from 1644 tolerance seems to be grantedto that delicacy of taste which rejected equal temperamentin favor of meantone. Mersennealways respectedexpertise, but it would be reasonableto surmisethat in his 60's he himself did not hear very keenly the intermittentbeating which no doubt earnedequal temperamentits 17th-centuryopponents. With due allowance for these points, Mersenne'sgeneral views on keyboard tuning may be outlined as follows. He liked the sound of purely intoned concords, and philosophicallyhe favored(at least until 1635) the idea of just intonation. To make just intonation feasible for organsand harpsichordshe thought that a keyboardwith 18 notes per octave, providingevery natural with all its concords, would be a practicableinnovation. He considered 1/4-comma meantone the "perfect" temperament for ordinary organs, citing the strong liking of 194 contemporaneousmusicians for its pure major thirds and large diatonic semitones. The step-by-stepinstructionsin Harmonieuniverselle for setting meantone, however, inadvertentlyallowed readersto infer an irregularscheme with the fifths E flat-B flat-F largerthan pure;and this as we shall see was a timely innovation because it renderedE flat serviceableas D sharpin its role of leadingtone in e-minor.Meanwhile, his interest in a "universaldiapason,"that is, a standardtuning for all instruments,promptedMersenneto suggestthat organbuildersconsider adopting equal temperament as practiced on lutes and viols. He appreciated the enharmonicflexibility of equal temperament,and positively enjoyed its mathematicalchallenge.The complexity and diffuse formulation of his views enabled opponents of equal temperament (Doni and Rousseau) as well as its advocates(Boulliauand MurrayBarbour) to cite him in support of their opinions. In this as in so many areas of inquiry he was, as Bailletwould put it, "plus propre a former des questionsqu'a les resoudre."74 e 195 NOTES 1. Accordingto J. MurrayBarbour,Tuningand Temperament, A HistoricalSurvey, 2d ed. (East Lansing,MichiganState College Press,1953), p. 7, "The most complete and importantdiscussionof tuning and temperamentoccurs in the works of Mersenne."Barbourhimself first becameinterestedin temperaments"when ProfessorCurt Sachs showed me his copy of Mersenne's HarmonieUniverselle"(ibid., pp. vii-viii). 2. MarinMersenne,HarmonicorumLibri . . . (Paris,1636; 2d ed., 1648), pp. 126-27 (in the last pagination).Each intervalis said to be equal to a certain number of commas. Some of the numbersare mutually inconsistent. (For example,G-B is said to form an imperfectthirdof 18 commasand not a perfect one of 17, even thoughA-B is a minorwhole tone (8 commas)and not a major one (9 commas)like G-A.) But the gist seems to have been to derive a 12-note scale in just intonation from an imperfectlyconceived53division of the octave. A year later Mersennegave, in HarmonieUniverselle (Paris,1636-37; facs. ed., Paris,CentreNationalde la RechercheScientifique, 1963), III, 308, quite a good account of the 53-division,which he credited to the militaryengineerJeanGall6: number nameandratio of of corresponding commas: interval: 3 4 5 8 9 10 14 17 22 31 36 39 53 Diese (25:24) Limma(135:128) demiton(16:15) ton mineur(10:9) ton maieur(9:8) ton superflu(256:225) Tiercemineure(6:5) Tiercemaieure(5:4) Quarte(4:3) Quinte(3:2) Sexte mineure(8:5) Sexte maieure(5:3) Octaue(2:1) (NB. This limma is, in modernterms,a schismasmallerthan the Pythagorean limma [256:243]). Galleseems to have been the first to formualtecorrectly the 53-part-division,which approximatesPythagoreanand also just intonation remarkablywell; however,the underlyingidea that the 9:8 whole tone can be dividedinto nine equalpartsgoes back to GiorgioAnselmo(1435). In referringto Salinashere in the HarmonicorumLibri, Mersennecites Book 3, chapter 13; but that chapter of Salinas'De MusicaLibri VII (Salamanca, 1577; facs. ed., Kassel,Barenreiter,1958, p. 139) containsvery little information apropos,and that little contradictsMersenne'sschemeby specifying a minorwhole tone for G-A. 3. Mersenne,Harmonie Universelle,III, 170, and QuestionesCeleberrimaein Genesim(Paris,1622), pp. 1527-28. 196 4. Mersenne,HarmonieUniverselle,III, 351-52, 354. 5. Joan Albert Ban, Kort Zangh-Bericht(Amsterdam,1643; facs. ed., Amsterdam, Frits Knupf, 1969), pp. 27-30. See also Ban's letter of October 16, 1639, to Constantin Huyghens, in Correspondancedu P. Mersenne,ed. Cornelisde Waardet al. (Paris,1932- ), VIII, 536, and the remarksof Descartes to Huyghenscited on pp. 537-38. The context of Ban'swork is set out by D. P. Walker,Studies in MusicalScience in the Late Renaissance (London,The WarburgInstitute, 1978), ch. 6. For the sakeof clarityI should perhapsmention that the system of 18 justly intoned pitch classes(or 19 marchesper octave) is quite different from the system of 19 equallyspaced scale degrees (entailing20 marchesper octave) which Titelouze used on his special harpsichord,accordingto Mersenne(HarmonieUniverselle,II, 439). The latter system, in which the diatonic semitonesare twice the size of the chromaticones, for all practicalpurposesamountsto the 1/3-commameantone systemof Salinas. 6. I.e. at the age of 31. Mersennehad attendedand then taught at the Jesuit Collegede la Fl&che,and throughouthis life defendedhis Christianbeliefs by opposing philosophicalskepticismand cabalisticsuperstitionwith equal constancy. 7. Mersenne,La Vgritedes Sciences(Paris,1625; facs.ed., Stuttgart,Frommann, 1969), p. 32. "La Musiquen'est rien qu'un apparence,puis que ce que ie treuue agreable,vn autre le treuue discordantselon l'esprit qu'il a, et ne donne t'on point de raison pour quoy l'octaue, la quinte, et la quart sont plustost consonances, qu'vne septieme, ou vne seconde, peut-estre que celle-cy sont les vrayes consonances,et que les autres sont des dissonances, car si ce nombre-laconuienta l'vn, celuy cy plairaa l'autre..." 8. Mersenne,QuestionsHarmoniques,pp. 80-83. QVESTIONII. "A sgavoirsi la Musiqueest une science, et si elle a des principescertainset evidens. Plvsievrscroyent que la Musiquen'est pas vne science, et qu'elle n'est qu'vn art mechanique,dont les reglessont fond6es sur les sens, et particulierement sur celuy de l'oreille;ce que l'on peut prouuerpar plusieursraisons,dont la plus puissanteest prise de l'incertitudede ses regleset de ses interualles,car l'on n'a pas encore demonstr6que la raisonde la quinte soit de 3. a 2. et l'on rencontred'excellensGeometres,qui composent tres-bienen Musique,qui nient toutes les raisonsdes consonances,et des dissonances,que les Pythagoriciens,Euclide,PtolomBe,Bo6ce, Zarlin,Salinas,et les autresont expliqu6es, et que croyent que les raisonsde tous les degrezet interuallessont inexplicables, ou sourdes,et irrationnelles,car ils mantiennentque tous les tons, et les demy-tonssont 6gaux:que trois ditons font l'octauejuste: que la quinte superflu6n'est point differentede la sexte mineure:que la fausse quinte et le triton sont vne mesme chose: que la pratiqueet la composition de la Musiqueest beaucoup meilleure,ou plus ais6e en suiuantl'egalit6des tons et des demy-tons,qu'en vsant de la theorie que met leur inegalit6:et finalement que les consonances,et les degrezqui se font surles luths, les violes, et les autres instrumens,et quand et quant que les oreilles,tesmoignentceste ' egalit6. Ie laisse plusieurs autres particularitezqui appartiennent cette et de ses ceux suiuent opinion d'aristoxene, positions, parce que i'en qui fais vn discoursparticulierdansle trait6de luth, et que ie responsen plusieurs 197 autres endroits a tout ce que l'on peut objecter contre les hypotheses de ceux qui ioignent perpetuellement le sens a la raison, et qui preferent l'esprit au corps, et l'intelligence a la sensation, c'est pourquoy ie donne seulement icy [vn discours sceptique sur la musique] afin que chacun le puisse lire auec le mesme contentement que i'en ay receu, et que le Musicien considere ce que l'on peut dire contre cet art, et consequemmet l'obligation qu'il a d'estudier et de se rendre assez scauant pout y repondre." 9. Some indirect links from Steven to Mersenne are cited by Walker, Studies in MusicalScience,p. 120. 10. Simon Stevin, Vande Spiegheling der Singconst, ed. A. D. Fikker in The Principal Works of Simon Stevin, V (Amsterdam, C. V. Swets and Zeitlinger, 1966), 440-41. "Ymant mocht nu achten na doude meyning hoe dattet soet gheluydt der vyfde in soo onuijtsprekelick, onredelick, ongeschickt ghetal bestonde, daer op wij int breede souden connen antwoorden maer want ons voornemen niet en is an donuytsprekelicke onredelicheyt ende ongeschicktheyt van silcken misverstant hier te leeren duytsprekelicheyt redelicheyt geschicktheijt ende natuerlicke constighe volmaecktheyt deser ghetalen, sullent, als elders bewesen hebbende, daer bij laten." See also the pertinent passage in Stevin's L'Arithmetique (Leiden, 1585), p. 83, which is included in his Principal Works,IIB (ed. D. J. Struik, 1958), 532. 11. Mersenne, Les Preludes de L'Harmonie Universelle (Paris, 1634), p. 169. ". .. sont necessaires pour trouuer les consonances iustes en plusieurs endroits de la main, ou de l'6chele de Musique, et du clauier des Orgues, et des Epinettes. Car encore que le temperament des Orgues, et des autres instruments approche si pres de la iustesse des accords, qu'il ne blesse pas l'oreille, qui souffre aysement les quintes diminuees, et les quartes augmentees des instrumens, l'on n'en reqoit pourtant pas tant de contentement que si tous les accords estoient parfaits." Cf. Harmonie Universelle, III, 107, where Mersenne says of the epinette that "its chords and tones approach more closely the just proportion of the harmony than they do on the lute." 12. Mersenne, Correspondance, IV, 51. 13. See Claude Palisca, "Giovanni Battista Doni," forthcoming in The New Grove. I acknowledge with pleasure my debt to Prof. Palisca and Prof. D. D. Walker for reading the present article and emending several of my translations. 14. Mersenne, Traite de L'Harmonie Universelle (Paris, 1627), pp. 198-300. ". .. on peut augmenter le ton mineur, et diminuer le majeur en plusieurs manieres, par la distribution du comma, qu'on peut premierement diuiser en trois parties . . . Secondement, on peut diuiser le comma en sept parties . .. En troisieme lieu, on peut diuiser le mesme comma en deux parties, afin d'augmenter le ton mineur d'une partie, et de diminuer le majeur de l'autre partie. .. Cette premiere maniere . . . est la plus imparfaite, comme je montreray clairement au liure des Instrumens, ou je parleray amplement de la seconde maniere, et de plusieurs autres, qui peuuent seruir ... sur l'epinette et sur le luth, et pour la fabrique des orgues." 15. Mersenne, Correspondance, VI, 80. 16. Mersenne, Correspondance, VII, 17-18; cf. Harmonie Universelle, II, 439. 17. Salinas, De Musica, p. 143; cf. Barbour, Tuning and Temperament, p. 34. Zarlino appears to have tried out this tuning and found it less satisfactory 198 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. (piCi languido) than 2/7-comma or 1/4-comma meantone; see Gioseffo Zarlino, De Tutte L'Opera . . . Dimostrationi Harmoniche (Venice, 1589), p. 145, cited by Kenneth J. Levy, "Costeley's Chromatic Chanson," Annales Musicologiques 3 (1955), p. 224. Mersenne, Harmonie Universelle, III, 104-05. Ibid., III, 109. Michael Praetorius, Syntagma Musicum, II, De Organographia (Wolfenbiittel, 1619; facs. ed., Kassel, Birenreiter, 1959), 132. Mersenne, loc. cit. Ibid., III, 364, 366. Ibid., III, 365. In Roger Chapman's translation (Harmonie Universelle: The Books on Instruments, The Hague, M. Nijhoff, 1957), p. 448, forte is mistranslated as "larger." The table de propositions and these corrigenda do not appear in the 1963 facsimile, because it is of Mersenne's own copy, from the first printing, and they were added subsequently. Mersenne, Harmonicorum Libri, p. 60 (in the last pagination). This point will be developed in a forthcoming article. Mersenne, Cogitata Physico-Mathematica (Paris, 1644), p. 338. Jean Denis, Traits de l'accord de l'espinette, 2d ed. (Paris, 1650; facs. ed., New York, Da Capo Press, 1969), p. 15. Mersenne, Harmonie Universelle, III, 159. Mersenne, Cogitata Physico-Mathematica, p. 335. "Licet autem nostri practici temperaturam suis systematibus adhibeant; haec tamen temperatura non est isomeres siue isemitoniaia, qualem Galeus desiderabat; quae licet nostrorum aures delicatiores, qualis est ingeniosissimus lyropoios Dionysius, vix illam temperaturam semitoniorum aequalium sustenere possent, aliis tamen musicis exercitatissiumis satisfaciebat." Mersenne, La Virite des Sciences, p. 495. "Il y a de certaines personnes qui feignent d'auoir les oreilles si delicates, et l'esprit se 6pure, et poli qu'ils ont horreur des termes de 1'Algebre, comme si ses difficultez 6toient si grandes qu'elles ne peussent estre comprises par vn homme qui a d'autres occupa." tions; mais leur crainte est puerile ... der AlgebraischenDenkweise im 17. JahrSee Michael S. Mahoney, "Die Anfainge derNaturwissenschaften 1 (1971), 15-30. hundert,"Rete:Strukturgeschichte 33. Arp~id Szab6, Anfadnge der Griechischen Mathematik (Budapest, 1969), pp. 148-49, 155-56. 34. Adrien Baillet, La Vie de Monsieur Descartes (Paris, 1691), I, 353. 35. Mersenne, Correspondance, V, 53-54. 36. Mersenne, Correspondance, V, 540. 37. Mersenne, Correspondance, I, 509 (November 2, 1637) and II, 39 (March 12, 1628). In a letter of February 8 Bredeau mentioned the same topic II, 18-19). (Correspondance, 38. Mersenne, Correspondance, II, 64 (April 8, 1628). 39. See Mark Lindley, "Early 16th-Century Keyboard Temperaments," Musica Disciplina 28 (1974), pp. 129-151, and, in reference to Verheyen, The PrincipalWorksof Simon Stevin,V, 465, and Barbour,Tuningand Temperament, p. 35. 199 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. Mersenne, Harmonie Universelle, III, 341. Ibid., I, eighth page of the first preface generale au lecteur. Ibid., III, 37. Ibid., III, 41. Ibid., iii, 39 (last sentence of proposition 14). Ibid., III, 40. Mersenne, Harmonicorum Libri, p. 129 (in the last pagination). Mersenne, Harmonie Universelle, III, 384. Ibid., III, 407-12. Mersenne probably intended to refer to chapter 32. Mersenne, Harmonie Universelle, III, 341-42. Barbour, Tuning and Temperament, pp. 146-48. The passage referred to by Barbour is in Giovanni Maria Artusi, Considerationi Mvsicale, i.e. the second section of the Seconda Parte dell'Artvsi (Venice, 1603), pp. 30-32. For a revision of Barbour's account, see my Lutes, Viols and Temperaments, (Lute Society, 1980). 52. Zarlino, Sopplimenti Mvsicali (Venice, 1588), p. 12. "Questo modo di diuider la Diapason, ouer'Ottaua in dodici Semituoni equali, ha lodato sopr'ogna'altra Diuisione, il molto R. P. Don Girolamo Roselli . . . come ' quella c'habbia da leuare ogni difficolta Cantori, Sonatori, & Compositori, communemente incominciare a cantare b sonare sopra qual delle per poter Dodici parti uoranno, Vt. Re. Mi. Fa. Sol. La. girando per tutte le Note, facendo (come ei dice)....la Musica sferica: . . . perche tutti gli instrumenti potranno tener le loro accordature & unirsi, & gli Organi (com'ei dice) non saranno ne troppo alti, ne troppo bassi di tuono." 53. JohannGeorgNeidhardt,GantzlichErschopfte,MathematischeAbtheilungen des Diatonisch-chromatishen, TemperirtenCanonis(K6nigsbergand Leipzig, 1932), pp. 40-41. ". .. die meisten finden doch an dieser Stimmung nicht, was sie suchen. Es fehlet (heisset es) ihren Tertiis maioribus an der Abwechselung der Schwebungen, und folglich mehrer Gemiiths-Bewegungen. In der triade harmonica lisset sich alles leidlich genug h6ren. Aber wenn die Tertiae maiores alleine, und die Tertiae minores auch alleine, angegeben werden, so wollen jene alzu hoch, diese alzu niedrig klingen.... Wiren denn die Hautbois, Fl6ten, u.d.g. wie auch die Violinen, Lauten, Viole di Gamba u.a.m. auch nach derselben eingerichtet, so miisste nothwendig Chor=und Cammer= Ton, durch und durch, auf das reinste zusammen stimmen: Und hierdurch konnte sich diese Stimmung auch alhier einschmeicheln.... Es Fiihrt also die gleich schwebende Temperatur ihre Bequemlichkeit mit sich, wie die liebe Ehestand." 54. Friedrich Wilhelm Marpurg, Versuch iOberdie MusikalischeTemperatur (Breslau, 1776), pp. 193-95. ". . . besser ist, bey einer aus verschiednen Instrumenten bestehenden Musik, eine gleich= als ungleichschwebende Temperatur zu gebrauchen, die verschiedne Characterisirung der Tonarten zu nichts weiterm dienen wird, als die Verschiedenheit der Misklinge in der Ausiibung zu vermehren. ... Wir wollen itzo das Clavier ausserhalb dem Zirkel eines Concerts fiir sich allein betrachten, und uns einen Componisten figuriren, welcher sich nach seiner eigenen Fantasie eine Temperatur schaffet, und fiir selbige ein Clavierstiick setzet.... Sobald das fiir diese oder jene Art 200 der Temperatur eingerichtete Tonstiick auf einem anders gestimmten Clavier gespielet wird, weg ist der durch die Art der Stimmung bef6rdert werden sollende Character. Wie zufillig sind also die Vortheile, welche die ungleichschwebende Stimmung darbietet. 55. Mersenne,Nouvelles ObservationsPhysiqueset Mathematiques(appendedto Harmonie Universelle, III), pp. 23. The discrepancies of intonation between fretted and keyboard instruments was a commonly recognized problem, mentioned for example by Doni (quoting Giovanni de'Bardi; cf. Oliver Strunk, Source Readings in Music History (New York, Norton, 1950), pp. 290, 297: " ... more than once I have felt like laughing when I saw musicians struggling to put a lute or viol into proper tune with a keyboard instrument") and by Titelouze in a letter to Mersenne dated March 2, 1622 (Correspondence, I, 76): "La diversit6 des instrumentz rendroit bien quelque effet pourveu qu'ilz fussent bien justment accordez et concertez. Mais il y a grande dificult6 de le faire pour la diversit6 des temperamentz qu'ils ont." 56. Mersenne, Nouvelles Observations. pp. 19-20. 57. Barbour,Tuningand Temperament,p. 98. 58. FrederickB. Hyde, The Position of MarinMersennein the History of Music (Ph.D. dissertation, Yale, 1954), pp. 4, 462. 59. Barbour,Tuningand Temperament,p. 7. 60. Ismael Boulliau, Notes sur Thbon de Smyrne, p. 269, cited in Hilarion de la Coste, La Vie du R. P. MarinMersenne,Theologen,Philosopheet Mathema- 61. 62. 63. 64. 65. 66. ticien (Paris, 1649), pp. 51-52. Experientia insuper docuit tonorum diuisionem in partes aequales symphoniam accuratiorem efficere et suauiorem: diuisaque tota octaua in duodecim aequalia semitonia in organorum systemate, melius inter se consonare sonos. De quibus legendus R. P. Marinus Mersennus in tractatu de organis, libro Harmoniae Vniuersalis. Mersenne, Correspondance, IV, 5. Mersenne, Correspondance, IX, 149 (February 29, 1640). Oskar Pollack, Die Kunsttaitigkeit unter Urban VIII (Vienna, 1928), I, 164, gives the following contemporaneous record: "Die Roma li 11 d'Agosto 1640 . . Hier mattina festa di San Lorenzo martire il Sign. Card. Barberino si trasferi con piui di 60 Prelati alla Chiesa de SS. Lorenzo e Damaso suo titolo, dove intervenne alla messa solenne, ... et con tale occasione si 6 scoperta la Tribuna fatta a spesa di detto Eminmo Franc. Barberino, dove vi sono 4 loggie, due per parte, con doi bellissimi organi, che per le vaga Architetture fa bellissima vista ..." In a forthcoming sequel to this article musical examples from Frescobaldi and Froberger will be discussed together with ancillary evidence relating to Froberger. Mersenne, Correspondance, VIII, 17-18. In this letter Doni said that Frescobaldi was overrated as a musician. Since Frescobaldi was organist at St. Peter's, however, Mersenne undoubtedly had a chance to form his own opinion during his Italian sojourns of 1639-40, 1641 and 1644-45 (listed by Baillet in his La Vie de Monsieur Des-Cartes (Paris, 1691), II, 586). Adriano Cavicchi in his preface to Luzzasco Luzzaschi, Madrigali per cantare e sonare a uno, due e tre soprani (L'Organo, Brescia, and Birenreiter, Kassel, 1965, p. 11) cites a relevant passage from Ercole Bottrigari, I Desiderio 201 (Bologna, 1599, p. 41 et seq.) which includes the following remark about Vicentino's archicembalo: "Luzzascho ... lo maneggia molto delicatamente, con alcune compositioni di musica fatte da lui a questo proposito." 67. Giovanni Battista Doni, De Praestantia Musicae Veteris Libri Tres (Florence, 1647), pp. 30-31. Meministis nuper cum pannosus quidam senex in hanc Vrbem venisset; qui nihil aliud sciret, quam modice polyplectrum pulsare; obtruderetque tamquam nouum vtilissimumque inuentum, eam semitoniorum aequalitatem quam vulgo Aristoxeneis, sed iniuria tribuunt ... nobilemque illum physauletem Psychogaurum, qui tunc Palatinae musicae praeerat, frequentibus et gratuitis compotationibus vsqueadeo demulserit, vt eum non puduerit, contra fidem aurium suarum, praeclarum hoc inuentum in caelum extollere apud optimum Principem;... eo iam rem perduxerat vt idem ille Princeps (qui tum forte vnam ex primarijs ac vetustissimus Vrbis basilicis reficiebat, apsidem potissimum atque odeum) ad dissonam illam temperaturae speciem, nobilem eius Ecclesiae physaulum redigi praeceperat: ac nisi Donius noster certissimis rationibus ei subostendisset, vanitatem operis, impensae iacturam, dedecoris infamiam, quae Romanis musicis parabatur, effectum res habuisset. 68. Denis, Traiti de l'Accord, p. 12. (C)eluy qui auoit accord6 l'Espinette me dit qu'il estoit bon pour en jouer, et detonner de semi-ton en semi-ton, et que tous les accords se trouuaient bons par tout, et qu'il s'accordoit mieux que le nostre auec le Luth et la Viole: ie luy dis qu'il auoit mauuaise raison de vouloir gaster le bon et parfait accord pour l'accomoder a des Instruments imparfaits, et qu'il falloit plustost chercher la perfection du Luth et de la Viole, et trouuer le moyen de faire que les semi-tons fussent majeurs et mineurs, comme nous les auons sur 'Espinette, ce qui ne se peut faire auec les touches des cordes dont on touche les Luths, pource qu'il faudroit qu'elles fussent faites en pieds de mousches; ce qui se peut faire par le moyen des touches d'yuoire. Maarten Vente's article "Severin" for The New Grove reports that at Liege in 1626 Jean Gall6, the engineer later cited by Mersenne as an advocate of equal temperament on keyboard instruments, had contracted to teach the young organ builder, Andr6 Severin (1605-1673): "la facon de faire orgues positives, regales, espinettes, et clavis, lesquelles par son invention se pourront haulser et abaisser, s'accordantes a touts tons avec une harmonie meilleure qu'a l'ordinaire, pouvant commencer UT par tout l'octave" ("the method of making positive organs, regals, virginals, and harpsichords such that by an invention of his they can raise or lower themselves to be in tune with any key, with a harmony better than the ordinary one, so that any note of the octave can become DO"). Evidently Gall6 had devised a shifting keyboard-for which equal temperament (or perhaps the 53-division; see note 2 above) would be better suited than any form of meantone. Severin became an eminent organ builder, but is not known to have built such an instrument or used equal temperament. 69. CharlesButler, The Principlesof Musik in Singing and Setting (London, 1636; facs. ed., N.Y., Da Capo Press, 1970), p. 22. Butler's spelling here is in keeping with the orthography proposed in his The English Grammar, or the Institution of Letters, Syllables,and Wordsin the EnglishTongue(Oxford, 1633). 202 70. Republished in Rousseau's Dictionnaire de Musique. "Le P. Mersenne assuroit qu'on disoit de son temps que les premiers qui practiquerent sur le clavier les semi-tons . . . accorderent d'abord toutes les quintes a peu pres selon l'accord egal propose par M. Rameau; mais que leur oreille ne pouvant souffrir la discordance des tierces majeures necessairement trop fortes, ils temperent l'accord en affoiblissant les premieres quintes pour baisser les tierces majeures. II paroit done que s'accoutumer ' cette maniere d'accord n'est pas pour une oreille exerc6e et sensible une habitude ais6e a prendre." 71. Mersenne, Nouvelles Observations, p. 22. 72. Barbour, Tuning and Temperament, pp. 5 (on Gaffurio), 45 (on Lanfranco). In his dissertation (Equal Temperament: Its History from Ramis [1482] to Rameau [1737], Cornell University, 1932) Barbour debunked the association of Ramos with equal temperament and expressed a more judicious opinion about Lanfranco. Lanfranco is discussed in my "Early 16th-Century Keyboard Temperaments," pp. 144-150, and Ramos and Gaffurio in my "15th-Century Evidence for Meantone Temperament," Proceedings of the Royal Musical Association, CII (1976). 73. Robert Lenoble, Mersenne: ou, La Naissance du Mecanisme (Paris, 1943). 74. Baillet, La Vie de Monsieur Des-Cartes, I, 410. ) 203