and Enharmonic Music Berger, Theoriesof Chromatic point I would hope the whole question of editions of this kind could be reconsidered. James Haar of North Carolina at University Chapel Hill CHROMATIC SYSTEMS VICENTINO (OR TO NON-SYSTEMS) FROM MONTEVERDI KAR 0 L B ERGE R, Theoriesof Chromaticand EnharmonicMusic in Late 16th CenturyItaly. UMI Research Press, 1980. vii + 178 pp. A lesser paradox of our times is that men and women are declared learned for having written long essays unfit to publish. Karol Berger is not one of these. His 1975 dissertation for Yale University, written under the guidance of an eminent scholar of sixteenth-century Italian music theory, Claude Palisca, has now appeared with but minor revisions in the 'Studies in Musicology' series published by University Microfilms International. The production of the book is sensibly economical. The author's musical examples have been reproduced in his own hand; the notes are all together at the end, but generously spaced. By setting the price at some 15 cents per octavo page, however, the publisher has shown less regard for the buyer than for the economic prospects of its parent corporation, Xerox. In the introduction and again in the concluding chapter, Dr Berger points out that he has not attempted a definitive treatise on late renaissance chromaticism. Rather, he has proposed 'a hypothesis which would have to be tested analytically to prove its usefulness. At present nothing more than just a brief example [the first phrase of the prologue to Lasso's ProphetiaeSibyllarum]can be given.' (p. 104) The gist of the analysis is that the prologue is in G major but the word 'chromatico' is set in E major. (Readers without access to the book may find the analysis in the October 1980 issue of The Musical Quarterly.) If not from the music, how has the hypothesis been derived? From a reading of contemporary theorists. They are treated, in the first three chapters, as three pairs of antagonists: Vicentino and Danckerts, Zarlino and Vincenzo Galilei, Bottrigari and Artusi. Vicentino and Zarlino receive the most attention; the section on Zarlino 377 Reviews includes some discussion - thirteen footnotes' worth - of Salinas. Dr Berger has a particularly interesting attitude toward the theorists, based upon his belief that 'a historian's task should be to recreate as much as possible the unconscious manner in which the music of a given era was experienced by the community which produced it' (p. 2). He says that the concepts of contemporary theorists may describe this musical experience inadequately, yet the historian should read them carefully and use their ideas to the extent that he deems them adequate: this will prevent him from subjecting the music to 'an arbitrary and ahistorical manner of explanation'. However, 'Chromaticism is an aspect of tonal organisation', and in any case 'the organisation of a sixteenth-century work is primarily tonal'. Therefore the sixteenth-century musical experience which it is the historian's task to reconstruct may rest entirely upon 'the basic principles of tonal organisation', which may in turn be reduced to 'a tonal system' (p. 3). One is tempted to object that the unconscious manner in which the music was experienced by the community must have had a good deal more to it than that. But no matter, this is a book about tonal coherence. The section on each theorist begins with a kind of hors d'oeuvresin the form of remarks outlining his life and career; these are fairly concise and extremely informative. There is always a fine dessert, a report of each theorist's opinion as to the value and purpose of chromaticism. (The last four and a half pages of the book recapitulate the theme of expressivity versus harmoniousness in a relatively informal manner which I found particularly stimulating.) Often there are interesting and substantial garnishments. But the main dish - the substance of the hypothesis - is the tonal system. The tonal system has two parts: first the 'gamut' or 'intervallic material available to a composer', and second the 'precompositional organisation' of this material (pp. 3-4). The second part has a 'generic level', where the distinctions between diatonic, chromatic and enharmonic are made, and an 'octave-species level', where the different modes are determined. The hierarchy of notes in each mode is as follows: the octave is divided into a 5th and 4th, each of these into the diatonic steps (the 5th being first divided into a pair of 3rds), and so on to the finer intervals (p. 102). All this exists prior to the music, but the gamut only metaphysically, inasmuch as 'various tuning systems may be used in practical application of a 378 and Enharmonic Music Berger, Theoriesof Chromatic given tonal system' (p. 100). As far as Dr Berger's reconstruction of the tonal system is concerned, 'Two intervals can be treated as truly different only when they are meant to be used differently by musicians.' (p. 55) This rule gives him a certain interpretative licence with intervallic quantities smaller than a semitone. He deduces that for Zarlino 'a whole tone consists of three dieses', whence it is but a short leap to taking the diesis as 'the largest common denominator of all the intervals of the three genera' in Zarlino's tonal system, which thus 'implies a division of the octave into 19 dieses' (pp. 47-51). Similarly, in Vicentino's tonal system 'There are thirty-one different intervals within an octave resulting from the division of the octave into thirty-one equal parts.' (p. 7) Unfortunately, Vicentino's enumerations of the various possible species of octave, 5th and 4th, from which his tonal system may be deduced, are 'full of mistakes', so a painstaking reconstruction of their intended forms has to be set out. For both Vicentino and Zarlino, and Bottrigari as well, the distinction between major and minor semitones is a feature of the tonal system and not just of some tuning system. This is established in Chapters 1, 2 and 3. But in Chapter 4, entitled 'Conclusion', Dr Berger asks, 'What is the gamut of sixteenth-century music?', and his answer is that since the pre-compositional principles embodied in the tonal system 'are common to all, or most, sixteenth-century compositions and are taken for granted by composers' (unlike precompositional decisions in serial music, which may vary from one work to another), and since certain sixteenth-century compositions treat Ebb as equivalent to D, Ab as equivalent to G#, and so on (the examples cited are Willaert's Quid non ebrietasand Marenzio's O voi che sospirate), therefore 'the perfect tuning system that is implied by the sixteenth-century tonal system is equal temperament' (pp. 98100). So much for the unequal semitones! We have been obliquely prepared for this coup de thedtreforty-four pages earlier, where Dr Berger remarks, at the end of the section entitled 'Zarlino: theory', that: 'the difference between the major and minor semitones is very real already on the level of the tonal system, since the major semitone, being diatonic, is allowed in composition, whereas the chromatic minor semitone is not. At this point, however, we have crossed the boundary between theory and practice.' (p. 56) I do think this is too wilful, and distorts the original musical thought. There are many sixteenth- and seventeenth-century com379 Reviews positions whose structure, unlike that of Quidnonebrietasor O voiche sospirate,depends upon the limitations of meantone temperament with its unequal semitones. Of course this is most clearly the case in music for keyboard instruments, where in any regular shade of meantone temperament each string or pipe from a chromatic note must be tuned either as a sharp or as a flat and will sound rather sour if used in the other capacity in an acoustically conspicuous way (the exact degree of sourness depending also upon the timbre of the instrument and the shade of meantone temperament used, for example 2/7-comma, ?4-comma or 1/5-comma).1 During the sixteenth century it seems to have become common in Italy to build instruments with split keys and thirteen or fourteen strings or pipes per octave to provide for both G? and Ab, or even for D? as well as Eb.2 If we leave aside this complication for the moment, and ignore the experimental use of keyboard instruments with more than fourteen pitch classes, we may readily see that the gamut (in Dr Berger's sense) of a twelve-note instrument, normally with two flats (Eb, Bb) and three sharps (F?, C?, G?), was well suited to maintaining the system of church modes with only a few privileged transpositions at a distance of a 4th or 5th. The Dorian could be transposed to G or A with impunity, but would want Ab if transposed to C, or D? if transposed to E; the Phrygian was often transposed to A but rarely to G; and so on. Tonal structure in the sixteenth century was materially affected by this consideration (which Dr Bergerignores in order to emphasise that 'only an equal division of the octave makes all transpositions These labels ('2/7 comma' and the like) say how much the 5ths and 4ths are tempered. They refer to some fraction of the syntonic comma (the amount by which the 3rds and 6ths will automatically be rendered impure if the 5ths and 4ths are tuned perfectly pure). If we call this fraction 't', then the amounts by which the major 3rds and 6ths are tempered will be 1-4t and 1-3t respectively (see Figures 5a and c below). If the major 3rd were tempered 7/11 comma (as in equal temperament), the diminished 4th - the difference between two major 3rds and an octave - would be, as it happens, the same size. (This is because three pure major 3rds fall short of an octave by an amount practically indistinguishable from of the syntonic comma.) Whatever amount 'm' the major 3rd is tempered less than 7/11 21/11 comma in any particular shade of meantone temperament, the diminished 4th will be 3m larger than the major 3rd, and will accordingly sound more or less inappropriate if used in its stead. 2 J. Barnes, 'The Specious Uniformity of Italian Harpsichords', Keyboard Instruments: Studies ed. E. M. Ripin (Edinburgh, 1971), shows that extant sixteenthin KeyboardOrganology, century instruments are liable to have been altered during the seventeenth and eighteenth centuries to eliminate the split keys. Ferdinando Tagliavini has assured me that the same is true of extant sixteenth-century Italian organs, for example, in Santa Maria Maggiore, Rome, and San Petronio, Bologna. I 380 and Enharmonic Music Berger, Theoriesof Chromatic possible' (p. 117)). Indeed I have often had the impression, reading through a late-sixteenth- or early-seventeenth-century keyboard composition, that the composer has judiciously timed the introduction of his limited number of chromatic pitch classes. A nice example delsettimotono,3in which F# (as well as F) is Andrea Gabrieli's Ricercar from the is present virtually outset, C# makes its first appearance in bar 13, Bb in bar 42, and G$ in bar 58, seven bars from the end (there is no D$ or Eb). A student of chromaticism might approach the matter by observing how often a composition will use all five chromatic pitch classes named above (Eb, Bb F#, C#, G#), and no others, regardless of its mode or tonal centre. That many pieces in the untransposed Phrygian use no D- or use it gingerly, perhaps as in Example 1 - might not seem very remarkable; but that some of them use Eb is noteworthy.4 The renaissance growth of interest in the chromatic genus did sometimes drive composers to think in terms of the equal-tempered gamut (familiar on fretted instruments but virtually never used on keyboard instruments), but sometimes not. An easy-to-find example of radical chromaticism tailored to the meantone gamut is Giovanni Macque's Consonanze stravaganti(transcribed towards the end of volume I of Davison and Apel's Historical Anthologyof Music). A whole class of examples may be evoked by pointing out that a 4th filled in chromatically - a familiar device at the time - cannot always be harmonised in the same manner. This is something Dr Berger's method, and particularly his use of the terms 'unconscious', 'pre-compositional' and 'tonal system', allowed him to miss. There may be other things - the concept of musica ficta is not treated, for instance - but this oversight by itself damages his hypothesis. As a matter of method, I should like to point Example 1 3 Andrea Gabrieli, Intonationedel quarto tono, conclusion; from Intonationi d'organo (Venice, 1593) From his Ricercariper ogni sortedi stromentida tasti (Venice, 1595); ed. in AndreaGabrieli, Ricercarifr Orgel,I, ed. P. Pidoux (Kassel, 1941). 4 The locus classicus is later: the Tocata cromatica for the Elevation in Frescobaldi's della domenica from Fiori musicali (Venice, 1635). 381 Messa Reviews out three ways a well-endowed scholar (such as Dr Berger) might have avoided it. 1. He might have taken note of the rule, commonplace among ethnomusicologists in recent years,5 that scales have no prior existence to the music but are induced after the fact. I grant that too many ethnomusicologists write badly, and some produce rubbish; still they have a broad view of certain issues germane to the history of Western music, and on these issues a scholar of Western art music really ought to take the best of them seriously. Had Dr Berger done so, he would not have presumed to say of Vicentino: 'If the tuning system was the first step in the process of translating the abstract tonal system into a concrete sound-medium usable in musical pracis the end-product of this process'6 without tice, the archicembalo giving a well-considered explanation why the instrument had five more notes per octave than the system. I am afraid the explanation he does give is vague and fanciful:7 'The keys of the sixth row are tuned a comma higher than the corresponding keys of the first row . . . to provide ... perfect fifths above some steps of the first row and below those of the fourth row . . . yet another proof that in Vicentino's mind the tonal system is independent of tuning.' (pp. 24-5) 2. He might have undertaken - or imagined, if his Greek were no better than mine - a systematically philological examination of the theorists' attempt to synthesise ancient and modern precepts in the light of modern practices. This exercise would have shown that a clear or consistent notion of a tonal system was the last thing most of them were likely to develop, and that the amount of space they gave to the effort shows rather that they failed to achieve the synthesis than that their musical or theoretical ideas were derived from a clearly conceived system or from an unconsciously apprehended one of such elegance as Dr Berger suggests. Zarlino's account of vocal intonation was criticised at the time as demonstrably incompatible Numerous books and articles could be cited apropos; the following are taken more or less at random: W. Kaufmann, The Ragas of North India (Bloomington, 1968), pp. 8-10; E. Sonis, ClassicalPersianMusic,an Introduction (Cambridge, Mass., 1973), pp. 52-3; Akira Music musicaleduNo (Paris, 1974), pp. 49-65; D. Morton, TheTraditional Tamba, La structure of Thailand(Berkeley, California, 1976), p. 28. See also H. S. Powers, 'Mode', TheNew GroveDictionaryof Music and Musicians,ed. S. Sadie, 20 vols. (London, 1980), xii, pp. 376-450, and its bibliography. 6 P. 24. The mixture of tenses is due to the inversion: 'was' refersto p. 22, not to Vincentino. 7 An account of Vicentino's tuning prescriptions is given below. 5 382 andEnharmonic Music Berger, Theoriesof Chromatic with any coherent system.8 His most articulate former pupils, Artusi and Vincenzo Galilei, were sometimes reduced to suggesting a mixture of two contradictory systems.9 Vicentino seems a particularly good example of an unsystematic theorist. It is true that he allotted five dieses to the whole-tone (three for the diatonic semitone: since the diatonic scale has five whole-tones and two semitones, one obtains a total of 5 x 5 + 2 x 3 = 31 for the octave). Anyone like Vicentino - a creative rebel, an unfastidious intellect, a microtonal composer - might have adapted to his own perhaps occasional uses this familiar but unorthodox rule from Marchetto's Lucidariummusicaeplanae (1317-18). Marchetto had given only one or two dieses to what we call the diatonic semitone. At the same time he had assigned the ratios 18:17 and 17:16 to the terms 'minor semitone' and 'major semitone', and also said that the 5th (3:2) comprised three 9:8 whole-tones and a minor semitone.l0 These precepts are incompatible with each other and also with what Dr Berger might call his '27-, 28- or 29-division of the octave'. I am not certain whether Vicentino's use of the five-diesis rule was more significant (of a system) than Marchetto's, but I observe that Dr Berger is only one of several scholars who have tried to reduce his thought to a scheme of thirty-one equal parts and had to declare that his explanations, or his compositions," are full of mistakes. The philological method would have yielded some unexpected insights. Dr Berger says that the most interesting aspect of Vicenti8 G. B. Benedetti, Diversarumspeculationem mathematicarum & physicorum liber (Turin, 1585), pp. 177-83. A good account of Benedetti's criticism, which goes awry only in attributing to Benedetti a wish to 'show the composer why a system of equal temperament was a necessity for modern music', is available in C. Palisca, 'Scientific Empiricism in Musical ScienceandtheArts,ed. H. H. Rhys (Provincetown, 1958), pp. Thought', Seventeenth Century 113-18. For an analysis of Benedetti's tuning prescription, see M. Lindley, 'Stimmung und Temperatur', Geschichte derMusiktheorie, vi, ed. F. Zaminer (Berlin, forthcoming). 9 Dialogo di VicentioGalileinobilefiorentino dellamusicaanticaet dellamoderna(Florence, 1581), p. 31: 'la Terza maggiore sia contenuta da vna proporzione irrationale assai vicina alla Sesquiquarta' ('The major 3rd is contained in an irrationalproportion quite close to 5:4'), but the 5ths and 4ths 'vengano accostarsi al Diatono Ditonieo' ('come to resemble those of della modernamusica(Venice, [Pythagorean intonation]'). L'artvsiovverodelle imperfettioni 1600), fol. 31: 'queste cose mi concludono, che la spetie Diatonica, che oggi si tiene per quella di Tolomeo, sia quella di Aristosseno; 6 almeno vna che sia mista dall'vna e l'altra'. 10 Martin Gerbert, Scriptores demusicasacrapotissimum, 3 vols. (St Blasien, 1784), 11m, ecclesiastici pp. 75, 77, 80. 11 An American academic composer, Easley Blackwood, has compiled a list of presumed mistakes in Vicentino's compositions; see his 'The 31 Note Equal Tuning of Nicola Vicentino', Proceedings of the 50th Anniversary Meeting of the National Association of Schools of Music (Houston, 1974). 383 Reviews no's theoryis that 'eachchromaticand enharmonicspeciesof fourth is a transformationof a diatonicone' (p. 10), and similarlyfor each speciesof 5th and octave. One of his tables (see Figure 1) summarises 'the generalformulafor the transformations of the species of fourths'(p. 11). Elsewherehe makesa shrewdpointwhen he says:" musicdoesnotknowanabsolutepitch-standard, Sincesixteenth-century a determined pitchis notdefinedin absoluteterms(asanabsolutely pointin the pitch-continuum), but only relatively, by its intervallic relations with otherpitches.To avoidconfusionwithmodernterminology, I haveused definedpitch.Italiantheorists thetermstepforsucha relatively oftheera in thissense.(p. 98) usedthe term'string'(corda) If he had inquired why they used 'corda' he would have found somethingpertinentto this matterof transformations(at one time the ancientkitharais said to have had only fourstringsof whichthe outer two were 'fixed'a 4th apart and the middle two would vary accordingto the genus).'3I trust other examplescould be found. 3. He might have investigatedhow chromaticmusic soundedin varioustuningsystemsof the day. This wouldhave persuadedhim with more to use Professor Barbour's Tuningand Temperament discrimination.14When he supportshis theory of Zarlino'stonal system by arguingthat of 1/4commanor the 2/7 Neitherthe ordinarymeantonetemperament commavarietyof meantonetemperament correspond perfectlyto the 19-divisionof an octave,but, inJ. MurrayBarbour'swords,'mostof these willhavea smallerdeviation[from varietiesofthemeantone temperament with19ormorenotes whenappliedto a keyboard theequaltemperament] to theoctavethanupontheusualkeyboard.' (p. 55) he goes astray,as ProfessorBarbour'sstatementis not reallyto the He followsProfessorBarbourin sayingthatVicentino'chose point.'15 12 Thispointstrengthens musicat whicheverpitch ourfreedomto performsixteenth-century See H. M. Brown,'Notes(andTransposingNotes)on levelbest suitsthe circumstances. the Viol in the Early Sixteenth Century', Musicin MedievalandModernEurope:Patronage, Sourcesand Texts,ed. I. Fenlon (Cambridge, 1981), pp. 61-78; and A. Mendel, 'Pitch in Western Music since 1500: a Re-examination', Acta Musicologica,1 (1978), p. 91. 13 See for instance Boethius, De institutionemusica,Book 4, Chap. 13. 14 15 This is, of course,J. M. Barbour, Tuning and Temperament,a Historical Survey (East Lansing, 1951), by far the best secondary source available to Berger at the time. The statement, as cited from Barbour, says that most shades of meantone temperament, when extended to twelve pitch classes with eleven uniform 5ths, will not yield as good an approximation to the 100-cent scale as they will contain when extended to some manageable number larger than eighteen. This has very little to do with the character of the non-uniform 5th which is bound to remain in all shades ofmeantone temperament (except when these are extended to nineteen pitch classes. V/2.98-comma) 384 and Enharmonic Music Berger, Theoriesof Chromatic CHROMATIC DIATONIC S T1 - S2 ENHARMONIC 6 s or m3 m3 -S- T3 d or M3 -4d M3 6j --s Figure 1 Berger's general formula (p. 11) for Vicentino's transformations of the species of 4ths. Here the delta sign stands for 'either D or dd', that is, either one 'major diesis' or two 'minor dieses'. the most obvious way to tune his gamut, the temperament dividing the octave into thirty-one equal parts' (except that Professor Barbour would not have implied that the properties of multiple divisions were obvious before the development of logarithms in the seventeenth century).16 Perhaps it would be expedient to outline here the tuning prescriptions in Book 5, Chapters 5 and 6 of Vicentino's treatise."7 The literature on Vicentino could benefit from such an exercise, and this journal seems to be an appropriate place in which to do it. The archicembalo had thirty-six notes per octave, but the tuning described in Chapter 5 has only thirty-one, omitting the last of the six ordini, or 'rows' as they are labelled in Dr Berger's helpful diagram (see Figure 2). The first two rows were to be tuned in a series of 5ths i i G A B 6th E D i Gb Ab Bb C row 5throw 2nd Db Eb keyboard E F 4th row I A6 Ax xI IEx 3rdrow st board st row F D E F G A B C Fx Gx • B E C 2nd row keyboard Figure 2 An octave section of the archicembalo's keyboards as shown diagrammatically by Berger (p. 24). 'I have discarded Vicentino's curious way of naming the keys since it does not correspond to his manner of notation and I have substituted one which does.' 16 '7 I have undertaken a fresh account of the role of logarithms in the history of tuning theory in Chapters 4 and 10 of 'Stimmung und Temperatur'. L'antica musica ridotta alla modernaprattica (Rome, 1555), fols. 103v-104v. 385 Reviews (and 4ths) from Eb to G#, and all these 5ths were to be 'alquanto spontate, secondo che fanno li buoni Maestri' in tuning 'Organi, Monochordi, Clavicemabli, Arpichordi, & altri simili stromenti' - in other words some shade of meantone temperament was to be used.'8 Then the third row was to be tuned by extending the chain of 5ths in two directions (this is set out note by note) to supply the enharmonic counterparts to the second row plus E# and B# (see the upper half of Figure 3). Vicentino said that B in the fourth row would make a 5th la below Gb in the third row ('G sol re ut terzo, & a questo si darar (5th row) 3rd row 1 lstand2nd rows t 7 I t t t t 3rd3rowr 4th and 5th t t 3rd row t / rows @ e (3rd row)( Figure 3 The thirty-one pitch classes in Vicentino's first tuning scheme. Each is represented as a hexagon facing all six of its possible triadic consonances. The entire matrix forms a twisted toroid (a tyre-like shape with a serpentine 'grain') in which the shaded hexagons are 'covered' and eliminated by their identical counterparts on the opposite side of this flat representation. In this kind of tuning, notes that would be enharmonic twins in equal temperament differ by an amount represented by Vicentino's dot; hence Vr, for example, is the same as Gb, but D is higher than D. 18 Not necessarily V4-comma:see M. Lindley, 'Early 16th-Century Keyboard Temperaments', MusicaDisciplina,28 (1974), p. 150. 386 and Enharmonic Music Berger, Theoriesof Chromatic quinta sotto, che sara B me in quarto ordine'), but he recommended tuning the fifth row before the fourth, starting from the fifth row's Gb a 5th above B#/Cb in the third row and proceeding by a chain of ascending 5ths (again he named them one by one) through the fifth and fourth rows (see the shaded area in the lower half of Figure 3). He concluded: cosi s'accordero 6 si temperera questo thus one will tune or temper this quarto ordine come fu fatto il primo, fourth row as the first was done, & sara accordato tutto lo stromento. and all the instrument will be tuned. In Chapter 1 he had said that on the archicembalo, 'in ogni tasta non li manca consonanza alcuna'. Chapter 7 confirms that the first tuning contains a circle of 5ths as shown in Figure 3. In this scheme the thirty-one 5ths must average 18/31octave, or 0.24 comma smaller than pure; and the major 3rds must average 0/31 octave, or larger than pure by some 0.04 comma - an inconsequential amount in itself, particularly on an Italian harpsichord,where the timbre of the individual notes is so lively as to camouflage such a tiny degree of impurity in an interval. But Vicentino did not say that the major 3rds would sound as pure as this average might suggest, nor that all thirty-one 5ths should be uniform. Meanwhile in Chapter 6 he described another 'modo d'accordare', in which the first three rows were tuned as before, but then il quarto dark le quinte perfette, al primo ascendenti, cioe al di sopra, & il quinto dark le quinte perfette al secondo ascendenti, et il sesto potra dar le quinte perfette al terzo, per6 ascendenti the fourth [row] will give perfect 5ths above the first, that is, higher; the fifth [row] will give perfect 5ths above the second; and the sixth [row] can give the perfect 5ths to the third [row], again higher I cannot imagine a scheme that answers literally to this prescription (for one thing, the sixth row does not contain enough notes to provide a pure 5th above every note in the third row), but perhaps Figure 4 represents the actual tuning. In any case, each note on the back keyboard must have been tuned higher than its (unison) counterpart on the front one by whatever amount the 5th below it was tempered - and this is a smaller order of magnitude than a diesis inasmuch as 0.24 comma is only some 131/2% of 1/31 octave. In discussing the second tuning Vicentino said in Chapter 6 that if 387 Reviews S3rd row and 2nd 14st cJ 0 S KEYBOARD 3rd row 6th row the onopposite side. The 5ths and 4ths are tempered within each keyboard, but are 6throw Figure 4 The thirty-six pitch classes in Vicentino's second tuning scheme, each represented as a hexagon facing its triadic consonances available on the same of (from lower left to keyboard. The matrix for each keyboard forms a spiral 5th3rds dark hexagons are covered by their identical counterparts upper right) in whichthtamente each keyboard, but are on the opposite side. The ths and ths within thosempered aremo purebetweenthetwokeyboards. ne played an octave front keyboard but the 3rd and 5th on obyan amthe the back, then nel i le quinte medesimi anchor che ordini, in perfet e le terze perfettamente che quel e I think this implies that in toccard si quel i ritrouera the maggiori, piu' also che perfectly within those usiamo same perfect accordate noi the the each 388 there 3rds major keyboard wil plays one one find more than tuned which which [in] rows 5ths, we use the major 3rds were and Enharmonic Music Berger, Theoriesof Chromatic 5ths were tempered: otherwise it would not improve, say, C-E to combine the back keyboard's slightly higher E with the front keyboard's slightly lower C.'9 For example: if the 5ths were tempered by 0.3 comma, then the major 3rds on the same keyboard would be 0.2 and the minor 3rds 0.1 comma smaller than pure (see Figure 5a: I might mention that this is fairly similar to the 2/7-comma scheme prescribed by Zarlino in 1558).2o Under these circumstances if one combined C on the front keyboard with its pure G on the back, then the back keyboard's E would be 0.1 comma higher than pure to the C as well as to the G (see Figure 5b), and thus the major 3rd would be only half as impure as if both its constituents were on the same keyboard. Then in the first tuning, if the seventeen 5ths tuned 'secondo che fanno li buoni Maestri' (that is, in the first, second and fourth rows) were tempered by an average of 0.3 comma, the tempering of the other fourteen would average about '/3 comma (for a total average of 0.24), and the major 3rds among these fourteen 5ths would average 1/3 comma larger than pure, while most of the other major 3rds would be intermediate between 0.2 comma smaller and 0.3 comma larger. This degree of variety among the 5ths and 3rds would not deprive the tuning of its euphony, but it would render Vicentino's thirty-one dieses somewhat unequal. Had Vicentino said that the minor 3rds (and not the major ones) in the second tuning were 'pidi perfettamente accordate' when the lower note was played on the front keyboard and the higher note on the back, then by substituting figures 5c and d for 5a and b, one might readily infer that in his first tuning the major 3rds were virtually pure and the thirty-one dieses virtually equal. But he didn't. More significant, however, is that the second tuning is very different indeed from the first one. Under the circumstances a particularly appropriate method of research into Vicentino's thought would be to reconstruct the instrument (he encourages us with measurements and full-size woodcuts) and explore its resources in the light of his tuning instructions and his compositions. Marco Tiella has played a 19 I assume that 'piuiperfetto' means not only 'better' (and 'larger') but also 'more nearly pure'. If not, the statement indicates merely that Vicentino preferred to use major 3rds somewhat larger than they were on either of his keyboards. 20 Inasmuch as 126 ff. 2/7 = 0.286, which is close to 0.3. Le istitutioni harmoniche(Venice, 389 1558), pp. Reviews t = 0.3 comma(seenote 1) -0.2 (a) 0.1 major6th or minor3rd tempered(1-3t) = 0.1 comma major3rd tempered(1-4t) = -0.2 comma 0.3 j0.3 0.1 (b) 0.30 I0.3 0.1 0 front keyboard G back keyboard t = 0.24 comma(seenote 1) major6th or minor3rd tempered(1-3t) = 0.28 comma major3rd tempered(1-4t) = 0.04 comma (c) 0.04 0.28 0.24 0.24 0.24 0 0.24 (d) 0.04 front keyboard 0.04 back keyboard Figure 5 Two models of regular meantone temperament (a and c), with ancillary models (b and d) showing how some of the 3rds might be improved by playing on two keyboards tempered alike but with the back keyboard tuned higher than the front onejust enough to make pure 5ths between them. In (a) and (c) the shaded Es match the unshaded ones. (c) extended to a chain of thirty-one 5ths and 4ths would divide the octave into equal microtones. fair amount of sixteenth-century music on an archicembalo which he has built to Vicentino's specifications, and has found the second tuning musically so much more telling that he now believes it gives us a better gamut for Vicentino's compositions. To judge by Mr Tiella's illustrations that I have heard, I might well agree. Certainly 390 and Enharmonic Music Berger, Theoriesof Chromatic Vicentino appears to have considered the second tuning indispensable: si potra far un'organo che sara diuino accordato con ii primo accordo senza quinte perfette, & poi s'aggiognera un registro con le quinte perfette accordate nel sopradetto modo one can make an organ that will be divine tuned with the first tuning [that is] without perfect 5ths, and then one will add a register with the 5ths perfect, tuned in the manner set out above as indeed he did the first. This is one of those labyrinthine problems for which an inordinate amount of time may be required to develop a new consensus among the experts. Whatever the outcome, let the sound of the music be taken into account. Dr Berger's finest work related to tuning is in Chapter 3. But even here he is not always as penetrating as the very intricate nature of the material would ideally require. He says: 'Probably the most interesting aspect of Artusi's theory was his clear realization that contemporary composers failed to differentiate either between the major and minor whole tones or between the major and minor semitones.' (p. 90) The discussion begins problematically with an assertion that not only were the whole-tones equal on instruments but 'according to Franciso Salinas to whom Artusi approvingly refers, the whole tones used in singing are also equal' (p. 90). Actually Salinas's considered opinion of the whole-tones in unaccompanied singing is somewhat of an embarrassment to this interpretation of Artusi's, though quite congenial to the main point at hand, which is that composers did not have to take into account a distinction between major and minor whole-tones: in cantu ...Ditonum in duos Tonos, alterum maiorem, & alterum minorem, semper diuidatur ... in arbitrio canentis est, utrum eorum ad graue, vel in acutum enuntiare malit, vt aliae partes, atque ipsa melodia videbitur postulare in song... the major 3rd is always to be divided into two whole-tones, one larger, the other smaller ... It is for the singer to judge which of them he prefers to execute below or above as the other [voice-]parts and the melody itself are seen to require"2 Now Dr Berger cites a very suggestive passage from Artusi, to the effect that: the most modern practitioners do not recognize any difference between whole tones nor between semitones ... [T]hey themselves do not know 21 De musica libri vii (Salamanca, 1577), p. 139. 391 Reviews what is the tuning system that is sung and played, although one can judge that they think they follow Aristoxenus, who had divided the whole tone exactly into two equal parts. The most certain proof of this will be provided by the composition for two parts by Mr. Adrian [Quidnonebrietas]. . . not far from it will be many madrigals by Porta, Cipriano, Gabrieli, and so many others.22 This leads Dr Berger to discuss Quid non ebrietasas 'the most important proof' that late-renaissance composers thought in terms of equal temperament. I think he might have done better to give examples illustrating the latter part of Artusi's sentence; but I certainly agree that Willaert's unusual and much discussed composition was conceived for equal temperament. An early authority, discounted by Professor Lowinsky in his article on Quid non ebrietas, was Marc Antonio Cavazzoni, cited in Giovanni Spataro's letter (1534) on Quid non ebrietashaving remarked apropos that 'el leuto ha tuti li soi tasti semitonii minorii'. Professor Lowinsky's gloss ('If Cavazzoni wrote that the frets of the lute were all placed in intervals of minor semitones, then he expressed himself inaccurately')23 shows an ignorance of the fact that late-medieval and renaissance theorists, following Boethius, often associated the term 'minor semitone' with the ratio 18:17,24 which happens to make a very good prescription for placing the frets down the neck of a lute for equal temperament. It puts the octave fret shy of the string's midpoint by some 1/3 of 1% of the total length (comparable on a tenor lute to the width of the fret itself). This might be considered a defect from a certain theoretical point of view, but in reality 18:17 works better no allowance for the string's greater than 1 as the latter makes V"2 when it is pressed down to the fret. On a good instrument tension (that is, with a low action) the 18:17 rule renders the stringjust about 22 L'artvsi,fol. 20v. I have abridged Dr Berger's translation (p. 90). 23 E. Lowinsky, 'Adrian Willaert's Chromatic "Duo" Re-examined', TijdschriftvoorMuziek18 (1956-9), pp. 17, 19. ProfessorLowinsky surmised, correctly,'What he must wetenschap, have meant is that they are placed in intervals of equal semitones.' 24 For a list, see M. Lindley, Lutes,ViolsandTemperaments (Cambridge, forthcoming). In Book 1, Chapter 16 of De institutionemusicaBoethius argued: '16 ac 18 collati, sesquioctavam retinent proportionem, atque idcirdo tonum. Sed hanc proportionem 17 numeris medius non in aequalia partitur . . . Est enim minor pars septimadecima, maior sextadecima ... inter haec unum maius semitonium nuncupatur, aliud minus.' ('16 and 18 together make a 9:8 proportion, and thus a whole-tone. But this proportion is not divided equally by the middle number 17 . . . For 1/17is the lesser part, and V16the greater ... Of these one is called the major semitone, the other minor.') 392 and Enharmonic Music Berger, Theoriesof Chromatic long enough to compensate.25 Nor is Cavazzoni's remark quite as isolated from Vincenzo Galilei's well-known prescription of the 18:17 rule (1581)26 as one might imagine: Martin Agricola said in 1545 that 'nearly the majority of lutenists and viol [players] make all the frets equal ... Each fret makes a minor semitone';27 and Girolamo Cardano, who probably played the lute and cittern, wrote in his De musica (first drafted c. 1546),28 'a whole-tone consists in the 9:8 proportion ... and either half consists in the proportion of 18 to 17 and is called a minor semitone'.29 Later Cardano remarked of the 18:17 and 17:16 semitones, 'It is wonderful how the minor semitone is so nicely suited to musical performance, but the major [one] never. '30 After setting out some additional evidence that for Artusi modern music presupposed equal temperament, Dr Berger takes care to show that notwithstanding all this, 'Artusi's attitude toward equal temperament was . . ambivalent.' In 1600 he denied its use to the harpsichord: If one could bring the harpsichord to the temperament of equal whole tones and similarly equal semitones, one would hear a strange [insolita]harmony. I recall having tempered a whole octave of a harpsichord with a lute as exactly as it was possible, but the sound, being outside of its natural temperament, offended the sense of hearing exceedingly.3' and in 1603 Artusi, as Dr Berger puts it, 'was no longer ready to accept the fact that ... vocal music could be conceived in terms of presupposed equal temperament and fell back upon the old Zarlinian doctrine of natural (presumablyjust) intonation of vocal music' (p. 92). The passage in question is the ninth considerationein the Considerationimusicali, which make up the second halfofArtusi's 1603 publication (itself entitled Secondaparte dell'artusi.. .). It is an impor25 For a modern calculation, see vonGitarre,Laute, F.Jahnel, Die GitarreundihrBau: Technologie Mandoline,Sister, TanburundSaite (Frankfurt-am-Main, 1962), pp. 150-3. 26 Dialogo, p. 49. 27 Musicainstrumentalis deudsch(4th edn, Wittemberg, 1545), fols. 53v-54: 'fast das groste part der Lautnisten und Geiger art Alle bund machen gleich von ein ... ein bund Der Semiton minus thut kind'. 28 See HieronymusCardanus,Writingson Music, trans. C. A. Miller (n.p., 1973), pp. 17-19. 29 Chap. 2, paragraph 26. Operumtomusdecimus(London, 1663), p. 107: 'sciendum, quod tonus consistit in sexquioctava vt demonstrabo & dimidium quodquod consistit in proportions 18. ad 17. & vocatur semitonium minus'. nvmerorvm 30 Opvsnovvmdeproportionibvs (Basle, 1570), p. 170: 'Hic subit admiratio quomodo semitonum minus aptent tam gratem in symphonijs, maius autem nequaquam.' 3' L'artvsi, fol. 27r-27v. Dr Berger's translation (p. 92). 393 Reviews tant passage, and more congenial to Dr Berger's hypothesis than he has realised. I shall conclude by discussing it in sufficient detail to bring out its significance. At the outset Artusi says that in his previous criticism (1600) of 'certain modern composers' he had not named anyone, hoping that this tact would induce them to admit their errors rationally. The examples of music cited by Artusi in 1600 have been identified and suggest that all these composers were really one and the same person, Claudio Monteverdi;32nevertheless Artusi says now that 'some' of them, guided by capricious humours, not only failed to change their views, but went from bad to worse. (In fact Artusi had been sending letters, 'full of benevolence and civility', to Monteverdi who 'instead of answering in the same manner, made reply through a third person, and letters without his own name'.33The letters were signed 'L'Ottuso Accademico' ('Obtuse Academic') - a wicked use of assonance and mock self-deprecation to stigmatise poor Artusi himself.) At this point the syntax breaks down as an elaborate simile is attempted about an artist who thinks he is depicting 'una figura ben fatta, & che nelle sue parti sia proportionata' but is really representing a monster such as described in Book 5 [sic] of the Aeneid.34 (The rhetorical effect would be more impressive if Artusi's own sentence were coherent!) On his way out of the simile Artusi gives us a concise summary of the musical monstrosities to be discovered in the work of these wretched composers: Il che imitano benissimo questi tali Which [monster] they imitate very well, quando nel mezo delle loro cantilene, when in the middle of their songs, nel principio, & nel fine ci rapportano at the beginning, and at the end they set out 32 See C. Palisca, 'The Artusi-Monteverdi Controversy', The MonteverdiCompanion,ed. D. Arnold and N. Fortune (London, 1968), pp. 134-5. Notice that in the passage cited on p. 392 above, Artusi did not disapprove of 'li pratici piuimoderni' ('the most modern practitioners'- Willaert, Porta, Rore et al.) even though he suggested that 'non conoscano qual sia quella spetia d'Harmonia, che si Canti, e Suoni' ('they do not know which is [really] the type of harmony that is sung and played'); yet now he attributes the defects of 'certi moderni compositori' (Monteverdi) to capriciousness even though 'they' have at least offered a theory to justify their simultaneous use of a sharp and a flat. mvsicaledi AntonioBraccinoda Todi(Venice, 1608), p. 6: 'Sig. Claudio hebbe 33 Discorsosecondo all'hora il torto lui; perche quando l'Artusi gli scrisse quelle lettere, gli le scrisse piene d'amoreuolezza e ciuilth; & egli in vece di rispondergli nella istessa maniera, gli fece rispondere per vna terza persona, & lettere senza nome proprio.' On the same page Artusi's fictitious spokesman 'Braccino' threatens 'on a better occasion' to publish 'le lettere, le copie di cui sono nelle mani mie, & debbono ancora essere nelle mani del Sig. Monteverde'. However the letters of 'Ottuso' were actually drafted, there is no evidence that the views they expressed were distinct from those of the composer they defended. 34 Artusi says 'quinto', but the lines he gives are from Book 3 (216-18). 394 Berger, Theoriesof Chromaticand EnharmonicMusic interualli sgarbatissimi da modulare; nella pouerth dell'Harmonia; nella lontananza tal uolta delle parti, gl'estremi di cui se ne giungono fino alle 23. Voci; nella poca osseruanza de modi; nella positione, & ordine delle consonantie, lontana dalle buone Regole; nella mala imitatione delle parole, come si pu6 uedere nel principio della Cantilena di sopra posta, che dice, Ma se con la piet', L'Harmonia di cui piui tosto moue a risa, che a pieth very ungraceful intervals to perform; in the poverty of the harmony; in having the voices sometimes too far apart (their extremes reach as far as 23 diatonic steps); in their slight regard for the modes; in placing and ordering the consonances [in a manner] alien to good rules; [and] in the poor imitation of the words, as one can see at the beginning of the song given above, which says 'Ma se con la pieta', the harmony of which would move [one] sooner to laughter than to pity 'Poor imitation of the words' - can thisbe Monteverdi? Indeed; the example cited is from the second half of EccoSilvio,later published in his fifth book (1606) of madrigals (see Example 2).35Now we reach the heart of the matter: the real cause of all these bad composers' faults is their reliance upon intuition rather than science (so venerable was Boethius's precept that the true musician, 'weighing Example 2 Monteverdi, Ma se con lapieta (second part of Ecco Silvio), opening; from II quintolibrode madrigali(Venice, 1605) Ma Ma se con la pie - t Ma Gen-ti-lez - za e va - lor va -lor e va la non e in te che te co chete-co nac lor che te non nac - - - e in te Gen-ti - lez spen-ta pie -tt - non pie -ta se con la e in spen-ta - za te spen-ta que, que, MI .W 1 1t E. Gen-ti-lez - za e se con - co nac - que, 35 In addition to the apparently balletto-like rhythm, I suppose Artusi disapproved of placing the weak syllables 'la' and '-za' on strong beats. For a modern discussion see F. Razzi, 'Polyphony of the seconda prattica: PerformancePractice in Italian Vocal Music of the Mannerist Era', Early Music, 8 (1980), pp. 306-9. 395 Reviews reason, claims the science of song not by the servitude of work but by the authority of speculation').36And the proof? Their absurd theory of certain chromatic intervals. According to the theory, the diminished 4th F$-Bb would be made up of two 16:15 semitones (F#-G and A-Bb) and a minor whole-tone (10:9 for G-A) were not the whole-tone rendered Pythagorean (9:8) by taking half a comma from each of the semitones (see Example 3). This would entail dividing the comma according to 'the doctrine of Ludovico Fogliano' (see Figure 6). Meanwhile the semitones, in giving away a 1/2comma each, would become exactly half the size of the 9:8 whole-tone. Example3 Intervalscomprisedin a diminished4th accordingto 'certainmodern composers'takento task by Artusi 16:15 10:9 16: 15 less comma 2 9:8 16:15 16: 15 less ; comma In modern terms, the theory's diminished 4th, with the ratio 2 512:405 (= 6/15 X 16/1-53X 1/9), would amount to 406 cents (log This is slightly less than the Pythagorean major 512/405 + -2). 3rd or 'ditone' (/8 x / = 8/64), which amounts to 408 cents (log The discrepancy is in the direction of the 400-cent 3rd of equal temperament (which also serves of course as major+-). a diminished 4th). The quantitative assertion to which Artusi takes exception is that 112 cents) were to give if the diatonic semitone (log 165 + 1 0 81/64 up a 1/2 comma (half of some 22 cents, as log /80 + 200 22), the resulting 101-cent semitone would amount to half of the 9:8 whole-tone, that is, half of 204 cents (log 9/8 + log 2/1200),which in fact it misses by only 1 cent, and again in the direction of the equal-temperament equivalent. Of course diminished 4ths were stock in trade for Monteverdi in 36 De institutione musica, Book 1, Chap. 34 'ratione perpensa canendi scientiam non servitio operis sed imperio speculationis adsumsit'. 396 .co I .? o FED t .D"'8 Figure 6 Ludovico Fogliano's division of the syntonic comma (Musica theorica (Venice, 1529), fol. 36). AB and BD represent the monochord string lengths for two pitches a comma apart; their perpendicular BC represents the length for a pitch half way between them musically.a The theoretical difference between this and the pitch produced by a string length of 801/2is only some 1/3ocent, b so in this case (dividing 81:80) the exercise is of purely metaphysical interest. aThisis becausethe ratiosof the lengthsBA:BCand BC:BDare equal,whichin turnwas known to have been proved by Euclid. If one drew the triangle ACD it would, being inscribed in a semicircle, make a right angle at C (according to a previously proved theorem), so the angles ACB and DCB would complement each other. Since the angle ACB would also be complementary to CAB, and DCB to CDB, the triangles ABC and DBC would have the same be similar, from which it follows that BA:BC::BC:BD. angles and therefore b (log 801/80 - 1/2log 81/80)+ log /1200= 1/29.9 Reviews his chromatic moments. One need only think of the concluding section ('Per far che moia') of Ah dolentepartite,the first madrigal in his Quartolibro(1603). Example 4 shows a diminished 4th in the peroration to that chromatic tourdeforce at the end of the fourth book Piagnee sospira. Example 4 Monteverdi, Piagnee sospira,first appearance of the concluding verse; from II quarto libro de madrigali (Venice, 1603) Spar- gea di pian - to le ver - mi - glie ver - mi-glie go Spar- gea di pian - to le ver - mi - glie go - te te - go - te The theory (as reported by Artusi) also gave a special status to C--Bb: not really a 7th, or a 6th, but nonetheless a good sound. Prominent diminished 7ths are rare in Monteverdi's music, but the few I have noticed are between C$ and Bb. Examples 5a and b are from Orfeoand the Lamentodi Arianna(first performed 1608). I should mention that in describing and criticising this theory Artusi wrote down a number of phrases and statements - 'Ludovico Fogliano's doctrine'; 'one must ascertain half of the 81:80 comma, because by such a quantity the 10:9 is smaller than the 9:8'; 'divide the whole-tone into two equal parts'; etc. - which seem to have given Professor Barbour the impression that Artusi himself was proposing to divide Fogliano's mean-tone (a 2 comma larger than 10:9 and smaller than 9:8) into two equal semitones.37Professor Barbour inferred that on lutes and viols the seven frets forming a major scale from the open string were to be set as in ?-comma meantone temperament (in French tablature these are frets c, e, f, etc.); but at the same time what we might call the five 'chromatic' frets (b, d, g, etc.) were to be set for a pitch halfway between the b and 4 alternatives (some 41 cents apart) in ?/-comma meantone. This means that within each group of frets every interval would be the same as in would ?/-comma meantone, but any interval involving both groups differ from its equivalent in ?/-comma meantone by some 201 cents a HistoricalSurvey(East Lansing, 1951), pp. 146-8. 37 Tuningand Temperament, 398 and Enharmonic Music Berger, Theoriesof Chromatic Example5 Diminished7ths in Monteverdi'sOrfeoand Arianna;(a) fromAct 2 scene 1 of Orfeo;(b) from O Teseo,the second part of the Lamento as d'Arianna, (Venice, 1614) publishedin Il sestolibrodemadrigali (a) [Messaggiero] e te chia-man-doOr-fe - spi - ro -o fra ques-te spi-ri Or-fe - o Do-poungra - ve so- brac - cia (b) ci ci - bo di - - fe bo di - fe - - re in so-li re di - - ta - spie-ta riea - te e - re - ne cru- de (which I shall reckon here as a comma since the syntonic comma is only about 1 cent more).38 If each open string were tuned at unison with the next lower string stopped at the appropriate fret - a standard procedure then as now - then between the two groups of frets any octaves, unisons or major 3rds would be impure by a comma, and any 5ths, 4ths, minor 3rds or major 6ths (all of which are tempered by V4comma in the regular scheme) would be impure by comma. I might add that some of these intervals are 3/4 or by 1?V4 so commonly encountered (see Example 6) that competent players, then or now, would move the frets in the course of tuning the instrument.39 Professor Barbour misunderstood. Artusi approved of equal temperament on the lute and chitarrone, but 'according to the view of Aristoxenus' - that is, without numbers.40 Elsewhere he had ex38 (log 81/80 - 1/2 log 128/125)+ log2/1200 = 0.98 39 For a more complete discussion, see Lindley, Lutes, Viols and Temperanents. The Harmonics of Aristoxenus, trans. H. S. Macran (Oxford, 1902), p. 189. 4 399 Reviews Example 6 Some sour octaves, 5ths and 4ths in the tuning Barbour to Artusi scheme attributed by pressed the hope that theorists might find a satisfactory formulation of equal temperament ('proportioni tali, che fra di loro siano eguali'),41 but his attack on what appears to have been Monteverdi's theory shows that he felt they must not do it with irrational numbers.42 (Of course that is the kind one needs, in this case V2. Here again Artusi was indebted to Boethius, who had assigned 'continuous quantities', alias 'magnitudes', to geometry and astronomy, and excluded them from the science of music, which he said dealt with 'discrete quantities' or 'multitudes' - by which he meant rational numbers.)43 Although Artusi could give no formula for equal temperament, he said that the intervals had a 'prefisso termine' on 'l'instromento fatto dall'arte'. He did not actually say whether they could be tuned by ear or must be set out by geometry, as his teacher Zarlino had shown how to do in 1588 (see Figure 7). But he did say - and this is the point which Dr Berger has (to his credit) cited - that the voice could not justly divide the whole-tone into two equal parts. He also described the diminished 7th and diminished 4th as 'false for singing' (though he approved implicitly of Marenzio's use of the diminished 7th in his madrigal Falsa credenza 41 L'artvsi, fol. 31. the result if not itself an integer will be a fraction or else an 42 When one divides two integers integer plus a fraction. But roots of integers are never fractional. One can easily understand why by taking the opposite perspective: all the powers of, say, 21/n (where 'n' is an integer) are bound to be fractional! Numbers that are neither integer nor fractional have been called 'irrational' (that is, divorced from ratios) since the ancient Greeks. A synonym is 'surd'. Of course there is nothing unreasonable or uncertain, pace Artusi, about these 'irrational' answers to root-extraction problems, just as there is nothing unwholesome or incomplete about fractional answers to division problems. It is just that the two categories are mutually exclusive. 43 De arithmetica, Book 1, Chap. 1. In this passage the quadrivium is introduced and divided as follows: stable magnitudes are to be studied in geometry; rotating magnitudes in astronomy; multitudes per se (like 3 or 4) in arithmetic; ratios between multitudes in the science of music. That Boethius saw this as more than just an academic distinction might be inferred from Book 3, Chapter 1 of De institutionemusica, where he criticised 'Aristoxenus musicus' for saying that the whole-tone comprised two equal semitones, and repeated in this connection the argument cited above in note 24. One infers that Aristoxenus was wrong to consider intervals as magnitudes. 400 onJonanfhqjfif n udmSmtn a 0- Figure 7 One ofZarlino's geometrical methods of fretting a lute for equal temperamusicali(Venice, 1588), p. 211), combining the Euclidian method ment (Sopplimenti for finding one geometrical mean (see Figure 6 and note a) with another for finding two geometrical means (A:x:y:B). Zarlino rejected the 18:17 rule because he knew that (17/18)12 > 1/2. Reviews havetedonna,opening; from the anthology I lieti Example 7 Marenzio, Falsacredenza amanti(Venice, 1586) - Fal vi ".I cre-den sa za ha -ve - ---- Fal r - sa cre-den - zaha-ve Fal - sa cre-den - za ha te, Don - - - te, Don - i - - e - Don te, - - na na na as shown in Example 7), and he declared that the natural voice could not negotiate so unnatural an interval as a diminished 4th by means of a natural major 3rd. A slippery word, that 'natural'; but clearly Artusi felt that singers could never gauge an interval by sheer magnitude, and he implied that the irrational numbers latent in the theory under attack - in modern terms the square roots of8 1/80and of 9/8 - prove that Monteverdi had no 'rational' understanding of music. Here, to conclude, is an abbreviated version of the passage, starting after the list of monstrosities translated above: Tutto questo disordine da altro non nasce, se non che non intendono altro che quello che gli capricij loro le dicono, che stij bene; perb ci rapportano interualli tall'hora, che loro stessi non li conoscono, dicono perb che sono cose noue, se ben sono piii uecchie, che il Cucco; come li seguenti, il primo de quali dicono, che non e ne sesta, ne settima, ma che consona benissimo alle sue orecchie, che sono purgate. All this disorder stems from nothing other than that they understand nothing other than that which their caprices tell them will be all right. For they sometimes set us out intervals which they themselves do not know, and say that they are something new even though they are older than the cuckoo-bird: like the following, the first of which, they say, is neither a 6th nor a 7th, but resounds very well to their ears, which are purged. Interualli Intervals per cantare falsi, false for singing; ma per sonare ne lauti buoni. rI - I , Il secondo uogliono, che sia una terza, ouero Decima ... contenuta de due semituoni ... di proportione sesquiquindecima, e'l tuono I but for playing on lutes, good. They hold that the second [interval] is a 3rd, or rather 10th, containing two semitones of a 16:15 proportion, and that the whole-tone 402 Berger, Theoriesof Chromaticand EnharmonicMusic che nel mezo uiene ad esserui posto, dia di proportione sesquinona, ma che per6 col mezo, & ordine della Dottrina di Ludouico Fogliani, uogliono leuare dall'uno e l'altro semituono tanta quantith, che il tuono per tale accrescimento diuenghi sesquiottauo, & gli semituoni restino fra di loro eguali, e per la meta del tuono. Quanto al primo interuallo, dico, non e cosa noua, perche fu usato da Luca Marenzio nel principio d'un suo madrigale, le parole di cui dicono: falsa credenza, per dimostrar apunto un'interuallo falso nelle voci, & nella modulatione, ma non e falso nel lauto, & nel chitarone . . . perche nel luogo istesso, che il Sonatore pone le dita per farci sentire una sesta, le pone ancora a farci intendere questo interuallo ... diuidendo il tuono in due semituoni eguali ... Quanto all Consideratione di questo secondo interuallo ... il Sonatore porra nello' istesso luogo le dita per farci sentire la terza naturale X.. . la uoce naturale non auezza a modulare simili interualli, non naturali, per interualli naturali . . . non hauendo prefisso termine come l'instromento fatto dall'Arte ... non pu6 giustamente diuidire il tuono in due parte eguali ... Ma intorno a quello che dicono di leuare tanto all'uno de semituoni, quanto all'altro per accomodare il tuono sesquinono, acci6 diuenti sesquiottauo, con certi, & determinati numeri rationali, bisognera prima ritrouare la meta del Comma sesquiottantessimo, perche di tanta quantita il sesquinono e minore del sesquiottauo. la qual meta conosciuta potrassi poi leuare dall'uno, e l'altro de semituoni, e aggiungerla al tuono, che all'hora fara il tuono sequinono diuentato sesquiottauo ... located in the middle has a 10:9 proportion - except that by the method and system of Ludovico Fogliani's doctrine they would remove from each of the two semitones a certain quantity so that the whole-tone by this increase would become 9:8, and the semitones remain equal to each other, and to half of the whole-tone. As for the first interval, I say it is not a new thing since it was used by Luca Marenzio at the beginning of a madrigal of his, the words of which say 'False Belief', to demonstrate indeed an interval false for voices and in harmony, but not false on the lute or chitarrone - because in the same place that the player puts his fingers to make us hear a 6th, he puts them again to make us perceive this interval ... dividing the whole-tone into two equal semitones... As for the second interval, the player will put his fingers in the same place to make us hear a natural 3rd ... But the natural voice is not suited to negotiate such unnatural intervals by means of natural ones, not having a preset stopping place like an artificial instrument. ... It cannot justly divide the whole-tone into two equal parts. ... And as for what they say about subtracting as much from one semitone as from the other, to accommodate the 10:9 whole-tone (thereby rendered 9:8) with known and specified rational numbers, one must first ascertain half of the 81:80 comma, because by such a quantity the 10:9 is smaller than the 9:8; when that half is known you can then subtract [it] from each of the semitones and add it to the whole-tone which will then make the 10:9 whole-tone converted to 9:8 ... [But] Due cosi quiui ci sono da considerare two things should be considered here: ... La prima, che gl'inuentori di ... first, that the inventors of cosi fatte spropositate such ill-conceived facts ... 403 Reviews ...non potranno mai diuidere la proportio sesquiottantessima ... in due parte eguali, con certi, & determinati numeri rationali. La seconda e, che . . . impossible e, che quel residuo delli due semituoni restino per la meta del tuono sesquiottauo; Essendo conclusione firmissima nelle Mathematiche, che nissuna proportione superparticolare possi essere diuisa in due parte eguali con certi & determinati numeri rationali ... Et perche io ho promesso di dimostrate vna sfilzata d'interualli forastieri, non conosciuti da quelli che essercitano questa moderna confusione, interualli inutili da cantare, con le voci nelle cantilene ordinarie, se bene sono e saranno conosciuti da quelli che suonano il Lauto, Chitarone, & altri cosi fatti instromenti; gli poner6 qua di sotto ordinatamente considerando il tuono diuiso in . due semituoni eguali . .. come si vede nel Lauto ... & e secondo la mente di Aristosseno appunto. Adunque la seconda minore, e lo istesso semituono. La seconda maggiore, e ... di due semituoni composto. La terza minore ... di tre ... will never be able to divide the 81:80 proportion into two parts with known and specified rational numbers, [and] secondly, that it is impossible that the residuum of [each of] the two semitones would amount to half of the 9:8 whole-tone. For it is a very firm conclusion in mathematics that no superparticular proportion can be divided into two equal parts with known and specified rational numbers." ... But since I have promised to demonstrate a series of surd intervals unknown to those who exercise this modern confusion - intervals useless for singing vocally in normal songs, even though they are and will remain known to those who play the lute, chitarrone, and other instruments made in that way I put them in order here below ... ... taking the whole-tone as divided into two equal semitones ... as one sees on the lute ... and this follows the view, indeed, of Aristoxenus. Now then, the minor 2nd is the semitone itself; the major 2nd is ... composed of two semitones; the minor 3rd ... of three [etc.] Mark Lindley Accademia Tartiniana, Padua 44 A weak argument, as the theory does not imply that any rational numbers shouldbe associated with a V2-commaor with the semitones in question. 404