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Simon Stevin's equal division of theoctaveH. Floris Cohen aa Department of Social History of Science and Technology, DeVrijhof, Twente University of Technology, P.O. Box 217, 7500 AE,Enschede, The Netherlands

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To cite this article: H. Floris Cohen (1987): Simon Stevin's equal division of the octave, Annals ofScience, 44:5, 471-488

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ANNALS OF SCIENCE, 44 (1987), 471-488

Simon Stevin's Equal Division of the Octavet

H. FLORIS COHEN

Twente University of Technology, Department of Social History of Science and Technology, De Vrijhof, P.O. Box 217, 7500 AE Euschede, The Netherlands

Received 3 N o v e m b e r 1986

Summary Many pioneers of the Scientific Revolution such as Galileo, Kepler, Stevin, Descartes, Mersenne, and others, wrote extensively about musical theory. This was not a chance interest of a few individual scientists. Rather, it reflects a continuing concern of scientists from Pythagorean times onwards to solve certain quantifiable problems in musical theory. One of the issues involved was technically known as 'the division of the octave', the problem, that is, of which notes to make music with. Simon Stevin's contribution to this issue, in his treatise Vande Spiegheling der Singconst ('On the Theory of Music'), is usually conceived of as a remarkably early plea for equal temperament, which is the tuning system we nowadays all take for granted. In this paper I show that, even though it is true that Stevin calculated the figures for what is now known as equal temperament, in fact the subject of temperament has almost nothing to do with his accompanying considerations, and that, therefore, his calculations served another purpose. A careful analysis of the problem situation in the science of music around 1600, reveals that Stevin's treatise highlights a particular stage in the history of what has always been the core issue of the science of music, namely, the problem of consonance. This is the search for an explanation, on scientific principles, of Pythagoras' law: 'Why is it that those few musical intervals which affect our ear in a sweet and pleasing manner, correspond to the ratios of the first few integers?' Through an analysis of the source material available (including contemporary comments on Stevin's ideas) we find that Stevin's theory, which makes no sense if interpreted as an early stage in the 'evolution' of equal temperament, was meant as a solution--as freshly original as it was wrongheaded--to this perennial problem of consonance, which has continued to baffle some of the best scientific minds from the very beginning of science to the present day.

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 2. What problems equal temperament was a solution to, and what not ... 475 3. Stevin's real problem situation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

1. Introduction E v e r y b o d y knows tha t p ianos are tuned in equal t emperament . Tha t is to say, all

twelve semi tones in the oc tave are m a d e equal. As a result , a p a r t f rom the octave itself, no consonan t interval , such as the fifth, or the m a j o r or m i n o r third, is left as ent i re ly

~" Part of the material that makes up the present paper has been used before in my book Quantifying Music. The Science of Music at the First Stage of the Scientific Revolution, 1580-1650 (Dordrecht, 1984). In this paper the focus is on whether or not Stevin's musical theories belong to the history of equal temperament, which issue is treated in a far more implicit way in the Stevin sections of Quantifying Music, which centre on the problem of consonance.

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472 H. Floris Cohen

pure. The aberrations from purity, however, are not so large as to annoy seriously other than quite sensitive ears, especially since we have grown used to equally-tempered intervals from early childhood on. In fact equal temperament has become so universal a tuning system that it is hard for us to imagine that there have been times when keyboard instruments were tuned according to principles other than that of making all semitones equal. In fact, in the seventeenth and eighteenth centuries fierce battles raged over what was to be regarded, and to be used, as the best temperament. The legend that Bach's 'Well-Tempered Clavier' was meant for, and worked indeed as the decisive contribution to, the victory of equal temperament is quite probably the only notion that the vaguely historically-oriented music lover has ever picked up regarding the battle in question.

The battle, although ultimately centring on the eminently practical issue of which notes should be used to make keyboard music, always carried a great deal of theoretical interest. The theory in question, known as the issue of the 'division of the octave', has never ceased to evoke the active participation of men of science, from ancient Greeks such as Pythagoras, Euclid, and Ptolemy, through early modern scientists such as Stevin, Kepler, Mersenne, Huygens, Newton, Euler, and Helmholtz, down to the present day. Thus, when in the 1930s serious historical interest arose in the 'battle of temperaments', not only historians of musical theory, but also historians of science were involved. In recent years the matter has acquired a great deal of added practical interest. The pioneers of the Early Music Movement, in their search for means to play music from the Renaissance and the Baroque eras as authentically as the historical data allow, came to realize that some really 'authentic' effects can only be achieved by adopting one or another of the many tuning systems that preceded equal temperament, and thus by tuning their harpsichords and organs in, for example, 'mean tone temperament'.

In the seventeenth century, quantitative musical theory, to which the issue of the division of the octave belongs, was still commonly regarded as one of the sciences. Thus it was not by chance that, as part of Simon Stevin's obligation to present the Dutch stadtholder, Prince Maurits, with a comprehensive overview of the state of science, he included among his treatises on hydrostatics, astronomy, etc., also one on the science of music. For reasons to be discussed later his treatise, called Vande Spiegheling der Singconst ('On the Theory of Music'), eventually was not included in his Hypomnemata Mathematica ('Mathematical Memoirs', 16054)8) for which it was intended, but remained in manuscript. As such it vanished soon after his death in 1620 (Christiaan Huygens was the last one to have been familiar with it), until it was rediscovered and published for the first time by Bierens de Haan in 1884. In the Netherlands it evoked some faint interest, mainly on the part of the editors of volume 20 of Christiaan Huygens' Oeuvres Complktes, which contains Huygens' musical theories (1940). They found in Stevin's treatise a highly accurate, and in fact the earliest, calculation of the values of what is known to us as equal temperament. Accordingly Stevin was hailed by them as an early defender of this temperament, far ahead of the great majority of his contemporaries who still clung to more old-fashioned modes of tempering. 1

i Christiaan Huygens, Oeuvres Completes .... 22 vols (The Hague. 1888-1950), xx (1940), 32, 144. The editor of volume xx was J. Volgraff. From his correspondence with E. J. Dijksterhuis (preserved at the Museum Boerhaave, Leiden) it appears that the musical information contained in the copious notes and numerous 'Avertissements' was in the main provided by the latter.

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Simon Stevin's equal division c~f the octave 473

This view has since become the current one, particularly since Barbour, in his comprehensive history of Tuning and Temperament (1951), adopted it and thus made it better known to the non-Dutch-speaking part of the world:

The first known appearance in print of the correct figures for equal temperament was in China . . . . More significant for European music, but buried in manuscript for nearly three centuries, was Stevin's solution. As important as this achievement was his contention that equal temperament was the only logical system for tuning instruments, including keyboard instruments. 2

This is still the accepted view, shared inside and outside Holland by music historians and historians of science alike? Thus Stevin's biographer, Dijksterhuis, wrote in 1961:

... the equal temperament system that is still in use.., was strongly championed at the beginning of the seventeenth century by Simon Stevin. 4

In addition, the Dutch physicist, Fokker, wrote in the introduction to his edition of Vande Spiegheling der Singconst and an English version of the treatise (to be found in volume 5 of Stevin's Principal Works (1966)):

Stevin boldly did away with all these subtleties [of contemporary musical theorists in devising temperaments]. In his view, all semitones had to be equal. 5

Also, among those who are now engaged in research in the field of music and psycho- acoustics, when they come to reflect on the history of their craft, the view of Stevin as a pioneer of equal temperament is still the first that comes to mind; witness this quotation from Sir Brian Pippard (1984):

Stevin's suggestion, enthusiastically championed a century later by Sebastian Bach and his son Emanuel, was to make all semitones equal, so that the frequency ratio for each successive semitone was the twelfth root of two, 1.0595. 6

However, unlike Sir Brian, the three first-mentioned authors were aware that there is something peculiar about Stevin's alleged defence of equal temperament. Having read the treatise for themselves, none of them failed to observe that Stevin regarded the values of his equal semitones, not as more convenient than others taken from rival systems of temperament, but rather as the true and natural values. In fact, they all state that the main thrust of the entire treatise appears to be a defence of such a viewpoint. Thus Barbour continues:

[Stevin's] contemporaries apologetically presented the equal system as a practical necessity, but Stevin held that its ratios, irrational though they may be, were "true" and that the simple, rational values such as 3 : 2 for the fifth were the

2 j. M. Barbour, Tuning and Temperament: A Historical Survey (East Lansing, Michigan, 1951: rcprint New York, 1972), p. 7.

3 The one and only exception I am aware of is the brief passage on Stevin in D. P. Walker, Studie.s in Musical Science in the Late Renaissance (London and Leiden: The Warburg Institute and Brill, 19781. pp. 120~22.

4 E. J. Dijksterhuis and R. J. Forbes, A History of Science and Technology (Harmondsworth, 1963), 11, 366. (The original Dutch version appeared in 1961.)

5 A. D. Fokker, 'Introduction. Simon Stevin's Views on Music', in The Principal Works of Simon Stevin, edited by E. Crone et al., 5 vols (Amsterdam, 1955-68), v (1968), p. 418.

6 B. Pippard, 'Pythagoras' Other Theorem', Times Literary Supplement, 21 December, 1984, p. 1469. This is a review of Quantifying Music, so the reviewer might perhaps have known better.

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474 H. Floris Cohen

approximations! In his day only a mathematician (and perhaps only a mathematician not fully cognizant of contemporary musical practice) could have made such a statement. It is refreshingly modern, agreeing completely with the views of Sch6nberg and other advanced theorists and composers of our day. 7

Dijksterhuis adds:

There is, however, one important difference . . . . Stevin's division of the octave into twelve equal intervals of a semitone each was not intended as a compromise in which true intonation is sacrificed to the exigencies of the mechanics of keyboard instruments, whereas in equal temperament it was. It was regarded by its author as the faithful representation of true intonation itself. 8

And Fokker goes on:

For [Stevin] the numbers resulting from the division of the ratio 2 : 1 into twelve equal ratios, twelve times the twelfth root of 2, are the true numbers. 9

Now all this is really quite strange. The essence of temperament is, after all, that it is indeed a compromise, forced on us by the mathematical impossibility of combining all consonant intervals as entirely pure in one musical scale. For instance, piling up seven pure octaves does not yield the same note as piling up twelve pure fifths. This is shown in Figure 1. The assertion is equivalent to stating that 2 7 is not equal to (3/2) 12. [NB musical intervals are added together by mult iply in9 their ratios.] As a result c differs slightly from b sharp, by a fraction known of old as the 'Pythagorean comma'. Similar, though quantitatively varying differences arise out of the incompatibility of true octaves and true major thirds, of true octaves and true minor thirds, or of true fifths and true major thirds. What the tempering of intervals ultimately comes down to is making concessions to purity, fortunately made possible by the willingness of our ear to put up with minor impurities in the consonances. How, then, could Stevin put forward his values for tempered musical intervals as more natural, more true, than the pure ones?

My concern in this paper is to show a way out of the apparent paradox, and thus to clear the way for a new conception of the state of the science of music in the first few decades of the seventeenth century. What we need at this point is a fairly radical change in historical perspective. The historians quoted above, however different the nature of their respective interests in the history of equal temperament, share one feature: Their interpretations of Stevin's treatise imply a conception of equal temperament as the crowning achievement of a long evolution of tempering practice and theory. They appear to regard equal temperament as the manifest and ultimate destiny of this early,

C G D A E B F" C = G" D" A" E" B ~ .,-- t i f ths I ! | I I I I I I i I I I I I I I i I

C C C C C C C C -*- octaves

Figure 1.

7 Barbour (footnote 2), p. 7. s E. J, Dijksterhuis, Simon Stevin: Science in the Netherlands around 1600 (The Hague, 1970), p. 121. This

book is a highly condensed version of the Dutch original Simon Stevin (The Hague, 1943). Chapter 13 of this book was for a long time the best account of Stevin's musical theories available, and in fact [ owe much to it, The more significant, then, that Dijksterhuis, too, missed the central point of Stevin's treatise, as I shall demonstrate further on in the present paper.

9Fokker (footnote 5), p. 418.

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Simon Stevin's equal division of the octave 475

and somewhat primitive, proliferation of modes of temperament. For Barbour, equal temperament served explicitly as the standard alongside which to measure all other temperaments (throughout his book he calculated for all the systems he reviewed their standard deviation from equal temperament). The Dutch physicist, Fokker, was primarily concerned with establishing as many priorities as he could for his defunct colleague, the Dutch physicist, Stevin. Finally, Dijksterhuis was certainly concerned, throughout his entire career as an historian of science, to place past scientists meticulously in the context of their times, yet he never swerved from his conception of the history of science as an evolutionary unfolding of truth. But this, I believe, is precisely the point of view we should attempt to give up.

Rather than regarding equal temperament as the timeless key to the perennial need for establishing the notes we can best make keyboard music with, we should see it as one possible solution to a specific problem that could arise only in the context of a given, although naturally changeable, problem situation. The issue, therefore, is not to trace the history of equal temperament back to the first clever person who hit upon the idea or seemed to elaborate it, but rather ask in the context of what specific historical problem situation equal temperament did, or did not, make sense. Not until we have managed to define the problem situation in which Stevin was working and to which he was implicitly responding, can we hope to establish whether his concern with 'equal temperament' had anything to do with what is commonly signified by it.~ 0 His strange contention that the equally-tempered intervals are truer than the pure ones is so foreign to the common conception of temperament as a system of concessions to purity that it should really be taken as a hint that Stevin may have had something entirely different in mind from equal temperament as we know it, or as some of Bach's contemporarics knew it. In order to find out, we must investigate what the overall problem situation in musical science in Stevin's day was like. Our first step is to give a brief overview of how the need for tempering arose in the course of the fifteenth and sixteenth centuries.

2. What problems equal temperament was a solution to, and what not Temperament is one offshoot of a more general issue, technically known as the

'division of the octave'? 1 This is, quite simply, the problem of what notes, out of the infinite number available, are to be used to make music with. This has always been determined primarily by the consonances. For example, the whole tone may be found by 'subtracting' the fourth D - G from the fifth C-G; the resulting tone C-D is then given by 2/3 : 3/4 = 8/9. Unfortunately, as has already been shown by means of the example in Figure 1, it is not possible to combine all consonances as pure in one scale. The scale that comes closest is so-called 'just intonation' (see Figure 2). One problem with this scale is that a few consonant intervals are not pure (for example, the fifth D-A is given by 27/40 as against 2/3). The 'just' scale could yield all consonances as pure only if a new note D* were added, one syntonic comma (80/81) below the original D, or if the piece were allowed now to gain, now to lose pitch somewhat throughout the entire composition in question.

0 The idea of reconstructing historical problem situations goes back, of course, to K. R. Popper. His most explicit discussions of the method is in his Objective Knowledge: An Evolutionary Approach (Oxford, 1972), pp. 186-90.

1 For a more detailed historical exposition of the issues of tuning and temperament, see my Quantifying Music (footnote 1), p. 34-45.

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c 8,9 D 9,,0 E%F 8,9 G % A 8,9 B%r

Figure 2.

Flexible as the human voice is, the singer may use either of these alternatives, or, as a third possibility, may in the course of a piece pragmatically adapt the purity of one or more consonant intervals somewhat. But a keyboard instrument lacks such flexibility: one cannot change its intonation while playing it. Keyboard instruments have to be tuned beforehand, and, since just intonation by itself is impossible, the need for some compromise imposes itself. This is where temperament comes in. Its invention, at some point in the fifteenth century, went hand in hand with the 'emancipation' of the keyboard instrument. The organ and the harpsichord were now being used as solo instruments in their own right, and thus stood in need of some form of tonal adjustment.

Tempering is the deliberate mistuning of one or more consonant intervals. As to the choice of intervals to be mistuned, and the extent to which they should be, an almost infinite range of possibilities presented itself. These were restricted by only a few considerations, the chief one being that the unison and the octave had to be kept pure by all means (these are the only consonances not to bear any tempering). The only other constraints are provided by the amount of tempering that our ear appears to permit (it accepts more in the thirds and the sixths than in the fifths and fourths), and, most importantly, by the musical style of the day. Thus the form of temperament that had come to prevail around 1550-1600 was so-called 'mean tone temperament'. Its main features are that its eight major thirds are kept pure, 12 whereas the fifths, fourths, and minor thirds are deliberately made somewhat, albeit tolerably, impure. The main drawback of mean tone temperament is that it permits only a small number of keys in which to play a piece. The only chromatic notes to be used in mean tone temperament are C sharp, E flat, F sharp, G sharp (or occasionally A flat, but not both at the same time), and B flat. Keys in which other chromatic notes occur cannot be used in mean tone temperament, and this sets strict limits to the amount of modulation (i.e., change from one key to another) a composer might wish to employ. However, and this is the salient point, at the time this was not felt to be a constraint at all. Music written in accordance with the precepts of what is now commonly known as the Renaissance style (roughly from Dufay until Lassus and early Monteverdi) moved quite easily within this limited range of available keys; the musical effects to be achieved by wide modulation were simply not aimed at by the highly contrapuntal style of writing that dominated European musical composition throughout the fifteenth and sixteenth centuries.

Thus mean tone temperament was duly described and canonized in the theoretical masterpieces of the time, notably those of Gioseffo Zarlino and Francisco Salinas. To be sure, both theorists also gave extensive descriptions of many other possible temperaments, none of which, however, was recommended as a practicable alternative. One of these was equal temperament. The essence of this temperament is that, in making all semitones equal, it removes the purity of all consonant intervals; in particular the major third is remarkably impure (far more so than the fifth in mean tone temperament). As against this major drawback equal temperament has one great

~2 More strictly speaking: that it has only eight major thirds, each in the pure ratio of 4:5, whereas four diminished fourths also occur, which were traditionally regarded as quite dissonaht.

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Simon Stet~in's equal division of the octat~e 477

advantage: in equalizing the semitones it eradicates all difference between C sharp and D fiat, between D sharp and E fiat, and so on. In other words, on an equally-tempered keyboard music can be played in any key whatever, and unlimited modulation becomes a practical possibility. However, Renaissance music was not at all in need o f such devices. There was one, and only one, type of instrument for which equal temperament made practical sense, and that was, a fretted instrument such as the lute. (Because the frets are placed across the strings, any temperament other than the equal one might yield false octaves.)

The conclusion here is that equal temperament was to remain a theoretical construction devoid of almost all practical interest as long as the stylistic ideals of Renaissance music prevailed. This continued indeed to be the case until around 1600. Then a major stylistic upheaval occurred, originating in Italy and only gradually spreading northwards. Such names are associated with its beginning as those of Peri, Caccini, Galilei (the father), and finally Monteverdi, who was the one to bring the pioneer efforts of the other, essentially second-rate composers to an artistic high-point. What they intended to do was to recreate what they fancied to be the Music of the Ancients. This, they believed, could be achieved by turning music into a sort of rhetoric: the aim of the musical composition should be to make the music express the 'affects' (a sort of standardized emotion, such as Love, or Hatred, or Mourning, or Joy) inherent in the text. The melody was to follow the words as closely as possible, and in order to do so it needed far more harmonic and melodic variety than was so far provided for by the intricate counterpoint of the Renaissance style, in which the harmonic progression was governed by the consonant chords on which the entire composition hinged. Thus one crucial element of this musical revolution was to liberate the melody of a great many contrapuntal constraints.

In order to achieve greater expressive variety, dissonances were used more freely and all kinds of previously unheard-of chords were experimented with. 13 Compo- sitions for keyboard followed the trend fairly rapidly. One unintended result was that the limits set to the number of keys in the scale to be used almost at once turned into a constraint that was not only theoretically present, but that was now becoming seriously felt in actual music making. Already in Frescobaldi's pioneer works for organ and harpsichord we find compositions that pass through the entire range of then available keys; a few attempts at overstepping the limits set by mean tone temperament can even be found in some. ~4 Thus it is not by chance that the very same Frescobaldi is known to have experimented with equal temperament. The one great advantage of this brand of temperament now became musically relevant for the first time. The very fact, however, that it was to take almost one century and a half for it to prevail illustrates how dearly its one and only asset was boughtr the price of sacrificing the purity of the major third (the essential interval in the major triad) was for a very long time felt by many to be too high to be paid. Nevertheless, mean tone temperament had now lost its privileged status; a feasible rival had emerged from developments in musical style that entailed the decline of mean tone temperament as a consequence as inevitable as it was unintended.

WasStevin aware of all this? That is, may he be regarded as the theorist to provide the mathematical elaboration and justification of what Frescobaldi et al. were

3 For an account of the onset of the Baroque style in music, see the first two chapters of M. F. Bukofzer, Music in the Baroque Era (New York, 1947).

14 See, for example, the 'Recercar con obligo del Basso' from the 'Messa delli Apostoli' (Fiori Musicali, 1635), or bars 28 and 30 of the 'Toccata per I'Elevazione' from the same Mass.

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478 H. Floris Cohen

beginning to find out in practice? Even chronology alone rules out such a link between theory and practice. Vande Spiegheling der Singconst was written at around 16054)8,15 whereas Frescobaldi's first, still fairly traditional collection of compositions for keyboard appeared in 1615, when he was just twenty-five years old. 16 Even if Stevin had been thoroughly knowledgeable about musical composition practice (which he was certainly not), and even if there had been no time delay in the news of the Italian avant-garde experiments reaching Northern Europe (which there certainly was), the first decade of the seventeenth century would still have been too early. And indeed, a careful reading of Stevin's treatise, guided by such clues as were indicated in the Introduction to the paper, reveals that no trace of the dramatic stylistic upheaval in Italian music is to be found in it, let alone of any Northern echo. The only possible application in practice of the equal division of the octave proposed by Stevin that he briefly and in passing refers to concerns the lu te- - the one instrument for which equal temperament had already been employed for some time. In fact, it will be shown in the next section that Stevin was almost entirely ignorant of temperament generally and of the theoretical considerations that made some form of temperament necessary for keyboard music.

In brief, Stevin's treatise, supposedly an early defence of equal temperament, appears not to be about temperament at all. Its real subject matter turns out to be far more abstruse; it has almost no links with any present-day musical practice or theory, and really fits into a quite different problem situation than the one that was ultimately to yield the victory of equal temperament. This problem situation car] be defined as the mathematical stage in the history of the problem of consonance.

3. Stevin's real problem situation So far we have tacitly regarded as given what 'purity ' of the consonant intervals

means. And, indeed, this is given in the sense that the human ear (at least the ear of those gifted with a modicum of musical sensitivity) is capable of telling, within quite narrow limits, a pure consonance from a slightly impure one, and certainly from outright false intervals, or dissonances. The basic discovery to be made regarding this perceptional basis of consonance is that it is matched exactly by certain elementary numerical relationships. 17 Pythagoras is credited with finding that the ratios of the string lengths which give the consonant intervals are those of the first few integers. For example, when a string and its half are sounded together the consonance called octave is heard; strings in the ratio of 2:3 yield the fifth, and so on. On the other hand, such an evidently dissonant interval as the second is given by the far more complicated ratio 8 : 9. Two related problems may be posed in this connection, and have in fact been posed in

15 There are three texts of Vande Spiegheling der Singconst. The manuscript is preserved in the Koninklijke Bibliotheek (Royal Library) in The Hague, where it is marked KA XLVII, pp. 624-705; it consists of two versions. Both were edited in 1884 by D. Bierens de Haan in S. Stevin, 'Vande spiegheling der singconst' et "Vande molens': Deux trait~s in~dits (Amsterdam, 1884). The early version, together with an English translation, was edited by A. D. Fokker in the work cited in footnote 5. Since Bierens de Haan's edition is both more accurate and far more complete, I refer to his edition rather than to Fokker's. For a more detailed treatment of the manuscript and its printing history, see Quantifying Music (footnote 1), note 46 to chapter 2.

16 There had been some experimenting before with 'extravagant consonances' (to use the title of some pieces by Giovanni de Macque); but Frescobaldi was the first really to explore the possibilities of the new style for the organ and the harpsichord. See also W. Apel, The History of Keyboard Music to 1700 (Bloomington, Indiana, 1972), p. 457.

17 For a more detailed exposition of the early history of the problem of consonance, see Quantifying Music (footnote 1), pp. 1-7.

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Simon Stez~in's equal division o f the octal~e 479

various guises through the twenty-five centuries that have since passed: "How is it that our experiencing the consonances, which takes place inside the seat of our perceptions, can be determined by a peculiar set of numerical relationships, which are given in the outside world?' And also: 'What is it that distinguishes the consonance-generating ratios from all other ratios?' i.e. 'What common property enables them to achieve this?'

These questions have repeatedly been reformulated through the ages, as a result of developments in musical style as well as in mathematics, in physics, and in anatomy and physiology. Stevin's treatise marks one such transition from one definition of the Pythagorean problem to the next: from the numerological (Zarlino, Salinas) through the overall mathematical (Stevin, Kepler, young Descartes) to the physical and- - modestly--physiological (Beeckman, Mersenne, Galileo, mature Descartes). A brief account of this transition has to be preceded by an even shorter summary of the solution to the problem against which Stevin, among quite a few others, rebelled. This solution was devised around the middle of the sixteenth century, in the heyday of the Renaissance style of music, by the two prominent musical theorists Zarlino (1558) and Salinas (1577) we have met before in connection with temperament.

In their time, the range of intervals that (within the compass of one octave) was admitted as consonant was as shown in the table. Now what property did these ratios have in common that all other ratios lacked? Zarlino's answer centred on the senario, that is, the range of the first six integers. Figures contained within the senario could make up consonance-generating ratios; figures outside it could not. Evidently, this explanation raises a difficulty for the minor sixth, but by declaring the number 8 to be 'potentially' part of the senario (that is, by regarding it, for this special occasion, as twice 4), Zarlino managed, to his own satisfaction, to get around it. Thus the number 6 that defines the senario indeed constituted the 'harmonic', the 'sonorous' number; the number that mediated between men's subjective perception of harmonic sound and the objective outside world. What was it that made the number 6 so privileged as to define the 'harmonic number'? It owed this privilege to facts such as: there are, after all, six planets, six directions, twice six apostles; God created the world in six days; 6 is the 'perfect' number (since 1 + 2 + 3 = 1 • 2 • 3); and so on. Thus 6 distinguishes itself from all other numbers and, in constituting the harmonic number, thus conveys the harmony of the consonances, through a corresponding harmony of the soul, to perceiving man. That is how Zarlino arid, with a few minor differences, Salinas solved the problem of consonance. For about half a century the senario indeed continued to be the reigning explanation. TM

Consonance ratio II 0r I Fifth I F~ [Maj~ 1 : 2 2 : 3 3 : /-, 3 : 5 ]Major third [ Minor third ] Minor sixth L " 5 5 " 6 5 B [I

It may come as no surprise that, however consonant this explanation was with the neo-Platonic temper of late Renaissance science and its tendency towards number mysticism and numerology in general, it held little value for the next generation of scientists, who subscribed to radically different scientific standards. Whatever the nature and origin of the Scientific Revolution may be judged to have been, it is certain that those who took part in it had no patience with this sort of argument. The one thing all those major scientists, who in the first few decades of the seventeenth century

18 The only contemporary dissenters were Benedetti and Vincenzo Galilei.

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devoted part of their efforts to musical theory, had in common was their universal rejection of the senario. What should take its place is what made them diverge.

The theory that was to replace the senario for more than a century was of a physical nature; it may be called the 'coincidence theory of consonance'. According to this theory musical sound is made up of regular little shocks that, after being produced by the voice or the musical instrument, are transmitted, in a wave-like manner, through the air to the ear. The frequency of these shocks, or pulses, determines pitch, and those intervals for which the shocks coincide regularly and often are consonant; e.g. for the octave in a frequency ratio of 2:1, and so on. Summarily anticipated by Benedetti around 1563, this theory was really developed by Beeckman (1614/15), communicated by him to Descartes (1618/19) and to Mersenne (1629), and proclaimed to the world by the latter (1636/37) and by Galileo (1638). From then on, it was to dominate all thinking on the problem of consonance for more than a century. What interests us now, however, is the brief period in between the end of the sway held by the senario and the onset of the coincidence theory. Between c. 1605 and 1619 three major scientists wrote treatises in which the problem of consonance was treated, no longer on a numerological, but not yet on a physical, acoustical basis. Neither Stevin, Kepler, nor young Descartes were willing even to consider an explanation based on the wondrous qualities of the number six, as had been attributed to it by Zarlino. But none of them, at the time, was sufficiently physics-oriented to look for a solution in the direction of the properties of sounding bodies either. The natural path to follow, for these three mathematicians, was shown, not by number theory in any form, but by mathematics generally, and, in particular, by geometry. Kepler had profound metaphysical reasons for his conviction that the key to no less than the entire creation was to be found in geometry:

Geomet ry . . . , coeternal with God, and radiating in the divine Mind, supplied God with the models. . , for establishing the World so as to make it the Best and the Finest, in short, the most similar to its Creator. 19

In his 1619 Harmonice Mundi Kepler replaced the senario by a geometric criterion distinguishing between consonance and dissonance that was based upon the arcs cut off circumscribed circles by the constructible regular polygons. A carefully axioma- tized, incredibly elaborate system underlay this criterion, that in its turn was made to serve a general account of the laws of the Harmony of the World. 2~ Descartes' early treatise Compendium Musicae (also 1619, but published posthumously) contains a far more superficial geometrization of the problem of consonance than Kepler had provided, in that Descartes was content with representing the ratios of the intervals through line segments of varying lengths, and deriving the consonant intervals through a process of continued bisection. His net result was that there are really three 'sonorous numbers': 2, 3 and 5 (2 giving the octave; 2 and 3 giving the fifth and the fourth; 2, 3, and 5 giving the thirds and sixths). Here, however, the term 'sonorous number' no longer stands for a numerological explanation, but rather for a sober description. No more detailed explanation of the problem of consonance is to be found in Compendium Musicae than the statement that only the first three bisections of a line segment are fit for generating consonances.

~gJohannes Kepler, Gesammelte Werke, edited by M. Caspar et al. (Munich, 1937-), Vl (1940), 104-05. z~, In Book Ill of ftarmonice Mundi, particularly.

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Even more than Kepler and Descartes, the slightly older Simon Stevin despised explanations based on numbers. In Kepler's view, the correspondence of the consonant intervals with the ratios made up of the first few integers certainly signalled an important hidden property of nature---even though not number mysticism, but only geometry could serve to bring it to light. But for Stevin number relationships could never serve as a clue to nature. Nothing special singled out the integers among the entire range of numbers. In considering, for instance, the incommensurability of 2 and the square root of 8 there was not the slightest reason to ascribe rationality to the one rather than to the other. 21

Looking at the problem of consonance from such a vantage point, for Stevin there was only one way out left. Feeling little pleased by Zarlino-like explanations, and being insufficiently physics-oriented to think up an account in terms of the properties of sounding bodies, 22 Stevin, led by his distrust of the capacity of numbers to suggest the presence of any meaningful natural laws, decided that, from Pythagorean times onwards, the problem of consonance had always been wrongly put. Vande Spiegheling der Singconst is really one sustained effort to prove that, and to make plausible why, the consonances are not given by ratios of the simple integers. One way of deriving the consonances had always been to show that they correspond with both the arithmetical and the harmonic divisions of the octave. For example, the harmonic series 6, 4, 3 represents the division of the octave into the fifth below and the fourth above (see Figure 3). Similarly, the arithmetical series 4, 3, 2 divides the octave into a fifth above and a fourth below (see Figure 4). In the same way the major and minor thirds can be derived through the harmonic and the arithmetical means of the fifth; and so on. In short, the traditionally accepted consonances are given by both harmonic and arithmetical proportionality. But Stevin disagreed. It follows from his theory of proportionality (as developed in his 1585 Arithmbtique) that both forms of proportion- ality are spurious. The only genuine type of proportionality is the one conventionally called 'geometric'. Only when the terms a, b, and c have a relationship of the form a : b = b:c can they be said to be proportional; neither the relationship a - b = b - c (our arithmetical proportionality), nor the relationship ( a - b ) / ( b - c ) = a / c (our harmonic proportionality) counts as proportionality proper. We need not dwell here on Stevin's reasons for this dissenting view, which go back partly to mathematical considerations, and partly to his peculiar views on language, in particular the Dutch one. 23 It suffices to understand that, from this view, the only proper division of the octave is generated by real, that is (in our terms) geometric, proportionality. Given that the octave contains twelve semitones, the value of each of them is given by the following series: 1, (1/2) 1/12, ((1/2)2)1/12 . . . . . ((1/2)11)1/12, 1/2. In other words, the notes making up the scale can be found by calculating eleven mean proportionals between C and c.

With a simple calculator this can now be done in a few seconds to a degree of precision of nine decimal places, but in Stevin's time it was a highly time-consuming chore. He performed it twice. For his first table he performed one extraction of roots,

21 Stevin develops this argument in his book L'Arithm~tique (reprinted in volume 2B of Stevin's Principal Works, footnote 5) around p. 512; in Vande Spiegheling der Singconst he refers to it on p. 69 (in Fokker's edition, on pp. 440-41).

22 The only link which Stevin establishes with the physical side of musical sound is his passing, and quite vague, use of the concept of 'coarseness' of a note as an unspecified measure for lowness of pitch: see Vande Spiegheling der Singconst, p. 22.

z3 Quantifying Music (footnote 1), pp. 58-60.

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3} octave 4

Figure 3.

fourth

filth

2 fifth octave 3

/0 fourth

Figure 4.

namely, for the fifth ((1/128)1/12), and derived the other figures through 'adding' and 'subtracting' intervals. The second time he performed three extractions of roots, namely for the augmented fourth, (1/2) 1/2, the major third ((1/2)1/3), and the minor third ((1/2)1/4). The results are given to four decimal places, and are fairly accurate, a4

Evidently, such a determination of the notes of the scale yields consonances with values remarkably different from the traditionally accepted ones. Essential here is not the extent of the difference (e.g., the value for the fifth of 0"6674 does not differ dramatically from the received value of 0.6667), but rather the very fact that they differ. 2s It did not matter to Stevin that in this scale the fifth was not represented by the "easy' fraction 2/3, but rather by the far more complicated term (1/128) 1/12 , since in his view such scales of comparative 'easiness' have no meaning to begin with, as mentioned before. But it should also be clear that this way of dividing the octave was incompatible with all preceding musical theory. Not only was it incompatible with any previous solution to the problem of consonance; it was even incompatible with the very way the problem had been stated from Pythagoras on. Stevin put forward his division, not as a suitable compromise, but rather as the one and only true definition of the musical intervals. His theory is not about temperament at all; it is a particular musical scale derived from a particular view of the nature of mathematical proportionality and based upon a highly peculiar conception of the problem of consonance.

Only one passage in Vande Spiegheling der Singconst seems to be in flagrant contradiction to this assertion. On pp. 3 If of his treatise Stevin, after having calculated the values of the equal division of the octave, remarked indeed that these values may be used for tuning organs and harpsichords. Although, in practice, this would of course amount to equal temperament, conceptually, from Stevin's own point of view, it does not. Advocating a temperament necessarily implies that one knows what temperament is about; but, as we shall see presently in detail, Stevin did not possess this knowledge. For him the tuning of an instrument is a neutral procedure; it is precisely what tuning for the musical lay-person has always been, namely: redressing falsities. If your piano goes out of tune: call the tuner! The lay-person takes for granted that the tuner does his tuning according to the only possible values for sufficiently purely-sounding musical notes, whereas in fact a large variety of values are in principle available, from which the tuner chooses just one. But Stevin did not know this. What makes the historical situation so ironic is that the one system present-day tuners choose as a matter of course corresponds exactly to the values advocated by Stevin. Yet when Stevin recommended his division of the octave for use in tuning keyboard instruments, he meant something different from temperament: I shall show that his only purpose was to present the tuner with the figures he needs in order to restore a false instrument to its lost purity.

24 Barbour (footnote 2), evaluates these figures on pp. 76-77. zs In cent values the difference is 702.(~700-0=2-0 cents.

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Two passages in Vande Spiegheling der Singconst indicate that Stevin really had no inkling of what temperament was about. At one point he interprets what we know to have been Zarlino's derivation of temperament as if Zarlino had tried by these means to get rid of certain inherently undesirable consequences of just intonation, by 'distribut- ing a certain comma.., over one note or another, wherever it served him right, but just gropingly'. 26 The other place that betrays Stevin's lack of understanding of tempera- ment is one of the attempted proofs of his equal division. 27 This proof amounts to showing that adding together two augmented fourths yields an octave, which is followed by the statement that experiment confirms this. The trouble is that two augmented fourths make an octave if, and only if, their value in string lengths is given by (1/2)1/2; adding together two augmented fourths as they appear, for example, in mean tone temperament yields an octave that is too narrow by more than half a chromatic semitone.

Such an enormous deviation from purity in the octave cannot possibly be missed even by the most tone-deaf theorist (a self-confessed quite insensitive listener as Descartes had no trouble in spotting the deviation at the far less critical interval of the minor sixth). 28 Thus either Stevin never carried out the experiment (an hypothesis completely at variance with what we know about his practice in other fields of science), or the harpsichords with which he checked his result had been tuned in equal temperament. The effect of which is that (1) his proof becomes worthless, as it is based on a petitio principii, and (2) we thus find one more confirmation of our allegation that Stevin did not know what temperament was about. For him tuning was really a value- neutral procedure. He did not realize that on a keyboard instrument no experimental check of the truth of his equal division is possible: if the instrument is tuned--however inadvertently--in equal temperament, the proposition appears true by definition; if in any other temperament, experience refutes it.

Thus we conclude that, since Stevin did not know what temperament was about, it makes no sense to ascribe to him the advocacy of one particular species of it, namely, equal temperament. The only thing he appears, in his treatise, to have picked up regarding equal temperament is that it is particularly suitable for tuning the lute. Again, he does not know why, but one may attribute some dim awareness of such a state of affairs to the calculation he gives, in Vande Spiegheling der Singconst, of the places for the frets when the lute is tuned according to the equal division, z9 Thus, as far as contemporary equal-tempering procedures are concerned Stevin in no way deviates from--highly limited---current practice; what is new and original--even though entirely mistaken--about his contribution to musical thought concerns the theory of the division of the octave as conceived of in the light of the contemporary problem situation in the explanation of the phenomenon of consonance and of its numerical peculiarities.

One other way to phrase the distinction between Stevin's equal division and equal temperament as everybody else has always conceived of it is to say that in the latter the semitones are deliberately made equal, whereas in the former they simply are equal,

26 Stevin, Vande Spiegheling der Singconst, p. 61. 27 Idem., p. 23. 28 Oeuvres de Descartes, edited by C. Adam and P. Tannery, second edition (Paris, 1971-74), I, pp. 142 and

295. In cents the difference between a pure octave and the interval that results from adding two mean tone augmented fourths is 41.1. I cannot possibly share Dijksterhuis' belief that Stevin must have been incapable of noticing this (Dijksterhuis, footnote 8, English version, p. 122).

29 Stevin, Vande Spiegheling der Singconst (footnote 15), pp. 3(~31.

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484 H. Floris Cohen

either through mathematical necessity or by nature. On this latter point Stevin is not quite consistent: in Vande Spiegheling der Sin#const he attempts both to find mathematical proofs for their equality (with all three proofs he can be shown unconsciously to have smuggled in the argument itself what he had set out to prove), and to make plausible that their equality is a natural given. But in the present context this is only a detail; the crucial point is that Stevin believes the semitones to be equal. He correctly observes that assuming the value for the fifth to be 2/3 inevitably entails two distinct values for two different types of semitones. For example, in the just scale the chromatic semitone C-C sharp is represented by the ratio 128/135, whereas the diatonic semitone C-D flat is given by 15/16 (and thus is the larger one). In other words, C sharp and D fiat are different notes, and this is in fact true of any twelve-tone scale other than Stevin's equal one. This is why in mean tone temperament such little modulation is possible: the black keys stand for either C sharp or D flat, for either D sharp or E fiat, and so on, but cannot be made to play the parts of both notes at once. Only the deliberate equalizing of the semitones, that is, making the black keys each perform dual functions, makes for unlimited modulation. The price to be paid is giving up a great deal of purity, and also much musical variety, since in temperaments other than the equal one all keys are characterized by somewhat different orders of the semitones.

Take, for example, two scales in mean tone temperament and the corresponding ones in equal temperament. In mean tone temperament the diatonic semitone is about 3/2 times as large as the chromatic semitone, whereas in equal temperament both are, by definition, equal. In mean tone temperament the scales of d and e are as shown in figure 5. Now in music-making two notes of great importance, especially in cadences, are the semitones below the tonic and below the fifth upon the tonic (the 'dominant'). Compare now both scales in mean tone temperament and the corresponding ones in equal temperament. In mean tone temperament cadences in d employ larger semitones

b m

D E E F F l l ! I I

G G~...~ A Bb B C C,,~,, , ,d I I I I I I I I

I l ; w I l I ; b I I CS l i i

E F G G" A B C D E b e

D D'/E b E F F'/G b G G'/A b A A'/Bb B C C'/D b d I I l I I I I I l I I I I

I I I I I I I I I I | I I

E F F"/G b G G~/A b A A'/B b B C C"/D b D D"/E b �9

Figure 5.

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Simon Stevin's equal division of the octave 485

(namely, C sharp-D and G sharp-A) than cadences in e (namely, E flat-E and B flat-B). As a result, cadences in e have 'something more plaintive and tender' than those in d (in the words of Christiaan Huygens, to whom this particular example is due). 3~ It should also be clear that in equal temperament such subtle shades of tonal variety are entirely destroyed, and, since there seems to be a musical law of decreasing tonal sensitivity over time, this particular form of musical variety has probably gone forever.

All of which goes to show that equal temperament is a man-made affair. One may prefer effects such as those indicated in the above, and thus opt for temperaments other than the equal one; or may attach most importance to unlimited modulation and thus choose equal temperament. Yet there would be no point in saying that one of these solutions was more 'progressive' that any other. Equal temperament has won only because more and more composers felt that unlimited modulation was more important than, for example, preserving the purity of the major third. One can point to various facts of musical history in order to make the final outcome plausible; yet the outcome provides no standard by which to judge previous solutions.

Coming back to Stevin, we can indeed conclude that his equal division of the octave had nothing to do with equal temperament. He was not aware (nor could he have been) of the developments in Italian musical style that, soon after he died in 1620, were to turn equal temperament into a possible solution to a new, practical problem that had scarcely existed before c. 1630. Rather he calculated the values that have since come to represent equally-tempered intervals as part of an exercise that derived from his urge to argue for the equal division as belonging to the natural scheme of things. He could do so only because at the time of his writing the standard explanation of consonance (the senario) had lost all credibility to scholars of his scientific temper, whereas a solution outside the mathematical realm had not yet been put forward. (Of course he might have done so himself; and we shall see presently that Beeckman, who started as an adherent of Stevin's conception of consonance, was soon to move from there to the first conception of the coincidence theory. But Stevin himself never made the step.) Thus Stevin's one really original contribution to musical theory is inextricably linked with one particular, brief, and rapidly passing stage in the history of the problem of consonance.

1 am not the only one to interpret thus those parts of Vande Spieghelin9 der Singconst that, to the modern eye, look so suspiciously like being centred on equal temperament. On inspection it turns out that none of Stevin's contemporaries and immediate successors mistook his thesis for that. His message was, quite rightly, interpreted as containing the idea that the geometric division of the octave in twelve equal steps defines, not a suitable temperament, but rather the true ratios of the consonances. And this proposition, that runs so entirely counter to the most elementary empirical evidence, has never been accepted by any musical theorist before or since.

Stevin's first critic was the Nijmegen city organist, Abraham Verheyen, to whom Stevin had sent a copy of his manuscript. Verheyen pointed out to him, in no uncertain terms, that experience shows that adding together three pure major thirds does not yield a pure octave (as would follow from Stevin's theory); rather, a pure octave can be produced by adding two pure major thirds to one diminished fourth (e.g. C-E, E - G

30 Huygens (footnote 1), xx, 73.

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486 H. Floris Cohen

sharp, G sharp-c), which is quite perceptibly larger than the pure major third. Therefore the value for the pure major third cannot possibly be (1/2) 1/3 , and thus Stevin's theory collapses.31 It is quite conceivable that Verheyen's friendly-worded, but in effect scathing critique is what made Stevin decide to withhold his treatise from publication. At any rate, Vande Spiegheling der Singconst did not appear in volume 2 of his bulky Wisconstighe Ghedachtenissen (1608, 'Mathematical Memoirs'), for which the treatise had been intended, and in the first volume of which (1605) he had briefly alluded to his theory of the natural equality of the semitones. Nor did he ever publish Vande Spiegheling der Singconst separately. The first to read the entire manuscript after Stevin's death was Isaac Beeckman, who, in 1624, borrowed it from his widow. Beeckman had originally been an adherent of Stevin's theory, with which he had become acquainted through the brief passage in the 'Mathematical Memoirs'; and possibly by word of mouth as well. But when, in 1614-15, Beeckman conceived of the coincidence theory of consonance, he must rapidly have realized that his new explanation of consonance, being based on the traditional sizes attributed to the consonant intervals, was in fact incompatible with Stevin's equal division. On reading, in 1624, the entire manuscript, Beeckman (in a note in his scientific diary) declared himself unimpressed, missing in particular any defence of Stevin's thesis that the fifth is really given by (1/128) 1/12. In a letter from 1629 to Mersenne, Beeckman even went so far as to call Stevin's theory a worthless play with numbers. 3z

Mersenne's direct knowledge of Stevin's theory was limited to the one allusion Stevin had made to it in volume 1 of the 'Mathematical Memoirs'. Significantly, in his encyclopaedic study of the science of music, Harmonie Universelle (1636-37), Mersenne did not mention it in any of the numerous and extensive passages on temperament that fill this book, but rather, in a quite different context, as an aside in order to help underpin his scepticism regarding the possibility for man of ever reaching any true conclusion in science. After all, not even so strange a theory as Stevin's can conclusively be disproved, writes Mersenne! 33

Such respectful vacillation when confronted with a palpable absurdity was nothing for Mersenne's friend Descartes. After probably having become acquainted with Stevin's theory as a result of his meeting with Beeckman in 1618-19, Descartes, in 1635, did not hesitate to write about 'the error of Stevin'; an error which implied the mistaken belief that two pure major thirds add up to form a pure minor sixth. But even Descartes himself could establish without any difficulty that this is simply not the case. 34

The last scholar to see Stevin's manuscript before it vanished (only to be rediscovered and published by Bierens de Haan in 1884) was Christiaan Huygens. He, too, rejected Stevin's theory out of hand: whoever, like Stevin, might believe that the fifth is given by any ratio other than 2/3 must have either poor ideas or poor ears, he wrote around 1661. 35

Note that both Mersenne and Huygens were quite familiar with equal temperament and its properties, even though they differed somewhat as to their assessment of its

31 Verheyen's critique was bound together with Stevin's treatise in the manuscript volume mentioned above in footnote 15 (pp. 674~80). In Bierens de Haan's edition it is to be found on pp. 87-97.

32 I. Beeckman, Journal tenu par Isaac Beeckman de 1604 d 1634, edited by C. de Waard 4 vols (The Hague, 1939-53). The relevant places are: If, 292; nl, 52, and Iv, 157 (letter to Mersenne, 1 October, 1629).

33 M. Mersenne, Harmonie Universelle (Paris, 1636/7), 'Preface, & advertissement au lecteur', preceding the 'Livre premier des consonances', second page.

3'*Descartes (footnote 28), I, 331 (cf. p. 295). 35 Huygens (footnote 1), x• 32.

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practical value. (Mersenne saw its advantages, but thought the drawbacks too serious, at least in the context of the musical composition habits of his own time; Huygens couldn't stand it, and went so far as to reject it even for the lute). But again, neither mistook Stevin's theory for having anything to do with the issue of temperament.

Mersenne's comments on equal temperament in Harmonie Universelle really fulfilled the pioneering role that present-century historians have wrongly attributed to Stevin's equal division of the octave. As noted before, equal temperament only became a practicable proposition in the third decade of the seventeenth century, as a result of the 'emancipation of the dissonance' that was one major characteristic of the new, Baroque style and its key experimental feature of daring modulations which necessitated the search for new, and more flexible modes of tempering than mean tone temperament. This is why Mersenne, who was quite aware of these recent musical developments, discussed equal temperament, not only in the 'Livre second des instrumens' of his Harmonie Universelle, that deals with the lute, but also--and this was really without precedent--at great length in the 'Livre sixiesme des Orgues'. Altogether he presented five different calculations of equal temperament--some far more accurate than Stevin's unpublished figures for the equal division of the octave--without, however, committing himself to the new system. These tables, then, were to serve as the starting point for a musical debate that went on for more than a century. Mean tone temperament was clearly on its way out, and ways and means were sought to circumvent the difficulties it presented without entirely losing its strong points: variety through differently-sized semitones, and pure major thirds. Attempts were made at key- splitting (thus providing both C sharp and D flat on one keyboard, but of course seriously complicating the job of the fingers and feet); at deriving multiple divisions in endless variety (from which then twelve suitable notes could be selected); or at creating 'well-tempered' systems (irregular scales that distributed the larger falsities amongst the keys that mattered least), and so on.

One factor that strongly worked in favour of equal temperament was the gradual coming to the fore of tonality, which, in replacing the medieval legacy of the modal system, reached maturity during the Middle and Late Baroque periods (about 1650- 1750). The new, sharp distinction between the major and the minor mode put a premium on systems in which the major and the minor third are wide apart, as is the case to a greater extent in equal temperament than in any other practicable tuning system.36 Around the middle of the eighteenth century a new keyboard instrument, the pianoforte, was introduced, which from the beginning was tuned in equal temperament. Its rapidly increasing popularity as a universal salon instrument settled the issue for good.

At the turn of the nineteenth century equal temperament had become the standard tuning, and by now we are so used to it that we are scarcely aware that alternative tuning systems are even conceivable, let alone that they had once prevailed. That is how, by a strange irony of history, the equally-tempered notes have eventually acquired a status of naturalness that was so wrongheadedly ascribed to them almost four centuries ago by the musically so ignorant but mathematically so inventive-- theorist, Simon Stevin.

36 W. Dupont, Geschichte der musikalischen Temperatur (N6rdlingen, 1935), pp. 83 84.

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488 Simon Stevin's equal division of the octave

Acknowledgments I wish to thank Nancy J. Nersessian for her help in demolishing the first draft of this

paper and in improving the second one. I also thank Penelope M. Gouk and Joella G. Yoder for their comments.

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