I argue that if we want to understand how the composer intended this music to work, it is not sufficient simply to dismiss the original notation and modernise it without comment in the way some earlier editors have done. Indeed, studying the original notation, and the theory surrounding it might provide the key to understanding these duple-triple relationships in practice.

First, let us consider some of the ways in which musicians of the seventeenth century conceived of the measuring of musical time, which in some ways is rather different to the modern approach.

The concept of tactus is fundamental to understanding proportional relationships both in theory and practice. The tactus was defined as a lowering and a raising of the hand; a down-beat and up-beat together comprising a single tactus.

Here is Zarlino in 1558. He uses the term battuta as synonymous with tactus:

‘If we consider the qualities that are found in the battuta and the pulse … we find many similarities between them. The pulse is … a certain broadening and tightening, or we could say raising and lowering, of the heart and the arteries. It is composed … of two movements and two still parts. Similarly, the battuta is composed of the same things: first of two movements … that are made by the hand … which are two opposite movements, and then two still parts … And just as medicine calls the first movement systole and the second diastole, music similarly calls the lowering, or beating, thesis and the raising arsis.’[4]

This is an especially useful definition, since it explains tactus as a motion of the hand and also gives us a sense of the strong-weak, stress-release, relationship between the down-beat and up-beat. It is important to note and understand that the down-beat and up-beat together comprise a single tactus.


In Renaissance mensural theory – at least according to some contemporary theorists – the tactus remains fixed throughout a given piece of music, and the relationship of individual note values to the tactus is governed by the mensuration signature. Depending on the signature, a tactus would measure the duration of a breve or a semibreve; most commonly it was applied to the semibreve by the early 17th Century. In mensural notation, a semibreve may subdivide into 2 or 3 minims, depending on the context. Under the theory of a constant tactus the duration of the semibreve remains the same, whether it subdivides into two or three. In duple time the tactus is divided into down-beat and up-beat of two equal parts, whereas in triple time the tactus is divided into two unequal parts – the first part slightly lengthened, the second shortened, but the duration of the tactus as a whole remains constant. This results in the most typical proportion during the Renaissance period: sequialtera (3 in the time of 2).

However, this principle of a constant tactus is by no means universally accepted by musicologists, or modern-day performers. As Roger M. Grant has observed, this has led to some diverse opinions expressed by eminent scholars, ranging from:

‘The whole system of mensural notation rests upon the theory of a fixed, i.e., unchangeable unit of time, the tactus.



The underlying explanation for this polarisation of scholarly opinion is essentially that the Renaissance theorists themselves often disagreed on these matters – perhaps unsurprisingly reflecting a diversity of localised practice over a long time-period and large geographical spread. Thus by being ‘selective among history’s wares’ (to paraphrase Taruskin) it is often possible to find theoretical support for both extremes as well as a variety of positions between these.

But whereas the editor might legitimately sit on the fence on this issue, the performer has necessarily to come down on one side or the other. Although I believe there are specific occasions when the tactus may be treated flexibly (some of which we will examine presently), on the whole I subscribe to the constant tactus theory. I do so primarily because of the extraordinary attention musicians of the early 17th century seem to have paid to their choice of mensuration signs governing the relationship between duple and triple time in their music. To be sure, their choice of notation may often seem confused and confusing, but the logical explanation for this state of affairs, I would argue, is that they were trying by these various means to express very specific relationships between the notated note values and underlying tactus.


A brief overview of the variety of mensural symbols we are likely to encounter in this period is perhaps in order here.

To quote Bowers once more,

‘To theorists of the time [early 17th Century], including Morley, Zacconi, Banchieri and Michael Praetorius, the primary system inherited from the past was still in good working order. Coloration, sesquialtera, tripla (and now sextupla) meant what they had always meant, and Praetorius in particular advocated a severe pruning of the symbols used in his day so as to clarify the basic simplicity underlying the system.’[6]


Praetorius summarises the confusing picture of contemporary notational practice for us in Syntagma musicum:[7] 

Understanding Triple Proportions in Music from Venice, 1629.

An earlier version of this paper was given at the Biennale Baroque Conference in Cremona, July 2018. 

Part 1: theoretical background

Through the process of editing the music for Venice 1629, I was struck by the variety of notational symbols used by the various composers to indicate triple time sections, and their relationship to the duple time passages they invariably follow or precede. Understanding the meaning of these diverse symbols was one of the biggest interpretative challenges of this project.

Indeed, the question of proportional relationships between duple and triple time is one of the most contested in early seventeenth-century performance practice. Perhaps the best known example of this is the now infamous musicological dispute between Roger Bowers and Jeffrey Kurtzman over the interpretation of duple- and triple-time relations in Monteverdi’s 1610 Vespers, which played out in the pages of Music & Letters in the early 1990s.[1] This was a bruising debate, which still has no definitive resolution.

Duple-triple relationships remain a hot topic among performers of early seventeenth-century repertoire; in my experience, discussions and practical experiments with proportions have resulted in some of the most profound musical revelations, and also some of the biggest bust-ups in the rehearsal room.


But it seems that confusion and debate about the meaning of proportional signatures is nothing new. Praetorius wrote, in 1618:

‘Mentioning the signatures used by earlier composers is hardly necessary since there is no particular use in their variety – but rather only confusion and bewilderment, by which not just school children but often professional vocal and instrumental chapel musicians are thrown into complete and utter consternation.’[2]

Praetorius was talking here about the difficulty of interpreting the proportional notation employed by composers of the Josquin-Obrecht generation of around a hundred years earlier, but his sentiments are not vastly different to those expressed by Roger Bowers, with reference to the music of Praetorius’s own time:

According to Bowers:

‘The performers and scholars who first disinterred this repertory [from the 1600s] in the early 20th Century found that the application of interpretations ostensibly correct for these proportions delivered results inconsistent with their aesthetic preconceptions for it. Consequently, the notation was conveniently dismissed as being in a state of ‘chaotic confusion’, and the evidence of the sources, both theoretical and musical, was disregarded.’[3]

A good example of such ‘convenient dismissal’ of notational evidence is found in Malipiero’s edition of Monteverdi’s Exulta, filia Sion (published in the supplementary 17th volume of his complete Monteverdi edition in 1966). As a nod to scholarly convention we can see at least the original clef is C1, but the piece is transcribed here in a modern 3/4 time signature with no further explanation.

Even a cursory glance at the original source, Lorenzo Calvi’s Quarta raccolta de sacri canti, is sufficient to demonstrate how unsatisfactory the modern edition is, in terms of information that is either missing or misleading with respect to time signatures and note values:

In other words, in spite of the great variety of symbols in contemporary usage, the proportions they are designed to represent are relatively few in number, and relatively straightforward in practice. To summarise:

  • The tactus may be equal or unequal.

  • If equal it may be slow (C, for madrigals), fast (cut C, for motets); and it may include the ‘new’ sextupla proportion (6 in the time of 1), which is essentially a compound duple meter.
  • If unequal (i.e. triple time) it may involve the proportions tripla (3 in the time of 1) or sesquialtera (3 in the time of 2).[9]


The reason, I believe, for the confusion in early seventeenth century notational practice is that there is often a disparity in note values between duple and triple time. Under the influence of late 16th century madrigal writing, the seventeenth century concertato motet is typically written in C, with a tactus dividing the semibreve into two minims, often featuring florid passaggi making frequent use of smaller note values in black notation. In triple-time sections, however, black notation is still used to indicate hemiola; triple time is thus typically still written in larger note values to avoid the possibility of confusion between crotchets and blackened minims. Therefore, triple-time passages often need to be understood in some kind of proportional diminution relative to the underlying tactus.

Part 2 will demonstrate the application of this theoretical understanding in practice. 



[1] Roger Bowers, ‘Some Reflection on Notation and Proportion in Monteverdi's Mass and Vespers of 1610’, Music & Letters 73 (1992), 347–98. See also correspondence, Music & Letters 74 (1993), 487–95; 75 (1994), 145–54.

[2] Michael Praetorius, Syntagma musicum, vol iii (1618). English translation Jeffrey T. Kite-Powell (New York: Oxford University Press, 2004), 74.

[3] Roger Bowers, ‘Proportional notation’, Grove Music Online <www.oxfordmusiconline.com>, accessed 1 May 2018.

[4] Le istitutioni harmoniche (1558), book 3, ch. 48 (“Della Battuta”), 207. Translated in Ruth I. DeFord, Tactus, Mensuration and Rhythm in Renaissance Music (Cambridge: Cambridge University Press, 2015), 79.

[5] Willi Apel, The Notation of Polyphonic Music: 900-1600, 5th ed. (Cambridge, MA: The Medieval Academy of America, 1961), 146-47. J.A. Bank, Tactus, Tempo and Notation in Mensural Music from the Thirteenth to the Seventeenth Century (Amsterdam, 1972), 259. Capitalization as in the original. Roger Matthew Grant draws attention to this scholarly polemic in Beating Time and Measuring Music in the Early Modern Era (New York: OUP, 2014), 41. 

[6] Roger Bowers, ‘Proportional notation’, Grove Music Online <www.oxfordmusiconline.com>, accessed 1 May 2018.

[7] Syntagma musicum, iii, p.52.

[8] Syntagma musicum, iii, p.79. 

[9] Further implications of this table are discussed in Paul Brainard, ‘Proportional Notation in the Music of Schütz and His Contemporaries’, Current Musicology 50 (1992), 21-46.

On this page he explains and demonstrates firstly the diversity of symbols used by his contemporaries to indicate tripla (3 in the time of 1), and then shows a similar variety used to indicate sesquialtera (3 in the time of 2). Note that there is some overlap between these symbols, which makes their interpretation even more problematic.

But he then explains that this diversity of notation belies the simplicity of the underlying system of proportional relationships. He says,

‘In my humble judgement one can avoid all the excessiveness and troublesome difficulties of meter signatures by contracting the entire treatment of signatures and meters in the following short and accurate table (subject to others’ corrections), and discarding the others entirely’:[8]