And here is Marini’s Canzon prima for four violins or cornetti, in which he gives us a straight 3/1 tripla proportion, with no additional symbol. The audio example begins in the fourth system.

Several printed collections from 1629 include dance music. Marini’s opus 8 is one of these, and it is interesting to observe that he thought of dance movements in similar proportional terms. The following four examples illustrate duple time, tripla, sextupla and sesquialtera proportions respecively.

And so I wondered when faced with this final example, Lilia convallium by Benedetto Rè, whether we might in fact be justified in interpreting 3/2 here as a modern orthochronic time signature, rather than as a strict proportion – and in fact we struggled in rehearsal to make this sound convincing under any other interpretation (strict proportional options would appear to be sesquialtera, or applying the logic of the Tarditi example, sextupla). And so in our interpretation of this piece the triple minim becomes duple crotchet.

In conclusion, I hope to have demonstrated through these worked examples that, far from being in a state of ‘chaotic confusion’, this was a period of notational transition that nevertheless operated within a common framework for understanding of proportional relationships. While some composers took pains to indicate their proportions with great precision, others relied rather more on a common-sense understanding, the obsolescence of which is the cause of our modern-day confusion. Although it is unlikely that we shall ever agree on ‘definitive’ solutions to these questions of rhythmic interpretation, I hope by giving an insight into the meeting of theory and practice in our interpretative process this may prove a useful contribution to what has hitherto proved an intractable debate in seventeenth-century performance practice.

[1] Jeffrey Kurtzman (responding to Roger Bowers), 'Notation and Proportions in Montveredi's Mass and Vespers of 1610', Music & Letters 74 (1993), 487-495, at 488.

Next, here is Maria virgo  by Donati, who uses a slightly different notation to indicate the same relationship – here he uses a circle, indicating tempus perfectum (but still with semibreve tactus) and then 3/1 as tripla proportion, three in the time of one. The audio example begins in the second system of the second page.

Now here’s something that on the surface suggests a rather more complex relationship. Carrone gives us C (semibreve tactus), followed by a circle (tempus perfectum) bisected by a vertical line (a sign of diminution, so the tactus now shifts to the breve), then a sesquialtera proportion of 3/2. So that’s three semibreves in the time of two, under a breve tactus for the triple section – which is equivalent to the semibreve tactus in duple time. The net result of that is the same tripla relationship as we had in the previous examples, just notated in a rather more convoluted way. But nevertheless, there is a logic to this sequence of symbols, and when we understand that logic it is clear that the composer was trying to indicate a very precise proportional relationship between the duple and triple sections.

But we also see the beginnings of modern, orthochronic time signatures by 1629. The following example is from Martino Pesenti’s second book of correnti alla francese in which all the pieces are written ostensibly in 3/4 (although the signature is missing a denominator to this effect).

Looking now at Monteverdi’s Exulta filia, Sion, we see this is written in C with an indication of 3/2 for the triple proportion. A true sesquialtera here would result in an extremely slow tempo that makes no sense at all of the joyful text. In fact the only meter that makes any sense here is compound duple – which Praetorius described as sextupla. In strict proportional theory, Monteverdi should have included a sign of double diminution here (a backward-facing C with a vertical stroke would have sufficed); but such attention to theoretical detail was not Monteverdi’s style; indeed, as Jeffrey Kurtzman has pointed out, Monteverdi had previously attracted criticism for his inconsistent use of 3/2 signatures, directly from Artusi, and obliquely from Banchieri.[1] Nevertheless, to Monteverdi (as to us) it was entirely obvious that the tempo should reflect the character of the text. The proportional relationship here is three semibreves per half tactus.

Understanting Triple Proportions in Music from Venice, 1629

Part 2: musical examples

Having established some of the basic theoretical precepts, let us now look at some examples of the application of theory in practice.

First, Schütz’s Exultavit cor meum. Below is an extract from Rudolf Gerber’s edition in the Neue Schütz Ausgabe (1957), following which is an extract from the original print. We can see that note values have been reduced in the modern edition and the time signatures changed, which obscures the intended proportional relationship indicated by the original. That’s a pity, because Schütz is entirely clear in his use of notation here.

He shows us C, which indicates a basic semibreve tactus (in duple time), followed by 3/1 for tripla, i.e. a relationship of three in the time of one. We have a regular alternation of triple and duple time in this passage and the audio example demonstrates the tripla relationship.

Another sextupla relationship is implied in the next example, by Tarditi. Here we have three minims per half tactus. The compound duple meter is emphasised by regular barlines. Note the frequent hemiola patterns indicated by blackened notation in the bass; it would not have been possible to indicate these shifts of accentuation in smaller note values. The audio example begins in the fourth system.

Now we come to Dario Castello, who is one of the first to use terms such as Allegro and Adagio to indicate variations in tempo. Castello was of course working very much in the same notational tradition, and so I would argue that these terms can best be understood as modifiers of the usual proportions in relation to tactus. In the following extract, the proportional relationship implied by the numerator ‘3’ (missing its denominator) is most likely sesquialtera (3 minims in the time of 2). Adagio seems to be an instruction to play a little slower than a strict sesquialtera; whereas the allegro which follows is an indication to play a little faster. The audio example begins in the sixth system.