Proportions – in search of a practical theory



It’s well-known that in music from around the year 1600, changes of metre are associated with Proportional notation. But how does this work in practice?

Proportions are like gear wheels – whilst the Tactus turns at a steady, slow pace (about one beat per second, as we will see), Proportional changes create fast notes: three times as fast, six times as fast, one and half times as fast. But how does the original notation show us how to select the correct gear?

George Houle’s Meter in Music 1600-1800 (Indiana, 1987) brings together evidence from many primary sources. The 17th century is a period of change, and there are differences between different decades, different countries. There are also contradictions between different writers within the same period and country. Part of the difficulty is that musicians were trying to reconcile contemporary usage with older authorities.

Nowadays, there is considerable debate about proportions amongst musicologists, with Roger Bowers (Emeritus Professor of Music at Cambridge) leading the arguments for a very conservative (almost ‘medieval’) interpretation of Monteverdi’s proportional notation.  In Bowers’ view, Monteverdi’s “time-signatures” give detailed information which can be decoded to reveal the intended proportion, applying rules that go back at least into the previous century.

Given the lack of consensus, most musicologists would agree that we do not have a complete understanding of proportions today. But I suggest that the situation is comparable to that of high-energy physics. We do not have a complete understanding, a Grand Unified Theory that brings together quantum mechanics and general relativity, uniting the various fundamental forces. But we do have a good idea of what such a theory would look like.

Grand Unified Theory

In this post, I’ll suggest what a practical understanding of Proportions should look like, and propose some tests that you can use to examine your theories. And yes, I’ll set out my own theory and explain why, with great respect, I disagree with Professor Bowers. That disagreement probably reflects the difference between my practical approach and his theoretical viewpoint, a difference that we see also in period sources.

But, just as with basso continuo, Proportions are not only theory, they are crucially necessary for practical music-making. So let’s begin our search for a ‘practical theory’.




I suggest that a practical theory of proportions should satisfy certain principles.

  • Proportions should be consistent, so that the same input produces the same result in whatever circumstances.

Of course, there are limits: for one composer, or for a certain period/place, perhaps as narrow as “within this one piece”. But even that most limited version of the Consistency principle is broken in many modern editions and performances (e.g. Monteverdi Vespers or Orfeo).

  • Proportions should be unambiguous, so that different people can reach the same answer.

Remember that most music circa 1600 was performed from part-books (not score), without conductor, and with minimal (if any) rehearsal. Each performer had to reach their own decision about each proportional change, and would not know what other parts were doing, until they heard the result.

  • Proportions should be simple, so that performers can reach an instantaneous decision

Since most music circa 1600 had minimal (if any) rehearsal, sight-reading performers should be able to turn a page, see a proportional change, and make an instantaneous decision. This is a comparable situation to continuo-playing, where performers similarly have to make instantaneous decisions of which harmony to play.

  • Proportions should be consistent with dance-types

Monteverdi’s balli, including the ballo and moresca within Orfeo, provide good examples. It’s worth remembering that Monteverdi recommended just 30 minutes of rehearsal for the Ballo di Tirsi & Clori. It’s a difficult piece, as he admits – most pieces would not have even this much rehearsal, whatever proportional challenges they might contain.

  • Proportions should be consistent with Tactus

Here musicologists have additional information, not taken into account by many of today’s performers. There is a very strong historical argument for assuming a consistent Tactus, around one beat per second, throughout a large-scale work, and indeed, across the entire repertoire in this period. The model is the pendulum swing, the limit of precision is human memory. This Tactus beat corresponds to a minim in C time. (Read more about Tactus here.)

  • Proportions should be consistent with the affetto of the text.

Such large-scale pieces as Monteverdi’s Vespers and Orfeo are good testing-grounds for any theory, since they provide a large number of cases, in full score, partial score, and part-books. In particular, the Magnificat from the Vespers has one set of proportional signs in the voice partbooks and a different set in the basso continuo book – obviously the result has to match between the two sets of notations. This hard case can test many a promising theory to destruction! There are many Proportional changes to consider in the Sonata, including one very hard case of black notation, discussed by Bowers here (and yes, once again I have to disagree. I’ll present my arguments in a later post).

The large-scale madrigals Altri canti di Marte and Altri canti d’Amor also provide challenging examples to consider.

One warning: be aware of the notation that creates triple-metre rhythms under a C time-signature, as in the first aria of Orfeo Act II: Ecco pur a voi ritorno (see below for the original notation). The minims here go at the same speed as in the preceding madrigal, Ecco Orfeo. 

And another warning: even supposedly reputable editions sometimes change original time-signatures without informing you. If you are not looking at the original notation, you are liable to be misled.

So at this point, I encourage you to consider your own theory of Proportions, and to test it against the principles and examples I’ve listed.

Grand unified theory 2


For anyone interested, here is my personal theory, which I have applied successfully across a wide selection of repertoire around the year 1600. You are welcome to use this, all the more so if you are polite enough to credit me for it in your writings or program notes. And if you disagree, I’d love to hear your counter-arguments, and to examine your theory in the light of my principles and test-pieces!

In common time, minims go with the Tactus, approx minim = 60.

In triple metre under a common time sign, there is no proportion, no change of speed, crotchet = crotchet. Dotted minim works out as ~ 40, but you still count the regular tactus  of minim = 60, even though the word-accents will not always coincide with the Tactus count. (This is rather like modern dancers, who count a Waltz in eight: ONE two three FOUR five six SEVEN eight one TWO three four… etc)

For proportional changes, you don’t have to worry about the time-signatures, which I consider to be obsolete remnants of an ancient system, now lost, and soon (i.e. in the second half of the 17th century) to be abandoned altogether. (Here is the crux of my disagreement with Bowers, who interprets proportional signs according to medieval principles). My approach is simple: just look at the note-values, whether in white or black notation.

In triple time,

minims represent Tripla, three minims to the Tactus, i.e. dotted semibreve = 60.

semibreves represent Sesquialtera, three semibreves to two Tactus, i.e. dotted breve = 30. Notice that dotted semibreve = 60 again, and minims have the same value as in tripla.

semi-minims (looking like crotchets) go very quickly, six to the Tactus. The default grouping is three groups of two, i.e. Tripla. Under a 6/4 mark, or with 3 marked over small groups of notes, the grouping becomes two groups of three (i.e. compound duple). Notice that minims, dotted minims, semibreves and dotted semibreves have consistent in any triple time notation.

This simple consistency ensures that different voices fit together in all situations, as well as allowing quick and unambiguous solving of proportion puzzles.


In triple time, semibreves can be white  or black. In certain circumstances (beyond the scope of this post, but the wiki article on mensural notation here  is a good introduction), white semibreves get “perfected” to contain three minims. What we regard today as the dot of prolongation (a dotted semibreve is worth three minims), the 17th century saw as the dot of perfection. Black semibreves do not get perfected.

Similarly, breves can be white or black.

In practice, black notation either shows a switch to triple proportion (easily noticeable, since in common time all semibreves and breves are white) or draws attention to a syncopation within triple proportion. (e.g. the rhythm short-long, rather than the customary long-short).

Minims can also be white or black. The problem is that a black minim (triple proportion, black notation) looks identical to a crotchet (common time). It also looks like a semi-minim (triple proportion, white notation). A black semi-minim (triple proportion, black notation) looks like a quaver (common time). We also see white semi-minims (triple proportion, white notation), which look strange to us, like white quavers.

But don’t worry too much. In most situations, it’s abundantly clear whether or not you are in triple proportion, and if the notation is generally white or black.

Changes in colouration draw attention to changes of metre and/or of short-time rhythm.


We end up with these five possible speeds of triple time (three different proportions, and two non-proportional notations):

Non-proportional triple metre

Very Slow & Steady: Three beats in 3 seconds (not proportional, minims under common-time sign)



Slow proportion: three beats in 2 seconds (semibreves in proportional notation, Sesquialtera)



Non-proportional triple metre 2

Medium Steady:  Three beats in 1.5 seconds, crotchets under duple-time sign



Medium proportion: three beats in 1 second (minims in proportional notation, Tripla)


In principle, there could be a fast steady notation, with groups of three quavers under common-time sign (non-proportional). These would go at three beats in three-quarters of a second. But I can’t think of an example from this period. Can anyone provide an example?


6 4 proportion

Fast proportion: compound duple, counting six beats in 1 second This example from “Altri canti d’amor” sets the text “le battaglie audaci” (‘daring battles’, hence the very fast declamation)


In any triple proportion, a semibreve lasts about two-thirds of a second. A minim lasts one-third of a second. A semi-minim lasts one-sixth of a second.

But whilst the notation, the metre and the short-term rhythms change, the Tactus remains constant.

Whatever the notation says, you maintain the same Tactus beat (around MM 60, i.e. one beat per second). But before a change into Sesquialtera, it helps to focus only on the Down-stroke of the Tactus (i.e. thinking semibreve = 30 rather than minim = 60), so that you can mentally subdivide the Down-strokes into three as the change to slow triple proportion takes effect.

Some period sources discuss this in terms of Tactus (semibreve = 30) and Semitactus (minim = 60), but most use Tactus interchangeably for the full Down-Up, or just for the Down stroke.

Of course, you need to avoid anachronistic rallentando as you approach a proportional change. Don’t use the brake and accelerator, just use the gears!

In a later post, I’ll discuss the application of this theory to balli in operas, in particular Anima e Corpo and Orfeo.

But one last thought… My theory is that the “time signatures” don’t matter, you can see everything from the note values. Roger Bowers believes the opposite is true, i.e. that the vital information is conveyed by Proportional signs. So what about the moresca that ends Monteverdi’s Orfeo? It is printed under a “time signature” of C. But clearly it must be proportional notation, you couldn’t continue in common time with minim = 60.

Obviously, this was a printer’s error, and the proportional signs were omitted. But – and here’s my point – if this error were highly significant (as for Professor Bowers’ theory, it would be), surely the printers would have restored the missing signs, perhaps during the print run (other errors were fixed along the way, as can be seen by comparing different copies of the 1609 edition), and certainly for the second edition (which fixes many errors, and introduces many new readings, some erroneous) of 1615.

But they didn’t fix it, not in 1609, not in 1615. I suggest that no-one (then or now) is confused by the lack of proportional signs, and there is broad agreement today that this lively dance goes at three minims to the Tactus. QED.

Moresca from Orfeo


PS Don’t forget to add this last killer example to the challenge tests for your own alternative theory!

Robot wars

Competing Theories of Historical Performance Practice?


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Opera, orchestra, vocal & ensemble director and early harpist, Andrew Lawrence-King is director of The Harp Consort and of Il Corago, and Senior Visiting Research Fellow at the Australian Research Council Centre for the History of Emotions.













One thought on “Proportions – in search of a practical theory

  1. Pingback: Quality Time: how does it feel? | Andrew Lawrence-King

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