Proportional Canons at Tokyo

Last week I travelled to Tokyo, Japan, to present a paper at the quinquennial congress of the International Musicological Society on some of my recent findings on proportional canons from c.1390 to c.1500. This research is part of a larger project that I am conducting with Denis Collins on Canonic Techniques and Musical Change, c. 1330–c.1530. While we have been doing much work last year on the fourteenth-century canon, these are relatively straight-forward examples in which voices imitated each other at the unison after a certain delay between voice entries. The latter is commonly called in technical parlance the interonset interval or IOI.

Proportional canons, on the other hand, present a different set of criteria from the earliest canons. Composers seem to have invented proportional canons at the end of the fourteenth century, possibly in response to other trends around the relationship between music notation and performance, especially the idea of transformation of musical elements into other duration or relationships by shifting them to different time signatures (or, to use historically correct terminology, mensurations) or proportions while using/thinking about the same notes. This is a phenomenon that Emily Zazulia (University of California, Berkeley) has been writing about recently. The BIG difference with the proportional canon (and, as I will soon argue, the mensuration canon) is that canonic voices start simultaneously and then slowly drift apart due to their respective proportional (or mensural) relationships.

So, I am interested in how composers (and probably musicians in general) might have worked out some “rules” for making proportional and mensuration canons. Even though the mensuration canon differs from the proportional canon in terms of process, the results that I was soon seeing in my computer assisted analysis suggested that they were in fact closely related to each other in terms of melodic and contrapuntal technique.

For this enquiry, I asked myself could have composers used principles of melodic design to control contrapuntal relationships between canonic voices. This was partly spurred on by scholarship by Julie Cumming, John Milsom (who was looking at this phenomenon more than 15 years ago) and Peter Schubert concerning the stretto fuga, that is canon in which the IOI between voice entries is no greater than one mensural unit (or tactus). These researchers had found that composers were limiting their choice of particular melodic intervals up or down according to the particular type of stretto fuga that they wished to compose. By particular type, I refer to the interval at which voices imitated each other. All they had to do was ensure was that the canonic voice preferred or avoided certain intervals, and the canon (stretto fuga) would just work! We get a good sense of this practice when it starts to be codified by Gioseffo Zarlino (1558) and then by William Bathe (1584). The latter even presented a table to be used for composing a stretto fuga at any interval, with a list of acceptable melodic intervals according to the interval of imitation. Composing by numbers!

Before 1558, no writer provides instructions for writing a canon (there isn’t really anything about proportional canons until the modern era), and so we are left with the question of how did earlier composers write canons. Cummings and Schubert have shown that stretto fuga techniques can be found in compositions as early as the 1430s and as late (or even later than) 1600.

In my paper at Tokyo I revealed some of the findings from computer-assisted analysis on the following proportional and mensuration canons:

Composer and Title Canon Type Interval between voices Sources
Anonymous, Blijfs mi doch bi, gheselle goet Proportional (2:1) (durations in comes double those of the dux), 2 in 1 Simultaneous unison; third voice possibly lost NL-Amsterdam, Universiteitsbibliotheek, Ms. ES 64, fol. 1r
Johannes Ciconia?, Le ray au soleil Proportional 4:3 and 3:1, 3 in 1 Simultaneous unison and octave Perugia, Biblioteca Comunale “Augusta”, MS 3065, 3r (part of Lucca 184 manuscript, LXXXIIIr). Lower half of page (under Una panthera)
Guillaume Du Fay, Bien veignés vous, amoureuse Proportional 1:2, 2 in 1 with Contratenor Simultaneous octave GB-Oxford, Bodleian Library, MS. Canon. Misc. 213, fol. 34v
Guillaume Du Fay, Inclita stella maris Mensuration, C and O, 2 in 1 with two free contratenor (concordans) Simultaneous unison I-Bologna, Museo internazionale e biblioteca della musica, Q.15, fols. A211v-212r (no. 173)
Anonymous, Eslongies suy de vous, belle mastresse Proportional 2:1 Simultaneous octave below I-Trento, Museo Provinciale d’Arte, Castello del Buonconsiglio, MS 1374 [87], “Trent 87”, fol. 34r
Josquin, Missa L’homme armé super voces musicales, Agnus dei II Mensuration, C, ¢ and 3 Simultaneous, fourth and octave below Numerous sources, from Vatican Library, Cappella Sisitina 197, fol. 9v to Glarean, Dodecachordon and beyond

Although the full extent of my findings will be reserved for a future publication (and reveal in a related blog post!), the following graphs highlight some of my findings. The first chart shows the melodic interval frequency of intervals across the selected canons.

Melodic interval frequency of proportion canons. CC-BY Jason Stoessel. Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

In some ways, this quite unsurprising – lots of conjunct melodic motion. But in other ways is it interest, particular the greater preference for thirds and fourths in some examples. Le ray au soleyl shows a preference for thirds, and as some readers would know, using a melody of thirds around a central tone is one of the easiest ways to write a canon.

Melodic 3rds abound in Ciconia’s(?) Le ray au soleyl, bb. 1-4

But more interesting is Josquin’s preference for fourths, something that is a quite obvious feature of the concluding section (bb.19-25) of the Agnus Dei II from his Missa L’homme armé super voces musicales.

Josquin-Agnus-MLHAsuper
Decorative melodic 4ths make canon at the 8ve easier in Josquin’s Agnus Dei II from his Missa L’homme armé super voces musicales

One of the ways of trying to make more sense of what is happing in the previous chart is to look at how melody and counterpoint relate. To do this, we can consider the succession of contrapuntal intervals between a pair of canonic voices as a series of trigrams which describe the movement of one interval (or dyad) to another interval, including the melodic step in one voice (here the lowest voice). To represent each trigram I use element notation of [x, y, z], where x is the first vertical interval, y is the second vertical interval and z is the melodic interval in the lower voice of the pair of voices forming this counterpoint. I use the interval indexing convention most well known from the contrapuntal theory of Sergei Taneyev, i.e. 0 is a unison, 1 is a step, 2 a third, 3 a fourth, etc. The results (which require further verification and testing) of trigram analysis of canonic voices in the selected canonic voices is shown in the following graph. For the purpose of this graph I have removed all dissonant vertical intervals, since they are by definition non-contrapuntal.

Trigram frequency (consonances only) new
Trigram frequency of selected proportional and mensuration canons. This work is licensed under a Creative Commons Attribution 4.0 International License.

This chart begins to demonstrate that some of the melodic interval preferences are related to contrapuntal preferences. For example, consider the highly individualised (at least in terms of this sample) trigram profile of Josquin’s Agnus Dei II. It shows a marked preference for thirds moving to unisons by a third in the upper voice [2,0,0], but more significantly are the spikes of descending or ascending melodic fourths in oblique counterpoint [7,4,0]. A quick glance to the above example showing bars 19-25 of this composition helps to understand what is going on here: Josquin is using the interval of the fourth in a quicker moving voice in the top staff over the longer notes of a slower moving voice in the middle staff. It’s a simple technique, but it creates variety and allows the canon at the octave between these two voices to work effortlessly.

Another interesting feature is the dominant use of 6ths to 5ths and vice versa as oblique contrapuntal progressions ([5,4,0] and [4,5,0]), as well as other individualising features like a preference for more parallel descending 6ths [5,5,-1] or 6th to octave [5,7,-1]  by contrary motion in Eslongies.

Another telling feature of the data is the closeness of the trigram profiles of Guillaume Du Fay’s two canons. One is a proportional canon, the other a mensuration canon, thus providing grist to my mill that there is little difference in the technique of both types of simultaneous canons. I could go on, but for now I will let this graph speak for itself. But at this stage, the data from these findings points to techniques similar to the stretto fuga, something that might not be too surprising given that there is only a small difference between close and simultaneous proportional/mensural imitation.

There’s more work to do. The perceptive reader would observe that there is a leap between the first and second chart presented above. The first chart considers all intervals regardless of whether or not they form dissonances with other voices. This type of melodic interval, mostly occurring as passing tones but sometimes are ornamenting dissonances (eg. in a suspension), needs to be eliminated in a model that can tell us more about the melodic skeleton of proportional canons that are often decorated by stepwise melodic motion. Similarly, I haven’t considered duration and rhythm in this present analysis. These are the next steps in my computer-assisted analysis of proportional canons which I hope to share soon.

Selected Bibliography

  • Milsom, John. “Hard Composing; Hard Performing; Hard Listening.” Early Music 41, no. 1 (2013): 108–12.
  • Schubert, Peter. “From Improvisation to Composition: Three 16th Century Case Studies.” In Improvising Early Music, edited by Dirk Moelants, 93–130. Leuven: Leuven University Press, 2014.
  • Schubert, Peter, and Julie Cumming. “Another Lesson from Lassus: Using Computers to Analyse Counterpoint.” Early Music 43, no. 4 (2015): 577–86.

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