Just a little mid-Summer check-in here! We’ll be back on a more regular schedule as we come into our season in the Fall, which kicks off mid-October.
I’ve just been totally engrossed by Ross Duffin’s fantastic article about the theory and practice of Just intonation in Renaissance music. While Ekmeles has performed music explicitly written for and formulated to work in Just intonation, I’ve always been fascinated with quantifying the more intuitive method of tuning employed by choirs and vocal ensembles performing older music.
Before we get into the nuts and bolts, a few terminology refreshers: The octave is divided into 1200 cents, making each half-step in 12 note equal temperament 100 cents. Just intonation is more easily expressed in terms of frequency ratios, which will appear as a ratio between 1:1 and 2:1, expressing an interval smaller than an octave.
Anyone who’s performed in an ensemble knows the sound of a just perfect fifth, and its solid, grounded feeling due to the elimination of beating caused by overtone dissonances. It can be expressed as the ratio 3:2, or 702 cents, putting it 2 cents, or 2 100ths of a halfstep wider than the fifth on the piano. Just thirds, the major’s ratio 5:4, the minor’s 6:5, are slightly less intuitive to modern ears, as they are 386 cents and 316 cents respectively, deviating by 14 and 16 cents from the piano’s tempered thirds. However, performers of Renaissance music are familiar with the sound of these mellower thirds. Why this discussion about applying Just intonation? Why not just sing these pure fifths and thirds and be done with it? The short answer: polyphony.
The long answer is dealt with beautifully in Duffin’s essay. Tuning purely homorhythmic vertical chords one after another is a simple matter, relatively speaking. When a composer writes notes that hold while other notes move, the moving notes have to tune to the held note (assuming the held note isn’t sliding around). This sometimes means pure fifths will have to be tuned downward from a held note that was tuned as, say, a pure major third. Now a third tuned above that lower note will be doubly low, and you can imagine how things go from there! Duffin deals with Renaissance theoretical puzzles in which the chains of intervals tend to push the pitch in one direction or another, and devises wonderfully musical solutions to the practical problems of keeping in tune.
As a practical example, Lassus’s Ave Regina coelorum is exceptional. If all the intervals were to be tuned justly, the pitch would droop by somewhat less than a quarter tone (43 cents) over the first 5 measures! This example is included in the text of Duffin’s essay, along with several possible tunings.
Below you can hear a synthesized version of this excerpt in a pure (and totally impractical, given the aforementioned drift) Just intonation tuning, followed by a different kind of mechanical solution, performed using an H-Pi Instruments keyboard. Incidentally, I use their exceptional Scordatura software for realizing microtonal scores. Below that video is a human performance of this beautiful piece which doesn’t drop a quarter tone – you can really hear them fighting that downward tendency though, when your ears are attuned to it!