Analyses of A cappella Tuning Practices

(That's a wonderfully ostentatious title for a study of how far off key people sing, don't you think?!)

This site is part of a discussion about whether unaccompanied singers tend to tune diatonic scales in 3-, 5-, or 7-limit JI (just intonation), or some other simple tuning, or tunings too dynamic to be described in relatively simple terms. Involved in the question is whether a long reverberation time in the acoustical environment increases the tendency toward tuning intervals according to higher harmonics. I estimate the RT-60 (standard measure of reverb time) for the recording here in question at just under 3 seconds.

The data in the table below were gathered from a spectrographic analysis of a monophonic chant; namely, "O gloriosa domina," the opening track from the CD The Lily and the Lamb by Anonymous 4, a wonderful collection of medieval English chant and polyphony. It has a simple and beautiful modal melody, and consists of 4 verses of 4 phrases each. The whole piece was analyzed so that verses may be compared to each other in addition to the whole set of adjacent-note intervalic analyses. Characteristically of English chant, intervals of a third are used here and there, and it could easily be harmonized triadically.

As an interesting aside, there are several layers of meticulous organization beneath the 4 groups of 4. For example, all together there are 99 intervals between distinctly pitched notes: 25 in each of the four verses, less 1 for the first note which has no previous one with which to form an interval. And I don't wanna hear no existential-math crap about that statement! ;-) The 25 are organized by phrase as 4 + 8 + 6 + 7 -- could that possibly be an accident?! Those wacky medieval composers, I tell ya! It's a thing of beauty, and no mistake.

It's probably in Dorian mode, or maybe Aolean Plagal? It's hard for me to tell without a 6th degree (the piece uses only 6 different tones). The perfect 5th A to D between the first and last notes of each verse indicates to me that D should be considered the 1/1 for purposes of a JI analysis. At first blush then, C would seem to need to be at 9/5. But since in the context it sounds like the 1/1 of a temporary Ionian mode (sort of the medieval equivalent of false modulation?), it's consistently sung nearer 16/9; that's confirmed by the fact that C to E is consistently sung as 81/64 or wider (!)

Since the overall pitch of a cappella music nearly always drifts somewhat over the course of a piece, 1/1 cannot be tied to an absolute frequency. For now I've chosen probably the simplest solution to this dilemma: consider only the intervals between immediately adjacent tones. This, of course, is overly simplistic but it's a place to start. For instance, once a tone goes "wide" the successive ones often stay "wide" for 3 or 4 in a row, indicating that the pitch before that string of "wide" ones is still being held in memory as the primary point of reference.

UPDATE! To my great delight, I received on June 25, 1999 an email from Susan Hellauer of Anonymous 4 giving their view of tuning in brief:

Here's how to interpret the chart: It's divided into verses and phrases indicated by the numbers in the left-most column, such as 1.1 for verse 1, phrase 1. NOTE = scale degree in terms of note name FREQ Hz = The center frequency in Herz of that note's pitch INTRVL CENTS = The interval in cents between that pitch and the one immediately previous. JI RATIO = The frequency ratio that I would expect to find for the interval between that note and the one immediately previous if the singing were tuned in 5-limit JI. DIFFERENCE = The difference in cents between the actual interval and the one expected. Negative values thus indicate an interval narrower than expected. ANALYSIS = A 5-limit analysis, where possible, of the tuning of that pitch with respect to the one immediately previous. Note that whenever an interval is further than 5 or 6 cents from any and all 5-limit intervals, one of three terms is used to describe it: Tempered = lies between two possible interpretations Wide = wider than the widest possible interpretation Narrow = narrower than the narrowest possible interpretation Repeated notes are not given separate consideration. Thankfully, the pitches within each note, and between repeated notes, in this recording are highly stable. Thank goodness for professionals! :-) Actual 3-limit intervals (between adjacent notes) are in green. Actual intervals (between adjacent notes) with 5 in the ratio are in red. Actual intervals (between adjacent notes) that match those expected of my 5-limit analysis are marked by an asterisk (*).

The first verse's spectrograph, and technical details of how the data were gathered, can be opened in a separate browser window for quick reference between the chart and the graph.

Out of a total of 99 intervals between adjacent distinct note names, I count

• 30 near-pure Pythagorean intervals
• 18 Just intervals with 5 in the ratio
• 48 Just 5-limit intervals, (including the 30 Pythagorean)
• 51 non-Just intervals

An aside: anyone see fractal patterns in these supposed deviations from Just tuning?

Further analysis to come!

Need a break? See my bio page! Or if you're interested in J. R. R. Tolkien's Middle-earth literature, peruse an early draft of my newest site, An Essay on the development of music for Middle-earth. It includes some microtonal references, charts, and MIDI and Real Audio files.