In my previous post on W. A. Mathieu’s idea of “virtual return,” we looked at a chord progression that drifts flat by a syntonic comma when rendered in just intonation. Here we’ll look at a shorter progression that drifts flat by an even greater amount.
Our present example is a sequence of major triads with their roots ascending in major thirds, with common tones sustained. Depending on whether we think of this sequence as a chromatic “expansion” of C major — starting and ending on the same chord — or as a progression from C to somewhere else in harmonic space, we could notate it as C – E – Ab – C or as C – E – G# – B#:
In equal temperament with no intonational liberties, C and B# are played at the same pitch, and it’s only the musical context that determines whether the listener hears the progression as a “return” or a “departure.” However, if we tune the passage in just intonation, so as to achieve pure major thirds and perfect fifths in each triad, with no pitch adjustment across ties, then all the pitches in the final chord will turn out flat of their counterparts in the opening chord by almost half a semitone — the diesis — and no matter whether the listener perceives the discrepancy, the sequence is technically not a return.
The diagram below shows how the progression is tuned so as to keep each triad just: we start at the bottom of the “ladder” and climb upwards. Our opening C-E-G, at the bottom, is tuned at 1/1, 5/4, and 3/2. The 5/4 E is sustained between measures 1 and 2, and if this E is to be used as the root of a just major triad in measure 2, we need to tune G# at 5/4×5/4=25/16 and B at 5/4×3/2=15/8 — a step upwards in the ladder. G# is then sustained between measures 2 and 3, and hence it dictates the tuning of B# and D# in measure 3. Finally, the sustained B# dictates the tuning of D## and F## in measure 4, the top of the ladder, where the pitches are flat of their opening counterparts by roughly 41 cents.
The following sound clips offer a few ways to examine what’s happening. There’s one clip using just intonation as described, and another using equal temperament. In each clip, you will first hear the four bar progression played once. After a pause, you’ll then hear eight bars where the progression is stated twice, back to back, without any modifications: listen for the contrast (or lack thereof) between the end of the first statement and the beginning of the second statement. After another pause, you will hear a comparison passage that plays the closing chord, the opening chord, the closing chord, and then both at once. The entire set of examples is then repeated an octave lower, so you can see whether the range affects how you hear it.
Diesis Progression — Just Intonation:
Diesis Progression — Equal Temperament: