Music, Tuning

Mathieu’s Virtual Return

This post is part of my ongoing effort to understand Harmonic Experience by W. A. Mathieu.

One of the major points of the book is that certain commas or intonational discrepancies affect how we hear equal-tempered harmony even though these commas are technically eliminated by the temperament.

Much of Mathieu’s text rests on the important assumption that when we listen to tempered harmony we hear it as an approximation of just intonation. Tempered pitches may be physically sounded, but they do not constitute the primary elements of our inner musical experience — tempered pitches are rather symbols that stand for their justly tuned counterparts.  When we hear a tempered major third between C and E, we are reminded, consciously or not, of a pure major third where E’s frequency is in a 5/4 ratio with C’s. The tempered E, which is 400 cents above C, evokes in our ear a lower “pure” E which is 386 cents above C, perhaps in the same way a blurred physical image might cause us to see, in our mind’s eye, the clearer picture it represents. I don’t want to discuss the strengths and weaknesses of this assumption here, since it’s a vast and controversial topic — I’m simply mentioning it as context.

Now, some chord progressions cause a subtle pitch drift in just intonation: if you maintain pure tuning throughout the progression, you wind up at a slightly flat (or sharp) version of the chord you started on. Mathieu suggests that when we hear such progressions in equal temperament, where the pitch drift does not actually occur, we still maintain an inner awareness that the drift would occur in pure tuning. Although equal temperament takes us squarely back to our starting pitches, we don’t experience a strong sense of “returning home” because we sense on some level that a pure tuning of the sequence would not have taken us home — rather, it would have taken us to the “comma siblings” of our starting pitches. Mathieu calls this experience a “virtual return” — it’s the feeling of arriving home in equal-tempered harmony without really being home, since an ideal tuning of the progression would have drifted sharp or flat.  The idea that our musical perceptions straddle two tuning systems — one approximate and the other ideal — is how Mathieu can devote all of Part Three of the book, roughly a hundred pages, to discussing “The Functional Commas of Equal-Tempered Tonal Harmony,” even though these commas don’t exist in equal temperament.

In this post, I’d like to share some audio examples I created in my efforts to make sense of Mathieu’s discussion of virtual return. We’ll look at the following chord progression which is inspired by Mathieu’s example 29.3. I’ve abstracted Mathieu’s example into a simple sequence of triads where adjacent triads share two common tones (this involved adding some intermediate triads that weren’t present in Mathieu’s original sequence). The progression is Cmaj Amin Fmaj Dmin Dmaj Bmin Gmaj Emin Cmaj or I vi IV ii II vii V iii I.


Looking at the portion of the 5-limit just intonation lattice depicted below, you can see what would happen if we adhered to a pure tuning of all major thirds and perfect fifths throughout the progression. In this diagram, each node represents a pitch and is labeled in three ways: by its ratio with respect to the home note C, by its cent value, and by its letter name. Moving up in the lattice corresponds to ascending by a pure major third (multiplying by 5/4), moving right corresponds to ascending by a pure perfect fifth (multiplying by 3/2), moving down corresponds to descending by a pure major third (multiplying by 8/5), and moving left corresponds to descending by a pure perfect fifth (multiplying by 4/3).


In strict just intonation, our progression starts on the nodes I’ve colored green (C-E-G) and moves left, introducing the blue nodes in succession — first the 5/3 A, then the 4/3 F, then the 10/9 D, until we finally reach the blue E-G-C at the far left.  The tuning of each successive note is mandated by the previous two notes that are being sustained. For example, although there are two “versions” of A show in the diagram, we must use the 5/3 A when moving from the opening Cmaj to Amin, since only the 5/3 A will make a pure interval with the 5/4 E; the following choices are similarly determined.  When the progression ends, each one of the final pitches is 22 cents lower than our initial C-E-G in green — that’s to say, the destination pitches are all flat by a syntonic comma.  (Although the leftmost C is marked at 1178 cents, you can think of that as 22 cents flat of 1200 cents, or an octave above the starting C.)

Example 1: Just Intonation With Downward Drift

In the following audio clip you’ll hear the progression tuned in just intonation, as described above; after a pause, you’ll hear the final E-G-C repeated, then the opening C-E-G, then the final E-G-C, and then both tunings played together. You can hear how they clash since the final E-G-C is flat with respect to the opening C-E-G.


Example 2: Just Intonation With Shift

Another way of rendering this passage is to bump the intonation up by a syntonic comma midway through, so that the sequence does return to the starting pitches precisely. In the lattice, this would correspond to jumping from left to right at some point before the progression ends.  In the clip below, you’ll hear a pause after the D major chord that’s built from the blue nodes on the left (10/9, 25/18, 5/3). Following the pause, the D major chord is repeated using the slightly higher pitches represented by the yellow nodes on the right (9/8, 45/32, 27/16).  From this slightly raised D major chord, the progression then continues left in the lattice, through the yellow nodes and back to the green starting nodes. At the end of the clip you’ll hear the final E-G-C contrasted with the opening C-E-G — this time they match.  After that, you’ll hear the two versions of the D major chord played in sequence and then simultaneously — you’ll hear that they clash as one’s a bit higher than the other.


Example 3: Just Intonation With Mixing

Instead of breaking the progression at D major and shifting the entire chord up, we could introduce the higher/yellow pitches one by one during the second half of the progression. This will give rise to some chords that are mistuned as they’ll contain a mixture of lower/blue pitches sustained from previous measures with the higher/yellow pitches being added. In this clip, we start introducing the yellow pitches right after the D major chord in bar 5; the mistuning occurs in the next two bars.  It’s particularly apparent in the G chord in bar 7, where the sustained 10/9 D is too flat to make a pure perfect fifth with the newly introduced 3/2 G.  However, in exchange for this intermediate mistuning we land back home on our exact starting pitches, and we achieve this without any abrupt transpositions.


Example 4: Equal Temperament

Finally, here’s how the progression sounds in equal temperament:

Now the question is whether your perception of the progression in equal temperament is somehow affected by the tuning phenomena we’ve exposed in the earlier clips. If you were hearing this progression for the first time, in equal temperament, would you sense that the intonation “should” shift downward as it does in the first clip, and would this leave you feeling unsettled at the final C chord, whether you knew why or not?  (To use Mathieu’s colorful metaphors, would your inner Pathfinder and your inner Homebody both feel they had been “zapped”?)  I don’t know how to answer such questions with any definitiveness, but they are fertile ground for exploration, and if nothing else, they promise to help us be more conscious users of equal temperament as opposed to being musicians who just work blindly in the system.

4 thoughts on “Mathieu’s Virtual Return

  1. You are extraordinarily clear. I look forward to moving backwards and forwards and reading the rest of your posts.
    As for your closing remarks, “I don’t know how to answer such questions with any definiteness….” I’d like to respond.
    I’m not differing with you in the smallest way. But, based on my own experience, I would frame the situation somewhat differently. Take a progression just intonation that moves from an open C chord to the same open C chord an octave higher. Harmonically you have come back where you started. Melodically, you have moved an octave. Your example is parallel. In equal temperament you can progress so that you end on the same C chord upon which you started, But harmonically you have moved. (down 4 fifths and up a major third). Whether I am aware of the difference in PITCH indicated by the comma is not the essence of my experience it is the sense of harmonic movement, analogous to sense of MELODIC movement in the parallel example. Let’s say that the situation were different and mathematically, just intonation were possible and the notes that now differ by a comma were in fact the same. In other words, let’s say that in the example you have supplied the pitches of the final chord were actually the same as the pitches of the initial chord.
    There would still be that sense of not returning home, even though the final pitches were the same. Am I making sense?
    I just feel that the fact of the comma is muddying the water so to speak about something that is more fundamental than that. I hope you reply. I really like the way you think and write.

  2. I’m not sure I was clear enough. I will try again. Just as it’s possible to move melodically without moving harmonically (such as in a pure shift of register as described above) It is also possible to move harmonically without moving melodically.
    (in the sense of moving to a from one “comma sibling” to another. My point is that even if you do not discern the slight difference in pitch, or even in the imaginary case where siblings have precisely the same pitch, there would still be a sense of change, of movement, illustrated by their different lattice positions. That is to say, what I hear is the change in harmonic relationship with the tonic. As I played around at the piano as a child, the single most memorable event that occured was the day I found that I could change the apparent pitch of a note on the keyboard by changing the notes that accompanied it. I changed the chord, keeping one note and ……..the note seemed to change pitch in my ear as I listened.I tried for years to make sense of this but never did to my satisfaction. Now as you so aptly put it, these things can never be nailed down definitively, nonetheless there seem to be two possibilities. 1. It could be that the change I hear is the change not in the keyboard sound itself, but in the pitch that it is representing. However there is another possiblitly. It could be that what I’m hearing is 2. Not a change in pitch, but a change in harmonic function with respect to the tonic, i.e. a change in its position on the lattice. In this latter case, I would hear a change even if the siblings literally had the same pitch. I hope this is clearer. Now the question is…. exactly how would I/you experiment with this to get a bead on which is happening.

  3. Very briefly, another approach to these problems. From the point of view of the writer of music, one can choose harmonic progressions that do NOT have comma drift — and that, I believe, is what earlier composers actually did, as they followed those ‘rules’ that so many people now say are simply “observations of what used to be done”.

    1. That’s a helpful way of putting it. The question is, when working in equal temperament, do we still find progressions more comfortable or “natural” when they don’t have comma drift (that is, when their just-intonation counterparts don’t have comma drift)? Or, once we inhabit the equal temperament world for a while, does it stop mattering so much? Does the mind/brain/ear lose track of what would have been going on in a just-intonation realization of the same music? I read Mathieu as implying we always maintain some subliminal awareness of the JI analogue of the music we might be hearing in ET. He seems to suggest that non-comma-drift progressions have a specially comfortable quality, even when the mechanism of ET is there to prevent us from ever hearing whether there would have been a comma drift or not. I’m not sure I’ve resolved this question for myself yet though — need to experiment and listen more. Thanks for the comment!

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