Music, Tuning

Overtone Scale?

There’s an idea floating around that the Lydian Dominant scale (1, 2, 3, #4, 5, 6, b7), defined in the context of twelve-tone equal temperament, is somehow special because it emulates a harmonic series. Sometimes the Lydian Dominant scale is called the “acoustic scale” or the “overtone scale.”  Those names strike me as misleading. The idea is that if we look at the harmonic series of C2, for example, we find something roughly similar to these notes from the piano: C2, C3, G3, C4, E4, G4, Bb4, C5, D5, E5, F#5, G5, A5. Arranging the distinct notes from that sequence into an octave starting at C gives us C, D, E, F#, G, A, Bb. This scale might have some wonderful properties, but can those properties really be explained by the scale’s resemblance to the harmonic series?

In fact it’s a stretch to say this scale is “based on the harmonic series” at all because the tuning discrepancies are so vast. The members of the harmonic series do not match the piano’s C, D, E, F#, G, A, Bb — you can’t play any portion of the harmonic series accurately on a conventionally tuned piano. The piano’s G and D are reasonably close to the third and ninth partials of C; E is far off from the fifth partial; Bb and F# are still further off from the seventh and eleventh partials; and A is even further off from the thirteenth partial (which is actually closer to Ab).

Alright then, even if we acknowledge the discrepancy between a true harmonic series and the notes we can access on the piano, might there still be something useful or interesting about approximating the harmonic series through our 1, 2, 3, #4, 5, 6, b7 scale — might the ear tolerate the discrepancies and still hear some special cohesion in those notes on the piano because of the way they mimic, if not exactly match the harmonic series?

One way to get at this question is to flip it around. We know that the ear does perceive a special cohesion in a set of simultaneous pitches arranged in an exact harmonic series with decaying amplitudes — we generally perceive this phenomenon as a single note. So what would happen if we were to take the harmonic series of C and adjust its pitches so that they aligned with the piano’s C, D, E, F#, G, A, Bb — would these “tempered harmonics” still seem to fuse into a single note?

Here is a sound composed of the first eleven partials of C2 (65.4 hz), where the amplitudes decay as 1/n where n is the number of the partial:

And here is a sound built from the equal-tempered pitches of the Lydian Dominant scale, arranged as an approximate harmonic series — C2, C3, G3, C4, E4, G4, Bb4, C5, D5, E5, F#5 — with the same decay pattern as before.  In other words, this how our C2 sounds if all its partials are tuned to be playable on a piano:

If your perceptions are at all like mine, you’ll hear the first example as a clearly defined steady pitch. The second example is less steady (there’s some wobbling or beating) and if you listen closely you can begin picking out individual components of the sound which don’t quite fuse into a seamless whole — the texture is “messy” and/or “chaotic.”

What can we conclude from this? I think it’s safe to say that the special perceptual properties of the harmonic series start breaking down when the harmonics are mistuned. If the mistuning is carried so far as to bring the harmonics in line with pitches of twelve-tone equal temperament, the difference in sound is quite drastic: the components cease to fuse.  There may be great creative value in taking inspiration from the harmonic series and trying to build structures in twelve-tone equal temperament that mimic it, but if the result sounds good, we shouldn’t be too quick to assume the goodness comes directly from some resemblance to the harmonic series; any such resemblance is limited by a very significant difference in tuning.

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A little meta-criticism

I grow suspicious when someone praises a work of art by saying “It’s not about X, it’s really about Y,” where X is the immediate subject matter and Y is some lofty theme that is supposedly embedded in the subject matter. The problem I have with this form of praise is that it can be applied to anything, anytime, because a critic can always claim to have identified a deeper theme anywhere; and that’s not to say that thematic connections are always fabricated by the critic, but simply that themes of sufficient generality do have apparent manifestations everywhere – that almost anything in life can be seen as comment on impermanence, or love, or hubris, or whatnot, if you really want to look at it that way. It seems to me that critics wax enthusiastic about the implicit themes in a work of art when they happen to like the work, but in pointing to such themes as the basis for their approval of the work, critics mix up cause and effect: it is because they like the work that they feel compelled to construct a justification for their belief in its aesthetic superiority, and in the making of this justification, they invariably discover a connection between the work and some theme that would confer importance upon it. For example, there could be a guy next to me and I hear him burp. Now, my previous interactions with this person, along with his impeccable reputation, may have convinced me that he is a great artist, and there may be something in his particular style of eructation, which when assessed with immense charity, sounds somehow captivating: I liked the burp. I could then go around saying that this burp wasn’t really about the release of gas from the stomach. Actually, it was about gender stereotypes. It was a parody of machismo, it was a daring provocation that challenged listeners to reconsider the very idea of crassness. Did it sound unpleasant or make you uncomfortable? It was supposed to do that. Good art pushes boundaries. Would you say the artist should have excused himself? You prude! You anti-belcher! Are you blind to the hidden meaning of this gaseous release? It was a deep burp! Now of course if the critic didn’t like the burp, the critic can just as easily marshal one of those negative tropes that may be applied anywhere, anytime: yes, the artist burped, but he didn’t burp in an interesting way. The burp was shallow, its style overwrought, its content conventional. The burp was not a paradigm-redefining burp – it didn’t tell us anything new about the act of burping, or about the world in which we burp. What theme did it grapple with beyond the narrow domain of dyspepsia? This was a burp that did not truly come from the artist’s gut, er…. well… Yes, yes, the artist burped; yes, in fact he’s a virtuoso of the art of burping, but he does it mechanically – there’s no feeling in it, no soul, only hot air!


Major 3rds Fretboard Layout

This diagram shows the fretboard layout for a 6-string guitar in All Major Thirds tuning, assuming the open strings are tuned to E, G#, C, E, G#, C as recommended by Ralph Patt.  I made the diagram because I’m beginning to learn this nonstandard tuning and I wanted a study aid that emphasized the amazing regularity of the system.

Notice that because the three open bass strings are tuned the same as the three open treble strings (modulo an octave), the entire pattern of notes among the bass strings is repeated among the trebles — the left and right halves of the diagram are identical.

Each of the four colors used in the diagram indicates one of four possible augmented triads (modulo inversion and enharmonic respelling).  Notice, for example, that two copies the F Augmented triad (F A C♯) occur along the first fret and are shown with a light green background.  At the fifth fret the same set of notes occurs in a different inversion — now the notes are ordered A C♯ F; again they are shown with a light green background.   Finally, at the ninth fret, the notes occur in the order C♯ F A.

The “fret dots” on the left are positioned according to most common inlay pattern for standard tuning.  For simplicity, notes are always spelled using sharps instead of flats, though of course all the notes in the diagram could be written in multiple ways.

Major 3rds Tuning For Guitar: Fretboard LayoutAn intriguing property of this layout is that any block of notes spanning three strings and four frets can be considered as a “tile” that repeats across the entire fretboard, in a way where the tiles don’t overlap and also don’t leave any gaps.  In the image directly below, I started by drawing a box around the notes across the bass strings at frets zero through three; next, I placed boxes around all other instances of that same pattern.  (The notes with a gray background aren’t actually on the fretboard, of course — I included them to make the pattern clear.)


This next diagram is similar to the previous one, except I outlined a different block of notes:


Other regular tunings like All Fourths also give rise to tiling patterns like the ones above, but in the case of All Fourths, non-overlapping tiles won’t form nice, simple rectangles, and it’s not possible to “fit” as many complete tiles on the fretboard.  Here’s one way of tiling a fretboard tuned to E, A, D, G, C, F:


Music, Voice

Thoughts on Practice

A recent voice lesson gave me the chance to reflect on my path in music and to notice how a major stumbling block of the past now seems easier to manage.

My teacher asked me to stop thinking about how I was singing and instead “let it come naturally.” We were working on Gute Nacht, the opening piece from Schubert’s Winterreise cycle.

In the many years that I listened to Winterreise as a teenager and then a twenty-something (collecting well over a dozen recordings along the way), I believed I would never be able to sing. I had played guitar since age 15, but my great secret wish was to make music with my own voice. Fear kept me from pursuing it: I thought I’d never be able to hold a steady pitch or make a pleasing tone.  But two years ago I finally signed up for voice lessons. These days it’s startling to realize that I’m singing the music I had once admired from afar, and it’s gratifying to see how a circumstance of self-doubt has turned into an opportunity to improve through practice.

My teacher read me a quote about running: according to research in sports psychology, the best runners don’t think at all while they’re running. They operate on auto-pilot.

Perhaps because my mind is so often abuzz, I sometimes receive this advice from teachers: “Don’t think so hard.” It’s good advice, but one faces a conundrum in applying it. Sure, the best athletes and musicians don’t need to think because they’ve practiced so long that good technique is now automatic, but what do you do before you reach that point? Sometimes you may have progressed further than you realize – all that’s needed to “cash in” on your practice is to step back, relinquish control, and let the good habits you’ve built now work for themselves. But there are other times when you try to “let go” and find that old habits come rushing back: without the oversight of your conscious mind, you regress. You can still benefit from taking a calmer approach, with less mental chatter, but you’re not quite ready for auto-pilot.

Of course, you don’t know what’s going to happen until you try. So, I took my teacher’s suggestion: I resolved to stop thinking about my singing and just enjoy the music. I let myself gesture freely and gave up my concerns about intonation and projection, jaw position and diction – all the things I had been studying in class. I was skeptical at first, but something magical happened within a few bars: I felt I had become a character in the play I had been watching all those years. Now I was that hapless wanderer, shivering as he departs his maiden’s house in that bleak snowy night. The music seemed to pour forth from me, and the dynamics fell into place: softer here, more forceful there – there was no need to consciously “interpret” the piece now that I was experiencing its drama first hand. The German text had become my own.

Fremd bin ich eingezogen,
Fremd zieh' ich wieder aus...

(A stranger I came,
and a stranger I depart...)

When the piece came to an end I felt I had said what I needed to say, nothing less, nothing more. The performance had been a transcendent moment for me – the reason I wanted to study music in the first place – and a terrifying moment too, as I had entered the psyche of Schubert and Müller’s frightful character.

There was silence as I came out of the “scene” and finally looked at my teacher. I was still swept up in the storm and cold of the piece and not quite ready to speak. She had been accompanying me on piano, and I thought she too might need a moment to rest after such intense music-making.

“That was nice…” she said. “It was nice… but… it might be time for you to take this piece to the next level… to make some more sensitive dramatic choices… to really start conveying the text. And also… I don’t want to nitpick, because the German was great overall, but there were a few places where the consonants got lost.”

My teacher is wonderfully encouraging, and she’s praised my Schubert before, so coming from her, this lukewarm response amounted to something like a C+.

The best way to vent my inner turmoil in that moment would have been to sing more Winterreise – but no, I thought, apparently my soulless Winterreise doesn’t convey any emotion so there would be no point in doing that!

While I experienced great emotional contrasts (from tenderness to rage) in the performance and thought I was communicating them, what actually came across to my teacher on this particular run-through was a narrow dramatic range, not enough variety between sections. Also, she thought my physicality could be more relaxed (less shifting back and forth and conducting with my hands) and I should take calmer breaths earlier, rather than gasping right before phrases began. And there were places where I could have rolled my German r’s with more vigor.

My teacher’s technical comments did not surprise me – these were precisely the things I had chosen not to worry about during my uninhibited performance, but I expected I’d still have to go back and work on them. The more confusing thing was that all the inner passion which I assumed must necessarily manifest in my singing just hadn’t come through. This was one of the most fervid moments I remembered having in the voice studio, and I felt I had taken a real risk in laying myself bare like that. For someone who’s usually reserved, these times of exposure don’t come often. How could my experience of performing the piece be so at odds with what my teacher perceived?

At one point in my life, a disconnect like this would have been more than confusing, it would have been crippling, sending me into a spiral of questioning and doubt. As an audience member – on the one hand – I’ve always reveled in the mystery of artistic expression. While we can analyze a performance and talk about its features, there’s no way to systematically predict what will move a listener, or how communication between artist and audience will unfold – and that keeps the game interesting. But when it comes to my own performances (whether in singing, playing guitar, speaking, or any other medium) I’ve always wanted there to be a clear causal relation between inner experience and external response. I want to know that what the audience hears will be somehow connected to what I feel, or at least that when I have a great inner moment, something rare and transcendent, when I think I’m at my very best, it won’t all turn out to be a fantasy! When that hope has failed, I’ve often become obsessed with trying to understand why. Where did the communication go wrong? Was I deluded, or was the listener in the wrong, or was something strange happening in the air between us?

In this particular context I began to wonder whether my teacher and I read the Schubert score differently — perhaps our interpretations were simply irreconcilable? Or maybe she was concentrating mainly on technical points as she listened? Or could it just have been that I was “off” without knowing it? But then how could it have felt so right? Endless questions sprung up, but I was able to walk away from them before too long, and that’s a choice that would have been difficult for me to make earlier in my life. It’s frustrating when there’s a disconnect and then… no, you don’t go and brood over it for hours… you go on and sing the next piece.

The way I look at things now is like this: as you perform, you might be moved by the music you’re making, or you might be unmoved, as if you’re executing the mechanics without true participation; likewise, the audience might be moved, or they might be unmoved. Of course, this is an extreme simplification of what’s possible. The important thing to realize is that all combinations of inner experience and external response can happen: you might be moved and the audience might be moved too – that’s great. Or you might be unmoved and the audience might be unmoved as well. That’s unfortunate, but at least it makes some kind of “sense.” In both cases, you and the audience appear to be in sync. But there are two other scenarios that make less intuitive sense and yet they happen all the time: you might be moved but the audience is unmoved, they just don’t “get it.” And on the other hand, you might be unmoved but the audience turns out to be deeply moved by what you’re doing – somehow! There’s really no way to know for certain, or to fully control, which combination will arise — the best you can do is influence it by practicing and trying your best every time.

And what do you know? In this particular class, we talked about a few other things I could work on, I thanked my teacher for the comments, and then went on to sing a couple of other pieces in a different vein, including Cole Porter’s So In Love and Donaudy’s O Del Mio Amato Ben, both of which she thought were spot on.

Guitar, Music

M3 Tuning For Guitar: A First Look

If you’ve followed my guitar posts here, you’ll know that I like the All Fourths tuning (E A D G C F) because it imposes a regularity on the fretboard that allows a player to shift chord and scale patterns across strings without fingering adjustments.  It’s also a comfortable tuning to explore if you’re familiar with standard tuning, since only the highest two strings are changed: you can reuse any chord or scale pattern that you’ve learned on the lower four strings, which are already tuned in fourths.

I had been cautiously avoiding other nonstandard tunings since switching to All Fourths around a year ago — I didn’t want to spread myself too thin.  But a blog visitor recently asked why I hadn’t considered All Major Thirds tuning here, and I couldn’t resist the invitation to experiment.  Now, after exploring M3 for a week, I’d like to share some initial observations.  (Alexandre, thanks again for the question that prompted this!)

One way to implement M3 is to keep the guitar’s lowest string at E and tune major thirds above that, giving E, G#, C, E, G#, C.  This is the 6-string setup recommended by Ralph Patt, who is considered to be the originator of M3 tuning.  Notice that while P4 tuning expands the guitar’s range by a semitone, M3 narrows it by a major third (the highest string drops from E down to C), which is why some M3 players prefer a 7-string guitar.  And while P4 only requires two strings to be retuned, M3 requires five retunings, which makes it a very different beast from standard tuning.

One of the first observations people make about M3 tuning is that you can play an entire 12-note chromatic scale in a span of four frets, with the same finger always playing the same fret, without shifting hand position.  Since any octave-repeating scale is a subset of the chromatic scale, this means you can play any scale you want without a position shift or stretch.  (If you’ve grown up adjusting to the shifts and stretches of standard tuning, it’s worth taking a moment to consider how remarkable this is.)

A nuance I haven’t seen emphasized elsewhere is that all this holds true regardless of which finger you use for the root note.  You can start the scale with your index finger or your pinky, and in each case there’s no need to move your hand or stretch beyond four frets.  The diagram below shows four different fingerings of the chromatic scale, corresponding to the four fingers you could use to play the root.  The numbers indicate which finger is used to play the notes at the corresponding fret — in the first example we play the root with the index finger (1), in the second example we play it with the middle finger (2), and so on.

M3ChromaticFingeringsJust as the chromatic scale can be played with any starting finger, without a position shift, so too can any scale be played with any starting finger, without a position shift.  What’s more, the four single-position fingerings of any given scale bear noticeable similarities to each other, as they are composed of the same building blocks — it’s easy to learn all four fingering patterns at the same time!  In the remainder of this post I’ll elaborate on this point with the major scale as an example.

In working out 7-note scales in M3 tuning I’ve found it useful to think of these scales as stacked tetrachords.  For our purposes a tetrachord is any sequence of four notes spanning a fourth (alternatively you could think of a tetrachord as a sequence of three intervals that add up to a fourth); here we’ll only look at tetrachords that span a perfect fourth but in a followup post we’ll consider tetrachords that span an augmented fourth.  The major scale is nicely regular in that it can be seen as two stacked copies of the same whole-whole-semi tetrachord pattern.  Starting at the root, say C, and traversing a whole tone, another whole tone, and finally a semitone gives us C, D, E, F.   Starting at the fifth, G, and applying the whole-whole-semi pattern again gives us G, A, B, C.  Put them together and you have the entire major scale: C, D, E, F, G, A, B, C.  The diagram below shows how the whole-whole-semi tetrachord pattern is fingered in M3 tuning, with all possible starting fingers.  Notice that the first two examples have the same shape though they employ the fingers differently.

M3WholeWholeSemiFingeringsNow we’re ready to finger the major scale itself, not just in one way but in four ways that are interrelated.  Since the major scale consists of two copies of the whole-whole-semi tetrachord, each single-position fingering of the major scale can be understood as a pairing of two of the whole-whole-semi fingerings we saw above.  Let’s say we want to play the major scale starting with our index finger on the root.  First we’d play the whole-whole-semi tetrachord pattern starting from the index finger (I’ve colored this pattern dark blue in the diagrams).  Notice how the pattern ends with the second finger playing the fourth degree of the scale.  Next we need to skip a whole step up to the fifth degree of the scale, which falls under the pinky.  Keeping the hand position fixed and applying the whole-whole-semi fingering starting from the pinky (I’ve colored this patten cyan) completes the scale.  What if we wanted to start playing the major scale with the pinky instead of the index finger on the root?  The reasoning is similar: first play the whole-whole-semi tetrachord pattern starting from the pinky; then, since the pattern ends on the first finger, skip a whole step up to the third finger and play the tetrachord patten that starts from the third finger (green). The diagram below shows all four fingerings of the major scale as combinations of the whole-whole-semi tetrachord fingerings from the previous diagram:

M3MajorFingeringsOf course it’s possible to conceive of scale fingerings in terms of tetrachords in other tuning systems, but M3 is the only system I know where it works so well — where you can mix and match tetrachord fingerings as we’ve seen to build scales that stay entirely within four frets.  In a followup post we’ll take a look at other tetrachord patterns (like semi-whole-whole, whole-semi-whole, etc.) and how they can be used in M3 to finger the natural minor scale, the melodic minor scale, and pretty much any scale you could imagine.

Music, Tuning

Diesis III

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#:

C-E-Ab-C C-E-Gsharp-Bsharp

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:

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.