The most challenging, time-consuming, and fascinating part of making my tanpura album, Uncommon Drones, was getting the instrument in tune for each track. I’d like to share some observations about the tuning process in a way that might be helpful to anyone who wants to know the “backstage story” of the album, as well as to those readers who play the tanpura or have an interest in the broader topic of just intonation.
In my previous post on the so called “overtone scale,” I questioned whether any scale defined in the context of 12-tone equal temperament can be said to mimic the harmonic series. I provided an audio example that shows the way accurately tuned harmonic partials seem to fuse into a single tone, whereas when those partials are altered to match equal temperament, the composite sound is rough and unstable. My point was that the special perceptual properties of the harmonic series depend on accurate intonation, an appropriate pattern of amplitude decay, and appropriate registration; we should be cautious about assuming that any scale constructed in equal temperament will somehow inherit the special properties of the harmonic series by virtue of an incomplete resemblance to it. My aim in the post was not to call into question the musical worth of any particular scale (and certainly not to address George Russell’s Lydian Chromatic Concept as a whole), but only to point out that the Lydian Dominant scale shouldn’t be called an “overtone scale” when played on the piano or any tempered instrument: that name is misleading.
In this post I’d like to share another simple audio example that might help readers form their own judgments on the matter. If the scale 1, 2, 3, #4, 5, 6, b7 really does invoke the harmonic series in our minds, even when we hear it on a tempered instrument, how would that scale sound if its pitches were brought into exact alignment with the harmonic series? Would that tuning bring us closer to the essence of the scale, or would it conflict with what we want to hear?
In the first audio example below, you will hear the Lydian Dominant scale rendered on an equal-tempered organ. The ascending and descending scale is followed by a short tune that I wrote in the scale. In the second audio example, you’ll hear the Lydian Dominant scale tuned so its pitches match the harmonic series, followed again by the example tune. The just intonation ratios used in the second clip are: 1/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4. You will hear that the second, third, and fifth degrees of the scale sound similar to what you hear in equal-temperament, while the flat seven, the sharp four, and the natural six are quite different. (The flat seven is tuned to the seventh partial, the sharp four is tuned to the eleventh partial, and the natural six is tuned to the thirteenth partial.) Which tuning do you prefer?
Clip 1 — Lydian Dominant Scale in Equal Temperament:
Clip 2 — Lydian Dominant Scale in Harmonic Series Tuning:
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.
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.
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 wouldnot 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.
This post offers an illustration of a discrepancy in musical tuning called the kleisma.
Technically, the kleisma is the amount by which a stack of six justly tuned minor thirds falls short of an octave plus a justly tuned perfect fifth. Wait, what?
Recall that in equal temperament a minor third is tuned as an exact quarter of an octave; as we stack minor thirds, we cycle through the notes of a fully diminished seventh chord, returning to the same pitch-class every four steps: C, Eb, Gb, Bbb, C, Eb, Gb, Bbb, and so on. This sequence never lands on the natural fifth degree above our starting note: a chain of tempered minor thirds starting on C will never arrive at G-natural. That’s why the kleisma is so interesting — it shows how a subtle tuning change alters the musical “logic” that we take as basic in equal temperament. If we tune our minor thirds wide, according to the 6/5 frequency ratio of just intonation, we can get from C to the vicinity of G-natural in six hops.
The just minor third is roughly 15.64 cents wide of a tempered minor third, so after traversing six just minor thirds we find ourselves roughly 94 cents (almost a semitone) sharp of a tempered Gb — which is to say we’re just a hair below G-natural, close enough for government work. What makes this startling is not only that it contradicts the equal-tempered musician’s expectation that minor thirds form a nice four-step cycle, but that these widened thirds sound so good. We’re not just arbitrarily retuning our minor thirds to 6/5 to wreak havoc on familiar musical logic, we’re actually tuning them the way the ear wants to hear them, or at least in a way that minimizes acoustic roughness. Of course, doing this changes not just how the thirds sound individually but how they behave when stacked: you might think you’re hearing a familiar arpeggiated diminished seventh chord, but it doesn’t repeat!
The sound clips contain a sequence of six ascending minor thirds, starting at C. At the end of the sequence you’ll hear the original C in the bass return while the top note is sustained. (In equal temperament you’ll hear the two voices form a compound tritone whereas in just tuning you’ll hear a wobbly compound fifth.) After a short pause, the whole sequence is repeated, this time with a C pedal sustained throughout. The notation below represents the tempered version (in the just version, the higher pitches are almost a semitone sharp of what’s written):
Kleisma — equal temperament:
Kleisma — pure intervals:
(The ratios used in the latter clip are as follows: C=1/1, Eb=6/5, Gb=36/25, Bbb=216/125, C’=648/625, Eb’=3888/3125, Gb’=23328/15625.)
It’s worth comparing this illustration with my example of the greater diesis, a stack of four minor thirds that’s sharp of an octave. In the diesis example, the destination note is heard as a sharp variant of the starting note; in the present kleisma example, the destination note is another scale degree altogether. And here, the aggregated difference between just and tempered tunings is so great that we land at a different degree in each case. Just to be clear though, that huge difference between landing points (Gb vs. G-natural) is not the kleisma. The kleisma is the small amount by which our derived G-natural falls short of a justly tuned twelfth above C — that gap is only about 8.1 cents. The example here doesn’t explicitly contrast the derived G-natural with a “correct” G-natural, but you can tell the derived G-natural is a bit flat from the way it wobbles against the C.
This post is a followup to my earlier example of the syntonic comma in musical tuning. In my original post, the goal was to arrive at the “comma sibling” of our starting note, C, by traversing a series of four ascending fifths followed by a descending major third. We saw how this sequence, when tuned using pure 3/2 fifths and 5/4 major thirds, leads us to a C’ that’s sharp of an octave above our starting note. In this post we’ll explore the same tuning discrepancy, but instead of considering two versions of C we’ll consider two versions of the sixth scale degree, A, differing by the same amount as our C’s.
This post is a followup to my previous post on the diesis in musical tuning. My earlier example of the so-called “greater diesis” involved a sequence of four ascending minor thirds. In equal temperament, four minor thirds add up to an octave, but when those thirds are tuned to a pure 6/5 ratio, the sequence lands quite sharp of an octave. In this post we’ll explore the same discrepancy, but we won’t traverse any minor thirds directly. To ascend by a minor third, we’ll first ascend by a perfect fifth and then descend by a major third. This way, we’ll hear perfect fifths and major thirds throughout the sequence, until the end when the destination pitch, Bbb, is contrasted against the starting A. Personally I find this example more shocking than the previous version, because the major thirds and perfect fifths are more “persuasive” to my ear than minor thirds. Although I sense that something funny might be happening as the sequence progresses, I’m convinced of the “rightness” of the destination pitch when it arrives, and when I then hear the original A, I just can’t believe that’s where the sequence started a little while earlier. The sound clips below use a sampled bassoon. (view score)
Ascending Fifths, Descending Major Thirds — Equal Temperament (bassoon):
Ascending Fifths, Descending Major Thirds — Pure Intervals (bassoon):
Here is an alternate set of clips using a sampled organ:
Ascending Fifths, Descending Major Thirds — Equal Temperament (organ):
Ascending Fifths, Descending Major Thirds — Pure Intervals (organ):
This post offers an audio example of the schisma in musical tuning. Since the schisma is such a tiny interval, it makes for a less dramatic illustration than the other commas I’ve written about. In my examples for the pythagorean comma, the greater and lesser dieses, the syntonic comma, and the diaschisma, we begin at a home note and follow a sequence of intervals that doesn’t get us back home; we arrive somewhere that’s quite audibly sharp or flat of our starting point, so much so that we can’t really accept it as the same note. In some of the examples, the pitch drifts incrementally sharper or flatter as the sequence progresses; in other examples, the pitch might drift sharp first, and then drift flat by a much greater amount, leaving us noticeably off target. The present example is somewhat different in that the upward and downward pitch drifts almost cancel out. Specifically, we start at C and traverse eight ascending pure fifths, each of which adds roughly 2 cents to our upward drift. At the end of that sequence we arrive at a G# that’s roughly 16 cents sharp of its tempered counterpart. Instead of continuing to ascend in fifths as we would if we were evoking the pythagorean comma, we cut the sequence short and skip up to B# via a pure major third. The pure major third is roughly 14 cents narrower than a tempered third, so it leaves us only 2 cents sharp of a tempered B#. (Technically, the schisma is around 1.95 cents). At the end of the example we hear this B# played above our initial C, and it sounds like a pretty decent octave, though there’s a slow and subtle beating effect.
Try comparing the equal-tempered and pure versions a few times, concentrating on the G#-B# major third near the end, and the following C-B#. In the equal-tempered version you may notice that the major third sounds harsher and busier, while the C-B# that follows sounds like a pristine octave (no matter that it’s notated as an augmented seventh). In the version using pure intervals, the major third should sound more restful, but the C-B# that follows is not quite pristine. It would be quite passable in many contexts, and the subtle mistuning probably wouldn’t be noticeable at all if vibrato had been in use throughout. Nevertheless, the discrepancy is there and it’s called the schisma!
Schisma — Equal Temperament:
Schisma — Pure Intervals:
Tuning: C — 1/1; G — 3/2; D — 9/8; A — 27/16; E — 81/64; B — 243/128; F# — 729/512; C# — 2187/2048; G# — 6561/4096; B# — 32805/16384.
In my previous post on the Pythagorean Comma, I wanted to provide slow-moving illustrations that would give the listener enough time to judge each interval traversed. Here I’d like to provide a simpler and faster-moving example. You’ll hear a rapid sequence of ascending fifths and descending octaves, starting on C and reaching B# an octave above, which is then contrasted with the initial C. These starting and ending notes will agree in equal temperament but will differ by a pythagorean comma if the fifths throughout the sequence are tuned justly.