Music, Tuning


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.

Music, Tuning

Syntonic Comma II

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.

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Music, Tuning

Diesis II

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): 


Librarian or Farmer?

I’m reading Daniel Kahneman’s Thinking, Fast and Slow, and I wasn’t quite satisfied by the author’s presentation of the “Librarian or Farmer” question (pp. 6-7 of the 2013 paperback).

As you consider the next question, please assume that Steve was selected at random from a representative sample. An individual has been described by a neighbor as follows: “Steve is very shy and withdrawn, invariably helpful but with little interest in people or in the world of reality. A meek and tidy soul, he has a need for order and structure, and a passion for detail.” Is Steve more likely to be a librarian or a farmer?

According to Kahneman, most people assume Steve is a librarian. That answer is wrong, because it depends on occupational stereotypes while ignoring “equally relevant statistical considerations.” The question is supposed to illustrate the shallowness of our intuitions about probability. “Did it occur to you that there are more than 20 male farmers for each male librarian in the United States? Because there are so many more farmers, it is almost certain that more ‘meek and tidy’ souls will be found on tractors than at library information desks.”

But the question itself is unclear: we are not told what “representative sample” Steve was selected from. Kahneman’s discussion of the “correct” answer suggests that he’s thinking of a representative sample of males throughout the United States, so let’s assume that. Now, to arrive at Kahneman’s correct answer we should reason as follows: Yes, it might be true that the proportion of shy individuals among male librarians is larger than the proportion of shy individuals among male farmers, but when we look at the U.S. population as a whole, the proportion of shy librarians among all U.S. males is in fact smaller than the proportion of shy farmers among all U.S. males. Choosing at random from U.S. males, we’re more likely to get a shy farmer because the “base rate” of farmers is so much higher.

I would argue that this “correct” answer is wrong if we consider the other significant details in the question, even assuming a 20:1 ratio of male farmers to librarians.

We’ve been told much more about Steve than that he’s shy. Shyness as a character trait is perfectly consistent with farming: you can be a shy farmer without having your disposition fundamentally challenged. But we’ve been told that Steve “has little interest in the world of reality,” which is quite different from being shy: it’s an extreme attitude that’s in direct conflict with the demands of farming, dependent as that profession is on the reality of weather, seasons, crops, and markets. A farmer with little interest in the world of reality wouldn’t survive long in the job (at least not without an attitude adjustment) while a librarian might do just fine (as long as he has enough tolerance for reality to handle book requests).

The question also gives us information about Steve’s neighbor: we get to hear how the neighbor speaks, and if we’re trying to make the very best guess we can, this linguistic information should figure into our thinking. If Steve’s neighbor speaks like a farmer, this increases the chance that Steve is a farmer too, since farmers tend to live in rural areas where there isn’t a big mix of professions. If Steve’s neighbor has a more urban style of speech, this suggests that they both live in a town or city where Steve is more likely to work indoors. So let’s listen to Steve’s neighbor. A phrase such as “A meek and tidy soul, he has a need for order and structure, and a passion for detailsounds like it was uttered by a psychologist or an author writing a character synopsis. We might expect that a real farmer describing another farmer would speak in a more vernacular style, and with more concreteness; instead of citing an abstract “passion for detail,” he might talk about how the other farmer cares for each seed he plants. I’m not a farmer myself, and I don’t want to be too stereotypical in my assumptions about dialect, but I’m pretty sure that what’s quoted there isn’t farm talk.

Let’s say that Property X means “being someone with little interest in the world of reality, whose neighbor would describe you with the exact words ‘a meek and tidy soul’”. Now even if the proportion of shy librarians among all U.S. males is smaller than the proportion of shy farmers among all U.S. males, I’m willing to bet that the proportion of Property X librarians among all U.S. males is larger than the proportion of Property X farmers among all U.S. males.

You could say I’m reading too much into the question, or that my reasoning is more elaborate than what happens in most people’s minds when they jump to the conclusion that Steve is a librarian. I would argue that my reasoning is not unusual at all. What might be unusual is the act of voicing it in such detail, but I think the sorts of inferences I’ve described here go on in people’s minds without their even being aware of it. Kahneman treats the Librarian or Farmer question as an example of the shallowness of our statistical intuitions because it shows that most people don’t consider base rates, and yet he ignores the possibility that our intuitions might depend on other important considerations which trump the base rate discrepancy: in this case, our intuitions might rely on a distinction between temperament (shyness) and an actively held attitude (disinterest in reality), an understanding of the link between attitude and profession (it’s hard to farm if you don’t care about reality), an understanding of speaking style and its implications (farmers tend to speak differently from city dwellers), and an understanding of neighbor relationships and geography (farmers tend to live near other farmers). I don’t disagree with Kahneman’s overall point, which is that people’s statistical intuitions are often shallow, but this may be one case where the shallowness is less in the intuition than in how the psychologist analyzes it.

Music, Tuning


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.

Music, Tuning

Pythagorean Comma II

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.

Equal Temperament:

Pure Intervals:


Music, Tuning

Note Derivations In Just Intonation

This post is intended as a resource for exploring the foundations of just intonation.

For musicians working in equal temperament it’s common to think of notes as being “built” from semitones.  What is sol?  It’s seven semitones above do.  You might also say it’s a “perfect fifth” above do, but what’s a perfect fifth?  Seven semitones.

In just intonation you don’t think of notes as being composed of some small indivisible unit.  The semitone is not primary.  Instead, notes are derived from each other, by applying certain “special” intervals in sequence: pure perfect fifths, pure major thirds, and their reciprocals, if you’re working in the “5-limit” variety of just intonation.  A note can be thought of as having intonational “relatives” with which it bears an affinity: a certain inflection of la might arise as the perfect fifth above a certain inflection of re, while another inflection of la might arise as the pure major third above fa.  (These relationships are acoustically significant, because they indicate that each la is an early member of the harmonic series of its parent: our first la is the 3rd harmonic of re while our second la is the 5th harmonic of fa.)  Our two la‘s both sound like a major sixth above do, and they are very close in pitch, but they have subtly different affects, as well as different lineages or derivations.

The genealogy of notes in just intonation is often summarized — with remarkable concision — in lattice diagrams.  Looking at the 5-limit lattice one might say that 27/16 la is three rungs right of do, while the 5/3 la is one rung left and one rung up (assuming an orientation where moving right corresponds to traversing a perfect fifth and moving up corresponds to traversing a major third).

One can spend a lot of time reading materials on just intonation and studying paths in the lattice without ever hearing these paths rendered in sound.  But every path in the lattice is more than a mathematical construct — it’s something you can listen to!  The goal of this post is to illustrate the lattice paths for some of the significant pitches in 5-limit just intonation through short musical examples.  Each example begins on a home note, C, and is designed so that you only hear pure fifths and thirds until you reach the target pitch, which is then sustained over C in the bass, and finally descends to C.  Where there are multiple paths to a note, we traverse fifths first, then thirds.  Notes in the sequence are held long enough so that you have time to judge the quality of each interval traversed.  Each example could be thought of a kind of auditory “proof.” The claim we prove is that a certain pitch can be reached through a sequence of pure, beat-free major thirds and fifths — all the pitches in the 5-limit lattice satisfy this claim, and pitches outside the lattice don’t.

If you do hear beating in the thirds or fifths it’s likely because some of the organ samples that I’ve used in creating these clips have a faint vibrato.  If you’d like to hear an alternate rendering using a different instrument, just let me know.

Also note that the examples assume octave equivalence: there are cases where the target pitch falls within the octave above the bass C, and other cases where the target pitch falls in an upper octave.  So, where I’ve indicated an interval like 3/2, the actual interval you hear against the bass might be 6/2. In addition, there are a few cases where a chain of fifths and thirds is transposed up an octave, midway through, to keep it in a manageable range.  However, octave transposition is never used in such a way that a descending fifth would be heard as an ascending fourth, or a descending third as an ascending sixth.  To keep things simple from the listener’s perspective, I’ve made sure only thirds and fifths are heard until the target pitch arrives.

Here are the scores and audio files for three groups of 5-limit ratios.  I’ve split them into these particular groups simply to make notation easier (I wanted to avoid having multiple tunings of the same letter name in one score).

Group 1:

Includes 2/1, 3/2, 9/8, 5/4, 15/8, 45/32, 8/5, 6/5, 4/3, 16/9, 16/15, 5/3.

(View Score)



Group 2:

Includes 27/16, 135/128, 27/20, 9/5, 36/25, 25/16, 75/64.

(View Score)



Group 3:

Includes 25/24, 10/9, 64/45, 32/27, 128/81, 256/135.

(View score)



For reference, here is a lattice diagram including all the notes in the examples (and more).  The bold numbers are cents, rounded to the nearest integer; below each cent value is the corresponding ratio.  Stepping right corresponds to traversing a perfect fifth (3/2) and stepping up corresponds to traversing a major third (5/4).  To get your bearings when reading cents notation, simply remember that a tempered semitone is 100 cents; other tempered pitches thus fall at multiples of 100.  So if you see 996 cents, for example, you know that it’s 4 cents flat of 10 tempered semitones, which is to say it’s slightly flat of a tempered minor seventh.

5-Limit Lattice including Cents and Ratios

Music, Tuning


This post offers an audio example of the diaschisma in musical tuning.

If we start at C and travel four perfect fifths upwards, we arrive at E.  To return to C, we could travel down one major third, or we could travel up two major thirds.  In equal temperament, either path will take us back to an octave equivalent of our initial C, but if we use just intervals — “3/2” fifths and “5/4” major thirds — we’ll land somewhere close, but not exactly on that C.  The first path, C->G->D->A->E->C’, exposes the syntonic comma, which I described in my previous post.  The second path, C->G->D->A->E->G#->B#, exposes the diaschisma.  In the case of the syntonic comma we land a little bit sharp of our starting point, because the final descending major third, when justly tuned, doesn’t span the full distance between our sharp Pythagorean E and the C below.  In the case of the diaschisma we land a little bit flat of our starting point, because even though the sequence of just fifths takes us to a sharp E, those two just major thirds at at the end are too narrow to get us all the way to an octave above our initial C.

My diaschisma example (view score) is set up so that the starting C and the “derived” B# are contrasted and then played simultaneously at the end.  In the equal-tempered version, they match; in the version using just intervals, they clash.  As you listen to the examples, also pay attention to the sound of the thirds before the end; the harsh tempered thirds guide us to a pure octave while the sweet just thirds belie the discord that follows!

Diaschisma — Equal Temperament:

Diaschisma — Pure Intervals:

As I experiment with these examples I keep returning to a synthesized timbre called Bright Pad (used above) as it makes the tuning discrepancies obvious to my ear, but I also made clips using a sampled bassoon:

Diaschisma — Equal Temperament:

Diaschisma — Pure Intervals:

Music, Tuning

Syntonic Comma

This post provides an illustration of the syntonic comma in musical tuning.

The audio example (view score) starts at C and ascends in a series of four perfect fifths, touching G, then D, then A, then E.  From this high E, the example descends by a major third, hitting an instance of C two octaves above the starting note.  This high C is restated, and then played together with the original C in the bass.

In equal temperament, the high C that we reach at the end of the progression forms a pure octave with the starting C — you’ll notice that the first clip ends without a trace of dissonance.

However, if we use pure fifths and thirds as opposed to tempered intervals, the progression doesn’t bring us to a pitch that matches our starting point.  At the end of the second clip, you can hear how the starting and ending “versions” of C create a mistuned compound octave that flutters or beats slighlty.  In some timbres, particularly where the sound has a bit of vibrato, the mistuning at the end is imperceptible, but I tried to use a plain sound that doesn’t mask the clash.

Here’s a way of thinking about the progression. A pure perfect fifth is slightly wider than a tempered fifth; obviously, traversing a wider fifth will take us to pitch that’s sharper than the one we would reach via a narrower tempered fifth.  As we traverse four of these pure fifths we end up at an E that’s sharper than a tempered E, which is itself sharper than a justly tuned E.  When we then descend by a just major third from our sharp “Pythagorean” E, we reach a C’ that’s in beautiful agreement with the E; but since that E is so sharp with respect to the original C, descending by a just major third cannot bring us all the way back home.  We land at a C’ that’s 21.51 cents higher than the starting point.

Syntonic Comma — Equal Temperament:

Syntonic Comma — Pure Intervals:

Music, Tuning


This post is a sequel to my previous post on the Pythagorean Comma.  Here I’d like to illustrate two discrepancies sometimes called the lesser diesis and the greater diesis, though it appears that several other things have been called “diesis” throughout history, so if someone walks up to you and says “diesis” you should probably ask what they mean and not immediately assume they’re talking about the contents of this blog post.

Our example of the lesser diesis (view score) starts on C and traverses three ascending major thirds.  We could spell these thirds as C-E, E-G#, and G#-B#.  The example ends with the B# that we’ve reached, played against the C that we started on.  In equal temperament, B# is the same pitch as C, and when they’re played together we hear an octave.  However, if we ascend using pure 5/4 major thirds, we reach a B# that’s significantly flat of C, and when this B# and our original C are then played together, the dissonance is jarring, especially in contrast to the pure thirds that led us there.

Ascending Major Thirds — Equal Temperament:

Ascending Major Thirds — “pure” 5/4 ratios:

Our example of the greater diesis (view score) starts on A and traverses four ascending minor thirds, which we’ll spell as A-C, C-Eb, Eb-Gb, and Gb-Bbb.  The example ends with the Bbb that we’ve reached, played against the A that we started on.  In equal temperament, Bbb is the same pitch as A, and we hear an octave.  However, if we ascend using pure 6/5 minor thirds, we reach a Bbb that’s quite sharp of A, and the notes clash.

Ascending Minor Thirds — Equal Temperament:

Ascending Minor Thirds — “pure” 6/5 ratios:

In summary, a chain of three pure major thirds falls short of an octave (by 41.06 cents, to be precise), and a chain of four pure minor thirds exceeds an octave by an even greater amount (62.57 cents), hence the qualifiers greater and lesser.

I’ve been thinking about why commas like these are so intriguing.  What comes to mind is an analogy between music and a logical argument.  When I hear notes connected by pure consonances like 5/4 and 6/5, it feels as though I’m being carried from one step of an “argument” to the next, via an impeccable line of reasoning.  At the end of a chain of pristine consonances, I’m inclined to expect that the landing point would “make sense” or fit nicely together with the the previous steps, but the note we reach after three iterations of 5/4 seems very much at odds with the place we started.  In some sense, it’s like listening to an argument that seems flawless and indisputable, only to find that that this perfect argument has whisked you along to a conclusion that seems… absurd?  Or that seems to contradict the premises you started with?  How can it be that such airtight logic leads to such apparent inconsistency?