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.

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