Music, Tuning

Diaschisma

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:

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