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.

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