Music, Tuning

Diesis

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?

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