[sdiy] Shifting just intonation

Donald Tillman don at till.com
Tue Nov 7 02:34:15 CET 2023

On Nov 5, 2023, at 12:36 PM, Didier Leplae <didierleplae at yahoo.com> wrote:
> Yes, this is what I’m suggesting, although maybe it doesn’t automatically recognize what you’re playing and automatically adjust (although that’s one possibility). Another might be  a pedal board, that the player uses to switch intonations as they are playing. 
> I’m just exploring the idea, and brainstorming, so that’s why I don’t really have a specific idea of what or how I’d want to do this.
> Also, i didn’t mean to disparage just intonation by calling it a problem. I’m a big enthusiast of microtonal music. But it is a limitation, maybe that would have been a better way to describe it.

This stuff gets interesting...

There's certainly a mathematical draw to a temperament with pure intervals, but is it really a good thing?  If you play two notes at once, does it end up sounding like a single note with a more complex set of harmonics?  And if you do play in perfect intervals, what's the first thing you're going to do with the sound?  That's right, add vibrato and chorus.

And what about string instruments, where the harmonics are naturally a little sharp anyway?

In a Hammond organ, the tonewheel signals leak a little, so the tonewheels are positioned in circle of fifths, and the leakage is harmonious.  But those pitches are equal tempered, and beat with harmonics of the fundamental, especially if you crank it a little.  And that's part of the animation of the sound we love so much.

But okay, let's say you build this contraption, and play... "Louie Louie".  

What key?  Oh right, it makes a difference.   The Kingsmen version is in A, but the original Richard Berry version is in G.

Whatever, let's say A.

Now, "Louie Louie" is in a mixolydian mode; the seventh is flatted.  What should the pitch of that G be relative to A?

The equal temperament folks say, 2^(10/12) => 1.781797.  (Followed by, "Now let's tap the keg and get this party started... doot-doot-doot, doot-doot, doot-doot-doot, doot-doot...")

The harmonic temperament folks would say this is an opportunity to use the 7th harmonic, so 7/4, or 1.75.

The Pythagorian temperament folks would say that the flatted seventh is two fourths, so that's 4/3 * 4/3 => 1.777778.

And others will say that the flatted 7th is really there because it is the minor 3rd of the 5th chord, so that's 3/2 * 6/5 => 1.8.

So which is best for "The Well-Tempered Louie Louie"?  I dunno, we'd have to try them them all.

(Meanwhile, the equal-temperament guy is making out with a cheerleader.)

(I'm just having fun here.  I'm not actually discounting the beauty of certain pieces with certain temperaments.  But it does get complicated, and the musical context is important.  And equal-temperament does work amazingly well.)

  -- Don
Donald Tillman, Palo Alto, California

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