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“Untempered” vs. “Just Intonation”

Even though I love just intonation, I have a couple of problems with the term itself. One is grammatical. It’s a noun, and sometimes I want an adjective, as in “the just intonation version compared with the equal tempered version.” Kind of awkward. How else would you say this? “Justly intonated”? “The version in just…

Mozart on the Lattice

In one of my favorite passages in Harmonic Experience (p. 104-105), W.A. Mathieu points out that by the time Mozart came around, equal temperament was well enough established that a D# and an Eb could be thought of as the same note. So when Mozart wanted his melody to go back and forth between an Eb and…

Untempered Vs. Tempered

I’ve been listening to yesterday’s chord progression showing off the b7. I think it offers an excellent opportunity to hear the difference between equal temperament and just intonation. Equal temperament works by implying or evoking a note rather than playing it exactly. There are dozens of singable notes per octave; ET represents them all with just…

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Names

Musical nomenclature has been cobbled together over the centuries like a medieval city. Different systems leave their imprint in convention, later developments try to be compatible with accepted names, and the whole thing ends up confusing and contradictory. Take enharmonic equivalents, for example. G# and Ab are the same note on the piano, the black…

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Mixed Messages

The harmonic lattice can be divided into four quadrants. Northeast is pure overtonal energy. All these notes are reached by multiplication alone. Powers of 3 and 5 are in the numerators, and the denominators are all powers of 2. Southwest is pure reciprocal energy. You get there by dividing. All the 3’s and 5’s are…

The Tonic Major Chord

The tonic is the center of the lattice. A drone note on the tonic establishes the center of that particular musical universe. Adding a major third and a perfect fifth (5/4 and 3/2) further reinforces the center and starts to carve out some territory on the map. This is the tonic major chord: In my…

The Major Third

Multiplying a note by 2 creates an octave, and multiplying it by 3 creates a perfect fifth. Multiplying by 5 gives yet another new note, the pure major third.5-1 5/1 is over two octaves above the original note, so you have to reduce it twice (divide by 4) to get it down into the same octave.5-4…

Between the Keys

I grew up thinking that music was made with a particular set of twelve notes, the ones on the piano keyboard. I had a vague sense that there were other scales in the world, but I thought of them as “more primitive” or perhaps subsets of the 12-tone scale, like that pseudo-Asian music you make…

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Untempered Music

For almost two years now, I’ve been exploring the nature of music almost full-time. I threw out everything I knew, started with the most basic thing I could think of, the number 1, the origin of the musical universe, and worked my way from there. My explorations quickly led to the underpinnings of musical harmony,…