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Posted by on Jan 29, 2013 in Background, Just Intonation | 0 comments

“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 intonation”? I haven’t found a construction that satisfies me.

The other reason is cultural. If you search “just intonation,” and start reading, you will get the distinct impression that just intonation is something avant-garde, esoteric, on the fringes. It’s as though equal temperament is the basic system of music, and just intonation is a modification of it. The word “microtonal” has similar connotations.

In fact, equal temperament is the newcomer, a development of a few hundred years ago that facilitated the flowering of a particular kind of music in Europe, and has spread, I think, because it makes so many things so much easier.

Equal temperament is built upon just intonation, not the other way around. If I put my music in the “just intonation” or “microtonal” category, I’m in great company — Harry Partch, Ben Johnston, Kyle Gann. These composers are exploring the edges of just intonation, picking up the trails that were abandoned when such music as Ars Nova was superseded by the slow growth to dominance of tempered scales. Ars Nova is amazing music, terribly neglected now. I like it better than either earlier or later European music — some of it sounds like jazz or bluegrass. Check out this exquisite piece by the group Ensemble PAN, performing some of the last of such music, from early 15th century Cyprus.

I’m not a classical composer, I’m a folk-pop singer-songwriter. I’m interested in such things as modulation, and exploring the edges (especially the world of the prime number 7). But my interest in JI comes from wanting to play music that is more accessible by virtue of being in tune, and thus having a more direct route to the heart and soul. My interest is in communication, and in musical joy. Untempered music simply speaks more directly to my heart.

Think of Ladysmith Black Mambazo on Paul Simon’s Graceland album. I get goosebumps even listening on these tiny computer speakers. Untempered music is not avant-garde at all. It’s the ancient miracle of resonance and joy that happens when we hear in-tune harmony.

Of course I still need a noun, and I’ll continue to use “just intonation” when it’s the word that works. But I have my adjective. I’m calling my music “untempered music.”

Next: The Untempered Major Scale

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Posted by on Jan 6, 2013 in The Lattice | 0 comments

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 an E, he wrote (spelled) the notes as D# and E, probably to make the music easier to read. In just intonation, D# is an entirely different note, and makes less harmonic sense than Eb.

I got curious as to how the two spellings would sound and look on the lattice. I also made one in equal temperament. They are quite different, and I think it’s fun to compare them. These are on YouTube with the following descriptions:


A passage from Sonata in F, K. 332. This is the spelling suggested by W.A. Mathieu, Harmonic Experience, p. 105. The melody moves between the major and minor third. Listen to the major/minor interplay in just intonation.


A passage from Sonata in F, K. 332. This is how Mozart spelled it, with a #2 instead of a b3. He likely did this to make it easier to read. This video shows how that spelling would look, and sound, in just intonation, if taken literally. The melody moves by a diatonic semitone (112 cents) rather than by a chromatic semitone (70 cents), and the effect is quite different.


A passage from Sonata in F, K. 332, in equal temperament. In ET, the b3 and #2 are both tuned to a compromise pitch, in between the two. I think the ear turns it into a b3 here, but the effect is off, the note is quite flat. The whole passage feels different than it does in either of the JI versions.

How beautiful Mozart’s music is! I have watched the first video, in JI with the major/minor pair, many times. Try following just the orange one a couple of times, then just the yellow one.

Next: Melodic Space, Harmonic Space

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Posted by on Dec 5, 2012 in Just Intonation | 0 comments

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 twelve tones.

Some ET notes are close to their just counterparts; the 4 and 5 are close enough to be essentially right on. The major third is not so great. It’s 0.8% sharp, enough to change the feeling it produces.

The ET b7 is even further off, a full 1% flat of the untempered note. For me, this is enough to change its flavor entirely, and dilute its resonance to the point where it’s just not the same note. I would contend that the real experience of the b7 is not actually available in equal temperament.

Here it is again:

And in ET:

To me, the real b7 sounds triumphant, like its arms are outstretched to the sky after a great victory.

The ET one sounds very different. It’s not unpleasant, but it sure is different. It it a little sad? The leaping dance is gone. The b3 is flat too. Poor minor, no wonder she’s sad! A mortal has seized the hem of her garment and made her earthbound, in order to put her in his power and make her a little better behaved.

Now go back and listen to the JI version. My experience is that I hear it a little sharp for a second, and then it settles in and wow. This is all subjective; you may hear entirely different things. But this example makes it pretty clear, I think, that JI and ET do not sound the same.

So here we have a note, with a distinct (and unique) personality, that produces a physiological sensation that just isn’t quite available in equal temperament. There are a lot more of these to come, with strange and beautiful colors. Really getting into JI and the lattice is like getting the 64-color Crayola box for Christmas. Orange-yellow and yellow-orange, what riches!

One of my favorite phrases in any song comes from the great Greg Brown. In Eugene, from The Evening Call, he says,

The Northwest is good, once you get off I-5 and wander up and down the Willamette dammit, on the back back roads. I know a few people who’d let me park in their drive, plug in for a night or two, stay up late, and talk about these crazy times — the blandification of our whole situation. And then back to the woods. A dog is bound to find me sooner or later. Sometimes you gotta not look too hard — just let the dog find you.

The blandification of our whole situation. Nice one, Mr. Brown. I recommend going back and forth between the last two vids a few times. Deblandification!

By the way, The Evening Call is packed with great lyrics and music. Top notch.

Next: The Minor Second

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Posted by on Nov 22, 2012 in The Lattice, The Notes | 0 comments


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 key between G and A. So why do you sometimes call that note by one name, and sometimes by the other? The answer actually leads to some deep realizations about music, and it comes back to just intonation. In untempered or just music, G# and Ab are not the same note, and which one you choose becomes important. It’s important in ET too — the music establishes a context, and the ear figures out which note it’s supposed to be. But if you grew up with ET, and have no idea that there used to be two different notes there, the names can be confusing. How do you imply one note or the other? Which one is right in a given situation? Why bother? It’s a huge part of writing chord progressions that make sense, but ET by itself isn’t going to tell you what to do. You have to dig deeper for that.

I’ve slowly evolved a personal system I’m very happy with. It’s based on the lattice.

The great advantage of this approach is that it’s entirely unambiguous. Every note on the infinite lattice has a unique name, and that name tells you exactly what its pitch is, and where it is on the map.

The seven notes I’ve covered so far form the core of the system. I’ve dropped all the word names and just use numbers:

1 — the tonic, 1/1

2 — the major second, 9/8

3 — the major third, 5/4

4 — the perfect fourth, 4/3

5 — the perfect fifth, 3/2

6 — the major sixth, 5/3

7 — the major seventh, 15/8

The rest of the notes are named by adding accidentals to modify the pitches. I’ll quantify these later, and explain how they work, but approximately they are:

b — flat by about 2/3 of an equal-tempered semitone

# — sharp by about 2/3 of a semitone

— flat by about 1/5 of a semitone

+ — sharp by 1/5 semitone

7 — flat by 1/2 semitone.

The basic notes occupy the center of the lattice. These seven notes form the major scale.

Next: The Major Scale

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Posted by on Nov 21, 2012 in The Lattice, The Notes | 2 comments

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 on the bottom of the ratio (or fraction), and the powers of 2 are on the top.

Northwest is mixed reciprocal and overtonal energy. You multiply by fives and divide by threes.

Southeast is also mixed. Multiply by threes and divide by fives.

The major sixth is in the northwest quadrant. First divide the tonic by 3, which gives you a perfect fourth. Then multiply that by 5, and you get the major sixth. Its ratio is 5/3. You can do this in any order, of course — multiplying by 5 gets you to the major third, and dividing by 3 brings you to the sixth again.

A different flavor yet! What do you hear? How does it make you feel?

Musical sensation can be related to other senses and emotions, but really these feelings have their own quality. It’s like a new sense. I had a friend once who had never tasted a peach. She didn’t like the feel of the skin, and she hadn’t dared bite into one. I was saddened, of course, but how to describe the taste of a ripe peach to someone who’s never tasted one? You can draw comparisons forever, but nothing will prepare them for the reality.

That was a long time ago. I hope that some time later, she found herself in the perfect place and went for it. There should be a First Peach ceremony. Yum!

The individual qualities of these notes are easier for me to hear in just intonation than in equal temperament. They are purer flavors, a more direct experience. Later, I’ll present some notes that are very close together in pitch, yet feel different. In just intonation, there are different notes for peach, nectarine and apricot. In equal temperament, a single note, close in pitch, will represent all three. Once you’ve tasted the actual fruits, equal temperament works better than it did before. You know what it is you’re supposed to be tasting.

OK, I’m out on a quivering limb of analogy now, and way ahead of my story. We have enough notes now to start playing with them.

Next: Names

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