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Posted by on Aug 16, 2017 in Background | 0 comments

The Book

Several readers have requested that I reorganize this blog so it’s easier to follow. Turns out the best way to read the entire thing is in forward chronological order — I had a book in mind the whole time I wrote it, and was careful to lay the groundwork for each concept before I presented it.

I’ve selected the relevant posts and listed them in order, on a new page. Here it is:

Book: Untempered Music

Enjoy! I think you will find your journey a lot smoother. I am interested in discussion — you may comment, or write me at the Contact page.

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Posted by on Sep 10, 2013 in Background, Equal Temperament, Just Intonation, Recordings, Septimal Harmony, The Lattice | 2 comments

Premature Nostalgia: Making Friends With Equal Temperament

I just recorded a new song, and it’s a perfect example of how equal temperament and just intonation can get along together.

Here’s the cut:


Reading this blog might give you the impression that I’m “against” equal temperament and “for” just intonation, or untempered music.

True, discovering untempered music has been like sailing to a new world. It’s delicious to have 20 or more notes to work with instead of 12, each with its own individual personality.

Equal temperament, however, is a fabulous invention. The lattice of fifths and thirds does not quite repeat. If you start with any note and go in any direction, you will soon encounter almost the same note again, but it will be off by a comma, a small interval, from the original note.

There are no two notes tuned exactly alike on the entire infinite lattice.

Equal temperament flattens out the lattice just a hair so it does repeat. Now there are only twelve notes to work with, and they imply the untempered ones in the ear. This innovation makes lots of things possible in music. Beethoven and Mozart could not exist without it.

It’s sometimes said that equal temperament and just intonation are incompatible with each other, because the notes will be out of tune. I say they can get along fine, you just have to show ’em who’s boss.

I submit for your consideration: Ray Charles.

Ray Charles’ piano is an equal tempered instrument. Ray Charles’ voice is most certainly not. He is singing the exact resonant notes, those blue notes, all tuned just like a gospel choir, which is what he grew up loving. Ray is boss. His voice establishes the tonality of the song. The backup singers, the horns and the standup bass all agree, this song is in the harmonic pocket, and it resonates.

That leaves the piano slightly out of tune, but who cares?

Notes that are slightly out of tune don’t necessarily sound bad — that’s the basis of the “chorus effect.” No two singers in a choir are exactly in tune with each other, and the resulting complexity is a huge part of the sound of the choir.

So if the tonality is established in the ear, maybe the equal tempered notes, which are only a bit off after all, will just enrich the sound a bit.

Listen to how “Hit the Road, Jack” starts off. First the piano intro. ET. Then the horns kick in, and they start to establish the soul of the tune. Then come the backup singers, that gospel choir. When Ray’s voice finally joins them, the pocket is waiting for him, and he proceeds to own it. The piano is now a background instrument.

I think that’s the secret. Put untempered instruments up front, and ET instruments more in the background. This asserts the untempered tonality in the ear.

Playing acoustic guitar and singing is a great playground for this. The acoustic guitar is, in its bones, an equally tempered instrument. Fretted instruments drove the adoption of ET in Europe, even before keyboards did. The voice is the archetypal untempered instrument. It can do anything.

If the guitar is boss, the song will be in equal temperament. If the voice is boss, you can establish any tonality you want (blues, Gypsy, whatever), and the guitar will tag along. You can retune it in the ear, just like Ray retunes his piano.

Here are some tricks for making friends with acoustic guitar (or any tempered instrument):

1) Sing solidly in tune, with the tonality coming from you, and not from the guitar. Don’t follow the guitar, lead it. The song is the melody, it is your voice, and you are accompanying that voice with guitar notes.

I like to think of the guitar as playing the grid lines on the map. The guitar notes are perfectly equally spaced, and are excellent reference points. The guitar tells me where I am. We completely agree on one note, the tonic. I use the tonic on the guitar as my true home base.

My voice is playing the actual territory.

2) Sing louder than the guitar.

This isn’t all that easy. The guitar is projecting outward, so it sounds louder to the audience than it does to me. The voice is right there in my head, so it sounds quieter to the audience than it does to me. If I sound balanced to myself, the audience will hear way more guitar than vocal. I hear this all the time at open mics.

I’ve found that in an acoustic setting, I have to sing twice as loud as my guitar (from my own point of view) for it to sound balanced out in front of me.

It gets easier with more JI instruments. In “Premature Nostalgia,” the fretless bass and backing vocals are all in strict just intonation. The guitar is truly a backing instrument, and the tonality of the song feels secure.

3) There is a third, more subtle thing you can do to bring the guitar closer to just intonation. The most clearly out-of-tune note on acoustic guitar is the major third. It’s already 14 cents sharp even when perfectly tuned, and the slightest unintentional string bend will take it into some really grating territory. Choose chord voicings that de-emphasize major thirds, and your guitar will sound a lot sweeter. I wrote an article illustrating this effect, here.

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Posted by on Aug 7, 2013 in Background | 0 comments

Why Can We Hear Harmony?



My friend Scott is an expert river rafter. I went down the American River with him once. We had about five crew. It was a big raft, and well-behaved, so he decided he’d leave the driving to us, sit in the bottom of the boat, and go through one of the rapids with his eyes closed.

He was grinning afterwards. He said that when he closed his eyes the temperature instantly got ten degrees cooler, and everything became three-dimensional.

I’ve thought about his experience a lot.

I think that Scott’s built-in orientation processor, which was already in high gear due to being in the rapids, suddenly lost an input channel (vision). The senses he had to work with were touch and hearing. They jumped up to do the job, and the model of the environment changed character according to what those senses had to offer.

We are constantly building a model of our environment in our minds, as a way of orienting ourselves in it. It’s a vital survival skill, and it takes in data from all the senses. Each sense contributes something different to the model.

I’m posting now from a restaurant, and as I walked here, I paid attention to what each sense does.

Vision gives me a really detailed view of the front surface of everything in my field of view. I have to move my eyes and head to get a wider picture. I can’t see through most things.

Touch gives me the surface texture and slope of the sidewalk, the feel of the breeze on my skin (which is a clue to my speed), the warmth of the sun which must be coming from the West at this time of day.

Balance gives me the direction of gravity, which is really important to know, in San Francisco.

I stop and close my eyes, and listen.

I have a lot less information now. A hazy picture starts to form, of what is all around me — above, below, and 360 degrees around. My ears are like dragonfly eyes, seeing in all directions at once.

The longer I listen, the more detail appears. Leaves are rustling above my head. A tree. There is a bird in it. Two birds. A couple walks by, in conversation, and I can tell exactly where they are as they come into earshot behind and above me, pass me on the right a few feet away, and fade away down the hill, below and in front of me. I can tell that the woman on the left is taller than the one on the right.

Somebody is hammering a couple of blocks away to my left, and there is sawing further away. A scooter goes by on a street behind me.

Hearing gives me three-dimensional X-ray vision, and more. I can hear far away, and through objects. I can even feel textures at a distance — the rustling of the leaves is almost tactile.

My hearing is building the big picture, the large-scale, full surround, 3D framework of the environmental model.

I think this is exactly why the organs of balance are located in the ears. If the reference frame of my world model is created by my ears, what happens when I turn my head? The balance input must be fed to the processor right along with the hearing data, or the model will spin and I will have vertigo.

Somewhere in us, we have an audio processor that is capable of doing the following, in a few milliseconds:

  • Start with two complex pressure waves, one for each ear. Each wave consists of hundreds or thousands of separate sound frequencies, made by many different objects, all mixed together into one single waveform.
  • Reverse-engineer the waves, sorting them back into individual sound sources.
  • Deduce from each sound the nature of the source.
  • Build a three-dimensional model of the environment within hearing range, including objects, their locations, movements and physical properties.

I don’t know how much computing power this represents, but I’m impressed. It’s like turning a smoothie back into peaches and strawberries. A quantum computer might be good at it — it could interpret the data in every possible way, all at once, and let the most likely, or lowest energy, scenario pop out.

Something in us, whatever is in that black box, takes in that huge mess of frequencies, sorts them into related bundles, and figures out what is making each bundle, in real time. I suggest that this processor is very, very good at recognizing and analyzing harmonic series.

Vibrating objects don’t just produce one frequency. They produce clusters of waves, with many frequencies that are small, whole number multiples of each other — 2x, 3x, 4x, 5x — the harmonic series. If the input sound contains frequencies of, say, 100, 200, 300 and 400 cycles per second, those waves almost certainly came from the same object.

And then, the harmonic content of a sound contains a huge amount of information about the object that made it. The overtones, and the way they change with time, are what tell us whether we’re hearing a barking dog, or a friend’s voice, or rustling leaves. They can give us information about what the source is made of, how big it is, its surface texture and motion.

This amazing harmonic processor must be built in at a deep level. I imagine any creature that hears can do it to some extent, and many better than we do. Imagine the hearing-model of a bat!


I think harmony feels good for the same reason sugar tastes good. Sugar is vital to our survival. The brain mostly runs on it. We usually have to work hard to get it, extracting small amounts from food, metabolizing it from starches. Our bodies are tuned to seek out sweetness, to find pleasure in it.

We humans have figured out how to concentrate and purify sugar, and the straight stuff tastes mighty good. It’s a focused shot to the pleasure center.

Same with harmony. Normally, our orientation processor gets a diet with a lot of roughage. There are dozens or hundreds of unrelated sound sources to sort out. The environment is full of noise, frequencies that aren’t in nice harmonic relationship to each other. The processor has to work hard to identify what’s making the noise.

When we are presented with music, we are getting a straight shot of undiluted harmonic information — a nutrient that is vital to our survival, in an easier-to-assimilate, more concentrated form than is found in nature.

Harmony is ear candy.

Next: A Mirror Quad

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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 18, 2013 in Background | 0 comments

Calendar Page

I’ve added a new page with places and times you can come hear and see this music live. If you’d like to be on my regular mailing list, please let me know through the contact page. I send out a few emails a month, mostly about upcoming shows, occasionally with other news.

I’m excited about the concerts with my friend Jody Mulgrew. And the SLO Down Pub showcase on January 24, with Loren Radis, and the Salty Suites, promises to be splendid. Hope to see you at a show!

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Posted by on Dec 14, 2012 in Background | 0 comments

Summary (So Far)

I messed around with electronics quite a bit as a kid. I’d put things together according to diagrams, and if they didn’t work, I’d change something and see what happened, and get a feeling for what was happening inside the black box.

When I started doing audio electronics in earnest, I found the oscilloscope. Here’s a cool handmade one by Andrew Smith.

An oscilloscope is a powerful tool, a visualizer, that lets you look right into the black box. It feels almost like cheating. All the energy that went into detective work can now be put to creative purposes. Electronics is much easier when you can directly see what’s happening in there.

I feel that hearing the notes in their untempered form, and learning their relationships on the lattice, has connected me with music in a similar way. Was blind, but now I see.

I’ve finished my first goal for this blog — to create and post the Flying Dream video, and post enough information for an interested person to understand it. I could go on for a long time about the uses of the lattice, and I imagine I will. It’s a fabulous tool.

There will be a slight pause in this blog as I write and rehearse for some upcoming shows. I intend to be back with some new subjects, especially an exploration of consonance and dissonance. If this work interests you, and you’d like to discuss it, you can reach me through the contact page.

Oh, and the matrix in which all of this is happening is love.

Next: Mozart on the Lattice

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