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Posted by on Feb 10, 2013 in Just Intonation, The Notes | 0 comments

An Easy Experiment To Try

It would be natural to read these posts and wonder why I’m so passionate about intonation, and why I’m going to so much trouble to explore it in this blog and in real life. After all, we’re talking about tiny differences in tuning here, why be so picky when it’s the heart that counts?

It’s true, the tuning differences are small, and hard to hear. Thing is, it’s not actually about the pitch. It’s about the way it feels, and in that realm, the difference is not subtle at all. It is profound, and once you hear (no, feel) it, I think you may be hooked, or at least understand more of why I’m so interested in this subject. I think it opens the door to music that truly moves both the performer and the listener, a recipe for audio joy. You bet it’s about the heart. This is not just an intellectual pursuit.

Here is an experiment you can do, to feel that difference in yourself. It uses a chord, and a melody, that you probably already know.

The open G chord is one of the most common chords in guitar music. It looks like this:

Open G

The notes, from left to right, are: G–B–D–G–B–G. If G is the tonic, these notes are the 1, 3, 5, 1, 3 and 1.

One of the best-known melodies in the world is Frère Jacques, or in English, Are You Sleeping (Brother John), a round that is hundreds of years old. It could be harmonized in several ways, but the melody is such that it sounds fine sung over just the tonic chord, over and over again.

Here’s the experiment. First, tune your guitar carefully. A tuner is best. When the open strings are in tune, double check the notes of the open G chord. I think Jody showed me this — it often sounds better if you tune to the tonic chord of the song instead of to the open strings.

Now play a full, open G chord as above. Make sure all the strings sound clearly. You are playing a chord with an unusual property: It has two equal-tempered major thirds in it. This chord is highly equal-tempered in character.

Strum away, and sing Frère Jacques over it, several times through. You may wish to capo and tune again, if this is not a comfortable key for you.

When you have a good sense of what this feels like, try fingering the G chord as follows:

Open G5

I’m a thumb-wrapper, so I finger the low G with my thumb, and mute the A string with more of my thumb. (This is heretical to some, but it’s a wonderfully useful technique when used at the right time. Here’s a beautiful explanation by guitar teacher Jim Bowley.) Then I finger the two high notes with my index. Any fingering will work as long as it mutes the A string.

Now you have a chord with no major thirds at all. It goes G—D–G–D–G, or 1—5–1–5–1.

Sing Frère Jacques over this chord, several times through and check out what happens.

I won’t tell you what to feel. Don’t worry about trying to hear or sing subtle tuning differences. Just pay attention to your singing, and to your body’s reaction.

Seriously, go do this now, or the next time you’re near a guitar. It works great with piano too, and in any key. First play a major chord, with a couple of thirds in it to really make the point. Then play only roots and fifths. Sing the song over each version of the chord, back and forth. The difference may surprise you.

I’ll check in tomorrow with my own conclusions.

Next: More Experimenting

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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 12, 2012 in Just Intonation, Septimal Harmony, The Notes | 2 comments


My favorite contemporary band is the Black Keys. I think Dan Auerbach is a harmonic genius. The video is funny.

Dinosaur? What dinosaur?

This song lives in the universe of 7. I just spent a half day taking apart the main riff and seeing how it works out on the lattice. It strictly uses a 5-note scale: 1, 7b3, 74+, 5, and 7b7. The notes are all in the universe of 3 and 7; there is no 5 energy at all, in this part of the song at least.

There are three new lattice notes in this video, the septimal minor third, or 7b3, the harmonic seventh, or septimal flatted seventh, labeled 7b7, and a crazy new note I’ll get to in a minute. The 7b3 is found (on the scale) between the 2 and the b3. It’s a lot flatter than the minor third. The 7b7 is pitched between the 6 and the b7, a lot flatter than the minor seventh. Neither of these can be played directly on the piano. Blues pianists can evoke them by trilling between the key above and the key below. Variable pitch instruments, notably voice and electric guitar, are capable of actually nailing these notes and delivering their full effect.

This song added a new note to my lattice! Mathieu writes that it is used in the blues, and I knew about it theoretically, but I hadn’t used it or observed it in the wild before. One source calls it the septimal narrow fourth. It is slightly flat of the 4. My name for it turns out to be 74+. (The + is a slight pitch adjustment to show exactly how it’s tuned in just intonation.) In Next Girl, it makes a harmony note with the 7b3 root — a nice interval of a ninth.

Cool, haven’t confirmed the existence of a new note in a while. The bestiary grows. Kind of like particle physics.

Septimal notes are essentially unknown in European classical music, but thanks to the blues, they thoroughly infuse the music of America and many other countries. Without them, some music just doesn’t sound the same. They are one reason the Beatles don’t translate well to elevator music. Check this out:

Listen to the signature riff, how it changes and morphs. Throughout the song, George is playing with the region between the septimal flatted third and the major third. As the chords change, the song moves around on the lattice. In response, he bends the note a little more, a little less, to evoke the septimal third, then the major, and maybe even the minor third, located between the other two notes.

By the way, this is a great little zone on guitar. It’s the second fret up from the tonic. You can play four distinct notes just by bending — the 2, 7b3, b3 and 3. George Harrison spends this whole song exploring the tension and resolution in that little melodic space.

John’s vocals are great blues, right in tune.

As I hear them, the Black Keys go even further by putting septimal notes in the roots. Great lyrics and a sense of musical history too, an excellent band.

The usable septimal notes are all close to the center. They just get too far out for me to hear, rather quickly. I personally have found three of them useful so far, and today I’ve been introduced to another.

Here are the septimal notes I have on my current lattice. I imagine I’ll add more as I explore.

7/4, the harmonic seventh, 7b7

7/6, the septimal minor third, 7b3

7/5, the septimal tritone, a staple of rock guitar and one of my personal favorite notes.

21/16, the septimal narrow fourth, or blu ma according to Mathieu. He has some great note names in his book, based on the Indian singing notes, sa – re – ga – ma and so on.

Next: Summary (So Far)

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Posted by on Dec 10, 2012 in The Notes | 0 comments

The Augmented Fourth

I’ve described eleven notes now, and each one has a piano key to go with it, an equal tempered equivalent.

The one remaining black key has a lot of names. It’s the note between the 4 and 5, right in the middle of the octave — the tritone, devil’s interval, flatted fifth, augmented fourth.

In ET there’s only one tritone, and it precisely splits the octave in half. In JI, there are several tritones, with different tunings, that sound and function differently from each other.

One tritone, that nicely fills out the set of 12 notes, is the augmented fourth:

This note is not like the other black keys. It’s completely overtonal, that is, it is generated entirely by multiplication — x3, x3, x5, or 45/1. It does appear in the Chord of Nature, but so far up that it wouldn’t be audible in the harmonics of a vibrating string. I think the fact that we can hear any harmony at all with this note shows that we can hear compounds of simple ratios, even when the numbers are getting pretty big. If pure ratios were all that mattered, 13/1 would be far more harmonious than 45/1 — the numbers are smaller. But 13/1 is almost nonexistent in the musics of the world, and even 11/1 is very rare.

So the harmonic connection with the tonic is tenuous, but it’s there. I hear a different kind of dissonance than the b6 or b2-, more harmonically distant, but without as much of that urgency-to-move that the reciprocal notes have.

It’s natural to resolve it melodically to the 5:

Or once again we can travel through harmonic space to get back home.

Can you hear yourself getting closer to home with each step?

We now have a set of 12 notes, one for each key of the keyboard. Next, the prime number 7, and then some notes between the keys. Oh, the places we’ll go!

Next: Prime Numbers and the Big Bang

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