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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 Jun 23, 2013 in Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes |

The Septimal Minor Third

At the very end of the chorusBe Love showcases one of my favorite notes, the 7b3.

Most of the notes of the inner lattice can be approximated on the piano, but not the septimal minor third. It’s in between the keys. Blues pianists can evoke it by trilling between 2 and b3, but only variable-pitch instruments can actually hit the note.

Here it is in context:

It’s fun to sing this part of the song, stretching out that septimal note and tasting its flavor.

The passage illustrates the harmonic function of the note. It’s the septimal flatted seventh of the 4, also called the harmonic seventh or barbershop seventh. This is a beautifully consonant note, a great addition to a major chord. It’s generated by multiplying by seven. I use it here as a harmonic seventh over the IV.

Relative to the 1, the 7b3 is a compound note. To get there, you divide by 3, and then multiply by 7. The ratio is 7/6, octave reduced. The pitch is 267 cents, between the 2 and the b3. Here it is on the scale. The colored notes are in just intonation, the black ones are in equal temperament.

Scale with 7b3

Over a I chord, the 7b3 sounds bluesy, restless, gutsy — it’s the insistent melody note in Taking Care of Business. It’s at the heart of the guitar riff in Dizzy Miss Lizzy (George often bends it up to the major third), it’s Jagger’s haunting first “ooooh” of Gimme Shelter. Gimme Shelter

The Stones’ music is a feast of 7b3’s. So is Led Zeppelin’s. These septimal notes are found everywhere the blues has left its impression.

There’s an old question: why do the minor melody notes of the blues sound good over major chords? The web is full of discussions as to why this is so.

I think it’s because the blue minor third is not the b3, but the 7b3. The regular minor third is a reciprocal third, and harmonically it doesn’t fit with major chords — it’s in a different part of the lattice.

But the 7b3 is an overtonal 7th, built on the 4, generated by multiplication. The major notes are made by multiplying by 5. Times 5 and times 7 go together very well. The harmonic seventh chord is a thing of beauty.

There’s an implication for blues guitar. You can’t play this note in the classic minor pentatonic blues box. You can play a b3 (bend it a little to tune it up), or a 3 (bend it harder), but not a 7b3, it’s flat of the b3, and you can’t bend down.

minor pentatonic

You can play a 7b3 by grabbing the 2, one fret below, and gently bending up to it. The following box works great for septimal notes. They’re all laid out under the ring finger. Bend them by less than a half step.

major pentatonic

This box makes it easy to play the classic bit of melody, 7b3 – 2 – 1. All three songs I linked to earlier have this melody in their bones — BTO, the Stones, the Beatles. It’s everywhere. Here it is in Be Love.



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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.

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Posted by on Nov 10, 2012 in Just Intonation | 0 comments

The Chord of Nature

When I first learned to play guitar, I would sit on the couch late at night and pluck the low E string, again and again, and just listen to the beauty of the sound as it died away.

That low E note is not just a simple vibration. The full length of the string is vibrating at about 82 Hz. But a pure 82 Hz note doesn’t sound like a guitar string at all. It sounds like this:

82 sine

The guitar sound is much more complex:

guitar low e

The difference comes from the fact that a string doesn’t just vibrate along its whole length — it also vibrates at twice the frequency, three times, four, and so on — all at the same time!

Maybe you did this as a kid. I did. When you get a jump rope going, you are essentially vibrating a big string. It has a characteristic frequency, maybe two cycles per second, set by the length of the rope and the amount of tension, just like a guitar string. This frequency, the natural vibration speed of the whole string, is called the fundamental.

But if one person holds their end still, and the person on the other end moves the rope twice as fast as usual, a funny thing happens. The rope divides in two, and the center point stays still, while each half does its own circle. Again, the length and tension determine the natural speed. Go three times as fast, and three sections will appear. These higher-mode vibrations are called harmonics.

Here are a couple of guys in lab coats to demonstrate:


This only works when you hit the right frequencies. Spin the rope at, say, 2 1/2 times the natural frequency and everything falls apart. The stable frequencies are the fundamental, 2x, 3x, 4x, 5x and so on. This video shows a string getting stable at 6x, 5x, 3x, and the chaos that happens in between.


When you pluck a real string, it will vibrate in all these modes, generating a complex sound. The particular recipe of added harmonics creates the timbre, or tone, of the note.

Here’s that same pure 82 Hz tone, with the harmonics 2x, 3x, 4x and 5x added successively:

Chord of Nature 2

This is the Chord of Nature. It is a sonic manifestation of number, and of the laws of the universe, and it’s very simple. If the fundamental frequency is 1, then the frequencies of the harmonics are 2, 3, 4, 5, 6 and so on. And somehow, our perception of sound is designed so that this sounds beautiful.

Here, in contrast, is the same demonstration but with the harmonics detuned randomly by less than two percent:

Chord from Hell 2

Yipe! Now go back and listen to the first one as a palate cleanser.

There is something deep inside us that recognizes the series of harmonics, and, for most of us, labels it “beautiful.” There is some connection between those small, whole numbers and musical beauty.


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