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Posted by on Feb 11, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

More Experimenting

Yesterday, I described a simple way to hear, and more importantly, feel, the difference between equal temperament and just intonation, by singing Frère Jacques over an open G chord with ET major thirds, and then over a G chord that has only roots and fifths in it.

The second half of the experiment is called singing over a drone, and it’s a great way to get acquainted with the resonance of the pure notes. The 1–5 drone is a bedrock foundation of North Indian (Hindustani) classical music. Check out this gorgeous song with Ravi Shankar’s two daughters, Norah Jones and Anoushka Shankar. It’s not drone music, but it’s a beautiful blend of East and West, and I feel the purity of the notes down to my bones. This is untempered music.

At this point I must pause again to acknowledge my enormous debt to W.A. Mathieu. I had been studying just intonation for several months when my friend Kay Ashley, a fine singer, guitarist and student of Hindustani music, loaned me her copy of Mathieu’s book Harmonic Experience. She did not get it back until I had my own copy.

The first part of the book introduces the pure notes by showing the reader how to sing them over a drone. I have found no better way to understand the notes of just intonation than to sing them. It isn’t just about hearing them, even though that can be beautiful and illuminating. It is about feeling the resonance in your body.

200px-Stable_equilibrium.svg copy

My own experience in singing over a drone of a root and perfect fifth is that it is much easier to sing in tune. It’s as though the drone sets up a sort of sonic field that has grooves or troughs in it, points of stable equilibrium into which my voice falls, and wants to stay.

The equal-tempered chord is not so friendly. The tempered major third is not in tune, that is, its natural resonance does not point exactly to the tonic. It’s as though it points to a different tonic, a little sharper than the one the root and fifth are pointing to. The groove is obscured, and there is a fight between the two worlds that makes it harder to know exactly what to sing. The reference is shaky.

I confess, yesterday’s experiment was a little unfair to equal temperament. I changed two things at once, which is not a good way to investigate nature. It’s much better to change only one thing at a time, so you know what causes what. When you sing over the straight G–D drone, you’re hearing two changes — the simpler chord (1–5 instead of 1–3–5), and the effect of removing the equal tempered third.

In the interest of scientific honesty, here’s one more exercise that shows only the effect of ET.

I’ve recorded some synthesized strings to sing along with. These are the same notes as the open G chord: G-B-D-G-B-G. Once again, sing Frère Jacques. Row, Row, Row Your Boat and Three Blind Mice are also excellent, I recommend trying them too.

Here is a G major chord in just intonation:

G chord JI

And in equal temperament.

G chord ET

If this is not in your most comfortable vocal range, here are some six-note chords in the key of C. I find these better for my own voice. These are note-for-note the same as the first position C chord on guitar, another common chord with two equal tempered thirds in it.

C major in just intonation:

C chord JI

And in ET.

C chord ET

I invite you to go back and forth between the JI and ET versions of the chord that is most comfortable for you, singing over each.

While you’re singing, pay attention to how the notes feel, in your body.

Also notice how easy, or how difficult, it is to hold your notes, to jump straight to the next note, to not waver when you hold a long one.

And perhaps most importantly, pay close attention to the emotion you feel while singing.

I have long experienced flashes of musical ecstasy — it’s why I make music, to experience and share that transcendence. But such experiences have been sporadic, and somewhat mysterious. Encountering, studying and internalizing the pure notes, and their relationships to each other (the lattice), has thrown open the double doors, and I am now in the long process of walking through them.

Next: Entrainment

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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 Feb 6, 2013 in Just Intonation, The Lattice, The Notes | 3 comments

The Untempered Chromatic Scale (Part 1)

The familiar 12-note scale is also called the chromatic scale. There are many ways to generate an untempered version of such a scale.

The oldest way seems to be Pythagorean tuning. When you multiply a frequency by 3, you get a new note, an octave plus a fifth above. Multiply by 3 again, and you get another new note, and so on. Dividing by 3 gives you another new note, the perfect fourth, dividing by 3 again delivers another new note, and so on. Do this enough times and you can generate a 12-tone scale.

This scale forms the central, horizontal spine of the lattice.

pythagorean lattice

 

The central six notes of the Pythagorean scale are highlighted. If you keep extending to the left and right you get all 12 notes, and then even more.

Trouble is, once you’re more than two or three steps away from the center, the notes are definitely dissonant. The ratios just get so big that the ear can no longer hear them as harmony. The Pythagorean major third, for example, is the next note to the right of the 6+, just off the central lattice. Its ratio is 3x3x3x3, or 81/1. The octave reduction trick allows you to divide by 2 until it’s in the same octave as the tonic, so the ratio becomes 81/64.

Harmony is built from small, whole number ratios, and these are not small numbers. Here is what a Pythagorean major third sounds like:

clarinet 81:64

Wow. Back when Pythagorean tuning was the norm, the third was considered a dissonant interval. I concur!

In late Medieval times, the 5/4 tuning became more popular, and major thirds began to be regarded as a consonant interval. Here’s the 5/4, for comparison:

clarinet 5:4

Ah. Using the prime number 5 allows a more consonant chromatic scale, with much smaller ratios. Here it is on the lattice:

12-tone lattice

 

All of these notes have been covered in previous posts. Each one has a unique personality. Some are more consonant, some more dissonant, but all twelve have small enough numbers in their ratios to be perceived as harmony by the ear.

Here are their ratios, and pitches in cents:

1 — 1/1 — 0

b2 — 16/15 — 112

2 — 9/8 — 204

b3 — 6/5 — 316

3 — 5/4 — 386

4 — 4/3 — 498

#4+ — 45/32 — 590

5 — 3/2 — 702

b6 — 8/5 — 814

6 — 5/3 — 884

b7 — 9/5 — 1018

7 — 15/8 — 1088

Here is the scale, compared with the 12 notes of equal temperament.

untempered chromatic

 

When I first drew this scale, on graph paper, I was startled by how narrow the major-minor half steps are, compared with the equal-tempered versions. Take a look at the b3 (minor third) and the 3 (major third). They are only 70 cents apart, not the 100 cents of the piano scale. Same with the b6-6 and b7-7. It’s such a small move in pitch, such a large move harmonically!

The Mozart movies I made a month ago or so illustrate the major-minor pair quite well. The blandification of equal temperament has obscured a beautiful detail of harmonic music. So much is gained with ET, and so much is lost or obfuscated.

Next: The Untempered Chromatic Scale (Part 2)

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Posted by on Feb 5, 2013 in Just Intonation | 0 comments

The Major Scale in Cents

The simplest untempered tuning of the major scale is:

1 — 0 cents

2 — 204

3 — 386

4 — 498

5 — 702

6 — 884

7 — 1088

Here’s how that tuning compares with the equal tempered scale:

Cents

 

The black numbers show the pitches of 12-tone equal temperament. They are equally spaced, like inches on a ruler.

The red numbers show the tuning of the untempered major scale. They are spaced in the way they naturally turn out when you generate them with small whole number ratios. As is so often the case with the natural world, they don’t line up too well with the nice human grid lines we love.

The way I see it, when you play in equal temperament, you’re playing the grid lines on the map.

When you sing or play the untempered notes, you are visiting the actual territory.

I’ve read that it’s not possible to combine the two, but I disagree. It’s a matter of showing the ET instrument who’s the boss. My favorite example is Ray Charles. Here’s a video from 1976. He’s playing the piano, laying down those grid lines, and the rest of the band is too, but when he sings, his voice owns the sound, and the sound becomes him. A great, dominant singer will infuse the whole combo with that soul.

Another great example is Ella Fitzgerald. Want some goosebumps? Check this video out.

Next: The Untempered Chromatic Scale (Part 1)

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Posted by on Feb 1, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

The Untempered Major Scale

In harmonic space, the clearest name for each note is its ratio — 5/3, 3/2, 4/3, etc. Precise and unambiguous. But the ratios don’t give a very good idea of how to go about playing or singing those notes.

There are a few instruments, such as the left hand keyboard on the accordion, that are organized for harmonic thinking.

800px-120-button_Stradella_chart.svg

The rows are arranged in fifths, and the first two rows are a major third apart, just like the lattice.

For most music making, though, we need to know the pitch of the note. Instruments and voices tend to live in the world of melodic space — scales and pitches.

Here’s what the simplest, most harmonically consonant major scale looks like on the lattice:

One can convert the ratios to pitches using the formula for cents. The ET values are in parentheses.

1 = 1/1 = 0 cents (0)

2 = 9/8 = 204 (200)

3 = 5/4 = 386 (400)

4 = 4/3 = 498 (500)

5 = 3/2 = 702 (700)

6 = 5/3 = 884 (900)

7 = 15/8 = 1088 (1100)

Note that the 1, 2, 4 and 5 are very close to their ET equivalents. Most ears would be unable to tell the difference.

The third, sixth and seventh, however, are all noticeably flat. Or perhaps I should say their ET namesakes are noticeably sharp.

I think this explains a lot about rock music, which depends heavily on power chords (roots and fifths with no thirds) and 1-4-5 chord progressions. The notes of a 1-4-5 power chord progression are 1, 4, 5 and 2!

As Eddie Van Halen said in his terrific Guitar Magazine interview, “Really, the best songs are still based on I-IV-V, which is so pleasing to the ear. Billy Gibbons [of ZZ Top] calls me now and then, and he always asks, ‘Eddie, have you found that fourth chord yet?’ [Laughs].”

Of course the I-IV-V is inherently satisfying, it’s that great rocking chair between reciprocal and overtonal territory, in the most consonant part of the lattice, as close to the tonic as you can get (harmonically).

But I think the special appeal for rock music is that, in ET (and guitar is essentially an equal-tempered instrument), the 1, 4 and 5, and also the fifth of the 5 (the 2) are all in tune. Whatever one might think of ZZ Top’s simplicity (and some do scoff), it’s undeniable that they are fiercely in tune, and harmonically their music strikes a deep chord in the psyche, pun intended.

If the bass and rhythm guitar stick to those roots and fifths, the voices and lead guitar can play all the other notes, because they can be bent and wiggled until they sound right.

This works just as well in traditional country music. Let the bass player nail down those roots and fifths, and the voices (and the fiddle) can sing in-tune harmonies as sweet as you please.

Next: The Major Scale in Cents

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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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