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Posted by on Mar 14, 2014 in Equal Temperament, Just Intonation, The Lattice | 3 comments

A Harmonic Journey: ET and JI Compared

The Harmonic Lattice can be viewed as a map of harmonic space. Music moves in harmonic space, just as it moves in melodic space (the world of scales and keyboards). The two spaces are very different from each other.

In melodic space, such as a piano keyboard, when two notes are close together, it means they are close in pitch.

In a harmonic space, such as the lattice, when two notes are close together, it means they are harmonically related.

“Harmonically related” means that one note can be converted into the other note by multiplying and dividing by small whole numbers. A note vibrating at 100 cycles per second is closely related to a note at 300 cycles per second. In melodic space, these two notes are far apart, but in harmonic terms they are right next door to each other — they harmonize.

In my video, Flying Dream, I animated the movement of one of my songs on the lattice. Now I’ve animated a composition of W. A. Mathieu’s.

Mathieu is the author of Harmonic Experience, an astonishing book that takes music back to its origins in resonance and pure harmony, and then uses the lattice concept to bring that harmonic understanding forward into the world of equal temperament. For me, the book opened the study of music like a flower.

The lattice, and my stop-motion animations, have given me a sort of musical oscilloscope. Instead of the music being some sort of black box, I can see inside it, get a visual image of what is going on harmonically. The new tool has made songwriting, improvising and arranging much easier.

I’ve animated Example 22.10 from the book. It’s intended to be an illustration of unambiguous harmony — the chord progression moves by short distances on the lattice, so it is clear to the eye and ear where you are. I think it’s a beautiful piece of music in its own right, a one-minute tour of a huge area of the lattice. It uses 28 different notes!

There are two versions of the video. The first one, in ET, has a soundtrack of Allaudin Mathieu playing the piece on his beautifully tuned piano. This is perfect equal temperament. It uses twelve notes to approximate the twenty-eight notes that the piece visits.

For the second one, in JI, I retuned the piano to the actual pitches of the lattice notes. Now, magically, the piano has all 28 notes. There is a whole new dimension to the music. In the JI version, I feel:

  • Slight vertigo when the music moves quickly
  • Satisfaction when a spread-out (tense) pattern collapses to a compact (resolved) one
  • A great sense of homecoming at the end
  • Stronger consonance and dissonance than in the ET version.

The four voices, from lowest to highest, are red, green, orange and yellow. It’s fun to follow one voice at a time.

This lattice is notated differently. It’s my usual system, but with letters instead of numbers. C is the tonic.

 

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Posted by on Sep 23, 2013 in Just Intonation, The Lattice | 0 comments

Chords on the Lattice

A chord is a collection of three or more notes sounded at the same time. Arpeggios, in which the notes are sounded one after the other, are considered chords too. Two notes sounded at once are generally called an interval rather than a chord.

Chords make patterns on the lattice. A given kind of chord will look the same no matter where it is.

The most common chords are the major and minor triads (a triad is a three-note chord that is a stack of major and/or minor thirds). Here is what a major triad looks and sounds like on the lattice:

The major triad is an upright triangle. It even looks stable. It’s made of three interlocking intervals — in this case, from 1 to 3 (a major third), from 3 to 5 (a minor third), and from 1 to 5 (a perfect fifth).

Anything that looks like this on the lattice is a perfectly-in-tune major chord.

A minor triad is an upside-down triangle. Minor triads look like this:

Major and minor triads interlock to form the hexagonal lattice of fifths and thirds. This generates another lattice, a lattice of chords. W.A. Mathieu goes into great detail in Harmonic Experience, extending the chord lattice a long ways out and showing how music wanders on it. Here is an illustration based on my own lattice:

Chord Lattice

I use roman numerals for chord names, because the relationships between chords stay the same no matter what key I’m in. For example, the progression C-F-G is exactly the same as the progression G-C-D, at a different pitch. Both are I-IV-V progressions. This convention uses capital letters for major chords, and lower case for minors. I add a little twist by adding + and – to show commas; this allows a unique name for every chord on the infinite lattice.

It’s illuminating to track a chord progression on this lattice. The famous “Heart and Soul” progression, I-vi-IV-V, is what Mathieu calls “Matchstick Harmony.” The lines move like the matches in those matchstick puzzles. Progressions that move by these small harmonic distances are intuitive and easy to follow. The last move, from IV to V, is also easy for the ear, making this chord progression as natural as breathing. Start playing it on the piano and you will instantly have a crowd. In the key of C, it goes C-Am-F-G.

The chord lattice adds another dimension to lattice thinking. Watch the Flying Dream video for a good example. The progression travels far afield, exploring many of these major and minor triangles before finally coming home.

Other chords make other shapes that also repeat all over the lattice. For example, there are at least three different kinds of minor seventh chord. Here’s an article distinguishing them.

Next: Three Flavors of Seventh Chord

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Posted by on Aug 6, 2013 in Consonance, The Lattice, Tonal Gravity | 0 comments

Polarity

The following video compares the perfect fifth with the perfect fourth.

These notes are the next-door neighbors of the tonic. They are equally close to the center. They are both harmonious. Yet there is a great difference in their character.

The difference between these two intervals is polarity.

I learned this term from W.A. Mathieu, in his amazing book “Harmonic Experience.”

Polarity is the main driver of tension and release in tonal music. I think it’s much more important than harmonic distance, the other component of consonance.

Here’s how I think polarity works:

Almost all sounds are actually a bundle of waves, of different frequencies. The frequencies in these bundles tend to vibrate at multiples of each other — 2x, 3x, 4x, 5x some base frequency — a harmonic series.

Our ears are highly attuned to such relationships. They help us figure out which frequencies belong together, so we can analyze them and identify the source. If we hear two frequencies in lockstep, three cycles to one, it is very likely they come from the same object.

I think this is why we can hear harmony. It’s a byproduct of our built-in orientation software. The overtone series of a sound is a powerful source of information about the object that made it. The harmonic content tells us whether it’s a barking dog or a friend or rustling leaves. Something in us sorts out and analyzes many harmonic series at once, in real time, identifying sound sources, locating them in space, and even sensing their texture at a distance. It’s a phenomenal processor.

So the processor recognizes the 3-wiggles-to-1 dance of the perfect fourth. But something is wrong — the 4 does not belong in the overtone series of the 1. The overtone series is generated by multiplication, not division.

This is a strange input for the mighty processor.

  • The 4 has to belong to the 1, because the two are in step with each other at three beats for one. In nature, that’s a dead giveaway.
  • But the 4 can’t belong to the 1, because the ratio is the wrong way around — natural sounds do not contain 1/3 in their overtone series.
  • Maybe the 1 actually belongs to the 4. The 1 is three times the 4, so if the 4 were the base, all would make sense.
  • But every other clue, all the harmonics in the drone (and, importantly, the listener’s memory), are pointing to the 1, hollering “This is the basic frequency!”

What’s a supercomputer to do?

I feel this sensation as unrest or instability, a need for something to change. Either the 4 needs to resolve to an overtonal note such as the 5 or major 3, or the root needs to change. Moving the root to the 4 will resolve the dissonance, introducing a new reciprocal tension — now the root “wants” to go to the 1, because the ear remembers. There are videos of this here.

On the lattice, there are two basic polarities, overtonal and reciprocal. Some notes combine both energies.

Overtonal notes are created by multiplying the base frequency. They appear in the natural harmonic content of sounds. They sound stable, restful, resolved — more so the closer they are to the center of the lattice, where tonal gravity is stronger. Pure overtonal notes are in the Northeast quadrant of the lattice.

Reciprocal notes are created by dividing the base frequency. The mind recognizes them as being related to the base frequency, but they do not appear in its natural overtone series. They sound unstable, restless, tense — and, like the overtonal ones, the effect is stronger closer to the center.

Polarity is our sense of the tonal gravity field. It is how we orient ourselves in harmonic space.

Next: More Mirror Twins

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Posted by on Jul 22, 2013 in Consonance, Just Intonation, The Lattice, The Notes, Tonal Gravity | 0 comments

Consonance and Dissonance

I just passed the 10,000 photo mark on the stop motion animations, good thing I’m not hand-drawing them like Winsor McCay!

The one I’m working on, Real Girl, has a lot of dissonant notes in it. The melody ranges far from the roots and makes some slightly dizzying harmonic jumps. I want to use it as a framework for discussing consonance and dissonance. While it’s in progress, I want to lay down some groundwork.

The Wikipedia article Consonance and Dissonance is really thorough. Here’s a quote from the introduction:

In music, a consonance (Latin con-, “with” + sonare, “to sound”) is a harmony, chord, or interval considered stable (at rest), as opposed to a dissonance (Latin dis-, “apart” + sonare, “to sound”), which is considered unstable (or temporary, transitional). In more general usage, a consonance is a combination of notes that sound pleasant to most people when played at the same time; dissonance is a combination of notes that sound harsh or unpleasant to most people.

This definition has two distinct concepts in it — the “stability” of a harmony, and whether the notes sound pleasant or unpleasant together. I used to think of consonance/dissonance as a linear spectrum, with consonant notes at one end and dissonant ones at the other.

After working with the lattice, and reading Mathieu, I now see consonance as having two distinct components, that do not necessarily track together:

  1. How the notes sound together, away from any musical context. The range would be from smooth and harmonious to rough and grating.
  2. The stability of the interval. Does it create a sensation of rest, or does it feel restless, ready to move?

I propose that these two qualities can be directly seen on the lattice as follows:

  1. The way the notes will sound when simply played together is a function of the distance between the notes in harmonic space (how far apart they are on the lattice). The farther apart the two notes are, the less harmonious they will sound when played together.
  2. The stability of the interval is a function of the direction of the interval on the lattice (whether it’s generated by multiplying, dividing, or a combination of the two). Intervals generated by multiplying (moving to the East and North on the lattice) are restful, those generated by dividing (moving West and South) are unstable and restless.

The interval quality is also powerfully affected by which primes (3, 5, 7) are used to generate the interval, but I hear this as a sort of flavor or color, rather than as consonance per se.

The first component, the sound of the notes simply played together, is a property of the interaction of those frequencies in the ear. It isn’t dependent on the musical context in which it appears.

The sense of stability or instability, on the other hand, depends entirely on context. This sensation comes from the direction of the interval, which implies that the interval must start somewhere (the tonic or root) and end somewhere (the harmony note), so as to have a direction. One note is home base, the other is an excursion from that base.

Here a couple of examples to show the difference.

The perfect fifth is the most consonant interval on the lattice that actually involves a distance. (Octaves and unisons are more consonant, but on the lattice, they cover no distance at all — multiplying the frequency of a note by 1 gives a unison, which is of course the same note, and multiplying or dividing by two gives an octave, which, by a miraculous quirk of human perception, also sounds like the same note, harmonically.)

To make a fifth, you multiply by 3. You can then then multiply or divide by 2 at will, (which doesn’t add any distance) to put it in the octave you desire. The frequencies of the two notes in this video are related by a ratio of 3:2. There is no context, just the two notes sounding together.

This is clearly a consonant interval. There is a smoothness, a harmoniousness to the sound that I imagine would be perceived as such by anyone in the world. Two notes in a ratio of 3:2 will sound like that no matter what the context.

So how do stability and instability enter in? It happens when there is a reference note, which can be the tonic (the main key center around which everything is arranged), a root (a local tonal center that changes from chord to chord), or even a bass note, which, if it is not the root of the chord, shifts the harmonic feel of the chord.

The music in this next video establishes that the tonal center is the 1, and then introduces the 1-5 interval.

The interval sounds stable; the ear does not crave a change. There is resolution.

In the next video, the music establishes a new tonal center in the ear. Now it sounds like the 5 is home. Listen to what happens when I introduce the very same 1-5 interval:

The interval is exactly the same, and the effect is quite different. There is tension. Something’s gotta move!

I can make this point more clearly by resolving the tension. Hear the unfinished quality, and how it resolves?

Aaaaah.

In the first video, home base is the 1, and the 5 is an overtonal note — that is, it is generated by multiplying the home note by 3. It sounds restful and stable.

In the second video, the tonal center is the 5, and the 1 is reciprocal, that is, it is generated by dividing by 3.

So the same exact interval can be stable or unstable according to harmonic context, even though the “degree of roughness” is the same. That’s why I think Wikipedia’s two-part definition is referring to two different things, which should be thought of separately.

Next: Consonance Experiment

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Posted by on Jul 18, 2013 in Just Intonation, The Lattice, The Notes |

Mixolydian Mode

I’ve been quiet lately because I’ve been working on an animation of my song Real Girl. It’s a complicated one, a dance of harmonic tension and resolution. The bass and melody chase each other around the lattice like courting butterflies.

Meanwhile, there’s more to be extracted from the Be Love video.

A mode is a type of scale, characterized by the pattern of intervals between its notes. The note spacings stay the same no matter what key it’s in. When we say a song is in A major, we mean the tonic is A, and the mode is major. The major mode, also called Ionian, is the familiar Do-re-mi-fa-sol-la-ti-do.

Any combination of notes, covering an octave and organized in pitch order, can be a mode. There is a particular set of modes, often called church modes, that can be played on the white keys of the piano. The different modes start on different notes. The major mode goes from C to C; Aeolian, or minor mode, runs from A to A. Stick to the white keys, and the notes will be right for that mode.

Ionian and Aeolian are the commonest modes in modern Western music, but Dorian (D to D) and Mixolydian (G to G) are popular too.

I wrote the chorus of Be Love first. It’s in major mode. I wanted the verse to have a different feel, so I decided to make it Mixolydian — a favorite for several reasons. I love the name. And, as Mathieu points out in his book Harmonic Experience, it’s particularly easy to improvise over. It’s common in rock music from the seventies on — see the BTO clip in this post. My song Driving is mostly in Mixolydian mode.

The big reason I wanted the change was to make the chorus more of an anthem, by contrast. Mixolydian has a dark, beefy quality to me, and when the chorus comes around it sounds like the sun is coming out.

In equal temperament, there is only one difference between major and Mixolydian scales. The seventh degree is minor instead of major. Starting with G, and going up the white keys, you get G-A-B-C-D-E-F-G. The G major scale goes G-A-B-C-D-E-F#-G. Only the seventh is different.

In just intonation, the situation is a bit different. There are three b7s in the inner lattice:  b7, b7-, and 7b7. Which to choose?

Here is the major scale.

P1110316c

One possibility is to just drop the 7 to the b7:

P1110316d

I like the interpretation below. It uses the b7-, and also changes the 2 to a 2-.

P1110316b

This gives me an in-tune major flatted seventh chord, which I love. In the key of G that’s F major — lots of rock music uses this chord.

The notes of this scale (Western Mixolydian?) are in the same relationship to each other as the notes of the major scale, shifted one space to the left. The intervals are just as consonant as the major scale ones, only arranged in a different order.

It’s easy to write chord progressions in this mode. Major and minor triads form triangles on the lattice (major triads point up, and minor ones point down) and there are five in-tune triads, just like the major scale. Since the triangles are all connected, moving from one to another feels natural and is easy for the ear to follow.

Again I refer to Mathieu’s book. Have I lately? In Harmonic Experience he writes about “matchstick harmony.” Cool stuff.

Here is the scale, animated against a drone.

Next: Mixolydian Matchsticks

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