For every note on the lattice (except the 1), there is another note, the same distance away from the center and exactly opposite it. The harmonic moves for the two notes are the same, but the directions are opposite.
Mirror twins are reciprocals of each other. Flipping a note’s ratio upside down will produce its twin.
Listening to these mirror twins helps demonstrate polarity.
The simplest ratios on the lattice, the ones with the smallest numbers in them, are 3:1 and 1:3, the perfect fifth and perfect fourth. Here they are:
Tonal gravity is strong here close to the sun. The fifth sounds remarkably resolved, like it’s part of the drone. In fact, this note shows up so strongly in the natural overtone series that it is part of the drone. If you listen carefully to that tonic drone by itself, you can hear it. The following video shows a 5 by itself, followed by the straight drone on the 1. I hear the fifth appear again, quietly, after the drone has a few seconds to settle in and bloom.
The fifth says, “You’re home, relax.”
The fourth also shows where home is, but in a very different way. Instead of saying, in effect, “Home is here, come on,” It is saying “Home is over there, now go.”
Our built-in audio processor is always looking for mathematical relationships between notes so we can tell which frequencies belong together, identify different sound sources and orient ourselves in our surroundings. I think this is why we hear harmony and why it sounds like a journey — it’s a part of our built-in orientation software.
Three cycles for every one is a “right” ratio. It strongly says, “These frequencies belong together, they are being made by the same thing.” But the ratio is upside down, 1/3, the exact opposite of the “right” sound, 3/1. It’s the shadow version of the 5, yin to its yang.
The 5 feels like this:
The 4 feels like this:
The beautiful stability of the 5, contrasted with the equally beautiful instability of the 4, is what I mean by polarity.
In my last post I raised the idea that the consonance of an interval is not just one thing, but has two distinct parts:
The sensation created by the sound of the two notes played together — smooth or rough, pleasant or unpleasant
The sense of stability of the note — does it feel restful, like it has arrived, or restless, unstable, like it “wants” to move?
I propose that #1 is correlated with the harmonic distance between the two notes of the interval, that is, the length of the solid lines that connect them on the lattice.
I think #2 is determined by the polarity of those connections, whether they consist of multiplying (stable) or dividing (unstable), or a combination of the two.
This suggests some experiments.
A mostly good rule in experiments is to change one thing at a time. The prime factors used in an interval’s ratio (3, 5 or 7) strongly affect the sound, so I’ll stick to one axis (x3), and only go in one direction (multiplying, overtonal, East).
This will all happen on the backbone of the lattice, the infinite horizontal line of fifths. The notes on this line are in Pythagorean tuning, which is any tuning that uses only 3 and 2 in its ratios. Pythagorean tuning is used in music, but many of the intervals are sharply dissonant, as we shall see.
If I’m right, according to #1 the notes should get rougher or more unpleasant as I go further from the center, and according to #2 there should be a sense of stability, that gets weaker as the tonal gravity field weakens with distance.
I’ll play all the notes against a drone, several octaves of the tonic. The notes will all be played in a single octave, in between two drone notes. I will octave reduce (divide by 2) as necessary, to put all the notes in this octave.
The first interval is a perfect fifth. To get this, you multiply the frequency of the tonic by 3.
No denying it, this is a consonant interval. Only the drone sounds smoother. It sounds stable too, as it should according to #2. If a song ends like this, I’m perfectly satisfied.
Next note up is the 2. This is two perfect fifths, a factor of 9.
To my ears, this note is still quite consonant. And yes, it still sounds stable. Lots of songs end on this note — I feel wistfulness, gentle yearning, perhaps a sweet sadness, but I don’t feel a powerful need for the note to change.
Next is 3x3x3, the Pythagorean sixth or 6+. I still hear this one as consonant, but just barely. For me the pattern continues — the 6+ is right on the edge of dissonance, and stable, but more weakly. I could hear it resolving down to the 5, but the urge is not strong.
Now 3x3x3x3, the 3+, Pythagorean major third.
I don’t think there’s much disagreement that this is a dissonant note. This major third is 22 cents, an entire comma, sharper than the sweet 5/4 third that appears as “3” on the lattice. It’s has the sharpness of the equal tempered third, on steroids.
For me, the major third is the Achilles Heel of Pythagorean tuning. Any tuning with dissonant major chords is going to be of limited usefulness. I’d go nuts if all songs ended on this note.
Yep, ugly. But still feels somewhat stable. Next?
Ow! And the stability is very weak now; I’d be fine with this note resolving to the 1.
Keep going! Next up is a suitably obnoxious tritone, a version of the good old Devil’s Interval. To my ear, the stability is now lost, and actually the dissonance itself is less profound than it is with the 3+ and the 7+. Tonal gravity is very weak here in the outer solar system.
Just for fun, here are the notes in reverse order. You can hear them coming back to consonance and stability.
To my ear, this experiment supports the idea that more harmonic distance equals more dissonance. The sheer obnoxiousness does seem to fade a bit past the major third or major seventh. I’d say that both consonance and dissonance get weaker as the note gets farther from the center. The ear seems to have less signal to work with as the tonal gravity field weakens.
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.
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:
How the notes sound together, away from any musical context. The range would be from smooth and harmonious to rough and grating.
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:
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.
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?
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.
If you move by as few matchsticks as possible when going from triad to triad, you will generate a chord progression that “makes sense” to the ear.
Here’s a rather artificial matchstick chord progression in Mixolydian mode. All I do is flip from each triangle to the one that borders it. It isn’t great music, but it shows how moving small distances on the lattice can draw the ear to a distant spot and bring it back again.
Actually this progression does drag the ear along rather fast. The roots move by major thirds (solid lines) and minor thirds (broken lines), which are not the shortest distances on the lattice. I like those equilateral triangles — they make visualizing easier for me — but if I wanted to accurately show harmonic distance, the horizontal lines, showing movement by fifths, should be the shortest, and the broken lines, the minor thirds, should be the longest, with the major thirds in between.
Progressions that move left or right, by fifths, are easier yet to follow.
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.
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.
One possibility is to just drop the 7 to the b7:
I like the interpretation below. It uses the b7-, and also changes the 2 to a 2-.
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.