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 that experiment, I kept the axis (3) and direction (multiplication, overtonal) the same, and increased the distance.
This time I’ll keep the axis and the distance the same, and switch direction. Each illustration will compare a note with its mirror twin, its reciprocal.
First up is the strongest polarity flip there is, the perfect fourth and fifth. One divides the tonic by 3, the other multiplies it by 3.
The 4 is clearly unstable, it wants to move. The 5 is clearly stable. If a song ends with this interval, I will feel completely satisfied.
The next matchup is the b7- and the 2. The b7- is the crucial note that provides the tension in dominant-type seventh chords and makes their resolution so satisfying. Here it is in undiluted form.
The 2 is fairly stable. Quite a few songs end on this note, and there is a pretty good sense of resolution, maybe with some wistfulness mixed in.
The two notes are about equally harmonious, and of opposite polarity. This is the same pattern as the 4 and 5, only weaker.
Moving outward, we get the b3- and 6+ pair:
The pattern continues — now both notes are rather dissonant, with the b3- weakly unstable and the 6+ weakly stable. It would be rather unsettling to end a song on the 6+, but maybe you could get away with it.
Here are the next two:
These are interesting. They are dissonant, all right, and the b6- is unstable and the 3+ is stable. But I actually hear the polarity a little more strongly than the last pair.
I think my ear is trying to interpret these notes as out-of-tune versions of the b6 (a strongly unstable note) and the 3 (strongly stable).
How is my ear to interpret this 3+ note, the Pythagorean major third? Can I even hear a ratio of 81/64? Maybe not well enough to really recognize it.
Perhaps the ear “decides” that it’s simpler to read this strange note as a badly tuned version of a simpler interval, one I am familiar with. So I hear it as an out-of-tune 5/4 instead of an in-tune 81/64.
This is why equal temperament works, as Mathieu demonstrates so well in Harmonic Experience. A painting doesn’t have to be exactly straight on the wall for the eye to interpret it as straight. Thank goodness! In the same way, a note doesn’t have to be exactly in tune to be heard as that note. The ear is willing to accept “close enough” and hear it as the real thing, though the consonance will not be as good.
Maybe the part of the mind that processes this stuff is like a quantum computer, taking in the sound, trying out all possibilities at once, and spitting out the “most likely” interpretation, which would be the solution with the lowest “potential energy,” the one that is closest to the center, just like real gravity.
We’re probably too far out now to really recognize these intervals as what they are, but for the heck of it:
Suitably nasty, and now the sense of polarity is pretty much gone, I can’t hear it.
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.
Almost all Western music, including my own, lives in the world of tonal harmony. This means:
There can be, and usually are, multiple notes playing at the same time.
There is a key center, or tonic, around which the notes are arranged. The tonic doesn’t always sound — it’s an intangible presence, the home from which you leave on your harmonic journey, and to which you will hopefully return.
The multiple notes can have different functions:
Roots are the fundamental notes of chords. A G chord has its root on G. Roots are local centers that move the ear around the lattice as they change.
Harmonies flesh out the chord. In a G major chord, the harmony notes are B and D. They stake out more lattice territory and add definition to the chord. Is it a G major, minor, seventh? The harmonies establish this.
Melodies dance in the harmonic field set up by the tonic, roots and harmonies. They have more freedom than the others. Melodies travel fast and light, and though they can sing the same notes as the others, they can also travel farther afield, further embellishing the chord, or leading the ear toward the next chord in the progression, or lingering on the last one after it has changed.
All this action is happening in two musical spaces at once.
Melodic space is the world of scales. It’s organized in order of pitch. The piano keyboard is a perfect representation of melodic space.
Harmonic space is the world of ratios. Multiply a note by a small whole number ratio, and you have moved a small distance in harmonic space. Multiply by large numbers, and you have moved a large distance. The lattice is a map of harmonic space.
The two worlds are not the same. Often, they are opposites. The perfect fifth is a small move harmonically but it’s a mile in the melody — bass singers have to jump all over the place in pitch. Small melodic moves tend to be big harmonic ones. A chromatic half step, the distance between the 3 and b3, is only 70 cents, less than the distance between neighboring keys on the piano. But on the lattice, it’s a long haul — down a third, down another third, and up a fifth.
Writing and arranging a song is sort of like designing (rather than solving) a crossword puzzle. There are two intersecting, independent universes, Up and Down. To design the puzzle, you work back and forth between the two, massaging them until they don’t conflict, and each one makes sense on its own.
All of the notes live in both harmonic and melodic space. They may have a foot in one more than the other — the roots tend to move small distances on the lattice, the melodies usually move small distances in pitch, and the harmonies tend to bridge the two, moving melodically while staking out the form of the music on the lattice. But every note moves in both spaces, all the time.
A great advantage of the lattice is that it serves as a sort of Rosetta Stone, a bridge or translator between the two worlds.
The Rosetta Stone was carved in 196 BC and rediscovered in 1799. It immediately became famous because it repeats the same text three times, in three different languages. It was the key that allowed scholars to decipher Egyptian hieroglyphs.
The lattice bridges the two musical spaces by means of the patterns it presents to the eye.
When two or more notes are plotted on the lattice, they will form a particular visual pattern. Any time you see this pattern, no matter where on the lattice it is, the relationship between the notes of the pattern will be exactly the same, inboth harmonic and melodic space.
For example, this pattern shows an interval of a major third. The ratio of the frequencies of these two notes is 5/4 (or 5/2, or 5/1 — twos don’t count, they just shift the note by an octave). Any time you see two notes in this formation, no matter where they are, you know they have the following relationship to each other:
Harmonic space: When the notes are sounded simultaneously, they will have the characteristic sound of a pure major third.
Melodic space: When you move from one note to the other, you are traveling a distance of 386 cents, or about four semitones on the piano.
Getting familiar with these patterns, and learning to recognize them wherever they are, has made it easier for me to think in harmonic and melodic space at the same time, which makes writing and arranging music much easier.