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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 harmonychord, 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.

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