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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.

Next: Consonance Experiment

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Posted by on Jun 21, 2013 in The Lattice, The Notes, Tonal Gravity |

The Power of the Seventh Chord

The V chord, the major chord based on the 5, is a powerful compositional tool. It points, very clearly and with a lot of tension, directly at the tonic. If you want to lead the ear to the I, the V chord is the top-of-the-line triad.

Why this is so is still a bit mysterious to me. It’s been discussed a lot. It seems to have both melodic and harmonic elements.

Melodies “like” to move short distances in pitch, and the move from the V to the I is elegant melodically. The 7, or major seventh, resolves up a half step to the 1. The major seventh is called a leading tone because of this very property. The 2 drops a whole step, also to the 1, and the 5 stays put.

In harmonic space, voices, especially roots, “like” to move short distances too. The shortest move of all is a fifth, and when the V goes to the I, the root moves down by a fifth. It seems natural that if the ear is anticipating the next chord, it will place its bet on the change that expends the least energy. All three notes could be seen as moving that same short distance, the easiest possible move.

I like to think of it in terms of tonal gravity. The tonic, the 1, is like a sun at the center of a solar system, and it exerts a gravitational pull. Moving away from it creates tension, collapsing into it creates resolution. Just as with gravity, the closer in you are, the stronger the force. The V is right next to the I, harmonically, so the tension is very strong.

The V chord isn’t the last word, however. It’s possible to crank it up, by adding another tense note.

P1080225

The 4 and the 5 are the closest notes to the 1, in harmonic space. These two notes have the strongest tonal gravity of all. Their effect is different — 5 is the strongest overtonal note, and 4 is the strongest reciprocal note. Both point straight at the tonic.

Melodically, the 4 is two half steps below the 5. This makes it a flatted or minor seventh, added to the V chord. So the final chord is called a V7.

Of all the notes we could add to the V chord, the 4 creates the most tension, and it’s pointed directly at the tonic. I say this is the source of the power of the dominant 7th chord.

In Be Love, I add even more tension before I’m through. The melody dances around, and right before the final resolution, it lands on the 6.

P1080237

I’ve added yet another tense note to the mix. It’s not as strong as the 4, but it jacks up the gravity another notch. The root is on 5, so the 6 is two half steps up from it melodically. This makes it a ninth chord — start with the basic major triad, and add a seventh and a ninth.

Now I’m set up as strongly as possible for a return to the tonic, and sure enough when the drop happens it lands with authority. I’m in major land now, and the chorus will feel entirely different from the verse.

Here’s the whole effect:

Next: To the Far Northwest

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Posted by on Jun 19, 2013 in Consonance, Just Intonation, Resonance, The Lattice, The Notes |

The Compass Points

There are two basic directions on the lattice: multiplication and division.

If I start with a note, and then multiply it by 3, or 5, or 7, I will get a harmony note with overtonal energy. Such a note is in the natural overtone series of the original note.

Overtonal energy is stable, restful, it belongs where it is and wouldn’t mind staying there.

If I divide by 3, 5 or 7, I get a completely different kind of note. I call this division energy “reciprocal,” after W.A. Mathieu’s suggestion in his amazing book Harmonic Experience.

Reciprocal energy is restless, unstable. The note wants to move, or for the music to come to it, until it is overtonal.

On the lattice of fifths and thirds, there are two axes, fifths and thirds, and two directions, overtonal and reciprocal.

This makes four total directions one can move on this lattice. Each direction has own characteristic flavor, or energy. I use the following names for these energies, mostly after Mathieu.

  • Dominant = East = Overtonal fifths
  • Subdominant = West = Reciprocal fifths
  • Major = North = Overtonal thirds
  • Minor = South = Reciprocal thirds

Compass Points

Every interval has its own unique recipe of moves in these four directions. The perfect fifth has pure dominant energy, the major third pure major. The minor third, b3 on the lattice, is a compound note — dominant and minor.

It’s interesting to look at the minor third (b3) from the viewpoint of tonal gravity. On the horizontal axis, dominant/subdominant, the b3 is overtonal, stable, restful. On the vertical axis, major/minor, the note is reciprocal, unstable, restless.

Tonal gravity is stronger the closer you are to the center. To make a minor third, you multiply by 3 (an overtonal jump of a fifth), and divide by 5 (a reciprocal jump of a third). I know, 3 generates fifths and 5 generates thirds, a confusing coincidence.

Fifths are closer to the center, harmonically, than thirds are, so the overtonal energy is stronger than the reciprocal.

This makes the minor third a stable note, although less stable than the major third. Songs can end on a tonic minor chord and they will still sound finished.

Next: Leading the Ear

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Posted by on Feb 26, 2013 in Just Intonation, The Lattice, The Notes | 0 comments

Another Major Second: The 10/9

When I started exploring the extended lattice beyond the central 12 notes, the first note that was really new to me was the 10/9 major second, also called the minor or lesser whole tone. Now I call it the 2-.
Other-major-2-latticeThe lattice extends forever in all directions. When you continue multiplying and dividing, generating new notes beyond the boundaries of the central zone, the notes start to repeat, but not quite. The notes in red, the 2 and the 2-, are very close in pitch. They are different flavors, if you will, of the interval of a major second, or whole tone — a distance of two half steps, two keys on the piano.

Even though they are so close in pitch (204 cents for the 2, 182 cents for the 2-, only 22 cents apart), the two major seconds are generated in different ways and have very different functions and characters.

The 2 is an entirely overtonal note, that is, generated by multiplying alone. Such notes can be found in the chord of nature, the harmonics of a vibrating string. The character of notes is somewhat subjective, but for me, overtonal notes have a stable, sort of upbeat or positive character, and even though the 2 is somewhat dissonant, it has a kind of peaceful sound, that shows up well in ninth chords. Its recipe is x3, x3, or x9, octave reduced to 9/8.

The 2- is a combination of reciprocal and overtonal energy. It’s farther from the center than the 2, and more dissonant. Its recipe is /3, /3, x5, or 5/9, which octave reduces (or expands, really) to 10/9. It is darker, bluesier perhaps, and functions differently in chord progressions.

These very similar ratios, 10/9 and 9/8, 182 and 204 cents, are in fact entirely different beasts. Equal temperament has obscured this difference over the years. In ET, both notes are played at the compromise pitch of 200 cents, but that does not change the functional difference. It is extremely useful when writing or arranging to know whether you are playing a 2 or a 2-.

I tried making a demo of how they sound, as with other notes, but I think that played by themselves, out of context, the 2 and 2- are hard to tell apart. To get the difference, I think you have to sing them against a drone (scroll down the linked page a bit and there’s a list of Indian drones to play around with, it’s really fun to improvise melodies over these) and feel them in your own body. Mathieu shows you how to sing the 10/9 note in Harmonic Experience.

The functional differences really show up when you’re designing chord progressions that make sense. A chord progression is a journey on the lattice, and if you’re roaming in western territory, that is, to the left of the center, you want to use the 2- in your chords and melodies, and if you’re in overtonal, eastern lands, to the right of center, the 2 is going to sound better. It’s a crucial distinction in just intonation. Not so much in ET, since the notes are tuned the same — but awareness of where you are on the lattice really helps when you’re writing ET chord progressions.

It’s an old puzzle. Why do some progressions feel “right,” and others “wrong”? Knowing the map of harmony, the lattice, helps a lot. Much more to come in later posts.

Next: Commas

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Posted by on Dec 9, 2012 in The Notes | 0 comments

The Minor Second

The last three notes (b6, b3 and b7) are related to each other. They all contain a reciprocal third. There is a family resemblance of sound and function. (They also all happen to be a little flat in equal temperament. On a guitar it’s a nice trick to bend them a little to sweeten them.)

Here is another note in the family, farther out harmonically, the minor second:

That’s a dissonant interval. The b6 is already tense with reciprocal third energy. Now this b2- (The minus is an accidental to show its exact pitch; more later) is another reciprocal fifth beyond (below?) that note. Its ratio is 1/15, which expands to 16/15 — just above 1. See how the ratios show where the pitch of the note is? 1/1 is the tonic, 2/1 is the octave. 16/15 is just a little bit greater than 1, so it’s just a little sharper than the tonic.

It’s not pitch so much that makes consonance and dissonance. It’s harmonic relationship.

Music is all about tension and resolution. Here’s a very tense note. How to resolve it?

One answer is just a half step away, a drop to the tonic.

That’s a move in melodic space. The tonic is right next door and it’s an easy drop.

On the lattice, the 1 is not a next door neighbor. How about going home through harmonic space instead?

Going to the 4 is an interesting experience for me. There’s still reciprocal tension, but I’m much closer to home — I can smell the stables. It’s as though I felt a bit lost at the b2-, the harmonic distance was too great to really get my bearings. But moving to the 4 allows me to figure out where I am, and where the tonic is, so that the final move home sounds really right. The 4 says to me, “There is home, now go.”

Then the melody moves to the 5, and there is resolution. The 5 sends just as strong a signal as the 4, but of opposite polarity. The 5 says, “Here is home. Now stay.”

It’s a little story, a journey on a microcosmic landscape of attraction, repulsion and beauty.

Next: The Augmented Fourth

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