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Posted by on Jul 27, 2013 in Consonance, Just Intonation, The Lattice, The Notes, Tonal Gravity | 0 comments

Consonance Experiment

In my last post I raised the idea that the consonance of an interval is not just one thing, but has two distinct parts:

  1. The sensation created by the sound of the two notes played together — smooth or rough, pleasant or unpleasant
  2. 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.

Next: Polarity Experiment

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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 Jul 20, 2013 in Consonance, Just Intonation, The Lattice | 0 comments

Mixolydian Matchsticks

In yesterday’s post I mentioned matchstick harmony.

This concept is from Mathieu’s book Harmonic Experience, which I’ve discussed a lot on this blog.

Matchstick harmony is governed by a rule: It’s easiest for the ear to follow harmonies that move short distances on the lattice.

Imagine that the lines of the lattice are matchsticks. The triangles that they make are triads, major ones pointing up and minors pointing down.

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.

Next: Consonance and Dissonance

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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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Posted by on Jun 23, 2013 in Equal Temperament, Just Intonation, Septimal Harmony, The Lattice, The Notes |

The Septimal Minor Third

At the very end of the chorusBe Love showcases one of my favorite notes, the 7b3.

Most of the notes of the inner lattice can be approximated on the piano, but not the septimal minor third. It’s in between the keys. Blues pianists can evoke it by trilling between 2 and b3, but only variable-pitch instruments can actually hit the note.

Here it is in context:

It’s fun to sing this part of the song, stretching out that septimal note and tasting its flavor.

The passage illustrates the harmonic function of the note. It’s the septimal flatted seventh of the 4, also called the harmonic seventh or barbershop seventh. This is a beautifully consonant note, a great addition to a major chord. It’s generated by multiplying by seven. I use it here as a harmonic seventh over the IV.

Relative to the 1, the 7b3 is a compound note. To get there, you divide by 3, and then multiply by 7. The ratio is 7/6, octave reduced. The pitch is 267 cents, between the 2 and the b3. Here it is on the scale. The colored notes are in just intonation, the black ones are in equal temperament.

Scale with 7b3

Over a I chord, the 7b3 sounds bluesy, restless, gutsy — it’s the insistent melody note in Taking Care of Business. It’s at the heart of the guitar riff in Dizzy Miss Lizzy (George often bends it up to the major third), it’s Jagger’s haunting first “ooooh” of Gimme Shelter. Gimme Shelter

The Stones’ music is a feast of 7b3’s. So is Led Zeppelin’s. These septimal notes are found everywhere the blues has left its impression.

There’s an old question: why do the minor melody notes of the blues sound good over major chords? The web is full of discussions as to why this is so.

I think it’s because the blue minor third is not the b3, but the 7b3. The regular minor third is a reciprocal third, and harmonically it doesn’t fit with major chords — it’s in a different part of the lattice.

But the 7b3 is an overtonal 7th, built on the 4, generated by multiplication. The major notes are made by multiplying by 5. Times 5 and times 7 go together very well. The harmonic seventh chord is a thing of beauty.

There’s an implication for blues guitar. You can’t play this note in the classic minor pentatonic blues box. You can play a b3 (bend it a little to tune it up), or a 3 (bend it harder), but not a 7b3, it’s flat of the b3, and you can’t bend down.

minor pentatonic

You can play a 7b3 by grabbing the 2, one fret below, and gently bending up to it. The following box works great for septimal notes. They’re all laid out under the ring finger. Bend them by less than a half step.

major pentatonic

This box makes it easy to play the classic bit of melody, 7b3 – 2 – 1. All three songs I linked to earlier have this melody in their bones — BTO, the Stones, the Beatles. It’s everywhere. Here it is in Be Love.

Next: Mixolydian Mode

 

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Posted by on Jun 22, 2013 in Just Intonation, Recordings, The Lattice, The Notes, Tonal Gravity |

To the Far Northwest

The chorus of Be Love is solidly in major territory. The chords are I, V, IV, the classic backbone chords of the major scale.

The verse, however, is going to be back in the Northwest — I, bVII-, IV. I have to get back over there somehow, and I want to lead the ear strongly so that the move feels right.

Once again I’ll use a dominant seventh type chord. Here’s the shape again:

P1080225

The ear expects a I chord to come after this. Perhaps it’s one of those “nature abhors a vacuum” things. All the notes have tonal gravity that is pointing at the center, yet there’s nothing there. It’s as though the planets are orbiting the sun, and the sun is missing. The ear wants to put it there.

This “dominant seventh” shape is named after the dominant chord, the V7, where it shows up so often. It can be used elsewhere, and is the main tool used in classical music to change keys.

When I started to work this out, I already had a climax in mind for the chorus. In the last line, I wanted to leap all the way to the far Northwest, the ii- chord (lower case roman numerals mean it’s a minor chord), and have the melody sit still while the chords revolve around it. Here’s the ii-:

P1080430

 

It’s a long leap, but I can make it with one transition chord, the I7.

P1080421

Adding a seventh to the I chord sends a strong signal to the ear: “We’re going west!” Normally the next chord would be a IV.

This move is used all the time in music. It’s common for the bridge of a song to start on a IV chord — it’s a shift on the lattice that makes the bridge sound different from the rest of the music, much like my shift to major for the chorus of Be Love. Putting a I7 right before the bridge tells the ear to expect this shift.

I do something a little different, though. There is also a huge vacuum where the ii- ought to be — look how the two chord shapes fit into each other. Sure enough, going to the ii- after the I7 is satisfying and dramatic. Here’s how the sequence looks and sounds.

Notice how the melody leads the way again, by going to the 4. There’s another lovely passing chord, a stack of perfect fifths, right before the change.

P1080425

 

Next: The Septimal Minor Third

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