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Posted by on Jul 30, 2013 in Consonance, The Lattice, The Notes, Tonal Gravity | 2 comments

Harmonic Distance

Harmonic distance is the total length of the connection between two notes on the lattice, as measured on the solid lines. The more tinkertoy sticks you traverse to get from one note to the other, the greater the harmonic distance.

It’s not the same thing as melodic distance, which is a difference in pitch. Two notes can be far apart in harmonic space, but close together in melodic space, or vice versa. This post has a demonstration.

Each solid line on the lattice is a prime factor — 3, 5 or 7. A simple way to put a number on harmonic distance is to multiply together all the prime factors used in the ratio of the interval. Doesn’t matter if you’re multiplying or dividing by the factor, the distance is the same. Twos don’t count; these are octaves and they don’t add distance on the lattice.

The closest intervals on the lattice are the perfect fifth and perfect fourth. To get these intervals, you multiply or divide the original note by 3. The ratio of the fifth is 3/1, and the ratio of the fourth is 1/3. The harmonic distance is 3, in both cases.

The major seventh, or 7, is a more distant interval. Its formula is x3, x5, or 15/1, so its harmonic distance is 15.

The b2- is the reciprocal of the 7. Its formula is ÷3, ÷5, or 1/15, and it is equally distant. The polarity is opposite, but it’s the same distance away from the center.

There are two other notes at this same distance of 15 — the 6 and the b3. Their ratios are 5/3 and 3/5 respectively. They are reciprocals of each other, and have opposite polarities.

Here is the inner lattice, showing the ratios (without any factors of 2), and harmonic distances instead of the note names. The ratio of an interval defines it completely; it would make perfect sense to name the notes by their ratios alone (it’s been done).Harmonic Distance central lattice

 

In the consonance experiment from a few posts ago, I played intervals in order of harmonic distance, and sure enough, as they got further out, they got more dissonant. I used the Pythagorean axis (multiples of 3) to keep it simple. Pythagorean tuning is somewhat limited musically; harmonic distance increases so fast that there are very few consonant notes.

On the lattice of thirds and fifths, there are more consonant notes to play with. How would that same experiment sound, when you add in these new intervals?

I’ll stick with the overtonal, Northeast quadrant of the lattice. Every ratio involves multiplication only, so there is no reciprocal energy, and I’m not comparing apples to oranges. My intention is to test only one ingredient of consonance, the harmonic distance. The intervals travel away from the center, and back again. Listen and watch a couple of times, and hear what happens.

I think the pattern holds very nicely. At the very end, the #4+ with its distance of 45, I think the dissonance has lost some of its obnoxiousness. It does appear that as the distance gets big enough, both consonance and dissonance start to weaken. The ear has less to go on, the signal is weaker.

Also note how the other component of consonance, stability/instability, changes as we roam farther out and come home again. All these intervals are stable, since they are all overtonal. This sense of stability gets stronger the closer we are to home, as though the ear is receiving a stronger signal and is more and more sure of itself. I start to clearly hear the stability at the major seventh (15/1), and it quickly gets stronger from there on in.

Next: Mirror Twins

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Posted by on Jul 29, 2013 in Consonance, Just Intonation, The Lattice, The Notes, Tonal Gravity | 0 comments

Polarity Experiment

In the last post I did a consonance experiment, listening to intervals with wider and wider spacing.

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.

Finally:

The Pythagorean spine, the sequence of fifths, has come full circle — almost. The two notes are 24 cents apart, a Pythagorean Comma. All that remains of tonal harmony at this distance is a generic sort of dissonance. I hear no polarity at all. The tonal gravity field is too weak to detect.

Here’s one more video to bring it all back home. I start to smell the stables at about the b3-/6+, and the sense of direction gets rapidly stronger from there.

Next: Harmonic Distance

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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 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:

  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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