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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?


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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Posted by on Jan 28, 2013 in Just Intonation, The Lattice, The Notes | 0 comments


Musical notes can be mapped onto many different spaces. The two I find useful so far are:

— Harmonic space, the space of the lattice, organized by harmonic connections (ratios of whole numbers).

— Melodic space, the space of the scale, organized by pitch, or frequency.

Both maps show the location of a note relative to a reference tone, the Tonic, the “do” of do-re-mi.


Distance on the lattice could be measured by the number and length of the connections to the tonic, sort of “how many Tinkertoy sticks away are we?”

How to measure distance in melodic space?

One of my favorite music theorists is Alexander Ellis. Ellis was an interesting character, a researcher in phonetics, and the prototype for Professor Henry Higgins of George Bernard Shaw’s Pygmalion (My Fair Lady). He wrote a huge appendix for Helmholz’s foundational book about psychoacoustics, On the Sensations of Tone, in which he laid out a version of the harmonic lattice that is very much like the one I’m using. The appendix was published in 1885.

Ellis proposed dividing each equal-tempered semitone into 100 equal parts, called cents. This gives 1200 cents to the octave. Cents have caught on almost universally as a way to describe and compare pitches of tones.

Cents are a logarithmic unit. Logarithms form a bridge between addition and multiplication. When you add logarithms, you are multiplying in the real world. Adding 1200 cents is the same as multiplying by 2. When you add one cent, you are multiplying by a small number, the same number each time. It’s the 1200th root of two, in fact, a very small number, about 1.0006. Multiply by 1.0006, 1200 times, and you get 2.

The ratios themselves show what the pitch of a note will be, and there’s a formula for translating from harmonic space (ratios, the lattice) to melodic space (cents, pitch). It is great fun, if you’re a geek like me, to plug this formula into a spreadsheet and start exploring the musical spectrum.

For any ratio, b/a, the pitch in cents is:

1200 x log2(b/a)

That’s log to the base 2. A good straightforward explanation of logarithms can be found here. They are a handy concept in the study of perception, since many human senses, including visual brightness, loudness and pitch, work in a logarithmic way. A 100-watt amplifier sounds louder than a 10-watt amp, but it’s nowhere near 10 times as loud. Maybe three times as loud, subjectively? A 10-watt amp is louder than a 1-watt by about the same amount. I have a 1-watt Vox tube amp that the neighbors have yelled at me about. For something to sound “twice as loud,” it has to be moving something like 4 or 5 times as much air.

So let’s run that formula. The untempered major third is a ratio of 5/4.

log2(5/4) = 0.32

x 1200 = 386.3 cents

The ET major third is at exactly 400 cents, 14 cents sharper. This is a clearly audible difference — the ear can distinguish a difference of about 5-10 cents.

Cents give us a language for comparing pitches, and quantifying the differences between them.

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Posted by on Nov 13, 2012 in The Notes | 0 comments

The Major Third

Multiplying a note by 2 creates an octave, and multiplying it by 3 creates a perfect fifth.

Multiplying by 5 gives yet another new note, the pure major third.5-1

5/1 is over two octaves above the original note, so you have to reduce it twice (divide by 4) to get it down into the same octave.5-4

Now we have four notes: 1/1, 5/4, 3/2 and 2/1 — enough for a scale.1-3-5-8

This scale is contained in the chord of nature, and it pops up all over the place. A clear example is the bugle.

Bugles have no valves or keys. So how can you play more than one note on one?

A bugle is a long tube full of air, curved so it fits in a small space. The player’s lips get the air column vibrating, and by changing the tightness of her lips, the player can coax the air column into vibrating along its whole length, or get it to break up into sections, just like the jump rope in the Chord of Nature demonstration.

Here are the bugle notes:bugle scale

Two sidebars before I go on.

1) Isn’t it strange that when you multiply by 3 you get a fifth and when you multiply by 5 you get a third? The note names come from their position in a seven-tone scale. Here’s how our new scale fits with the standard do-re-mi. The notes we’ve explored are played louder to set them apart.five notes in do re mi

The 5/4 note pops up third in the scale and the 3/2 note comes up fifth. It’s just a confusing coincidence, based on our fondness for seven-tone scales.

2) Here’s a sneak preview of why I’m going to all this trouble. The equal-tempered major third that we’ve been hearing all these years is not tuned to the 5/4 ratio. It’s tuned sharp, by almost 1%. This isn’t enough to make the note sound obviously sour, but it’s certainly enough to change the feel of it.

Try listening to the following example a few times, and pay attention to how you feel while listening.JI3 vs ET3

The first note you hear is the tonic with a pure major third. The second note is with an equal tempered major third. Then it goes back to the pure 5/4 note. The pitch difference is small, but I perceive an uneasiness, almost a queasiness about the equal-tempered version. Do you hear a difference, and if so, how does it feel to you?


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Posted by on Nov 12, 2012 in Just Intonation, The Notes | 0 comments

Octave Reduction

Doubling the frequency of a note certainly changes it. The ear hears a higher-pitched note. But there is something in the essence of the note that does not change, a character that stays consistent through the octaves.

This allows a process called octave reduction. When you’re working with notes as ratios, it’s convenient to multiply or divide the raw ratio by 2, as many times as is necessary to bring it into the same octave as the tonic.

3/1 generates a perfect fifth. 3-1

This note is actually an octave plus a fifth above the tonic. Now divide by 2 and you have 3/2, one and a half times the original frequency, and just a fifth above. 3-2

The reference frequency is 1, the octave is 2, so what you want to achieve with octave reduction is a ratio, or fraction, between 1 and 2.

These are the beginnings of a scale, a collection of notes within a single octave. Such a scale can be repeated up and down the octaves to cover the whole range of hearing.

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Posted by on Nov 4, 2012 in Background, Just Intonation | 0 comments

Pythagoras’ Epiphany

Pythagoras was a Greek philosopher who lived about a century before Athens’ golden age. Some time before 530 BC, he had an epiphany. He had been investigating vibrating strings, and found that when you cut the length of the string in half, the note it makes is an octave higher.

Sound is vibration. When one sound is vibrating twice as fast as another, it still sounds the same in some crucial way. The pitch is higher, or lower, but somehow we perceive it as having the same essential character. A C note, multiplied or divided by two as many times as you like, still sounds like a C.

Here are all eight C’s on the piano. They are different in pitch, but all have the same character. Eight Cs

Pythagoras also found that when you shorten the string to a third of its original length, it vibrates three times as fast. The note this creates is different in character from the C. Today it is called a perfect fifth. If you’re in the key of C (that is, if the full string sounds a C), this note will be a G.

This observation led him to what must have been a terrific epiphany — math, particularly number, is at the heart of all things. I sometimes envy those early Greek thinkers — what joy, to come across something basic for the first time!

But you know, everyone, everywhere, has lived in modern times. A thousand years after Pythagoras, Galileo was the first to find out that the Milky Way is made of stars. Can you imagine how he felt? And we are still on the cutting edge — civilization is in its infancy. Future generations will envy us our discoveries while smiling at their primitiveness. “A keyboard, how quaint!”

Pythagoras’ epiphany still has merit. Cosmologists have imagined many alternate universes, with different basic physical constants and laws, curved space, more dimensions — but it’s pretty tough to imagine a universe without number. I believe the integers — 1, 2, 3 and so on, are the most basic things we know about for sure.

Pythagoras actually founded a religion based on this insight. The inner circle were called the mathematikoi, and they lived a monastic life of study. The order had many rules, including a ban on eating beans. Perhaps they worked in close quarters. They also had a rule against picking something up when you dropped it. Cluttered, close quarters! But they found out a lot about math.


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