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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.

Next: Untempered vs. Just Intonation

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Posted by on Jan 16, 2013 in Just Intonation, The Lattice | 0 comments

Melodic Space, Harmonic Space

Throughout my musical education, I’ve been taught that music happens in a linear space. This is the space so beautifully laid out on the piano keyboard.


Music teaching is organized around scales. In most Western music, the full scale consists of twelve notes, equally spaced. Other scales, such as the seven-note minor and major scales, are subsets of this full, “chromatic” scale. Due to octave magic, a mysterious and crucial aspect of our inner perception, when we get to the thirteenth note, we have multiplied the original note by two, and the sequence starts over again.

So, fortunately for musical analysis, melodic space can be described in one octave. It takes about ten of these octaves to cover the range of human hearing.

On the piano keyboard, melodies look the way they sound. When the pitch goes up, you move up the scale, and when the pitch goes down, you move down the scale. Short distances (the shortest is from one key to the next, a half step), feel short. Long distances (more than about three half steps) feel long. This is a good and useful space for visualizing melody.

Harmony, not so much.

Musical nomenclature, as I’ve pointed out before, has grown like an old city over the years. As music theory changes, bits and pieces of the old terminology are appropriated and redefined by new thinkers. The result is a cobbled-together mass that has a lot of weird contradictions and misleading names.

I think one of the most regrettable bits of confusion comes from the word interval.

The distance between two notes on the keyboard is called an interval. When my melody moves by an interval of a minor third, it has covered a distance of three half steps. When I move by a major third, I’ve covered four half steps. The major interval is bigger than the minor one — that’s why it’s called “major.” No problem! The move feels bigger when you sing it.

The problem comes when you start to think about harmony — two or more notes sounding simultaneously. The word “interval,” with the same connotation of pitch difference, is also used to describe the distance between harmony notes. Yet in the world of harmony, the interval, or pitch distances don’t make any intuitive sense at all.

For example, two notes a fifth apart (seven half steps) sound wonderful when played together. C and G are two such notes. They are closely related to each other, harmonically. So are C and F, which are a fourth apart (five half steps). These are the best consonances there are, except for unisons and octaves.

So what about the note in between them, an interval of six half steps?

Yep, none other than the dreaded tritone, the devil’s interval, definitely a dissonant note.

If the linear scale were the best way to think about harmony, wouldn’t the tritone be between the fourth and fifth in consonance? Why would three notes in a row, next-door neighbors on the scale, be so drastically different from each other harmonically? The scale gives no clue. You just have to remember.

Perhaps there is a more intuitive way to visualize harmony, one that puts harmonically related notes closer to each other, and puts the notes that are harmonically farther apart … farther apart?

I think there is indeed a harmonic space as distinct from a melodic space. This space can be illustrated on the lattice. It’s not a good model for melody — scales do a much better job. But it’s a great model for visualizing harmony — what you see corresponds intuitively to what you hear.

The interplay between these two spaces creates the beautiful dance that is harmonized music.

Next: Cents

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Posted by on Nov 6, 2012 in Just Intonation, The Notes | 0 comments

Notes and Intervals

A note, in music, is a sound with a particular pitch. Pitch is frequency, measured in cycles per second, or Hertz (Hz). The faster the vibration, the higher the pitch.

A vibration, at, say, 220 Hz, all by itself is a note by that general definition. But the note doesn’t acquire its distinct personality until it’s considered in relation to some other note. That relationship is called an interval.

Here is that 220 Hz note, played on a cello, all by itself: 220 Hz

Here it is in relation to a note an octave below, vibrating half as fast, at 110 Hz: 220 and 110

It still sounds like the same note. But now play it with a note vibrating at 1/3 of its frequency, or 73.33 Hz. The 220 Hz note acquires a very different character: 220 and 73

And now with a note at 1/5 its frequency, 44 Hz: 220 and 44

Even though the 220 Hz note always has the same pitch, in a different context it has a different personality and function.

The lattice of the Flying Dream video does not show absolute pitch. Each intersection, or node, represents a note, named according to its relationship to one special note: the Tonic.

Next: The Tonic

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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.

Next: Beauty is Truth

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