Notes are pitched sounds. A given note means little by itself. It could be the tonic of a key, or some member of a scale based on a different tonic. By itself, it generates no tension, resolution or sense of place on the harmonic map.
So when I name a note in this blog, I’ll usually be referring to a ratio, the relationship between the note and a reference note — the tonic, or the root of a chord, or another note in the harmony or melody.
Ratios are fractions. The first number is divided by the second number to give the value of the ratio.
If the tonic is, say, 100 Hz, then another 100 Hz note is related to the tonic by the ratio 1/1. This is the interval of a unison, two identical notes.
Each note name on the lattice represents a unique ratio, relative to the tonic. The 1, at the center, stands for 1/1.
Next: Octave Reduction
When I first learned to play guitar, I would sit on the couch late at night and pluck the low E string, again and again, and just listen to the beauty of the sound as it died away.
That low E note is not just a simple vibration. The full length of the string is vibrating at about 82 Hz. But a pure 82 Hz note doesn’t sound like a guitar string at all. It sounds like this:
The guitar sound is much more complex:
guitar low e
The difference comes from the fact that a string doesn’t just vibrate along its whole length — it also vibrates at twice the frequency, three times, four, and so on — all at the same time!
Maybe you did this as a kid. I did. When you get a jump rope going, you are essentially vibrating a big string. It has a characteristic frequency, maybe two cycles per second, set by the length of the rope and the amount of tension, just like a guitar string. This frequency, the natural vibration speed of the whole string, is called the fundamental.
But if one person holds their end still, and the person on the other end moves the rope twice as fast as usual, a funny thing happens. The rope divides in two, and the center point stays still, while each half does its own circle. Again, the length and tension determine the natural speed. Go three times as fast, and three sections will appear. These higher-mode vibrations are called harmonics.
Here are a couple of guys in lab coats to demonstrate:
This only works when you hit the right frequencies. Spin the rope at, say, 2 1/2 times the natural frequency and everything falls apart. The stable frequencies are the fundamental, 2x, 3x, 4x, 5x and so on. This video shows a string getting stable at 6x, 5x, 3x, and the chaos that happens in between.
When you pluck a real string, it will vibrate in all these modes, generating a complex sound. The particular recipe of added harmonics creates the timbre, or tone, of the note.
Here’s that same pure 82 Hz tone, with the harmonics 2x, 3x, 4x and 5x added successively:
Chord of Nature 2
This is the Chord of Nature. It is a sonic manifestation of number, and of the laws of the universe, and it’s very simple. If the fundamental frequency is 1, then the frequencies of the harmonics are 2, 3, 4, 5, 6 and so on. And somehow, our perception of sound is designed so that this sounds beautiful.
Here, in contrast, is the same demonstration but with the harmonics detuned randomly by less than two percent:
Chord from Hell 2
Yipe! Now go back and listen to the first one as a palate cleanser.
There is something deep inside us that recognizes the series of harmonics, and, for most of us, labels it “beautiful.” There is some connection between those small, whole numbers and musical beauty.
Next: Notes As Ratios
I grew up thinking that music was made with a particular set of twelve notes, the ones on the piano keyboard. I had a vague sense that there were other scales in the world, but I thought of them as “more primitive” or perhaps subsets of the 12-tone scale, like that pseudo-Asian music you make if you play around on the black keys of the piano. I certainly didn’t know that those 12 notes, now so unconsciously established that hardly anyone in Western culture even questions them, are a relatively recent invention. In Europe, where they first caught on, they were fought bitterly for a century or so before they became the norm. Even now, much of the world still does not tune to these notes, although they are still spreading.
But I also grew up deeply aware of blues singers, and that notes sung “blue” could not be duplicated on the piano, or on the guitar without bending strings. Something was always different about rock, country and other blues-influenced music. All my favorite music had this quality in common — somehow richer in sound, with more heart, and it wasn’t just feel. And it wasn’t just blues either — almost all vocal harmony had “it” too, regardless of genre.
When I was a teenager, I heard a tiny phrase that hit me like Sirius falling from the sky in the Truman Show. Here it is, fair use excerpt:
Hear it there, at the end? In the right channel, George Harrison plays something you absolutely cannot play on a piano, yet it is perfectly in tune. There is a wealth of information in that little phrase — it points to a whole world living there, in between the keys. That lick has stuck with me for all these years, a sign in the sky, that there was a lot more to know about music than I had been taught in textbooks.
Next: The Chord of Nature
Over the next series of posts, I’m going to explain how the lattice in the Flying Dream video works. Before I do, I want to take time to mention a terrific book.
I started investigating just intonation in earnest in early 2011. A couple of months in, my friend Kay Ashley loaned me her copy of Harmonic Experience, by W. A. Mathieu. Thank you so much, Kay!
I spent a few weeks with Kay’s copy and very soon knew I had to have my own. I devoured the book almost daily for at least a year. I still pull it out often, lug it to a cafe for browsing over breakfast, do bibliomancy with it if I’m stuck creatively, take it on vacations.
Harmonic Experience is the only music theory book I’ve read so far that actually increases my understanding of music, rather than obfuscating it. It’s huge, which could be intimidating. But I found it to be immediately accessible and entertaining. Mathieu has a great, light sense of humor. The concepts are introduced at a beautiful pace. And the ideas he presents are enlightening. “Aha” experiences abound.
Much of what I’ll present in this blog is heavily influenced and inspired by Mathieu. The lattice itself goes back to Euler in the 1700’s, but Mathieu expands on the idea enormously, arranging it so it corresponds to traditional musical staff notation, using it as a means to understand equal temperament, harmony, melody, chord progressions, world music, and much more.
Mathieu uses the term “positional analysis” to describe his system. For me, positional analysis opens the black box. It shows what’s happening in there. When my music is informed by the lattice, it makes more sense. I have more control over the effect it has on me and my audience. And it’s way more fun, because I know more about what I’m doing and why, rather than flailing around finding good sounds by instinct. And when I do compose by instinct (which is essential), I understand better why it sounds good, and can expand on my inspirations in a rewarding way.
‘Nuff said! If music theory has been frustrating for you in the past, as it has been for me, I can’t recommend this book highly enough.
Next: Between the Keys
The heart of the lattice is the note called 1. This note is the tonic.
Almost all the music you hear — pop, rock, classical — has one note that is at the center, a master note against which all other notes are measured. That note is the tonic. It’s the Do of Do Re Mi. When you call a scale “G Major,” or say that a song is in the key of G, the G is the tonic.
A single note means little by itself. But when it’s considered in relation to the tonic, it acquires meaning. The examples in yesterday’s post show how a note changes character when played against different tonics.
The tonic establishes the framework for the rest of the notes in a piece. It’s the anvil on which the music is forged.
The tonic can be any note. When you tune your guitar by the campfire, without a tuner, just tuning it to itself, you’ve chosen a reference frame that will make perfect sense, regardless of whether it’s the same frame as a piano or orchestra back home. You can happily play great music in the key of G-and-a-half, if you’re playing solo.
Once you’ve established the tonic, the rest of the notes are tuned, and named, relative to that note. The tonic is the center, the Big Bang of that particular musical universe. The rest of the structure comes from the interplay between the tonic and small, whole numbers — mainly 2, 3, 5 and 7.
The tonic is Home. The lattice shows how music is a journey, away from home and back again, through different lands, each with its own scenery and feeling.
Next: Harmonic Experience
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