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.
Music is the thread that stitches my life together. I was born singing, as are we all. My parents loved to sing and they both had beautiful voices. They met in a musical play in college.
I remember our house as being filled with music, especially jazz and harmony vocals. Stan Kenton, the Mills Brothers, Sons of the Pioneers, and all the big bands. Sing Sing Sing at Carnegie Hall. Cool Water. All from the Curtis Mathes stereo, literally a piece of furniture, as big as a couple of dressers. I loved that sound, LPs coming out of that big resonant wood cabinet.
With my sister, we had four good voices, and singing harmony in the car was a highlight of growing up. My sister and I started writing songs together when we were kids.
I troubadoured randomly around the country with a guitar and the clothes on my back (hey kid, can you play that thing? hop in!), formed a rock’n’roll band, sang Christmas carols professionally, did singing telegrams in a chicken suit, and continued to write and record. I spent most of my “serious” career designing electronics for the remarkable L.R. Baggs Corporation in California. They make beautiful, innovative equipment for amplifying acoustic instruments.
I’ve become increasingly focused on music over the past five years or so. About two years ago, I began working full time on music and almost nothing else. Soon after, I had a huge epiphany, the best intellectual and spiritual experience I’ve ever had.
I’ve added a few new tracks to the Audio page today. Two of them are live cuts with Jody Mulgrew, my great musical pal. The Cove is my song and Jack of Hearts is his. They were both sung at the Greco’s in Shell Beach, a couple of years apart. We plan to do some shows together next year, and I’m excited, it’s been a long time.
The last cut is one of my favorite songs, Nature Boy, by eden ahbez. I knew the melody, but I didn’t know the chords, so as an exercise, I harmonized it from scratch using the lattice. The melody wanders around the map, and the harmonies support and accent it in different ways as they follow (and lead) it around. When I finished, I looked up the chords online, and was pleased to find that my progression was different than the official one — I had made the song my own. Good exercise.
This may have been the most intense art project of my life. Some time early in 2012, I got it into my head to create a stop-motion animation of my song Flying Dream, moving in harmonic space. I’ve spent the past five months working like crazy on it. The song is carefully arranged using the lattice of fifths and thirds, sung and played in just intonation, and animated using colored lenses, rice paper and a flashlight.
If it doesn’t play smoothly, don’t hesitate to lower the resolution. It survives scaling well, and the timing is crucial.
This video encompasses almost everything I’ve learned in the past couple of years. I will be writing much more about it. Meanwhile, here are a couple of external links about the basic subjects:
For almost two years now, I’ve been exploring the nature of music almost full-time. I threw out everything I knew, started with the most basic thing I could think of, the number 1, the origin of the musical universe, and worked my way from there.
My explorations quickly led to the underpinnings of musical harmony, the natural notes that can be expressed as ratios of small whole numbers. These are the notes people generally played and sang, before twelve-tone equal temperament (12ET) came along. 12ET is a clever tuning system, a collection of 12 notes that are slightly retuned from the natural ones, mathematically fudged so that you can play fixed-pitch instruments in any key, and change keys without retuning.
Before 2011, I’d never seriously questioned those 12 notes. They are “The Notes,” after all. They’re the ones on the piano, and they’re what your guitar tuner tunes to. They’re right, right? Well, not really. In the past two years, I have met and made friends with a whole color palette of new, untempered notes, no two alike, each with its own function and personality. In the process, I’ve discovered new ways of thinking about and visualizing music that have greatly increased my enjoyment of it. I have even made friends with equal temperament again, after a long journey away. This website will tell about the journey. Welcome!