Quest: A Short Story
"There's something about Pink Floyd," Eddie mused, "that reminds me of church music."
"Yeah, right." I closed my eyes and listened to Dark Side of the Moon and thought of a cathedral. Mary-with-the-circle-around-her-head frowned at the straining guitars.
"No, really," he insisted, "I don't know anything about music, but I hear things in Pink Floyd's harmonies or progression or something that just feels the way a big church organ feels. In my gut."
"If you say so." I wasn't especially interested.
He and I did this often, sitting in my apartment with the stereo turned up loud, soaking up the rich resonances, drinking beer, "spacing out" as he called it.
But every time we listened to Pink Floyd he brought it up. "Maybe it's just that they're using the same chords. Or the chord changes, like when the organ goes from one key to another."
I couldn't say much about it. Neither of us had learned to play music, even though both of us got immersed in listening to it at times, and had toyed with instruments in the past. We had been close friends since grade school, had been through innumerable crises together, financial, romantic and philosophical. Now we got together once or twice a month to drink and share our records. And talk.
Other people seem to talk about different things than we did. Women talk about men. Or, maybe it's about relationships, which is usually the same thing. He and I talked about relationships, too, when one of us was getting involved with somebody or breaking up with somebody. But mostly we talked about discoveries one of us was having. Like this music thing.
He asked another friend—actually, he asked most of his friends who knew anything at all about music, "Why is it we play certain notes together? Everybody knows how to sing do re mi fa so la ti do and they can move up a note or two and sing the same sequence without thinking about it, and it still makes sense to the ear."
This friend, who was working on a Ph.D. in music composition, started to say something, then stopped. "Let's get together sometime and talk about it," he finally said. "I can't tell you much until I know where you're coming from."
So Eddie kept getting put off, either because nobody seemed to know where to start, or else they didn't have the foggiest idea what he was talking about. One musician tried to show him on a piano what different chords sounded like. "Yeah, that's what I'm talking about—now, why do those go together? They just sound right!"
The musician just kept playing more chords, waiting for that clue from Eddie that could put them in the same intellectual ball park.
Others he asked put it down to culture. "Our ears are attuned to these notes together. Eastern music isn't the same thing at all."
The subject kept getting pushed into the background as he did his work as an illustrator of machinery and puttered with computers in his spare time.
One evening he called me just after I had gone to bed. "Hey, you won't believe this." He had been browsing through a computer magazine and came across a table of frequencies in an article about programming a computer to play music.
He read me the list:
"That's not much help," I offered. I had no idea what the numbers meant.
"I wrote a Basic program to subtract each frequency from its next higher neighbor, and plotted a graph. It was a smooth curve, but I didn't know what to do with it. Somehow it has to be put into numbers." I could hear his computer keyboard clicking in the background. I pictured him sitting there, his head cocked to one side holding the phone with his shoulder while he typed away at the computer.
"Wait a minute," he said. "The A, which is 880, divided by G sharp, which is 830.608, is 1.059. And that same G sharp—830.608 divided by G natural—783.990 is also 1.059!" More clicking, while he proceeded through the list.
I racked my brain trying to remember something from my math classes fifteen years ago. "If you can express a relationship as a simple 'A times B equals C,' then their logarithms relate as 'log A plus log B equals log C. See what that does."
I was wide awake by then. "And if you can relate them as A plus B equals C then you can draw them on a straight line graph—or you can make a slide rule of them."
"Yeah," he said, still clicking away. The computer made the job easy. Converting the frequencies to their logarithms, he repeated the subtraction of each from its neighbor. It worked. "Each note, expressed as the logarithm of its frequency, is exactly .30103 more than the note below it and .30103 less than the note above it. I should have seen that right away, with an octave being double the frequency."
"But why are there twelve notes in an octave?" I asked. "Doesn't octave have something to do with eight?"
"If you take out the sharps and flats, there are eight—A, B, C, D, E, F, G, and then A again."
"That's only seven different notes."
"Well, you got to name the eighth note—the octave—the same name, so it keeps on going the same way."
"Are those eight notes equally spaced?"
"No. As I see it the scale was done by a committee. They had to agree on some notes, and some guy from Italy held out for sixteen but the rest of them wanted eight, so they finally threw out some."
"B sharp and E sharp, and—"
"Lessee, there's twelve notes in the octave. If you add B sharp and E sharp, you have fourteen. Add the octave note if you want, that's fifteen. Where's the sixteenth?" I was tired and just a bit hostile.
"Dummy, look at the piano keyboard." He was picking up on my annoyance. "Eight white keys in the octave. In between you can fit seven black keys. Except the committee threw out two of them in negotiating with the Italian."
"Okay," I said, "you put in a call to the Vatican. It's coffee break time there. I'm going to sleep."
I didn't hear from him for a week. We were both out a lot, and left each other cryptic messages on the machines. "Why the hell would they leave out those keys?" "Cause guitars only have six strings." "What the hell's that got to do with it?" "As long as they're evenly spaced, what difference is it what they name them?" "Maybe there were originally sixteen on the committee, and two guys missed the crucial meeting. They dropped their keys off the list."
Saturday afternoon he showed up with a six-pack, his little Casio keyboard and a grin. "I want to show you something."
He set up the keyboard on the coffee table and spread his long fingers over the keys. "This is an octave, right? Well, if you were going to play these notes together plus one or two others, just to make a quartet, say, what ones would you play?"
"Who says you have to play the octave?"
"Cause they sound good together, dummy! Now what other keys sound good with them?"
We tried several combinations. Using a C at both ends and playing only the white keys, the only notes that didn't fit were notes right next to each other. "C, E, G, C," he said, "they work. Or C, F, A, C. You can't play C, E, A, C. It doesn't sound right."
"So if you drop off one of the C's, you've got C, E, G..."
"Or F, A, C. Why?!!" He pounded the chord over and over. His intensity was scary, sometimes.
I leaned over the end of the couch and retrieved one of my guitar books from the magazine stand. There was a chart of chords in the back.
"Okay," I said, "According to this, the C Major chord is C, E, G. The F Major chord is F, A, C. There's nothing I see here that goes C, F, A. But they seem to sound good together. Now what?"
"I don't know." He was again dejected and confused. Gathering up his keyboard and notes, he prepared to leave. "Can I borrow that book?"
"Sure. I haven't looked at it in six months—obviously."
As I held the door for him, I thought, I wish I had his drive. I could do anything. But all I said was, "Let me know what you find out."
Compulsions are strange phenomena. Psychologists talk about compulsive personalities, people who count all the steps they take, or arrange their potato chips on their plates. Most compulsions are useless, wasted energy. But Thomas Edison must have been pretty compulsive, to go through all those thousands of materials before he found one that would make a light bulb filament. Eddie sometimes got on these kicks about something, and just couldn't give up. He didn't have enough education to do what he wanted to do, but that didn't seem to stop him. He had this stubbornness inside him. I suppose I was the only person who was willing to listen to him as his brain went through its gyrations.
The next night I came home to find a message on my machine. "I got something," he said, "Give me a call." I couldn't deal with him right then, so I waited a day to return his call.
When he heard my voice, he said, "Whole step, whole step, half step, whole step, whole step, whole step, half step."
"Oh, good," I responded. "You switched over to ballet."
"No, wait—that's a major scale. Do re mi fa so la ti do."
"Uh, right. Of course. Why didn't I see that?" Sarcasm was kind of fun with Eddie. We both did it, and neither of us got pushed out of shape when it was directed at us. It was a game, I guess, a guy-type of game. When women do it, they draw blood.
"Each note is a half-step from the one below it. Okay, that's what they call it. C to D is a whole step. D to E is a whole step. C is do, and D is re. But E to F is a half step. F is mi, and that's . . ."
"Hold it," I broke in. "I can't follow this. I gotta see something on paper. Should I come over there?"
"Yeah, c'mon. I'll show you."
Part of me saw him as a kid, learning a puzzle. Another part of me was fascinated with what he was doing. The more questions he raised, the more interesting it became to me. I figured we could look all this up in the library in a fraction of the time it was taking him, but part of it was that neither of us knew where it was going. You take a course and the instructor not only knows where he's going, but he leads you by the nose to get there, and it's so much work to learn that way. What Eddie was up to was discovery, and that's a whole different thing.
I didn't ring his bell when I got there, because I knew he was waiting. His folks were probably out, anyway.
"Look at this," he said, without so much as a greeting. "I copied these out of your guitar book." He had drawn a diagram of a dozen scales, the do re mi fa so la ti do, beginning with each letter. But there were some sharps and flats there, in all the scales except C. "Here," he said, pointing, "The missing notes, between B and C, and between E and F. You know why? Read all the scales, without the sharps and flats. B, C, D, E, F, G, A, B. Then D, E, F, G, A, B, C, D. All the letters are in each one. But the half steps are there, anyway. It all makes sense!"
I sat down and looked at the chart. "Wait a minute. Wait a minute. What makes sense? Nothing makes sense."
"E, F, G, A, B, . . ."
"I know, I know, I see it. So each scale has all the letters, and each scale has the same sequence—what do you call it—the same intervals between notes, so . . ."
"So that's why some notes are missing." He looked at me, beaming. "There's no B sharp. There's no E sharp."
I shook my head. "Okay, I understand that. It's very clever. They left out the notes they didn't need. But why the whole-step half-step thing?"
"Those are the intervals that sound right. Look." He turned the little keyboard on, and played a C, E, and G. "There's a C Major chord. Now here's a D Major chord, D, F sharp, and A." He alternated the two chords. "Sound alike?"
"If you say so." I tried the combination D, F, and A, simply moving my hand up one white key from the C chord. It had a different feel from the one with F sharp, as though you wanted to go somewhere with it. The chord he played felt complete, somehow, like you could end there. "Interesting."
"Now, here's the really cool part." He picked up another paper from the table. It was full of numbers. "I did some comparing of frequencies. Remember I said there's a constant interval between adjacent notes? Each one is one point oh five nine times the one below it. Equally spaced, but logarithmically. Right?"
"Okay." I looked at the paper, uncomprehending.
"Look at this," he said, pointing. "If we start with A, which we know is a whole number, 440 Hertz, and go up four notes, to E. That's 659.254." He picked up a hand calculator. "Divide 440 by 659.254 and you get .667421. A is almost exactly two thirds of E!" Moving to the keyboard, he played the two notes, one after the other, then together.
"Hey, cool. They sound, like-they sound like they belong together!" I tried it. The relationship between the notes was unmistakable. I tried several other combinations of the same interval: C and G, D and A, E and B. There was the same feel to them. "Two to three, eh?" Then I played A and D together. There was a similar feel to them. "What about this? A and D? What's their frequency relationship?"
He punched in the numbers on the calculator. "440 divided by 587.330 is .749153—almost exactly three fourths!"
"So our eardrums are beating in a clear pattern of multiple frequencies when they are played together. The A beats three times while the E beats twice. Or four times while the D beats three times. But wait, listen to this." I sounded the A and B together. "Something's wrong here."
He was punching in the numbers. ".94428"
"What kind of fraction is that?"
"Seventeen eighteenths, about."
"The A beats eighteen times while the B beats seventeen. No wonder it sounds strange—there's no easy pattern. So why have these notes at all?"
He was suddenly energized. "Wait. Wait. . . Wait. I got something." He pounded his knuckles on the edge of the table, his eyes closed. "We're going backwards. What if we started with the problem of finding notes that sounded good together before we gave them names? What if we knew that we wanted an octave, which is a ratio of one to two, right? And filled in between them." He grabbed a clean sheet of paper and wrote A at the top and another A at the bottom. Beside the bottom A, he wrote 2:1.
I was catching on. "Okay, we already have approximately 2 to 3, with E, right?"
"Right." He inserted the E about midway on the paper, with 2:3 beside it. Then he wrote D above that, with 3:4.
I picked up the calculator and divided 440 by 554.364. "C sharp is .79370"
"What's the fraction?"
"Four fifths! Within a thousandth of a Hertz."
"Okay, try one the other way—what frequency is six-fifths of 440?"
"528. Five Hertz away from C."
"See," he began, gesturing wildly, "Notes that sound good together fall into simple fractional relationships with each other—one to two, two to three, three to four. But how do you get all these to fit together so you can go up the scale beyond an octave?" He had this big grin on his face. "You take the ones that sound good, and you transpose them up and down!"
He tore some lined paper out of a notebook. On the top line of the first one, he wrote A, then counted down eleven spaces and wrote A again. In between, he filled in the remaining notes, including the sharps and flats. On the second sheet, he did the same thing, except beginning and ending with B flat. Then on the third, he began with B, and continued until he had twelve columns on twelve sheets of paper.
Then he took a highlighter and colored the top and bottom rows on all twelve sheets (the 1:2 ratio), and then the eighth, sixth and fifth rows: the 2:3 ratio notes, the 3:4 ratio notes, and the 4:5 ratio notes. Finally, he arranged the sheets so that the same notes on all the sheets were lined up in the same rows.
"Wow." I was impressed.
"Now," he said, looking up at me with a grin, "How do we know which notes, from these twelve, sound good together? Why does do re mi fa so la ti do use the notes it does?" He picked out the scale on the white keys, beginning with C. "C, D, E, F, G, A, B, C—C Major scale."
"Okay, we know we're going to use the low-order fractions, right?"
"Right. So from C we know we have E, F and G."
I looked at the series. "We've got this big hole between C and E. How about D?"
Eddie punched in some numbers on the calculator. "Well," he said finally, "that's a ratio of about 9 to 10 or 10 to 11."
"Not very good. But look at this!" I said, suddenly seeing the light. "The D is in a 2 to 3 ratio to the G, which is one of our first choices!"
"And the B flat is a 3 to 4 ratio to F, another of our first choices." His voice went flat as he said it.
I saw the problem. "Where do we get A and B for the C Major scale?"
He studied the sheets. "I don't know. Dog poop. Let's have a beer."
We started Doom on the computer, and forgot about missing notes as we fought the bad guys on the screen. Eventually, I went home, feeling a little agitated. I was catching his obsession.
I talked to him once during the next week, and neither of us mentioned music. It felt like a brick wall, and there was no way around it. Do re mi fa so la ti do was make-shift music, even if it did sound so familiar that no variation of the notes felt right.
On Friday evening, I was at his place again, furiously thrashing around with the joystick, trying to avoid being killed in Doom. He was picking at the Casio keyboard. All our paperwork from the previous work session was still strewn around, some on the floor of his room.
"An ordinary major chord is just three notes," he said, absent-mindedly. I made some sound in response, just as absent-mindedly, while I dodged missiles. "One, three and five." He played a chord on the keyboard. "Actually, they are one, five and eight out of the twelve. One, three and five in the scale. They call it a triad."
I turned away from Doom. "So, they don't use the high notes we were trying to justify, anyway?"
He sat with his hands clasped, almost as if he were praying, except that his teeth were propped against his two thumbnails. He took his thumbs out of his mouth. "That's only if you play a chord." He played the C Major chord a couple of times. "What's the point of having scales?" He played the C Major scale. Do re mi fa so la ti do. And then he began to sing, softly. "Do, a deer, a fe-male deer, re a drop of golden sun . . ."
"Play that," I said. "Play it on the keys." I was wondering if a song used only the notes in its scale. Sure enough, as he played, every note was on the white keys—until he got to the part, "La, a note that follows so . . ." when he had to play a black key-F sharp. "Wait," I interrupted him. "Play that part again."
He played the line again, and again stumbled on the note for the word "that."
"In the line before, you played all white keys: C, D, E, F, G, A. That's all C Major. But the next line is the same structure, except raised one note. Only, to sound right it needs an F sharp instead of an F."
His voice was loud with excitement. "And that's all D Major. They used a key change to make it different." He played the two lines again, then the following line. "And the next line is in E Major!"
"And ends on C, where the song started, except an octave up."
"And the next line simply descends to the original C, where it starts all over again on the next verse." We grinned at each other. "Okay," he said, "that one's done." He leaned back in his chair and took a big swig from his beer. "Now for Pink Floyd."
"How about Shostakovitch?"