Mark de Silva - Square Wave

He rolled off her with an athlete’s grace and pulled the sheaths from his cock. He flushed them, taking care not to leave any semen on the bathroom floor or the rim of the toilet. It was five a.m. and the radiator had just come crackling to life. He dressed before her overturned body and the bent couch.

She moaned softly, whether from orgasm, alcohol, or simple blunt force, he couldn’t tell. That was the point, to leave without knowing the meaning of her quivers and coos. The beast had been felled by cock, or fist, or bottle. How she would rise again, in what state, how much she would remember, how she would explain this night — these were questions for other men: brother, father, officer.

Jen had been felled by a man of privilege too, it seemed, someone idle enough for such lovingly scripted cruelties — even if he hadn’t actually fucked her. Though that did make the story odder. They always fucked them. But not this man, a good-looking man at that.

“Okay.” Stagg finished his summary and flipped the page in the pad. “How many girls have left over the last months, would you say, because of these assaults?”

“Left Halsley?”

Stagg nodded. “Or the business.”

“A lot. Half maybe. More all the time.”

“And are they still sex workers, as far as you know?”

“I can’t really say. I don’t know these people, this community, if you want to call it that, very well. But I don’t see why not.”

“Right — but you would think there’d be a lingering fear. Do you know of any who’ve definitely gone in another direction? Away from prostitution.”

She swept the hair from her face as that word came from his mouth. It kept cascading down when she shook her head, yes or no. “Sure. A few. When you go as far as moving, you consider a lot of things.”

“And of the victims themselves?”

“I think at least half are doing other things. Back with family or friends, some in NA or halfway houses, some working regular jobs, retail, that sort of thing.”

“And that’s pretty well known to the other women, what they’ve chosen, you would guess?”

Jen paused and squinted, searching the apartments across the street. “Yes. That I know, actually.”

He stood. “A lot of people are working on this, Ms. Best,” he said. “I’m hoping there’s something to tell you soon.”

She yawned as he spoke. “Can you get me my pills? They’re on the counter.”

He picked up the translucent orange bottle from the kitchen counter. Percocet. He thought to bring it to her but paused between kitchen and couch. He twisted off the white cap and tapped a single oblong pill into his hand and set the bottle on the counter.

“They gave me so few,” she said.

The glass of water Mariela had set out for him remained on the coffee table, untouched. He picked up the chilly glass and condensed water dripped across the table. The last of the ice, just milky slivers now, was dissolving. He held out the pill and the glass to her, one in each hand, as if she might receive them the same way, though she had only one good hand at this point. She took the pill from his palm and pushed it into her closed mouth, between her lower lip and teeth, like chewing tobacco. She returned with the same hand, the good one, for the glass. Before he could lower his own, she put the glass back in it. Only after all this did she look him in the eye, and then only briefly, in a sweep of much else.

Her phone chirped twice.

“Your friend?”

“Mariela. She’s here almost.”

“Good.”

She dropped the phone on the couch and put her feet up over the arm. “Bye,” she said in something just above a whisper.

“Should I wait for Mariela? I can.”

Jen lay there with her eyes half open, willfully oblivious to him, with the very mien the opioid would anyway force on her shortly. Perhaps she learned this at the hospital, he thought, that it was better to adopt the look of lassitude than to wait for it to seize you. Just as well.

He drank the rest of the water and left.

Larent sawed at the string, the open E. A continuous tone rose from the double bass, and from it sprang further tones, harmonics, an infinite ascending series, growing ever fainter. He’d trained himself to hear it, though, a portion of it at least, as Stockhausen could in even the roar of taxiing airplanes.

The series came as a mix of ratios to the fundamental, E, all whole numbers, and small. Loudest, most resonant, early in the series, were the superparticulars: (n+1)/n. Two to one, the octave; three to two, the fifth; four to three, the fourth. Larent followed these tones up through the registers, fixing the intervals with his ear, tracing an elemental order. The first thirty-one harmonics, more than he could resolve, produced a pure , a just —a Ptolemaic — version of the common twelve-note chromatic scale.

He stopped the string a pure fifth above the open E. He held the B against the E still ringing through the amplifier on a delay pedal. The dyad was glassy, luminous, and fragile. Shaving just a fiftieth of a semitone off of it, as the usual tempering did, managed to shatter its coherence, sending waves of sound beating in and out of phase, canceling and strengthening each other by turns.

He carried on forming dyads this way, twelve of them, taking each rising fifth as the new root, locking it in place with the pedal, and bowing a fifth above. He made his way through the circle of fifths, climbing seven octaves this way, seven and a remainder. The E at the top was not an E, could not be. It overshot E by a Pythagorean comma —less than a quarter of a semitone, and nowhere close to superparticular: 531,441 to 524,288.

This wasn’t his mistake. The glitch was in the mathematics itself. You couldn’t return to the root pitch through pure fifths. The circle wouldn’t close. Instead it spiraled upward, a comma for every twelve fifths. To close the loop, to make E meet E, you had to narrow that last fifth by a comma. This was how a wolf was born, a howling, beating fifth.

The spiral wreaked other havoc. The major third you got from building the chromatic scale by stacking pure fifths, the way Pythagoras did, was much wider than the pure one (5:4) drawn directly from the harmonic series. So, to capture those pure major thirds, they tried tempering Pythagoras’s scale. This was two millennia later, during the Renaissance.

That just relocated the problem. Tempering the scale to achieve pure thirds meant that some of those previously pure fifths had to be narrowed, which is to say coarsened, not so much as to breed vicious wolves, but enough to steal the brilliance from the scale you were left with. And even then, a true wolf sat there at the tail, making most of the keys unusable for their dissonance.

If instead you tempered the twelve fifths by different amounts, so that the interval between notes varied throughout the scale, you could make even more keys playable, as Bach did in his Well-Tempered Clavier . All twelve keys become usable to one degree or another, and each takes on a distinctive character, depending on the precise spacing of intervals to be found in it. You have fewer true wolves this way, but then, you also have fewer pure intervals.

What took hold in the nineteenth century, what still reigned today, equal temperament, went the whole distance with tempering. All the keys became equally playable, because all of them became identical. Each of the twelve fifths was tempered by the same small amount. Flattening the spiral in this way made harmonic motion, modulation, effortless. But nothing from the harmonic series — the very origin of the scale — remained, less the octave. So you’d chased away all the wolves, yes, but then you’d done the same to everything pure.