Inharmonicity

The Equal Temperament

Chapter 11: Inharmonicity

Inharmonicity is the deviation of a real string's partials from the theoretical harmonic series. A theoretically perfect string — infinitely thin, infinitely flexible, vibrating in one dimension only — would produce overtones at exact integer multiples of the fundamental: the second partial at twice, the third at three times, the fourth at four times, a series of frequencies as regular as the rungs of a ladder, evenly spaced, mathematically clean. But no string is theoretical. Every string has mass and stiffness and thickness and these physical properties cause the partials to deviate from the theoretical positions, each partial slightly sharper than the integer multiple would predict, the deviation increasing with each successive partial, so that the second partial is almost where it should be but the tenth partial is noticeably sharp and the twentieth partial, if the ear could hear it, would be sharper still, and this progressive sharpening is inharmonicity, the imperfection of the real, the gap between what mathematics predicts and what physics produces.

Clara had been thinking about inharmonicity for weeks — not the acoustic phenomenon, which she had understood for twenty-eight years, but the metaphorical resonance of it, the way the concept described not just the behavior of strings but the behavior of everything that existed in the real world rather than the theoretical world, everything that deviated from the ideal by virtue of being actual, everything that was shaped by its physical properties into something slightly different from what it was supposed to be, and the thinking was not idle but purposeful, was Clara's mind working on the problem of her hearing loss the way her ear worked on the problem of a temperament — circling it, testing it, listening for the pattern, the structure, the meaning.

The inharmonicity of Clara's ear was her hearing loss. A theoretically perfect ear would hear all frequencies with equal sensitivity, would resolve all pitch differences with equal precision, would not age, would not decline, would be the auditory equivalent of the theoretically perfect string. But no ear was theoretical. Every ear had physical properties — the mass of the ossicles, the stiffness of the basilar membrane, the number and condition of the hair cells — and these properties caused the ear to deviate from the theoretical ideal, and the deviation increased with age, the way the string's inharmonicity increased with higher partials, the real ear departing further and further from the ideal ear, the gap widening, the deviation growing.

She was thinking about this on a Tuesday afternoon in May as she drove to the Songbird Recording Studio in the Pearl District, a studio she had been tuning for seven years, the studio that owned the Steinway B on which she had first noticed the difficulty in the upper treble eight months ago, the piano that had been the first to reveal what the audiogram would later confirm, and the piano was due for its monthly tuning, and Clara was driving toward it with the particular mixture of anticipation and dread that the Steinway B now produced in her, the anticipation of doing the work she loved on an excellent instrument and the dread of confronting, in the upper octave, the evidence of her decline.

The studio was on the third floor of a converted warehouse on Northwest Thirteenth Avenue, a building that had been a furniture factory in the 1920s and that now housed creative businesses — a graphic design firm, an architecture studio, a podcast production company — and the recording studio occupied the northeast corner, a suite of rooms that had been acoustically treated by a designer from Los Angeles, the walls lined with absorption panels and diffusion surfaces, the control room separated from the live room by a double-paned window, the live room where the Steinway B sat on a riser, surrounded by microphone stands and baffles and the cables that connected the microphones to the console in the control room.

The studio owner, a man named Paul Dreyfus, was forty-three, a recording engineer who had worked in Nashville and New York before moving to Portland with the declared intention of recording acoustic music with analog equipment, a declaration that was partly ideological — Paul believed that digital recording sacrificed the "warmth" and "presence" that analog tape captured, a belief that Clara found sympathetic because it paralleled her own belief about electronic tuners — and partly commercial, the analog aesthetic being fashionable among musicians who wanted their recordings to sound "vintage," which meant warm, which meant slightly saturated, which meant imperfect in a way that was perceived as more authentic than the clinical perfection of digital recording.

Paul met Clara at the studio door. "The treble is bugging me," he said.

Clara felt a contraction in her stomach, a tightening that was the body's response to a statement that could mean many things but that, coming from Paul Dreyfus, whose ear was trained to hear recording artifacts at a level of detail that few listeners could match, probably meant something specific, something audible, something that Paul's ear had detected and that Clara's ear might or might not be able to confirm.

"Bugging you how," Clara said.

"The upper register. Above — I don't know, above C7 maybe. It sounds — tight. Harsh. Like the octaves are wide."

Clara set her bag on the console and walked into the live room and stood beside the Steinway B and looked at it, at the nine-foot case, at the satin ebony finish, at the open lid showing the strings and the hammers, and she felt what she always felt in this room — the professional recognition of an excellent instrument in an excellent acoustic space, the pleasure of working with a piano that was well-maintained and well-positioned and that responded to the tuning lever with the sensitivity of a thoroughbred — and she also felt the dread, the specific dread of the upper treble, the knowledge that the territory Paul was complaining about was the territory where her hearing was compromised.

She sat at the bench and played a chromatic scale from C6 to C8, slowly, each note sustained, each note listened to, and the notes sounded — she listened carefully, with the full attention of her trained ear — the notes sounded correct to her. The octaves did not sound wide. The treble did not sound harsh. The piano sounded like the Steinway B always sounded — bright in the treble, warm in the bass, with the particular Steinway tone that was neither the richness of a Bösendorfer nor the brilliance of a Yamaha but something between, something quintessentially American, a sound that was democratic, versatile, suitable for Chopin or Coltrane or Clara playing a chromatic scale on a Tuesday afternoon.

But Paul had heard something. Paul's ear, trained by twenty years of recording engineering, attuned to the subtleties of microphone placement and room acoustics and the coloration that different preamps and compressors and equalizers introduced, Paul's ear had heard something in the upper treble that he described as "tight" and "harsh" and "wide," and Paul's ear was reliable, was discriminating, was an ear that Clara respected, and if Paul said the treble was wrong, the treble might be wrong.

Clara struck the fork and set the A and checked the temperament, which was clean, had held from last month's tuning, the intervals correctly tempered, the beat rates consistent. She checked the octaves in the mid-range — clean. She checked the octaves in the upper treble, above C6, and she listened for the wideness that Paul had described, listened for the beats that would indicate the octaves were stretched too far, and she heard — nothing. The octaves sounded clean. The beats were not present. The intervals were correct.

Or the beats were present and she could not hear them.

She took out the Korg.

She had never used the Korg in front of Paul before. She had always tuned this piano by ear, had always presented herself to this client as an aural tuner, a tuner who did not need the device, who relied on the ear alone, and the ear alone had been sufficient, had been reliable, had produced tunings that Paul accepted without complaint, and the act of taking out the Korg in the live room of the Songbird Recording Studio felt like a concession, a confession, an admission that the ear could no longer do what the client expected the ear to do.

She turned on the Korg and checked C7. The display showed C7 at minus one cent. One cent flat. Within tolerance. Fine.

C-sharp 7. Plus two cents. Two cents sharp. Acceptable, arguably, for a concert piano with the stretch that the Steinway required, but two cents sharp was on the edge — if every note in the top octave was two cents sharp, the cumulative effect would be a treble that was "tight," that was "harsh," that was "wide," exactly as Paul had described.

D7. Plus three cents.

E7. Plus two cents.

F7. Plus four cents.

Clara looked at the numbers. The top octave was consistently sharp — two to four cents above the target, a systematic error, a bias, and the bias was in the direction that her hearing loss would predict, because when the ear could not hear the beats in the upper octave, the ear tended to accept octaves that were slightly wide, because wide octaves beat less audibly than narrow octaves at high frequencies, because the beating was in the territory where the hearing was diminished, and the tuner, unable to hear the wideness, accepted it as correct.

She had tuned this piano last month. The error was hers.

She corrected the top octave, bringing each note down by two to four cents, the Korg confirming each correction, the device doing what the ear could not, and the corrections took fifteen minutes, and when she was done she played the chromatic scale again, from C6 to C8, and the treble sounded — different. Not dramatically different. The difference between a top octave that was three cents sharp and one that was correct was subtle, was at the edge of perception for most listeners, but for Paul Dreyfus, whose ear was tuned to this piano's sound, who heard this piano through microphones and monitors every day, the difference was audible, was significant, was the difference between a piano that sounded right and a piano that sounded tight.

Paul came into the live room and sat at the bench and played a passage from a Debussy prelude — the Piano has always been his testing piece, the repeated octaves and the shimmering treble a test of the upper register's quality — and he played it and listened and nodded.

"Better," he said. "Much better. What was it."

Clara could have said "the stretch was slightly wide," which was technically true and which would have been accepted without further question, because the stretch was a technical matter that clients accepted as part of the tuner's expertise, the way a patient accepts a doctor's diagnosis without demanding to see the lab results. She could have said this and left and Paul would have been satisfied.

"The top octave was sharp," she said. "By two to four cents. I missed it last month."

Paul looked at her. His expression was not accusatory — Paul was not an accusatory person, was a listener by nature and by profession, a person whose default response to information was attention rather than judgment — but his expression contained the mild surprise of a client hearing a tuner admit an error, the surprise that came from the assumption, held by most clients, that the tuner's ear was infallible, that the tuning was correct by definition because the tuner said it was correct, and Clara had just said that the tuning had not been correct, had said that she had missed it, had said the words "I missed it," which were words that a tuner almost never said, because saying them undermined the foundation of the tuner-client relationship, which was the assumption of competence, the trust in the ear.

"It happens," Paul said.

"It shouldn't happen."

"Everyone misses things. I miss things in recordings all the time. You catch it on the second pass."

"On a recording, you can do a second pass. On a piano, the second pass is next month. The pianist who recorded here last week played on a piano with a sharp top octave."

"Nobody noticed. I didn't notice until yesterday, when I was reviewing the tracks and the treble sounded — just slightly off. If I hadn't been listening for it, I wouldn't have caught it."

Clara packed her tools. She replaced the Korg in the bag, in the pocket where it lived, next to the felt mutes and the lever, and the placement of the Korg in the bag — the fact that it had a pocket, a designated space, a regular position in the bag's geography — was evidence of how central it had become to her work, how far it had traveled from the optional accessory it had been a year ago to the necessary tool it was now, the training wheels that had become the wheels.

She left the studio and sat in her car in the parking garage beneath the building and gripped the steering wheel and did not start the engine and did not move and did not think, for a moment, about anything except the fact that Paul Dreyfus had heard something she had not heard, that a client had detected an error in her tuning, that the tuning she had delivered last month had been wrong in the top octave and she had not known it was wrong because she could not hear the wrongness, and this was the thing she had been afraid of since October, the thing the audiogram had predicted, the thing that had been approaching with the inevitability of a tide, and it had arrived, it was here, it had happened — a client had heard what Clara had not heard, and the gap between the client's ear and Clara's ear, in this one register, in this one octave, had been closed, and the client's ear had won.

She started the car. She drove home. She sat at the kitchen table and took out the audiogram — the second one, from January, which she had been carrying in the bag since January although the bag was not where it belonged, the bag was for tools, the audiogram belonged in the filing cabinet — and she looked at the numbers. 30 decibels at 6,000 hertz. 45 at 8,000. The loss was in the range where the top octave's partials lived, in the range where the beats between the octaves were produced, in the range that Paul Dreyfus could hear and Clara Resnikoff could not.

She put the audiogram back in the bag. She picked up the phone and called Dr. Chau's office and made an appointment for the following week. The third audiogram. The third measurement. The third point on the curve that was descending, that was tracing the trajectory of Clara's ear through the frequencies it was losing.

She sat at the table for a long time. The fork was in front of her. She did not strike it. The fork was in the range of frequencies that were fine, that were normal, that were not the problem, and striking it would confirm what she already knew — that 440 hertz was still present, still clear, still the one fixed point — and the confirmation would not help, would not address the problem, which was not at 440 but at 8,000, not at the center but at the edge, not in the reference tone but in the territory that the reference tone organized.

She picked up the fork and held it and looked at the steel, at the tines, at the stamped A and 440, and she thought about Leonard. Leonard had known when to stop. Leonard had had a clear metric — the seventh partial of C8 — and when the metric was met, he stopped. Clara did not have a clear metric. Clara's loss was not a single missing frequency but a gradient, a slope, a continuous decline in resolution across a range of frequencies, and the slope did not provide a clear stopping point, did not say "here, at this frequency, at this decibel level, you are no longer able to do the work," but said instead "the work is getting harder, and it will get harder, and at some point it will be too hard, and you will not know exactly when that point arrives because the point is not a point but a region, a zone, a gradual transition from can to cannot."

She thought about Paul Dreyfus saying "nobody noticed." She thought about the pianist who had recorded on a piano with a sharp top octave and had not noticed. She thought about all the clients who had not noticed the errors she had caught with the Korg, the errors that the Korg had corrected, the errors that existed for a moment between the end of the ear's tuning and the beginning of the Korg's correction, errors that were real but that no one had perceived, that had existed in the gap between Clara's diminished hearing and the clients' untrained hearing, the gap that had been, until today, wide enough to contain the errors without consequence.

Today the gap had closed. Paul Dreyfus had noticed. Paul's ear had reached into the gap and found the error and pulled it out and presented it to Clara, and Clara had confirmed it and corrected it and admitted it, and the admission was — Clara searched for a word — the admission was a turning point, not because the error was catastrophic or the admission was public but because the event confirmed what the audiogram had predicted and what Clara had been trying to deny, which was that her hearing loss was not merely a medical fact but a professional fact, not merely a line on a graph but a flaw in her work, not merely a personal loss but a loss that affected other people, that affected the pianos she tuned and the musicians who played them and the audiences who heard them, and the loss was real and it was here and it was hers.

She struck the fork.

A440. Clear. True. Present.

She held it to her ear and listened to the partials — the fundamental, the second, the third, climbing upward through the harmonic series — and somewhere above the tenth partial the harmonics faded into the background, blurred into the noise floor of the kitchen, disappeared into the territory where Clara's hearing could no longer resolve them, and she listened to the disappearance, to the transition from audible to inaudible, and the transition was not a cliff but a slope, not a line but a smear, the sound fading rather than stopping, diminishing rather than disappearing, and the gradualness of the transition was the problem, because a cliff you could stand at the edge of and see, a cliff had a location, had coordinates, had a point where you were on solid ground and a point where you were in the air, but a slope had no such clarity, offered no such certainty, presented only the continuous question of whether you were still on the solid part or had begun the slide.

She set the fork on the table. She opened the appointment book and looked at tomorrow's schedule. Three pianos. Three houses. Three instruments that needed her ear.

She would give them what she had. She would use the Korg for the top octave. She would lean closer. She would strike harder. She would compensate, accommodate, adjust.

She would be right enough.

For now.