The Imperfect Scale
Jeon Hye-min, an after-school piano instructor, had actually given up on opening a class at Hanyoung Elementary School this term.
A class was only confirmed if at least five students signed up, and the numbers had been precariously low.
But fortunately, two students signed up at the very last minute, barely meeting the quota to confirm the class.
While public schools let you know in advance, independently run private elementary school after-school programs sometimes only gave confirmation on the very first day.
Well, it was what it was. She just had to roll with it.
The pay was better than public schools anyway, so she wasn't complaining.
In truth, the skill level of children in after-school piano classes was usually mediocre at best.
Kids who wanted to learn the piano seriously usually took private lessons or attended specialized academies.
For that reason, rather than students who genuinely wanted to learn, most were kids who had signed up simply because other classes were full and they needed to do something.
"Alright, who here is playing the piano for the very first time? Raise your hand!"
Three out of the five raised their hands.
First, she decided to gauge the level of the experienced ones who hadn't raised their hands.
"Oh, how far has our friend here studied?"
"...Czerny 100."
A boy answered hesitantly.
Next was a pretty-looking girl.
"And you?"
"Czerny 40."
One of Jeon Hye-min's eyebrows twitched.
It was extremely rare to encounter that level in an after-school piano class.
At Czerny 40, they could already handle competition-level pieces to some extent.
The after-school curriculum was tailored entirely for beginners.
It would be incredibly boring for her.
"Alright, even if you two already know most of this, we'll be focusing heavily on the basics. You understand, right?"
"Yes!"
Jeon Hye-min walked over to the upright piano.
And then, she lightly swept her hand across the keys.
The sound was excellent.
A Steinway K-132.
As expected of a private elementary school, they had provided a high-end piano.
Furthermore, digital pianos equipped with headphones were set up to match the number of registered students.
The teaching environment was exceptionally top-tier.
She usually started the first day of class with a demonstration.
Hearing a live performance always completely changed the children's level of focus.
Mozart's Piano Sonata No. 11 in A major, K. 331, 3rd movement: Rondo alla Turca.
It was familiar to their ears, yet flashy enough to make them go "Wow!"
If she started the beginning with a slow-tempo beginner's arrangement and then suddenly unleashed a dazzling, full-difficulty performance of the original piece halfway through, the kids were usually blown away.
Jeon Hye-min smiled warmly and sat down at the piano.
"Now, before we begin the lesson, shall we listen to what kind of sound this piano makes?"
"Yes!"
At least the kids in this term's class were highly responsive.
They answered enthusiastically.
Jeon Hye-min's long, slender fingers rested on the keys.
Soon, the performance for her five young audience members began.
Wow.
Cheol-jun's eyes widened as he stared at the teacher's fingers.
It was a piece he had never heard before.
But it was incredibly exciting.
It started out calm and cheerful, but later it accelerated so much that it made him want to bounce in his seat.
It was just as he had expected.
Hearing it played live in person was entirely different from the videos he had watched yesterday.
When the teacher finished playing and the children began to clap, Cheol-jun quickly joined in, clapping his hands together.
Jeon Hye-min stood up from the piano bench and bowed.
"My name is Jeon Hye-min! It's nice to meet you all."
Jeon Hye-min scanned the children's faces.
"How was it? Playing the piano seems fun, right?"
"Yes!"
"Alright, who wants to try playing?"
There were two experienced students.
And one of them had even reached Czerny 40.
It was the perfect age to want to show off.
Hoping to gauge her skills, she subtly made eye contact with the girl, but—
"...Huh?"
The tiny boy sitting right next to the girl shot his hand straight up into the air.
"Would you like to try?"
"Yes!"
"Oh, alright. Why don't you come over and sit down?"
This was a first.
A child with absolutely zero experience volunteering to play the piano in front of his peers.
It happened occasionally with younger kids, but this was a class for fifth and sixth graders.
At that age, kids started becoming self-conscious and rarely stepped forward unless they were highly confident. Yet, this boy looked completely bright and carefree.
Jeon Hye-min hid her bewilderment and watched the boy walk over.
He was much smaller than the other kids his age, and he looked incredibly adorable.
"Um, this is your first time playing, right?"
"Yes!"
"Since today is your first day, just do whatever you like. Is there any song you know?"
The boy smiled brightly.
"The song you just played, Teacher!"
"Pardon...?"
The Turkish March she had just performed was her own custom arrangement designed specifically for the kids.
The beginning was a beginner's version, so it might have sounded relatively simple.
Even so, it was absolutely not at a level where someone could mimic it after hearing it just once.
Even kids with some piano experience would need sheet music to play anything close to it, unless they had practiced it beforehand.
While she was lost in thought, the boy sat down in front of the piano without a moment's hesitation, his bright eyes blinking.
And then, he placed his hands on the keys.
Jeon Hye-min's brow furrowed slightly.
His wrists were drooping, and his fingers weren't naturally curved; they were flat, meaning the pads of his fingers, rather than the tips, were touching the keys.
On top of that, his left hand was resting at his side, and he was only using his right.
He had placed his index finger, not his thumb, on the starting note.
It was the posture of a complete, absolute beginner who had truly never touched a piano before.
Jeon Hye-min chuckled inwardly, leaning down slightly as she spoke.
"What's your name?"
"Lee Cheol-jun!"
"Alright, Cheol-jun. You can start whenever you're ready."
She straightened her back, already mentally preparing what comforting words to say in case the other children started laughing.
The boy smiled brightly and began to move his fingers.
Cheol-jun recalled the piece he had just heard.
He remembered every single sound.
The structure was simple, and because the theme repeated, he had memorized the entire pattern during that single listen.
The rhythm was regular, too.
And since he had closely watched the teacher's fingers producing those sounds, he had a rough idea of which key produced which note.
The problem was that he couldn't move his fingers as freely as the teacher.
Cheol-jun knew exactly what he could and couldn't do.
So, he simplified the structure.
He gave up on the left hand entirely from the start, and even with the right hand, there were quite a few notes he had to abandon once the rhythm sped up.
He understood the music in his head, but his fingers wouldn't be able to keep up like the teacher's.
But still...
I think I can make it sound similar...
Cheol-jun smiled innocently and began to press the keys.
The boy's performance began, and Jeon Hye-min's eyes went wide.
...Huh?
First of all, the notes and the rhythm were correct.
Did he already know the Turkish March?
No, even if he did, he shouldn't be able to play it like that.
Yet, with clumsy finger movements, the boy was producing the notes and rhythm with remarkable accuracy.
Of course, his fingers occasionally got tangled, causing adjacent keys to be pressed together.
But that only made it more shocking.
Because no matter how she looked at it, he was definitely a beginner.
His finger positioning was a mess, and his placement on the keys was completely haphazard.
And yet, somehow, the song was actually coming together.
Even though he was only playing the melody line with his right hand, for an absolute beginner to do this was extraordinary.
The feeling of his playing was uniquely strange.
It was hard to define precisely, but it felt less like he was following a pre-existing path and more like he was discovering the path step-by-step as he went.
One thing was certain.
He was absolutely not playing from memory.
He was cautious when encountering a measure for the first time, but when that same measure repeated, he became much bolder.
It felt as though he explored it on the first pass and then gained confidence on the second.
Even his finger positions were finding more optimal spots than at the start.
By the time the introductory section was nearly over, his finger movements pressing the keys had actually become quite precise.
For a kid who didn't even know proper posture, the amount of growth he displayed in less than a minute was utterly absurd.
And when the beginner's section ended and the piece reached the middle section—where the tempo accelerated and the difficulty spiked—Jeon Hye-min unconsciously let her mouth fall open.
Wow... He's doing that?
Jeon Hye-min had spent years watching how beginners made mistakes when tackling difficult pieces.
Their fingers would get tangled, causing them to hit wrong notes or hesitate constantly, with no distinction between important and unimportant notes.
But the way this boy made mistakes was different.
No way...
It's not that he made a mistake, but that he calculatedly abandoned those notes?
The section she had arranged for her demonstration was not only fast but also contained quite a bit of competition-level ornamentation.
Yet, the only notes the boy missed were those decorative ornaments and passing tones.
The skeletal melody remained perfectly intact.
This meant he was accurately distinguishing between the core melody and the embellishments, intentionally omitting the latter.
In all her years of teaching children, Jeon Hye-min had never encountered a child like this.
Of course, there were still plenty of sections where he genuinely made mistakes.
But the way he handled those mistakes was extraordinary.
When a beginner made a mistake, their instinct was to freeze in panic.
But when this boy hit a wrong note, he simply moved on to the next one.
He didn't stop, and his expression didn't even flicker.
Because of that, the rhythm never faltered, and the melody quickly found its way back on track.
It was only possible because he clearly recognized his mistakes and was able to instantly let them go.
Even kids who had played the piano for a while tended to lose their rhythm the moment they made a mistake.
Yet, even when this boy made mistakes, the song never collapsed.
He didn't lose the rhythm right up until the very end.
It was absolutely astonishing.
Having finished his performance, the boy scratched his head sheepishly and chuckled.
"Wow, playing it in real life is actually pretty hard... Hehe."
At the boy's muttered words, Jeon Hye-min swallowed hard.
Cheol-jun smiled awkwardly as the other children clapped for him.
Arin stared at Cheol-jun with wide eyes.
He played really badly, but... he played well?
It seemed like a mess, but it was definitely the Turkish March, and he had indeed played it just like the teacher had.
Arin could tell, too.
That Cheol-jun's posture was completely terrible.
When Cheol-jun first raised his hand, Arin had sighed inwardly.
What is he so confident about to step forward like that?
That was what she had thought back then.
Even she, the best piano player in the room, was staying quiet, yet a kid who had never even played before was stepping up.
So she had felt embarrassed for him.
She didn't know why, but she had felt incredibly embarrassed.
Even when Cheol-jun sat down and placed his hands on the keys, Arin had slightly lowered her gaze.
Because if he had even a single piano lesson, he would never have had posture like that.
But the moment the performance started, Arin's head had snapped right back up.
It was incredibly clumsy, yet he was playing it properly, which was mind-boggling.
Honestly, if she were asked to play that piece, could she have done it without sheet music?
No, even with sheet music, she would have had to practice the latter half quite a bit to play it accurately.
Of course, he had only played the melody line with his right hand, but it was a performance that was utterly unbelievable for someone playing the piano for the very first time.
Arin stared blankly at Cheol-jun as he returned to his seat with a bright, innocent face.
Th-that was insane...
Their eyes met, and Cheol-jun smiled warmly at her.
Arin quickly composed her expression and whipped her head away.
Wow! That was so much fun!
His first piano lesson had been truly delightful.
Once he learned the proper angle of the wrists and the correct placement of the fingers, he understood exactly why he had been so clumsy earlier.
Moreover, the textbook he received, the Beyer Piano Method, was incredibly fascinating.
This was Cheol-jun's very first encounter with sheet music.
Well, not quite the first.
He did have a music class last Wednesday.
But back then, his mind had been so consumed by Baduk that he hadn't paid any attention.
But now that he had been introduced to music, it felt entirely different.
The notes drawn on the five-line staff were mesmerizing.
The connections to mathematics that he had vaguely sensed while listening to music yesterday were now becoming visually intuitive.
He loved it.
First of all, the fact that it could be explained through mathematics put his mind at ease.
Time signatures were fractions, which was beautifully clean.
The musical staff was a coordinate system, a two-dimensional graph.
Mapping the notes to the keys was also incredibly fun.
Seven white keys, five black keys—twelve in an octave.
As he walked toward the main gate of his apartment complex, happily calculating and linking sheet music to math, Cheol-jun suddenly tilted his head and stopped in his tracks.
Why twelve?
On a piano, it was seven steps from C to G. Twelve steps from C to C. He knew that G was 3/2 times the frequency of C.
However, if you moved another 3/2 times starting from G, a different note was produced.
Since an octave consisted of twelve semitones, repeating this cycle should eventually bring you back to C, but something was slightly off.
The piano keys pointed to C, but no matter how he calculated it, the math didn't yield a perfect answer.
What if I multiply 3/2 twelve times?
(3/2)¹² = 129.746...
The total number of keys on a piano was 88.
From the first C to the last C, there were 84 keys, which meant 7 octaves of 12 semitones each.
An octave was a doubling of frequency, so 2⁷.
Which was 128.
Just as I thought...
129.746 ≠ 128.
The piano keys returned to C, but the mathematical value did not reach the exact octave.
In that case, the ratio of each semitone had to be adjusted slightly downward.
If all semitones were adjusted equally, multiplying that value twelve times had to equal exactly 2.
12 semitones = 1 octave = 2 times the frequency.
x¹² = 2
He tried multiplying 1.05 twelve times, but it was less than 2.
He tried 1.06, and it was greater than 2.
He tried 1.059, and it was still slightly less.
1.0595...
No matter how much he narrowed it down, a clean, terminating number never appeared.
He tried to express it as a fraction.
13/12, 21/20.........
No fraction, when multiplied twelve times, would ever equal exactly 2.
Cheol-jun stood frozen in place, his brow deeply furrowed.
The more he calculated, the more he realized that the answer was not a clean number.
It was clear that a single semitone on the piano, which produced such beautiful sounds, was mathematically messy.
But what bothered him even more was that, with his current mathematical knowledge, he couldn't mathematically prove why it could never be clean.
He had never felt this way before.
He knew it was impossible, but not fully understanding the underlying proof of why it had to be so was incredibly frustrating.
Cheol-jun's eyebrow twitched.
To resolve this frustration, he needed a level of mathematics far higher than what he currently possessed.
Cheol-jun was capable of objectively assessing his own knowledge.
He couldn't solve this alone.
He needed someone's help.
Among his acquaintances, there was only one person with whom he had ever discussed mathematics.
He quickly pulled out his phone.
"Alright, so let's prime factorize 36. What is the smallest prime number that can divide 36?"
"Um..."
"We just did this three times together. Think carefully."
"..."
"Hmm, okay. You do understand what a prime number is, right?"
The child shook their head side to side.
It seemed they still didn't get it, even though she had explained prime numbers for the fourth time now.
It had been a very long time since she had encountered a sixth grader who couldn't comprehend the sixth-grade curriculum.
Especially in this neighborhood, where advanced learning was the norm.
Even though it was a situation that should have been incredibly frustrating, Cha Se-young smiled warmly.
Ah, this is so therapeutic.
Cha Se-young ran a hand through her hair.
The hair that had fallen out in patches had begun to grow back a few days ago, and she could feel the slightly prickly new growth.
She absolutely loved that feeling.
The doctor at the clinic had told her she didn't need to worry anymore.
"Alright~ Teacher will explain it again~ A prime number is a number that cannot be divided any further..."
Just as Cha Se-young was continuing her explanation with a deeply peaceful mind, the smartwatch on her wrist vibrated.
...Huh?
Cha Se-young checked her smartwatch, and her shoulders suddenly flinched.
