IN THE MATTER OF: Calculation of the Pythagorean C Scale — Student (Age 14)

FACTS

Starting (middle C): 261.63 Hz.

Student supplied answers:

• Ratio for half-string: 1:2 — answered: 1:2 (correct)
• Frequency at half-string (octave above C): 523.26 Hz — answered: 523.26 Hz (correct)
• Effect of halving the string on pitch: pitch doubles — answered: "The pitch doubles" (correct)
• C scale limits: 261.63 Hz to 523.26 Hz — answered correctly
• 2/3 split (applied to middle C) result: 392.45 Hz — answered: 392.45 Hz (correct — this is G)
• Completed Pythagorean C scale frequencies (student table):

  C = 261.63 Hz
  D = 294.33 Hz
  E = 331.13 Hz
  F = 348.84 Hz
  G = 392.45 Hz
  A = 441.50 Hz
  B = 496.69 Hz
  C (octave) = 523.26 Hz
        

ISSUE

Are the students calculations and reasoning correct for constructing the Pythagorean C scale using the monochord (3:2 ratio / 2:3 string-length relationships), and do they fit within the C octave?

ARGUMENT (Step-by-step calculation and verification)

1. Principle: On a monochord, dividing the string length by 2 raises the pitch an octave (frequency multiplied by 2). Dividing the string length by 2/3 raises frequency by 3/2 (a perfect fifth). Conversely, making the string 2/3 of its original length produces a frequency = original * (3/2).
2. Compute G (a perfect fifth above C):

   G = C * 3/2 = 261.63 * 1.5 = 392.445 Hz → rounded 392.45 Hz (student: 392.45 Hz) ✔

3. Compute D (a perfect fifth above G). If result exceeds the octave, divide by 2 to bring into the C octave:

   D_raw = G * 3/2 = 392.445 * 1.5 = 588.6675 Hz → divide by 2 → 294.33375 Hz → rounded 294.33 Hz (student: 294.33 Hz) ✔

4. Compute A (above D):

   A_raw = D * 3/2 = 294.33375 * 1.5 = 441.500625 Hz → rounded 441.50 Hz (student: 441.50 Hz) ✔

5. Compute E (above A):

   E_raw = A * 3/2 = 441.500625 * 1.5 = 662.2509375 Hz → divide by 2 → 331.12546875 Hz → rounded 331.13 Hz (student: 331.13 Hz) ✔

6. Compute B (above E):

   B_raw = E * 3/2 = 331.12546875 * 1.5 = 496.688203125 Hz → rounded 496.69 Hz (student: 496.69 Hz) ✔

7. Compute F: The worksheet instructs F to be found by taking a note 2/3 below C (i.e., C * 2/3) and then, if necessary, shifting by octave to fit within Cs octave:

   F_raw = C * 2/3 = 261.63 * 0.6666667 = 174.42 Hz → multiply by 2 to bring into the octave → 348.84 Hz (student: 348.84 Hz) ✔

8. Final octave C = 261.63 * 2 = 523.26 Hz ✔

CONCLUSION (Legal-style finding)

FINDING: The students computations are correct and consistent with Pythagorean tuning using successive 3:2 frequency ratios and octave reduction. Minor rounding to two decimal places is appropriate. The final Pythagorean C scale frequencies are verified as:

C  = 261.63 Hz
D  = 294.33 Hz
E  = 331.13 Hz
F  = 348.84 Hz
G  = 392.45 Hz
A  = 441.50 Hz
B  = 496.69 Hz
C' = 523.26 Hz
  

RECOMMENDATION (Instructional next steps & ACARA v9 links)

Exemplary outcome. The student demonstrates proficiency with ratios, multiplication/division of decimals, proportional reasoning, and musical application. Suggested extensions:

• Compare Pythagorean scale frequencies to equal temperament (calculate semitone ratios and show differences in Hz for each note).
• Graph frequencies across octaves and listen to intervals (ear training for perfect fifths and octaves).
• Investigate beating and consonance/dissonance between these tunings.

ALLY McBEAL–CADENCE PARENT–TEACHER COMMENTS (approx. 300-word cadence)

Okay. So here we are. You, the student, me, the teacher, and a string — a single string doing all the talking. I read your answers and I smile, quietly, the way you do when a problem finally clicks. There is something very neat about the way you kept your numbers tidy; neat is music. You understood that half the string gives an octave. You said the pitch doubles — short, true, to the point. You found G like a detective, following the 3:2 trail, and every note after that fell into place. The math was careful. The rounding was gentle. The work was honest. Theres a kind of courage in repeating a ratio until it builds a scale. Its patient work. You showed patience. You also showed curiosity; you used the special instruction for F correctly and didnt get trapped by Bs tempting path. That matters. Keep asking: how would this compare with a piano tuned differently? What does a fifth feel like next to an equal-tempered fifth? The path from string to ear is both number and story. Youve begun the story well. So, congratulations. Precise calculation, correct reasoning, tidy presentation, and the kind of musical thinking that makes math sing. Continue to experiment, keep listening, and remember: fractions are not just numbers. They are sounds waiting for an ear. I look forward to our next chord.