Pythagorean Scale & Interval Ratios for a 13-year-old — Monochord, 2/3 splits and converting decimals into simple fractions

Imagine Ally McBeal onstage, a monochord across her lap, dramatic pause, then: "Listen!" Short sentences. Big drama. Simple math. Music comes alive.

Lesson 1 — The monochord, halves and 2/3 splits (in a theatrical cadence)

Pythagoras used a single string. Shorter string = higher pitch. Important rule: frequency is inversely proportional to length. If you make the string half as long, the frequency doubles. If you make the string two-thirds as long, the frequency multiplies by 3/2 (because f_new = f_old / (2/3) = f_old * 3/2).

1. Question 1a. Ratio of one part to total when cut in half? Answer: 1:2.
2. Question 1b. Middle C = 261.63 Hz. Half the string => frequency doubles: 261.63 x 2 = 523.26 Hz.
3. Question 1c. What happens to pitch when string is halved? The pitch goes up one octave; frequency doubles.

So a C scale must live between 261.63 Hz and 523.26 Hz (one octave).

Now the 2/3 split (the magic that makes a perfect fifth)

If you shorten the string to 2/3 of its length, frequency becomes 3/2 of the original. That is a perfect fifth above. Starting at middle C (261.63 Hz):

1. G = C * 3/2 = 261.63 * 1.5 = 392.445 Hz (rounded 392.45 Hz).
2. Next note (D) = G * 3/2 = 392.445 * 1.5 = 588.6675 Hz, but this is above the octave, so divide by 2 to bring into the C octave: 588.6675 / 2 = 294.33375 ≈ 294.34 Hz.
3. Continue stacking fifths (multiply by 3/2) and fold into the octave by doubling or halving by 2 when needed.

Pythagorean C scale (frequencies, rounded to 2 decimal places)

Why F is 348.84 Hz: the instructions said to find 2/3 below C for F. That means length = 2/3 of C length but below so frequency = C * (2/3) = 261.63 * 0.666666... = 174.42 Hz; then multiply by 2 to bring F into the C octave: 174.42 * 2 = 348.84 Hz.

Lesson 2 — Interval ratios (make decimals into neat fractions)

We compare the root note C to each other note. Do C divided by the other note (C/note). That decimal often equals a simple fraction because Pythagorean tuning uses powers of 3/2 and factors of 2.

Exact interval fractions from Pythagorean tuning (C compared to each note)

How we found those tidy fractions:

1. Notice patterns in the decimal: 0.888888... is repeating 8/9; 0.66666... is 2/3; 0.75 is 3/4.
2. Know the Pythagorean fractions: D = 9/8 above C, so C:D = 8/9. G = 3/2, so C:G = 2/3. E = 81/64 above C, so C:E = 64/81, and so on. These come from stacking 3:2 fifths and adjusting by powers of 2 to stay inside the octave.
3. If you must, divide the decimal and use fraction recognition or a calculator fraction tool, or derive using the algebra of (3/2)^n times powers of 2.

Corrections to the student answers

• Student put C:D as 4/5 — the correct simplified interval ratio is 8/9. The decimal 0.888888... signals 8/9.
• Student left C:G blank — correct ratio is 2/3 because G is a perfect fifth above C (frequency 3/2 times C).
• C:E is not 4/5. The correct ratio is 64/81 (about 0.790123...).
• C:A should be 16/27 (about 0.592593), not 3/5.
• C:B is 128/243 (about 0.52713), not 13/25 or 53/100.

Quick how-to for students (steps)

1. Build each new note by multiplying the previous note by 3/2 (when using the 2/3 string length trick). If the frequency goes outside the target octave, divide or multiply by 2 until it fits between C and C'.
2. To find C to other-note ratio: compute decimal = C / note.
3. Recognise a repeating decimal or match to known small fractions (8/9, 2/3, 3/4, etc.). If needed, use algebra with powers of 3 and 2: every Pythagorean interval is of the form 2^a * 3^b.

Final takeaway (Ally voice, short and strong): You did the musical math — now read the decimals like a detective and write them as neat little fractions. Drama. Clarity. Music.