Hey — imagine geometry that sings

While geometry makes proportions visible, harmony makes them audible. The monochord is like the ruler and compass of music: a single stretched string that shows how simple fractions of length make musical intervals (octave, fifth, fourth, etc.). We will build and use a simple monochord, explore ratios that make pleasant sounds, and — time permitting — do a paper-folding/calculation experiment that explains why musicians sometimes use tempered tuning.

What you need

• A board about 50–80 cm long (a cutting board or plank works)
• One string (guitar string or strong twine) and two fixed points to attach it (nails or screws)
• A moveable bridge (a small block of wood or even a pencil) to change the vibrating length
• A ruler (cm) and a calculator (or phone) for ratios
• Paper strip about 1 m long and a pencil for the folding experiment
• Optional: a tuner app to measure pitch in Hz

Part 1 — Build and use the monochord (step-by-step)

1. Attach the string to one end of the board, stretch it tight and attach the other end so the string is taut. The portion that vibrates between the two fixed points is the full length; call that length L. Pick L = 60 cm if you want easy numbers.
2. Pluck the whole string and listen. This is the base note (call its frequency f).
3. To get an octave above that note (which has frequency 2f), place the bridge at L/2 (for L = 60 cm, place it at 30 cm). Pluck the shorter segment; it sounds one octave higher. Why? Because frequency is roughly inversely proportional to length: halving the length doubles the frequency.
4. To make a perfect fifth (ratio 3:2 in frequency), place the bridge so the sounding length is 2/3 of L. For L = 60 cm, 2/3 L = 40 cm. Pluck it — that interval (the base note up to that new pitch) is a fifth.
5. Other simple ratios:
   • Perfect fourth: frequency 4:3 → length = 3/4 L (for 60 cm, 45 cm)
   • Major third (just): frequency 5:4 → length = 4/5 L (for 60 cm, 48 cm)
6. Try these: place the bridge at those lengths and compare how the notes sound together. You should notice simple ratios (2:1, 3:2, 4:3, 5:4) sound especially pleasant. That’s because the wave cycles line up periodically — geometry producing harmony.

Quick explanation — why simple ratios sound good

If two notes have frequencies in a ratio of small integers, their waves meet in regular, predictable ways (their peaks and troughs line up periodically). For example, with a 3:2 frequency ratio (perfect fifth), every 3 cycles of the higher note line up with every 2 cycles of the lower note. This regular alignment gives a clear, consonant sound.

Part 2 — The problem: stacking perfect intervals

People long ago used perfect fifths (3:2) to build scales. But if you stack 12 perfect fifths you do not land exactly on 7 octaves. Mathematically:

• Frequency after 12 perfect fifths: (3/2)^12
• Frequency after 7 octaves: 2^7

These two numbers are close but not equal. The ratio (3/2)^12 / 2^7 ≈ 1.01364 — a small but real mismatch called the Pythagorean comma. It means if you tune every fifth perfectly, you will slowly drift out of tune when you cycle around many notes.

Part 3 — Paper-folding/calculation experiment (simple, visual)

We can see the mismatch with a simple paper-and-calculator activity. We will work with lengths because length on the monochord is inverse to frequency. Pick a straight strip of paper and pretend the whole strip is the full string length.

1. Label the full strip length as 100 cm (or use your real measured L, but 100 makes percentages easy).
2. Fold once in half and mark 50 cm to show an octave (length = 1/2).
3. Fold the paper to find one third (fold carefully to make three equal parts) and mark the 2/3 point — this corresponds to the perfect fifth length (2/3 of the string).
4. Now calculate (using a calculator) how short the string becomes if you keep taking 2/3 twelve times: length factor = (2/3)^12. For a 100 cm start this gives about 0.77 cm. That represents the length you would get after stacking twelve perfect fifths, measured relative to the original full string.
5. Next calculate halving seven times, i.e. 1/128 (because 2^7 = 128). For 100 cm this is 100/128 ≈ 0.78125 cm.
6. Compare the two results: (2/3)^12 × 100 ≈ 0.7709 cm and 1/128 × 100 ≈ 0.78125 cm. They are very close but not identical — there is that small Pythagorean comma. If you try to fold the paper repeatedly to get these tiny lengths you’ll see they almost line up but won’t perfectly match.

That tiny mismatch is why just-intonation systems (built from perfect ratios) don’t allow every key to sound perfectly in tune. Musicians needed a practical solution.

Part 4 — Tempered tuning: the fix

Equal temperament spreads the tiny mismatch out over all 12 semitones, so each semitone increases frequency by the same factor: 2^(1/12) ≈ 1.059463. After 12 of those equal steps the frequency is exactly doubled (one octave). That means the perfect fifth in equal temperament is slightly smaller than a pure 3:2 fifth. Numerically:

• Pure fifth = 3/2 = 1.500000
• Equal-tempered fifth = 2^(7/12) ≈ 1.498307 (about 1.955 cents lower)

The change is tiny — often barely noticeable — but it solves the problem: every key is usable and the octave remains exact. Musicians trade the absolute purity of some intervals so that music can be played in any key without accumulating tuning errors.

Try this by ear

1. Tune the monochord to a reference note (use the tuner app).
2. Place the bridge for a pure fifth (2/3 L) and listen. Then tune the string to that pure fifth by ear (so the ratio is exactly 3:2).
3. Now try to go around the circle of fifths by moving the bridge 12 times (or retuning each time). After several steps you will begin to notice slight mismatches when you come back to the start — that’s the comma showing up.
4. Finally, use the tuner and retune the notes to equal temperament or just compare the pure fifth and the tempered fifth (use the tuner to measure Hz or cents). Listen for the tiny difference.

Curriculum connections (ACARA v9, quick map)

• Math: ratios and proportional reasoning — using fractions like 1/2, 2/3, 3/4 and interpreting them on a number line.
• Science (waves): relation between string length and frequency, and how wave alignment creates consonance.
• Music: intervals, scale construction, temperament vs just intonation; listening and tuning skills.

Short summary

The monochord turns abstract ratios into sounds you can hear: halves give octaves, two-thirds give fifths, etc. But stacking perfect intervals causes a small mismatch (the Pythagorean comma). Tempered tuning spreads that mismatch across all notes so the octave stays exact and we can play in any key. Your paper-folding/calculation experiment will show how very small that mismatch is, and your ears can explore where pure and tempered intervals differ.

Want a printable worksheet or step-by-step cardboard monochord plan to build in class? I can make one next.