Finger-hole positions follow from the ratio of the notes wanted: a pipe’s pitch is inversely proportional to its effective length, so a note a ratio r above the lowest needs an effective air column of the full length divided by r. A 1 mm error on a 300 mm column shifts the pitch about 5.8 cents, where a semitone is 100 cents, and a hole drilled too large cannot be made smaller, so drill small, play, and enlarge only the holes that sound flat. Warm air raises every note, so tune at playing temperature.
Part of Music & Simple Instrument Making
Calculating where finger holes go from a chosen scale, the size of the errors that matter, and how to tune an open-hole flute by stages.
Pitch Is Inverse To Effective Length
For an open pipe the lowest note is f = v ÷ 2L. Because v is fixed by the air, the ratio of two notes is the inverse ratio of their effective lengths. A note r times higher than the lowest needs an effective length of L ÷ r. In equal temperament each semitone is the ratio 2^(1/12) = 1.0595, which is 100 cents by definition. In cents, pitch difference = 1200 × log₂(f₂ ÷ f₁).
Tolerance: what 1 mm costs. An error of 1 mm in effective length L shifts pitch by 1200 × log₂(L ÷ (L − 1 mm)):
- L = 300 mm: 1200 × log₂(300 ÷ 299) = 5.8 cents
- L = 250 mm: 6.9 cents
- L = 150 mm: 11.6 cents
So the same slip matters twice as much on a short pipe. Whether 6-12 cents is audible depends on the listener and the tune; I have no sourced figure for a tolerance, so test by playing the note against a reference.
Worked Example: A Five-Hole Pentatonic Flute
Choose a major pentatonic scale in just intonation, ratios 1, 9/8, 5/4, 3/2, 5/3 and 2 above the lowest note. Let the effective open-pipe length (all holes closed) be L = 300 mm and v = 343 m/s.
- Lowest note: f₀ = 343 ÷ (2 × 0.300) = 571.7 Hz, which is 47 cents flat of D5 (587.3 Hz). The note is set by L, not by a standard pitch, so decide whether to match another instrument.
- Effective length for each note = L ÷ ratio. Measured from the blowing end to the centre of the hole:
| Ratio | Cents above lowest | Hole centre from blowing end (mm) | Distance from foot (mm) |
|---|---|---|---|
| 9/8 | 203.9 | 300 ÷ 1.125 = 266.7 | 33.3 |
| 5/4 | 386.3 | 300 ÷ 1.25 = 240.0 | 60.0 |
| 3/2 | 702.0 | 300 ÷ 1.5 = 200.0 | 100.0 |
| 5/3 | 884.4 | 300 ÷ 1.667 = 180.0 | 120.0 |
| 2 | 1200.0 | 300 ÷ 2 = 150.0 | 150.0 |
- Check the gaps: 266.7, 240.0, 200.0, 180.0, 150.0 differ by 26.7, 40.0, 20.0 and 30.0 mm, so the smallest spacing is 20 mm, which must still fit a fingertip. If the player’s fingers are wider than 20 mm centre to centre, move to a longer pipe or accept a narrower scale. The finger width needed here is from general experience (roughly 16-20 mm for an adult fingertip; recalled, not measured).
- To play the note for a hole, open it and every hole below it toward the foot. All holes closed gives the lowest note.
Just versus equal temperament. The just ratios differ from equal-tempered ones: 5/4 is 386.3 cents against 400 cents, 13.7 cents flat; 9/8 is 203.9 against 200; 3/2 is 702.0 against 700; 5/3 is 884.4 against 900, 15.6 cents flat. If the flute plays with a voice or a string, tune to the voice, and if the community has no tradition of equal temperament do not force it. Choose the scale by ear and then measure it.
Why Holes Never Land Exactly
The calculation is for an ideal pipe. Three effects shift the real pitch:
- End correction. An open end acts as if the pipe were a little longer, by roughly 0.6 times the bore radius at an unflanged open end (a recalled textbook value). For a 7 mm radius bore that is about 4 mm at one open end, which on L = 300 mm lowers the pitch by about 1200 × log₂(304 ÷ 300) = 23 cents.
- Hole size. A hole smaller than the bore does not end the air column cleanly; the pipe behaves as if it were longer than the hole’s position. A small hole therefore sounds lower than the calculated note. I did not rederive the size correction, which is why the method below is by trial.
- Temperature. v rises with temperature, from 343 m/s at 20 °C to 352 m/s at 35 °C, computed from v = √(γRT ÷ M) with γ = 1.4, R = 8.314 J/mol·K, M = 0.02897 kg/mol. The pitch shift is 1200 × log₂(351.9 ÷ 343.2) = 43 cents. Breath warms the air in the tube toward body temperature as you play, so a flute tuned cold is sharp when played; I took 35 °C as an illustrative playing temperature and did not source it.
Method: Tune In Stages
Step 1: Make the lowest note first. Cut the tube slightly long (10 mm over the target), blow it, and compare with a reference. Trim the foot to raise the pitch.
Step 2: Mark the hole centres from the table, or from your own scale.
Step 3: Drill each hole at about half its final diameter. A small hole sounds flat of the target, so start flat and move up.
Step 4: Enlarge in small steps, one hole at a time, testing after each. Enlarging a hole raises its note; you cannot put material back, so stop early.
Step 5: Use a reference. A one-string instrument (see One-String Diddley Bow) can serve as a monochord: its fret positions for any semitone are L(1 − 2^(−n/12)).
Step 6: Recheck warm. Play for a few minutes, then recheck the notes.
Common Mistakes
- Drilling the final size first. A hole too big cannot be repaired.
- Ignoring finger spacing. A scale that cannot be fingered is unusable.
- Tuning cold. The flute sounds flat in the morning and sharp when played.
- Forgetting that holes interact. Opening one hole changes how the others behave; check every note after each change.
Safety
Drilling and mouthed instruments
Hold the work in a vice or against a block, drill away from your hand, and wear eye protection for bone and wood chips. The mouthpiece goes in a mouth: finish only with wax or plain oil, never paints, and never use salvaged tube of unknown material. See Material Selection.
What’s Next
- Learn next: Bone Flutes — applying this to a bone tube
- Combine with: Panpipes — a flute with one tube per note instead of holes