The short answer
A semitone is about 2.8% of a tine’s free length. Measure from the bridge to the tip, take 2.8% of it, and that is roughly how far the tine has to slide to move one semitone — about 1.7 mm on a 17-key’s centre C4, 0.8 mm on the short E6 at the edge, and 2.1 mm on a 21-key’s F3. One cent — the resolution you are tuning to — is about a fiftieth of that, which is 18 thousandths of a millimetre.
That percentage does not depend on what the tine is made of, how thick it is or how wide. It is the same 2.8% on brass, on a hand-forged mbira key and on a 0.8 mm treble tine. Everything below is why.
- One cent is 14 thousandths of a millimetre on a tine this long — which is why the discipline is one tap, then pluck, then read.
- This number needs no knowledge of what your tine is made of or how thick it is. It is a ratio of the length you measured, and nothing else.
The equation
A tine is a cantilever: clamped at the bridge, free at the tip. Its fundamental transverse mode is the standard Euler–Bernoulli result for a clamped-free beam, and for a uniform rectangular cross-section it reduces to this:
f = 0.16154 × ( h / L² ) × √( E / ρ )
f fundamental frequency Hz
h thickness of the bar metres
L free length, bridge to tip metres
E Young's modulus pascals
ρ density kg/m³
0.16154 = β₁² / (4π√3), where β₁ = 1.8751 is the first
root of cos(β)·cosh(β) = −1 — the clamped-free eigenvalue.Solved the other way, for the length that gives a wanted note:
L = √( 0.16154 × h × √(E/ρ) / f )
The direction that matters
Nobody with a kalimba in their hands wants to know what note a 61.2 mm tine plays. They want to know how far to move the tine they have. That question has a much better answer than the first one, and the reason is that almost everything cancels.
Frequency goes as 1/L², so for one tine — one thickness, one material, one piece of steel — the ratio of two lengths depends only on the ratio of two frequencies:
f ∝ 1 / L²
L_target = L_current × √( f_current / f_target )
f_target / f_current = 2^(cents / 1200)
⇒ L_target = L_current × 2^(−cents / 2400)No modulus. No density. No thickness, and no width. The only input is the free length you can measure with a ruler, and the interval you want. That is why the retune planner gives millimetres per tine and means them, while the note prediction above comes with an error bar the size of a semitone.
What a semitone costs, tine by tine
Every note on a 21-key in C major, at 1.2 mm carbon spring steel — the commonest stock. The bottom four rows are the tines only a 21-key has; everything from C4 down the list is a 17-key as well.
| Note | Position | Frequency | Free length | One semitone | One cent |
|---|---|---|---|---|---|
| F3 21-key only | centre | 174.61 Hz | 74.9 mm | 2.13 mm | 21 µm |
| G3 21-key only | 1st left | 196.00 Hz | 70.7 mm | 2.01 mm | 20 µm |
| A3 21-key only | 1st right | 220.00 Hz | 66.7 mm | 1.90 mm | 19 µm |
| B3 21-key only | 2nd left | 246.94 Hz | 62.9 mm | 1.79 mm | 18 µm |
| C4 | 2nd right | 261.63 Hz | 61.2 mm | 1.74 mm | 17 µm |
| D4 | 3rd left | 293.66 Hz | 57.7 mm | 1.64 mm | 16 µm |
| E4 | 3rd right | 329.63 Hz | 54.5 mm | 1.55 mm | 16 µm |
| F4 | 4th left | 349.23 Hz | 52.9 mm | 1.51 mm | 15 µm |
| G4 | 4th right | 392.00 Hz | 50.0 mm | 1.42 mm | 14 µm |
| A4 | 5th left | 440.00 Hz | 47.2 mm | 1.34 mm | 13 µm |
| B4 | 5th right | 493.88 Hz | 44.5 mm | 1.27 mm | 13 µm |
| C5 | 6th left | 523.25 Hz | 43.2 mm | 1.23 mm | 12 µm |
| D5 | 6th right | 587.33 Hz | 40.8 mm | 1.16 mm | 12 µm |
| E5 | 7th left | 659.26 Hz | 38.5 mm | 1.10 mm | 11 µm |
| F5 | 7th right | 698.46 Hz | 37.4 mm | 1.07 mm | 11 µm |
| G5 | 8th left | 783.99 Hz | 35.3 mm | 1.01 mm | 10 µm |
| A5 | 8th right | 880.00 Hz | 33.3 mm | 0.95 mm | 9 µm |
| B5 | 9th left | 987.77 Hz | 31.5 mm | 0.90 mm | 9 µm |
| C6 | 9th right | 1046.50 Hz | 30.6 mm | 0.87 mm | 9 µm |
| D6 | 10th left | 1174.66 Hz | 28.9 mm | 0.82 mm | 8 µm |
| E6 | 10th right | 1318.51 Hz | 27.2 mm | 0.78 mm | 8 µm |
The free-length column is the predicted one and carries the error bar below. The last two columns do not: they are 2.8% and 0.028% of whatever the length actually is, so if your F3 measures 71 mm rather than 74.9, scale the last two columns by the same proportion and they are correct for your instrument.
Materials, and where the numbers come from
What the equation actually reads is √(E/ρ) — the speed of extensional waves in the material — so neither the modulus nor the density decides anything on its own. Brass has about half the stiffness of steel and more density, which is why a brass bar of identical geometry sounds a long way lower.
| Material | E | ρ | √(E/ρ) | Where you meet it |
|---|---|---|---|---|
| Carbon spring steel (AISI 1095, blue-tempered) AZoM, “AISI 1095 Carbon Steel (UNS G10950)” | 200 GPa (190–210) | 7850 kg/m³ | 5048 m/s | The default. Nearly every factory kalimba tine, and what replacement-tine sets are sold as. |
| Stainless steel (AISI 304) MakeItFrom, “AISI 304 (S30400) Stainless Steel” | 200 GPa | 7800 kg/m³ | 5064 m/s | Sold on humid-climate and “rust-proof” models. Behaves almost identically to carbon steel here. |
| Cartridge brass (C26000) MakeItFrom, “UNS C26000 (CW505L) Cartridge Brass” | 110 GPa | 8200 kg/m³ | 3663 m/s | Decorative tines and some hand-made instruments. Softer, darker, and it moves further per tap. |
| Phosphor bronze (C51000) MakeItFrom, “UNS C26000 Cartridge Brass vs. UNS C51000 Phosphor Bronze” | 110 GPa | 8800 kg/m³ | 3536 m/s | Occasional on hand-built lamellophones, and the usual stock for mbira keys that were not forged from scrap. |
The two steels differ by six cents in this model, which is less than a tine holds overnight — so if you do not know which one your instrument has, it does not matter. Brass and phosphor bronze are a different instrument entirely: the same geometry sounds roughly a fifth lower, so a brass tine has to be substantially shorter to reach the same note.
What the model assumes, and what a real tine does
The equation describes an ideal bar. A kalimba tine is not one, and the differences all push in knowable directions:
| The model assumes | A real tine |
|---|---|
| A uniform rectangular cross-section along the whole free length. | Most tines are ground thinner toward the tip, and many are rounded at the end. A tapered bar is lighter where it moves most, which raises the pitch above what a uniform bar of the same length would give. |
| An ideal clamp: the tine is perfectly rigid at the bridge and free beyond it. | A kalimba tine is held by friction between a curved bridge rod and a pressure bar. It flexes a little inside the clamp, which makes the effective free length slightly longer than the one you can measure with a ruler — the single biggest source of error here, and it gets worse as the clamp loosens. |
| The bar vibrates alone. | It is bolted to a wooden box that resonates back. The coupling pulls a little energy into low-order harmonics and shifts the fundamental slightly, which is part of why a tine sounds different mounted than in a vice. |
| Room temperature, and steel of the published stiffness. | Young’s modulus falls a little as steel warms, which is a real part of why a kalimba drifts on a hot day. And the published range for one grade of spring steel is already ±5%, which is ±43 cents on its own. |
| Small vibrations — the bar stays in its linear range. | A hard thumbnail pluck bends the tine far enough to read a few cents sharp for the first instant. That is why every guide on this site says to read the dial in the sustain rather than on the attack. |
The clamp is the big one. A tine flexes a little inside the bridge before it is truly held, so the length that vibrates is slightly longer than the length you can measure — and it gets longer as the clamp loosens with age. That is one of the reasons a well-used kalimba drifts flat rather than randomly.
Using it
- Measure, do not predict. A ruler against the tine you are about to move replaces every uncertain term in the first direction. Bridge to tip.
- Take the travel figure. 2.8% of that length per semitone, or use the calculator above for an exact interval.
- Move less than the figure says, then re-read. The number is where you are going, not how hard to hit. Tines overshoot.
- Finish on the tuner. The kalimba tuner is what tells you when you have arrived. Nothing on this page is a substitute for that, and it is not trying to be.
Frequently asked questions
How far does a kalimba tine move for one semitone?
Roughly 2.8% of its free length, whichever tine it is. On a 17-key in C that works out at about 1.7 mm for the centre C4 tine and about 0.8 mm for the short E6 at the edge; a 21-key’s F3 bass tine needs about 2.1 mm. The percentage is the useful part, because it needs nothing but a ruler: measure the free length of the tine you are about to move, take 2.8% of it, and that is your semitone. One cent is about a fiftieth of that — 17 thousandths of a millimetre on a middle tine — which is why one tap and a re-read beats two taps.
Why does the tine move further than I expect?
Because pitch goes as one over the length squared, so the same millimetre buys more pitch on a short tine than on a long one, and the tine you are usually fighting with is a long one. It is also why the direction people get wrong is the small one: a tap that feels like nothing on a bass tine is a real move on a treble tine, and the trebles are where an overshoot is hardest to walk back.
Does the width of the tine change its pitch?
Not in this model, and the reason is worth knowing rather than taking on trust. Frequency depends on the bar’s stiffness divided by its mass, and widening a rectangular bar increases both in exactly the same proportion — the width cancels out of the equation algebraically. Two tines of the same length, thickness and steel sound the same note whether they are 3 mm wide or 6 mm. Width does change how loud the tine is, how well it resists being knocked sideways into its neighbour, and how much a real tine departs from the ideal bar this equation describes. It does not change the note.
How accurate is the predicted note?
About a semitone, and no better. The published range for the Young’s modulus of one grade of spring steel is 190–210 GPa, which on its own is 87 cents of pitch — most of a semitone, from a number a datasheet presents as settled. Add a tine that is ground thinner toward the tip, a clamp that flexes, and a wooden body coupled to the whole thing, and “about a semitone” is the honest claim. Use it to cut a blank a little long and tune down onto the note. Do not use it as a cut list.
Then why trust the millimetre figure for a retune?
Because that direction is a different calculation, and every uncertain term cancels out of it. The note prediction needs the modulus, the density and the thickness. The travel figure needs only the ratio of two frequencies and the length you measured, because both the before and the after are the same piece of steel: L_target = L_current × 2^(−cents/2400), and nothing about the material survives into it. That is the whole reason this page leads with the move direction and the retune planner is built on it.
Can I use this for an mbira or another lamellophone?
The physics, yes — a key is the same object, a steel bar clamped at one end, and the travel formula is as true of an mbira key as of a kalimba tine. The pitches are a different matter. An mbira has no standard pitch: its tunings are named sets of intervals that vary from family to family, so there is no target note to calculate toward. What transfers is the ratio: to raise any key by a known number of cents, shorten it by 2^(−cents/2400) of its free length.
Related
- Kalimba tuner — the dial, and the full tine map for every size.
- Retune planner — a whole key change as a work list, in millimetres.
- Buzz diagnostic — when the pitch is right and the sound is not.
- How to tune a kalimba — the hammer technique these numbers are for.
- Retuning a kalimba to another key — why a key signature is a price list.
- Playable kalimba — the same tine model, synthesised.