How To Make a Transverse (Side-Blown) Flute.md

How To Make a Transverse (Side-Blown) Flute

A transverse, side-blown flute is a cylindrical aeroacoustic resonator whose playable pitches arise from selective shortening of the effective air-column length through tone-hole venting, modified by embouchure geometry, end correction, chimney height, and head-end boundary conditions imposed by the cork.

The main takeaway is that the note is set by the effective vibrating length of the air column, which must resonate; within reason* [A]; in accordance with the wavelength of the note in equal temperament (12-TET) — and that the tone holes shorten that column by allowing air to escape.

1. Base tube length for the fundamental

For an open cylindrical tube, the ideal relationship is:

$$ f = \frac{v}{2L_{\text{eff}}} $$

So:

$$ L_{\text{eff}} = \frac{v}{2f} $$

Where: • f = target frequency in Hz • v = speed of sound, about 343 m/s at room temp • $L_{\text{eff}}$ = effective acoustic length

That is the effective length, not the exact physical cut length.

Because the flute has end effects, the physical tube is a little shorter than the ideal acoustic length.

A common rough correction is:

$$ L_{\text{phys}} \approx L_{\text{eff}} - \Delta_{\text{emb}} - \Delta_{\text{end}} $$

Where those correction terms depend on bore and embouchure geometry.

For a rough first pass on a simple flute, people often use something like:

$$ \Delta_{\text{end}} \approx 0.3D $$

with $D$ = inside bore diameter.

The embouchure correction is trickier and more empirical.

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1. Hole location idea

Each open hole acts like a new end of the tube.

So for a note with one specific hole acting as the first open hole, the distance from the acoustic origin near the embouchure to that hole should be about:

$$ x_n \approx \frac{v}{2f_n} $$

Where: • x_n = effective length needed for note n • f_n = frequency of that note

But that is still only the effective acoustic location.

The actual drilled hole center has to be adjusted because the hole is not a perfect open end.

So a more useful working idea is:

$$ x_{n,\text{physical}} = x_{n,\text{effective}} + \text{hole correction} $$

And the correction depends mostly on: • bore diameter • hole diameter • wall thickness • how strongly that hole vents

Small holes vent weakly, so the acoustic end acts a bit farther down the tube than the hole center. That means small holes often need to be placed a bit higher or enlarged to behave right.

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1. Frequency of each note

In 12-TET, if you know the semitone offset N from A4 = 440 Hz:

$$ f = 440 \cdot 2^{N/12} $$

Examples: • A4: N=0 • B4: N=2 • C5: N=3 • D5: N=5

Then for each target note:

$$ L_n = \frac{v}{2f_n} $$

That gives the ideal acoustic length from embouchure to first open vent.

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1. Practical hole size relationship

There is no single perfect closed-form equation for hole diameter that works for every homemade flute, because the venting depends on geometry. But a common design relationship is that the hole must vent strongly enough relative to the bore.

A rough rule is:

$$ d_h \propto D $$

Where: • d_h = hole diameter • D = bore diameter

Typical finger holes on simple flutes are often somewhere around:

$$ d_h \approx 0.3D \text{ to } 0.6D $$

depending on note, ergonomics, and tuning needs.

Upper holes often need to be larger than beginners expect, because they must act more like a true vent.

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1. A more useful approximate hole-placement formula

A common simplified model for a tone hole is:

$$ L_{\text{needed}} = x_h + \delta_h $$

So:

$$ x_h = L_{\text{needed}} - \delta_h $$

Where: • x_h = physical hole-center distance from embouchure reference • $L_{\text{needed}} = \frac{v}{2f}$ • $\delta_h$ = tone-hole end correction

A rough approximation is that $\delta_h$ gets larger when: • hole diameter is smaller • wall thickness is thicker

One intuitive approximation is that:

$$ \delta_h \propto \frac{t}{d_h^2} $$

where: • t = wall thickness • d_h = hole diameter

That is not a precision build formula by itself, but it captures the behavior: • thicker wall → weaker vent • smaller hole → weaker vent • weaker vent → acoustic end moves farther away than you want

Which is exactly the “can’t decide which octave” mess you were describing.

⸻

1. Practical step-by-step design equations

If you want a usable design workflow, it looks like this.

Step A: pick your notes

Suppose your scale notes are $f_1, f_2, f_3, \dots$

Step B: compute ideal acoustic lengths

$$ L_n = \frac{v}{2f_n} $$

Step C: choose a starting bore

Say inside diameter is $D$

Step D: estimate physical hole centers

$$ x_n \approx L_n - \delta_n $$

Where $\delta_n$ is a guessed correction based on hole size and wall thickness.

Step E: start with conservative hole sizes

Pick smaller pilot holes first.

Step F: enlarge gradually upward

Because enlarging a hole generally: • raises the note • strengthens venting • stabilizes response

⸻

1. What enlarging a hole does mathematically

A bigger hole reduces the tone-hole correction.

In other words:

$$ d_h \uparrow \Rightarrow \delta_h \downarrow $$

and therefore the hole behaves more like a true open end.

That makes the effective length shorter, so the pitch rises and usually stabilizes better.

So when your upper holes are too small:

$$ \delta_h \text{ is too large} $$

which means the flute doesn’t strongly commit to the intended shorter air column.

⸻

1. A very practical design approximation

For a homemade flute, a decent first-pass approach is:

Hole positions

$$ x_n \approx \frac{v}{2f_n} - C_n $$

Where C_n is a small empirical correction.

Hole sizes

Start around:

$$ d_h \approx 0.35D \text{ to } 0.45D $$

for middle holes, then adjust by testing.

Upper holes often end up a bit larger acoustically than you think they should.

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1. Example

Let’s say you want a note at 523.25 Hz, about C5.

$$ L = \frac{343}{2 \cdot 523.25} $$

$$ L \approx 0.3277 \text{ m} $$

So the effective acoustic length is about:

$$ 327.7 \text{ mm} $$

If your tone hole is small and thick-walled, the actual center of that hole may need to be somewhat above that effective point, or the hole may need to be enlarged until it behaves like that point.

⸻

1. The honest version

The core math is clean:

$$ f = \frac{v}{2L} $$

But tone hole math is only partly clean. After that, flute building becomes: • acoustics • venting efficiency • end corrections • ergonomics • tiny adjustments by ear

So the real build equation is basically:

$$ \text{ideal length from frequency} + \text{corrections from geometry} + \text{final tuning by enlargement} $$

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1. The main equations to keep

These are the big ones:

Note frequency

$$ f = 440 \cdot 2^{N/12} $$

Ideal open-tube length

$$ L = \frac{v}{2f} $$

Physical hole center estimate

$$ x_h \approx \frac{v}{2f} - \delta_h $$

Hole correction trend

$$ \delta_h \uparrow \text{ when } d_h \downarrow \text{ or } t \uparrow $$

Practical hole-size relation

$$ d_h \approx kD \quad\text{with } k \text{ often around } 0.3\text{–}0.6 $$

⸻

If you want, give me: • inside diameter • wall thickness • overall tube length • embouchure location • the scale you want

and I’ll run a real hole chart for your flute in inches or mm.

Extended Theory

The fuller derivation below restates the same flute model in a more formal way, with the same Markdown-safe math formatting.

Side-Blown Flute Acoustics: Core Equations

1. Fundamental open-pipe model

For a cylindrical flute acting approximately like an open-open air column:

$$ f_n = \frac{n v}{2L_{\mathrm{eff}}} \qquad n = 1,2,3,\dots $$

Where: • $f_n$ is the $n$-th resonance frequency • $v$ is the speed of sound • $L_{\mathrm{eff}}$ is the effective acoustic length

For the fundamental:

$$ f_1 = \frac{v}{2L_{\mathrm{eff}}} $$

So:

$$ L_{\mathrm{eff}} = \frac{v}{2f_1} $$

2. Speed of sound

At temperature $T$ in Celsius, a common approximation is:

$$ v \approx 331.3 + 0.6T $$

At room temperature ($20^\circ\mathrm{C}$):

$$ v \approx 343.3\ \mathrm{m/s} $$

A more thermodynamic-looking form is:

$$ v = \sqrt{\gamma R T_{\mathrm{K}}} $$

Where: • $\gamma$ is the adiabatic index for air • $R$ is the specific gas constant • $T_{\mathrm{K}}$ is temperature in Kelvin

3. Equal temperament note frequency

If you want any pitch from 12-TET relative to A4 = 440 Hz:

$$ f = 440 \cdot 2^{N/12} $$

Where $N$ is the number of semitones from A4.

Examples:

$$ f_{\mathrm{C5}} = 440 \cdot 2^{3/12} $$

$$ f_{\mathrm{D5}} = 440 \cdot 2^{5/12} $$

$$ f_{\mathrm{E5}} = 440 \cdot 2^{7/12} $$

4. Ideal acoustic length for each note

For each target note $f_i$, the required effective tube length is:

$$ L_i = \frac{v}{2f_i} $$

This is the ideal acoustic length from the embouchure reference to the first acoustically dominant open vent.

Real Flute Corrections

Now we leave “tube with holes” territory and enter “the tube is lying to you.”

5. Effective length vs physical length

The actual physical distance is not exactly the same as the acoustic length:

$$ L_{\mathrm{eff}} = L_{\mathrm{phys}} + \Delta_{\mathrm{emb}} + \Delta_{\mathrm{end}} + \Delta_{\mathrm{hole}} $$

Where: • $\Delta_{\mathrm{emb}} = embouchure correction$ • $\Delta_{\mathrm{end}} = open-end correction$ • $\Delta_{\mathrm{hole}} = tone-hole correction$

So:

$$ L_{\mathrm{phys}} = L_{\mathrm{eff}} - \Delta_{\mathrm{emb}} - \Delta_{\mathrm{end}} - \Delta_{\mathrm{hole}} $$

6. Open-end correction

For an unflanged open cylindrical end, a standard approximation is:

$$ \Delta_{\mathrm{end}} \approx 0.61r $$

where $r$ is the bore radius.

Equivalently:

$$ \Delta_{\mathrm{end}} \approx 0.305D $$

where $D$ is the bore diameter.

So the acoustic length extends slightly beyond the literal end of the tube.

7. Embouchure correction

This one is messier and more empirical. A simplified model treats the embouchure as contributing its own correction:

$$ \Delta_{\mathrm{emb}} \approx C_{\mathrm{emb}} $$

where $C_{\mathrm{emb}}$ depends on: • embouchure hole size • chimney height / wall thickness • cut angle • bore diameter

There is no universal one-line closed form that works perfectly for homemade flutes, which is why people end up tuning by enlargement and cork adjustment.

Tone Hole Geometry

8. Ideal hole position

If a given tone hole is intended to produce note $f_i$, then the ideal acoustic location of that first open hole is approximately:

$$ x_{i,\mathrm{eff}} = \frac{v}{2f_i} $$

But the actual physical hole center must account for venting inefficiency:

$$ x_{i,\mathrm{phys}} = x_{i,\mathrm{eff}} - \delta_i $$

where $\delta_i$ is the tone-hole correction term.

This means the physical hole center often sits a bit upstream of the pure ideal acoustic point, especially when holes are small or thick-walled.

9. Tone-hole correction trend

Tone-hole correction grows when the hole vents weakly.

Qualitatively:

$$ \delta_i \uparrow \quad \text{as} \quad d_i \downarrow $$

$$ \delta_i \uparrow \quad \text{as} \quad t \uparrow $$

Where: • $d_i = hole diameter$ • $t = wall thickness$

A rough proportional trend is often written conceptually like:

$$ \delta_i \propto \frac{t}{d_i^2} $$

This is not a full design law by itself, but it captures the behavior correctly: • smaller hole $\to$ weaker vent • thicker wall $\to$ weaker vent • weaker vent $\to$ note acts longer than expected • longer-than-expected note $\to$ unstable pitch and register ambiguity

That’s your “doesn’t know which octave to play” problem in a necktie.

10. Hole diameter scaling

A practical starting rule is to scale finger-hole diameter to bore diameter:

$$ d_i \approx kD $$

with

$$ 0.3 \lesssim k \lesssim 0.6 $$

depending on: • desired note • ergonomics • tuning flexibility • hole order on the flute

Upper holes often need to be effectively larger than intuition suggests because they must vent strongly enough to define the shorter air column cleanly.

11. Acoustic consequence of enlarging a hole

When you enlarge a hole:

$$ d_i \uparrow \implies \delta_i \downarrow $$

Therefore:

$$ x_{i,\mathrm{eff}} \approx x_{i,\mathrm{phys}} + \delta_i $$

shrinks acoustically, which means the pitch rises.

So:

$$ d_i \uparrow \implies f_i \uparrow $$

and usually response stability improves too.

That is why tuning generally proceeds by: • drill slightly undersized hole • test pitch • enlarge gradually • stop before overshooting

Because once you hog out the hole, the universe says “congratulations, that mistake is permanent.”

Cork / Stopper Placement

12. Cork placement rule

For a side-blown flute with a stopper upstream of the embouchure, a standard first approximation is:

$$ x_c \approx D $$

Where: • $x_c = distance from cork face to center of embouchure hole$ • $D = bore diameter$

A practical working range is:

$$ 0.8D \lesssim x_c \lesssim 1.2D $$

13. Why cork matters

The cork adjusts the head-end boundary condition. It does not set finger-hole notes directly, but it changes how the instrument speaks and how the registers line up.

In boundary-condition language, cork placement affects the headjoint correction term:

$$ L_{\mathrm{eff}} = L_{\mathrm{body}} + \Delta_{\mathrm{head}} $$

and

$$ \Delta_{\mathrm{head}} = f(x_c, \text{embouchure geometry}, D) $$

If the cork is misplaced, symptoms can include: • fuzzy upper register • octave instability • uneven tuning between registers • delayed or over-eager response

So yes, the humble cork is also part of the “tube with holes” conspiracy.

Example Calculation

Let’s say you want one note to be C_5, roughly:

$$ f = 523.25\ \mathrm{Hz} $$

Assume:

$$ v = 343\ \mathrm{m/s} $$

Then the ideal acoustic length is:

$$ L = \frac{343}{2 \cdot 523.25} $$

$$ L \approx 0.3277\ \mathrm{m} $$

$$ L \approx 327.7\ \mathrm{mm} $$

If your bore diameter is:

$$ D = 19\ \mathrm{mm} $$

then open-end correction at the foot is approximately:

$$ \Delta_{\mathrm{end}} \approx 0.305D $$

$$ \Delta_{\mathrm{end}} \approx 0.305 \cdot 19 $$

$$ \Delta_{\mathrm{end}} \approx 5.8\ \mathrm{mm} $$

If the tone hole contributes an estimated correction of, say:

$$ \delta_i \approx 4\ \mathrm{mm} $$

and embouchure/head correction contributes another:

$$ \Delta_{\mathrm{emb}} \approx 6\ \mathrm{mm} $$

then a rough physical location becomes:

$$ x_{i,\mathrm{phys}} \approx 327.7 - 5.8 - 4 - 6 $$

$$ x_{i,\mathrm{phys}} \approx 311.9\ \mathrm{mm} $$

That gives you a starting point for the hole center, after which reality shows up and asks for sanding.

Register / Harmonic Behavior

14. Octave jumping

Because the flute supports multiple resonances:

$$ f_n = \frac{nv}{2L_{\mathrm{eff}}} $$

the second resonance is:

$$ f_2 = 2f_1 $$

If a fingering doesn’t vent strongly enough, the instrument may ambiguously support both:

$$ f_1 \quad \text{and} \quad f_2 $$

That's why a poorly vented hole can feel like:

"Fo̷o̴͐̎͝͝l̵̉͂̃͠ì̴̼̊s̴̮͖͑̽ḧ̵̛͚͎̯̪̀͒͒ ̷̨͔̫̹̫̺̰̖̩̔m̶̳̪͉̪͔̭̖͑́͐̊̀͘o̴̡̹͗́͋́̊͐́̋ř̴̟͕̈́͊̿͆̉͗̅͘͘ẗ̵͇̺̦́́͌̽̋a̶̡̢̭̪̪̣̪͒̀̓̾́̈̈́͜ͅl̶̢͇̞͊͋͑͗͊̌̎͆͂̕!̴͍̺̳͈̼̈́̍̈́̋̀͘͝"
(╯°□°)╯︵ ┻━┻

Thanks,
- Mgmt.

Mathematically, the hole has failed to create a strong enough impedance discontinuity to suppress the longer effective mode. As you can see, if the hole cannot vent, the steam ends up coming out somewhere.

Design Workflow Formula Set

15. Full quick-reference set

Step 1: choose target notes

$$ f_i = 440 \cdot 2^{N_i/12} $$

Step 2: compute ideal acoustic lengths

$$ L_i = \frac{v}{2f_i} $$

Step 3: estimate physical hole centers

$$ x_{i,\mathrm{phys}} = L_i - \Delta_{\mathrm{emb}} - \delta_i $$

Step 4: choose initial hole diameters

$$ d_i \approx k_i D $$

with $k_i$ chosen empirically

Step 5: account for end correction

$$ \Delta_{\mathrm{end}} \approx 0.305D $$

Step 6: place cork

$$ x_c \approx D $$

Step 7: tune by enlargement

$$ d_i \uparrow \implies f_i \uparrow $$

$$ d_i \uparrow \implies \delta_i \downarrow $$

Reference

[A] — Aside from the referrential concert pitch — A4 at 440 Hz — the other eleven pitches in 12-tone equal temperament generally have irrational frequencies. https://en.wikipedia.org/wiki/Equal_temperament#:~:text=irrational