# What variables determine the pitches produced by a flute and its intonation?

I recently read about and heard recordings of the bansuri, a bamboo flute used in Indian (Hindu) music. I decided to try and make one myself, and according to the tutorial I read it shouldn't be too difficult, but there are some points I need to get clear about:

The tutorial said to find a bamboo stalk with an internal diameter of between 3/4" to 7/8", according to the scale you want the flute to be tuned to. So I figure both the length AND the internal diameter of the pipe determine the base pitch it produces, yes? If that's the case, because you can't have very specific sizes when picking a random bamboo, am I required to make specific calculations against the measured internal diameter in order to find the exact length the bamboo should be cut to if I want to reach a specific concert pitch (A, for example)?

And then there's the question about intonation (accuracy of all pitches along a scale): if the internal diameter of the bamboo stalk is not consistent along its entire length, will the pitches be off-scale if I were to make the finger holes positioned without correction as dictated by simple perfect-scenario (symmetrical bamboo) calculations? If it is so, should I rather find a bamboo stalk with a smaller internal diameter and sand it on the inside to a specific, symmetrical diameter?

And the final question — is there a way to make such a simple instrument chromatic rather than just major-scale? It seems like flutes don't use an excessive amount of finger holes for this (if that could even reach the desired result) but rather use keys that allow for secondary in-between notes, so I figure.

The vibration of an object (either the air inside the flute, a string, or a membrane) is formed by the superposition of various "normal modes" of vibration, each with a different frequency. The lowest frequency is typically obtained when the whole object moves in the same direction; e.g. a string makes an arc, not an S-shaped curve. This frequency is the "fundamental". In 1-dimensional objects, such as a string, the higher frequencies are typically multiples of the fundamental: they are called the "armonics". Such sounds are the "notes". 2- or 3-dimensional objects generate more complex superpositions of frequencies, that we hear as "noise": an example is the drum. This does not mean that a "noise" is not nice, nor that it does not have a specific pitch and cannot be evaluated as a note.

A flute generates a "note", so it should be 1-dimensional. This is almost true, since its vibration is mostly along the length. If the diameter is small compared to the length, what determines the sound is basically the length; the diameter and the possible variation of diameter only determine the harmonics, and hence the particular "voice" of the instrument. However, if the diameter is large, you start hearing a "noise", corresponding to transversal vibrations. It is, e.g., the particular sound similar to the wind that you hear in the Pan flute. If the diameter is very large, you get a real "noise", such as in the didjeridoo: although it is a noise, it can be nice!

So, summarizing: the diameter does not define the frequency; what matters is only the length. Also if there are variation in internal diameter, the "keys" must not be displaced. However, the particular "voice" that you will get depends on such details.

Finally, this is what physics teaches us. The experts in music can teach you much more.

• I play guitar (and a bit of other instruments) and read about *harmonics/overtones (which are basically involved in the timbre of a note), but I don't really understand your definitions involving "1/2/3 dimensional" objects. In what way are strings or a flute 1 dimensional? And if the width of the pipe doesn't matter I suppose it makes the manufacture of a playable flute much easier than I suspected, but I would still like to know what should one do to make a flute chromatic (that is, play all 12 semitones in the Western octave)?
– TLSO
Jun 30, 2019 at 21:45
• @TLSO It is explained in the answer. If the flute's diameter is small compared to it's diameter, then we really only need to focus on vibrations along the length (1D). The exact value diameter shouldn't matter in this regime. Jun 30, 2019 at 22:48
• @Aaron Stevens I suppose you wanted to say "compared to its *length", but either way even if you define the parameters involved in the fundamental pitch production as "dimensions", and in a flute you say it's one dimensional because only (to an extent) the pipe's length is involved, the string of a stringed instrument on the other hand is also affected by its diameter and tension. I'm still trying to understand how can I also turn a simple flute into a chromatic-capable flute; what additional modifications to the embouchure + finger holes construction could be done to add all possible tones?
– TLSO
Jun 30, 2019 at 23:35
• @TLSO yes sorry for that typo. For the guitar the vibrations only propagate in one dimension (along the length of the string). We don't consider vibrations along the string's thickness. Tension isn't a dimension. Jun 30, 2019 at 23:49
• @TLSO in other words. For the guitar string you can express the wave amplitude as a function of just one spatial dimension and time. Jun 30, 2019 at 23:56