How many tone holes are there actually in different orchestral woodwind instruments?

Question

I'm trying to make my own bamboo flute or perhaps "saxophone" of sorts, and before wasting any material I want to have the right construction plans.

I've seen some examples of simple types of bamboo flutes, but they usually seem to be built for one key only, and if you want to play at other keys (or basically playing chromatic notes) you have to either half-hole or use cross-fingering (or both even).

Now, as I've been trying to study the physics behind the sound production of the woodwinds, what I understood is that playing chromatic notes using cross-fingering (=tone holes being closed below the first open tone hole, rather than keeping a continuous series of open tone holes) isn't perfect in terms of intonation, so the chromatic notes are off from what's expected (to my understanding, flat) and thus you can't modulate accurately to all keys differing from the instrument's base key.

To my understanding, the issue for a long time of these instruments' histories is the physiological limitation of the number of fingers humans have, such that we can't place a finger to control each dedicated tone hole in a truly chromatic tone hole-configuration. So I then understood that the mechanical key systems developed for these instruments in their later stages involved enabling each finger the control more than one tone hole in some convoluted interconnected way, and thus I figured there now should be a dedicated tone hole for each chromatic note.

But when I read about the concert flute, I can't figure how many tone holes are actually being controlled. Different places cite different numbers, but on Wikipedia the concert flute is said to usually have 16 tone holes, while the range is 3 octaves. I know it overblows to an octave, so 16+12 (overblow in semitones) is only 28, not 36. Does that mean in the modern concert flute still not all chromatic notes are played without actual (I disregard how the fingers look when pressing the key mechanism) cross-fingering? Does that mean some notes within a flute of a specific keys are known to have somewhat impaired intonation or tone?

And how does the saxophone have it, on the other hand?

The layout I imagined would "only" involve 12 tone holes each a semitone apart (so opening all 12 plays the octave of the bass note), so with the overblow you'll have a range of 2 fully-chromatic octaves with no cross-fingering patterns. I'm still trying to figure out the possible mechanism for operating that.

Thanks in advance.

Answer 1

Every wind instrument has a theoretically infinite range, which is only limited by the player's skill. The first octave is the fundamental, and then all higher notes are played as overtones of that first octave. The second octave is by far the most stable, it's just the entire first octave again but on the second harmonic. The third octave could theoretically be played as the fourth harmonic, but this is harder than it needs to be. On flute, the first half of the third octave is the third harmonic, and the second half is a variety of harmonics. Beyond three octaves, the harmonics become too thin, the realities of physics (edge effects, etc.) start to create chaos, and the fingerings required to tame all of that get too impractical. Look up a fingering chart for altissimo notes and try to imagine playing a scale.

So the minimal number of tone holes is 12. Modern instruments have more, for a variety of reasons:

• They are extended downwards. The history woodwinds led us to a fingering system where the first octave is D-C#, but it's sorta awkward for an instrument's lowest note to be D because of how often we would like to play in C. So flute is extended to C or B, saxophone to Bb or A, etc.
• They are extended upwards. There's a somewhat awkward break between the octaves where to go up slightly in pitch requires adding all of the fingers and jumping to the second harmonic. A trill across this break would be extremely difficult. So modern instruments have a few extra keys to extend the range above the base octave so you can do trills or otherwise avoid crossing the break in certain passages. On saxophone, these are the palm keys, and they extend the top of the second octave up to F, and sometimes further.
• There are duplicate tone holes, used to offer alternate fingerings. Flute doesn't have any of these, but saxophone has an alternate F#, as well as "side" Bb and C, which are required for certain trills and make other passages more fluid.

Answer 2

To add to MattPutnam’s general answer, here are some specifics for the standard (professional) concert flute:

A professional concert flute almost always has a B foot, and so has 17 (rather than 16) tone holes. Many beginner flutes are still made with C foots and 16 tone holes, however.

After some experimentation, the cross-fingering is required because of the way the mechanism is designed. Not all keys are directly actuated, and some fingers close holes that are above the key pressed. The lower closed hole doesn’t affect the air column (at least in the lowest octave), so I don’t know whether you would consider this “actual” cross-fingering or not. The F# and Bb in particular have more closed holes than are actually required. Here is the first octave of my favorite fingering guide - also see the fourth octave chart, which is the (insanely difficult, frankly) altissimo referred to in MattPutnam’s answer.

On most if not all flutes, the “default” intonation is well-removed from true equal temperament. Some notes are known to be more out of tune than others, averaged over all flutes, but almost every note is going to be a few cents out from perfectly in tune, ranging from an average of 5-10 to the worst being 20-30 out (on a decent instrument). Every player learns their own instrument’s tendencies and how to compensate to play in tune (though note that “in tune” isn’t always in equal temperament - unlike pianos, wind instruments can adjust for individual keys and chords so that they’re playing in just intonation, with perfectly consonant intervals).