“Registers” are a tricky concept in woodwind playing. Here’s how they work.
For simplicity’s sake, let’s say I am playing a flute with a C footjoint. If I finger a low C, that closes all the instrument’s toneholes and produces a C4:
As I work my way up the chromatic scale, I gradually open more and more of the flute’s holes. When I reach C-sharp 5, I run out of holes to open. (Using standard fingerings, that is.) To continue upward, the fingerings sort of restart—I close a bunch of toneholes again, fingering D5 in almost the same way I played D4. By the time I get to E5 I am using fingerings identical to the ones from an octave lower.
I can play higher notes using the same fingerings I used for lower ones because I have moved into a higher register, which in this case is an octave above the lower register. On flute, I do this by changing something about my tone production; on reed instruments I get extra help from octave/register/”whisper” keys.
If I continue my scale up to C-sharp 6, I run out of toneholes again, and move up to the third register, which is a fifth higher than the second. To play D6, I use a fingering that looks similar to G5, but sounds a fifth higher.
So, for typical flute playing situations, we can consider the second register to begin at D5, and the third to begin at D6.
But this doesn’t paint a complete picture in terms of the instrument’s acoustical properties. When the fingerings “start over” at D5, that’s not really starting over—I have left out the low C and C-sharp fingerings. And it turns out I can in fact play C5 and C-sharp 5 using those fingerings. The reason that flutists typically don’t is that the “standard” fingerings (with most of the toneholes opened) happen to work better for most situations, with regard to pitch, tone, and/or response. Likewise, when I reach C-sharp 5 and C-sharp 6, I haven’t completely run out of toneholes to open. If I open everything I have left (both trill keys, plus maybe the G-sharp key) I can get up to about D-sharp in either octave. But I usually don’t do that unless I have a special reason, because the standard fingerings are more usable.
And starting the third register on D6 with an adapted “G” fingering raises this question again, but with an even larger gap. What about third-register notes using the fingerings from “low C” up to “F-sharp?” The answer is that those fingerings work, too (producing G up to C-sharp, an octave plus a fifth above the corresponding low-register fingerings). But, again, they aren’t as useful because of their pitch, tone, and response characteristics.
So from an acoustical standpoint the first and second registers overlap in the C5-D-sharp 5 range, and the second and third overlap in the G5-D-sharp 6 range.
When there’s overlap, there are fingering options available. The “standard” fingerings are the ones that have been chosen over the centuries by flutists as the ones best suited to most situations, but the others (sometimes called “overtone” fingerings or “harmonic” fingerings) can be used to good musical effect at times.
The flute and most of the reed instruments follow the same pattern of registers: the second register is an octave above the first, and the third is a fifth above that. Additional registers above those are also used sometimes, spaced with increasingly small intervals. This series of intervals is a naturally-occurring phenomenon known as the harmonic series.
The clarinet is an exception; due to its acoustical characteristics it uses only every other harmonic. This is why the clarinet doesn’t have an “octave” key, it has a “register” key that skips the octave register and goes straight to an octave plus a fifth.
Understanding registers is helpful in navigating between them and in finding alternate fingerings for special situations. Happy practicing!