FFT (Fast Fourier transform) analysis using SignalScope Pro Software

In this snapshot (47 seconds into the piece), Yoshinobu Taniguchi is playing in second octave (it’s amplitude being dominant) and all the harmonic series up to the seventh are fairly prominent. After that, all (except the twelfth) harmonics up to the fourteenth are in evidence. It looks like the cutoff frequency of this flute is around 1300hz, anyway, that's about where the harmonic series is depressed.

Every shakuhachi bore has the potential to produce the harmonic series—and that's any length, i.e. any note played. Yet, the series only appears as prominently as in the graphic above, somewhat fleetingly. It's definitely under the players control as to when the series is amplified.

With the cutoff frequency being what it is, the potential for harmonics is greater with low pitched flutes and notes. Playing a high note on a D flute (~600hz) puts the next step in the harmonic series at ~1200hz, and the next at ~1800hz--usually beyond cutoff, so we'd get the first two in the series (maybe three) but the rest would be dampened.

In general, the cutoff frequency is controlled by hole size. The closer hole size is to bore diameter the higher the cutoff frequency. That's why silver Boehm flutes sound different from simple bamboo/wooden flutes. It isn't the flute material, it's that Boehm flutes have huge holes compared to shakuhachi and correspondingly have a higher cutoff frequency which means more harmonics in the sound.

What's required of a flute to produce the harmonic series? The biggest (and perhaps only) requirement is a geometry which can sustain some backpressure. When a player increases pressure, the series is amplified and appears. Is anything required of the bore other than the ability to sustain backpressure? I don't know, but think it'd be of far less importance than the backpressure requirement. And if we're talking backpressure we're talking aspect ratio.

The harmonic series appears any time a player pushes the flute. What we perceive as the flute ‘ringing’ is the sound of higher harmonics. I'm sure there's a Japanese word for this, but to me it sounds like I imagine wind-shear sounds. I hear it as having a tearing quality—something being torn or ripped. Since the harmonics only appear at higher pressure there's usually some wind-rush associated with them. Played at low pressure, flutes tend to suppress everything except a single frequency and have a sweet, pure, sin-wave sound.

Want more harmonics? Engineer in more backpressure and make the holes bigger.