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A Faster S.S. Head?

I never did get to analyze saxes.

True that doubling or tripling square waves with slight detune and drift would make a pseudo pulse wave. Yeah even my VSTi's have drift options.

Saxes, Clarinets and about every woodwind except flutes, and every brass are closed pipes (at one end) so you start with a square wave.

Recorders are often used for flute sounds but they're a closed pipe also.

It's kind of all the extra artifacts that define the "sound"

Pipe organs for example have all kinds of different pipes but they're all closed pipes. They unison pipes for chorusing and mix pipes for harmonics, but they all start with a closed pipe.
 
Brass instruments like trumpet, trombone, tuba, flügelhorn, French horn, et al, don't use a vibrating reed but instead the vibration of the player's lips in the mouthpiece, the embouchure. The result is more akin to a pulse or sawtooth waveform than a square.