I think the drop in tone has more to do with the temperature of the air column within the sax. The speed of sound varies from substance to substance: sound travels most slowly in gases; it travels faster in liquids; and faster still in solids. The higher the density of the medium, the faster the sound travels. Colder air is more dense so the sound travels faster, extending the reed’s wavelength of oscillation, thus lowering its frequency. Just like the siren on a moving vehicle, the faster the vehicle travels away, the lower the siren’s perceived frequency becomes.
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