Publications of Dr. Martin Rothenberg:

Cosi Fan Tutte and What it Means
Nonlinear Source-Tract Acoustic interaction in the Soprano Voice and Some Implications for the Definition of Vocal Efficiency


The decreased average flow brought about by proper vocal tract tuning should come as no surprise to an electronic engineer familiar with radio transmitter amplifiers or to an experienced ham radio operator. It would only be necessary to point out that a soprano singing in the upper part of her range is analogous to the final amplifying stage of a radio transmitter. This amplifier must supply a maximum electrical power to the antenna while drawing a minimum average electrical current from the power supply. In the so-called class C amplifier, commonly used for this purpose, the power supply current is allowed to flow to a tuned electrical circuit, and thence to the antenna, for only a short interval during each cycle of the transmitter signal (Terman, 1947). The transmitter engineer can check for the proper tuning (proper resonance frequency) of this antenna circuit by adjusting the tuning for a minimum average power supply current, just as the trained soprano can adjust her vocal tract tuning for a minimum average expenditure of lung air.

From a mathematical-not an aesthetic-point of view, the primary difference in operation between the transmitter amplifier and the soprano is that the amplifier, for proper operation, uses a duty cycle (an open quotient) much smaller than 50 percent. Looking to the amplifier analogy for lessons, a soprano might well note that an output amplifier that operates over a period of time during which it is improperly tuned can overheat and blow its fuse.


This work was supported by a research grant from the National Institutes of Health.

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