Hmm, I just realized that the AVR family OSCCAL oscillator calibration register can adjust their internal oscillator frequency, normally 8MHz or 16MHz, by fine degrees. The 12th harmonic of an 8MHz square wave would be 96MHz, which is within the FM radio band (87.5 MHz–108.0 MHz). The maximum permitted deviation from a nominal carrier frequency is ±75 kHz, which would be ±780 ppm at 96 MHz, ±860 ppm at 87.5 MHz, and ±690 ppm at 108 MHz. So, if "fine degrees" is about 1900 ppm or less (0.2%), then I ought to be able to transmit at least impulses and square waves over FM radio in this way. In fact, since FM radio stations have about 100 kHz of bandwidth after demodulation, that would be sufficient to generate PWM audio by spending different percentages of the time at each different wave.
In the ATTiny2313 OSCCAL is 7 bits, selecting one of 128 frequencies in order to calibrate down from a nominal ±10%, so the situation doesn't look that great. Maybe a better plan is the thing I'd wanted to try previously: put a resistor on the AVR's VCC pin and modulate its current consumption in order to affect its operating frequency an therefore the frequency of the waves generated.
Also there's an ominous note in the datasheet saying that the processor needs to remain in RESET while OSCCAL is being written.
On p. 200 the datasheet says that with user calibration the oscillator can range from 7.3–9.1 MHz, with ±2% error at the given voltage and temperature. On p. 230 we have a chart suggesting that OSCCAL is capable of moving the frequency anywhere from 3 MHz or so up to 12 MHz! It looks like an approximate exponential, too, suggesting that each step of OSCCAL is about 1.1% of frequency variation. So that idea probably won't work, unless there are two adjacent OSCCAL values that are anomalously close.