As is shown in chapter 4, see “Effects of Harmonics on f vs. T,” the f vs. T of the fundamental mode of a resonator is different from that of the third and higher overtones. This fact is exploited for “self-temperature sensing” in the microcomputer compensated crystal oscillator (MCXO).  The fundamental (f1) and third overtone (f3) frequencies are excited simultaneously (“dual mode” excitation) and a beat frequency f is generated such that f = 3f1 - f3  (or f =  f1 - f3/3). The f is a monotonic and nearly linear function of temperature, as is shown above for a 10 MHz 3rd overtone (3.3. MHz fundamental mode) SC-cut resonator.  This resonator was 14 mm in diameter, plano-convex, and had a 3 diopter contour.
	The f is a measure of the resonator’s temperature exactly where the resonator is vibrating, thereby eliminating the need for a thermometer other than the resonator.  Because the SC-cut is thermal transient compensated, the thermal transient effects are also eliminated, as are the effects of temperature gradients between the thermometer and the resonator.
	For temperature compensation purposes, the f vs. T need not be used; the calibration can consist of f vs. f only.  The role of a thermometer during calibration is then only to insure that the specified temperature range is covered.
	See also “Mode Spectrograph of an SC-cut” and the page that follows it in Chapter 3.
	S. Schodowski, "Resonator Self-Temperature-Sensing Using a Dual-Harmonic-Mode Crystal Oscillator," Proc. 43rd Annual Symposium on Frequency Control, pp. 2-7, 1989, IEEE Catalog No. 89CH2690-6.
	R. Filler and J. Vig, "Resonators for the Microcomputer-Compensated Crystal Oscillator," Proc. 43rd Annual Symposium on Frequency Control, pp. 8-15, 1989, IEEE Catalog No. 89CH2690-6.