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I think it is a special crystal that provides a relative high harmonic at the desired frequency. When I ordered crystals, I specified it was fundamental or 3rd tone.
I think every crystal can oscillate on the third harmonic. But loop gain on this frequency should be higer than unity. And you should suppress fundamental harmonic by filter so that the loop gain on the fundamental frequency should be less than unity.
There is a difference between the crystal being used on an overtone and the oscillator circuit output being tuned to a harmonic.
The crystal manufacturer controls the two angles of the cut to optimize the parameters at the fundamental and the overtones. They all cannot be optimized at once. Therefore it is best to use the crystal in the mode that it is optimized for.
In overtone operation the signal can be a pure sine wave. In harmonic it has to be nonlinear to have harmonics.
One further thing is that the overtones are not the same frequencies as the harmonics because of what can be thought of as the end effects. The mechanical acoustic wave going through the crystal is controlled by the velocity of propagation. The part of the crystal material that is next to the outside has a slightly different velocity.
Also, the numbering system is different. Overtone numbers are one count lower than the harmonic number. 2x in frequency is the second harmonic and the first overtone. We owe the overtone terminology to the music world.
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