I don't know, but I'm no expert. I can calculate the frequency comparisons, however. Using the even-tempered frequency of 32.68828 Hz for the low C of the 16' register, the 11th harmonic is a frequency of 359.51173 Hz, which is almost exactly halfway between the F and F# that are 3 1/2 octaves higher. I believe that this would create a dissonance and might even be unpleasant. On the other hand, it could just add to the total harmonic structure of the tone and make it a little stringier.
I made a wild guess and found this entry in the on-line Encyclopedia of Organ Stops:
A very rare mutation stop that speaks approximately an F when played from the C key, one octave and a fourth above. It is found at 2 10/11' pitch in the manuals, where it is part of the 32' harmonic series. In the pedal it is found at 5 9/11' pitch, and is part of the 64' harmonic series.
This stop is listed only by Irwin, who incorrectly claims that it is “not a natural harmonic of any series”, and goes on to say: “It gives a dissonant effect and a close-toned brilliance to any loud Reeds or a flue combination, and also makes some clang-tone which may be brass-like in effect.”
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Note that the Quarte in question plays an octave above the 2 10/11' listed above, so one would expect it to affect the 16' series much the same as the ones listed affect their parent series.
David