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I visited a "small loop antenna" calculator web site today and fed in my horizontal loop gepmetry, curious to see what it would say. 2" dia wire, 72' circumference, 3.935MHz. Interesting results, though it mentioned avoiding lambda/4 lengths due to self resonance and recommended wire circumference be under about 60'.
Why is self resonance bad?
FWIW, the antenna seems okay-ish at 3.935 MHz. There is a net on that frequency and on a good day I can sometimes check in.
VK1OD compared several small transmitting loop antenna calculators. The author concludes that there are serious errors in the ARRL analysis which have made their way into some (most?) of these calculators. He buttresses these claims with a detailed NEC model to estimate the structure, capacitor and ground losses which are so crucial to proper calculation of Q, efficiency, etc. He goes on to compare his modeling results with measurements made on a commercial antenna, lending credence to the models. These articles are all worth reading.
While it is certainly true that the main lobe moves from "end fire" to "broadside" as one crosses the quarter-wave circumference boundary, evidence of flaws in the theoretical basis for the "calculators" may indicate that it would be worthwhile to re-evaluate the "quarter wave circumference" advice, particularly with respect to efficiency and Q.
"self-resonance" sounds like not quite the way to describe the issue.
The problem is more simply that if the circumference is not very small relative to the wavelength, the antenna is no longer a small loop. As a rule of thumb, electrical engineers consider anything smaller than $\lambda/10$ to be "small". When things are "small" the analysis can be simplified with an approximation: current is equal everywhere in the loop.
There is no hard boundary where this approximation suddenly becomes invalid: in only becomes progressively worse. By $\lambda / 4$ it's certainly no longer a very good approximation.
As the loop becomes less "small", it becomes more like a folded dipole, and thus will have a substantially different feedpoint impedance and radiation pattern than a "small loop". It's possible to match such an antenna, but calculation of the necessary capacitance is different since the approximation of constant current is no longer valid.
As the circumference approaches 1/4 wavelength, it becomes increasingly difficult to tune it, as it is no longer a small loop and starts taking on properties of a a resonant loop, which is tuned differently and has a different radiation pattern. The primary principle of a small loop is that it is small enough that the RF in the loop is nearly all the same phase. As you approach 1/4wl, it will be 90 degrees out of phase by the time it goes around the loop.
A resonant monopole is 1/4wl long. A typical (resonant) open loop (like a halo) is 1/2wl, and a resonant closed loop is 1wl or more.
