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Nuclear magnetic resonance (NMR) probes constructed using High Temperature Superconducting (HTS) materials have been proven to achieve higher sensitivities than conventional probes which use cold normal metal. The high sensitivity is due to the high Q-values possible with HTS materials. Planar multi-turn spiral resonators are extremely useful as low frequency transmit-receive coils (60 - 150 MHz) in such probes. However, these spirals have higher order modes which can interfere with the performance of other coils in the probe. These modes might be expected to occur at integer multiples of the fundamental resonance. Our simulations and experiments show that the resonance frequencies of planar spirals are generally linear with respect to the mode number; however the spacing between the modes is generally not equal to the fundamental frequency. Knowing just the mode number and resonance frequencies of two modes, we can predict the entire spectrum. The “sheath-helix” model approximates propagation on a solenoid by assuming that the wave travels down the axis of the solenoid instead of along the wires. We find that similarly treating the spiral as a slow-wave structure carrying a wave traveling radially outward provides an alternative for predicting the current distribution null and peak locations. While the resonant frequencies of the modes are attainable through simulation, analysis of this nature can lead to insights which accelerate the design process. A better understanding of planar spiral resonators will have a wide reaching impact because of their utility in varied applications.