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Unscreened (ungated) plasmons in large-area grating-gate AlGaN/GaN heterostructures were studied experimentally by Fourier-transform spectroscopy. Special attention was paid to the recently discovered THz plasmonic crystal modes observed at totally depleted gated regions when plasma oscillations were localized only in ungated parts of the grating-gate structures. The frequency of these modes is still gate voltage-dependent in the limited range due to the depletion of the ungated parts located close to the gate edges. Double gate structures with an additional bottom gate were fabricated and studied to improve the gate voltage tunability of the unscreened plasmons. Since this gate is located deep below the channel, the plasmons behaved as ungated ones, but their frequency still could be tuned by this bottom gate. We show that the combined effect of the top and bottom gates allows the efficient tuning of terahertz frequencies of unscreened modes in the grating-gate AlGaN/GaN plasmonic crystals. [ABSTRACT FROM AUTHOR]