Convective Quasi-equilibrium (QE) and weak temperature gradient (WTG) balances are frequently employed to study the tropical atmosphere. This study uses linearized equatorial beta-plane solutions to examine the relevant regimes for these balances. Wave solutions are characterized by moisture-temperature ratio (q-T ratio), and dominant thermodynamic balances. An empirically-constrained precipitation closure assigns different sensitivities of convection to temperature (εt) and moisture (εq). Longwave equatorial Kelvin and Rossby waves tend toward the QE balance with q-T ratios of εt : εq, which can be ~ 1-3. Departures from strict QE, essential to both precipitation and wave dynamics, grow with wavenumber. The propagating QE modes have reduced phase speeds due to the effective gross moist stability (meff) with a further reduction when εt > 0. Moisture modes obeying the WTG balance, and with large q-T ratios (> 10) emerge in the shortwave regime; these modes exist with both Kelvin and Rossby wave meridional structures. In the v = 0 case, long propagating gravity waves are absent and only emerge beyond a cutoff wavenumber. Two bifurcations in the wave solutions are identified and used to locate the spatial scales for QE-WTG transition and gravity wave emergence. These scales are governed by the competition between the convection and gravity wave adjustment times, and are modulated by meff. Near-zero values of meff shift the QE-WTG transition wavenumber toward zero. Continuous transitions replace the bifurcations when meff <0 or moisture advection/WISHE mechanisms are included, but the wavenumber-dependent QE and WTG balances remain qualitatively unaltered. Rapidly decaying convective/gravity-wave modes adjust to the slowly evolving QE/WTG state in the longwave/shortwave regimes, respectively.