Spectrum analyzer?

The spectrum analyser in the Behringer DEQ2496 will do the job for about $350 (including the microphone). 1/3 octave.
What's more, it can fix your problems once you have figured out what they are.

1/10 octave or better can be achieved by both TrueRTA and by REW.

The reason that RTA is not useful is that it has little memory while the major room influence is in the delayed reflections and their interactions. For that, you need to see "waterfall" plots or similar that show how the sounds at different frequencies decay in the room. Generally, one wants uniform decay of all frequencies.

For more info, I recommend Toole's new book "Sound Reproduction - Loudspeakers and Rooms." I am about half way through it now.

Kal

If your equalization requirement at some frequency is a "cut" rather than a "boost" the "feedback destroyer" function of the DEQ2496 can do 1/10 octave resolution.

Actually the DEQ2496 can simultaneously implement several sharp (1/10 octave) notches, several broad (multi-octave parametric equalizer) curves, and the 1/3 octave (31 band) graphic equalizer. It does not chew gum.