yoink wrote on Thu, 07 June 2007 06:41 
@Wes (or anyone who can answer): the one thing that stands out in my current(lacking) grasp of the theory involved, is your comment about modal densities. I'm not sure if I follow that the minor 7th is better than the minor 6th (8ft versus 8.75 feet) and what the ideal height of the ceiling should be. And perhaps I even got that backwards.

Yoink,
Modal density is an important concept, especially in the bottom two octaves where room resonances are hardest to absorb effectively. The idea is that if you must have resonances (every room has them), let's at least have them evenly spread across the frequency spectrum, rather than have them all bunched up together. So if the room's longest dimension is 19', the corresponding resonance is around 30Hz, and whole number multiples of 30Hz. Two octaves above 30Hz is 120 Hz (two doublings), so if you play around with various room ratios, you will find that the toughest problem is how to get a reasonably even modal progression from 30 to 120, where it really counts. This is roughly from the lowest note on a fivestring bass to the 3rd fret on the bass G string.
Ratio sets that have all three dimension falling within the same octave are naturally denser, since you get nine resonances to play with in the first two octaves as opposed to six. An example would be 1 : 1.26 : 1.59. Using 30Hz as the lowest resonance, your spread would be 30, 37.8, 47.7, 60, 75.6, 90, 95.4, 113.4, 120. This is as even and dense as is mathematically possible for three dimensions, in the lowest octaves.
On the other hand, where the long dimension is outside one octave from the shortest dimension, like say 1 : 1.59 : 2.53, you'd get 30, 47.7, 60, 75.9, 95.4, 120. See how the results are less dense than in the first example?
In musical terms*, the denser set is, starting with the lowest Bflat on the piano: Bb, D, F#, Bb, D, F, F#, B, Bb. The less dense example goes Bb, F#, Bb, D, F#, Bb.
Modal density is a good thing, but with small rooms, you have to consider the tradeoffs involved, or we'd just make all small room ratios fall within the first octave. The trade off is that if the smallest dimension is around 8' or so, the longest good long dimension would be around 15.2' (1 : 1.9), and the room would start to roll off steeply at 37Hz, and you really want your room to easily go down to 30Hz in this day and age. So a ratio set like the aformentioned 1 : 1.59 : 2.52 gives you that 19' dimension and extends the natural bass response of the room. But there is no free lunch, as we have traded modal density (bass smoothness) for longer dimensions (bass extension). The only free lunch is for those who sit in the first class cabin, where the wine is flowing and the headroom is somewhere around 11'. But the ticket price is steep...
Wes
*There has been some rounding for the sake of clarity. Think of it as "tuning".