I'm not sure this is what Josh meant, but the 'space and depth' I love is more about the song arrangements, and how the sounds themselves have been recorded. On a number of tracks on the album there's something distance miked, or has some of what mostly sounds like "real" ambience/effect captured in the recording. It gives a great character, and those tracks save everything from being flat and in your face. From the drums and bells opening of No-one loves me, the main riff guitar sounds in Elephants and Bandoliers... at least, thats the depth I enjoy on the album (as well as some great playing!
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I saw the Vultures out here in Melbourne a while back, and it was an AWESOME show. As per the other comments, Dave was definitely the highlight for me and his live sounds were pretty much what you expect to hear after the album (not sure, but a rack of Distressors looked to be his). The rest of the band were also great, and being a QOTSA fan also it was sort of nice seeing Josh up front but not being 'the main guy'.
Best rock album and show I've heard for a long time.