The G# major part occurs in part of the song where it ventures from C# to G#, but without a key signature change. The F double sharp is just notated as such. From a practical, as well as music theory standpoint, it’s easier to write it out that way. Same thing happens briefly in Chopin’s 1st Piano Concerto, second movement, where the key goes from E to G# major.
So it would be unusual to write the key signature with a double flat, and some would say that it’s just wrong. I wouldn’t be surprised if it’s occasionally been done, but in any case it’s a matter of convention.
Notational convention is really beside the point. G# major is apparently considered to be a genuine key. I presume that in that context people would refer to F## rather than G. It would certainly be reasonable to do so. I thought it was relevant to the OP. It would connect G/F## to other cases where enharmonically equivalent notes have different names, which needn’t involve notes that are “out of key”. But maybe it’s not helpful. Either way, let’s not get into a fight over it.
Bartok’s Mikrokosmos is a good example of this. I no longer have it handy (and it looks like the Wikipedia article you mentioned has this), but there’s pieces with different key signatures in each hand, mixed sharps and flats in the same key signature (I seem to recall one with F# and B flat in the same signature, but it could have been something else) as well as unusual orderings of key signatures (like two sharps, but the F# and the G#, not the usual F# and C#, sharpened. Once again, not positive those were the two notes, but same idea.)
You can find some examples of written out enharmonic key signatures here. I’ve never actually seen these in use, but I wouldn’t be surprised if some composers have used them. I didn’t think time signatures over anything but a multiple of two were possible, but I’ve seen examples of such irrational time signatures in the work of Brian Ferneyhough and Thomas Ades, who use meters like 2/6, 3/10, 9/14, and 5/24.
Weird, but not irrational. Now pi time, that would be irrational. Hard to play, but easy to conduct (you just go in a circle).
There was a thread on this subject a couple of years ago. I read it and didn’t get it. Later when i tried to search I came up empty handed. I’d love it if someone could explain. IIRC the lower numbr was not a fractional subdivision of a semibrieve in the same sense as in 2/4 time 3/4 time/ 4/4 time etc. But beyond that the details escaped me.
Let’s agree that you can put a double-sharp in a key signature. That doesn’t make 8 sharps. That makes 6 sharps + 1 double-sharp = 7 symbols in the key sig. The double-sharp (or double-flat) is a single entity/symbol, not two. So in our G# Major key, there are seven tones, six of them sharp and one of them double-sharp.
Doesn’t “avant garde” generally indicate “unconventional” and/or “rulebreaking”?
IMHO, that’s just silly. How is a 14th note written?
Surely you’ve seen a septuplet marking before?
Page 2 of the second thread I linked to has an explanation by GorillaMan, as well as a link to a PDF file with an explanation of Ferneyhough’s notation and logic. Here is the link to the PDF, for your convenience.
edit: And Here’s a passage in 2/10. Damned if I could sight read that. Also note the additional accidentals. I believe, but I’m not sure, that those are quarter tone sharps and flats.
Tchah! Now he’s just trying to piss us off! 
100 years ago, it was. Now it’s just music history.
haha! LOVE IT! teehee!
After seeing how it’s used and reading the composer’s explanation, it makes a bit more sense, and actually matches up with the rationale I came up with while turning the idea over in my head last night and earlier today. Basically, the only “sensible” way I could figure to use something like 4/10 or 5/14 would be as a single bar inserted between two bars with “conventional” time sigs (after all, if the entire piece was in 5/14, then the /14 would indeed just be silly and pretentious - if there are 5 beats to the measure, every measure, then the actual value of a beat is kind of irrelevant), and the “irrational” time signature functions to alter the tempo. Though, at the same time, it seems to me that simply designating a more familiar ritard or accelerando would accomplish the same thing without unduly confusing the musicians tasked with actually playing the piece
(I should note that, as a Rush fan, I love odd time signatures. It’s fun trying to transcribe a section that has 7-beat patterns, and having to carefully listen to how everything fits together in order to decide whether it should be transcribed as 7/4, or alternating bars of 4/4 and 3/4, or 5/4 and 2/4. Of course, even at their most complex Rush sticks with “standard” note values.)
Fair enough.
Now that’s just sad. I think those are indeed quarter tone symbols, but there are some conventional symbols that are placed in locations (not on the following note’s pitch) that are either wrong, sloppy or have some other meaning not disclosed. In my mind, this casts doubt on the veracity of the entire excerpt.
Mister Rik, your “single bar” comment reminds me of a film score orchestral piece I was hired to write a lead sheet on once. In conventional 4/4 time mostly – and definitely not rubato – but I was bothered by one spot that seemed to stumble and throw my counting off. Close examination showed there was indeed one bar with one eighth note more than the others, so I notated it exactly (4/4…1/8…4/4…). Someone spotted that on my lead sheet and called me on it – “We didn’t play it that way.” I played the recording back to him and showed where the 1/8 bar was. He said, “Oh, that was where the editor cut the tape and spliced in another take. He just didn’t match the beats!”
So I guess you should never give a non-musician the task of editing music or bear the consequences.
And for the record, when you’re playing a string instrument, there’s a sort of mental difference between, say, A# and Bb, particularly if the A# resolves up by a half step or the Bb resolves downward by a half step. I’ve never been sure, given that my ear is not a machine, if the two notes are truly enharmonic, but it does feel a little different playing those notes. Likewise, something feels different when you go from F## to G#. I know F## is “really” G, but in that context, it sometimes feels a little higher, and I think I tend to play it a little higher.
I think “a little higher” is right, at least from one point of view. If you go all the way around the circle of fifths from G to F##, using justly tuned perfect fifths (ratios of 3:2), then F## is higher than G by a Pythagorean comma, which is a little under a quarter of a semitone ((3/2)^12 is 129.75, whereas 2^7 is 128).
Pretty sure that’s authentic. The notations I’ve seen of his work are aneurysm inducing. Which incorrect conventional symbols are you referring to? The only thing I see that looks weird is the natural symbol for the A that’s placed a bit high. Look here for some other notations of Ferneyhough’s. It’s not 2/10 time, but it’s got some of the same quarter tone accidentals and difficult-to-parse rhythmic notation.