I'm very familiar with Scriabin's harmonic style, and he is one of my favourite composers - so I feel I can offer a few thoughts on this - largely in response to comments by rachmaninoff_forever.
I can assure you that Scriabin's system of notation as regards the choice of accidentals and enharmonic notation is entirely logical, and there is nothing arbitrary or "mad" about it. If he writes E-natural, then immediately after that Fb, you can be sure there is a reason for it - very probably that the underlying harmony changes, and thus the root of the chords changes, and that can change the enharmonic spelling of other notes, because the enharmonic notation is chosen to have the right relationship with the root.
I just looked up the Op. 67, no. 1 Prelude, because I couldn't recall the exact situation from memory. To begin with, we have a chord based on C7, essentially, with F# added immediately above the root (C). E makes perfect sense there. On the fourth quaver, the harmony changes - essentially to a slightly similar chord based on Eb7, with other notes added. The Fb played by the left thumb there is a semitone (and an octave, but that doesn't matter in a sense) above the root, Eb, so needs to be an F, to show the 2nd (or 9th) interval. And of course it has to be Fb, because F-natural would be a *major* 2nd (9th) higher. So the movement from E to Fb makes perfect sense, and it would be incorrect to change either of these.
I do disagree with a couple of other notes, though: consideration of the horizontal movement in the melody would have prompted me to write the Eb in the right hand as a D#, and the Gb as an F# - but that's entirely separate from the Fb-E issue you raised. I do occasionally disagree with Scriabin's enharmonic choices, but these usually concern passing notes or decorative notes, and very rarely the essential notes that define the harmony, which without exception are rigorously logical.
You also objected to putting a natural sign in parentheses (a cautionary natural) before a subsequent E that appeared immediately after an Fb, which you presumably thought should be notated as E anyway. This is an entirely orthodox and correct thing to do. Cautionary accidentals are defined as those which are not needed by the normal rules governing the scope of accidentals, but which are provided as a guard against misinterpretation of accidentals that may occur otherwise. After Fb, there might be a tendency to think an E following it should be Eb, even though it doesn't have a flat sign in front of it: so Scriabin carefully and correctly placed the cautionary natural in front of the E to guard against that. While an application of the rules governing accidentals would not dictate an Eb in that position, one of the quirks or frailties of the human mind might lead some performers to unconsciously make that E flat; so it is actually very thoughtful of composers to be aware of that and put in cautionary accidentals to steer you away from that.
Whether Scriabin's world-view and spiritual philosophy were crazy or not may be open to debate - I think his supposed "craziness" in real life has coloured some people's opinion of his music, and caused them to think his music is a touch mad or insane; but there is nothing in the least bit crazy or irrational about his music or his notation of it. In fact, he is one of the most careful, consistent, and logical of all composers when it comes to the details of notation.
You also grudgingly (I thought) accepted the use of double-sharps or -flats in tonal pieces with a key signature, but thought them totally irrelevant in music that uses no key signature and does not use traditional tonality. I completely disagree here. Possibly (I won't say certainly) they make no sense in serial or 12-tone music, and some other atonal types of music; but Scriabin's music, and much music not in traditional tonality, is not atonal either, much less serial or 12-tone, and in many such pieces double accidentals can still be logical and indeed correct. You will see frequent double accidentals in Scriabin's most abstruse music - completely correctly so, in my opinion. Scriabin was rarely if ever truly atonal (without any tonal centre), but he grew away from traditional tonality, and developed or invented a totally new tonal system, with its own rules governing things like accidentals.
You will even see a major triad in the upper reaches of some complex, multi-tiered chord notated something like this: E# A C. This may seem like a needlessly obscure F-major triad, but the notation is entirely correct, and precisely what I would do in the same harmonic situation. For instance if, in the bass, you had a B7 chord sustained (or treated like arpeggiated figuration) for a bar or two, and in the upper voices you had those notes E# A C, the logic of the situation would require you to notate them just like that. The reason is that each note is related to the B7 harmony underpinning it - and, technically, the upper notes are not at all an F-major triad, or any kind of triad. The E# is an augmented 4th above the root (a fundamental relationship in Scriabin's system of harmony - exactly parallel to the F# above the C at the start of the Op. 67, no. 1 Prelude discussed above - F-natural would be incorrect here - which would be more obvious if Scriabin included an F# in the lower harmony, as he occasionally does, but usually doesn't. The A is obvious enough: the min. 7th that is an essential component of an ordinary B7 chord. The C-natural is a minor 9th (plus 1 or more extra octaves) above the root, and so has to be C-natural. (If the C-natural were in a melodic voice that moved up to C#, you might then write it as B# instead - so your "F major" triad would then be E#-A-B# - entirely correctly and properly. So much depends on the surrounding context.)
That is a simple example: other examples in Scriabin are far more complex, but can essentially be analyzed in a similar way, and, almost without exception, his apparently odd notation can be rigorously analyzed in an analogous way. That often explains why chords in late Scriabin appear to have a bunch of grapes hanging in front of them, combining sharps, double-sharps, flats, naturals, etc., sometimes even in the same chord or in succeeding ones.
This is a radically opposite approach to the one that doesn't take much notice of such things, and where an Eb7 is just that, regardless of whether it should theoretically be a D#7, etc. I guess one can take that more casual approach and consider it acceptable and simpler. It does make many chords look more familiar, but does tend to obscure the harmonic structure of the music, which can be very important. Scriabin obviously chose not to take that approach and worked out logically correct ways of notating everything - an approach I favour myself, too, with equal rigour.
(Alkan was notorious for apparently refusing to notate anything enharmonically differing from the norm - but in fact I have seen scores by him which do, so he did not always apply this policy fully. Scriabin almost always did, however - to the nth degree. If you think Scriabin was bad, try looking at some scores by Nikolai Roslavets on imslp.org - makes Scriabin's notation look like that of a suckling babe! I don't understand Roslavets' style well enough to determine if he had as logical reasons for his thickets of double accidentals and even occasionally triple ones - but the music has the appearance of having a logical system behind it. Listen to him on YouTube, too (the score displays as the music plays) - it's fascinating - truly intriguing music, quite beautiful in a very alien sort of way.)
In the Bartok example given further on, the second chord is simply an E# major triad - it's just a remote key, but a perfectly orthodox major triad otherwise. That he notated this chord as E# major here, and then as F major a few bars later, is presumably because the context (the preceding and following chords) is different, and that can affect your enharmonic choices, especially if you have a melodic or linear or horizontal interpretation of harmonic progressions. Actually, in that case, I think the first of those chords could have been written either as E# major or F major, with no obvious "more correct" option (even if you take care to be rigorous, sometimes the better choice can still be ambiguous); the second could only be F major, I think.
When you have a notational system that was designed to cater only for 7-note-per-octave scales, with the rest treated as accidentals of lesser importance, and then you try to adapt that system to modern styles where all 12 chromatic degrees are of potentially equal importance, you are always going to have enharmonic anomalies somewhere or other, points of ambiguous choice of notation here and there, no matter how logical and rigorous you try to be. As for the "random" naturals - again, they are cautionary naturals provided for the same reason I mentioned earlier in connection with Scriabin.
Regards, Michael.
Topic: E# F flat B# C flat and double naturals