Diminished and augmented key signature
Now I know that augmented chords and diminished chords are dissonant normally. Arpeggiate it and it is much more consonant. 3rd octave and it is much more consonant.
But can you actually have something in the key of C augmented? If so would it be based on this scale: C C# D# E F G G# B C which is an octotonic scale similar to the diminished scale except it is based off of augmented chords instead of diminished?
This would have all of the first four sharps except for F#.
Likewise can you have something in the key of C diminished? If so would it be based off of the diminished scale?
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Short answer: yes and no.
No, it's not at all a standard usage. There are no augmented and diminished key signatures.
Yes, you can do whatever you want. Microtones, even... it's all cool. So if you have a need for this, you certainly can do it.
Some scales will fit these chords better. I don't quite follow your augmented scale idea... it would make more sense written in flats perhaps... but I'm not sure why you have F,G against the G# of the augmented chord... just seems a bit odd to my ear but I'm just imagining not playing it aloud. Lydian can work over an augmented chord, but be careful of that 5th.
on the other hand, a whole tone scale would fit the bill easily... for the diminished, locrian will make sense. Both of these scales leave you out in the middle of nowhere tonally because there's no perfect fifth, but that's also the idea.
Key understanding: diminished chords are highly dissonant and demand resolution. Augmented chords are very ambiguous. Two very different animals.
It's not a standard usage. There are 12 keys (or a few more due to treating C# and Db as different.) Keys have two modes, major and minor. Other tonal organizations are possible but the usual (CPP) terminology isn't used in their descriptions.
The term "key" is not synonymous with scale. Scale (from the Italian for "ladder") is a collection of notes taken in ascending (or descending) order. Key tends to refer to the harmonic relationships used in a composition. The usual terminology doesn't really apply well to music outside the Common Practice Period.
Olivier Messiaen (1908-1992) did exactly that. If you add A to your scale, you complete the augmented triad F A C#. Your scale now contains 9 pitches per octave. This set of 9 is exactly the same as that of one of Messiaen's modes of limited transposition, specifically mode 3. The set of 9 pitches yields 3 modes: which mode you get depends on which pitch you start a scale on. In your example, with A added to your scale, the intervals go STSSTSSTS. The example in wikipedia goes TSSTSSTSS, and the other mode goes SSTSSTSST.
Messiaen used it a lot in Un reflet dans le vent from his early collection of Preludes for piano.
As for your idea of a scale of C diminished: if you combine diminished chords on C and G you get the scale C C# D# E F# G A Bb. This is Messiaen's mode 2. Messiaen used this in many of his works (including that Prelude, briefly). It also crops up in works by some earlier composers, e.g. this chord-sequence over a pedal G in "Le Gibet" from Ravel's Gaspard de la nuit.
For theory which deals with music using the common harmonic vocabulary from about 1700 to mid 1800s, the keys are: C major, D major, E major, F major, G major, A major, B major, F# major, C# major, F major, Bb major, Eb major, Ab major, Db major, Gb major and Cb major. Also the minor keys of all the above, that is, C minor, D minor, etc. Total 32 keys.
There is a scale corresponding with each of the major keys and 2 standard versions with each of the minors (harmonic and melodic). Each scale has a basic triad (3 note chord) formed on each note of the scale and the triads are either major, minor, diminished or augmented.
The vast majority of modern popular music sticks with this system because it is what most people generally grow up hearing. Most people never hear anything else. You CAN do anything you want, but the further from the common practice vocabulary you go, the smaller your audience becomes.
"C diminished" could be Locrian C (C Db Eb F Gb Ab Bb) which has a lowered 5th (probably tricky to use, on account of <C Eb Gb> being dissonant) and lowered 2nd scale degree (easier to handle, see the Neapolitan chord for common uses, or consider the Db as borrowed from the subdominant Aeolian F) as compared to the Aeolian C mode (C D Eb F G Ab Bb a.k.a. C minor).
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