Is A? major a closely related key to C? major?
I'm trying to list the closely related keys to C? major, and here is what I get:
C? major, the original key
D? minor
(???)
F? major
(???)
A? minor
x (B? diminished is not a key!)
For the points that have (???) next to it, I get some keys that are not part of standard usage: E? minor and G? major. Sometimes I think about turning them into enharmonic equivalents which makes F minor and A? major, but they do not seem to fit the definition. Are they close to C? major?
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If you try to identify the keys related to C# ... all you have to do is to find the related keys of C and then ad a sharp to the related keys of C and you've got the related keys to C#.
As the dominant key of C is G -> the dominant key of C# is G#
So you'll find that Ab is no related key of C#.
The key immediately clockwise is the dominant key of the key immediately counterclockwise, and features either one more sharp or one less flat.
en.wikipedia.org/wiki/Dominant_(music)#Dominant_key
III of C-major is Em -> III of C#-major is E#-minor.
same game with flats:
A is related to D -> Ab is related to Db
etc.
I think about turning them into enharmonic equivalents which makes F minor and A? major, but they do not seem to fit the definition.
In modern notation (pop and jazz and in arrangements for bands or even orchestra parts the enharmonic exchange (equivalents) is quite usual and then the keys are violated as flat keys to be related to sharp keys. Since the well-tempered tuning this points are pure theoretical but it makes sense to respect them and to keep them in mind.
No, they're nowhere near! For a start, A? has 4 flats, and C? has 7 sharps. How could they be related?
However, if we instead change the notes that sound the same into D?, then we're talking. D? has one extra flat to A?, so it's quite close.
You mention the diatonic chords that are usually associated with a key - I, ii, iii, IV, V, vi, viio. Those are not really keys but chords that are made with the notes from a particular key. Then you say they're not usual. C? isn't necessarily a usual key! Some composers loved it, others never ever used it - except perhaps to write for transposing instruments, but that would be rare to use C?.
So, basic answer to the question as it stands - a big fat NO.
It depends what you mean by 'closely related'.
Personally, I would say that one perfectly reasonable definition of a 'closely related' key is one that has many notes in common, regardless of spelling (i.e. treating enharmonic equivalents as equivalent). You can't (usually?) hear the spellings, after all. And C# major actually only has one note different to A? Major.
This can be seen easily if we 'spell' the notes in C# major according to their enharmonic equivalents in D? major: D?, E?, F, G?, A?, B?, C.
This collection of notes only has one note different - a flattened G - from the notes in A? Major: A?, B?, C, D?, E?, F, G.
So by that definition, the two keys are very closely-related.
Yes, Ab major is as closely related to C# major as two major keys can be: it's just one fifth away. Ab major is (in equal temperament) the same as G# major, one fifth higher than C# major.
Not if you spell it that way, no! But Ab major has four flats, Db major has five. So yes, pretty close!
But it sounds as if you're listing the triads that can be made from the notes of C# major rather than listing related keys.
Accepting that, yes E# minor and G# major are theoretically correct. And you've discovered what happens if you choose a key name with lots of sharps (C# major has 7 sharps) rather than one with rather fewer flats (Db major has 4 flats). You get some correct but somewhat unwieldy spellings!
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