The distinctions between D-sharp and E-flat only make sense in the context of a key.
For instance, you construct a major key in E-flat and not D-sharp for the practical need to represent the scale nicely on the staff - so each tone in the scale should have a unique place in the staff.
You can construct the E-flat major scale with just three flat tones (Eb F G Ab Bb C D), whereas you would need four sharp and two double-sharp(!) tones if you started with D# (D# E# F## G# A# B# C##). And having to use F## and C## to refer to G and D tones is just ugly.
(I had made a mistake in the earlier version of this comment.)
I don’t think of it as ugly. It’s just what happens sometimes. Like if you start in G# minor and then use the leading tone. It’s way better to see F-double sharp than to see two different Gs fighting each other on the page. And it’s even worse to have to decipher those awful chromatic systems that are all painful to read.
Technically speaking that’s a Ab minor. New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s. Adding #’s are used for deriving new major scales. At least that’s how I understand it.
You can see this on the Wikipedia article on various minor[1] and major scales[2].
G# minor is used all the time as the relative minor of B major (5 sharps in the key signature). Ab minor is the relative minor of Cb minor (7 flats) and thus is almost never used except perhaps in passing for a modulation.
> New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s. Adding #’s are used for deriving new major scales.
No, a major scale can have both flats and sharps and the same for minor scales. In fact major scales often start on a flat while minor scales often start on a sharp. Major scales use Db Eb F# Gb Ab Bb as the roots of the scales (rarely C# and Cb), plus the white keys; while minor uses C# D# Eb F# G# Bb (rarely Ab and A#), plus the white keys.
> New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s.
I think you're confusing two different ways of constructing the minor scales.
One way is to start with the A minor scale (which has no sharps or flats) and to go around the circle of fifths[1] adding sharps or flats. Whether you add sharps or flats depends on whether you're going clockwise or counter-clockwise: for example, D minor[2] (one step from A minor going counter-clockwise) has one flat, and E minor[3] (one step from A minor going clockwise) has one sharp.
Another way to construct a minor scale is to start with its parallel major[4] and add a flat to the 3rd, 6th, and 7th. But note that the result can still have sharps (like in the E example above, where E major has 4 sharps).
In any case, G# minor is definitely a key that is used[5].
Technically speaking, if they said it's G# it's G#.
G# minor is a much better use of the key signature system than Ab: 5 sharps versus 7 flats. In practical terms, that's a proxy for it being more common.
Your vision that minor scales are constructed from A minor is valid; thinking it's by adding flats exclusively is misguided.
I'm not going to go out on a limb and defend the fact that sharps-based minor scales could be more common than flat-based, as that's likely not the case. A much easier argument against your logic is that flats-based major scales are used all the time.
Any given key signature can be either major or minor, be it made of sharps or of flats. It can be seen as altering C major or A minor indeed, but the alteration is allowed to go either way.
> awful chromatic systems that are all painful to read
The worst ones are the ones that petulantly stick to some theoretically-correct framework and produce a mishmash of accidentals that are canceled on the next note. If it's ascending, add sharps (or cancel the flat) on the second note. Let the key signature do the work instead of making me read all that to discover it's a simple chromatic run.
Of course people use double-sharp keys. And like you said, it is usually done in cases where it is the simpler notation to describe what is happening musically.
Simplicity is beautiful and construction of the major key in E-flat is decidedly simpler than in D-sharp.
Sorry, I had made a mistake. Wanting to create a major key starting with D# would end up looking like D# E# F## G# A# B# C##. The ugliness is even more stark.
- two double sharp keys
- F## to refer to G, C## to refer to D, B# to refer to C
If we keep the constraint that each letter has to be used exactly once when naming the notes of a major scale, but drop the constraint that the tonic has to be named using the same letter as the scale name (e.g., we can write G major starting at F##) then that pattern of sharps and flats generalizes nicely.
Number the 12 tones of 12-TET starting with C=0, C#/Db=1, ..., B=11. Then if you write a major scale starting at note N, the sum of all the accidentals counting sharps as +1 and flats as -1 will be equal to 7N mod 12.
For example G is note 7. G major then should have an accidental sum of 7 x 7 = 1 mod 12. We get that writing it G A B C D E F#. But it could also be written with a sum of 13, as F## G## A## B# C## D## E##, or with a sum of -11 as Abb Bbb Cb Dbb Ebb Fb Gb.
Note that because 7 x 7 = 1 mod 12, if we have to answer the question what scale N would have an accidental sum of K mod 12, we can solve 7N = K mod 12 by multiplying both sides by 7, giving N = 7K mod 12.
E.g., what major scale as 3 flats? 7 x -3 = -21 = 3 mod 12, which gives us the major scale starting at Eb.
Personally I find this approach a lot easier than memorizing the circle of 5ths to find key signatures given the key or to find the key given the signature.
A couple of questions naturally arise at this point. Why 7N? Why mod 12. The 12 part is easy to guess--it is because we are picking our major scale out of an underlying 12 tone scale. The major scale has 7 notes out of those underlying 12 notes, so a reasonable guess is that is where the 7 comes from.
But if you think about starting with C major (all white keys) and going up half a step, because the white keys are 0 2 4 5 7 9 11 12 (I've included the octave at 12 to make things clearer), and two of those (4 and 11) are white keys that do not have a black key immediately to the right, it might seem that how many accidentals get added or removed each time you go up in key half a step is going to vary a lot. Going from C to C#, every position goes black except those two. Those two will go black when you go C# to D, and all the ones on black will go to white.
The way the white and black keys are distributed gives you some different regions of the keyboard, each of which has a distinct pattern of adding and removing accidentals as you step through, and the overall pattern of accidentals is a result of those different patterns interacting. So maybe the 7 depends on those regions, and would be different if you had a 7 tone major scale chosen from 12 underlying tones but did not have the same pattern of white/black that we have.
I spent a while trying to show that the patterns would interact in such a way as to make 7N mod 12 work, but utterly failed.
To check that out we can try imagining alien music. Maybe some aliens who also use a 12-TET underlying scale and also have a 7 tone major scale have picked 0 2 3 4 7 9 10 as their major scale. Quite a different pattern. However, it turns out that 7N mod 12 works for that too. It also works even with alien music whose major scale is 0 1 2 3 4 5 6. You can have to use a crazy number of sharps or flats in that system!
What the pattern of white/black keys affects is which notes get accidentals when, not the total number of accidentals. By having the white and black keys spread out about as evenly as you can for a 7 white/5 black system we can write every key using the "right" starting note without needing any note to have more than one sharp or flat. Less even distributions of the black keys make it so you need multiple sharps and flats on some notes, but don't change the total number of accidentals mod 12.
Once you realize it really doesn't have anything to do with the pattern of black/white but only on the number of white keys, it is then not too hard to prove that it does indeed only depend on the number.
This can be further generalized. If aliens used a 5 note major scale, then the accidental sum of key N would be 5N mod 12. Since 5x5 = 1 mod 12, they could also go the other way and find the key from the accidental count K via 5K mod 12.
In general for a M note major scale from a T tone underlying scale, transposing that scale to note N uses NM mod T accidentals.
how would you use this approach in practical application?
i haven’t met many working musicians who had much difficulty learning the relationships between different keys, how they connect to the circle of fifths (fourths), and key signatures.
i get that it can seem overwhelming and non-intuitive, but it’s really not that complicated once you spend time playing and practicing music that illuminates these relationships (like playing ii-V-I progressions in every key, going around the circle of fifths). very little memorization involved; moreso muscle memory and an accumulation of applied theory in context.
most of the musicians i know are jazz players, where being able to play in any key is a critical aspect of mastering the genre. all the classical musicians i know are professional orchestral musicians, and they don’t seem to have any difficulty either.
For instance, you construct a major key in E-flat and not D-sharp for the practical need to represent the scale nicely on the staff - so each tone in the scale should have a unique place in the staff.
You can construct the E-flat major scale with just three flat tones (Eb F G Ab Bb C D), whereas you would need four sharp and two double-sharp(!) tones if you started with D# (D# E# F## G# A# B# C##). And having to use F## and C## to refer to G and D tones is just ugly.
(I had made a mistake in the earlier version of this comment.)