Equivalent Key Signatures

Question

I've noticed that a variety of pieces in the classical repetoire (certainly at least the piano repetoire), label what seems to be effectively the same key signature differently. That is, I am not aware of any differences in terms of key between a piece in "G Sharp major" and "A flat major" (or their equivalent minors). Is there any reason that sometimes a sharped key signature is used while at other times the equivalent flatted key signature is used?

As a related point, I have undoubtedly noticed the preference of certain "equivalent" key signatures over others. For example, one encounters "E flat" significantly more than "D sharp" and likewise "C sharp" significantly more than "D flat". Is there any particular reason for this, or is it simply a convention that has been handed down to use through musical history?

Answer 1

The short (and oversimplified) answer is: Because Ab Major has fewer flats than G# Major has sharps, and thus it's easier for musicians to read. This becomes especially apparent with keys such as D# Major, which has a double-sharp in it---the seventh note of the D# Major scale is not D, but C^x (that's "C double-sharp").

The longer and more accurate answer is that these keys are not in fact equivalent. Only in an equal-tempered tuning system do G# and Ab have the same pitch, Eb and D# have the same pitch, etc. That is, only in equal temperament do enharmonically equivalent notes actually have the same pitch. In every other tuning system, the enharmonically equivalent notes are slightly (but definitely noticeably) different from each other. If you've only been exposed to fixed-pitch instruments such as the piano and the guitar, this concept can be challenging at first, but violinists, wind, and brass players know that their intonation depends on the context of the key in which they're playing.

This in turn leads to different keys having different sonic characteristics. In the modern, equal-tempered era, we've mostly lost these distinctions, but to composers of two hundred years ago and more, the keys of Ab and G# didn't sound the same at all, and they would compose to a specific key in order to take advantage of its particular character.

Update: In response to Brian's comment, I thought it would be helpful to post an example. Here are links to YouTube videos, all three of which were posted by the same person, using the same synthesizer, all three playing Bach's Air on the G String. But each link uses a different tuning system:

• Equal Temperament: http://www.youtube.com/watch?v=c6XkgNT20Eg
• Just Intonation: http://www.youtube.com/watch?v=gdL8aPQUOk0
• Pythagorean Tuning: http://www.youtube.com/watch?v=hq-SrgV7_Ow

Answer 2

There is a strong case for certain enharmonic key uses depending on the mode (minor/Major).

These four little tables show that Eb is significantly easier to write than D# for a major scale as the latter would involve double sharps in the signature.

But F# and Gb Major are really close in complexity.

Major scales

Number of #s (upper lines) versus enharmonic number of bs

Number of bs (upper lines) versus enharmonic number of #s

Likewise C# minor is to be prefered to Db minor but D# and Eb are interchangeable and this is indeed what Bach did in its Well Tempered Keyboard (Preludium 8 and Fugue 8, BWV 853). He used one for the prelude and the other for the fugue (See Alex Basson's answer about the validity of this substitution).

Minor scales

Number of #s (upper lines) versus enharmonic number of bs

Number of bs (upper lines) versus enharmonic number of #s

It explains that one favor keys that are practical in both modes especially in styles when it is frequent to make modulations or to switch between modes, such as variations.

Answer 3

This is a great question that is subtlely different from the usual "what is the difference between A♭ and G♯?" Apart from frequency differences in non-12-et tunings, that question relates to the function of the note within the chord/scale/key.

But if we're talking about choice of key, then there are other factors at work. Obviously in non-12-et tunings, the choice of A♭ major vs G♯ major does have to do with how it will sound. But let's assume 12-et for the rest of this answer.

There are three general factors that, I would have thought (but see below about Bach), led a composer in the common practice era to choose one key over another enharmonic one:

• avoid double sharps and double flats in the key signature itself
• reduce double sharps and double flats in the keys to be modulated to
• reduce double sharps and double flats in the keys used by transposed instruments

As others have mentioned, the Well-Tempered Clavier is interesting here. The point of the piece was to demonstrate the 12-major and 12-minor keys on an instrument where enharmonic notes have the same frequency. So frequency-wise there really can only be 12 major keys.

But from a naming point of view (assuming you wouldn't be as crazy as to have double sharps or double flats in your tonic) you have 21 possible key names for each of major and minor.

How would one go about choosing which 12 key names out of 21 to use? First step might be to avoid double sharps and double flats in the key signature itself. Note that the result is different between major and minor keys (as WTC demonstrates). That reduces your choice from 21 to 15 (so G♯ D♯ A♯ E♯ B♯ F♭ majors are out)

Then it likely just comes down to what you're going to modulate too. If you are going to modulate to V, you might want to avoid having to introduce the double-sharp to do so (which means C♯ major is out). If you are going to modulate to V of V, then F♯ might be out too).

All that said, it makes it particular interesting that WTC does use C♯ major and F♯ major and instead eschews D♭ G♭ C♭ to get from 15 possibilities to 12. The C♯ major prelude and fugue does indeed need a double-sharp in modulated passages, even though the note is diatonic.

In the case of ensemble pieces with transposing instruments: imagine a clarinet part in concert C♯ major :-)

Answer 4

Some of the answers have touched upon convenience for the player, but have missed one point about this. In making arrangements band and orchestral arrangements, I have found that wind players often prefer sharps to flats, preferring for example 7 sharps over 5 flats. This is noticeable particularly with transposing instruments. String players, on the other hand, seem to prefer flat keys to sharp ones.