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I'm trying to figure out in which situations say, a G# is preferred to an Ab. I know that the two spellings wont necessarily represent tones of identical frequency, and that they only coincide in equal temperament tuning. But within equal temperament, most resources I've seen online insist that the note G# is not the same as Ab, despite them having the same pitch. So I'm assuming that it's not that the notation is wasteful, but that the two spellings somehow signify different things to musicians.

I get that within scales we only want one note of each letter A-G, so that G major scale consists of G A B C D E F# and not G A B C D E Gb. But say I'm considering the note G# (Ab, F###, Bbbb etc.) in the context of a melody within the key of C Major. I can't see any reason to prefer any one enharmonic spelling to another. I guess that requiring maximal simplicity would reduce the number of possible enharmonic spellings to just two (i.e. we could discount F### etc.) but I can't see a reason to prefer G# to Ab or vice-versa.

Another case I find problematic is that the Augmented C triad is written (C E G#) and not (C E Ab). I understand that it is called an augmented chord because the fifth was sharpened, but why don't we call it a diminished chord because the sixth was flattened? I understand that there's already a pre-existing diminished chord; I'm questioning why the chord is named and spelled in terms of "what happened to the fifth interval?" instead of "what happened to the sixth (or any other)?"

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Direction. Tonal context.

When you introduce a chromatic note, you are usually adding a tendency tone, which will imply movement either upwards or downwards, depending on where the note is being introduced. The tendency can imply movement by whole tone or semitone. (Generally a chromatic note that moves away by a whole tone moves into a chromatic leading tone, such as the standard ♮6-♮7 substitution in the minor mode.)

In most cases, whether you sharp a note, or use its enharmonic equivalent will depend on the direction of movement. Also, in most cases, the introduction of a chromatic note in any manner other than as a trivial passing note or auxiliary implies the introduction of a foreign mode or tonality. (This is true, for instance, of the ♮6-♮7 substitution in the minor mode: it implies a major mode substitution at that spot.)

The augmented triad is particularly ambiguous because its constituents divide the octave equally so that the augmented triad on A♭ uses the same notes as the one on C, which uses the same notes as the one on E. Your C E A♭ could easily be an inversion of the augmented triad on A♭, so the direction of movement will matter. For example:

enter image description here

(Augmented chords are marked with the cross-in-circle symbol. The chord beneath the x-in-circle is an augmented sixth, specifically a French sixth chord.)

In example (a), it wouldn't make sense to use G♯: we've inserted a descending minor segment into C major. B♭ leads into A♭, which leads as a descending leading tone into G. The augmented triad acts as a predominant in C.

In example (b), the tendency tones, through the sharping of F and G, suggest a move by ascending leading tone into A minor. The augmented triad acts as a predominant here as well, but to a different tonality's dominant.

However, although it would be very easy to resolve the augmented triad to an E7, I've inserted an augmented sixth instead to illustrate the combination of two concurrent modal substitutions: the descant is outlining an ascending major segment in A, while the bass is outlining a descending Phrygian segment, also in A. The net effect is that both the ascending and descending leading tones are in use at the same time, hence the augmented sixth, which is correctly written with both 7 and ♭2. The French sixth is a mixed-function chord, acting as both a dominant and a Phrygian cadence chord. (It is often used as a predominant as well: if I had finished with A major rather than A minor, it would be simple matter to cadence to D minor.)

So, as I said earlier, direction and tonal context: that's what determines which spelling you use.

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Whenever enharmonic equivalence comes up and a note can have multiple names, you need to look at the context the note is in. There's a lot to look at and depending on what kind of harmony you are going for and what's going on.

Let's stay in C for simplicity. The augmented C chord would be one scenario where you would use G# over Ab because the G# is part of the spelling and definition of the chord (i.e. augmented chords have augmented 5ths, not minor 6ths). There's another chord that would utilize G# over Ab which is if you were using a E major chord as a secondary dominant to take you to an A minor chord as the third of that chord is G# Another scenario is if you want to build a major chord off G#/Ab it would make more sense to call that chord an Ab major because the chord can easily be viewed as borrowed from C minor which is the parallel minor.

If the note is just a passing tone and not part of the harmony however, the naming is a lot more straightforward. If you are chromatically ascending you would prefer sharps over flats while still adhering to the key and if you are chromatically descending you would prefer flats over sharps while still adhering to the key.

Why this is the case is easy to see let's look at the the spelling C-E-G# vs C-E-Ab:

enter image description here

They are equivalent enharmonically, but you can see the one with G# is easily identified as a triad (a 3 note chord), while in the other chord it is not easy to see. This is a huge difference when sight reading and makes much more harmonic sense.

  • Ah ok, that makes sense given that the definition of the augmented chord is that it has an augmented 5th. I'm still not quite sure why the combination of notes that is C E G# (or C E Ab) is given the name augmented (viewing G# as augmented fifth) rather than minor (viewing Ab as a minor 6th)! – James Machin Aug 3 '15 at 18:28
  • @JamesMachin because we build chords in 3rds it doesn't make sense to say. I added a little bit to help explain. – Dom Aug 3 '15 at 18:37
  • Also passing tones' direction of resolution can hint whether it's a sharp or flat, all other things being equal. If it resolves up, favor sharp, if it resolves down, favor flat. – Josiah Aug 3 '15 at 23:05
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Regarding your example in C. Yes, if the chord is augmented, the G gets stretched to G# - same name, augmented interval. If the chord changes to F minor, the A from F major gets changed to Ab, for a minor 3rd interval. Same note, different names for different reasons. The diminished notion doesn't work, as the 6th note is not the one to be diminished - it's actually the 7th, so the note becomes Bbb, sounding like an A. The Ab would be a minor 6th against C- you don't diminish an interval unless it's a 5th or 4th, both perfect intervals which when made smaller (diminished!) by a semitone get called 'diminished'.

  • Thanks for the answer! So is the idea to always compare to the major triad and see what changed? I mean I'm guessing it's wrong to spell the augmented C chord C E Ab, but I can't see why! Right I think I understand what you're saying as to why I can't call it a diminished 6th chord, but why not then, for example, C minor 6th or C diminished 7th, rather than C Augmented 5th? – James Machin Aug 3 '15 at 17:45
  • In general, chords in common-practice harmony (which is probably what you are talking about when you said "a context of within the key of C major") are built by stacking up thirds. A C major 6th chord C E G A is exactly the same notes as an A minor 7th chord A C G E which is built from thirds, except that voicing it with C in the bass instead of A gives an ambiguous sound somewhere between C major and A minor. As a general rule you only write notes one semitone sharp or flat from the key itself, so you wouldn't want to write an "ultra-flattened-7th chord" in C major as C E G B-double-flat. – user19146 Aug 4 '15 at 0:26
  • In your augmented chord example, bear in mind that augmented and diminished chords are often used to modulate between fairly remote keys. For example you might well re-spell C E G# as Ab C E (or even E G# B# or Fb Ab C) if it was used as a "pivot chord" to get to a new key. And if the music is oscillating between the keys of say E major (4 sharps) and Bb minor (5 flats) the spelling of the notes may be chosen more by pragmatism than by pedantry. – user19146 Aug 4 '15 at 0:34
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@Dom and @Tim both both make strong points. Here is a more philosophical take on things.

Music and language share many common features. Music is like language in that is has both subjective and objective components.

In language, the subjective experience of hearing the word "cat" may result from hearing someone read the word "kat", or "khat", or even "cat". By when we look at language objectively, there is only one correct way to write the word "cat".

The same is true in music. We hear CEG# the same as CEAb, or even CEBbbb, but the rules of musical grammar require that the composer express his/her intent in one and only one way - the correct way. If the intent is to augment the 5th, as in a C augmented triad, then we write CEG#. The correct spelling will depend on the composer's intention and the context of the chord or note - i.e., how it relates to the previous chords, or phrase.

I hope that is not too wishy-washy. I'm just having my first coffee of the day.

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Another factor that comes into play, and is part of the "readability" doctrine: If you're in a "sharps" key (#), tend first to write a #. If you're in a "flats" key (b), tend to write a b. In both cases, tend toward consistency.

If you're in the key of E, and you have to represent the note "Bb/A#", you will prefer to use the A#, because it's more intuitive to add a sharp to a sharp key than a flat. HOWEVER, if the harmonic function of the note is clearly as a diminished 5th, then it should be notated as such (Bb).

Enharmonic spelling should be used to clarify as much as possible, but it should NEVER obscure the actual underlying harmony...i.e. calling something an augmented 2nd when it's clearly functioning as a minor 3rd.

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For me gramatical correctness is way more important than readability!

Because at the end of the day when you play a piece of music you use the score only as a rough reminder for what comes next. What a score is actually for is to preserve musical intent for the future. And here an approach as precise as possible is absolutely necessary. How could a musical director tell an orchestra which voices have to be more dominant and which not, if he didn't understand the functional structure and the 'grammatical' intent of the composer or editor.

Look at all these Skrjiabin or Chopin pieces with a lot of flats and sharps - they all strive for grammatical correctness and not for readability.

And talking 'bout language: ICU, UCme2 or this is 4U is also readable and very economic... ;-)
This is why the english orthography is such a mess - there have never been rules - no academy - we are all so free and independent!

And here I am talking about NOTATION and not about music and composing itself, where we definitely need all the liberty we can get...

Just my two cents...

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