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This question asks if the accidentals are not "additive" (i.e. placing a sharp on F where the key signature already contains an F sharp would result in an F double sharp, and placing a flat on F in the same key would mean F natural), but it does not explain why they are not.

At least in my opinion, additive accidentals would have the following advantages:

  • there would be no need for a separate natural sign, since cancellation could be expressed with the opposite accidental.
  • double alterations would be used much less often (in the most cases, double sharps and flats are used to raise/lower a note which has already been sharpened/flattened, not to raise/lower a natural note by 2 semitones).
  • accidentals would stay the same upon transposing. In particular, all scales would be written in the same manner (e.g. a harmonic minor scale would always have a sharp on its 7th degree, regardless of the key signature).
  • the notation system would become more logically coherent (for example, a note with a sharp near it will always sound a semitone higher than the same note without this sharp - i.e. if we erase this sharp, the sound will always be a semitone lower)

So, why was it "decided" to stick to the current system?

P.S. I know that "why" may not be (or at least sound as) a correct question for the StackExchange format (because it associates with primarily opinion-based questions; so I apologise in advance for the question being out of format), but I believe that the objective reasons for this particular topic should exist (because the notation system is (almost) worldwide-spread).

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    Interesting concept. I'm sure various derivatives have been mooted and tried over time, but the 'old' method was the one that won through. Similar idea to both # and b in the same key sig. Never caught on.
    – Tim
    Jan 18 '19 at 11:08
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    It would be way too confusing. Notation should be crystal-clear, not "absolute minimum to be correct." Jan 18 '19 at 13:36
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    I bet it depends on the instrument. As a singer without perfect pitch, I think I'd prefer trolley813's system. As a singer with perfect pitch, or a player of an instrument that confers perfect pitch via muscle memory, I'd probably prefer the current system. It's worth noting that some early vocal music notation (e.g. Pammelia, I think) did use a sharp sign to cancel a flat in the key signature.
    – benrg
    Jan 18 '19 at 19:26
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    Given the frequency with which amateur choristers will already sing a G when there is a 'reminder' F# printed for the accompanist in a key that is already sharp (usually because there was previously a natural in another voice)... oh please no :) :)
    – Affe
    Jan 18 '19 at 22:41
  • Is your idea to make accidentals additive with respect to the key signature only, or also with respect to previous accidentals? E.g., if I'm in the key of Db, and I see a D# and then another D#, are the notes I play (in standard notation) D D#, or D D?
    – user9480
    Jan 19 '19 at 16:10
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A lot of this also has to do with convention. Once we learn certain patterns, we can respond to them quickly and efficiently, even if there are more logical ways to notate them. We see "of" and say "uv", without blinking an eye. "Once" = "wunce". But if we were to change the standard spelling of such words, it would take quite a bit of getting used to.

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Hmm.

Lets take an example of how this would work in practice.

Currently, when I see a sharp sign in front of a note (lets say F as an example) I know that the note required is an F sharp. It may be in the key signature already but that does not matter: it is an F sharp, always - no question.

Under your system when I see a sharp sign in front of an F what note is it? F sharp probably but perhaps it is F double sharp (because there is already a sharp in the key signature) or perhaps it is F natural (because there is an F flat in the key signature or there was an F flat earlier in the bar). I have to do a lot more work to know what note to play.

Can you see the problem with that? It is perhaps not the reason that the notation works the way it does but it makes me think that the current system is easier than your proposal

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    Actually when you see an F without any accidental you have to do almost the same amount of work (to see or recall the key signature and the accidentals early in the bar) - it is probably F natural, but may be an F sharp or F flat (or even double sharp or flat in some extreme cases). So, it's probably not the main reason for it. (Upvoted nevertheless, but did not accept, waiting for other answers)
    – trolley813
    Jan 18 '19 at 11:23
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    This is what I was going to say too. Holding the key in your head as a starting point and then altering it bar by bar is much easier than holding the key plus whatever alterations have occurred thus far. Also since we are not perfect, reminder or courtesy accidental are used to help us out.
    – b3ko
    Jan 18 '19 at 13:08
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    @trolley813 that's not the way any of us musicians see it. Knowing key signatures and scales is fundamental to playing. Further, music scores are chock-full of "reminder" sharp, flat, natural indicators to avoid uncertainty while playing. Jan 18 '19 at 13:38
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    @CarlWitthoft - Agreed. I don't see two sharps in the key sig and think "okay that's an F# and a C#" for every measure. I glance at the start of the line, see vaguely two sharps and know where my fingers go. That is - I know the patterns of the key from the signature and how it relates to my instrument. (Even sitting here now thinking about D Major I feel the extension of my third finger on the C and G strings.)
    – Geoff
    Jan 18 '19 at 15:49
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    The point is that you only have to look back until you find the last alteration, and then that tells you everything you need to know. If accidentals were additive, you would need to look back and collect every single accidental and then add them together. So you see an A, look back and there's a sharp, then a flat, then a sharp, and there's a flat in the key signature. You have to collapse all of that to see that everything cancels.
    – MattPutnam
    Jan 18 '19 at 16:44
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Situations can arise, especially in multi-voice music where a performer would have difficulty determining whether a note has an accidental or not. Consider, for example, choral piece on F major with two vocal parts on the same staff; one sings a marked B natural early in the measure and the other sings an unmarked B at the same staff position later in the same measure. A vocalist playing off that score would be unlikely to see the accidental on the other part, and thus sing Bb, but a pianist who is sight reading the score would be unlikely to notice that the later note wasn't in the same part that had the accidental, thus playing Bnat.

If accidental markings indicate absolute pitches, the later note may be explicitly marked as Bb or Bnat (depending upon which pitch is needed), ensuring that it will be played consistently whether or not the performer noticed the previous accidental. If accidentals indicate relative pitches, however, then it would be difficult to notate the piece in such a way as to facilitate sight reading by both singers and pianists.

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Additive accidentals? Let's take this to the ultimate! Here's a C major scale.

C | ##C | ##C | #C | ##C | ##C | ##C | #C

Not at all confusing!

The reason it isn't used is that the memory load is impossible for a human. A computer wouldn't have a problem as long as it always began from the start of the score and looked at every note on the way. Starting in the middle of a piece would be indeterminate for human or machine.

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As mentioned in the other answers, it's partially a matter of convention and partly of convenience. A problem could arise in cases where, for example in D major, there is a neutral sign on one F or C. Were the accidentals additive, it would be difficult to return to the default. An accidental on a note continues on that note (and not its octave equivalents) to the end of the measure for the same reason.

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It could be hard to keep track of the accumulation over a long time. Ex. m.5 add a sharp, later m. 24 add a sharp. Depending on how many times the pitch was replayed you might forget what it's current value is.

You couldn't read an isolated part of the score and know a notes actual pitch without backtracking in the score... all the way to the beginning. That would make rehearsal difficult.

So, why it was "decided" to stick to the current system?

I think the reason comes from a fundamental practice of composition: a piece starts and ends in one main key while the unfolding of the music contain digressions way from and returns to the main key.

The use of accidentals matches that fundamental concept. The key signature sets the main key, then accidentals indicate temporary pitch changes for relatively brief durations before returning to the main key. Contrast that with the cumulative concept which might make more sense if the music constantly modulated without returning to a main key.

The relative brevity of pitch changes away from the main key is reflected in the basic practice of accidental only applying for the measure in which they appear. If a change in key is very long, or coincides with a large thematic area, the key signature may simply be re-defined in the score.

It's best to think about this stuff with a historical view. Think of the time prior to Bach's Well Tempered Clavier when all 24 major/minor keys were not equally available. For a long time a lot of music used fairly simple key signatures and often only a few sharps, flats, and naturals were needed. The system is pretty good under those conditions. When music becomes very chromatic - Debussy, atonal, jazz, etc. - spellings with accidentals become hard to read. The system is strongly diatonic and easier when the tonic is natural (not sharp or flat.)

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