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Recent questions led me to discussions of theoretical keys, which are defined by Wikipedia as keys with at least one double accidental in the key signature. (Unfortunately, the source of that definition is not given.)

The key signatures on Wikipedia (and most other sources I've seen) put the double accidental in the spot where the single accidental would go. In other words, for G♯ major, the F𝄪 is in the spot where F♯ would normally go (i.e., at the beginning). This, I admit, seems intuitive to me:

enter image description here

Although the majority of online sources notate it that way, at least one other source puts the double sharp at the end. But this is hardly the greatest source, so I was skeptical that this notation would be correct. Imagine my surprise when I saw that LilyPond's default behavior (version 2.19.80) seems to notate it the same way:

enter image description here

While I can understand the logic of this latter key signature—the last added accidental appears last in the key signature—it's odd to see a key signature that begins with a C♯. (With that said, I guess it's no more odd than seeing a key signature with an F𝄪!)

Is there an official rule for notating key signatures with double accidentals? Or is there no rule since these are just "theoretical" entities?

Note that, according to this discussion among the MuseScore community, the top key signature (with the F𝄪 first) was custom made. The LilyPond key signature, however, exists by default.

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    Very interesting question. To me the second way is 'more intuitive' : the last added accidental appears last in the key signature - the order is not determined by the note values, but by the order of appearance. F is the first # because F is the first sharp, not because it's F .
    – Stinkfoot
    Jan 20, 2018 at 19:32
  • To be fair, you're second source was generated by Lilypond so it's not a surprise that Lilypond matches that (see attribute at the bottom of the source).
    – Dom ♦
    Jan 20, 2018 at 19:37
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    @Richard nope it's created using musescore (see bottom of link) en.wikipedia.org/wiki/Theoretical_key#/media/…
    – Dom ♦
    Jan 20, 2018 at 19:40
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    Wow, I was sure Gardner’s notation book would at least mention this sort of thing, but I don’t see it. Anyone have a copy of Stone’s notation book?
    – Pat Muchmore
    Jan 20, 2018 at 20:02
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    There is something that shows a scan of the G# signature with F## at the beginning. I don't know what the original source is though. translate.google.com/…
    – yo'
    Feb 4, 2018 at 23:09

5 Answers 5

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+50

I cannot give you any reference, but I'll try to give a typographer's insight into the problem and state my opinion. I hope it's fine.

Typography has some basic rules, and they are in general applied in this order:

  1. Deliver the message.
  2. Be consistent.
  3. Be invisible (meaning smooth, not "shocking", eye-catching at the wrong places etc.)

Applying these rules, I would go with the double sharp in the end. The reason is that primarily you want to focus on the delivered message, and that is that the key is significantly odd. Basically, we want to break the "invisibility" rule, we want the signature to stand out. And this is what you do. First, it starts with C# rather than the usual F#, and second, I believe (I can't support this by any study though) that if you only count the sharps, the one at the end is more noticeable than the one at the beginning.

Actually, purely from my mathematical point of view, I would also consider the following option viable. It best delivers the message that the key is G# major / E# minor, but the first F# is or at least could be confusing.

enter image description here

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    I think this makes a ton of sense, especially in light of the final example you provide. If we're dealing with theoretical keys, then why choose a notation that rules out the possibility of F# and F##? Placing the double sharp at the end (as you suggest) avoids that issue. If a satisfactorily authoritative citation doesn't exist, then I think this answer deserves the bounty because it provides the very logic one would seek in an authoritative source. If a satisfactorily authoritative citation does exist, then I think it should have reasons as strong as the ones you mention!
    – jdjazz
    Feb 5, 2018 at 23:53
  • The bottom example was my original guess, but I didn't mention it in the original question because I never saw it anywhere!
    – Richard
    Feb 6, 2018 at 16:03
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    I'd go along with this one. The purpose of a double-sharp is not to alter a natural note by two semitones, but rather to raise an already-sharp note by one additional semitone.
    – supercat
    May 2, 2018 at 15:13
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    @supercat, in this context of theoretical keys, I wonder if we need to define the purpose of a double-sharp that narrowly.
    – jdjazz
    May 2, 2018 at 22:22
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    I like the final, 'mathematical' option as it show the tonic ascending in fifths 8 times.
    – Michael Curtis
    Jul 8, 2019 at 16:24
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Gould's engraving style guide Behind bars (Key signatures, p. 91) says only the following:

The order of accidentals follows the 'cycle of fifths'.

While there is no direct instruction on what to do in case of double sharps, it seems that the only two readings that are consistent with this principle are:

  • Add the double sharp last, cf. the second example of the OP.
  • Add a single sharp at the beginning and a double sharp at the end, cf. yo's answer.
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Here are some gifs I made that show the process of gradually adding sharps in the key signature until there are twelve sharps, or adding flats until there are 12 flats. The cool thing is that when you have five double-sharps in the key signature, the remaining two single sharps are E# and B#, which are enharmonically the same as F natural and C. So 12 sharps is enharmonically the same as zero sharps (and likewise 12 flats)

enter image description here enter image description here

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  • This is really useful for teaching! you should an emoticon that gradually fades from smile to fear with each added flat/sharp....
    – nuggethead
    Feb 10 at 13:02
  • No fear - I wanted to push the notation as far as I could - and it doesn't break - it's all beautifully consistent whether you have 11 sharps in the key signature or just 1. Lovely.
    – Brian THOMAS
    Feb 10 at 13:03
  • agreed - our notation system, while flawed, is amazingly complete and capable of capturing just about anything
    – nuggethead
    Feb 10 at 14:18
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Personally, I can't see any logic in having both an F sharp AND an F double sharp. It can't be both.

I would put the F double sharp at the beginning for convenience. You are going to notice it there more than at the end of a group of sharps. We need to make life as uncomplicated as possible for the person reading the music.

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    While this answer makes sense, can you back it up with a reference?
    – Dom ♦
    Feb 4, 2018 at 22:09
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    No reference. I was just giving my opinion on this theoretical question. In practice, the composer would never use such a key. He/she would simply add the double accidental within the appropriate bar/s.
    – Jomiddnz
    Feb 4, 2018 at 22:16
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    The question is "Is there an official rule for notating such key signatures?" so your answer should address this. These keys do come up although are very rare so having a reference here rather than just an opinion will greatly help the community.
    – Dom ♦
    Feb 4, 2018 at 22:20
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    This question specifically asks for "an official rule for notating such key signatures." Others have offered their opinions on the matter in answers which have since been removed. It seems clear that OP is perfectly capable of forming an opinion, but is looking for more.
    – user39614
    Feb 4, 2018 at 22:26
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    In defense of this answer, I think it's worth noting that the accepted answer provides no hard reference either! Still, that answer got my upvote just like this one because they both stress a point I agree with.
    – wizclown
    Mar 1, 2019 at 21:18
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Up to this question I would not have considered the double sharp as legal in the key signature. On the the other hand, I never read about a limit on the number of accidentals in the key signature and so I would have happily added an eighth (single) sharp, again at the f line.

This keeps the well-established order and appearance in place.

Update with all sparse references, which I could find:

This page lists under "extreme accidentals" 8 flats for Victor Ewald's brass quintet, (admittedly it also list John Foulds World Requiem with 6 sharps and a double sharp, so also no clear strategy here) and the German wikipedia offers under "double sharp":

Opposed to the simple sharp, the double sharp is typically not used in key signatures.

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  • So would you first have all seven sharps, and then the F𝄪 as the eighth and final accidental?
    – Richard
    Jan 24, 2018 at 15:34
  • Or, based on your edit, it sounds you wouldn't have any 𝄪, but rather you'd start the signature off with F♯ and end with another F♯? (I might have misunderstood you.)
    – Richard
    Jan 24, 2018 at 16:28
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    I downvoted this because of the lack of source. The Question has two sources that do very different things and without a source, this is very much just a personal preference.
    – Dom ♦
    Feb 4, 2018 at 17:09

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