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Let's say there is a song that starts at middle C and the next note is always the previous note raised by a perfect fifth or lowered by a perfect fourth to keep all notes within audible ranges. By working out the math, we can see that every seven notes, we add a sharp accidental (so that the notes cycle C, G, D, A, E, B, F), so the eighty-fifth note should be a (ridiculous) C♯♯♯♯♯♯♯♯♯♯♯♯, with twelve sharps. Is it correct to write this as a C in the next higher octave or replace a perfect fifth up to C♯♯♯♯♯♯♯♯♯♯♯♯ with a perfect fourth down to C, or do we have to keep it as C♯♯♯♯♯♯♯♯♯♯♯♯ (and let the ridiculousness grow)?

P.S. The thirteenth note of this song is not C, but B♯. To get the standard "circle of fifths" in this song, one of the fifths needs to be replaced by a diminished sixth (for example, A♯ to F) instead of the enharmonically equivalent perfect fifth (A♯ to E♯).

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    This question is so hypothetical and not connected to real-world composition or performance of music. As one answer says, when communicating with actual musicians, we wouldn’t go behind double sharps. Enharmonic equivalents are preferred over preserving the “theory” because theory is not music. Jan 11, 2023 at 0:42
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    I think there's a much better question hiding inside this one. Starting with your theoretical upward movement by fifths, a much better question is, "When we get to B#, should we just replace it with C and consider the circle to be closed?" The answer to that question is an emphatic yes. Once you get to the 13th note and it's B# AKA C, you're done. You've gone all the way around and hit every note. The 13th note is C, the 14th is G again, the 15th is D again, etc. You never even get to C## much less 12 sharps. Jan 11, 2023 at 3:42
  • Reality check: if you give me a score with twelve # I may play it, but I may also charge more $. Dec 28, 2023 at 12:27

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Is it correct to write C♯♯♯♯♯♯♯♯♯♯♯♯ (twelve sharps) as C?

It depends on the context. Even the context given in the question isn't sufficient for an answer:

the next note is always the previous note raised by a perfect fifth

How do you define perfect fifth here?

For most European music (that is, for most music using the European classical notation system regardless of the actual style of the music or its place of origin), an ascending perfect fifth must be interpreted in the context of the twelve-tone system, where it means "increase the note's letter name by four and adjust the accidental so the resulting interval comprises seven semitones." Note the absence of any explicit mention of frequency. The frequencies are determined by the temperament, but the choice of temperament does not change the definition of "perfect fifth."

On top of that, there's a somewhat flexible rule along the lines of "adjust the spelling of the pitch enharmonically for the convenience of the performer." Different composers may apply the rule differently. The same composer may apply it differently in different contexts or at different times. Jazz and popular music tend to apply it more liberally than classical music. But in every context, the rule would be applied long before you reach C triple-sharp, much less C duodecuple-sharp.

This enharmonic respelling rule is available in all twelve-tone temperaments, not only in equal temperament. In unequal temperaments, a particular tone may serve better in one enharmonic identity than in another, and nominally identical chords will therefore sound different, but the enharmonic respelling is nonetheless theoretically available. In the example song given in the question, therefore, some perfect fifths are indeed going to be spelled as diminished sixths (or perfect fourths as augmented thirds) -- if you use this definition of "perfect fifth."

The next time someone asserts that C♯ is different from D♭, you can reply "except when it isn't." For example, Beethoven's Moonlight Sonata is in C♯ minor but its middle movement is in D♭ major. Beethoven clearly did not intend for the tonal center of the middle movement to be microtonally distinct from the tonal center of the outer movements. Rather, he used D♭ major as the parallel major of C♯ minor because the notation is simpler that way. Similarly, look at the choices of keys in the 48 preludes and fugues in Bach's WTC. There are three movements in D♭ minor and one in C♯ minor.

In this context, therefore, the answer is "it's wrong to use C duodecuple-sharp, but if you really must then it's acceptable to respell it as C. Or maybe B♯. Or maybe D♭♭."

If, however, your definition of perfect fifth is "3:2 ratio of frequency no matter what" then you're not using the 12-tone system. In this case, you're going beyond what traditional staff notation can reasonably express. You're also going beyond what human musicians can reasonably play. You have a few options for notating this song, therefore.

The first is simply to describe it, as you have in the question. This would be especially useful if the song is intended as a piece of electronic music. It could then be taken, for example, as the specification for a computer program that realizes the song through audio synthesis or something like that.

Another approach would be to have several keyboards or other instruments detuned from one another by a Pythagorean comma. The first would play the first twelve notes, and the second would play the next twelve. The second and subsequent instruments would have transposed parts, so where the second plays B♯, the score shows C.

In this context, therefore, the answer is "you probably shouldn't be using staff notation for this, but if you are you should write it with a transposition to keep it legible."

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Whether it's correct or not doesn't matter; musicians (and their instruments) aren't used to accidentals beyond double-sharps and double-flats. It seems your song is using some kind of modulation, and it should change key once in a while, as to avoid uncommon numbers of sharps.

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  • How does a key change reduce the number of sharps?
    – mathlander
    Jan 10, 2023 at 21:07
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    When you reach C♯♯, I'd change the key to D major and write it as D. It's very common for songs with modulation - here's an example. Of course, at some point (e.g. G♯ -> Ab), you'd have to replace the sharps with flats.
    – Glorfindel
    Jan 10, 2023 at 21:09
  • To write C♯♯ as D, the previous note would still be F♯♯, and F♯♯ --> D is a diminished sixth. Won't this make the interval serve a different purpose from a perfect fifth?
    – mathlander
    Jan 10, 2023 at 21:41
  • Well, yes. But the point is that once you have established the C## as an "important" note in the music, it becomes harder to justify thinking of notes in the "current scale" with reference to C. At some point the previously-marked C## becomes the tonic, and there's no longer a point to naming it C## when D will do. Unless, of course, you are specifically concerned with the difference between the perfect fifth (i.e., a 3:2 pitch ratio) and the seventh degree of a 12edo scale, and don't accept C## as enharmonic to D. Jan 11, 2023 at 5:37
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    @mathlander if you've got an F-double-sharp followed by a D-double-sharp then you should not begin the enharmonic respelling by changing the D-double-sharp to C. You should instead begin earlier in the piece so you retain the perfect fifth as G to C.
    – phoog
    Jan 11, 2023 at 10:16
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It all depends on how rigorous you require the theory to be. Remember, it's all just a model. It's all made up. And if the goal is a meaningful analysis or notation that musicians can understand, you might be going down a needless rabbit hole. However, these are the sorts of rabbit holes that composers might explore in search of an interesting idea or new way to think about music.

It comes down to this: Do you believe that a C♯♯ is equivalent to a D? If so, then there's no problem. C♯♯ = D, C♯♯♯♯ = E, etc. If not, then at what point do we reach an equivalency? In the extreme case, just as B♯ and C sound the same but are regarded as separate pitches in certain simpler theoretic contexts than this one, you could demand that C♯♯♯♯♯♯♯♯♯♯♯♯ is NOT equivalent to C, because in your theory a C♯♯♯♯♯♯♯♯♯♯♯♯ has a different function than C. Dare to dream!

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    It seems to me that either the question is subjective and therefore off topic, or this answer is incorrect. If there’s any objectivity to this topic, then the clear, correct answer is “no”. There is no real-world or academic context in which a note followed by more than a double sharp is the correct way to write music. Jan 11, 2023 at 0:45
  • @ToddWilcox A B♯ augmented triad has third degree F♯♯♯.
    – mathlander
    Jan 11, 2023 at 1:59
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    @mathlander I'm trying to say that there's no reason for anyone to ever write a B# augmented triad. However you got to B# augmented you should just write a C augmented chord. Otherwise you're making things much too difficult for musicians, teachers, students, whomever you want to communicate with. I hope you would acknowledge that your question is a kind of thought experiment. While it can be a bit interesting to wonder about such things, thinking beyond B# aug. becomes less and less interesting. Thinking about C### is one thing, thinking about 12 sharps is not helpful, as far as I can tell. Jan 11, 2023 at 3:38
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    @ToddWilcox consider, for example, microtonal music intended specifically to explore the consonance of "far" intervals created while a comma pump is ongoing. Actually, that sounds pretty much exactly like what OP had in mind. There is surely no reason for anyone to write a B# augmented triad within the context of music written using a 12-tone scale, specifically for instruments tuned in 12TET, but many would find that rather limiting. Jan 11, 2023 at 5:33
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    @KarlKnechtel using traditional staff notation for such music and requiring microtonally distinct F-triple-sharp or C-duodecuple-sharp goes beyond the abilities of mere humans and requires computers or other electronics. It's better to invent a new system of notation or to have multiple keyboards tuned in microtonal relation to one another but with their parts transposed to a normal number of accidentals. It's hard enough to play in F-sharp major, never mind D-undecuple-sharp major.
    – phoog
    Jan 11, 2023 at 10:13
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It's a bit hard to answer since the premise itself is a bit of a "stunt." If this is an extension of my comment here, that you add the sharp when the melody note is "raised," maybe I need to expand. It's not so much because of the melody, as because the melody is "stretching" the underlying chord. The proposed circle-of-fifths melody doesn't really accommodate traditional tonality (i.e. it's going to have trouble "being in a key,") so perhaps it's an experimental work made to prove its own point. In which case, why not highlight it with a bizarre notation! It wouldn't exactly be a matter of "correct" or "have to," though.

(The mention of a progression according to repeating intervals is not so far fetched, though: check out "Giant Steps"!)

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Is it correct to write C♯♯♯♯♯♯♯♯♯♯♯♯ (twelve sharps) as C?

It seems strange to me to ask this question about correctness, give the pitches with letters rather than notation, and then not use some octave indicator.

C♯♯♯♯♯♯♯♯♯♯♯♯4 is C5

...at least that looks like adding 12 sharps did something to the pitch.

But more importantly there is a question of "correctness" in regard to practical notation.

This is the beginning of a familiar tune...

enter image description here

...which through an impractical application of accidentals is enharmonically equal to this...

enter image description here

Is it correct to write it the second way with multiple sharps?

No.

There are multiple lines on a staff for a reason.

There are key signatures for a reason.

There are accidentals for a reason.

When you don't use staff notation the way it was intended to be used that use is incorrect.

I can use a butter knife as a screwdriver, but it is an incorrect use of a butter knife.

...always the previous note raised by a perfect fifth or lowered by a perfect fourth to keep all notes within audible ranges.

So, there is a practical concern for audibility, but not for readability?

In order to write your series of P5/P4 all you need to do is introduce an enharmonically equal tone somewhere is the series, you could do for the twelfth tone and just write what would be a B# as a C natural...

enter image description here

Or, you might do it sooner and use Bb for the A#...

enter image description here

It's bizarre and impractical to envoke enharmonic equivalence to write C♯♯♯♯♯♯♯♯♯♯♯♯ while overlooking the simple, practical solution of using enharmonic equivalence to have the thirteenth tone return back to a C natural.


The point of key signatures and accidentals is to transpose the diatonic gamut of pitches ABCDEFG up and down to 7 tonics and alter those tones up and down by half steps. In that system, the system that staff notation was developed for, the practical limit of multiple accidentals is only double sharps and flats. Actually, the system even accommodates 12-tone atonal music without much trouble.

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  • In the case of my atonal song, the perfect fifth up to C♯♯♯♯♯♯♯♯♯♯♯♯ can be replaced by a fourth down to C. It still preserves the idea of a cycle of notes.
    – mathlander
    Jan 11, 2023 at 16:32
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    I really don't understand what you are doing or why. It would be a lot clearer if you posted your notation in your question. Jan 11, 2023 at 16:37
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    @mathlander atonal or not, a perfect 5th up and perfect 4th down will be at the same note from a set perspective. A P5 up from C is G, a P4 down from C is G. To me it makes even less sense atonally as you shouldn't be able to get 12 steps away since in most case you are treating the notes as enums from 0 to 11 so adding 12 gets you to the same value.
    – Dom
    Jan 12, 2023 at 0:09
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Yes! Of course it's possible! BUT why on Earth would it ever be used?

I thought the objective (followed by 99.9% of writers) is to write down or publish music that is in its easiest form to read. Even a double sharp is eschewed in some music - technically incorrect, but replaced with the far easier to read note a tone above.

It stands to reason that this idea is good and works well, otherwise we'd have multiple sharps/flats - but we don't - because we don't need them. They would become superfluous, and any pieces written with them resigned to the bin, rightly so.

The counter question arises - why bother, when there's a far better acceptable alternative?

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Generally not. If we define # as what "adds" a minor semitone to the original note, then it depends on the minor semitone. There are a lot of them. See https://en.wikipedia.org/wiki/Semitone#In_other_temperaments for a plot. Only one of them, the 12-tet (100 cents) adds up to a 2 factor of the frequency (i.e. a pure octave) when applied 12 times.

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    Check out @Phoog's answer which mentions tuning systems. Jan 11, 2023 at 16:02

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