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Here's what I did:

Starting from the lowest C on a grand piano, I went up to the last 88th key C in perfect fifths to see how many sharps there were in all sharped major keys, then from the lowest C I went up perfect fourths to see flat major keys there were. Then I did the same for the minor keys starting from the first A key on the piano going up fifths and then going up fourths (you can also go down fourths from the top key for the flat major or minor keys). Here's the chart I came up with while doing this:

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So major keys: C G D A E B F# C# G# D# A# E# F Bb Eb Ab Db Gb Cb Fb Bbb Ebb Abb which adds up to 23 major keys.

For minor keys: A E B F# C# G# D# A# E# B# F## C## D G C F Bb Eb Ab Db Gb Cb Fb which also adds up to 23 minor keys.

23 + 23 = 46

Although a lot of the keys that have double sharps and flats and odd keys like E# Major and Fb minor are theoretical, very far from practical, I would never imagine anyone using C## minor instead of D minor unless you're going into C## minor from a key that has a bunch of sharps and double sharps, which are still unlikely to be used I'd imagine.

Is this correct? Are there theoretically 46 keys? Or did I miscalculate or misinterpret something?

MarcLikesMath
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2 Answers2

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What's wrong with the D major scale? It goes D E F♭ G A B C♭. Basically, you can go on as long as you don't run out of note names. Whether the scales end up just as renames of simpler scales depends on your tuning system. 12-tone equal temperament (12-TET) typical for a lot of current Western music wraps around after 12 scales, but there is also 19-TET or 31-TET dividing a diatonic whole note into 3 and 5 steps, respectively (with an accidental causing a change by 1 or 2 steps, respectively, and the "natural" half steps E-F B-C being 2 or 3 steps, respectively).

And of course there are non-equal tuning systems where ending up at your starting point might never be the case, strictly speaking.

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First you need to decide whether you are equating enharmonic notes e.g. F# / Gb and C/ Dbb.

If you are then there will be a maximum of 12 notes. Count how many distinct keys in an octave of your piano. As you go around the cycle of fifths, you will at some point need to switch to an enharmonic equivalent. The most common place is at F# / Gb but other locations are possible. Double flats and double sharps can be avoided as there are simpler ways to spell these notes.

If you are not equating enharmonic notes then you move into double sharps, triple sharps etc without limit. You have "math" in your name so I will add this comment. Going up a well tempered fifth raises the frequency by 1.5. This will never exactly equal a whole number of octaves. Try to find m and n such that 1.5^m = 2^n. It can't be done.

I will now assume that you are using equal temperament and equating enharmonics.

In your major key list:

  1. You have Bbb but this is the same as the A that you previously listed. The following double flats are also duplicates.

  2. You have E# and F. These are the same so another duplicate.

  3. You follow F with Bb but a fifth from F is C (so the cycle should end here).

Here is a corrected version: C G D A E B F# C# G# D# A# F with just 12. However, I would switch to enharmonics sooner: C G D A E B F# Db Ab Eb Bb F. F#/Gb is the trickiest choice since their key signatures are equally complex (6 sharps / 6 flats). Switching later just causes unnecessarily complex key signatures e.g. C# with 7 sharps rather than Db with just 5 flats.

The minor version will just be a rotation of this.

Look at Bach's choices in the Well Temprered Clavier. 12 major and 12 minor were enough for him. Several other composers have done "all the keys" and only found 12.

Substantially edited based on comments. For example, I now see that the range of the piano was not relevant. However, I also noticed an error in your list.

badjohn
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  • What I did was go up 5ths like in the circle of 5ths. When I was at E# Major (F) and went up a 5th I ended up back at C. If you go up 5ths again you'll do the same pattern, so there's no way you can have triple or quadruple flats to continue infinitely. Usually on a circle of 5ths, the major sharped scales end on C# Major or Cb Major on the flats side, but I kept on going further up 5th just out of curiosity and googled the keys I came across and they're theoretically real on basicmusictheory.com. Obviously not very practical. – MarcLikesMath Apr 17 '18 at 23:48
  • Regardless of the size of the piano, I ended up back at C, so the size of the instrument doesn't matter cause you'll end up doing the same patterns. – MarcLikesMath Apr 17 '18 at 23:53
  • @MarcLikesMath But you have double flats in your list and, for an unexplained reason, you stop at Abb. Why end there? Looking more carefully I see a couple of other oddities. You have E# and F in your list so you are double counting that note. Also, after F you have Bb, a fifth up from F is C. If you are equating enharmonic notes then you should have returned to C after just 12 steps. – badjohn Apr 18 '18 at 07:00
  • I have edited my answer substantially based on these comments. Also, I see that the question has been marked as duplicate. The linked question has good answers. I see 12, 30, and infinity but not 23. – badjohn Apr 18 '18 at 10:09
  • You're completely right. The reason why I thought of this question was because I went up perfect fifths from the first C on the piano all the way to the last C. I thought this pattern might have meant something just as I thought of the sharps for major keys ending at 7 sharps on the circle of fifths and remembering keys like G# major that have a double sharp. So on C# major, I decided to "go further". A fifth above G# major is D# major which has 2 double sharps, then A# major that has 3 double sharps and I found it interesting that they kept adding up when going up a fifth. – MarcLikesMath May 17 '18 at 12:11
  • Right until you get back to C with 0 sharps. – MarcLikesMath May 17 '18 at 12:11
  • So I applied the same logic on the flats, then the same for sharps and flats for minor scales. I guess it's just a different way of seeing it but this pattern is not relevant nor pragmatic to the way music is actually composed, used, ect. – MarcLikesMath May 17 '18 at 12:14
  • One of my points was that if you equate enharmonic notes e.g. F# = Gb then the circle of fifths ends after 12 steps in either direction. If you do not equate enharmonic notes then it will never end and you will have to deal with ridiculous numbers of sharps and flats. Any ending point between 12 and infinity will be arbitrary. This is because there is no power of 1.5 (perfect fifth in just temperament) which is exactly equal to a power of 2 (octave). If you use equal temperament then the circle of fifths will close but its "perfect fifth" is the irrational 2^(7/12). – badjohn May 17 '18 at 12:19
  • I asked a question on unusual keys myself. You might find it and its responses interesting: https://music.stackexchange.com/questions/70588/how-many-sharps-in-d-major . – badjohn May 17 '18 at 12:25