The term "musical" in my title refers to the sharp and flat keys of the "Western" chromatic ("twelve-tone") scale. The "arithmetic" refers to an easy way of deriving them and understanding them.

The typical treatment, taught in schools and talked up elsewhere, is confusing and depends on rote memorization because it uses diatonic ("seven-tone") labels for chromatic ("twelve-tone") intervals of pitch. It refers (diatonically) to the the cycle of sharps as "the cycle of fifths", and to the cycle of flats as "the cycle of fourths", providing no understanding as to why "fifths" and "fourths" are cyclical: running through all twelve possible chromatic keys and returning to "the beginning key". On the other hand, the chromatic labels for these as, respectively, "sevenths" and "fifths" can be easily explained by primary school arithmetic.

The sharp keys -- from zero sharps to eleven sharps are: C (0), G(7), D(14 = 12 + 2), A (21 = 12 + 9), E (28 = 12 + 12 + 4), B (37 = 12 + 12 + 12 + 1), F# (44 = 12 + 12 + 12 + 8), C# (51 = 12 + 12 + 12 + 12 + 3), D# (58 = 12 + 12 + 12 + 12 + 10), A# (65 = 12 + 12 + 12 + 12, 12 + 5), F (72 = 12 + 12 + 12 + 12 + 12 + 12), then back to C. The parentheses after each lettering shows a "jumping along by sevens", known as "an arithmetic progression".

The flat keys -- from 0 flats to eleven flats are: C (0), F (5), B-flat (10), E-flat (15 = 12 + 3), A-flat (20 = 12 + 8), D-flat (25 = 12 + 12 + 1), G-flat (30 = 12 + 12 + 6), C-flat (35 = 12 + 12 + 11), F-flat (40 = 12 + 12 + 12 + 4), x (45 = 12 + 12 + 12 + 9) ..., then back to C. (I didn't bother to write the other five flat keys because musicians, ordinarily, encounter only the first seven because the other five are confusing in the standard diatonic jargon. But composers have been known to go beyond the usual seven sharps or flats.) The parentheses after each lettering shows a "jumping along by fives", another arithmetic progression.

Why these? Consider the natural numbers (positive integers): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. Let's rewrite these as 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 . Please note the numbers 2, 3, 5, 7, 11. The mathematicians calls these "prime numbers". Why? Because of factoring. Thus, 12 = 2 x 6 displays 12 as product of factors 2 and 6. Now, every natural number has at least two factors: number one and itself: thus, n = 1 x n. The mathematician calls these "improper factors", because they are trivial or obvious. But in the factoring, 6 = 1 x 6 = 2 x 3 -- whereas the first factoring is improper -- the second factoring is proper, because nontrivial.

DEFINITION: A NATURAL NUMBER IS PRIME, IF, AND ONLY IF, IT HAS NO PROPER FACTORS. Thus, 2, 3, 5, 7, 11 are proper factors of 12, which enumerates tones of the chromatic scale. Please note, within that prime-list, the 5, 7 we considered above in the cycle of flats and sharps. But what about primes 2, 3, 11?

The problem with 2 is that, while being a prime, it is also a factor of 12: 12 = 2 x 6. So what? Ah, look as this arithmetic progression of "jumping along by twos": 2, 4, 6, 8, 10, 12, 14 (12 + 2), 16 (12 + 4), 18 (12 + 6) 18 (12 + 8), 20 (12 + 10), 24 (12 + 12), .... See? After six distinct keys, it starts repeating. And NATURALLY ONLY SIX SINCE 12 = 2 x 6. A similar problem arises with 3, as in the arithmetic progression of "jumping along by threes": 3, 6, 9, 12, 15 (12 + 3), 18 (12 + 6), 21 (12 + 9), 24 (12 + 12), .... After four distinct keys, it starts repeating. NATURALLY, SINCE 12 = 3 x 4. So, primes 2, 3 fail to give us the TWELVE CHROMATIC KEYS.

What about prime 11. I'll leave to you to discover that 11, 22 (12 + 10), 33 = 12 + 12 + 9), 44 (12 + 12 + 12 + 8), 55 (12 + 12 + 12 + 12 + 7), ... results merely in a descending of the chromatic scale, an almost trivial result.

Hence, the only prime factors of twelve which yield full nontrivial cycles are those of the fifths and sevenths, confusingly labeled, respectively, as "fourths" and "fifths" in nonprimal and primal labeling. (I leave it to you to see that these repeat only after twelve keys.)

Once we adopt useful labels, the explanation arises via elementary school arithmetic!