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There are $12$ months in a year and $12$ notes in the chromatic scale. Moreover, there are $7$ long and $5$ short months and there are $7$ white and $5$ black keys in each octave on the piano keyboard. The patterns of long/short months and white/black keys match as well:

  1. January (long) — $F$ (white)
  2. February (short) — $F^\#$ (black)
  3. March (long) — $G$ (white)
  4. April (short) — $G^\#$ (black)
  5. May (long) — $A$ (white)
  6. June (short) — $A^\#$ (black)
  7. July (long) — $B$ (white)
  8. August (long) — $C$ (white)
  9. September (short) — $C^\#$ (black)
  10. October (long) — $D$ (white)
  11. November (short) — $D^\#$ (black)
  12. December (long) — $E$ (white)

The choice of $12 = 7 + 5$ notes for the music scale is not arbitrary. It is dictated by the desire to have an interval close to a perfect fifth (3:2 frequency ratio) in an equidistant scale. Since there are very few good rational approximations to $\mathrm{log}_2(3/2)$ with a small denominator, the $12$ tone scale is rather unique having the $A-E$ interval with the frequency ratio of $2^{7/12} \approx 1.4983$ which is indistinguishable from $3/2=1.5$ for most people.

I wonder if there is a similar mathematical fact that led to the choices made in the Julian calendar, or this match in the patterns is a pure coincidence.

Even if this parallel is deemed to be a pure coincidence, I still would like to know whether it can be used to restate music-theoretical facts in terms of calendar calculations and vice versa. For example, is there a music-theoretic analog of the Zeller Congruence or the calendar analog of the Circle of Fifths?

Update

Apparently, I was not the first to ask this question and not the only one who believes it is relevant to mathematics. Here are the drawings from a 1985 article by a future Fields medal laureate Maxim Kontsevich.

FD_bfa
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Alexander Belopolsky
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    I personally like this question; it's creative. It's probably just the "strong law of small numbers," which would be hard to confirm, but it's a fun question anyway! – pjs36 Oct 06 '15 at 19:11
  • The month lengths were partially chosen so that each quarter has the same number of days, supposedly. – Chappers Oct 06 '15 at 19:12
  • @Bye_World: I believe the last two months to be added were January and February, but my question is not about history. I doubt Andreas Werckmeister made any use of binary logarithms or continuous fractions when he invented the well temperament. I think he just wanted his musical instruments to sound pleasant. It was only much later discovered that the solution that he found was somewhat unique mathematically. Similarly, there might be mathematical facts that make $7+5$ months pattern inevitable. – Alexander Belopolsky Oct 06 '15 at 19:39
  • @Chappers: As far as I know, month lengths approximate the moon cycle which is about 29 days. However, this relationship is too imprecise to explain the 7+5 pattern. – Alexander Belopolsky Oct 07 '15 at 00:24
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    The pattern of months with 30 (or 28) days, that is, $0,2,4,6,7,9,11$ is simply $\left\lceil \frac{12}{7}i\right\rceil$ for $i=0,1,\ldots 6$. Also ${0,2,4,6,7,9,11}={7i \bmod 12 : i=0,1,\ldots 6}$ (the circle of fifths). These facts might be useful. – Joshua Meyers Jul 30 '17 at 10:42
  • This will help you!!!!! http://ehess.modelisationsavoirs.fr/atiam/biblio/Jeff%20Pressing%20-%20Cognitive%20Isomorphisms%20between%20Pitch%20and%20%20Rhythm83.pdf – Joshua Meyers Aug 17 '17 at 19:44
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    I believe musicians were using 12-note octaves long before anyone thought of logarithms and/or equal temperament. 12 fifths is very close to 7 octaves, and that leads in a natural way to 12 notes in an octave. Also, the arrangment of month lengths is partly based on historical accidents, such as taking a day away from February to make August longer in honor of Emperor Augustus (or something like that). I wish that the scale could explain the calendar, or that the calendar could explain the scale, but I think it's all accidental. Neat, but accidental. <– no pun intended – Gerry Myerson Mar 17 '18 at 06:47
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    I updated the question and believe it deserves to be reopened. While the choice of 5 out of 12 for the number of short months and black keys was made for different reasons, it is not a coincidence that the spacings of short months and black keys are the same. In both cases the spacing is the most uniform spacing. The same pattern can be obtained as follows: take the face of the 12-hour clock and inscribe a regular pentagon so that the vertices fall inside the hour arcs. The hours in which the five vertices will appear will follow the same pattern. See Kontsevich (1985). – Alexander Belopolsky Jan 17 '19 at 23:33
  • @FD_bfa - why did you delete the drawings that I referred to in the last paragraph? The reference is still there, but not pictures. – Alexander Belopolsky Feb 03 '23 at 09:41
  • The drawings are still linked, but they don’t add anything essential to the question. If readers are curious, they can view them at the link, but it unnecessarily lengthens the question to keep them in – FD_bfa Feb 03 '23 at 11:18

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