The leap-year patterns of the Jewish and Islamic calendars are Euclidean necklaces

The Jewish calendar distributes 7 leap years across a 19-year cycle as evenly as possible. Bjorklund’s algorithm on (7,19) yields E(7,19), and started at the 7th pulse it matches exactly the traditional leap years 3,6,8,11,14,17,19. The Islamic calendar distributes 11 leap years in a 30-year cycle; E(11,30) started at the 11th pulse matches positions 2,5,7,10,13,16,18,21,24,26,29. The same ‘distribute k events as evenly as possible over n slots’ principle that produces world rhythm timelines therefore also appears, independently, in calendar design — evidence that maximal evenness is a general structural solution, not a musical accident.

Examples

Jewish leap years: E(7,19) rotated to the 7th pulse = years 3,6,8,11,14,17,19. Islamic: E(11,30) rotated to the 11th pulse = 2,5,7,10,13,16,18,21,24,26,29.

Assessment

Explain why the Jewish leap-year pattern being a Euclidean necklace supports the claim that Euclidean rhythms are ‘natural’. If a calendar needed 5 leap years in 13 years, what pattern would E(5,13) give?