just intonation: , correct to about 0.1%. The just fifth is the basis of Pythagorean tuning; the difference between twelve just fifths and seven octaves is the Pythagorean comma. The coincidence permitted the development of meantone temperament, in which just perfect fifths (ratio ) and major thirds () are "tempered" so that four 's is approximately equal to , or a major third up two octaves. The difference () between these stacks of intervals is the syntonic comma. The coincidence leads to the rational version of 12-TET, as noted by Johann Kirnberger. The coincidence leads to the rational version of quarter-comma meantone temperament. The coincidence of powers of 2, above, leads to the approximation that three major thirds concatenate to an octave, . This and similar approximations in music are called dieses. Numerical expressions Concerning powers of correct to about 1.32%. This can be understood in terms of the formula for the zeta function This coincidence was used in the design of slide rules, where the "folded" scales are folded on rather than because it is a more useful number and has the effect of folding the scales in about the same place. correct to about 0.086%. correct to 4 parts per million. correct to 0.02%. correct to about 0.002% and can be seen as a combination of the above coincidences. or accurate to 8 decimal places (due to Ramanujan: Quarterly Journal of Mathematics, XLV, 1914, pp. 350–372). Ramanujan states that this "curious approximation" to was "obtained empirically" and has no connection with the theory developed in the remainder of the paper. Some near-equivalences, which hold to a high degree of accuracy, are not actually coincidences. For example, The two sides of this expression only differ after the 42nd decimal place; this is not a coincidence. Containing both and e , to about 7 decimal places. Equivalently, . , to about 8 decimal places. , to about 9 decimal places. , to about 4 decimal places. (Conway, Sloane, Plouffe, 1988); this is equivalent to , within 4 parts per million. , to about 5 decimal places. That is, , within 0.0002%. , within 0.02%. . In fact, this generalizes to the approximate identity: which can be explained by the Jacobian theta functional identity. Ramanujan's constant: , within , discovered in 1859 by Charles Hermite. This very close approximation is not a typical sort of accidental mathematical coincidence, where no mathematical