Sqrt(25/24)

Sqrt(25/24), the classical semichroma or ptolemaic semichroma, is a the difference between a 5-limit major or minor third and a neutral third sqrt(3/2) dividing the perfect fifth in two.

Interval information

Expression	[math]\displaystyle{ \sqrt{25/24} }[/math]
Size in cents	35.336¢
Names	classical semichroma, ptolemaic semichroma
Special properties	reduced

This is an interval that allows to pass from a just major third (5/4) to a just minor third (6/5) by equal contrary motion, and vice versa. Let be two voices forming a 5/4 interval. If the lower voice goes up by a sqrt(25/24) and the upper voice goes down by the same interval, the next interval formed by the two voices will be a 6/5 interval. Any edo that maps 25/24 to an even number of steps, and thus any edo that contains a true neutral third, contains a representation of this interval.

Rational intervals that are close to this interval include 49/48 and 50/49, which stack to 25/24. The difference between 49/48 and 50/49 is 2401/2400, and tempering it out leads to the super-accurate breed temperament.

This interval, when stacked, yields a tuning system close to 34edo, which consistently represents it as one step.

Listen

Just major third and just minor third alternating by equal contrary motion

Just major seventh chord and just minor seventh chord alternating by equal contrary motion

Approximations

EDOs that have both a good 5-odd-limit and a sqrt(25/24) distinct from 25/24 include (among others) 24, 27, 31 and 34.

34edo has such an excellent sqrt(25/24) that the next EDO to have a better one is 441.