Rhodium

Rhodium is a rank-2 temperament which splits the octave into 45 parts and has a generator that is mapped to 55/32. It is named after the 45th element.

For technical data see: Ragismic microtemperaments#Rhodium

Theory

From a classical regular temperament theory perspective, rhodium is just an extension of the ennealimmal temperament, and respectively has the same mappings for the 9edo subset - 27/25 for 1\9, 7/6 for 2\9, 63/50 for third-octave and 49/36 for 4\9.

However, it also provides a mapping of the Alpharabian quarter-tone 33/32 to 2\45 and thereby also tempers out the quartisma. Since rhodium tempers out 4225/4224, a single period is mapped to an interval that stands both for ~66/65 and ~65/64 and two of them make 33/32, and ten thus make 7/6.

1665edo is the optimal patent val for rhodium both in the 11-limit and in the 13-limit. Other notable equal temperaments which support rhodium include 270edo and 1395edo, which are zeta edos, and 1125edo, which while is not zeta, is still a multiple of 45 consistent in the 15-odd-limit.

Relationship to Ben Johnston's notation

Rhodium tempers 65/64 to 1\45 and 33/32 to 2\45. These intervals are used in Ben Johnston notation to translate a nearby simple interval into a usually more complex tridecimal (65/64) or undecimal (33/32) version. As such, using symbols 11 and 13 while playing in rhodium enables modulation by various steps of 45edo.