Barium

Barium is a rank-2 temperament defined in the 5-limit by tempering out the comma which sets 56 syntonic commas equal to the octave. Extensions exist in the 7-limit and the 11-limit. It is named after the 56th chemical element.

For technical data see: 56th-octave temperaments#Barium

Theory

An octave is equal to $\frac{1}{\log{2}{\frac{81}{80}}} \approx 55.79763$ syntonic commas, which when rounded to the closest integer yields 56. The associated comma in the 5-limit is [-225 224 -56, and therefore is tempered if and only if the EDO divides 56. The comma is about 4 cents wide, but since each 81/80 is flattened by only about 0.07 cents as a consequence, barium is a very precise microtemperament.

Because the period is set to 81/80, interval stacking scheme works the same way as in meantone, with the only difference being that the resulting intervals are represented in different 56ths of the octave. When the interval 3/2 is stacked 4 times, it also mirrors the pattern in every 1/56th of the octave, reaching 5/4 in 4 steps just as meantone would.

In the 7-limit, the reduced generator of barium is equal to the 126/125, a comma which together with the syntonic comma completes the basis for the septimal meantone. As such, barium can be interpreted this way as an "unfolding" of the septimal meantone into the fractional-octave temperament where one comma (81/80) is the period and the other (126/125) is the generator. Reading directly from the mapping, 7/4 is attained in 5 stacked intervals.

Barium in the 7-limit also tempers out the akjaysma, meaning that 40 periods are set to 105/64.