Tungsten meantone

Tungsten meantone is the name given by Mason Green in 2016 to the variant of meantone in which the size of the tempered whole tone is equal to 1/S₆ of an octave, where S₆ is the sixth metallic mean, equal to 3 + sqrt (10) or about 6.162. This gives a whole tone of ~194.733 cents, and a generator width of ~697.367 cents.

Tungsten meantone has the unique property that the ratio between the sizes of its fifth and fourth is the same as the ratio between the sizes of its diatonic and chromatic semitones; therefore, the whole tone may be divided into two chromatic semitones and a diesis in a manner exactly proportional to that in which the octave divides into two fourths and a whole tone. Successive MOSes thus become self-similar, with the whole tone looking like a scaled-down version of the the octave, and the diesis eventually looking like a scaled-down version of the whole tone, etc. Tungsten meantone is thus a cousin to golden meantone, since in each case the same ratios between interval sizes appear repeatedly and the scale is thus self-similar.

Successively better approximations of tungsten meantone are given by the following edos: 2, 12, 74, 456… These form a Lucas sequence, 2 × U_n (6, −1).

74edo has a fifth only ~0.04 cent narrower than that of tungsten meantone, and thus is a close-enough approximation for all practical purposes. Hence the name tungsten (which has atomic number 74).

Tungsten has a similar density to gold, but is stronger and harder; likewise, tungsten meantone's sound is brighter and less soft than golden meantone, as a consequence of having a sharper generator; it's also more familiar as it is closer to 12edo. Yet it still has very high 5- and 7-limit accuracy, and can also be extended to the 13-limit via grosstone (whereas golden meantone instead supports 13-limit meanpop).

The diesis of tungsten meantone is almost exactly 31.6 cents; this is slightly larger than a septimal comma. It's small enough that it would never be mistaken for a semitone, yet large enough to be perceivable to all but the most amusia-challenged. Melodically it could be considered a "shamble" or tone flux (smaller than a step, yet still a definite motion).

"Enharmonicism" (raising or lowering a note by one diesis) is analogous to chromaticism and can also be used to build memorable melodies.