Meansquared

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Meansquared is a nonoctave regular temperament repeating at 4/1 based on a chain of tempered 9/4 major ninths. It tempers out 6561/6400 (or 81/80²) in the 4.9.25 subgroup. The name was first coined by CompactStar in 2023. Meansquared in the 4.9.25 subgroup is an insane restriction of 4.9.5 subgroup meantone, because it includes the interval of 100/81~81/64 which is effectively 5/4.

This temperament is the precise logarithmic stretching of meantone temperament by a factor of 2, and it follows that it is assosicated with the macrodiatonic scale 5L 2s⟨4/1⟩ and the more melodically usable macrochromatic scale 7L 5s ⟨4/1⟩. It also follows that the Ed4s which support meansquad have the same number of tones as the EDOs which support meantone – 7ed4, 12ed4 (6edo), 19ed4, 26ed4 (13edo), 31ed4 and so on.

Meansquared has a nearly identical structure to meantone temperament, but it sounds very much unrecognizable and xenharmonic, due to the extreme stretching involved and the lack of octaves. The 5-limit major and minor triads are stretched out into the macro-major triad (16:25:36) and macro-minor triad (100:144:225). Both of these have a somewhat minor-like sound, with the 25/16 coming off as a subminor sixth.