Meansquared

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 meantone temperament with all intervals (including octaves) squared. It follows that it generates the macrodiatonic scale 5L 2s⟨4/1⟩ and the macrochromatic scale 7L 5s ⟨4/1⟩ (a very xenharmonic variety of detempered whole tone scale). It also follows that the Ed4s which support meansquared have the same number of tones as the EDOs which support meantone: 7ed4, 12ed4 (6edo), 19ed4, 26ed4 (13edo), 31ed4 and so on. Meansquared is supported by both nonoctave odd Ed4s and even Ed4s (EDOs), including ones without conventional meantone temperament (like the previously mentioned 6edo and 13edo). It has extensions in the 4.9.25.49 subgroup and beyond corresponding to the extensions of meantone, including counterparts of septimal meantone, dominant, flattone and so on.

Meansquared has the same structure, MOSes and interval chain as meantone temperament, but with every interval squared, which stretches them out so much that they are in completely different size categories. The perfect fifth becomes a major ninth (9/4), the major third becomes an augmented fifth (25/16), and the minor third becomes a diminished fifth (36/25). The two tritone intervals are stretched out to compressed and stretched pseudo-octaves, but these are pulled closer to major sevenths and minor ninths in the flattone equivalents, while in 6edo these two are conflated with each other to produce the pure octave, like how in 12edo the tritones are conflated to produce the 2edo tritone. In flatter tunings like 19ed4 and 31ed4, this system is very xenharmonic with it lacking single octaves, and the stretched versions of the standard 5-limit major and minor chords (1-25/16-9/4 and 1-36/25-9/4 respectively) are also exotic and tense-sounding. However, the 12-tone MOS is resemblant in many ways to the familiar 6edo or whole tone scale, being a detempered version of it, and it gets closer and closer to it with a more and more accurate pseudo-octave as L and s become more similar.

Interval chain

#	Cents*	Approximate ratios
0	0.000	1/1
1	1394.429	9/4, (20/9)
2	388.858	(5/4), 81/64, 100/81
3	1783.287	25/9
4	777.716	25/16
5	2172.145	225/64
6	1166.574	625/324
7	161.003	625/576
8	1555.432	625/256

* In 4.9.25 subgroup CTE tuning