Meansquared: Difference between revisions

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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 [[81/80]]<sup>2</sup>) in the 4.9.25 subgroup. The name was first coined by [[User:CompactStar|CompactStar]] in 2023. Meansquared in the 4.9.25 subgroup is an [[sane and insane temperaments|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) stretched by a ~~stretch~~ factor of exactly ~~2, and it~~ follows that it ~~is assosicated with~~ the [[macrodiatonic and microdiatonic scales|macrodiatonic]] scale [[5L 2s (4/1-equivalent)|5L 2s&lang;4/1&rang;]] and the ~~more melodically usable~~ macrochromatic scale [[7L 5s (4/1-equivalent)|7L 5s &lang;4/1&rang;]]~~, which constitutes~~ a very xenharmonic variety of [[detempering|detempered]] whole tone scale. It also follows that the [[Ed4]]s which [[support]] meansquared have the same number of tones as the [[EDO]]s which support [[meantone]] – [[7ed4]], 12ed4 ([[6edo]]), [[19ed4]], 26ed4 ([[13edo]]), [[31ed4]] and so on. Meansquared is supported by both nonoctave odd Ed4s and even Ed4s ([[EDO]]s), including ones without conventional meantone temperament (like the previously mentioned 6edo and 13edo). 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 [[2/1|octave]], like how in 12edo the tritones are conflated to produce the [[2edo]] tritone.

This temperament is [[meantone]] temperament with all intervals (including octaves) stretched by a factor of exactly ] It follows that it generates the [[macrodiatonic and microdiatonic scales|macrodiatonic]] scale [[5L 2s (4/1-equivalent)|5L 2s&lang;4/1&rang;]] and the macrochromatic scale [[7L 5s (4/1-equivalent)|7L 5s &lang;4/1&rang;]] (a very xenharmonic variety of [[detempering|detempered]] whole tone scale). It also follows that the [[Ed4]]s which [[support]] meansquared have the same number of tones as the [[EDO]]s which support [[meantone]] – [[7ed4]], 12ed4 ([[6edo]]), [[19ed4]], 26ed4 ([[13edo]]), [[31ed4]] and so on. Meansquared is supported by both nonoctave odd Ed4s and even Ed4s ([[EDO]]s), including ones without conventional meantone temperament (like the previously mentioned 6edo and 13edo).

== Structure ==

Meansquared has an identical structure and interval chain to meantone temperament, but with every interval squared, which stretches them 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 [[2/1|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 minor and major chords are also exotic and tense-sounding. However, it is resemblant in many ways to the somewhat-common whole tone scale, and becomes identical to it in the tuning of [[6edo]].

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.

[[Category:Temperaments]]

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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 [[81/80]]<sup>2</sup>) in the 4.9.25 subgroup. The name was first coined by [[User:CompactStar|CompactStar]] in 2023.  Meansquared in the 4.9.25 subgroup is an [[sane and insane temperaments|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) stretched by a stretch factor of exactly 2, and it follows that it is assosicated with the [[macrodiatonic and microdiatonic scales|macrodiatonic]] scale [[5L 2s (4/1-equivalent)|5L 2s&lang;4/1&rang;]] and the more melodically usable macrochromatic scale [[7L 5s (4/1-equivalent)|7L 5s &lang;4/1&rang;]], which constitutes a very xenharmonic variety of [[detempering|detempered]] whole tone scale. It also follows that the [[Ed4]]s which [[support]] meansquared have the same number of tones as the [[EDO]]s which support [[meantone]] – [[7ed4]], 12ed4 ([[6edo]]), [[19ed4]], 26ed4 ([[13edo]]), [[31ed4]] and so on. Meansquared is supported by both nonoctave odd Ed4s and even Ed4s ([[EDO]]s), including ones without conventional meantone temperament (like the previously mentioned 6edo and 13edo). 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 [[2/1|octave]], like how in 12edo the tritones are conflated to produce the [[2edo]] tritone.
+This temperament is  [[meantone]] temperament with all intervals (including octaves) stretched by a factor of exactly ] It follows that it generates the [[macrodiatonic and microdiatonic scales|macrodiatonic]] scale [[5L 2s (4/1-equivalent)|5L 2s&lang;4/1&rang;]] and the macrochromatic scale [[7L 5s (4/1-equivalent)|7L 5s &lang;4/1&rang;]] (a very xenharmonic variety of [[detempering|detempered]] whole tone scale). It also follows that the [[Ed4]]s which [[support]] meansquared have the same number of tones as the [[EDO]]s which support [[meantone]] – [[7ed4]], 12ed4 ([[6edo]]), [[19ed4]], 26ed4 ([[13edo]]), [[31ed4]] and so on. Meansquared is supported by both nonoctave odd Ed4s and even Ed4s ([[EDO]]s), including ones without conventional meantone temperament (like the previously mentioned 6edo and 13edo).
+== Structure ==
+Meansquared has an identical structure and interval chain to meantone temperament, but with every interval squared, which stretches them 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 [[2/1|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 minor and major chords are also exotic and tense-sounding. However, it is resemblant in many ways to the somewhat-common whole tone scale, and becomes identical to it in the tuning of [[6edo]].
-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.
 [[Category:Temperaments]]
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