Hemimean clan: Difference between revisions

[[Optimal tuning]] ([[POTE]]): 2 = 1\1, ~28/25 = 193.772

=== Mediantone ===

In the no-3's [[11-limit]], there is a natural extension to prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. Mediantone is named after its whole tone generator serving as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[10/9]][[~]][[9/8]]. We will cover related logics as they are relevant to the subgroup shown, such as undecimal meantone being the result of tempering 81/80 to find prime 3 so that the complexity of all mappings except 2 (the period) and 3 (the new gen) doubles.

Subgroup: 2.5.7.11 (no-3's [[11-limit]])

Comma list: [[3136/3125]], [[176/175]]

{{Mapping|legend=2| 1 0 -3 -7 | 0 2 5 9 }}

Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.428

{{Optimal ET sequence|legend=1| 6, 19e, 25, 31, 37 }}

==== 2.5.7.11.13 ====

In the no-3's [[13-limit]], this temperament is identical to [[roulette]], so its higher-limit extensions may be seen as extensions of roulette if one prefers. It is also the same as [[hemiwur]] without a mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators, and not by an insignificant amount, but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned considerably flat, it makes sense to equate two of it with [[13/10]], as this is also how we find [[22/17]] in the no-3's 17-limit. The mapping of 13 increases the [[badness]] of the temperament as a result.

Subgroup: 2.5.7.11.13 (no-3's [[13-limit]])

Comma list: [[3136/3125]], [[176/175]], [[640/637]]

{{Mapping|legend=2| 1 0 -3 -7 13 | 0 2 5 9 -8 }}

Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.569

{{Optimal ET sequence|legend=1| 6, 25, 31, 37 }}

==== 2.5.7.11.13.17 ====

Subgroup: 2.5.7.11.13.17 (no-3's [[17-limit]])

Comma list: [[3136/3125]], [[176/175]], [[640/637]], [[221/220]]

{{Mapping|legend=2| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}

Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.887

{{Optimal ET sequence|legend=1| 6h, 31gh, 37, 80, 117d }}

==== 2.5.7.11.13.17.19 ====

In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank 2 version of [[orion]] with a mapping for 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of [[~]][[19/17]][[~]][[28/25]] as ([[~]][[3/2]])² / [[2/1|2]]. (It doesn't work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway, a notable patent tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.)

Subgroup: 2.5.7.11.13.17.19 (no-3's [[19-limit]])

Comma list: [[3136/3125]], [[176/175]], [[640/637]], [[221/220]], [[476/475]]

{{Mapping|legend=2| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}

Optimal tuning ([[CWE]]): 2 = 1\1, ~19/17 = 194.927

{{Optimal ET sequence|legend=1| 6h, 31gh, 37, 80 }}