Hemimean clan: Difference between revisions

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=== Roulette ===

In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[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. This whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]]. We will cover related logics as they are relevant to the subgroup shown, such as undecimal meantone tempering out 81/80 to find prime 3 so that the complexity of all mappings except 2 (the period) and 3 (the new gen) doubles.

In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[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. This whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]].

In the no-3's [[13-limit]], this temperament is [[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.

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 === Roulette ===
-In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[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. This whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]]. We will cover related logics as they are relevant to the subgroup shown, such as undecimal meantone tempering out 81/80 to find prime 3 so that the complexity of all mappings except 2 (the period) and 3 (the new gen) doubles.
+In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[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. This whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]].
 In the no-3's [[13-limit]], this temperament is [[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.