Hemimean clan: Difference between revisions

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=== 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 3 doubles.

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]])

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 === 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 3 doubles.
+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]])