Hemimean clan: Difference between revisions

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== Didacus ==

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 is equivalent to finding [[11/4]] as ([[7/5]])3 In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in didacus and its extension to the no-3's 13-limit called roulette only the latter interpretation is relevant.

[[Subgroup]]: 2.5.7

[[Comma list]]: 3136/3125

[[Comma list]]: [[3136/3125]]

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[[Badness]] (Dirichlet): 0.091

=== ~~Roulette ===~~

=== 2.5.7.11 subgroup ===

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.

Subgroup: 2.5.7.11

Comma list: 176/175, 1375/1372

Comma list: [[176/175]], [[1375/1372]]

Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}

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Badness (Dirichlet): 0.195

==== ~~2.5~~.7.11.~~13 subgroup ====~~

==== Roulette ====

Roulette is essentially [[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 extension called mediantone. The mapping of 13 increases the [[badness]] of the temperament as a result.

Subgroup: 2.5.7.11.13

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==== Mediantone ====

Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[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]].

Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[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]].

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 primes 11 and 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 does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway, a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.

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 primes 11 and 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 does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val 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

Comma list: 176/175, 640/637, 221/220, 1375/1372

Comma list: [[176/175]], [[640/637]], [[221/220]], [[1375/1372]]

Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}

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Badness (Dirichlet): 0.612

==== 2.5.7.11.13.17.19 subgroup ====

===== 2.5.7.11.13.17.19 subgroup =====

Subgroup: 2.5.7.11.13.17.19

Comma list: 176/175, 640/637, 221/220, 476/475, 1375/1372

Comma list: [[176/175]], [[640/637]], [[221/220]], [[476/475]], [[1375/1372]]

Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}

@@ Line 25: / Line 25: @@
 == Didacus ==
+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 is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup> In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in didacus and its extension to the no-3's 13-limit called roulette only the latter interpretation is relevant.
 [[Subgroup]]: 2.5.7
-[[Comma list]]: 3136/3125
+[[Comma list]]: [[3136/3125]]
 {{Mapping|legend=2| 1 0 -3 | 0 2 5 }}
@@ Line 45: / Line 47: @@
 [[Badness]] (Dirichlet): 0.091
-=== Roulette ===
+=== 2.5.7.11 subgroup ===
-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.
 Subgroup: 2.5.7.11
-Comma list: 176/175, 1375/1372
+Comma list: [[176/175]], [[1375/1372]]
 Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}
@@ Line 67: / Line 65: @@
 Badness (Dirichlet): 0.195
-==== 2.5.7.11.13 subgroup ====
+==== Roulette ====
+Roulette is essentially [[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 extension called mediantone. The mapping of 13 increases the [[badness]] of the temperament as a result.
 Subgroup: 2.5.7.11.13
@@ Line 87: / Line 87: @@
 ==== Mediantone ====
-Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[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]].
+Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[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]].
-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 primes 11 and 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]])<sup>2</sup>/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway, a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.
+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 primes 11 and 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]])<sup>2</sup>/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val 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
-Comma list: 176/175, 640/637, 221/220, 1375/1372
+Comma list: [[176/175]], [[640/637]], [[221/220]], [[1375/1372]]
 Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}
@@ Line 105: / Line 105: @@
 Badness (Dirichlet): 0.612
-==== 2.5.7.11.13.17.19 subgroup ====
+===== 2.5.7.11.13.17.19 subgroup =====
 Subgroup: 2.5.7.11.13.17.19
-Comma list: 176/175, 640/637, 221/220, 476/475, 1375/1372
+Comma list: [[176/175]], [[640/637]], [[221/220]], [[476/475]], [[1375/1372]]
 Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}