Hemimean clan: Difference between revisions

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[[~~Gencom~~]]: [2 56/25; 3136/3125]

: [[gencom]]: [2 56/25; 3136/3125]

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

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{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}

Badness (Dirichlet): 0.091

[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents

[[Badness]] (Dirichlet): 0.091

=== 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 [[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.

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]])<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.

Subgroup: 2.5.7.11

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==== 2.5.7.11.13 subgroup ====

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: 2.5.7.11.13

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==== 2.5.7.11.13.17.19 subgroup ====

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]])<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.

Subgroup: 2.5.7.11.13.17.19

@@ Line 35: / Line 35: @@
 {{Mapping|legend=3| 1 0 0 -3 | 0 0 2 5 }}
-[[Gencom]]: [2 56/25; 3136/3125]
+: [[gencom]]: [2 56/25; 3136/3125]
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772
@@ Line 41: / Line 41: @@
 {{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}
-Badness (Dirichlet): 0.091
+[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
+[[Badness]] (Dirichlet): 0.091
 === 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 [[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.
+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]])<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.
 Subgroup: 2.5.7.11
@@ Line 64: / Line 70: @@
 ==== 2.5.7.11.13 subgroup ====
-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: 2.5.7.11.13
@@ Line 100: / Line 104: @@
 ==== 2.5.7.11.13.17.19 subgroup ====
-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]])<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.
 Subgroup: 2.5.7.11.13.17.19