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strong, weak, and subgroup; the trinity
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* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]


= 2.5.7 subgroup =
== Didacus ==
== Didacus ==
{{main|Didacus}}
{{main|Didacus}}
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[[Badness]] (Dirichlet): 0.091
[[Badness]] (Dirichlet): 0.091


=== Undecimal didacus ===
= Strong extensions =
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 undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
 
Subgroup: 2.5.7.11
 
Comma list: [[176/175]], [[1375/1372]]
 
Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}
 
: sval mapping generators: ~2, ~56/25
 
Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428
 
Optimal ET sequence: {{Optimal ET sequence| 6, 19e, 25, 31, 37 }}
 
RMS error: 0.5567 cents
 
Badness (Dirichlet): 0.195
 
==== Tridecimal didacus ====
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no 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 a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
 
Subgroup: 2.5.7.11.13
 
Comma list: 176/175, 640/637, 1375/1372
 
Sval mapping: {{mapping| 1 0 -3 -7 13 | 0 2 5 9 -8 }}
 
: sval mapping generators: ~2, ~56/25
 
Gencom mapping: {{mapping| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}
 
: gencom: [2 28/25; 176/175 1375/1372 640/637]
 
Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594
 
Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
 
Badness (Dirichlet): 0.324
 
===== 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]].
 
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]]
 
Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}
 
: sval mapping generators: ~2, ~56/25
 
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.887
 
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d }}
 
Badness (Dirichlet): 0.612
 
====== 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]]
 
Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}
 
: sval mapping generators: ~2, ~56/25
 
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
 
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80 }}
 
Badness (Dirichlet): 0.618
 
===== Roulette =====
{{See also | Chromatic pairs #Roulette }}
 
Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.
 
Subgroup: 2.5.7.11.13.17
 
Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]
 
Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}
 
: sval mapping generators: ~2, ~28/25
 
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285
 
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
 
Badness (Dirichlet): 0.685
 
====== 2.5.7.11.13.17.19 subgroup ======
Subgroup: 2.5.7.11.13.17.19
 
Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]
 
Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}
 
: sval mapping generators: ~2, ~28/25
 
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259
 
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
 
Badness (Dirichlet): 0.676
 
=== Rectified hebrew ===
{{Main| Rectified hebrew }}
 
Rectified hebrew (37 &amp; 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three.
 
Subgroup: 2.5.7.13
 
Comma list: 3136/3125, 4394/4375
 
Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}
 
: sval mapping generators: ~2, ~26/25
 
Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086
 
{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}


== Hemiwürschmidt ==
== Hemiwürschmidt ==
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Badness: 0.041603
Badness: 0.041603
= Weak extensions =


== Semisept ==
== Semisept ==
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Badness: 0.042744
Badness: 0.042744
= Subgroup extensions =
== Undecimal 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 undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
Subgroup: 2.5.7.11
Comma list: [[176/175]], [[1375/1372]]
Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}
: sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428
Optimal ET sequence: {{Optimal ET sequence| 6, 19e, 25, 31, 37 }}
RMS error: 0.5567 cents
Badness (Dirichlet): 0.195
=== Tridecimal didacus ===
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no 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 a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.
Subgroup: 2.5.7.11.13
Comma list: 176/175, 640/637, 1375/1372
Sval mapping: {{mapping| 1 0 -3 -7 13 | 0 2 5 9 -8 }}
: sval mapping generators: ~2, ~56/25
Gencom mapping: {{mapping| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}
: gencom: [2 28/25; 176/175 1375/1372 640/637]
Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594
Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
Badness (Dirichlet): 0.324
==== 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]].
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]]
Sval mapping: {{mapping| 1 0 -3 -7 13 -18 | 0 2 5 9 -8 19 }}
: sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.887
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d }}
Badness (Dirichlet): 0.612
===== 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]]
Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}
: sval mapping generators: ~2, ~56/25
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927
Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80 }}
Badness (Dirichlet): 0.618
==== Roulette ====
{{See also | Chromatic pairs #Roulette }}
Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)<sup>2</sup> to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])<sup>3</sup> is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.
Subgroup: 2.5.7.11.13.17
Comma list: [[170/169]], [[176/175]], [[640/637]], [[1375/1372]]
Sval mapping: {{mapping| 1 2 2 2 5 7 | 0 2 5 9 -8 -18 }}
: sval mapping generators: ~2, ~28/25
Optimal tuning (CWE): ~2 = 1\1, ~28/25 = 194.285
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
Badness (Dirichlet): 0.685
===== 2.5.7.11.13.17.19 subgroup =====
Subgroup: 2.5.7.11.13.17.19
Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]
Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}
: sval mapping generators: ~2, ~28/25
Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
Badness (Dirichlet): 0.676
== Rectified hebrew ==
{{Main| Rectified hebrew }}
Rectified hebrew (37 &amp; 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three.
Subgroup: 2.5.7.13
Comma list: 3136/3125, 4394/4375
Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}
: sval mapping generators: ~2, ~26/25
Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086
{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}


== Isra ==
== Isra ==
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