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The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}. The head of this clan is the 2.5.7 [[subgroup temperament]] didacus, generated by a tempered hemithird of [[28/25]]. Two generator steps make [[5/4]] and five make [[7/4]].  
{{Technical data page}}
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].


The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.  
The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.  
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Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.  
Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.  


Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. A notable subgroup extension of didacus is [[#Roulette|roulette]]. Discussed elsewhere are
Didacus has canonical subgroup extensions to primes 11 and 13, at [[#Undecimal didacus|undecimal didacus]]. Other subgroup extensions include rectified hebrew and isra.
 
Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. Discussed elsewhere are
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
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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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[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents


[[Badness]] (Dirichlet): 0.091
[[Badness]] (Sintel): 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.
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong extensions
|-
! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*
|-
! Temperament !! 5-limit generator location
|-
| [[#Hemiwürschmidt|Hemiwürschmidt]] || [[Würschmidt family#Würschmidt|Würschmidt]] || +2 || +16 || ↓ [[31edo|31]]
|-
| [[#Hemithirds|Hemithirds]] || [[Luna family#Luna|Luna]] || +1 || -15 || ↑ 31 <br /> ↓ [[25edo|25]]  
|-
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
<nowiki />* Defined by intersection with other documented extensions


Subgroup: 2.5.7.11
== Hemiwürschmidt ==
 
''[[#Strong extensions|Return to the map]]''
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 ==
{{See also| Würschmidt family }}
{{See also| Würschmidt family }}


'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, {{multival| 16 2 5 40 -39 -49 -48 28 … }}.
'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Mapping generators: ~2, ~25/14
Mapping generators: ~2, ~25/14
{{Multival|legend=1| 16 2 5 -34 -37 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
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{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}


{{todo|verify this & get badness|inline=1}}
Badness (Sintel): 0.304


=== 11-limit ===
=== 11-limit ===
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=== Quadrawürschmidt ===
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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== Hemithirds ==
== Hemithirds ==
''[[#Strong extensions|Return to the map]]''
{{Main| Hemithirds }}
{{Main| Hemithirds }}


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{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
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* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
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== Spell ==
== Spell ==
''[[#Strong extensions|Return to the map]]''
{{See also| Magic family }}
{{See also| Magic family }}


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{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
{{Multival|legend=1| 10 2 5 -20 -20 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
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Badness: 0.041603
Badness: 0.041603
= Weak extensions =


== Semisept ==
== Semisept ==
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: mapping generators: ~2, ~75/49
: mapping generators: ~2, ~75/49
{{Multival|legend=1| 17 6 15 -30 -24 18 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
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: mapping generators: ~2, ~48/35
: mapping generators: ~2, ~48/35
{{Multival|legend=1| 27 8 20 -50 -44 24 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
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{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
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{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Multival|legend=1| 29 16 40 -42 -18 48 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
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{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Multival|legend=1| 15 22 55 0 45 66 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
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{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
{{Multival|legend=1| 31 24 60 -34 8 72 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
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{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Multival|legend=1| 11 30 75 22 88 90 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
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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 (Sintel): 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 (Sintel): 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: 2.5.7.11.13.17.19
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, 117d* }}
<nowiki />* 117d only appears without prime 19
Badness (Sintel): 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: 2.5.7.11.13.17.19
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 (Sintel): 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 ==
Line 1,103: Line 1,087:


[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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