@@ Line 1: / Line 1: @@
-This page discusses miscellaneous rank-2 temperaments tempering out {{monzo|-5 2 2 -1}} = [[225/224]], the marvel comma or septimal kleisma.
+{{Technical data page}}
+This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] [[225/224]], the marvel comma or septimal kleisma.
 Temperaments considered in families and clans are:
-* ''[[Pelogic]]'' → [[Pelogic family #Pelogic|Pelogic family]] (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3)
+* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
-* [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3)
+* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
-* [[Garibaldi temperament|Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+3125/3087, generated by the fifth with 5/4 mapped to the d4)
+* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
-* [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]] (+50/49 or 64/63, generated by the fifth with a semioctave period)
+* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
-* ''[[Sharp]]'' → [[Dicot family #Sharp|Dicot family]] (+25/24 or 28/27, fifth sliced in two)
+* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
-* ''[[Immune]]'' → [[Immunity family #Immune|Immunity family]] (+781250/750141, twelfth sliced in two)
+* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
-* ''[[August]]'' → [[Augmented family #August|Augmented family]] (+36/35 or 128/125, generated by the fifth with a 1/3-octave period)
+* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
-* ''[[Fog]]'' → [[Misty family #Fog|Misty family]] (+156250/151263, generated by the fifth with a 1/3-octave period)
+* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
-* [[Negri]] → [[Slendro clan #Negri|Slendro clan]] (+49/48, fourth sliced in four)
+* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
-* [[Magic]] → [[Magic family #Magic|Magic family]] (+245/243, twelfth sliced in five)
+* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
-* ''[[Passive]]'' → [[Passion family #Passive|Passion family]] (+256/245, fourth sliced in five)
+* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
-* ''[[Quintapole]]'' → [[Quintaleap family #Quintapole|Quintaleap family]] (+7812500/7411887, fourth sliced in five)
+* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
-* ''[[Houborizic]]'' → [[Amity family #Houborizic|Amity family]] (+1250000/1240029, eleventh sliced in five)
+* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
-* ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
+* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
-* [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six)
+* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
-* [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six)
+* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
-* ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six)
+* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
-* [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven)
+* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
-* ''[[Snipes]]''  → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven)
+* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
-* ''[[Escapade]]'' → [[Escapade family #Escapade|Escapade family]] (+65625/65536, fourth sliced in nine)
+* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
-* ''[[Decic]]'' → [[Cloudy clan #Decic|Cloudy clan]] (+16807/16384, generated by the fifth with a 1/10-octave period)
+* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
-* ''[[Amavil]]'' → [[Mabila family #Amavil|Mabila family]] (+17496/16807, four octaves and a fourth sliced in ten)
+* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
-* ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] (+1071875/1062882, fifth sliced in eleven)
+* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
-* ''[[Compton]]'' → [[Compton family #Compton|Compton family]] (+250047/250000, generated by the classical major third with a 1/12-octave period)
+* [[Wizard]] (+118098/117649) → [[Stearnsmic clan #Wizard|Stearnsmic clan]]
-* ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen)
+* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
-* ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
+* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
-* ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+94143178827/91913281250, fifth sliced in twenty)
+* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
+* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
-Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.
+Considered below are tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]].
-Since (5/4)<sup>2</sup> = 225/224 × 14/9, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
+Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
 The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
-Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
+Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is {{nowrap|(16/9)/(5/4) {{=}} 64/45}}. Setting this equal to 10/7 gives {{nowrap|(10/7)/(64/45) {{=}} 225/224}}.)
-== Wizard ==
+== Tritonic ==
-{{Main| Wizard }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
+Tritonic tempers out [[50421/50000]] and may be described as the {{nowrap| 29 & 31 }} temperament. It splits the [[6/1|6th]] [[harmonic]] into five generators of [[~]][[10/7]] [[tritone]]s, hence the name. Its [[ploidacot]] is beta-pentacot. [[60edo]] may be used as a tuning, which in the 11-limit entails the 60e val.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 118098/117649
+[[Comma list]]: 225/224, 50421/50000
-{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
+{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
+: mapping generators: ~2, ~10/7
-: mapping generators: ~1225/864, ~245/216
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
+: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
+: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}
-[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~5/4 = 383.256 (~245/216 = 216.744)
+{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
-{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
-[[Badness]]: 0.040846
-Scales: [[wizard22]]
+[[Badness]] (Sintel): 1.20
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 4000/3993
+Comma list: 121/120, 225/224, 441/440
-Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
+{{Mapping|legend=0| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.232 (~25/22 = 216.768)
+Optimal tunings:
+* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
-Badness: 0.018539
+Badness (Sintel): 0.782
-Scales: [[wizard22]]
+==== 13-limit ====
-==== Lizard ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 351/350, 364/363, 385/384
+Comma list: 105/104, 121/120, 196/195, 275/273
-Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
-Optimal tuning (POTE): ~55/39 = 1\2, ~5/4 = 383.389 (~25/22 = 216.711)
+Optimal tunings:
+* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 122, 194df }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e }}
-Badness: 0.021781
+Badness (Sintel): 0.950
-===== 17-limit =====
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
+Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
-Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.381 (~17/15 = 216.619)
+Optimal tunings:
+* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness: 0.014536
+Badness (Sintel): 0.973
-===== 19-limit =====
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
+Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
-Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.477 (~17/15 = 216.523)
+Optimal tunings:
+* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
-{{Optimal ET sequence|legend=1| 22h, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness: 0.015702
+Badness (Sintel): 1.03
-==== Gizzard ====
+==== 23-limit ====
-Subgroup: 2.3.5.7.11.13
+Subgroup: 2.3.5.7.11.13.17.19.23
-Comma list: 225/224, 325/324, 385/384, 1573/1568
+Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
-Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
+{{Mapping|legend=0| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.170 (~25/22 = 216.830)
+Optimal tunings:
+* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
-{{Optimal ET sequence|legend=1| 72, 166, 238cf }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness: 0.020252
+Badness (Sintel): 1.04
-===== 17-limit =====
+=== Tritoni ===
-Subgroup: 2.3.5.7.11.13.17
+Subgroup: 2.3.5.7.11
+Comma list: 225/224, 385/384, 27783/27500
+{{Mapping|legend=0| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
-Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
+Optimal tunings:
+* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
-Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
+{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.175 (~25/22 = 216.825)
+Badness (Sintel): 1.50
-{{Optimal ET sequence|legend=1| 72, 166g, 238cfg }}
+== Septimin ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
-Badness: 0.013624
+Septimin may be described as the {{nowrap| 41 & 50 }} temperament. It is generated by a septimal minor third ([[7/6]]), which gives rise to the name, but the generator can be taken to be the [[octave complement]], [[12/7]], such that eleven of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus eta-hendecacot. [[91edo]] may be recommended as a tuning.
-===== 19-limit =====
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
+[[Comma list]]: 225/224, 84035/82944
-Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
+{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
+: mapping generators: ~2, ~12/7
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.138 (~17/15 = 216.862)
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.2452{{c}}, ~12/7 = 937.3394{{c}}
+: [[error map]]: {{val| +1.245 +0.062 -1.633 -1.658 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~12/7 = 936.4036{{c}}
+: error map: {{val| 0.000 -1.516 -4.735 -4.790 }}
-{{Optimal ET sequence|legend=1| 72, 94, 166g }}
+{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}
-Badness: 0.014810
+[[Badness]] (Sintel): 1.38
-=== Mage ===
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 99/98, 176/175, 1331/1296
+Comma list: 225/224, 245/242, 385/384
+{{Mapping|legend=0| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
+Optimal tunings:
+* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
+{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
+Badness (Sintel): 1.04
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 105/104, 144/143, 196/195, 245/242
+{{Mapping|legend=0| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
-Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
+Optimal tunings:
+* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
-Optimal tuning (POTE): ~77/54 = 1\2, ~5/4 = 383.124 (~55/48 = 216.876)
+{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
-{{Optimal ET sequence|legend=1| 22, 50e, 72ee, 94ee }}
+Badness (Sintel): 0.955
-Badness: 0.057799
+== Merman ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
-== Tritonic ==
+Merman may be described as the {{nowrap| 41 & 43 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, seven generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is gamma-heptacot.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
+The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 50421/50000
+[[Comma list]]: 225/224, 2500000/2470629
-{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}
+{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
+: [[error map]]: {{val| +0.390 -0.435 -1.630 +1.634 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 614.4073{{c}}
+: error map: {{val| 0.000 -1.104 -2.423 +0.657 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 580.286
+{{Optimal ET sequence|legend=1| 41, 84, 125 }}
-{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
-[[Badness]]: 0.047578
+[[Badness]] (Sintel): 1.39
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 121/120, 225/224, 441/440
+Comma list: 225/224, 441/440, 1344/1331
-Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}
+{{Mapping|legend=0| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.267
+Optimal tunings:
+* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
-{{Optimal ET sequence|legend=1| 29, 31, 60e }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e }}
-Badness: 0.023659
+Badness (Sintel): 1.20
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 121/120, 196/195, 275/273
+Comma list: 144/143, 225/224, 364/363, 441/440
-Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}
+{{Mapping|legend=0| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.108
+Optimal tunings:
+* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
-{{Optimal ET sequence|legend=1| 29, 31, 60e, 151cde }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
-Badness: 0.022993
+Badness (Sintel): 1.14
-=== Tritoni ===
+=== Mermaid ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 27783/27500
+Comma list: 225/224, 385/384, 532400/531441
+{{Mapping|legend=0| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
+Optimal tunings:
+* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
+{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
+Badness (Sintel): 1.46
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
+Comma list: 225/224, 325/324, 385/384, 10648/10647
+{{Mapping|legend=0| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
-Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}
+Optimal tunings:
+* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.389
+{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
-{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}
+Badness (Sintel): 1.47
-Badness: 0.045456
+== Slender ==
+Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.
-== Septimin ==
+The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 84035/82944
+[[Comma list]]: 225/224, 589824/588245
-{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}
+{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
+: mapping generators: ~2, ~49/48
-{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3816{{c}}, ~49/48 = 38.4256{{c}}
+: [[error map]]: {{val| +0.382 -0.725 -1.295 +1.765 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 38.4079{{c}}
+: error map: {{val| 0.000 -1.257 -2.235 +0.727 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 263.632
+{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
-{{Optimal ET sequence|legend=1| 41, 91, 132d }}
+[[Badness]] (Sintel): 1.44
-[[Badness]]: 0.054502
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 2401/2376
+Comma list: 225/224, 385/384, 1331/1323
-Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}
+{{Mapping|legend=0| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.634
+Optimal tunings:
+* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
-{{Optimal ET sequence|legend=1| 41, 91, 223cdef }}
+{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
-Badness: 0.031309
+Badness (Sintel): 0.838
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 144/143, 196/195, 245/242
+Comma list: 225/224, 275/273, 385/384, 1331/1323
+{{Mapping|legend=0| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
-Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}
+Optimal tunings:
+* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.700
+{{Optimal ET sequence|legend=0| 31, 63, 94 }}
-{{Optimal ET sequence|legend=1| 41, 91 }}
+Badness (Sintel): 1.07
-Badness: 0.023117
+== Triton ==
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
-== Merman ==
+Triton may be described as the {{nowrap| 19 & 21 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, three generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is alpha-tricot.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 2500000/2470629
+[[Comma list]]: 225/224, 1029/1000
-{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}
+{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
+: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
+: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 585.585
+{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
-{{Optimal ET sequence|legend=1| 41, 84, 125 }}
-[[Badness]]: 0.055078
+[[Badness]] (Sintel): 1.50
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 441/440, 1344/1331
+Comma list: 45/44, 56/55, 1029/1000
-Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}
+{{Mapping|legend=0| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.606
+Optimal tunings:
+* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
-{{Optimal ET sequence|legend=1| 41, 84, 125e }}
+{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
-Badness: 0.036383
+Badness (Sintel): 1.51
-=== 13-limit ===
-Subgroup: 2.3.5.7.11.13
-Comma list: 144/143, 225/224, 364/363, 441/440
-Mapping: {{mapping| 1 5 -5 -5 2 12 | 0 -7 15 16 3 -17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.657
-{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}
-Badness: 0.027544
-== Slender ==
-Slender (31 &amp; 32) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma. This temperament has a generator of [[49/48]], 3 of which equal marvel's 16/15~15/14, and 10 generators is 5/4.
+== Marvolo ==
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 589824/588245
+[[Comma list]]: 225/224, 156250000/155649627
-{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
+{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
+: mapping generators: ~2, ~21/20
-{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/48 = 38.413
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.7714{{c}}, ~21/20 = 83.4014{{c}}
+: [[error map]]: {{val| +0.772 -0.820 -0.916 +0.381 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.3640{{c}}
+: error map: {{val| 0.000 -2.139 -2.398 -1.362 }}
-{{Optimal ET sequence|legend=1| 31, 94, 125 }}
+{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
-[[Badness]]: 0.056934
+[[Badness]] (Sintel): 2.11
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 1331/1323
+Comma list: 225/224, 441/440, 4000/3993
-Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
+{{Mapping|legend=0| 1 2 1 1 2 | 0 -6 19 26 21 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.387
+Optimal tunings:
+* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}
-{{Optimal ET sequence|legend=1| 31, 63, 94, 125 }}
+{{Optimal ET sequence|legend=0| 29, 43, 72 }}
-Badness: 0.025342
+Badness (Sintel): 0.958
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 275/273, 385/384, 1331/1323
+Comma list: 169/168, 225/224, 364/363, 441/440
-Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
+{{Mapping|legend=0| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.314
+Optimal tunings:
+* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
-{{Optimal ET sequence|legend=1| 31, 63, 94 }}
+{{Optimal ET sequence|legend=0| 29, 43, 72 }}
-Badness: 0.025913
+Badness (Sintel): 0.887
-== Triton ==
+=== 17-limit ===
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
-[[Subgroup]]: 2.3.5.7
+{{Mapping|legend=0| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
-[[Comma list]]: 225/224, 1029/1000
+Optimal tunings:
+* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
-{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
+Badness (Sintel): 0.760
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 568.865
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
+Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
-[[Badness]]: 0.059245
+{{Mapping|legend=0| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
-=== 11-limit ===
+Optimal tunings:
-Subgroup: 2.3.5.7.11
+* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
-Comma list: 45/44, 56/55, 1029/1000
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
+Badness (Sintel): 0.895
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 569.144
+== Decic ==
+{{Main| Decic }}
-{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}
+Named by [[Xenllium]] in 2021, decic tempers out 16807/16384, the [[cloudy comma]], and {{monzo| 11 -10 -10 10 }}, the [[linus comma]], in addition to the marvel comma. It may be described as the {{nowrap| 50 & 60 }} temperament, with a period of 1/10 octave and a [[ploidacot]] signature of decaploid monocot. It is [[support]]ed by [[10edo|10-]], [[50edo|50-]], and [[60edo]].
-Badness: 0.045675
+It can be extended to the 11-, 13-, and 17-limit by adding [[385/384]], [[105/104]], and [[170/169]] to the comma list in this order.
-== Submajor ==
+[[Subgroup]]: 2.3.5.7
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 69198046875/68719476736
+[[Comma list]]: 225/224, 16807/16384
-{{Mapping|legend=1| 1 4 -1 | 0 -8 11 }}
+{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
+: mapping generators: ~15/14, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10125/8192 = 362.321
+[[Optimal tuning]]s:
+* [[WE]]: ~15/14 = 120.1837{{c}}, ~3/2 = 699.7654{{c}} (~49/48 = 21.3366{{c}})
+: [[error map]]: {{val| +1.837 -0.353 -0.753 -3.683 }}
+* [[CWE]]: ~15/14 = 120.0000{{c}}, ~3/2 = 698.8236{{c}} (~49/48 = 21.1764{{c}})
+: error map: {{val| 0.000 -3.131 -5.137 -8.826 }}
-{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
+{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
-[[Badness]]: 0.130236
+[[Badness]] (Sintel): 2.26
-=== 7-limit ===
+=== 11-limit ===
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11
-[[Comma list]]: 225/224, 51200/50421
+Comma list: 225/224, 385/384, 3087/3025
-{{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }}
+{{Mapping|legend=0| 10 0 39 28 3 | 0 1 -1 0 2 }}
-{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1406{{c}}, ~3/2 = 697.6075{{c}} (~56/55 = 23.2360{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.0142{{c}} (~56/55 = 22.9858{{c}})
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }}
+Badness (Sintel): 2.11
-[[Badness]]: 0.060533
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-=== 11-limit ===
+Comma list: 105/104, 144/143, 196/195, 2200/2197
-Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 6655/6561
+{{Mapping|legend=0| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
-Mapping: {{mapping| 1 4 -1 1 11 | 0 -8 11 6 -25 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1166{{c}}, ~3/2 = 697.6705{{c}} (~78/77 = 23.0289{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.1492{{c}} (~78/77 = 22.8508{{c}})
-Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cde }}
+Badness (Sintel): 1.52
-Badness: 0.050582
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-==== 13-limit ====
+Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
-Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 225/224, 275/273, 385/384
+{{Mapping|legend=0| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 1 }}
-Mapping: {{mapping| 1 4 -1 1 11 4 | 0 -8 11 6 -25 -1 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1262{{c}}, ~3/2 = 697.8185{{c}} (~78/77 = 22.9388{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 697.2757{{c}} (~78/77 = 22.7243{{c}})
-Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cdef }}
+Badness (Sintel): 1.28
-Badness: 0.027689
+=== Splendecic ===
+Splendecic (50 & 60e) is an alternative extension of decic, tempering out 1617/1600, 2401/2376 and 4375/4356 in the 11-limit. As a temperament of the [[fantastic]] rank-3 temperament, its name is a portmanteau of ''splendid'' and ''decic''.
-=== Interpental ===
 Subgroup: 2.3.5.7.11
-Comma list: 99/98, 176/175, 51200/50421
+Comma list: 225/224, 1617/1600, 2401/2376
-Mapping: {{mapping| 1 4 -1 1 -5 | 0 -8 11 6 28 }}
+{{Mapping|legend=0| 10 0 39 28 82 | 0 1 -1 0 -3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418
+Optimal tunings:
+* WE: ~15/14 = 120.1874{{c}}, ~3/2 = 699.6085{{c}} (~99/98 = 21.5156{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3531{{c}} (~99/98 = 21.6469{{c}})
-{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
+{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de, 170cddee }}
-Badness: 0.051806
+Badness (Sintel): 1.98
 ==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 99/98, 169/168, 176/175, 640/637
+Comma list: 105/104, 196/195, 1001/1000, 1188/1183
-Mapping: {{mapping| 1 4 -1 1 -5 4 | 0 -8 11 6 28 -1 }}
+{{Mapping|legend=0| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402
+Optimal tunings:
+* WE: ~15/14 = 120.1565{{c}}, ~3/2 = 699.2756{{c}} (~91/90 = 21.6631{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.2480{{c}} (~91/90 = 21.7520{{c}})
-{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
+{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}
-Badness: 0.029680
+Badness (Sintel): 1.57
-== Marvolo ==
+==== 17-limit ====
-[[Subgroup]]: 2.3.5.7
+Subgroup: 2.3.5.7.11.13.17
-[[Comma list]]: 225/224, 156250000/155649627
+Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
-{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
+{{Mapping|legend=0| 10 0 39 28 82 37 25 | 0 1 -1 0 -3 0 1 }}
-{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1571{{c}}, ~3/2 = 699.2892{{c}} (~91/90 = 21.6536{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 698.3144{{c}} (~91/90 = 21.6856{{c}})
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/20 = 83.348
+{{Optimal ET sequence|legend=0| 10e, 50, 60e, 110deg }}
-{{Optimal ET sequence|legend=1| 29, 43, 72, 619bcd, 691bcd }}
+Badness (Sintel): 1.33
-[[Badness]]: 0.083338
+=== Prodecic ===
+Prodecic (50e & 60e) is an alternative extension of decic, tempering out 441/440, 1375/1372 and 4375/4356 in the 11-limit. As a temperament of the [[prodigy]] rank-3 temperament, its name is a portmanteau of ''prodigy'' and ''decic''.
-=== 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 441/440, 4000/3993
+Comma list: 225/224, 441/440, 5929/5832
-Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
+{{Mapping|legend=0| 10 0 39 28 -13 | 0 1 -1 0 3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.340
+Optimal tunings:
+* WE: ~15/14 = 120.2024{{c}}, ~3/2 = 701.3908{{c}} (~55/54 = 19.8237{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.5235{{c}} (~55/54 = 19.4765{{c}})
-{{Optimal ET sequence|legend=1| 29, 43, 72 }}
+{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
-Badness: 0.028965
+Badness (Sintel): 2.20
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 225/224, 364/363, 441/440
+Comma list: 105/104, 196/195, 275/273, 5929/5832
-Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
+{{Mapping|legend=0| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330
+Optimal tunings:
+* WE: ~15/14 = 120.1654{{c}}, ~3/2 = 701.4683{{c}} (~91/90 = 19.5242{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.7175{{c}} (~91/90 = 19.2825{{c}})
-{{Optimal ET sequence|legend=1| 29, 43, 72, 115f }}
+{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
-Badness: 0.021470
+Badness (Sintel): 1.73
-=== 17-limit ===
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
-Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
+Comma list: 105/104, 154/153, 170/169, 196/195, 289/288
-Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330
+{{Mapping|legend=0| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
-{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
+Optimal tunings:
+* WE: ~15/14 = 120.1577{{c}}, ~3/2 = 701.3950{{c}} (~91/90 = 19.5514{{c}})
+* CWE: ~15/14 = 120.0000{{c}}, ~3/2 = 700.6932{{c}} (~91/90 = 19.3068{{c}})
-=== 19-limit ===
+{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
+Badness (Sintel): 1.41
-Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330
-{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
 == Enneaportent ==
@@ Line 530: / Line 624: @@
 {{Mapping|legend=1| 9 0 28 11 | 0 2 -1 2 }}
+: mapping generators: ~2592/2401, ~12005/6912
-{{Multival|legend=1| 18 -9 18 -56 -22 67 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2592/2401 = 133.4174{{c}}, ~12005/6912 = 950.7667{{c}} (~1728/1715 = 16.8452{{c}})
+: [[error map]]: {{val| +0.756 -0.422 -1.395 +0.298 }}
+* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~12005/6912 = 950.2969{{c}} (~1728/1715 = 16.9636{{c}})
+: error map: {{val| 0.000 -1.361 -3.277 -1.565 }}
-[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~5/4 = 383.165
+{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
-{{Optimal ET sequence|legend=1| 9, 63, 72, 495bcd }}
+[[Badness]] (Sintel): 2.37
-[[Badness]]: 0.093679
 === 11-limit ===
@@ Line 544: / Line 641: @@
 Comma list: 225/224, 385/384, 12005/11979
-Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
+{{Mapping|legend=0| 9 0 28 11 24 | 0 2 -1 2 1 }}
-Optimal tuning (POTE): ~121/112 = 1\9, ~5/4 = 383.146
+Optimal tunings:
+* WE: ~121/112 = 133.4071{{c}}, ~210/121 = 950.7131{{c}} (~99/98 = 16.8633{{c}})
+* CWE: ~121/112 = 133.3333{{c}}, ~210/121 = 950.2994{{c}} (~99/98 = 16.9661{{c}})
-{{Optimal ET sequence|legend=1| 9, 63, 72, 423cd, 495bcd }}
+{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
-Badness: 0.030426
+Badness (Sintel): 1.01
 === 13-limit ===
@@ Line 557: / Line 656: @@
 Comma list: 169/168, 225/224, 364/363, 1716/1715
-Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
+{{Mapping|legend=0| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
-Optimal tuning (POTE): ~14/13 = 1\9, ~5/4 = 383.047
+Optimal tunings:
+* WE: ~14/13 = 133.4245{{c}}, ~26/15 = 950.9362{{c}} (~105/104 = 16.9650{{c}})
+* CWE: ~14/13 = 133.3333{{c}}, ~26/15 = 950.4364{{c}} (~99/98 = 17.1031{{c}})
-{{Optimal ET sequence|legend=1| 9, 63, 72, 279cf }}
+{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
-Badness: 0.022322
+Badness (Sintel): 0.922
 == Gracecordial ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Gracecordial]].''
+: ''For the 5-limit version, see [[Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit)]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 573: / Line 674: @@
 {{Mapping|legend=1| 1 0 34 63 | 0 1 -20 -38 }}
+: mapping generators: ~2, ~3
-{{Multival|legend=1| 1 -20 -38 -34 -63 -32 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.4904{{c}}, ~3/2 = 701.1103{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 700.824
+: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.8112{{c}}
+: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}
-{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
+{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
-[[Badness]]: 0.096279
+[[Badness]] (Sintel): 2.44
 === 11-limit ===
@@ Line 587: / Line 691: @@
 Comma list: 225/224, 385/384, 236328125/234365481
-Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
+{{Mapping|legend=0| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.834
+Optimal tunings:
+* WE: ~2 = 1200.5571{{c}}, ~3/2 = 701.1589{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8328{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125, 238c }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}
-Badness: 0.089588
+Badness (Sintel): 2.96
 ==== 13-limit ====
@@ Line 600: / Line 706: @@
 Comma list: 225/224, 325/324, 385/384, 831875/830466
-Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841
+Optimal tunings:
+* WE: ~2 = 1200.6282{{c}}, ~3/2 = 701.2080{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8421{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness: 0.052235
+Badness (Sintel): 2.16
 ==== 17-limit ====
@@ Line 613: / Line 721: @@
 Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.841
+Optimal tunings:
+* WE: ~2 = 1200.5058{{c}}, ~3/2 = 701.1360{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8414{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness: 0.038565
+Badness (Sintel): 1.96
 ==== 19-limit ====
@@ Line 626: / Line 736: @@
 Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842
+Optimal tunings:
+* WE: ~2 = 1200.4418{{c}}, ~3/2 = 701.0999{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8425{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness: 0.028165
+Badness (Sintel): 1.71
 ==== 23-limit ====
@@ Line 639: / Line 751: @@
 Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.843
+Optimal tunings:
+* WE: ~2 = 1200.4641{{c}}, ~3/2 = 701.1145{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8444{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
-Badness: 0.021879
+Badness (Sintel): 1.57
 ==== 29-limit ====
@@ Line 652: / Line 766: @@
 Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.842
+Optimal tunings:
+* WE: ~2 = 1200.4400{{c}}, ~3/2 = 701.0986{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8428{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cde, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
-Badness: 0.018011
+Badness (Sintel): 1.50
 ==== 31-limit ====
@@ Line 665: / Line 781: @@
 Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
-Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
+{{Mapping|legend=0| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.838
+Optimal tunings:
+* WE: ~2 = 1200.4178{{c}}, ~3/2 = 701.0822{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8396{{c}}
-{{Optimal ET sequence|legend=1| 12e, 101cdek, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
-Badness: 0.016007
+Badness (Sintel): 1.53
 === Gracecord ===
@@ Line 678: / Line 796: @@
 Comma list: 225/224, 441/440, 109375/107811
-Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
+{{Mapping|legend=0| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.885
+Optimal tunings:
+* WE: ~2 = 1200.6064{{c}}, ~3/2 = 701.2398{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8718{{c}}
-{{Optimal ET sequence|legend=1| 12, 101cd, 113 }}
+{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
-Badness: 0.066964
+Badness (Sintel): 2.21
 ==== 13-limit ====
@@ Line 691: / Line 811: @@
 Comma list: 225/224, 364/363, 441/440, 6125/6084
-Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
+{{Mapping|legend=0| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.890
+Optimal tunings:
+* WE: ~2 = 1200.6225{{c}}, ~3/2 = 701.2539{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8781{{c}}
-{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
-Badness: 0.044196
+Badness (Sintel): 1.83
 ==== 17-limit ====
@@ Line 704: / Line 826: @@
 Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
-Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
+{{Mapping|legend=0| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.870
+Optimal tunings:
+* WE: ~2 = 1200.3308{{c}}, ~3/2 = 701.0632{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8654{{c}}
-{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness: 0.036637
+Badness (Sintel): 1.87
 ==== 19-limit ====
@@ Line 717: / Line 841: @@
 Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
-Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
+{{Mapping|legend=0| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 700.866
+Optimal tunings:
+* WE: ~2 = 1200.2658{{c}}, ~3/2 = 701.0213{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.8629{{c}}
-{{Optimal ET sequence|legend=1| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness: 0.027559
+Badness (Sintel): 1.68
 == Alphorn ==
@@ Line 730: / Line 856: @@
 [[Comma list]]: 225/224, 5764801/5668704
-{{Mapping|legend=1| 1 9 0 13 | 0 -16 5 -22 }}
+{{Mapping|legend=1| 1 -7 5 -9 | 0 16 -5 22 }}
+: mapping generators: ~2, ~35/24
-{{Multival|legend=1| 16 -5 22 -45 -10 65 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 556.221
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3004{{c}}, ~35/24 = 644.4767{{c}}
+: [[error map]]: {{val| +1.300 +0.569 -2.195 -2.043 }}
+* [[CWE]]: ~2 = 1200.3333{{c}}, ~35/24 = 643.8137{{c}}
+: error map: {{val| 0.000 -0.936 -5.382 -4.924 }}
-{{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
+{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
-[[Badness]]: 0.129258
+[[Badness]] (Sintel): 3.27
 === 11-limit ===
@@ Line 745: / Line 874: @@
 Comma list: 225/224, 385/384, 12250/11979
-Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}
+{{Mapping|legend=0| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 556.144
+Optimal tunings:
+* WE: ~2 = 1200.5123{{c}}, ~16/11 = 644.1307{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 643.8662{{c}}
-{{Optimal ET sequence|legend=1| 28d, 41, 315cde }}
+{{Optimal ET sequence|legend=0| 13d, 28d, 41 }}
-Badness: 0.073459
+Badness (Sintel): 2.43
 == Misneb ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Misneb]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Misneb]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 760: / Line 891: @@
 [[Comma list]]: 225/224, 4194304/4117715
-{{Mapping|legend=1| 1 3 1 3 | 0 -15 14 -2 }}
+{{Mapping|legend=1| 1 -12 15 1 | 0 15 -14 2 }}
+: mapping generators: ~2, ~15/8
-{{Multival|legend=1| 15 -14 2 -57 -39 44 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~16/15 = 113.235
+: [[error map]]: {{val| -0.236 -0.856 -1.569 +4.041 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/8 = 1086.7633{{c}}
+: error map: {{val| 0.000 -0.506 -0.999 +4.701 }}
 {{Optimal ET sequence|legend=1| 21, 32, 53 }}
-[[Badness]]: 0.140970
+[[Badness]] (Sintel): 3.57
 === 11-limit ===
@@ Line 775: / Line 909: @@
 Comma list: 99/98, 176/175, 1310720/1294139
-Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}
+{{Mapping|legend=0| 1 -12 15 1 27 | 0 15 -14 2 -26 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.323
+Optimal tunings:
+* WE: ~2 = 1200.1654{{c}}, ~15/8 = 1086.8269{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6766{{c}}
-{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
-Badness: 0.085390
+Badness (Sintel): 2.82
 ==== 13-limit ====
@@ Line 788: / Line 924: @@
 Comma list: 99/98, 176/175, 640/637, 847/845
-Mapping: {{mapping| 1 3 1 3 1 2 | 0 -15 14 -2 26 18 }}
+{{Mapping|legend=0| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.323
+Optimal tunings:
+* WE: ~2 = 1200.1687{{c}}, ~15/8 = 1086.8295{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.6757{{c}}
-{{Optimal ET sequence|legend=1| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
-Badness: 0.045569
+Badness (Sintel): 1.88
 === Musneb ===
@@ Line 801: / Line 939: @@
 Comma list: 225/224, 385/384, 66550/64827
-Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
+{{Mapping|legend=0| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/15 = 113.142
+Optimal tunings:
+* WE: ~2 = 1200.0839{{c}}, ~15/8 = 1086.9343{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/8 = 1086.8593{{c}}
-{{Optimal ET sequence|legend=1| 32, 53, 191de, 244cddee, 297cddee }}
+{{Optimal ET sequence|legend=0| 21e, 32, 53 }}
-Badness: 0.087333
+Badness (Sintel): 2.89
 == Untriton ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Untriton]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Untriton]].''
+Named by [[Petr Pařízek]] in 2011<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>, untriton may be described as the {{nowrap| 51 & 53 }} temperament. Like [[#Tritonic|tritonic]], it is generated by a [[~]][[10/7]] [[tritone]], but here, nine generator steps give the [[interval class]] of [[3/1|3]]. The [[ploidacot]] for this temperament is delta-enneacot.
 [[Subgroup]]: 2.3.5.7
@@ Line 816: / Line 958: @@
 [[Comma list]]: 225/224, 125000000/121060821
-{{Mapping|legend=1| 1 6 -7 -7 | 0 -9 19 20 }}
+{{Mapping|legend=1| 1 -3 12 13 | 0 9 -19 -20 }}
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 9 -19 -20 -51 -57 7 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.8275{{c}}, ~10/7 = 611.2710{{c}}
+: [[error map]]: {{val| -0.172 +0.002 -2.533 +3.511 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 611.3614{{c}}
+: error map: {{val| 0.000 +0.298 -2.181 +3.946 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 588.641
+{{Optimal ET sequence|legend=1| 51, 53 }}
-{{Optimal ET sequence|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
+[[Badness]] (Sintel): 3.64
-[[Badness]]: 0.143976
 === 11-limit ===
@@ Line 831: / Line 976: @@
 Comma list: 121/120, 225/224, 22000/21609
-Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}
+{{Mapping|legend=0| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.626
+Optimal tunings:
+* WE: ~2 = 1200.3591{{c}}, ~10/7 = 611.5569{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3690{{c}}
-{{Optimal ET sequence|legend=1| 51, 53 }}
+{{Optimal ET sequence|legend=0| 51, 53 }}
-Badness: 0.074295
+Badness (Sintel): 2.46
 === 13-limit ===
@@ Line 844: / Line 991: @@
 Comma list: 121/120, 225/224, 275/273, 1040/1029
-Mapping: {{mapping| 1 6 -7 -7 1 -12 | 0 -9 19 20 5 32 }}
+{{Mapping|legend=0| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
+Optimal tunings:
+* WE: ~2 = 1200.4078{{c}}, ~10/7 = 611.5536{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 611.3392{{c}}
+{{Optimal ET sequence|legend=0| 51f, 53 }}
+Badness (Sintel): 1.96
+== Naiadical ==
+Named by [[Xenllium]] in 2026, naiadical may be described as the {{nowrap| 21 & 29 }} temperament.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 225/224, 823543/800000
+{{Mapping|legend=1| 1 -4 11 9 | 0 9 -14 -10 }}
+: mapping generators: ~2, ~32/21
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1202.1198{{c}}, ~32/21 = 745.4675{{c}}
+: [[error map]]: {{val| +2.120 -1.227 +0.459 -4.423 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~32/21 = 744.1318{{c}}
+: error map: {{val| 0.000 -4.769 -4.159 -10.144 }}
+{{Optimal ET sequence|legend=1| 21, 29, 50, 79d, 129cdd, 179bcddd }}
+[[Badness]] (Sintel): 3.67
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 225/224, 245/242, 1617/1600
+Mapping: {{Mapping| 1 -4 11 9 14 | 0 9 -14 -10 -17 }}
+Optimal tunings:
+* WE: ~2 = 1201.9008{{c}}, ~21/16 = 745.3867{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~32/21 = 744.1777{{c}}
+{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
+Badness (Sintel): 2.00
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 105/104, 196/195, 245/242, 1001/1000
+Mapping: {{Mapping| 1 -4 11 9 14 13 | 0 9 -14 -10 -17 -15 }}
+Optimal tunings:
+* WE: ~2 = 1201.7863{{c}}, ~20/13 = 745.3344{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1931{{c}}
+{{Optimal ET sequence|legend=0| 21, 29, 50, 79d }}
+Badness (Sintel): 1.43
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
+Mapping: {{Mapping| 1 -4 11 9 14 13 14 | 0 9 -14 -10 -17 -15 -16 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.654
+Optimal tunings:
+* WE: ~2 = 1201.9208{{c}}, ~20/13 = 745.3976{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 744.1669{{c}}
-{{Optimal ET sequence|legend=1| 51f, 53 }}
+{{Optimal ET sequence|legend=0| 21, 29g, 50, 79dg }}
-Badness: 0.047441
+Badness (Sintel): 1.26
 == Quintannic ==
+Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 858: / Line 1,074: @@
 {{Mapping|legend=1| 1 1 5 7 | 0 5 -23 -36 }}
+: mapping generators: ~2, ~10000/9261
-{{Multival|legend=1| 5 -23 -36 -48 -71 -19 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10000/9261 = 139.838
+: [[error map]]: {{val| +0.980 -1.214 -0.313 -0.243 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10000/9261 = 139.8184{{c}}
+: error map: {{val| 0.000 -2.863 -2.136 -2.287 }}
 {{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
-[[Badness]]: 0.150565
+[[Badness]] (Sintel): 3.81
 === 11-limit ===
@@ Line 872: / Line 1,091: @@
 Comma list: 225/224, 441/440, 43923/43750
-Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
+{{Mapping|legend=0| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
-Optimal tuning (POTE): ~2 = 1\1, ~320/297 = 139.827
+Optimal tunings:
+* WE: ~2 = 1201.0031{{c}}, ~320/297 = 139.9435{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~320/297 = 139.8053{{c}}
-{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}
-Badness: 0.052590
+Badness (Sintel): 1.74
 === 13-limit ===
@@ Line 885: / Line 1,106: @@
 Comma list: 225/224, 441/440, 1001/1000, 1188/1183
-Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
+{{Mapping|legend=0| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.812
+Optimal tunings:
+* WE: ~2 = 1200.8354{{c}}, ~13/12 = 139.9095{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.7997{{c}}
-{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness: 0.032730
+Badness (Sintel): 1.35
 === 17-limit ===
@@ Line 898: / Line 1,121: @@
 Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
-Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
+{{Mapping|legend=0| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
+Optimal tunings:
+* WE: ~2 = 1200.7402{{c}}, ~13/12 = 139.9015{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.8038{{c}}
+{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
+Badness (Sintel): 1.17
+== Hendeca ==
+{{Distinguish| Hendec }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Hendecatonic]].''
+Hendeca tempers out the same 5-limit comma as [[porwell temperaments #Hendecatonic|hendecatonic]], and has a period of 1/11 octave. However, in this temperament, nine periods represent [[7/4]], the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. It was named by [[Xenllium]] in 2025 as a low-accuracy variant of hendecatonic.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 225/224, 122880/117649
-Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.815
+{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
+: mapping generators: ~16/15, ~3
-{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}}
+: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }}
+* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}}
+: error map: {{val| 0.000 +1.252 +1.388 +12.992 }}
-Badness: 0.023038
+{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }}
+[[Badness]] (Sintel): 4.37
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 121/120, 225/224, 352/343
+{{Mapping|legend=0| 11 0 43 31 38 | 0 1 -1 0 0 }}
+Optimal tunings:
+* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}}
+* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}}
+{{Optimal ET sequence|legend=0| 22, 55d }}
+Badness (Sintel): 2.46
 == Gwazy ==
-{{See also| Very high accuracy temperaments #Kwazy }}
+: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''
+Named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, gwazy may be described as the {{nowrap| 22 & 74 }} temperament.
 [[Subgroup]]: 2.3.5.7
@@ Line 914: / Line 1,179: @@
 {{Mapping|legend=1| 2 1 6 4 | 0 8 -5 6 }}
+: mapping generators: ~2401/1728, ~35/32
-{{Multival|legend=1| 16 -10 12 -53 -26 56 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2401/1728 = 599.7132{{c}}, ~35/32 = 162.5806{{c}}
-[[Optimal tuning]] ([[POTE]]): 1\2, ~35/32 = 162.658
+: [[error map]]: {{val| -0.574 -1.597 -0.937 +5.510 }}
+* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~35/32 = 162.6388{{c}}
+: error map: {{val| 0.000 -0.844 +0.492 +7.007 }}
 {{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
-[[Badness]]: 0.178826
+[[Badness]] (Sintel): 4.53
 === 11-limit ===
@@ Line 928: / Line 1,196: @@
 Comma list: 99/98, 176/175, 65536/65219
-Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
+{{Mapping|legend=0| 2 1 6 4 8 | 0 8 -5 6 -4 }}
-Optimal tuning (POTE): 1\2, ~11/10 = 162.592
+Optimal tunings:
+* WE: ~363/256 = 599.8517{{c}}, ~11/10 = 162.5518{{c}}
+* CWE: ~363/256 = 600.0000{{c}}, ~11/10 = 162.5863{{c}}
-{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
+{{Optimal ET sequence|legend=0| 22, 74, 96 }}
-Badness: 0.068410
+Badness (Sintel): 2.26
 == Tertiosec ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tertiosec]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tertiosec]].''
+Tertiosec may be described as the {{nowrap| 21 & 75 }} temperament. It was initially named ''tertiomar'' by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, but was changed to ''tertiosec'' in 2012<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>.
 [[Subgroup]]: 2.3.5.7
@@ Line 943: / Line 1,215: @@
 [[Comma list]]: 225/224, 14495514624/13841287201
-{{Mapping|legend=1| 3 7 5 9 | 0 -8 7 -2 }}
+{{Mapping|legend=1| 3 -1 12 7 | 0 8 -7 2 }}
+: mapping generators: ~3072/2401, ~2048/1715
-{{Multival|legend=1| 24 -21 6 -89 -58 73 }}
-[[Optimal tuning]] ([[POTE]]): 1\3, ~15/14 = 112.283
+[[Optimal tuning]]s:
+* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
+: [[error map]]: {{val| -0.523 -1.044 -1.549 +5.138 }}
+* [[CWE]]: ~3072/2401 = 400.0000{{c}}, ~2048/1715 = 287.7088{{c}}
+: error map: {{val| 0.000 -0.284 -0.276 +6.592 }}
 {{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
-[[Badness]]: 0.431636
+[[Badness]] (Sintel): 10.9
 === 11-limit ===
@@ Line 958: / Line 1,233: @@
 Comma list: 225/224, 3840/3773, 12005/11979
-Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}
+{{Mapping|legend=0| 3 -1 12 7 14 | 0 8 -7 2 -5 }}
+Optimal tunings:
+* WE: ~44/35 = 399.6550{{c}}, ~33/28 = 287.5803{{c}}
+* CWE: ~44/35 = 400.0000{{c}}, ~33/28 = 287.8224{{c}}
-Optimal tuning (POTE): 1\3, ~15/14 = 112.171
+{{Optimal ET sequence|legend=0| 21, 54, 75e }}
-{{Optimal ET sequence|legend=1| 21, 54, 75e }}
+Badness (Sintel): 5.74
-Badness: 0.173485
+== References ==
+[[Category:Marvel temperaments| ]] <!-- main article -->
 [[Category:Temperament collections]]
-[[Category:Marvel temperaments| ]] <!-- main article -->
+[[Category:Catalogs of rank-2 temperaments]]
-[[Category:Marvel| ]] <!-- key article -->
-[[Category:Rank 2]]

Marvel temperaments: Difference between revisions