@@ 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)
+* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
-* ''[[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]]
-Considered below are wizard, tritonic, septimin, slender, triton, merman, marvolo, amavil, enneaportent, submajor, alphorn, tertiosec, gwazy, and gracecordial.
+Considered below are wizard, 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 ==
 {{Main| Wizard }}
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
+Wizard has a [[semi-octave]] period and is generated by an interval that can be treated as [[~]][[17/15]]. The semi-octave complement of this interval is ~[[5/4]]. Wizard can be described as {{nowrap| 22 & 72 }}. Its [[ploidacot]] is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the [[3/1|perfect twelfth]]. [[72edo]], [[94edo]], and especially [[166edo]] are good tunings for it.
 [[Subgroup]]: 2.3.5.7
@@ Line 47: / Line 50: @@
 {{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
 : mapping generators: ~1225/864, ~245/216
-{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
-[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~5/4 = 383.256 (~245/216 = 216.744)
+: [[error map]]: {{val| +0.688 -0.403 -1.463 +0.541 }}
+* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~245/216 = 216.7977{{c}}
+: error map: {{val| 0.000 -1.169 -3.111 -0.849 }}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c, 382c }}
+{{Optimal ET sequence|legend=1| 22, 50, 72, 238c, 310c, 382c, 454bccd }}
-[[Badness]]: 0.040846
+[[Badness]] (Sintel): 1.03
-Scales: [[wizard22]]
 === 11-limit ===
@@ Line 67: / Line 69: @@
 Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.232 (~25/22 = 216.768)
+Optimal tunings:
+* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c }}
+{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
-Badness: 0.018539
+Badness (Sintel): 0.613
-Scales: [[wizard22]]
 ==== Lizard ====
@@ Line 82: / Line 84: @@
 Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
-Optimal tuning (POTE): ~55/39 = 1\2, ~5/4 = 383.389 (~25/22 = 216.711)
+Optimal tunings:
+* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 122, 194df }}
+{{Optimal ET sequence|legend=0| 22, 50, 72 }}
-Badness: 0.021781
+Badness (Sintel): 0.900
 ===== 17-limit =====
@@ Line 95: / Line 99: @@
 Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.381 (~17/15 = 216.619)
+Optimal tunings:
+* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 22, 50, 72 }}
-Badness: 0.014536
+Badness (Sintel): 0.741
 ===== 19-limit =====
@@ Line 108: / Line 114: @@
 Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.477 (~17/15 = 216.523)
+Optimal tunings:
+* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
-{{Optimal ET sequence|legend=1| 22h, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
-Badness: 0.015702
+Badness (Sintel): 0.955
 ==== Gizzard ====
@@ Line 121: / Line 129: @@
 Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
-Optimal tuning (POTE): ~99/70 = 1\2, ~5/4 = 383.170 (~25/22 = 216.830)
+Optimal tunings:
+* WE: ~99/70 = 600.2896{{c}}, ~25/22 = 216.9343{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.8501{{c}}
-{{Optimal ET sequence|legend=1| 72, 166, 238cf }}
+{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
-Badness: 0.020252
+Badness (Sintel): 0.837
 ===== 17-limit =====
@@ Line 134: / Line 144: @@
 Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.175 (~25/22 = 216.825)
+Optimal tunings:
+* WE: ~17/12 = 600.3227{{c}}, ~17/15 = 216.9414{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8469{{c}}
-{{Optimal ET sequence|legend=1| 72, 166g, 238cfg }}
+{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
-Badness: 0.013624
+Badness (Sintel): 0.694
 ===== 19-limit =====
@@ Line 147: / Line 159: @@
 Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
-Optimal tuning (POTE): ~17/12 = 1\2, ~5/4 = 383.138 (~17/15 = 216.862)
+Optimal tunings:
+* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
-{{Optimal ET sequence|legend=1| 72, 94, 166g }}
+{{Optimal ET sequence|legend=0| 72, 94, 166g }}
-Badness: 0.014810
+Badness (Sintel): 0.901
 === Mage ===
@@ Line 160: / Line 174: @@
 Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
-Optimal tuning (POTE): ~77/54 = 1\2, ~5/4 = 383.124 (~55/48 = 216.876)
+Optimal tunings:
+* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
+* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
-{{Optimal ET sequence|legend=1| 22, 50e, 72ee, 94ee }}
+{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
-Badness: 0.057799
+Badness (Sintel): 1.91
 == Tritonic ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
+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
@@ Line 173: / Line 191: @@
 [[Comma list]]: 225/224, 50421/50000
-{{Mapping|legend=1| 1 4 -3 -3 | 0 -5 11 12 }}
+{{Mapping|legend=1| 1 -1 8 9 | 0 5 -11 -12 }}
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 5 -11 -12 -29 -33 3 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 580.286
+: [[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 ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
-[[Badness]]: 0.047578
+[[Badness]] (Sintel): 1.20
 === 11-limit ===
@@ Line 188: / Line 209: @@
 Comma list: 121/120, 225/224, 441/440
-Mapping: {{mapping| 1 4 -3 -3 2 | 0 -5 11 12 3 }}
+Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
-{{Multival|legend=1| 5 -11 -12 -3 -29 -33 -22 3 31 33 }}
+Optimal tunings:
+* WE: ~2 = 1201.7116{{c}}, ~10/7 = 620.6166{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6890{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.267
+{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
-{{Optimal ET sequence|legend=1| 29, 31, 60e }}
+Badness (Sintel): 0.782
-Badness: 0.023659
 ==== 13-limit ====
@@ Line 203: / Line 224: @@
 Comma list: 105/104, 121/120, 196/195, 275/273
-Mapping: {{mapping| 1 4 -3 -3 2 -5 | 0 -5 11 12 3 18 }}
+Mapping: {{mapping| 1 -1 8 9 5 13 | 0 5 -11 -12 -3 -18 }}
+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=0| 29, 31, 60e }}
+Badness (Sintel): 0.950
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
+Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
+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=0| 29g, 31, 60e }}
+Badness (Sintel): 0.973
+==== 19-limit ====
+Subgroup: 2.3.5.7.11.13.17.19
+Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
+Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
+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=0| 29g, 31, 60e }}
+Badness (Sintel): 1.03
+==== 23-limit ====
+Subgroup: 2.3.5.7.11.13.17.19.23
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.108
+Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
-{{Optimal ET sequence|legend=1| 29, 31, 60e, 151cde }}
+Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
-Badness: 0.022993
+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=0| 29g, 31, 60e }}
+Badness (Sintel): 1.04
 === Tritoni ===
@@ Line 216: / Line 284: @@
 Comma list: 225/224, 385/384, 27783/27500
-Mapping: {{mapping| 1 4 -3 -3 17 | 0 -5 11 12 -28 }}
+Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
-{{Multival|legend=1| 5 -11 -12 28 -29 -33 27 3 103 120 }}
+Optimal tunings:
+* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 580.389
+{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
-{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}
+Badness (Sintel): 1.50
-Badness: 0.045456
+== Septimin ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
-== Septimin ==
+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.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
 [[Subgroup]]: 2.3.5.7
@@ Line 233: / Line 303: @@
 [[Comma list]]: 225/224, 84035/82944
-{{Mapping|legend=1| 1 4 1 5 | 0 -11 6 -10 }}
+{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
+: mapping generators: ~2, ~12/7
-{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
+[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~7/6 = 263.632
+{{Optimal ET sequence|legend=1| 9, 23, 32, 41, 91, 132d }}
-{{Optimal ET sequence|legend=1| 41, 91, 132d }}
+[[Badness]] (Sintel): 1.38
-[[Badness]]: 0.054502
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 2401/2376
+Comma list: 225/224, 245/242, 385/384
-Mapping: {{mapping| 1 4 1 5 5 | 0 -11 6 -10 -7 }}
+Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.634
+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=1| 41, 91, 223cdef }}
+{{Optimal ET sequence|legend=0| 9, 23, 32, 41, 91, 223cdef }}
-Badness: 0.031309
+Badness (Sintel): 1.04
 === 13-limit ===
@@ Line 261: / Line 336: @@
 Comma list: 105/104, 144/143, 196/195, 245/242
-Mapping: {{mapping| 1 4 1 5 5 7 | 0 -11 6 -10 -7 -15 }}
+Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/6 = 263.700
+Optimal tunings:
+* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
-{{Optimal ET sequence|legend=1| 41, 91 }}
+{{Optimal ET sequence|legend=0| 9, 32f, 41, 91 }}
-Badness: 0.023117
+Badness (Sintel): 0.955
 == Merman ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
+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.
+The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
 [[Subgroup]]: 2.3.5.7
@@ Line 276: / Line 357: @@
 [[Comma list]]: 225/224, 2500000/2470629
-{{Mapping|legend=1| 1 5 -5 -5 | 0 -7 15 16 }}
+{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.3898{{c}}, ~10/7 = 614.6413{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 585.585
+: [[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 ET sequence|legend=1| 41, 84, 125 }}
-[[Badness]]: 0.055078
+[[Badness]] (Sintel): 1.39
 === 11-limit ===
@@ Line 291: / Line 375: @@
 Comma list: 225/224, 441/440, 1344/1331
-Mapping: {{mapping| 1 5 -5 -5 2 | 0 -7 15 16 3 }}
+Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.606
+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| 41, 84, 125e }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e }}
-Badness: 0.036383
+Badness (Sintel): 1.20
-=== 13-limit ===
+==== 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 }}
+Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
+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=0| 41, 84, 125e, 209ef, 293ef }}
+Badness (Sintel): 1.14
+=== Mermaid ===
+Subgroup: 2.3.5.7.11
+Comma list: 225/224, 385/384, 532400/531441
+Mapping: {{mapping| 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: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 585.657
+Optimal tunings:
+* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
-{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}
+{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
-Badness: 0.027544
+Badness (Sintel): 1.47
 == 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.
+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.
+The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
 [[Subgroup]]: 2.3.5.7
@@ Line 320: / Line 440: @@
 {{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
+: mapping generators: ~2, ~49/48
-{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
+[[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, ~49/48 = 38.413
+{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
-{{Optimal ET sequence|legend=1| 31, 94, 125 }}
+[[Badness]] (Sintel): 1.44
-[[Badness]]: 0.056934
 === 11-limit ===
@@ Line 336: / Line 459: @@
 Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.387
+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| 31, 63, 94, 125 }}
+{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
-Badness: 0.025342
+Badness (Sintel): 0.838
 === 13-limit ===
@@ Line 349: / Line 474: @@
 Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/48 = 38.314
+Optimal tunings:
+* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
-{{Optimal ET sequence|legend=1| 31, 63, 94 }}
+{{Optimal ET sequence|legend=0| 31, 63, 94 }}
-Badness: 0.025913
+Badness (Sintel): 1.07
 == Triton ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
+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.
 [[Subgroup]]: 2.3.5.7
@@ Line 363: / Line 492: @@
 {{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
+: mapping generators: ~2, ~10/7
-{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 568.865
+: [[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 ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
-[[Badness]]: 0.059245
+[[Badness]] (Sintel): 1.50
 === 11-limit ===
@@ Line 379: / Line 511: @@
 Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 569.144
+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=0| 2, 17d, 19 }}
+Badness (Sintel): 1.51
+== Marvolo ==
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 225/224, 156250000/155649627
+{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
+: mapping generators: ~2, ~21/20
+[[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| 29, 43, 72, 619bbccd, 691bbccd }}
+[[Badness]] (Sintel): 2.11
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 225/224, 441/440, 4000/3993
+Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
+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=0| 29, 43, 72 }}
+Badness (Sintel): 0.958
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 169/168, 225/224, 364/363, 441/440
+Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
+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=0| 29, 43, 72 }}
+Badness (Sintel): 0.887
-{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
-Badness: 0.045675
+Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
-== Submajor ==
+Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
-{{ See also | Greater tendoneutralic }}
-[[Subgroup]]: 2.3.5
-[[Comma list]]: 69198046875/68719476736
+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 4 -1 | 0 -8 11 }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10125/8192 = 362.321
+Badness (Sintel): 0.760
-{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
+=== 19-limit ===
+Subgroup: 2.3.5.7.11.13.17.19
-[[Badness]]: 0.130236
+Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
+Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
+Optimal tunings:
+* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
+Badness (Sintel): 0.895
+== Decic ==
+{{Main| Decic }}
+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]].
+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.
-=== 7-limit ===
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 225/224, 51200/50421
+[[Comma list]]: 225/224, 16807/16384
-{{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }}
-{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
+{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
+: mapping generators: ~15/14, ~3
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255
+[[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 }}
+{{Optimal ET sequence|legend=1| 10, 30b, 40, 50, 60, 110d, 170cdd }}
-[[Badness]]: 0.060533
+[[Badness]] (Sintel): 2.26
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 6655/6561
+Comma list: 225/224, 385/384, 3087/3025
-Mapping: {{mapping| 1 4 -1 1 11 | 0 -8 11 6 -25 }}
+Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
-Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101
+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 ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cde }}
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-Badness: 0.050582
+Badness (Sintel): 2.11
 ==== 13-limit ====
-{{ See also | Greater tendoneutralic }}
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 225/224, 275/273, 385/384
+Comma list: 105/104, 144/143, 196/195, 2200/2197
-Mapping: {{mapping| 1 4 -1 1 11 4 | 0 -8 11 6 -25 -1 }}
+Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105
+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 ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cdef }}
+{{Optimal ET sequence|legend=0| 10, 40, 50 }}
-Badness: 0.027689
+Badness (Sintel): 1.52
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 105/104, 144/143, 170/169, 196/195, 221/220
+Mapping: {{mapping| 10 0 39 28 3 37 25 | 0 1 -1 0 2 0 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 ET sequence|legend=0| 10, 40, 50 }}
+Badness (Sintel): 1.28
+=== 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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: {{mapping| 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 }}
+Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/20 = 83.330
+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}})
-{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
+{{Optimal ET sequence|legend=0| 10, 50e, 60e }}
-=== 19-limit ===
+Badness (Sintel): 1.41
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
-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 536: / Line 766: @@
 {{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}})
-[[Optimal tuning]] ([[POTE]]): ~2592/2401 = 1\9, ~12005/6912 = 950.1680 (~1728/1715 = 16.8347)
+: [[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 ET sequence|legend=1| 9, 63, 72, 495bcd }}
+{{Optimal ET sequence|legend=1| 9, 54, 63, 72, 495bccd, 567bcccd }}
-[[Badness]]: 0.093679
+[[Badness]] (Sintel): 2.37
 === 11-limit ===
@@ Line 554: / Line 785: @@
 Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
-Optimal tuning (POTE): ~121/112 = 1\9, ~210/121 = 950.1873 (~99/98 = 16.8540)
+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 567: / Line 800: @@
 Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
-Optimal tuning (POTE): ~14/13 = 1\9, ~26/15 = 950.2867 (~105/104 = 16.9534)
+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 581: / Line 816: @@
 {{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 597: / Line 835: @@
 Mapping: {{mapping| 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 610: / Line 850: @@
 Mapping: {{mapping| 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 623: / Line 865: @@
 Mapping: {{mapping| 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 636: / Line 880: @@
 Mapping: {{mapping| 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 649: / Line 895: @@
 Mapping: {{mapping| 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 662: / Line 910: @@
 Mapping: {{mapping| 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 675: / Line 925: @@
 Mapping: {{mapping| 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 688: / Line 940: @@
 Mapping: {{mapping| 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 701: / Line 955: @@
 Mapping: {{mapping| 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 714: / Line 970: @@
 Mapping: {{mapping| 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 727: / Line 985: @@
 Mapping: {{mapping| 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 738: / Line 998: @@
 [[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 753: / Line 1,016: @@
 Comma list: 225/224, 385/384, 12250/11979
-Mapping: {{mapping| 1 9 0 13 3 | 0 -16 5 -22 1 }}
+Mapping: {{mapping| 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 768: / Line 1,033: @@
 [[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]] ([[POTE]]): ~2 = 1\1, ~16/15 = 113.235
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.7642{{c}}, ~15/8 = 1086.5513{{c}}
+: [[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 783: / Line 1,051: @@
 Comma list: 99/98, 176/175, 1310720/1294139
-Mapping: {{mapping| 1 3 1 3 1 | 0 -15 14 -2 26 }}
+Mapping: {{mapping| 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 796: / Line 1,066: @@
 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: {{mapping| 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 811: / Line 1,083: @@
 Mapping: {{mapping| 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 824: / Line 1,100: @@
 [[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 839: / Line 1,118: @@
 Comma list: 121/120, 225/224, 22000/21609
-Mapping: {{mapping| 1 6 -7 -7 1 | 0 -9 19 20 5 }}
+Mapping: {{mapping| 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 852: / Line 1,133: @@
 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: {{mapping| 1 -3 12 13 6 20 | 0 9 -19 -20 -5 -32 }}
-Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 588.654
+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=1| 51f, 53 }}
+{{Optimal ET sequence|legend=0| 51f, 53 }}
-Badness: 0.047441
+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 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=0| 21, 29g, 50, 79dg }}
+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 866: / Line 1,216: @@
 {{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 882: / Line 1,235: @@
 Mapping: {{mapping| 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 895: / Line 1,250: @@
 Mapping: {{mapping| 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 908: / Line 1,265: @@
 Mapping: {{mapping| 1 1 5 7 8 3 7 | 0 5 -23 -36 -39 6 -25 }}
-Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.815
+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 }}
-{{Optimal ET sequence|legend=1| 43, 60e, 103 }}
+Badness (Sintel): 1.17
-Badness: 0.023038
+== 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
+{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }}
+: mapping generators: ~16/15, ~3
+[[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 }}
+{{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: {{mapping| 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 922: / Line 1,321: @@
 {{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]]): ~2401/1728 = 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 938: / Line 1,340: @@
 Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
-Optimal tuning (POTE): ~363/256 = 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 951: / Line 1,357: @@
 [[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]]s:
+* [[WE]]: ~3072/2401 = 399.8257{{c}}, ~2048/1715 = 287.5920{{c}}
-[[Optimal tuning]] ([[POTE]]): ~3072/2401 = 1\3, ~15/14 = 112.283
+: [[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 966: / Line 1,375: @@
 Comma list: 225/224, 3840/3773, 12005/11979
-Mapping: {{mapping| 3 7 5 9 9 | 0 -8 7 -2 5 }}
+Mapping: {{mapping| 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): ~44/35 = 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:Temperament collections]]
 [[Category:Marvel temperaments| ]] <!-- main article -->
-[[Category:Marvel| ]] <!-- key article -->
 [[Category:Rank 2]]

Marvel temperaments: Difference between revisions