@@ 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:
-* ''[[Sharp]]'', {25/24, 28/27} → [[Dicot family #Sharp|Dicot family]]
+* ''[[Pelogic]]'' (+21/20 or 135/128) → [[Mavila family #Pelogic|Mavila family]]
-* ''[[Pelogic]]'', {21/20, 135/128} → [[Pelogic family #Pelogic|Pelogic family]]
+* [[Meantone]] (+81/80 or 126/125) → [[Meantone family #Septimal meantone|Meantone family]]
-* ''[[August]]'', {36/35, 128/125} → [[Augmented family #August|Augmented family]]
+* [[Garibaldi]] (+3125/3087) → [[Schismatic family #Garibaldi|Schismatic family]]
-* [[Negri]], {49/48, 225/224} → [[Slendro clan #Negri|Slendro clan]]
+* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
-* [[Pajara]], {50/49, 64/63} → [[Diaschismic family #Pajara|Diaschismic family]]
+* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
-* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
+* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
-* [[Magic]], {225/224, 245/243} → [[Magic family #Magic|Magic family]]
+* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
-* ''[[Passive]]'', {225/224, 256/245} → [[Passion family #Passive|Passion family]]
+* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
-* [[Miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle|Gamelismic clan]]
+* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
-* [[Orwell]], {225/224, 1728/1715} → [[Semicomma family #Orwell|Semicomma family]]
+* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
-* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
+* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
-* [[Catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic|Kleismic family]]
+* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
-* ''[[Snipes]]'', {225/224, 6125/5832} → [[Wesley family #Snipes|Wesley family]]
+* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
-* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic|Cloudy clan]]
+* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
-* ''[[Amavil]]'', {225/224, 17496/16807} → [[Mabila family #Amavil|Mabila family]]
+* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
-* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade|Escapade family]]
+* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
-* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog|Misty family]]
+* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
-* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
+* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
-* ''[[Immune]]'', {225/224, 781250/750141} → [[Immunity family #Immune|Immunity family]]
+* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
-* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
+* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
-* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
+* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
-* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
+* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
-* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
+* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
-* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
+* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
-* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity family]]
+* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
-* ''[[Gammy]]'', {225/224, 94143178827/91913281250} → [[Gammic family #Gammy|Gammic family]]
+* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
+* ''[[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}}
+{{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]].''
-Subgroup: 2.3.5.7
+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
 [[Comma list]]: 225/224, 118098/117649
-[[Mapping]]: [{{val| 2 1 5 2 }}, {{val| 0 6 -1 10 }}]
+{{Mapping|legend=1| 2 1 5 2 | 0 6 -1 10 }}
+: mapping generators: ~1225/864, ~245/216
-Mapping generators: ~1225/864, ~245/216
-[[POTE generator]]: ~5/4 = 383.256
-{{Multival|legend=1| 12 -2 20 -31 -2 52 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1225/864 = 600.3438{{c}}, ~245/216 = 216.8680{{c}}
+: [[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 64: / Line 67: @@
 Comma list: 225/224, 385/384, 4000/3993
-Mapping: [{{val| 2 1 5 2 8 }}, {{val| 0 6 -1 10 -3 }}]
+Mapping: {{mapping| 2 1 5 2 8 | 0 6 -1 10 -3 }}
-Mapping generators: ~99/70, ~25/22
+Optimal tunings:
+* WE: ~99/70 = 600.3051{{c}}, ~25/22 = 216.8782{{c}}
+* CWE: ~99/70 = 600.0000{{c}}, ~25/22 = 216.7961{{c}}
-POTE generator: ~5/4 = 383.232
+{{Optimal ET sequence|legend=0| 22, 50, 72, 166, 238c, 310c }}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 166, 238c, 310c }}
+Badness (Sintel): 0.613
-Badness: 0.018539
-Scales: [[wizard22]]
 ==== Lizard ====
@@ Line 81: / Line 82: @@
 Comma list: 225/224, 351/350, 364/363, 385/384
-Mapping: [{{val| 2 1 5 2 8 11 }}, {{val| 0 6 -1 10 -3 -10 }}]
+Mapping: {{mapping| 2 1 5 2 8 11 | 0 6 -1 10 -3 -10 }}
-Mapping generators: ~99/70, ~25/22
+Optimal tunings:
+* WE: ~55/39 = 600.4824{{c}}, ~25/22 = 216.7852{{c}}
+* CWE: ~55/39 = 600.0000{{c}}, ~25/22 = 216.6247{{c}}
-POTE generator: ~5/4 = 383.389
+{{Optimal ET sequence|legend=0| 22, 50, 72 }}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 122, 194df }}
+Badness (Sintel): 0.900
-Badness: 0.021781
 ===== 17-limit =====
@@ Line 96: / Line 97: @@
 Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
-Mapping: [{{val| 2 1 5 2 8 11 6 }}, {{val| 0 6 -1 10 -3 -10 6 }}]
+Mapping: {{mapping| 2 1 5 2 8 11 6 | 0 6 -1 10 -3 -10 6 }}
-Mapping generators: ~17/12, ~17/15
+Optimal tunings:
+* WE: ~17/12 = 600.5032{{c}}, ~17/15 = 216.8002{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.6361{{c}}
-POTE generator: ~5/4 = 383.381
+{{Optimal ET sequence|legend=0| 22, 50, 72 }}
-{{Optimal ET sequence|legend=1| 22, 50, 72, 122g, 194dfg }}
+Badness (Sintel): 0.741
-Badness: 0.014536
 ===== 19-limit =====
@@ Line 111: / Line 112: @@
 Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
-Mapping: [{{val| 2 1 5 2 8 11 6 2 }}, {{val| 0 6 -1 10 -3 -10 6 18 }}]
+Mapping: {{mapping| 2 1 5 2 8 11 6 2 | 0 6 -1 10 -3 -10 6 18 }}
-Mapping generators: ~17/12, ~17/15
+Optimal tunings:
+* WE: ~17/12 = 600.4698{{c}}, ~17/15 = 216.6925{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.5434{{c}}
-POTE generator: ~5/4 = 383.477
+{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
-{{Optimal ET sequence|legend=1| 22h, 50, 72, 122g, 194dfg }}
+Badness (Sintel): 0.955
-Badness: 0.015702
 ==== Gizzard ====
@@ Line 126: / Line 127: @@
 Comma list: 225/224, 325/324, 385/384, 1573/1568
-Mapping: [{{val| 2 1 5 2 8 -2 }}, {{val| 0 6 -1 10 -3 26 }}]
+Mapping: {{mapping| 2 1 5 2 8 -2 | 0 6 -1 10 -3 26 }}
-Mapping generators: ~99/70, ~25/22
-POTE generator: ~5/4 = 383.170
+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 141: / Line 142: @@
 Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
-Mapping: [{{val| 2 1 5 2 8 -2 6 }}, {{val| 0 6 -1 10 -3 26 6 }}]
+Mapping: {{mapping| 2 1 5 2 8 -2 6 | 0 6 -1 10 -3 26 6 }}
-Mapping generators: ~17/12, ~17/15
-POTE generator: ~5/4 = 383.175
+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 156: / Line 157: @@
 Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
-Mapping: [{{val| 2 1 5 2 8 -2 6 15 }}, {{val| 0 6 -1 10 -3 26 6 -18 }}]
+Mapping: {{mapping| 2 1 5 2 8 -2 6 15 | 0 6 -1 10 -3 26 6 -18 }}
-Mapping generators: ~17/12, ~17/15
+Optimal tunings:
+* WE: ~17/12 = 600.2637{{c}}, ~17/15 = 216.9570{{c}}
+* CWE: ~17/12 = 600.0000{{c}}, ~17/15 = 216.8687{{c}}
-POTE generator: ~5/4 = 383.138
+{{Optimal ET sequence|legend=0| 72, 94, 166g }}
-{{Optimal ET sequence|legend=1| 72, 94, 166g }}
+Badness (Sintel): 0.901
-Badness: 0.014810
 === Mage ===
@@ Line 171: / Line 172: @@
 Comma list: 99/98, 176/175, 1331/1296
-Mapping: [{{val| 2 1 5 2 4 }}, {{val| 0 6 -1 10 8 }}]
+Mapping: {{mapping| 2 1 5 2 4 | 0 6 -1 10 8 }}
-Mapping generators: ~77/54, ~55/48
+Optimal tunings:
+* WE: ~77/54 = 600.6486{{c}}, ~55/48 = 217.1099{{c}}
+* CWE: ~77/54 = 600.0000{{c}}, ~55/48 = 216.9841{{c}}
-POTE generator: ~5/4 = 383.124
+{{Optimal ET sequence|legend=0| 22, 50e, 72ee }}
-{{Optimal ET sequence|legend=1| 22, 50e, 72ee, 94ee }}
+Badness (Sintel): 1.91
-Badness: 0.057799
+== Tritonic ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritonic]].''
-== 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.
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Tritonic]].''
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 50421/50000
-[[Mapping]]: [{{val| 1 4 -3 -3 }}, {{val| 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}}
-[[POTE generator]]: ~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 203: / Line 209: @@
 Comma list: 121/120, 225/224, 441/440
-Mapping: [{{val| 1 4 -3 -3 2 }}, {{val| 0 -5 11 12 3 }}]
+Mapping: {{mapping| 1 -1 8 9 5 | 0 5 -11 -12 -3 }}
-POTE generator: ~7/5 = 580.267
+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| 29, 31, 60e }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
-Badness: 0.023659
+Badness (Sintel): 0.782
 ==== 13-limit ====
@@ Line 216: / Line 224: @@
 Comma list: 105/104, 121/120, 196/195, 275/273
-Mapping: [{{val| 1 4 -3 -3 2 -5 }}, {{val| 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
-POTE generator: ~7/5 = 580.108
+Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
-{{Optimal ET sequence|legend=1| 29, 31, 60e, 151cde }}
+Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
-Badness: 0.022993
+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
+Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
+Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
+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 229: / Line 284: @@
 Comma list: 225/224, 385/384, 27783/27500
-Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
+Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
-POTE generator: ~7/5 = 580.389
+Optimal tunings:
+* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
-{{Optimal ET sequence|legend=1| 31, 91, 122, 153d }}
+{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
-Badness: 0.045456
+Badness (Sintel): 1.50
 == Septimin ==
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
-Subgroup: 2.3.5.7
+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.
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 84035/82944
-[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 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 }}
-[[POTE generator]]: ~7/6 = 263.632
+[[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| 41, 91, 132d }}
-[[Badness]]: 0.054502
+[[Badness]] (Sintel): 1.38
 === 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: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
+Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
-POTE generator: ~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| 41, 91, 223cdef }}
-Badness: 0.031309
+Badness (Sintel): 1.04
 === 13-limit ===
@@ Line 272: / Line 336: @@
 Comma list: 105/104, 144/143, 196/195, 245/242
-Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
+Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
-POTE generator: ~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| 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.
-Subgroup: 2.3.5.7
+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, 2500000/2470629
-[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 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 }}
-[[POTE generator]]: ~7/5 = 585.585
+[[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 ET sequence|legend=1| 41, 84, 125 }}
-[[Badness]]: 0.055078
+[[Badness]] (Sintel): 1.39
 === 11-limit ===
@@ Line 302: / Line 375: @@
 Comma list: 225/224, 441/440, 1344/1331
-Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
+Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
-POTE generator: ~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: [{{val| 1 5 -5 -5 2 12 }}, {{val| 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 }}
-POTE generator: ~7/5 = 585.657
+Badness (Sintel): 1.14
-{{Optimal ET sequence|legend=1| 41, 84, 125e, 209ef, 293ef }}
+=== Mermaid ===
+Subgroup: 2.3.5.7.11
-Badness: 0.027544
+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 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=0| 41, 84ef, 125f, 166 }}
+Badness (Sintel): 1.47
 == Slender ==
-Slender (31&amp;32) tempers out the [[Hewuermera temperaments|hewuermera comma]] in addition to the marvel comma, this has a generator [[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.
-Subgroup: 2.3.5.7
+The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 589824/588245
-[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
+{{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 }}
-[[POTE generator]]: ~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 345: / Line 457: @@
 Comma list: 225/224, 385/384, 1331/1323
-Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
+Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
-POTE generator: ~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 358: / Line 472: @@
 Comma list: 225/224, 275/273, 385/384, 1331/1323
-Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
+Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
-POTE generator: ~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]].''
-Subgroup: 2.3.5.7
+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
 [[Comma list]]: 225/224, 1029/1000
-[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
+{{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}}
-[[POTE generator]]: ~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 388: / Line 509: @@
 Comma list: 45/44, 56/55, 1029/1000
-Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
+Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
+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
+=== 17-limit ===
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
+Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
-POTE generator: ~7/5 = 569.144
+Optimal tunings:
+* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
-{{Optimal ET sequence|legend=1| 2, 17d, 19, 59bde, 78bde, 97bde }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-Badness: 0.045675
+Badness (Sintel): 0.760
-== Submajor ==
+=== 19-limit ===
-Subgroup: 2.3.5
+Subgroup: 2.3.5.7.11.13.17.19
-[[Comma list]]: 69198046875/68719476736
+Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
-[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
+Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
-[[POTE generator]]: ~10125/8192 = 362.321
+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=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
-[[Badness]]: 0.130236
+Badness (Sintel): 0.895
-=== 7-limit ===
+== Decic ==
-Subgroup: 2.3.5.7
+{{Main| Decic }}
-[[Comma list]]: 225/224, 51200/50421
+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]].
-[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
+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.
-{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
+[[Subgroup]]: 2.3.5.7
-[[POTE generator]]: ~49/40 = 362.255
+[[Comma list]]: 225/224, 16807/16384
-{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }}
+{{Mapping|legend=1| 10 0 39 28 | 0 1 -1 0 }}
+: mapping generators: ~15/14, ~3
-[[Badness]]: 0.060533
+[[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, 30b, 40, 50, 60, 110d, 170cdd }}
+[[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: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
+Mapping: {{mapping| 10 0 39 28 3 | 0 1 -1 0 2 }}
-POTE generator: ~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 ====
 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: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
+Mapping: {{mapping| 10 0 39 28 3 37 | 0 1 -1 0 2 0 }}
-POTE generator: ~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
-POTE generator: ~49/40 = 362.418
-Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
+Mapping: {{mapping| 10 0 39 28 82 | 0 1 -1 0 -3 }}
-POTE generator: ~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
-POTE generator: ~16/13 = 362.402
+Mapping: {{mapping| 10 0 39 28 82 37 | 0 1 -1 0 -3 0 }}
-Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
+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}})
-POTE generator: ~16/13 = 362.402
+{{Optimal ET sequence|legend=0| 10e, 40e, 50, 60e, 110de }}
-{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
+Badness (Sintel): 1.57
-Badness: 0.029680
+==== 17-limit ====
+Subgroup: 2.3.5.7.11.13.17
-== Marvolo ==
-Subgroup: 2.3.5.7
-[[Comma list]]: 225/224, 156250000/155649627
+Comma list: 105/104, 170/169, 196/195, 289/288, 375/374
-[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 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}})
-[[POTE generator]]: ~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: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
+Mapping: {{mapping| 10 0 39 28 -13 | 0 1 -1 0 3 }}
-POTE generator: ~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: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
+Mapping: {{mapping| 10 0 39 28 -13 37 | 0 1 -1 0 3 0 }}
-POTE generator: ~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: [{{val| 1 2 1 1 2 3 2 }}, {{val| 0 -6 19 26 21 10 30 }}]
+Mapping: {{mapping| 10 0 39 28 -13 37 25 | 0 1 -1 0 3 0 1 }}
-POTE generator: ~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: [{{val| 1 2 1 1 2 3 2 3 }}, {{val| 0 -6 19 26 21 10 30 18 }}]
-POTE generator: ~21/20 = 83.330
-{{Optimal ET sequence|legend=1| 29g, 43, 72 }}
 == Enneaportent ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 40353607/40310784
-[[Mapping]]: [{{val| 9 0 28 11 }}, {{val| 0 2 -1 2 }}]
+{{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 }}
-[[POTE generator]]: ~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 563: / Line 783: @@
 Comma list: 225/224, 385/384, 12005/11979
-Mapping: [{{val| 9 0 28 11 24 }}, {{val| 0 2 -1 2 1 }}]
+Mapping: {{mapping| 9 0 28 11 24 | 0 2 -1 2 1 }}
-POTE generator: ~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 576: / Line 798: @@
 Comma list: 169/168, 225/224, 364/363, 1716/1715
-Mapping: [{{val| 9 0 28 11 24 19 }}, {{val| 0 2 -1 2 1 2 }}]
+Mapping: {{mapping| 9 0 28 11 24 19 | 0 2 -1 2 1 2 }}
-POTE generator: ~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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 781250000/771895089
-[[Mapping]]: [{{val| 1 0 34 63 }}, {{val| 0 1 -20 -38 }}]
+{{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}}
+: [[error map]]: {{val| +0.490 -0.354 -1.655 +1.241 }}
+* [[CWE]]: ~2 = 1200.3333{{c}}, ~3/2 = 700.8112{{c}}
+: error map: {{val| 0.000 -1.144 -2.537 +0.349 }}
-[[POTE generator]]: ~3/2 = 700.824
+{{Optimal ET sequence|legend=1| 12, …, 113, 125, 238c, 363c }}
-{{Optimal ET sequence|legend=1| 12, 113, 125, 238c, 363c }}
+[[Badness]] (Sintel): 2.44
-[[Badness]]: 0.096279
 === 11-limit ===
@@ Line 606: / Line 833: @@
 Comma list: 225/224, 385/384, 236328125/234365481
-Mapping: [{{val| 1 0 34 63 -90 }}, {{val| 0 1 -20 -38 59 }}]
+Mapping: {{mapping| 1 0 34 63 -90 | 0 1 -20 -38 59 }}
-POTE generator: ~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 619: / Line 848: @@
 Comma list: 225/224, 325/324, 385/384, 831875/830466
-Mapping: [{{val| 1 0 34 63 -90 -66 }}, {{val| 0 1 -20 -38 59 44 }}]
+Mapping: {{mapping| 1 0 34 63 -90 -66 | 0 1 -20 -38 59 44 }}
-POTE generator: ~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 632: / Line 863: @@
 Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
-Mapping: [{{val| 1 0 34 63 -90 -66 -7 }}, {{val| 0 1 -20 -38 59 44 7 }}]
+Mapping: {{mapping| 1 0 34 63 -90 -66 -7 | 0 1 -20 -38 59 44 7 }}
-POTE generator: ~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 645: / Line 878: @@
 Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
-Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 }}, {{val| 0 1 -20 -38 59 44 7 -3 }}]
+Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 | 0 1 -20 -38 59 44 7 -3 }}
-POTE generator: ~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 658: / Line 893: @@
 Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
-Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 }}]
+Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 | 0 1 -20 -38 59 44 7 -3 30 }}
-POTE generator: ~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 671: / Line 908: @@
 Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
-Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 34 }}]
+Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 | 0 1 -20 -38 59 44 7 -3 30 34 }}
-POTE generator: ~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 684: / Line 923: @@
 Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
-Mapping: [{{val| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 }}, {{val| 0 1 -20 -38 59 44 7 -3 30 34 53 }}]
+Mapping: {{mapping| 1 0 34 63 -90 -66 -7 9 -43 -49 -79 | 0 1 -20 -38 59 44 7 -3 30 34 53 }}
-POTE generator: ~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 697: / Line 938: @@
 Comma list: 225/224, 441/440, 109375/107811
-Mapping: [{{val| 1 0 34 63 89 }}, {{val| 0 1 -20 -38 -54 }}]
+Mapping: {{mapping| 1 0 34 63 89 | 0 1 -20 -38 -54 }}
-POTE generator: ~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 710: / Line 953: @@
 Comma list: 225/224, 364/363, 441/440, 6125/6084
-Mapping: [{{val| 1 0 34 63 89 113 }}, {{val| 0 1 -20 -38 -54 -69 }}]
+Mapping: {{mapping| 1 0 34 63 89 113 | 0 1 -20 -38 -54 -69 }}
-POTE generator: ~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 723: / Line 968: @@
 Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
-Mapping: [{{val| 1 0 34 63 89 113 -7 }}, {{val| 0 1 -20 -38 -54 -69 7 }}]
+Mapping: {{mapping| 1 0 34 63 89 113 -7 | 0 1 -20 -38 -54 -69 7 }}
-POTE generator: ~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 736: / Line 983: @@
 Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
-Mapping: [{{val| 1 0 34 63 89 113 -7 9 }}, {{val| 0 1 -20 -38 -54 -69 7 -3 }}]
+Mapping: {{mapping| 1 0 34 63 89 113 -7 9 | 0 1 -20 -38 -54 -69 7 -3 }}
-POTE generator: ~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 ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 5764801/5668704
-[[Mapping]]: [{{val| 1 9 0 13 }}, {{val| 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]]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 }}
-[[POTE generator]]: ~48/35 = 556.221
+{{Optimal ET sequence|legend=1| 13d, 28d, 41, 151cd, 192cdd, 233ccdd }}
-{{Optimal ET sequence|legend=1| 28d, 41, 151cd, 192cd, 233cd }}
+[[Badness]] (Sintel): 3.27
-[[Badness]]: 0.129258
 === 11-limit ===
@@ Line 764: / Line 1,016: @@
 Comma list: 225/224, 385/384, 12250/11979
-Mapping: [{{val| 1 9 0 13 3 }}, {{val| 0 -16 5 -22 1 }}]
+Mapping: {{mapping| 1 -7 5 -9 4 | 0 16 -5 22 -1 }}
-POTE generator: ~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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 4194304/4117715
-[[Mapping]]: [{{Val|1 3 1 3}}, {{Val|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}}
-[[POTE generator]]: ~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 794: / Line 1,051: @@
 Comma list: 99/98, 176/175, 1310720/1294139
-Mapping: [{{Val|1 3 1 3 1}}, {{Val|0 -15 14 -2 26}}]
+Mapping: {{mapping| 1 -12 15 1 27 | 0 15 -14 2 -26 }}
-POTE generator: ~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 807: / Line 1,066: @@
 Comma list: 99/98, 176/175, 640/637, 847/845
-Mapping: [{{Val|1 3 1 3 1 2}}, {{Val|0 -15 14 -2 26 18}}]
+Mapping: {{mapping| 1 -12 15 1 27 20 | 0 15 -14 2 -26 -18 }}
-POTE generator: ~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 820: / Line 1,081: @@
 Comma list: 225/224, 385/384, 66550/64827
-Mapping: [{{Val|1 3 1 3 6}}, {{Val|0 -15 14 -2 -27}}]
+Mapping: {{mapping| 1 3 1 3 6 | 0 -15 14 -2 -27 }}
-POTE generator: ~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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 125000000/121060821
-[[Mapping]]: [{{Val|1 6 -7 -7}}, {{Val|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 }}
-[[POTE generator]]: ~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 850: / Line 1,118: @@
 Comma list: 121/120, 225/224, 22000/21609
-Mapping: [{{Val|1 6 -7 -7 1}}, {{Val|0 -9 19 20 5}}]
+Mapping: {{mapping| 1 -3 12 13 6 | 0 9 -19 -20 -5 }}
-POTE generator: ~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 863: / Line 1,133: @@
 Comma list: 121/120, 225/224, 275/273, 1040/1029
-Mapping: [{{Val|1 6 -7 -7 1 -12}}, {{Val|0 -9 19 20 5 32}}]
+Mapping: {{mapping| 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.
-POTE generator: ~7/5 = 588.654
+[[Subgroup]]: 2.3.5.7
-{{Optimal ET sequence|legend=1| 51f, 53 }}
+[[Comma list]]: 225/224, 823543/800000
-Badness: 0.047441
+{{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 ==
-Subgroup: 2.3.5.7
+Named by [[Scott Dakota]], quintannic may be described as the {{nowrap| 43 & 60 }} temperament.
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 9805926501/9765625000
-[[Mapping]]: [{{val| 1 1 5 7 }}, {{val| 0 5 -23 -36 }}]
+{{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}}
-[[POTE generator]]: ~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 891: / Line 1,233: @@
 Comma list: 225/224, 441/440, 43923/43750
-Mapping: [{{val| 1 1 5 7 8 }}, {{val| 0 5 -23 -36 -39 }}]
+Mapping: {{mapping| 1 1 5 7 8 | 0 5 -23 -36 -39 }}
-POTE generator: ~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 904: / Line 1,248: @@
 Comma list: 225/224, 441/440, 1001/1000, 1188/1183
-Mapping: [{{val| 1 1 5 7 8 3 }}, {{val| 0 5 -23 -36 -39 6 }}]
+Mapping: {{mapping| 1 1 5 7 8 3 | 0 5 -23 -36 -39 6 }}
-POTE generator: ~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 917: / Line 1,263: @@
 Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
-Mapping: [{{val| 1 1 5 7 8 3 7 }}, {{val| 0 5 -23 -36 -39 6 -25 }}]
+Mapping: {{mapping| 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
+{{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 }}
-POTE generator: ~13/12 = 139.815
+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=1| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 22, 55d }}
-Badness: 0.023038
+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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 5971968/5764801
-[[Mapping]]: [{{val| 2 1 6 4 }}, {{val| 0 8 -5 6 }}]
+{{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}}
-[[POTE generator]]: ~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 947: / Line 1,338: @@
 Comma list: 99/98, 176/175, 65536/65219
-Mapping: [{{val| 2 1 6 4 8 }}, {{val| 0 8 -5 6 -4 }}]
+Mapping: {{mapping| 2 1 6 4 8 | 0 8 -5 6 -4 }}
-POTE generator: ~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
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 14495514624/13841287201
-[[Mapping]]: [{{val| 3 7 5 9 }}, {{val| 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}}
-[[POTE generator]]: ~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 977: / Line 1,375: @@
 Comma list: 225/224, 3840/3773, 12005/11979
-Mapping: [{{val| 3 7 5 9 9 }}, {{val| 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}}
-POTE generator: ~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]]
 [[Category:Marvel temperaments| ]] <!-- main article -->
 [[Category:Rank 2]]

Marvel temperaments: Difference between revisions