@@ 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]]
-* [[Pajara]], {50/49, 64/63} → [[Diaschismic family #Pajara|Diaschismic family]]
+* [[Pajara]] (+50/49 or 64/63) → [[Diaschismic family #Pajara|Diaschismic family]]
-* [[Meantone]], {81/80, 126/125} → [[Meantone family #Septimal meantone|Meantone family]]
+* ''[[Sharpie]]'' (+25/24 or 28/27) → [[Dicot family #Sharpie|Dicot family]]
-* [[Magic]], {225/224, 245/243} → [[Magic family #Magic|Magic family]]
+* ''[[Immune]]'' (+781250/750141) → [[Immunity family #Immune|Immunity family]]
-* ''[[Passive]]'', {225/224, 256/245} → [[Passion family #Passive|Passion family]]
+* ''[[August]]'' (+36/35 or 128/125) → [[Augmented family #August|Augmented family]]
-* [[Miracle]], {225/224, 1029/1024} → [[Gamelismic clan #Miracle|Gamelismic clan]]
+* ''[[Fog]]'' (+156250/151263) → [[Misty family #Fog|Misty family]]
-* [[Orwell]], {225/224, 1728/1715} → [[Semicomma family #Orwell|Semicomma family]]
+* [[Bunya]] (+15625/15309) → [[Tetracot family #Bunya|Tetracot family]]
-* [[Garibaldi temperament|Garibaldi]], {225/224, 3125/3087} → [[Schismatic family #Garibaldi|Schismatic family]]
+* [[Negri]] (+49/48) → [[Semaphoresmic clan #Negri|Semaphoresmic clan]]
-* [[Catakleismic]], {225/224, 4375/4374} → [[Kleismic family #Catakleismic|Kleismic family]]
+* [[Magic]] (+245/243) → [[Magic family #Magic|Magic family]]
-* ''[[Snipes]]'', {225/224, 6125/5832} → [[Wesley family #Snipes|Wesley family]]
+* ''[[Passive]]'' (+256/245) → [[Passion family #Passive|Passion family]]
-* ''[[Decic]]'', {225/224, 16807/16384} → [[Cloudy clan #Decic|Cloudy clan]]
+* ''[[Quintapole]]'' (+7812500/7411887) → [[Quintaleap family #Quintapole|Quintaleap family]]
-* ''[[Amavil]]'', {225/224, 17496/16807} → [[Mabila family #Amavil|Mabila family]]
+* ''[[Houborizic]]'' (+1250000/1240029) → [[Amity family #Houborizic|Amity family]]
-* ''[[Escapade]]'', {225/224, 65625/65536} → [[Escapade family #Escapade|Escapade family]]
+* ''[[Qintosec]]'' (+2560000/2470629) → [[Quintosec family #Qintosec|Quintosec family]]
-* ''[[Fog]]'', {225/224, 156250/151263} → [[Misty family #Fog|Misty family]]
+* [[Miracle]] (+1029/1024) → [[Gamelismic clan #Miracle|Gamelismic clan]]
-* ''[[Compton]]'', {225/224, 250047/250000} → [[Compton family #Compton|Compton family]]
+* [[Catakleismic]] (+4375/4374) → [[Kleismic family #Catakleismic|Kleismic family]]
-* ''[[Immune]]'', {225/224, 781250/750141} → [[Immunity family #Immune|Immunity family]]
+* ''[[Marvo]]'' (+78125000/78121827) → [[Gravity family #Marvo|Gravity family]]
-* ''[[Betic]]'', {225/224, 1071875/1062882} → [[Sycamore family #Betic|Sycamore family]]
+* [[Orwell]] (+1728/1715) → [[Semicomma family #Orwell|Semicomma family]]
-* ''[[Houborizic]]'' {225/224, 1250000/1240029} → [[Amity family #Houborizic|Amity family]]
+* ''[[Snipes]]'' (+6125/5832)  → [[Wesley family #Snipes|Wesley family]]
-* ''[[Qintosec]]'', {225/224, 2560000/2470629} → [[Qintosec family #Qintosec|Qintosec family]]
+* ''[[Demibuzzard]]'' (+65536/64827) → [[Buzzardsmic clan #Demibuzzard|Buzzardsmic clan]]
-* ''[[Quintapole]]'', {225/224, 7812500/7411887} → [[Quintaleap family #Quintapole|Quintaleap family]]
+* ''[[Escapist]]'' (+65625/65536) → [[Escapade family #Escapist|Escapade family]]
-* ''[[Maquila]]'', {225/224, 30233088/28824005} → [[Maquila family #Septimal maquila|Maquila family]]
+* ''[[Decic]]'' (+16807/16384) → [[Cloudy clan #Decic|Cloudy clan]]
-* ''[[Marvo]]'', {225/224, 78125000/78121827} → [[Gravity family #Marvo|Gravity family]]
+* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]]
-* ''[[Gammy]]'', {225/224, 94143178827/91913281250} → [[Gammic family #Gammy|Gammic family]]
+* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]]
+* ''[[Hendeca]]'' (+122880/117649) → [[11th-octave temperaments #Hendeca|11th-octave temperaments]]
+* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]]
+* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]]
+* ''[[Maquila]]'' (+30233088/28824005) → [[Maquila family #Septimal maquila|Maquila family]]
+* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]]
-Considered below are negri, 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, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, 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}}.)
-== Negri ==
+== Wizard ==
-{{main| Negri }}
+{{Main| Wizard }}
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Wizard]].''
-Negri tempers out the [[negri comma]] in the 5-limit, [[49/48]] and [[225/224]] in the 7-limit. It can be extended naturally to the 2.3.5.7.13 subgroup by adding [[91/90]] to the comma list; this will be discussed below under the title of negra.
-Subgroup: 2.3.5
-[[Comma list]]: 16875/16384
-[[Mapping]]: [{{val| 1 2 2 }}, {{val| 0 -4 3 }}]
-{{Multival|legend=1| 4 -3 -14 }}
-[[POTE generator]]: ~16/15 = 125.7549
-{{Val list|legend=1| 9, 10, 19, 67c, 86c, 105c }}
-[[Badness]]: 0.086856
-=== 7-limit ===
-Subgroup: 2.3.5.7
-[[Comma list]]: 49/48, 225/224
-[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -4 3 -2 }}]
-{{Multival|legend=1| 4 -3 2 -14 -8 13 }}
-[[POTE generator]]: ~15/14 = 125.608
-{{Val list|legend=1| 9, 10, 19, 48d, 67cdd, 86cdd }}
-[[Badness]]: 0.026483
-==== Negra ====
-This is the 2.3.5.7.13 extension of negri.
-Subgroup: 2.3.5.7.13
-[[Comma list]]: 49/48, 65/64, 91/90
-[[Sval]] [[mapping]]: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -3 }}]
-[[Gencom]] [[mapping]]: [{{val| 1 2 2 3 0 4 }}, {{val| 0 -4 3 -2 0 -3 }}]
-[[Gencom]]: [2 14/13; 49/48 65/64 91/90]
-[[POTE generator]]: ~14/13 = 125.567
-{{Val list|legend=1| 9, 10, 19, 48df, 67cddf, 86cddff }}
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 49/48, 56/55
-Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -4 3 -2 -5 }}]
-POTE generator: ~15/14 = 126.474
-Optimal GPV sequence: {{Val list| 9, 10, 19 }}
-Badness: 0.026190
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 45/44, 49/48, 56/55, 78/77
-Mapping: [{{val| 1 2 2 3 4 4 }}, {{val| 0 -4 3 -2 -5 -3 }}]
-POTE generator: ~14/13 = 126.431
-Optimal GPV sequence: {{Val list| 9, 10, 19 }}
-Badness: 0.017833
-=== Negril ===
-Subgroup: 2.3.5.7.11
-Comma list: 49/48, 100/99, 225/224
-Mapping: [{{val| 1 2 2 3 2 }}, {{val| 0 -4 3 -2 14 }}]
-POTE generator: ~15/14 = 124.767
-Optimal GPV sequence: {{Val list| 19, 29, 48d, 77cdd }}
+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.
-Badness: 0.038679
+[[Subgroup]]: 2.3.5.7
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 49/48, 65/64, 91/90, 875/858
-Mapping: [{{val| 1 2 2 3 2 4 }}, {{val| 0 -4 3 -2 14 -3 }}]
-POTE generator: ~14/13 = 124.716
-Optimal GPV sequence: {{Val list| 19, 29, 48df, 77cddf }}
-Badness: 0.024383
-=== Negric ===
-Subgroup: 2.3.5.7.11
-Comma list: 33/32, 49/48, 77/75
-Mapping: [{{val| 1 2 2 3 3 }}, {{val| 0 -4 3 -2 4 }}]
-POTE generator: ~15/14 = 127.039
-Optimal GPV sequence: {{Val list| 9, 19e }}
-Badness: 0.030617
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 33/32, 49/48, 65/64, 91/90
-Mapping: [{{val| 1 2 2 3 3 4 }}, {{val| 0 -4 3 -2 4 -3 }}]
-POTE generator: ~14/13 = 127.039
-Optimal GPV sequence: {{Val list| 9, 19e }}
-Badness: 0.020205
-=== Negroni ===
-Subgroup: 2.3.5.7.11
-Comma list: 49/48, 55/54, 225/224
-Mapping: [{{val| 1 2 2 3 5 }}, {{val| 0 -4 3 -2 -15 }}]
-POTE generator: ~15/14 = 124.539
-Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddee }}
-Badness: 0.035296
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 49/48, 55/54, 65/64, 91/90
-Mapping: [{{val| 1 2 2 3 5 4 }}, {{val| 0 -4 3 -2 -15 -3 }}]
-POTE generator: ~14/13 = 124.545
-Optimal GPV sequence: {{Val list| 10, 19e, 29, 77cddeef }}
-Badness: 0.021559
-=== Wilsec ===
-Subgroup: 2.3.5.7.11
-Comma list: 49/48, 121/120, 225/224
-Mapping: [{{val| 1 6 -1 5 4 }}, {{val| 0 -8 6 -4 -1 }}]
-POTE generator: ~11/8 = 537.186
-Optimal GPV sequence: {{Val list| 9, 20, 29, 38d, 67cdde }}
-Badness: 0.041886
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 49/48, 65/64, 91/90, 121/120
-Mapping: [{{val| 1 6 -1 5 4 7 }}, {{val| 0 -8 6 -4 -1 -6 }}]
-POTE generator: ~11/8 = 537.208
-Optimal GPV sequence: {{Val list| 9, 20, 29, 38df, 67cddef }}
-Badness: 0.025192
-==== 17-limit ====
-Subgroup: 2.3.5.7.11.13.17
-Comma list: 49/48, 65/64, 91/90, 121/120, 154/153
-Mapping: [{{val| 1 6 -1 5 4 7 -2 }}, {{val| 0 -8 6 -4 -1 -6 11 }}]
-POTE generator: ~11/8 = 537.230
-Optimal GPV sequence: {{Val list| 9, 20g, 29g, 38df, 67cddefg }}
-Badness: 0.021778
-==== 19-limit ====
-Subgroup: 2.3.5.7.11.13.17.19
-Comma list: 49/48, 65/64, 77/76, 91/90, 121/120, 154/153
-Mapping: [{{val| 1 6 -1 5 4 7 -2 7 }}, {{val| 0 -8 6 -4 -1 -6 11 -5 }}]
-POTE generator: ~11/8 = 537.214
-Optimal GPV sequence: {{Val list| 9, 20g, 29g, 38df, 67cddefgh }}
-Badness: 0.016828
-== Wizard ==
-{{Main|Wizard}}
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Wizard]].''
-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 }}
-{{Val list|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 270: / Line 70: @@
 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 GPV sequence: {{Val list| 22, 50, 72, 166, 238c, 310c }}
+Badness (Sintel): 0.613
-Badness: 0.018539
-Scales: [[wizard22]]
 ==== Lizard ====
@@ Line 287: / Line 85: @@
 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
-POTE generator: ~5/4 = 383.389
+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 GPV sequence: {{Val list| 22, 50, 72, 122, 194df }}
+{{Optimal ET sequence|legend=0| 22, 50, 72 }}
-Badness: 0.021781
+Badness (Sintel): 0.900
 ===== 17-limit =====
@@ Line 302: / Line 100: @@
 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
-POTE generator: ~5/4 = 383.381
+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 GPV sequence: {{Val list| 22, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 22, 50, 72 }}
-Badness: 0.014536
+Badness (Sintel): 0.741
 ===== 19-limit =====
@@ Line 317: / Line 115: @@
 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
-POTE generator: ~5/4 = 383.477
+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 GPV sequence: {{Val list| 22h, 50, 72, 122g, 194dfg }}
+{{Optimal ET sequence|legend=0| 22h, 50, 72, 122g, 194dfg }}
-Badness: 0.015702
+Badness (Sintel): 0.955
 ==== Gizzard ====
@@ Line 332: / Line 130: @@
 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 GPV sequence: {{Val list| 72, 166, 238cf }}
+{{Optimal ET sequence|legend=0| 22f, 72, 166, 238cf }}
-Badness: 0.020252
+Badness (Sintel): 0.837
 ===== 17-limit =====
@@ Line 347: / Line 145: @@
 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 GPV sequence: {{Val list| 72, 166g, 238cfg }}
+{{Optimal ET sequence|legend=0| 22f, 72, 166g, 238cfg }}
-Badness: 0.013624
+Badness (Sintel): 0.694
 ===== 19-limit =====
@@ Line 362: / Line 160: @@
 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
-POTE generator: ~5/4 = 383.138
+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 GPV sequence: {{Val list| 72, 94, 166g }}
+{{Optimal ET sequence|legend=0| 72, 94, 166g }}
-Badness: 0.014810
+Badness (Sintel): 0.901
 === Mage ===
@@ Line 377: / Line 175: @@
 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 GPV sequence: {{Val list| 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 }}
-[[POTE generator]]: ~7/5 = 580.286
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1201.3539{{c}}, ~10/7 = 620.4131{{c}}
+: [[error map]]: {{val| +1.354 -1.243 -0.027 -1.598 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.6778{{c}}
+: error map: {{val| 0.000 -3.566 -2.769 -4.959 }}
-{{Val list|legend=1| 29, 31, 60, 91, 122, 213bcd }}
+{{Optimal ET sequence|legend=1| 29, 31, 60, 91, 122, 213bcd }}
-[[Badness]]: 0.047578
+[[Badness]] (Sintel): 1.20
 === 11-limit ===
@@ Line 409: / Line 212: @@
 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 GPV sequence: {{Val list| 29, 31, 60e }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e, 91e, 213bcdeee }}
-Badness: 0.023659
+Badness (Sintel): 0.782
 ==== 13-limit ====
@@ Line 422: / Line 227: @@
 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 }}
-POTE generator: ~7/5 = 580.108
+Optimal tunings:
+* WE: ~2 = 1201.5355{{c}}, ~10/7 = 620.6855{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8469{{c}}
-Optimal GPV sequence: {{Val list| 29, 31, 60e, 151cde }}
+{{Optimal ET sequence|legend=0| 29, 31, 60e }}
-Badness: 0.022993
+Badness (Sintel): 0.950
-=== Tritoni ===
+==== 17-limit ====
-Subgroup: 2.3.5.7.11
+Subgroup: 2.3.5.7.11.13.17
-Comma list: 225/224, 385/384, 27783/27500
+Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
-Mapping: [{{val| 1 4 -3 -3 17 }}, {{val| 0 -5 11 12 -28 }}]
+Mapping: {{mapping| 1 -1 8 9 5 13 17 | 0 5 -11 -12 -3 -18 -25 }}
-POTE generator: ~7/5 = 580.389
+Optimal tunings:
+* WE: ~2 = 1201.5260{{c}}, ~10/7 = 620.7330{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.8986{{c}}
-Optimal GPV sequence: {{Val list| 31, 91, 122, 153d }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-Badness: 0.045456
+Badness (Sintel): 0.973
-== Septimin ==
+==== 19-limit ====
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Septimin]].''
+Subgroup: 2.3.5.7.11.13.17.19
-Subgroup: 2.3.5.7
+Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
-[[Comma list]]: 225/224, 84035/82944
+Mapping: {{mapping| 1 -1 8 9 5 13 17 12 | 0 5 -11 -12 -3 -18 -25 -15 }}
-[[Mapping]]: [{{val| 1 4 1 5 }}, {{val| 0 -11 6 -10 }}]
+Optimal tunings:
+* WE: ~2 = 1201.3100{{c}}, ~10/7 = 620.6509{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9328{{c}}
-{{Multival|legend=1| 11 -6 10 -35 -15 40 }}
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-[[POTE generator]]: ~7/6 = 263.632
+Badness (Sintel): 1.03
-{{Val list|legend=1| 41, 91, 132d }}
+==== 23-limit ====
+Subgroup: 2.3.5.7.11.13.17.19.23
-[[Badness]]: 0.054502
+Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
-=== 11-limit ===
+Mapping: {{mapping| 1 -1 8 9 5 13 17 12 4 | 0 5 -11 -12 -3 -18 -25 -15 1 }}
-Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 2401/2376
+Optimal tunings:
+* WE: ~2 = 1201.4074{{c}}, ~10/7 = 620.7185{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.9548{{c}}
-Mapping: [{{val| 1 4 1 5 5 }}, {{val| 0 -11 6 -10 -7 }}]
+{{Optimal ET sequence|legend=0| 29g, 31, 60e }}
-POTE generator: ~7/6 = 263.634
+Badness (Sintel): 1.04
-Optimal GPV sequence: {{Val list| 41, 91, 223cdef }}
+=== Tritoni ===
+Subgroup: 2.3.5.7.11
-Badness: 0.031309
+Comma list: 225/224, 385/384, 27783/27500
-=== 13-limit ===
+Mapping: {{mapping| 1 -1 8 9 -11 | 0 5 -11 -12 28 }}
-Subgroup: 2.3.5.7.11.13
-Comma list: 105/104, 144/143, 196/195, 245/242
+Optimal tunings:
+* WE: ~2 = 1201.0888{{c}}, ~10/7 = 620.1733{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.6146{{c}}
-Mapping: [{{val| 1 4 1 5 5 7 }}, {{val| 0 -11 6 -10 -7 -15 }}]
+{{Optimal ET sequence|legend=0| 31, 91, 122, 153d }}
-POTE generator: ~7/6 = 263.700
+Badness (Sintel): 1.50
-Optimal GPV sequence: {{Val list| 41, 91 }}
+== Septimin ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Septimin]].''
-Badness: 0.023117
+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.
-== Merman ==
+[[Subgroup]]: 2.3.5.7
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Merman]].''
-Subgroup: 2.3.5.7
+[[Comma list]]: 225/224, 84035/82944
-[[Comma list]]: 225/224, 2500000/2470629
-[[Mapping]]: [{{val| 1 5 -5 -5 }}, {{val| 0 -7 15 16 }}]
-{{Multival|legend=1| 7 -15 -16 -40 -45 5 }}
+{{Mapping|legend=1| 1 -7 7 -5 | 0 11 -6 10 }}
+: mapping generators: ~2, ~12/7
-[[POTE generator]]: ~7/5 = 585.585
+[[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 }}
-{{Val list|legend=1| 41, 84, 125 }}
+{{Optimal ET sequence|legend=1| 41, 91, 132d }}
-[[Badness]]: 0.055078
+[[Badness]] (Sintel): 1.38
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 441/440, 1344/1331
+Comma list: 225/224, 245/242, 385/384
-Mapping: [{{val| 1 5 -5 -5 2 }}, {{val| 0 -7 15 16 3 }}]
+Mapping: {{mapping| 1 -7 7 -5 -2 | 0 11 -6 10 7 }}
-POTE generator: ~7/5 = 585.606
+Optimal tunings:
+* WE: ~2 = 1200.8059{{c}}, ~12/7 = 936.9952{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3906{{c}}
-Optimal GPV sequence: {{Val list| 41, 84, 125e }}
+{{Optimal ET sequence|legend=0| 41, 91, 223cdef }}
-Badness: 0.036383
+Badness (Sintel): 1.04
 === 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 144/143, 225/224, 364/363, 441/440
+Comma list: 105/104, 144/143, 196/195, 245/242
+Mapping: {{mapping| 1 -7 7 -5 -2 -8 | 0 11 -6 10 7 15 }}
-Mapping: [{{val| 1 5 -5 -5 2 12 }}, {{val| 0 -7 15 16 3 -17 }}]
+Optimal tunings:
+* WE: ~2 = 1200.5990{{c}}, ~12/7 = 936.7670{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~12/7 = 936.3196{{c}}
-POTE generator: ~7/5 = 585.657
+{{Optimal ET sequence|legend=0| 41, 91 }}
-Optimal GPV sequence: {{Val list| 41, 84, 125e, 209ef, 293ef }}
+Badness (Sintel): 0.955
-Badness: 0.027544
+== Merman ==
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Merman]].''
-== Slender ==
+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.
-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.
-Subgroup: 2.3.5.7
+The name was likely derived from {{w|Triton (mythology)|''Triton''}}, which was in turn derived from ''tritonic''.
-[[Comma list]]: 225/224, 589824/588245
+[[Subgroup]]: 2.3.5.7
-[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -13 10 -6 }}]
+[[Comma list]]: 225/224, 2500000/2470629
-{{Multival|legend=1| 13 -10 6 -46 -27 42 }}
+{{Mapping|legend=1| 1 -2 10 11 | 0 7 -15 -16 }}
+: mapping generators: ~2, ~10/7
-[[POTE generator]]: ~49/48 = 38.413
+[[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 }}
-{{Val list|legend=1| 31, 94, 125 }}
+{{Optimal ET sequence|legend=1| 41, 84, 125 }}
-[[Badness]]: 0.056934
+[[Badness]] (Sintel): 1.39
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 1331/1323
+Comma list: 225/224, 441/440, 1344/1331
-Mapping: [{{val| 1 2 2 3 4 }}, {{val| 0 -13 10 -6 -17 }}]
+Mapping: {{mapping| 1 -2 10 11 5 | 0 7 -15 -16 -3 }}
-POTE generator: ~49/48 = 38.387
+Optimal tunings:
+* WE: ~2 = 1199.9578{{c}}, ~10/7 = 614.3720{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3943{{c}}
-Optimal GPV sequence: {{Val list| 31, 63, 94, 125 }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e }}
-Badness: 0.025342
+Badness (Sintel): 1.20
-=== 13-limit ===
+==== 13-limit ====
 Subgroup: 2.3.5.7.11.13
-Comma list: 225/224, 275/273, 385/384, 1331/1323
+Comma list: 144/143, 225/224, 364/363, 441/440
-Mapping: [{{val| 1 2 2 3 4 3 }}, {{val| 0 -13 10 -6 -17 22 }}]
+Mapping: {{mapping| 1 -2 10 11 5 -5 | 0 7 -15 -16 -3 17 }}
-POTE generator: ~49/48 = 38.314
+Optimal tunings:
+* WE: ~2 = 1199.7422{{c}}, ~10/7 = 614.2110{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.3442{{c}}
-Optimal GPV sequence: {{Val list| 31, 63, 94 }}
+{{Optimal ET sequence|legend=0| 41, 84, 125e, 209ef, 293ef }}
-Badness: 0.025913
+Badness (Sintel): 1.14
-== Triton ==
+=== Mermaid ===
-: ''For the 5-limit version of this temperament, see [[High badness temperaments #Stump]].''
+Subgroup: 2.3.5.7.11
-Subgroup: 2.3.5.7
+Comma list: 225/224, 385/384, 532400/531441
-[[Comma list]]: 225/224, 1029/1000
+Mapping: {{mapping| 1 -2 10 11 -16 | 0 7 -15 -16 38 }}
-[[Mapping]]: [{{val| 1 0 6 7 }}, {{val| 0 3 -7 -8 }}]
+Optimal tunings:
+* WE: ~2 = 1199.4973{{c}}, ~10/7 = 614.7004{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4470{{c}}
-{{Multival|legend=1| 3 -7 -8 -18 -21 1 }}
+{{Optimal ET sequence|legend=0| 41, 84e, 125, 166 }}
-[[POTE generator]]: ~7/5 = 568.865
+Badness (Sintel): 1.46
-{{Val list|legend=1| 2, 17d, 19, 78bd, 97bd }}
+==== 13-limit ====
+Subgroup: 2.3.5.7.11.13
-[[Badness]]: 0.059245
+Comma list: 225/224, 325/324, 385/384, 10648/10647
-=== 11-limit ===
+Mapping: {{mapping| 1 -2 10 11 22 32 | 0 7 -15 -16 38 58 }}
-Subgroup: 2.3.5.7.11
-Comma list: 45/44, 56/55, 1029/1000
+Optimal tunings:
+* WE: ~2 = 1200.5126{{c}}, ~10/7 = 614.7152{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 614.4562{{c}}
-Mapping: [{{val| 1 0 6 7 4 }}, {{val| 0 3 -7 -8 -1 }}]
+{{Optimal ET sequence|legend=0| 41, 84ef, 125f, 166 }}
-POTE generator: ~7/5 = 569.144
+Badness (Sintel): 1.47
-Optimal GPV sequence: {{Val list| 2, 17d, 19, 59bde, 78bde, 97bde }}
+== Slender ==
+Slender tempers out the [[hewuermera comma]] in addition to the marvel comma, and may be described as the {{nowrap| 31 & 32 }} temperament. This temperament has a generator of [[49/48]], three of which equal marvel's [[16/15]][[~]][[15/14]], and ten generators give [[5/4]]. Its [[ploidacot]] is omega-13-cot.
-Badness: 0.045675
+The name was likely derived from ''slendro diesis'', one of the names for the interval 49/48.
-== Submajor ==
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5
-[[Comma list]]: 69198046875/68719476736
+[[Comma list]]: 225/224, 589824/588245
-[[Mapping]]: [{{val| 1 4 -1 }}, {{val| 0 -8 11 }}]
-[[POTE generator]]: ~10125/8192 = 362.321
+{{Mapping|legend=1| 1 2 2 3 | 0 -13 10 -6 }}
+: mapping generators: ~2, ~49/48
-{{Val list|legend=1| 10, 33, 43, 53, 202, 255, 308, 361, 414, 775, 1189bc }}
+[[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 }}
-[[Badness]]: 0.130236
+{{Optimal ET sequence|legend=1| 31, 94, 125, 406c }}
-=== 7-limit ===
+[[Badness]] (Sintel): 1.44
-Subgroup: 2.3.5.7
-[[Comma list]]: 225/224, 51200/50421
-[[Mapping]]: [{{val| 1 4 -1 1 }}, {{val| 0 -8 11 6 }}]
-{{Multival|legend=1| 8 -11 -6 -36 -32 17 }}
-[[POTE generator]]: ~49/40 = 362.255
-{{Val list|legend=1| 10, 33, 43, 53 }}
-[[Badness]]: 0.060533
 === 11-limit ===
 Subgroup: 2.3.5.7.11
-Comma list: 225/224, 385/384, 6655/6561
+Comma list: 225/224, 385/384, 1331/1323
-Mapping: [{{val| 1 4 -1 1 11 }}, {{val| 0 -8 11 6 -25 }}]
+Mapping: {{mapping| 1 2 2 3 4 | 0 -13 10 -6 -17 }}
-POTE generator: ~27/22 = 362.101
+Optimal tunings:
+* WE: ~2 = 1199.4983{{c}}, ~49/48 = 38.4030{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3775{{c}}
-Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cde }}
+{{Optimal ET sequence|legend=0| 31, 63, 94, 125 }}
-Badness: 0.050582
+Badness (Sintel): 0.838
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
-Comma list: 169/168, 225/224, 275/273, 385/384
+Comma list: 225/224, 275/273, 385/384, 1331/1323
-Mapping: [{{val| 1 4 -1 1 11 4 }}, {{val| 0 -8 11 6 -25 -1 }}]
+Mapping: {{mapping| 1 2 2 3 4 3 | 0 -13 10 -6 -17 22 }}
-POTE generator: ~16/13 = 362.105
+Optimal tunings:
+* WE: ~2 = 1200.1728{{c}}, ~49/48 = 38.3192{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 38.3129{{c}}
-Optimal GPV sequence: {{Val list| 10, 43e, 53, 116, 169de, 285cdef }}
+{{Optimal ET sequence|legend=0| 31, 63, 94 }}
-Badness: 0.027689
+Badness (Sintel): 1.07
-=== Interpental ===
+== Triton ==
-Subgroup: 2.3.5.7.11
+: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Stump]].''
-Comma list: 99/98, 176/175, 51200/50421
+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.
-POTE generator: ~49/40 = 362.418
+[[Subgroup]]: 2.3.5.7
-Mapping: [{{val| 1 4 -1 1 -5 }}, {{val| 0 -8 11 6 28 }}]
+[[Comma list]]: 225/224, 1029/1000
-POTE generator: ~49/40 = 362.418
+{{Mapping|legend=1| 1 0 6 7 | 0 3 -7 -8 }}
+: mapping generators: ~2, ~10/7
-Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1203.3828{{c}}, ~10/7 = 632.9137{{c}}
+: [[error map]]: {{val| +3.383 -3.214 +3.587 -8.457 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 630.9827{{c}}
+: error map: {{val| 0.000 -9.007 -3.192 -16.687 }}
-Badness: 0.051806
+{{Optimal ET sequence|legend=1| 2, 17d, 19, 78bd, 97bd }}
-==== 13-limit ====
+[[Badness]] (Sintel): 1.50
-Subgroup: 2.3.5.7.11.13
-Comma list: 99/98, 169/168, 176/175, 640/637
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
-POTE generator: ~16/13 = 362.402
+Comma list: 45/44, 56/55, 1029/1000
-Mapping: [{{val| 1 4 -1 1 -5 4 }}, {{val| 0 -8 11 6 28 -1 }}]
+Mapping: {{mapping| 1 0 6 7 4 | 0 3 -7 -8 -1 }}
-POTE generator: ~16/13 = 362.402
+Optimal tunings:
+* WE: ~2 = 1201.3875{{c}}, ~10/7 = 631.5852{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 630.8007{{c}}
-Optimal GPV sequence: {{Val list| 43, 53, 96, 149d }}
+{{Optimal ET sequence|legend=0| 2, 17d, 19 }}
-Badness: 0.029680
+Badness (Sintel): 1.51
 == Marvolo ==
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[Comma list]]: 225/224, 156250000/155649627
-[[Mapping]]: [{{val| 1 2 1 1 }}, {{val| 0 -6 19 26 }}]
+{{Mapping|legend=1| 1 2 1 1 | 0 -6 19 26 }}
+: mapping generators: ~2, ~21/20
-{{Multival|legend=1| 6 -19 -26 -44 -58 -7 }}
-[[POTE generator]]: ~21/20 = 83.348
+[[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 }}
-{{Val list|legend=1| 29, 43, 72, 619bcd, 691bcd }}
+{{Optimal ET sequence|legend=1| 29, 43, 72, 619bbccd, 691bbccd }}
-[[Badness]]: 0.083338
+[[Badness]] (Sintel): 2.11
 === 11-limit ===
@@ Line 706: / Line 545: @@
 Comma list: 225/224, 441/440, 4000/3993
-Mapping: [{{val| 1 2 1 1 2 }}, {{val| 0 -6 19 26 21 }}]
+Mapping: {{mapping| 1 2 1 1 2 | 0 -6 19 26 21 }}
-POTE generator: ~21/20 = 83.340
+Optimal tunings:
+* WE: ~2 = 1200.7075{{c}}, ~21/20 = 83.3888{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3564{{c}}
-Optimal GPV sequence: {{Val list| 29, 43, 72 }}
+{{Optimal ET sequence|legend=0| 29, 43, 72 }}
-Badness: 0.028965
+Badness (Sintel): 0.958
 === 13-limit ===
@@ Line 719: / Line 560: @@
 Comma list: 169/168, 225/224, 364/363, 441/440
-Mapping: [{{val| 1 2 1 1 2 3 }}, {{val| 0 -6 19 26 21 10 }}]
+Mapping: {{mapping| 1 2 1 1 2 3 | 0 -6 19 26 21 10 }}
-POTE generator: ~21/20 = 83.330
+Optimal tunings:
+* WE: ~2 = 1200.9467{{c}}, ~21/20 = 83.3956{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3516{{c}}
-Optimal GPV sequence: {{Val list| 29, 43, 72, 115f }}
+{{Optimal ET sequence|legend=0| 29, 43, 72 }}
-Badness: 0.021470
+Badness (Sintel): 0.887
 === 17-limit ===
@@ Line 732: / Line 575: @@
 Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
-Mapping: [{{val| 1 2 1 1 2 3 2 }}, {{val| 0 -6 19 26 21 10 30 }}]
+Mapping: {{mapping| 1 2 1 1 2 3 2 | 0 -6 19 26 21 10 30 }}
-POTE generator: ~21/20 = 83.330
+Optimal tunings:
+* WE: ~2 = 1200.9606{{c}}, ~21/20 = 83.4030{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3594{{c}}
-Optimal GPV sequence: {{Val list| 29g, 43, 72 }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
+Badness (Sintel): 0.760
 === 19-limit ===
@@ Line 743: / Line 590: @@
 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 }}]
+Mapping: {{mapping| 1 2 1 1 2 3 2 3 | 0 -6 19 26 21 10 30 18 }}
-POTE generator: ~21/20 = 83.330
+Optimal tunings:
+* WE: ~2 = 1200.7625{{c}}, ~21/20 = 83.3895{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.3551{{c}}
-Optimal GPV sequence: {{Val list| 29g, 43, 72 }}
+{{Optimal ET sequence|legend=0| 29g, 43, 72 }}
+Badness (Sintel): 0.895
 == 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 }}
-[[POTE generator]]: ~5/4 = 383.165
+[[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 }}
-{{Val list|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 769: / Line 623: @@
 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 GPV sequence: {{Val list| 9, 63, 72, 423cd, 495bcd }}
+{{Optimal ET sequence|legend=0| 9, 54, 63, 72 }}
-Badness: 0.030426
+Badness (Sintel): 1.01
 === 13-limit ===
@@ Line 782: / Line 638: @@
 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 GPV sequence: {{Val list| 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 }}
-[[POTE generator]]: ~3/2 = 700.824
+[[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 }}
-{{Val list|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 812: / Line 673: @@
 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 GPV sequence: {{Val list| 12e, 101cde, 113, 125, 238c }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125, 238c }}
-Badness: 0.089588
+Badness (Sintel): 2.96
 ==== 13-limit ====
@@ Line 825: / Line 688: @@
 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 GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness: 0.052235
+Badness (Sintel): 2.16
 ==== 17-limit ====
@@ Line 838: / Line 703: @@
 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 GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness: 0.038565
+Badness (Sintel): 1.96
 ==== 19-limit ====
@@ Line 851: / Line 718: @@
 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 GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cf }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cf }}
-Badness: 0.028165
+Badness (Sintel): 1.71
 ==== 23-limit ====
@@ Line 864: / Line 733: @@
 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 GPV sequence: {{Val list| 12e, 101cde, 113, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 238cfi }}
-Badness: 0.021879
+Badness (Sintel): 1.57
 ==== 29-limit ====
@@ Line 877: / Line 748: @@
 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 GPV sequence: {{Val list| 12e, 101cde, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
-Badness: 0.018011
+Badness (Sintel): 1.50
 ==== 31-limit ====
@@ Line 890: / Line 763: @@
 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 GPV sequence: {{Val list| 12e, 101cdek, 113, 125f, 238cfi }}
+{{Optimal ET sequence|legend=0| 12e, 113, 125f, 238cfi }}
-Badness: 0.016007
+Badness (Sintel): 1.53
 === Gracecord ===
@@ Line 903: / Line 778: @@
 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 GPV sequence: {{Val list| 12, 101cd, 113 }}
+{{Optimal ET sequence|legend=0| 12, …, 101cd, 113 }}
-Badness: 0.066964
+Badness (Sintel): 2.21
 ==== 13-limit ====
@@ Line 916: / Line 793: @@
 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 GPV sequence: {{Val list| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, …, 101cdf, 113 }}
-Badness: 0.044196
+Badness (Sintel): 1.83
 ==== 17-limit ====
@@ Line 929: / Line 808: @@
 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 GPV sequence: {{Val list| 12f, 101cdf, 113 }}
+{{Optimal ET sequence|legend=0| 12f, 101cdf, 113 }}
-Badness: 0.036637
+Badness (Sintel): 1.87
 ==== 19-limit ====
@@ Line 942: / Line 823: @@
 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 GPV sequence: {{Val list| 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 }}
-[[POTE generator]]: ~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 }}
-{{Val list|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 970: / Line 856: @@
 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 GPV sequence: {{Val list| 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}}
-[[POTE generator]]: ~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 }}
-{{Val list|legend=1| 21, 32, 53 }}
+{{Optimal ET sequence|legend=1| 21, 32, 53 }}
-[[Badness]]: 0.140970
+[[Badness]] (Sintel): 3.57
 === 11-limit ===
@@ Line 1,000: / Line 891: @@
 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 GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
-Badness: 0.085390
+Badness (Sintel): 2.82
 ==== 13-limit ====
@@ Line 1,013: / Line 906: @@
 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 GPV sequence: {{Val list| 21, 32e, 53, 127, 180de }}
+{{Optimal ET sequence|legend=0| 21, 32e, 53, 127 }}
-Badness: 0.045569
+Badness (Sintel): 1.88
 === Musneb ===
@@ Line 1,026: / Line 921: @@
 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 GPV sequence: {{Val list| 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}}
-[[POTE generator]]: ~7/5 = 588.641
+[[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 }}
-{{Val list|legend=1| 51, 53, 316cd, 369cdd, 422cdd }}
+{{Optimal ET sequence|legend=1| 51, 53 }}
-[[Badness]]: 0.143976
+[[Badness]] (Sintel): 3.64
 === 11-limit ===
@@ Line 1,056: / Line 958: @@
 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 GPV sequence: {{Val list| 51, 53 }}
+{{Optimal ET sequence|legend=0| 51, 53 }}
-Badness: 0.074295
+Badness (Sintel): 2.46
 === 13-limit ===
@@ Line 1,069: / Line 973: @@
 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 }}
-POTE generator: ~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 GPV sequence: {{Val list| 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 ==
-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 }}
-[[POTE generator]]: ~10000/9261 = 139.838
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.9803{{c}}, ~10000/9261 = 139.9522{{c}}
+: [[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 }}
-{{Val list|legend=1| 43, 60, 103, 266bcd, 369bcd }}
+{{Optimal ET sequence|legend=1| 43, 60, 103, 266bcd, 369bcd }}
-[[Badness]]: 0.150565
+[[Badness]] (Sintel): 3.81
 === 11-limit ===
@@ Line 1,097: / Line 1,073: @@
 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 GPV sequence: {{Val list| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103, 369bcdeee, 472bbcddeee }}
-Badness: 0.052590
+Badness (Sintel): 1.74
 === 13-limit ===
@@ Line 1,110: / Line 1,088: @@
 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 GPV sequence: {{Val list| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness: 0.032730
+Badness (Sintel): 1.35
 === 17-limit ===
@@ Line 1,123: / Line 1,103: @@
 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 }}
-POTE generator: ~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 GPV sequence: {{Val list| 43, 60e, 103 }}
+{{Optimal ET sequence|legend=0| 43, 60e, 103 }}
-Badness: 0.023038
+Badness (Sintel): 1.17
 == 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}}
+: [[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 }}
-[[POTE generator]]: ~35/32 = 162.658
+{{Optimal ET sequence|legend=1| 22, 74, 96, 118d }}
-{{Val list|legend=1| 22, 74, 96, 118d }}
+[[Badness]] (Sintel): 4.53
-[[Badness]]: 0.178826
 === 11-limit ===
@@ Line 1,153: / Line 1,140: @@
 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 GPV sequence: {{Val list| 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]].''
-Subgroup: 2.3.5.7
+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
 [[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}}
+: [[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 }}
-[[POTE generator]]: ~15/14 = 112.283
+{{Optimal ET sequence|legend=1| 21, 54, 75, 96, 171d }}
-{{Val list|legend=1| 21, 54, 75, 96, 171d }}
+[[Badness]] (Sintel): 10.9
-[[Badness]]: 0.431636
 === 11-limit ===
@@ Line 1,183: / Line 1,177: @@
 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 GPV sequence: {{Val list| 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