2023/2000: Difference between revisions

VisualWikitext

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 '''2023/2000''', the '''pentagoth comma''', is the interval between a stack of two [[20/17]] minor thirds and one [[7/5]] tritone.
-[[Tempering out]] this comma alone in the 2.5.7.17 subgroup leads to the [[pentagoth family|pentagoth]] temperament.
+[[Tempering out]] this comma alone in the 2.5.7.17 subgroup leads to the '''pentagoth''' temperament.
+== Temperaments ==
+Note: groundfault uses extensions that identify the ~20/17~13/11 with ~19/16 and tempering out 209/208; however, this leads to considerable damage to prime 19.
+=== Pentagoth ===
+[[Subgroup]]: 2.5.7.17
+[[Comma list]]: 2023/2000
+{{Mapping|legend=1| 1 2 2 4 | 0 1 1 1 | 0 0 2 -1 }}
+: mapping generators: ~2, ~5/4, ~20/17
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1200.566{{c}}, ~5/4 = 387.475{{c}}, ~20/17 = 289.545{{c}}
+: [[error map]]: {{val| +0.566 2.294 -1.127 -4.760}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 388.049{{c}}, ~5/4 = 289.369{{c}}
+: error map: {{val| 0.000 1.735 -2.038 -6.275 }}
+[[Badness]] (Sintel): 0.509
+==== 2.5.7.13/11.17.23 ====
+[[Subgroup]]: 2.5.7.13/11.17.23
+[[Comma list]]: 221/220, 161/160, 1309/1300
+{{Mapping|legend=2|1 2 2 0 4 5|0 1 1 0 1 0|0 0 2 1 -1 -2}}
+[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~5/4 = 387.513, ~[[13/11]] = 288.786
+[[Badness]] (Sintel): 0.191
+===== No-3 no-19 23-limit =====
+[[Subgroup]]: 2.5.7.11.13.17.23
+[[Comma list]]: 176/175, 221/220, 161/160, 1309/1300
+{{Mapping|legend=2|1 2 2 2 2 4 5|0 1 1 3 3 1 0|0 0 2 2 3 -1 -2}}
+[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~5/4 = 390.453, ~[[13/11]] = 288.844
+[[Badness]] (Sintel): 0.497
+=== Pentagoth clan ===
+Rank-2 temperaments supporting pentagoth
+==== Vengeance ====
+{{Main| Vengeance }}
+A lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].
+[[Subgroup]]: 2.5.7.17
+[[Comma list]]: 2023/2000, 78608/78125
+{{Mapping|legend=2|1 1 5 1|0 3 -5 7}}
+[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[34/25]] = 527.718
+{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}
+[[Badness]] (Sintel): 0.720
+==== Sidewalk ====
+Sidewalk, 2.5.7.17[21 & 25], tempers out 823543/800000 in the 2.5.7 subgroup.
+[[Subgroup]]: 2.5.7.17
+[[Comma list]]: [[343/340]], 2023/2000
+{{Mapping|legend=2|1 4 4 6|0 -7 -5 -8}}
+[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[20/17]] = 287.377
+[[Badness]] (Sintel): 0.447
+===== 2.5.7.13/11.17.23 =====
+[[Subgroup]]: 2.5.7.13/11.17.23
+[[Comma list]]: 221/220, 161/160, 1309/1300, 343/340
+{{Mapping|legend=2|1 4 4 0 6 5|0 -7 -5 1 -8 -2}}
+[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[13/11]] = 287.412
+[[Badness]] (Sintel): 0.256
+===== No-3 no-19 23-limit =====
+[[Subgroup]]: 2.5.7.11.13.17.23
+[[Comma list]]: 176/175, 221/220, 161/160, 1309/1300, 343/340
+{{Mapping|legend=2|1 4 4 8 8 6 5|0 -7 -5 -19 -18 -8 -2}}
+[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[13/11]] = 286.821
+[[Badness]] (Sintel): 0.671
 == Etymology ==
-As of June 6, 2026, pentagoth is the name given to the rank-3 temperament 2.5.7.17[2023/2000] by [[groundfault]]. It originally referred to a temperament now called [[vengeance]], which as a 2.5.17 temperament tempers out this comma.
+As of June 6, 2026, pentagoth is the name given to the rank-3 temperament 2.5.7.17[2023/2000] by [[groundfault]]. It originally referred to a rank-2 temperament now called [[vengeance]], which as a 2.5.7.17 temperament tempers out this comma.
 [[Category:Pentagoth]]
+[[Category:Rank 2]]