2023/2000: Difference between revisions
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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. | '''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 | [[Tempering out]] this comma alone in the 2.5.7.17 subgroup leads to the '''pentagoth''' temperament. | ||
== Temperaments == | == 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 === | === Pentagoth === | ||
[[Subgroup]]: 2.5.7.17 | [[Subgroup]]: 2.5.7.17 | ||
| Line 25: | Line 26: | ||
[[Badness]] (Sintel): 0.509 | [[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 == | == 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:Pentagoth]] | ||
[[Category:Rank 2]] | |||