Garischismic clan: Difference between revisions

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 Like [[#Cotoneum|cotoneum]], newt can be notated in the same way as [[cassaschismic]], but with half-sharps and half-flats and the extra equivalence that two comma steps and an aberschisma step make a half-apotome step. In other words, C–^↑↑E = C–v↓↓E = C–Ed.
-Newt continues to be significant as an [[11-limit]] temperament, where it tempers out the [[lehmerisma]] (3025/3024). This extends into a very strong [[13-limit]] temperament and eventually a very strong no-17 [[19-limit]] temperament, a.k.a. ''neonewt''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] are even better.
+Newt continues to be significant as an [[11-limit]] temperament, where it tempers out the lehmerisma ([[3025/3024]]). This extends into a very strong [[13-limit]] temperament and eventually a very strong no-17 [[19-limit]] temperament, a.k.a. ''neonewt''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] are more accurate.
 [[Subgroup]]: 2.3.5.7
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 : ''For the 5-limit version, see [[Schismic–commatic equivalence continuum #Heptacot (5-limit)]].''
-Heptacot tempers out the [[meter]] and may be described as the {{nowrap| 12 & 311 }} temperament.
+Heptacot tempers out the [[meter]] and may be described as the {{nowrap| 12 & 311 }} temperament, splitting the perfect fifth into seven semitones. It is the natural 7-limit extension of the 5-limit temperament named by [[Tristan Bay]] in 2024. [[311edo]] and [[323edo]] are obvious tuning choices, as well as anything in between such as [[634edo]].
+Heptacot extends to the 11-limit in the same way as does gary, which best preserves its accuracy, though it should be noted that {{nowrap| 299 & 311 }} and {{nowrap| 323 & 335d }} make for simpler but less accurate alternative extensions.
 [[Subgroup]]: 2.3.5.7
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 [[Badness]] (Sintel): 3.06
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 19712/19683, 41503/41472, 703125/702464
+Mapping: {{mapping| 1 1 6 11 -10 | 0 7 -44 -98 161 }}
+: mapping generators: ~2, ~1323/1250
+Optimal tunings:
+* WE: ~2 = 1199.9981{{c}}, ~1323/1250 = 100.3174{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~1323/1250 = 100.3176{{c}}
+{{Optimal ET sequence|legend=0| 12e, 311, 634, 945 }}
+Badness (Sintel): 3.21
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 2080/2079, 4096/4095, 19712/19683, 31250/31213
+Mapping: {{mapping| 1 1 6 11 -10 -7 | 0 7 -44 -98 161 128 }}
+: mapping generators: ~2, ~1323/1250
+Optimal tunings:
+* WE: ~2 = 1199.9938{{c}}, ~675/637 = 100.3169{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~675/637 = 100.3174{{c}}
+{{Optimal ET sequence|legend=0| 12e, 311, 634, 945 }}
+Badness (Sintel): 1.89
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
+Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 31250/31213
+Mapping: {{mapping| 1 1 6 11 -10 -7 5 | 0 7 -44 -98 161 128 -9 }}
+: mapping generators: ~2, ~1323/1250
+Optimal tunings:
+* WE: ~2 = 1200.0076{{c}}, ~675/637 = 100.3179{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~675/637 = 100.3173{{c}}
+{{Optimal ET sequence|legend=0| 12e, 311, 634, 945 }}
+Badness (Sintel): 1.38
 == Garitritonic ==
 : ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''
-Garitritonic may be described as the {{nowrap| 53 & 581 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[634edo]] makes for a strong tuning.
+Garitritonic may be described as the {{nowrap| 53 & 581 }} temperament, splitting the [[24/1|24th harmonic]] into nine tritone generators; its [[ploidacot]] is thus delta-enneacot. [[634edo]] makes for a strong 7-limit tuning, though in the higher limits one may prefer sticking to [[581edo]].
 Garitritonic was named by [[Flora Canou]] in 2026 as a contraction of ''gary'' and ''tritonic''.
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 [[Badness]] (Sintel): 6.12
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 19712/19683, 41503/41472, 1953125/1948617
+Mapping: {{mapping| 1 -3 -15 67 -102 | 0 9 34 -126 207 }}
+Optimal tunings:
+* WE: ~2 = 1199.9795{{c}}, ~4375/3072 = 611.3485{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~4375/3072 = 611.3589{{c}}
+{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}
+Badness (Sintel): 3.60
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 2080/2079, 4096/4095, 19712/19683, 78125/78078
+Mapping: {{mapping| 1 -3 -15 67 -102 -34 | 0 9 34 -126 207 74 }}
+Optimal tunings:
+* WE: ~2 = 1199.9813{{c}}, ~500/351 = 611.3494{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3589{{c}}
+{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}
+Badness (Sintel): 1.73
+=== 2.3.5.7.11.13.19 subgroup ===
+Subgroup: 2.3.5.7.11.13.19
+Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 59375/59319
+Mapping: {{mapping| 1 -3 -15 67 -102 -34 -36 | 0 9 34 -126 207 74 79 }}
+Optimal tunings:
+* WE: ~2 = 1199.9884{{c}}, ~500/351 = 611.3531{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~500/351 = 611.3590{{c}}
+{{Optimal ET sequence|legend=0| 53, 528, 581, 1796, 2377b }}
+Badness (Sintel): 1.22
 [[Category:Temperament clans]]
 [[Category:Garischismic clan| ]] <!-- main article -->
 [[Category:Rank 2]]