@@ Line 28: / Line 28: @@
 The second comma of the comma list determines which full 7-limit or 11-limit family member we are looking at. Garibaldi adds the [[schisma]], or equivalently [[225/224]] and finds 5/4 at the diminished fourth. Cotoneum adds [[10976/10935]] and finds 5/4 at the septuple-diminished octave. These are generated by the fifth as is gary.
-Gariwizmic adds the [[wizma]] with a 1/2-octave period. Newt adds [[2401/2400]], halving the fifth. Alphatrident adds [[6144/6125]], slicing the twelfth in three. Satin adds [[2100875/2097152]], slicing the fourth in three. Vulture adds [[4375/4374]], slicing the twelfth in four. Sextile adds [[250047/250000]] with a 1/6-octave period. World calendar adds [[390625/388962]] with a 1/4-octave period as well as a halved fifth. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five.
+Newt adds [[2401/2400]], halving the fifth. Gariwizmic adds the [[wizma]] with a 1/2-octave period. Alphatrident adds [[6144/6125]], slicing the twelfth in three. Satin adds [[2100875/2097152]], slicing the fourth in three. Vulture adds [[4375/4374]], slicing the twelfth in four. Sextile adds [[250047/250000]] with a 1/6-octave period. World calendar adds [[390625/388962]] with a 1/4-octave period as well as a halved fifth. Quintagar adds [[3136/3125]], slicing the fourth in five. Paramity adds [[65625/65536]], slicing the eleventh in five. Heptacot adds [[703125/702464]], slicing the fifth in seven. Finally, garitritonic adds 95703125/95551488 ({{monzo| -17 -6 9 2 }}), slicing the 24th harmonic in nine.
 Temperaments discussed elsewhere are:
 * [[Garibaldi]] (+225/224) → [[Schismatic family #Garibaldi|Schismatic family]]
-* ''[[Newt]]'' (+2401/2400) → [[Breedsmic temperaments #Newt|Breedsmic temperaments]]
 * ''[[Alphatrident]]'' (+6144/6125) → [[Alphatricot family #Alphatrident|Alphatricot family]]
 * ''[[Vulture]]'' (+4375/4374) → [[Vulture family #Vulture|Vulture family]]
@@ Line 39: / Line 38: @@
 * ''[[Garistearn]]'' (+118098/117649) → [[Stearnsmic clan #Garistearn|Stearnsmic clan]]
-Considered below are cotoneum, gariwizmic, satin, sextile, and world calendar.
+Considered below are cotoneum, newt, gariwizmic, satin, sextile, and world calendar.
 ==== Subgroup extensions ====
@@ Line 61: / Line 60: @@
 == Cotoneum ==
 {{Main| Cotoneum }}
+: ''For the 5-limit version, see [[Schismic–countercommatic equivalence continuum #Cotoneum (5-limit)]].''
-The cotoneum temperament tempers out 10976/10935 ([[hemimage comma]]), and 823543/819200 ([[quince comma]]) in addition to the garischisma. This temperament can be described as {{nowrap| 41 & 217 }}, and is supported by [[176edo|176-]], [[217edo|217-]], and [[258edo]]. 5/4 is found at the septuple-diminished octave (C-Cbbbbbbb) or equivalently at the perfect fourth minus four Pythagorean commas. It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.
+Cotoneum tempers out 10976/10935 ([[hemimage comma]]), and 823543/819200 ([[quince comma]]) in addition to the garischisma. This temperament can be described as {{nowrap| 41 & 217 }}, and is supported by [[176edo|176-]], [[217edo|217-]], and [[258edo]]. 5/4 is found -49 generators away. In terms of chain-of-fifths notation, this is a sextuple-diminished octave, or a perfect fourth minus four generic commas.
+However, cotoneum can be notated like [[cassaschismic]], where 5/4 is conceptualized as an aberschisma-up comma-down major third (C–^↓E), but with the extra equivalence that the generic aberschisma is identical to the [[41-comma]]. In other words, we have C–^↑↑E = C–↓↓E.
+It can be extended to the 11-, 13-, 17-, and 19-limit by adding 441/440, 364/363, 595/594, and 343/342 to the comma list in this order.
 [[Subgroup]]: 2.3.5.7
@@ Line 161: / Line 165: @@
 Badness (Sintel): 1.33
+== Newt ==
+: ''For the 5-limit version, see [[Schismic–countercommatic equivalence continuum #Newt (5-limit)]].''
+Newt tempers out the [[breedsma]] and may be described as the {{nowrap| 41 & 270 }} temperament. It has a generator of a neutral third (0.2 cents flat of [[49/40]]) with a [[ploidacot]] signature of dicot. 41 generator steps fall short of 12 octaves by a generic aberschisma step of a [[schisma]]~[[aberschisma]]. From there the intervals of 5 and 7 can be derived.
+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 more accurate.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 2401/2400, 33554432/33480783
+{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}
+: mapping generators: ~2, ~49/40
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9315{{c}}, ~49/40 = 351.0932{{c}}
+: [[error map]]: {{val| -0.068 +0.163 +0.075 -0.188 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.1141{{c}}
+: error map: {{val| 0.000 +0.273 +0.180 -0.022 }}
+{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}
+[[Badness]] (Sintel): 1.06
+=== 11-limit ===
+Subgroup: 2.3.5.7.11
+Comma list: 2401/2400, 3025/3024, 19712/19683
+Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}
+Optimal tunings:
+* WE: ~2 = 1199.9603{{c}}, ~49/40 = 351.1038{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1155{{c}}
+{{Optimal ET sequence|legend=0| 41, 188, 229, 270, 581, 851, 1121, 1972 }}
+Badness (Sintel): 0.643
+=== 13-limit ===
+Subgroup: 2.3.5.7.11.13
+Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095
+Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}
+Optimal tunings:
+* WE: ~2 = 1199.9747{{c}}, ~49/40 = 351.1094{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1168{{c}}
+{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851, 2283b }}
+Badness (Sintel): 0.571
+=== 2.3.5.7.11.13.19 subgroup (neonewt) ===
+Subgroup: 2.3.5.7.11.13.19
+Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400
+Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}
+Optimal tunings:
+* WE: ~2 = 1199.9782{{c}}, ~49/40 = 351.1102{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1166{{c}}
+{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851 }}
+Badness (Sintel): 0.438
 == Gariwizmic ==
@@ Line 184: / Line 259: @@
 [[Badness]] (Sintel): 2.22
-==== 11-limit ====
+=== 11-limit ===
 Subgroup: 2.3.5.7.11
@@ Line 201: / Line 276: @@
 Badness (Sintel): 1.01
-==== 13-limit ====
+=== 13-limit ===
 Subgroup: 2.3.5.7.11.13
@@ Line 218: / Line 293: @@
 Badness (Sintel): 0.822
-==== 2.3.5.7.11.13.19 subgroup ====
+=== 2.3.5.7.11.13.19 subgroup ===
 Subgroup: 2.3.5.7.11.13.19
@@ Line 448: / Line 523: @@
 Badness (Sintel): 1.82
+== Heptacot ==
+: ''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, 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
+[[Comma list]]: 703125/702464, 33554432/33480783
+{{Mapping|legend=1| 1 1 6 11 | 0 7 -44 -98 }}
+: mapping generators: ~2, ~1323/1250
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9434{{c}}, ~1323/1250 = 100.3096{{c}}
+: [[error map]]: {{val| -0.057 +0.155 -0.274 +0.215 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1250 = 100.3148{{c}}
+: error map: {{val| 0.000 +0.249 -0.165 +0.324 }}
+{{Optimal ET sequence|legend=1| 12, …, 299, 311, 323, 634, 957, 1591 }}
+[[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 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''.
+[[Subgroup]]: 2.3.5.7
+[[Comma list]]: 33554432/33480783, 95703125/95551488
+{{Mapping|legend=1| 1 -3 -15 67 | 0 9 34 -126 }}
+: mapping generators: ~2, ~4375/3072
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9678{{c}}, ~4375/3072 = 611.3417{{c}}
+: [[error map]]: {{val| -0.032 +0.217 -0.213 -0.036 }}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~4375/3072 = 611.3582{{c}}
+: error map: {{val| 0.000 +0.268 -0.136 +0.045 }}
+{{Optimal ET sequence|legend=1| 53, 422d, 475, 528, 581, 634, 1215 }}
+[[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]]

Garischismic clan: Difference between revisions