Garischismic clan: Difference between revisions

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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.

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.

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:

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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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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 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}}

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}}

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}}

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 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}}

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}}

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}}

Badness (Sintel): 1.22

[[Category:Temperament clans]]

[[Category:Garischismic clan| ]]

[[Category:Rank 2]]

@@ 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.
-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.
+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:
@@ Line 173: / Line 173: @@
 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
@@ Line 523: / 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]]