Septischismic 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.

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.

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]]

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* ''[[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 ====

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Badness (Sintel): 1.33

== Newt ==

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 [[schisma]]~[[aberschisma]]. From there the intervals of 5 and 7 can be derived.

Newt continues to be significant as an [[11-limit]] temperament, where it tempers out the [[lehmerisma]] (3025/3024). This extends into a strong [[13-limit]] temperament and eventually a 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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 33554432/33480783

: 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}}

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}}

Badness (Sintel): 0.438

== Gariwizmic ==

@@ 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.
 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 165: / Line 164: @@
 Badness (Sintel): 1.33
+== Newt ==
+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 [[schisma]]~[[aberschisma]]. From there the intervals of 5 and 7 can be derived.
+Newt continues to be significant as an [[11-limit]] temperament, where it tempers out the [[lehmerisma]] (3025/3024). This extends into a strong [[13-limit]] temperament and eventually a 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.
+[[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 ==