7th-octave temperaments: Difference between revisions

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-{{Fractional-octave navigation|7}}
+{{Technical data page}}
-a 7th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 7. The most notable 7th-octave family is the [[whitewood family]] – tempering out [[2187/2048]] and associating 4\7 to [[3/2]].
+{{Infobox fractional-octave|7}}
+A 7th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 7. The most notable 7th-octave family is the [[whitewood family]] – tempering out [[2187/2048]] and associating 4\7 to [[3/2]].
 A comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets [[105/64]] to be equal to 5\7.
 Temperaments discussed elsewhere include:
-* ''Septant →'' [[Schismatic family #Septant|Schismatic family]]
+* ''[[Septant]]'' → [[Schismatic family #Septant|Schismatic family]]
-* ''Brahmagupta →'' [[Ragismic microtemperaments #Brahmagupta|Ragismic microtemperaments]]
+* ''[[Brahmagupta]]'' → [[Ragismic microtemperaments #Brahmagupta|Ragismic microtemperaments]]
-* ''Absurdity'' ''→'' [[Syntonic-chromatic equivalence continuum #Absurdity|Syntonic chromatic equivalence continuum]]
+* ''[[Absurdity]]'' → [[Porwell temperaments #Absurdity|Porwell temperaments]]
-== Jamesbond ==
+== Nitrogen ==
-This temperament uses exactly the same 5-limit as 7et, but the harmonic 7 is mapped to an independent generator. It is so named because its [[wedgie]] starts with {{multival| 0 0 7 … }}.
+Nitrogen may be described as the {{nowrap| 140 & 1407 }} temperament in the 7-limit. It was named after the 7th element for having a 7th-octave period and also because 140 and 1407 only contain numbers 7 and 14, atomic number and atomic weight of nitrogen respectively. On top of this connection to the number 7, it also reaches the 7th harmonic 7 generators down.
 [[Subgroup]]: 2.3.5.7
-[[Comma list]]: 25/24, 81/80
+[[Comma list]]: 3955078125/3954653486, {{monzo| 47 -7 -7 -7 }}
-{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }}
-{{Multival|legend=1| 0 0 7 0 11 16 }}
-[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~7/4 = 941.861
-{{Optimal ET sequence|legend=1| 7, 14c }}
-[[Badness]]: 0.041714
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 25/24, 33/32, 45/44
-Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}
-Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.090
-{{Optimal ET sequence|legend=1| 7, 14c }}
-Badness: 0.023524
-==== 13-limit ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 25/24, 27/26, 33/32, 40/39
-Mapping: {{mapping| 7 11 16 0 24 26 | 0 0 0 1 0 0 }}
-Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 949.236
-{{Optimal ET sequence|legend=1| 7, 14c }}
-Badness: 0.023003
-==== Austinpowers ====
-Subgroup: 2.3.5.7.11.13
-Comma list: 25/24, 33/32, 45/44, 65/63
-Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }}
-Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 952.555
-{{Optimal ET sequence|legend=1| 7, 14cf }}
-Badness: 0.022569
-== Akjaysmic (rank-3) ==
-{{Main| Akjaysma }}
-Subgroup: 2.3.5.7
-[[Comma list]]: {{monzo| 47 -7 -7 -7 }}
-[[Mapping]]: [{{val| 7 0 0 47 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -1 }}]
-Mapping generators: ~1157625/1048576, ~3, ~5
-[[POTE generator]]s: ~3/2 = 701.965, ~5/4 = 386.330
-{{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }}
-=== 11-limit ===
-Subgroup: 2.3.5.7.11
-Comma list: 184549376/184528125, 199297406/199290375
-Mapping: [{{val| 7 0 0 47 -168 }}, {{val| 0 1 0 -1 10 }}, {{val| 0 0 1 -1 5 }}]
-Mapping generators: ~29160/26411, ~3, ~5
-POTE generators: ~3/2 = 701.968, ~5/4 = 386.332
-{{Optimal ET sequence|legend=1| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }}
-== Nitrogen ==
-Described as 140 & 1407 temperament in the 7-limit, named after the 7th element for being period-7 and also because 140 and 1407 only contain numbers 7 and 14, atomic number and atomic weight of nitrogen respectively. On top of this connection to the number 7, it also reaches 7th harmonic 7 generators down.
-Subgroup: 2.3.5.7
-Comma list: 3955078125/3954653486, 140737488355328/140710042265625
-Mapping: [{{val|7 10 17 20}}, {{val|0 22 -15 -7}}]
-Mapping generators: ~1157625/1048576, ~1029/1024
-Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531
-{{Optimal ET sequence|legend=1|140, 1407, 1547}}, ...
-== Jackpot ==
-Jackpot identifies [[29/16]] with 6\7.
-[[Subgroup]]: 2.3.29
-[[Comma list]]: 17249876309/17179869184
+{{Mapping|legend=1| 7 10 17 20 | 0 22 -15 -7 }}
+: mapping generators: ~1157625/1048576, ~1029/1024
-{{mapping|legend=1| 7 0 34 | 0 1 0 }}
+[[Optimal tuning]]s:
+* [[WE]]: ~1157625/1048576 = 171.4278{{c}}, ~1029/1024 = 8.5308{{c}}
+: [[error map]]: {{val| -0.005 +0.001 -0.002 +0.015 }}
+* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~1029/1024 = 8.5308{{c}}
+: error map: {{val| 0.000 +0.008 +0.010 +0.030 }}
-: mapping generators: ~32/29, ~3
+{{Optimal ET sequence|legend=1| 140, 847, 987, 1127, 1267, 1407, 1547, 2954 }}
-[[Optimal tuning]] ([[CTE]]): ~32/29 = 1\7, ~3/2 = 701.955 (~24576/24389 = 16.239)
+[[Badness]] (Sintel): 1.50
-[[Support]]ing [[ET]]s: {{EDOs|7, 77, 70, 147, 224, 84, 63, 301, 217, 371, 56, 161, 91, 378}}
+{{Navbox fractional-octave}}
 [[Category:7edo]]
-[[Category:Temperament collections]]