7th-octave temperaments: Difference between revisions

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7th octave temperament ~~occurs through~~ temperament merging ~~any two~~ edos whose greatest common divisor is 7. ~~A comma that frequently appears in~~ 7th-octave ~~temps~~ is [[~~akjaysma~~]]~~, which sets 105~~/~~64 to be equal to 5~~\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]].

~~Temperaments discussed elsewhere include:~~

A comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets [[105/64]] to be equal to 5\7.

* ''Septant →'' [[Schismatic family#Septant|Schismatic family]]

Temperaments discussed elsewhere include:

* ''Brahmagupta →'' [[Ragismic microtemperaments#Brahmagupta|Ragismic microtemperaments]]

* ''[[Sevond]]'' → [[Keemic temperaments #Sevond|Keemic temperaments]]

* ''Absurdity'' ''→'' [[~~Syntonic-chromatic equivalence continuum~~#~~Absurdity~~|~~Syntonic chromatic equivalence continuum~~]]

* ''[[Septant]]'' → [[Schismatic family #Septant|Schismatic family]]

~~==Akjaysmic (rank-3)==~~

* ''[[Brahmagupta]]'' → [[Ragismic microtemperaments #Brahmagupta|Ragismic microtemperaments]]

~~{{Main|Akjaysma}}~~

* ''[[Absurdity]]'' → [[Porwell temperaments #Absurdity|Porwell temperaments]]

~~Subgroup: 2.3.5.7~~

* ''[[Neutron]]'' → [[Mistismic temperaments #Neutron|Mistismic temperaments]]

~~[[Comma list]]:~~ {{~~monzo~~| 47 -7 -7 -7 }}

== Nitrogen ==

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.

[[~~Mapping~~]]: ~~[{{val|~~ 7 ~~0 0 47 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -1 }}]~~

[[Subgroup]]: 2.3.5.7

~~Mapping generators~~: ~~~1157625~~/~~1048576~~, ~~~3, ~5~~

[[Comma list]]: 3955078125/3954653486, {{monzo| 47 -7 -7 -7 }}

~~[[POTE generator]]s~~: ~3/~~2 = 701.965~~, ~5/~~4 = 386.330~~

: mapping generators: ~1157625/1048576, ~1029/1024

{{~~Optimal ET sequence|legend~~=1| ~~140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802~~ }}

[[Optimal tuning]]s:

==~~= 11-limit ===~~

* [[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 }}

~~Subgroup: 2.3.5.7.11~~

{{Optimal ET sequence|legend=1| 140, 847, 987, 1127, 1267, 1407, 1547, 2954 }}

~~Comma list~~: ~~184549376/184528125, 199297406/199290375~~

[[Badness]] (Sintel): 1.50

~~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}}, ...~~

[[Category:7edo]]

~~[[Category:Temperament collections]]~~

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-th octave temperament occurs through temperament merging any two edos whose greatest common divisor is 7. A comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets 105/64 to be equal to 5\7.
+{{Technical data page}}
+{{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]].
-Temperaments discussed elsewhere include:
+A comma that frequently appears in 7th-octave temps is [[akjaysma]], which sets [[105/64]] to be equal to 5\7.
-* ''Septant →'' [[Schismatic family#Septant|Schismatic family]]
+Temperaments discussed elsewhere include:
-* ''Brahmagupta →'' [[Ragismic microtemperaments#Brahmagupta|Ragismic microtemperaments]]
+* ''[[Sevond]]'' → [[Keemic temperaments #Sevond|Keemic temperaments]]
-* ''Absurdity'' ''→'' [[Syntonic-chromatic equivalence continuum#Absurdity|Syntonic chromatic equivalence continuum]]
+* ''[[Septant]]'' → [[Schismatic family #Septant|Schismatic family]]
-==Akjaysmic (rank-3)==
+* ''[[Brahmagupta]]'' → [[Ragismic microtemperaments #Brahmagupta|Ragismic microtemperaments]]
-{{Main|Akjaysma}}
+* ''[[Absurdity]]'' → [[Porwell temperaments #Absurdity|Porwell temperaments]]
-Subgroup: 2.3.5.7
+* ''[[Neutron]]'' → [[Mistismic temperaments #Neutron|Mistismic temperaments]]
-[[Comma list]]: {{monzo| 47 -7 -7 -7 }}
+== Nitrogen ==
+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.
-[[Mapping]]: [{{val| 7 0 0 47 }}, {{val| 0 1 0 -1 }}, {{val| 0 0 1 -1 }}]
+[[Subgroup]]: 2.3.5.7
-Mapping generators: ~1157625/1048576, ~3, ~5
+[[Comma list]]: 3955078125/3954653486, {{monzo| 47 -7 -7 -7 }}
-[[POTE generator]]s: ~3/2 = 701.965, ~5/4 = 386.330
+{{Mapping|legend=1| 7 10 17 20 | 0 22 -15 -7 }}
+: mapping generators: ~1157625/1048576, ~1029/1024
-{{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }}
+[[Optimal tuning]]s:
-=== 11-limit ===
+* [[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 }}
-Subgroup: 2.3.5.7.11
+{{Optimal ET sequence|legend=1| 140, 847, 987, 1127, 1267, 1407, 1547, 2954 }}
-Comma list: 184549376/184528125, 199297406/199290375
+[[Badness]] (Sintel): 1.50
-Mapping: [{{val| 7 0 0 47 -168 }}, {{val| 0 1 0 -1 10 }}, {{val| 0 0 1 -1 5 }}]
+{{Navbox fractional-octave}}
-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}}, ...
 [[Category:7edo]]
-[[Category:Temperament collections]]