7th-octave temperaments: Difference between revisions

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== Jamesbond ==

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]]" (a kind of mathematical object representing the temperament) starts with {{multival| 0 0 7 … }} (in fact, it is {{~~Multival|legend=~~| 0 0 7 0 11 16 }})

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]]" (a kind of mathematical object representing the temperament) starts with {{multival| 0 0 7 … }} (in fact, it is {{multival| 0 0 7 0 11 16 }})

[[Subgroup]]: 2.3.5.7

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: mapping generators: ~10/9, ~7

[[Optimal tuning]] ([[~~POTE~~]]): ~10/9 = 1\7, ~7/4 = ~~941~~.~~861~~

[[Optimal tuning]]s:

* [[WE]]: ~10/9 = 172.790{{c}}, ~7/4 = 949.343{{c}}

: [[error map]]: {{val| +9.533 -1.261 -21.668 -0.418 }}

* [[CWE]]: ~10/9 = 171.429{{c}}, ~7/4 = 948.499{{c}}

: error map: {{val| -0.000 -16.241 -43.457 -20.327 }}

[[Badness]]: 0.~~041714~~

[[Badness]] (Sintel): 1.06

=== 11-limit ===

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Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}

Optimal ~~tuning (POTE)~~: ~10/9 = 1\7, ~7/4 = ~~941~~.~~090~~

Optimal tunings:

* WE: ~10/9 = 172.830{{c}}, ~7/4 = 948.784{{c}}

* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 946.554{{c}}

Badness: 0.~~023524~~

Badness (Sintel): 0.778

==== 13-limit ====

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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 tunings:

* WE: ~10/9 = 172.390{{c}}, ~7/4 = 954.559{{c}}

* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 952.367{{c}}

Badness: 0.~~023003~~

Badness (Sintel): 0.951

==== Austinpowers ====

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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 tunings:

* WE: ~10/9 = 172.873{{c}}, ~7/4 = 960.581{{c}}

* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 958.793{{c}}

Badness: 0.~~022569~~

Badness (Sintel): 0.933

== Akjaysmic (rank-3) ==

Subgroup: 2.3.5.7

[[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

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

[[Optimal tuning]]s:

* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}

: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}

* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}

: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}

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

{{Optimal ET sequence|legend=1| 56, 63, 77, 84, 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}

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

[[Badness]] (Sintel): 2.22

=== 11-limit ===

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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: {{mapping| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}

: mapping generators: ~29160/26411, ~3, ~5

: ~~Mapping generators~~: ~29160/26411, ~3, ~5

Optimal tunings:

* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}

* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}

~~POTE generators: ~3/2~~ = ~~701.968~~, ~~~5/4 = 386.332~~

{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}

~~{{Optimal ET sequence|legend=~~1~~| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }}~~

Badness (Sintel): 1.32

== Nitrogen ==

~~Described~~ as 140 &~~amp;~~ 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.

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

[[Subgroup]]: 2.3.5.7

Comma list: 3955078125/3954653486, ~~140737488355328/140710042265625~~

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

~~Mapping: [~~{{~~val~~|7 10 17 20~~}}, {{val~~|0 22 -15 -7}}]

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

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

[[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 }}

Optimal ~~tuning (CTE): ~1157625/1048576~~ = 1\7, ~~~1029/1024 = 8.531~~

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

~~{{Optimal ET sequence|legend=~~1~~|140, 1407, 1547}}, ..~~.

[[Badness]] (Sintel): 1.50

[[Category:7edo]]

@@ Line 11: / Line 11: @@
 == Jamesbond ==
-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]]" (a kind of mathematical object representing the temperament) starts with {{multival| 0 0 7 … }} (in fact, it is {{Multival|legend=| 0 0 7 0 11 16 }})
+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]]" (a kind of mathematical object representing the temperament) starts with {{multival| 0 0 7 … }} (in fact, it is {{multival| 0 0 7 0 11 16 }})
 [[Subgroup]]: 2.3.5.7
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 {{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }}
+: mapping generators: ~10/9, ~7
-[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~7/4 = 941.861
+[[Optimal tuning]]s:
+* [[WE]]: ~10/9 = 172.790{{c}}, ~7/4 = 949.343{{c}}
+: [[error map]]: {{val| +9.533 -1.261 -21.668 -0.418 }}
+* [[CWE]]: ~10/9 = 171.429{{c}}, ~7/4 = 948.499{{c}}
+: error map: {{val| -0.000 -16.241 -43.457 -20.327 }}
-{{Optimal ET sequence|legend=1| 7, 14c }}
+{{Optimal ET sequence|legend=1| 7(d), 14c }}
-[[Badness]]: 0.041714
+[[Badness]] (Sintel): 1.06
 === 11-limit ===
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 Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}
-Optimal tuning (POTE): ~10/9 = 1\7, ~7/4 = 941.090
+Optimal tunings:
+* WE: ~10/9 = 172.830{{c}}, ~7/4 = 948.784{{c}}
+* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 946.554{{c}}
-{{Optimal ET sequence|legend=1| 7, 14c }}
+{{Optimal ET sequence|legend=1| 7(d), 14c }}
-Badness: 0.023524
+Badness (Sintel): 0.778
 ==== 13-limit ====
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 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 tunings:
+* WE: ~10/9 = 172.390{{c}}, ~7/4 = 954.559{{c}}
+* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 952.367{{c}}
-{{Optimal ET sequence|legend=1| 7, 14c }}
+{{Optimal ET sequence|legend=1| 7(d), 14c }}
-Badness: 0.023003
+Badness (Sintel): 0.951
 ==== Austinpowers ====
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 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 tunings:
+* WE: ~10/9 = 172.873{{c}}, ~7/4 = 960.581{{c}}
+* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 958.793{{c}}
-{{Optimal ET sequence|legend=1| 7, 14cf }}
+{{Optimal ET sequence|legend=1| 7(df), 14cf }}
-Badness: 0.022569
+Badness (Sintel): 0.933
 == Akjaysmic (rank-3) ==
-{{Main| Akjaysma }}
+{{See also| Akjaysma }}
-Subgroup: 2.3.5.7
+[[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|legend=1| 7 0 0 47 | 0 1 0 -1 | 0 0 1 -1 }}
+: mapping generators: ~1157625/1048576, ~3, ~5
-: Mapping generators: ~1157625/1048576, ~3, ~5
+[[Optimal tuning]]s:
+* [[WE]]: ~1157625/1048576 = 171.427811{{c}}, ~3/2 = 701.962313{{c}}, ~5/4 = 386.328628{{c}}
+: [[error map]]: {{val| -0.0053 +0.0020 +0.0043 +0.0062 }}
+* [[CWE]]: ~1157625/1048576 = 171.428571{{c}}, ~3/2 = 701.964859{{c}}, ~5/4 = 386.330310{{c}}
+: error map: {{val| 0.0000 +0.0099 +0.0166 +0.0218 }}
-[[POTE generator]]s: ~3/2 = 701.965, ~5/4 = 386.330
+{{Optimal ET sequence|legend=1| 56, 63, 77, 84, 140, 217, 224, 301, 441, 966, 1106, 1407, 1547, 1848, 2513, 2954, 6349, 9303, 11151, 14105, 17500, 20454 }}
-{{Optimal ET sequence|legend=1| 140, 224, 301, 441, 665, 742, 966, 1106, 1407, 1547, 1848, 2289, 2513, 2954, 3395, 4802 }}
+[[Badness]] (Sintel): 2.22
 === 11-limit ===
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 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: {{mapping| 7 0 0 47 -168 | 0 1 0 -1 10 | 0 0 1 -1 5 }}
+: mapping generators: ~29160/26411, ~3, ~5
-: Mapping generators: ~29160/26411, ~3, ~5
+Optimal tunings:
+* WE: ~29160/26411 = 171.427802{{c}}, ~3/2 = 701.964561{{c}}, ~5/4 = 386.329837{{c}}
+* CWE: ~29160/26411 = 171.428571{{c}}, ~3/2 = 701.967291{{c}}, ~5/4 = 386.331624{{c}}
-POTE generators: ~3/2 = 701.968, ~5/4 = 386.332
+{{Optimal ET sequence|legend=0| 301, 441, 665e, 742, 1106, 1547, 1848, 3395, 4501, 5243, 6349, 17941, 24290, 30639, 45185cde, 63126bcde, 69475bccdde, 75824bccddee }}
-{{Optimal ET sequence|legend=1| 301, 364, 441, 742, 805, 1043, 1106, 1407, 1547, 1848, 2289, 2653, 2954, 3395, 4501, 5243, 6349, 8197 }}
+Badness (Sintel): 1.32
 == Nitrogen ==
-Described as 140 &amp; 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.
+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
+[[Subgroup]]: 2.3.5.7
-Comma list: 3955078125/3954653486, 140737488355328/140710042265625
+[[Comma list]]: 3955078125/3954653486, {{monzo| 47 -7 -7 -7 }}
-Mapping: [{{val|7 10 17 20}}, {{val|0 22 -15 -7}}]
+{{Mapping|legend=1| 7 10 17 20 | 0 22 -15 -7 }}
+: mapping generators: ~1157625/1048576, ~1029/1024
-Mapping generators: ~1157625/1048576, ~1029/1024
+[[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 }}
-Optimal tuning (CTE): ~1157625/1048576 = 1\7, ~1029/1024 = 8.531
+{{Optimal ET sequence|legend=1| 140, 847, 987, 1127, 1267, 1407, 1547, 2954 }}
-{{Optimal ET sequence|legend=1|140, 1407, 1547}}, ...
+[[Badness]] (Sintel): 1.50
 {{Navbox fractional-octave}}
 [[Category:7edo]]