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Gariboh family: Difference between revisions

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{{Technical data page}}<br><br>
{{Technical data page}}
The '''gariboh family''' of [[Rank-3 temperament|rank-3]] [[temperament]]s [[Tempering out|tempers out]] the gariboh comma, [[3125/3087]].  
The '''gariboh family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperament]]s [[tempering out|tempers out]] the gariboh comma, [[3125/3087]].  


== Gariboh ==
== Gariboh ==
Gariboh is generated by a perfect fifth and a narrow minor third of ~25/21, three make ~5/3 and five make ~7/3. [[94edo]] makes for an excellent tuning for it.
Another notable tuning is given by [[TE]], [[CTE]] and [[POTE]], all coinciding at 702.4482{{c}}, 293.7397{{c}} with pure octaves since prime 2 is not involved in the comma to begin with, though its difference from [[CWE]] is practically unnoticeable.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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{{Mapping|legend=1| 1 0 0 0 | 0 1 1 1 | 0 0 3 5 }}
{{Mapping|legend=1| 1 0 0 0 | 0 1 1 1 | 0 0 3 5 }}
: mapping generators: ~2, ~3, ~25/21


: Mapping generators: ~2, ~3, ~25/21
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.9983{{c}}, ~3/2 = 702.4472{{c}}, ~25/21 = 293.7393{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3 = 702.4482, ~25/21 = 293.7397
: [[error map]]: {{val| -0.002 +0.490 -2.650 +2.316 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.4468{{c}}, ~25/21 = 293.7395{{c}}
: error map: {{val| 0.000 +0.492 -2.648 +2.319 }}


{{Optimal ET sequence|legend=1| 8d, 12, 29, 37, 41, 53, 94 }}
{{Optimal ET sequence|legend=1| 8d, 12, 29, 37, 41, 53, 94 }}


[[Badness]]: 0.614 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.71


[[Projection pair]]s: 5 – 15625/3087, 7 – 9765625/1361367 to 2.3.25/7
[[Projection pair]]s: 5 – 15625/3087, 7 – 9765625/1361367 to 2.3.25/7
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{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 1 2 | 0 0 3 5 -7 }}
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 1 2 | 0 0 3 5 -7 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3 = 702.7001, ~25/21 = 293.5587
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9983{{c}}, ~3/2 = 702.4472{{c}}, ~25/21 = 293.7393{{c}}
: [[error map]]: {{val| +0.086 +0.881 -2.738 +1.909 -0.531 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.7792{{c}}, ~25/21 = 293.5634{{c}}
: error map: {{val| 0.000 +0.824 -2.844 +1.770 -0.703 }}


{{Optimal ET sequence|legend=1| 8d, 12e, 25bccdd, 33cd, 37ee, 41, 53, 94, 229c }}
{{Optimal ET sequence|legend=1| 8d, 12e, 25bccdd, 33cd, 37ee, 41, 53, 90e, 94, 229c }}


[[Badness]]: 1.75 × 10<sup>-3</sup>
[[Badness]] (Sintel): 2.10


=== 13-limit ===
=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
 
Comma list: 275/273, 325/324, 540/539


[[Comma list]]: 275/273, 325/324, 540/539
Mapping: {{mapping| 1 0 0 0 2 2 | 0 1 1 1 2 2 | 0 0 3 5 -7 -6 }}


{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 1 1 2 2 | 0 0 3 5 -7 -6 }}
Optimal tunings:
* WE: ~2 = 1199.9929{{c}}, ~3/2 = 702.5339{{c}}, ~13/11 = 293.6861{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5314{{c}}, ~13/11 = 293.6875{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3 = 702.5381, ~13/11 = 293.6878
{{Optimal ET sequence|legend=0| 8d, 12e, 25bccddf, 33cd, 41, 53, 94 }}


{{Optimal ET sequence|legend=1| 8d, 12e, 33cd, 41, 53, 94 }}
Badness (Sintel): 1.22


== Androboh ==
== Androboh ==
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{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 1 0 | 0 0 3 5 6 }}
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 1 0 | 0 0 3 5 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3 = 705.0554, ~25/21 = 293.1102
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2176{{c}}, ~3/2 = 704.5957{{c}}, ~25/21 = 292.9191{{c}}
: [[error map]]: {{val| -0.782 +1.858 -3.743 -0.417 +4.632 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5845{{c}}, ~25/21 = 292.9597{{c}}
: error map: {{val| 0.000 +2.630 -2.850 +0.557 +6.440 }}


{{Optimal ET sequence|legend=1| 8d, 12, 29, 37, 41, 90e, 131e }} *
{{Optimal ET sequence|legend=1| 8d, 12, 29, 37, 41, 90e, 131e }} *
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<nowiki>*</nowiki> [[optimal patent val]]: [[119edo|119]]
<nowiki>*</nowiki> [[optimal patent val]]: [[119edo|119]]


[[Badness]]: 1.20 × 10<sup>-3</sup>
[[Badness]] (Sintel): 1.44


== Trismegiboh ==
== Trismegiboh ==
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{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 1 1 | 0 0 6 10 -1 }}
{{Mapping|legend=1| 1 0 0 0 2 | 0 1 1 1 1 | 0 0 6 10 -1 }}
: mapping generators: ~2, ~3, ~12/11


: Mapping generators: ~2, ~3, ~12/11
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1199.5499{{c}}, ~3/2 = 702.2924{{c}}, ~12/11 = 146.9551{{c}}
: [[error map]]: {{val| -0.450 -0.113 -2.741 +2.568 +2.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1698{{c}}, ~12/11 = 146.9960{{c}}
: error map: {{val| 0.000 +0.215 -2.168 +3.304 +3.856 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.5559, ~12/11 = 147.0103
{{Optimal ET sequence|legend=1| 8d, 16, 24d, 33cd, 41, 65d, 90e, 106, 147 }}


{{Optimal ET sequence|legend=1| 8d, 16, 33cd, 41, 49 }}
[[Badness]] (Sintel): 3.38


=== 13-limit ===
=== 13-limit ===
[[Subgroup]]: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 144/143, 275/273, 847/845
Comma list: 144/143, 275/273, 847/845


{{Mapping|legend=1| 1 0 0 0 2 2 | 0 1 1 1 1 1 | 0 0 6 10 -1 1 }}
Mapping: {{mapping| 1 0 0 0 2 2 | 0 1 1 1 1 1 | 0 0 6 10 -1 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 702.9404, ~12/11 = 147.1132
Optimal tunings:
* WE: ~2 = 1198.9833{{c}}, ~3/2 = 702.3448{{c}}, ~12/11 = 146.9886{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0567{{c}}, ~12/11 = 147.0902{{c}}


{{Optimal ET sequence|legend=1| 8d, 16, 33cd, 41, 49f }}
{{Optimal ET sequence|legend=0| 8d, 16, 24d, 33cd, 41, 65d, 106f }}
 
Badness (Sintel): 1.83


=== 17-limit ===
=== 17-limit ===
[[Subgroup]]: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 120/119, 144/143, 170/169, 847/845


[[Comma list]]: 120/119, 144/143, 170/169, 847/845
Mapping: {{mapping| 1 0 0 0 2 2 3 | 0 1 1 1 1 1 1 | 0 0 6 10 -1 1 -4 }}


{{Mapping|legend=1| 1 0 0 0 2 2 3 | 0 1 1 1 1 1 1 | 0 0 6 10 -1 1 -4 }}
Optimal tunings:
* WE: ~2 = 1198.6534{{c}}, ~3/2 = 702.2909{{c}}, ~12/11 = 147.0794{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.8744{{c}}, ~12/11 = 147.2433{{c}}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 703.0799, ~12/11 = 147.2446
{{Optimal ET sequence|legend=0| 8d, 16, 24d, 33cd, 41, 49fg, 65d, 106fg }}


{{Optimal ET sequence|legend=1| 8d, 16, 33cd, 41, 49fg }}
Badness (Sintel): 1.55


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Gariboh family| ]] <!-- main article -->
[[Category:Gariboh family| ]] <!-- main article -->
[[Category:Gariboh| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]
Retrieved from "https://en.xen.wiki/w/Gariboh_family"