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Whitewood family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The '''whitewood family''' of [[temperament]]s [[tempering out|tempers out]] the apotome, [[2187/2048]]. Consequently the [[3/2|fifth]]s are always 4/7 of an [[octave]], a distinctly flat 685.714 [[cent]]s. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.
The '''whitewood family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the Pythagorean apotome, [[2187/2048]]. Consequently the [[3/2|fifth]]s are always 4/7 of an [[octave]], a distinctly flat 685.714 [[cent]]s. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.
 
Temperaments discussed elsewhere include:
* ''[[Jamesbond]]'' → [[7th-octave temperaments#Jamesbond|7th-octave temperaments]]


== Whitewood ==
== Whitewood ==
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Scales: [[7L 7s in 140edo]]
Scales: [[7L 7s in 140edo]]
=== Overview to extensions ===
Temperaments discussed elsewhere include:
* ''[[Sept]]'' → [[Very low accuracy temperaments #Sept|Very low accuracy temperaments]]
Considered below are septimal whitewood, redwood, greenwood, and jamesbond.


== Septimal whitewood ==
== Septimal whitewood ==
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Badness (Sintel): 2.59
Badness (Sintel): 2.59


== Mujannab ==
== Greenwood ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 54/49, 64/63
[[Comma list]]: 405/392, 1323/1280
 
{{Mapping|legend=1| 7 11 1 12 | 0 0 2 1 }}


{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}
: mapping generators: ~9/8, ~15/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~9/8 = 170.8233{{c}}, ~5/4 = 393.7921{{c}} (~15/14 = 52.1453{{c}})
* [[WE]]: ~9/8 = 172.1073{{c}}, ~15/14 = 101.7681{{c}} (~21/20 = 70.3391{{c}})
: [[error map]]: {{val| -4.236 -22.898 -0.994 +47.642 }}
: [[error map]]: {{val| +4.751 -8.775 -1.169 +2.980 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 392.7194{{c}} (~15/14 = 49.8622{{c}})
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~15/14 = 103.3802{{c}} (~21/20 = 68.0484{{c}})
: error map: {{val| 0.000 -16.241 +6.406 +59.746 }}
: error map: {{val| 0.000 -16.241 -8.125 -8.303 }}


{{Optimal ET sequence|legend=1| 7, 14d }}
{{Optimal ET sequence|legend=1| 7c, 14c, 21, 35, 84bbccd }}


[[Badness]] (Sintel): 2.68
[[Badness]] (Sintel): 3.08


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 54/49, 64/63
Comma list: 45/44, 99/98, 1323/1280


Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}
Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~11/10 = 170.8171{{c}}, ~5/4 = 393.2529{{c}} (~33/32 = 51.6187{{c}})
* WE: ~11/10 = 172.0795{{c}}, ~15/14 = 100.5259{{c}} (~21/20 = 71.5536{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 391.8401{{c}} (~33/32 = 48.9830{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 102.1866{{c}} (~21/20 = 69.2419{{c}})


{{Optimal ET sequence|legend=0| 7, 14de }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35, 49bcde }}


Badness (Sintel): 2.02
Badness (Sintel): 1.90


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 45/44, 52/49, 64/63
Comma list: 27/26, 45/44, 99/98, 640/637


Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}
Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~11/10 = 170.7953{{c}}, ~5/4 = 393.6112{{c}} (~33/32 = 52.0206{{c}})
* WE: ~11/10 = 171.6777{{c}}, ~15/14 = 104.4016{{c}} (~21/20 = 67.2761{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 392.7250{{c}} (~33/32 = 49.8678{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 104.8518{{c}} (~21/20 = 66.5768{{c}})
 
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}


{{Optimal ET sequence|legend=0| 7, 14de }}
Badness (Sintel): 2.23


Badness (Sintel): 1.77
== 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| 0 0 7 0 11 16 }})


== Greenwood ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 405/392, 1323/1280
[[Comma list]]: 25/24, 81/80


{{Mapping|legend=1| 7 11 1 12 | 0 0 2 1 }}
{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }}
 
: mapping generators: ~10/9, ~7
: mapping generators: ~9/8, ~15/7


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[WE]]: ~9/8 = 172.1073{{c}}, ~15/14 = 101.7681{{c}} (~21/20 = 70.3391{{c}})
* [[WE]]: ~10/9 = 172.790{{c}}, ~7/4 = 949.343{{c}}
: [[error map]]: {{val| +4.751 -8.775 -1.169 +2.980 }}
: [[error map]]: {{val| +9.533 -1.261 -21.668 -0.418 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~15/14 = 103.3802{{c}} (~21/20 = 68.0484{{c}})
* [[CWE]]: ~10/9 = 171.429{{c}}, ~7/4 = 948.499{{c}}
: error map: {{val| 0.000 -16.241 -8.125 -8.303 }}
: error map: {{val| -0.000 -16.241 -43.457 -20.327 }}


{{Optimal ET sequence|legend=1| 7c, 14c, 21, 35, 84bbccd }}
{{Optimal ET sequence|legend=1| 7(d), 14c }}


[[Badness]] (Sintel): 3.08
[[Badness]] (Sintel): 1.06


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 45/44, 99/98, 1323/1280
Comma list: 25/24, 33/32, 45/44
 
Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}
 
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=0| 7(d), 14c }}
 
Badness (Sintel): 0.778
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 27/26, 33/32, 40/39


Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}
Mapping: {{mapping| 7 11 16 0 24 26 | 0 0 0 1 0 0 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~11/10 = 172.0795{{c}}, ~15/14 = 100.5259{{c}} (~21/20 = 71.5536{{c}})
* WE: ~10/9 = 172.390{{c}}, ~7/4 = 954.559{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 102.1866{{c}} (~21/20 = 69.2419{{c}})
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 952.367{{c}}


{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35, 49bcde }}
{{Optimal ET sequence|legend=0| 7(d), 14c }}


Badness (Sintel): 1.90
Badness (Sintel): 0.951


=== 13-limit ===
==== Austinpowers ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 27/26, 45/44, 99/98, 640/637
Comma list: 25/24, 33/32, 45/44, 65/63


Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}
Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~11/10 = 171.6777{{c}}, ~15/14 = 104.4016{{c}} (~21/20 = 67.2761{{c}})
* WE: ~10/9 = 172.873{{c}}, ~7/4 = 960.581{{c}}
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 104.8518{{c}} (~21/20 = 66.5768{{c}})
* CWE: ~10/9 = 171.429{{c}}, ~7/4 = 958.793{{c}}


{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}
{{Optimal ET sequence|legend=0| 7(df), 14cf }}


Badness (Sintel): 2.23
Badness (Sintel): 0.933


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Whitewood family| ]] <!-- main article -->
[[Category:Whitewood family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Whitewood_family"