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Whitewood family: Difference between revisions

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The '''apotome family''' or '''whitewood family''' of temperaments tempers out the apotome, [[2187/2048]]. Consequently the fifths 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.
{{Technical data page}}
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.


The 5-limit version of this temperament is called ''whitewood'', to serve in contrast with the "blackwood" temperament which tempers out [[256/243]], the pythagorean limma. Whereas blackwood temperament can be thought of as a closed chain of 5 fifths and a major third generator, whitewood is a closed chain of 7 fifths and a major third generator. This means that blackwood is generally supported by 5''n''-edos, and whitewood is supported by 7''n''-edos, and the [[mos]] of both scales follow a similar pattern.
== Whitewood ==
{{Main| Whitewood }}


The 14-note mos of whitewood, like the 10-note mos of blackwood, shares a number of interesting properties which derive from the relatively small circle of fifths common to both. From any major or minor triad in the scale, one can always move away by ~3/2 or ~4/3 to reach another triad of the same type. This contrasts with the diatonic scale, in which one will eventually "hit a wall" if one moves by perfect fifth for long enough; the chain of fifths will eventually "stop" and make the next fifth a diminished fifth. This means that this scale is, in a sense, "pantonal", since resolutions that work in one key will work in all other keys in the scale, at least keys that share the same chord quality.
Whitewood is the natural counterpart of [[blackwood]]: whereas blackwood can be thought of as a closed chain of five fifths and a [[5/4]] major third generator, whitewood is a closed chain of seven fifths and a 5/4 major third generator. This means that blackwood is generally supported by 5''n''-edos, and whitewood is supported by 7''n''-edos, and the [[mos]] of both scales follow a similar pattern.


Another interesting property is that it becomes possible to construct "super-linked" 5-limit chords. In whitewood[14], or blackwood[10], if one stacks alternating major and minor thirds on top of one another, one will eventually come back to the root without ever hitting a wall, and hence the pattern can continue forever. Since all of the diatonic modes can be thought of as a stacked chain of 7 alternating thirds, placed in inversion, this means that whitewood[14] and blackwood[10] also make for excellent "panmodal" scales, in which you can construct "modal" sounding sonorities in one key that will work in all keys.
[[Subgroup]]: 2.3.5


Lastly, while blackwood fifths are sharp and thus necessitate the tuning as a whole to be sharp-leaning, whitewood fifths are flat and thus this tuning is generally flat-leaning.
[[Comma list]]: 2187/2048


== Whitewood ==
{{Mapping|legend=1| 7 11 0 | 0 0 1 }}
[[Subgroup]]: 2.3.5
: mapping generators: ~9/8, ~5


[[Comma list]]: 2187/2048
[[Optimal tuning]]s:  
* [[WE]]: ~9/8 = 172.1541{{c}}, ~5/4 = 376.0535{{c}} (~80/81 = 31.7453{{c}})
: [[error map]]: {{val| +5.079 -8.260 -0.102 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 378.3830{{c}} (~80/81 = 35.5258{{c}})
: error map: {{val| 0.000 -16.241 -7.931 }}


[[Mapping]]: [{{val| 7 11 16 }}, {{val| 0 0 1 }}]
{{Optimal ET sequence|legend=1| 7, 21, 28, 35, 77bbc }}


Mapping generators: ~9/8, ~5
[[Badness]] (Sintel): 3.63


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 374.469
Scales: [[7L 7s in 140edo]]


{{Optimal ET sequence|legend=1| 7, 21, 28, 35, 77bb }}
=== Overview to extensions ===
Temperaments discussed elsewhere include:
* ''[[Sept]]'' → [[Very low accuracy temperaments #Sept|Very low accuracy temperaments]]


[[Badness]]: 0.154651
Considered below are septimal whitewood, redwood, greenwood, and jamesbond.  


== Septimal whitewood ==
== Septimal whitewood ==
{{Main| Whitewood }}
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 36/35, 2187/2048
[[Comma list]]: 36/35, 2187/2048


[[Mapping]]: [{{val| 7 11 16 20 }}, {{val| 0 0 1 -1 }}]
{{Mapping|legend=1| 7 11 0 36 | 0 0 1 -1 }}
 
{{Multival|legend=1| 0 7 -7 11 -11 -36 }}


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 392.700
[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 171.5524{{c}}, ~5/4 = 392.9834{{c}} (~64/63 = 49.8786{{c}})
: [[error map]]: {{val| +0.867 -14.879 +8.403 +12.343 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 392.7412{{c}} (~64/63 = 49.8841{{c}})
: error map: {{val| 0.000 -16.241 +6.428 +9.861 }}


{{Optimal ET sequence|legend=1| 7, 14, 21, 28, 49b }}
{{Optimal ET sequence|legend=1| 7, 14, 21, 28, 49b }}


[[Badness]]: 0.113987
[[Badness]] (Sintel): 2.88


=== 11-limit ===
=== 11-limit ===
Line 44: Line 56:
Comma list: 36/35, 45/44, 2079/2048
Comma list: 36/35, 45/44, 2079/2048


Mapping: [{{val| 7 11 16 20 24 }}, {{val| 0 0 1 -1 1 }}]
Mapping: {{mapping| 7 11 0 36 8 | 0 0 1 -1 1 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 389.968
Optimal tunings:
* WE: ~11/10 = 171.4451{{c}}, ~5/4 = 390.0053{{c}} (~64/63 = 47.1151{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 389.9864{{c}} (~64/63 = 47.1293{{c}})


{{Optimal ET sequence|legend=1| 7, 14e, 21, 28, 49b }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28 }}


Badness: 0.060908
Badness (Sintel): 2.01


=== 13-limit ===
=== 13-limit ===
Line 57: Line 71:
Comma list: 27/26, 36/35, 45/44, 512/507
Comma list: 27/26, 36/35, 45/44, 512/507


Mapping: [{{val| 7 11 16 20 24 26 }}, {{val| 0 0 1 -1 1 0 }}]
Mapping: {{mapping| 7 11 0 36 8 26 | 0 0 1 -1 1 0 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 390.735
Optimal tunings:
* WE: ~11/10 = 171.3236{{c}}, ~5/4 = 390.4957{{c}} (~64/63 = 47.8484{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 390.6336{{c}} (~64/63 = 47.7765{{c}})


{{Optimal ET sequence|legend=1| 7, 14e, 21, 28, 49bf }}
{{Optimal ET sequence|legend=0| 7, 14e, 21, 28 }}


Badness: 0.039956
Badness (Sintel): 1.65


== Redwood ==
== Redwood ==
Line 70: Line 86:
[[Comma list]]: 525/512, 729/700
[[Comma list]]: 525/512, 729/700


[[Mapping]]: [{{val| 7 11 16 20 }}, {{val| 0 0 1 -2 }}]
{{Mapping|legend=1| 7 11 0 52 | 0 0 1 -2 }}
 
{{Multival|legend=1| 0 7 -14 11 -22 -52 }}


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 378.152
[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 172.0521{{c}}, ~5/4 = 379.5277{{c}} (~36/35 = 35.4234{{c}})
: [[error map]]: {{val| +4.365 -9.382 +1.944 +1.370 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 377.7903{{c}} (~36/35 = 34.9331{{c}})
: error map: {{val| 0.000 -16.241 -8.523 -10.121 }}


{{Optimal ET sequence|legend=1| 7, 21d, 28d, 35 }}
{{Optimal ET sequence|legend=1| 7, 28d, 35 }}


[[Badness]]: 0.165257
[[Badness]] (Sintel): 4.18


=== 11-limit ===
=== 11-limit ===
Line 85: Line 103:
Comma list: 45/44, 385/384, 729/700
Comma list: 45/44, 385/384, 729/700


Mapping: [{{val| 7 11 16 20 24 }}, {{val| 0 0 1 -2 1 }}]
Mapping: {{mapping| 7 11 0 52 8 | 0 0 1 -2 1 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 376.711
Optimal tunings:
* WE: ~11/10 = 171.9390{{c}}, ~5/4 = 377.8321{{c}} (~36/35 = 33.9542{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~5/4 = 376.7162{{c}} (~36/35 = 33.8590{{c}})


{{Optimal ET sequence|legend=1| 7, 21d, 28d, 35 }}
{{Optimal ET sequence|legend=0| 7, 28d, 35 }}


Badness: 0.078193
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]]: [{{val| 7 11 16 20 }}, {{val| 0 0 1 0 }}]
{{Mapping|legend=1| 7 11 1 12 | 0 0 2 1 }}


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


[[Optimal tuning]] ([[POTE]]): ~9/8 = 1\7, ~5/4 = 395.187
[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 172.1073{{c}}, ~15/14 = 101.7681{{c}} (~21/20 = 70.3391{{c}})
: [[error map]]: {{val| +4.751 -8.775 -1.169 +2.980 }}
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~15/14 = 103.3802{{c}} (~21/20 = 68.0484{{c}})
: error map: {{val| 0.000 -16.241 -8.125 -8.303 }}


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


[[Badness]]: 0.105820
[[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: [{{val| 7 11 16 20 24 }}, {{val| 0 0 1 0 1 }}]
Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 394.661
Optimal tunings:
* WE: ~11/10 = 172.0795{{c}}, ~15/14 = 100.5259{{c}} (~21/20 = 71.5536{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 102.1866{{c}} (~21/20 = 69.2419{{c}})


{{Optimal ET sequence|legend=1| 7, 14de, 21dd }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35, 49bcde }}


Badness: 0.060985
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: [{{val| 7 11 16 20 24 26 }}, {{val| 0 0 1 0 1 0 }}]
Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}


Optimal tuning (POTE): ~9/8 = 1\7, ~5/4 = 395.071
Optimal tunings:
* WE: ~11/10 = 171.6777{{c}}, ~15/14 = 104.4016{{c}} (~21/20 = 67.2761{{c}})
* CWE: ~11/10 = 171.4286{{c}}, ~15/14 = 104.8518{{c}} (~21/20 = 66.5768{{c}})


{{Optimal ET sequence|legend=1| 7, 14de, 21dd }}
{{Optimal ET sequence|legend=0| 7ce, 14c, 21, 35 }}


Badness: 0.042830
Badness (Sintel): 2.23


== Greenwood ==
== Jamesbond ==
{{See also| Greenwoodmic temperaments #Greenwood }}
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
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 405/392, 1323/1280
[[Comma list]]: 25/24, 81/80
 
{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }}
: mapping generators: ~10/9, ~7
 
[[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(d), 14c }}
 
[[Badness]] (Sintel): 1.06


[[Mapping]]: [{{val| 7 11 1 12 }}, {{val| 0 0 2 1 }}]
=== 11-limit ===
Subgroup: 2.3.5.7.11


Mapping generators: ~9/8, ~15/7
Comma list: 25/24, 33/32, 45/44


{{Multival|legend=1| 0 14 7 22 11 -23 }}
Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}


[[Optimal tuning]] ([[CTE]]): ~9/8 = 1\7, ~15/14 = 108.062
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| 14c, 21, 35 }}
{{Optimal ET sequence|legend=0| 7(d), 14c }}


[[Badness]]: 0.121752
Badness (Sintel): 0.778


=== 11-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11.13


Comma list: 45/44, 99/98, 1323/1280
Comma list: 25/24, 27/26, 33/32, 40/39


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


Optimal tuning (CTE): ~9/8 = 1\7, ~15/14 = 106.997
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| 14c, 21, 35, 49bcde, 84bbccde }}
{{Optimal ET sequence|legend=0| 7(d), 14c }}


Badness: 0.057471
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: [{{val| 7 11 1 12 9 26 }}, {{val| 0 0 2 1 2 0 }}]
Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }}


Optimal tuning (CTE): ~9/8 = 1\7, ~15/14 = 106.997
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| 14c, 21, 35 }}
{{Optimal ET sequence|legend=0| 7(df), 14cf }}


Badness: 0.054009
Badness (Sintel): 0.933


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