Whitewood family: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Technical data page}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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.
: This revision was by author [[User:guest|guest]] and made on <tt>2011-01-02 12:44:54 UTC</tt>.<br>
: The original revision id was <tt>190488166</tt>.<br>
: The revision comment was: <tt></tt><br>
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
<h4>Original Wikitext content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">This family of temperaments tempers out the apotome, 2187/2048. Consequently the fifths are always 4/7 of an octave, a distinctly flat 685.714 cents. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.


[[POTE tuning|POTE generator]]: 374.469
== Whitewood ==
{{Main| Whitewood }}


Map: [&lt;7 11 0|, &lt;0 0 1|]
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.
EDOs: 7, 35, 77


==Whitewood==
[[Subgroup]]: 2.3.5
Commas: 36/35, 2187/2048


[[POTE tuning|POTE generator]]: 392.700
[[Comma list]]: 2187/2048


Map: [&lt;7 11 00 36|, &lt;0 0 1 -1|]
{{Mapping|legend=1| 7 11 0 | 0 0 1 }}
Wedgie: &lt;&lt;7 -7 11 -11 -36||
: mapping generators: ~9/8, ~5
EDOs: 7, 14, 21, 28, 35, 49


==Mujannab==  
[[Optimal tuning]]s:
Commas: 54/49, 64/63
* [[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 }}


[[POTE tuning|POTE generator]]: 395.187
{{Optimal ET sequence|legend=1| 7, 21, 28, 35, 77bbc }}


Map: [&lt;7 11 0 20|, &lt;0 0 1 0|]
[[Badness]] (Sintel): 3.63
Wedgie: &lt;&lt;0 7 0 11 0 -20||
 
EDOs: 7, 21, 70, 91</pre></div>
Scales: [[7L 7s/13:7]] (140edo)
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Apotome family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This family of temperaments tempers out the apotome, 2187/2048. Consequently the fifths are always 4/7 of an octave, a distinctly flat 685.714 cents. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.&lt;br /&gt;
=== Overview to extensions ===
&lt;br /&gt;
Temperaments discussed elsewhere include:
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 374.469&lt;br /&gt;
* ''[[Sept]]'' → [[Very low accuracy temperaments #Sept|Very low accuracy temperaments]]
&lt;br /&gt;
 
Map: [&amp;lt;7 11 0|, &amp;lt;0 0 1|]&lt;br /&gt;
Considered below are septimal whitewood, redwood, greenwood, and jamesbond.
EDOs: 7, 35, 77&lt;br /&gt;
 
&lt;br /&gt;
== Septimal whitewood ==
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc0"&gt;&lt;a name="x-Whitewood"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Whitewood&lt;/h2&gt;
{{Main| Whitewood }}
Commas: 36/35, 2187/2048&lt;br /&gt;
 
&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 392.700&lt;br /&gt;
 
&lt;br /&gt;
[[Comma list]]: 36/35, 2187/2048
Map: [&amp;lt;7 11 00 36|, &amp;lt;0 0 1 -1|]&lt;br /&gt;
 
Wedgie: &amp;lt;&amp;lt;7 -7 11 -11 -36||&lt;br /&gt;
{{Mapping|legend=1| 7 11 0 36 | 0 0 1 -1 }}
EDOs: 7, 14, 21, 28, 35, 49&lt;br /&gt;
 
&lt;br /&gt;
[[Optimal tuning]]s:
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc1"&gt;&lt;a name="x-Mujannab"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Mujannab&lt;/h2&gt;
* [[WE]]: ~9/8 = 171.5524{{c}}, ~5/4 = 392.9834{{c}} (~64/63 = 49.8786{{c}})
Commas: 54/49, 64/63&lt;br /&gt;
: [[error map]]: {{val| +0.867 -14.879 +8.403 +12.343 }}
&lt;br /&gt;
* [[CWE]]: ~9/8 = 171.4286{{c}}, ~5/4 = 392.7412{{c}} (~64/63 = 49.8841{{c}})
&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: 395.187&lt;br /&gt;
: error map: {{val| 0.000 -16.241 +6.428 +9.861 }}
&lt;br /&gt;
 
Map: [&amp;lt;7 11 0 20|, &amp;lt;0 0 1 0|]&lt;br /&gt;
{{Optimal ET sequence|legend=1| 7, 14, 21, 28, 49b }}
Wedgie: &amp;lt;&amp;lt;0 7 0 11 0 -20||&lt;br /&gt;
 
EDOs: 7, 21, 70, 91&lt;/body&gt;&lt;/html&gt;</pre></div>
[[Badness]] (Sintel): 2.88
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 36/35, 45/44, 2079/2048
 
Mapping: {{mapping| 7 11 0 36 8 | 0 0 1 -1 1 }}
 
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=0| 7, 14e, 21, 28 }}
 
Badness (Sintel): 2.01
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 36/35, 45/44, 512/507
 
Mapping: {{mapping| 7 11 0 36 8 26 | 0 0 1 -1 1 0 }}
 
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=0| 7, 14e, 21, 28 }}
 
Badness (Sintel): 1.65
 
== Redwood ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 525/512, 729/700
 
{{Mapping|legend=1| 7 11 0 52 | 0 0 1 -2 }}
 
[[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, 28d, 35 }}
 
[[Badness]] (Sintel): 4.18
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 385/384, 729/700
 
Mapping: {{mapping| 7 11 0 52 8 | 0 0 1 -2 1 }}
 
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=0| 7, 28d, 35 }}
 
Badness (Sintel): 2.59
 
== Greenwood ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 405/392, 1323/1280
 
{{Mapping|legend=1| 7 11 1 12 | 0 0 2 1 }}
 
: mapping generators: ~9/8, ~15/7
 
[[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| 7c, 14c, 21, 35, 84bbccd }}
 
[[Badness]] (Sintel): 3.08
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 45/44, 99/98, 1323/1280
 
Mapping: {{mapping| 7 11 1 12 9 | 0 0 2 1 2 }}
 
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=0| 7ce, 14c, 21, 35, 49bcde }}
 
Badness (Sintel): 1.90
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 27/26, 45/44, 99/98, 640/637
 
Mapping: {{mapping| 7 11 1 12 9 26 | 0 0 2 1 2 0 }}
 
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=0| 7ce, 14c, 21, 35 }}
 
Badness (Sintel): 2.23
 
== 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 }})
 
[[Subgroup]]: 2.3.5.7
 
[[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
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
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 16 0 24 26 | 0 0 0 1 0 0 }}
 
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=0| 7(d), 14c }}
 
Badness (Sintel): 0.951
 
==== Austinpowers ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 25/24, 33/32, 45/44, 65/63
 
Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }}
 
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=0| 7(df), 14cf }}
 
Badness (Sintel): 0.933
 
[[Category:Whitewood family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 12:26, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The whitewood family of temperaments tempers out the Pythagorean apotome, 2187/2048. Consequently the fifths are always 4/7 of an octave, a distinctly flat 685.714 cents. While quite flat, this is close enough to a just fifth to serve as one, and some people are fond of it.

Whitewood

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 5n-edos, and whitewood is supported by 7n-edos, and the mos of both scales follow a similar pattern.

Subgroup: 2.3.5

Comma list: 2187/2048

Mapping[7 11 0], 0 0 1]]

mapping generators: ~9/8, ~5

Optimal tunings:

  • WE: ~9/8 = 172.1541 ¢, ~5/4 = 376.0535 ¢ (~80/81 = 31.7453 ¢)
error map: +5.079 -8.260 -0.102]
  • CWE: ~9/8 = 171.4286 ¢, ~5/4 = 378.3830 ¢ (~80/81 = 35.5258 ¢)
error map: 0.000 -16.241 -7.931]

Optimal ET sequence7, 21, 28, 35, 77bbc

Badness (Sintel): 3.63

Scales: 7L 7s/13:7 (140edo)

Overview to extensions

Temperaments discussed elsewhere include:

Considered below are septimal whitewood, redwood, greenwood, and jamesbond.

Septimal whitewood

Subgroup: 2.3.5.7

Comma list: 36/35, 2187/2048

Mapping[7 11 0 36], 0 0 1 -1]]

Optimal tunings:

  • WE: ~9/8 = 171.5524 ¢, ~5/4 = 392.9834 ¢ (~64/63 = 49.8786 ¢)
error map: +0.867 -14.879 +8.403 +12.343]
  • CWE: ~9/8 = 171.4286 ¢, ~5/4 = 392.7412 ¢ (~64/63 = 49.8841 ¢)
error map: 0.000 -16.241 +6.428 +9.861]

Optimal ET sequence7, 14, 21, 28, 49b

Badness (Sintel): 2.88

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 2079/2048

Mapping: [7 11 0 36 8], 0 0 1 -1 1]]

Optimal tunings:

  • WE: ~11/10 = 171.4451 ¢, ~5/4 = 390.0053 ¢ (~64/63 = 47.1151 ¢)
  • CWE: ~11/10 = 171.4286 ¢, ~5/4 = 389.9864 ¢ (~64/63 = 47.1293 ¢)

Optimal ET sequence: 7, 14e, 21, 28

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 512/507

Mapping: [7 11 0 36 8 26], 0 0 1 -1 1 0]]

Optimal tunings:

  • WE: ~11/10 = 171.3236 ¢, ~5/4 = 390.4957 ¢ (~64/63 = 47.8484 ¢)
  • CWE: ~11/10 = 171.4286 ¢, ~5/4 = 390.6336 ¢ (~64/63 = 47.7765 ¢)

Optimal ET sequence: 7, 14e, 21, 28

Badness (Sintel): 1.65

Redwood

Subgroup: 2.3.5.7

Comma list: 525/512, 729/700

Mapping[7 11 0 52], 0 0 1 -2]]

Optimal tunings:

  • WE: ~9/8 = 172.0521 ¢, ~5/4 = 379.5277 ¢ (~36/35 = 35.4234 ¢)
error map: +4.365 -9.382 +1.944 +1.370]
  • CWE: ~9/8 = 171.4286 ¢, ~5/4 = 377.7903 ¢ (~36/35 = 34.9331 ¢)
error map: 0.000 -16.241 -8.523 -10.121]

Optimal ET sequence7, 28d, 35

Badness (Sintel): 4.18

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 385/384, 729/700

Mapping: [7 11 0 52 8], 0 0 1 -2 1]]

Optimal tunings:

  • WE: ~11/10 = 171.9390 ¢, ~5/4 = 377.8321 ¢ (~36/35 = 33.9542 ¢)
  • CWE: ~11/10 = 171.4286 ¢, ~5/4 = 376.7162 ¢ (~36/35 = 33.8590 ¢)

Optimal ET sequence: 7, 28d, 35

Badness (Sintel): 2.59

Greenwood

Subgroup: 2.3.5.7

Comma list: 405/392, 1323/1280

Mapping[7 11 1 12], 0 0 2 1]]

mapping generators: ~9/8, ~15/7

Optimal tunings:

  • WE: ~9/8 = 172.1073 ¢, ~15/14 = 101.7681 ¢ (~21/20 = 70.3391 ¢)
error map: +4.751 -8.775 -1.169 +2.980]
  • CWE: ~9/8 = 171.4286 ¢, ~15/14 = 103.3802 ¢ (~21/20 = 68.0484 ¢)
error map: 0.000 -16.241 -8.125 -8.303]

Optimal ET sequence7c, 14c, 21, 35, 84bbccd

Badness (Sintel): 3.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 45/44, 99/98, 1323/1280

Mapping: [7 11 1 12 9], 0 0 2 1 2]]

Optimal tunings:

  • WE: ~11/10 = 172.0795 ¢, ~15/14 = 100.5259 ¢ (~21/20 = 71.5536 ¢)
  • CWE: ~11/10 = 171.4286 ¢, ~15/14 = 102.1866 ¢ (~21/20 = 69.2419 ¢)

Optimal ET sequence: 7ce, 14c, 21, 35, 49bcde

Badness (Sintel): 1.90

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 99/98, 640/637

Mapping: [7 11 1 12 9 26], 0 0 2 1 2 0]]

Optimal tunings:

  • WE: ~11/10 = 171.6777 ¢, ~15/14 = 104.4016 ¢ (~21/20 = 67.2761 ¢)
  • CWE: ~11/10 = 171.4286 ¢, ~15/14 = 104.8518 ¢ (~21/20 = 66.5768 ¢)

Optimal ET sequence: 7ce, 14c, 21, 35

Badness (Sintel): 2.23

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 ⟨⟨ 0 0 7 … ]] (in fact, it is ⟨⟨ 0 0 7 0 11 16 ]])

Subgroup: 2.3.5.7

Comma list: 25/24, 81/80

Mapping[7 11 16 0], 0 0 0 1]]

mapping generators: ~10/9, ~7

Optimal tunings:

  • WE: ~10/9 = 172.790 ¢, ~7/4 = 949.343 ¢
error map: +9.533 -1.261 -21.668 -0.418]
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 948.499 ¢
error map: -0.000 -16.241 -43.457 -20.327]

Optimal ET sequence7(d), 14c

Badness (Sintel): 1.06

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 45/44

Mapping: [7 11 16 0 24], 0 0 0 1 0]]

Optimal tunings:

  • WE: ~10/9 = 172.830 ¢, ~7/4 = 948.784 ¢
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 946.554 ¢

Optimal ET sequence: 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: [7 11 16 0 24 26], 0 0 0 1 0 0]]

Optimal tunings:

  • WE: ~10/9 = 172.390 ¢, ~7/4 = 954.559 ¢
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 952.367 ¢

Optimal ET sequence: 7(d), 14c

Badness (Sintel): 0.951

Austinpowers

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 33/32, 45/44, 65/63

Mapping: [7 11 16 0 24 6], 0 0 0 1 0 1]]

Optimal tunings:

  • WE: ~10/9 = 172.873 ¢, ~7/4 = 960.581 ¢
  • CWE: ~10/9 = 171.429 ¢, ~7/4 = 958.793 ¢

Optimal ET sequence: 7(df), 14cf

Badness (Sintel): 0.933