Fifive 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 '''fifive family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[fifive comma]] ({{monzo|legend=1| -1 -14 10 }}, [[ratio]]: 9765625/9565938).
: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2017-12-19 21:08:25 UTC</tt>.<br>
: The original revision id was <tt>624085645</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 tempers out the fifive comma, 9765625/9565938 = (-1, -14, 10).


=Fifive=  
== Fifive ==
Comma: 9765625/9565938
The head of this family is fifive, which splits the [[3/2|perfect fifth]] into five [[27/25]]'s, and [[5/4]] is found as seven generators minus a half-octave period. Its [[ploidacot]] is diploid pentacot, and it is a member of the [[diaschismic–gothmic equivalence continuum]] with equivalence number ''n'' = 5/2.


POTE generator: ~27/25 = 140.624
The name ''fifive'' was given by [[Petr Pařízek]] in 2011 for it splits the perfect fifth in five.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>


Map: [&lt;2 2 3|, &lt;0 5 7|]
[[Subgroup]]: 2.3.5
EDOs: 8, 26, 34, 94, 128
Badness: 0.2058


=Crepuscular=
[[Comma list]]: 9765625/9565938
Commas: 50/49 4375/4374


POTE generator: ~27/25 = 140.349
{{Mapping|legend=1| 2 2 3 | 0 5 7 }}
: mapping generators: ~78125/50421, ~27/25


Map: [&lt;2 2 3 4|, &lt;0 5 7 7|]
[[Optimal tuning]]s:
Wedgie: &lt;&lt;10 14 14 -1 -6 -7||
* [[WE]]: ~78125/50421 = 600.0168{{c}}, ~27/25 = 140.6276{{c}}
EDOs: 26, 34d, 60d, 94d
: [[error map]]: {{val| +0.034 +1.217 -1.870 }}
Badness: 0.0867
* [[CWE]]: ~78125/50421 = 600.0000{{c}}, ~27/25 = 140.6291{{c}}
: error map: {{val| 0.000 +1.191 -1.910 }}


==11-limit==
{{Optimal ET sequence|legend=1| 8, 18bc, 26, 34, 94, 128 }}
Commas: 50/49, 99/98, 1944/1925


POTE generator: ~12/11 = ~27/25 = 140.587
[[Badness]] (Sintel): 4.83


Map: [&lt;2 2 3 4 6|, &lt;0 5 7 7 4|]
=== Overview to extensions ===
EDOs: 26, 34d, 60d, 94de
The second comma in the comma list defines which 7-limit family member we are looking at. Crepuscular (26 & 34d) adds [[50/49]]. Fifives (26 & 34) adds [[875/864]]. Both use the same generators as fifive. The weak extension fourfives (60 & 68) adds [[245/243]]. All are considered below.
Badness: 0.0408


==13-limit==
The fifive family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].
Commas: 50/49, 78/77, 99/98, 144/143


POTE generator: ~12/11 = 140.554
== Crepuscular ==
[[Subgroup]]: 2.3.5.7


Map: [&lt;2 2 3 4 6 6|, &lt;0 5 7 7 4 6|]
[[Comma list]]: 50/49, 4375/4374
EDOs: 26, 34d, 60d, 94de
Badness: 0.0244


=Fifives=
{{Mapping|legend=1| 2 2 3 4 | 0 5 7 7 }}
Commas: 875/864, 83349/81920


=Fourfives=  
[[Optimal tuning]]s:
Commas: 245/243, 235298/234375</pre></div>
* [[WE]]: ~7/5 = 600.000{{c}}, ~27/25 = 140.349{{c}}
<h4>Original HTML content:</h4>
: [[error map]]: {{val| -0.787 -1.458 -5.696 +11.398 }}
<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;Fifive family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This tempers out the fifive comma, 9765625/9565938 = (-1, -14, 10).&lt;br /&gt;
* [[CWE]]: ~7/5 = 600.000{{c}}, ~27/25 = 140.349{{c}}
&lt;br /&gt;
: error map: {{val| 0.000 -0.876 -4.803 +12.685 }}
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Fifive"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Fifive&lt;/h1&gt;
 
Comma: 9765625/9565938&lt;br /&gt;
{{Optimal ET sequence|legend=1| 8d, 18bcd, 26, 34d, 60d }}
&lt;br /&gt;
 
POTE generator: ~27/25 = 140.624&lt;br /&gt;
[[Badness]] (Sintel): 2.19
&lt;br /&gt;
 
Map: [&amp;lt;2 2 3|, &amp;lt;0 5 7|]&lt;br /&gt;
=== 11-limit ===
EDOs: 8, 26, 34, 94, 128&lt;br /&gt;
Subgroup: 2.3.5.7.11
Badness: 0.2058&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 50/49, 99/98, 864/847
&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Crepuscular"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Crepuscular&lt;/h1&gt;
 
Commas: 50/49 4375/4374&lt;br /&gt;
Mapping: {{mapping| 2 2 3 4 6 | 0 5 7 7 4 }}
&lt;br /&gt;
 
POTE generator: ~27/25 = 140.349&lt;br /&gt;
Optimal tunings:  
&lt;br /&gt;
* WE: ~7/5 = 599.1577{{c}}, ~12/11 = 140.3893{{c}}
Map: [&amp;lt;2 2 3 4|, &amp;lt;0 5 7 7|]&lt;br /&gt;
* CWE: ~7/5 = 600.0000{{c}}, ~12/11 = 140.3244{{c}}
Wedgie: &amp;lt;&amp;lt;10 14 14 -1 -6 -7||&lt;br /&gt;
 
EDOs: 26, 34d, 60d, 94d&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 18bcd, 26, 34d, 60d }}
Badness: 0.0867&lt;br /&gt;
 
&lt;br /&gt;
Badness (Sintel): 1.35
&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Crepuscular-11-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;11-limit&lt;/h2&gt;
 
Commas: 50/49, 99/98, 1944/1925&lt;br /&gt;
=== 13-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13
POTE generator: ~12/11 = ~27/25 = 140.587&lt;br /&gt;
 
&lt;br /&gt;
Comma list: 50/49, 78/77, 99/98, 144/143
Map: [&amp;lt;2 2 3 4 6|, &amp;lt;0 5 7 7 4|]&lt;br /&gt;
 
EDOs: 26, 34d, 60d, 94de&lt;br /&gt;
Mapping: {{mapping| 2 2 3 4 6 6 | 0 5 7 7 4 6 }}
Badness: 0.0408&lt;br /&gt;
 
&lt;br /&gt;
Optimal tunings:  
&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc3"&gt;&lt;a name="Crepuscular-13-limit"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;13-limit&lt;/h2&gt;
* WE: ~7/5 = 599.2598{{c}}, ~13/12 = 140.3805{{c}}
Commas: 50/49, 78/77, 99/98, 144/143&lt;br /&gt;
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.3237{{c}}
&lt;br /&gt;
 
POTE generator: ~12/11 = 140.554&lt;br /&gt;
{{Optimal ET sequence|legend=0| 8d, 18bcdf, 26, 34d, 60d }}
&lt;br /&gt;
 
Map: [&amp;lt;2 2 3 4 6 6|, &amp;lt;0 5 7 7 4 6|]&lt;br /&gt;
Badness (Sintel): 1.01
EDOs: 26, 34d, 60d, 94de&lt;br /&gt;
 
Badness: 0.0244&lt;br /&gt;
=== 17-limit ===
&lt;br /&gt;
Subgroup: 2.3.5.7.11.13.17
&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc4"&gt;&lt;a name="Fifives"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;Fifives&lt;/h1&gt;
 
Commas: 875/864, 83349/81920&lt;br /&gt;
Comma list: 50/49, 78/77, 85/84, 99/98, 144/143
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc5"&gt;&lt;a name="Fourfives"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Fourfives&lt;/h1&gt;
Mapping: {{mapping| 2 2 3 4 6 6 7 | 0 5 7 7 4 6 5 }}
Commas: 245/243, 235298/234375&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Optimal tunings:
* WE: ~7/5 = 599.5348{{c}}, ~13/12 = 140.2961{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~13/12 = 140.2668{{c}}
 
{{Optimal ET sequence|legend=0| 8d, 18bcdfg, 26, 34d, 60d }}
 
Badness (Sintel): 0.946
 
== Fifives ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 875/864, 83349/81920
 
{{Mapping|legend=1| 2 2 3 7 | 0 5 7 -6 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~567/400 = 600.9312{{c}}, ~27/25 = 140.1261{{c}}
: [[error map]]: {{val| +1.862 +0.538 -2.637 -3.064 }}
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~27/25 = 139.9826{{c}}
: error map: {{val| 0.000 -2.042 -6.435 -8.722 }}
 
{{Optimal ET sequence|legend=1| 8, 26, 34, 60, 266bcccddd }}
 
[[Badness]] (Sintel): 3.30
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 100/99, 385/384, 3969/3872
 
Mapping: {{mapping| 2 2 3 7 6 | 0 5 7 -6 4 }}
 
Optimal tunings:
* WE: ~63/44 = 600.5091{{c}}, ~12/11 = 140.0024{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~12/11 = 139.9267{{c}}
 
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
 
Badness (Sintel): 2.65
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 100/99, 105/104, 144/143, 1352/1331
 
Mapping: {{mapping| 2 2 3 7 6 6 | 0 5 7 -6 4 6 }}
 
Optimal tunings:
* WE: ~55/39 = 600.4600{{c}}, ~12/11 = 139.9737{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 139.9092{{c}}
 
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
 
Badness (Sintel): 1.83
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 100/99, 105/104, 144/143, 170/169, 221/220
 
Mapping: {{mapping| 2 2 3 7 6 6 7 | 0 5 7 -6 4 6 5 }}
 
Optimal tunings:
* WE: ~17/12 = 600.4903{{c}}, ~12/11 = 139.9825{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~12/11 = 139.9150{{c}}
 
Optimal tuning (POTE): ~17/12 = 600.000{{c}}, ~12/11 = 139.868{{c}}
 
{{Optimal ET sequence|legend=0| 8, 26, 34, 60 }}
 
Badness (Sintel): 1.50
 
== Fourfives ==
{{See also| Sensamagic clan }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 245/243, 235298/234375
 
{{Mapping|legend=1| 4 4 6 7 | 0 5 7 9 }}
: mapping generators: ~25/21, ~27/25
 
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 300.0011{{c}}, ~27/25 = 140.7547{{c}}
: [[error map]]: {{val| +0.004 +1.823 -1.024 -2.026 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~27/25 = 140.7549{{c}}
: error map: {{val| 0.000 +1.819 -1.030 -2.032 }}
 
{{Optimal ET sequence|legend=1| 8d, …, 60, 68, 128, 196 }}
 
[[Badness]] (Sintel): 2.89
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 245/243, 385/384, 235298/234375
 
Mapping: {{mapping| 4 4 6 7 19 | 0 5 7 9 -11 }}
 
Optimal tunings:
* WE: ~25/21 = 299.9901{{c}}, ~27/25 = 140.7659{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/25 = 140.7693{{c}}
 
{{Optimal ET sequence|legend=0| 60, 68, 128, 196 }}
 
Badness (Sintel): 3.97
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 196/195, 245/243, 385/384, 20000/19773
 
Mapping: {{mapping| 4 4 6 7 19 12 | 0 5 7 9 -11 6 }}
 
Optimal tunings:
* WE: ~25/21 = 299.9488{{c}}, ~13/12 = 140.7359{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7539{{c}}
 
{{Optimal ET sequence|legend=0| 60, 68, 128, 196f }}
 
Badness (Sintel): 2.78
 
=== Quadrafives ===
Subgroup: 2.3.5.7.11
 
Comma list: 121/120, 245/243, 1375/1372
 
Mapping: {{mapping| 4 4 6 7 11 | 0 5 7 9 6 }}
 
Optimal tunings:
* WE: ~25/21 = 300.1673{{c}}, ~27/25 = 140.7084{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~27/25 = 140.7353{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68, 128e }}
 
Badness (Sintel): 1.89
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 121/120, 196/195, 245/243, 275/273
 
Mapping: {{mapping| 4 4 6 7 11 12 | 0 5 7 9 6 6 }}
 
Optimal tunings:
* WE: ~25/21 = 300.0500{{c}}, ~13/12 = 140.7516{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7590{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68 }}
 
Badness (Sintel): 1.49
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 121/120, 154/153, 170/169, 196/195, 245/243
 
Mapping: {{mapping| 4 4 6 7 11 12 14 | 0 5 7 9 6 6 5 }}
 
Optimal tunings:
* WE: ~25/21 = 300.0593{{c}}, ~13/12 = 140.7457{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~13/12 = 140.7520{{c}}
 
Optimal tuning (POTE): ~25/21 = 300.000{{c}}, ~13/12 = 140.718{{c}}
 
{{Optimal ET sequence|legend=0| 8d, …, 60e, 68 }}
 
Badness (Sintel): 1.26
 
== Subgroup extensions ==
=== Fifive (2.3.5.13) ===
As the [[~]][[27/25]] generator of fifive is so close to [[13/12]], one may temper out their difference, [[325/324]], to obtain this extension in the 2.3.5.13 subgroup. It is also not unreasonable to equate both [[17/12]] and [[24/17]] with the semi-octave period given its overall level of precision, tempering out [[289/288]] and leading to a 2.3.5.13.17-subgroup temperament.
 
Subgroup: 2.3.5.13
 
Comma list: 325/324, 20000/19773
 
Subgroup-val mapping: {{mapping| 2 2 3 6 | 0 5 7 6 }}
 
Optimal tunings:
* WE: ~351/250 = 599.8593{{c}}, ~13/12 = 140.6362{{c}}
* CWE: ~351/250 = 600.0000{{c}}, ~13/12 = 140.6232{{c}}
 
{{Optimal ET sequence|legend=0| 8, 18bcf, 26, 34, 94, 128 }}
 
Badness (Sintel): 0.800
 
==== 2.3.5.13.17 subgroup ====
Subgroup: 2.3.5.13.17
 
Comma list: 170/169, 289/288, 325/324
 
Subgroup-val mapping: {{mapping| 2 2 3 6 7 | 0 5 7 6 5 }}
 
Optimal tunings:
* WE: ~17/12 = 599.9773{{c}}, ~13/12 = 140.6075{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~13/12 = 140.6057{{c}}
 
{{Optimal ET sequence|legend=0| 8, 18bcfg, 26, 34, 94, 128 }}
 
Badness (Sintel): 0.488
 
== References ==
 
[[Category:Fifive family| ]] <!-- main article -->
[[Category:Fifive| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 02:49, 27 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 fifive family of temperaments tempers out the fifive comma (monzo: [-1 -14 10⟩, ratio: 9765625/9565938).

Fifive

The head of this family is fifive, which splits the perfect fifth into five 27/25's, and 5/4 is found as seven generators minus a half-octave period. Its ploidacot is diploid pentacot, and it is a member of the diaschismic–gothmic equivalence continuum with equivalence number n = 5/2.

The name fifive was given by Petr Pařízek in 2011 for it splits the perfect fifth in five.[1]

Subgroup: 2.3.5

Comma list: 9765625/9565938

Mapping: [⟨2 2 3], ⟨0 5 7]]

mapping generators: ~78125/50421, ~27/25

Optimal tunings:

  • WE: ~78125/50421 = 600.0168 ¢, ~27/25 = 140.6276 ¢
error map: ⟨+0.034 +1.217 -1.870]
  • CWE: ~78125/50421 = 600.0000 ¢, ~27/25 = 140.6291 ¢
error map: ⟨0.000 +1.191 -1.910]

Optimal ET sequence: 8, 18bc, 26, 34, 94, 128

Badness (Sintel): 4.83

Overview to extensions

The second comma in the comma list defines which 7-limit family member we are looking at. Crepuscular (26 & 34d) adds 50/49. Fifives (26 & 34) adds 875/864. Both use the same generators as fifive. The weak extension fourfives (60 & 68) adds 245/243. All are considered below.

The fifive family boasts a very remarkable extension to the 2.3.5.13 subgroup, which has further extensions with higher primes. These are listed at the bottom of this page, in #Subgroup extensions.

Crepuscular

Subgroup: 2.3.5.7

Comma list: 50/49, 4375/4374

Mapping: [⟨2 2 3 4], ⟨0 5 7 7]]

Optimal tunings:

  • WE: ~7/5 = 600.000 ¢, ~27/25 = 140.349 ¢
error map: ⟨-0.787 -1.458 -5.696 +11.398]
  • CWE: ~7/5 = 600.000 ¢, ~27/25 = 140.349 ¢
error map: ⟨0.000 -0.876 -4.803 +12.685]

Optimal ET sequence: 8d, 18bcd, 26, 34d, 60d

Badness (Sintel): 2.19

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 99/98, 864/847

Mapping: [⟨2 2 3 4 6], ⟨0 5 7 7 4]]

Optimal tunings:

  • WE: ~7/5 = 599.1577 ¢, ~12/11 = 140.3893 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~12/11 = 140.3244 ¢

Optimal ET sequence: 8d, 18bcd, 26, 34d, 60d

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 99/98, 144/143

Mapping: [⟨2 2 3 4 6 6], ⟨0 5 7 7 4 6]]

Optimal tunings:

  • WE: ~7/5 = 599.2598 ¢, ~13/12 = 140.3805 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~13/12 = 140.3237 ¢

Optimal ET sequence: 8d, 18bcdf, 26, 34d, 60d

Badness (Sintel): 1.01

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 85/84, 99/98, 144/143

Mapping: [⟨2 2 3 4 6 6 7], ⟨0 5 7 7 4 6 5]]

Optimal tunings:

  • WE: ~7/5 = 599.5348 ¢, ~13/12 = 140.2961 ¢
  • CWE: ~7/5 = 600.0000 ¢, ~13/12 = 140.2668 ¢

Optimal ET sequence: 8d, 18bcdfg, 26, 34d, 60d

Badness (Sintel): 0.946

Fifives

Subgroup: 2.3.5.7

Comma list: 875/864, 83349/81920

Mapping: [⟨2 2 3 7], ⟨0 5 7 -6]]

Optimal tunings:

  • WE: ~567/400 = 600.9312 ¢, ~27/25 = 140.1261 ¢
error map: ⟨+1.862 +0.538 -2.637 -3.064]
  • CWE: ~567/400 = 600.0000 ¢, ~27/25 = 139.9826 ¢
error map: ⟨0.000 -2.042 -6.435 -8.722]

Optimal ET sequence: 8, 26, 34, 60, 266bcccddd

Badness (Sintel): 3.30

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 3969/3872

Mapping: [⟨2 2 3 7 6], ⟨0 5 7 -6 4]]

Optimal tunings:

  • WE: ~63/44 = 600.5091 ¢, ~12/11 = 140.0024 ¢
  • CWE: ~63/44 = 600.0000 ¢, ~12/11 = 139.9267 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 2.65

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 1352/1331

Mapping: [⟨2 2 3 7 6 6], ⟨0 5 7 -6 4 6]]

Optimal tunings:

  • WE: ~55/39 = 600.4600 ¢, ~12/11 = 139.9737 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~12/11 = 139.9092 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 1.83

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 144/143, 170/169, 221/220

Mapping: [⟨2 2 3 7 6 6 7], ⟨0 5 7 -6 4 6 5]]

Optimal tunings:

  • WE: ~17/12 = 600.4903 ¢, ~12/11 = 139.9825 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~12/11 = 139.9150 ¢

Optimal tuning (POTE): ~17/12 = 600.000 ¢, ~12/11 = 139.868 ¢

Optimal ET sequence: 8, 26, 34, 60

Badness (Sintel): 1.50

Fourfives

See also: Sensamagic clan

Subgroup: 2.3.5.7

Comma list: 245/243, 235298/234375

Mapping: [⟨4 4 6 7], ⟨0 5 7 9]]

mapping generators: ~25/21, ~27/25

Optimal tunings:

  • WE: ~25/21 = 300.0011 ¢, ~27/25 = 140.7547 ¢
error map: ⟨+0.004 +1.823 -1.024 -2.026]
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7549 ¢
error map: ⟨0.000 +1.819 -1.030 -2.032]

Optimal ET sequence: 8d, …, 60, 68, 128, 196

Badness (Sintel): 2.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/243, 385/384, 235298/234375

Mapping: [⟨4 4 6 7 19], ⟨0 5 7 9 -11]]

Optimal tunings:

  • WE: ~25/21 = 299.9901 ¢, ~27/25 = 140.7659 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7693 ¢

Optimal ET sequence: 60, 68, 128, 196

Badness (Sintel): 3.97

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 245/243, 385/384, 20000/19773

Mapping: [⟨4 4 6 7 19 12], ⟨0 5 7 9 -11 6]]

Optimal tunings:

  • WE: ~25/21 = 299.9488 ¢, ~13/12 = 140.7359 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7539 ¢

Optimal ET sequence: 60, 68, 128, 196f

Badness (Sintel): 2.78

Quadrafives

Subgroup: 2.3.5.7.11

Comma list: 121/120, 245/243, 1375/1372

Mapping: [⟨4 4 6 7 11], ⟨0 5 7 9 6]]

Optimal tunings:

  • WE: ~25/21 = 300.1673 ¢, ~27/25 = 140.7084 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~27/25 = 140.7353 ¢

Optimal ET sequence: 8d, …, 60e, 68, 128e

Badness (Sintel): 1.89

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 245/243, 275/273

Mapping: [⟨4 4 6 7 11 12], ⟨0 5 7 9 6 6]]

Optimal tunings:

  • WE: ~25/21 = 300.0500 ¢, ~13/12 = 140.7516 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7590 ¢

Optimal ET sequence: 8d, …, 60e, 68

Badness (Sintel): 1.49

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 154/153, 170/169, 196/195, 245/243

Mapping: [⟨4 4 6 7 11 12 14], ⟨0 5 7 9 6 6 5]]

Optimal tunings:

  • WE: ~25/21 = 300.0593 ¢, ~13/12 = 140.7457 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~13/12 = 140.7520 ¢

Optimal tuning (POTE): ~25/21 = 300.000 ¢, ~13/12 = 140.718 ¢

Optimal ET sequence: 8d, …, 60e, 68

Badness (Sintel): 1.26

Subgroup extensions

Fifive (2.3.5.13)

As the ~27/25 generator of fifive is so close to 13/12, one may temper out their difference, 325/324, to obtain this extension in the 2.3.5.13 subgroup. It is also not unreasonable to equate both 17/12 and 24/17 with the semi-octave period given its overall level of precision, tempering out 289/288 and leading to a 2.3.5.13.17-subgroup temperament.

Subgroup: 2.3.5.13

Comma list: 325/324, 20000/19773

Subgroup-val mapping: [⟨2 2 3 6], ⟨0 5 7 6]]

Optimal tunings:

  • WE: ~351/250 = 599.8593 ¢, ~13/12 = 140.6362 ¢
  • CWE: ~351/250 = 600.0000 ¢, ~13/12 = 140.6232 ¢

Optimal ET sequence: 8, 18bcf, 26, 34, 94, 128

Badness (Sintel): 0.800

2.3.5.13.17 subgroup

Subgroup: 2.3.5.13.17

Comma list: 170/169, 289/288, 325/324

Subgroup-val mapping: [⟨2 2 3 6 7], ⟨0 5 7 6 5]]

Optimal tunings:

  • WE: ~17/12 = 599.9773 ¢, ~13/12 = 140.6075 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~13/12 = 140.6057 ¢

Optimal ET sequence: 8, 18bcfg, 26, 34, 94, 128

Badness (Sintel): 0.488

References

  1. ↑ Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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