Xenharmonic Wiki

Breed family: Difference between revisions

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**Imported revision 160331723 - Original comment: **
 
m Restore the category of the corresponding temp
 
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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 '''breed family''' of [[rank-3 temperament]]s are [[microtemperament]]s which [[tempering out|temper out]] [[2401/2400]].  
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-09-03 20:14:10 UTC</tt>.<br>
: The original revision id was <tt>160331723</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">Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2578et will certainly do the trick. Breed has generators of a 49/40 neutral third, and 10/7.


===Jove, aka Wonder[[#wonder]]===
== Breed ==
Jove, formerly known as wonder, tempers out 243/242 and 441/440. Wonder has been deprecated as a name due to conflict with another temperament also given that name. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72 and 130 are good edos for jove, and if that doesn't suit there's 476edo.</pre></div>
Breed has the same lattice structure as the [[2.5.7 subgroup]] of JI. In terms of extension from the 5-limit, it splits 25/6 into four equal parts representing 10/7. While it is so accurate it hardly matters what is used to temper it, or whether it is tempered at all, the [[optimal patent val]] [[2749edo|2749et]] will certainly do the trick.
<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;Breed family&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Breed is a 7-limit microtemperament which tempers out 2401/2400. While it is so accurate it hardly matters what is used to temper it, or whether it is even tempered at all, 2578et will certainly do the trick. Breed has generators of a 49/40 neutral third, and 10/7.&lt;br /&gt;
[[Subgroup]]: 2.3.5.7
&lt;br /&gt;
 
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Jove, aka Wonder"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Jove, aka Wonder&lt;!-- ws:start:WikiTextAnchorRule:2:&amp;lt;img src=&amp;quot;/i/anchor.gif&amp;quot; class=&amp;quot;WikiAnchor&amp;quot; alt=&amp;quot;Anchor&amp;quot; id=&amp;quot;wikitext@@anchor@@wonder&amp;quot; title=&amp;quot;Anchor: wonder&amp;quot;/&amp;gt; --&gt;&lt;a name="wonder"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextAnchorRule:2 --&gt;&lt;/h3&gt;
[[Comma list]]: 2401/2400
Jove, formerly known as wonder, tempers out 243/242 and 441/440. Wonder has been deprecated as a name due to conflict with another temperament also given that name. Jove converts breed into an 11-limit temperament via 441/440, which equates 49/40 with 11/9, and 243/242, which tells us 11/9 can serve as a neutral third. While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. 72 and 130 are good edos for jove, and if that doesn't suit there's 476edo.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
{{Mapping|legend=1| 1 1 1 2 | 0 2 1 1 | 0 0 2 1 }}
: Mapping generators: ~2, ~49/40, ~10/7
 
[[Mapping to lattice]]: [{{val| 0 2 -1 0 }}, {{val| 0 0 -2 -1 }}]
 
Lattice basis:
: 49/40 length = 0.7858, 8/7 length = 1.1241
: Angle (49/40, 8/7) = 107.367°
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0206{{c}}, ~49/40 = 350.9724{{c}}, ~10/7 = 617.6826{{c}}
: [[error map]]: {{val| +0.021 +0.010 +0.044 -0.130 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9743{{c}}, ~10/7 = 617.6807{{c}}
: error map: {{val| 0.000 -0.006 +0.022 -0.171 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.5
 
{{Optimal ET sequence|legend=1| 27, 31, 41, 58, 68, 72, 99, 171, 441, 612, 1966, 2308, 2578, 2749, 3361d }}
 
[[Badness]] (Sintel): 0.0674
 
[[Projection pair]]: <code>3 2401/800</code> to 2.5.7
 
Scales: [[breed11]]
 
; Music
* ''Bodacious Breed'' (archived 2010) by [[Gene Ward Smith]] – [http://www.archive.org/details/BodaciousBreed details] | [http://www.archive.org/download/BodaciousBreed/Genewardsmith-BodaciousBreed.mp3 play] – breed in 441edo tuning
 
== Jove ==
{{Main| Jove }}
 
Jove (formerly known as wonder) tempers out [[243/242]], [[441/440]], and [[540/539]]. Jove converts breed into an 11-limit temperament via 441/440, which equates [[49/40]] with [[11/9]], and 243/242, which tells us 11/9 can serve as a neutral third (as in exactly half of [[~]][[3/2]]). While jove is no longer a super-accurate microtemperament like breed, it has the advantage of adjusting its tuning to deal with the 11-limit. [[72edo|72]], [[130edo|130]], [[171edo|171]] and [[202edo|202]] are good edo tunings for jove.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 243/242, 441/440
 
{{Mapping|legend=1| 1 1 1 2 2 | 0 2 1 1 5 | 0 0 2 1 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0985{{c}}, ~11/9 = 350.5322{{c}}, ~10/7 = 617.8782{{c}}
: [[error map]]: {{val| +0.098 -0.792 +0.073 -0.218 +1.540 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~11/9 = 350.5374{{c}}, ~10/7 = 617.8712{{c}}
: error map: {{val| 0.000 -0.880 -0.034 -0.417 +1.369 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5.11
 
{{Optimal ET sequence|legend=1| 27e, 31, 41, 58, 72, 130, 202 }}
 
[[Badness]] (Sintel): 0.290
 
[[Projection pair]]s: <code>3 242/81, 5 2200/441, 7 440/63, 11 644204/59049</code> to 2.7/5.11/9
 
=== Jovial ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 364/363, 441/440
 
{{Mapping|legend=0| 1 1 1 2 2 1 | 0 2 1 1 5 11 | 0 0 2 1 0 -1 }}
 
Optimal tunings:
* WE: ~2 = 1199.9775{{c}}, ~11/9 = 350.7113{{c}}, ~10/7 = 617.8170{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.7130{{c}}, ~10/7 = 617.8178{{c}}
 
Minimax tuning:
* 13-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/5.13/9
* 15-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/5.15/13
 
{{Optimal ET sequence|legend=0| 27eff, 31f, 41, 58, 72, 130, 243, 301e, 373e }}
 
Badness (Sintel): 0.583
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 441/440, 595/594
 
{{Mapping|legend=0| 1 1 1 2 2 1 3 | 0 2 1 1 5 11 9 | 0 0 2 1 0 -1 -3 }}
 
Optimal tunings:
* WE: ~2 = 1200.0876{{c}}, ~11/9 = 350.7203{{c}}, ~10/7 = 617.5766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.7138{{c}}, ~10/7 = 617.5610{{c}}
 
Minimax tuning:
* 17-odd-limit unchanged-interval (eigenmonzo) basis: 2.5/3.17/9
 
{{Optimal ET sequence|legend=0| 27effg, 31fg, 41, 58, 72, 130, 171, 243 }}
 
Badness (Sintel): 0.704
 
==== Heartlandia ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 441/440, 1452/1445
 
{{Mapping|legend=0| 1 1 1 2 2 1 3 | 0 4 0 1 10 23 12 | 0 0 2 1 0 -1 -1 }}
: Mapping generators: ~2, ~119/108, ~27/17
 
Optimal tunings:
* WE: ~2 = 1199.5594{{c}}, ~119/108 = 175.3533{{c}}, ~27/17 = 793.6847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~119/108 = 175.3707{{c}}, ~27/17 = 793.7559{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g }}
 
Badness (Sintel): 2.84
 
===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 171/170, 243/242, 324/323, 364/363, 441/440
 
{{Mapping|legend=0| 1 1 1 2 2 1 3 3 | 0 4 0 1 10 23 12 4 | 0 0 2 1 0 -1 -1 1 }}
 
Optimal tunings:
* WE: ~2 = 1199.7333{{c}}, ~21/19 = 175.3472{{c}}, ~19/12 = 793.7794{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3584{{c}}, ~19/12 = 793.8194{{c}}
 
{{Optimal ET sequence|legend=0| 14cf, 27effg, 41, 75ce, 89, 116cefg, 130g }}
 
Badness (Sintel): 2.29
 
=== Jovis ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440
 
{{Mapping|legend=0| 1 1 1 2 2 2 | 0 2 1 1 5 -3 | 0 0 2 1 0 5 }}
 
Optimal tunings:
* WE: ~2 = 1200.1435{{c}}, ~11/9 = 350.4354{{c}}, ~10/7 = 618.1775{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4402{{c}}, ~10/7 = 618.1759{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 31, 45ef, 58, 72, 103, 130, 233, 363 }}
 
Badness (Sintel): 0.507
 
=== Jofur ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242
 
{{Mapping|legend=0| 1 1 1 2 2 4 | 0 2 1 1 5 -1 | 0 0 2 1 0 0 }}
 
Optimal tunings:
* WE: ~2 = 1198.9534{{c}}, ~11/9 = 351.1412{{c}}, ~10/7 = 618.3494{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.2536{{c}}, ~10/7 = 618.5932{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 31, 41, 58, 99ef, 157eff }}
 
Badness (Sintel): 0.701
 
=== Semijove ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 676/675
 
{{Mapping|legend=0| 2 0 1 3 -1 -1 | 0 2 1 1 5 4 | 0 0 2 1 0 2 }}
: mapping generators: ~55/39, ~26/15, ~10/7
 
Optimal tunings:
* WE: ~55/39 = 599.9886{{c}}, ~26/15 = 950.6645{{c}}, ~10/7 = 618.1458{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~26/15 = 950.6755{{c}}, ~10/7 = 618.1489{{c}}
 
{{Optimal ET sequence|legend=0| 48c, 58, 72, 130 }}
 
Badness (Sintel): 0.948
 
== Agni ==
Agni tempers out [[385/384]] and [[1375/1372]], so that a stack of three 10/7 generators is identified with [[16/11]] plus an octave. [[140edo]], [[212edo]] and [[284edo]] show how it can be tuned distinctly from [[#Jove|jove]].
 
This temperament suggests at least two extensions to the 13-limit: pavaka (31 & 68 & 72), named after an alias of Agni, tempers out [[625/624]]; and svaha (41 & 68 & 72), named after Agni's wife, tempers out [[325/324]]. The edos above can be used to tune both, which reduce them to [[quadritikleismic]] without much additional damage, but note pavaka is a detemperament of [[benediction]] whereas svaha is a detemperament of [[manna]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 385/384, 1375/1372
 
{{Mapping|legend=1| 1 1 1 2 5 | 0 2 1 1 0 | 0 0 2 1 -3 }}
 
Mapping to lattice: [{{val| 0 2 1 1 0 }}, {{val| 0 0 2 1 -3 }}]
 
Lattice basis:
: 49/40 length = 0.756, 10/7 length = 0.819
: Angle (49/40, 10/7) = 106.460 degrees
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4168{{c}}, ~49/40 = 350.8363{{c}}, ~10/7 = 617.2187{{c}}
: [[error map]]: {{val| +0.417 +0.134 -0.623 +0.063 -0.890 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.8369{{c}}, ~10/7 = 616.9907{{c}}
: error map: {{val| 0.000 -0.281 -1.495 -0.998 -2.290 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 23/10 3/10 2/5 0 -2/5 }}, {{monzo| 12/5 2/5 1/5 0 -1/5 }}, {{monzo| 23/10 3/10 -3/5 0 3/5 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.11/5
 
{{Optimal ET sequence|legend=1| 31, 41, 68, 72, 140, 171e, 212, 284, 496ce, 527cee, 739cdeee, 811ccdeee, 1023ccdeee }}
 
[[Badness]] (Sintel): 0.593
 
=== Pavaka ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 625/624, 1375/1372
 
{{Mapping|legend=0| 1 1 1 2 5 -1 | 0 2 1 1 0 2 | 0 0 2 1 -3 8 }}
 
Optimal tunings:
* WE: ~2 = 1200.4413{{c}}, ~49/40 = 350.8092{{c}}, ~10/7 = 617.3718{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.8046{{c}}, ~10/7 = 617.1572{{c}}
 
{{Optimal ET sequence|legend=0| 31, 68, 72, 103, 140, 212, 243e, 315ef, 455eef, 770cdeeeff }}
 
Badness (Sintel): 0.863
 
=== Svaha ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 325/324, 385/384, 1375/1372
 
{{Mapping|legend=0| 1 1 1 2 5 4 | 0 2 1 1 0 6 | 0 0 2 1 -3 -4 }}
 
Optimal tunings:
* WE: ~2 = 1200.3582{{c}}, ~49/40 = 351.0784{{c}}, ~10/7 = 617.0054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0491{{c}}, ~10/7 = 616.8317{{c}}
 
{{Optimal ET sequence|legend=0| 41, 68, 72, 140, 212, 393ce, 465cef, 677cdeeff }}
 
Badness (Sintel): 0.997
 
== Zisa ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 5632/5625
 
{{Mapping|legend=1| 1 1 1 2 -3 | 0 2 1 1 8 | 0 0 2 1 8 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9178{{c}}, ~49/40 = 350.0519{{c}}, ~10/7 = 617.8150{{c}}
: [[error map]]: {{val| -0.082 +0.067 +0.286 -0.123 -0.135 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.0615{{c}}, ~10/7 = 617.8571{{c}}
: error map: {{val| 0.000 +0.168 +0.462 +0.093 +0.032 }}
 
{{Optimal ET sequence|legend=1| 31, 68e, 99e, 130, 239, 270, 670, 940, 1210, 2150c }}
 
[[Badness]] (Sintel): 0.769
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 4096/4095
 
{{Mapping|legend=0| 1 1 1 2 -3 7 | 0 2 1 1 8 -6 | 0 0 2 1 8 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9787{{c}}, ~49/40 = 350.9976{{c}}, ~10/7 = 617.8701{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0039{{c}}, ~10/7 = 617.8795{{c}}
 
{{Optimal ET sequence|legend=0| 31, 78f, 99e, 109, 130, 239, 270, 571, 701, 841, 971, 1241 }}
 
Badness (Sintel): 0.776
 
== Lif ==
Lif tempers out the [[olympia]] and the [[lifthrasirsma]], so that the [[rastma]] is split in halves with each part standing in not only for [[441/440]][[~]][[540/539]], but the [[32805/32768|schisma]]. Lif readily extends to the [[13-limit]] and the no-17 [[19-limit]], where the [[513/512|undevicesimal schisma]] is also equated with the schisma.
 
This temperament was named by [[Flora Canou]] in 2023 along with the [[weak restriction]], [[lifthrasir]], and the corresponding comma, lifthrasirsma, after the human survivors of Ragnarök.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 131072/130977
 
{{Mapping|legend=1| 1 1 1 2 8 | 0 2 1 1 -12 | 0 0 2 1 -2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9386{{c}}, ~49/40 = 351.0586{{c}}, ~10/7 = 617.7065{{c}}
: [[error map]]: {{val| -0.061 +0.101 +0.096 -0.184 +0.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.0806{{c}}, ~10/7 = 617.7228{{c}}
: error map: {{val| 0.000 +0.206 +0.213 -0.022 +0.269 }}
 
{{Optimal ET sequence|legend=1| 41, 89, 130, 229, 270, 581, 670, 711, 981, 1251, 2232e }}
 
[[Badness]] (Sintel): 0.953
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 4096/4095
 
{{Mapping|legend=0| 1 1 1 2 8 7 | 0 2 1 1 -12 -6 | 0 0 2 1 -2 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9597{{c}}, ~49/40 = 351.0718{{c}}, ~10/7 = 617.6543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0856{{c}}, ~10/7 = 617.6715{{c}}
 
{{Optimal ET sequence|legend=0| 41, 89, 99, 130, 270, 581, 711, 981, 1292, 1562 }}
 
Badness (Sintel): 0.542
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1729/1728, 2080/2079, 2401/2400
 
{{Mapping|legend=2| 1 1 1 2 8 7 0 | 0 2 1 1 -12 -6 11 | 0 0 2 1 -2 -3 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9752{{c}}, ~49/40 = 351.0908{{c}}, ~10/7 = 617.6473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0984{{c}}, ~10/7 = 617.6588{{c}}
 
{{Optimal ET sequence|legend=0| 41, 89, 130, 229, 270, 581, 851, 1562, 1832, 2413 }}
 
Badness (Sintel): 0.488
 
== Freyr ==
Freyr tempers out the [[symbiotic comma]], so that the [[rastma]] is split in halves with each part standing in not only for [[441/440]][[~]][[540/539]], but the [[5120/5103|aberschisma]]. Freyr easily extends to the [[13-limit]], equating the [[352/351|major minthma]] with the aberschisma, and the no-17 [[19-limit]], equating the [[513/512|undevicesimal schisma]] with it too.
 
It can be notated with [[neutral chain-of-fifths notation]] with two additional sets of accidentals, one for the generic comma step, and the other for the generic aberschisma step.
 
This temperament was named by [[Flora Canou]] in 2026, under the impression that it functions as a counterpart of [[#Freya|freya]].
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 19712/19683
 
{{Mapping|legend=1| 1 1 1 2 -1 | 0 2 1 1 17 | 0 0 2 1 -1 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9474{{c}}, ~49/40 = 351.1064{{c}}, ~10/7 = 617.6483{{c}}
: [[error map]]: {{val| -0.053 +0.205 +0.037 -0.176 -0.104 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.1156{{c}}, ~10/7 = 617.6499{{c}}
: error map: {{val| 0.000 +0.276 +0.102 -0.060 -0.003 }}
 
{{Optimal ET sequence|legend=1| 41, 58, 99e, 171e, 212, 270, 581, 851, 1121, 1972, 2881bc }}
 
[[Badness]] (Sintel): 1.23
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 10648/10647
 
{{Mapping|legend=0| 1 1 1 2 -1 -2 | 0 2 1 1 17 23 | 0 0 2 1 -1 -2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9432{{c}}, ~49/40 = 351.1128{{c}}, ~10/7 = 617.6419{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1232{{c}}, ~10/7 = 617.6431{{c}}
 
{{Optimal ET sequence|legend=0| 41, 58, 99ef, 154cef, 171ef, 212, 270, 581, 851, 1490, 1760, 2071be, 2341b }}
 
Badness (Sintel): 0.760
 
=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1216/1215, 1540/1539, 2080/2079, 2401/2400
 
{{Mapping|legend=0| 1 1 1 2 -1 -2 0 | 0 2 1 1 17 23 11 | 0 0 2 1 -1 -2 2 }}
 
Optimal tunings:
* WE: ~2 = 1199.9435{{c}}, ~49/40 = 351.1127{{c}}, ~10/7 = 617.6404{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1236{{c}}, ~10/7 = 617.6459{{c}}
 
{{Optimal ET sequence|legend=0| 41, 58h, 99ef, 171ef, 212h, 270, 581, 851, 1490h, 1760h, 2071beh, 2341bh }}
 
Badness (Sintel): 0.523
 
== Baldur ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 9801/9800
 
{{Mapping|legend=1| 2 0 1 3 7 | 0 2 1 1 -2 | 0 0 2 1 3 }}
: Mapping generators: ~99/70, ~343/198, ~10/7
 
[[Optimal tuning]]s:
* [[WE]]: ~99/70 = 600.0149{{c}}, ~343/198 = 950.9717{{c}}, ~10/7 = 617.7007{{c}}
: [[error map]]: {{val| +0.030 -0.012 +0.074 -0.109 -0.055 }}
* [[CWE]]: ~99/70 = 600.0000{{c}}, ~343/198 = 950.9565{{c}}, ~10/7 = 617.7017{{c}}
: error map: {{val| 0.000 -0.042 +0.046 -0.168 -0.126 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 3/4 0 1/2 1/2 -1/2 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 23/16 0 5/8 1/8 -1/8 }}, {{monzo| 23/16 0 5/8 -7/8 7/8 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5.11/7
 
{{Optimal ET sequence|legend=1| 58, 72, 130, 198, 212, 270, 342, 612, 954, 1084, 1354, 1696, 4004de, 5700de }}
 
[[Badness]] (Sintel): 0.200
 
[[Projection pair]]s: <code>2 9801/4900, 3 117649/39204, 7 9801/1400, 11 913517247483640899/83082326424002500</code> to 5.7/2.99/4
 
=== Greenland ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 676/675, 1001/1000, 1716/1715
 
{{Mapping|legend=0| 2 0 1 3 7 -1 | 0 2 1 1 -2 4 | 0 0 2 1 3 2 }}
 
Optimal tunings:
* WE: ~99/70 = 599.9868{{c}}, ~26/15 = 951.0538{{c}}, ~10/7 = 617.8062{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.0689{{c}}, ~10/7 = 617.8067{{c}}
 
{{Optimal ET sequence|legend=0| 58, 72, 130, 198, 270, 940, 1210f }}
 
Badness (Sintel): 0.415
 
Projection pairs: <code>2 19600/9801, 3 676/225, 5 10400/2079, 7 20384000/2910897, 11 19208000000000000/1750211597736459, 13 5026736/385875</code> to 10/7.200/99.26/15
 
==== Midnatssol ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 442/441, 561/560, 676/675
 
{{Mapping|legend=0| 2 0 1 3 7 -1 5 | 0 2 1 1 -2 4 2 | 0 0 2 1 3 2 0 }}
 
Optimal tunings:
* WE: ~17/12 = 600.1118{{c}}, ~26/15 = 951.1907{{c}}, ~10/7 = 617.6100{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.0626{{c}}, ~10/7 = 617.5842{{c}}
 
{{Optimal ET sequence|legend=0| 58, 72, 130, 140, 198, 212g, 270g, 342fg, 482fgg }}
 
Badness (Sintel): 0.720
 
== Freya ==
Freya tempers out [[3025/3024]] and [[41503/41472]], so it splits breed's [[28/27]] into two [[55/54]]~[[56/55]]'s and [[8/7]] into two [[77/72]]'s. It also takes the [[225/224]][[~]][[1029/1024]] [[kleisma #As an interval region|kleisma]] to represent [[243/242]], and splits it in half to get [[385/384]], [[441/440]], and [[540/539]]. Since these are the commas tempered out by [[miracle]], the simple relations between them make freya an elegant miracle detemperament.
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 3025/3024
 
{{Mapping|legend=1| 1 1 -1 1 5 | 0 2 3 2 1 | 0 0 4 2 -3 }}
: Mapping generators: ~2, ~49/40, ~84/55
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0491{{c}}, ~49/40 = 350.9820{{c}}, ~84/55 = 733.3514{{c}}
: [[error map]]: {{val| +0.030 -0.012 +0.074 -0.109 -0.055 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9890{{c}}, ~84/55 = 733.3160{{c}}
: error map: {{val| 0.000 -0.042 +0.046 -0.168 -0.126 }}
 
[[Minimax tuning]]:
* [[11-odd-limit]] [[eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.3.11/7
 
{{Optimal ET sequence|legend=1| 31, 41, 72, 167, 188, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}
 
[[Badness]] (Sintel): 0.204
 
[[Projection pair]]s: <code>3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024</code> to 2.49/5.77/3
 
=== 13-limit ===
Tridecimal freya maps [[352/351]] to the half-kleisma. It has [[miraculous]] as a sub-temperament.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 2401/2400, 3025/3024, 4096/4095
 
{{Mapping|legend=0| 1 1 -1 1 5 10 | 0 2 3 2 1 -9 | 0 0 4 2 -3 -6 }}
 
Optimal tunings:
* WE: ~2 = 1199.9870{{c}}, ~49/40 = 351.0502{{c}}, ~84/55 = 733.2969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0534{{c}}, ~84/55 = 733.3055{{c}}
 
{{Optimal ET sequence|legend=0| 31, 41, 72f, 198f, 229, 239, 270, 571, 581, 851, 882, 1152, 1463, 1733, 2615 }}
 
Badness (Sintel): 0.800
 
Projection pairs: <code>3 2401/800, 5 22880495169/4575312500, 7 1058841/151250, 11 33275/3024, 13 1814078464000000000000000/139662717676432916098329</code> to 2.49/5.77/3
 
=== Eir ===
Eir maps [[325/324]], [[364/363]], and [[729/728]] to the half-kleisma and has [[manna]] as a sub-temperament. {{U|VIxen}} named this extension after a healer goddess or valkyrie from the Norse mythology, as it extends freya via the [[ibnsinma]], which evokes associations with medicine.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 3025/3024
 
{{Mapping|legend=0| 1 1 -1 1 5 5 | 0 2 3 2 1 6 | 0 0 4 2 -3 -5 }}
 
Optimal tunings:
* WE: ~2 = 1199.9994{{c}}, ~49/40 = 351.0880{{c}}, ~84/55 = 733.2478{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0880{{c}}, ~84/55 = 733.2482{{c}}
 
{{Optimal ET sequence|legend=0| 41, 72, 157, 185cf, 198, 270, 581, 851, 1504, 1774f, 2085, 2355f }}
 
Badness (Sintel): 0.543
 
==== 2.3.5.7.11.13.19 subgroup ====
Subgroup: 2.3.5.7.11.13.19
 
Comma list: 1540/1539, 2080/2079, 2401/2400, 3025/3024
 
{{Mapping|legend=2| 1 1 -1 1 5 5 3 | 0 2 3 2 1 6 -2 | 0 0 4 2 -3 -5 3 }}
 
Optimal tunings:
* WE: ~2 = 1199.9951{{c}}, ~49/40 = 351.0906{{c}}, ~84/55 = 733.2475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.0913{{c}}, ~84/55 = 733.2473{{c}}
 
{{Optimal ET sequence|legend=0| 41, 72, 113, 157, 198, 270, 581, 851, 2085, 2355f }}
 
Badness (Sintel): 0.500
 
=== Heimlaug ===
Heimlaug maps [[351/350]] and [[625/624]] to the half-kleisma and has [[benediction]] as a subtemperament. VIxen named this extension after a völva (seeress) from the Gull-Þóris saga of Icelanders, as it extends freya via the [[1001/1000|fairytale comma]], adding to the mystical theme.
 
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024
 
{{Mapping|legend=0| 1 1 -1 1 5 -6 | 0 2 3 2 1 6 | 0 0 4 2 -3 13 }}
 
Optimal tunings:
* WE: ~2 = 1200.0601{{c}}, ~49/40 = 350.9562{{c}}, ~84/55 = 733.4471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9646{{c}}, ~84/55 = 733.4049{{c}}
 
{{Optimal ET sequence|legend=0| 31, 72, 103, 167, 198, 270, 571, 643, 913f }}
 
Badness (Sintel): 0.562
 
== Vili ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 391314/390625
 
{{Mapping|legend=1| 1 1 -1 1 -4 | 0 2 3 2 6 | 0 0 6 3 14 }}
: Mapping generators: ~2, ~49/40, ~175/132
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9872{{c}}, ~49/40 = 350.9740{{c}}, ~175/132 = 488.9418{{c}}
: [[error map]]: {{val| -0.013 -0.020 +0.272 -0.065 -0.238 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9728{{c}}, ~175/132 = 488.9475{{c}}
: error map: {{val| 0.000 -0.009 +0.290 -0.038 -0.216 }}
 
{{Optimal ET sequence|legend=1| 27e, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 670, 913, 1043, 1313, 1583 }}
 
[[Badness]] (Sintel): 1.51
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 4225/4224
 
{{Mapping|legend=0| 1 1 -1 1 -4 3 | 0 2 3 2 6 1 | 0 0 6 3 14 1 }}
 
Optimal tunings:
* WE: ~2 = 1200.0597{{c}}, ~49/40 = 350.9419{{c}}, ~65/49 = 488.9741{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9441{{c}}, ~65/49 = 488.9475{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 37, 64be, 76e, 93, 103, 130, 233, 243e, 270, 643, 913f }}
 
Badness (Sintel): 0.690
 
== Frigg ==
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 644204/643125
 
{{Mapping|legend=1| 1 1 -7 -2 -7 | 0 2 3 2 4 | 0 0 10 5 11 }}
: Mapping generators: ~2, ~49/40, ~88/49
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0183{{c}}, ~49/40 = 350.9854{{c}}, ~88/49 = 1013.3709{{c}}
: [[error map]]: {{val| +0.018 +0.034 +0.223 -0.037 -0.425 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 350.9871{{c}}, ~88/49 = 1013.3556{{c}}
: error map: {{val| 0.000 +0.019 +0.203 -0.074 -0.458 }}
 
{{Optimal ET sequence|legend=1| 45e, 58, 103, 161, 212, 270, 643, 913, 1183e }}
 
[[Badness]] (Sintel): 2.15
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 10648/10647
 
{{Mapping|legend=0| 1 1 -7 -2 -7 -9 | 0 2 3 2 4 3 | 0 0 10 5 11 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.0387{{c}}, ~49/40 = 350.9502{{c}}, ~70/39 = 1013.4065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 350.9526{{c}}, ~70/39 = 1013.3747{{c}}
 
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161, 212, 270, 643, 913f, 1614ef }} *
 
<nowiki/>* optimal patent val: [[1241edo|1241]]
 
Badness (Sintel): 0.873
 
== Ennealimmic ==
{{Distinguish|Ennealimnic}}
 
[[Subgroup]]: 2.3.5.7.11
 
[[Comma list]]: 2401/2400, 4375/4374
 
{{Mapping|legend=1| 9 1 1 12 0 | 0 2 3 2 0 | 0 0 0 0 1 }}
: Mapping generators: ~27/25, ~5/3, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~27/25 = 133.3357{{c}}, ~5/3 = 884.3288{{c}}, ~11/8 = 551.2529{{c}}
: [[error map]]: {{val| +0.021 +0.038 +0.009 -0.139 -0.000 }}
* [[CWE]]: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3215{{c}}, ~11/8 = 551.2775{{c}}
: error map: {{val| 0.000 +0.021 -0.016 -0.183 -0.040 }}
 
{{Optimal ET sequence|legend=1| 72, 171(e), 198, 270, 342, 612, 954, 1323, 1395, 1665, 2007, 2277, 2619, 4284d, 6561dd, 6903dd }}
 
[[Badness]] (Sintel): 0.330
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2080/2079, 2401/2400, 4375/4374
 
{{Mapping|legend=0| 9 1 1 12 0 -31 | 0 2 3 2 0 5 | 0 0 0 0 1 1 }}
 
Optimal tunings:
* WE: ~27/25 = 133.3276{{c}}, ~5/3 = 884.3660{{c}}, ~11/8 = 551.7221{{c}}
* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.3905{{c}}, ~11/8 = 551.7085{{c}}
 
{{Optimal ET sequence|legend=0| 72, 171(ef), 198, 270, 639, 711, 981, 1692e }}
 
Badness (Sintel): 0.706
 
[[Category:Breed family| ]] <!-- main article -->
[[Category:Breed| ]] <!-- key article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-3 temperaments]]
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