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Breedsmic temperaments: 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>
This page discusses miscellaneous [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the [[breedsma]] ({{monzo|legend=1| -5 -1 -2 4 }}, [[ratio]]: 2401/2400). This is the amount by which two [[49/40]] intervals exceed [[3/2]], and by which two [[60/49]] intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system ([[12edo]], for example) which does not possess a neutral third cannot be tempering out the breedsma.
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-02 22:07:42 UTC</tt>.<br>
: The original revision id was <tt>151418309</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">Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.


It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.
The breedsma is also the amount by which four stacked [[10/7]] intervals exceed 25/6: (10000/2401)⋅(2401/2400) = 10000/2400 = 25/6, which is two octaves above the classic chromatic semitone, [[25/24]]. We might note also that (49/40)⋅(10/7) = 7/4 and (49/40)⋅(10/7)<sup>2</sup> = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40~60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.


===Tertiaseptal===
Temperaments discussed elsewhere include:
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. [[171edo]] makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.</pre></div>
* ''[[Beatles]]'' (+64/63) → [[Archytas clan #Beatles|Archytas clan]]
<h4>Original HTML content:</h4>
* ''[[Newt]]'' (+33554432/33480783) → [[Garischismic clan #Beatles|Garischismic clan]]
<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;Breedsmic temperaments&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Breedsmic temperaments are rank two temperaments tempering out the breedsma, 2401/2400. This is the amount by which two 49/40 intervals exceed 3/2, and by which two 60/49 intervals fall short. Either of these represent a neutral third interval which is highly characteristic of breedsmic tempering; any tuning system (12edo, for example) which does not possess a neutral third cannot be tempering out the breedsma.&lt;br /&gt;
* [[Decimal]] (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]
&lt;br /&gt;
* [[Squares]] (+81/80) → [[Meantone family #Squares|Meantone family]]
It is also the amount by which four stacked 10/7 intervals exceed 25/6: 10000/2401 * 2401/2400 = 10000/2400 = 25/6, which is two octaves above the chromatic semitone, 25/24. We might note also that 49/40 * 10/7 = 7/4 and 49/40 * (10/7)^2 = 5/2, relationships which will be significant in any breedsmic temperament. As a consequence of these facts, the 49/40+60/49 neutral third and the 7/5 and 10/7 intervals tend to have relatively low complexity in a breedsmic system.&lt;br /&gt;
* ''[[Sesquiquartififths]]'' (+32805/32768) → [[Schismatic family #Sesquiquartififths|Schismatic family]]
&lt;br /&gt;
* [[Miracle]] (+225/224) → [[Gamelismic clan #Miracle|Gamelismic clan]]
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc0"&gt;&lt;a name="x--Tertiaseptal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Tertiaseptal&lt;/h3&gt;
* ''[[Octacot]]'' (+245/243) → [[Tetracot family #Octacot|Tetracot family]]
Aside from the breedsma, tertiaseptal tempers out 65625/65536, the horwell comma, 703125/702464, the meter, and 2100875/2097152. It can be described as the 140&amp;amp;171 temperament, and 256/245, 1029/1024 less than 21/20, serves as its generator. Three of these fall short of 8/7 by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. &lt;a class="wiki_link" href="/171edo"&gt;171edo&lt;/a&gt; makes for an excellent tuning. The 15 or 16 note MOS can be used to explore no-threes harmony, and the 31 note MOS gives plenty of room for those as well.&lt;/body&gt;&lt;/html&gt;</pre></div>
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]
* [[Harry]] (+19683/19600) → [[Gravity family #Harry|Gravity family]]
* ''[[Quasitemp]]'' (+875/864) → [[Keemic temperaments #Quasitemp|Keemic temperaments]]
* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]
* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]
* [[Ennealimmal]] (+4375/4374) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Quadrimage]]'' (+3125/3072) → [[Magic family #Quadrimage|Magic family]]
* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]
* ''[[Decoid]]'' (+67108864/66976875) → [[Quintosec family #Decoid|Quintosec family]]
* ''[[Quadritikleismic]]'' (+15625/15552) → [[Kleismic family #Quadritikleismic|Kleismic family]]
* ''[[Neptune]]'' (+48828125/48771072) → [[Gammic family #Neptune|Gammic family]]
* ''[[Tertiseptisix]]'' (+390625000/387420489) → [[Quartonic family #Tertiseptisix|Quartonic family]]
* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Whitewood family #Greenwood|Whitewood family]]
 
Considered below are tertiaseptal, emmthird, hemififths, osiris, quasiorwell, quinmite, septidiasemi, subneutral, maviloid, lockerbie, unthirds, neominor, catafourth, cotritone, fibo, quasimoha, mintone, gorgik, hemigoldis, and surmarvelpyth, in the order of increasing [[badness]].  
 
== Tertiaseptal ==
{{Main| Tertiaseptal }}
 
Aside from the breedsma, tertiaseptal tempers out [[65625/65536]], the horwell comma, [[703125/702464]], the meter, and [[2100875/2097152]], the rainy comma. It can be described as the {{nowrap| 31 & 171 }} temperament, and [[256/245]], [[1029/1024]] less than [[21/20]], serves as its generator. Three of these fall short of [[8/7]] by 2100875/2097152, and the generator can be taken as 1/3 of an 8/7 flattened by a fraction of a cent. The [[ploidacot]] for this temperament is 20-sheared 22-cot (or pentaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure).  
 
[[171edo]] makes for an excellent tuning, although [[140edo]] ({{nowrap| {{=}} 171 - 31 }}) also makes sense, and in very high limits [[311edo]] ({{nowrap| {{=}} 140 + 171 }}) is especially notable. The 15- or 16-note [[mos]] can be used to explore no-threes harmony, and the 31-note mos gives plenty of room for those as well.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 65625/65536
 
{{Mapping|legend=1| 1 -19 7 0 | 0 22 -5 3 }}
: mapping generators: ~2, ~245/128
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979{{c}})
: [[error map]]: {{val| +0.100 -0.008 -0.123 -0.119 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899{{c}})
: error map: {{val| 0.000 -0.133 -0.364 -0.396 }}
 
{{Optimal ET sequence|legend=1| 31, 109, 140, 171 }}
 
[[Badness]] (Sintel): 0.329
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 65625/65536
 
Mapping: {{mapping| 1 -19 7 0 -48 | 0 22 -5 3 55 }}
 
Optimal tunings:
* WE: ~2 = 1200.1034{{c}}, ~245/128 = 1122.8694{{c}} (~256/245 = 77.2340{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.7743{{c}} (~256/245 = 77.2257{{c}})
 
{{Optimal ET sequence|legend=0| 31, 109e, 140e, 171, 202 }}
 
Badness (Sintel): 1.18
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 -19 7 0 -48 43 | 0 22 -5 3 55 -42 }}
 
Optimal tunings:
* WE: ~2 = 1199.8783{{c}}, ~224/117 = 1122.6835{{c}} (~117/112 = 77.1948{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.7968{{c}} (~117/112 = 77.2032{{c}})
 
{{Optimal ET sequence|legend=0| 31, 140e, 171, 373ef }}
 
Badness (Sintel): 1.52
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 375/374, 441/440, 625/624, 3584/3575
 
Mapping: {{mapping| 1 -19 7 0 -48 43 49 | 0 22 -5 3 55 -42 -48 }}
 
Optimal tunings:
* WE: ~2 = 1199.8677{{c}}, ~65/34 = 1122.6748{{c}} (~68/65 = 77.1929{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.7985{{c}} (~68/65 = 77.2015{{c}})
 
{{Optimal ET sequence|legend=0| 31, 140e, 171 }}
 
Badness (Sintel): 1.40
 
=== Tertia ===
Subgroup:2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 1375/1372
 
Mapping: {{mapping| 1 -19 7 0 -19 | 0 22 -5 3 24 }}
 
Optimal tunings:
* WE: ~2 = 1200.2336{{c}}, ~21/11 = 1123.0454{{c}} (~22/21 = 77.1882{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8311{{c}} (~22/21 = 77.1689{{c}})
 
{{Optimal ET sequence|legend=0| 31, 109, 140, 171e, 311e }}
 
Badness (Sintel): 0.997
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 625/624, 1331/1323
 
Mapping: {{mapping| 1 -19 7 0 -19 43 | 0 22 -5 3 24 -42 }}
 
Optimal tunings:
* WE: ~2 = 1200.1395{{c}}, ~21/11 = 1122.9727{{c}} (~22/21 = 77.1669{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8426{{c}} (~22/21 = 77.1574{{c}})
 
{{Optimal ET sequence|legend=0| 31, 78f, 109, 140 }}
 
Badness (Sintel): 1.17
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 352/351, 385/384, 561/560, 625/624, 715/714
 
Mapping: {{mapping| 1 -19 7 0 -19 43 49 | 0 22 -5 3 24 -42 -48 }}
 
Optimal tunings:
* WE: ~2 = 1200.1655{{c}}, ~21/11 = 1122.9926{{c}} (~22/21 = 77.1729{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~21/11 = 1122.8376{{c}} (~22/21 = 77.1624{{c}})
 
{{Optimal ET sequence|legend=0| 31, 78fg, 109g, 140 }}
 
Badness (Sintel): 1.14
 
=== Tertiaseptia ===
This extension was considered by [[Gene Ward Smith]] as a 41-limit temperament<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_9274.html Yahoo! Tuning Group | ''A 41-limit temperament'']</ref>. It can be extended as such by tempering out 875/874, 714/713, 703/702 and 697/696, and mapping 19, 31, 37 and 41 to 94, 105, -81 and +10 steps, respectively.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 6250/6237, 65625/65536
 
Mapping: {{mapping| 1 -19 7 0 112 | 0 22 -5 3 -116 }}
 
Optimal tunings:
* WE: ~2 = 1200.0053{{c}}, ~245/128 = 1122.8357{{c}} (~256/245 = 77.1696{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8308{{c}} (~256/245 = 77.1692{{c}})
 
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}
 
Badness (Sintel): 1.88
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 2080/2079, 2200/2197, 2401/2400
 
Mapping: {{mapping| 1 -19 7 0 112 43 | 0 22 -5 3 -116 -42 }}
 
Optimal tunings:
* WE: ~2 = 1199.9823{{c}}, ~224/117 = 1122.8150{{c}} (~117/112 = 77.1673{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~224/117 = 1122.8316{{c}} (~117/112 = 77.1684{{c}})
 
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311, 1073 }}
 
Badness (Sintel): 1.14
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 595/594, 625/624, 833/832, 1156/1155, 2200/2197
 
Mapping: {{mapping| 1 -19 7 0 112 43 49 | 0 22 -5 3 -116 -42 -48 }}
 
Optimal tunings:
* WE: ~2 = 1200.0092{{c}}, ~65/34 = 1122.8392{{c}} (~68/65 = 77.1700{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~65/34 = 1122.8305{{c}} (~68/65 = 77.1695{{c}})
 
{{Optimal ET sequence|legend=0| 31e, 140, 171, 311 }}
 
Badness (Sintel): 0.956
 
==== 2.3.5.7.11.13.17.23 subgroup ====
Subgroup: 2.3.5.7.11.13.17.23
 
Comma list: 595/594, 625/624, 833/832, 1105/1104, 1156/1155, 2200/2197
 
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 | 0 22 -5 3 -116 -42 -48 -117 }}
 
Optimal tunings:
* WE: ~2 = 1200.0047{{c}}, ~44/23 = 1122.8363{{c}} (~23/22 = 77.1684{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8319{{c}} (~23/22 = 77.1681{{c}})
 
{{Optimal ET sequence|legend=0| 31ei, 140, 171, 311 }}
 
Badness (Sintel): 0.944
 
==== 2.3.5.7.11.13.17.23.29 subgroup ====
Subgroup: 2.3.5.7.11.13.17.23.29
 
Comma list: 595/594, 625/624, 784/783, 833/832, 1015/1014, 1105/1104, 1156/1155
 
Mapping: {{mapping| 1 -19 7 0 112 43 49 114 61 | 0 22 -5 3 -116 -42 -48 -117 -60 }}
 
Optimal tunings:
* WE: ~2 = 1199.9945{{c}}, ~44/23 = 1122.8270{{c}} (~23/22 = 77.1675{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8322{{c}} (~23/22 = 77.1678{{c}})
 
{{Optimal ET sequence|legend=0| 31ei, 140, 311, 762g }}
 
Badness (Sintel): 0.858
 
=== Hemitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 65625/65536
 
Mapping: {{mapping| 1 -41 12 -3 -73 | 0 44 -10 6 79 }}
: mapping generators: ~2, ~88/45
 
Optimal tunings:
* WE: ~2 = 1200.1008{{c}}, ~88/45 = 1161.5020{{c}} (~45/44 = 38.5988{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4053{{c}} (~45/44 = 38.5947{{c}})
 
{{Optimal ET sequence|legend=0| 31, …, 280, 311, 342, 2021cde, 2363cde, …, 3389ccddee, 3731ccddee }}
 
Badness (Sintel): 0.517
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2401/2400, 4096/4095
 
Mapping: {{mapping| 1 -41 12 -3 -73 85 | 0 44 -10 6 79 -84 }}
 
Optimal tunings:
* WE: ~2 = 1199.9822{{c}}, ~88/45 = 1161.3952{{c}} (~45/44 = 38.5871{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4123{{c}} (~45/44 = 38.5877{{c}})
 
{{Optimal ET sequence|legend=0| 31, 280, 311 }}
 
Badness (Sintel): 1.39
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 625/624, 833/832, 1225/1224, 1575/1573, 4096/4095
 
Mapping: {{mapping| 1 -41 12 -3 -73 85 97| 0 44 -10 6 79 -84 -96 }}
 
Optimal tunings:
* WE: ~2 = 1200.0042{{c}}, ~88/45 = 1161.4149{{c}} (~45/44 = 38.5893{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~88/45 = 1161.4109{{c}} (~45/44 = 38.5891{{c}})
 
{{Optimal ET sequence|legend=0| 31, 280, 311, 653f }}
 
Badness (Sintel): 1.29
 
=== Semitert ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 9801/9800, 65625/65536
 
Mapping: {{mapping| 2 -16 9 3 47 | 0 22 -5 3 -46 }}
: mapping generators: ~99/70, ~693/512
 
Optimal tunings:
* WE: ~99/70 = 600.0548{{c}}, ~693/512 = 522.8547{{c}} (~256/245 = 77.2002{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~693/512 = 522.8069{{c}} (~256/245 = 77.1931{{c}})
 
{{Optimal ET sequence|legend=0| 62e, 140, 202, 342 }}
 
Badness (Sintel): 0.853
 
== Emmthird ==
Emmthird tempers out the [[scheme comma]] and may be described as the {{nowrap| 58 & 171 }} temperament. The generator for emmthird is flatter than [[81/64]] by a lee comma, [[177147/175616]], and sharper than [[5/4]] by the hemimage comma, [[10976/10935]]. The [[ploidacot]] for this temperament is delta-14-cot.
 
The [[11-limit]] version, which tempers out [[243/242]] and [[441/440]], has much lower accuracy and is [[support]]ed by much fewer equal temperaments.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 14348907/14336000
 
{{Mapping|legend=1| 1 -3 -17 -8 | 0 14 59 33 }}
: mapping generators: ~2, ~2744/2187
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0435{{c}}, ~2744/2187 = 393.0021{{c}}
: [[error map]]: {{val| +0.043 -0.057 +0.069 -0.106 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2744/2187 = 392.9887{{c}}
: error map: {{val| 0.000 -0.113 +0.022 -0.197 }}
 
{{Optimal ET sequence|legend=1| 58, 113, 171, 742, 913, 1084, 1255, 2681d, 3936d }}
 
[[Badness]] (Sintel): 0.424
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 1792000/1771561
 
Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}
 
Optimal tunings:
* WE: ~2 = 1199.8090{{c}}, ~1372/1089 = 392.9286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1372/1089 = 392.9870{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.73
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 364/363, 441/440, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}
 
Optimal tunings:
* WE: ~2 = 1199.7756{{c}}, ~180/143 = 392.9154{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~180/143 = 392.9840{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.11
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 364/363, 441/440, 595/594, 2200/2197
 
Mapping: {{mapping| 1 -3 -17 -8 -8 -13 9 | 0 14 59 33 35 51 -15 }}
 
Optimal tunings:
* WE: ~2 = 1199.8396{{c}}, ~64/51 = 392.9322{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~64/51 = 392.9826{{c}}
 
{{Optimal ET sequence|legend=0| 58, 113, 171 }}
 
Badness (Sintel): 1.18
 
== Hemififths ==
{{Main| Hemififths }}
 
Hemififths may be described as the {{nowrap| 41 & 58 }} temperament, tempering out [[5120/5103]], the hemifamity comma, and [[10976/10935]], hemimage. It has a neutral third as a generator; its [[ploidacot]] is dicot. [[99edo]] and [[140edo]] provides good tunings, and [[239edo]] an even better one; and other possible tunings are 160<sup>(1/25)</sup>, giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14<sup>(1/13)</sup>, giving just 7's. It requires 25 generator steps to get to the class for the harmonic 5, whereas the 7 is half as complex, and hence hemififths makes for a good no-fives temperament, to which the 17- and 24-note mos are suited. The full force of this highly accurate temperament can be found using the 41-note mos or even the 34-note 2mos{{clarify}}.
 
By adding [[243/242]] (which also means [[441/440]], [[540/539]] and [[896/891]]) to the commas, hemififths extends to a less accurate 11-limit version, but one where 11/4 is only five generator steps. 99edo is an excellent tuning; one which loses little of the accuracy of the 7-limit but improves the 11-limit a bit. Now adding [[144/143]] brings in the 13-limit with less accuracy yet, but with very low complexity, as the generator can be taken to be [[16/13]]. 99 remains a good tuning choice.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 5120/5103
 
{{Mapping|legend=1| 1 1 -5 -1 | 0 2 25 13 }}
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7412{{c}}, ~49/40 = 351.4016{{c}}
: [[error map]]: {{val| -0.259 +0.590 +0.021 -0.346 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4671{{c}}
: error map: {{val| 0.000 +0.979 +0.364 +0.246 }}
 
[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}
: {{monzo list| 1 0 0 0 | 7/5 0 2/25 0 | 0 0 1 0 | 8/5 0 13/25 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
 
[[Algebraic generator]]: (2 + sqrt(2))/2
 
{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}
 
[[Badness]] (Sintel): 0.563
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 896/891
 
Mapping: {{mapping| 1 1 -5 -1 2 | 0 2 25 13 5 }}
 
Optimal tunings:
* WE: ~2 = 1199.2845{{c}}, ~11/9 = 351.3110{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.4956{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99e }}
 
Badness (Sintel): 0.777
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 144/143, 196/195, 243/242, 364/363
 
Mapping: {{mapping| 1 1 -5 -1 2 4 | 0 2 25 13 5 -1 }}
 
Optimal tunings:
* WE: ~2 = 1198.8875{{c}}, ~11/9 = 351.2475{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.5438{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 41, 58, 99ef, 157eff }}
 
Badness (Sintel): 0.789
 
=== Semihemi ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3388/3375, 5120/5103
 
Mapping: {{mapping| 2 0 -35 -15 -47 | 0 2 25 13 34 }}
: mapping generators: ~99/70, ~400/231
 
Optimal tunings:
* WE: ~99/70 = 599.8556{{c}}, ~400/231 = 951.2757{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~400/231 = 951.4939{{c}}
 
{{Optimal ET sequence|legend=0| 58, 140, 198 }}
 
Badness (Sintel): 1.40
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 676/675, 847/845, 1716/1715
 
Mapping: {{mapping| 2 0 -35 -15 -47 -37 | 0 2 25 13 34 28 }}
 
Optimal tunings:
* WE: ~99/70 = 599.8513{{c}}, ~26/15 = 951.2662{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/15 = 951.4905{{c}}
 
{{Optimal ET sequence|legend=0| 58, 140, 198, 536f }}
 
Badness (Sintel): 0.876
 
=== Quadrafifths ===
This has been catalogued as ''semihemififths'' in Graham Breed's temperament finder, but ''quadrafifths'' arguably makes more sense because it straight-up splits the fifth in four.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5120/5103
 
Mapping: {{mapping| 1 1 -5 -1 8 | 0 4 50 26 -31 }}
: mapping generators: ~2, ~243/220
 
Optimal tunings:
* WE: ~2 = 1199.7520{{c}}, ~243/220 = 175.7015{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/220 = 175.7360{{c}}
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 239, 676b, 915be }}
 
Badness (Sintel): 1.33
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 2401/2400, 3025/3024
 
Mapping: {{mapping| 1 1 -5 -1 8 10 | 0 4 50 26 -31 -43 }}
 
Optimal tunings:
* WE: ~2 = 1199.6502{{c}}, ~72/65 = 175.6957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.7461{{c}}
 
{{Optimal ET sequence|legend=0| 41, 157, 198, 437f, 635bcff }}
 
Badness (Sintel): 1.29
 
=== Cutefourths ===
This extension splits the neutral third plus an octave in three, with a ploidacot signature of beta-hexacot. The generator is an acute fourth in size (but not representing [[27/20]]), hence the name.
 
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 4000/3993, 5120/5103
 
Mapping: {{mapping| 1 -1 -30 -14 -28 | 0 6 75 39 73 }}
: mapping generators: ~2, ~66/49
 
Optimal tunings:
* WE: ~2 = 1199.7345{{c}}, ~66/49 = 517.0436{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1543{{c}}
 
{{Optimal ET sequence|legend=0| 58, 181, 239, 1014bcee }}
 
Badness (Sintel): 1.71
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 847/845, 1575/1573, 2401/2400
 
Mapping: {{mapping| 1 -1 -30 -14 -28 -20 | 0 6 75 39 73 55 }}
 
Optimal tunings:
* WE: ~2 = 1199.6427{{c}}, ~66/49 = 517.0035{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~66/49 = 517.1524{{c}}
 
{{Optimal ET sequence|legend=0| 58, 181, 239f }}
 
Badness (Sintel): 1.45
 
== Osiris ==
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 31381059609/31360000000
 
{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}
: mapping generators: ~2, ~2187/1400
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0285{{c}}, ~2187/1400 = 771.9522{{c}}
: [[error map]]: {{val| +0.028 -0.025 +0.068 -0.117 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2187/1400 = 771.9343{{c}}
: error map: {{val| 0.000 -0.056 +0.039 -0.175 }}
 
{{Optimal ET sequence|legend=1| 157, 171, 1012, 1183, 1354, 1525, 1696 }}
 
[[Badness]] (Sintel): 0.716
 
== Quasiorwell ==
In addition to 2401/2400, quasiorwell tempers out the quasiorwellisma, 29360128/29296875 ({{monzo| 22 -1 -10 1 }}). It has a generator 1024/875, which is [[6144/6125]] more than [[7/6]]. It may be described as the {{nowrap| 31 & 270 }} temperament, and its [[ploidacot]] is eta-38-cot (or omega-triseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] structure). As one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)<sup>1/8</sup>, giving just 7's, or 384<sup>1/38</sup>, giving pure fifths.
 
Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 29360128/29296875
 
{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}
: mapping generators: ~2, ~1024/875
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9403{{c}}, ~1024/875 = 271.0935{{c}}
: [[error map]]: {{val| -0.060 +0.018 +0.226 -0.137 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1024/875 = 271.1064{{c}}
: error map: {{val| 0.000 +0.087 +0.367 +0.025 }}
 
{{Optimal ET sequence|legend=1| 31, …, 177, 208, 239, 270, 571, 841, 1111 }}
 
[[Badness]] (Sintel): 0.907
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 5632/5625
 
Mapping: {{mapping| 1 -7 3 1 -11 | 0 38 -3 8 64 }}
 
Optimal tunings:
* WE: ~2 = 1199.9484{{c}}, ~90/77 = 271.0989{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1099{{c}}
 
{{Optimal ET sequence|legend=0| 31, …, 177e, 208, 239, 270 }}
 
Badness (Sintel): 0.580
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095
 
Mapping: {{mapping| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}
 
Optimal tunings:
* WE: ~2 = 1199.9916{{c}}, ~90/77 = 271.1051{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~90/77 = 271.1070{{c}}
 
{{Optimal ET sequence|legend=0| 31, 239, 270, 571, 841, 1111 }}
 
Badness (Sintel): 0.741
 
== Quinmite ==
Quinmite may be described as the {{nowrap| 99 & 103 }} temperament. The generator for quinmite is the quasi-tempered minor third [[25/21]], sharper than [[32/27]] by the marvel comma, [[225/224]]. It is also generated by 1/5 of the minor tenth [[12/5]], and its name is a play on the words "quintans" (Latin for "one fifth") and "minor tenth", given by [[Petr Pařízek]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_104268.html Yahoo! Tuning Group | ''2D temperament names, part I -- reclassified temperaments from message #101780'']</ref>. Its [[ploidacot]] is eta-34-cot.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1959552/1953125
 
{{Mapping|legend=1| 1 -7 -5 -3 | 0 34 29 23 }}
: mapping generators: ~2, ~25/21
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9361{{c}}, ~25/21 = 302.9808{{c}}
: [[error map]]: {{val| -0.064 -0.162 +0.448 -0.077 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 302.9953{{c}}
: error map: {{val| 0.000 -0.116 +0.549 +0.065 }}
 
{{Optimal ET sequence|legend=1| 99, 202, 301, 400, 701, 1101c, 1802c }}
 
[[Badness]] (Sintel): 0.945
 
== Septidiasemi ==
{{Main| Septidiasemi }}
 
Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit, and may be described as the {{nowrap| 10 & 171 }} temperament. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]), with a [[ploidacot]] of beta-26-cot. It is an excellent temperament for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 2152828125/2147483648
 
{{Mapping|legend=1| 1 -1 6 4 | 0 26 -37 -12 }}
: mapping generators: ~2, ~15/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1043{{c}}, ~15/14 = 119.3076{{c}}
: [[error map]]: {{val| +0.104 -0.061 -0.070 -0.100 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 119.2971{{c}}
: error map: {{val| 0.000 -0.230 -0.307 -0.391 }}
 
{{Optimal ET sequence|legend=1| 10, 151, 161, 171, 3581bcdd, 3752bcdd, …, 5633bbccddd, 5804bbccddd }}
 
[[Badness]] (Sintel): 1.12
 
=== Sedia ===
The sedia temperament (10 & 161) is an 11-limit extension of the septidiasemi, which tempers out [[243/242]] and [[441/440]].
 
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 939524096/935859375
 
Mapping: {{mapping| 1 -1 6 4 -3 | 0 26 -37 -12 65 }}
 
Optimal tunings:
* WE: ~2 = 1199.9635{{c}}, ~15/14 = 119.2755{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2791{{c}}
 
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}
 
Badness (Sintel): 3.00
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 441/440, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 -1 6 4 -3 4 | 0 26 -37 -12 65 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.8922{{c}}, ~15/14 = 119.2700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2804{{c}}
 
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332 }}
 
Badness (Sintel): 1.89
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 441/440, 833/832, 2200/2197, 3584/3575
 
Mapping: {{mapping| 1 -1 6 4 -3 4 2 | 0 26 -37 -12 65 -3 21 }}
 
Optimal tunings:
* WE: ~2 = 1199.9088{{c}}, ~15/14 = 119.2719{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 119.2808{{c}}
 
{{Optimal ET sequence|legend=0| 10, 151, 161, 171, 332, 503ef }}
 
Badness (Sintel): 1.39
 
== Subneutral ==
{{See also| Luna family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 274877906944/274658203125
 
{{Mapping|legend=1| 1 -41 8 -5 | 0 60 -8 11 }}
: mapping generators: ~2, ~46875/28672
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9998{{c}}, ~46875/28672 = 851.6994 (~57344/46875 = 348.3005{{c}})
: [[error map]]: {{val| -0.000 +0.013 +0.090 -0.132 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~46875/28672 = 851.6995{{c}} (~57344/46875 = 348.3005{{c}})
: error map: {{val| 0.000 +0.014 +0.090 -0.132 }}
 
{{Optimal ET sequence|legend=1| 31, …, 348, 379, 410, 441, 1354, 1795, 2236 }}
 
[[Badness]] (Sintel): 1.16
 
== Maviloid ==
{{See also| Ragismic microtemperaments #Parakleismic }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 1224440064/1220703125
 
{{Mapping|legend=1| 1 -21 -22 -15 | 0 52 56 41 }}
: mapping generators: ~2, ~875/648
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9863{{c}}, ~875/648 = 521.1837{{c}}
: [[error map]]: {{val| -0.014 -0.115 +0.274 -0.089 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/648 = 521.1894{{c}}
: error map: {{val| 0.000 -0.106 +0.293 -0.060 }}
 
{{Optimal ET sequence|legend=1| 76, 99, 274, 373, 472, 571, 1043, 1614 }}
 
[[Badness]] (Sintel): 1.46
 
== Lockerbie ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Lockerbie]].''
 
Lockerbie can be described as the {{nowrap| 103 & 270 }} temperament. Its generator is [[~]][[77/60]] from the 11-limit onwards, and 74 generator steps give the interval class of [[3/1|3]]; its [[ploidacot]] is 26-sheared 74-cot. An obvious tuning is given by 270edo, but [[373edo]] and especially [[643edo]] work as well.
 
The temperament derives its name from the {{w|Lockerbie|Scottish town}}, where a {{w|Pan Am Flight 103|flight numbered 103}} crashed with 270 casualties, and the temperament has a [[temperament merging|join]] 103 & 270, hence the name. The name was proposed in 2022 by [[Eliora]], who favours it due to simplicity, ease of pronunciation and relation to numbers 103 and 270.
 
Lockerbie also has a unique extension that adds the [[41/1|41st]] [[harmonic]] such that the generator is also on the same step in 103 or 270 as [[41/32]], which means that [[616/615]] is tempered out.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, {{monzo| 24 13 -18 -1 }}
 
{{Mapping|legend=1| 1 -25 -16 -13 | 0 74 51 44 }}
: mapping generators: ~2, ~3828125/2985984
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9950{{c}}, ~3828125/2985984 = 431.1055{{c}}
: [[error map]]: {{val| -0.005 -0.024 +0.146 -0.120 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3828125/2985984 = 431.1072{{c}}
: error map: {{val| 0.0000 -0.020 +0.155 -0.108 }}
 
{{Optimal ET sequence|legend=1| 103, 167, 270, 643, 913, 1183 }}
 
[[Badness]] (Sintel): 1.51
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 766656/765625
 
Mapping: {{mapping| 1 -25 -16 -13 -26 | 0 74 51 44 82 }}
 
Optimal tunings:
* WE: ~2 = 1200.0199{{c}}, ~77/60 = 431.1147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1078{{c}}
 
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913, 1183e }}
 
Badness (Sintel): 0.865
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 1001/1000, 1716/1715, 3025/3024, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 | 0 74 51 44 82 27 }}
 
Optimal tunings:
* WE: ~2 = 1200.0707{{c}}, ~77/60 = 431.1316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1069{{c}}
 
{{Optimal ET sequence|legend=0| 103, 167, 270, 643, 913f }}
 
Badness (Sintel): 0.662
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 715/714, 936/935, 1001/1000, 1225/1224, 4225/4224
 
Mapping: {{mapping| 1 -25 -16 -13 -26 -6 -11 | 0 74 51 44 82 27 42 }}
 
Optimal tunings:
* WE: ~2 = 1199.9639{{c}}, ~77/60 = 431.0957{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/60 = 431.1083{{c}}
 
{{Optimal ET sequence|legend=0| 103, 167, 270 }}
 
Badness (Sintel): 1.07
 
== Unthirds ==
Despite the complexity of its mapping, unthirds is an important temperament to the structure of the [[11-limit]]; this is hinted at by unthirds' representation as the [[72edo|72]] & [[311edo|311]] temperament, the [[temperament merging|join]] of two tuning systems well-known for their high accuracy. It is generated by the interval of [[14/11]] (<u>un</u>decimal major <u>third</u>, hence the name) tuned less than a cent flat, 42 of which [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 14-sheared 42-cot. The 23-note [[mos]] from the generator serves as a well temperament of, of all things, [[23edo]]. The 49-note mos is needed to access the [[3/1|3rd]], [[5/1|5th]], [[7/1|7th]], and [[11/1|11th]] [[harmonic]]s.
 
The commas it tempers out in the 11-limit include the [[lehmerisma]] (3025/3024), the [[pine comma]] (4000/3993), the [[unisquary comma]] (12005/11979), the [[argyria]] (41503/41472), and 42875/42768, all of which appear individually in various 11-limit systems. It is also notable that there is a [[restriction]] of the temperament to the 2.5/3.7/3.11/3 [[fractional subgroup]] that tempers out 3025/3024 and 12005/11979, which is of considerably less complexity, and which is shared with [[sqrtphi]] (whose generator is tuned flat of 72edo's).
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 68359375/68024448
 
{{Mapping|legend=1| 1 -13 -14 -9 | 0 42 47 34 }}
: mapping generators: ~2, ~3969/3125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0859{{c}}, ~3969/3125 = 416.7465{{c}}
: [[error map]]: {{val| +0.086 +0.281 -0.431 -0.218 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3969/3125 = 416.7184{{c}}
: error map: {{val| 0.000 +0.220 -0.547 -0.399 }}
 
{{Optimal ET sequence|legend=1| 72, 167, 239, 311, 694, 1005c }}
 
[[Badness]] (Sintel): 1.90
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 3025/3024, 4000/3993
 
Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}
 
Optimal tunings:
* WE: ~2 = 1200.0246{{c}}, ~14/11 = 416.7270{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}
 
{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}
 
Badness (Sintel): 0.758
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 625/624, 1575/1573, 2080/2079, 2401/2400
 
Mapping: {{mapping| 1 -13 -14 -9 -8 -47 | 0 42 47 34 33 146 }}
 
Optimal tunings:
* WE: ~2 = 1200.0536{{c}}, ~14/11 = 416.7343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7164{{c}}
 
{{Optimal ET sequence|legend=0| 72, 239f, 311, 694, 1005c }}
 
Badness (Sintel): 0.863
 
== Neominor ==
Neominor tempers out [[177147/175616]] and may be described as the {{nowrap| 72 & 89 }} temperament. The generator is a neogothic minor third, which represents [[13/11]][[~]][[20/17]], or its [[octave complement]], which represents [[17/10]]~[[22/13]]. The latter stacked six times [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]], and the temperament has a [[ploidacot]] of delta-hexacot. [[72edo]] and [[89edo]] can be used as tunings.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 177147/175616
 
{{Mapping|legend=1| 1 -3 -29 -14 | 0 6 41 22 }}
: mapping generators: ~2, ~320/189
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4276{{c}}, ~320/189 = 917.0471{{c}}
: [[error map]]: {{val| +0.428 -0.955 +0.216 +0.224 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~320/189 = 916.7320{{c}}
: error map: {{val| 0.000 -1.563 -0.301 -0.722 }}
 
{{Optimal ET sequence|legend=1| 17c, 55c, 72, 161, 233, 305 }}
 
[[Badness]] (Sintel): 2.23
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 35937/35840
 
Mapping: {{mapping| 1 -3 -29 -14 -8 | 0 6 41 22 15 }}
 
Optimal tunings:
* WE: ~2 = 1200.3466{{c}}, ~56/33 = 916.9889{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~56/33 = 916.7330{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 55c, 72, 161, 233, 305 }}
 
Badness (Sintel): 0.924
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 243/242, 364/363, 441/440
 
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 | 0 6 41 22 15 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.6874{{c}}, ~22/13 = 917.2313{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}
 
{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}
 
Badness (Sintel): 1.11
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 221/220, 243/242, 273/272, 364/363
 
Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}
 
Optimal tunings:
* WE: ~2 = 1200.6905{{c}}, ~17/10 = 917.2356{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 916.7252{{c}}
 
{{Optimal ET sequence|legend=0| 17cg, 55cfg, 72 }}
 
Badness (Sintel): 0.918
 
== Catafourth ==
{{See also| Sensipent family }}
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 78732/78125
 
{{Mapping|legend=1| 1 -15 -19 -12 | 0 28 36 25 }}
: mapping generators: ~2, ~189/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9278{{c}}, ~189/125 = 710.7220{{c}}
: [[error map]]: {{val| -0.072 -0.656 +1.050 +0.091 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~189/125 = 710.7626{{c}}
: error map: {{val| 0.000 -0.603 +1.139 +0.238 }}
 
{{Optimal ET sequence|legend=1| 27, 76, 103, 130 }}
 
[[Badness]] (Sintel): 2.01
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 78408/78125
 
Mapping: {{mapping| 1 -15 -19 -12 -38 | 0 28 36 25 70 }}
 
Optimal tunings:
* WE: ~2 = 1200.0219{{c}}, ~189/125 = 710.7610{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~189/125 = 710.7487{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363, 493e }}
 
Badness (Sintel): 1.22
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 10985/10976
 
Mapping: {{mapping| 1 -15 -19 -12 -38 -4 | 0 28 36 25 70 13 }}
 
Optimal tunings:
* WE: ~2 = 1200.1023{{c}}, ~98/65 = 710.8043{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~98/65 = 710.7459{{c}}
 
{{Optimal ET sequence|legend=0| 27e, 76e, 103, 130, 233, 363 }}
 
Badness (Sintel): 0.896
 
== Cotritone ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 390625/387072
 
{{Mapping|legend=1| 1 -13 -4 -4 | 0 30 13 14 }}
: mappping generators: ~2, ~7/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9278{{c}}, ~7/5 = 583.5994{{c}}
: [[error map]]: {{val| +0.441 +0.289 -1.287 -0.200 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/5 = 583.3956{{c}}
: error map: {{val| 0.000 -0.086 -2.170 -1.287 }}
 
{{Optimal ET sequence|legend=1| 35, 37, 72, 181, 253, 325c }}
 
[[Badness]] (Sintel): 2.49
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 4000/3993
 
Mapping: {{mapping| 1 -13 -4 -4 2 | 0 30 13 14 3 }}
 
Optimal tunings:
* WE: ~2 = 1200.4058{{c}}, ~7/5 = 583.5845{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 583.3950{{c}}
 
{{Optimal ET sequence|legend=0| 35, 37, 72, 181, 253, 325c }}
 
Badness (Sintel): 1.07
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 364/363, 385/384, 625/624
 
Mapping: {{mapping| 1 -13 -4 -4 2 -7 | 0 30 13 14 3 22 }}
 
Optimal tunings:
* WE: ~2 = 1200.6111{{c}}, ~7/5 = 583.6837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/5 = 583.3987{{c}}
 
{{Optimal ET sequence|legend=0| 35f, 37, 72, 181f, 253ff }}
 
Badness (Sintel): 1.19
 
== Fibo ==
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 341796875/339738624
 
{{Mapping|legend=1| 1 -27 -7 -9 | 0 46 15 19 }}
: mapping generators: ~2, ~192/125
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2050{{c}}, ~192/125 = 745.8170{{c}}
: [[error map]]: {{val| +0.205 +0.094 -0.493 -0.147 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/125 = 745.6927{{c}}
: error map: {{val| 0.000 -0.092 -0.924 -0.665 }}
 
{{Optimal ET sequence|legend=1| 37, 66b, 103, 140, 243, 383, 1009cd, 1392ccd }}
 
Badness (Sintel): 2.54
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1375/1372, 43923/43750
 
Mapping: {{mapping| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}
 
Optimal tunings:
* WE: ~2 = 1200.4064{{c}}, ~77/50 = 745.9349{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/50 = 745.6876{{c}}
 
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
 
Badness (Sintel): 1.87
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 385/384, 625/624, 847/845, 1375/1372
 
Mapping: {{mapping| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}
 
Optimal tunings:
* WE: ~2 = 1200.3728{{c}}, ~20/13 = 745.9152{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~20/13 = 745.6879{{c}}
 
{{Optimal ET sequence|legend=0| 37, 66b, 103, 140, 243e }}
 
Badness (Sintel): 1.13
 
== Quasimoha ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasimoha]].''
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 3645/3584
 
{{Mapping|legend=1| 1 1 9 6 | 0 2 -23 -11 }}
: mapping generators: ~2, ~49/40
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5059{{c}}, ~49/40 = 348.0409{{c}}
: [[error map]]: {{val| +1.506 -2.367 -0.702 +0.759 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 348.5582{{c}}
: error map: {{val| 0.000 -4.839 -3.152 -2.966 }}
 
{{Optimal ET sequence|legend=1| 24c, 31, 117c, 148bc, 179bcd }}
 
[[Badness]] (Sintel): 2.80
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 1815/1792
 
Mapping: {{mapping| 1 1 9 6 2 | 0 2 -23 -11 5 }}
 
Optimal tunings:
* WE: ~2 = 1201.7630{{c}}, ~11/9 = 349.1510{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.6050{{c}}
 
{{Optimal ET sequence|legend=0| 24c, 31, 86ce, 117ce, 148bce }}
 
Badness (Sintel): 1.53
 
== Mintone ==
In addition to 2401/2400, mintone tempers out 177147/175000 ({{monzo| -3 11 -5 -1 }}) in the 7-limit; [[243/242]], [[441/440]], and [[43923/43750]] in the 11-limit. It may be described as the {{nowrap| 58 & 103 }} temperament. It has a generator of [[~]][[10/9]], tuned to around [[49/44]]. Note that in the data below, the generator is its [[octave complement]], ~[[9/5]], so that 22 of them [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]. Its [[ploidacot]] is 18-sheared 22-cot. As one might expect, 25\161 makes for an excellent tuning choice.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 177147/175000
 
{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}
: mapping generators: ~2, ~9/5
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1458{{c}}, ~9/5 = 1013.7798{{c}}
: [[error map]]: {{val| +0.146 -1.277 +1.263 +0.314 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6611{{c}}
: error map: {{val| 0.000 -1.410 +1.116 +0.025 }}
 
{{Optimal ET sequence|legend=1| 45, 58, 103, 161 }}
 
[[Badness]] (Sintel): 3.18
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 243/242, 441/440, 43923/43750
 
Mapping: {{mapping| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}
 
Optimal tunings:
* WE: ~2 = 1200.1491{{c}}, ~9/5 = 1013.7809{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6593{{c}}
 
{{Optimal ET sequence|legend=0| 45e, 58, 103, 161, 425b }}
 
Badness (Sintel): 1.32
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 243/242, 351/350, 441/440, 847/845
 
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}
 
Optimal tunings:
* WE: ~2 = 1200.0928{{c}}, ~9/5 = 1013.7311{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6556{{c}}
 
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}
 
Badness (Sintel): 0.903
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 243/242, 351/350, 441/440, 561/560, 847/845
 
Mapping: {{mapping| 1 -17 -34 -20 -43 -36 10 | 0 22 43 27 55 47 -7 }}
 
Optimal tunings:
* WE: ~2 = 1200.1085{{c}}, ~9/5 = 1013.7433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1013.6537{{c}}
 
{{Optimal ET sequence|legend=0| 45ef, 58, 103, 161 }}
 
Badness (Sintel): 1.03
 
== Gorgik ==
{{See also| Llywelynsmic clan }}
 
Gorgik may be described as the {{nowrap| 21 & 37 }} temperament, with a [[ploidacot]] of 14-sheared 18-cot (or alpha-heptaseph due to a much simpler [[2.5.7 subgroup|2.5.7-subgroup]] [[restriction]]). [[58edo]] makes for a strong tuning for this temperament. 
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 28672/28125
 
{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}
: mapping generators: ~2, ~7/4
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.5503{{c}}, ~7/4 = 971.3132{{c}} (~8/7 = 227.2371{{c}})
: [[error map]]: {{val| -1.450 +0.528 +2.896 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 972.4675{{c}} (~8/7 = 227.5325{{c}})
: error map: {{val| 0.000 +2.460 +6.414 +3.642 }}
 
{{Optimal ET sequence|legend=1| 21, 37, 58, 153bc, 211bccd, 269bccd }}
 
[[Badness]] (Sintel): 4.01
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 176/175, 2401/2400, 2560/2541
 
Mapping: {{mapping| 1 -13 8 2 14 | 0 18 -7 1 -13 }}
 
Optimal tunings:
* WE: ~2 = 1198.4615{{c}}, ~7/4 = 971.2535{{c}} (~8/7 = 227.2079{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.4918{{c}} (~8/7 = 227.5082{{c}})
 
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bce, 211bccdee, 269bccdee }}
 
Badness (Sintel): 1.96
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 176/175, 196/195, 364/363, 512/507
 
Mapping: {{mapping| 1 -13 8 2 14 11 | 0 18 -7 1 -13 -9 }}
 
Optimal tunings:
* WE: ~2 = 1198.4012{{c}}, ~7/4 = 971.2110{{c}} (~8/7 = 227.1903{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 972.5030{{c}} (~8/7 = 227.4970{{c}})
 
{{Optimal ET sequence|legend=0| 21, 37, 58, 153bcef, 211bccdeeff }}
 
Badness (Sintel): 1.33
 
== Hemigoldis ==
Hemigoldis may be described as the {{nowrap| 68 & 89 }} temperament. Though fairly complex in the [[7-limit]], it does a lot better in badness metrics than pure [[5-limit]] [[goldis]], and yet again has many possible extensions to higher primes. For example, two periods minus six generators yields a "tetracot second" which can be interpreted as ~[[21/19]] to add prime 19 or perhaps more accurately ~[[31/28]] to add prime 7, or even simply as ~[[32/29]] to add prime 29, though the other two have the benefit of clearly connecting to the 7-limit representation. Note that again [[89edo]] is a possible tuning for combining it with flat nestoria and not appearing in the optimal ET sequence.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, 549755813888/533935546875
 
{{Mapping|legend=1| 1 21 -9 2 | 0 24 -14 -1 }}
: mapping generators: ~2, ~8/7
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2264{{c}}, ~8/7 = 229.1679{{c}}
: [[error map]]: {{val| -0.774 +0.394 +1.468 -0.314 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.3103{{c}}
: error map: {{val| 0.000 +1.491 +3.343 +1.864 }}
 
{{Optimal ET sequence|legend=1| 21, 47b, 68, 157, 382bccd, 529bccd }}
 
[[Badness]] (Sintel): 4.40
 
== Surmarvelpyth ==
Surmarvelpyth can be described as the {{nowrap| 311 & 431 }} temperament, starting with the 7-limit to the 19-limit. Its [[ploidacot]] is 28-sheared 70-cot. It was named by [[Eliora]] in 2022 for the generator fifth, 675/448 being 225/224 (marvel comma) sharp of 3/2.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 2401/2400, {{monzo| 93 -32 -17 -1 }}
 
{{Mapping|legend=1| 1 -27 55 22 | 0 70 -129 -47 }}
: mapping generators: ~2, ~896/675
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0051{{c}}, ~896/675 = 490.0303{{c}}
: [[error map]]: {{val| +0.005 +0.025 +0.063 -0.136 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~896/675 = 490.0282{{c}}
: error map: {{val| 0.000 +0.017 +0.052 -0.150 }}
 
{{Optimal ET sequence|legend=1| 120, 191, 311, 742, 1053, 2848, 3901 }}
 
[[Badness]] (Sintel): 5.12
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 2401/2400, 820125/819896, 2097152/2096325
 
Mapping: {{mapping| 1 -27 55 22 -19 | 0 70 -129 -47 55 }}
 
Optimal tunings:
* WE: ~2 = 1199.9901{{c}}, ~896/675 = 490.0239{{c}}
* CWE: ~2 = 1200.000{{c}}, ~896/675 = 490.0279{{c}}
 
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795 }}
 
Badness (Sintel): 1.73
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 2401/2400, 4096/4095, 6656/6655, 24192/24167
 
Mapping: {{mapping| 1 -27 55 22 -19 -11 | 0 70 -129 -47 55 36 }}
 
Optimal tunings:
* WE: ~2 = 1199.9701{{c}}, ~65/49 = 490.0155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0277{{c}}
 
{{Optimal ET sequence|legend=0| 120, 191, 311, 742, 1053, 1795f }}
 
Badness (Sintel): 1.34
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 2401/2400, 2601/2600, 4096/4095, 6656/6655, 8624/8619
 
Mapping: {{mapping| 1 -27 55 22 -19 -11 78 | 0 70 -129 -47 55 36 -181 }}
 
Optimal tunings:
* WE: ~2 = 1199.9726{{c}}, ~65/49 = 490.0164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0276{{c}}
 
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness (Sintel): 1.07
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 2401/2400, 2601/2600, 2926/2925, 3136/3135, 3213/3211, 5985/5984
 
Mapping: {{mapping| 1 -27 55 22 -19 -11 78 41 | 0 70 -129 -47 55 36 -181 -90 }}
 
Optimal tunings:
* WE: ~2 = 1199.9756{{c}}, ~65/49 = 490.0176{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~65/49 = 490.0276{{c}}
 
{{Optimal ET sequence|legend=0| 120g, 191g, 311, 431, 742, 1795f }}
 
Badness (Sintel): 0.838
 
== References ==
 
[[Category:Temperament collections]]
[[Category:Breedsmic temperaments| ]] <!-- main article -->
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Breedsmic_temperaments"