Breedsmic temperaments: Difference between revisions

This page discusses miscellaneous [[Rank-2 temperament|rank-2]] [[temperament]]s [[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.

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)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.

* ''[[Decimal]]'' (+25/24, 49/48 or 50/49) → [[Dicot family #Decimal|Dicot family]]

* ''[[Beatles]]'' (+64/63 or 686/675) → [[Archytas clan #Beatles|Archytas clan]]

* [[Myna]] (+126/125) → [[Starling temperaments #Myna|Starling temperaments]]

* ''[[Greenwood]]'' (+405/392 or 1323/1280) → [[Greenwoodmic temperaments #Greenwood|Greenwoodmic temperaments]]

* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]

* ''[[Hemiwürschmidt]]'' (+3136/3125 or 6144/6125) → [[Hemimean clan #Hemiwürschmidt|Hemimean clan]]

* [[Ennealimmal]] (+4375/4374) → [[Ragismic microtemperaments #Ennealimmal|Ragismic microtemperaments]]

* ''[[Amicable]]'' (+1600000/1594323) → [[Amity family #Amicable|Amity family]]

* ''[[Eagle]]'' (+10485760000/10460353203) → [[Vulture family #Eagle|Vulture family]]

 

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^(1/25), giving just 5's, the 7- and 9-odd-limit minimax tuning, or 14^(1/13), 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.

 

 

[[Comma list]]: 2401/2400, 5120/5103

 

: mapping generators: ~2, ~49/40

 

* [[WE]]: ~2 = 1199.7412{{c}}, ~49/40 = 351.4016{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.4671{{c}}

 

* [[7-odd-limit|7-]] and [[9-odd-limit]] minimax: ~49/40 = {{monzo| 1/5 0 1/25 }}

 

 

{{Optimal ET sequence|legend=1| 17c, 41, 58, 99, 239, 338 }}

 

[[Badness]] (Sintel): 0.563

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness (Sintel): 1.29

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. [[171edo]] makes for an excellent tuning, although 171edo - [[31edo]] = [[140edo]] also makes sense, and in very high limits 140edo + 171edo = [[311edo]] 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.

* [[WE]]: ~2 = 1200.1004{{c}}, ~245/128 = 1122.9024{{c}} (~256/245 = 77.1979)

* [[CWE]]: ~2 = 1200.0000{{c}}, ~245/128 = 1122.8101{{c}} (~256/245 = 77.1899)

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

 

 

 

 

 

 

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

 

 

 

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8323{{c}} (~23/22 = 77.1677{{c}})

 

Badness (Sintel): 1.08

 

 

Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 | 0 22 -5 3 -116 -42 -48 105 -117 -60 }}

* WE: ~2 = 1200.0007{{c}}, ~44/23 = 1122.8332{{c}} (~23/22 = 77.1675{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8326{{c}} (~23/22 = 77.1674{{c}})

{{Optimal ET sequence|legend=0| 140, 311, 762g }}

Badness (Sintel): 1.02

==== 31-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 595/594, 625/624, 714/713, 784/783, 833/832, 875/874, 900/899, 931/930, 1015/1014

Mapping: {{mapping| 1 -19 7 0 112 43 49 -94 114 61 -83 | 0 22 -5 3 -116 -42 -48 105 -117 -60 94 }}

* WE: ~2 = 1199.9721{{c}}, ~44/23 = 1122.8047{{c}} (~23/22 = 77.1673{{c}})

* CWE: ~2 = 1200.0000{{c}}, ~44/23 = 1122.8309{{c}} (~23/22 = 77.1691{{c}})

{{Optimal ET sequence|legend=0| 140, 171, 311 }}

Badness (Sintel): 1.18

 

 

 

 

 

 

 

 

 

 

 

{{Optimal ET sequence|legend=0| 140, 171, 311 }}

 

Badness (Sintel): 1.20

== 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 31 & 270 temperament, and as one might expect, 61\270 makes for an excellent tuning choice. Other possibilities are (7/2)^1/8, giving just 7's, or 384^1/38, giving pure fifths.

Adding 3025/3024 extends to the 11-limit and as expected, 270 remains an excellent tuning.

[[Comma list]]: 2401/2400, 29360128/29296875

{{Mapping|legend=1| 1 -7 3 1 | 0 38 -3 8 }}

: mapping generators: ~2, ~1024/875

* [[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

Comma list: 2401/2400, 3025/3024, 5632/5625

Mapping: {{mapping| 1 -7 3 1 -11 | 0 38 -3 8 64 }}

* 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

Comma list: 1001/1000, 1716/1715, 3025/3024, 4096/4095

Mapping: {{mapping| 1 -7 3 1 -11 22 | 0 38 -3 8 64 -81 }}

* 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

== Neominor ==

The generator for neominor temperament is tridecimal minor third [[13/11]], also known as ''Neo-gothic minor third''.

[[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

* [[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 }}

=== 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:  

* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 916.7228{{c}}

{{Optimal ET sequence|legend=0| 17c, 55cf, 72 }}

Badness (Sintel): 1.11

 

 

Mapping: {{mapping| 1 -3 -29 -14 -8 -7 -28 | 0 6 41 22 15 14 42 }}

* 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

== Emmthird ==

The generator for emmthird is the hemimage third, sharper than 5/4 by the hemimage comma, 10976/10935.

[[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

* [[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 }}

 

=== 11-limit ===

Comma list: 243/242, 441/440, 1792000/1771561

Mapping: {{mapping| 1 -3 -17 -8 -8 | 0 14 59 33 35 }}

* 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 ===

Comma list: 243/242, 364/363, 441/440, 2200/2197

Mapping: {{mapping| 1 -3 -17 -8 -8 -13 | 0 14 59 33 35 51 }}

* 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

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 }}

* 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

== Quinmite ==

The generator for quinmite is quasi-tempered minor third [[25/21]], flatter than 6/5 by the starling comma, [[126/125]]. It is also generated by 1/5 of 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>.

[[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

* [[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

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 in the 11-limit and [[41-limit]] respectively. It is generated by the interval of [[14/11]] ('''un'''decimal major '''third''', hence the name) tuned less than a cent flat, and the 23-note [[MOS]] this interval generates serves as a well temperament of, of all things, [[23edo]]. The 49-note MOS is needed to access the 3rd, 5th, 7th, and 11th harmonics, however.

The commas it tempers out include the [[breedsma]] (2401/2400), 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 }}

[[Optimal tuning]]s:  

: 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 ===

Comma list: 2401/2400, 3025/3024, 4000/3993

Mapping: {{mapping| 1 -13 -14 -9 -8 | 0 42 47 34 33 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~14/11 = 416.7190{{c}}

{{Optimal ET sequence|legend=0| 72, 167, 239, 311 }}

Badness (Sintel): 0.758

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

Newt has a generator of a neutral third (0.2 cents flat of [[49/40]]) and tempers out the [[garischisma]]. It can be described as the 41 & 270 temperament, and extends naturally to the no-17 19-limit, a.k.a. '''neonewt'''. [[270edo]] and [[311edo]] are obvious tuning choices, but [[581edo]] and especially [[851edo]] work much better.

[[Comma list]]: 2401/2400, 33554432/33480783

{{Mapping|legend=1| 1 1 19 11 | 0 2 -57 -28 }}

: mapping generators: ~2, ~49/40

* [[WE]]: ~2 = 1199.9315{{c}}, ~49/40 = 351.0932{{c}}

: [[error map]]: {{val| -0.068 +0.163 +0.075 -0.188 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/40 = 351.1141{{c}}

: error map: {{val| 0.000 +0.273 +0.180 -0.022 }}

{{Optimal ET sequence|legend=1| 41, 147c, 188, 229, 270, 1121, 1391, 1661, 1931, 2201 }}

[[Badness]] (Sintel): 1.06

Comma list: 2401/2400, 3025/3024, 19712/19683

Mapping: {{mapping| 1 1 19 11 -10 | 0 2 -57 -28 46 }}

* WE: ~2 = 1199.9603{{c}}, ~49/40 = 351.1038{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1155{{c}}

{{Optimal ET sequence|legend=0| 41, 188, 229, 270, 581, 851, 1121, 1972 }}

Badness (Sintel): 0.643

Comma list: 2080/2079, 2401/2400, 3025/3024, 4096/4095

Mapping: {{mapping| 1 1 19 11 -10 -20 | 0 2 -57 -28 46 81 }}

* WE: ~2 = 1199.9747{{c}}, ~49/40 = 351.1094{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1168{{c}}

{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851, 2283b }}

Badness (Sintel): 0.571

=== 2.3.5.7.11.13.19 subgroup (neonewt) ===

Subgroup: 2.3.5.7.11.13.19

Comma list: 1216/1215, 1540/1539, 1729/1728, 2080/2079, 2401/2400

Mapping: {{mapping| 1 1 19 11 -10 -20 18 | 0 2 -57 -28 46 81 -47 }}

Optimal tunings:  

* WE: ~2 = 1199.9782{{c}}, ~49/40 = 351.1102{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~49/40 = 351.1166{{c}}

{{Optimal ET sequence|legend=0| 41, 229, 270, 581, 851 }}

Badness (Sintel): 0.438

Aside from 2401/2400, [[septidiasemi]] tempers out 2152828125/2147483648 in the 7-limit. It is so named because the generator is a "septimal diatonic semitone" (0.15 cents flat of [[15/14]]). It is an excellent tuning for 2.3.5.7.13 and 2.3.5.7.13.17 subgroups rather than full 13- and 17-limit.

: mpping generators: ~2, ~15/14

The ''sedia'' temperament (10 & 161) is an 11-limit extension of the septidiasemi, which tempers out 243/242 and 441/440.

== Subneutral ==

 

[[Subgroup]]: 2.3.5.7

 

 

{{Mapping|legend=1| 1 -41 8 -5 | 0 60 -8 11 }}

: mapping generators: ~2, ~46875/28672

 

[[Optimal tuning]]s:

: [[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

{{See also| Metric microtemperaments #Geb }}

[[Comma list]]: 2401/2400, 31381059609/31360000000

{{Mapping|legend=1| 1 13 33 21 | 0 32 86 51 }}

: mapping generators: ~2, ~2187/1400

* [[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

== Gorgik ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 28672/28125

{{Mapping|legend=1| 1 -13 8 2 | 0 18 -7 1 }}

[[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 }}

* 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 }}

* 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

[[Comma list]]: 2401/2400, 341796875/339738624

{{Mapping|legend=1| 1 -27 -7 -9 | 0 46 15 19 }}

: mapping generators: ~2, ~192/125

* [[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

Comma list: 385/384, 1375/1372, 43923/43750

Mapping: {{mapping| 1 -27 -7 -9 -4 | 0 46 15 19 12 }}

* 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

Comma list: 385/384, 625/624, 847/845, 1375/1372

Mapping: {{mapping| 1 -27 -7 -9 -4 -5 | 0 46 15 19 12 14 }}

* 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

== 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 has a generator tuned around 49/44. It may be described as the 58 & 103 temperament, and as one might expect, 25\161 makes for an excellent tuning choice.

[[Comma list]]: 2401/2400, 177147/175000

{{Mapping|legend=1| 1 -17 -34 -20 | 0 22 43 27 }}

: mapping generators: ~2, ~9/5

* [[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

Comma list: 243/242, 441/440, 43923/43750

Mapping: {{mapping| 1 -17 -34 -20 -43 | 0 22 43 27 55 }}

* 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

Comma list: 243/242, 351/350, 441/440, 847/845

Mapping: {{mapping| 1 -17 -34 -20 -43 -36 | 0 22 43 27 55 47 }}

* 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

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 }}

* 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