Xenharmonic Wiki - User contributions [en]

User:Xenllium

2026-07-08T08:45:19Z

Xenllium: /* Temperaments which this user proposed names */

'''Xenllium''' joined Xenharmonic Wiki on 15 November 2018. The name "Xenllium" is a portmanteau of "Xenharmonic" and "''[[Wikipedia:Trillium|Trillium]]''".

== Temperaments which this user proposed names ==
This user proposed new names for following temperaments.

===Proposed in 2019===
'''March 2019'''
* [[Icositritonic]]
* [[Hemicocomp|Hemicounterpyth]] (→ Hemicocomp)
* [[Hemermacomp|Hemermapyth]] (→ Hemermacomp)
* [[Pentacontatritonic]]
* Hectaheptacontahenatonic (171 & 17442)
* [[Meridic]]
* Tetracontaheptatonic (94 & 705)
* [[Palladium]]
'''April 2019'''
* [[Septisemitonic]]
* [[Tertiathirds]]
* [[Tertiaschis]]

===Proposed in 2021===
'''March 2021'''
* [[Trident]] (→ Alphatrident)
* [[Trillium]] (→ Alphatrillium)
* [[Pseudotrillium]]
* [[Tritricot]]
* [[Decic]]
* [[Cloudtone]]
'''April 2021'''
* [[Hemipental]]
* [[Sengasec]]
* [[Qinto]]
* [[Monzism]]
* [[Houborizic]]
* [[Houbor]]
* [[Gamity]]
* [[Familia]]
* [[Sabric]]
* [[Quadrawell]]
* [[Sfour]]
* [[Rainwell]]
* [[Sengage]]
* [[Marfifths]]
* [[Diatessic]]
* [[Marf]]
'''May 2021'''
* [[Garistearn]]
* [[Decistearn]]
* [[Quadrant]]
* [[Quintilipyth]]
* [[Helenoid]]
* [[Ponta]]
* [[Pontic]]
* [[Bipont]]
* [[Counterbipont]]
* [[Quadrapont]]
* [[Decimetra]]
* [[Stockhausenic]]
* [[Decavish]]
* Bronzoid (109 & 142)
* [[Ennealimmia]]
* [[Paralytic]]
* [[Cotoneum]]
* [[Fog]]
* [[Countermiracle]]
* [[Paraklein]]
* [[Countermiraculous]]
* [[Counterbenediction]]
* [[Counterrevelation]]
* [[Septant]]
* [[Terminal]]
* [[Semiterm]]
* [[Sedia]]
* [[Marthirds]]
* [[Counterkleismic]]
* [[Counterlytic]]
* [[Geb]]
* [[Hemidimfi]]
* [[Maquiloid]]
* [[Paramity]]
'''June 2021'''
* [[Schisennealimmal]]
* [[Schisennealimmic]]
* [[Decades]] (→ Gamelstearn)
* [[Undecentic]]
* [[Pentadecal]]
* [[Quinkee]]
* [[Coditonic]]
* [[Terture]]
* [[Semivulture]]
* [[Eagle]]
* [[Quindromeda]]
* [[Quintakwai]]
* [[Quintagar]]
* [[Quintoneum]]
* [[Quinsandra]]
* [[Quinsandric]]
* [[Quinsandro]]
* [[Semiquindromeda]]
* [[Quintoneoid]]
'''July 2021'''
* [[Mineral]]
* [[Ore]]
* [[Quintaleap]]
* [[Quintupole]]
* [[Quinticosiennic]]
* [[Quintapole]]
* [[Hemiseptisix]]
'''August 2021'''
* [[Goldmine]]
* [[Gamelacomp|Gamelapyth]] (→ Gamelacomp)
* [[Mermacomp|Mermapyth]] (→ Mermacomp)
* [[Undesemi]]
* [[Decipentic]]
* [[Quasijerome]]
* [[Septendesemi]]
* [[Witcher]] (99 & 166, formerly ''witch'', revised in August 2023)
* [[Semiwitcher]] (166 & 198, formerly ''semiwitch'', revised in August 2023)
* [[Hitch]]
* [[Hitcher]]
* [[Quinwell]]
* [[Septiquarschis]]
* [[Quinbetter]]
* [[Ultramarine]]
* [[Mowglic]]
'''Septemper 2021'''
* [[Terminator]]
* [[Quintaschis]]
* [[Quintahelenic]]
* [[Quintahelenoid]]
* [[Semihemiwür]]
* [[Semihemiwürat]]
* [[Beetle]]
* [[Subsedia]]
* [[Quinkwai]]
* [[Quintakwoid]]
* [[Passionate]]
* [[Pash]]
* [[Majvamic]]
* [[Majvamoid]]
* [[Narayana]]
* [[Turkey]]
'''October 2021'''
* [[Lunennealimmal]]
* [[Hexadecoid]]
* [[Undetrita]]
* [[Rubidium]]
* [[Terbium]]
* [[Parakleischis]]
* [[Triacontaheptoid]]
* [[Octogintic]]
* [[Senator]]
* [[Hemivalentine]]
* [[Hemivalentino]]
* [[Hemivalentoid]]
* [[Undetritoid]]
* [[Biquartonic]]
* [[Quindro]]
'''November 2021'''
* [[Tridecatonic]]
* [[Trideci]]
* [[Decimaleap]]
* [[Hemiquindromeda]]
* [[Triboh]]
'''December 2021'''
* [[Hemimabila]]
* [[Sedecic]]
* [[Hyperkleismic]]

===Proposed in 2022===
'''January 2022'''
* [[Cobalt]]
* [[Cobaltous]]
* [[Cobaltic]]
* [[Cobaltite]]
* [[Pontoid]]
* [[Grackle]] (11, 13, 17 and 19-limit extension)
* [[Grackloid]]
* [[Grack]]
* [[Catahelenic]]
* [[Countriton]]
* [[Aufic]]
* [[Pajarina]]
* [[Muggloid]]
* [[Gracecord]]
* [[Gaster]]
* [[Tuskaloosa]]
* [[Muscogee]]
* [[Cohemimabila]]
* [[Trimabila]]
* [[Immune]]
* [[Meantritone]]
* [[Aurora]]
* [[Leapweeker]]
* [[Interaufo]]
* [[Helena]]
'''February 2022'''
* [[Cohendecatonic]]
* [[Nessa]]
* [[Splendecic]]
* [[Prodecic]]
* [[Semimeantone]]
* [[Keenic]]
* [[Warrior]]
* [[Trisensory]]
* [[Bisensi]]
* [[Browser]]
* [[Grazer]]
* [[Quadrasruta]]
* [[Quadrafourths]]
* [[Pajarita]]
'''March 2022'''
* [[Stalagmite|Catamite]] (→ Stalagmite)
* [[Hemikwai]]
* [[Yarm]]
* [[Cohedim]]
* [[October]]
* [[Dichotomic]]
* [[Seminar]]
* [[Chromat]] (11, 13 and 17-limit extension)
* [[Catachrome]]
* [[Chromic]]
* [[Trinity]]
* [[Poseidon]]
* [[Salacia]]
* [[Gammy]]
* [[Tertiaseptia]]
* [[Semitert]]
'''April 2022'''
* <s>[[Meantonic]]</s>
* <s>[[Meantoid]]</s>
* [[Huygens]] (→ Fokkertone)
* <s>[[Meridetonic]]</s>
* <s>[[Meridetoid]]</s>
* <s>[[Sauveuric]]</s>
* [[Meanpoid]]
* <s>[[Meanplop]] (17 and 19-limit extension)</s>
* <s>[[Meanploid]]</s>
* <s>[[Meanenneadecoid]]</s>
* <s>[[Vincenzoid]]</s>
* [[Andromeda]] (17 and 19-limit extension)
* [[Schisicosiennic]]
* [[Schisicosiennoid]]
* [[Cassandra]] (Cassie)
* [[Cassandric]]
* [[Cassander]]
'''May 2022'''
* [[Supersensi]]
* [[Quadrafives]]
'''June 2022'''
* [[Trimean]]

===Proposed in 2023===
'''June 2023'''
* [[Galileic]]
* [[Catagali]]
* [[Quintain]]
'''July 2023'''
* [[Devisemi]]
* [[Garibaldia]] (2.3.5.7.17.19 extension of [[garibaldi]])
* [[Pontia]] (2.3.5.7.19 extension of [[pontiac]])
'''August 2023'''
* [[Heartland]] (2.3.7.11.19 14&41& 48)
* [[Hearts]] (2.3.7.11.19 41&48)
* [[Heartia]]
* [[Heartful]] (2.3.7.11.19 41&75e)
* [[Doublehearted]] (2.3.7.11.19 14&41)
* [[Weasel|Byhearted]] (→ Weasel)
'''September 2023'''
* [[Heartlandia]] (14cf & 41 & 89)
* [[Darkstone]]
* [[Brightstone]]
'''October 2023'''
* [[Magical]]
* [[Magica]]
* [[Magia]]
* [[Sesquartia]]
* [[Charismic]]
* [[Quinmage]]
* [[Hearty]]
* [[Nonant]]
* [[Hemiandromeda]] (29 & 82)
* [[Hemidromeda]] (29 & 111)
'''November 2023'''
* [[Trismegiboh]]
* [[Baldanders]] (2.9.5.7.11 6&35)
* [[Neomirkwai]] (2.3.5.7.11.19 31&41&80)
* [[Dhaivatic]]
* [[Triquart]]
* [[Tertiseptisix]]
* [[Quartiquart]]
* [[Quintiquart]]
* [[Neovulture]] (2.3.5.7.11.13.19 53&58&217)
* [[Neovish]] (2.3.5.7.11.19 50&118&152)
* [[Neobreed]] (41 & 130 & 270)
* [[Neonewt]] (2.3.5.7.11.13.19 41&270)
'''December 2023'''
* [[Heartlander]] (14cf & 41 & 130)
* [[Neosatanic]] (665 & 3325)
* [[Monzosatanic]] (665 & 7980)
* [[Dog]] (2.3.19 7&9)
* [[Armodog]] (2.3.5.7.11.13.19 7&9)

===Proposed in 2024===
'''January 2024'''
* [[Quinbaldi]] (65d & 135)
* [[Quintant]] (65d & 200)
* [[Hemiquintant]] (130 & 135)

'''May 2024'''
* [[Raccoon]] (11 & 32)

'''June 2024'''
* [[Piracy]] (84 & 155)
* [[Pirateship]] (84 & 239)
* [[Unisquary]] (11-limit 72&118&121&130)
* [[Hensquary]] (13-limit 46&72&121&130)
* [[Ekasquary]] (13-limit 72&121&130&190)

===Proposed in 2025===
'''January 2025'''
* [[Pentadecoid]] (2.3.7.11 120&135)
* [[Slendscape]] (255 & 270)
* [[Triacontoid]] (120 & 150)

'''February 2025'''
* [[Devichrome]] (2.3.5.7.19 27&31&41&53)

'''March 2025'''
* [[Octasand]] (11 and 13-limit 27e&41&53)
* [[Devimarv]] (2.3.5.7.19 31&41&53)
** [[Neomarv]] (2.3.5.7.11.19 31&41&53)
* [[Deva]] (2.3.5.7.19 27&41&53)
** [[Neonakika]] (2.3.5.7.11.19 15&27e&41)
** [[Neoctasand]] (2.3.5.7.11.19 27e&41&53)
* [[Nutrient]] (2.3.5.7.19 27&31&41)
** [[Neojovia]] (2.3.5.7.11.19 27e&31&41)
* [[Passkey]] (2.3.5.7.19 41&53&99)
** [[Neolakia]] (2.3.5.7.11.19 41&53&99)

'''April 2025'''
* [[Magikarp]] (2.3.43 5&7)
* [[Hendeca]] (11&22)
* [[Char]] (2.3.5.7.17 19&31&41)
* [[Elven]]

'''June 2025'''
* [[Reversed marvel]] (2.3.5.113 22&31&34)

===Proposed in 2026===
'''January 2026'''
* [[Mendel]] (11-limit 12&31&152)
* [[Moctpine]] (11-limit 29&72&80)
* [[Biscordial]] (12 & 190)
* [[Triscordial]] (12 & 231)
* [[Naiad]]
* [[Naiadical]]

'''February 2026'''
* [[Naian]] (8d&13b)
* [[Semihemimean]] (80 & 118 & 130)
* [[Marvelpine]]
* [[Seminaiad]]
** [[Seminaiad|Seminaiadia]]
* <s>[[Seesaw]]</s> (1c & 2)
** <s>[[Heavy windmill]]</s> (1bd & 2)
** <s>[[Light windmill]]</s> (1ce & 2)
* [[Antonian]]
* [[Antonym (temperament)|Antonym]]
* [[Antony]]
* [[No-threes subgroup temperaments #Argument|Argument]]
* [[Landy]]
* [[Septarium]]

'''March 2026'''
* [[Augenic]]
* [[Sida]]
* [[Bohlenic]] (no-twos temperament)

'''May 2026'''
* [[Xenial]] (19 & 70)
** [[Xenia]] (19 & 51c, 19 & 70d)
* [[Hemigoth]] (34d & 68)
* [[Paraguay]] (19 & 23bcf, 19 & 42ef)
** [[Uruguay]] (19 & 23bc, 19 & 42e)

'''June 2026'''
* <s>[[Sixseven]]</s> (2.3.7 2d & 3)
** <s>[[Heaven]]</s> (2.3.7.11 2de & 3)

== Commas which this user proposed names ==
This user proposed new names for following commas.

===Proposed in 2021===
'''April 2021'''
* [[Hewuermera temperaments|hewuermera]]: 589824/588245, satribiru-agu
* [[Triwellismic temperaments|triwellisma]]: 235298/234375, tribizo-asepgu
* [[quadrawellisma]]: 67528125/67108864, lala-quadzo-aquinyo
* [[catasyc comma]]: 390625/387072, laruquadbiyo
* [[Mistismic temperaments|mistisma]]: 458752/455625, sazoquadgu

'''May 2021'''
* [[Hemfiness temperaments|hemfiness comma]]: 4096000/4084101, saquinru-atriyo
* [[Marvelfifths temperaments|marvelfifths comma]]: 158203125/157351936, laquadru-atritriyo
* [[Chromatisma|chromatisma]]: 640000000000000000 / 635585924776181463, trisa-triru-aquadquadyo
* [[Bronzismic temperaments|bronzisma]]: 2097152/2083725, satriru-agugu
* <s>[[Breeze comma|sesquiquartisma]]</s>: 2460375/2458624, laquadru-atruyo (→ breezema, breeze comma)
* [[Mitonismic temperaments|mitonisma]]: 5250987/5242880, laquadzo-agu
* [[Sensamagicfifths temperaments|sensamagicfifths comma]]: {{monzo|-11 42 -7 -14}}, trila-seprurugu
* [[Secanticorn temperaments|secanticornisma]]: 177147/175000, laruquingu
* [[Supermatertismic temperaments|supermatertisma]]: 52734375/52706752, lasepru-atritriyo
* [[Supermasesquartismic temperaments|supermasesquartisma]]: 184528125/184473632, laquadbiru-aquinyo
* [[Septimirod temperaments|septimirod comma]]: 84035/82944, quinzo-ayo

'''June 2021'''
* [[Vorwell clan|vorwell comma]]: 134217728/133984375, sasatriru-aquadbigu
* [[Maitri clan|maitri comma]]: 40353607/40000000, tritrizo-asepgu
* [[compass comma]]: 9765625/9680832, quinruyoyo

'''July 2021'''
* [[4375/4356|fantares comma]]: 4375/4356, luluzoquadyo
* [[Septiness clan|septiness comma]]: 67108864/66706983, sasasepru
* [[decovulture comma]]: 67108864/66976875, sasa-birugugu
* [[Pontiqak temperaments|pontiqak comma]]: 95703125/95551488, lazozotritriyo

'''August 2021'''
* [[Slither clan|slither comma]]: 40960000/40353607, satritriru-aquadyo
* [[Mermismic temperaments|mermisma]]: 2500000/2470629, sepruyo
* [[Valenwuer temperaments|valenwuer comma]]: 110592/109375, sarutribigu
* [[Negricorn temperaments|negricorn comma]]: 153664/151875, saquadzogu
* [[Parkleiness temperaments|parkleiness comma]]: 1959552/1953125, zotritrigu
* [[Mynaslendic temperaments|mynaslender comma]]: 829440/823543, sepru-ayo

'''September 2021'''
* [[65536/64827|buzzardisma]]: 65536/64827, saquadru (→ buzzardsma, buzzard comma)
* [[Neptunismic temperaments|neptunisma]]: 48828125/48771072, laruruleyo

'''October 2021'''
* [[Domicomp temperaments|domicomp comma]]: 413343/409600, lazogugu

'''November 2021'''
* [[Trienstonisma|trienstonisma]]: 43046721/43025920, laquinru-agu
* [[Lokismic temperaments|lokisma]]: 102760448/102515625, sasazozo-atribigu

===Proposed in 2022===
'''January 2022'''
* [[Countriton comma|countriton comma]]: 16777216000000000 / 16677181699666569, quadsa-tritriyo
* [[3042/3025|diagassormisma]]: 3042/3025, bitholugu
* [[gaster comma]]: {{monzo|-70 72 -19}}, quadbila-negu
* [[Mabilismic clan|mabilisma]]: 1071875/1048576, latrizo-aquinyo
* [[pentonisma]]: 281974669312/281950621875, saquinthuzogu

'''February 2022'''
* [[8281/8192|diahuntmisma]]: 8281/8192, labithozo
* [[3645/3584|schismean comma]]: 3645/3584, laruyo

'''March 2022'''
* [[Acrodec temperaments|acrodec comma]]: 7710244864/7688671875, sasa-tribizo-aquadbigu
* [[120/119|lynchisma]]: 120/119, suruyo

'''May 2022'''
* [[Augmist temperaments|augmist comma]]: 1438646272/1423828125, sasa-trizotrigu
* [[Starling temperaments|supersensi comma]]: 33059881728/30517578125, quintrigu

===Proposed in 2023===
'''11 May 2023'''
* [[Syntonoschisma|syntonisma]]: {{monzo|30 -27 7 0 -1}}, trisa-lusepyo (→ syntonoschisma, syntonic schisma)

'''27 May 2023'''
* [[170/169|major naiadma]]: 170/169, sothuthuyo
* [[221/220|minor naiadma]]: 221/220, sotholugu
* <s>[[256/255|diasemisma]]</s>: 256/255, sugu (→ charisma, charic comma)
* [[442/441|seminaiadma]]: 442/441, sothoruru

'''18 July 2023'''
* [[grossmisma]]: {{monzo|-30 -13 -1 1 7 7}}, septholo-azogu

'''21 July 2023'''
* [[3971/3969|heartlandisma]]: 3971/3969, nonoloruru

'''23 July 2023'''
* [[171/170|malcolmisma]]: 171/170, nosugu
* [[190/189|cotylisma]]: 190/189, noruyo
* [[324/323|photisma]]: 324/323, nusu
* [[343/342|nutrisma]]: 343/342, nutrizo
* [[400/399|devichroma]]: 400/399, nuruyoyo
* [[devicisma]]: 633556/633555, nunusothutrilo-azogu

'''15 August 2023'''
* [[133/132|minithirdma]]: 133/132, noluzo
* <s>[[136/135|diatonisma]]</s>: 136/135, sogu (→ diatisma)
* [[154/153|augustma]]: 154/153, sulozo
* [[161/160|major kirnbergerisma]]: 161/160, twethozogu
* [[162/161|minor kirnbergerisma]]: 162/161, twethuru
* [[286/285|chthonisma]]: 286/285, nuthologu
* [[1275/1274|cimbrisma]]: 1275/1274, sothubiruyo
* [[galileisma]]: 171382426877952 / 171359481538165, lalesu-agu

'''16 August 2023'''
* [[85/84|monk comma]]: 85/84, soruyo
* [[456/455|abnobisma]]: 456/455, nothulugu
* [[476/475|hedwigma]]: 476/475, nusozogugu
* [[495/494|eulalisma]]: 495/494, nuthuloyo

'''21 August 2023'''
* [[363/361|godzillisma]]: 363/361, binulo
* [[2431/2430|heptacircle comma]]: 2431/2430, sothologu

'''26 August 2023'''
* [[1083/1078|bihendrixma]]: 1083/1078, nonolururu

'''27 August 2023'''
* [[No-fives subgroup temperaments #Doublehearted|doublehearted comma]]: 5764801/5668704, quadbizo
* [[No-fives subgroup temperaments #Hearts|hearts comma]]: 34451725707/34359738368, trila-quadzo

'''17 September 2023'''
* [[4914/4913|baladisma]]: 4914/4913, trisu-athozo

'''5 November 2023'''
* [[969/968|buddhisma]]: 969/968, nosolulu (→ kingfisher comma)
* [[1331/1330|solvejgsma]]: 1331/1330, nutrilo-arugu
* [[1540/1539|kevolisma]]: 1540/1539, nulozoyo
* [[2376/2375|trichthonisma]]: 2376/2375, nulotrigu
* [[2926/2925|neovulturisma]]: 2926/2925, nothulozogugu
* [[3136/3135|neomirkwaisma]]: 3136/3135, nuluzozogu
* [[3250/3249|martebisma]]: 3250/3249, nunuthotriyo
* [[4200/4199|neosatanisma]]: 4200/4199, nusuthuzoyoyo
* [[5776/5775|neovishma]]: 5776/5775, nonolurugugu
* [[5985/5984|neogrendelisma]]: 5985/5984, nosuluzoyo
* [[6175/6174|neonewtisma]]: 6175/6174, nothotriru-ayoyo

'''12 November 2023'''
* [[460/459|scanisma]]: 460/459, twethosuyo
* [[484/483|pittsburghisma]]: 484/483, twethuloloru
* [[507/506|laodicisma]]: 507/506, twethuthotholu
* [[529/528|preziosisma]]: 529/528, bitwetho-alu
* [[736/735|harvardisma]]: 736/735, twethorurugu
* [[760/759|squadronisma]]: 760/759, twethunoluyo
* [[875/874|nymphisma]]: 875/874, twethunuzotriyo
* [[897/896|lysistratisma]]: 897/896, twethothoru
* [[1105/1104|fragarisma]]: 1105/1104, twethusothoyo
* [[1197/1196|rhodesisma]]: 1197/1196, twethunothuzo
* [[1288/1287|santisma]]: 1288/1287, twethothuluzo
* [[1496/1495|turkisma]]: 1496/1495, twethusothulogu
* [[1863/1862|antinousisma]]: 1863/1862, twethonururu
* [[2024/2023|insincere comma]]: 2024/2023, twethosusuloru
* <s>[[2025/2024|fokkerisma]]</s>: 2025/2024, latwethuluyoyo (→ cupcakesma, cupcake comma)
* [[2185/2184|guangdongisma]]: 2185/2184, twethonothuruyo
* [[2300/2299|travellisma]]: 2300/2299, twethonubiluyo
* [[2646/2645|biyativice comma]]: 2646/2645, bitwethuzo-agu
* [[2737/2736|kotkisma]]: 2737/2736, twethonusozo
* [[3060/3059|vicious comma]]: 3060/3059, twethunusoruyo
* [[3381/3380|seminaiadvice comma]]: 3381/3380, twethobithuzo-agu (→ mikkolisma)
* [[3520/3519|vicedim comma]]: 3520/3519, twethusuloyo
* [[3888/3887|vicetride comma]]: 3888/3887, twethuthuthu
* [[4693/4692|viceaug comma]]: 4693/4692, twethunonosutho
* [[4761/4760|demiquartervice comma]]: 4761/4760, bitwetho-asurugu
* [[vicetertisma]]: 12168/12167, tritwethu-athotho

'''13 November 2023'''
* [[208/207|vicetone comma]]: 208/207, twethutho
* [[231/230|major neutravicema]]: 231/230, twethulozogu
* [[253/252|middle neutravicema]]: 253/252, twetholoru
* [[276/275|minor neutravicema]]: 276/275, twetholugugu
* [[300/299|major naiadvicema]]: 300/299, twethuthuyoyo
* [[323/322|major semivicema]]: 323/322, twethunosoru
* [[391/390|minor naiadvicema]]: 391/390, twethosothugu
* [[392/391|minor semivicema]]: 392/391, twethusuzozo
* [[460/459|vicewolf comma]]: 460/459, twethosuyo
* [[5083/5082|broadviewsma]]: 5083/5082, twethosotholuluru

'''15 November 2023'''
* [[92/91|undinisma]]: 92/91, twethothuru
* [[115/114|yarmanisma]]: 115/114, twethonuyo

'''26 November 2023'''
* [[E-shaped comma]]: 31213/31212, susuthoquadzo

'''1 December 2023'''
* <s>[[kilobyte comma]]</s>: 1024/1023, thiwulu (→ kibisma)
* <s>[[megabyte comma]]</s>: 1048576/1048575, safowuthiwulugugu (→ mebisma)

'''10 December 2023'''
* [[81/76|dog comma]]: 81/76, inu
* [[primate comma]]: 16384/15309, saru

'''23 December 2023'''
* [[841/840|arabellisma]]: 841/840, bitweno-arugu
* [[4901/4900|large grapevine comma]]: 4901/4900, twenothothobirugu, twenobithorugu
* [[6877/6875|grossvice comma]]: 6877/6875, bitwetho-atholuquadgu
* [[7425/7424|small grapevine comma]]: 7425/7424, latwenuloyoyo
* [[mynucuvine comma]]: 10935/10933, labitwenu-athuyo

===Proposed in 2024===
'''2 January 2024'''
* <s>[[slendroschisma|slendrisma]]</s>: 68719476736/68641485507, sasa-quinbiru (→ slendroschisma)

'''27 January 2024'''
* [[116/115|sironisma]]: 116/115, twenotwethugu
* [[117/116|lomisma]]: 117/116, twenutho
* [[175/174|major chthonovinema]]: 175/174, twenuzoyoyo
* [[204/203|kallistisma]]: 204/203, twenusoru
* [[232/231|major paravinema]]: 232/231, twenoluru
* [[261/260|major vinetonema]]: 261/260, twenothugu
* [[290/289|brunisma]]: 290/289, twenosusuyo
* [[320/319|minor paravinema]]: 320/319, twenuluyo
* [[378/377|major semivinema]]: 378/377, twenuthuzo
* [[406/405|minor semivinema]]: 406/405, twenozogu
* [[494/493|minor vinetonema]]: 494/493, twenunosutho
* [[551/550|minor chthonovinema]]: 551/550, twenonolugugu
* [[609/608|vineyard comma]]: 609/608, twenonuzo

'''3 May 2024'''
* [[729/722|undevicesimal diaschisma]]: 729/722, lanunu

'''1 June 2024'''
* [[unisquarisma]]: 12005/11979, trilu-aquadzo-ayo

'''22 December 2024'''
* [[429/425|middle semisixthma]]: 429/425, suthologugu
* [[507/500|large semisixthma]]: 507/500, thothotrigu
* [[1452/1445|small semisixthma]]: 1452/1445, susulologu
* [[semisixthmisma]]: 63888/63869, trisu-athutrilo

===Proposed in 2025===
'''20 March 2025'''
* [[6860/6859|devicubisma]]: 6860/6859, trinuzo-ayo
* [[tricecubisma]]: 29792/29791, trithiwu-anozozo

'''21 March 2025'''
* [[8625/8624|beerglass comma]]: 8625/8624, twetholururutriyo
* [[10626/10625|demiglace comma]]: 10626/10625, twethosulozoquadgu

'''29 March 2025'''
* [[145/144|adeonisma]]: 145/144, twenoyo
* [[552/551|sigelindisma]]: 552/551, twenutwethonu
* [[638/637|moirisma]]: 638/637, twenothuloruru
* [[726/725|joellisma]]: 726/725, twenubilogu
* [[783/782|norisma]]: 783/782, twenotwethusu
* [[1015/1014|christisma]]: 1015/1014, twenothuthuzoyo
* [[1045/1044|michelisma]]: 1045/1044, twenunoloyo
* [[1276/1275|ucclisma]]: 1276/1275, twenosulogugu
* [[1450/1449|raimondisma]]: 1450/1449, twenotwethuruyoyo
* [[1596/1595|itzigsohnisma]]: 1596/1595, twenunoluzogu
* [[1625/1624|norcma]]: 1625/1624, twenuthorutriyo
* [[1683/1682|castafiorisma]]: 1683/1682, bitwenu-asolo
* [[2176/2175|donarisma]]: 2176/2175, twenusogugu
* [[2205/2204|glinkisma]]: 2205/2204, twenunuzozoyo
* [[2262/2261|mitidikisma]]: 2262/2261, twenonusuthoru
* [[5929/5928|manzanisma]]: 5929/5928, nuthubilozo

'''6 April 2025'''
* [[6144/6137|langwisma]]: 6144/6137, nunusu

'''10 April 2025'''
* <s>[[129/128|Magikarp comma]]</s>: 129/128, fotho
* <s>[[130/129|Gyarados comma]]</s>: 130/129, fothuthoyo
* [[2640/2639|hällströmisma]]: 2640/2639, twenuthuloruyo
* [[2784/2783|domeykisma]]: 2784/2783, twenotwethululu
* [[3249/3248|musashinisma]]: 3249/3248, twenunonoru
* [[3451/3450|mentorisma]]: 3451/3450, twenotwethusozogugu
* [[4785/4784|petrovsma]]: 4785/4784, twenotwethuthuloyo
* [[5916/5915|woudisma]]: 5916/5915, twenosothuthurugu

'''11 April 2025'''
* [[10557/10556|rowlandisma]]: 10557/10556, twenutwethosothuru

'''15 April 2025'''
* [[371293/262144|tridecimal antidilemma]]: 371293/262144, laquintho 5th
* [[524288/371293|tridecimal dilemma]]: 524288/371293, saquinthu 4th
* [[1610510/1594323|fifthchromisma]]: 1610510/1594323, saquinlo-ayo

'''17 April 2025'''
* [[2001/2000|major discoverisma]]: 2001/2000, twenotwethotrigu
* [[2002/2001|minor discoverisma]]: 2002/2001, twenutwethutholozo

'''26 April 2025'''
* [[2465/2464|laservine comma]]: 2465/2464, twenosoluruyo

'''27 April 2025'''
* [[280/279|tricetone comma]]: 280/279, thiwuzoyo

'''2 May 2025'''
* [[6912/6859|deviaug comma]]: 6912/6859, trinu

'''3 May 2025'''
* [[93/92|tricema]]: 93/92, thiwotwethu
* [[155/154|scyllisma]]: 155/154, thiwoluruyo
* [[156/155|xanthippisma]]: 156/155, thiwuthogu
* [[187/186|lambertisma]]: 187/186, thiwusolo
* [[217/216|eudorisma]]: 217/216, thiwozo
* [[248/247|lameisma]]: 248/247, thiwonuthu
* [[341/340|californisma]]: 341/340, thiwosulogu
* [[342/341|endymisma]]: 342/341, thiwunolu
* [[435/434|chinthisma]]: 435/434, thiwutwenoruyo
* [[465/464|alektisma]]: 465/464, thiwotwenuyo
* [[528/527|rezisma]]: 528/527, thiwusulo
* [[589/588|croatisma]]: 589/588, thiwonoruru
* [[651/650|antiklisma]]: 651/650, thiwothuzogugu
* [[714/713|ululisma]]: 714/713, thiwutwethusozo

'''4 May 2025'''
* [[2755/2754|avicema]]: 2755/2754, twenonosuyo
* [[4641/4640|vinecute comma]]: 4641/4640, twenusothozogu
* [[5888/5887|vinocular comma]]: 5888/5887, bitwenu-atwethoru
* [[8671/8670|vinous comma]]: 8671/8670, twenotwethosusuthogu

'''9 May 2025'''
* [[6076/6075|large ricegrain comma]]: 6076/6075, sathiwobizogu
* [[6728/6727|sushi comma]]: 6728/6727, bithiwutweno-aru
* [[8960/8959|small ricegrain comma]]: 8960/8959, thiwususuzoyo

'''10 May 2025'''
* [[12636/12635|padriellisma]]: 12636/12635, nunuthorugu
* [[13690/13689|lesleymartin comma]]: 13690/13689, bithisothu-ayo
* [[16337/16335|brown rice comma]]: 16337/16335, bithiwo-asolulugu
* [[23275/23273|sunrise comma]]: 23275/23273, bithisu-anosubizoyo

'''18 May 2025'''
* [[navicular comma]]: 24192/24167, trithu-aluzo
* [[48013/48000|small tridevisemma]]: 48013/48000, trino-azotrigu
* [[93347/93312|trilute comma]]: 93347/93312, notriso
* [[116640/116603|large tridevisemma]]: 116640/116603, trinu-asuyo
* [[176000/175959|triseptichrome comma]]: 176000/175959, nulotriruyo
* [[471648/471625|small triquarterisma]]: 471648/471625, triso-alutrirugu
* [[1257795/1257728|large triquarterisma]]: 1257795/1257728, latrisulo-azoyo
* [[quinticular comma]]: 1419857/1417176, saquinso

'''28 May 2025'''
* [[7942/7935|brigade comma]]: 7942/7935, bitwethuno-alogu

'''29 May 2025'''
* [[barbadisma]]: 114383962274805 / 114349209288704, quadla-thuyo
* [[undevicesimal counterschisma]]: 2.3.19 {{monzo|-75 50 -1}}, seplanu

'''30 May 2025'''
* [[septendecimal quasithirdma]]: 19683/19652, latrisu
* [[septendecimal quasiminthma]]: 6975757441 / 6973568802, sasa-quadbiso

'''1 June 2025'''
* [[undevicesimal quasitonema]]: 34359738368 / 34296447249, sasa-tribinu
* [[deviquasisma]]: 17594077438737 / 17592186044416, lala-sepno

'''9 June 2025'''
* [[226/225|reversed marvel comma]]: 226/225, 113ogg2

'''5 July 2025'''
* [[386/385|siegenisma]]: 386/385, 193o1urg1
* [[57122/57121|phocitremisma]]: 57122/57121, 239uu3o42

===Proposed in 2026===
'''10 January 2026'''
* [[septimagic comma]]: 537824/531441, saquinzo
* [[acromagic comma]]: 26985857024 / 26904200625, sasa-sepzo-aquadgu

'''16 January 2026'''
* [[septrongisma]]: 205891132094649 / 205885750000000, lala-sepru-atritrigu

'''17 January 2026'''
* [[sensibeta comma]]: 1071875/1062882, satrizo-aquinyo

'''20 January 2026'''
* [[undetritisma]]: 205891132094649 / 204800000000000, trila-legu

'''31 January 2026'''
* [[naiadisma]]: 161414428 / 158203125, satritrizogu

'''23 February 2026'''
* [[septarium comma]]: {{monzo|73 77 -84}}, sasepquadtrigu

'''1 May 2026'''
* [[129140163/125000000|xenial comma]]: 129140163 / 125000000, latritrigu

== Scales which this user proposed names ==
This user proposed new names for following scales.

===Proposed in May 2023===
'''16 May 2023'''
* [[3L 16s|magicaltonic]]: 3L_16s MOS scale
* [[16L 3s|muggletonic]]: 16L_3s MOS scale
* [[Nicetone|zarlinoid]]: 3L_2m_2s rank-3 scale (MV3, LMLSLMS and LMLSMLS)
* [[sephipechroid]]: 3L_5m_5s rank-3 scale (MV3, LMSMLSMSLMSMS and LMSMLSMSMLSMS)
* [[magicaloid]]: 3L_8m_8s rank-3 scale (MV3, LMSMSMLSMSMSLMSMSMS and LMSMSMLSMSMSMLSMSMS)

'''18 May 2023'''
* [[Omnidiatonic|interdia]]: 2L_3m_2s rank-3 scale (MV3, LMSLMSM and LMSMLSM)
* [[antinicetone]]: 2L_2m_3s rank-3 scale (MV3, LMSLSMS and LSMLSMS)

'''20 May 2023'''
* [[villoid]]: 7L_2m_2s rank-3 scale (MV3, LLMLLSLLMLS and LLMLLSLMLLS)
* [[intervilla]]: 2L_7m_2s rank-3 scale (MV3, LMMSMLMMSMM and LMMSMMLMSMM)
* [[kleistoid]]: 2L_2m_7s rank-3 scale (MV3, LSMSSLSSMSS and LSSMSLSSMSS)
* [[interoneichro]]: 5L_3m_5s rank-3 scale (MV3, LMSLSLMSLSMLS and LMSLSMLSLSMLS)
* [[sephimechroid]]: 5L_5m_3s rank-3 scale (MV3, LMLMSLMLSMLMS and LMLSMLMLSMLMS)
* [[moharoid]]: 7L_5m_5s rank-3 scale (MV3, LMLSLMSLMLSMLSLMS and LMLSMLSLMLSMLSLMS)
* [[enharoid]]: 5L_7m_5s rank-3 scale (MV3, LMSLMSMLMSLMSMLSM and LMSLMSMLSMLMSMLSM)
* [[beatloid]]: 5L_5m_7s rank-3 scale (MV3, LMSLSMLSMSLMSLSMS and LMSLSMLSMSLSMLSMS)

'''21 May 2023'''
* [[Dualdye|archeoid]]: 5L_1m_1s rank-3 scale (MV3, LLLMLLS and LLLSLLM)
* [[interarcheo]]: 1L_5m_1s rank-3 scale (MV3, LMMMSMM and LMMSMMM)
* [[antarcheoid]]: 1L_1m_5s rank-3 scale (MV3, LSSMSSS and LSSSMSS)

===Proposed in July 2023===
'''2 July 2023'''
* [[jarachroid]]: 5L_5m_2s rank-3 scale (MV3, LMLMLSMLMLMS)
* [[Diachrome|interchroid]]: 5L_2m_5s rank-3 scale (MV3, LMSLSLSMLSLS)
* [[jerachroid]]: 2L_5m_5s rank-3 scale (MV3, LMSMSMLSMSMS)

== Documents ==
Notation:
* [[User:Xenllium/Xenllium's microtonal notation|Xenllium's microtonal notation]]
Tempered scales:
* [[User:Xenllium/Xenllium's circulating scales|Xenllium's circulating scales]]
External links:
* [https://docs.google.com/spreadsheets/d/18CRD8KRk9Gsg_tWSxbDJ_GjprgNvjGTx8ZWcpOEA-0I/edit?usp=sharing Rank-two regular temperament]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://drive.google.com/file/d/1Eov2FSp5Y4X4HPaaHV8nvqHX21LE7AQa/view?usp=drive_link 純正音程と平均律] (日本語/Japanese)
* [https://drive.google.com/file/d/1oBpqNqqIYCZaPuTFEt4HYdIULOMbX6sE/view?usp=drive_link 平均律 (2～31)] (日本語/Japanese)

== Music ==
* ''[https://youtube.com/watch?v=5iBXKRWE-_U Magical life]'' – in 19-tone pure-fifth magic scale
* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning
* [https://youtube.com/watch?v=bbJ0HjPAuaA 【オリジナル曲】かなしいこと / The sadness]

User:Xenllium

2026-07-08T08:42:28Z

Xenllium:

'''Xenllium''' joined Xenharmonic Wiki on 15 November 2018. The name "Xenllium" is a portmanteau of "Xenharmonic" and "''[[Wikipedia:Trillium|Trillium]]''".

== Temperaments which this user proposed names ==
This user proposed new names for following temperaments.

===Proposed in 2019===
'''March 2019'''
* [[Icositritonic]]
* [[Hemicocomp|Hemicounterpyth]] (→ Hemicocomp)
* [[Hemermacomp|Hemermapyth]] (→ Hemermacomp)
* [[Pentacontatritonic]]
* Hectaheptacontahenatonic (171 & 17442)
* [[Meridic]]
* Tetracontaheptatonic (94 & 705)
* [[Palladium]]
'''April 2019'''
* [[Septisemitonic]]
* [[Tertiathirds]]
* [[Tertiaschis]]

===Proposed in 2021===
'''March 2021'''
* [[Trident]] (→ Alphatrident)
* [[Trillium]] (→ Alphatrillium)
* [[Pseudotrillium]]
* [[Tritricot]]
* [[Decic]]
* [[Cloudtone]]
'''April 2021'''
* [[Hemipental]]
* [[Sengasec]]
* [[Qinto]]
* [[Monzism]]
* [[Houborizic]]
* [[Houbor]]
* [[Gamity]]
* [[Familia]]
* [[Sabric]]
* [[Quadrawell]]
* [[Sfour]]
* [[Rainwell]]
* [[Sengage]]
* [[Marfifths]]
* [[Diatessic]]
* [[Marf]]
'''May 2021'''
* [[Garistearn]]
* [[Decistearn]]
* [[Quadrant]]
* [[Quintilipyth]]
* [[Helenoid]]
* [[Ponta]]
* [[Pontic]]
* [[Bipont]]
* [[Counterbipont]]
* [[Quadrapont]]
* [[Decimetra]]
* [[Stockhausenic]]
* [[Decavish]]
* Bronzoid (109 & 142)
* [[Ennealimmia]]
* [[Paralytic]]
* [[Cotoneum]]
* [[Fog]]
* [[Countermiracle]]
* [[Paraklein]]
* [[Countermiraculous]]
* [[Counterbenediction]]
* [[Counterrevelation]]
* [[Septant]]
* [[Terminal]]
* [[Semiterm]]
* [[Sedia]]
* [[Marthirds]]
* [[Counterkleismic]]
* [[Counterlytic]]
* [[Geb]]
* [[Hemidimfi]]
* [[Maquiloid]]
* [[Paramity]]
'''June 2021'''
* [[Schisennealimmal]]
* [[Schisennealimmic]]
* [[Decades]] (→ Gamelstearn)
* [[Undecentic]]
* [[Pentadecal]]
* [[Quinkee]]
* [[Coditonic]]
* [[Terture]]
* [[Semivulture]]
* [[Eagle]]
* [[Quindromeda]]
* [[Quintakwai]]
* [[Quintagar]]
* [[Quintoneum]]
* [[Quinsandra]]
* [[Quinsandric]]
* [[Quinsandro]]
* [[Semiquindromeda]]
* [[Quintoneoid]]
'''July 2021'''
* [[Mineral]]
* [[Ore]]
* [[Quintaleap]]
* [[Quintupole]]
* [[Quinticosiennic]]
* [[Quintapole]]
* [[Hemiseptisix]]
'''August 2021'''
* [[Goldmine]]
* [[Gamelacomp|Gamelapyth]] (→ Gamelacomp)
* [[Mermacomp|Mermapyth]] (→ Mermacomp)
* [[Undesemi]]
* [[Decipentic]]
* [[Quasijerome]]
* [[Septendesemi]]
* [[Witcher]] (99 & 166, formerly ''witch'', revised in August 2023)
* [[Semiwitcher]] (166 & 198, formerly ''semiwitch'', revised in August 2023)
* [[Hitch]]
* [[Hitcher]]
* [[Quinwell]]
* [[Septiquarschis]]
* [[Quinbetter]]
* [[Ultramarine]]
* [[Mowglic]]
'''Septemper 2021'''
* [[Terminator]]
* [[Quintaschis]]
* [[Quintahelenic]]
* [[Quintahelenoid]]
* [[Semihemiwür]]
* [[Semihemiwürat]]
* [[Beetle]]
* [[Subsedia]]
* [[Quinkwai]]
* [[Quintakwoid]]
* [[Passionate]]
* [[Pash]]
* [[Majvamic]]
* [[Majvamoid]]
* [[Narayana]]
* [[Turkey]]
'''October 2021'''
* [[Lunennealimmal]]
* [[Hexadecoid]]
* [[Undetrita]]
* [[Rubidium]]
* [[Terbium]]
* [[Parakleischis]]
* [[Triacontaheptoid]]
* [[Octogintic]]
* [[Senator]]
* [[Hemivalentine]]
* [[Hemivalentino]]
* [[Hemivalentoid]]
* [[Undetritoid]]
* [[Biquartonic]]
* [[Quindro]]
'''November 2021'''
* [[Tridecatonic]]
* [[Trideci]]
* [[Decimaleap]]
* [[Hemiquindromeda]]
* [[Triboh]]
'''December 2021'''
* [[Hemimabila]]
* [[Sedecic]]
* [[Hyperkleismic]]

===Proposed in 2022===
'''January 2022'''
* [[Cobalt]]
* [[Cobaltous]]
* [[Cobaltic]]
* [[Cobaltite]]
* [[Pontoid]]
* [[Grackle]] (11, 13, 17 and 19-limit extension)
* [[Grackloid]]
* [[Grack]]
* [[Catahelenic]]
* [[Countriton]]
* [[Aufic]]
* [[Pajarina]]
* [[Muggloid]]
* [[Gracecord]]
* [[Gaster]]
* [[Tuskaloosa]]
* [[Muscogee]]
* [[Cohemimabila]]
* [[Trimabila]]
* [[Immune]]
* [[Meantritone]]
* [[Aurora]]
* [[Leapweeker]]
* [[Interaufo]]
* [[Helena]]
'''February 2022'''
* [[Cohendecatonic]]
* [[Nessa]]
* [[Splendecic]]
* [[Prodecic]]
* [[Semimeantone]]
* [[Keenic]]
* [[Warrior]]
* [[Trisensory]]
* [[Bisensi]]
* [[Browser]]
* [[Grazer]]
* [[Quadrasruta]]
* [[Quadrafourths]]
* [[Pajarita]]
'''March 2022'''
* [[Stalagmite|Catamite]] (→ Stalagmite)
* [[Hemikwai]]
* [[Yarm]]
* [[Cohedim]]
* [[October]]
* [[Dichotomic]]
* [[Seminar]]
* [[Chromat]] (11, 13 and 17-limit extension)
* [[Catachrome]]
* [[Chromic]]
* [[Trinity]]
* [[Poseidon]]
* [[Salacia]]
* [[Gammy]]
* [[Tertiaseptia]]
* [[Semitert]]
'''April 2022'''
* <s>[[Meantonic]]</s>
* <s>[[Meantoid]]</s>
* [[Huygens]] (→ Fokkertone)
* <s>[[Meridetonic]]</s>
* <s>[[Meridetoid]]</s>
* <s>[[Sauveuric]]</s>
* [[Meanpoid]]
* <s>[[Meanplop]] (17 and 19-limit extension)</s>
* <s>[[Meanploid]]</s>
* <s>[[Meanenneadecoid]]</s>
* <s>[[Vincenzoid]]</s>
* [[Andromeda]] (17 and 19-limit extension)
* [[Schisicosiennic]]
* [[Schisicosiennoid]]
* [[Cassandra]] (Cassie)
* [[Cassandric]]
* [[Cassander]]
'''May 2022'''
* [[Supersensi]]
* [[Quadrafives]]
'''June 2022'''
* [[Trimean]]

===Proposed in 2023===
'''June 2023'''
* [[Galileic]]
* [[Catagali]]
* [[Quintain]]
'''July 2023'''
* [[Devisemi]]
* [[Garibaldia]] (2.3.5.7.17.19 extension of [[garibaldi]])
* [[Pontia]] (2.3.5.7.19 extension of [[pontiac]])
'''August 2023'''
* [[Heartland]] (2.3.7.11.19 14&41& 48)
* [[Hearts]] (2.3.7.11.19 41&48)
* [[Heartia]]
* [[Heartful]] (2.3.7.11.19 41&75e)
* [[Doublehearted]] (2.3.7.11.19 14&41)
* [[Weasel|Byhearted]] (→ Weasel)
'''September 2023'''
* [[Heartlandia]] (14cf & 41 & 89)
* [[Darkstone]]
* [[Brightstone]]
'''October 2023'''
* [[Magical]]
* [[Magica]]
* [[Magia]]
* [[Sesquartia]]
* [[Charismic]]
* [[Quinmage]]
* [[Hearty]]
* [[Nonant]]
* [[Hemiandromeda]] (29 & 82)
* [[Hemidromeda]] (29 & 111)
'''November 2023'''
* [[Trismegiboh]]
* [[Baldanders]] (2.9.5.7.11 6&35)
* [[Neomirkwai]] (2.3.5.7.11.19 31&41&80)
* [[Dhaivatic]]
* [[Triquart]]
* [[Tertiseptisix]]
* [[Quartiquart]]
* [[Quintiquart]]
* [[Neovulture]] (2.3.5.7.11.13.19 53&58&217)
* [[Neovish]] (2.3.5.7.11.19 50&118&152)
* [[Neobreed]] (41 & 130 & 270)
* [[Neonewt]] (2.3.5.7.11.13.19 41&270)
'''December 2023'''
* [[Heartlander]] (14cf & 41 & 130)
* [[Neosatanic]] (665 & 3325)
* [[Monzosatanic]] (665 & 7980)
* [[Dog]] (2.3.19 7&9)
* [[Armodog]] (2.3.5.7.11.13.19 7&9)

===Proposed in 2024===
'''January 2024'''
* [[Quinbaldi]] (65d & 135)
* [[Quintant]] (65d & 200)
* [[Hemiquintant]] (130 & 135)

'''May 2024'''
* [[Raccoon]] (11 & 32)

'''June 2024'''
* [[Piracy]] (84 & 155)
* [[Pirateship]] (84 & 239)
* [[Unisquary]] (11-limit 72&118&121&130)
* [[Hensquary]] (13-limit 46&72&121&130)
* [[Ekasquary]] (13-limit 72&121&130&190)

===Proposed in 2025===
'''January 2025'''
* [[Pentadecoid]] (2.3.7.11 120&135)
* [[Slendscape]] (255 & 270)
* [[Triacontoid]] (120 & 150)

'''February 2025'''
* [[Devichrome]] (2.3.5.7.19 27&31&41&53)

'''March 2025'''
* [[Octasand]] (11 and 13-limit 27e&41&53)
* [[Devimarv]] (2.3.5.7.19 31&41&53)
** [[Neomarv]] (2.3.5.7.11.19 31&41&53)
* [[Deva]] (2.3.5.7.19 27&41&53)
** [[Neonakika]] (2.3.5.7.11.19 15&27e&41)
** [[Neoctasand]] (2.3.5.7.11.19 27e&41&53)
* [[Nutrient]] (2.3.5.7.19 27&31&41)
** [[Neojovia]] (2.3.5.7.11.19 27e&31&41)
* [[Passkey]] (2.3.5.7.19 41&53&99)
** [[Neolakia]] (2.3.5.7.11.19 41&53&99)

'''April 2025'''
* [[Magikarp]] (2.3.43 5&7)
* [[Hendeca]] (11&22)
* [[Char]] (2.3.5.7.17 19&31&41)
* [[Elven]]

'''June 2025'''
* [[Reversed marvel]] (2.3.5.113 22&31&34)

===Proposed in 2026===
'''January 2026'''
* [[Mendel]] (11-limit 12&31&152)
* [[Moctpine]] (11-limit 29&72&80)
* [[Biscordial]] (12&190)
* [[Triscordial]] (12&231)
* [[Naiad]]
* [[Naiadical]]

'''February 2026'''
* [[Naian]] (8d&13b)
* [[Semihemimean]] (80&118&130)
* [[Marvelpine]]
* [[Seminaiad]]
** [[Seminaiad|Seminaiadia]]
* <s>[[Seesaw]]</s> (1c & 2)
** <s>[[Heavy windmill]]</s> (1bd & 2)
** <s>[[Light windmill]]</s> (1ce & 2)
* [[Antonian]]
* [[Antonym (temperament)|Antonym]]
* [[Antony]]
* [[Argument (temperament)|Argument]]
* [[Landy]]
* [[Septarium]]

'''March 2026'''
* [[Augenic]]
* [[Sida]]
* [[Bohlenic]] (no-twos temperament)

'''May 2026'''
* [[Xenial]] (19 & 70)
** [[Xenia]] (19 & 51c, 19 & 70d)
* [[Hemigoth]] (34d & 68)
* [[Paraguay]] (19 & 23bcf, 19 & 42ef)
** [[Uruguay]] (19 & 23bc, 19 & 42e)

'''June 2026'''
* <s>[[Sixseven]]</s> (2.3.7 2d & 3)
** <s>[[Heaven]]</s> (2.3.7.11 2de & 3)

== Commas which this user proposed names ==
This user proposed new names for following commas.

===Proposed in 2021===
'''April 2021'''
* [[Hewuermera temperaments|hewuermera]]: 589824/588245, satribiru-agu
* [[Triwellismic temperaments|triwellisma]]: 235298/234375, tribizo-asepgu
* [[quadrawellisma]]: 67528125/67108864, lala-quadzo-aquinyo
* [[catasyc comma]]: 390625/387072, laruquadbiyo
* [[Mistismic temperaments|mistisma]]: 458752/455625, sazoquadgu

'''May 2021'''
* [[Hemfiness temperaments|hemfiness comma]]: 4096000/4084101, saquinru-atriyo
* [[Marvelfifths temperaments|marvelfifths comma]]: 158203125/157351936, laquadru-atritriyo
* [[Chromatisma|chromatisma]]: 640000000000000000 / 635585924776181463, trisa-triru-aquadquadyo
* [[Bronzismic temperaments|bronzisma]]: 2097152/2083725, satriru-agugu
* <s>[[Breeze comma|sesquiquartisma]]</s>: 2460375/2458624, laquadru-atruyo (→ breezema, breeze comma)
* [[Mitonismic temperaments|mitonisma]]: 5250987/5242880, laquadzo-agu
* [[Sensamagicfifths temperaments|sensamagicfifths comma]]: {{monzo|-11 42 -7 -14}}, trila-seprurugu
* [[Secanticorn temperaments|secanticornisma]]: 177147/175000, laruquingu
* [[Supermatertismic temperaments|supermatertisma]]: 52734375/52706752, lasepru-atritriyo
* [[Supermasesquartismic temperaments|supermasesquartisma]]: 184528125/184473632, laquadbiru-aquinyo
* [[Septimirod temperaments|septimirod comma]]: 84035/82944, quinzo-ayo

'''June 2021'''
* [[Vorwell clan|vorwell comma]]: 134217728/133984375, sasatriru-aquadbigu
* [[Maitri clan|maitri comma]]: 40353607/40000000, tritrizo-asepgu
* [[compass comma]]: 9765625/9680832, quinruyoyo

'''July 2021'''
* [[4375/4356|fantares comma]]: 4375/4356, luluzoquadyo
* [[Septiness clan|septiness comma]]: 67108864/66706983, sasasepru
* [[decovulture comma]]: 67108864/66976875, sasa-birugugu
* [[Pontiqak temperaments|pontiqak comma]]: 95703125/95551488, lazozotritriyo

'''August 2021'''
* [[Slither clan|slither comma]]: 40960000/40353607, satritriru-aquadyo
* [[Mermismic temperaments|mermisma]]: 2500000/2470629, sepruyo
* [[Valenwuer temperaments|valenwuer comma]]: 110592/109375, sarutribigu
* [[Negricorn temperaments|negricorn comma]]: 153664/151875, saquadzogu
* [[Parkleiness temperaments|parkleiness comma]]: 1959552/1953125, zotritrigu
* [[Mynaslendic temperaments|mynaslender comma]]: 829440/823543, sepru-ayo

'''September 2021'''
* [[65536/64827|buzzardisma]]: 65536/64827, saquadru (→ buzzardsma, buzzard comma)
* [[Neptunismic temperaments|neptunisma]]: 48828125/48771072, laruruleyo

'''October 2021'''
* [[Domicomp temperaments|domicomp comma]]: 413343/409600, lazogugu

'''November 2021'''
* [[Trienstonisma|trienstonisma]]: 43046721/43025920, laquinru-agu
* [[Lokismic temperaments|lokisma]]: 102760448/102515625, sasazozo-atribigu

===Proposed in 2022===
'''January 2022'''
* [[Countriton comma|countriton comma]]: 16777216000000000 / 16677181699666569, quadsa-tritriyo
* [[3042/3025|diagassormisma]]: 3042/3025, bitholugu
* [[gaster comma]]: {{monzo|-70 72 -19}}, quadbila-negu
* [[Mabilismic clan|mabilisma]]: 1071875/1048576, latrizo-aquinyo
* [[pentonisma]]: 281974669312/281950621875, saquinthuzogu

'''February 2022'''
* [[8281/8192|diahuntmisma]]: 8281/8192, labithozo
* [[3645/3584|schismean comma]]: 3645/3584, laruyo

'''March 2022'''
* [[Acrodec temperaments|acrodec comma]]: 7710244864/7688671875, sasa-tribizo-aquadbigu
* [[120/119|lynchisma]]: 120/119, suruyo

'''May 2022'''
* [[Augmist temperaments|augmist comma]]: 1438646272/1423828125, sasa-trizotrigu
* [[Starling temperaments|supersensi comma]]: 33059881728/30517578125, quintrigu

===Proposed in 2023===
'''11 May 2023'''
* [[Syntonoschisma|syntonisma]]: {{monzo|30 -27 7 0 -1}}, trisa-lusepyo (→ syntonoschisma, syntonic schisma)

'''27 May 2023'''
* [[170/169|major naiadma]]: 170/169, sothuthuyo
* [[221/220|minor naiadma]]: 221/220, sotholugu
* <s>[[256/255|diasemisma]]</s>: 256/255, sugu (→ charisma, charic comma)
* [[442/441|seminaiadma]]: 442/441, sothoruru

'''18 July 2023'''
* [[grossmisma]]: {{monzo|-30 -13 -1 1 7 7}}, septholo-azogu

'''21 July 2023'''
* [[3971/3969|heartlandisma]]: 3971/3969, nonoloruru

'''23 July 2023'''
* [[171/170|malcolmisma]]: 171/170, nosugu
* [[190/189|cotylisma]]: 190/189, noruyo
* [[324/323|photisma]]: 324/323, nusu
* [[343/342|nutrisma]]: 343/342, nutrizo
* [[400/399|devichroma]]: 400/399, nuruyoyo
* [[devicisma]]: 633556/633555, nunusothutrilo-azogu

'''15 August 2023'''
* [[133/132|minithirdma]]: 133/132, noluzo
* <s>[[136/135|diatonisma]]</s>: 136/135, sogu (→ diatisma)
* [[154/153|augustma]]: 154/153, sulozo
* [[161/160|major kirnbergerisma]]: 161/160, twethozogu
* [[162/161|minor kirnbergerisma]]: 162/161, twethuru
* [[286/285|chthonisma]]: 286/285, nuthologu
* [[1275/1274|cimbrisma]]: 1275/1274, sothubiruyo
* [[galileisma]]: 171382426877952 / 171359481538165, lalesu-agu

'''16 August 2023'''
* [[85/84|monk comma]]: 85/84, soruyo
* [[456/455|abnobisma]]: 456/455, nothulugu
* [[476/475|hedwigma]]: 476/475, nusozogugu
* [[495/494|eulalisma]]: 495/494, nuthuloyo

'''21 August 2023'''
* [[363/361|godzillisma]]: 363/361, binulo
* [[2431/2430|heptacircle comma]]: 2431/2430, sothologu

'''26 August 2023'''
* [[1083/1078|bihendrixma]]: 1083/1078, nonolururu

'''27 August 2023'''
* [[No-fives subgroup temperaments #Doublehearted|doublehearted comma]]: 5764801/5668704, quadbizo
* [[No-fives subgroup temperaments #Hearts|hearts comma]]: 34451725707/34359738368, trila-quadzo

'''17 September 2023'''
* [[4914/4913|baladisma]]: 4914/4913, trisu-athozo

'''5 November 2023'''
* [[969/968|buddhisma]]: 969/968, nosolulu (→ kingfisher comma)
* [[1331/1330|solvejgsma]]: 1331/1330, nutrilo-arugu
* [[1540/1539|kevolisma]]: 1540/1539, nulozoyo
* [[2376/2375|trichthonisma]]: 2376/2375, nulotrigu
* [[2926/2925|neovulturisma]]: 2926/2925, nothulozogugu
* [[3136/3135|neomirkwaisma]]: 3136/3135, nuluzozogu
* [[3250/3249|martebisma]]: 3250/3249, nunuthotriyo
* [[4200/4199|neosatanisma]]: 4200/4199, nusuthuzoyoyo
* [[5776/5775|neovishma]]: 5776/5775, nonolurugugu
* [[5985/5984|neogrendelisma]]: 5985/5984, nosuluzoyo
* [[6175/6174|neonewtisma]]: 6175/6174, nothotriru-ayoyo

'''12 November 2023'''
* [[460/459|scanisma]]: 460/459, twethosuyo
* [[484/483|pittsburghisma]]: 484/483, twethuloloru
* [[507/506|laodicisma]]: 507/506, twethuthotholu
* [[529/528|preziosisma]]: 529/528, bitwetho-alu
* [[736/735|harvardisma]]: 736/735, twethorurugu
* [[760/759|squadronisma]]: 760/759, twethunoluyo
* [[875/874|nymphisma]]: 875/874, twethunuzotriyo
* [[897/896|lysistratisma]]: 897/896, twethothoru
* [[1105/1104|fragarisma]]: 1105/1104, twethusothoyo
* [[1197/1196|rhodesisma]]: 1197/1196, twethunothuzo
* [[1288/1287|santisma]]: 1288/1287, twethothuluzo
* [[1496/1495|turkisma]]: 1496/1495, twethusothulogu
* [[1863/1862|antinousisma]]: 1863/1862, twethonururu
* [[2024/2023|insincere comma]]: 2024/2023, twethosusuloru
* <s>[[2025/2024|fokkerisma]]</s>: 2025/2024, latwethuluyoyo (→ cupcakesma, cupcake comma)
* [[2185/2184|guangdongisma]]: 2185/2184, twethonothuruyo
* [[2300/2299|travellisma]]: 2300/2299, twethonubiluyo
* [[2646/2645|biyativice comma]]: 2646/2645, bitwethuzo-agu
* [[2737/2736|kotkisma]]: 2737/2736, twethonusozo
* [[3060/3059|vicious comma]]: 3060/3059, twethunusoruyo
* [[3381/3380|seminaiadvice comma]]: 3381/3380, twethobithuzo-agu (→ mikkolisma)
* [[3520/3519|vicedim comma]]: 3520/3519, twethusuloyo
* [[3888/3887|vicetride comma]]: 3888/3887, twethuthuthu
* [[4693/4692|viceaug comma]]: 4693/4692, twethunonosutho
* [[4761/4760|demiquartervice comma]]: 4761/4760, bitwetho-asurugu
* [[vicetertisma]]: 12168/12167, tritwethu-athotho

'''13 November 2023'''
* [[208/207|vicetone comma]]: 208/207, twethutho
* [[231/230|major neutravicema]]: 231/230, twethulozogu
* [[253/252|middle neutravicema]]: 253/252, twetholoru
* [[276/275|minor neutravicema]]: 276/275, twetholugugu
* [[300/299|major naiadvicema]]: 300/299, twethuthuyoyo
* [[323/322|major semivicema]]: 323/322, twethunosoru
* [[391/390|minor naiadvicema]]: 391/390, twethosothugu
* [[392/391|minor semivicema]]: 392/391, twethusuzozo
* [[460/459|vicewolf comma]]: 460/459, twethosuyo
* [[5083/5082|broadviewsma]]: 5083/5082, twethosotholuluru

'''15 November 2023'''
* [[92/91|undinisma]]: 92/91, twethothuru
* [[115/114|yarmanisma]]: 115/114, twethonuyo

'''26 November 2023'''
* [[E-shaped comma]]: 31213/31212, susuthoquadzo

'''1 December 2023'''
* <s>[[kilobyte comma]]</s>: 1024/1023, thiwulu (→ kibisma)
* <s>[[megabyte comma]]</s>: 1048576/1048575, safowuthiwulugugu (→ mebisma)

'''10 December 2023'''
* [[81/76|dog comma]]: 81/76, inu
* [[primate comma]]: 16384/15309, saru

'''23 December 2023'''
* [[841/840|arabellisma]]: 841/840, bitweno-arugu
* [[4901/4900|large grapevine comma]]: 4901/4900, twenothothobirugu, twenobithorugu
* [[6877/6875|grossvice comma]]: 6877/6875, bitwetho-atholuquadgu
* [[7425/7424|small grapevine comma]]: 7425/7424, latwenuloyoyo
* [[mynucuvine comma]]: 10935/10933, labitwenu-athuyo

===Proposed in 2024===
'''2 January 2024'''
* <s>[[slendroschisma|slendrisma]]</s>: 68719476736/68641485507, sasa-quinbiru (→ slendroschisma)

'''27 January 2024'''
* [[116/115|sironisma]]: 116/115, twenotwethugu
* [[117/116|lomisma]]: 117/116, twenutho
* [[175/174|major chthonovinema]]: 175/174, twenuzoyoyo
* [[204/203|kallistisma]]: 204/203, twenusoru
* [[232/231|major paravinema]]: 232/231, twenoluru
* [[261/260|major vinetonema]]: 261/260, twenothugu
* [[290/289|brunisma]]: 290/289, twenosusuyo
* [[320/319|minor paravinema]]: 320/319, twenuluyo
* [[378/377|major semivinema]]: 378/377, twenuthuzo
* [[406/405|minor semivinema]]: 406/405, twenozogu
* [[494/493|minor vinetonema]]: 494/493, twenunosutho
* [[551/550|minor chthonovinema]]: 551/550, twenonolugugu
* [[609/608|vineyard comma]]: 609/608, twenonuzo

'''3 May 2024'''
* [[729/722|undevicesimal diaschisma]]: 729/722, lanunu

'''1 June 2024'''
* [[unisquarisma]]: 12005/11979, trilu-aquadzo-ayo

'''22 December 2024'''
* [[429/425|middle semisixthma]]: 429/425, suthologugu
* [[507/500|large semisixthma]]: 507/500, thothotrigu
* [[1452/1445|small semisixthma]]: 1452/1445, susulologu
* [[semisixthmisma]]: 63888/63869, trisu-athutrilo

===Proposed in 2025===
'''20 March 2025'''
* [[6860/6859|devicubisma]]: 6860/6859, trinuzo-ayo
* [[tricecubisma]]: 29792/29791, trithiwu-anozozo

'''21 March 2025'''
* [[8625/8624|beerglass comma]]: 8625/8624, twetholururutriyo
* [[10626/10625|demiglace comma]]: 10626/10625, twethosulozoquadgu

'''29 March 2025'''
* [[145/144|adeonisma]]: 145/144, twenoyo
* [[552/551|sigelindisma]]: 552/551, twenutwethonu
* [[638/637|moirisma]]: 638/637, twenothuloruru
* [[726/725|joellisma]]: 726/725, twenubilogu
* [[783/782|norisma]]: 783/782, twenotwethusu
* [[1015/1014|christisma]]: 1015/1014, twenothuthuzoyo
* [[1045/1044|michelisma]]: 1045/1044, twenunoloyo
* [[1276/1275|ucclisma]]: 1276/1275, twenosulogugu
* [[1450/1449|raimondisma]]: 1450/1449, twenotwethuruyoyo
* [[1596/1595|itzigsohnisma]]: 1596/1595, twenunoluzogu
* [[1625/1624|norcma]]: 1625/1624, twenuthorutriyo
* [[1683/1682|castafiorisma]]: 1683/1682, bitwenu-asolo
* [[2176/2175|donarisma]]: 2176/2175, twenusogugu
* [[2205/2204|glinkisma]]: 2205/2204, twenunuzozoyo
* [[2262/2261|mitidikisma]]: 2262/2261, twenonusuthoru
* [[5929/5928|manzanisma]]: 5929/5928, nuthubilozo

'''6 April 2025'''
* [[6144/6137|langwisma]]: 6144/6137, nunusu

'''10 April 2025'''
* <s>[[129/128|Magikarp comma]]</s>: 129/128, fotho
* <s>[[130/129|Gyarados comma]]</s>: 130/129, fothuthoyo
* [[2640/2639|hällströmisma]]: 2640/2639, twenuthuloruyo
* [[2784/2783|domeykisma]]: 2784/2783, twenotwethululu
* [[3249/3248|musashinisma]]: 3249/3248, twenunonoru
* [[3451/3450|mentorisma]]: 3451/3450, twenotwethusozogugu
* [[4785/4784|petrovsma]]: 4785/4784, twenotwethuthuloyo
* [[5916/5915|woudisma]]: 5916/5915, twenosothuthurugu

'''11 April 2025'''
* [[10557/10556|rowlandisma]]: 10557/10556, twenutwethosothuru

'''15 April 2025'''
* [[371293/262144|tridecimal antidilemma]]: 371293/262144, laquintho 5th
* [[524288/371293|tridecimal dilemma]]: 524288/371293, saquinthu 4th
* [[1610510/1594323|fifthchromisma]]: 1610510/1594323, saquinlo-ayo

'''17 April 2025'''
* [[2001/2000|major discoverisma]]: 2001/2000, twenotwethotrigu
* [[2002/2001|minor discoverisma]]: 2002/2001, twenutwethutholozo

'''26 April 2025'''
* [[2465/2464|laservine comma]]: 2465/2464, twenosoluruyo

'''27 April 2025'''
* [[280/279|tricetone comma]]: 280/279, thiwuzoyo

'''2 May 2025'''
* [[6912/6859|deviaug comma]]: 6912/6859, trinu

'''3 May 2025'''
* [[93/92|tricema]]: 93/92, thiwotwethu
* [[155/154|scyllisma]]: 155/154, thiwoluruyo
* [[156/155|xanthippisma]]: 156/155, thiwuthogu
* [[187/186|lambertisma]]: 187/186, thiwusolo
* [[217/216|eudorisma]]: 217/216, thiwozo
* [[248/247|lameisma]]: 248/247, thiwonuthu
* [[341/340|californisma]]: 341/340, thiwosulogu
* [[342/341|endymisma]]: 342/341, thiwunolu
* [[435/434|chinthisma]]: 435/434, thiwutwenoruyo
* [[465/464|alektisma]]: 465/464, thiwotwenuyo
* [[528/527|rezisma]]: 528/527, thiwusulo
* [[589/588|croatisma]]: 589/588, thiwonoruru
* [[651/650|antiklisma]]: 651/650, thiwothuzogugu
* [[714/713|ululisma]]: 714/713, thiwutwethusozo

'''4 May 2025'''
* [[2755/2754|avicema]]: 2755/2754, twenonosuyo
* [[4641/4640|vinecute comma]]: 4641/4640, twenusothozogu
* [[5888/5887|vinocular comma]]: 5888/5887, bitwenu-atwethoru
* [[8671/8670|vinous comma]]: 8671/8670, twenotwethosusuthogu

'''9 May 2025'''
* [[6076/6075|large ricegrain comma]]: 6076/6075, sathiwobizogu
* [[6728/6727|sushi comma]]: 6728/6727, bithiwutweno-aru
* [[8960/8959|small ricegrain comma]]: 8960/8959, thiwususuzoyo

'''10 May 2025'''
* [[12636/12635|padriellisma]]: 12636/12635, nunuthorugu
* [[13690/13689|lesleymartin comma]]: 13690/13689, bithisothu-ayo
* [[16337/16335|brown rice comma]]: 16337/16335, bithiwo-asolulugu
* [[23275/23273|sunrise comma]]: 23275/23273, bithisu-anosubizoyo

'''18 May 2025'''
* [[navicular comma]]: 24192/24167, trithu-aluzo
* [[48013/48000|small tridevisemma]]: 48013/48000, trino-azotrigu
* [[93347/93312|trilute comma]]: 93347/93312, notriso
* [[116640/116603|large tridevisemma]]: 116640/116603, trinu-asuyo
* [[176000/175959|triseptichrome comma]]: 176000/175959, nulotriruyo
* [[471648/471625|small triquarterisma]]: 471648/471625, triso-alutrirugu
* [[1257795/1257728|large triquarterisma]]: 1257795/1257728, latrisulo-azoyo
* [[quinticular comma]]: 1419857/1417176, saquinso

'''28 May 2025'''
* [[7942/7935|brigade comma]]: 7942/7935, bitwethuno-alogu

'''29 May 2025'''
* [[barbadisma]]: 114383962274805 / 114349209288704, quadla-thuyo
* [[undevicesimal counterschisma]]: 2.3.19 {{monzo|-75 50 -1}}, seplanu

'''30 May 2025'''
* [[septendecimal quasithirdma]]: 19683/19652, latrisu
* [[septendecimal quasiminthma]]: 6975757441 / 6973568802, sasa-quadbiso

'''1 June 2025'''
* [[undevicesimal quasitonema]]: 34359738368 / 34296447249, sasa-tribinu
* [[deviquasisma]]: 17594077438737 / 17592186044416, lala-sepno

'''9 June 2025'''
* [[226/225|reversed marvel comma]]: 226/225, 113ogg2

'''5 July 2025'''
* [[386/385|siegenisma]]: 386/385, 193o1urg1
* [[57122/57121|phocitremisma]]: 57122/57121, 239uu3o42

===Proposed in 2026===
'''10 January 2026'''
* [[septimagic comma]]: 537824/531441, saquinzo
* [[acromagic comma]]: 26985857024 / 26904200625, sasa-sepzo-aquadgu

'''16 January 2026'''
* [[septrongisma]]: 205891132094649 / 205885750000000, lala-sepru-atritrigu

'''17 January 2026'''
* [[sensibeta comma]]: 1071875/1062882, satrizo-aquinyo

'''20 January 2026'''
* [[undetritisma]]: 205891132094649 / 204800000000000, trila-legu

'''31 January 2026'''
* [[naiadisma]]: 161414428 / 158203125, satritrizogu

'''23 February 2026'''
* [[septarium comma]]: {{monzo|73 77 -84}}, sasepquadtrigu

'''1 May 2026'''
* [[129140163/125000000|xenial comma]]: 129140163 / 125000000, latritrigu

== Scales which this user proposed names ==
This user proposed new names for following scales.

===Proposed in May 2023===
'''16 May 2023'''
* [[3L 16s|magicaltonic]]: 3L_16s MOS scale
* [[16L 3s|muggletonic]]: 16L_3s MOS scale
* [[Nicetone|zarlinoid]]: 3L_2m_2s rank-3 scale (MV3, LMLSLMS and LMLSMLS)
* [[sephipechroid]]: 3L_5m_5s rank-3 scale (MV3, LMSMLSMSLMSMS and LMSMLSMSMLSMS)
* [[magicaloid]]: 3L_8m_8s rank-3 scale (MV3, LMSMSMLSMSMSLMSMSMS and LMSMSMLSMSMSMLSMSMS)

'''18 May 2023'''
* [[Omnidiatonic|interdia]]: 2L_3m_2s rank-3 scale (MV3, LMSLMSM and LMSMLSM)
* [[antinicetone]]: 2L_2m_3s rank-3 scale (MV3, LMSLSMS and LSMLSMS)

'''20 May 2023'''
* [[villoid]]: 7L_2m_2s rank-3 scale (MV3, LLMLLSLLMLS and LLMLLSLMLLS)
* [[intervilla]]: 2L_7m_2s rank-3 scale (MV3, LMMSMLMMSMM and LMMSMMLMSMM)
* [[kleistoid]]: 2L_2m_7s rank-3 scale (MV3, LSMSSLSSMSS and LSSMSLSSMSS)
* [[interoneichro]]: 5L_3m_5s rank-3 scale (MV3, LMSLSLMSLSMLS and LMSLSMLSLSMLS)
* [[sephimechroid]]: 5L_5m_3s rank-3 scale (MV3, LMLMSLMLSMLMS and LMLSMLMLSMLMS)
* [[moharoid]]: 7L_5m_5s rank-3 scale (MV3, LMLSLMSLMLSMLSLMS and LMLSMLSLMLSMLSLMS)
* [[enharoid]]: 5L_7m_5s rank-3 scale (MV3, LMSLMSMLMSLMSMLSM and LMSLMSMLSMLMSMLSM)
* [[beatloid]]: 5L_5m_7s rank-3 scale (MV3, LMSLSMLSMSLMSLSMS and LMSLSMLSMSLSMLSMS)

'''21 May 2023'''
* [[Dualdye|archeoid]]: 5L_1m_1s rank-3 scale (MV3, LLLMLLS and LLLSLLM)
* [[interarcheo]]: 1L_5m_1s rank-3 scale (MV3, LMMMSMM and LMMSMMM)
* [[antarcheoid]]: 1L_1m_5s rank-3 scale (MV3, LSSMSSS and LSSSMSS)

===Proposed in July 2023===
'''2 July 2023'''
* [[jarachroid]]: 5L_5m_2s rank-3 scale (MV3, LMLMLSMLMLMS)
* [[Diachrome|interchroid]]: 5L_2m_5s rank-3 scale (MV3, LMSLSLSMLSLS)
* [[jerachroid]]: 2L_5m_5s rank-3 scale (MV3, LMSMSMLSMSMS)

== Documents ==
Notation:
* [[User:Xenllium/Xenllium's microtonal notation|Xenllium's microtonal notation]]
Tempered scales:
* [[User:Xenllium/Xenllium's circulating scales|Xenllium's circulating scales]]
External links:
* [https://docs.google.com/spreadsheets/d/18CRD8KRk9Gsg_tWSxbDJ_GjprgNvjGTx8ZWcpOEA-0I/edit?usp=sharing Rank-two regular temperament]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://drive.google.com/file/d/1Eov2FSp5Y4X4HPaaHV8nvqHX21LE7AQa/view?usp=drive_link 純正音程と平均律] (日本語/Japanese)
* [https://drive.google.com/file/d/1oBpqNqqIYCZaPuTFEt4HYdIULOMbX6sE/view?usp=drive_link 平均律 (2～31)] (日本語/Japanese)

== Music ==
* ''[https://youtube.com/watch?v=5iBXKRWE-_U Magical life]'' – in 19-tone pure-fifth magic scale
* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning
* [https://youtube.com/watch?v=bbJ0HjPAuaA 【オリジナル曲】かなしいこと / The sadness]

2nd-octave temperaments

2026-07-04T02:40:39Z

Xenllium: Redirected page to 2nd- to 4th-octave temperaments

#redirect [[2nd- to 4th-octave temperaments]]

33rd-octave temperaments

2026-07-04T02:39:25Z

Xenllium: Created page with "{{Infobox fractional-octave|33}} 33edo might not be immediately recognizable for fractional-octave potential, but its step is in direct proximity to 48/47. == Arsenic == Arsenic is named after the 33rd chemical element. Conceptualized as a no-threes subgroup temperament. Subgroup: 2.11.13.17.19 Comma list: 2432/2431, 7086244/7054567, 33698267/33554432 {{Mapping|legend=2| 33 0 8 249 26 | 0 1 1 -1 1 }} : mapping generators: ~247/242, ~11 Optimal tun..."

{{Infobox fractional-octave|33}}
[[33edo]] might not be immediately recognizable for fractional-octave potential, but its step is in direct proximity to [[48/47]].

== Arsenic ==
Arsenic is named after the 33rd chemical element. Conceptualized as a no-threes subgroup temperament.

[[Subgroup]]: 2.11.13.17.19

[[Comma list]]: 2432/2431, 7086244/7054567, 33698267/33554432

{{Mapping|legend=2| 33 0 8 249 26 | 0 1 1 -1 1 }}

: mapping generators: ~247/242, ~11

[[Optimal tuning]]s:
* [[CTE]]: ~247/242 = 36.3636¢ (1 ⧵ 33), ~11/8 = 550.632¢
* [[CWE]]: ~247/242 = 36.3636¢ (1 ⧵ 33), ~11/8 = 550.419¢

[[Optimal ET sequence]]: {{EDOs| 33, 132fg, 165, 198, 231 }}

[[Badness]] (Sintel): 2.300

{{Navbox fractional-octave}}

2nd- to 4th-octave temperaments

2026-07-04T02:13:02Z

Xenllium:

[[2edo|2-]], [[3edo|3-]] and [[4edo]] are much less used as scales, rather as chords. In many [[5L 2s|diatonic-based]] [[interval region]] schemes:
* One step of 2edo is known as a tritone.
* One step of 3edo is known as a major third, and the stacking of them is the augmented triad.
* One step of 4edo is known as a minor third, and the stacking of them is the diminished seventh chord.

Usage of [[7/5]] or [[10/7]] as one step of 2edo by tempering out [[50/49]], and therefore using 2edo as a tritone is one of the features of standard Western music theory, and is supported by [[12edo]]. See [[Jubilismic clan]] or [[Jubilismic family]] for a collection of such temperaments. Other interpretations as one step of 2edo include [[Diaschismic family|diaschismic]] (tempering out [[2048/2025]]), [[semitonismic]] (tempering out [[289/288]]), and [[kalismic]] (tempering out [[9801/9800]]).

Usage of the [[5/4]] major third as one step of 3edo by tempering out [[128/125]], and therefore using 3edo as an augmented triad produced by stacking two major thirds is one of the features of standard Western music theory, and is supported by [[12edo]]. See [[Augmented family]] for a collection of such temperaments. [[24/19]] is much closer to 1/3-octave than 5/4, and while it is not a microtemperament, a lot of equal divisions support it. An interval closer to 1\3 is [[63/50]], with the associated comma being the landscape comma. See [[Landscape family]] for the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in three.

Usage of the [[6/5]] minor third as one step of 4edo by tempering out [[648/625]], and therefore using 4edo as a diminished seventh chord produced by stacking three minor thirds is one of the features of standard Western music theory, and is supported by [[12edo]]. See [[Diminished family]] for a collection of such temperaments. [[19/16]], the 19th harmonic octave-reduced, is much closer to quarter-octave than 6/5, and while it is not a microtemperament, a lot of equal divisions support it. An interval closer to 1\4 is [[25/21]], with the associated comma being the dimcomp comma. See [[Dimcomp family]] for the family of rank-3 temperaments tempering it out. There are nonetheless other less common temperaments which divide the octave in four.

2nd-octave temperaments:
* [[Diaschismic family|Diaschismic temperaments]]
* [[Vishnuzmic family|Vishnuzmic temperaments]]
* [[Jubilismic clan|Jubilismic temperaments]]
* [[Varunismic temperaments]]
* [[Lokismic temperaments]]
* [[Nimona|Nimona temperament]]
* [[Very low accuracy temperaments #Antitonic|Antitonic]]

3rd-octave temperaments:
* [[Augmented family]]
* [[Misty family]]
* [[Landscape microtemperaments]]
* [[Very low accuracy temperaments #Alteraugment|Alteraugment]]
* [[Very low accuracy temperaments #Ternary|Ternary]]
* [[No-fives subgroup temperaments #Deviaug|Deviaug]]

4th-octave temperaments:
* [[Diminished family]]
* [[Undim family]]
* [[Very low accuracy temperaments #Quad|Quad]]
* [[No-threes subgroup temperaments #Berylic|Berylic]]

[[Category:Fractional-octave temperaments|#]]

Mabilic and trismegistus

2026-06-21T05:40:55Z

Xenllium:

{{Infobox regtemp
| Title = Mabilic; Trismegistus
| Subgroups = 2.5.7; 2.3.5.7
| Comma basis = 1071875/1048576 (2.5.7); 1029/1024, 3125/3072 (2.3.5.7)
| Edo join 1 = 16 | Edo join 2 = 25
| Mapping = 1; -15 -3 5
| Generators = 175/128
| Generators tuning = 526.7
| Optimization method = CWE
| MOS scales = [[7L 2s]], [[9L 7s]]
| Ploidacot = alpha-triseph
| Odd limit 1 = 2.5.7 7 | Mistuning 1 = 4.76 | Complexity 1 = 9
| Odd limit 2 = 2.5.7 25 | Mistuning 2 = 6.92 | Complexity 2 = 16
| Odd limit 3 = add-15, 21 7 | Mistuning 3 = 5.92 | Complexity 3 = 25
|}}
'''Mabilic''' is a [[regular temperament|temperament]] in the [[2.5.7 subgroup|2.5.7]] [[subgroup]] where [[5/2]] is split into three [[generator]]s, five of which octave-reduced reach [[8/7]]. The generator is a sharpened fourth (or, conversely, a flattened fifth) in size, best tuned around 528 [[cent]]s. Mabilic, as a result, tempers out the [[mabilisma]] 1071875/1048576. Mabilic is arrived at by removing the inaccurate 3/2 mapping from [[armodue (temperament)|armodue]] temperament. As such, mabilic can be associated with [[mos scale]]s like [[antidiatonic]] and [[armotonic]]. Thus, while the generating interval is not [[3/2]], [[2L 5s #Notation|melodic]] [[chain-of-fifths notation]] makes some amount of sense to use here.

Mabilic can be extended into a full [[7-limit]] temperament called '''trismegistus''' in which [[3/1|3]] is found at 15 steps. Since [[5/1|5]] is at 3 steps, that makes trismegistus a [[magic family|magic temperament]]. Similarly, since 8/7 is at 5 steps, trismegistus is a [[gamelismic clan|slendric temperament]]. Trismegistus is associated with the mos scale [[9L 7s]], where 3/2 is found as the "augmented" version of the generator, and [[25edo]], [[41edo]], and [[66edo]] make for good tunings.

There is an alternative extension, '''semabila''' ([[Mabila family #Semabila]]), which is a [[semaphoresmic clan|semaphore temperament]] (hence its name) and thus finds [[4/3]] at 10 generators. It is best tuned sharper than trismegistus.

Making the generator itself 4/3 leads to the [[exotemperament]] [[mavila]], after which mabilic is named.

The tuning optimum of mabilic is 527.2 cents, which is almost exactly the [[Golden sequences and tuning|golden]] antidiatonic generator.

For technical data, see [[No-threes subgroup temperaments #Mabilic]] and [[Magic family #Trismegistus]].

== Intervals ==
In the following tables, odd harmonics and subharmonics 1–15 are labeled in '''bold'''.
{| class="wikitable center-1 right-2 right-4"
|+
!
! colspan="2" | Generators up
! colspan="2" | Generators down
|-
! #
! Cents
! Approximate ratios
! Cents
! Approximate ratios
|-
| 0
| 0.000
| '''1/1'''
| 1200.000
| '''2/1'''
|-
| 1
| 527.236
|
| 672.764
|
|-
| 2
| 1054.472
| 64/35
| 145.528
| 35/32
|-
| 3
| 381.708
| '''5/4'''
| 818.292
| '''8/5'''
|-
| 4
| 908.944
| 42/25
| 291.056
| 25/21
|-
| 5
| 236.18
| '''8/7'''
| 963.82
| '''7/4'''
|-
| 6
| 763.416
| 25/16
| 436.584
| 32/25
|-
| 7
| 90.652
|
| 1109.348
|
|-
| 8
| 617.888
| 10/7
| 582.112
| 7/5
|-
| 9
| 1145.124
|
| 54.876
|
|-
| 10
| 472.36
| 21/16
| 727.64
| 32/21
|-
| 11
| 999.596
| 25/14
| 200.404
| 28/25
|-
| 12
| 326.832
| 6/5
| 873.168
| 5/3
|-
| 13
| 854.068
|
| 345.932
|
|-
| 14
| 181.304
|
| 1018.696
|
|-
| 15
| 708.54
| '''3/2'''
| 491.46
| '''4/3'''
|-
| 16
| 35.776
|
| 1164.224
|
|}

{{Todo| unify precision }}
[[Category:Rank-2 temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Magic family]]
[[Category:Gamelismic clan]]
[[Category:Gariboh clan]]

Heaven

2026-06-13T09:14:02Z

Xenllium: Redirected page to Very low accuracy temperaments#Heaven

#redirect [[Very low accuracy temperaments #Heaven]]

User:Xenllium

2026-06-13T09:12:20Z

Xenllium: /* Proposed in 2026 */

'''Xenllium''' joined Xenharmonic Wiki on 15 November 2018. The name "Xenllium" is a portmanteau of "Xenharmonic" and "''[[Wikipedia:Trillium|Trillium]]''".

== Temperaments which this user proposed names ==
This user proposed new names for following temperaments.

===Proposed in 2019===
'''March 2019'''
* [[Icositritonic]]
* [[Hemicocomp|Hemicounterpyth]] (→ Hemicocomp)
* [[Hemermacomp|Hemermapyth]] (→ Hemermacomp)
* [[Pentacontatritonic]]
* Hectaheptacontahenatonic (171 & 17442)
* [[Meridic]]
* Tetracontaheptatonic (94 & 705)
* [[Palladium]]
'''April 2019'''
* [[Septisemitonic]]
* [[Tertiathirds]]
* [[Tertiaschis]]

===Proposed in 2021===
'''March 2021'''
* [[Trident]] (→ Alphatrident)
* [[Trillium]] (→ Alphatrillium)
* [[Pseudotrillium]]
* [[Tritricot]]
* [[Decic]]
* [[Cloudtone]]
'''April 2021'''
* [[Hemipental]]
* [[Sengasec]]
* [[Qinto]]
* [[Monzism]]
* [[Houborizic]]
* [[Houbor]]
* [[Gamity]]
* [[Familia]]
* [[Sabric]]
* [[Quadrawell]]
* [[Sfour]]
* [[Rainwell]]
* [[Sengage]]
* [[Marfifths]]
* [[Diatessic]]
* [[Marf]]
'''May 2021'''
* [[Garistearn]]
* [[Decistearn]]
* [[Quadrant]]
* [[Quintilipyth]]
* [[Helenoid]]
* [[Ponta]]
* [[Pontic]]
* [[Bipont]]
* [[Counterbipont]]
* [[Quadrapont]]
* [[Decimetra]]
* [[Stockhausenic]]
* [[Decavish]]
* Bronzoid (109 & 142)
* [[Ennealimmia]]
* [[Paralytic]]
* [[Cotoneum]]
* [[Fog]]
* [[Countermiracle]]
* [[Paraklein]]
* [[Countermiraculous]]
* [[Counterbenediction]]
* [[Counterrevelation]]
* [[Septant]]
* [[Terminal]]
* [[Semiterm]]
* [[Sedia]]
* [[Marthirds]]
* [[Counterkleismic]]
* [[Counterlytic]]
* [[Geb]]
* [[Hemidimfi]]
* [[Maquiloid]]
* [[Paramity]]
'''June 2021'''
* [[Schisennealimmal]]
* [[Schisennealimmic]]
* [[Decades]] (→ Gamelstearn)
* [[Undecentic]]
* [[Pentadecal]]
* [[Quinkee]]
* [[Coditonic]]
* [[Terture]]
* [[Semivulture]]
* [[Eagle]]
* [[Quindromeda]]
* [[Quintakwai]]
* [[Quintagar]]
* [[Quintoneum]]
* [[Quinsandra]]
* [[Quinsandric]]
* [[Quinsandro]]
* [[Semiquindromeda]]
* [[Quintoneoid]]
'''July 2021'''
* [[Mineral]]
* [[Ore]]
* [[Quintaleap]]
* [[Quintupole]]
* [[Quinticosiennic]]
* [[Quintapole]]
* [[Hemiseptisix]]
'''August 2021'''
* [[Goldmine]]
* [[Gamelacomp|Gamelapyth]] (→ Gamelacomp)
* [[Mermacomp|Mermapyth]] (→ Mermacomp)
* [[Undesemi]]
* [[Decipentic]]
* [[Quasijerome]]
* [[Septendesemi]]
* [[Witcher]] (99 & 166, formerly ''witch'', revised in August 2023)
* [[Semiwitcher]] (166 & 198, formerly ''semiwitch'', revised in August 2023)
* [[Hitch]]
* [[Hitcher]]
* [[Quinwell]]
* [[Septiquarschis]]
* [[Quinbetter]]
* [[Ultramarine]]
* [[Mowglic]]
'''Septemper 2021'''
* [[Terminator]]
* [[Quintaschis]]
* [[Quintahelenic]]
* [[Quintahelenoid]]
* [[Semihemiwür]]
* [[Semihemiwürat]]
* [[Beetle]]
* [[Subsedia]]
* [[Quinkwai]]
* [[Quintakwoid]]
* [[Passionate]]
* [[Pash]]
* [[Majvamic]]
* [[Majvamoid]]
* [[Narayana]]
* [[Turkey]]
'''October 2021'''
* [[Lunennealimmal]]
* [[Hexadecoid]]
* [[Undetrita]]
* [[Rubidium]]
* [[Terbium]]
* [[Parakleischis]]
* [[Triacontaheptoid]]
* [[Octogintic]]
* [[Senator]]
* [[Hemivalentine]]
* [[Hemivalentino]]
* [[Hemivalentoid]]
* [[Undetritoid]]
* [[Biquartonic]]
* [[Quindro]]
'''November 2021'''
* [[Tridecatonic]]
* [[Trideci]]
* [[Decimaleap]]
* [[Hemiquindromeda]]
* [[Triboh]]
'''December 2021'''
* [[Hemimabila]]
* [[Sedecic]]
* [[Hyperkleismic]]

===Proposed in 2022===
'''January 2022'''
* [[Cobalt]]
* [[Cobaltous]]
* [[Cobaltic]]
* [[Cobaltite]]
* [[Pontoid]]
* [[Grackle]] (11, 13, 17 and 19-limit extension)
* [[Grackloid]]
* [[Grack]]
* [[Catahelenic]]
* [[Countriton]]
* [[Aufic]]
* [[Pajarina]]
* [[Muggloid]]
* [[Gracecord]]
* [[Gaster]]
* [[Tuskaloosa]]
* [[Muscogee]]
* [[Cohemimabila]]
* [[Trimabila]]
* [[Immune]]
* [[Meantritone]]
* [[Aurora]]
* [[Leapweeker]]
* [[Interaufo]]
* [[Helena]]
'''February 2022'''
* [[Cohendecatonic]]
* [[Nessa]]
* [[Splendecic]]
* [[Prodecic]]
* [[Semimeantone]]
* [[Keenic]]
* [[Warrior]]
* [[Trisensory]]
* [[Bisensi]]
* [[Browser]]
* [[Grazer]]
* [[Quadrasruta]]
* [[Quadrafourths]]
* [[Pajarita]]
'''March 2022'''
* [[Stalagmite|Catamite]] (→ Stalagmite)
* [[Hemikwai]]
* [[Yarm]]
* [[Cohedim]]
* [[October]]
* [[Dichotomic]]
* [[Seminar]]
* [[Chromat]] (11, 13 and 17-limit extension)
* [[Catachrome]]
* [[Chromic]]
* [[Trinity]]
* [[Poseidon]]
* [[Salacia]]
* [[Gammy]]
* [[Tertiaseptia]]
* [[Semitert]]
'''April 2022'''
* <s>[[Meantonic]]</s>
* <s>[[Meantoid]]</s>
* [[Huygens]] (→ Fokkertone)
* <s>[[Meridetonic]]</s>
* <s>[[Meridetoid]]</s>
* <s>[[Sauveuric]]</s>
* [[Meanpoid]]
* <s>[[Meanplop]] (17 and 19-limit extension)</s>
* <s>[[Meanploid]]</s>
* <s>[[Meanenneadecoid]]</s>
* <s>[[Vincenzoid]]</s>
* [[Andromeda]] (17 and 19-limit extension)
* [[Schisicosiennic]]
* [[Schisicosiennoid]]
* [[Cassandra]] (Cassie)
* [[Cassandric]]
* [[Cassander]]
'''May 2022'''
* [[Supersensi]]
* [[Quadrafives]]
'''June 2022'''
* [[Trimean]]

===Proposed in 2023===
'''June 2023'''
* [[Galileic]]
* [[Catagali]]
* [[Quintain]]
'''July 2023'''
* [[Devisemi]]
* [[Garibaldia]] (2.3.5.7.17.19 extension of [[garibaldi]])
* [[Pontia]] (2.3.5.7.19 extension of [[pontiac]])
'''August 2023'''
* [[Heartland]] (2.3.7.11.19 14&41& 48)
* [[Hearts]] (2.3.7.11.19 41&48)
* [[Heartia]]
* [[Heartful]] (2.3.7.11.19 41&75e)
* [[Doublehearted]] (2.3.7.11.19 14&41)
* [[Weasel|Byhearted]] (→ Weasel)
'''September 2023'''
* [[Heartlandia]] (14cf & 41 & 89)
* [[Darkstone]]
* [[Brightstone]]
'''October 2023'''
* [[Magical]]
* [[Magica]]
* [[Magia]]
* [[Sesquartia]]
* [[Charismic]]
* [[Quinmage]]
* [[Hearty]]
* [[Nonant]]
* [[Hemiandromeda]] (29 & 82)
* [[Hemidromeda]] (29 & 111)
'''November 2023'''
* [[Trismegiboh]]
* [[Baldanders]] (2.9.5.7.11 6&35)
* [[Neomirkwai]] (2.3.5.7.11.19 31&41&80)
* [[Dhaivatic]]
* [[Triquart]]
* [[Tertiseptisix]]
* [[Quartiquart]]
* [[Quintiquart]]
* [[Neovulture]] (2.3.5.7.11.13.19 53&58&217)
* [[Neovish]] (2.3.5.7.11.19 50&118&152)
* [[Neobreed]] (41 & 130 & 270)
* [[Neonewt]] (2.3.5.7.11.13.19 41&270)
'''December 2023'''
* [[Heartlander]] (14cf & 41 & 130)
* [[Neosatanic]] (665 & 3325)
* [[Monzosatanic]] (665 & 7980)
* [[Dog]] (2.3.19 7&9)
* [[Armodog]] (2.3.5.7.11.13.19 7&9)

===Proposed in 2024===
'''January 2024'''
* [[Quinbaldi]] (65d & 135)
* [[Quintant]] (65d & 200)
* [[Hemiquintant]] (130 & 135)

'''May 2024'''
* [[Raccoon]] (11 & 32)

'''June 2024'''
* [[Piracy]] (84 & 155)
* [[Pirateship]] (84 & 239)
* [[Unisquary]] (11-limit 72&118&121&130)
* [[Hensquary]] (13-limit 46&72&121&130)
* [[Ekasquary]] (13-limit 72&121&130&190)

===Proposed in 2025===
'''January 2025'''
* [[Pentadecoid]] (2.3.7.11 120&135)
* [[Slendscape]] (255 & 270)
* [[Triacontoid]] (120 & 150)

'''February 2025'''
* [[Devichrome]] (2.3.5.7.19 27&31&41&53)

'''March 2025'''
* [[Octasand]] (11 and 13-limit 27e&41&53)
* [[Devimarv]] (2.3.5.7.19 31&41&53)
** [[Neomarv]] (2.3.5.7.11.19 31&41&53)
* [[Deva]] (2.3.5.7.19 27&41&53)
** [[Neonakika]] (2.3.5.7.11.19 15&27e&41)
** [[Neoctasand]] (2.3.5.7.11.19 27e&41&53)
* [[Nutrient]] (2.3.5.7.19 27&31&41)
** [[Neojovia]] (2.3.5.7.11.19 27e&31&41)
* [[Passkey]] (2.3.5.7.19 41&53&99)
** [[Neolakia]] (2.3.5.7.11.19 41&53&99)

'''April 2025'''
* [[Magikarp]] (2.3.43 5&7)
* [[Hendeca]] (11&22)
* [[Char]] (2.3.5.7.17 19&31&41)
* [[Elven]]

'''June 2025'''
* [[Reversed marvel]] (2.3.5.113 22&31&34)

===Proposed in 2026===
'''January 2026'''
* [[Mendel]] (11-limit 12&31&152)
* [[Moctpine]] (11-limit 29&72&80)
* [[Biscordial]] (12&190)
* [[Triscordial]] (12&231)
* [[Naiad]]
* [[Naiadical]]

'''February 2026'''
* [[Naian]] (8d&13b)
* [[Semihemimean]] (80&118&130)
* [[Marvelpine]]
* [[Seminaiad]]
** [[Seminaiad|Seminaiadia]]
* [[Seesaw]]
** [[Heavy windmill]]
** [[Light windmill]]
* [[Antonian]]
* [[Antonym (temperament)|Antonym]]
* [[Antony]]
* [[Argument (temperament)|Argument]]
* [[Landy]]
* [[Septarium]]

'''March 2026'''
* [[Augenic]]
* [[Sida]]
* [[Bohlenic]] (no-twos temperament)

'''May 2026'''
* [[Xenial]] (19 & 70)
** [[Xenia]] (19 & 51c, 19 & 70d)
* [[Hemigoth]] (34d & 68)
* [[Paraguay]] (19 & 23bcf, 19 & 42ef)
** [[Uruguay]] (19 & 23bc, 19 & 42e)

'''June 2026'''
* [[Sixseven]] (2.3.7 2d & 3)
** [[Heaven]] (2.3.7.11 2de & 3)

== Commas which this user proposed names ==
This user proposed new names for following commas.

===Proposed in 2021===
'''April 2021'''
* [[Hewuermera temperaments|hewuermera]]: 589824/588245, satribiru-agu
* [[Triwellismic temperaments|triwellisma]]: 235298/234375, tribizo-asepgu
* [[quadrawellisma]]: 67528125/67108864, lala-quadzo-aquinyo
* [[catasyc comma]]: 390625/387072, laruquadbiyo
* [[Mistismic temperaments|mistisma]]: 458752/455625, sazoquadgu

'''May 2021'''
* [[Hemfiness temperaments|hemfiness comma]]: 4096000/4084101, saquinru-atriyo
* [[Marvelfifths temperaments|marvelfifths comma]]: 158203125/157351936, laquadru-atritriyo
* [[Chromatisma|chromatisma]]: 640000000000000000 / 635585924776181463, trisa-triru-aquadquadyo
* [[Bronzismic temperaments|bronzisma]]: 2097152/2083725, satriru-agugu
* <s>[[Breeze comma|sesquiquartisma]]</s>: 2460375/2458624, laquadru-atruyo (→ breezema, breeze comma)
* [[Mitonismic temperaments|mitonisma]]: 5250987/5242880, laquadzo-agu
* [[Sensamagicfifths temperaments|sensamagicfifths comma]]: {{monzo|-11 42 -7 -14}}, trila-seprurugu
* [[Secanticorn temperaments|secanticornisma]]: 177147/175000, laruquingu
* [[Supermatertismic temperaments|supermatertisma]]: 52734375/52706752, lasepru-atritriyo
* [[Supermasesquartismic temperaments|supermasesquartisma]]: 184528125/184473632, laquadbiru-aquinyo
* [[Septimirod temperaments|septimirod comma]]: 84035/82944, quinzo-ayo

'''June 2021'''
* [[Vorwell clan|vorwell comma]]: 134217728/133984375, sasatriru-aquadbigu
* [[Maitri clan|maitri comma]]: 40353607/40000000, tritrizo-asepgu
* [[compass comma]]: 9765625/9680832, quinruyoyo

'''July 2021'''
* [[4375/4356|fantares comma]]: 4375/4356, luluzoquadyo
* [[Septiness clan|septiness comma]]: 67108864/66706983, sasasepru
* [[decovulture comma]]: 67108864/66976875, sasa-birugugu
* [[Pontiqak temperaments|pontiqak comma]]: 95703125/95551488, lazozotritriyo

'''August 2021'''
* [[Slither clan|slither comma]]: 40960000/40353607, satritriru-aquadyo
* [[Mermismic temperaments|mermisma]]: 2500000/2470629, sepruyo
* [[Valenwuer temperaments|valenwuer comma]]: 110592/109375, sarutribigu
* [[Negricorn temperaments|negricorn comma]]: 153664/151875, saquadzogu
* [[Parkleiness temperaments|parkleiness comma]]: 1959552/1953125, zotritrigu
* [[Mynaslendic temperaments|mynaslender comma]]: 829440/823543, sepru-ayo

'''September 2021'''
* [[65536/64827|buzzardisma]]: 65536/64827, saquadru (→ buzzardsma, buzzard comma)
* [[Neptunismic temperaments|neptunisma]]: 48828125/48771072, laruruleyo

'''October 2021'''
* [[Domicomp temperaments|domicomp comma]]: 413343/409600, lazogugu

'''November 2021'''
* [[Trienstonisma|trienstonisma]]: 43046721/43025920, laquinru-agu
* [[Lokismic temperaments|lokisma]]: 102760448/102515625, sasazozo-atribigu

===Proposed in 2022===
'''January 2022'''
* [[Countriton comma|countriton comma]]: 16777216000000000 / 16677181699666569, quadsa-tritriyo
* [[3042/3025|diagassormisma]]: 3042/3025, bitholugu
* [[gaster comma]]: {{monzo|-70 72 -19}}, quadbila-negu
* [[Mabilismic clan|mabilisma]]: 1071875/1048576, latrizo-aquinyo
* [[pentonisma]]: 281974669312/281950621875, saquinthuzogu

'''February 2022'''
* [[8281/8192|diahuntmisma]]: 8281/8192, labithozo
* [[3645/3584|schismean comma]]: 3645/3584, laruyo

'''March 2022'''
* [[Acrodec temperaments|acrodec comma]]: 7710244864/7688671875, sasa-tribizo-aquadbigu
* [[120/119|lynchisma]]: 120/119, suruyo

'''May 2022'''
* [[Augmist temperaments|augmist comma]]: 1438646272/1423828125, sasa-trizotrigu
* [[Starling temperaments|supersensi comma]]: 33059881728/30517578125, quintrigu

===Proposed in 2023===
'''11 May 2023'''
* [[Syntonoschisma|syntonisma]]: {{monzo|30 -27 7 0 -1}}, trisa-lusepyo (→ syntonoschisma, syntonic schisma)

'''27 May 2023'''
* [[170/169|major naiadma]]: 170/169, sothuthuyo
* [[221/220|minor naiadma]]: 221/220, sotholugu
* <s>[[256/255|diasemisma]]</s>: 256/255, sugu (→ charisma, charic comma)
* [[442/441|seminaiadma]]: 442/441, sothoruru

'''18 July 2023'''
* [[grossmisma]]: {{monzo|-30 -13 -1 1 7 7}}, septholo-azogu

'''21 July 2023'''
* [[3971/3969|heartlandisma]]: 3971/3969, nonoloruru

'''23 July 2023'''
* [[171/170|malcolmisma]]: 171/170, nosugu
* [[190/189|cotylisma]]: 190/189, noruyo
* [[324/323|photisma]]: 324/323, nusu
* [[343/342|nutrisma]]: 343/342, nutrizo
* [[400/399|devichroma]]: 400/399, nuruyoyo
* [[devicisma]]: 633556/633555, nunusothutrilo-azogu

'''15 August 2023'''
* [[133/132|minithirdma]]: 133/132, noluzo
* <s>[[136/135|diatonisma]]</s>: 136/135, sogu (→ diatisma)
* [[154/153|augustma]]: 154/153, sulozo
* [[161/160|major kirnbergerisma]]: 161/160, twethozogu
* [[162/161|minor kirnbergerisma]]: 162/161, twethuru
* [[286/285|chthonisma]]: 286/285, nuthologu
* [[1275/1274|cimbrisma]]: 1275/1274, sothubiruyo
* [[galileisma]]: 171382426877952 / 171359481538165, lalesu-agu

'''16 August 2023'''
* [[85/84|monk comma]]: 85/84, soruyo
* [[456/455|abnobisma]]: 456/455, nothulugu
* [[476/475|hedwigma]]: 476/475, nusozogugu
* [[495/494|eulalisma]]: 495/494, nuthuloyo

'''21 August 2023'''
* [[363/361|godzillisma]]: 363/361, binulo
* [[2431/2430|heptacircle comma]]: 2431/2430, sothologu

'''26 August 2023'''
* [[1083/1078|bihendrixma]]: 1083/1078, nonolururu

'''27 August 2023'''
* [[No-fives subgroup temperaments #Doublehearted|doublehearted comma]]: 5764801/5668704, quadbizo
* [[No-fives subgroup temperaments #Hearts|hearts comma]]: 34451725707/34359738368, trila-quadzo

'''17 September 2023'''
* [[4914/4913|baladisma]]: 4914/4913, trisu-athozo

'''5 November 2023'''
* [[969/968|buddhisma]]: 969/968, nosolulu (→ kingfisher comma)
* [[1331/1330|solvejgsma]]: 1331/1330, nutrilo-arugu
* [[1540/1539|kevolisma]]: 1540/1539, nulozoyo
* [[2376/2375|trichthonisma]]: 2376/2375, nulotrigu
* [[2926/2925|neovulturisma]]: 2926/2925, nothulozogugu
* [[3136/3135|neomirkwaisma]]: 3136/3135, nuluzozogu
* [[3250/3249|martebisma]]: 3250/3249, nunuthotriyo
* [[4200/4199|neosatanisma]]: 4200/4199, nusuthuzoyoyo
* [[5776/5775|neovishma]]: 5776/5775, nonolurugugu
* [[5985/5984|neogrendelisma]]: 5985/5984, nosuluzoyo
* [[6175/6174|neonewtisma]]: 6175/6174, nothotriru-ayoyo

'''12 November 2023'''
* [[460/459|scanisma]]: 460/459, twethosuyo
* [[484/483|pittsburghisma]]: 484/483, twethuloloru
* [[507/506|laodicisma]]: 507/506, twethuthotholu
* [[529/528|preziosisma]]: 529/528, bitwetho-alu
* [[736/735|harvardisma]]: 736/735, twethorurugu
* [[760/759|squadronisma]]: 760/759, twethunoluyo
* [[875/874|nymphisma]]: 875/874, twethunuzotriyo
* [[897/896|lysistratisma]]: 897/896, twethothoru
* [[1105/1104|fragarisma]]: 1105/1104, twethusothoyo
* [[1197/1196|rhodesisma]]: 1197/1196, twethunothuzo
* [[1288/1287|santisma]]: 1288/1287, twethothuluzo
* [[1496/1495|turkisma]]: 1496/1495, twethusothulogu
* [[1863/1862|antinousisma]]: 1863/1862, twethonururu
* [[2024/2023|insincere comma]]: 2024/2023, twethosusuloru
* <s>[[2025/2024|fokkerisma]]</s>: 2025/2024, latwethuluyoyo (→ cupcakesma, cupcake comma)
* [[2185/2184|guangdongisma]]: 2185/2184, twethonothuruyo
* [[2300/2299|travellisma]]: 2300/2299, twethonubiluyo
* [[2646/2645|biyativice comma]]: 2646/2645, bitwethuzo-agu
* [[2737/2736|kotkisma]]: 2737/2736, twethonusozo
* [[3060/3059|vicious comma]]: 3060/3059, twethunusoruyo
* [[3381/3380|seminaiadvice comma]]: 3381/3380, twethobithuzo-agu (→ mikkolisma)
* [[3520/3519|vicedim comma]]: 3520/3519, twethusuloyo
* [[3888/3887|vicetride comma]]: 3888/3887, twethuthuthu
* [[4693/4692|viceaug comma]]: 4693/4692, twethunonosutho
* [[4761/4760|demiquartervice comma]]: 4761/4760, bitwetho-asurugu
* [[vicetertisma]]: 12168/12167, tritwethu-athotho

'''13 November 2023'''
* [[208/207|vicetone comma]]: 208/207, twethutho
* [[231/230|major neutravicema]]: 231/230, twethulozogu
* [[253/252|middle neutravicema]]: 253/252, twetholoru
* [[276/275|minor neutravicema]]: 276/275, twetholugugu
* [[300/299|major naiadvicema]]: 300/299, twethuthuyoyo
* [[323/322|major semivicema]]: 323/322, twethunosoru
* [[391/390|minor naiadvicema]]: 391/390, twethosothugu
* [[392/391|minor semivicema]]: 392/391, twethusuzozo
* [[460/459|vicewolf comma]]: 460/459, twethosuyo
* [[5083/5082|broadviewsma]]: 5083/5082, twethosotholuluru

'''15 November 2023'''
* [[92/91|undinisma]]: 92/91, twethothuru
* [[115/114|yarmanisma]]: 115/114, twethonuyo

'''26 November 2023'''
* [[E-shaped comma]]: 31213/31212, susuthoquadzo

'''1 December 2023'''
* <s>[[kilobyte comma]]</s>: 1024/1023, thiwulu (→ kibisma)
* <s>[[megabyte comma]]</s>: 1048576/1048575, safowuthiwulugugu (→ mebisma)

'''10 December 2023'''
* [[81/76|dog comma]]: 81/76, inu
* [[primate comma]]: 16384/15309, saru

'''23 December 2023'''
* [[841/840|arabellisma]]: 841/840, bitweno-arugu
* [[4901/4900|large grapevine comma]]: 4901/4900, twenothothobirugu, twenobithorugu
* [[6877/6875|grossvice comma]]: 6877/6875, bitwetho-atholuquadgu
* [[7425/7424|small grapevine comma]]: 7425/7424, latwenuloyoyo
* [[mynucuvine comma]]: 10935/10933, labitwenu-athuyo

===Proposed in 2024===
'''2 January 2024'''
* <s>[[slendroschisma|slendrisma]]</s>: 68719476736/68641485507, sasa-quinbiru (→ slendroschisma)

'''27 January 2024'''
* [[116/115|sironisma]]: 116/115, twenotwethugu
* [[117/116|lomisma]]: 117/116, twenutho
* [[175/174|major chthonovinema]]: 175/174, twenuzoyoyo
* [[204/203|kallistisma]]: 204/203, twenusoru
* [[232/231|major paravinema]]: 232/231, twenoluru
* [[261/260|major vinetonema]]: 261/260, twenothugu
* [[290/289|brunisma]]: 290/289, twenosusuyo
* [[320/319|minor paravinema]]: 320/319, twenuluyo
* [[378/377|major semivinema]]: 378/377, twenuthuzo
* [[406/405|minor semivinema]]: 406/405, twenozogu
* [[494/493|minor vinetonema]]: 494/493, twenunosutho
* [[551/550|minor chthonovinema]]: 551/550, twenonolugugu
* [[609/608|vineyard comma]]: 609/608, twenonuzo

'''3 May 2024'''
* [[729/722|undevicesimal diaschisma]]: 729/722, lanunu

'''1 June 2024'''
* [[unisquarisma]]: 12005/11979, trilu-aquadzo-ayo

'''22 December 2024'''
* [[429/425|middle semisixthma]]: 429/425, suthologugu
* [[507/500|large semisixthma]]: 507/500, thothotrigu
* [[1452/1445|small semisixthma]]: 1452/1445, susulologu
* [[semisixthmisma]]: 63888/63869, trisu-athutrilo

===Proposed in 2025===
'''20 March 2025'''
* [[6860/6859|devicubisma]]: 6860/6859, trinuzo-ayo
* [[tricecubisma]]: 29792/29791, trithiwu-anozozo

'''21 March 2025'''
* [[8625/8624|beerglass comma]]: 8625/8624, twetholururutriyo
* [[10626/10625|demiglace comma]]: 10626/10625, twethosulozoquadgu

'''29 March 2025'''
* [[145/144|adeonisma]]: 145/144, twenoyo
* [[552/551|sigelindisma]]: 552/551, twenutwethonu
* [[638/637|moirisma]]: 638/637, twenothuloruru
* [[726/725|joellisma]]: 726/725, twenubilogu
* [[783/782|norisma]]: 783/782, twenotwethusu
* [[1015/1014|christisma]]: 1015/1014, twenothuthuzoyo
* [[1045/1044|michelisma]]: 1045/1044, twenunoloyo
* [[1276/1275|ucclisma]]: 1276/1275, twenosulogugu
* [[1450/1449|raimondisma]]: 1450/1449, twenotwethuruyoyo
* [[1596/1595|itzigsohnisma]]: 1596/1595, twenunoluzogu
* [[1625/1624|norcma]]: 1625/1624, twenuthorutriyo
* [[1683/1682|castafiorisma]]: 1683/1682, bitwenu-asolo
* [[2176/2175|donarisma]]: 2176/2175, twenusogugu
* [[2205/2204|glinkisma]]: 2205/2204, twenunuzozoyo
* [[2262/2261|mitidikisma]]: 2262/2261, twenonusuthoru
* [[5929/5928|manzanisma]]: 5929/5928, nuthubilozo

'''6 April 2025'''
* [[6144/6137|langwisma]]: 6144/6137, nunusu

'''10 April 2025'''
* <s>[[129/128|Magikarp comma]]</s>: 129/128, fotho
* <s>[[130/129|Gyarados comma]]</s>: 130/129, fothuthoyo
* [[2640/2639|hällströmisma]]: 2640/2639, twenuthuloruyo
* [[2784/2783|domeykisma]]: 2784/2783, twenotwethululu
* [[3249/3248|musashinisma]]: 3249/3248, twenunonoru
* [[3451/3450|mentorisma]]: 3451/3450, twenotwethusozogugu
* [[4785/4784|petrovsma]]: 4785/4784, twenotwethuthuloyo
* [[5916/5915|woudisma]]: 5916/5915, twenosothuthurugu

'''11 April 2025'''
* [[10557/10556|rowlandisma]]: 10557/10556, twenutwethosothuru

'''15 April 2025'''
* [[371293/262144|tridecimal antidilemma]]: 371293/262144, laquintho 5th
* [[524288/371293|tridecimal dilemma]]: 524288/371293, saquinthu 4th
* [[1610510/1594323|fifthchromisma]]: 1610510/1594323, saquinlo-ayo

'''17 April 2025'''
* [[2001/2000|major discoverisma]]: 2001/2000, twenotwethotrigu
* [[2002/2001|minor discoverisma]]: 2002/2001, twenutwethutholozo

'''26 April 2025'''
* [[2465/2464|laservine comma]]: 2465/2464, twenosoluruyo

'''27 April 2025'''
* [[280/279|tricetone comma]]: 280/279, thiwuzoyo

'''2 May 2025'''
* [[6912/6859|deviaug comma]]: 6912/6859, trinu

'''3 May 2025'''
* [[93/92|tricema]]: 93/92, thiwotwethu
* [[155/154|scyllisma]]: 155/154, thiwoluruyo
* [[156/155|xanthippisma]]: 156/155, thiwuthogu
* [[187/186|lambertisma]]: 187/186, thiwusolo
* [[217/216|eudorisma]]: 217/216, thiwozo
* [[248/247|lameisma]]: 248/247, thiwonuthu
* [[341/340|californisma]]: 341/340, thiwosulogu
* [[342/341|endymisma]]: 342/341, thiwunolu
* [[435/434|chinthisma]]: 435/434, thiwutwenoruyo
* [[465/464|alektisma]]: 465/464, thiwotwenuyo
* [[528/527|rezisma]]: 528/527, thiwusulo
* [[589/588|croatisma]]: 589/588, thiwonoruru
* [[651/650|antiklisma]]: 651/650, thiwothuzogugu
* [[714/713|ululisma]]: 714/713, thiwutwethusozo

'''4 May 2025'''
* [[2755/2754|avicema]]: 2755/2754, twenonosuyo
* [[4641/4640|vinecute comma]]: 4641/4640, twenusothozogu
* [[5888/5887|vinocular comma]]: 5888/5887, bitwenu-atwethoru
* [[8671/8670|vinous comma]]: 8671/8670, twenotwethosusuthogu

'''9 May 2025'''
* [[6076/6075|large ricegrain comma]]: 6076/6075, sathiwobizogu
* [[6728/6727|sushi comma]]: 6728/6727, bithiwutweno-aru
* [[8960/8959|small ricegrain comma]]: 8960/8959, thiwususuzoyo

'''10 May 2025'''
* [[12636/12635|padriellisma]]: 12636/12635, nunuthorugu
* [[13690/13689|lesleymartin comma]]: 13690/13689, bithisothu-ayo
* [[16337/16335|brown rice comma]]: 16337/16335, bithiwo-asolulugu
* [[23275/23273|sunrise comma]]: 23275/23273, bithisu-anosubizoyo

'''18 May 2025'''
* [[navicular comma]]: 24192/24167, trithu-aluzo
* [[48013/48000|small tridevisemma]]: 48013/48000, trino-azotrigu
* [[93347/93312|trilute comma]]: 93347/93312, notriso
* [[116640/116603|large tridevisemma]]: 116640/116603, trinu-asuyo
* [[176000/175959|triseptichrome comma]]: 176000/175959, nulotriruyo
* [[471648/471625|small triquarterisma]]: 471648/471625, triso-alutrirugu
* [[1257795/1257728|large triquarterisma]]: 1257795/1257728, latrisulo-azoyo
* [[quinticular comma]]: 1419857/1417176, saquinso

'''28 May 2025'''
* [[7942/7935|brigade comma]]: 7942/7935, bitwethuno-alogu

'''29 May 2025'''
* [[barbadisma]]: 114383962274805 / 114349209288704, quadla-thuyo
* [[undevicesimal counterschisma]]: 2.3.19 {{monzo|-75 50 -1}}, seplanu

'''30 May 2025'''
* [[septendecimal quasithirdma]]: 19683/19652, latrisu
* [[septendecimal quasiminthma]]: 6975757441 / 6973568802, sasa-quadbiso

'''1 June 2025'''
* [[undevicesimal quasitonema]]: 34359738368 / 34296447249, sasa-tribinu
* [[deviquasisma]]: 17594077438737 / 17592186044416, lala-sepno

'''9 June 2025'''
* [[226/225|reversed marvel comma]]: 226/225, 113ogg2

'''5 July 2025'''
* [[386/385|siegenisma]]: 386/385, 193o1urg1
* [[57122/57121|phocitremisma]]: 57122/57121, 239uu3o42

===Proposed in 2026===
'''10 January 2026'''
* [[septimagic comma]]: 537824/531441, saquinzo
* [[acromagic comma]]: 26985857024 / 26904200625, sasa-sepzo-aquadgu

'''16 January 2026'''
* [[septrongisma]]: 205891132094649 / 205885750000000, lala-sepru-atritrigu

'''17 January 2026'''
* [[sensibeta comma]]: 1071875/1062882, satrizo-aquinyo

'''20 January 2026'''
* [[undetritisma]]: 205891132094649 / 204800000000000, trila-legu

'''31 January 2026'''
* [[naiadisma]]: 161414428 / 158203125, satritrizogu

'''23 February 2026'''
* [[septarium comma]]: {{monzo|73 77 -84}}, sasepquadtrigu

'''1 May 2026'''
* [[129140163/125000000|xenial comma]]: 129140163 / 125000000, latritrigu

== Scales which this user proposed names ==
This user proposed new names for following scales.

===Proposed in May 2023===
'''16 May 2023'''
* [[3L 16s|magicaltonic]]: 3L_16s MOS scale
* [[16L 3s|muggletonic]]: 16L_3s MOS scale
* [[Nicetone|zarlinoid]]: 3L_2m_2s rank-3 scale (MV3, LMLSLMS and LMLSMLS)
* [[sephipechroid]]: 3L_5m_5s rank-3 scale (MV3, LMSMLSMSLMSMS and LMSMLSMSMLSMS)
* [[magicaloid]]: 3L_8m_8s rank-3 scale (MV3, LMSMSMLSMSMSLMSMSMS and LMSMSMLSMSMSMLSMSMS)

'''18 May 2023'''
* [[Omnidiatonic|interdia]]: 2L_3m_2s rank-3 scale (MV3, LMSLMSM and LMSMLSM)
* [[antinicetone]]: 2L_2m_3s rank-3 scale (MV3, LMSLSMS and LSMLSMS)

'''20 May 2023'''
* [[villoid]]: 7L_2m_2s rank-3 scale (MV3, LLMLLSLLMLS and LLMLLSLMLLS)
* [[intervilla]]: 2L_7m_2s rank-3 scale (MV3, LMMSMLMMSMM and LMMSMMLMSMM)
* [[kleistoid]]: 2L_2m_7s rank-3 scale (MV3, LSMSSLSSMSS and LSSMSLSSMSS)
* [[interoneichro]]: 5L_3m_5s rank-3 scale (MV3, LMSLSLMSLSMLS and LMSLSMLSLSMLS)
* [[sephimechroid]]: 5L_5m_3s rank-3 scale (MV3, LMLMSLMLSMLMS and LMLSMLMLSMLMS)
* [[moharoid]]: 7L_5m_5s rank-3 scale (MV3, LMLSLMSLMLSMLSLMS and LMLSMLSLMLSMLSLMS)
* [[enharoid]]: 5L_7m_5s rank-3 scale (MV3, LMSLMSMLMSLMSMLSM and LMSLMSMLSMLMSMLSM)
* [[beatloid]]: 5L_5m_7s rank-3 scale (MV3, LMSLSMLSMSLMSLSMS and LMSLSMLSMSLSMLSMS)

'''21 May 2023'''
* [[Dualdye|archeoid]]: 5L_1m_1s rank-3 scale (MV3, LLLMLLS and LLLSLLM)
* [[interarcheo]]: 1L_5m_1s rank-3 scale (MV3, LMMMSMM and LMMSMMM)
* [[antarcheoid]]: 1L_1m_5s rank-3 scale (MV3, LSSMSSS and LSSSMSS)

===Proposed in July 2023===
'''2 July 2023'''
* [[jarachroid]]: 5L_5m_2s rank-3 scale (MV3, LMLMLSMLMLMS)
* [[Diachrome|interchroid]]: 5L_2m_5s rank-3 scale (MV3, LMSLSLSMLSLS)
* [[jerachroid]]: 2L_5m_5s rank-3 scale (MV3, LMSMSMLSMSMS)

== Documents ==
Notation:
* [[User:Xenllium/Xenllium's microtonal notation|Xenllium's microtonal notation]]
Tempered scales:
* [[User:Xenllium/Xenllium's circulating scales|Xenllium's circulating scales]]
External links:
* [https://docs.google.com/spreadsheets/d/18CRD8KRk9Gsg_tWSxbDJ_GjprgNvjGTx8ZWcpOEA-0I/edit?usp=sharing Rank-two regular temperament]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://drive.google.com/file/d/1Eov2FSp5Y4X4HPaaHV8nvqHX21LE7AQa/view?usp=drive_link 純正音程と平均律] (日本語/Japanese)
* [https://drive.google.com/file/d/1oBpqNqqIYCZaPuTFEt4HYdIULOMbX6sE/view?usp=drive_link 平均律 (2～31)] (日本語/Japanese)

== Music ==
* ''[https://youtube.com/watch?v=5iBXKRWE-_U Magical life]'' – in 19-tone pure-fifth magic scale
* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning
* [https://youtube.com/watch?v=bbJ0HjPAuaA 【オリジナル曲】かなしいこと / The sadness]

Very low accuracy temperaments

2026-06-13T09:10:51Z

Xenllium:

{{Technical data page}}
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.

== Antitonic ==
This temperament is characterized by [[9/8]] being tempered out and has been termed a "troll temperament" by its namers. Its [[ploidacot]] is diploid acot. It is named on account of 4/3 and 3/2 both being represented by the 600-cent half octave, which, in terms of diatonic function, serves as an antitonic. The 7-limit extension tempers out 15/14 and 21/20, equating 5/4 with 7/6 and 6/5 with 8/7. The 11-limit extension tempers out 12/11 and 33/32. The original 5-limit is basically the 3-limit music of [[2edo]] with the addition of harmonic 5 represented by an independent generator.

[[Subgroup]]: 2.3.5

[[Comma list]]: [[9/8]]

{{Mapping|legend=1| 2 3 0 | 0 0 1 }}

: mapping generators: ~3/2, ~5

[[Optimal tuning]]s:
* [[WE]]: ~3/2 = 615.125{{c}}, ~5/4 = 321.597{{c}} (~6/5 = 293.528{{c}})
: [[error map]]: {{val| +30.250 -56.581 -4.217 }}
* [[CWE]]: ~3/2 = 600.000{{c}}, ~5/4 = 336.527{{c}} (~6/5 = 263.473{{c}})
: error map: {{val| 0.000 -101.955 -49.787 }}

{{Optimal ET sequence|legend=1| 2, 4 }}

[[Badness]] (Sintel): 0.508

; Music
* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.

=== Septimal antitonic ===
Subgroup: 2.3.5.7

Comma list: 9/8, 15/14

Mapping: {{mapping| 2 3 0 1 | 0 0 1 1 }}

Optimal tunings:
* WE: ~3/2 = 614.759{{c}}, ~7/6 = 309.426{{c}} (~6/5 = 305.334{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~7/6 = 326.047{{c}} (~6/5 = 273.953{{c}})

{{Optimal ET sequence|legend=0| 2, 4 }}

Badness (Sintel): 0.490

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 15/14

Mapping: {{mapping| 2 3 0 1 7 | 0 0 1 1 0 }}

Optimal tunings:
* WE: ~3/2 = 606.293{{c}}, ~5/4 = 343.862{{c}} (~8/7 = 262.431{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 348.102{{c}} (~8/7 = 251.898{{c}})

{{Optimal ET sequence|legend=0| 2, 4 }}

Badness (Sintel): 0.721

==== Antietam ====
Subgroup: 2.3.5.7.11

Comma list: 9/8, 11/10, 15/14

Mapping: {{mapping| 2 3 0 1 2 | 0 0 1 1 1 }}

Optimal tunings:
* WE: ~3/2 = 616.135{{c}}, ~5/4 = 330.741{{c}} (~6/5 = 285.393{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 349.843{{c}} (~8/7 = 250.157{{c}})

{{Optimal ET sequence|legend=0| 2, 4e }}

Badness (Sintel): 0.628

=== Antaeus ===
Subgroup: 2.3.5.7

Comma list: 9/8, 35/32

Mapping: {{mapping| 2 3 0 10 | 0 0 1 -1 }}

Optimal tunings:
* WE: ~3/2 = 614.854{{c}}, ~5/4 = 323.784{{c}} (~6/5 = 291.070{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 317.349{{c}} (~6/5 = 282.651{{c}})

{{Optimal ET sequence|legend=0| 2d, 4 }}

Badness (Sintel): 0.950

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 35/32

Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}

Optimal tunings:
* WE: ~3/2 = 609.311{{c}}, ~5/4 = 323.110{{c}} (~6/5 = 286.200{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 318.904{{c}} (~6/5 = 281.096{{c}})

{{Optimal ET sequence|legend=0| 2d, 4 }}

Badness (Sintel): 1.12

== Alteraugment ==
Alteraugment tempers out the [[32/27|Pythagorean minor third (32/27)]]. It is like [[augmented (temperament)|augmented]], but the period represents 4/3 instead of 5/4, and the generator in turn provides 5/4 instead of 3/2. Its ploidacot is triploid acot. [[User:VectorGraphics|Vector Graphics]] suggests the name ''kinsborough'' for this temperament.

[[Subgroup]]: 2.3.5

[[Comma list]]: [[32/27]]

{{Mapping|legend=1| 3 5 0 | 0 0 1 }}

: mapping generators: ~4/3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~4/3 = 389.212{{c}}, ~5/4 = 447.530{{c}} (~10/9 = 58.318{{c}})
: [[error map]]: {{val| -32.364 +44.105 -3.512 }}
* [[CWE]]: ~4/3 = 400.000{{c}}, ~5/4 = 434.191{{c}} (~15/16 = 34.191{{c}})
: error map: {{val| 0.000 +98.045 +47.878 }}

{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}

[[Badness]] (Sintel): 1.83

== Antonian ==
{{Main| Antonian }}
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].

[[Subgroup]]: 2.3.5

[[Comma list]]: [[10/9]]

{{Mapping|legend=1| 1 0 -1 | 0 1 2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1187.236{{c}}, ~3/2 = 767.716{{c}}
: [[error map]]: {{val| -12.764 +52.997 -63.645 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 767.718{{c}}
: error map: {{val| 0.000 +65.763 -50.877 }}

{{Optimal ET sequence|legend=1| 2c, 3 }}

[[Badness]] (Sintel): 0.392

=== Septimal antonian ===
{{See also| Trienstonic clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 15/14

Mapping: {{mapping| 1 0 -1 -2 | 0 1 2 3 }}

Optimal tunings:
* WE: ~2 = 1193.691{{c}}, ~3/2 = 742.509{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.086{{c}}

{{Optimal ET sequence|legend=0| 2cd, 3d, 5c }}

Badness (Sintel): 0.606

=== Antonym ===
Subgroup: 2.3.5.7

Comma list: 7/6, 10/9

Mapping: {{mapping| 1 0 -1 1 | 0 1 2 1 }}

Optimal tunings:
* WE: ~2 = 1209.795{{c}}, ~3/2 = 765.995{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 765.949{{c}}

{{Optimal ET sequence|legend=0| 2cd, 3 }}

Badness (Sintel): 0.568

=== Antony ===
Subgroup: 2.3.5.7

Comma list: 8/7, 10/9

Mapping: {{mapping| 1 0 -1 3 | 0 1 2 0 }}

Optimal tunings:
* WE: ~2 = 1151.235{{c}}, ~3/2 = 789.399{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}

{{Optimal ET sequence|legend=0| 1c, 3d }}

Badness (Sintel): 0.720

=== Brutus ===
Subgroup: 2.3.5.7

Comma list: 10/9, 28/25

Mapping: {{mapping| 1 0 -1 -4 | 0 1 2 4 }}

Optimal tunings:
* WE: ~2 = 1158.982{{c}}, ~3/2 = 819.228{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 831.346{{c}}

{{Optimal ET sequence|legend=0| 3, 7bc }}

Badness (Sintel): 1.35

=== Phlegyas ===
{{See also| Archytas clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 35/32

Mapping: {{mapping| 1 0 -1 6 | 0 1 2 -2 }}

Optimal tunings:
* WE: ~2 = 1206.510{{c}}, ~3/2 = 747.166{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.797{{c}}

{{Optimal ET sequence|legend=0| 3, 5c, 8c }}

Badness (Sintel): 1.30

=== Charon ===
{{See also| Jubilismic clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
: mapping generators: ~7/5, ~3

Optimal tunings:
* WE: ~7/5 = 593.832{{c}}, ~3/2 = 774.559{{c}} (~15/14 = 180.726{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 774.466{{c}} (~15/14 = 174.466{{c}})

{{Optimal ET sequence|legend=0| 4bcd, 6 }}

Badness (Sintel): 1.43

=== Nessus ===
{{See also| Semaphoresmic clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 49/48

Mapping: {{mapping| 1 0 -1 2 | 0 2 4 1 }}
: mapping generators: ~2, ~7/4

Optimal tunings:
* WE: ~2 = 1189.201{{c}}, ~7/4 = 978.002{{c}}
* CWE: ~2 = 1200.000{{c}}, ~7/4 = 983.918{{c}}

{{Optimal ET sequence|legend=0| 5c, 6 }}

Badness (Sintel): 1.49

== Ternary ==
Ternary is identical to [[3edo|3et]] in the 5-limit, but has an independent generator for prime 7.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10/9, 16/15

{{Mapping|legend=1| 3 5 7 0 | 0 0 0 1 }}

: mapping generators: ~5/4, ~7

[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 391.796{{c}}, ~7/4 = 1012.806{{c}} (~15/14 = 162.582{{c}})
: [[error map]]: {{val| -24.612 +57.026 -43.741 -5.243 }}
* [[CWE]]: ~5/4 = 400.000{{c}}, ~7/4 = 1016.378{{c}} (~8/7 = 183.622{{c}})
: error map: {{val| 0.000 +98.045 +13.686 +47.552 }}

{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}

[[Badness]] (Sintel): 0.726

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 10/9, 16/15, 22/21

Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}

Optimal tunings:
* WE: ~5/4 = 391.788{{c}}, ~7/4 = 1011.942{{c}} (~12/11 = 163.422{{c}})
* CWE: ~5/4 = 400.000{{c}}, ~7/4 = 1013.973{{c}} (~12/11 = 186.027{{c}})

{{Optimal ET sequence|legend=0| 3de, 6 }}

Badness (Sintel): 0.846

== Quad ==
Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 9/8, 25/24

{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 308.074{{c}}, ~7/4 = 963.670{{c}}
: [[error map]]: {{val| +32.295 -53.513 -13.650 -5.150 }}
* [[CWE]]: ~6/5 = 300.000{{c}}, ~7/4 = 897.589{{c}}
: error map: {{val| 0.000 -101.955 -86.314 -71.236 }}

{{Optimal ET sequence|legend=1| 4 }}

[[Badness]] (Sintel): 1.16

== Quint ==
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16/15, 27/25

{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}

: Mapping generators: ~6/5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) 
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}

{{Optimal ET sequence|legend=1| 5, 15ccd }}

[[Badness]] (Sintel): 1.22

== Sept ==
{{See also| Whitewood family }}

Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what [[jamesbond]] does.

This temperament used to be known as ''mujannab''.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 54/49, 64/63

{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}

[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 170.823{{c}}, ~5/4 = 393.792{{c}} (~15/14 = 52.145{{c}})
: [[error map]]: {{val| -4.236 -22.898 -0.994 +47.642 }}
* [[CWE]]: ~9/8 = 171.429{{c}}, ~5/4 = 392.719{{c}} (~15/14 = 49.862{{c}})
: error map: {{val| 0.000 -16.241 +6.406 +59.746 }}

{{Optimal ET sequence|legend=1| 7, 14d }}

[[Badness]] (Sintel): 2.68

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 54/49, 64/63

Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}

Optimal tunings:
* WE: ~11/10 = 170.817{{c}}, ~5/4 = 393.252{{c}} (~33/32 = 51.619{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 391.840{{c}} (~33/32 = 48.983{{c}})

{{Optimal ET sequence|legend=0| 7, 14de }}

Badness (Sintel): 2.02

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 52/49, 64/63

Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}

Optimal tunings:
* WE: ~11/10 = 170.795{{c}}, ~5/4 = 393.611{{c}} (~33/32 = 52.021{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 392.725{{c}} (~33/32 = 49.868{{c}})

{{Optimal ET sequence|legend=0| 7, 14de }}

Badness (Sintel): 1.77

== Geryon ==
{{See also| Dicot family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 8/7, 25/21

{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1164.885{{c}}, ~5/4 = 374.162{{c}}
: [[error map]]: {{val| -35.115 +11.253 -82.382 +125.830 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}

{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}

[[Badness]] (Sintel): 1.29

== Malacoda ==
{{See also| Semaphoresmic clan }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 35/32

{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}

: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1223.542{{c}}, ~7/4 = 941.838{{c}}
: [[error map]]: {{val| +23.542 -18.278 -57.528 +20.096 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 927.096{{c}}
: error map: {{val| 0.000 -47.763 -113.410 -41.730 }}

{{Optimal ET sequence|legend=1| 1, 3b, 4, 9c, 13bcc }}

[[Badness]] (Sintel): 0.942

== Ugolino ==
{{See also| Bug family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 27/25

{{Mapping|legend=1| 1 0 0 -1 | 0 2 3 5 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.628{{c}}, ~7/4 = 926.730{{c}}
: [[error map]]: {{val| +6.628 -48.494 -6.122 +58.198 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 923.776{{c}}
: error map: {{val| 0.000 -54.403 -14.986 +50.054 }}

{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}

[[Badness]] (Sintel): 1.11

== Medusa ==
{{See also| Archytas clan | Mavila family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 64/63

{{Mapping|legend=1| 1 0 7 6 | 0 1 -3 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.960{{c}}, ~3/2 = 686.181{{c}}
: [[error map]]: {{val| +0.960 -14.814 -41.014 +62.655 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 685.511{{c}}
: error map: {{val| 0.000 -16.443 -42.849 +60.150 }}

{{Optimal ET sequence|legend=1| 2, 5c, 7 }}

[[Badness]] (Sintel): 1.08

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 33/32

Mapping: {{mapping| 1 0 7 6 5 | 0 1 -3 -2 -1 }}

Optimal tunings:
* WE: ~2 = 1202.757{{c}}, ~3/2 = 687.384{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 685.462{{c}}

{{Optimal ET sequence|legend=0| 2, 5c, 7 }}

Badness (Sintel): 0.887

== Wallaby ==
{{See also| Trienstonic clan | Mavila family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 28/27, 35/32

{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1216.024{{c}}, ~3/2 = 700.583{{c}}
: [[error map]]: {{val| +16.024 +14.652 -23.967 -51.053 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 691.757{{c}}
: error map: {{val| 0.000 -10.198 -61.585 -93.555 }}

{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}

[[Badness]] (Sintel): 1.48

== Plutus ==
{{Distinguish| Pluto }}
{{See also| Meantone family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 81/80

{{Mapping|legend=1| 1 0 -4 -5 | 0 1 4 5 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.936{{c}}, ~3/2 = 685.135{{c}}
: [[error map]]: {{val| +3.936 -12.884 -45.774 +56.849 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 683.935{{c}}
: error map: {{val| 0.000 -18.020 -50.573 +50.850 }}

{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}

[[Badness]] (Sintel): 1.14

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 81/80

Mapping: {{mapping| 1 0 -4 -5 -6 | 0 1 4 5 6 }}

Optimal tunings:
* WE: ~2 = 1203.293{{c}}, ~3/2 = 687.114{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 686.078{{c}}

{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}

Badness (Sintel): 1.08

== Involution ==
Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.

[[Subgroup]]: 2.3.5

[[Comma list]]: [[45/32]]

{{Mapping|legend=1| 1 0 5 | 0 1 -2 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: [[error map]]: {{val| +65.405 -84.203 -94.789 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
: error map: {{val| 0.000 -181.329 -227.565 }}

{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}

[[Badness]] (Sintel): 3.12

=== 7-limit ===
Subgroup: 2.3.5.7

Comma list: 8/7, 45/28

Mapping: {{mapping| 1 0 5 3 | 0 1 -2 0 }}

Optimal tunings:
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}

Optimal ET sequence: none

Badness (Sintel): 3.01

== Devolution ==
Devolution was named by [[User:Akselai|Akselai]] in 2024.

[[Subgroup]]: 3.5.7

[[Comma list]]: [[35/27]]

{{Mapping|legend=1| 1 0 3 | 0 1 -1 }}

: mapping generators: ~3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
: [[error map]]: {{val| +86.594 -78.085 -111.407 }}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
: error map: {{val| 0.000 -192.891 -256.384 }}

[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]

[[Badness]] (Sintel): 0.751

== Codex ==
{{See also| Bug family }}
Codex was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid dicot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[27/25]], [[50/49]]

{{Mapping|legend=1| 2 0 0 1 | 0 2 3 3 }}

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
: [[error map]]: {{val| -2.821 -31.998 +18.621 +34.699 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~5/3 = 936.030{{c}}
: error map: {{val| 0.000 -29.895 +21.776 +39.264 }}

{{Optimal ET sequence|legend=1| 4, 10cd, 14d }}

[[Badness]] (Sintel): 1.95

== Oxygen ==
{{See also| Porcupine family }}
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent either of those intervals with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 21/20, 175/162

{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1213.695{{c}}, ~10/9 = 171.042{{c}}
: error map: {{val| +13.695 +12.309 -0.438 -69.825 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.042{{c}}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}

{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*

<nowiki/>*[[Optimal patent val]]: [[8edo|8]]

[[Badness]] (Sintel): 1.52

== Bixby ==
{{Main| Bixby }}

[[Subgroup]]: 2.3.5

[[Comma list]]: [[4/3]]

{{Mapping|legend=1| 1 2 0 | 0 0 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1020.058{{c}}, ~5/4 = 674.394{{c}}
: [[error map]]: {{val| -179.942 +138.161 -71.803 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 629.521{{c}}
: error map: {{val| 0.000 +498.045 +243.208 }}

{{Optimal ET sequence|legend=1| 1c, 2b, 3bbcc }}

[[Badness]] (Sintel): 0.424

== Archon ==
{{Main| Archon }}

[[Subgroup]]: 2.3.5

[[Comma list]]: [[5/4]]

{{Mapping|legend=1| 1 0 2 | 0 1 0 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1268.274{{c}}, ~3/2 = 612.921{{c}}
: [[error map]]: {{val| +68.274 -20.760 -249.765 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 614.055{{c}}
: error map: {{val| 0.000 -87.900 -386.314 }}

{{Optimal ET sequence|legend=1| 2c }}

[[Badness]] (Sintel): 0.474

== Seesaw ==
{{Main| Seesaw }}
Seesaw tempers out the [[6/5|classic minor third (6/5)]], equating the [[5/1|fifth]] and [[6/1|sixth harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.

[[Subgroup]]: 2.3.5

[[Comma list]]: 6/5

{{Mapping|legend=1| 1 0 1 | 0 1 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1155.569{{c}}, ~3/2 = 643.349{{c}}
: [[error map]]: {{val| -44.431 -103.037 +168.173 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 627.511{{c}}
: error map: {{val| 0.000 -74.444 +241.197 }}

{{Optimal ET sequence|legend=1| 2 }}

[[Badness]] (Sintel): 0.367

=== 2.3.5.11 subgroup ===
This temperament is extended to the 2.3.5.11 subgroup naturally, tempering out both [[11/10]] and [[12/11]], undecimal neutral seconds.

Subgroup: 2.3.5.11

Comma list: 6/5, 11/10

Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}

Optimal tunings:
* WE: ~2 = 1156.418{{c}}, ~3/2 = 643.202{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 627.023{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.499

=== Heavy windmill ===
Heavy windmill tempers out [[9/7]] and [[15/14]] in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 6/5, 9/7

Mapping: {{mapping| 1 0 1 0 | 0 1 1 2 }}

Optimal tunings:
* WE: ~2 = 1161.600{{c}}, ~3/2 = 571.169{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 559.563{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.676

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 6/5, 9/7, 11/10

Mapping: {{mapping| 1 0 1 0 2 | 0 1 1 2 1 }}

Optimal tunings:
* WE: ~2 = 1166.584{{c}}, ~3/2 = 568.073{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 558.941{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.774

=== Light windmill ===
Light windmill tempers out [[8/7]] and [[21/20]] in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 6/5, 8/7

Mapping: {{mapping| 1 0 1 3 | 0 1 1 0 }}

Optimal tunings:
* WE: ~2 = 1134.018{{c}}, ~3/2 = 670.285{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 667.893{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.629

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 6/5, 8/7, 11/10

Mapping: {{mapping| 1 0 1 3 2 | 0 1 1 0 1 }}

Optimal tunings:
* WE: ~2 = 1136.109{{c}}, ~3/2 = 672.403{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 668.374{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.681

== Sixseven ==
Sixseven tempers out the [[7/6|septimal minor third (7/6)]], equating the [[6/1|sixth]] and [[7/1|seventh harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.

[[Subgroup]]: 2.3.7

[[Comma list]]: 7/6

{{Mapping|legend=1| 1 0 1 | 0 1 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1218.135{{c}}, ~3/2 = 734.187{{c}}
: [[error map]]: {{val| +18.135 +50.367 -198.368 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 738.927{{c}}
: error map: {{val| 0.000 +36.972 -229.899 }}

{{Optimal ET sequence|legend=1| 1, 2d, 3, 5dd }}

[[Badness]] (Sintel): 0.329

=== 2.3.7.13 subgroup ===
This temperament is extended to the 2.3.7.13 subgroup naturally, tempering out both [[13/12]] and [[14/13]], tridecimal neutral seconds.

Subgroup: 2.3.7.13

Comma list: 7/6, 13/12

Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}

Optimal tunings:
* WE: ~2 = 1220.937{{c}}, ~3/2 = 734.003{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 739.780{{c}}

{{Optimal ET sequence|legend=0| 1, 2d, 3, 5ddf }}

Badness (Sintel): 0.422

=== Heaven ===
Heaven tempers out [[11/9]] and [[22/21]] in the 2.3.7.11 subgroup.

Subgroup: 2.3.7.11

Comma list: 7/6, 11/9

Mapping: {{mapping| 1 0 1 0 | 0 1 1 2 }}

Optimal tunings:
* WE: ~2 = 1210.810{{c}}, ~3/2 = 783.079{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 785.173{{c}}

{{Optimal ET sequence|legend=0| 1e, 2de, 3 }}

Badness (Sintel): 0.577

==== 2.3.7.11.13 subgroup ====
Subgroup: 2.3.7.11.13

Comma list: 7/6, 11/9, 13/12

Mapping: {{mapping| 1 0 1 0 2 | 0 1 1 2 1 }}

Optimal tunings:
* WE: ~2 = 1210.595{{c}}, ~3/2 = 783.176{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 784.989{{c}}

{{Optimal ET sequence|legend=0| 1e, 2de, 3 }}

Badness (Sintel): 0.627

[[Category:Temperament collections]]

Very low accuracy temperaments

2026-06-13T05:39:55Z

Xenllium:

{{Technical data page}}
Below are listed some '''very low accuracy temperaments'''. Temperaments with exceedingly low accuracy such as these tend to [[tempering out|temper out]] very large intervals such as [[9/8]], [[10/9]], [[32/27]], or [[15/14]], equating wildly different interval sizes with [[semitone (interval size measure)|semitone]]-level or even greater [[error]]s, and often swapping the sizes of simple ratios compared to just intonation. As a result, all of them are right on or even beyond the edge of what can be sensibly called a temperament at all; that is to say, they are [[exotemperament]]s.

== Antitonic ==
This temperament is characterized by [[9/8]] being tempered out and has been termed a "troll temperament" by its namers. Its [[ploidacot]] is diploid acot. It is named on account of 4/3 and 3/2 both being represented by the 600-cent half octave, which, in terms of diatonic function, serves as an antitonic. The 7-limit extension tempers out 15/14 and 21/20, equating 5/4 with 7/6 and 6/5 with 8/7. The 11-limit extension tempers out 12/11 and 33/32. The original 5-limit is basically the 3-limit music of [[2edo]] with the addition of harmonic 5 represented by an independent generator.

[[Subgroup]]: 2.3.5

[[Comma list]]: [[9/8]]

{{Mapping|legend=1| 2 3 0 | 0 0 1 }}

: mapping generators: ~3/2, ~5

[[Optimal tuning]]s:
* [[WE]]: ~3/2 = 615.125{{c}}, ~5/4 = 321.597{{c}} (~6/5 = 293.528{{c}})
: [[error map]]: {{val| +30.250 -56.581 -4.217 }}
* [[CWE]]: ~3/2 = 600.000{{c}}, ~5/4 = 336.527{{c}} (~6/5 = 263.473{{c}})
: error map: {{val| 0.000 -101.955 -49.787 }}

{{Optimal ET sequence|legend=1| 2, 4 }}

[[Badness]] (Sintel): 0.508

; Music
* [[:File:Antitonic(8).mp3|''Antitonic(8)'']] (2024) – short composition by [[Wensik]] in POTE-tuned 5-limit antitonic using an 8-note ternary scale.

=== Septimal antitonic ===
Subgroup: 2.3.5.7

Comma list: 9/8, 15/14

Mapping: {{mapping| 2 3 0 1 | 0 0 1 1 }}

Optimal tunings:
* WE: ~3/2 = 614.759{{c}}, ~7/6 = 309.426{{c}} (~6/5 = 305.334{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~7/6 = 326.047{{c}} (~6/5 = 273.953{{c}})

{{Optimal ET sequence|legend=0| 2, 4 }}

Badness (Sintel): 0.490

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 15/14

Mapping: {{mapping| 2 3 0 1 7 | 0 0 1 1 0 }}

Optimal tunings:
* WE: ~3/2 = 606.293{{c}}, ~5/4 = 343.862{{c}} (~8/7 = 262.431{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 348.102{{c}} (~8/7 = 251.898{{c}})

{{Optimal ET sequence|legend=0| 2, 4 }}

Badness (Sintel): 0.721

==== Antietam ====
Subgroup: 2.3.5.7.11

Comma list: 9/8, 11/10, 15/14

Mapping: {{mapping| 2 3 0 1 2 | 0 0 1 1 1 }}

Optimal tunings:
* WE: ~3/2 = 616.135{{c}}, ~5/4 = 330.741{{c}} (~6/5 = 285.393{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 349.843{{c}} (~8/7 = 250.157{{c}})

{{Optimal ET sequence|legend=0| 2, 4e }}

Badness (Sintel): 0.628

=== Antaeus ===
Subgroup: 2.3.5.7

Comma list: 9/8, 35/32

Mapping: {{mapping| 2 3 0 10 | 0 0 1 -1 }}

Optimal tunings:
* WE: ~3/2 = 614.854{{c}}, ~5/4 = 323.784{{c}} (~6/5 = 291.070{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 317.349{{c}} (~6/5 = 282.651{{c}})

{{Optimal ET sequence|legend=0| 2d, 4 }}

Badness (Sintel): 0.950

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 9/8, 12/11, 35/32

Mapping: {{mapping| 2 3 0 10 7 | 0 0 1 -1 0 }}

Optimal tunings:
* WE: ~3/2 = 609.311{{c}}, ~5/4 = 323.110{{c}} (~6/5 = 286.200{{c}})
* CWE: ~3/2 = 600.000{{c}}, ~5/4 = 318.904{{c}} (~6/5 = 281.096{{c}})

{{Optimal ET sequence|legend=0| 2d, 4 }}

Badness (Sintel): 1.12

== Alteraugment ==
Alteraugment tempers out the [[32/27|Pythagorean minor third (32/27)]]. It is like [[augmented (temperament)|augmented]], but the period represents 4/3 instead of 5/4, and the generator in turn provides 5/4 instead of 3/2. Its ploidacot is triploid acot. [[User:VectorGraphics|Vector Graphics]] suggests the name ''kinsborough'' for this temperament.

[[Subgroup]]: 2.3.5

[[Comma list]]: [[32/27]]

{{Mapping|legend=1| 3 5 0 | 0 0 1 }}

: mapping generators: ~4/3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~4/3 = 389.212{{c}}, ~5/4 = 447.530{{c}} (~10/9 = 58.318{{c}})
: [[error map]]: {{val| -32.364 +44.105 -3.512 }}
* [[CWE]]: ~4/3 = 400.000{{c}}, ~5/4 = 434.191{{c}} (~15/16 = 34.191{{c}})
: error map: {{val| 0.000 +98.045 +47.878 }}

{{Optimal ET sequence|legend=1| 3, 12bcc, 15bbcc }}

[[Badness]] (Sintel): 1.83

== Antonian ==
{{Main| Antonian }}
This temperament family is characterized by the [[color notation|yo 2nd]] ([[10/9]]) being tempered out. It identifies [[3/2]] with [[5/3]], [[4/3]] with [[6/5]], and [[5/4]] with [[9/8]].

[[Subgroup]]: 2.3.5

[[Comma list]]: [[10/9]]

{{Mapping|legend=1| 1 0 -1 | 0 1 2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1187.236{{c}}, ~3/2 = 767.716{{c}}
: [[error map]]: {{val| -12.764 +52.997 -63.645 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 767.718{{c}}
: error map: {{val| 0.000 +65.763 -50.877 }}

{{Optimal ET sequence|legend=1| 2c, 3 }}

[[Badness]] (Sintel): 0.392

=== Septimal antonian ===
{{See also| Trienstonic clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 15/14

Mapping: {{mapping| 1 0 -1 -2 | 0 1 2 3 }}

Optimal tunings:
* WE: ~2 = 1193.691{{c}}, ~3/2 = 742.509{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.086{{c}}

{{Optimal ET sequence|legend=0| 2cd, 3d, 5c }}

Badness (Sintel): 0.606

=== Antonym ===
Subgroup: 2.3.5.7

Comma list: 7/6, 10/9

Mapping: {{mapping| 1 0 -1 1 | 0 1 2 1 }}

Optimal tunings:
* WE: ~2 = 1209.795{{c}}, ~3/2 = 765.995{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 765.949{{c}}

{{Optimal ET sequence|legend=0| 2cd, 3 }}

Badness (Sintel): 0.568

=== Antony ===
Subgroup: 2.3.5.7

Comma list: 8/7, 10/9

Mapping: {{mapping| 1 0 -1 3 | 0 1 2 0 }}

Optimal tunings:
* WE: ~2 = 1151.235{{c}}, ~3/2 = 789.399{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 800.996{{c}}

{{Optimal ET sequence|legend=0| 1c, 3d }}

Badness (Sintel): 0.720

=== Brutus ===
Subgroup: 2.3.5.7

Comma list: 10/9, 28/25

Mapping: {{mapping| 1 0 -1 -4 | 0 1 2 4 }}

Optimal tunings:
* WE: ~2 = 1158.982{{c}}, ~3/2 = 819.228{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 831.346{{c}}

{{Optimal ET sequence|legend=0| 3, 7bc }}

Badness (Sintel): 1.35

=== Phlegyas ===
{{See also| Archytas clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 35/32

Mapping: {{mapping| 1 0 -1 6 | 0 1 2 -2 }}

Optimal tunings:
* WE: ~2 = 1206.510{{c}}, ~3/2 = 747.166{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 743.797{{c}}

{{Optimal ET sequence|legend=0| 3, 5c, 8c }}

Badness (Sintel): 1.30

=== Charon ===
{{See also| Jubilismic clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

Mapping: {{mapping| 2 0 -2 -1 | 0 1 2 2 }}
: mapping generators: ~7/5, ~3

Optimal tunings:
* WE: ~7/5 = 593.832{{c}}, ~3/2 = 774.559{{c}} (~15/14 = 180.726{{c}})
* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 774.466{{c}} (~15/14 = 174.466{{c}})

{{Optimal ET sequence|legend=0| 4bcd, 6 }}

Badness (Sintel): 1.43

=== Nessus ===
{{See also| Semaphoresmic clan }}

Subgroup: 2.3.5.7

Comma list: 10/9, 49/48

Mapping: {{mapping| 1 0 -1 2 | 0 2 4 1 }}
: mapping generators: ~2, ~7/4

Optimal tunings:
* WE: ~2 = 1189.201{{c}}, ~7/4 = 978.002{{c}}
* CWE: ~2 = 1200.000{{c}}, ~7/4 = 983.918{{c}}

{{Optimal ET sequence|legend=0| 5c, 6 }}

Badness (Sintel): 1.49

== Ternary ==
Ternary is identical to [[3edo|3et]] in the 5-limit, but has an independent generator for prime 7.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10/9, 16/15

{{Mapping|legend=1| 3 5 7 0 | 0 0 0 1 }}

: mapping generators: ~5/4, ~7

[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 391.796{{c}}, ~7/4 = 1012.806{{c}} (~15/14 = 162.582{{c}})
: [[error map]]: {{val| -24.612 +57.026 -43.741 -5.243 }}
* [[CWE]]: ~5/4 = 400.000{{c}}, ~7/4 = 1016.378{{c}} (~8/7 = 183.622{{c}})
: error map: {{val| 0.000 +98.045 +13.686 +47.552 }}

{{Optimal ET sequence|legend=1| 3d, 6, 9bd }}

[[Badness]] (Sintel): 0.726

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 10/9, 16/15, 22/21

Mapping: {{mapping| 3 5 7 0 2 | 0 0 0 1 1 }}

Optimal tunings:
* WE: ~5/4 = 391.788{{c}}, ~7/4 = 1011.942{{c}} (~12/11 = 163.422{{c}})
* CWE: ~5/4 = 400.000{{c}}, ~7/4 = 1013.973{{c}} (~12/11 = 186.027{{c}})

{{Optimal ET sequence|legend=0| 3de, 6 }}

Badness (Sintel): 0.846

== Quad ==
Quad is identical to [[4edo|4et]] in the 5-limit, but has an independent generator for prime 7.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 9/8, 25/24

{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 308.074{{c}}, ~7/4 = 963.670{{c}}
: [[error map]]: {{val| +32.295 -53.513 -13.650 -5.150 }}
* [[CWE]]: ~6/5 = 300.000{{c}}, ~7/4 = 897.589{{c}}
: error map: {{val| 0.000 -101.955 -86.314 -71.236 }}

{{Optimal ET sequence|legend=1| 4 }}

[[Badness]] (Sintel): 1.16

== Quint ==
Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16/15, 27/25

{{Mapping|legend=1| 5 8 12 0 | 0 0 0 1 }}

: Mapping generators: ~6/5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) 
: [[error map]]: {{val| -17.921 -10.628 +50.676 -1.966 }}
* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
: error map: {{val| 0.000 +18.045 +93.686 +36.309 }}

{{Optimal ET sequence|legend=1| 5, 15ccd }}

[[Badness]] (Sintel): 1.22

== Sept ==
{{See also| Apotome family }}

Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what [[jamesbond]] does.

This temperament used to be known as ''mujannab''.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 54/49, 64/63

{{Mapping|legend=1| 7 11 0 20 | 0 0 1 0 }}

[[Optimal tuning]]s:
* [[WE]]: ~9/8 = 170.823{{c}}, ~5/4 = 393.792{{c}} (~15/14 = 52.145{{c}})
: [[error map]]: {{val| -4.236 -22.898 -0.994 +47.642 }}
* [[CWE]]: ~9/8 = 171.429{{c}}, ~5/4 = 392.719{{c}} (~15/14 = 49.862{{c}})
: error map: {{val| 0.000 -16.241 +6.406 +59.746 }}

{{Optimal ET sequence|legend=1| 7, 14d }}

[[Badness]] (Sintel): 2.68

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 54/49, 64/63

Mapping: {{mapping| 7 11 0 20 8 | 0 0 1 0 1 }}

Optimal tunings:
* WE: ~11/10 = 170.817{{c}}, ~5/4 = 393.252{{c}} (~33/32 = 51.619{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 391.840{{c}} (~33/32 = 48.983{{c}})

{{Optimal ET sequence|legend=0| 7, 14de }}

Badness (Sintel): 2.02

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 45/44, 52/49, 64/63

Mapping: {{mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}

Optimal tunings:
* WE: ~11/10 = 170.795{{c}}, ~5/4 = 393.611{{c}} (~33/32 = 52.021{{c}})
* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 392.725{{c}} (~33/32 = 49.868{{c}})

{{Optimal ET sequence|legend=0| 7, 14de }}

Badness (Sintel): 1.77

== Geryon ==
{{See also| Dicot family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 8/7, 25/21

{{Mapping|legend=1| 1 1 2 3 | 0 2 1 0 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1164.885{{c}}, ~5/4 = 374.162{{c}}
: [[error map]]: {{val| -35.115 +11.253 -82.382 +125.830 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 375.277{{c}}
: error map: {{val| 0.000 +48.600 -11.036 +231.174 }}

{{Optimal ET sequence|legend=1| 1b, 2b, 3d }}

[[Badness]] (Sintel): 1.29

== Malacoda ==
{{See also| Semaphoresmic clan }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 35/32

{{Mapping|legend=1| 1 0 3 2 | 0 2 -1 1 }}

: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1223.542{{c}}, ~7/4 = 941.838{{c}}
: [[error map]]: {{val| +23.542 -18.278 -57.528 +20.096 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 927.096{{c}}
: error map: {{val| 0.000 -47.763 -113.410 -41.730 }}

{{Optimal ET sequence|legend=1| 1, 3b, 4, 9c, 13bcc }}

[[Badness]] (Sintel): 0.942

== Ugolino ==
{{See also| Bug family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 27/25

{{Mapping|legend=1| 1 0 0 -1 | 0 2 3 5 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.628{{c}}, ~7/4 = 926.730{{c}}
: [[error map]]: {{val| +6.628 -48.494 -6.122 +58.198 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~7/4 = 923.776{{c}}
: error map: {{val| 0.000 -54.403 -14.986 +50.054 }}

{{Optimal ET sequence|legend=1| 1cdd, 3bcdd, 4, 9d }}

[[Badness]] (Sintel): 1.11

== Medusa ==
{{See also| Archytas clan | Mavila family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 64/63

{{Mapping|legend=1| 1 0 7 6 | 0 1 -3 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.960{{c}}, ~3/2 = 686.181{{c}}
: [[error map]]: {{val| +0.960 -14.814 -41.014 +62.655 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 685.511{{c}}
: error map: {{val| 0.000 -16.443 -42.849 +60.150 }}

{{Optimal ET sequence|legend=1| 2, 5c, 7 }}

[[Badness]] (Sintel): 1.08

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 33/32

Mapping: {{mapping| 1 0 7 6 5 | 0 1 -3 -2 -1 }}

Optimal tunings:
* WE: ~2 = 1202.757{{c}}, ~3/2 = 687.384{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 685.462{{c}}

{{Optimal ET sequence|legend=0| 2, 5c, 7 }}

Badness (Sintel): 0.887

== Wallaby ==
{{See also| Trienstonic clan | Mavila family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 28/27, 35/32

{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1216.024{{c}}, ~3/2 = 700.583{{c}}
: [[error map]]: {{val| +16.024 +14.652 -23.967 -51.053 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 691.757{{c}}
: error map: {{val| 0.000 -10.198 -61.585 -93.555 }}

{{Optimal ET sequence|legend=1| 2d, 5c, 7d, 19ccdd }}

[[Badness]] (Sintel): 1.48

== Plutus ==
{{Distinguish| Pluto }}
{{See also| Meantone family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 15/14, 81/80

{{Mapping|legend=1| 1 0 -4 -5 | 0 1 4 5 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.936{{c}}, ~3/2 = 685.135{{c}}
: [[error map]]: {{val| +3.936 -12.884 -45.774 +56.849 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 683.935{{c}}
: error map: {{val| 0.000 -18.020 -50.573 +50.850 }}

{{Optimal ET sequence|legend=1| 2cd, 5d, 7 }}

[[Badness]] (Sintel): 1.14

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 81/80

Mapping: {{mapping| 1 0 -4 -5 -6 | 0 1 4 5 6 }}

Optimal tunings:
* WE: ~2 = 1203.293{{c}}, ~3/2 = 687.114{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 686.078{{c}}

{{Optimal ET sequence|legend=0| 2cde, 5de, 7 }}

Badness (Sintel): 1.08

== Involution ==
Involution tempers out the [[45/32|ptolemaic augmented fourth (45/32)]]. Its ploidacot is monocot. It was named by [[User:CompactStar|CompactStar]] in 2023.

[[Subgroup]]: 2.3.5

[[Comma list]]: [[45/32]]

{{Mapping|legend=1| 1 0 5 | 0 1 -2 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1265.406{{c}}, ~3/2 = 552.346{{c}}
: [[error map]]: {{val| +65.405 -84.203 -94.789 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 520.626{{c}}
: error map: {{val| 0.000 -181.329 -227.565 }}

{{Optimal ET sequence|legend=1| 2c, 5bc, 7bbcc }}

[[Badness]] (Sintel): 3.12

=== 7-limit ===
Subgroup: 2.3.5.7

Comma list: 8/7, 45/28

Mapping: {{mapping| 1 0 5 3 | 0 1 -2 0 }}

Optimal tunings:
* WE: ~2 = 1205.230{{c}}, ~3/2 = 517.557{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 515.099{{c}}

Optimal ET sequence: none

Badness (Sintel): 3.01

== Devolution ==
Devolution was named by [[User:Akselai|Akselai]] in 2024.

[[Subgroup]]: 3.5.7

[[Comma list]]: [[35/27]]

{{Mapping|legend=1| 1 0 3 | 0 1 -1 }}

: mapping generators: ~3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~3 = 1988.549{{c}}, ~5/3 = 719.679{{c}}
: [[error map]]: {{val| +86.594 -78.085 -111.407 }}
* [[CWE]]: ~3 = 1901.955{{c}}, ~5/3 = 691.468{{c}}
: error map: {{val| 0.000 -192.891 -256.384 }}

[[Optimal ET sequence]]: [[1edt|b1]], [[2edt|b2d]], [[3edt|b3]], [[8edt|b8cdd]]

[[Badness]] (Sintel): 0.751

== Codex ==
{{See also| Bug family }}
Codex was named by [[User:Jerdle|Jerdle]] as an abbreviation of "co-decimal", as it resembles [[decimal]] in many ways, but exchanges the roles of 5's and 7's. While decimal equates [[8/7]] and [[7/6]], as well as [[6/5]] and [[5/4]], this equates [[10/9]] and [[6/5]], as well as [[7/6]] and [[9/7]]. It is an extension of [[bug]] and [[54/49|mujannabic]] in the same way decimal is of [[semaphore]] and [[dicot]]. Its ploidacot is diploid dicot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[27/25]], [[50/49]]

{{Mapping|legend=1| 2 0 0 1 | 0 2 3 3 }}

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.589{{c}}, ~5/3 = 934.978{{c}}
: [[error map]]: {{val| -2.821 -31.998 +18.621 +34.699 }}
* [[CWE]]: ~7/5 = 600.000{{c}}, ~5/3 = 936.030{{c}}
: error map: {{val| 0.000 -29.895 +21.776 +39.264 }}

{{Optimal ET sequence|legend=1| 4, 10cd, 14d }}

[[Badness]] (Sintel): 1.95

== Oxygen ==
{{See also| Porcupine family }}
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent either of those intervals with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 21/20, 175/162

{{Mapping|legend=1| 1 2 3 3 | 0 -3 -5 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1213.695{{c}}, ~10/9 = 171.042{{c}}
: error map: {{val| +13.695 +12.309 -0.438 -69.825 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 166.042{{c}}
: error map: {{val| 0.000 -0.083 -16.526 -100.911 }}

{{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*

<nowiki/>*[[Optimal patent val]]: [[8edo|8]]

[[Badness]] (Sintel): 1.52

== Bixby ==
{{Main| Bixby }}

[[Subgroup]]: 2.3.5

[[Comma list]]: [[4/3]]

{{Mapping|legend=1| 1 2 0 | 0 0 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1020.058{{c}}, ~5/4 = 674.394{{c}}
: [[error map]]: {{val| -179.942 +138.161 -71.803 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 629.521{{c}}
: error map: {{val| 0.000 +498.045 +243.208 }}

{{Optimal ET sequence|legend=1| 1c, 2b, 3bbcc }}

[[Badness]] (Sintel): 0.424

== Archon ==
{{Main| Archon }}

[[Subgroup]]: 2.3.5

[[Comma list]]: [[5/4]]

{{Mapping|legend=1| 1 0 2 | 0 1 0 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1268.274{{c}}, ~3/2 = 612.921{{c}}
: [[error map]]: {{val| +68.274 -20.760 -249.765 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 614.055{{c}}
: error map: {{val| 0.000 -87.900 -386.314 }}

{{Optimal ET sequence|legend=1| 2c }}

[[Badness]] (Sintel): 0.474

== Seesaw ==
{{Main| Seesaw }}
Seesaw tempers out the [[6/5|classic minor third (6/5)]], equating the [[5/1|fifth]] and [[6/1|sixth harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.

[[Subgroup]]: 2.3.5

[[Comma list]]: 6/5

{{Mapping|legend=1| 1 0 1 | 0 1 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1155.569{{c}}, ~3/2 = 643.349{{c}}
: [[error map]]: {{val| -44.431 -103.037 +168.173 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 627.511{{c}}
: error map: {{val| 0.000 -74.444 +241.197 }}

{{Optimal ET sequence|legend=1| 2 }}

[[Badness]] (Sintel): 0.367

=== 2.3.5.11 subgroup ===
This temperament is extended to the 2.3.5.11 subgroup naturally, tempering out both [[11/10]] and [[12/11]], undecimal neutral seconds.

Subgroup: 2.3.5.11

Comma list: 6/5, 11/10

Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}

Optimal tunings:
* WE: ~2 = 1156.418{{c}}, ~3/2 = 643.202{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 627.023{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.499

=== Heavy windmill ===
Heavy windmill tempers out [[9/7]] and [[15/14]] in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 6/5, 9/7

Mapping: {{mapping| 1 0 1 0 | 0 1 1 2 }}

Optimal tunings:
* WE: ~2 = 1161.600{{c}}, ~3/2 = 571.169{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 559.563{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.676

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 6/5, 9/7, 11/10

Mapping: {{mapping| 1 0 1 0 2 | 0 1 1 2 1 }}

Optimal tunings:
* WE: ~2 = 1166.584{{c}}, ~3/2 = 568.073{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 558.941{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.774

=== Light windmill ===
Light windmill tempers out [[8/7]] and [[21/20]] in the 7-limit.

Subgroup: 2.3.5.7

Comma list: 6/5, 8/7

Mapping: {{mapping| 1 0 1 3 | 0 1 1 0 }}

Optimal tunings:
* WE: ~2 = 1134.018{{c}}, ~3/2 = 670.285{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 667.893{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.629

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 6/5, 8/7, 11/10

Mapping: {{mapping| 1 0 1 3 2 | 0 1 1 0 1 }}

Optimal tunings:
* WE: ~2 = 1136.109{{c}}, ~3/2 = 672.403{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 668.374{{c}}

{{Optimal ET sequence|legend=0| 2 }}

Badness (Sintel): 0.681

== Sixseven ==
{{Main| Sixseven }}
Sixseven tempers out the [[7/6|septimal minor third (7/6)]], equating the [[6/1|sixth]] and [[7/1|seventh harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.

[[Subgroup]]: 2.3.7

[[Comma list]]: 7/6

{{Mapping|legend=1| 1 0 1 | 0 1 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1218.135{{c}}, ~3/2 = 734.187{{c}}
: [[error map]]: {{val| +18.135 +50.367 -198.368 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 738.927{{c}}
: error map: {{val| 0.000 +36.972 -229.899 }}

{{Optimal ET sequence|legend=1| 1, 2d, 3, 5dd }}

[[Badness]] (Sintel): 0.329

=== 2.3.7.13 subgroup ===
This temperament is extended to the 2.3.7.13 subgroup naturally, tempering out both [[13/12]] and [[14/13]], tridecimal neutral seconds.

Subgroup: 2.3.7.13

Comma list: 7/6, 13/12

Mapping: {{mapping| 1 0 1 2 | 0 1 1 1 }}

Optimal tunings:
* WE: ~2 = 1220.937{{c}}, ~3/2 = 734.003{{c}}
* CWE: ~2 = 1200.000{{c}}, ~3/2 = 739.780{{c}}

{{Optimal ET sequence|legend=0| 1, 2d, 3, 5ddf }}

Badness (Sintel): 0.422

[[Category:Temperament collections]]

User:Xenllium

2026-06-12T23:23:07Z

Xenllium: /* Temperaments which this user proposed names */

'''Xenllium''' joined Xenharmonic Wiki on 15 November 2018. The name "Xenllium" is a portmanteau of "Xenharmonic" and "''[[Wikipedia:Trillium|Trillium]]''".

== Temperaments which this user proposed names ==
This user proposed new names for following temperaments.

===Proposed in 2019===
'''March 2019'''
* [[Icositritonic]]
* [[Hemicocomp|Hemicounterpyth]] (→ Hemicocomp)
* [[Hemermacomp|Hemermapyth]] (→ Hemermacomp)
* [[Pentacontatritonic]]
* Hectaheptacontahenatonic (171 & 17442)
* [[Meridic]]
* Tetracontaheptatonic (94 & 705)
* [[Palladium]]
'''April 2019'''
* [[Septisemitonic]]
* [[Tertiathirds]]
* [[Tertiaschis]]

===Proposed in 2021===
'''March 2021'''
* [[Trident]] (→ Alphatrident)
* [[Trillium]] (→ Alphatrillium)
* [[Pseudotrillium]]
* [[Tritricot]]
* [[Decic]]
* [[Cloudtone]]
'''April 2021'''
* [[Hemipental]]
* [[Sengasec]]
* [[Qinto]]
* [[Monzism]]
* [[Houborizic]]
* [[Houbor]]
* [[Gamity]]
* [[Familia]]
* [[Sabric]]
* [[Quadrawell]]
* [[Sfour]]
* [[Rainwell]]
* [[Sengage]]
* [[Marfifths]]
* [[Diatessic]]
* [[Marf]]
'''May 2021'''
* [[Garistearn]]
* [[Decistearn]]
* [[Quadrant]]
* [[Quintilipyth]]
* [[Helenoid]]
* [[Ponta]]
* [[Pontic]]
* [[Bipont]]
* [[Counterbipont]]
* [[Quadrapont]]
* [[Decimetra]]
* [[Stockhausenic]]
* [[Decavish]]
* Bronzoid (109 & 142)
* [[Ennealimmia]]
* [[Paralytic]]
* [[Cotoneum]]
* [[Fog]]
* [[Countermiracle]]
* [[Paraklein]]
* [[Countermiraculous]]
* [[Counterbenediction]]
* [[Counterrevelation]]
* [[Septant]]
* [[Terminal]]
* [[Semiterm]]
* [[Sedia]]
* [[Marthirds]]
* [[Counterkleismic]]
* [[Counterlytic]]
* [[Geb]]
* [[Hemidimfi]]
* [[Maquiloid]]
* [[Paramity]]
'''June 2021'''
* [[Schisennealimmal]]
* [[Schisennealimmic]]
* [[Decades]] (→ Gamelstearn)
* [[Undecentic]]
* [[Pentadecal]]
* [[Quinkee]]
* [[Coditonic]]
* [[Terture]]
* [[Semivulture]]
* [[Eagle]]
* [[Quindromeda]]
* [[Quintakwai]]
* [[Quintagar]]
* [[Quintoneum]]
* [[Quinsandra]]
* [[Quinsandric]]
* [[Quinsandro]]
* [[Semiquindromeda]]
* [[Quintoneoid]]
'''July 2021'''
* [[Mineral]]
* [[Ore]]
* [[Quintaleap]]
* [[Quintupole]]
* [[Quinticosiennic]]
* [[Quintapole]]
* [[Hemiseptisix]]
'''August 2021'''
* [[Goldmine]]
* [[Gamelacomp|Gamelapyth]] (→ Gamelacomp)
* [[Mermacomp|Mermapyth]] (→ Mermacomp)
* [[Undesemi]]
* [[Decipentic]]
* [[Quasijerome]]
* [[Septendesemi]]
* [[Witcher]] (99 & 166, formerly ''witch'', revised in August 2023)
* [[Semiwitcher]] (166 & 198, formerly ''semiwitch'', revised in August 2023)
* [[Hitch]]
* [[Hitcher]]
* [[Quinwell]]
* [[Septiquarschis]]
* [[Quinbetter]]
* [[Ultramarine]]
* [[Mowglic]]
'''Septemper 2021'''
* [[Terminator]]
* [[Quintaschis]]
* [[Quintahelenic]]
* [[Quintahelenoid]]
* [[Semihemiwür]]
* [[Semihemiwürat]]
* [[Beetle]]
* [[Subsedia]]
* [[Quinkwai]]
* [[Quintakwoid]]
* [[Passionate]]
* [[Pash]]
* [[Majvamic]]
* [[Majvamoid]]
* [[Narayana]]
* [[Turkey]]
'''October 2021'''
* [[Lunennealimmal]]
* [[Hexadecoid]]
* [[Undetrita]]
* [[Rubidium]]
* [[Terbium]]
* [[Parakleischis]]
* [[Triacontaheptoid]]
* [[Octogintic]]
* [[Senator]]
* [[Hemivalentine]]
* [[Hemivalentino]]
* [[Hemivalentoid]]
* [[Undetritoid]]
* [[Biquartonic]]
* [[Quindro]]
'''November 2021'''
* [[Tridecatonic]]
* [[Trideci]]
* [[Decimaleap]]
* [[Hemiquindromeda]]
* [[Triboh]]
'''December 2021'''
* [[Hemimabila]]
* [[Sedecic]]
* [[Hyperkleismic]]

===Proposed in 2022===
'''January 2022'''
* [[Cobalt]]
* [[Cobaltous]]
* [[Cobaltic]]
* [[Cobaltite]]
* [[Pontoid]]
* [[Grackle]] (11, 13, 17 and 19-limit extension)
* [[Grackloid]]
* [[Grack]]
* [[Catahelenic]]
* [[Countriton]]
* [[Aufic]]
* [[Pajarina]]
* [[Muggloid]]
* [[Gracecord]]
* [[Gaster]]
* [[Tuskaloosa]]
* [[Muscogee]]
* [[Cohemimabila]]
* [[Trimabila]]
* [[Immune]]
* [[Meantritone]]
* [[Aurora]]
* [[Leapweeker]]
* [[Interaufo]]
* [[Helena]]
'''February 2022'''
* [[Cohendecatonic]]
* [[Nessa]]
* [[Splendecic]]
* [[Prodecic]]
* [[Semimeantone]]
* [[Keenic]]
* [[Warrior]]
* [[Trisensory]]
* [[Bisensi]]
* [[Browser]]
* [[Grazer]]
* [[Quadrasruta]]
* [[Quadrafourths]]
* [[Pajarita]]
'''March 2022'''
* [[Stalagmite|Catamite]] (→ Stalagmite)
* [[Hemikwai]]
* [[Yarm]]
* [[Cohedim]]
* [[October]]
* [[Dichotomic]]
* [[Seminar]]
* [[Chromat]] (11, 13 and 17-limit extension)
* [[Catachrome]]
* [[Chromic]]
* [[Trinity]]
* [[Poseidon]]
* [[Salacia]]
* [[Gammy]]
* [[Tertiaseptia]]
* [[Semitert]]
'''April 2022'''
* <s>[[Meantonic]]</s>
* <s>[[Meantoid]]</s>
* [[Huygens]] (→ Fokkertone)
* <s>[[Meridetonic]]</s>
* <s>[[Meridetoid]]</s>
* <s>[[Sauveuric]]</s>
* [[Meanpoid]]
* <s>[[Meanplop]] (17 and 19-limit extension)</s>
* <s>[[Meanploid]]</s>
* <s>[[Meanenneadecoid]]</s>
* <s>[[Vincenzoid]]</s>
* [[Andromeda]] (17 and 19-limit extension)
* [[Schisicosiennic]]
* [[Schisicosiennoid]]
* [[Cassandra]] (Cassie)
* [[Cassandric]]
* [[Cassander]]
'''May 2022'''
* [[Supersensi]]
* [[Quadrafives]]
'''June 2022'''
* [[Trimean]]

===Proposed in 2023===
'''June 2023'''
* [[Galileic]]
* [[Catagali]]
* [[Quintain]]
'''July 2023'''
* [[Devisemi]]
* [[Garibaldia]] (2.3.5.7.17.19 extension of [[garibaldi]])
* [[Pontia]] (2.3.5.7.19 extension of [[pontiac]])
'''August 2023'''
* [[Heartland]] (2.3.7.11.19 14&41& 48)
* [[Hearts]] (2.3.7.11.19 41&48)
* [[Heartia]]
* [[Heartful]] (2.3.7.11.19 41&75e)
* [[Doublehearted]] (2.3.7.11.19 14&41)
* [[Weasel|Byhearted]] (→ Weasel)
'''September 2023'''
* [[Heartlandia]] (14cf & 41 & 89)
* [[Darkstone]]
* [[Brightstone]]
'''October 2023'''
* [[Magical]]
* [[Magica]]
* [[Magia]]
* [[Sesquartia]]
* [[Charismic]]
* [[Quinmage]]
* [[Hearty]]
* [[Nonant]]
* [[Hemiandromeda]] (29 & 82)
* [[Hemidromeda]] (29 & 111)
'''November 2023'''
* [[Trismegiboh]]
* [[Baldanders]] (2.9.5.7.11 6&35)
* [[Neomirkwai]] (2.3.5.7.11.19 31&41&80)
* [[Dhaivatic]]
* [[Triquart]]
* [[Tertiseptisix]]
* [[Quartiquart]]
* [[Quintiquart]]
* [[Neovulture]] (2.3.5.7.11.13.19 53&58&217)
* [[Neovish]] (2.3.5.7.11.19 50&118&152)
* [[Neobreed]] (41 & 130 & 270)
* [[Neonewt]] (2.3.5.7.11.13.19 41&270)
'''December 2023'''
* [[Heartlander]] (14cf & 41 & 130)
* [[Neosatanic]] (665 & 3325)
* [[Monzosatanic]] (665 & 7980)
* [[Dog]] (2.3.19 7&9)
* [[Armodog]] (2.3.5.7.11.13.19 7&9)

===Proposed in 2024===
'''January 2024'''
* [[Quinbaldi]] (65d & 135)
* [[Quintant]] (65d & 200)
* [[Hemiquintant]] (130 & 135)

'''May 2024'''
* [[Raccoon]] (11 & 32)

'''June 2024'''
* [[Piracy]] (84 & 155)
* [[Pirateship]] (84 & 239)
* [[Unisquary]] (11-limit 72&118&121&130)
* [[Hensquary]] (13-limit 46&72&121&130)
* [[Ekasquary]] (13-limit 72&121&130&190)

===Proposed in 2025===
'''January 2025'''
* [[Pentadecoid]] (2.3.7.11 120&135)
* [[Slendscape]] (255 & 270)
* [[Triacontoid]] (120 & 150)

'''February 2025'''
* [[Devichrome]] (2.3.5.7.19 27&31&41&53)

'''March 2025'''
* [[Octasand]] (11 and 13-limit 27e&41&53)
* [[Devimarv]] (2.3.5.7.19 31&41&53)
** [[Neomarv]] (2.3.5.7.11.19 31&41&53)
* [[Deva]] (2.3.5.7.19 27&41&53)
** [[Neonakika]] (2.3.5.7.11.19 15&27e&41)
** [[Neoctasand]] (2.3.5.7.11.19 27e&41&53)
* [[Nutrient]] (2.3.5.7.19 27&31&41)
** [[Neojovia]] (2.3.5.7.11.19 27e&31&41)
* [[Passkey]] (2.3.5.7.19 41&53&99)
** [[Neolakia]] (2.3.5.7.11.19 41&53&99)

'''April 2025'''
* [[Magikarp]] (2.3.43 5&7)
* [[Hendeca]] (11&22)
* [[Char]] (2.3.5.7.17 19&31&41)
* [[Elven]]

'''June 2025'''
* [[Reversed marvel]] (2.3.5.113 22&31&34)

===Proposed in 2026===
'''January 2026'''
* [[Mendel]] (11-limit 12&31&152)
* [[Moctpine]] (11-limit 29&72&80)
* [[Biscordial]] (12&190)
* [[Triscordial]] (12&231)
* [[Naiad]]
* [[Naiadical]]

'''February 2026'''
* [[Naian]] (8d&13b)
* [[Semihemimean]] (80&118&130)
* [[Marvelpine]]
* [[Seminaiad]]
** [[Seminaiad|Seminaiadia]]
* [[Seesaw]]
** [[Heavy windmill]]
** [[Light windmill]]
* [[Antonian]]
* [[Antonym (temperament)|Antonym]]
* [[Antony]]
* [[Argument (temperament)|Argument]]
* [[Landy]]
* [[Septarium]]

'''March 2026'''
* [[Augenic]]
* [[Sida]]
* [[Bohlenic]] (no-twos temperament)

'''May 2026'''
* [[Xenial]] (19 & 70)
** [[Xenia]] (19 & 51c, 19 & 70d)
* [[Hemigoth]] (34d & 68)
* [[Paraguay]] (19 & 23bcf, 19 & 42ef)
** [[Uruguay]] (19 & 23bc, 19 & 42e)

'''June 2026'''
* [[Sixseven]] (2.3.7 2d & 3)

== Commas which this user proposed names ==
This user proposed new names for following commas.

===Proposed in 2021===
'''April 2021'''
* [[Hewuermera temperaments|hewuermera]]: 589824/588245, satribiru-agu
* [[Triwellismic temperaments|triwellisma]]: 235298/234375, tribizo-asepgu
* [[quadrawellisma]]: 67528125/67108864, lala-quadzo-aquinyo
* [[catasyc comma]]: 390625/387072, laruquadbiyo
* [[Mistismic temperaments|mistisma]]: 458752/455625, sazoquadgu

'''May 2021'''
* [[Hemfiness temperaments|hemfiness comma]]: 4096000/4084101, saquinru-atriyo
* [[Marvelfifths temperaments|marvelfifths comma]]: 158203125/157351936, laquadru-atritriyo
* [[Chromatisma|chromatisma]]: 640000000000000000 / 635585924776181463, trisa-triru-aquadquadyo
* [[Bronzismic temperaments|bronzisma]]: 2097152/2083725, satriru-agugu
* <s>[[Breeze comma|sesquiquartisma]]</s>: 2460375/2458624, laquadru-atruyo (→ breezema, breeze comma)
* [[Mitonismic temperaments|mitonisma]]: 5250987/5242880, laquadzo-agu
* [[Sensamagicfifths temperaments|sensamagicfifths comma]]: {{monzo|-11 42 -7 -14}}, trila-seprurugu
* [[Secanticorn temperaments|secanticornisma]]: 177147/175000, laruquingu
* [[Supermatertismic temperaments|supermatertisma]]: 52734375/52706752, lasepru-atritriyo
* [[Supermasesquartismic temperaments|supermasesquartisma]]: 184528125/184473632, laquadbiru-aquinyo
* [[Septimirod temperaments|septimirod comma]]: 84035/82944, quinzo-ayo

'''June 2021'''
* [[Vorwell clan|vorwell comma]]: 134217728/133984375, sasatriru-aquadbigu
* [[Maitri clan|maitri comma]]: 40353607/40000000, tritrizo-asepgu
* [[compass comma]]: 9765625/9680832, quinruyoyo

'''July 2021'''
* [[4375/4356|fantares comma]]: 4375/4356, luluzoquadyo
* [[Septiness clan|septiness comma]]: 67108864/66706983, sasasepru
* [[decovulture comma]]: 67108864/66976875, sasa-birugugu
* [[Pontiqak temperaments|pontiqak comma]]: 95703125/95551488, lazozotritriyo

'''August 2021'''
* [[Slither clan|slither comma]]: 40960000/40353607, satritriru-aquadyo
* [[Mermismic temperaments|mermisma]]: 2500000/2470629, sepruyo
* [[Valenwuer temperaments|valenwuer comma]]: 110592/109375, sarutribigu
* [[Negricorn temperaments|negricorn comma]]: 153664/151875, saquadzogu
* [[Parkleiness temperaments|parkleiness comma]]: 1959552/1953125, zotritrigu
* [[Mynaslendic temperaments|mynaslender comma]]: 829440/823543, sepru-ayo

'''September 2021'''
* [[65536/64827|buzzardisma]]: 65536/64827, saquadru (→ buzzardsma, buzzard comma)
* [[Neptunismic temperaments|neptunisma]]: 48828125/48771072, laruruleyo

'''October 2021'''
* [[Domicomp temperaments|domicomp comma]]: 413343/409600, lazogugu

'''November 2021'''
* [[Trienstonisma|trienstonisma]]: 43046721/43025920, laquinru-agu
* [[Lokismic temperaments|lokisma]]: 102760448/102515625, sasazozo-atribigu

===Proposed in 2022===
'''January 2022'''
* [[Countriton comma|countriton comma]]: 16777216000000000 / 16677181699666569, quadsa-tritriyo
* [[3042/3025|diagassormisma]]: 3042/3025, bitholugu
* [[gaster comma]]: {{monzo|-70 72 -19}}, quadbila-negu
* [[Mabilismic clan|mabilisma]]: 1071875/1048576, latrizo-aquinyo
* [[pentonisma]]: 281974669312/281950621875, saquinthuzogu

'''February 2022'''
* [[8281/8192|diahuntmisma]]: 8281/8192, labithozo
* [[3645/3584|schismean comma]]: 3645/3584, laruyo

'''March 2022'''
* [[Acrodec temperaments|acrodec comma]]: 7710244864/7688671875, sasa-tribizo-aquadbigu
* [[120/119|lynchisma]]: 120/119, suruyo

'''May 2022'''
* [[Augmist temperaments|augmist comma]]: 1438646272/1423828125, sasa-trizotrigu
* [[Starling temperaments|supersensi comma]]: 33059881728/30517578125, quintrigu

===Proposed in 2023===
'''11 May 2023'''
* [[Syntonoschisma|syntonisma]]: {{monzo|30 -27 7 0 -1}}, trisa-lusepyo (→ syntonoschisma, syntonic schisma)

'''27 May 2023'''
* [[170/169|major naiadma]]: 170/169, sothuthuyo
* [[221/220|minor naiadma]]: 221/220, sotholugu
* <s>[[256/255|diasemisma]]</s>: 256/255, sugu (→ charisma, charic comma)
* [[442/441|seminaiadma]]: 442/441, sothoruru

'''18 July 2023'''
* [[grossmisma]]: {{monzo|-30 -13 -1 1 7 7}}, septholo-azogu

'''21 July 2023'''
* [[3971/3969|heartlandisma]]: 3971/3969, nonoloruru

'''23 July 2023'''
* [[171/170|malcolmisma]]: 171/170, nosugu
* [[190/189|cotylisma]]: 190/189, noruyo
* [[324/323|photisma]]: 324/323, nusu
* [[343/342|nutrisma]]: 343/342, nutrizo
* [[400/399|devichroma]]: 400/399, nuruyoyo
* [[devicisma]]: 633556/633555, nunusothutrilo-azogu

'''15 August 2023'''
* [[133/132|minithirdma]]: 133/132, noluzo
* <s>[[136/135|diatonisma]]</s>: 136/135, sogu (→ diatisma)
* [[154/153|augustma]]: 154/153, sulozo
* [[161/160|major kirnbergerisma]]: 161/160, twethozogu
* [[162/161|minor kirnbergerisma]]: 162/161, twethuru
* [[286/285|chthonisma]]: 286/285, nuthologu
* [[1275/1274|cimbrisma]]: 1275/1274, sothubiruyo
* [[galileisma]]: 171382426877952 / 171359481538165, lalesu-agu

'''16 August 2023'''
* [[85/84|monk comma]]: 85/84, soruyo
* [[456/455|abnobisma]]: 456/455, nothulugu
* [[476/475|hedwigma]]: 476/475, nusozogugu
* [[495/494|eulalisma]]: 495/494, nuthuloyo

'''21 August 2023'''
* [[363/361|godzillisma]]: 363/361, binulo
* [[2431/2430|heptacircle comma]]: 2431/2430, sothologu

'''26 August 2023'''
* [[1083/1078|bihendrixma]]: 1083/1078, nonolururu

'''27 August 2023'''
* [[No-fives subgroup temperaments #Doublehearted|doublehearted comma]]: 5764801/5668704, quadbizo
* [[No-fives subgroup temperaments #Hearts|hearts comma]]: 34451725707/34359738368, trila-quadzo

'''17 September 2023'''
* [[4914/4913|baladisma]]: 4914/4913, trisu-athozo

'''5 November 2023'''
* [[969/968|buddhisma]]: 969/968, nosolulu (→ kingfisher comma)
* [[1331/1330|solvejgsma]]: 1331/1330, nutrilo-arugu
* [[1540/1539|kevolisma]]: 1540/1539, nulozoyo
* [[2376/2375|trichthonisma]]: 2376/2375, nulotrigu
* [[2926/2925|neovulturisma]]: 2926/2925, nothulozogugu
* [[3136/3135|neomirkwaisma]]: 3136/3135, nuluzozogu
* [[3250/3249|martebisma]]: 3250/3249, nunuthotriyo
* [[4200/4199|neosatanisma]]: 4200/4199, nusuthuzoyoyo
* [[5776/5775|neovishma]]: 5776/5775, nonolurugugu
* [[5985/5984|neogrendelisma]]: 5985/5984, nosuluzoyo
* [[6175/6174|neonewtisma]]: 6175/6174, nothotriru-ayoyo

'''12 November 2023'''
* [[460/459|scanisma]]: 460/459, twethosuyo
* [[484/483|pittsburghisma]]: 484/483, twethuloloru
* [[507/506|laodicisma]]: 507/506, twethuthotholu
* [[529/528|preziosisma]]: 529/528, bitwetho-alu
* [[736/735|harvardisma]]: 736/735, twethorurugu
* [[760/759|squadronisma]]: 760/759, twethunoluyo
* [[875/874|nymphisma]]: 875/874, twethunuzotriyo
* [[897/896|lysistratisma]]: 897/896, twethothoru
* [[1105/1104|fragarisma]]: 1105/1104, twethusothoyo
* [[1197/1196|rhodesisma]]: 1197/1196, twethunothuzo
* [[1288/1287|santisma]]: 1288/1287, twethothuluzo
* [[1496/1495|turkisma]]: 1496/1495, twethusothulogu
* [[1863/1862|antinousisma]]: 1863/1862, twethonururu
* [[2024/2023|insincere comma]]: 2024/2023, twethosusuloru
* <s>[[2025/2024|fokkerisma]]</s>: 2025/2024, latwethuluyoyo (→ cupcakesma, cupcake comma)
* [[2185/2184|guangdongisma]]: 2185/2184, twethonothuruyo
* [[2300/2299|travellisma]]: 2300/2299, twethonubiluyo
* [[2646/2645|biyativice comma]]: 2646/2645, bitwethuzo-agu
* [[2737/2736|kotkisma]]: 2737/2736, twethonusozo
* [[3060/3059|vicious comma]]: 3060/3059, twethunusoruyo
* [[3381/3380|seminaiadvice comma]]: 3381/3380, twethobithuzo-agu (→ mikkolisma)
* [[3520/3519|vicedim comma]]: 3520/3519, twethusuloyo
* [[3888/3887|vicetride comma]]: 3888/3887, twethuthuthu
* [[4693/4692|viceaug comma]]: 4693/4692, twethunonosutho
* [[4761/4760|demiquartervice comma]]: 4761/4760, bitwetho-asurugu
* [[vicetertisma]]: 12168/12167, tritwethu-athotho

'''13 November 2023'''
* [[208/207|vicetone comma]]: 208/207, twethutho
* [[231/230|major neutravicema]]: 231/230, twethulozogu
* [[253/252|middle neutravicema]]: 253/252, twetholoru
* [[276/275|minor neutravicema]]: 276/275, twetholugugu
* [[300/299|major naiadvicema]]: 300/299, twethuthuyoyo
* [[323/322|major semivicema]]: 323/322, twethunosoru
* [[391/390|minor naiadvicema]]: 391/390, twethosothugu
* [[392/391|minor semivicema]]: 392/391, twethusuzozo
* [[460/459|vicewolf comma]]: 460/459, twethosuyo
* [[5083/5082|broadviewsma]]: 5083/5082, twethosotholuluru

'''15 November 2023'''
* [[92/91|undinisma]]: 92/91, twethothuru
* [[115/114|yarmanisma]]: 115/114, twethonuyo

'''26 November 2023'''
* [[E-shaped comma]]: 31213/31212, susuthoquadzo

'''1 December 2023'''
* <s>[[kilobyte comma]]</s>: 1024/1023, thiwulu (→ kibisma)
* <s>[[megabyte comma]]</s>: 1048576/1048575, safowuthiwulugugu (→ mebisma)

'''10 December 2023'''
* [[81/76|dog comma]]: 81/76, inu
* [[primate comma]]: 16384/15309, saru

'''23 December 2023'''
* [[841/840|arabellisma]]: 841/840, bitweno-arugu
* [[4901/4900|large grapevine comma]]: 4901/4900, twenothothobirugu, twenobithorugu
* [[6877/6875|grossvice comma]]: 6877/6875, bitwetho-atholuquadgu
* [[7425/7424|small grapevine comma]]: 7425/7424, latwenuloyoyo
* [[mynucuvine comma]]: 10935/10933, labitwenu-athuyo

===Proposed in 2024===
'''2 January 2024'''
* <s>[[slendroschisma|slendrisma]]</s>: 68719476736/68641485507, sasa-quinbiru (→ slendroschisma)

'''27 January 2024'''
* [[116/115|sironisma]]: 116/115, twenotwethugu
* [[117/116|lomisma]]: 117/116, twenutho
* [[175/174|major chthonovinema]]: 175/174, twenuzoyoyo
* [[204/203|kallistisma]]: 204/203, twenusoru
* [[232/231|major paravinema]]: 232/231, twenoluru
* [[261/260|major vinetonema]]: 261/260, twenothugu
* [[290/289|brunisma]]: 290/289, twenosusuyo
* [[320/319|minor paravinema]]: 320/319, twenuluyo
* [[378/377|major semivinema]]: 378/377, twenuthuzo
* [[406/405|minor semivinema]]: 406/405, twenozogu
* [[494/493|minor vinetonema]]: 494/493, twenunosutho
* [[551/550|minor chthonovinema]]: 551/550, twenonolugugu
* [[609/608|vineyard comma]]: 609/608, twenonuzo

'''3 May 2024'''
* [[729/722|undevicesimal diaschisma]]: 729/722, lanunu

'''1 June 2024'''
* [[unisquarisma]]: 12005/11979, trilu-aquadzo-ayo

'''22 December 2024'''
* [[429/425|middle semisixthma]]: 429/425, suthologugu
* [[507/500|large semisixthma]]: 507/500, thothotrigu
* [[1452/1445|small semisixthma]]: 1452/1445, susulologu
* [[semisixthmisma]]: 63888/63869, trisu-athutrilo

===Proposed in 2025===
'''20 March 2025'''
* [[6860/6859|devicubisma]]: 6860/6859, trinuzo-ayo
* [[tricecubisma]]: 29792/29791, trithiwu-anozozo

'''21 March 2025'''
* [[8625/8624|beerglass comma]]: 8625/8624, twetholururutriyo
* [[10626/10625|demiglace comma]]: 10626/10625, twethosulozoquadgu

'''29 March 2025'''
* [[145/144|adeonisma]]: 145/144, twenoyo
* [[552/551|sigelindisma]]: 552/551, twenutwethonu
* [[638/637|moirisma]]: 638/637, twenothuloruru
* [[726/725|joellisma]]: 726/725, twenubilogu
* [[783/782|norisma]]: 783/782, twenotwethusu
* [[1015/1014|christisma]]: 1015/1014, twenothuthuzoyo
* [[1045/1044|michelisma]]: 1045/1044, twenunoloyo
* [[1276/1275|ucclisma]]: 1276/1275, twenosulogugu
* [[1450/1449|raimondisma]]: 1450/1449, twenotwethuruyoyo
* [[1596/1595|itzigsohnisma]]: 1596/1595, twenunoluzogu
* [[1625/1624|norcma]]: 1625/1624, twenuthorutriyo
* [[1683/1682|castafiorisma]]: 1683/1682, bitwenu-asolo
* [[2176/2175|donarisma]]: 2176/2175, twenusogugu
* [[2205/2204|glinkisma]]: 2205/2204, twenunuzozoyo
* [[2262/2261|mitidikisma]]: 2262/2261, twenonusuthoru
* [[5929/5928|manzanisma]]: 5929/5928, nuthubilozo

'''6 April 2025'''
* [[6144/6137|langwisma]]: 6144/6137, nunusu

'''10 April 2025'''
* <s>[[129/128|Magikarp comma]]</s>: 129/128, fotho
* <s>[[130/129|Gyarados comma]]</s>: 130/129, fothuthoyo
* [[2640/2639|hällströmisma]]: 2640/2639, twenuthuloruyo
* [[2784/2783|domeykisma]]: 2784/2783, twenotwethululu
* [[3249/3248|musashinisma]]: 3249/3248, twenunonoru
* [[3451/3450|mentorisma]]: 3451/3450, twenotwethusozogugu
* [[4785/4784|petrovsma]]: 4785/4784, twenotwethuthuloyo
* [[5916/5915|woudisma]]: 5916/5915, twenosothuthurugu

'''11 April 2025'''
* [[10557/10556|rowlandisma]]: 10557/10556, twenutwethosothuru

'''15 April 2025'''
* [[371293/262144|tridecimal antidilemma]]: 371293/262144, laquintho 5th
* [[524288/371293|tridecimal dilemma]]: 524288/371293, saquinthu 4th
* [[1610510/1594323|fifthchromisma]]: 1610510/1594323, saquinlo-ayo

'''17 April 2025'''
* [[2001/2000|major discoverisma]]: 2001/2000, twenotwethotrigu
* [[2002/2001|minor discoverisma]]: 2002/2001, twenutwethutholozo

'''26 April 2025'''
* [[2465/2464|laservine comma]]: 2465/2464, twenosoluruyo

'''27 April 2025'''
* [[280/279|tricetone comma]]: 280/279, thiwuzoyo

'''2 May 2025'''
* [[6912/6859|deviaug comma]]: 6912/6859, trinu

'''3 May 2025'''
* [[93/92|tricema]]: 93/92, thiwotwethu
* [[155/154|scyllisma]]: 155/154, thiwoluruyo
* [[156/155|xanthippisma]]: 156/155, thiwuthogu
* [[187/186|lambertisma]]: 187/186, thiwusolo
* [[217/216|eudorisma]]: 217/216, thiwozo
* [[248/247|lameisma]]: 248/247, thiwonuthu
* [[341/340|californisma]]: 341/340, thiwosulogu
* [[342/341|endymisma]]: 342/341, thiwunolu
* [[435/434|chinthisma]]: 435/434, thiwutwenoruyo
* [[465/464|alektisma]]: 465/464, thiwotwenuyo
* [[528/527|rezisma]]: 528/527, thiwusulo
* [[589/588|croatisma]]: 589/588, thiwonoruru
* [[651/650|antiklisma]]: 651/650, thiwothuzogugu
* [[714/713|ululisma]]: 714/713, thiwutwethusozo

'''4 May 2025'''
* [[2755/2754|avicema]]: 2755/2754, twenonosuyo
* [[4641/4640|vinecute comma]]: 4641/4640, twenusothozogu
* [[5888/5887|vinocular comma]]: 5888/5887, bitwenu-atwethoru
* [[8671/8670|vinous comma]]: 8671/8670, twenotwethosusuthogu

'''9 May 2025'''
* [[6076/6075|large ricegrain comma]]: 6076/6075, sathiwobizogu
* [[6728/6727|sushi comma]]: 6728/6727, bithiwutweno-aru
* [[8960/8959|small ricegrain comma]]: 8960/8959, thiwususuzoyo

'''10 May 2025'''
* [[12636/12635|padriellisma]]: 12636/12635, nunuthorugu
* [[13690/13689|lesleymartin comma]]: 13690/13689, bithisothu-ayo
* [[16337/16335|brown rice comma]]: 16337/16335, bithiwo-asolulugu
* [[23275/23273|sunrise comma]]: 23275/23273, bithisu-anosubizoyo

'''18 May 2025'''
* [[navicular comma]]: 24192/24167, trithu-aluzo
* [[48013/48000|small tridevisemma]]: 48013/48000, trino-azotrigu
* [[93347/93312|trilute comma]]: 93347/93312, notriso
* [[116640/116603|large tridevisemma]]: 116640/116603, trinu-asuyo
* [[176000/175959|triseptichrome comma]]: 176000/175959, nulotriruyo
* [[471648/471625|small triquarterisma]]: 471648/471625, triso-alutrirugu
* [[1257795/1257728|large triquarterisma]]: 1257795/1257728, latrisulo-azoyo
* [[quinticular comma]]: 1419857/1417176, saquinso

'''28 May 2025'''
* [[7942/7935|brigade comma]]: 7942/7935, bitwethuno-alogu

'''29 May 2025'''
* [[barbadisma]]: 114383962274805 / 114349209288704, quadla-thuyo
* [[undevicesimal counterschisma]]: 2.3.19 {{monzo|-75 50 -1}}, seplanu

'''30 May 2025'''
* [[septendecimal quasithirdma]]: 19683/19652, latrisu
* [[septendecimal quasiminthma]]: 6975757441 / 6973568802, sasa-quadbiso

'''1 June 2025'''
* [[undevicesimal quasitonema]]: 34359738368 / 34296447249, sasa-tribinu
* [[deviquasisma]]: 17594077438737 / 17592186044416, lala-sepno

'''9 June 2025'''
* [[226/225|reversed marvel comma]]: 226/225, 113ogg2

'''5 July 2025'''
* [[386/385|siegenisma]]: 386/385, 193o1urg1
* [[57122/57121|phocitremisma]]: 57122/57121, 239uu3o42

===Proposed in 2026===
'''10 January 2026'''
* [[septimagic comma]]: 537824/531441, saquinzo
* [[acromagic comma]]: 26985857024 / 26904200625, sasa-sepzo-aquadgu

'''16 January 2026'''
* [[septrongisma]]: 205891132094649 / 205885750000000, lala-sepru-atritrigu

'''17 January 2026'''
* [[sensibeta comma]]: 1071875/1062882, satrizo-aquinyo

'''20 January 2026'''
* [[undetritisma]]: 205891132094649 / 204800000000000, trila-legu

'''31 January 2026'''
* [[naiadisma]]: 161414428 / 158203125, satritrizogu

'''23 February 2026'''
* [[septarium comma]]: {{monzo|73 77 -84}}, sasepquadtrigu

'''1 May 2026'''
* [[129140163/125000000|xenial comma]]: 129140163 / 125000000, latritrigu

== Scales which this user proposed names ==
This user proposed new names for following scales.

===Proposed in May 2023===
'''16 May 2023'''
* [[3L 16s|magicaltonic]]: 3L_16s MOS scale
* [[16L 3s|muggletonic]]: 16L_3s MOS scale
* [[Nicetone|zarlinoid]]: 3L_2m_2s rank-3 scale (MV3, LMLSLMS and LMLSMLS)
* [[sephipechroid]]: 3L_5m_5s rank-3 scale (MV3, LMSMLSMSLMSMS and LMSMLSMSMLSMS)
* [[magicaloid]]: 3L_8m_8s rank-3 scale (MV3, LMSMSMLSMSMSLMSMSMS and LMSMSMLSMSMSMLSMSMS)

'''18 May 2023'''
* [[Omnidiatonic|interdia]]: 2L_3m_2s rank-3 scale (MV3, LMSLMSM and LMSMLSM)
* [[antinicetone]]: 2L_2m_3s rank-3 scale (MV3, LMSLSMS and LSMLSMS)

'''20 May 2023'''
* [[villoid]]: 7L_2m_2s rank-3 scale (MV3, LLMLLSLLMLS and LLMLLSLMLLS)
* [[intervilla]]: 2L_7m_2s rank-3 scale (MV3, LMMSMLMMSMM and LMMSMMLMSMM)
* [[kleistoid]]: 2L_2m_7s rank-3 scale (MV3, LSMSSLSSMSS and LSSMSLSSMSS)
* [[interoneichro]]: 5L_3m_5s rank-3 scale (MV3, LMSLSLMSLSMLS and LMSLSMLSLSMLS)
* [[sephimechroid]]: 5L_5m_3s rank-3 scale (MV3, LMLMSLMLSMLMS and LMLSMLMLSMLMS)
* [[moharoid]]: 7L_5m_5s rank-3 scale (MV3, LMLSLMSLMLSMLSLMS and LMLSMLSLMLSMLSLMS)
* [[enharoid]]: 5L_7m_5s rank-3 scale (MV3, LMSLMSMLMSLMSMLSM and LMSLMSMLSMLMSMLSM)
* [[beatloid]]: 5L_5m_7s rank-3 scale (MV3, LMSLSMLSMSLMSLSMS and LMSLSMLSMSLSMLSMS)

'''21 May 2023'''
* [[Dualdye|archeoid]]: 5L_1m_1s rank-3 scale (MV3, LLLMLLS and LLLSLLM)
* [[interarcheo]]: 1L_5m_1s rank-3 scale (MV3, LMMMSMM and LMMSMMM)
* [[antarcheoid]]: 1L_1m_5s rank-3 scale (MV3, LSSMSSS and LSSSMSS)

===Proposed in July 2023===
'''2 July 2023'''
* [[jarachroid]]: 5L_5m_2s rank-3 scale (MV3, LMLMLSMLMLMS)
* [[Diachrome|interchroid]]: 5L_2m_5s rank-3 scale (MV3, LMSLSLSMLSLS)
* [[jerachroid]]: 2L_5m_5s rank-3 scale (MV3, LMSMSMLSMSMS)

== Documents ==
Notation:
* [[User:Xenllium/Xenllium's microtonal notation|Xenllium's microtonal notation]]
Tempered scales:
* [[User:Xenllium/Xenllium's circulating scales|Xenllium's circulating scales]]
External links:
* [https://docs.google.com/spreadsheets/d/18CRD8KRk9Gsg_tWSxbDJ_GjprgNvjGTx8ZWcpOEA-0I/edit?usp=sharing Rank-two regular temperament]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://drive.google.com/file/d/1Eov2FSp5Y4X4HPaaHV8nvqHX21LE7AQa/view?usp=drive_link 純正音程と平均律] (日本語/Japanese)
* [https://drive.google.com/file/d/1oBpqNqqIYCZaPuTFEt4HYdIULOMbX6sE/view?usp=drive_link 平均律 (2～31)] (日本語/Japanese)

== Music ==
* ''[https://youtube.com/watch?v=5iBXKRWE-_U Magical life]'' – in 19-tone pure-fifth magic scale
* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning
* [https://youtube.com/watch?v=bbJ0HjPAuaA 【オリジナル曲】かなしいこと / The sadness]

Jubilismic family

2026-06-08T23:32:37Z

Xenllium: Added 17-limit subgroup temperaments

{{Technical data page}}
The '''jubilismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] [[50/49]] in the full [[7-limit]]. It therefore identifies the two septimal tritones [[7/5]] and [[10/7]], an identification familiar from [[12edo]]. While many rank-3 temperaments are planar, a jubilismic temperament divides the [[2/1|octave]] in two. Related to this is the 2.5.7-subgroup {50/49} temperament [[jubilic]].

== Jubilismic ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49

{{Mapping|legend=1| 2 0 0 1 | 0 1 0 0 | 0 0 1 1 }}

: mapping generators: ~7/5, ~3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6673{{c}}, ~3/2 = 702.5906{{c}}, ~5/4 = 380.6287{{c}}
: [[error map]]: {{val| -0.665 -0.030 -7.016 +10.139 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 702.4574{{c}}, ~5/4 = 380.0086{{c}}
: error map: {{val| 0.000 +0.502 -6.305 +11.183 }}

[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: {{monzo list| 1 0 0 0 | 0 1 0 0 | -1/4 0 1/2 1/2 | 1/4 0 1/2 1/2 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.35

{{Optimal ET sequence|legend=1| 4, 8d, 10, 12, 22, 34d, 48, 60d }}

[[Badness]] (Sintel): 0.561

Scales: [[jubilismic10]], [[jubilismic12]]

=== Jubilife ===
There is a natural extension of jubilismic to the 2.3.5.7.17 subgroup, which tempers out [[85/84]] and [[120/119]], and identifies 7/5 with 10/7, 17/12 and 24/17 as the semioctave.

Subgroup: 2.3.5.7.17

Comma list: 50/49, 85/84

Mapping: {{Mapping| 2 0 0 1 5 | 0 1 0 0 1 | 0 0 1 1 0 }}

Optimal tunings:
* WE: ~7/5 = 599.8903{{c}}, ~3/2 = 702.6008{{c}}, ~5/4 = 379.6441{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 702.5503{{c}}, ~5/4 = 379.4777{{c}}

{{Optimal ET sequence|legend=0| 4, 8d, 10, 12, 22, 34d, 48 }}

Badness (Sintel): 0.357

== Undecimal jubilismic ==
Formerly known as ''jubilee'', undecimal jubilismic tempers out [[99/98]] and [[100/99]], and is the most natural extension of jubilismic to the 11-limit. [[11/8]] is found by a stack of two [[7/6]]'s or two [[5/3]]'s octave reduced.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 50/49, 99/98

{{Mapping|legend=1| 2 0 0 1 4 | 0 1 0 0 -2 | 0 0 1 1 2 }}

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 599.6219{{c}}, ~3/2 = 702.9722{{c}}, ~5/4 = 380.4574{{c}}
: [[error map]]: {{val| -0.756 +0.261 -7.369 +9.741 +0.628 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 702.9249{{c}}, ~5/4 = 379.6796{{c}}
: error map: {{val| 0.000 +0.970 -6.634 +10.854 +2.192 }}

{{Optimal ET sequence|legend=1| 4, 8d, 10e, 12, 22, 34d, 48 }}

[[Badness]] (Sintel): 0.673

=== Jubilee ===
Jubilismic temperament works on the [[No-thirteens subgroup temperaments|no-13s]] 17-limit subgroup very well, where the semioctave stands in for ~7/5, ~17/12, ~99/70 and their [[octave complement]]s.

Subgroup: 2.3.5.7.11.17

Comma list: 50/49, 85/84, 99/98

Mapping: {{Mapping| 2 0 0 1 4 5 | 0 1 0 0 -2 1 | 0 0 1 1 2 0 }}

Optimal tunings:
* WE: ~7/5 = 599.8463{{c}}, ~3/2 = 702.9469{{c}}, ~5/4 = 379.5011{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 702.9274{{c}}, ~5/4 = 379.2350{{c}}

{{Optimal ET sequence|legend=0| 4, 8d, 10e, 12, 22, 34d, 48 }}

Badness (Sintel): 0.578

== Festival ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 45/44, 50/49

{{Mapping|legend=1| 2 0 0 1 -4 | 0 1 0 0 2 | 0 0 1 1 1 }}

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.8501{{c}}, ~3/2 = 694.6084{{c}}, ~5/4 = 371.7918{{c}}
: [[error map]]: {{val| +1.700 -5.646 -11.121 +7.216 +13.091 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 694.5001{{c}}, ~5/4 = 372.9799{{c}}
: error map: {{val| 0.000 -7.455 -13.334 +4.154 +10.662 }}

{{Optimal ET sequence|legend=1| 10, 12, 22e, 26 }}

[[Badness]] (Sintel): 0.827

== Fiesta ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 50/49, 56/55

{{Mapping|legend=1| 2 0 0 1 7 | 0 1 0 0 0 | 0 0 1 1 0 }}

[[Optimal tuning]]:
* [[WE]]: ~7/5 = 596.3068{{c}}, ~3/2 = 709.1930{{c}}, ~5/4 = 395.2472{{c}}
: [[error map]]: {{val| -7.386 -0.148 -5.839 +7.955 +22.830 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 709.6819{{c}}, ~5/4 = 391.4911{{c}}
: error map: {{val| 0.000 +7.727 +5.177 +22.665 +48.682 }}

{{Optimal ET sequence|legend=1| 8d, 10, 12, 22e, 30dee, 42ddeee }}

[[Badness]] (Sintel): 0.861

== Jamboree ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 50/49, 55/54

{{Mapping|legend=1| 2 0 0 1 2 | 0 1 0 0 3 | 0 0 1 1 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.0436{{c}}, ~3/2 = 706.7073{{c}}, ~5/4 = 376.8582{{c}}
: [[error map]]: {{val| +0.087 +4.839 -9.281 +8.250 -7.880 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 706.7334{{c}}, ~5/4 = 376.9332{{c}}
: error map: {{val| 0.000 +4.778 -9.381 +8.107 -8.051 }}

{{Optimal ET sequence|legend=1| 8d, 10, 12e, 14c, 22, 58ce }}

[[Badness]] (Sintel): 0.938

[[Category:Temperament families]]
[[Category:Jubilismic family| ]] 
[[Category:Rank 3]]

User:Xenllium

2026-06-08T22:59:25Z

Xenllium: /* Proposed in 2026 */

'''Xenllium''' joined Xenharmonic Wiki on 15 November 2018. The name "Xenllium" is a portmanteau of "Xenharmonic" and "''[[Wikipedia:Trillium|Trillium]]''".

== Temperaments which this user proposed names ==
This user proposed new names for following temperaments.

===Proposed in 2019===
'''March 2019'''
* [[Icositritonic]]
* [[Hemicocomp|Hemicounterpyth]] (→ Hemicocomp)
* [[Hemermacomp|Hemermapyth]] (→ Hemermacomp)
* [[Pentacontatritonic]]
* Hectaheptacontahenatonic (171 & 17442)
* [[Meridic]]
* Tetracontaheptatonic (94 & 705)
* [[Palladium]]
'''April 2019'''
* [[Septisemitonic]]
* [[Tertiathirds]]
* [[Tertiaschis]]

===Proposed in 2021===
'''March 2021'''
* [[Trident]] (→ Alphatrident)
* [[Trillium]] (→ Alphatrillium)
* [[Pseudotrillium]]
* [[Tritricot]]
* [[Decic]]
* [[Cloudtone]]
'''April 2021'''
* [[Hemipental]]
* [[Sengasec]]
* [[Qinto]]
* [[Monzism]]
* [[Houborizic]]
* [[Houbor]]
* [[Gamity]]
* [[Familia]]
* [[Sabric]]
* [[Quadrawell]]
* [[Sfour]]
* [[Rainwell]]
* [[Sengage]]
* [[Marfifths]]
* [[Diatessic]]
* [[Marf]]
'''May 2021'''
* [[Garistearn]]
* [[Decistearn]]
* [[Quadrant]]
* [[Quintilipyth]]
* [[Helenoid]]
* [[Ponta]]
* [[Pontic]]
* [[Bipont]]
* [[Counterbipont]]
* [[Quadrapont]]
* [[Decimetra]]
* [[Stockhausenic]]
* [[Decavish]]
* Bronzoid (109 & 142)
* [[Ennealimmia]]
* [[Paralytic]]
* [[Cotoneum]]
* [[Fog]]
* [[Countermiracle]]
* [[Paraklein]]
* [[Countermiraculous]]
* [[Counterbenediction]]
* [[Counterrevelation]]
* [[Septant]]
* [[Terminal]]
* [[Semiterm]]
* [[Sedia]]
* [[Marthirds]]
* [[Counterkleismic]]
* [[Counterlytic]]
* [[Geb]]
* [[Hemidimfi]]
* [[Maquiloid]]
* [[Paramity]]
'''June 2021'''
* [[Schisennealimmal]]
* [[Schisennealimmic]]
* [[Decades]] (→ Gamelstearn)
* [[Undecentic]]
* [[Pentadecal]]
* [[Quinkee]]
* [[Coditonic]]
* [[Terture]]
* [[Semivulture]]
* [[Eagle]]
* [[Quindromeda]]
* [[Quintakwai]]
* [[Quintagar]]
* [[Quintoneum]]
* [[Quinsandra]]
* [[Quinsandric]]
* [[Quinsandro]]
* [[Semiquindromeda]]
* [[Quintoneoid]]
'''July 2021'''
* [[Mineral]]
* [[Ore]]
* [[Quintaleap]]
* [[Quintupole]]
* [[Quinticosiennic]]
* [[Quintapole]]
* [[Hemiseptisix]]
'''August 2021'''
* [[Goldmine]]
* [[Gamelacomp|Gamelapyth]] (→ Gamelacomp)
* [[Mermacomp|Mermapyth]] (→ Mermacomp)
* [[Undesemi]]
* [[Decipentic]]
* [[Quasijerome]]
* [[Septendesemi]]
* [[Witcher]] (99 & 166, formerly ''witch'', revised in August 2023)
* [[Semiwitcher]] (166 & 198, formerly ''semiwitch'', revised in August 2023)
* [[Hitch]]
* [[Hitcher]]
* [[Quinwell]]
* [[Septiquarschis]]
* [[Quinbetter]]
* [[Ultramarine]]
* [[Mowglic]]
'''Septemper 2021'''
* [[Terminator]]
* [[Quintaschis]]
* [[Quintahelenic]]
* [[Quintahelenoid]]
* [[Semihemiwür]]
* [[Semihemiwürat]]
* [[Beetle]]
* [[Subsedia]]
* [[Quinkwai]]
* [[Quintakwoid]]
* [[Passionate]]
* [[Pash]]
* [[Majvamic]]
* [[Majvamoid]]
* [[Narayana]]
* [[Turkey]]
'''October 2021'''
* [[Lunennealimmal]]
* [[Hexadecoid]]
* [[Undetrita]]
* [[Rubidium]]
* [[Terbium]]
* [[Parakleischis]]
* [[Triacontaheptoid]]
* [[Octogintic]]
* [[Senator]]
* [[Hemivalentine]]
* [[Hemivalentino]]
* [[Hemivalentoid]]
* [[Undetritoid]]
* [[Biquartonic]]
* [[Quindro]]
'''November 2021'''
* [[Tridecatonic]]
* [[Trideci]]
* [[Decimaleap]]
* [[Hemiquindromeda]]
* [[Triboh]]
'''December 2021'''
* [[Hemimabila]]
* [[Sedecic]]
* [[Hyperkleismic]]

===Proposed in 2022===
'''January 2022'''
* [[Cobalt]]
* [[Cobaltous]]
* [[Cobaltic]]
* [[Cobaltite]]
* [[Pontoid]]
* [[Grackle]] (11, 13, 17 and 19-limit extension)
* [[Grackloid]]
* [[Grack]]
* [[Catahelenic]]
* [[Countriton]]
* [[Aufic]]
* [[Pajarina]]
* [[Muggloid]]
* [[Gracecord]]
* [[Gaster]]
* [[Tuskaloosa]]
* [[Muscogee]]
* [[Cohemimabila]]
* [[Trimabila]]
* [[Immune]]
* [[Meantritone]]
* [[Aurora]]
* [[Leapweeker]]
* [[Interaufo]]
* [[Helena]]
'''February 2022'''
* [[Cohendecatonic]]
* [[Nessa]]
* [[Splendecic]]
* [[Prodecic]]
* [[Semimeantone]]
* [[Keenic]]
* [[Warrior]]
* [[Trisensory]]
* [[Bisensi]]
* [[Browser]]
* [[Grazer]]
* [[Quadrasruta]]
* [[Quadrafourths]]
* [[Pajarita]]
'''March 2022'''
* [[Stalagmite|Catamite]] (→ Stalagmite)
* [[Hemikwai]]
* [[Yarm]]
* [[Cohedim]]
* [[October]]
* [[Dichotomic]]
* [[Seminar]]
* [[Chromat]] (11, 13 and 17-limit extension)
* [[Catachrome]]
* [[Chromic]]
* [[Trinity]]
* [[Poseidon]]
* [[Salacia]]
* [[Gammy]]
* [[Tertiaseptia]]
* [[Semitert]]
'''April 2022'''
* <s>[[Meantonic]]</s>
* <s>[[Meantoid]]</s>
* [[Huygens]] (→ Fokkertone)
* <s>[[Meridetonic]]</s>
* <s>[[Meridetoid]]</s>
* <s>[[Sauveuric]]</s>
* [[Meanpoid]]
* <s>[[Meanplop]] (17 and 19-limit extension)</s>
* <s>[[Meanploid]]</s>
* <s>[[Meanenneadecoid]]</s>
* <s>[[Vincenzoid]]</s>
* [[Andromeda]] (17 and 19-limit extension)
* [[Schisicosiennic]]
* [[Schisicosiennoid]]
* [[Cassandra]] (Cassie)
* [[Cassandric]]
* [[Cassander]]
'''May 2022'''
* [[Supersensi]]
* [[Quadrafives]]
'''June 2022'''
* [[Trimean]]

===Proposed in 2023===
'''June 2023'''
* [[Galileic]]
* [[Catagali]]
* [[Quintain]]
'''July 2023'''
* [[Devisemi]]
* [[Garibaldia]] (2.3.5.7.17.19 extension of [[garibaldi]])
* [[Pontia]] (2.3.5.7.19 extension of [[pontiac]])
'''August 2023'''
* [[Heartland]] (2.3.7.11.19 14&41& 48)
* [[Hearts]] (2.3.7.11.19 41&48)
* [[Heartia]]
* [[Heartful]] (2.3.7.11.19 41&75e)
* [[Doublehearted]] (2.3.7.11.19 14&41)
* [[Weasel|Byhearted]] (→ Weasel)
'''September 2023'''
* [[Heartlandia]] (14cf & 41 & 89)
* [[Darkstone]]
* [[Brightstone]]
'''October 2023'''
* [[Magical]]
* [[Magica]]
* [[Magia]]
* [[Sesquartia]]
* [[Charismic]]
* [[Quinmage]]
* [[Hearty]]
* [[Nonant]]
* [[Hemiandromeda]] (29 & 82)
* [[Hemidromeda]] (29 & 111)
'''November 2023'''
* [[Trismegiboh]]
* [[Baldanders]] (2.9.5.7.11 6&35)
* [[Neomirkwai]] (2.3.5.7.11.19 31&41&80)
* [[Dhaivatic]]
* [[Triquart]]
* [[Tertiseptisix]]
* [[Quartiquart]]
* [[Quintiquart]]
* [[Neovulture]] (2.3.5.7.11.13.19 53&58&217)
* [[Neovish]] (2.3.5.7.11.19 50&118&152)
* [[Neobreed]] (41 & 130 & 270)
* [[Neonewt]] (2.3.5.7.11.13.19 41&270)
'''December 2023'''
* [[Heartlander]] (14cf & 41 & 130)
* [[Neosatanic]] (665 & 3325)
* [[Monzosatanic]] (665 & 7980)
* [[Dog]] (2.3.19 7&9)
* [[Armodog]] (2.3.5.7.11.13.19 7&9)

===Proposed in 2024===
'''January 2024'''
* [[Quinbaldi]] (65d & 135)
* [[Quintant]] (65d & 200)
* [[Hemiquintant]] (130 & 135)

'''May 2024'''
* [[Raccoon]] (11 & 32)

'''June 2024'''
* [[Piracy]] (84 & 155)
* [[Pirateship]] (84 & 239)
* [[Unisquary]] (11-limit 72&118&121&130)
* [[Hensquary]] (13-limit 46&72&121&130)
* [[Ekasquary]] (13-limit 72&121&130&190)

===Proposed in 2025===
'''January 2025'''
* [[Pentadecoid]] (2.3.7.11 120&135)
* [[Slendscape]] (255 & 270)
* [[Triacontoid]] (120 & 150)

'''February 2025'''
* [[Devichrome]] (2.3.5.7.19 27&31&41&53)

'''March 2025'''
* [[Octasand]] (11 and 13-limit 27e&41&53)
* [[Devimarv]] (2.3.5.7.19 31&41&53)
** [[Neomarv]] (2.3.5.7.11.19 31&41&53)
* [[Deva]] (2.3.5.7.19 27&41&53)
** [[Neonakika]] (2.3.5.7.11.19 15&27e&41)
** [[Neoctasand]] (2.3.5.7.11.19 27e&41&53)
* [[Nutrient]] (2.3.5.7.19 27&31&41)
** [[Neojovia]] (2.3.5.7.11.19 27e&31&41)
* [[Passkey]] (2.3.5.7.19 41&53&99)
** [[Neolakia]] (2.3.5.7.11.19 41&53&99)

'''April 2025'''
* [[Magikarp]] (2.3.43 5&7)
* [[Hendeca]] (11&22)
* [[Char]] (2.3.5.7.17 19&31&41)
* [[Elven]]

'''June 2025'''
* [[Reversed marvel]] (2.3.5.113 22&31&34)

===Proposed in 2026===
'''January 2026'''
* [[Mendel]] (11-limit 12&31&152)
* [[Moctpine]] (11-limit 29&72&80)
* [[Biscordial]] (12&190)
* [[Triscordial]] (12&231)
* [[Naiad]]
* [[Naiadical]]

'''February 2026'''
* [[Naian]] (8d&13b)
* [[Semihemimean]] (80&118&130)
* [[Marvelpine]]
* [[Seminaiad]]
** [[Seminaiad|Seminaiadia]]
* [[Seesaw]]
** [[Heavy windmill]]
** [[Light windmill]]
* [[Antonian]]
* [[Antonym (temperament)|Antonym]]
* [[Antony]]
* [[Argument (temperament)|Argument]]
* [[Landy]]
* [[Septarium]]

'''March 2026'''
* [[Augenic]]
* [[Sida]]
* [[Bohlenic]] (no-twos temperament)

'''May 2026'''
* [[Xenial]] (19 & 70)
** [[Xenia]] (19 & 51c, 19 & 70d)
* [[Hemigoth]] (34d & 68)
* [[Paraguay]] (19 & 23bcf, 19 & 42ef)
** [[Uruguay]] (19 & 23bc, 19 & 42e)

'''June 2026'''
* [[Jubilife]] (2.3.5.7.17 4 & 10 & 12)
* [[Jubilee]] (2.3.5.7.11.17 4 & 10e & 12)

== Commas which this user proposed names ==
This user proposed new names for following commas.

===Proposed in 2021===
'''April 2021'''
* [[Hewuermera temperaments|hewuermera]]: 589824/588245, satribiru-agu
* [[Triwellismic temperaments|triwellisma]]: 235298/234375, tribizo-asepgu
* [[quadrawellisma]]: 67528125/67108864, lala-quadzo-aquinyo
* [[catasyc comma]]: 390625/387072, laruquadbiyo
* [[Mistismic temperaments|mistisma]]: 458752/455625, sazoquadgu

'''May 2021'''
* [[Hemfiness temperaments|hemfiness comma]]: 4096000/4084101, saquinru-atriyo
* [[Marvelfifths temperaments|marvelfifths comma]]: 158203125/157351936, laquadru-atritriyo
* [[Chromatisma|chromatisma]]: 640000000000000000 / 635585924776181463, trisa-triru-aquadquadyo
* [[Bronzismic temperaments|bronzisma]]: 2097152/2083725, satriru-agugu
* <s>[[Breeze comma|sesquiquartisma]]</s>: 2460375/2458624, laquadru-atruyo (→ breezema, breeze comma)
* [[Mitonismic temperaments|mitonisma]]: 5250987/5242880, laquadzo-agu
* [[Sensamagicfifths temperaments|sensamagicfifths comma]]: {{monzo|-11 42 -7 -14}}, trila-seprurugu
* [[Secanticorn temperaments|secanticornisma]]: 177147/175000, laruquingu
* [[Supermatertismic temperaments|supermatertisma]]: 52734375/52706752, lasepru-atritriyo
* [[Supermasesquartismic temperaments|supermasesquartisma]]: 184528125/184473632, laquadbiru-aquinyo
* [[Septimirod temperaments|septimirod comma]]: 84035/82944, quinzo-ayo

'''June 2021'''
* [[Vorwell clan|vorwell comma]]: 134217728/133984375, sasatriru-aquadbigu
* [[Maitri clan|maitri comma]]: 40353607/40000000, tritrizo-asepgu
* [[compass comma]]: 9765625/9680832, quinruyoyo

'''July 2021'''
* [[4375/4356|fantares comma]]: 4375/4356, luluzoquadyo
* [[Septiness clan|septiness comma]]: 67108864/66706983, sasasepru
* [[decovulture comma]]: 67108864/66976875, sasa-birugugu
* [[Pontiqak temperaments|pontiqak comma]]: 95703125/95551488, lazozotritriyo

'''August 2021'''
* [[Slither clan|slither comma]]: 40960000/40353607, satritriru-aquadyo
* [[Mermismic temperaments|mermisma]]: 2500000/2470629, sepruyo
* [[Valenwuer temperaments|valenwuer comma]]: 110592/109375, sarutribigu
* [[Negricorn temperaments|negricorn comma]]: 153664/151875, saquadzogu
* [[Parkleiness temperaments|parkleiness comma]]: 1959552/1953125, zotritrigu
* [[Mynaslendic temperaments|mynaslender comma]]: 829440/823543, sepru-ayo

'''September 2021'''
* [[65536/64827|buzzardisma]]: 65536/64827, saquadru (→ buzzardsma, buzzard comma)
* [[Neptunismic temperaments|neptunisma]]: 48828125/48771072, laruruleyo

'''October 2021'''
* [[Domicomp temperaments|domicomp comma]]: 413343/409600, lazogugu

'''November 2021'''
* [[Trienstonisma|trienstonisma]]: 43046721/43025920, laquinru-agu
* [[Lokismic temperaments|lokisma]]: 102760448/102515625, sasazozo-atribigu

===Proposed in 2022===
'''January 2022'''
* [[Countriton comma|countriton comma]]: 16777216000000000 / 16677181699666569, quadsa-tritriyo
* [[3042/3025|diagassormisma]]: 3042/3025, bitholugu
* [[gaster comma]]: {{monzo|-70 72 -19}}, quadbila-negu
* [[Mabilismic clan|mabilisma]]: 1071875/1048576, latrizo-aquinyo
* [[pentonisma]]: 281974669312/281950621875, saquinthuzogu

'''February 2022'''
* [[8281/8192|diahuntmisma]]: 8281/8192, labithozo
* [[3645/3584|schismean comma]]: 3645/3584, laruyo

'''March 2022'''
* [[Acrodec temperaments|acrodec comma]]: 7710244864/7688671875, sasa-tribizo-aquadbigu
* [[120/119|lynchisma]]: 120/119, suruyo

'''May 2022'''
* [[Augmist temperaments|augmist comma]]: 1438646272/1423828125, sasa-trizotrigu
* [[Starling temperaments|supersensi comma]]: 33059881728/30517578125, quintrigu

===Proposed in 2023===
'''11 May 2023'''
* [[Syntonoschisma|syntonisma]]: {{monzo|30 -27 7 0 -1}}, trisa-lusepyo (→ syntonoschisma, syntonic schisma)

'''27 May 2023'''
* [[170/169|major naiadma]]: 170/169, sothuthuyo
* [[221/220|minor naiadma]]: 221/220, sotholugu
* <s>[[256/255|diasemisma]]</s>: 256/255, sugu (→ charisma, charic comma)
* [[442/441|seminaiadma]]: 442/441, sothoruru

'''18 July 2023'''
* [[grossmisma]]: {{monzo|-30 -13 -1 1 7 7}}, septholo-azogu

'''21 July 2023'''
* [[3971/3969|heartlandisma]]: 3971/3969, nonoloruru

'''23 July 2023'''
* [[171/170|malcolmisma]]: 171/170, nosugu
* [[190/189|cotylisma]]: 190/189, noruyo
* [[324/323|photisma]]: 324/323, nusu
* [[343/342|nutrisma]]: 343/342, nutrizo
* [[400/399|devichroma]]: 400/399, nuruyoyo
* [[devicisma]]: 633556/633555, nunusothutrilo-azogu

'''15 August 2023'''
* [[133/132|minithirdma]]: 133/132, noluzo
* <s>[[136/135|diatonisma]]</s>: 136/135, sogu (→ diatisma)
* [[154/153|augustma]]: 154/153, sulozo
* [[161/160|major kirnbergerisma]]: 161/160, twethozogu
* [[162/161|minor kirnbergerisma]]: 162/161, twethuru
* [[286/285|chthonisma]]: 286/285, nuthologu
* [[1275/1274|cimbrisma]]: 1275/1274, sothubiruyo
* [[galileisma]]: 171382426877952 / 171359481538165, lalesu-agu

'''16 August 2023'''
* [[85/84|monk comma]]: 85/84, soruyo
* [[456/455|abnobisma]]: 456/455, nothulugu
* [[476/475|hedwigma]]: 476/475, nusozogugu
* [[495/494|eulalisma]]: 495/494, nuthuloyo

'''21 August 2023'''
* [[363/361|godzillisma]]: 363/361, binulo
* [[2431/2430|heptacircle comma]]: 2431/2430, sothologu

'''26 August 2023'''
* [[1083/1078|bihendrixma]]: 1083/1078, nonolururu

'''27 August 2023'''
* [[No-fives subgroup temperaments #Doublehearted|doublehearted comma]]: 5764801/5668704, quadbizo
* [[No-fives subgroup temperaments #Hearts|hearts comma]]: 34451725707/34359738368, trila-quadzo

'''17 September 2023'''
* [[4914/4913|baladisma]]: 4914/4913, trisu-athozo

'''5 November 2023'''
* [[969/968|buddhisma]]: 969/968, nosolulu (→ kingfisher comma)
* [[1331/1330|solvejgsma]]: 1331/1330, nutrilo-arugu
* [[1540/1539|kevolisma]]: 1540/1539, nulozoyo
* [[2376/2375|trichthonisma]]: 2376/2375, nulotrigu
* [[2926/2925|neovulturisma]]: 2926/2925, nothulozogugu
* [[3136/3135|neomirkwaisma]]: 3136/3135, nuluzozogu
* [[3250/3249|martebisma]]: 3250/3249, nunuthotriyo
* [[4200/4199|neosatanisma]]: 4200/4199, nusuthuzoyoyo
* [[5776/5775|neovishma]]: 5776/5775, nonolurugugu
* [[5985/5984|neogrendelisma]]: 5985/5984, nosuluzoyo
* [[6175/6174|neonewtisma]]: 6175/6174, nothotriru-ayoyo

'''12 November 2023'''
* [[460/459|scanisma]]: 460/459, twethosuyo
* [[484/483|pittsburghisma]]: 484/483, twethuloloru
* [[507/506|laodicisma]]: 507/506, twethuthotholu
* [[529/528|preziosisma]]: 529/528, bitwetho-alu
* [[736/735|harvardisma]]: 736/735, twethorurugu
* [[760/759|squadronisma]]: 760/759, twethunoluyo
* [[875/874|nymphisma]]: 875/874, twethunuzotriyo
* [[897/896|lysistratisma]]: 897/896, twethothoru
* [[1105/1104|fragarisma]]: 1105/1104, twethusothoyo
* [[1197/1196|rhodesisma]]: 1197/1196, twethunothuzo
* [[1288/1287|santisma]]: 1288/1287, twethothuluzo
* [[1496/1495|turkisma]]: 1496/1495, twethusothulogu
* [[1863/1862|antinousisma]]: 1863/1862, twethonururu
* [[2024/2023|insincere comma]]: 2024/2023, twethosusuloru
* <s>[[2025/2024|fokkerisma]]</s>: 2025/2024, latwethuluyoyo (→ cupcakesma, cupcake comma)
* [[2185/2184|guangdongisma]]: 2185/2184, twethonothuruyo
* [[2300/2299|travellisma]]: 2300/2299, twethonubiluyo
* [[2646/2645|biyativice comma]]: 2646/2645, bitwethuzo-agu
* [[2737/2736|kotkisma]]: 2737/2736, twethonusozo
* [[3060/3059|vicious comma]]: 3060/3059, twethunusoruyo
* [[3381/3380|seminaiadvice comma]]: 3381/3380, twethobithuzo-agu (→ mikkolisma)
* [[3520/3519|vicedim comma]]: 3520/3519, twethusuloyo
* [[3888/3887|vicetride comma]]: 3888/3887, twethuthuthu
* [[4693/4692|viceaug comma]]: 4693/4692, twethunonosutho
* [[4761/4760|demiquartervice comma]]: 4761/4760, bitwetho-asurugu
* [[vicetertisma]]: 12168/12167, tritwethu-athotho

'''13 November 2023'''
* [[208/207|vicetone comma]]: 208/207, twethutho
* [[231/230|major neutravicema]]: 231/230, twethulozogu
* [[253/252|middle neutravicema]]: 253/252, twetholoru
* [[276/275|minor neutravicema]]: 276/275, twetholugugu
* [[300/299|major naiadvicema]]: 300/299, twethuthuyoyo
* [[323/322|major semivicema]]: 323/322, twethunosoru
* [[391/390|minor naiadvicema]]: 391/390, twethosothugu
* [[392/391|minor semivicema]]: 392/391, twethusuzozo
* [[460/459|vicewolf comma]]: 460/459, twethosuyo
* [[5083/5082|broadviewsma]]: 5083/5082, twethosotholuluru

'''15 November 2023'''
* [[92/91|undinisma]]: 92/91, twethothuru
* [[115/114|yarmanisma]]: 115/114, twethonuyo

'''26 November 2023'''
* [[E-shaped comma]]: 31213/31212, susuthoquadzo

'''1 December 2023'''
* <s>[[kilobyte comma]]</s>: 1024/1023, thiwulu (→ kibisma)
* <s>[[megabyte comma]]</s>: 1048576/1048575, safowuthiwulugugu (→ mebisma)

'''10 December 2023'''
* [[81/76|dog comma]]: 81/76, inu
* [[primate comma]]: 16384/15309, saru

'''23 December 2023'''
* [[841/840|arabellisma]]: 841/840, bitweno-arugu
* [[4901/4900|large grapevine comma]]: 4901/4900, twenothothobirugu, twenobithorugu
* [[6877/6875|grossvice comma]]: 6877/6875, bitwetho-atholuquadgu
* [[7425/7424|small grapevine comma]]: 7425/7424, latwenuloyoyo
* [[mynucuvine comma]]: 10935/10933, labitwenu-athuyo

===Proposed in 2024===
'''2 January 2024'''
* <s>[[slendroschisma|slendrisma]]</s>: 68719476736/68641485507, sasa-quinbiru (→ slendroschisma)

'''27 January 2024'''
* [[116/115|sironisma]]: 116/115, twenotwethugu
* [[117/116|lomisma]]: 117/116, twenutho
* [[175/174|major chthonovinema]]: 175/174, twenuzoyoyo
* [[204/203|kallistisma]]: 204/203, twenusoru
* [[232/231|major paravinema]]: 232/231, twenoluru
* [[261/260|major vinetonema]]: 261/260, twenothugu
* [[290/289|brunisma]]: 290/289, twenosusuyo
* [[320/319|minor paravinema]]: 320/319, twenuluyo
* [[378/377|major semivinema]]: 378/377, twenuthuzo
* [[406/405|minor semivinema]]: 406/405, twenozogu
* [[494/493|minor vinetonema]]: 494/493, twenunosutho
* [[551/550|minor chthonovinema]]: 551/550, twenonolugugu
* [[609/608|vineyard comma]]: 609/608, twenonuzo

'''3 May 2024'''
* [[729/722|undevicesimal diaschisma]]: 729/722, lanunu

'''1 June 2024'''
* [[unisquarisma]]: 12005/11979, trilu-aquadzo-ayo

'''22 December 2024'''
* [[429/425|middle semisixthma]]: 429/425, suthologugu
* [[507/500|large semisixthma]]: 507/500, thothotrigu
* [[1452/1445|small semisixthma]]: 1452/1445, susulologu
* [[semisixthmisma]]: 63888/63869, trisu-athutrilo

===Proposed in 2025===
'''20 March 2025'''
* [[6860/6859|devicubisma]]: 6860/6859, trinuzo-ayo
* [[tricecubisma]]: 29792/29791, trithiwu-anozozo

'''21 March 2025'''
* [[8625/8624|beerglass comma]]: 8625/8624, twetholururutriyo
* [[10626/10625|demiglace comma]]: 10626/10625, twethosulozoquadgu

'''29 March 2025'''
* [[145/144|adeonisma]]: 145/144, twenoyo
* [[552/551|sigelindisma]]: 552/551, twenutwethonu
* [[638/637|moirisma]]: 638/637, twenothuloruru
* [[726/725|joellisma]]: 726/725, twenubilogu
* [[783/782|norisma]]: 783/782, twenotwethusu
* [[1015/1014|christisma]]: 1015/1014, twenothuthuzoyo
* [[1045/1044|michelisma]]: 1045/1044, twenunoloyo
* [[1276/1275|ucclisma]]: 1276/1275, twenosulogugu
* [[1450/1449|raimondisma]]: 1450/1449, twenotwethuruyoyo
* [[1596/1595|itzigsohnisma]]: 1596/1595, twenunoluzogu
* [[1625/1624|norcma]]: 1625/1624, twenuthorutriyo
* [[1683/1682|castafiorisma]]: 1683/1682, bitwenu-asolo
* [[2176/2175|donarisma]]: 2176/2175, twenusogugu
* [[2205/2204|glinkisma]]: 2205/2204, twenunuzozoyo
* [[2262/2261|mitidikisma]]: 2262/2261, twenonusuthoru
* [[5929/5928|manzanisma]]: 5929/5928, nuthubilozo

'''6 April 2025'''
* [[6144/6137|langwisma]]: 6144/6137, nunusu

'''10 April 2025'''
* <s>[[129/128|Magikarp comma]]</s>: 129/128, fotho
* <s>[[130/129|Gyarados comma]]</s>: 130/129, fothuthoyo
* [[2640/2639|hällströmisma]]: 2640/2639, twenuthuloruyo
* [[2784/2783|domeykisma]]: 2784/2783, twenotwethululu
* [[3249/3248|musashinisma]]: 3249/3248, twenunonoru
* [[3451/3450|mentorisma]]: 3451/3450, twenotwethusozogugu
* [[4785/4784|petrovsma]]: 4785/4784, twenotwethuthuloyo
* [[5916/5915|woudisma]]: 5916/5915, twenosothuthurugu

'''11 April 2025'''
* [[10557/10556|rowlandisma]]: 10557/10556, twenutwethosothuru

'''15 April 2025'''
* [[371293/262144|tridecimal antidilemma]]: 371293/262144, laquintho 5th
* [[524288/371293|tridecimal dilemma]]: 524288/371293, saquinthu 4th
* [[1610510/1594323|fifthchromisma]]: 1610510/1594323, saquinlo-ayo

'''17 April 2025'''
* [[2001/2000|major discoverisma]]: 2001/2000, twenotwethotrigu
* [[2002/2001|minor discoverisma]]: 2002/2001, twenutwethutholozo

'''26 April 2025'''
* [[2465/2464|laservine comma]]: 2465/2464, twenosoluruyo

'''27 April 2025'''
* [[280/279|tricetone comma]]: 280/279, thiwuzoyo

'''2 May 2025'''
* [[6912/6859|deviaug comma]]: 6912/6859, trinu

'''3 May 2025'''
* [[93/92|tricema]]: 93/92, thiwotwethu
* [[155/154|scyllisma]]: 155/154, thiwoluruyo
* [[156/155|xanthippisma]]: 156/155, thiwuthogu
* [[187/186|lambertisma]]: 187/186, thiwusolo
* [[217/216|eudorisma]]: 217/216, thiwozo
* [[248/247|lameisma]]: 248/247, thiwonuthu
* [[341/340|californisma]]: 341/340, thiwosulogu
* [[342/341|endymisma]]: 342/341, thiwunolu
* [[435/434|chinthisma]]: 435/434, thiwutwenoruyo
* [[465/464|alektisma]]: 465/464, thiwotwenuyo
* [[528/527|rezisma]]: 528/527, thiwusulo
* [[589/588|croatisma]]: 589/588, thiwonoruru
* [[651/650|antiklisma]]: 651/650, thiwothuzogugu
* [[714/713|ululisma]]: 714/713, thiwutwethusozo

'''4 May 2025'''
* [[2755/2754|avicema]]: 2755/2754, twenonosuyo
* [[4641/4640|vinecute comma]]: 4641/4640, twenusothozogu
* [[5888/5887|vinocular comma]]: 5888/5887, bitwenu-atwethoru
* [[8671/8670|vinous comma]]: 8671/8670, twenotwethosusuthogu

'''9 May 2025'''
* [[6076/6075|large ricegrain comma]]: 6076/6075, sathiwobizogu
* [[6728/6727|sushi comma]]: 6728/6727, bithiwutweno-aru
* [[8960/8959|small ricegrain comma]]: 8960/8959, thiwususuzoyo

'''10 May 2025'''
* [[12636/12635|padriellisma]]: 12636/12635, nunuthorugu
* [[13690/13689|lesleymartin comma]]: 13690/13689, bithisothu-ayo
* [[16337/16335|brown rice comma]]: 16337/16335, bithiwo-asolulugu
* [[23275/23273|sunrise comma]]: 23275/23273, bithisu-anosubizoyo

'''18 May 2025'''
* [[navicular comma]]: 24192/24167, trithu-aluzo
* [[48013/48000|small tridevisemma]]: 48013/48000, trino-azotrigu
* [[93347/93312|trilute comma]]: 93347/93312, notriso
* [[116640/116603|large tridevisemma]]: 116640/116603, trinu-asuyo
* [[176000/175959|triseptichrome comma]]: 176000/175959, nulotriruyo
* [[471648/471625|small triquarterisma]]: 471648/471625, triso-alutrirugu
* [[1257795/1257728|large triquarterisma]]: 1257795/1257728, latrisulo-azoyo
* [[quinticular comma]]: 1419857/1417176, saquinso

'''28 May 2025'''
* [[7942/7935|brigade comma]]: 7942/7935, bitwethuno-alogu

'''29 May 2025'''
* [[barbadisma]]: 114383962274805 / 114349209288704, quadla-thuyo
* [[undevicesimal counterschisma]]: 2.3.19 {{monzo|-75 50 -1}}, seplanu

'''30 May 2025'''
* [[septendecimal quasithirdma]]: 19683/19652, latrisu
* [[septendecimal quasiminthma]]: 6975757441 / 6973568802, sasa-quadbiso

'''1 June 2025'''
* [[undevicesimal quasitonema]]: 34359738368 / 34296447249, sasa-tribinu
* [[deviquasisma]]: 17594077438737 / 17592186044416, lala-sepno

'''9 June 2025'''
* [[226/225|reversed marvel comma]]: 226/225, 113ogg2

'''5 July 2025'''
* [[386/385|siegenisma]]: 386/385, 193o1urg1
* [[57122/57121|phocitremisma]]: 57122/57121, 239uu3o42

===Proposed in 2026===
'''10 January 2026'''
* [[septimagic comma]]: 537824/531441, saquinzo
* [[acromagic comma]]: 26985857024 / 26904200625, sasa-sepzo-aquadgu

'''16 January 2026'''
* [[septrongisma]]: 205891132094649 / 205885750000000, lala-sepru-atritrigu

'''17 January 2026'''
* [[sensibeta comma]]: 1071875/1062882, satrizo-aquinyo

'''20 January 2026'''
* [[undetritisma]]: 205891132094649 / 204800000000000, trila-legu

'''31 January 2026'''
* [[naiadisma]]: 161414428 / 158203125, satritrizogu

'''23 February 2026'''
* [[septarium comma]]: {{monzo|73 77 -84}}, sasepquadtrigu

'''1 May 2026'''
* [[129140163/125000000|xenial comma]]: 129140163 / 125000000, latritrigu

== Scales which this user proposed names ==
This user proposed new names for following scales.

===Proposed in May 2023===
'''16 May 2023'''
* [[3L 16s|magicaltonic]]: 3L_16s MOS scale
* [[16L 3s|muggletonic]]: 16L_3s MOS scale
* [[Nicetone|zarlinoid]]: 3L_2m_2s rank-3 scale (MV3, LMLSLMS and LMLSMLS)
* [[sephipechroid]]: 3L_5m_5s rank-3 scale (MV3, LMSMLSMSLMSMS and LMSMLSMSMLSMS)
* [[magicaloid]]: 3L_8m_8s rank-3 scale (MV3, LMSMSMLSMSMSLMSMSMS and LMSMSMLSMSMSMLSMSMS)

'''18 May 2023'''
* [[Omnidiatonic|interdia]]: 2L_3m_2s rank-3 scale (MV3, LMSLMSM and LMSMLSM)
* [[antinicetone]]: 2L_2m_3s rank-3 scale (MV3, LMSLSMS and LSMLSMS)

'''20 May 2023'''
* [[villoid]]: 7L_2m_2s rank-3 scale (MV3, LLMLLSLLMLS and LLMLLSLMLLS)
* [[intervilla]]: 2L_7m_2s rank-3 scale (MV3, LMMSMLMMSMM and LMMSMMLMSMM)
* [[kleistoid]]: 2L_2m_7s rank-3 scale (MV3, LSMSSLSSMSS and LSSMSLSSMSS)
* [[interoneichro]]: 5L_3m_5s rank-3 scale (MV3, LMSLSLMSLSMLS and LMSLSMLSLSMLS)
* [[sephimechroid]]: 5L_5m_3s rank-3 scale (MV3, LMLMSLMLSMLMS and LMLSMLMLSMLMS)
* [[moharoid]]: 7L_5m_5s rank-3 scale (MV3, LMLSLMSLMLSMLSLMS and LMLSMLSLMLSMLSLMS)
* [[enharoid]]: 5L_7m_5s rank-3 scale (MV3, LMSLMSMLMSLMSMLSM and LMSLMSMLSMLMSMLSM)
* [[beatloid]]: 5L_5m_7s rank-3 scale (MV3, LMSLSMLSMSLMSLSMS and LMSLSMLSMSLSMLSMS)

'''21 May 2023'''
* [[Dualdye|archeoid]]: 5L_1m_1s rank-3 scale (MV3, LLLMLLS and LLLSLLM)
* [[interarcheo]]: 1L_5m_1s rank-3 scale (MV3, LMMMSMM and LMMSMMM)
* [[antarcheoid]]: 1L_1m_5s rank-3 scale (MV3, LSSMSSS and LSSSMSS)

===Proposed in July 2023===
'''2 July 2023'''
* [[jarachroid]]: 5L_5m_2s rank-3 scale (MV3, LMLMLSMLMLMS)
* [[Diachrome|interchroid]]: 5L_2m_5s rank-3 scale (MV3, LMSLSLSMLSLS)
* [[jerachroid]]: 2L_5m_5s rank-3 scale (MV3, LMSMSMLSMSMS)

== Documents ==
Notation:
* [[User:Xenllium/Xenllium's microtonal notation|Xenllium's microtonal notation]]
Tempered scales:
* [[User:Xenllium/Xenllium's circulating scales|Xenllium's circulating scales]]
External links:
* [https://docs.google.com/spreadsheets/d/18CRD8KRk9Gsg_tWSxbDJ_GjprgNvjGTx8ZWcpOEA-0I/edit?usp=sharing Rank-two regular temperament]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://drive.google.com/file/d/1Eov2FSp5Y4X4HPaaHV8nvqHX21LE7AQa/view?usp=drive_link 純正音程と平均律] (日本語/Japanese)
* [https://drive.google.com/file/d/1oBpqNqqIYCZaPuTFEt4HYdIULOMbX6sE/view?usp=drive_link 平均律 (2～31)] (日本語/Japanese)

== Music ==
* ''[https://youtube.com/watch?v=5iBXKRWE-_U Magical life]'' – in 19-tone pure-fifth magic scale
* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning
* [https://youtube.com/watch?v=bbJ0HjPAuaA 【オリジナル曲】かなしいこと / The sadness]

Jubilismic chords

2026-06-08T22:24:01Z

Xenllium:

A '''jubilismic chord''' is an [[essentially tempered chord]] tempered by the jubilisma, [[50/49]]. There are two jubilismic chords available in the [[7-odd-limit]], and an additional one in the [[9-odd-limit]], all being palindromic [[tetrad]]s involving the [[semi-octave]] which stands in for [[7/5]][[~]][[10/7]]:
* 1–5/4–7/5–7/4 with steps 5/4, 8/7, 5/4, 8/7;
* 1–6/5–7/5–12/7 with steps 6/5, 7/6, 6/5, 7/6;
* 1–9/7–7/5–9/5 with steps 9/7, 10/9, 9/7, 10/9.

See [[Jubilismic family #Jubilismic]] for a list of [[equal temperament]]s with these chords.

== Undecimal jubilismic chords ==
An '''undecimal jubilismic chord''' is an essentially tempered chord of undecimal jubilismic (formerly known as ''jubilee''), with intervals in the tonality diamond of the 11-odd-limit. Since 50/49 and [[99/98]] are tempered out, undecimal jubilismic temperament supports jubilismic chords, [[mothwellsmic chords]], and [[ptolemismic chords]]. There are also undecimal jubilismic chords which are none of these, the essentially undecimal jubilismic chords:

* 1–7/6–11/8–5/3 with steps 7/6, 7/6, 6/5, 6/5;
* 1–7/6–11/8–3/2–5/3 with steps 7/6, 7/6, 12/11, 10/9, 6/5, and its inverse
* 1–12/11–9/7–3/2–9/5 with steps 12/11, 7/6, 7/6, 6/5, 10/9;
* 1–7/6–11/8–3/2–5/3–11/6 with steps 7/6, 7/6, 12/11, 10/9, 10/9, 12/11.

[[Category:7-odd-limit chords]]
[[Category:9-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Tetrads]]
[[Category:Jubilismic]]

Dyadic chord

2026-06-07T08:07:02Z

Xenllium: Merged list of innate chords to the list of essentially tempered dyadic chords

{{Distinguish| Dyad }}

A '''dyadic chord''', also described as a ''dyadically consonant chord'' or ''pairwise consonant chord'', is a [[chord]] each of whose [[interval]]s belongs to a specified set of intervals considered to be [[consonance and dissonance|consonant]]; it is therefore relative to the set of intervals in question.

For example, the [[tetrad]]
* 1 – 6/5 – 7/5 – 8/5

is a dyadic chord in the [[7-odd-limit]] since every interval involved in it is an element of the 7-odd-limit [[tonality diamond]]. Now if we replace 7/5 with 10/7:
* 1 – 6/5 – 10/7 – 8/5

is ''not'' a dyadic chord in the 7-odd-limit. Although each note is 7-odd-limit over the bass, the interval between 10/7 and 6/5 is 25/21, and that between 10/7 and 8/5 is 28/25 – these are not 7-odd-limit.

The significance of dyadic chords and of the paradigm where all interval pairs are examined in the chord has the psychoacoustic basis of timbral fusion and emergence of the [[virtual fundamental]]. In the above examples, it can be shown that the lower harmonics of each note in the first chord blends better than in the second. Meanwhile, the virtual fundamental of the first chord appears 5/1 below the bass, whereas that of the second appears much lower, at 35/1 below the bass as the denominators "fight" each other. For these reasons we tend to find the first chord more consonant than the second.

== Essentially tempered dyadic chord ==
In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in just intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. An ''essentially just'' dyadic chord is a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[4:5:6|1 – 5/4 – 3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0 – 10 – 18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1 – 5/4 – 3/2.

[[File:Essentially tempered chord.png|400px|thumb|right|A more in-depth work-through of the starling 1-6/5-10/7 essentially tempered chord example]]

An '''essentially tempered dyadic chord''', often simply written as '''essentially tempered chord''', or '''dyadically essentially tempered chord''' ('''DET chord'''){{idio}}, is a chord defined in a [[regular temperament]] and relative to a set of consonances such that each interval is within that consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1 – 6/5 – 10/7, when mapped by starling temperament, which makes [[126/125]] [[vanish]], has each of its intervals in the set of [[7-odd-limit]] consonances which is the tempering of the 7-odd-limit diamond by 126/125 (this is because 10/7 is off from 36/25 by 126/125, and therefore 10/7 and 36/25 are tempered together in starling temperament, and since 36/25 = (6/5)², the interval from 6/5 to 10/7 in starling may be heard as a second move by 6/5). However, (10/7)/(6/5) = 25/21 is [[25-odd-limit]], and there is no other 7-odd-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord.

Essentially tempered dyadic chords are a related notion to [[comma pump]]s, and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.

== Innate comma chord ==
An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit. However, instead of specifying the targeted JI ratios as in an [[#Essentially tempered dyadic chord|essentially tempered chord]], an innate comma chord describes only the general chord shape: the [[mapped interval]] representations or the [[interval span|sizes]] of its constituent intervals.

For example, the [[augmented triad]] in 5-limit JI is an innate comma chord below the 25-odd-limit, because it is impossible to tune all three major thirds (in the four-note chord doubling the root up an octave) to [[5/4]] or any other 5-limit interval with odd limit below 25: the innate comma here is 128/125 (41¢). In practice, it might be sung or played justly but with a large odd limit (containing [[wolf interval]]s): for example, 1–5/4–8/5–2, or 1–5/4–25/16–2 (or even 1–5/4–25/16–125/64). Or it might be tempered, e.g. in 12edo as 0¢–400¢–800¢–1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented triad, because the middle [[11/9]] interval is a neutral third rather than a major third.)

== Anomalous saturated suspension ==
An '''anomalous saturated suspension''' ('''ASS'''), introduced by [[Graham Breed]]<ref>[https://www.webcitation.org/60VBgPSUS ''Anomalous Saturated Suspensions'']</ref>, is a ''q''-odd-limit just dyadic chord to which no pitch ''q''-odd-limit pitch class can be added while keeping it in the ''q''-odd-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5:…:''q'' chord or the 1:1/3:1/5:…:1/''q'' chord. The existence of such chords was discovered by [[Paul Erlich]]<ref>[https://www.webcitation.org/60VCUHe6d ''Anomalous Saturated Suspensions -- Paul Erlich's post'']</ref>. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.

* [[12:15:18:20]]
* [[12:14:18:21]]

For a complete list of ASS chords through the 23-odd-limit see [[List of anomalous saturated suspensions]].

== List of just intonation tetrads ==
* [[Seven limit tetrads]]
* [[Nine limit tetrads]]
* [[1-3-7-11 tetrads]]
* [[Thirteen limit tetrads]]
* [[Fifteen limit tetrads]]
* [[Seventeen limit tetrads]]

== List of essentially tempered dyadic chords ==
Here are some pages on certain essentially tempered dyadic chords, sorted by the lowest odd limit in which they are available (shaded rows mark [[octave-splitting comma]] chords, which are induced by octave-splitting commas). See [[Dyadic chord/Pattern of essentially tempered chords]] for some notable abstract chord patterns.

=== 5-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated Temperament !! Associated Commas
|-
| bgcolor="#D4D4D4" | [[Augmented triad]] || bgcolor="#D4D4D4" | [[Augmented (temperament)|Augmented]] || bgcolor="#D4D4D4" | [[128/125]]
|}

=== 7-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| bgcolor="#D4D4D4" | [[Jubilismic chords]] || bgcolor="#D4D4D4" | [[Jubilismic]] || bgcolor="#D4D4D4" | [[50/49]]
|-
| [[Archytas chords]] || [[Archytas]] || [[64/63]]
|-
| [[Starling chords]] || [[Starling]] || [[126/125]]
|-
| bgcolor="#D4D4D4" | [[5edo|Blackwood pentad]] || bgcolor="#D4D4D4" | [[Blackwood]] || bgcolor="#D4D4D4" | [[28/27]], [[49/48]]
|-
| bgcolor="#D4D4D4" | [[Diminished seventh chord]] || bgcolor="#D4D4D4" | [[Diminished (temperament)|Diminished]] || bgcolor="#D4D4D4" | [[36/35]], [[50/49]]
|}

=== 9-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Didymic chords]] || [[Meantone]] || [[81/80]]
|-
| [[Marvel chords]] || [[Marvel]] || [[225/224]]
|-
| [[Sensamagic chords]] || [[Sensamagic]] || [[245/243]]
|}

=== 11-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Mothwellsmic chords]] || [[Mothwellsmic]] || [[99/98]]
|-
| [[Ptolemismic chords]] || [[Ptolemismic]] || [[100/99]]
|-
| [[Biyatismic chords]] || [[Biyatismic]] || [[121/120]]
|-
| [[Valinorsmic chords]] || [[Valinorsmic]] || [[176/175]]
|-
| [[Rastmic chords]] || [[Rastmic]] || [[243/242]]
|-
| [[Frostmic chords]] || [[Frostmic]] || [[245/242]]
|-
| [[Keenanismic chords]] || [[Keenanismic]] || [[385/384]]
|-
| [[Werckismic chords]] || [[Werckismic]] || [[441/440]]
|-
| [[Swetismic chords]] || [[Swetismic]] || [[540/539]]
|-
| [[Pentacircle chords]] || [[Pentacircle]] || [[896/891]]
|-
| bgcolor="#D4D4D4" | [[Jubilismic chords|Undecimal jubilismic chords]] || bgcolor="#D4D4D4" | [[Jubilismic family|Undecimal jubilismic]] || bgcolor="#D4D4D4" | [[50/49]], [[99/98]]
|-
| [[Undecimal marvel chords]] || [[Marvel]] || 225/224, 385/384
|-
| [[Prodigy chords]] || [[Prodigy]] || 225/224, 441/440
|-
| [[Undecimal sensamagic chords]] || [[Sensamagic]] || 245/243, 385/384
|-
| [[Jove chords]] || [[Jove]] || 243/242, 441/440
|-
| [[Miracle chords]] || [[Miracle]] || 225/224, 243/242, 385/384
|-
| [[Magic chords]] || [[Magic]] || 100/99, 225/224, 245/243
|-
| [[Keemic chords]] || [[Keemic]] || 100/99, 385/384
|-
| [[Orwell tetrad]] || [[Orwellian]] || 99/98, 385/384
|-
| [[Tutonic hexad|Tutonic hexads]] || [[Meantone]] || 81/80, 99/98, 126/125
|-
| [[Baldanders chords|Baldanders hexads]] || [[Andromeda]] || 100/99, 225/224, 245/242
|-
| [[Porcupine heptad]] || [[Porkypine]] || 55/54, 100/99
|}

=== 13-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Buzurgismic chords]] || [[Buzurgismic]] || [[169/168]]
|-
| [[Mynucumic chords]] || [[Mynucumic]] || [[196/195]]
|-
| [[Gassormic chords]] || [[Gassormic]] || [[275/273]]
|-
| [[Marveltwin chords]] || [[Marveltwin]] || [[325/324]]
|-
| [[Sooty fox chords]] || [[Sooty fox]] || [[343/338]]
|-
| [[Ratwolfsmic chords]] || [[Ratwolfsmic]] || [[351/350]]
|-
| [[Major minthmic chords]] || [[Major minthmic]] || [[352/351]]
|-
| [[Minor minthmic chords]] || [[Minor minthmic]] || [[364/363]]
|-
| [[Huntmic chords]] || [[Huntmic]] || [[640/637]]
|-
| [[Squbemic chords]] || [[Squbemic]] || [[729/728]]
|-
| [[Cuthbert chords]] || [[Cuthbert]] || [[847/845]]
|-
| [[Sinbadmic chords]] || [[Sinbadmic]] || [[1001/1000]]
|-
| [[Kestrel chords]] || [[Kestrel]] || [[1188/1183]]
|-
| [[Catadictmic chords]] || [[Catadictmic]] || [[1287/1280]]
|-
| [[Lambeth chords]] || [[Lambeth]] || [[1573/1568]]
|-
| [[Petrmic chords]] || [[Petrmic]] || [[2200/2197]]
|-
| [[Rastmic chords #Rastgross heptad|Rastgross heptad]] || [[Namo]] || 144/143, 243/242
|-
| [[Parapyth chords]] || [[Parapyth]], [[gentle]] || 352/351, 364/363
|-
| [[Hecate hexad]] || [[Hecate]] || 225/224, 325/324, 385/384
|-
| [[Woodpecker octad]] || [[Woodpecker]] || 66/65, 121/120, 126/125
|-
| [[Miraculous decad]] || [[Miraculous]], [[Revelation]] || 105/104, 196/195, 512/507
|}

=== 15-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Island chords]] || [[The Archipelago|Island]] || [[676/675]]
|-
| [[Nicolic chords]] || [[Nicolic]] || [[1575/1573]]
|-
| [[Myhemiwell chords]] || [[Myhemiwell]] || [[3388/3375]]
|-
| [[Battaglia chord]] || [[Marvel]] || 225/224
|-
| [[Tetracot chords]] || [[Tetracot]] || 100/99, 243/242
|-
| [[Orwell tetrad|Orwell ennead]] || [[Guanyin]] || 176/175, 540/539
|}

=== 17-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Augustmic chords]] || [[Augustmic]] || [[154/153]]
|-
| [[Charismic chords]] || [[Charismic]] || [[256/255]]
|-
| [[Prototannismic chords]] || [[Prototannismic]] || [[273/272]]
|-
| [[Ursulismic chords]] || [[Ursulismic]] || [[375/374]]
|-
| [[Seminaiadmic chords]] || [[Seminaiadmic]] || [[442/441]]
|-
| [[Monardismic chords]] || [[Monardismic]] || [[561/560]]
|-
| [[Dakotismic chords]] || [[Dakotismic]] || [[595/594]]
|-
| [[September chords]] || [[September]] || [[715/714]]
|-
| [[Horizmic chords]] || [[Horizmic]] || [[833/832]]
|-
| [[Ainismic chords]] || [[Ainismic]] || [[936/935]]
|-
| [[Twosquare chords]] || [[Twosquare]] || [[1089/1088]]
|-
| [[Quadrantonismic chords]] || [[Quadrantonismic]] || [[1156/1155]]
|-
| [[Cimbrismic chords]] || [[Cimbrismic]] || [[1275/1274]]
|-
| [[Fidesmic chords]] || [[Fidesmic]] || [[2025/2023]]
|-
| [[Heptacircle chords]] || [[Heptacircle]] || [[2431/2430]]
|-
| [[Sextantonismic chords]] || [[Sextantonismic]] || [[2601/2600]]
|}

=== 19-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Photismic chords]] || [[Photismic]] || [[324/323]]
|-
| [[Nutrismic chords]] || [[Nutrismic]] || [[343/342]]
|-
| [[Dudon chords]] || [[Dudon]] || [[361/360]]
|-
| [[Devichromic chords]] || [[Devichromic]] || [[400/399]]
|-
| [[Abnobismic chords]] || [[Abnobismic]] || [[456/455]]
|-
| [[Hedwigmic chords]] || [[Hedwigmic]] || [[476/475]]
|-
| [[Eulalismic chords]] || [[Eulalismic]] || [[495/494]]
|-
| [[Boethius chords]] || [[Boethius]] || [[513/512]]
|-
| [[Kingfisher chords]] || [[Kingfisher]] || [[969/968]]
|-
| [[Bihendrixmic chords]] || [[Bihendrixmic]] || [[1083/1078]]
|-
| [[Eratosthenes chords]] || [[Eratosthenes]] || [[1216/1215]]
|-
| [[Solvejgsmic chords]] || [[Solvejgsmic]] || [[1331/1330]]
|-
| [[Aureusmic chords]] || [[Aureusmic]] || [[1445/1444]]
|-
| [[Pinkanberry chords]] || [[Pinkanberry]] || [[1521/1520]]
|-
| [[Kevolismic chords]] || [[Kevolismic]] || [[1540/1539]]
|-
| [[Ramanujanismic chords]] || [[Ramanujanismic]] || [[1729/1728]]
|-
| [[Blumeyer chords]] || [[Blumeyer]] || [[2432/2431]]
|-
| [[Neovulturismic chords]] || [[Neovulturismic]] || [[2926/2925]]
|-
| [[Neomirkwaismic chords]] || [[Neomirkwaismic]] || [[3136/3135]]
|-
| [[Neosatanismic chords]] || [[Neosatanismic]] || [[4200/4199]]
|-
| [[Langwismic chords]] || [[Langwismic]] || [[6144/6137]]
|}

=== 21-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Slendric pentad]] || [[Gamelismic]] || [[1029/1024]]
|-
| [[Palingenetic chords]] || [[Palingenetic]] || [[1701/1700]]
|-
| [[Xenismic chords]] || [[Xenismic]] || [[2058/2057]]
|-
| [[Ibnsinmic chords]] || [[Ibnsinmic]] || [[2080/2079]]
|-
| [[Heartlandismic chords]] || [[Heartlandismic]] || [[3971/3969]]
|-
| [[Minismic chords]] || [[Minismic]] || [[4096/4095]]
|-
| [[Baladismic chords]] || [[Baladismic]] || [[4914/4913]]
|-
| [[Neogrendelismic chords]] || [[Neogrendelismic]] || [[5985/5984]]
|-
| [[Heartland #Chords|Heartland chords]] || [[Heartland]] || 243/242, 1083/1078
|}

=== 23-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Scanismic chords]] || [[Scanismic]] || [[460/459]]
|-
| [[Pittsburghismic chords]] || [[Pittsburghismic]] || [[484/483]]
|-
| [[Laodicismic chords]] || [[Laodicismic]] || [[507/506]]
|-
| [[Preziosismic chords]] || [[Preziosismic]] || [[529/528]]
|-
| [[Worcester chords]] || [[Worcester]] || [[576/575]]
|-
| [[Harvardismic chords]] || [[Harvardismic]] || [[736/735]]
|-
| [[Squadronismic chords]] || [[Squadronismic]] || [[760/759]]
|-
| [[Lysistratismic chords]] || [[Lysistratismic]] || [[897/896]]
|-
| [[Fragarismic chords]] || [[Fragarismic]] || [[1105/1104]]
|-
| [[Rodessismic chords]] || [[Rodessismic]] || [[1197/1196]]
|-
| [[Triaphonismic chords]] || [[Triaphonismic]] || [[1288/1287]]
|-
| [[Turkismic chords]] || [[Turkismic]] || [[1496/1495]]
|-
| [[Antinousismic chords]] || [[Antinousismic]] || [[1863/1862]]
|-
| [[Artifismic chords]] || [[Artifismic]] || [[2024/2023]]
|-
| [[Cupcake chords]] || [[Cupcake]] || [[2025/2024]]
|-
| [[Guangdongismic chords]] || [[Guangdongismic]] || [[2185/2184]]
|-
| [[Travellismic chords]] || [[Travellismic]] || [[2300/2299]]
|-
| [[Biyativice chords]] || [[Biyativice]] || [[2646/2645]]
|-
| [[Kotkismic chords]] || [[Kotkismic]] || [[2737/2736]]
|-
| [[Vicious chords]] || [[Vicious]] || [[3060/3059]]
|-
| [[Mikkolismic chords]] || [[Mikkolismic]] || [[3381/3380]]
|-
| [[Vicedim chords]] || [[Vicedim]] || [[3520/3519]]
|-
| [[Shoalmic chords]] || [[Shoalmic]] || [[3888/3887]]
|-
| [[Viceaug chords]] || [[Viceaug]] || [[4693/4692]]
|-
| [[Demiquartervice chords]] || [[Demiquartervice]] || [[4761/4760]]
|-
| [[Broadviewsmic chords]] || [[Broadviewsmic]] || [[5083/5082]]
|-
| [[Vicetertismic chords]] || [[Vicetertismic]] || [[Vicetertisma|12168/12167]]
|}

=== 25-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Tunbarsmic chords]] || [[Tunbarsmic]] || [[625/624]]
|-
| [[Nymphismic chords]] || [[Nymphismic]] || [[875/874]]
|-
| [[Noellismic chords]] || [[Noellismic]] || [[1225/1224]]
|-
| [[Trichthonismic chords]] || [[Trichthonismic]] || [[2376/2375]]
|-
| [[Sperasmic chords]] || [[Sperasmic]] || [[2500/2499]]
|-
| [[Lehmerismic chords]] || [[Lehmerismic temperaments|Lehmerismic]] || [[3025/3024]]
|-
| [[Martebismic chords]] || [[Martebismic]] || [[3250/3249]]
|-
| [[Leprechaun chords]] || [[Leprechaun]] || [[4225/4224]]
|-
| [[Neovishmic chords]] || [[Neovishmic]] || [[5776/5775]]
|-
| [[Neonewtismic chords]] || [[Neonewtismic]] || [[6175/6174]]
|}

=== 27-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Ragismic chords]] || [[Ragismic]] || [[4375/4374]]
|-
| [[Hagendorfismic chords]] || [[Hagendorfismic]] || [[8075/8073]]
|}

=== 29-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Vinecute chords]] || [[Vinecute]] || [[4641/4640]]
|-
| [[Large grapevine chords]] || [[Large grapevine]] || [[4901/4900]]
|-
| [[Small grapevine chords]] || [[Small grapevine]] || [[7425/7424]]
|-
| [[Vinous chords]] || [[Vinous]] || [[8671/8670]]
|}

=== 31-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Kibismic chords]] || [[Kibismic]] || [[1024/1023]]
|-
| [[Acronymismic chords]] || [[Acronymismic]] || [[Acronymisma|17577/17576]]
|-
| [[Tricecubismic chords]] || [[Tricecubismic]] || [[Tricecubisma|29792/29791]]
|}

=== 33-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Wizardharry chords]] || [[Wizardharry]] || [[4000/3993]]
|-
| [[Flashmic chords]] || [[The Flashmob|Flashmic]] || [[12376/12375]]
|}

=== 35-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Lummic chords]] || [[Lummic]] || [[1716/1715]]
|}

=== 37-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Bullionismic chords]] || [[Bullionismic]] || [[5292/5291]]
|}

=== 39-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Harmonismic chords]] || [[Harmonismic]] || [[10648/10647]]
|}

=== 55-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Jacobin chords]] || [[The Jacobins|Jacobin]] || [[6656/6655]]
|}

== List of essentially just dyadic chords ==
As chords that are unambiguous counterparts to common JI chords are not of particular relevance to this page, most of the entries here will be what [[User:Kaiveran|Kaiveran]] calls '''plurichords''', where there are multiple sets of consonances that a given chord can be mapped to. Note that this can still lead to ambiguous tonality in the case of otonal and utonal intervals being identified together.
{| class="wikitable"
|+
! Chords
! Equivalent Mappings
! Associated Commas
|-
| [[Hendrix chord]]
| 8:10:14:19 ~ 12:15:21:28
| [[57/56]]
|-
| [[Rootsubminor triad]]
| 6:7:9 ~ 16:19:24
| 57/56
|-
| [[The Biosphere|Biosphere]] triads
| 6:7:9 ~ 26:30:39 10:13:15 ~ 14:18:21
| [[91/90]]
|-
| [[Rootminor triad]]
| 10:12:15 ~ 16:19:24
| [[96/95]]
|-
| [[Lynchismic chords|Lynchismic plurichords]]
| 12:14:17:20 ~ 1/(20:17:14:12) 6:7:8:10 ~ 1/(20:17:15:12)
| [[120/119]]
|-
| [[Augustmic chords|Augustmic plurichords]]
| 14:17:18:22 ~ 1/(22:18:17:14)
| [[154/153]]
|}

== See also ==
* [[Essential tempering comma]]

== Notes ==

[[Category:Terms]]
[[Category:Regular temperament theory]]
[[Category:Dyadic chords| ]] 
[[Category:Lists of chords]]
<references />

Fractional-octave temperaments

2026-06-07T07:09:43Z

Xenllium:

'''Fractional-octave temperaments''' are [[temperament]]s which have a [[period]] which corresponds to a [[just]] [[interval]] mapped to a fraction of the [[octave]], that is one step of an [[edo]].

== Theory ==
Fractional-octave temperaments are valuable with regards to [[Polysystemic|polysystemicism]] and polychromatics. They are acoustically significant with regards to containing modes of limited transposition, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory.

=== Terminology ===
The terminology was developed by [[Eliora]]. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a ''wireframe'', and individual notes of that equal division are called ''hinges''. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y)-ET which repeats y times within that period, its "wireframe" is y-ET. If an equal division is a subset of a temperament, it is said to ''subtend'' the temperament, just how hinges on a ferris wheel subtend the structure to make it rotate and function.

The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, [[compton family]] tempers out the Pythagorean comma and maps 7 steps of 12edo to [[3/2]]. Likewise, a lot of 10th-octave temperaments have a [[13/8]] as 7\10, and 26th-octave temperaments often have a [[7/4]] for 21\26.

However, an equal division does not have to be harmonically decent to be a wireframe for a fractional-octave temperament. If an equal division has multiples which are high in consistency or are zeta equal divisions or otherwise harmonically strong, it can produce a lot of such temperaments—notable examples being [[20edo]] or [[32edo]]. Likewise, proximity of a step of equal division to a comma is often a source of these temperaments—for example [[56edo]]'s step being directly close to [[81/80]], and 44edo's step being extremely close to [[64/63]].

=== Disagreement between temperament catalog strategy and fractional-octave practice ===
Traditional regular temperament perspective on periods and generators has a shortcoming when it comes to handling fractional-octave temperaments, as it treats divisions of periods (for example, what [[hemiennealimmal]] is to [[ennealimmal]]) as [[extension]]s of a temperament with a subset period. However, fractional-octave temperaments and scales are sought for being able to treat an each equal division as an entity in its own right, so a composer might find hemiennealimmal to be a drastically different system to ennealimmal in line with [[18edo]] being very different from [[9edo]]. This facet is reflected by the distinction of strong and weak extensions.

A particularly strong offender of this is the [[landscape microtemperaments]] list, which features temperaments which are all supersets of 3edo, but from a composer's perspective it contains wildly different temperaments due to the fact that edo multiples of 3 themselves are different. For example, magnesium (12), and zinc (30), are both landscape systems due to being multiples of 3, but 30edo is drastically different from 12edo in terms of composition, and therefore such temperaments are not alike at all.

== Octave-splitting comma ==
An '''octave-splitting comma''' is a rational [[interval]] which induces a fractional-octave temperament. If tempering out the comma leads to splitting the [[octave]] into ''N'' equal parts (''N'' > 1), the comma is an octave-splitting comma which induces an ''N''th-octave temperament. This terminology was developed by [[Xenllium]].

=== Property ===
While a comma is given in the ''p''-limit [[monzo]] form {{monzo| ''a''2 ''a''3 ''a''5 … ''a''''p'' }}, as a rational interval, it is an octave-splitting comma if and only if GCD(''a''2, ''a''3, ''a''5, …, ''a''''p'') = 1 and ''N'' = GCD(''a''3, ''a''5, …, ''a''''p'') > 1, and leads to splitting the octave into ''N'' equal parts.

=== Examples ===
Below is a list of octave-splitting commas for common temperaments:

{| class="wikitable"
|-
! Comma !! Associated temperament !! Harmonic limit !! Splitting order
|-
| [[256/243]] || [[Limmic temperaments|Blackwood]] || 3 || 5
|-
| [[2187/2048]] || [[Apotome family|Whitewood]] || 3 || 7
|-
| [[531441/524288]] || [[Compton family|Compton]] || 3 || 12
|-
| [[2048/2025]] || [[Diaschismic family|Diaschismic]] || 5 || 2
|-
| [[128/125]] || [[Augmented family|Augmented]] || 5 || 3
|-
| [[648/625]] || [[Diminished family|Diminished]] || 5 || 4
|-
| [[50/49]] || [[Jubilismic clan|Jubilismic]] || 7 || 2
|-
| [[250047/250000]] || [[Landscape microtemperaments|Landscape]] || 7 || 3
|-
| [[9801/9800]] || [[Kalismic temperaments|Kalismic]] || 11 || 2
|-
| [[289/288]] || [[Semitonismic]] || 17 || 2
|}

== Individual pages of temperaments by subtending equal division ==
=== 2 to 100 ===
{| class="wikitable center-all"
|+
|
| [[2nd-octave temperaments|2]]
| [[3rd-octave temperaments|3]]
| [[4th-octave temperaments|4]]
| [[5th-octave temperaments|5]]
| [[6th-octave temperaments|6]]
| [[7th-octave temperaments|7]]
| [[8th-octave temperaments|8]]
| [[9th-octave temperaments|9]]
| [[10th-octave temperaments|10]]
|-
| [[11th-octave temperaments|11]]
| [[12th-octave temperaments|12]]/[[Compton family|C]]
| [[13th-octave temperaments|13]]
| [[14th-octave temperaments|14]]
| [[15th-octave temperaments|15]]
| [[16th-octave temperaments|16]]
| [[17th-octave temperaments|17]]
| [[18th-octave temperaments|18]]
| [[19th-octave temperaments|19]]
| [[20th-octave temperaments|20]]
|-
| [[21st-octave temperaments|21]]
| [[22nd-octave temperaments|22]]
| [[23rd-octave temperaments|23]]
| [[24th-octave temperaments|24]]
| [[25th-octave temperaments|25]]
| [[26th-octave temperaments|26]]
| [[27th-octave temperaments|27]]
| [[28th-octave temperaments|28]]
| [[29th-octave temperaments|29]]
| [[30th-octave temperaments|30]]
|-
| [[31st-octave temperaments|31]]
| [[32nd-octave temperaments|32]]
| [[33rd-octave temperaments|33]]
| [[34th-octave temperaments|34]]
| [[35th-octave temperaments|35]]
| [[36th-octave temperaments|36]]
| [[37th-octave temperaments|37]]
| [[38th-octave temperaments|38]]
| [[39th-octave temperaments|39]]
| [[40th-octave temperaments|40]]
|-
| [[41st-octave temperaments|41]]/[[Countercomp family|CC]]
| [[42nd-octave temperaments|42]]
| [[43rd-octave temperaments|43]]
| [[44th-octave temperaments|44]]
| [[45th-octave temperaments|45]]
| [[46th-octave temperaments|46]]
| [[47th-octave temperaments|47]]
| [[48th-octave temperaments|48]]
| [[49th-octave temperaments|49]]
| [[50th-octave temperaments|50]]
|-
| [[51st-octave temperaments|51]]
| [[52nd-octave temperaments|52]]
| [[53rd-octave temperaments|53]]/[[Mercator family|M]]
| [[54th-octave temperaments|54]]
| [[55th-octave temperaments|55]]
| [[56th-octave temperaments|56]]
| [[57th-octave temperaments|57]]
| [[58th-octave temperaments|58]]
| [[59th-octave temperaments|59]]
| [[60th-octave temperaments|60]]
|-
| [[61st-octave temperaments|61]]
| [[62nd-octave temperaments|62]]
| [[63rd-octave temperaments|63]]
| [[64th-octave temperaments|64]]
| [[65th-octave temperaments|65]]
| [[66th-octave temperaments|66]]
| [[67th-octave temperaments|67]]
| [[68th-octave temperaments|68]]
| [[69th-octave temperaments|69]]
| [[70th-octave temperaments|70]]
|-
| [[71st-octave temperaments|71]]
| [[72nd-octave temperaments|72]]
| [[73rd-octave temperaments|73]]
| [[74th-octave temperaments|74]]
| [[75th-octave temperaments|75]]
| [[76th-octave temperaments|76]]
| [[77th-octave temperaments|77]]
| [[78th-octave temperaments|78]]
| [[79th-octave temperaments|79]]
| [[80th-octave temperaments|80]]
|-
| [[81st-octave temperaments|81]]
| [[82nd-octave temperaments|82]]
| [[83rd-octave temperaments|83]]
| [[84th-octave temperaments|84]]
| [[85th-octave temperaments|85]]
| [[86th-octave temperaments|86]]
| [[87th-octave temperaments|87]]
| [[88th-octave temperaments|88]]
| [[89th-octave temperaments|89]]
| [[90th-octave temperaments|90]]
|-
| [[91st-octave temperaments|91]]
| [[92nd-octave temperaments|92]]
| [[93rd-octave temperaments|93]]
| [[94th-octave temperaments|94]]
| [[95th-octave temperaments|95]]
| [[96th-octave temperaments|96]]
| [[97th-octave temperaments|97]]
| [[98th-octave temperaments|98]]
| [[99th-octave temperaments|99]]
| [[100th-octave temperaments|100]]
|}

=== 101 and up ===
[[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]]

* C = compton family
* CC = countercomp family
* M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments

== Temperaments discussed elsewhere ==
Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:

* 1\2 period temperaments
** [[Diaschismic family|Diaschismic temperaments]]
** [[Vishnuzmic family|Vishnuzmic temperaments]]
** [[Jubilismic clan|Jubilismic temperaments]]
** [[Varunismic temperaments]]
** [[Lokismic temperaments]]
** [[Nimona|Nimona temperament]]
* 1\3 period temperaments
** [[Augmented family|Augmented temperaments]]
** [[Misty family|Misty temperaments]]
** [[Landscape microtemperaments|Landscape temperaments]]
* 1\4 period temperaments
** [[Diminished family|Diminished temperaments]]
** [[Undim family|Undim temperaments]]
* 1\5 period temperaments
** [[Quintile family|Quintile temperaments]]
** [[Quintosec family|Quintosec temperaments]]
** [[Trisedodge family|Trisedodge temperaments]]
** [[Cloudy clan|Cloudy temperaments]]
** [[Limmic temperaments]]
* 1\6 period temperaments
** [[Augmented family #Hexe|Hexe]]
** [[Landscape microtemperaments #Sextile|Sextile]]
** [[Stearnsmic clan #Stearnscape|Stearnscape]]
* [[Akjaysma|Akjaysmic temperaments]] (1\7 period)
** [[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
** [[Schismatic family #Septant|Septant]]
** [[Whitewood family #Whitewood|Whitewood temperaments]]
** [[Keemic temperaments #Sevond|Sevond]]
** [[Mistismic temperaments #Neutron|Neutron]]
* [[Ragismic microtemperaments #Octoid|Octoid]], [[Schismatic family #Octant|octant]] (1\8 period)
* [[Septiennealimmal clan|Septiennealimmal temperaments]] (1\9 period)
** [[Ragismic microtemperaments #Ennealimmal|Ennealimmal]]
** [[Augmented family #Niner|Niner]]
** [[Marvel temperaments #Enneaportent|Enneaportent]]
** [[Kleismic family #Novemkleismic|Novemkleismic]]
* [[Linus]] temperaments (1\10 period)
** [[Breedsmic temperaments #Decoid|Decoid]]
** [[Ragismic microtemperaments #Deca|Deca]]
** [[Cloudy clan #Decic|Decic]]
** [[Stearnsmic clan #Decistearn|Decistearn]]
** [[Quintile family #Decile|Decile]]
** [[Vishnuzmic family #Decavish|Decavish]]
** [[Metric microtemperaments #Decimetra|Decimetra]]
* [[Porwell temperaments #Hendecatonic|Hendecatonic]] (1\11 period)
* [[Compton family|Compton]], [[Very high accuracy temperaments #Atomic|atomic]] (1\12 period)
* [[Orwellismic temperaments #Triskaidekic|Triskaidekic]], [[Octagar temperaments #Tridecatonic|tridecatonic]], [[Ragismic microtemperaments #Trideci|trideci]], [[aluminium]] (1\13 period)
* [[Silicon]] (1\14 period)
* [[Cloudy clan #Pentadecal|Pentadecal]], [[Trienstonic clan #Quindecic|quindecic]] (1\15 period)
* [[Ragismic microtemperaments #Octoid|Hexadecoid]], [[Jubilismic clan #Sedecic|sedecic]] (1\16 period)
* [[Ragismic microtemperaments #Chlorine|Chlorine]] (1\17 period)
* [[Septiennealimmal clan #Ennealimmal|Hemiennealimmal]] (1\18 period)
* [[Ragismic microtemperaments #Enneadecal|Enneadecal]], [[Meantone family #Meanmag|meanmag]] (1\19 period)
* [[Hemimage temperaments #Degrees|Degrees]] (1\20 period)
* [[Akjayland]] (1\21 period)
* [[Porwell temperaments #Hendecatonic|Icosidillic]] (1\22 period)
* [[Porwell temperaments #Icositritonic|Icositritonic]] (1\23 period)
* [[Compton family #Hours|Hours]], [[chromium]] (1\24 period)
* [[Septiennealimmal clan #Ennealimmal|Trinealimmal]], [[Tritrizo clan #Cobalt|cobalt]] (1\27 period)
* [[Horwell temperaments #Oquatonic|Oquatonic]] (1\28 period)
* [[Hemifamity temperaments #Mystery|Mystery]], [[Copper comma|copper]] (1\29 period)
* [[31st-octave temperaments|Birds]] (1\31 period)
* [[Compton family #Gamelstearn|Gamelstearn]] (1\36 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]], [[semihemienneadecal]] (1\38 period)
* [[Countercomp family|Countercomp temperaments]], [[niobium]] (1\41 period)
* [[Mitonismic temperaments #Meridic|Meridic]] (1\43 period)
* [[Ragismic microtemperaments #Palladium|Palladium]] (1\46 period)
* [[Compton family #Omicronbeta|Omicronbeta]] (1\72 period)
* [[The Flashmob#Iridium|Iridium]] (1\77 period)
* [[Parkleiness temperaments #Octogintic|Octogintic]] (1\80 period)
* [[Stearnsmic clan #Garistearn|Garistearn]] (1\94 period)
* [[Septiennealimmal clan #Undecentic|Undecentic]] (1\99 period)
* [[Septiennealimmal clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
* [[Septiennealimmal clan #Lunennealimmal|Lunennealimmal]] (1\441 period)

== See also ==
* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.

[[Category:Temperament collections]]

September chords

2026-06-07T06:13:23Z

Xenllium:

'''September chords''' are [[Dyadic chord|essentially tempered dyadic chords]] tempered by the september comma, [[715/714]].

There are 20 triads, 76 tetrads, 60 pentads, and 10 hexads as [[17-odd-limit]] essentially tempered chords.

For triads, there are 10 pairs of chords in inverse relationship:

{| class="wikitable center-all"
|-
! colspan="2" | Inversely related pairs of triads
|-
| 1 – 14/11 – 5/3 (steps 14/11, 17/13, 6/5) || 1 – 6/5 – 11/7 (steps 6/5, 17/13, 14/11)
|-
| 1 – 12/11 – 10/7 (steps 12/11, 17/13, 7/5) || 1 – 7/5 – 11/6 (steps 7/5, 17/13, 12/11)
|-
| 1 – 14/13 – 5/3 (steps 14/13, 17/11, 6/5) || 1 – 6/5 – 13/7 (steps 6/5, 17/11, 14/13)
|-
| 1 – 13/12 – 7/5 (steps 13/12, 22/17, 10/7) || 1 – 10/7 – 24/13 (steps 10/7, 22/17, 13/12)
|-
| 1 – 13/12 – 14/11 (steps 13/12, 20/17, 11/7) || 1 – 11/7 – 24/13 (steps 11/7, 20/17, 13/12)
|-
| 1 – 12/11 – 13/7 (steps 12/11, 17/10, 14/13) || 1 – 14/13 – 11/6 (steps 14/13, 17/10, 12/11)
|-
| 1 – 11/9 – 17/13 (steps 11/9, 15/14, 26/17) || 1 – 15/14 – 17/13 (steps 15/14, 11/9, 26/17)
|-
| 1 – 13/9 – 17/11 (steps 13/9, 15/14, 22/17) || 1 – 15/14 – 17/11 (steps 15/14, 13/9, 22/17)
|-
| 1 – 14/11 – 13/9 (steps 14/11, 17/15, 18/13) || 1 – 18/13 – 11/7 (steps 18/13, 17/15, 14/11)
|-
| 1 – 18/11 – 13/7 (steps 18/11, 17/15, 14/13) || 1 – 14/13 – 11/9 (steps 14/13, 17/15, 18/11)
|}

For tetrads, there are 38 pairs of chords in inverse relationship:

{| class="wikitable center-all"
|-
! colspan="2" | Inversely related pairs of tetrads
|-
| 1 – 13/12 – 7/5 – 3/2 (steps 13/12, 22/17, 15/14, 4/3) || 1 – 15/14 – 18/13 – 3/2 (steps 15/14, 22/17, 13/12, 4/3)
|-
| 1 – 7/5 – 3/2 – 11/6 (steps 7/5, 15/14, 11/9, 12/11) || 1 – 15/14 – 3/2 – 18/11 (steps 15/14, 7/5, 12/11, 11/9)
|-
| 1 – 18/13 – 3/2 – 30/17 (steps 18/13, 13/12, 20/17, 17/15) || 1 – 13/12 – 3/2 – 17/10 (steps 13/12, 18/13, 17/15, 20/17)
|-
| 1 – 17/15 – 11/9 – 4/3 (steps 17/15, 14/13, 12/11, 3/2) || 1 – 12/11 – 20/17 – 4/3 (steps 12/11, 14/13, 17/15, 3/2)
|-
| 1 – 14/11 – 14/9 – 5/3 (steps 14/11, 11/9, 15/14, 6/5) || 1 – 6/5 – 9/7 – 11/7 (steps 6/5, 15/14, 11/9, 14/11)
|-
| 1 – 14/11 – 17/11 – 5/3 (steps 14/11, 17/14, 14/13, 6/5) || 1 – 6/5 – 22/17 – 11/7 (steps 6/5, 14/13, 17/14, 14/11)
|-
| 1 – 14/11 – 13/9 – 5/3 (steps 14/11, 17/15, 15/13, 6/5) || 1 – 6/5 – 18/13 – 11/7 (steps 6/5, 15/13, 17/15, 14/11)
|-
| 1 – 14/11 – 15/11 – 5/3 (steps 14/11, 15/14, 11/9, 6/5) || 1 – 6/5 – 22/15 – 11/7 (steps 6/5, 11/9, 15/14, 14/11)
|-
| 1 – 9/7 – 18/13 – 11/7 (steps 9/7, 14/13, 17/15, 14/11) || 1 – 14/11 – 13/9 – 14/9 (steps 14/11, 17/15, 14/13, 9/7)
|-
| 1 – 14/11 – 13/9 – 17/11 (steps 14/11, 17/15, 15/14, 22/17) || 1 – 22/17 – 18/13 – 11/7 (steps 22/17, 15/14, 17/15, 14/11)
|-
| 1 – 17/14 – 10/7 – 17/11 (steps 17/14, 20/17, 13/12, 22/17) || 1 – 17/14 – 11/7 – 17/10 (steps 17/14, 22/17, 13/12, 20/17)
|-
| 1 – 13/11 – 13/9 – 17/11 (steps 13/11, 11/9, 15/14, 22/17) || 1 – 15/14 – 17/13 – 17/11 (steps 15/14, 11/9, 13/11, 22/17)
|-
| 1 – 17/15 – 13/9 – 17/11 (steps 17/15, 14/11, 15/14, 22/17) || 1 – 17/15 – 22/15 – 11/7 (steps 17/15, 22/17, 15/14, 14/11)
|-
| 1 – 17/13 – 17/12 – 11/6 (steps 17/13, 13/12, 22/17, 12/11) || 1 – 13/12 – 17/12 – 17/11 (steps 13/12, 17/13, 12/11, 22/17)
|-
| 1 – 17/13 – 17/11 – 5/3 (steps 17/13, 13/11, 14/13, 6/5) || 1 – 6/5 – 22/17 – 26/17 (steps 6/5, 14/13, 13/11, 17/13)
|-
| 1 – 17/13 – 20/13 – 5/3 (steps 17/13, 20/17, 13/12, 6/5) || 1 – 6/5 – 13/10 – 26/17 (steps 6/5, 13/12, 20/17, 17/13)
|-
| 1 – 17/13 – 10/7 – 5/3 (steps 17/13, 12/11, 7/6, 6/5) || 1 – 6/5 – 7/5 – 26/17 (steps 6/5, 7/6, 12/11, 17/13)
|-
| 1 – 17/13 – 17/12 – 5/3 (steps 17/13, 13/12, 20/17, 6/5) || 1 – 6/5 – 24/17 – 26/17 (steps 6/5, 20/17, 13/12, 17/13)
|-
| 1 – 17/13 – 5/3 – 17/9 (steps 17/13, 14/11, 17/15, 18/17) || 1 – 17/13 – 18/13 – 11/7 (steps 17/13, 18/17, 17/15, 14/11)
|-
| 1 – 17/13 – 5/3 – 11/6 (steps 17/13, 14/11, 11/10, 12/11) || 1 – 17/13 – 10/7 – 11/7 (steps 17/13, 12/11, 11/10, 14/11)
|-
| 1 – 17/13 – 17/10 – 11/6 (steps 17/13, 13/10, 14/13, 12/11) || 1 – 14/13 – 7/5 – 11/6 (steps 14/13, 13/10, 17/13, 12/11)
|-
| 1 – 17/13 – 22/13 – 11/6 (steps 17/13, 22/17, 13/12, 12/11) || 1 – 13/12 – 7/5 – 11/6 (steps 13/12, 22/17, 17/13, 12/11)
|-
| 1 – 15/11 – 17/11 – 5/3 (steps 15/11, 17/15, 14/13, 6/5) || 1 – 6/5 – 22/17 – 22/15 (steps 6/5, 14/13, 17/15, 15/11)
|-
| 1 – 13/11 – 14/11 – 13/9 (steps 13/11, 14/13, 17/15, 18/13) || 1 – 17/15 – 11/9 – 13/9 (steps 17/15, 14/13, 13/11, 18/13)
|-
| 1 – 7/5 – 17/10 – 11/6 (steps 7/5, 17/14, 14/13, 12/11) || 1 – 17/14 – 17/10 – 13/7 (steps 17/14, 7/5, 12/11, 14/13)
|-
| 1 – 17/12 – 17/10 – 11/6 (steps 17/12, 6/5, 14/13, 12/11) || 1 – 6/5 – 17/10 – 13/7 (steps 6/5, 17/12, 12/11, 14/13)
|-
| 1 – 6/5 – 22/17 – 7/5 (steps 6/5, 14/13, 13/12, 10/7) || 1 – 13/12 – 7/6 – 7/5 (steps 13/12, 14/13, 6/5, 10/7)
|-
| 1 – 11/10 – 22/17 – 7/5 (steps 11/10, 20/17, 13/12, 10/7) || 1 – 13/12 – 14/11 – 7/5 (steps 13/12, 20/17, 11/10, 10/7)
|-
| 1 – 13/12 – 13/10 – 7/5 (steps 13/12, 6/5, 14/13, 10/7) || 1 – 14/13 – 22/17 – 7/5 (steps 14/13, 6/5, 13/12, 10/7)
|-
| 1 – 13/9 – 17/11 – 17/9 (steps 13/9, 15/14, 11/9, 18/17) || 1 – 11/9 – 17/13 – 17/9 (steps 11/9, 15/14, 13/9, 18/17)
|-
| 1 – 13/9 – 17/11 – 13/7 (steps 13/9, 15/14, 6/5, 14/13) || 1 – 15/14 – 17/11 – 5/3 (steps 15/14, 13/9, 14/13, 6/5)
|-
| 1 – 13/9 – 17/11 – 5/3 (steps 13/9, 15/14, 14/13, 6/5) || 1 – 14/13 – 15/13 – 5/3 (steps 14/13, 15/14, 13/9, 6/5)
|-
| 1 – 17/15 – 11/9 – 17/13 (steps 17/15, 14/13, 15/14, 26/17) || 1 – 15/14 – 15/13 – 17/13 (steps 15/14, 14/13, 17/15, 26/17)
|-
| 1 – 14/13 – 11/9 – 17/13 (steps 14/13, 17/15, 15/,14 26/17) || 1 – 15/14 – 17/14 – 17/13 (steps 15/14, 17/15, 14/13, 26/17)
|-
| 1 – 17/11 – 5/3 – 17/9 (steps 17/11, 14/13, 17/15, 18/17) || 1 – 17/15 – 11/9 – 17/9 (steps 17/15, 14/13, 17/11, 18/17)
|-
| 1 – 17/11 – 17/10 – 13/7 (steps 17/11, 11/10, 12/11, 14/13) || 1 – 12/11 – 6/5 – 13/7 (steps 12/11, 11/10, 17/11, 14/13)
|-
| 1 – 11/7 – 17/10 – 13/7 (steps 11/7, 13/12, 12/11, 14/13) || 1 – 13/12 – 17/10 – 11/6 (steps 13/12, 11/7, 14/13, 12/11)
|-
| 1 – 11/7 – 17/10 – 11/6 (steps 11/7, 13/12, 14/13, 12/11) || 1 – 13/12 – 17/10 – 13/7 (steps 13/12, 11/7, 12/11, 14/13)
|}

For pentads, there are 30 pairs of chords in inverse relationship:

{| class="wikitable center-all"
|-
! colspan="2" | Inversely related pairs of pentads
|-
| 1 – 13/12 – 13/10 – 7/5 – 3/2 (steps 13/12, 6/5, 14/13, 15/14, 4/3) || 1 – 15/14 – 15/13 – 18/13 – 3/2 (steps 15/14, 14/13, 6/5, 13/12, 4/3)
|-
| 1 – 13/12 – 7/6 – 7/5 – 3/2 (steps 13/12, 14/13, 6/5, 15/14, 4/3) || 1 – 15/14 – 9/7 – 18/13 – 3/2 (steps 15/14, 6/5, 14/13, 13/12, 4/3)
|-
| 1 – 7/6 – 7/5 – 3/2 – 11/6 (steps 7/6, 6/5, 15/14, 11/9, 12/11) || 1 – 15/14 – 9/7 – 3/2 – 18/11 (steps 15/14, 6/5, 7/6, 12/11, 11/9)
|-
| 1 – 15/13 – 18/13 – 3/2 – 30/17 (steps 15/13, 6/5, 13/12, 20/17, 17/15) || 1 – 13/12 – 13/10 – 3/2 – 17/10 (steps 13/12, 6/5, 15/13, 17/15, 20/17)
|-
| 1 – 11/10 – 7/5 – 3/2 – 11/6 (steps 11/10, 14/11, 15/14, 11/9, 12/11) || 1 – 15/14 – 15/11 – 3/2 – 18/11 (steps 15/14, 14/11, 11/10, 12/11, 11/9)
|-
| 1 – 13/12 – 17/12 – 3/2 – 17/10 (steps 13/12, 17/13, 18/17, 17/15, 20/17) || 1 – 18/17 – 18/13 – 3/2 – 30/17 (steps 18/17, 17/13, 13/12, 20/17, 17/15)
|-
| 1 – 13/12 – 7/5 – 3/2 – 11/6 (steps 13/12, 22/17, 15/14, 11/9, 12/11) || 1 – 15/14 – 18/13 – 3/2 – 18/11 (steps 15/14, 22/17, 13/12, 12/11, 11/9)
|-
| 1 – 13/12 – 7/5 – 3/2 – 17/10 (steps 13/12, 22/17, 15/14, 17/15, 20/17) || 1 – 15/14 – 18/13 – 3/2 – 30/17 (steps 15/14, 22/17, 13/12, 20/17, 17/15)
|-
| 1 – 17/12 – 3/2 – 17/10 – 11/6 (steps 17/12, 18/17, 17/15, 14/13, 12/11) || 1 – 18/17 – 3/2 – 18/11 – 30/17 (steps 18/17, 17/12, 12/11, 14/13, 17/15)
|-
| 1 – 7/5 – 3/2 – 17/10 – 11/6 (steps 7/5, 15/14, 17/15, 14/13, 12/11) || 1 – 15/14 – 3/2 – 18/11 – 30/17 (steps 15/14, 7/5, 12/11, 14/13, 17/15)
|-
| 1 – 18/13 – 3/2 – 18/11 – 30/17 (steps 18/13, 13/12, 12/11, 14/13, 17/15) || 1 – 13/12 – 3/2 – 17/10 – 11/6 (steps 13/12, 18/13, 17/15, 14/13, 12/11)
|-
| 1 – 15/11 – 3/2 – 18/11 – 30/17 (steps 15/11, 11/10, 12/11, 14/13, 17/15) || 1 – 11/10 – 3/2 – 17/10 – 11/6 (steps 11/10, 15/11, 17/15, 14/13, 12/11)
|-
| 1 – 13/12 – 14/11 – 17/11 – 5/3 (steps 13/12, 20/17, 17/14, 14/13, 6/5) || 1 – 14/13 – 17/13 – 20/13 – 5/3 (steps 14/13, 17/14, 20/17, 13/12, 6/5)
|-
| 1 – 14/13 – 14/11 – 14/9 – 5/3 (steps 14/13, 13/11, 11/9, 15/14, 6/5) || 1 – 15/14 – 17/13 – 17/11 – 5/3 (steps 15/14, 11/9, 13/11, 14/13, 6/5)
|-
| 1 – 17/14 – 17/13 – 10/7 – 11/7 (steps 17/14, 14/13, 12/11, 11/10, 14/11) || 1 – 11/10 – 6/5 – 22/17 – 11/7 (steps 11/10, 12/11, 14/13, 17/14, 14/11)
|-
| 1 – 6/5 – 22/17 – 22/15 – 11/7 (steps 6/5, 14/13, 17/15, 15/14, 14/11) || 1 – 15/14 – 17/14 – 17/13 – 11/7 (steps 15/14, 17/15, 14/13, 6/5, 14/11)
|-
| 1 – 6/5 – 22/17 – 18/13 – 11/7 (steps 6/5, 14/13, 15/14, 17/15, 14/11) || 1 – 17/15 – 17/14 – 17/13 – 11/7 (steps 17/15, 15/14, 14/13, 6/5, 14/11)
|-
| 1 – 6/5 – 9/7 – 18/13 – 11/7 (steps 6/5, 15/14, 14/13, 17/15, 14/11) || 1 – 17/15 – 11/9 – 17/13 – 11/7 (steps 17/15, 14/13, 15/14, 6/5, 14/11)
|-
| 1 – 17/15 – 11/9 – 22/15 – 11/7 (steps 17/15, 14/13, 6/5, 15/14, 14/11) || 1 – 15/14 – 9/7 – 18/13 – 11/7 (steps 15/14, 6/5, 14/13, 17/15, 14/11)
|-
| 1 – 17/15 – 6/5 – 22/15 – 11/7 (steps 17/15, 18/17, 11/9, 15/14, 14/11) || 1 – 15/14 – 17/13 – 18/13 – 11/7 (steps 15/14, 11/9, 18/17, 17/15, 14/11)
|-
| 1 – 17/14 – 17/13 – 10/7 – 17/11 (steps 17/14, 14/13, 12/11, 13/12, 22/17) || 1 – 13/12 – 13/11 – 14/11 – 17/11 (steps 13/12, 12/11, 14/13, 17/14, 22/17)
|-
| 1 – 13/11 – 14/11 – 13/9 – 17/11 (steps 13/11, 14/13, 17/15, 15/14, 22/17) || 1 – 15/14 – 17/14 – 17/13 – 17/11 (steps 15/14, 17/15, 14/13, 13/11, 22/17)
|-
| 1 – 17/13 – 17/11 – 5/3 – 17/9 (steps 17/13, 13/11, 14/13, 17/15, 18/17) || 1 – 13/11 – 17/11 – 18/11 – 13/7 (steps 13/11, 17/13, 18/17, 17/15, 14/13)
|-
| 1 – 17/13 – 17/12 – 5/3 – 11/6 (steps 17/13, 13/12, 20/17, 11/10, 12/11) || 1 – 13/12 – 17/12 – 17/11 – 17/10 (steps 13/12, 17/13, 12/11, 11/10, 20/17)
|-
| 1 – 17/13 – 10/7 – 17/11 – 5/3 (steps 17/13, 12/11, 13/12, 14/13, 6/5) || 1 – 14/13 – 7/6 – 14/11 – 5/3 (steps 14/13, 13/12, 12/11, 17/13, 6/5)
|-
| 1 – 17/13 – 10/7 – 20/13 – 5/3 (steps 17/13, 12/11, 14/13, 13/12, 6/5) || 1 – 13/12 – 7/6 – 14/11 – 5/3 (steps 13/12, 14/13, 12/11, 17/13, 6/5)
|-
| 1 – 17/13 – 17/12 – 17/11 – 5/3 (steps 17/13, 13/12, 12/11, 14/13, 6/5) || 1 – 14/13 – 20/17 – 14/11 – 5/3 (steps 14/13, 12/11, 13/12, 17/13, 6/5)
|-
| 1 – 13/12 – 17/12 – 17/11 – 5/3 (steps 13/12, 17/13, 12/11, 14/13, 6/5) || 1 – 14/13 – 20/17 – 20/13 – 5/3 (steps 14/13, 12/11, 17/13, 13/12, 6/5)
|-
| 1 – 13/9 – 17/11 – 5/3 – 17/9 (steps 13/9, 15/14, 14/13, 17/15, 18/17) || 1 – 17/15 – 11/9 – 17/13 – 17/9 (steps 17/15, 14/13, 15/14, 13/9, 18/17)
|-
| 1 – 11/10 – 6/5 – 22/17 – 7/5 (steps 11/10, 12/11, 14/13, 13/12, 10/7) || 1 – 13/12 – 7/6 – 14/11 – 7/5 (steps 13/12, 14/13, 12/11, 11/10, 10/7)
|}

For hexads, there are 5 pairs of chords in inverse relationship:

{| class="wikitable center-all"
|-
! colspan="2" | Inversely related pairs of hexads
|-
| 1 – 13/12 – 13/10 – 7/5 – 3/2 – 17/10 (steps 13/12, 6/5, 14/13, 15/14, 17/15, 20/17) || 1 – 15/14 – 15/13 – 18/13 – 3/2 – 30/17 (steps 15/14, 14/13, 6/5, 13/12, 20/17, 17/15)
|-
| 1 – 13/12 – 7/6 – 7/5 – 3/2 – 11/6 (steps 13/12, 14/13, 6/5, 15/14, 11/9, 12/11) || 1 – 15/14 – 9/7 – 18/13 – 3/2 – 18/11 (steps 15/14, 6/5, 14/13, 13/12, 12/11, 11/9)
|-
| 1 – 11/10 – 7/5 – 3/2 – 17/10 – 11/6 (steps 11/10, 14/11, 15/14, 17/15, 14/13, 12/11) || 1 – 15/14 – 15/11 – 3/2 – 18/11 – 30/17 (steps 15/14, 14/11, 11/10, 12/11, 14/13, 17/15)
|-
| 1 – 13/12 – 17/12 – 3/2 – 17/10 – 11/6 (steps 13/12, 17/13, 18/17, 17/15, 14/13, 12/11) || 1 – 18/17 – 18/13 – 3/2 – 18/11 – 30/17 (steps 18/17, 17/13, 13/12, 12/11, 14/13, 17/15)
|-
| 1 – 13/12 – 7/5 – 3/2 – 17/10 – 11/6 (steps 13/12, 22/17, 15/14, 17/15, 14/13, 12/11) || 1 – 15/14 – 18/13 – 3/2 – 18/11 – 30/17 (steps 15/14, 22/17, 13/12, 12/11, 14/13, 17/15)
|}

Equal temperaments with september chords include {{EDOs| 31, 41, 46, 72, 87, 94, 103, 111, 121, 140, 183, 224 and 270 }}.

[[Category:17-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:September]]

Ploidacot

2026-06-06T06:21:59Z

Xenllium: Added notation and relationship to pergens

The '''ploidacot''' system is a classification of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] based on how a temperament divides the intervals of [[Pythagorean tuning]]. A particularly simple case is if a temperament divides its [[3/2]] interval into ''n'' steps, it can be called an ''n''-cot tuning. More generally, ploidacots are written as ''m''-ploid ''s''-sheared ''n''-cot, with ''m''- and ''n''- often replaced by greek numeral prefixes, such as mono-, di-, tri-, etc. (and ''m''-ploid omitted entirely if the [[2/1|octave]] is not split), and "''s''-sheared" replaced by a greek letter, such as alpha-, beta-, etc. (or omitted entirely if ''s'' = 0).

The "ploid" number of a temperament refers to how many equal parts, or [[period]]s the octave is divided into, and the "cot" number refers to how many [[generator]] steps of the temperament are needed to reach the third harmonic. Cots are generally presumed to reach 3/2 in a nonnegative number of generators. Temperaments where 3/2 is a whole number of ploids are written as ''acot''. However, stacking ''n'' cots sometimes doesn't reach 3/2, but instead an interval ''s'' ploids above 3/2. There are infinitely many possible values of ''s'', but for the sake of ploidacot, ''s'' takes its residue modulo ''n'' (which is the same for all possible cots), and is an integer between 0 and {{nowrap| ''n'' - 1 }} inclusive.

For example, [[meantone]] is monocot because it is does not split the octave, and is generated by the perfect fifth. [[Kleismic]] is alpha-hexacot, since it does not split the octave, but splits [[3/1]], which is one octave above 3/2, into six equal parts (~317{{c}} each). [[Pajara]] is diploid monocot, since it is generated by the fifth and splits the octave in two 600{{c}} halves. [[Shrutar]] is diploid alpha-dicot, since it splits the octave in half, and splits the interval 600{{c}} above 3/2 (~1300{{c}}) into two ~650{{c}} halves. Note that in shrutar the interval one ploid above 3/2 is ~1300{{c}} and not 3/1, since the octave is split into two 600{{c}} ploids.

It is similar to the [[pergen]], and is a canonical naming scheme for pergens of rank-2 temperaments of 2.3.(…) [[subgroup]]s in that every such pergen corresponds to a unique name in the ploidacot system.

The ploidacot system was developed by [[Praveen Venkataramana]].

== Specification ==
=== Ploids ===
Any rank-2 temperament of the 2.3.… subgroup has an octave, and it may split the octave into a number of parts, or [[period]]s, called '''ploids'''. The temperament's number of ploids per octave is specified by a Greek numeral prefix (di-, tri-, etc.) and -ploid. For instance, pajara divides the octave into two, so it is diploid. Temperaments that do not divide the octave are called haploid (''not'' *monoploid), which can be omitted.

=== Cots ===
If [[3/2]] is represented by a linearly independent element to the ploid, there is a number of ploids which when added to 3/2 gives the interval which is split into the largest number of parts, namely [[generator]]s, by the temperament. Each of these parts is called a '''cot''' or '''cotyledon'''. The ploidacot system uses Greek letters (alpha-, beta-, etc.) to describe the smallest nonnegative number of ploids that should be added to 3/2 to form a whole number of cots. If the number is zero, it is left empty. The number of cots is then indicated by a Greek numeral prefix. Temperaments that do not divide the fifth are called ''monocot'' (''not'' *haplocot). The full specification of cots is thus a (possibly empty) Greek letter prefix, followed by a Greek numeral prefix, and -cot.

Temperaments where the image of 3/2 is a whole number of ploids are called '''acot'''.

==== Greek letter prefixes ====
The Greek letter prefixes follow the ancient gematria/isopsephic system, detailed below:

{| class="wikitable"
|+ style="font-size: 105%;" | Greek letter prefixes in ploidacot
|-
! colspan="2" | Number ''n''
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
|-
! rowspan="3" | Prefix
! ''n''
| alpha
| beta
| gamma
| delta
| epsilon
| digamma/wau
| zeta
| eta
| theta
|-
! 10''n''
| iota
| kappa
| lambda
| mu
| nu
| xi
| omicron
| pi
| qoppa
|-
! {{nowrap|''n'' + 10}}
| iota-alpha
| iota-beta
| iota-gamma
| iota-delta
| iota-epsilon
| iota-digamma/iota-wau
| iota-zeta
| iota-eta
| iota-theta
|}

Prefixes for numbers between 21 and 99 are constructed the same way as number words in English, for instance 21 is kappa-alpha and 99 is qoppa-theta.

Alternatively, Arabic numerals may be used in place of the Greek alphabetical and numeric prefixes, with the word "sheared" or its equivalent in other languages used in place of the alphabetic prefixes, so a diploid epsilon-heptacot system may be referred to as a 2-ploid 5-sheared 7-cot system.

== Properties ==
* For ''n''-cot systems there are exactly ''n'' settings of shear, or number of ploids to add to the step that represents the interval class of 3. The possible values of shear are 0, 1, 2, …, {{nowrap|(''n'' − 1)}}. For example, the tricot systems are tricot (0-sheared), alpha-tricot (1-sheared), and beta-tricot (2-sheared). There is not a *gamma-tricot since that would be equivalent to tricot.

== Extensions ==
=== Omega extension ===
The Greek letter omega, proposed by [[User:Godtone|Godtone]], is used for −1. ("Contra" has also been used in place of omega.) This simplifies the classification of certain temperaments, e.g. porcupine, which instead of beta-tricot can be omega-tricot, as splitting the interval 4/3 into three is arguably more intuitive than splitting the interval 6. This effectively shifts the possible values of shear to -1, 0, 1, …, {{nowrap|(''n'' − 2)}} if ''n'' ≥ 3.

Note that omega should only be used with {{nowrap| ''n'' ≥ 3 }}. When {{nowrap| ''n'' {{=}} 1 }}, there is only monocot. When {{nowrap| ''n'' {{=}} 2 }}, alpha-dicot is preferred over omega-dicot. Omega-based names are also not preferred when dealing with temperaments that split the octave, as they may be confusing - for instance, diploid alpha-tricot splits 4/3 in three while diploid beta-tricot splits 3/1 in three.

=== No-twos or no-threes temperaments ===
The ploidacot system, similarly to [[pergen]]s, relies on the presence of a [[3-limit]], i.e. 2.3 subgroup, spine, but its defining principles can be easily applied to a 2.5, 3.5, 3.7, etc. spine instead, and in the case of ploidacot, the "cot" suffix is simply replaced with a different suffix indicating the family of intervals being cloven. The existing extensions are "seph" for [[5/4]] with octave equivalence, and "gem" for [[7/3]] with tritave equivalence (note that 3.7 is preferred over 3.5 since [[9/7]] and 7/3 generate a much more commonly used structure in tritave systems, i.e. [[4L 5s (3/1-equivalent)|Lambda]], than [[5/3]] and [[9/5]]).

For instance, in the 2.5.7 subgroup, [[didacus]] can be labeled as "diseph", because its generator divides 5/4 in two, and [[llywelyn]] can be labeled as "alpha-heptaseph" because seven generators make up [[5/2]]. In the tritave world, [[BPS]] (3.5.7) is "monogem" as its generator is 9/7, while [[mintaka]] (3.7.11) is alpha-trigem as its generator (of ~[[21/11]]) splits [[7/1]] in three.

Even if 3 is included in a given temperament, the ploidaseph framework may occasionally be more useful than the ploidacot framework, in cases where the mapping of 3 is very complex and the structure of the temperament therefore deprioritizes prime 3. [[Hemiwürschmidt]], a [[strong extension]] of the aforementioned didacus, has a ploidacot of beta-hexadecacot as it divides 6/1 into sixteen generators; while [[trismegistus]] has a ploidacot of epsilon-pentadecacot as it maps [[96/1]] to fifteen generators. Each of these has a more intuitizable expression in terms of 2.5 intervals, which are much simpler in the respective temperaments: hemiwürschmidt is diseph and trismegistus is alpha-triseph (one-third 5/2).

Combining ploidacots and ploidasephs determines its [[5-limit]] properties; for instance, meantone can be labeled as "monocot beta-tetraseph" because four generators make up [[5/1]] while the generator represents [[3/2]], and valentine can be labeled as "enneacot pentaseph" because five generators make up [[5/4]] and nine of them make up [[3/2]].

== Relationship to pergens ==
Each ploidacot has one pergen. The numbers of ploid (''p''), shear (''s''), and cot (''c'') are given, its pergen form has following features:
* Every ''p''-ploid has a form of (P8/''p'', X).
* Haploids (''p'' = 1) are of the form (P8, X/''c'') since the octave is unsplit.
* Monocots (''c'' = 1) are of the form (P8/''p'', P5) since the fifth and its compounds are unsplit.
* If ''p'' and ''c'' are coprime, the ploidacot has a perfect pergen, of the form (P8/''p'', X/''c'').
* If ''s'' mod GCD(''p'', ''c'') = 0, the ploidacot has a perfect pergen, of the form (P8/''p'', X/''c'').
* If ''s'' mod GCD(''p'', ''c'') is not 0, the ploidacot has an imperfect pergen.

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Pergen forms of ploidacot
|-
! colspan="2" | Ploids
! Haploid
! Diploid
! Triploid
! Tetraploid
! Pentaploid
! Hexaploid
! Heptaploid
! Octaploid
! Enneaploid
! Decaploid
! Hendecaploid
|-
! colspan="2" | Monocot
| (P8, P5)
| (P8/2, P5)
| (P8/3, P5)
| (P8/4, P5)
| (P8/5, P5)
| (P8/6, P5)
| (P8/7, P5)
| (P8/8, P5)
| (P8/9, P5)
| (P8/10, P5)
| (P8/11, P5)
|-
! rowspan="2" | Dicots
! dicot
| (P8, P5/2)
| (P8/2, P4/2)
| (P8/3, P5/2)
| (P8/4, P4/2)
| (P8/5, P5/2)
| (P8/6, P4/2)
| (P8/7, P5/2)
| (P8/8, P4/2)
| (P8/9, P5/2)
| (P8/10, P4/2)
| (P8/11, P5/2)
|-
! alpha-dicot
| (P8, P4/2)
| (P8/2, M2/4)
| (P8/3, P4/2)
| (P8/4, m6/8)
| (P8/5, P4/2)
| (P8/6, M2/4)
| (P8/7, P4/2)
| (P8/8, d4/16)
| (P8/9, P4/2)
| (P8/10, M2/4)
| (P8/11, P4/2)
|-
! rowspan="3" | Tricots
! tricot
| (P8, P5/3)
| (P8/2, P5/3)
| (P8/3, P4/3)
| (P8/4, P5/3)
| (P8/5, P5/3)
| (P8/6, P4/3)
| (P8/7, P5/3)
| (P8/8, P5/3)
| (P8/9, P4/3)
| (P8/10, P5/3)
| (P8/11, P5/3)
|-
! alpha-tricot
| (P8, P11/3)
| (P8/2, P4/3)
| (P8/3, m3/9)
| (P8/4, P11/3)
| (P8/5, P4/3)
| (P8/6, M6/9)
| (P8/7, P11/3)
| (P8/8, P4/3)
| (P8/9, A2/27)
| (P8/10, P11/3)
| (P8/11, P4/3)
|-
! beta-tricot
| (P8, P4/3)
| (P8/2, P11/3)
| (P8/3, M6/9)
| (P8/4, P4/3)
| (P8/5, P11/3)
| (P8/6, m3/9)
| (P8/7, P4/3)
| (P8/8, P11/3)
| (P8/9, A9/27)
| (P8/10, P4/3)
| (P8/11, P11/3)
|-
! rowspan="4" | Tetracots
! tetracot
| (P8, P5/4)
| (P8/2, P5/4)
| (P8/3, P5/4)
| (P8/4, P4/4)
| (P8/5, P5/4)
| (P8/6, P5/4)
| (P8/7, P5/4)
| (P8/8, P4/4)
| (P8/9, P5/4)
| (P8/10, P5/4)
| (P8/11, P5/4)
|-
! alpha-tetracot
| (P8, P12/4)
| (P8/2, cm7/8)
| (P8/3, P4/4)
| (P8/4, m6/16)
| (P8/5, P12/4)
| (P8/6, M2/8)
| (P8/7, P4/4)
| (P8/8, A12/32)
| (P8/9, P12/4)
| (P8/10, cm7/8)
| (P8/11, P4/4)
|-
! beta-tetracot
| (P8, P11/4)
| (P8/2, P4/4)
| (P8/3, P11/4)
| (P8/4, M2/8)
| (P8/5, P11/4)
| (P8/6, P4/4)
| (P8/7, P11/4)
| (P8/8, m6/16)
| (P8/9, P11/4)
| (P8/10, P4/4)
| (P8/11, P11/4)
|-
! gamma-tetracot
| (P8, P4/4)
| (P8/2, M2/8)
| (P8/3, P12/4)
| (P8/4, M10/16)
| (P8/5, P4/4)
| (P8/6, cm7/8)
| (P8/7, P12/4)
| (P8/8, d4/32)
| (P8/9, P4/4)
| (P8/10, M2/8)
| (P8/11, P12/4)
|-
! rowspan="5" | Pentacots
! pentacot
| (P8, P5/5)
| (P8/2, P5/5)
| (P8/3, P5/5)
| (P8/4, P5/5)
| (P8/5, P4/5)
| (P8/6, P5/5)
| (P8/7, P5/5)
| (P8/8, P5/5)
| (P8/9, P5/5)
| (P8/10, P4/5)
| (P8/11, P5/5)
|-
! alpha-pentacot
| (P8, P12/5)
| (P8/2, P11/5)
| (P8/3, ccP4/5)
| (P8/4, P4/5)
| (P8/5, m9/25)
| (P8/6, P12/5)
| (P8/7, P11/5)
| (P8/8, ccP4/5)
| (P8/9, P4/5)
| (P8/10, M7/25)
| (P8/11, P12/5)
|-
! beta-pentacot
| (P8, ccP4/5)
| (P8/2, P12/5)
| (P8/3, P4/5)
| (P8/4, P11/5)
| (P8/5, m2/25)
| (P8/6, ccP4/5)
| (P8/7, P12/5)
| (P8/8, P4/5)
| (P8/9, P11/5)
| (P8/10, m9/25)
| (P8/11, ccP4/5)
|-
! gamma-pentacot
| (P8, P11/5)
| (P8/2, P4/5)
| (P8/3, P12/5)
| (P8/4, ccP4/5)
| (P8/5, M7/25)
| (P8/6, P11/5)
| (P8/7, P4/5)
| (P8/8, P12/5)
| (P8/9, ccP4/5)
| (P8/10, cM7/25)
| (P8/11, P11/5)
|-
! delta-pentacot
| (P8, P4/5)
| (P8/2, ccP4/5)
| (P8/3, P11/5)
| (P8/4, P12/5)
| (P8/5, cM7/25)
| (P8/6, P4/5)
| (P8/7, ccP4/5)
| (P8/8, P11/5)
| (P8/9, P12/5)
| (P8/10, m2/25)
| (P8/11, P4/5)
|-
! rowspan="6" | Hexacots
! hexacot
| (P8, P5/6)
| (P8/2, P5/6)
| (P8/3, P5/6)
| (P8/4, P5/6)
| (P8/5, P5/6)
| (P8/6, P4/6)
| (P8/7, P5/6)
| (P8/8, P5/6)
| (P8/9, P5/6)
| (P8/10, P5/6)
| (P8/11, P5/6)
|-
! alpha-hexacot
| (P8, P12/6)
| (P8/2, ccM2/12)
| (P8/3, ccM6/18)
| (P8/4, ccm6/24)
| (P8/5, P4/6)
| (P8/6, d12/36)
| (P8/7, P12/6)
| (P8/8, d4/48)
| (P8/9, A2/54)
| (P8/10, M2/12)
| (P8/11, P4/6)
|-
! beta-hexacot
| (P8, ccP5/6)
| (P8/2, P11/6)
| (P8/3, ccm3/18)
| (P8/4, P4/6)
| (P8/5, P11/6)
| (P8/6, m3/18)
| (P8/7, ccP5/6)
| (P8/8, P11/6)
| (P8/9, A9/54)
| (P8/10, P4/6)
| (P8/11, P11/6)
|-
! gamma-hexacot
| (P8, ccP4/6)
| (P8/2, cm7/12)
| (P8/3, P4/6)
| (P8/4, m6/24)
| (P8/5, ccP4/6)
| (P8/6, M2/12)
| (P8/7, ccP4/6)
| (P8/8, A12/48)
| (P8/9, P4/6)
| (P8/10, cm7/12)
| (P8/11, ccP4/6)
|-
! delta-hexacot
| (P8, P11/6)
| (P8/2, P4/6)
| (P8/3, m3/18)
| (P8/4, P11/6)
| (P8/5, ccP5/6)
| (P8/6, M6/18)
| (P8/7, P11/6)
| (P8/8, P4/6)
| (P8/9, ccd7/54)
| (P8/10, P11/6)
| (P8/11, ccP5/6)
|-
! epsilon-hexacot
| (P8, P4/6)
| (P8/2, M2/12)
| (P8/3, M6/18)
| (P8/4, M10/24)
| (P8/5, P12/6)
| (P8/6, ccA4/36)
| (P8/7, P4/6)
| (P8/8, ccd4/48)
| (P8/9, cd7/54)
| (P8/10, ccM2/12)
| (P8/11, P12/6)
|-
! rowspan="7" | Heptacots
! heptacot
| (P8, P5/7)
| (P8/2, P5/7)
| (P8/3, P5/7)
| (P8/4, P5/7)
| (P8/5, P5/7)
| (P8/6, P5/7)
| (P8/7, P4/7)
| (P8/8, P5/7)
| (P8/9, P5/7)
| (P8/10, P5/7)
| (P8/11, P5/7)
|-
! alpha-heptacot
| (P8, P12/7)
| (P8/2, ccP4/7)
| (P8/3, P11/7)
| (P8/4, ccP5/7)
| (P8/5, c3P4/7)
| (P8/6, P4/7)
| (P8/7, cd8/49)
| (P8/8, P12/7)
| (P8/9, ccP4/7)
| (P8/10, P11/7)
| (P8/11, ccP5/7)
|-
! beta-heptacot
| (P8, ccP5/7)
| (P8/2, P12/7)
| (P8/3, c3P4/7)
| (P8/4, ccP4/7)
| (P8/5, P4/7)
| (P8/6, P11/7)
| (P8/7, d8/49)
| (P8/8, ccP5/7)
| (P8/9, P12/7)
| (P8/10, c3P4/7)
| (P8/11, ccP4/7)
|-
! gamma-heptacot
| (P8, c3P4/7)
| (P8/2, P11/7)
| (P8/3, P12/7)
| (P8/4, P4/7)
| (P8/5, ccP5/7)
| (P8/6, ccP4/7)
| (P8/7, A1/49)
| (P8/8, c3P4/7)
| (P8/9, P11/7)
| (P8/10, P12/7)
| (P8/11, P4/7)
|-
! delta-heptacot
| (P8, ccP4/7)
| (P8/2, ccP5/7)
| (P8/3, P4/7)
| (P8/4, P12/7)
| (P8/5, P11/7)
| (P8/6, c3P4/7)
| (P8/7, A8/49)
| (P8/8, ccP4/7)
| (P8/9, ccP5/7)
| (P8/10, P4/7)
| (P8/11, P12/7)
|-
! epsilon-heptacot
| (P8, P11/7)
| (P8/2, P4/7)
| (P8/3, ccP4/7)
| (P8/4, c3P4/7)
| (P8/5, P12/7)
| (P8/6, ccP5/7)
| (P8/7, ccA1/49)
| (P8/8, P11/7)
| (P8/9, P4/7)
| (P8/10, ccP4/7)
| (P8/11, c3P4/7)
|-
! wau-heptacot
| (P8, P4/7)
| (P8/2, c3P4/7)
| (P8/3, ccP5/7)
| (P8/4, P11/7)
| (P8/5, ccP4/7)
| (P8/6, P12/7)
| (P8/7, c3A1/49)
| (P8/8, P4/7)
| (P8/9, c3P4/7)
| (P8/10, ccP5/7)
| (P8/11, P11/7)
|-
! rowspan="8" | Octacots
! octacot
| (P8, P5/8)
| (P8/2, P5/8)
| (P8/3, P5/8)
| (P8/4, P5/8)
| (P8/5, P5/8)
| (P8/6, P5/8)
| (P8/7, P5/8)
| (P8/8, P4/8)
| (P8/9, P5/8)
| (P8/10, P5/8)
| (P8/11, P5/8)
|-
! alpha-octacot
| (P8, P12/8)
| (P8/2, ccM2/16)
| (P8/3, c3P5/8)
| (P8/4, c3M3/32)
| (P8/5, ccP4/8)
| (P8/6, c3m7/16)
| (P8/7, P4/8)
| (P8/8, ccd4/64)
| (P8/9, P12/8)
| (P8/10, cm7/16)
| (P8/11, c3P5/8)
|-
! beta-octacot
| (P8, ccP5/8)
| (P8/2, P12/8)
| (P8/3, P11/8)
| (P8/4, cm7/16)
| (P8/5, ccP5/8)
| (P8/6, P4/8)
| (P8/7, P11/8)
| (P8/8, m6/32)
| (P8/9, ccP5/8)
| (P8/10, P12/8)
| (P8/11, P11/8)
|-
! gamma-octacot
| (P8, c3P5/8)
| (P8/2, c3m7/16)
| (P8/3, P12/8)
| (P8/4, ccm6/32)
| (P8/5, P4/8)
| (P8/6, ccM2/16)
| (P8/7, ccP4/8)
| (P8/8, d4/64)
| (P8/9, c3P5/8)
| (P8/10, M2/16)
| (P8/11, P12/8)
|-
! delta-octacot
| (P8, c3P4/8)
| (P8/2, P11/8)
| (P8/3, c3P4/8)
| (P8/4, P4/8)
| (P8/5, c3P4/8)
| (P8/6, P11/8)
| (P8/7, c3P4/8)
| (P8/8, M2/16)
| (P8/9, c3P4/8)
| (P8/10, P11/8)
| (P8/11, c3P4/8)
|-
! epsilon-octacot
| (P8, ccP4/8)
| (P8/2, cm7/16)
| (P8/3, P4/8)
| (P8/4, m6/32)
| (P8/5, P12/8)
| (P8/6, M2/16)
| (P8/7, c3P5/8)
| (P8/8, A12/64)
| (P8/9, ccP4/8)
| (P8/10, ccM2/16)
| (P8/11, P4/8)
|-
! wau-octacot
| (P8, P11/8)
| (P8/2, P4/8)
| (P8/3, ccP5/8)
| (P8/4, M2/16)
| (P8/5, P11/8)
| (P8/6, P12/8)
| (P8/7, ccP5/8)
| (P8/8, M10/32)
| (P8/9, P11/8)
| (P8/10, P4/8)
| (P8/11, ccP5/8)
|-
! zeta-octacot
| (P8, P4/8)
| (P8/2, M2/16)
| (P8/3, ccP4/8)
| (P8/4, M10/16)
| (P8/5, c3P5/8)
| (P8/6, cm7/16)
| (P8/7, P12/8)
| (P8/8, c3A5/64)
| (P8/9, P4/8)
| (P8/10, c3m7/16)
| (P8/11, ccP4/8)
|-
! rowspan="9" | Enneacots
! enneacot
| (P8, P5/9)
| (P8/2, P5/9)
| (P8/3, P5/9)
| (P8/4, P5/9)
| (P8/5, P5/9)
| (P8/6, P5/9)
| (P8/7, P5/9)
| (P8/8, P5/9)
| (P8/9, P4/9)
| (P8/10, P5/9)
| (P8/11, P5/9)
|-
! alpha-enneacot
| (P8, P12/9)
| (P8/2, c3P4/9)
| (P8/3, ccM6/27)
| (P8/4, P11/9)
| (P8/5, ccP5/9)
| (P8/6, ccm3/27)
| (P8/7, c4P4/9)
| (P8/8, P4/9)
| (P8/9, ccd7/81)
| (P8/10, P12/9)
| (P8/11, c3P4/9)
|-
! beta-enneacot
| (P8, ccP5/9)
| (P8/2, P12/9)
| (P8/3, c3M6/27)
| (P8/4, c3P4/9)
| (P8/5, c4P4/9)
| (P8/6, ccM6/27)
| (P8/7, P4/9)
| (P8/8, P11/9)
| (P8/9, cd7/81)
| (P8/10, ccP5/9)
| (P8/11, P12/9)
|-
! gamma-enneacot
| (P8, c3P5/9)
| (P8/2, ccP4/9)
| (P8/3, P11/9)
| (P8/4, c3P5/9)
| (P8/5, ccP4/9)
| (P8/6, P4/9)
| (P8/7, c3P5/9)
| (P8/8, ccP4/9)
| (P8/9, m3/27)
| (P8/10, c3P5/9)
| (P8/11, ccP4/9)
|-
! delta-enneacot
| (P8, c4P4/9)
| (P8/2, ccP5/9)
| (P8/3, c3m3/27)
| (P8/4, P12/9)
| (P8/5, P4/9)
| (P8/6, c3M6/27)
| (P8/7, P11/9)
| (P8/8, c3P4/9)
| (P8/9, A2/81)
| (P8/10, c4P4/9)
| (P8/11, ccP5/9)
|-
! epsilon-enneacot
| (P8, c3P4/9)
| (P8/2, P11/9)
| (P8/3, ccm3/27)
| (P8/4, P4/9)
| (P8/5, P12/9)
| (P8/6, m3/27)
| (P8/7, ccP5/9)
| (P8/8, c4P4/9)
| (P8/9, A9/81)
| (P8/10, c3P4/9)
| (P8/11, P11/9)
|-
! wau-enneacot
| (P8, ccP4/9)
| (P8/2, c3P5/9)
| (P8/3, P4/9)
| (P8/4, ccP4/9)
| (P8/5, c3P5/9)
| (P8/6, P11/9)
| (P8/7, ccP4/9)
| (P8/8, c3P5/9)
| (P8/9, M6/27)
| (P8/10, ccP4/9)
| (P8/11, c3P5/9)
|-
! zeta-enneacot
| (P8, P11/9)
| (P8/2, P4/9)
| (P8/3, m3/27)
| (P8/4, c4P4/9)
| (P8/5, c3P4/9)
| (P8/6, M6/27)
| (P8/7, P12/9)
| (P8/8, ccP5/9)
| (P8/9, c3A2/81)
| (P8/10, P11/9)
| (P8/11, P4/9)
|-
! eta-enneacot
| (P8, P4/9)
| (P8/2, c4P4/9)
| (P8/3, M6/27)
| (P8/4, ccP5/9)
| (P8/5, P11/9)
| (P8/6, c3m3/27)
| (P8/7, c3P4/9)
| (P8/8, P12/9)
| (P8/9, c4A2/81)
| (P8/10, P4/9)
| (P8/11, c4P4/9)
|-
! rowspan="10" | Decacots
! decacot
| (P8, P5/10)
| (P8/2, P5/10)
| (P8/3, P5/10)
| (P8/4, P5/10)
| (P8/5, P5/10)
| (P8/6, P5/10)
| (P8/7, P5/10)
| (P8/8, P5/10)
| (P8/9, P5/10)
| (P8/10, P4/10)
| (P8/11, P5/10)
|-
! alpha-decacot
| (P8, P12/10)
| (P8/2, ccM2/20)
| (P8/3, ccP4/10)
| (P8/4, c3M3/40)
| (P8/5, c3M7/50)
| (P8/6, cm7/20)
| (P8/7, c3P5/10)
| (P8/8, c4d4/80)
| (P8/9, P4/10)
| (P8/10, c3d3/100)
| (P8/11, P12/10)
|-
! beta-decacot
| (P8, ccP5/10)
| (P8/2, P12/10)
| (P8/3, c4P5/10)
| (P8/4, P11/10)
| (P8/5, c4M7/50)
| (P8/6, ccP4/10)
| (P8/7, c3P4/10)
| (P8/8, P4/10)
| (P8/9, P11/10)
| (P8/10, m9/50)
| (P8/11, ccP5/10)
|-
! gamma-decacot
| (P8, c3P5/10)
| (P8/2, c4M2/20)
| (P8/3, P12/10)
| (P8/4, c4m6/40)
| (P8/5, c4m2/50)
| (P8/6, ccM2/20)
| (P8/7, P4/10)
| (P8/8, ccd4/80)
| (P8/9, ccP4/10)
| (P8/10, d10/100)
| (P8/11, c3P5/10)
|-
! delta-decacot
| (P8, c4P5/10)
| (P8/2, ccP4/10)
| (P8/3, P11/10)
| (P8/4, P12/10)
| (P8/5, c3m2/50)
| (P8/6, P4/10)
| (P8/7, ccP5/10)
| (P8/8, P11/10)
| (P8/9, c3P4/10)
| (P8/10, m2/50)
| (P8/11, c4P5/10)
|-
! epsilon-decacot
| (P8, c4P4/10)
| (P8/2, c3m7/20)
| (P8/3, c4P4/10)
| (P8/4, ccm6/40)
| (P8/5, P4/10)
| (P8/6, c3m7/20)
| (P8/7, c4P4/10)
| (P8/8, d4/80)
| (P8/9, c4P4/10)
| (P8/10, M2/20)
| (P8/11, c4P4/10)
|-
! wau-decacot
| (P8, c3P4/10)
| (P8/2, P11/10)
| (P8/3, ccP5/10)
| (P8/4, P4/10)
| (P8/5, m9/50)
| (P8/6, P12/10)
| (P8/7, P11/10)
| (P8/8, ccP4/10)
| (P8/9, c4P5/10)
| (P8/10, M7/50)
| (P8/11, c3P4/10)
|-
! zeta-decacot
| (P8, ccP4/10)
| (P8/2, cm7/20)
| (P8/3, P4/10)
| (P8/4, m6/40)
| (P8/5, m2/50)
| (P8/6, M2/20)
| (P8/7, P12/10)
| (P8/8, A12/80)
| (P8/9, c3P5/10)
| (P8/10, ccA6/100)
| (P8/11, ccP4/10)
|-
! eta-decacot
| (P8, P11/10)
| (P8/2, P4/10)
| (P8/3, c3P4/10)
| (P8/4, ccP4/10)
| (P8/5, M7/50)
| (P8/6, P11/10)
| (P8/7, c4P5/10)
| (P8/8, P12/10)
| (P8/9, ccP5/10)
| (P8/10, cM7/50)
| (P8/11, P11/10)
|-
! theta-decacot
| (P8, P4/10)
| (P8/2, M2/20)
| (P8/3, c3P5/10)
| (P8/4, M10/40)
| (P8/5, cM7/50)
| (P8/6, c4M2/20)
| (P8/7, ccP4/10)
| (P8/8, c3A5/80)
| (P8/9, P12/10)
| (P8/10, c4A6/100)
| (P8/11, P4/10)
|-
! rowspan="11" | Hendecacots
! hendecacot
| (P8, P5/11)
| (P8/2, P5/11)
| (P8/3, P5/11)
| (P8/4, P5/11)
| (P8/5, P5/11)
| (P8/6, P5/11)
| (P8/7, P5/11)
| (P8/8, P5/11)
| (P8/9, P5/11)
| (P8/10, P5/11)
| (P8/11, P4/11)
|-
! alpha-hendecacot
| (P8, P12/11)
| (P8/2, c4P4/11)
| (P8/3, c4P5/11)
| (P8/4, c3P5/11)
| (P8/5, P11/11)
| (P8/6, ccP5/11)
| (P8/7, ccP4/11)
| (P8/8, c3P4/11)
| (P8/9, c5P4/11)
| (P8/10, P4/11)
| (P8/11, c3d6/121)
|-
! beta-hendecacot
| (P8, ccP5/11)
| (P8/2, P12/11)
| (P8/3, ccP4/11)
| (P8/4, c4P4/11)
| (P8/5, c3P4/11)
| (P8/6, c4P5/11)
| (P8/7, c5P4/11)
| (P8/8, c3P5/11)
| (P8/9, P4/11)
| (P8/10, P11/11)
| (P8/11, ccd6/121)
|-
! gamma-hendecacot
| (P8, c3P5/11)
| (P8/2, c3P4/11)
| (P8/3, P12/11)
| (P8/4, P11/11)
| (P8/5, c5P4/11)
| (P8/6, c4P4/11)
| (P8/7, ccP5/11)
| (P8/8, P4/11)
| (P8/9, c4P5/11)
| (P8/10, ccP4/11)
| (P8/11, cd6/121)
|-
! delta-hendecacot
| (P8, c4P5/11)
| (P8/2, ccP5/11)
| (P8/3, c5P4/11)
| (P8/4, P12/11)
| (P8/5, c3P5/11)
| (P8/6, ccP4/11)
| (P8/7, P4/11)
| (P8/8, c4P4/11)
| (P8/9, P11/11)
| (P8/10, c3P4/11)
| (P8/11, d6/121)
|-
! epsilon-hendecacot
| (P8, c5P4/11)
| (P8/2, ccP4/11)
| (P8/3, P11/11)
| (P8/4, c4P5/11)
| (P8/5, P12/11)
| (P8/6, P4/11)
| (P8/7, c3P4/11)
| (P8/8, ccP5/11)
| (P8/9, c3P5/11)
| (P8/10, c4P4/11)
| (P8/11, A3/121)
|-
! wau-hendecacot
| (P8, c4P4/11)
| (P8/2, c3P5/11)
| (P8/3, ccP5/11)
| (P8/4, c3P4/11)
| (P8/5, P4/11)
| (P8/6, P12/11)
| (P8/7, c4P5/11)
| (P8/8, P11/11)
| (P8/9, ccP4/11)
| (P8/10, c5P4/11)
| (P8/11, A10/121)
|-
! zeta-hendecacot
| (P8, c3P4/11)
| (P8/2, P11/11)
| (P8/3, c4P4/11)
| (P8/4, P4/11)
| (P8/5, ccP4/11)
| (P8/6, c3P5/11)
| (P8/7, P12/11)
| (P8/8, c5P4/11)
| (P8/9, ccP5/11)
| (P8/10, c4P5/11)
| (P8/11, ccA3/121)
|-
! eta-hendecacot
| (P8, ccP4/11)
| (P8/2, c4P5/11)
| (P8/3, P4/11)
| (P8/4, ccP5/11)
| (P8/5, c4P4/11)
| (P8/6, c5P4/11)
| (P8/7, P11/11)
| (P8/8, P12/11)
| (P8/9, c3P4/11)
| (P8/10, c3P5/11)
| (P8/11, c3A3/121)
|-
! theta-hendecacot
| (P8, P11/11)
| (P8/2, P4/11)
| (P8/3, c3P5/11)
| (P8/4, c5P4/11)
| (P8/5, c4P5/11)
| (P8/6, c3P4/11)
| (P8/7, c4P4/11)
| (P8/8, ccP4/11)
| (P8/9, P12/11)
| (P8/10, ccP5/11)
| (P8/11, c4A3/121)
|-
! iota-hendecacot
| (P8, P4/11)
| (P8/2, c5P4/11)
| (P8/3, c3P4/11)
| (P8/4, ccP4/11)
| (P8/5, ccP5/11)
| (P8/6, P11/11)
| (P8/7, c3P5/11)
| (P8/8, c4P5/11)
| (P8/9, c4P4/11)
| (P8/10, P12/11)
| (P8/11, c5A3/121)
|}

== Notation ==
While there are no agreed-upon notation system for many ploidacots, some of which can be notated by [[Kite's ups and downs notation|ups and downs notation system]]. For example, [[Ploidacot/Tricot|tricot]] is based on interpreting the generator as a supermajor second, allowing for an ^ or v to stand for 1/3 of a diatonic semitone, and [[Ploidacot/Tetracot|tetracot]] is based on interpreting the generator as a submajor second, allowing for an ^ or v to stand for 1/4 of a chromatic semitone, and [[Ploidacot/Triploid monocot|triploid monocot]] is based on interpreting the period as a submajor third, allowing for an ^ or v to stand for 1/3 of an ''inversed'' diminished second (the difference between diatonic semitone and chromatic semitone, equivalent to the [[Pythagorean comma]]). Certain ploidacots (such as [[Ploidacot/Diploid dicot|diploid dicot]]) require another additional pair, such as lifts and drops, written / and ⧵ .

== Examples ==
The ploidacots of most common temperaments can be intuitively derived from a basic understanding of its mapping. [[Meantone]] and [[Helmholtz (temperament)|helmholtz]] are monocot since they have a period of a whole octave and are generated by the perfect fifth. Dicot is dicot since it has a period of a whole octave and splits the perfect fifth in two. Semaphore has a period of a whole octave and splits the perfect twelfth in two. It requires one period to add to the fifth to make it a twelfth, and one is alpha. So it is alpha-dicot.

For a more complex example, let us consider sensi and its weak extension bison. Sensi splits 6/1 in seven. It requires two periods to the fifth to reach 6/1, and two is beta. So it is beta-heptacot. Bison splits the period of sensi in two. As a result, it now requires four periods to the fifth to reach 6/1, and four is delta. So it is diploid delta-heptacot.

Below is a list of ploidacots for common temperaments
* [[Meantone]] and [[Helmholtz (temperament)|helmholtz]] are haploid monocot
* [[Mohajira]] and [[dicot]] are dicot
* [[Bug]] and [[semaphore]] are alpha-dicot
* [[Shrutar]] is diploid alpha-dicot
* [[Ennealimmal]] is enneaploid dicot
* [[Hemiennealimmal]] is octodecaploid (18-ploid) dicot
* [[Slendric]], [[mothra]], and [[rodan]] are tricot
* [[Alphatricot]] is alpha-tricot
* [[Porcupine]] is beta-tricot
* [[Hedgehog]] is diploid alpha-tricot
* [[Tetracot]] is tetracot
* [[Squares]] is beta-tetracot
* [[Bleu]] is pentacot
* [[Magic]] is alpha-pentacot
* [[Amity]] is gamma-pentacot
* [[Miracle]] is hexacot
* [[Hanson]] is alpha-hexacot
* [[Harry]] is diploid delta-hexacot
* [[Orwell]] is alpha-heptacot
* [[Sensi]] is beta-heptacot
* [[Vishnu]] is diploid epsilon-heptacot
* [[Octacot]] is octacot
* [[Würschmidt]] is beta-octacot
* [[Valentine]] is enneacot
* [[Sycamore]] is hendecacot
* [[Chromo]] is tridecacot
* [[Pajara]] and [[injera]] are diploid
* [[Antitonic]] is diploid acot
* [[Augene]] is triploid
* [[Diminished (temperament)|Diminished]] is tetraploid
* [[Blackwood]] is pentaploid acot
* [[Whitewood]] is heptaploid acot
* [[Compton]] is dodecaploid acot

== List of ploidacots ==
=== Acot ===
* Pentaploid acot ([[blackwood]], [[5edo]])
* Heptaploid acot ([[whitewood]], [[7edo]])
* Dodecaploid acot ([[compton]], [[12edo]])

=== Monocot ===
* [[Ploidacot/Monocot|Monocot]]
* [[Ploidacot/Diploid monocot|Diploid monocot]]
* [[Ploidacot/Triploid monocot|Triploid monocot]]
* [[Ploidacot/Tetraploid monocot|Tetraploid monocot]]
* [[Ploidacot/Pentaploid monocot|Pentaploid monocot]]
* [[Ploidacot/Hexaploid monocot|Hexaploid monocot]]
* [[Ploidacot/Heptaploid monocot|Heptaploid monocot]]

=== Dicot ===
* [[Ploidacot/Dicot|Dicot]]
* [[Ploidacot/Alpha-dicot|Alpha-dicot]]
* [[Ploidacot/Diploid dicot|Diploid dicot]]
* [[Ploidacot/Diploid alpha-dicot|Diploid alpha-dicot]]
* [[Ploidacot/Triploid dicot|Triploid dicot]]
* [[Ploidacot/Triploid alpha-dicot|Triploid alpha-dicot]]

=== Tricot ===
* [[Ploidacot/Tricot|Tricot]]
* [[Ploidacot/Alpha-tricot|Alpha-tricot]]
* [[Ploidacot/Omega-tricot|Omega-tricot]]
* [[Ploidacot/Diploid tricot|Diploid tricot]]
* [[Ploidacot/Diploid alpha-tricot|Diploid alpha-tricot]]
* [[Ploidacot/Diploid beta-tricot|Diploid beta-tricot]]
* [[Ploidacot/Triploid tricot|Triploid tricot]]

=== Tetracot ===
* [[Ploidacot/Tetracot|Tetracot]]
* [[Ploidacot/Alpha-tetracot|Alpha-tetracot]]
* [[Ploidacot/Beta-tetracot|Beta-tetracot]]
* [[Ploidacot/Omega-tetracot|Omega-tetracot]]

=== Pentacot ===
* [[Ploidacot/Pentacot|Pentacot]]
* [[Ploidacot/Alpha-pentacot|Alpha-pentacot]]
* [[Ploidacot/Beta-pentacot|Beta-pentacot]]
* [[Ploidacot/Gamma-pentacot|Gamma-pentacot]]
* [[Ploidacot/Omega-pentacot|Omega-pentacot]]

=== Hexacot ===
* [[Ploidacot/Hexacot|Hexacot]]
* [[Ploidacot/Alpha-hexacot|Alpha-hexacot]]
* [[Ploidacot/Beta-hexacot|Beta-hexacot]]
* [[Ploidacot/Gamma-hexacot|Gamma-hexacot]]
* [[Ploidacot/Delta-hexacot|Delta-hexacot]]
* [[Ploidacot/Omega-hexacot|Omega-hexacot]]

=== Heptacot ===
* [[Ploidacot/Heptacot|Heptacot]]
* [[Ploidacot/Alpha-heptacot|Alpha-heptacot]]
* [[Ploidacot/Beta-heptacot|Beta-heptacot]]
* [[Ploidacot/Gamma-heptacot|Gamma-heptacot]]
* [[Ploidacot/Delta-heptacot|Delta-heptacot]]
* [[Ploidacot/Epsilon-heptacot|Epsilon-heptacot]]
* [[Ploidacot/Omega-heptacot|Omega-heptacot]]

=== Octacot ===
* [[Ploidacot/Octacot|Octacot]]
* [[Ploidacot/Beta-octacot|Beta-octacot]]
* [[Ploidacot/Epsilon-octacot|Epsilon-octacot]]
* [[Ploidacot/Omega-octacot|Omega-octacot]]

=== Enneacot ===
* [[Ploidacot/Enneacot|Enneacot]]
* [[Ploidacot/Delta-enneacot|Delta-enneacot]]
* [[Ploidacot/Omega-enneacot|Omega-enneacot]]

=== Decacot ===
* [[Ploidacot/Decacot|Decacot]]
* [[Ploidacot/Beta-decacot|Beta-decacot]]
* [[Ploidacot/Epsilon-decacot|Epsilon-decacot]]

=== >10 cots ===
* [[Ploidacot/Hendecacot|Hendecacot]]
* [[Ploidacot/Icosacot|Icosacot]]

== See also ==
* [[Wedgie]] – a mathematical generalization of the concept of ploidacots that uniquely characterizes a temperament

[[Category:Temperament naming]]

Godzilla extensions

2026-05-23T01:29:48Z

Xenllium: Created page with "{{Breadcrumb|Semaphore and godzilla}} Godzilla has various competing extensions to the 11- and 13-limit. The following temperaments are discussed in this article: * '''Tridecimal godzilla''' ({{nowrap| 14cf & 19 }}) – tempering out 45/44, 49/48, 78/77, and 81/80; * '''Semafour''' ({{nowrap| 14cf & 19e }}) – tempering out 33/32, 49/48, 55/54, and 91/90; * '''Varan''' ({{nowrap| 19e & 24 }}) – tempering out 49/48, 66/65, 77/75, and 81/80;..."

{{Breadcrumb|Semaphore and godzilla}}

[[Godzilla]] has various competing [[extension]]s to the [[11-limit|11-]] and [[13-limit]]. The following temperaments are discussed in this article:

* '''Tridecimal godzilla''' ({{nowrap| 14cf & 19 }}) – tempering out 45/44, 49/48, 78/77, and 81/80;
* '''Semafour''' ({{nowrap| 14cf & 19e }}) – tempering out 33/32, 49/48, 55/54, and 91/90;
* '''Varan''' ({{nowrap| 19e & 24 }}) – tempering out 49/48, 66/65, 77/75, and 81/80;
* '''Baragon''' ({{nowrap| 19 & 24 }}) – tempering out 49/48, 56/55, 81/80, and 91/90.

The most important of these is tridecimal godzilla, where primes 11 and 13 are mapped such that 10/9 and 11/10 are tempered together and that 7/6, 8/7, 13/11, and 15/13 are tempered together. Semafour takes the same mapping of 13 as godzilla, but swaps in a less complex and less accurate mapping for 11, so that 4/3 and 11/8 are tempered together. Similarly, varan takes the same mapping of 13 as godzilla, but swaps in a more complex mapping for 11, so that 6/5 and 13/11 are tempered together. Finally, baragon swaps in a less complex mapping for prime 11, so that 7/5 and 11/8 are tempered together.

== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! rowspan="3" | #
! rowspan="3" | Cents*
! colspan="5" | Approximate ratios
|-
! rowspan="2" | 2.3.5.7.13 subgroup
! colspan="4" | Full 13-limit extensions
|-
! Godzilla
! Semafour
! Varan
! Baragon
|-
| 0
| 0.0
| 1/1
|
|
|
|
|-
| 1
| 948.0
| 7/4, 12/7, 26/15
| 22/13
|
|
|
|-
| 2
| 696.0
| 3/2
| 22/15
| 16/11
|
|
|-
| 3
| 444.0
| 9/7, 13/10, 21/16
|
| 14/11
|
|
|-
| 4
| 192.0
| 9/8, 10/9
| 11/10
| 12/11
|
|
|-
| 5
| 1140.0
| 25/13, 27/14, 39/20, 40/21
|
|
|
|
|-
| 6
| 888.0
| 5/3
|
| 18/11
| 22/13
|
|-
| 7
| 636.0
| 10/7, 13/9
|
|
| 22/15
| 16/11
|-
| 8
| 384.0
| 5/4
| 11/9
|
|
| 14/11
|-
| 9
| 132.0
| 13/12, 15/14
|
|
| 11/10
| 12/11
|-
| 10
| 1080.0
| 13/7, 15/8
| 11/6
| 20/11
|
|
|-
| 11
| 828.0
| 13/8
| 11/7
|
|
| 18/11
|-
| 12
| 576.0
| 25/18
| 11/8
| 15/11
|
|
|-
| 13
| 324.0
|
|
| 13/11
| 11/9
|
|-
| 14
| 72.1
| 25/24
|
|
|
|
|-
| 15
| 1020.1
| 25/14
|
|
| 11/6
| 20/11
|-
| 16
| 768.1
| 25/16
|
|
| 11/7
|
|-
| 17
| 516.1
|
|
|
| 11/8
| 15/11
|-
| 18
| 264.1
|
|
|
|
| 13/11
|}
<nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]]

[[Category:Godzilla]]
[[Category:Temperament extensions]]
[[Category:Rank-2 temperaments]]

Semaphore and godzilla

2026-05-23T00:10:52Z

Xenllium:

{{interwiki
| en = Semaphore and godzilla
| de = Semiphor, Semaphor, Godzilla
| es =
| ja =
}}
{{Infobox regtemp
| Title = {{nowrap|Semaphore; Godzilla}}
| Subgroups = 2.3.7, 2.3.5.7, 2.3.5.7.13
| Comma basis = [[49/48]] (2.3.7); [[49/48]], [[81/80]] (2.3.5.7); [[49/48]], [[81/80]], [[91/90]] (L7.13)
| Edo join 1 = 5 | Edo join 2 = 19
| Mapping = 1; 2 8 1 11
| Generators = 7/4
| Generators tuning = 947.8
| Optimization method = CWE
| Pergen = (P8, P4/2)
| Color name = Zozoti
| MOS scales = [[4L 1s]], [[5L 4s]], [[5L 9s]], [[5L 14s]]
| Odd limit 1 = 9 | Mistuning 1 = 20.5 | Complexity 1 = 9
| Odd limit 2 = 2.3.5.7.13 15 | Mistuning 2 = 20.5 | Complexity 2 = 14
}}
'''Semaphore''', of the [[semaphoresmic clan]], is characterized by [[49/48]] being [[tempering out|tempered out]], so the [[generator]] represents [[7/4]] and [[12/7]] (or [[8/7]] and [[7/6]]) equally. This results in a very low [[complexity]] 2.3.7-[[subgroup]] [[regular temperament|temperament]], with the drawback that most intervals of 7 must be out of tune by at least half of the comma 49/48, or about 18 [[cent]]s. ''Semaphore'' is a play on the words "semi-" and "fourth".

If the [[5/1|5th harmonic]]'s intervals are desired, [[5/4]] can be sensibly mapped to +8 generators by tempering out [[81/80]], making it a [[Meantone family #Extensions|meantone temperament]]. This temperament is '''godzilla'''. Moreover, the generator can be taken to be [[26/15]], which maps [[13/8]] to +11 generators by tempering out [[91/90]] and [[105/104]]. This extends the temperament to the 2.3.5.7.13 subgroup, with an abundance of harmonic resource and little additional damage.

A more accurate but complex mapping of 5 can be found in [[immunity]], or 5/4 itself can be made the period by tempering out [[128/125]], resulting in [[triforce]].

For technical information, see [[Semaphoresmic clan #Semaphore]] and [[Semaphoresmic clan #Godzilla|#Godzilla]]. For a discussion on 11- and 13-limit extensions, see [[Godzilla extensions]].

== Interval chains ==
In the following tables, odd harmonics 1–13 and their inverses are in '''bold'''.

=== Semaphore ===
{| class="wikitable center-1 right-2"
|-
! # !! Cents* !! Approximate ratios
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 950.7 || '''7/4''', 12/7
|-
| 2 || 701.4 || '''3/2'''
|-
| 3 || 452.1 || 9/7, 21/16
|-
| 4 || 202.8 || '''9/8'''
|-
| 5 || 1153.4 || 27/14, 63/32
|}
<nowiki/>* In 2.3.7-subgroup CWE tuning, octave reduced

=== Godzilla ===
{| class="wikitable center-1 right-2"
|-
! # !! Cents* !! Approximate ratios
|-
| 0 || 0.0 || '''1/1'''
|-
| 1 || 948.0 || '''7/4''', 12/7, 26/15
|-
| 2 || 696.0 || '''3/2'''
|-
| 3 || 444.0 || 9/7, 13/10, 21/16
|-
| 4 || 192.0 || '''9/8''', 10/9
|-
| 5 || 1140.0 || 27/14, 39/20, 40/21, 52/27, 63/32
|-
| 6 || 888.0 || 5/3
|-
| 7 || 636.0 || 10/7, 13/9
|-
| 8 || 384.0 || '''5/4'''
|-
| 9 || 132.0 || 13/12, 15/14
|-
| 10 || 1080.0 || 13/7, 15/8
|-
| 11 || 828.0 || '''13/8'''
|-
| 12 || 576.0 || 25/18, 39/28, 45/32
|-
| 13 || 324.0 || 39/32
|-
| 14 || 72.1 || 25/24, 50/49
|}
<nowiki/>* In 2.3.5.7.13-subgroup CWE tuning, octave reduced

== Scales ==
Scala files:
* [[Semaphore5]]
* [[Semaphore9]]
* [[Semaphore14]]

=== 5-note (proper) ===
{| class="wikitable center-all"
|-
! Small ("minor") interval
| 202.8
| 452.1
| 701.4
| 950.7
|-
! [[JI]] intervals represented
| 9/8
| 9/7~13/10
| 3/2
| 7/4~12/7
|-
! Large ("major") interval
| 249.3
| 498.6
| 747.9
| 997.2
|-
! JI intervals represented
| 7/6~8/7
| 4/3
| 14/9~20/13
| 16/9
|}

=== 9-note (improper) ===
{{Main| 5L 4s }}

{| class="wikitable center-all"
|-
! Small ("minor") interval
| 60.0
| 252.0
| 312.0
| 504.0
| 564.0
| 756.0
| 816.0
| 1008.0
|-
! JI intervals represented
|
| 7/6~8/7
| 6/5
| 4/3
| 7/5~18/13
| 14/9~20/13
| 8/5~13/8
| 9/5~16/9
|-
! Large ("major") interval
| 192.0
| 384.0
| 444.0
| 636.0
| 696.0
| 888.0
| 948.0
| 1140.0
|-
! JI intervals represented
| 9/8~10/9
| 5/4
| 9/7~13/10
| 10/7~13/9
| 3/2
| 5/3
| 7/4~12/7
|
|}

In 19edo, Godzilla[9] has steps 3 3 1 3 1 3 1 3 1, and contains the following useful scales as subsets:
* Meantone pentic (5 3 5 3 3)
* Altered diatonic I (3 4 3 1 3 4 1)
* Altered diatonic II (3 4 3 1 4 3 1)
* Altered diatonic III (4 3 3 1 4 3 1)
* Altered diatonic IV (3 3 4 1 3 4 1)

It does not, however, contain the ordinary diatonic scale. Godzilla[9] thus expands on the pentic scale, but in a different way than diatonic scales do.

The four heptatonic subsets can be regarded as chromatic alterations of the diatonic scale, or alternatively as variants of Archytas' septimal diatonic scale, but with a greatly exaggerated difference between the two different whole tone sizes. All five of these subsets are very expressive melodically. Godzilla[9] combines all of these and is expressive in its own right; it could even be thought of as 19edo's answer to the well-loved Supra[7] diatonic scale of [[17edo]], as both are improper and made up of whole-tones and third-tones.

Like Supra[7], Godzilla[9] is well stocked with subminor and supermajor triads; in this case they can be viewed as 6:7:9 and 10:13:15 since 19edo is a [[The Biosphere|biome]] temperament. Godzilla[9] has only ''one'' each of the more stable 5-limit major and minor triads, which might be considered a drawback, but could also be considered a strength for helping to establish a clearer tonal center (since all triads other than the tonic have tension in them).

== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.7-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/4 = 952.2948{{c}}
| CWE: ~7/4 = 950.6890{{c}}
| POTE: ~7/4 = 949.6154{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/4 = 948.7959{{c}}
| CWE: ~7/4 = 947.8216{{c}}
| POTE: ~7/4 = 947.3650{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~7/4 = 948.9311{{c}}
| CWE: ~7/4 = 948.0037{{c}}
| POTE: ~7/4 = 947.5708{{c}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 7/6
| 933.129
|
|-
| [[9edo|7\9]]
|
| 933.333
| 9cff val
|-
| [[14edo|11\14]]
|
| 942.857
| 14cf val, lower bound of 7- and 9-odd-limit diamond monotone
|-
|
| 9/7
| 945.028
|
|-
|
| 7/5
| 945.355
|
|-
|
| 13/7
| 947.170
|
|-
| [[19edo|15\19]]
|
| 947.368
| Lower bound of {{nowrap|no-11}} 13-odd-limit diamond monotone {{nowrap|No-11}} 15-odd-limit diamond monotone (singleton)
|-
|
| 5/3
| 947.393
|
|-
|
| 13/9
| 948.088
|
|-
|
| 5/4
| 948.289
| 7-, 9-odd-limit, {{nowrap|no-11}} 13- and 15-odd-limit minimax
|-
|
| 13/12
| 948.730
|
|-
|
| 13/8
| 949.139
|
|-
| [[24edo|19\24]]
|
| 950.000
|
|-
|
| 3/2
| 950.978
|
|-
|
| 13/10
| 951.405
|
|-
| [[5edo|4\5]]
|
| 960.000
| Upper bound of 7-, 9-odd-limit, and {{nowrap|no-11}} 13-odd-limit diamond monotone
|-
|
| 7/4
| 968.826
|
|}
<nowiki/>* Besides the octave

== Music ==
; [[Cameron Bobro]]
* [https://web.archive.org/web/20201127014130/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Bobro/GodzillaExample.mp3 ''Godzilla Example'']

; [[Igliashon Jones]]
* [http://tinyurl.com/4uyumk9 "Change is on the Wind"]{{dead link}} in Godzilla[9]

; [[Roncevaux]]
* [https://web.archive.org/web/20201127013241/http://micro.soonlabel.com/gene_ward_smith/Others/Roncevaux/S__no_Contratempo_by_Roncevaux_on_SoundCloud___Hear_the_world_s_sounds.mp3 ''Só no Contratempo'']
* [https://web.archive.org/web/20201127013653/http://micro.soonlabel.com/gene_ward_smith/Others/Roncevaux/__O_que_a_gente_quer__em_19_TET__temperamento_Godzilla__9___by_Roncevaux.mp3 ''O que a gente quer'']

; [[Starshine]]
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] (2020) – in Semaphore[9], 19edo tuning

== See also ==
* [[Diasem]], a [[maximum variety|max-variety-3]] JI [[detempering]] of semaphore
* [[Semaphore–chromatic equivalence continuum]]

[[Category:Semaphore| ]] 
[[Category:Godzilla]] 
[[Category:Rank-2 temperaments]]
[[Category:Semaphoresmic clan]]
[[Category:Meantone family]]
[[Category:Sensamagic clan]]

Sycamore family

2026-05-22T22:49:52Z

Xenllium:

{{Technical data page}}
The head of the '''sycamore family''' is [[5-limit]] sycamore, which tempers out (25/24)6/(5/4) = {{monzo| -16 -6 11 }} = 48828125/47775744, the [[sycamore comma]]. Its [[generator]] is a [[25/24 | classic chromatic semitone]], and stacking six of these gives 5/4 (and hence five 6/5) and eleven give 3/2. [[94edo]] [[support]]s sycamore, and 5\94 is recommendable as a generator. It can be described as the 19 & 94 temperament, and uses a decidedly flat version of the chromatic semitone as a generator. [[mos]] of 18 or 19 notes to the octave give enough room for sycamore's triads, but 37 notes can be tried by the adventurous.

Another possible tuning uses a generator which is a near pure 3/2 at 702.162258 [[cent]]s divided into 11 parts, and this makes the generator chain of sycamore exactly the same as [[Carlos Beta]]. In fact, Carlos Beta is characterized by Carlos as taking five steps to reach 6/5 and six to reach 5/4, which means it tempers out the sycamore comma. It can be described as the generator chain of sycamore, or sycamore can be called Carlos Beta with octaves.

== Sycamore ==
[[Subgroup]]: 2.3.5

[[Comma list]]: 48828125/47775744

{{Mapping|legend=1| 1 1 2 | 0 11 6 }}

: mapping generators: ~2, ~25/24

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6031{{c}}, ~25/24 = 63.8108{{c}}
: [[error map]]: {{val| +0.603 +0.567 -2.242 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/24 = 63.8234{{c}}
: error map: {{val| 0.000 +0.103 -3.373 }}

{{Optimal ET sequence|legend=1| 18, 19, 56, 75, 94, 207c, 301c }}

[[Badness]] (Sintel): 4.925

== Septimal sycamore ==
{{main| Sycamore and betic }}

The second element of the [[Normal lists #Normal interval list|normal comma list]] for septimal sycamore is [[875/864]], the keema, and it also tempers out [[686/675]], the senga, and [[3136/3125]], hemimean. It may also be called the 19 & 56 temperament. This may also be used as the name for the temperament obtained by adding [[100/99]] to sycamore's commas, giving unidecimal sycamore, where 10 generator steps reaches 16/11, 11 reach 3/2, and 15 give 7/4, adding a considerable dose of 11-limit harmonies to the 19-note MOS. [[75edo]] is an excellent tuning for 7-limit sycamore, and [[56edo]] for the 11-limit version.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 686/675, 875/864

{{Mapping|legend=1| 1 1 2 2 | 0 11 6 15 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7208, ~25/24 = 64.0334
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 64.0496

{{Optimal ET sequence|legend=1| 18, 19, 56, 75d }}

[[Badness]] (Sintel): 1.569

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 385/384, 686/675

Mapping: {{mapping| 1 1 2 2 4 | 0 11 6 15 -10 }}

Optimal tunings:
* WE: ~2 = 1199.4126, ~25/24 = 64.2363
* CWE: ~2 = 1200.0000, ~25/24 = 64.2505

{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}

Badness (Sintel): 1.849

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 169/168, 385/384

Mapping: {{mapping| 1 1 2 2 4 3 | 0 11 6 15 -10 13 }}

Optimal tunings:
* WE: ~2 = 1199.6597, ~25/24 = 64.2778
* CWE: ~2 = 1200.0000, ~25/24 = 64.2853

{{Optimal ET sequence|legend=0| 18, 19, 37, 56 }}

Badness (Sintel): 1.417

== Betic ==
{{main| Sycamore and betic }}

Septimal sycamore sharpens the fifth from where it stands in the 5-limit, and lowers accuracy in order to reach 7-limit harmonies. If we retain tunings approximately (e.g. 94edo) or exactly those of Carlos Beta, we get the 19 & 94 temperament, betic, for the 7-limit. This adds [[225/224]] to the sycamore comma. The Carlos Beta tuning, with pure fifths, is a good tuning choice, but 94 or 113 equal are as well. Betic extends to the 11-limit upon addition of [[385/384]] or [[540/539]] to the list of commas, which means it supports both 7 and 11-limit marvel.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 1071875/1062882

{{Mapping|legend=1| 1 1 2 1 | 0 11 6 34 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6891, ~25/24 = 63.7773
* [[CWE]]: ~2 = 1200.0000, ~25/24 = 63.7683

{{Optimal ET sequence|legend=1| 19, 56d, 75, 94, 113, 320cc, 433ccd }}

[[Badness]] (Sintel): 1.765

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 218750/216513

Mapping: {{mapping| 1 1 2 1 5 | 0 11 6 34 -29 }}

Optimal tunings:
* WE: ~2 = 1200.4466, ~25/24 = 63.7993
* CWE: ~2 = 1200.0000, ~25/24 = 63.7796

{{Optimal ET sequence|legend=0| 19, 75, 94, 207c }}

Badness (Sintel): 1.880

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 325/324, 385/384, 1875/1859

Mapping: {{mapping| 1 1 2 1 5 2 | 0 11 6 34 -29 32 }}

Optimal tunings:
* WE: ~2 = 1200.3946, ~25/24 = 63.7867
* CWE: ~2 = 1200.0000, ~25/24 = 63.7702

{{Optimal ET sequence|legend=0| 19, 75, 94, 113, 207c }}

Badness (Sintel): 1.342

[[Category:Temperament families]]
[[Category:Sycamore family ]] 
[[Category:Sycamore| ]] 
[[Category:Rank 2]]

Fractional-octave temperaments

2026-05-20T09:29:13Z

Xenllium:

'''Fractional-octave temperaments''' are [[temperament]]s which have a [[period]] which corresponds to a [[just]] [[interval]] mapped to a fraction of the [[octave]], that is one step of an [[edo]].

== Theory ==
Fractional-octave temperaments are valuable with regards to [[Polysystemic|polysystemicism]] and polychromatics. They are acoustically significant with regards to containing modes of limited transposition, as well as their ability to expand on the harmony of the equal division they are a superset of. Such temperaments are also a way of introducing less common and harmonically less performing equal divisions into music that prefers consonance and is based on regular temperament theory.

=== Terminology ===
The terminology was developed by [[Eliora]]. The equal division containing the mos scale of such a temperament, starting from the tonic, is referred to as a ''wireframe'', and individual notes of that equal division are called ''hinges''. Thus in this context, the wireframe is the tuning consisting of only stacks of the period and no stacks of the generator. Temperament-agnostically, this can be used to refer to any structure embedded in an (x,y)-ET which repeats y times within that period, its "wireframe" is y-ET. If an equal division is a subset of a temperament, it is said to ''subtend'' the temperament, just how hinges on a ferris wheel subtend the structure to make it rotate and function.

The most common way to produce a fractional-octave temperament is through an excellent approximation of an interval relative to the size of the wireframe edo. For example, [[compton family]] tempers out the Pythagorean comma and maps 7 steps of 12edo to [[3/2]]. Likewise, a lot of 10th-octave temperaments have a [[13/8]] as 7\10, and 26th-octave temperaments often have a [[7/4]] for 21\26.

However, an equal division does not have to be harmonically decent to be a wireframe for a fractional-octave temperament. If an equal division has multiples which are high in consistency or are zeta equal divisions or otherwise harmonically strong, it can produce a lot of such temperaments—notable examples being [[20edo]] or [[32edo]]. Likewise, proximity of a step of equal division to a comma is often a source of these temperaments—for example [[56edo]]'s step being directly close to [[81/80]], and 44edo's step being extremely close to [[64/63]].

=== Disagreement between temperament catalog strategy and fractional-octave practice ===
Traditional regular temperament perspective on periods and generators has a shortcoming when it comes to handling fractional-octave temperaments, as it treats divisions of periods (for example, what [[hemiennealimmal]] is to [[ennealimmal]]) as [[extension]]s of a temperament with a subset period. However, fractional-octave temperaments and scales are sought for being able to treat an each equal division as an entity in its own right, so a composer might find hemiennealimmal to be a drastically different system to ennealimmal in line with [[18edo]] being very different from [[9edo]]. This facet is reflected by the distinction of strong and weak extensions.

A particularly strong offender of this is the [[landscape microtemperaments]] list, which features temperaments which are all supersets of 3edo, but from a composer's perspective it contains wildly different temperaments due to the fact that edo multiples of 3 themselves are different. For example, magnesium (12), and zinc (30), are both landscape systems due to being multiples of 3, but 30edo is drastically different from 12edo in terms of composition, and therefore such temperaments are not alike at all.

== Individual pages of temperaments by subtending equal division ==

=== 2 to 100 ===
{| class="wikitable center-all"
|+
|
| [[2nd-octave temperaments|2]]
| [[3rd-octave temperaments|3]]
| [[4th-octave temperaments|4]]
| [[5th-octave temperaments|5]]
| [[6th-octave temperaments|6]]
| [[7th-octave temperaments|7]]
| [[8th-octave temperaments|8]]
| [[9th-octave temperaments|9]]
| [[10th-octave temperaments|10]]
|-
| [[11th-octave temperaments|11]]
| [[12th-octave temperaments|12]]/[[Compton family|C]]
| [[13th-octave temperaments|13]]
| [[14th-octave temperaments|14]]
| [[15th-octave temperaments|15]]
| [[16th-octave temperaments|16]]
| [[17th-octave temperaments|17]]
| [[18th-octave temperaments|18]]
| [[19th-octave temperaments|19]]
| [[20th-octave temperaments|20]]
|-
| [[21st-octave temperaments|21]]
| [[22nd-octave temperaments|22]]
| [[23rd-octave temperaments|23]]
| [[24th-octave temperaments|24]]
| [[25th-octave temperaments|25]]
| [[26th-octave temperaments|26]]
| [[27th-octave temperaments|27]]
| [[28th-octave temperaments|28]]
| [[29th-octave temperaments|29]]
| [[30th-octave temperaments|30]]
|-
| [[31st-octave temperaments|31]]
| [[32nd-octave temperaments|32]]
| [[33rd-octave temperaments|33]]
| [[34th-octave temperaments|34]]
| [[35th-octave temperaments|35]]
| [[36th-octave temperaments|36]]
| [[37th-octave temperaments|37]]
| [[38th-octave temperaments|38]]
| [[39th-octave temperaments|39]]
| [[40th-octave temperaments|40]]
|-
| [[41st-octave temperaments|41]]/[[Countercomp family|CC]]
| [[42nd-octave temperaments|42]]
| [[43rd-octave temperaments|43]]
| [[44th-octave temperaments|44]]
| [[45th-octave temperaments|45]]
| [[46th-octave temperaments|46]]
| [[47th-octave temperaments|47]]
| [[48th-octave temperaments|48]]
| [[49th-octave temperaments|49]]
| [[50th-octave temperaments|50]]
|-
| [[51st-octave temperaments|51]]
| [[52nd-octave temperaments|52]]
| [[53rd-octave temperaments|53]]/[[Mercator family|M]]
| [[54th-octave temperaments|54]]
| [[55th-octave temperaments|55]]
| [[56th-octave temperaments|56]]
| [[57th-octave temperaments|57]]
| [[58th-octave temperaments|58]]
| [[59th-octave temperaments|59]]
| [[60th-octave temperaments|60]]
|-
| [[61st-octave temperaments|61]]
| [[62nd-octave temperaments|62]]
| [[63rd-octave temperaments|63]]
| [[64th-octave temperaments|64]]
| [[65th-octave temperaments|65]]
| [[66th-octave temperaments|66]]
| [[67th-octave temperaments|67]]
| [[68th-octave temperaments|68]]
| [[69th-octave temperaments|69]]
| [[70th-octave temperaments|70]]
|-
| [[71st-octave temperaments|71]]
| [[72nd-octave temperaments|72]]
| [[73rd-octave temperaments|73]]
| [[74th-octave temperaments|74]]
| [[75th-octave temperaments|75]]
| [[76th-octave temperaments|76]]
| [[77th-octave temperaments|77]]
| [[78th-octave temperaments|78]]
| [[79th-octave temperaments|79]]
| [[80th-octave temperaments|80]]
|-
| [[81st-octave temperaments|81]]
| [[82nd-octave temperaments|82]]
| [[83rd-octave temperaments|83]]
| [[84th-octave temperaments|84]]
| [[85th-octave temperaments|85]]
| [[86th-octave temperaments|86]]
| [[87th-octave temperaments|87]]
| [[88th-octave temperaments|88]]
| [[89th-octave temperaments|89]]
| [[90th-octave temperaments|90]]
|-
| [[91st-octave temperaments|91]]
| [[92nd-octave temperaments|92]]
| [[93rd-octave temperaments|93]]
| [[94th-octave temperaments|94]]
| [[95th-octave temperaments|95]]
| [[96th-octave temperaments|96]]
| [[97th-octave temperaments|97]]
| [[98th-octave temperaments|98]]
| [[99th-octave temperaments|99]]
| [[100th-octave temperaments|100]]
|}

=== 101 and up ===
[[111th-octave temperaments|111]], [[118th-octave temperaments|118]], [[159th-octave temperaments|159]], [[400th-octave temperaments|400]], [[665th-octave temperaments|665]]

* C = compton family
* CC = countercomp family
* M = mercator family equated with 53rd-octave temperaments until otherwise documented, also contains 106th-octave temperaments

== Temperaments discussed elsewhere ==
Temperaments discussed as a part of a commatic family, or otherwise in temperament lists unrelated to fractional-octave theory include:

* 1\2 period temperaments
** [[Diaschismic family|Diaschismic temperaments]]
** [[Vishnuzmic family|Vishnuzmic temperaments]]
** [[Jubilismic clan|Jubilismic temperaments]]
** [[Varunismic temperaments]]
** [[Lokismic temperaments]]
** [[Nimona|Nimona temperament]]
* 1\3 period temperaments
** [[Augmented family|Augmented temperaments]]
** [[Misty family|Misty temperaments]]
** [[Landscape microtemperaments|Landscape temperaments]]
* 1\4 period temperaments
** [[Diminished family|Diminished temperaments]]
** [[Undim family|Undim temperaments]]
* 1\5 period temperaments
** [[Quintile family|Quintile temperaments]]
** [[Quintosec family|Quintosec temperaments]]
** [[Trisedodge family|Trisedodge temperaments]]
** [[Cloudy clan|Cloudy temperaments]]
** [[Limmic temperaments]]
* 1\6 period temperaments
** [[Augmented family #Hexe|Hexe]]
** [[Landscape microtemperaments #Sextile|Sextile]]
** [[Stearnsmic clan #Stearnscape|Stearnscape]]
* [[Akjaysma|Akjaysmic temperaments]] (1\7 period)
** [[Ragismic microtemperaments #Brahmagupta|Brahmagupta]]
** [[Schismatic family #Septant|Septant]]
** [[Whitewood family #Whitewood|Whitewood temperaments]]
** [[Keemic temperaments #Sevond|Sevond]]
** [[Mistismic temperaments #Neutron|Neutron]]
* [[Ragismic microtemperaments #Octoid|Octoid]], [[Schismatic family #Octant|octant]] (1\8 period)
* [[Septiennealimmal clan|Septiennealimmal temperaments]] (1\9 period)
** [[Ragismic microtemperaments #Ennealimmal|Ennealimmal]]
** [[Augmented family #Niner|Niner]]
** [[Marvel temperaments #Enneaportent|Enneaportent]]
** [[Kleismic family #Novemkleismic|Novemkleismic]]
* [[Linus]] temperaments (1\10 period)
** [[Breedsmic temperaments #Decoid|Decoid]]
** [[Ragismic microtemperaments #Deca|Deca]]
** [[Cloudy clan #Decic|Decic]]
** [[Stearnsmic clan #Decistearn|Decistearn]]
** [[Quintile family #Decile|Decile]]
** [[Vishnuzmic family #Decavish|Decavish]]
** [[Metric microtemperaments #Decimetra|Decimetra]]
* [[Porwell temperaments #Hendecatonic|Hendecatonic]] (1\11 period)
* [[Compton family|Compton]], [[Very high accuracy temperaments #Atomic|atomic]] (1\12 period)
* [[Orwellismic temperaments #Triskaidekic|Triskaidekic]], [[Octagar temperaments #Tridecatonic|tridecatonic]], [[Ragismic microtemperaments #Trideci|trideci]], [[aluminium]] (1\13 period)
* [[Silicon]] (1\14 period)
* [[Cloudy clan #Pentadecal|Pentadecal]], [[Trienstonic clan #Quindecic|quindecic]] (1\15 period)
* [[Ragismic microtemperaments #Octoid|Hexadecoid]], [[Jubilismic clan #Sedecic|sedecic]] (1\16 period)
* [[Ragismic microtemperaments #Chlorine|Chlorine]] (1\17 period)
* [[Septiennealimmal clan #Ennealimmal|Hemiennealimmal]] (1\18 period)
* [[Ragismic microtemperaments #Enneadecal|Enneadecal]], [[Meantone family #Meanmag|meanmag]] (1\19 period)
* [[Hemimage temperaments #Degrees|Degrees]] (1\20 period)
* [[Akjayland]] (1\21 period)
* [[Porwell temperaments #Hendecatonic|Icosidillic]] (1\22 period)
* [[Porwell temperaments #Icositritonic|Icositritonic]] (1\23 period)
* [[Compton family #Hours|Hours]], [[chromium]] (1\24 period)
* [[Septiennealimmal clan #Ennealimmal|Trinealimmal]], [[Tritrizo clan #Cobalt|cobalt]] (1\27 period)
* [[Horwell temperaments #Oquatonic|Oquatonic]] (1\28 period)
* [[Hemifamity temperaments #Mystery|Mystery]], [[Copper comma|copper]] (1\29 period)
* [[31st-octave temperaments|Birds]] (1\31 period)
* [[Compton family #Gamelstearn|Gamelstearn]] (1\36 period)
* [[Ragismic microtemperaments #Enneadecal|Hemienneadecal]], [[semihemienneadecal]] (1\38 period)
* [[Countercomp family|Countercomp temperaments]], [[niobium]] (1\41 period)
* [[Mitonismic temperaments #Meridic|Meridic]] (1\43 period)
* [[Ragismic microtemperaments #Palladium|Palladium]] (1\46 period)
* [[Compton family #Omicronbeta|Omicronbeta]] (1\72 period)
* [[The Flashmob#Iridium|Iridium]] (1\77 period)
* [[Parkleiness temperaments #Octogintic|Octogintic]] (1\80 period)
* [[Stearnsmic clan #Garistearn|Garistearn]] (1\94 period)
* [[Septiennealimmal clan #Undecentic|Undecentic]] (1\99 period)
* [[Septiennealimmal clan #Schisennealimmal|Schisennealimmal]] (1\171 period)
* [[Septiennealimmal clan #Lunennealimmal|Lunennealimmal]] (1\441 period)

== See also ==
* [[Map of rank-2 temperaments]]: Visual map of many of the temperaments listed here.
* [[Octave-splitting comma]]

[[Category:Temperament collections]]

Octave-splitting comma

2026-05-20T09:02:33Z

Xenllium: Created page with "'''Octave-splitting commas''' are rational intervals which induce fractional-octave temperaments. If tempering out the comma leads to splitting the octave into ''N'' equal parts (''N'' > 1), the comma is an octave-splitting comma which induces an ''N''th-octave temperament. This terminology was developed by Xenllium. == Examples == Below is a list of octave-splitting commas for common temperaments: {| class="wikitable" |- ! Comma !!..."

'''Octave-splitting commas''' are rational [[interval]]s which induce [[fractional-octave temperaments]]. If tempering out the comma leads to splitting the [[octave]] into ''N'' equal parts (''N'' > 1), the comma is an octave-splitting comma which induces an ''N''th-octave temperament.

This terminology was developed by [[User:Xenllium|Xenllium]].

== Examples ==
Below is a list of octave-splitting commas for common temperaments:

{| class="wikitable"
|-
! Comma !! Associated temperament !! Harmonic limit !! Splitting order
|-
| [[256/243]] || [[Limmic temperaments|Blackwood]] || 3 || 5
|-
| [[2187/2048]] || [[Apotome family|Whitewood]] || 3 || 7
|-
| [[531441/524288]] || [[Compton family|Compton]] || 3 || 12
|-
| [[2048/2025]] || [[Diaschismic family|Diaschismic]] || 5 || 2
|-
| [[128/125]] || [[Augmented family|Augmented]] || 5 || 3
|-
| [[648/625]] || [[Diminished family|Diminished]] || 5 || 4
|-
| [[50/49]] || [[Jubilismic clan|Jubilismic]] || 7 || 2
|-
| [[250047/250000]] || [[Landscape microtemperaments|Landscape]] || 7 || 3
|-
| [[9801/9800]] || [[Kalismic temperaments|Kalismic]] || 11 || 2
|-
| [[289/288]] || [[Semitonismic]] || 17 || 2
|}

== See also ==
* [[Fractional-octave temperaments]]

[[Category:Comma]]

Comma

2026-05-20T03:47:59Z

Xenllium:

{{interwiki
| de = Komma
| en = Comma
| es =
| ja =
| ro = Come raționale
}}
{{Wikipedia|Comma (music)}}
In just intonation, a '''comma''' is a small [[interval]] that occurs between two intervals which are close in [[pitch]]. Commas are always greater than [[1/1]] (0 [[cent]]s) and not a power such as a square or cube of any other just interval, and generally in a low [[prime limit]].

Commas are often considered [[dissonant]] due to their small but noticeable [[interval size measure|size]] which induces an audible [[beat]]. In addition, certain chord progressions are [[comma pump]]s, which may cause the [[tonal center]] of a piece to drift up or down in pitch over time. These effects can be treated either as features to be desired or as problems to be solved. Examples of approaches that try to solve these problems include [[adaptive just intonation]], [[temperament]]s, and [[fudging]].

In [[regular temperament theory]], a comma is something to be ''[[tempered out]]'', equating two mathematically distinct intervals. For example, the [[syntonic comma]] (81/80), which occurs between [[10/9]] and [[9/8]] as well as between [[81/64]] and [[5/4]], is tempered out by [[meantone]] temperament and thus conflates these two pairs of intervals.
[[File:EufaJICommasFinalV2.svg|thumb|A visualization of intervals commonly used as commas, built from stacked intervals, to scale.]]
Commas are usually written as [[frequency ratio]]s, but they can also be written as products of primes, sometimes called [[monzo]]s or '''unison vectors'''. The [[color name]] refers to both the comma and the temperament created when it is tempered out, except for 3-limit commas, which create [[edo]]s, commonly called n-commas, such as the [[29-comma]], or the [[41-comma]].

== As an interval region ==
{{Main| Comma and diesis }}

''Comma'' can be used to refer to any interval bigger than 0 [[cents]], but smaller than around 30 cents.

== Lists of commas by size ==
Commas can theoretically have any size, but in practice most are much smaller than a [[12edo]] semitone (100{{cent}}). The following categories, while arbitrary, are used on the Xenharmonic Wiki to classify commas by size.
* [[Unnoticeable comma]]: under 3.5 cents in size; below the rough boundary of melodic [[just-noticeable difference]] and thus imperceptible.
* [[Small comma]]: between 3.5 and 30 cents.
* [[Medium comma]]: between 30 and 100 cents.
* [[Large comma]]: over 100 cents in size; much wider than what is typically considered to be a comma.

== See also ==
* [[PIFE comma]]
* [[TIFE comma]]
* [[Superparticular ratio]]
* [[Wolf interval]]
* [[Octave-splitting comma]]

== External links ==
* [http://www.tonalsoft.com/enc/c/comma.aspx Comma] on [[Tonalsoft Encyclopedia]]

[[Category:Comma| ]]

Uniwiz comma

2026-05-20T03:34:44Z

Xenllium:

{{Infobox interval
| Ratio = 1500625/1492992
| Monzo = -11 -6 4 4
| Name = uniwiz comma, uniwizma
| Color name = quadzoyo 2nd, z^4y^42
| Comma = true
}}
The '''uniwiz comma''' ({{monzo|legend=1| -11 -6 4 4 }}, [[ratio]]: 1500625/1492992), is the small interval that separates a stack of four [[35/24]] subfifths with the just major ninth [[9/2]]. It also is the difference between [[9/8]] and a stack of four [[36/35]] quartertones.

== Approximations ==
It is very well approximated by [[145edo|1\145]] and [[270edo|2\270]], and consistently represented in edos such as [[171edo|171]], [[224edo|224]], and [[441edo|441]].

== Temperaments ==
When tempered out alone, it leads to the 7-limit version of the rank-3 [[uniwiz]] temperament, which has ~36/35 (or ~35/24) and ~5/4 as generators, and a [[semi-octave]] period. This has the effect of splitting the whole tone in four equal parts. Because it has 2 periods per octave, odd-numbered edos cannot be tempering out this comma.

When the [[schisma]] is added, it leads to the rank-2 [[kleischismic]], for which [[94edo|94]] and [[118edo|118]] are great edos. Despite the name ''kleischismic'', it does not temper out the [[15625/15552|kleisma]]. It finds ~3/2 at 2 generators plus a semioctave. By tempering out [[1225/1224]], it also finds ~[[17/16]] at two generators.

[[Category:Uniwiz]]

Neovulturismic chords

2026-05-16T02:16:40Z

Xenllium: Created page with "'''Neovulturismic chords''' are essentially tempered chords tempered by the neovulturisma, 2926/2925. Neovulturismic chords are of pattern 2 in the 2.3.5.7.11.13.19 subgroup 19-odd-limit, meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures. For triads, there are three pairs of chords in inverse relationship: * 1–15/11–19/13 with s..."

'''Neovulturismic chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the neovulturisma, [[2926/2925]].

Neovulturismic chords are of [[Dyadic chord/Pattern of essentially tempered chords|pattern 2]] in the 2.3.5.7.11.13.19 [[subgroup]] [[19-odd-limit]], meaning that there are 6 triads, 15 tetrads and 6 pentads, for a total of 27 distinct chord structures.

For triads, there are three pairs of chords in inverse relationship:
* 1–15/11–19/13 with steps of 15/11, 15/14, 26/19, and its inverse
* 1–15/14–19/13 with steps of 15/14, 15/11, 26/19;
* 1–19/15–15/11 with steps of 19/15, 14/13, 22/15, and its inverse
* 1–14/13–15/11 with steps of 14/13, 19/15, 22/15;
* 1–13/11–19/15 with steps of 13/11, 15/14, 30/19, and its inverse
* 1–15/14–19/15 with steps of 15/14, 13/11, 30/19.

For tetrads, there are three palindromic chords and six pairs of chords in inverse relationship. The palindromic chords are
* 1–19/15–15/11–19/11 with steps of 19/15, 14/13, 19/15, 22/19;
* 1–15/14–19/15–19/14 with steps of 15/14, 13/11, 15/14, 28/19;
* 1–15/14–15/11–19/13 with steps of 15/14, 14/11, 15/14, 26/19.

The inversely related pairs of chords are
* 1–19/15–15/11–19/13 with steps of 19/15, 14/13, 15/14, 26/19, and its inverse
* 1–15/14–15/13–19/13 with steps of 15/14, 14/13, 19/15, 26/19;
* 1–14/13–15/11–19/13 with steps of 14/13, 19/15, 15/14, 26/19, and its inverse
* 1–15/14–19/14–19/13 with steps of 15/14, 19/15, 14/13, 26/19;
* 1–15/13–15/11–19/13 with steps of 15/13, 13/11, 15/14, 26/19, and its inverse
* 1–15/14–19/15–19/13 with steps of 15/14, 13/11, 15/13, 26/19;
* 1–15/11–19/13–19/11 with steps of 15/11, 15/14, 13/11, 22/19, and its inverse
* 1–13/11–19/15–19/11 with steps of 13/11, 15/14, 15/11, 22/19;
* 1–13/11–19/15–15/11 with steps of 13/11, 15/14, 14/13, 22/15, and its inverse
* 1–14/13–15/13–15/11 with steps of 14/13, 15/14, 13/11, 22/15;
* 1–15/14–19/15–15/11 with steps of 15/14, 13/11, 14/13, 22/15, and its inverse
* 1–14/13–14/11–15/11 with steps of 14/13, 13/11, 15/14, 22/15.

For pentads, there are three pairs of chords in inverse relationship:
* 1–14/13–15/13–15/11–19/13 with steps of 14/13, 15/14, 13/11, 15/14, 26/19, and its inverse
* 1–15/14–19/15–19/14–19/13 with steps of 15/14, 13/11, 15/14, 14/13, 26/19;
* 1–15/14–19/15–15/11–19/13 with steps of 15/14, 13/11, 14/13, 15/14, 26/19, and its inverse
* 1–15/14–15/13–15/11–19/13 with steps of 15/14, 14/13, 13/11, 15/14, 26/19;
* 1–19/15–15/11–19/13–19/11 with steps of 19/15, 14/13, 15/14, 13/11, 22/19, and its inverse
* 1–13/11–19/15–15/11–19/11 with steps of 13/11, 15/14, 14/13, 19/15, 22/19.

Equal temperaments with neovulturismic chords include {{Optimal ET sequence| 41, 53, 87, 94, 111, 130, 140, 159, 190, 217, 270, 311, 422, 460, 581, 742, 795 and 1012 }}.

[[Category:19-odd-limit]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Neovulturismic]]

Starling temperaments

2026-05-14T10:17:22Z

Xenllium:

{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.

Temperaments discussed in families and clans are:
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]

Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].

Since {{nowrap|(6/5)3 {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.

== Myna ==
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''

7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). [[11-limit]] myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]). In that sense, it is opposed to [[keemic temperaments]], in particular [[quasitemp]], where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.

In terms of vanishing commas, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.

[[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round cent values may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11- and 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 1728/1715

{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}
: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}

[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}

[[Badness]] (Sintel): 0.684

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}

Optimal tunings:
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}

Badness (Sintel): 0.557

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 196/195

Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}

Optimal tunings:
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}

Badness (Sintel): 0.708

==== Minah ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 126/125, 176/175

Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}

Optimal tunings:
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}

{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}

Badness (Sintel): 1.14

==== Maneh ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 126/125, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}

Optimal tunings:
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}

{{Optimal ET sequence|legend=0| 27eff, 31 }}

Badness (Sintel): 1.23

=== Myno ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 385/384

Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}

Optimal tunings:
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}

{{Optimal ET sequence|legend=0| 27, 31 }}

Badness (Sintel): 1.11

=== Coleto ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 1728/1715

Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}

Optimal tunings:
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}

{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}

Badness (Sintel): 1.61

== Nusecond ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''

Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 70 }}. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.

[[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2430/2401

{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
: mapping generators: ~2, ~49/27

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}
: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}
: error map: {{val| 0.000 -2.655 +3.768 +2.819 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}

[[Badness]] (Sintel): 1.28

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}

Optimal tunings:
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}

Minimax tuning:
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of 15''x''2 - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}

Badness (Sintel): 0.847

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}

{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}

Badness (Sintel): 0.964

== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 117649/116640

{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}
: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}

{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}

[[Badness]] (Sintel): 1.86

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 26411/26244

Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.88

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}

Optimal tunings:
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.47

== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''

Vines may be described as the {{nowrap| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 84035/82944

{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
: [[error map]]: {{val| +0.487 -0.363 +3.036 -4.448 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}
: error map: {{val| 0.000 -0.717 +2.269 -5.552 }}

{{Optimal ET sequence|legend=1| 46, 96d, 142d }}

[[Badness]] (Sintel): 1.98

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2401/2376

Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}

Optimal tunings:
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}

{{Optimal ET sequence|legend=0| 46, 96d, 142d }}

Badness (Sintel): 1.47

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}

Optimal tunings:
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}

{{Optimal ET sequence|legend=0| 46, 96d }}

Badness (Sintel): 1.23

== Xenial ==
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''

Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 177147/175616

{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}

{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}

[[Badness]] (Sintel): 2.13

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 16384/16335

Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}

Optimal tunings:
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.31

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 729/728

Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}

Optimal tunings:
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 1.98

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}

Optimal tunings:
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.06

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}

Optimal tunings:
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 2.03

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}

Optimal tunings:
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 1.93

== Kumonga ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 12288/12005

{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0653{{c}}, ~7/4 = 975.6277{{c}}
: [[error map]]: {{val| -1.935 -1.382 +4.009 +2.932 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 977.1096{{c}}
: error map: {{val| 0.000 +0.470 +7.673 +8.284 }}

{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}

[[Badness]] (Sintel): 2.21

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}

Optimal tunings:
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}

Badness (Sintel): 1.43

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}

Optimal tunings:
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}

Badness (Sintel): 1.19

== Paraguay ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''

Named by [[User:Xenllium|Xenllium]] in 2026, paraguay tempers out [[12005/11664]] and may be described as the {{nowrap| 19 & 61 }} temperament. It is a variant of [[parakleismic]], mapping 7th harmonic to 16 generators.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 12005/11664

{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}

{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}

[[Badness]] (Sintel): 2.47

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 12005/11664

Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}

Optimal tunings:
* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}

{{Optimal ET sequence|legend=0| 19, 42e, 61e }}

Badness (Sintel): 2.49

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99, 343/338

Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}

Optimal tunings:
* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}

{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}

Badness (Sintel): 1.86

==== Uruguay ====
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 100/99, 1183/1152

Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}

Optimal tunings:
* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}

{{Optimal ET sequence|legend=0| 19, 42e }}

Badness (Sintel): 2.51

== Bisemidim ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 118098/117649

{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45

[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}

{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}

[[Badness]] (Sintel): 2.47

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}

Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}

Badness (Sintel): 1.36

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}

Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}

Badness (Sintel): 0.987

== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 19683/19208

{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}

{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}

[[Badness]] (Sintel): 2.53

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}

Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}

{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}

Badness (Sintel): 1.41

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}

Optimal tunings:
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}

{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}

Badness (Sintel): 1.56

== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''

Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.

If we add 385/384 to the list of commas, 48/35 is identified with [[11/8]], and casablanca is revealed as an [[11-limit]] temperament with a very low complexity for [[11/1|11]] and not too high a one for [[7/1|7]]; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit [[meantone]].

Marrakesh, named by [[Herman Miller]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, is a more accurate 11-limit extension where the generator is identified with [[15/11]] as opposed to 11/8 in casablanca.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 589824/588245

{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}
: [[error map]]: {{val| -0.371 -1.087 +3.370 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 542.1684{{c}}
: error map: {{val| 0.000 -0.756 +4.044 -0.152 }}

{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}

[[Badness]] (Sintel): 2.56

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}

Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.22

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}

Optimal tunings:
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.31

=== Marrakesh ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}

Optimal tunings:
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}

Badness (Sintel): 1.34

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}

Optimal tunings:
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}

Badness (Sintel): 1.68

==== Murakuc ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}

Optimal tunings:
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}

{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}

Badness (Sintel): 1.71

== Amigo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2097152/2083725

{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}
: [[error map]]: {{val| -0.565 -0.811 +3.467 -1.206 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.0937{{c}}
: error map: {{val| 0.000 +0.076 +4.780 +0.393 }}

{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}

[[Badness]] (Sintel): 2.81

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 16384/16335

Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}

Optimal tunings:
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}

Badness (Sintel): 1.44

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 176/175, 364/363

Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}

Optimal tunings:
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89 }}

Badness (Sintel): 1.27

== Gilead ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 343/324

{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4516{{c}}, ~5/3 = 879.6394{{c}}
: [[error map]]: {{val| +1.452 +7.542 +2.823 -21.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.7223{{c}}
: error map: {{val| 0.000 +6.545 +0.909 -24.159 }}

{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}

[[Badness]] (Sintel): 2.92

== Supersensi ==
Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 17496/16807

{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}
: [[error map]]: {{val| -0.859 -4.800 +3.337 +6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/270 = 446.5163{{c}}
: error map: {{val| 0.000 -4.210 +4.464 +8.017 }}

{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}

[[Badness]] (Sintel): 3.76

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 864/847

Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}

Optimal tunings:
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}

Badness (Sintel): 1.97

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}

Optimal tunings:
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.46

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}

Optimal tunings:
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.32

== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''

Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.

Cobalt was named by [[Xenllium]] in 2022 after the 27th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 40353607/40310784

{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3

[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 44.4363{{c}}, ~3/2 = 701.1154{{c}}
: [[error map]]: {{val| -0.221 -1.060 +3.307 -1.534 }}
* [[CWE]]: ~36/35 = 44.4444{{c}}, ~3/2 = 701.0414{{c}}
: error map: {{val| 0.000 -0.914 +3.617 -1.118 }}

{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}

[[Badness]] (Sintel): 4.39

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108 }}

Badness (Sintel): 2.58

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}

Optimal tunings:
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}

Badness (Sintel): 2.36

===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445

Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 2.14

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968

Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 1.85

===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968

Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}

Optimal tunings:
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}

Badness (Sintel): 2.40

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083

Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}

Badness (Sintel): 2.08

==== Cobaltite ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 975/968

Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}

{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}

Badness (Sintel): 2.18

== References ==

[[Category:Temperament collections]]
[[Category:Starling temperaments| ]] 
[[Category:Rank 2]]

Ragismic microtemperaments

2026-05-14T10:06:09Z

Xenllium:

{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].

Since {{nowrap|(10/9)4 {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)2 }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:
* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]
* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]
* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]
* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]
* [[Flattone]] (+81/80) → [[Meantone family #Flattone|Meantone family]]
* [[Sensi]] (+126/125 or 245/243) → [[Sensipent family #Sensi|Sensipent family]]
* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]
* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]
* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]
* ''[[Srutal]]'' (+2048/2025) → [[Diaschismic family #Srutal|Diaschismic family]]
* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]
* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]
* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]
* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]
* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]
* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]
* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortonic family #Mitonic|Minortonic family]]
* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnuzmic family #Septimal vishnu|Vishnuzmic family]]
* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]
* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]
* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]
* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]
* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]
* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]
* ''[[Dzelic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Dzelic|37th-octave temperaments]]

== Supermajor ==
The generator for supermajor temperament is a supermajor third, [[9/7]], tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its [[octave complement]]. 37 of these give 3/222, 46 give 5/227, and 75 give 7/245. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: [[1106edo]] or [[1277edo]] can be used as tunings, leading to accuracy even greater than that of [[ennealimmal]]. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 52734375/52706752

{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}
: mapping generators: ~2, ~14/9

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0067{{c}}, ~14/9 = 764.9222{{c}}
: [[error map]]: {{val| +0.007 +0.019 -0.074 +0.037 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 764.9181{{c}}
: error map: {{val| 0.000 +0.013 -0.083 +0.029 }}

{{Optimal ET sequence|legend=1| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}

[[Badness]] (Sintel): 0.274

=== Semisupermajor ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 35156250/35153041

Mapping: {{mapping| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}
: mapping generators: ~99/70, ~11/10

Optimal tunings:
* WE: ~99/70 = 600.0103{{c}}, ~11/10 = 164.9205{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 164.9180{{c}}

{{Optimal ET sequence|legend=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}

Badness (Sintel): 0.422

== Enneadecal ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal]].''

Enneadecal tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen [[6/5|just minor thirds]] fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)1/3. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones.

[[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 703125/702464

{{Mapping|legend=1| 19 0 14 -37 | 0 1 1 3 }}
: mapping generators: ~28/27, ~3

[[Optimal tuning]]s:
* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})
: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}
* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})
: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}

{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}

[[Badness]] (Sintel): 0.277

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 16384/16335

Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}

Optimal tunings:
* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})
* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})

{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}

Badness (Sintel): 1.45

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2205/2197

Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}

Optimal tunings:
* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})
* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})

{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}

Badness (Sintel): 1.39

=== Hemienneadecal ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 234375/234256

Mapping: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}
: mapping generators: ~55/54, ~3

Optimal tunings:
* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}

Badness (Sintel): 0.330

==== Hemienneadecalis ====
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256

Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}

Optimal tunings:
* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})

{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}

Badness (Sintel): 0.859

==== Hemienneadec ====
Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213

Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}

Optimal tunings:
* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}

Badness (Sintel): 1.26

==== Semihemienneadecal ====
Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}
: mapping generators: ~55/54, ~429/250

Optimal tunings:
* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})
* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})

{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}

Badness (Sintel): 0.607

=== Kalium ===
Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344

Mapping: {{mapping| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}

Optimal tunings:
* WE: ~28/27 = 63.1582{{c}}, ~6545/5928 = 171.2448{{c}}
* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}

{{Optimal ET sequence|legend=0| 855, 988, 1843 }}

Badness (Sintel): 3.15

== Semidimi ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''

The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 3955078125/3954653486

{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}
: mapping generators: ~2, ~35/27

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}
: [[error map]]: {{val| +0.002 +0.031 -0.040 -0.012 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}
: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}

{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}

[[Badness]] (Sintel): 0.382

== Brahmagupta ==
The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}), and may be described as the {{nowrap| 217 & 224 }} temperament.

Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|Keenan]] found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}

{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}
: mapping generators: ~1157625/1048576, ~27/20

[[Optimal tuning]]s:
* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}
: [[error map]]: {{val| -0.007 +0.037 -0.034 -0.004 }}
* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~27/20 = 519.7156{{c}}
: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}

{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}

[[Badness]] (Sintel): 0.737

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 4375/4374, 131072/130977

Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}

Optimal tunings:
* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}

Badness (Sintel): 1.73

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374

Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}

Optimal tunings:
* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}
* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}

Badness (Sintel): 0.956

== Abigail ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Abigail]].''

Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit, and may be described as the {{nowrap| 46 & 224 }} temperament, with a [[ploidacot]] signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. [[494edo]], [[764edo]] and [[1258edo]] are among the possible tunings.

Abigail was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref>

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2147483648/2144153025

{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}
: mapping generators: ~46305/32768, ~1536/1225

[[Optimal tuning]]s:
* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}
: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}
* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}
: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}

{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}

[[Badness]] (Sintel): 0.936

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 131072/130977

Mapping: {{mapping| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}

Optimal tunings:
* WE: ~99/70 = 599.9782{{c}}, ~1536/1225 = 391.0852{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}

{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}

Badness (Sintel): 0.425

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Mapping: {{mapping| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}

Optimal tunings:
* WE: ~99/70 = 599.9862{{c}}, ~351/280 = 391.0879{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}

{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}

Badness (Sintel): 0.366

== Gamera ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 589824/588245

{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}
: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}

{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}

[[Badness]] (Sintel): 0.953

=== Hemigamera ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 589824/588245

Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}
: mapping generators: ~99/70, ~99/80

Optimal tunings:
* WE: ~99/70 = 599.9323{{c}}, ~99/80 = 369.6212{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}

{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}

Badness (Sintel): 1.35

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

Mapping: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}

Optimal tunings:
* WE: ~99/70 = 599.9207{{c}}, ~26/21 = 369.6139{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}

{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}

Badness (Sintel): 0.844

=== Semigamera ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 14641/14580, 15488/15435

Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}
: mapping generators: ~2, ~144/77

Optimal tunings:
* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}

{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}

Badness (Sintel): 2.59

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

Mapping: {{mapping| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}

Optimal tunings:
* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}

{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}

Badness (Sintel): 1.82

== Crazy ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''

Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, with a [[ploidacot]] of diploid alpha-octacot. [[1106edo]] gives a strong tuning.

Crazy was named by [[Flora Canou]] in 2025 by removing the mutation from ''kwazy'', the name for the 5-limit microtemperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}

{{Mapping|legend=1| 2 1 6 -15 | 0 8 -5 76 }}
: mapping generators: ~332150625/234881024, ~1125/1024

[[Optimal tuning]]s:
* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}
: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}
* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}
: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}

{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}

[[Badness]] (Sintel): 0.998

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}

Optimal tunings:
* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}

{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}

Badness (Sintel): 0.562

== Orga ==
Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}

{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}
: mapping generators: ~7411887/5242880, ~8/7

[[Optimal tuning]]s:
* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}
: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}
* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}
: error map: {{val| 0.000 +0.053 -0.023 +0.070 }}

{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}

[[Badness]] (Sintel): 1.02

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5767168/5764801

Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}

Optimal tunings:
* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}

{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}

Badness (Sintel): 0.535

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360

Mapping: {{mapping| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}

Optimal tunings:
* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}

{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836f, 1106f }}

Badness (Sintel): 0.899

== Seniority ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].

Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the [[senior comma]] ({{monzo| -17 62 -35 }}) is tempered out.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 201768035/201326592

{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}
: mapping generators: ~2, ~5120/3087

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}
: [[error map]]: {{val| +0.075 +0.008 -0.016 -0.203 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}
: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}

{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}

[[Badness]] (Sintel): 1.14

=== Senator ===
Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 65536/65219

Mapping: {{mapping| 1 -24 -43 5 2 | 0 35 62 -3 2 }}

Optimal tunings:
* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}

{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}

Badness (Sintel): 3.05

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 2200/2197, 4375/4374

Mapping: {{mapping| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}

Optimal tunings:
* WE: ~2 = 1199.7136{{c}}, ~108/65 = 877.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}

{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}

Badness (Sintel): 1.85

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197

Mapping: {{mapping| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}

Optimal tunings:
* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}

{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}

Badness (Sintel): 1.35

== Monzismic ==
: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].

Monzismic tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}

{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}
: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}
: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}
: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}

{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}

[[Badness]] (Sintel): 1.18

=== Monzism ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 184549376/184528125

Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}

Optimal tunings:
* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}

{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}

Badness (Sintel): 1.89

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625

Mapping: {{mapping| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}

Optimal tunings:
* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}

{{Optimal ET sequence|legend=0| 53, 559, 612 }}

Badness (Sintel): 2.22

== Semidimfourth ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''

The semidimfourth temperament is featured by a semidiminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, [[235298/234375]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 235298/234375

{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}
: mapping generators: ~2, ~35/27

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}
: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}}
: error map: {{val| 0.000 +0.165 +0.361 -0.685 }}

{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}

[[Badness]] (Sintel): 1.40

=== Neusec ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 235298/234375

Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}
: mapping generators: ~99/70, ~35/27

Optimal tunings:
* WE: ~99/70 = 600.0381{{c}}, ~35/27 = 448.4812{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}

Badness (Sintel): 1.95

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374

Mapping: {{mapping| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}

Optimal tunings:
* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}

Badness (Sintel): 1.28

== Acrokleismic ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2202927104/2197265625

{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}
: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}
: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}

{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}

[[Badness]] (Sintel): 1.42

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 172032/171875

Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}

Optimal tunings:
* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}

{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}

Badness (Sintel): 1.22

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976

Mapping: {{mapping| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}

Optimal tunings:
* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}

{{Optimal ET sequence|legend=0| 19, 251, 270 }}

Badness (Sintel): 1.11

=== Counteracro ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 117649/117612

Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}

Optimal tunings:
* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}

Badness (Sintel): 1.41

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374

Mapping: {{mapping| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}

Optimal tunings:
* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}

Badness (Sintel): 1.08

== Quasithird ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''

Quasithird may be described as the {{nowrap| 224 & 388 }} temperament, featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows it to temper out the ragisma and {{monzo| -60 29 0 5 }}. Its [[ploidacot]] is tetraploid delta-pentacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}

{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}
: mapping generators: ~65536/55125, ~5103/4096

[[Optimal tuning]]s:
* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}
: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}
* [[CWE]]: ~65536/55125 = 300.0000{{c}}, ~5103/4096 = 380.3884{{c}}
: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}

{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}

[[Badness]] (Sintel): 1.56

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 4296700485/4294967296

Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}

Optimal tunings:
* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})
* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})

{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}

Badness (Sintel): 0.698

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374

Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}

Optimal tunings:
* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}})
* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}})

{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}

Badness (Sintel): 1.22

== Deca ==
: ''For 5-limit version, see [[10th-octave temperaments #Neon]].''

Deca has a period of 1/10 octave and tempers out the neon comma {{monzo| 21 60 -50 }} in the 5-limit, the [[linus comma]]{{monzo| 11 -10 -10 10 }} and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 165288374272/164794921875

{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}
: mapping generators: ~15/14, ~460992/390625

[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 119.9966{{c}}, ~460992/390625 = 284.4150{{c}} (5625/5488 = 44.4219{{c}})
: [[error map]]: {{val| -0.034 +0.106 +0.145 -0.268 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~460992/390625 = 284.4182{{c}} (5625/5488 = 44.4182{{c}})
: error map: {{val| 0.000 +0.136 +0.195 -0.226 }}

{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}

[[Badness]] (Sintel): 2.04

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 391314/390625

Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}

Optimal tunings:
* WE: ~15/14 = 120.0004{{c}}, ~33/28 = 284.4193{{c}} (77/75 = 44.4185{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})

{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}

Badness (Sintel): 0.804

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}

Optimal tunings:
* WE: ~15/14 = 120.0067{{c}}, ~33/28 = 284.4139{{c}} (~40/39 = 44.4006{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})

{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}

Badness (Sintel): 0.695

=== 2.3.5.7.11.13.19 subgroup ===
Subgroup: 2.3.5.7.11.13.19

Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}

Optimal tunings:
* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}})
* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}})

{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}

Badness (Sintel): 0.556

== Keenanose ==
Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}

{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}
: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}}
: [[error map]]: {{val| +0.007 +0.031 -0.035 -0.032 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}}
: error map: {{val| 0.000 +0.025 -0.043 -0.050 }}

{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}

[[Badness]] (Sintel): 2.17

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 117649/117612, 67110351/67108864

Mapping: {{mapping| 1 2 3 3 3 | 0 -112 -183 -52 124 }}

Optimal tunings:
* WE: ~2 = 1199.9970{{c}}, ~385/384 = 4.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}

{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}

Badness (Sintel): 1.02

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612

Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}

Optimal tunings:
* WE: ~2 = 1200.0065{{c}}, ~385/384 = 4.4467{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~385/384 = 4.4467{{c}}

{{Optimal ET sequence|legend=0| 270, 1079, 1349, 1619, 1889, 4048 }}

Badness (Sintel): 0.879

== Aluminium ==
: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''

Aluminium was named by [[Eliora]] in 2023 after the 13th element. It tempers out the {{monzo| 92 -39 -13 }} comma, which sets [[135/128]] interval to be equal to 1/13th of the octave.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}

[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}
: Mapping generators: ~135/128, ~3

[[Optimal tuning]]s:
* [[WE]]: ~135/128 = 92.3072{{c}}, ~3/2 = 701.9995{{c}}
: [[error map]]: {{val| -0.006 +0.038 -0.030 -0.013 }}
* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0030{{c}}
: error map: {{val| 0.000 +0.048 -0.015 +0.001 }}

{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}

[[Badness]] (Sintel): 3.20

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 234375/234256, 2097152/2096325

Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}

Optimal tunings:
* WE: ~135/128 = 92.3062{{c}}, ~3/2 = 701.9946{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0056{{c}}

{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}

Badness (Sintel): 1.39

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078

Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}

Optimal tunings:
* WE: ~135/128 = 92.3055{{c}}, ~3/2 = 701.9928{{c}}
* CWE: ~135/128 = 92.3077{{c}}, ~3/2 = 702.0098{{c}}

{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}

Badness (Sintel): 1.18

== Countritonic ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic]].''

Countritonic (''co-un-tritonic'') may be described as the {{nowrap| 53 & 422 }} temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit. It was named by [[Flora Canou]] in 2023 as a variant of [[countriton]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 68719476736/68356598625

{{Mapping|legend=1| 1 -3 -15 40 | 0 9 34 -73 }}
: mapping generators: ~2, ~65536/45927

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8189{{c}}, ~65536/45927 = 611.2850{{c}}
: [[error map]]: {{val| -0.181 +0.153 +0.094 +0.123 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~65536/45927 = 611.3775{{c}}
: error map: {{val| 0.000 +0.443 +0.522 +0.615 }}

{{Optimal ET sequence|legend=1| 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd }}

[[Badness]] (Sintel): 3.37

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 2621440/2614689

Mapping: {{mapping| 1 -3 -15 40 -75 | 0 9 34 -73 154 }}

Optimal tunings:
* WE: ~2 = 1199.8147{{c}}, ~768/539 = 611.2822{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~768/539 = 611.3762{{c}}

{{Optimal ET sequence|legend=0| 53, 316e, 369, 422, 791e, 1213cde }}

Badness (Sintel): 2.34

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625

Mapping: {{mapping| 1 -3 -15 40 -75 -34 | 0 9 34 -73 154 74 }}

Optimal tunings:
* WE: ~2 = 1199.7916{{c}}, ~91/64 = 611.2698{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/64 = 611.3754{{c}}

{{Optimal ET sequence|legend=0| 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff }}

Badness (Sintel): 1.51

== Quatracot ==
{{See also| Stratosphere }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -32 5 14 -3 }}

{{Mapping|legend=1| 2 -6 -1 -36 | 0 13 8 59 }}
: mapping generators: ~2278125/1605632, ~7168/5625

[[Optimal tuning]]s:
* [[WE]]: ~2278125/1605632 = 600.0888{{c}}, ~7168/5625 = 423.2574{{c}}
: [[error map]]: {{val| +0.178 -0.141 -0.343 +0.165 }}
* [[CWE]]: ~2278125/1605632 = 600.0000{{c}}, ~7168/5625 = 423.1986{{c}}
: error map: {{val| 0.000 -0.374 -0.725 -0.111 }}

{{Optimal ET sequence|legend=1| 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc }}

[[Badness]] (Sintel): 4.45

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 1265625/1261568

Mapping: {{mapping| 2 -6 -1 -36 -22 | 0 13 8 59 41 }}

Optimal tunings:
* WE: ~99/70 = 600.0847{{c}}, ~225/176 = 423.2536{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~225/176 = 423.1977{{c}}

{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638, 1052c }}

Badness (Sintel): 1.36

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 2200/2197

Mapping: {{mapping| 2 -6 -1 -36 -22 -6 | 0 13 8 59 41 19 }}

Optimal tunings:
* WE: ~99/70 = 600.0571{{c}}, ~143/112 = 423.2366{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~143/112 = 423.1987{{c}}

{{Optimal ET sequence|legend=0| 34d, 156de, 190, 224, 414, 638 }}

Badness (Sintel): 0.936

== Moulin ==
Moulin can be described as the {{nowrap| 494 & 1619 }} temperament. It has a generator of ~[[22/13]], and it was named by [[Eliora]] in 2022 after the ''Law & Order: Special Victims Unit'' episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~[[13/11]], and 73 of them octave reduced reach the [[3/2|perfect fifth]]. Since [[11/8]] is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -88 2 45 -7 }}

{{Mapping|legend=1| 1 -16 -9 -75 | 0 73 47 323 }}
: mapping generators: ~2, ~3796875/3211264

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0272{{c}}, ~3796875/3211264 = 289.0675{{c}}
: [[error map]]: {{val| +0.027 +0.007 -0.084 +0.013 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3796875/3211264 = 289.0675{{c}}
: error map: {{val| 0.000 -0.029 -0.142 -0.029 }}

{{Optimal ET sequence|legend=1| 494, 1125, 1619, 8589cc, 10208cc }}

[[Badness]] (Sintel): 5.93

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 759375/758912, 100663296/100656875

Mapping: {{mapping| 1 -16 -9 -75 9 | 0 73 47 323 -23 }}

Optimal tunings:
* WE: ~2 = 1200.0043{{c}}, ~605/512 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~605/512 = 289.0677{{c}}

{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}

Badness (Sintel): 2.24

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078

Mapping: {{mapping| 1 -16 -9 -75 9 9 | 0 73 47 323 -23 -22 }}

Optimal tunings:
* WE: ~2 = 1200.0043{{c}}, ~13/11 = 289.0687{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/11 = 289.0677{{c}}

{{Optimal ET sequence|legend=0| 494, 1125, 1619, 2113 }}

Badness (Sintel): 1.12

== Palladium ==
: ''For the 5-limit version, see [[46th-octave temperaments #Palladium]]''.

The name of the ''palladium'' temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, {{monzo| -39 92 -46 }}, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as {{nowrap| 46 & 414 }} temperament, which tempers out {{monzo| -51 8 2 12 }} as well as the ragisma.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -51 8 2 12 }}

{{Mapping|legend=1| 46 0 -39 202 | 0 1 2 -1 }}
: mapping generators: ~83349/81920, ~3

[[Optimal tuning]]s:
* [[WE]]: ~83349/81920 = 26.0910{{c}}, ~3/2 = 701.7155{{c}}
: [[error map]]: {{val| +0.185 -0.055 -0.061 +0.349 }}
* [[CWE]]: ~83349/81920 = 26.0870{{c}}, ~3/2 = 701.6491{{c}}
: error map: {{val| 0.000 -0.306 -0.407 -0.910 }}

{{Optimal ET sequence|legend=1| 46, …, 368, 414, 460, 874d }}

[[Badness]] (Sintel): 7.81

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: {{mapping| 46 0 -39 202 232 | 0 1 2 -1 -1 }}

Optimal tunings:
* WE: ~8192/8085 = 26.0912{{c}}, ~3/2 = 701.7082{{c}}
* CWE: ~8192/8085 = 26.0870{{c}}, ~3/2 = 701.6173{{c}}

{{Optimal ET sequence|legend=0| 46, …, 368, 414, 460, 874de }}

Badness (Sintel): 2.44

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: {{mapping| 46 0 -39 202 232 316 | 0 1 2 -1 -1 -2 }}

Optimal tunings:
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7411{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6465{{c}}

{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334dde }}

Badness (Sintel): 1.68

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: {{mapping| 46 0 -39 202 232 316 188 | 0 1 2 -1 -1 -2 0 }}

Optimal tunings:
* WE: ~65/64 = 26.0906{{c}}, ~3/2 = 701.7399{{c}}
* CWE: ~65/64 = 26.0870{{c}}, ~3/2 = 701.6464{{c}}

{{Optimal ET sequence|legend=0| 46, 368, 414, 460, 874de, 1334ddeg }}

Badness (Sintel): 1.14

== Counterorson ==
Counterorson tempers out the {{monzo| 147 -103 7 }} comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the [[semicomma family]], 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on [[53edo]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| 154 -54 -21 -7 }}

{{Mapping|legend=1| 1 0 -21 85 | 0 7 103 -363 }}
: mapping generators: ~2, ~{{monzo| 66 -23 -9 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0040{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7122{{c}}
: [[error map]]: {{val| +0.004 -0.303 -0.041 -0.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 66 -23 -9 -3 }} = 271.7113{{c}}
: error map: {{val| 0.000 +0.024 -0.051 -0.025 }}

{{Optimal ET sequence|legend=1| 53, …, 1612, 1665, 1718 }}

[[Badness]] (Sintel): 7.92

== Oviminor ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor]].''

Oviminor was named by [[Eliora]] in 2022 after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[4/3]], the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -100 53 48 -34 }}

{{Mapping|legend=1| 1 -134 -134 -401 | 0 184 185 548 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}
: [[error map]]: {{val| +0.019 +0.010 -0.085 +0.032 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}
: error map: {{val| 0.000 -0.011 -0.120 +0.008 }}

{{Optimal ET sequence|legend=1| 19, …, 1600, 1619, 4838, 6457c }}

[[Badness]] (Sintel): 14.7

== Octoid ==
: {{Main| Octoid }}
: ''For the 5-limit version, see [[8th-octave temperaments #Octoid]].''

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].

The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 16875/16807

{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}
: mapping generators: ~49/45, ~7/5

[[Optimal tuning]]s:
* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}
: [[error map]]: {{val| +0.002 -0.130 -0.547 +0.883 }}
* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}
: error map: {{val| 0.000 -0.132 -0.549 +0.880 }}

[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
* 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
* 7-odd-limit [[diamond tradeoff]]: ~7/5 = [582.512, 584.359]
* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}

[[Badness]] (Sintel): 1.08

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}

Optimal tunings:
* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}

Badness (Sintel): 0.466

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}

Optimal tunings:
* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}

{{Optimal ET sequence|legend=0| 72, 152f, 224 }}

Badness (Sintel): 0.631

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}

Optimal tunings:
* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}

{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}

Badness (Sintel): 0.729

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714

Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}

Optimal tunings:
* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}

{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}

Badness (Sintel): 0.975

==== Octopus ====
A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{c}}.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}

Optimal tunings:
* WE: ~12/11 = 150.0313{{c}}, ~7/5 = 584.0134{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}

Badness (Sintel): 0.896

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}

Optimal tunings:
* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}

Badness (Sintel): 0.795

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399

Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}

Optimal tunings:
* WE: ~12/11 = 150.0049{{c}}, ~7/5 = 584.0833{{c}}
* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}

{{Optimal ET sequence|legend=0| 8d, 72, 152 }}

Badness (Sintel): 0.993

Scales: [[Octoid72]], [[Octoid80]]

==== Hexadecoid ====
{{See also| 16th-octave temperaments }}

Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}
: mapping generators: ~448/429, ~7/5

Optimal tunings:
* WE: ~448/429 = 74.9943{{c}}, ~7/5 = 583.9408{{c}}
* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224 }}

Badness (Sintel): 1.27

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}

Optimal tunings:
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9626{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0463{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}

Badness (Sintel): 1.46

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}

Optimal tunings:
* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}
* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}

Badness (Sintel): 1.44

== Parakleismic ==
{{Main| Parakleismic }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''

In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, {{monzo| 8 14 -13 }}, with the [[118edo]] tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat [[6/5]], 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding [[3136/3125]] and 4375/4374, for which [[99edo]], 118edo, and especially [[217edo]] are accurate tunings.

Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 4375/4374

{{Mapping|legend=1| 1 -8 -8 -23 | 0 13 14 35 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7820{{c}}, ~5/3 = 884.6581{{c}}
: [[error map]]: {{val| -0.218 +0.344 +0.644 -0.779 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8088{{c}}
: error map: {{val| 0.000 +0.560 +1.010 -0.516 }}

{{Optimal ET sequence|legend=1| 19, 61d, 80, 99, 217, 316, 415 }}

[[Badness]] (Sintel): 0.694

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: {{mapping| 1 -8 -8 -23 30 | 0 13 14 35 -36 }}

Optimal tunings:
* WE: ~2 = 1200.3296{{c}}, ~5/3 = 884.9921{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7519{{c}}

{{Optimal ET sequence|legend=0| 19, 99, 118 }}

Badness (Sintel): 1.64

=== Paralytic ===
Paralytic (99e & 118) tempers out [[441/440]], [[5632/5625]], and [[19712/19683]]. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: {{mapping| 1 -8 -8 -23 -57 | 0 13 14 35 82 }}

Optimal tunings:
* WE: ~2 = 1199.9940{{c}}, ~5/3 = 884.7757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7800{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 99e, 118, 217, 335 }}

Badness (Sintel): 1.19

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: {{mapping| 1 -8 -8 -23 -57 59 | 0 13 14 35 82 -75 }}

Optimal tunings:
* WE: ~2 = 1199.9218{{c}}, ~5/3 = 884.7285{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7858{{c}}

{{Optimal ET sequence|legend=0| 99e, 118, 217 }}

Badness (Sintel): 1.85

==== Paraklein ====
Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates [[13/11]] with [[32/27]], [[14/13]] with [[15/14]], [[25/24]] with [[26/25]], and [[27/26]] with [[28/27]].

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: {{mapping| 1 -8 -8 -23 -57 -28 | 0 13 14 35 82 43 }}

Optimal tunings:
* WE: ~2 = 1199.8239{{c}}, ~5/3 = 884.6449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.7709{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 99ef, 118 }}

Badness (Sintel): 1.55

=== Parkleismic ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: {{mapping| 1 -8 -8 -23 -43 | 0 13 14 35 63 }}

Optimal tunings:
* WE: ~2 = 1199.1848{{c}}, ~5/3 = 884.3386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9158{{c}}

{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179, 259cd }}

Badness (Sintel): 1.85

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: {{mapping| 1 -8 -8 -23 -43 -14 | 0 13 14 35 63 24 }}

Optimal tunings:
* WE: ~2 = 1199.5318{{c}}, ~5/3 = 884.5800{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9118{{c}}

{{Optimal ET sequence|legend=0| 19e, 61de, 80, 179 }}

Badness (Sintel): 1.51

=== Paradigmic ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: {{mapping| 1 -8 -8 -23 16 | 0 13 14 35 -17 }}

Optimal tunings:
* WE: ~2 = 1199.0616{{c}}, ~5/3 = 884.2124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.8877{{c}}

{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e, 179e, 457bcddeeee }}

Badness (Sintel): 1.38

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: {{mapping| 1 -8 -8 -23 16 -14 | 0 13 14 35 -17 24 }}

Optimal tunings:
* WE: ~2 = 1199.2683{{c}}, ~5/3 = 884.3805{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.9061{{c}}

{{Optimal ET sequence|legend=0| 19, 61d, 80, 99e }}

Badness (Sintel): 1.48

=== Semiparakleismic ===
Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: {{mapping| 2 -3 -2 -11 -4 | 0 13 14 35 23 }}
: mapping generators: ~99/70, ~33/28

Optimal tunings:
* WE: ~99/70 = 599.9270{{c}}, ~33/28 = 284.7841{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8119{{c}}

{{Optimal ET sequence|legend=0| 80, 118, 198, 316, 514c }}

Badness (Sintel): 1.13

==== Semiparamint ====
This extension was named ''semiparakleismic'' in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: {{mapping| 2 -3 -2 -11 -4 15 | 0 13 14 35 23 -16 }}

Optimal tunings:
* WE: ~99/70 = 599.8253{{c}}, ~33/28 = 284.7608{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~33/28 = 284.8366{{c}}

{{Optimal ET sequence|legend=0| 80, 118, 198 }}

Badness (Sintel): 1.40

==== Semiparawolf ====
This extension was named ''gentsemiparakleismic'' in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: {{mapping| 2 -3 -2 -11 -4 -4 | 0 13 14 35 23 24 }}

Optimal tunings:
* WE: ~99/70 = 600.0569{{c}}, ~13/11 = 284.8431{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~13/11 = 284.8216{{c}}

{{Optimal ET sequence|legend=0| 80, 118f, 198f }}

Badness (Sintel): 1.67

== Counterkleismic ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Counterhanson]].''

In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, {{monzo| -20 -24 25 }}, the amount by which six [[648/625|major dieses]] ((648/625)6) fall short of the [[5/4|classic major third (5/4)]]. It can be described as {{nowrap| 19 & 224 }} temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to [[catakleismic]] and parakleismic)

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 158203125/157351936

{{Mapping|legend=1| 1 -5 -4 -18 | 0 25 24 79 }}
: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1778{{c}}, ~6/5 = 316.1065{{c}}
: [[error map]]: {{val| +0.178 -0.181 -0.469 +0.388 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0631{{c}}
: error map: {{val| 0.000 -0.377 -0.799 +0.161 }}

{{Optimal ET sequence|legend=1| 19, …, 205, 224, 243, 467 }}

[[Badness]] (Sintel): 2.29

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 2097152/2096325

Mapping: {{mapping| 1 -5 -4 -18 19 | 0 25 24 79 -59 }}

Optimal tunings:
* WE: ~2 = 1199.9944{{c}}, ~6/5 = 316.0690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0705{{c}}

{{Optimal ET sequence|legend=0| 19, 205, 224 }}

Badness (Sintel): 2.35

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 10985/10976

Mapping: {{mapping| 1 -5 -4 -18 19 -15 | 0 25 24 79 -59 71 }}

Optimal tunings:
* WE: ~2 = 1199.9827{{c}}, ~6/5 = 316.0650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0695{{c}}

{{Optimal ET sequence|legend=0| 19, 205, 224 }}

Badness (Sintel): 1.40

=== Counterlytic ===
Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4374, 496125/495616

Mapping: {{mapping| 1 -5 -4 -18 -40 | 0 25 24 79 165 }}

Optimal tunings:
* WE: ~2 = 1200.1247{{c}}, ~6/5 = 316.0976{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0660{{c}}

{{Optimal ET sequence|legend=1| 19e, 205e, 224, 467e, 691, 915c }}

Badness (Sintel): 2.16

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1375/1372, 10985/10976

Mapping: {{mapping| 1 -5 -4 -18 -40 -15 | 0 25 24 79 165 71 }}

Optimal tunings:
* WE: ~2 = 1200.0987{{c}}, ~6/5 = 316.0908{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.0658{{c}}

{{Optimal ET sequence|legend=0| 19e, 205e, 224, 467e, 691, 915c }}

Badness (Sintel): 1.23

== Quincy ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 823543/819200

{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}
: mapping generators: ~2, ~1728/1715

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}}
: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}}
: error map: {{val| 0.000 -0.205 -0.122 -1.343 }}

{{Optimal ET sequence|legend=1| 72, 217, 289, 650d, 939dd }}

[[Badness]] (Sintel): 2.02

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 441/440, 4000/3993, 4375/4374

Mapping: {{mapping| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}

Optimal tunings:
* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}

{{Optimal ET sequence|legend=0| 72, 217, 289 }}

Badness (Sintel): 1.02

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 676/675, 4375/4374

Mapping: {{mapping| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}

Optimal tunings:
* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}

{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}

Badness (Sintel): 0.986

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155

Mapping: {{mapping| 1 2 3 3 4 5 5 | 0 -30 -49 -14 -39 -94 -66 }}

Optimal tunings:
* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}

{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}

Badness (Sintel): 0.751

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675

Mapping: {{mapping| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}

Optimal tunings:
* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}

{{Optimal ET sequence|legend=0| 72, 145, 217 }}

Badness (Sintel): 0.924

== Sfourth ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Sfourth]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 64827/64000

{{Mapping|legend=1| 1 2 3 3 | 0 -19 -31 -9 }}
: mapping generators: ~2, ~49/48

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8332{{c}}, ~49/48 = 26.3053{{c}}
: [[error map]]: {{val| +0.833 -0.090 +0.721 -3.074 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/48 = 26.2590{{c}}
: error map: {{val| 0.000 -0.876 -0.343 -5.157 }}

{{Optimal ET sequence|legend=1| 45, 46, 91, 137d }}

[[Badness]] (Sintel): 3.12

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 4375/4374

Mapping: {{mapping| 1 2 3 3 4 | 0 -19 -31 -9 -25 }}

Optimal tunings:
* WE: ~2 = 1201.1486{{c}}, ~49/48 = 26.3112{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2461{{c}}

{{Optimal ET sequence|legend=0| 45e, 46, 91e, 137de }}

Badness (Sintel): 1.78

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -19 -31 -9 -25 -14 }}

Optimal tunings:
* WE: ~2 = 1201.4956{{c}}, ~49/48 = 26.3423{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2614{{c}}

{{Optimal ET sequence|legend=0| 45ef, 46, 91ef, 137def, 228ddeeefff }}

Badness (Sintel): 1.37

=== Sfour ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 4375/4374

Mapping: {{mapping| 1 2 3 3 3 | 0 -19 -31 -9 21 }}

Optimal tunings:
* WE: ~2 = 1200.4402{{c}}, ~49/48 = 26.2557{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2403{{c}}

{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}

Badness (Sintel): 2.53

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 4375/4374

Mapping: {{mapping| 1 2 3 3 3 3 | 0 -19 -31 -9 21 32 }}

Optimal tunings:
* WE: ~2 = 1200.3796{{c}}, ~49/48 = 26.2473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~49/48 = 26.2372{{c}}

{{Optimal ET sequence|legend=0| 45, 46, 91, 137d, 183d }}

Badness (Sintel): 2.14

== Trideci ==
: ''For the 5-limit version, see [[13th-octave temperaments #Tridecatonic]].''

The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the [[Octagar temperaments #Tridecatonic|tridecatonic temperament]], but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name ''trideci'' comes from ''tridecim'' (Latin for "thirteen").

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 83349/81920

{{Mapping|legend=1| 13 0 -11 57 | 0 1 2 -1 }}
: mapping generators: ~256/245, ~3

[[Optimal tuning]]s:
* [[WE]]: ~256/245 = 92.4141{{c}}, ~3/2 = 699.9466{{c}}
: [[error map]]: {{val| +1.383 -0.626 -0.210 -2.554 }}
* [[CWE]]: ~256/245 = 92.3077{{c}}, ~3/2 = 699.4521{{c}}
: error map: {{val| 0.000 -2.503 -2.794 -6.740 }}

{{Optimal ET sequence|legend=1| 26, 65, 91 }}

[[Badness]] (Sintel): 4.67

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 4375/4374

Mapping: {{mapping| 13 0 -11 57 45 | 0 1 2 -1 0 }}

Optimal tunings:
* WE: ~22/21 = 92.3729{{c}}, ~3/2 = 700.1118{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.7703{{c}}

{{Optimal ET sequence|legend=0| 26, 65, 91 }}

Badness (Sintel): 2.80

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/242, 325/324, 385/384

Mapping: {{mapping| 13 0 -11 57 45 48 | 0 1 2 -1 0 0 }}

Optimal tunings:
* WE: ~22/21 = 92.4003{{c}}, ~3/2 = 699.9983{{c}}
* CWE: ~22/21 = 92.3077{{c}}, ~3/2 = 699.4772{{c}}

{{Optimal ET sequence|legend=0| 26, 65f, 91f }}

Badness (Sintel): 2.16

== References ==

[[Category:Temperament collections]]
[[Category:Ragismic microtemperaments| ]] 
[[Category:Rank 2]]

Syntonic–kleismic equivalence continuum

2026-05-14T09:57:05Z

Xenllium:

{{Technical data page}}
The '''syntonic–kleismic equivalence continuum''' (or '''syntonic–enneadecal equivalence continuum''') is a [[equivalence continuum|continuum]] of [[5-limit]] [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[19-comma]] ({{monzo| -30 19 }}).

All temperaments in the continuum satisfy {{nowrap|(81/80)''n'' ~ {{monzo| -30 19 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments [[support]]ed by [[19edo]] (due to it being the unique equal temperament that tempers out both commas and thus tempers out all combinations of them). The just value of ''n'' is approximately 6.376…, and temperaments having ''n'' near this value tend to be the most accurate ones.

This continuum can also be expressed as the relationship between 81/80 and the [[enneadeca]] ({{monzo| -14 -19 19 }}). That is, {{nowrap|(81/80)''k'' ~ {{monzo| -14 -19 19 }}}}. In this case, {{nowrap| ''k'' {{=}} 3''n'' − 19 }}.

{| class="wikitable center-1 center-2"
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| 0
| [[Graywood]]
| [[19-comma|1162261467/1073741824]]
| {{Monzo| -30 19 }}
|-
| 1
| 7c & 12c
| [[71744535/67108864]]
| {{Monzo| -26 15 1 }}
|-
| 2
| [[Hogzilla]]
| [[4428675/4194304]]
| {{monzo|-22 11 2}}
|-
| 3
| [[Stump]]
| [[273375/262144]]
| {{Monzo| -18 7 3 }}
|-
| 4
| [[Negri]]
| [[16875/16384]]
| {{Monzo| -14 3 4 }}
|-
| 5
| [[Magic]]
| [[3125/3072]]
| {{Monzo| -10 -1 5 }}
|-
| 6
| [[Hanson]]
| [[15625/15552]]
| {{Monzo| -6 -5 6 }}
|-
| 7
| [[Sensipent]]
| [[78732/78125]]
| {{Monzo| 2 9 -7 }}
|-
| 8
| [[Unicorn]]
| [[1594323/1562500]]
| {{Monzo| -2 13 -8 }}
|-
| 9
| [[Xenial]]
| [[129140163/125000000]]
| {{Monzo| -6 17 -9 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Meantone]]
| [[81/80]]
| {{Monzo| -4 4 -1 }}
|}

Examples of temperaments with fractional values of ''k'':

{| class="wikitable"
|+ style="font-size: 105%;" | Notable temperaments of fractional ''n''
|-
! Temperament !! ''n'' !! Comma
|-
| [[Unsmate]] || 9/2 = 4.5 || {{monzo| -24 2 9 }}
|-
| [[Sycamore]] || 11/2 = 5.5 || {{monzo| -16 -6 11 }}
|-
| [[Counterhanson]] || 25/4 = 6.25 || {{monzo| -20 -24 25 }}
|-
| [[Enneadecal]] || 19/3 = 6.{{overline|3}} || {{monzo| -14 -19 19 }}
|-
| [[Egads]] || 51/8 = 6.375 || {{monzo| -36 -52 51 }}
|-
| [[Acrokleismic]] || 32/5 = 6.4 || {{monzo| 22 33 -32 }}
|-
| [[Parakleismic]] || 13/2 = 6.5 || {{monzo| 8 14 -13 }}
|-
| [[Countermeantone]] || 20/3 = 6.{{overline|6}} || {{monzo| 10 23 -20 }}
|-
| [[Mowgli]] || 15/2 = 7.5 || {{monzo| 0 22 -15 }}
|}

== Graywood ==
Named by [[CompactStar]] in 2024, graywood tempers out the [[19-comma]], corresponding to {{nowrap| ''n'' {{=}} 0 }}. It takes [[19edo]]'s closed [[circle of fifths]], but adds an independent generator for [[prime interval|prime]] [[5/1|5]]. 19 is the only equal temperament that makes it to the optimal ET sequence as all the small edo tunings, e.g. [[38edo|38c-edo]] or [[57edo|57c-edo]], are not nearly as accurate as 19 itself.

[[Subgroup]]: 2.3.5

[[Comma list]]: 1162261467/1073741824

{{Mapping|legend=1| 19 30 0 | 0 0 1 }}
: mapping generators: ~2187/2048, ~5

[[Optimal tuning]]s:
* [[WE]]: ~2187/2048 = 63.2773{{c}}, ~5/4 = 381.7568{{c}}
: [[error map]]: {{val| +2.268 -3.637 -0.020 }}
* [[CWE]]: ~2187/2048 = 63.1579{{c}}, ~5/4 = 382.7889{{c}}
: error map: {{val| 0.000 -7.218 -3.525 }}

{{Optimal ET sequence|legend=1| 19 }}

[[Badness]] (Sintel): 32.4

== Hogzilla ==
: ''For extensions, see [[Semaphoresmic clan #Helayo]].''

Hogzilla is similar to [[godzilla]] in that it is generated by a [[semitwelfth]]. It corresponds to {{nowrap| ''n'' {{=}} 2 }}.

[[Subgroup]]: 2.3.5

[[Comma list]]: 4428675/4194304

{{Mapping|legend=1| 1 0 11 | 0 2 -11 }}
: mapping generators: ~2, ~2048/1215

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5490{{c}}, ~2048/1215 = 949.3637{{c}}
: [[error map]]: {{val| +2.549 -3.228 -1.275 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2048/1215 = 947.2462{{c}}
: error map: {{val| 0.000 -7.463 -6.022 }}

{{Optimal ET sequence|legend=1| 14, 19 }}

[[Badness]] (Sintel): 9.96

== Stump ==
: ''For extensions, see [[Marvel temperaments #Triton]] and [[Sensamagic clan #Pycnic]].''

Stump splits the [[3/1|3rd]] [[harmonic]] into three equal parts, each for [[~]][[64/45]]. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.

[[Subgroup]]: 2.3.5

[[Comma list]]: 273375/262144

{{Mapping|legend=1| 1 0 6 | 0 3 -7 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.6288{{c}}, ~64/45 = 633.1214{{c}}
: [[error map]]: {{val| +2.629 -2.591 -2.391 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~64/45 = 631.6779{{c}}
: error map: {{val| 0.000 -6.921 -8.059 }}

{{Optimal ET sequence|legend=1| 17, 19, 207bbccc }}

[[Badness]] (Sintel): 4.71

== Negri (5-limit) ==
{{Main| Negri }}
: ''For extensions, see [[Semaphoresmic clan #Negri]].''

The 5-limit version of negri tempers out the [[negri comma]], spliting a perfect fourth into four ~16/15 generators. It corresponds to {{nowrap| ''n'' {{=}} 4 }}. The only 7-limit extension that make any sense to use is to map the hemifourth to 7/6~8/7.

[[Subgroup]]: 2.3.5

[[Comma list]]: 16875/16384

{{Mapping|legend=1| 1 2 2 | 0 -4 3 }}
: mapping generators: ~2, ~16/15

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.3403{{c}}, ~16/15 = 126.0002{{c}}
: [[error map]]: {{val| +2.340 -1.275 -3.633 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~16/15 = 125.6610{{c}}
: error map: {{val| 0.000 -4.599 -9.331 }}

{{Optimal ET sequence|legend=1| 9, 10, 19, 67c, 86c, 105c }}

[[Badness]] (Sintel): 2.04

== Xenial ==
: ''For extensions, see [[Starling temperaments #Xenial]] and [[Sensamagic clan #Xenia]].''

Named by [[User:Xenllium|Xenllium]] in 2026, xenial splits the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. It corresponds to {{nowrap| ''n'' {{=}} 9 }}. Its [[ploidacot]] is zeta-enneacot, and from this it derives its name.

[[Subgroup]]: 2.3.5

[[Comma list]]: 129140163/125000000

{{Mapping|legend=1| 1 -6 -12 | 0 9 17 }}
: mapping generators: ~2, ~9/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2802{{c}}, ~9/5 = 1011.2914{{c}}
: [[error map]]: {{val| +0.280 -2.013 +2.278 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.0762{{c}}
: error map: {{val| 0.000 -2.269 +1.982 }}

{{Optimal ET sequence|legend=1| 19, 70, 89, 108, 127 }}

[[Badness]] (Sintel): 8.84

== Lalasepyo (8c & 11) ==
[[Subgroup]]: 2.3.5

[[Comma list]]: 4613203125/4294967296

{{Mapping|legend=1| 1 -1 6 | 0 7 -10 }}
: mapping generators: ~2, ~675/512

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.5641{{c}}, ~675/512 = 443.2124{{c}}
: [[error map]]: {{val| +2.564 -2.033 -3.053 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~675/512 = 442.2692{{c}}
: error map: {{val| 0.000 -6.071 -9.006 }}

{{Optimal ET sequence|legend=1| 8c, 11, 19 }}

[[Badness]] (Sintel): 24.9

[http://x31eq.com/cgi-bin/rt.cgi?ets=19_8c&limit=5 The temperament finder - 5-limit 19 & 8c]

== Unsmate ==
[[Subgroup]]: 2.3.5

[[Comma list]]: 17578125/16777216

{{Mapping|legend=1| 1 -6 4 | 0 9 -2 }}
: mapping generators: ~2, ~1875/1024

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.8892{{c}}, ~1875/1024 = 1012.5428{{c}}
: [[error map]]: {{val| +1.889 -0.405 -3.843 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1875/1024 = 1011.0348{{c}}
: error map: {{val| 0.000 -2.642 -8.383 }}

{{Optimal ET sequence|legend=1| 6b, 13, 19, 89c, 108c, 127c, 146cc }}

[[Badness]] (Sintel): 10.8

== Parakleismic ==
{{Main| Parakleismic }}
: ''For extensions, see [[Ragismic microtemperaments #Parakleismic]] and [[Starling temperaments #Paraguay]].''

The 5-limit version of parakleismic tempers out the [[parakleisma]]. It corresponds to {{nowrap| ''n'' {{=}} 13/2 }}, and 13 generator steps give the interval class of [[3/1|3]].

[[Subgroup]]: 2.3.5

[[Comma list]]: 1224440064/1220703125

{{Mapping|legend=1| 1 -8 -8 | 0 13 14 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.971{{c}}, ~5/3 = 884.7383{{c}}
: [[error map]]: {{val| -0.029 -0.127 +0.253 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/3 = 884.8576{{c}}
: error map: {{val| 0.000 -0.106 +0.293 }}

{{Optimal ET sequence|legend=1| 19, 61, 80, 99, 118, 453, 571, 689, 1496 }}

[[Badness]] (Sintel): 1.02

== Mowgli ==
: ''For extensions, see [[Hemimean clan #Mowglic]].''

[[TE]], [[CTE]] and [[POTE]] coincide at 126.7237{{c}} with pure octaves since prime 2 is not involved in the comma to begin with.

[[Subgroup]]: 2.3.5

[[Comma list]]: 31381059609/30517578125

{{Mapping|legend=1| 1 0 0 | 0 15 22 }}
: mapping generators: ~2, ~27/25

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9478{{c}}, ~27/25 = 126.7236{{c}}
: [[error map]]: {{val| -0.001 -1.100 +1.606 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 126.7237{{c}}
: error map: {{val| 0.000 -1.100 +1.607 }}

{{Optimal ET sequence|legend=1| 19, 85c, 104c, 123, 142, 161, 303 }}

[[Badness]] (Sintel): 15.3

== Enneadecal (5-limit) ==
: ''For extensions, see [[Ragismic microtemperaments #Enneadecal]].''

The 5-limit version of enneadecal tempers out the [[enneadeca]], which simply equates a stack of nineteen [[6/5]] minor thirds with five [[2/1|octaves]]. It corresponds to {{nowrap| ''n'' {{=}} 19/3 }}, with a 19th-octave period and a generator of a [[3/2|perfect fifth]].

[[Subgroup]]: 2.3.5

[[Comma list]]: 19073486328125/19042491875328

{{Mapping|legend=1| 19 0 14 | 0 1 1 }}
: mapping generators: ~648/625, ~3

[[Optimal tuning]]s:
* [[WE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9861{{c}}
: [[error map]]: {{val| +0.013 +0.044 -0.095 }}
* [[CWE]]: ~648/625 = 63.1579{{c}}, ~3/2 = 701.9900{{c}}
: error map: {{val| 0.000 -0.035 -0.113 }}

{{Optimal ET sequence|legend=1| 19, 95, 114, 133, 152, 171, 323, 494, 665, 1159, 1824, 2983, 7125c }}

[[Badness]] (Sintel): 1.12

== Countermeantone ==
[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| 10 23 -20 }}

{{Mapping|legend=1| 1 -10 -11 | 0 20 23 }}
: mapping generators: ~2, ~78125/52488

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9478{{c}}, ~78125/52488 = 695.0566{{c}}
: [[error map]]: {{val| -0.052 -0.301 +0.562 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~78125/52488 = 695.0846{{c}}
: error map: {{val| 0.000 -0.264 +0.631 }}

{{Optimal ET sequence|legend=1| 19, …, 126, 145, 164, 183, 713, 896c, 1079c, 1262c, 1445c }}

[[Badness]] (Sintel): 8.76

== Counterhanson ==
: ''For extensions, see [[Ragismic microtemperaments #Counterkleismic]].''

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -20 -24 25 }}

{{Mapping|legend=1| 1 -5 -4 | 0 25 24 }}
: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0419{{c}}, ~6/5 = 316.0916{{c}}
: [[error map]]: {{val| +0.042 +0.126 -0.282 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 316.0021{{c}}
: error map: {{val| 0.000 +0.097 -0.344 }}

{{Optimal ET sequence|legend=1| 19, …, 148, 167, 186, 205, 224, 429, 653, 1082, 1735c }}

[[Badness]] (Sintel): 7.45

== Oviminor ==
{{See also| Ragismic microtemperaments #Oviminor }}

Oviminor is named after the facts that it takes 184 minor thirds of 6/5 to reach 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.

[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -134 -185 184 }}

{{Mapping|legend=1| 1 -134 -134 | 0 184 185 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0094{{c}}, ~5/3 = 884.2568{{c}}
: [[error map]]: {{val| +0.009 +0.033 -0.069 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2499{{c}}
: error map: {{val| 0.000 +0.026 -0.083 }}

{{Optimal ET sequence|legend=1| 19, …, 1600, 3219, 4819 }}

[[Badness]] (Sintel): 751

[[Category:19edo]]
[[Category:Equivalence continua]]

Starling temperaments

2026-05-14T09:54:08Z

Xenllium:

{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.

Temperaments discussed in families and clans are:
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]

Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].

Since {{nowrap|(6/5)3 {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.

== Myna ==
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''

7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). [[11-limit]] myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]). In that sense, it is opposed to [[keemic temperaments]], in particular [[quasitemp]], where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.

In terms of vanishing commas, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.

[[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round cent values may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11- and 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 1728/1715

{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}
: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}

[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}

[[Badness]] (Sintel): 0.684

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}

Optimal tunings:
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}

Badness (Sintel): 0.557

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 196/195

Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}

Optimal tunings:
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}

Badness (Sintel): 0.708

==== Minah ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 126/125, 176/175

Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}

Optimal tunings:
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}

{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}

Badness (Sintel): 1.14

==== Maneh ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 126/125, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}

Optimal tunings:
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}

{{Optimal ET sequence|legend=0| 27eff, 31 }}

Badness (Sintel): 1.23

=== Myno ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 385/384

Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}

Optimal tunings:
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}

{{Optimal ET sequence|legend=0| 27, 31 }}

Badness (Sintel): 1.11

=== Coleto ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 1728/1715

Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}

Optimal tunings:
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}

{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}

Badness (Sintel): 1.61

== Nusecond ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''

Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 70 }}. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.

[[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2430/2401

{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
: mapping generators: ~2, ~49/27

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}
: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}
: error map: {{val| 0.000 -2.655 +3.768 +2.819 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}

[[Badness]] (Sintel): 1.28

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}

Optimal tunings:
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}

Minimax tuning:
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of 15''x''2 - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}

Badness (Sintel): 0.847

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}

{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}

Badness (Sintel): 0.964

== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 117649/116640

{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}
: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}

{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}

[[Badness]] (Sintel): 1.86

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 26411/26244

Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.88

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}

Optimal tunings:
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.47

== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''

Vines may be described as the {{nowrap| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 84035/82944

{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
: [[error map]]: {{val| +0.487 -0.363 +3.036 -4.448 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}
: error map: {{val| 0.000 -0.717 +2.269 -5.552 }}

{{Optimal ET sequence|legend=1| 46, 96d, 142d }}

[[Badness]] (Sintel): 1.98

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2401/2376

Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}

Optimal tunings:
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}

{{Optimal ET sequence|legend=0| 46, 96d, 142d }}

Badness (Sintel): 1.47

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}

Optimal tunings:
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}

{{Optimal ET sequence|legend=0| 46, 96d }}

Badness (Sintel): 1.23

== Xenial ==
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''

Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 177147/175616

{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}

{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}

[[Badness]] (Sintel): 2.13

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 16384/16335

Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}

Optimal tunings:
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.31

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 729/728

Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}

Optimal tunings:
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 1.98

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}

Optimal tunings:
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.06

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}

Optimal tunings:
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 2.03

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}

Optimal tunings:
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 1.93

== Kumonga ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 12288/12005

{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0653{{c}}, ~7/4 = 975.6277{{c}}
: [[error map]]: {{val| -1.935 -1.382 +4.009 +2.932 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 977.1096{{c}}
: error map: {{val| 0.000 +0.470 +7.673 +8.284 }}

{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}

[[Badness]] (Sintel): 2.21

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}

Optimal tunings:
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}

Badness (Sintel): 1.43

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}

Optimal tunings:
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}

Badness (Sintel): 1.19

== Paraguay ==
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Parakleismic]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 12005/11664

{{Mapping|legend=1| 1 -8 -8 -9 | 0 13 14 16 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6421{{c}}, ~5/3 = 885.3232{{c}}
: [[error map]]: {{val| +0.642 +2.110 +3.074 -9.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.8949{{c}}
: error map: {{val| 0.000 +1.678 +2.214 -10.508 }}

{{Optimal ET sequence|legend=1| 19, 61, 80d, 99d }}

[[Badness]] (Sintel): 2.47

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 12005/11664

Mapping: {{mapping| 1 -8 -8 -9 2 | 0 13 14 16 2 }}

Optimal tunings:
* WE: ~2 = 1197.7783{{c}}, ~5/3 = 883.6140{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1383{{c}}

{{Optimal ET sequence|legend=0| 19, 42e, 61e }}

Badness (Sintel): 2.49

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99, 343/338

Mapping: {{mapping| 1 -8 -8 -9 2 -14 | 0 13 14 16 2 24 }}

Optimal tunings:
* WE: ~2 = 1197.7848{{c}}, ~5/3 = 883.6431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.1623{{c}}

{{Optimal ET sequence|legend=0| 19, 42ef, 61e }}

Badness (Sintel): 1.86

==== Uruguay ====
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 100/99, 1183/1152

Mapping: {{mapping| 1 -8 -8 -9 2 0 | 0 13 14 16 2 5 }}

Optimal tunings:
* WE: ~2 = 1199.6132{{c}}, ~5/3 = 884.7325{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 885.0005{{c}}

{{Optimal ET sequence|legend=0| 19, 42e }}

Badness (Sintel): 2.51

== Bisemidim ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 118098/117649

{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45

[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}

{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}

[[Badness]] (Sintel): 2.47

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}

Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}

Badness (Sintel): 1.36

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}

Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}

Badness (Sintel): 0.987

== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 19683/19208

{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}

{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}

[[Badness]] (Sintel): 2.53

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}

Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}

{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}

Badness (Sintel): 1.41

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}

Optimal tunings:
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}

{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}

Badness (Sintel): 1.56

== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''

Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.

If we add 385/384 to the list of commas, 48/35 is identified with [[11/8]], and casablanca is revealed as an [[11-limit]] temperament with a very low complexity for [[11/1|11]] and not too high a one for [[7/1|7]]; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit [[meantone]].

Marrakesh, named by [[Herman Miller]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, is a more accurate 11-limit extension where the generator is identified with [[15/11]] as opposed to 11/8 in casablanca.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 589824/588245

{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}
: [[error map]]: {{val| -0.371 -1.087 +3.370 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 542.1684{{c}}
: error map: {{val| 0.000 -0.756 +4.044 -0.152 }}

{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}

[[Badness]] (Sintel): 2.56

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}

Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.22

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}

Optimal tunings:
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.31

=== Marrakesh ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}

Optimal tunings:
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}

Badness (Sintel): 1.34

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}

Optimal tunings:
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}

Badness (Sintel): 1.68

==== Murakuc ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}

Optimal tunings:
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}

{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}

Badness (Sintel): 1.71

== Amigo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2097152/2083725

{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}
: [[error map]]: {{val| -0.565 -0.811 +3.467 -1.206 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.0937{{c}}
: error map: {{val| 0.000 +0.076 +4.780 +0.393 }}

{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}

[[Badness]] (Sintel): 2.81

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 16384/16335

Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}

Optimal tunings:
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}

Badness (Sintel): 1.44

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 176/175, 364/363

Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}

Optimal tunings:
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89 }}

Badness (Sintel): 1.27

== Gilead ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 343/324

{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4516{{c}}, ~5/3 = 879.6394{{c}}
: [[error map]]: {{val| +1.452 +7.542 +2.823 -21.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.7223{{c}}
: error map: {{val| 0.000 +6.545 +0.909 -24.159 }}

{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}

[[Badness]] (Sintel): 2.92

== Supersensi ==
Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 17496/16807

{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}
: [[error map]]: {{val| -0.859 -4.800 +3.337 +6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/270 = 446.5163{{c}}
: error map: {{val| 0.000 -4.210 +4.464 +8.017 }}

{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}

[[Badness]] (Sintel): 3.76

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 864/847

Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}

Optimal tunings:
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}

Badness (Sintel): 1.97

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}

Optimal tunings:
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.46

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}

Optimal tunings:
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.32

== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''

Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.

Cobalt was named by [[Xenllium]] in 2022 after the 27th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 40353607/40310784

{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3

[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 44.4363{{c}}, ~3/2 = 701.1154{{c}}
: [[error map]]: {{val| -0.221 -1.060 +3.307 -1.534 }}
* [[CWE]]: ~36/35 = 44.4444{{c}}, ~3/2 = 701.0414{{c}}
: error map: {{val| 0.000 -0.914 +3.617 -1.118 }}

{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}

[[Badness]] (Sintel): 4.39

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108 }}

Badness (Sintel): 2.58

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}

Optimal tunings:
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}

Badness (Sintel): 2.36

===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445

Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 2.14

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968

Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 1.85

===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968

Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}

Optimal tunings:
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}

Badness (Sintel): 2.40

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083

Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}

Badness (Sintel): 2.08

==== Cobaltite ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 975/968

Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}

{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}

Badness (Sintel): 2.18

== References ==

[[Category:Temperament collections]]
[[Category:Starling temperaments| ]] 
[[Category:Rank 2]]

User:Xenllium

2026-05-14T03:51:25Z

Xenllium:

'''Xenllium''' joined Xenharmonic Wiki on 15 November 2018. The name "Xenllium" is a portmanteau of "Xenharmonic" and "''[[Wikipedia:Trillium|Trillium]]''".

== Temperaments which this user proposed names ==
This user proposed new names for following temperaments.

===Proposed in 2019===
'''March 2019'''
* [[Icositritonic]]
* [[Hemicocomp|Hemicounterpyth]] (→ Hemicocomp)
* [[Hemermacomp|Hemermapyth]] (→ Hemermacomp)
* [[Pentacontatritonic]]
* Hectaheptacontahenatonic (171 & 17442)
* [[Meridic]]
* Tetracontaheptatonic (94 & 705)
* [[Palladium]]
'''April 2019'''
* [[Septisemitonic]]
* [[Tertiathirds]]
* [[Tertiaschis]]

===Proposed in 2021===
'''March 2021'''
* [[Trident]] (→ Alphatrident)
* [[Trillium]] (→ Alphatrillium)
* [[Pseudotrillium]]
* [[Tritricot]]
* [[Decic]]
* [[Cloudtone]]
'''April 2021'''
* [[Hemipental]]
* [[Sengasec]]
* [[Qinto]]
* [[Monzism]]
* [[Houborizic]]
* [[Houbor]]
* [[Gamity]]
* [[Familia]]
* [[Sabric]]
* [[Quadrawell]]
* [[Sfour]]
* [[Rainwell]]
* [[Sengage]]
* [[Marfifths]]
* [[Diatessic]]
* [[Marf]]
'''May 2021'''
* [[Garistearn]]
* [[Decistearn]]
* [[Quadrant]]
* [[Quintilipyth]]
* [[Helenoid]]
* [[Ponta]]
* [[Pontic]]
* [[Bipont]]
* [[Counterbipont]]
* [[Quadrapont]]
* [[Decimetra]]
* [[Stockhausenic]]
* [[Decavish]]
* Bronzoid (109 & 142)
* [[Ennealimmia]]
* [[Paralytic]]
* [[Cotoneum]]
* [[Fog]]
* [[Countermiracle]]
* [[Paraklein]]
* [[Countermiraculous]]
* [[Counterbenediction]]
* [[Counterrevelation]]
* [[Septant]]
* [[Terminal]]
* [[Semiterm]]
* [[Sedia]]
* [[Marthirds]]
* [[Counterkleismic]]
* [[Counterlytic]]
* [[Geb]]
* [[Hemidimfi]]
* [[Maquiloid]]
* [[Paramity]]
'''June 2021'''
* [[Schisennealimmal]]
* [[Schisennealimmic]]
* [[Decades]] (→ Gamelstearn)
* [[Undecentic]]
* [[Pentadecal]]
* [[Quinkee]]
* [[Coditonic]]
* [[Terture]]
* [[Semivulture]]
* [[Eagle]]
* [[Quindromeda]]
* [[Quintakwai]]
* [[Quintagar]]
* [[Quintoneum]]
* [[Quinsandra]]
* [[Quinsandric]]
* [[Quinsandro]]
* [[Semiquindromeda]]
* [[Quintoneoid]]
'''July 2021'''
* [[Mineral]]
* [[Ore]]
* [[Quintaleap]]
* [[Quintupole]]
* [[Quinticosiennic]]
* [[Quintapole]]
* [[Hemiseptisix]]
'''August 2021'''
* [[Goldmine]]
* [[Gamelacomp|Gamelapyth]] (→ Gamelacomp)
* [[Mermacomp|Mermapyth]] (→ Mermacomp)
* [[Undesemi]]
* [[Decipentic]]
* [[Quasijerome]]
* [[Septendesemi]]
* [[Witcher]] (99 & 166, formerly ''witch'', revised in August 2023)
* [[Semiwitcher]] (166 & 198, formerly ''semiwitch'', revised in August 2023)
* [[Hitch]]
* [[Hitcher]]
* [[Quinwell]]
* [[Septiquarschis]]
* [[Quinbetter]]
* [[Ultramarine]]
* [[Mowglic]]
'''Septemper 2021'''
* [[Terminator]]
* [[Quintaschis]]
* [[Quintahelenic]]
* [[Quintahelenoid]]
* [[Semihemiwür]]
* [[Semihemiwürat]]
* [[Beetle]]
* [[Subsedia]]
* [[Quinkwai]]
* [[Quintakwoid]]
* [[Passionate]]
* [[Pash]]
* [[Majvamic]]
* [[Majvamoid]]
* [[Narayana]]
* [[Turkey]]
'''October 2021'''
* [[Lunennealimmal]]
* [[Hexadecoid]]
* [[Undetrita]]
* [[Rubidium]]
* [[Terbium]]
* [[Parakleischis]]
* [[Triacontaheptoid]]
* [[Octogintic]]
* [[Senator]]
* [[Hemivalentine]]
* [[Hemivalentino]]
* [[Hemivalentoid]]
* [[Undetritoid]]
* [[Biquartonic]]
* [[Quindro]]
'''November 2021'''
* [[Tridecatonic]]
* [[Trideci]]
* [[Decimaleap]]
* [[Hemiquindromeda]]
* [[Triboh]]
'''December 2021'''
* [[Hemimabila]]
* [[Sedecic]]
* [[Hyperkleismic]]

===Proposed in 2022===
'''January 2022'''
* [[Cobalt]]
* [[Cobaltous]]
* [[Cobaltic]]
* [[Cobaltite]]
* [[Pontoid]]
* [[Grackle]] (11, 13, 17 and 19-limit extension)
* [[Grackloid]]
* [[Grack]]
* [[Catahelenic]]
* [[Countriton]]
* [[Aufic]]
* [[Pajarina]]
* [[Muggloid]]
* [[Gracecord]]
* [[Gaster]]
* [[Tuskaloosa]]
* [[Muscogee]]
* [[Cohemimabila]]
* [[Trimabila]]
* [[Immune]]
* [[Meantritone]]
* [[Aurora]]
* [[Leapweeker]]
* [[Interaufo]]
* [[Helena]]
'''February 2022'''
* [[Cohendecatonic]]
* [[Nessa]]
* [[Splendecic]]
* [[Prodecic]]
* [[Semimeantone]]
* [[Keenic]]
* [[Warrior]]
* [[Trisensory]]
* [[Bisensi]]
* [[Browser]]
* [[Grazer]]
* [[Quadrasruta]]
* [[Quadrafourths]]
* [[Pajarita]]
'''March 2022'''
* [[Stalagmite|Catamite]] (→ Stalagmite)
* [[Hemikwai]]
* [[Yarm]]
* [[Cohedim]]
* [[October]]
* [[Dichotomic]]
* [[Seminar]]
* [[Chromat]] (11, 13 and 17-limit extension)
* [[Catachrome]]
* [[Chromic]]
* [[Trinity]]
* [[Poseidon]]
* [[Salacia]]
* [[Gammy]]
* [[Tertiaseptia]]
* [[Semitert]]
'''April 2022'''
* <s>[[Meantonic]]</s>
* <s>[[Meantoid]]</s>
* [[Huygens]] (→ Fokkertone)
* <s>[[Meridetonic]]</s>
* <s>[[Meridetoid]]</s>
* <s>[[Sauveuric]]</s>
* [[Meanpoid]]
* <s>[[Meanplop]] (17 and 19-limit extension)</s>
* <s>[[Meanploid]]</s>
* <s>[[Meanenneadecoid]]</s>
* <s>[[Vincenzoid]]</s>
* [[Andromeda]] (17 and 19-limit extension)
* [[Schisicosiennic]]
* [[Schisicosiennoid]]
* [[Cassandra]] (Cassie)
* [[Cassandric]]
* [[Cassander]]
'''May 2022'''
* [[Supersensi]]
* [[Quadrafives]]
'''June 2022'''
* [[Trimean]]

===Proposed in 2023===
'''June 2023'''
* [[Galileic]]
* [[Catagali]]
* [[Quintain]]
'''July 2023'''
* [[Devisemi]]
* [[Garibaldia]] (2.3.5.7.17.19 extension of [[garibaldi]])
* [[Pontia]] (2.3.5.7.19 extension of [[pontiac]])
'''August 2023'''
* [[Heartland]] (2.3.7.11.19 14&41& 48)
* [[Hearts]] (2.3.7.11.19 41&48)
* [[Heartia]]
* [[Heartful]] (2.3.7.11.19 41&75e)
* [[Doublehearted]] (2.3.7.11.19 14&41)
* [[Weasel|Byhearted]] (→ Weasel)
'''September 2023'''
* [[Heartlandia]] (14cf & 41 & 89)
* [[Darkstone]]
* [[Brightstone]]
'''October 2023'''
* [[Magical]]
* [[Magica]]
* [[Magia]]
* [[Sesquartia]]
* [[Charismic]]
* [[Quinmage]]
* [[Hearty]]
* [[Nonant]]
* [[Hemiandromeda]] (29 & 82)
* [[Hemidromeda]] (29 & 111)
'''November 2023'''
* [[Trismegiboh]]
* [[Baldanders]] (2.9.5.7.11 6&35)
* [[Neomirkwai]] (2.3.5.7.11.19 31&41&80)
* [[Dhaivatic]]
* [[Triquart]]
* [[Tertiseptisix]]
* [[Quartiquart]]
* [[Quintiquart]]
* [[Neovulture]] (2.3.5.7.11.13.19 53&58&217)
* [[Neovish]] (2.3.5.7.11.19 50&118&152)
* [[Neobreed]] (41 & 130 & 270)
* [[Neonewt]] (2.3.5.7.11.13.19 41&270)
'''December 2023'''
* [[Heartlander]] (14cf & 41 & 130)
* [[Neosatanic]] (665 & 3325)
* [[Monzosatanic]] (665 & 7980)
* [[Dog]] (2.3.19 7&9)
* [[Armodog]] (2.3.5.7.11.13.19 7&9)

===Proposed in 2024===
'''January 2024'''
* [[Quinbaldi]] (65d & 135)
* [[Quintant]] (65d & 200)
* [[Hemiquintant]] (130 & 135)

'''May 2024'''
* [[Raccoon]] (11 & 32)

'''June 2024'''
* [[Piracy]] (84 & 155)
* [[Pirateship]] (84 & 239)
* [[Unisquary]] (11-limit 72&118&121&130)
* [[Hensquary]] (13-limit 46&72&121&130)
* [[Ekasquary]] (13-limit 72&121&130&190)

===Proposed in 2025===
'''January 2025'''
* [[Pentadecoid]] (2.3.7.11 120&135)
* [[Slendscape]] (255 & 270)
* [[Triacontoid]] (120 & 150)

'''February 2025'''
* [[Devichrome]] (2.3.5.7.19 27&31&41&53)

'''March 2025'''
* [[Octasand]] (11 and 13-limit 27e&41&53)
* [[Devimarv]] (2.3.5.7.19 31&41&53)
** [[Neomarv]] (2.3.5.7.11.19 31&41&53)
* [[Deva]] (2.3.5.7.19 27&41&53)
** [[Neonakika]] (2.3.5.7.11.19 15&27e&41)
** [[Neoctasand]] (2.3.5.7.11.19 27e&41&53)
* [[Nutrient]] (2.3.5.7.19 27&31&41)
** [[Neojovia]] (2.3.5.7.11.19 27e&31&41)
* [[Passkey]] (2.3.5.7.19 41&53&99)
** [[Neolakia]] (2.3.5.7.11.19 41&53&99)

'''April 2025'''
* [[Magikarp]] (2.3.43 5&7)
* [[Hendeca]] (11&22)
* [[Char]] (2.3.5.7.17 19&31&41)
* [[Elven]]

'''June 2025'''
* [[Reversed marvel]] (2.3.5.113 22&31&34)

===Proposed in 2026===
'''January 2026'''
* [[Mendel]] (11-limit 12&31&152)
* [[Moctpine]] (11-limit 29&72&80)
* [[Biscordial]] (12&190)
* [[Triscordial]] (12&231)
* [[Naiad]]
* [[Naiadical]]

'''February 2026'''
* [[Naian]] (8d&13b)
* [[Semihemimean]] (80&118&130)
* [[Marvelpine]]
* [[Seminaiad]]
** [[Seminaiad|Seminaiadia]]
* [[Seesaw]]
** [[Heavy windmill]]
** [[Light windmill]]
* [[Antonian]]
* [[Antonym (temperament)|Antonym]]
* [[Antony]]
* [[Argument (temperament)|Argument]]
* [[Landy]]
* [[Septarium]]

'''March 2026'''
* [[Augenic]]
* [[Sida]]
* [[Bohlenic]] (no-twos temperament)

'''May 2026'''
* [[Xenial]] (19 & 70)
** [[Xenia]] (19 & 51c, 19 & 70d)
* [[Hemigoth]] (34d & 68)
* [[Paraguay]] (19 & 23bcf, 19 & 42ef)
** [[Uruguay]] (19 & 23bc, 19 & 42e)

== Commas which this user proposed names ==
This user proposed new names for following commas.

===Proposed in 2021===
'''April 2021'''
* [[Hewuermera temperaments|hewuermera]]: 589824/588245, satribiru-agu
* [[Triwellismic temperaments|triwellisma]]: 235298/234375, tribizo-asepgu
* [[quadrawellisma]]: 67528125/67108864, lala-quadzo-aquinyo
* [[catasyc comma]]: 390625/387072, laruquadbiyo
* [[Mistismic temperaments|mistisma]]: 458752/455625, sazoquadgu

'''May 2021'''
* [[Hemfiness temperaments|hemfiness comma]]: 4096000/4084101, saquinru-atriyo
* [[Marvelfifths temperaments|marvelfifths comma]]: 158203125/157351936, laquadru-atritriyo
* [[Chromatisma|chromatisma]]: 640000000000000000 / 635585924776181463, trisa-triru-aquadquadyo
* [[Bronzismic temperaments|bronzisma]]: 2097152/2083725, satriru-agugu
* <s>[[Breeze comma|sesquiquartisma]]</s>: 2460375/2458624, laquadru-atruyo (→ breezema, breeze comma)
* [[Mitonismic temperaments|mitonisma]]: 5250987/5242880, laquadzo-agu
* [[Sensamagicfifths temperaments|sensamagicfifths comma]]: {{monzo|-11 42 -7 -14}}, trila-seprurugu
* [[Secanticorn temperaments|secanticornisma]]: 177147/175000, laruquingu
* [[Supermatertismic temperaments|supermatertisma]]: 52734375/52706752, lasepru-atritriyo
* [[Supermasesquartismic temperaments|supermasesquartisma]]: 184528125/184473632, laquadbiru-aquinyo
* [[Septimirod temperaments|septimirod comma]]: 84035/82944, quinzo-ayo

'''June 2021'''
* [[Vorwell clan|vorwell comma]]: 134217728/133984375, sasatriru-aquadbigu
* [[Maitri clan|maitri comma]]: 40353607/40000000, tritrizo-asepgu
* [[compass comma]]: 9765625/9680832, quinruyoyo

'''July 2021'''
* [[4375/4356|fantares comma]]: 4375/4356, luluzoquadyo
* [[Septiness clan|septiness comma]]: 67108864/66706983, sasasepru
* [[decovulture comma]]: 67108864/66976875, sasa-birugugu
* [[Pontiqak temperaments|pontiqak comma]]: 95703125/95551488, lazozotritriyo

'''August 2021'''
* [[Slither clan|slither comma]]: 40960000/40353607, satritriru-aquadyo
* [[Mermismic temperaments|mermisma]]: 2500000/2470629, sepruyo
* [[Valenwuer temperaments|valenwuer comma]]: 110592/109375, sarutribigu
* [[Negricorn temperaments|negricorn comma]]: 153664/151875, saquadzogu
* [[Parkleiness temperaments|parkleiness comma]]: 1959552/1953125, zotritrigu
* [[Mynaslendic temperaments|mynaslender comma]]: 829440/823543, sepru-ayo

'''September 2021'''
* [[65536/64827|buzzardisma]]: 65536/64827, saquadru (→ buzzardsma, buzzard comma)
* [[Neptunismic temperaments|neptunisma]]: 48828125/48771072, laruruleyo

'''October 2021'''
* [[Domicomp temperaments|domicomp comma]]: 413343/409600, lazogugu

'''November 2021'''
* [[Trienstonisma|trienstonisma]]: 43046721/43025920, laquinru-agu
* [[Lokismic temperaments|lokisma]]: 102760448/102515625, sasazozo-atribigu

===Proposed in 2022===
'''January 2022'''
* [[Countriton comma|countriton comma]]: 16777216000000000 / 16677181699666569, quadsa-tritriyo
* [[3042/3025|diagassormisma]]: 3042/3025, bitholugu
* [[gaster comma]]: {{monzo|-70 72 -19}}, quadbila-negu
* [[Mabilismic clan|mabilisma]]: 1071875/1048576, latrizo-aquinyo
* [[pentonisma]]: 281974669312/281950621875, saquinthuzogu

'''February 2022'''
* [[8281/8192|diahuntmisma]]: 8281/8192, labithozo
* [[3645/3584|schismean comma]]: 3645/3584, laruyo

'''March 2022'''
* [[Acrodec temperaments|acrodec comma]]: 7710244864/7688671875, sasa-tribizo-aquadbigu
* [[120/119|lynchisma]]: 120/119, suruyo

'''May 2022'''
* [[Augmist temperaments|augmist comma]]: 1438646272/1423828125, sasa-trizotrigu
* [[Starling temperaments|supersensi comma]]: 33059881728/30517578125, quintrigu

===Proposed in 2023===
'''11 May 2023'''
* [[Syntonoschisma|syntonisma]]: {{monzo|30 -27 7 0 -1}}, trisa-lusepyo (→ syntonoschisma, syntonic schisma)

'''27 May 2023'''
* [[170/169|major naiadma]]: 170/169, sothuthuyo
* [[221/220|minor naiadma]]: 221/220, sotholugu
* <s>[[256/255|diasemisma]]</s>: 256/255, sugu (→ charisma, charic comma)
* [[442/441|seminaiadma]]: 442/441, sothoruru

'''18 July 2023'''
* [[grossmisma]]: {{monzo|-30 -13 -1 1 7 7}}, septholo-azogu

'''21 July 2023'''
* [[3971/3969|heartlandisma]]: 3971/3969, nonoloruru

'''23 July 2023'''
* [[171/170|malcolmisma]]: 171/170, nosugu
* [[190/189|cotylisma]]: 190/189, noruyo
* [[324/323|photisma]]: 324/323, nusu
* [[343/342|nutrisma]]: 343/342, nutrizo
* [[400/399|devichroma]]: 400/399, nuruyoyo
* [[devicisma]]: 633556/633555, nunusothutrilo-azogu

'''15 August 2023'''
* [[133/132|minithirdma]]: 133/132, noluzo
* <s>[[136/135|diatonisma]]</s>: 136/135, sogu (→ diatisma)
* [[154/153|augustma]]: 154/153, sulozo
* [[161/160|major kirnbergerisma]]: 161/160, twethozogu
* [[162/161|minor kirnbergerisma]]: 162/161, twethuru
* [[286/285|chthonisma]]: 286/285, nuthologu
* [[1275/1274|cimbrisma]]: 1275/1274, sothubiruyo
* [[galileisma]]: 171382426877952 / 171359481538165, lalesu-agu

'''16 August 2023'''
* [[85/84|monk comma]]: 85/84, soruyo
* [[456/455|abnobisma]]: 456/455, nothulugu
* [[476/475|hedwigma]]: 476/475, nusozogugu
* [[495/494|eulalisma]]: 495/494, nuthuloyo

'''21 August 2023'''
* [[363/361|godzillisma]]: 363/361, binulo
* [[2431/2430|heptacircle comma]]: 2431/2430, sothologu

'''26 August 2023'''
* [[1083/1078|bihendrixma]]: 1083/1078, nonolururu

'''27 August 2023'''
* [[No-fives subgroup temperaments #Doublehearted|doublehearted comma]]: 5764801/5668704, quadbizo
* [[No-fives subgroup temperaments #Hearts|hearts comma]]: 34451725707/34359738368, trila-quadzo

'''17 September 2023'''
* [[4914/4913|baladisma]]: 4914/4913, trisu-athozo

'''5 November 2023'''
* [[969/968|buddhisma]]: 969/968, nosolulu (→ kingfisher comma)
* [[1331/1330|solvejgsma]]: 1331/1330, nutrilo-arugu
* [[1540/1539|kevolisma]]: 1540/1539, nulozoyo
* [[2376/2375|trichthonisma]]: 2376/2375, nulotrigu
* [[2926/2925|neovulturisma]]: 2926/2925, nothulozogugu
* [[3136/3135|neomirkwaisma]]: 3136/3135, nuluzozogu
* [[3250/3249|martebisma]]: 3250/3249, nunuthotriyo
* [[4200/4199|neosatanisma]]: 4200/4199, nusuthuzoyoyo
* [[5776/5775|neovishma]]: 5776/5775, nonolurugugu
* [[5985/5984|neogrendelisma]]: 5985/5984, nosuluzoyo
* [[6175/6174|neonewtisma]]: 6175/6174, nothotriru-ayoyo

'''12 November 2023'''
* [[460/459|scanisma]]: 460/459, twethosuyo
* [[484/483|pittsburghisma]]: 484/483, twethuloloru
* [[507/506|laodicisma]]: 507/506, twethuthotholu
* [[529/528|preziosisma]]: 529/528, bitwetho-alu
* [[736/735|harvardisma]]: 736/735, twethorurugu
* [[760/759|squadronisma]]: 760/759, twethunoluyo
* [[875/874|nymphisma]]: 875/874, twethunuzotriyo
* [[897/896|lysistratisma]]: 897/896, twethothoru
* [[1105/1104|fragarisma]]: 1105/1104, twethusothoyo
* [[1197/1196|rhodesisma]]: 1197/1196, twethunothuzo
* [[1288/1287|santisma]]: 1288/1287, twethothuluzo
* [[1496/1495|turkisma]]: 1496/1495, twethusothulogu
* [[1863/1862|antinousisma]]: 1863/1862, twethonururu
* [[2024/2023|insincere comma]]: 2024/2023, twethosusuloru
* <s>[[2025/2024|fokkerisma]]</s>: 2025/2024, latwethuluyoyo (→ cupcakesma, cupcake comma)
* [[2185/2184|guangdongisma]]: 2185/2184, twethonothuruyo
* [[2300/2299|travellisma]]: 2300/2299, twethonubiluyo
* [[2646/2645|biyativice comma]]: 2646/2645, bitwethuzo-agu
* [[2737/2736|kotkisma]]: 2737/2736, twethonusozo
* [[3060/3059|vicious comma]]: 3060/3059, twethunusoruyo
* [[3381/3380|seminaiadvice comma]]: 3381/3380, twethobithuzo-agu (→ mikkolisma)
* [[3520/3519|vicedim comma]]: 3520/3519, twethusuloyo
* [[3888/3887|vicetride comma]]: 3888/3887, twethuthuthu
* [[4693/4692|viceaug comma]]: 4693/4692, twethunonosutho
* [[4761/4760|demiquartervice comma]]: 4761/4760, bitwetho-asurugu
* [[vicetertisma]]: 12168/12167, tritwethu-athotho

'''13 November 2023'''
* [[208/207|vicetone comma]]: 208/207, twethutho
* [[231/230|major neutravicema]]: 231/230, twethulozogu
* [[253/252|middle neutravicema]]: 253/252, twetholoru
* [[276/275|minor neutravicema]]: 276/275, twetholugugu
* [[300/299|major naiadvicema]]: 300/299, twethuthuyoyo
* [[323/322|major semivicema]]: 323/322, twethunosoru
* [[391/390|minor naiadvicema]]: 391/390, twethosothugu
* [[392/391|minor semivicema]]: 392/391, twethusuzozo
* [[460/459|vicewolf comma]]: 460/459, twethosuyo
* [[5083/5082|broadviewsma]]: 5083/5082, twethosotholuluru

'''15 November 2023'''
* [[92/91|undinisma]]: 92/91, twethothuru
* [[115/114|yarmanisma]]: 115/114, twethonuyo

'''26 November 2023'''
* [[E-shaped comma]]: 31213/31212, susuthoquadzo

'''1 December 2023'''
* <s>[[kilobyte comma]]</s>: 1024/1023, thiwulu (→ kibisma)
* <s>[[megabyte comma]]</s>: 1048576/1048575, safowuthiwulugugu (→ mebisma)

'''10 December 2023'''
* [[81/76|dog comma]]: 81/76, inu
* [[primate comma]]: 16384/15309, saru

'''23 December 2023'''
* [[841/840|arabellisma]]: 841/840, bitweno-arugu
* [[4901/4900|large grapevine comma]]: 4901/4900, twenothothobirugu, twenobithorugu
* [[6877/6875|grossvice comma]]: 6877/6875, bitwetho-atholuquadgu
* [[7425/7424|small grapevine comma]]: 7425/7424, latwenuloyoyo
* [[mynucuvine comma]]: 10935/10933, labitwenu-athuyo

===Proposed in 2024===
'''2 January 2024'''
* <s>[[slendroschisma|slendrisma]]</s>: 68719476736/68641485507, sasa-quinbiru (→ slendroschisma)

'''27 January 2024'''
* [[116/115|sironisma]]: 116/115, twenotwethugu
* [[117/116|lomisma]]: 117/116, twenutho
* [[175/174|major chthonovinema]]: 175/174, twenuzoyoyo
* [[204/203|kallistisma]]: 204/203, twenusoru
* [[232/231|major paravinema]]: 232/231, twenoluru
* [[261/260|major vinetonema]]: 261/260, twenothugu
* [[290/289|brunisma]]: 290/289, twenosusuyo
* [[320/319|minor paravinema]]: 320/319, twenuluyo
* [[378/377|major semivinema]]: 378/377, twenuthuzo
* [[406/405|minor semivinema]]: 406/405, twenozogu
* [[494/493|minor vinetonema]]: 494/493, twenunosutho
* [[551/550|minor chthonovinema]]: 551/550, twenonolugugu
* [[609/608|vineyard comma]]: 609/608, twenonuzo

'''3 May 2024'''
* [[729/722|undevicesimal diaschisma]]: 729/722, lanunu

'''1 June 2024'''
* [[unisquarisma]]: 12005/11979, trilu-aquadzo-ayo

'''22 December 2024'''
* [[429/425|middle semisixthma]]: 429/425, suthologugu
* [[507/500|large semisixthma]]: 507/500, thothotrigu
* [[1452/1445|small semisixthma]]: 1452/1445, susulologu
* [[semisixthmisma]]: 63888/63869, trisu-athutrilo

===Proposed in 2025===
'''20 March 2025'''
* [[6860/6859|devicubisma]]: 6860/6859, trinuzo-ayo
* [[tricecubisma]]: 29792/29791, trithiwu-anozozo

'''21 March 2025'''
* [[8625/8624|beerglass comma]]: 8625/8624, twetholururutriyo
* [[10626/10625|demiglace comma]]: 10626/10625, twethosulozoquadgu

'''29 March 2025'''
* [[145/144|adeonisma]]: 145/144, twenoyo
* [[552/551|sigelindisma]]: 552/551, twenutwethonu
* [[638/637|moirisma]]: 638/637, twenothuloruru
* [[726/725|joellisma]]: 726/725, twenubilogu
* [[783/782|norisma]]: 783/782, twenotwethusu
* [[1015/1014|christisma]]: 1015/1014, twenothuthuzoyo
* [[1045/1044|michelisma]]: 1045/1044, twenunoloyo
* [[1276/1275|ucclisma]]: 1276/1275, twenosulogugu
* [[1450/1449|raimondisma]]: 1450/1449, twenotwethuruyoyo
* [[1596/1595|itzigsohnisma]]: 1596/1595, twenunoluzogu
* [[1625/1624|norcma]]: 1625/1624, twenuthorutriyo
* [[1683/1682|castafiorisma]]: 1683/1682, bitwenu-asolo
* [[2176/2175|donarisma]]: 2176/2175, twenusogugu
* [[2205/2204|glinkisma]]: 2205/2204, twenunuzozoyo
* [[2262/2261|mitidikisma]]: 2262/2261, twenonusuthoru
* [[5929/5928|manzanisma]]: 5929/5928, nuthubilozo

'''6 April 2025'''
* [[6144/6137|langwisma]]: 6144/6137, nunusu

'''10 April 2025'''
* <s>[[129/128|Magikarp comma]]</s>: 129/128, fotho
* <s>[[130/129|Gyarados comma]]</s>: 130/129, fothuthoyo
* [[2640/2639|hällströmisma]]: 2640/2639, twenuthuloruyo
* [[2784/2783|domeykisma]]: 2784/2783, twenotwethululu
* [[3249/3248|musashinisma]]: 3249/3248, twenunonoru
* [[3451/3450|mentorisma]]: 3451/3450, twenotwethusozogugu
* [[4785/4784|petrovsma]]: 4785/4784, twenotwethuthuloyo
* [[5916/5915|woudisma]]: 5916/5915, twenosothuthurugu

'''11 April 2025'''
* [[10557/10556|rowlandisma]]: 10557/10556, twenutwethosothuru

'''15 April 2025'''
* [[371293/262144|tridecimal antidilemma]]: 371293/262144, laquintho 5th
* [[524288/371293|tridecimal dilemma]]: 524288/371293, saquinthu 4th
* [[1610510/1594323|fifthchromisma]]: 1610510/1594323, saquinlo-ayo

'''17 April 2025'''
* [[2001/2000|major discoverisma]]: 2001/2000, twenotwethotrigu
* [[2002/2001|minor discoverisma]]: 2002/2001, twenutwethutholozo

'''26 April 2025'''
* [[2465/2464|laservine comma]]: 2465/2464, twenosoluruyo

'''27 April 2025'''
* [[280/279|tricetone comma]]: 280/279, thiwuzoyo

'''2 May 2025'''
* [[6912/6859|deviaug comma]]: 6912/6859, trinu

'''3 May 2025'''
* [[93/92|tricema]]: 93/92, thiwotwethu
* [[155/154|scyllisma]]: 155/154, thiwoluruyo
* [[156/155|xanthippisma]]: 156/155, thiwuthogu
* [[187/186|lambertisma]]: 187/186, thiwusolo
* [[217/216|eudorisma]]: 217/216, thiwozo
* [[248/247|lameisma]]: 248/247, thiwonuthu
* [[341/340|californisma]]: 341/340, thiwosulogu
* [[342/341|endymisma]]: 342/341, thiwunolu
* [[435/434|chinthisma]]: 435/434, thiwutwenoruyo
* [[465/464|alektisma]]: 465/464, thiwotwenuyo
* [[528/527|rezisma]]: 528/527, thiwusulo
* [[589/588|croatisma]]: 589/588, thiwonoruru
* [[651/650|antiklisma]]: 651/650, thiwothuzogugu
* [[714/713|ululisma]]: 714/713, thiwutwethusozo

'''4 May 2025'''
* [[2755/2754|avicema]]: 2755/2754, twenonosuyo
* [[4641/4640|vinecute comma]]: 4641/4640, twenusothozogu
* [[5888/5887|vinocular comma]]: 5888/5887, bitwenu-atwethoru
* [[8671/8670|vinous comma]]: 8671/8670, twenotwethosusuthogu

'''9 May 2025'''
* [[6076/6075|large ricegrain comma]]: 6076/6075, sathiwobizogu
* [[6728/6727|sushi comma]]: 6728/6727, bithiwutweno-aru
* [[8960/8959|small ricegrain comma]]: 8960/8959, thiwususuzoyo

'''10 May 2025'''
* [[12636/12635|padriellisma]]: 12636/12635, nunuthorugu
* [[13690/13689|lesleymartin comma]]: 13690/13689, bithisothu-ayo
* [[16337/16335|brown rice comma]]: 16337/16335, bithiwo-asolulugu
* [[23275/23273|sunrise comma]]: 23275/23273, bithisu-anosubizoyo

'''18 May 2025'''
* [[navicular comma]]: 24192/24167, trithu-aluzo
* [[48013/48000|small tridevisemma]]: 48013/48000, trino-azotrigu
* [[93347/93312|trilute comma]]: 93347/93312, notriso
* [[116640/116603|large tridevisemma]]: 116640/116603, trinu-asuyo
* [[176000/175959|triseptichrome comma]]: 176000/175959, nulotriruyo
* [[471648/471625|small triquarterisma]]: 471648/471625, triso-alutrirugu
* [[1257795/1257728|large triquarterisma]]: 1257795/1257728, latrisulo-azoyo
* [[quinticular comma]]: 1419857/1417176, saquinso

'''28 May 2025'''
* [[7942/7935|brigade comma]]: 7942/7935, bitwethuno-alogu

'''29 May 2025'''
* [[barbadisma]]: 114383962274805 / 114349209288704, quadla-thuyo
* [[undevicesimal counterschisma]]: 2.3.19 {{monzo|-75 50 -1}}, seplanu

'''30 May 2025'''
* [[septendecimal quasithirdma]]: 19683/19652, latrisu
* [[septendecimal quasiminthma]]: 6975757441 / 6973568802, sasa-quadbiso

'''1 June 2025'''
* [[undevicesimal quasitonema]]: 34359738368 / 34296447249, sasa-tribinu
* [[deviquasisma]]: 17594077438737 / 17592186044416, lala-sepno

'''9 June 2025'''
* [[226/225|reversed marvel comma]]: 226/225, 113ogg2

'''5 July 2025'''
* [[386/385|siegenisma]]: 386/385, 193o1urg1
* [[57122/57121|phocitremisma]]: 57122/57121, 239uu3o42

===Proposed in 2026===
'''10 January 2026'''
* [[septimagic comma]]: 537824/531441, saquinzo
* [[acromagic comma]]: 26985857024 / 26904200625, sasa-sepzo-aquadgu

'''16 January 2026'''
* [[septrongisma]]: 205891132094649 / 205885750000000, lala-sepru-atritrigu

'''17 January 2026'''
* [[sensibeta comma]]: 1071875/1062882, satrizo-aquinyo

'''20 January 2026'''
* [[undetritisma]]: 205891132094649 / 204800000000000, trila-legu

'''31 January 2026'''
* [[naiadisma]]: 161414428 / 158203125, satritrizogu

'''23 February 2026'''
* [[septarium comma]]: {{monzo|73 77 -84}}, sasepquadtrigu

'''1 May 2026'''
* [[129140163/125000000|xenial comma]]: 129140163 / 125000000, latritrigu

== Scales which this user proposed names ==
This user proposed new names for following scales.

===Proposed in May 2023===
'''16 May 2023'''
* [[3L 16s|magicaltonic]]: 3L_16s MOS scale
* [[16L 3s|muggletonic]]: 16L_3s MOS scale
* [[Nicetone|zarlinoid]]: 3L_2m_2s rank-3 scale (MV3, LMLSLMS and LMLSMLS)
* [[sephipechroid]]: 3L_5m_5s rank-3 scale (MV3, LMSMLSMSLMSMS and LMSMLSMSMLSMS)
* [[magicaloid]]: 3L_8m_8s rank-3 scale (MV3, LMSMSMLSMSMSLMSMSMS and LMSMSMLSMSMSMLSMSMS)

'''18 May 2023'''
* [[Omnidiatonic|interdia]]: 2L_3m_2s rank-3 scale (MV3, LMSLMSM and LMSMLSM)
* [[antinicetone]]: 2L_2m_3s rank-3 scale (MV3, LMSLSMS and LSMLSMS)

'''20 May 2023'''
* [[villoid]]: 7L_2m_2s rank-3 scale (MV3, LLMLLSLLMLS and LLMLLSLMLLS)
* [[intervilla]]: 2L_7m_2s rank-3 scale (MV3, LMMSMLMMSMM and LMMSMMLMSMM)
* [[kleistoid]]: 2L_2m_7s rank-3 scale (MV3, LSMSSLSSMSS and LSSMSLSSMSS)
* [[interoneichro]]: 5L_3m_5s rank-3 scale (MV3, LMSLSLMSLSMLS and LMSLSMLSLSMLS)
* [[sephimechroid]]: 5L_5m_3s rank-3 scale (MV3, LMLMSLMLSMLMS and LMLSMLMLSMLMS)
* [[moharoid]]: 7L_5m_5s rank-3 scale (MV3, LMLSLMSLMLSMLSLMS and LMLSMLSLMLSMLSLMS)
* [[enharoid]]: 5L_7m_5s rank-3 scale (MV3, LMSLMSMLMSLMSMLSM and LMSLMSMLSMLMSMLSM)
* [[beatloid]]: 5L_5m_7s rank-3 scale (MV3, LMSLSMLSMSLMSLSMS and LMSLSMLSMSLSMLSMS)

'''21 May 2023'''
* [[Dualdye|archeoid]]: 5L_1m_1s rank-3 scale (MV3, LLLMLLS and LLLSLLM)
* [[interarcheo]]: 1L_5m_1s rank-3 scale (MV3, LMMMSMM and LMMSMMM)
* [[antarcheoid]]: 1L_1m_5s rank-3 scale (MV3, LSSMSSS and LSSSMSS)

===Proposed in July 2023===
'''2 July 2023'''
* [[jarachroid]]: 5L_5m_2s rank-3 scale (MV3, LMLMLSMLMLMS)
* [[Diachrome|interchroid]]: 5L_2m_5s rank-3 scale (MV3, LMSLSLSMLSLS)
* [[jerachroid]]: 2L_5m_5s rank-3 scale (MV3, LMSMSMLSMSMS)

== Documents ==
Notation:
* [[User:Xenllium/Xenllium's microtonal notation|Xenllium's microtonal notation]]
Tempered scales:
* [[User:Xenllium/Xenllium's circulating scales|Xenllium's circulating scales]]
External links:
* [https://docs.google.com/spreadsheets/d/18CRD8KRk9Gsg_tWSxbDJ_GjprgNvjGTx8ZWcpOEA-0I/edit?usp=sharing Rank-two regular temperament]
* [https://docs.google.com/spreadsheets/d/1vnZJTEGOG4FhnGyOwXdpo1KHg73e0KwzgtgbayhT4y0/edit?usp=sharing 11-limit comma lists of selected microtonal EDOs]
* [https://docs.google.com/spreadsheets/d/1NSuACLto7egh8rqDCQ-DwQFZBqdOiYHdo180tDRP740/edit?usp=sharing 171EDO / 171平均律]
* [https://drive.google.com/file/d/1Eov2FSp5Y4X4HPaaHV8nvqHX21LE7AQa/view?usp=drive_link 純正音程と平均律] (日本語/Japanese)
* [https://drive.google.com/file/d/1oBpqNqqIYCZaPuTFEt4HYdIULOMbX6sE/view?usp=drive_link 平均律 (2～31)] (日本語/Japanese)

== Music ==
* ''[https://youtube.com/watch?v=5iBXKRWE-_U Magical life]'' – in 19-tone pure-fifth magic scale
* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning
* [https://youtube.com/watch?v=bbJ0HjPAuaA 【オリジナル曲】かなしいこと / The sadness]

Parakleismic

2026-05-12T22:12:31Z

Xenllium:

{{Infobox regtemp
| Title = Parakleismic
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[Parakleisma|1224440064/1220703125]] (5-limit); [[3136/3125]], [[4375/4374]] (7-limit)
| Edo join 1 = 19 | Edo join 2 = 99
| Mapping = 1; 13 14 35
| Generators = 5/3
| Generators tuning = 884.81
| Optimization method = CWE
| MOS scales = [[4L 15s]], [[19L 4s]], [[19L 23s]]
| Odd limit 1 = 9 | Mistuning 1 = 1.34 | Complexity 1 = 42
}}
'''Parakleismic''' is a [[microtemperament]] generated by a [[~]][[6/5]] minor third, much like [[catakleismic]] but a good tuning has the generator flat, instead of sharp, than just. The sixth generator step is half a [[syntonic comma]] flat of the [[3/1|3rd]] [[harmonic]]. Consequently, the 12th generator step after [[octave reduction]] represents [[10/9]] instead of [[9/8]], and the 13th generator step after octave reduction represents [[4/3]]. This results in the [[parakleisma]] being [[tempering out|tempered out]].

Note that [[5/4]] is found at the 14th generator step (as the [[octave complement]] of [[8/5]]) and is already split in halves. Letting each part represent [[28/25]] gives rise to the canonical [[7-limit]] [[extension]] where it tempers out [[3136/3125]] and [[4375/4374]].

Extensions for harmonic 11 include ''undecimal parakleismic'', mapping it to +36 steps, ''paralytic'', to -82 steps, ''parkleismic'', to -63 steps, and ''paradigmic'', to +17 steps.

See [[Ragismic microtemperaments #Parakleismic]] for technical data. See [[Parakleismic extensions]] for a discussion on [[11-limit|11-]] and [[13-limit]] extensions.

== Interval chain ==
In the following table, odd harmonics and subharmonics 1–9 are in '''bold'''.

{| class="wikitable center-1 right-2"
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.00
| '''1/1'''
|-
| 1
| 315.19
| 6/5
|-
| 2
| 630.38
| 36/25
|-
| 3
| 945.57
| 140/81
|-
| 4
| 60.76
| 28/27
|-
| 5
| 375.96
| 56/45
|-
| 6
| 691.15
| 112/75
|-
| 7
| 1006.34
| 25/14
|-
| 8
| 121.53
| 15/14
|-
| 9
| 436.72
| 9/7
|-
| 10
| 751.91
| 54/35
|-
| 11
| 1067.10
| 50/27
|-
| 12
| 182.29
| 10/9
|-
| 13
| 497.49
| '''4/3'''
|-
| 14
| 812.68
| '''8/5'''
|-
| 15
| 1127.87
| 48/25
|-
| 16
| 243.06
| 144/125
|-
| 17
| 558.25
| 112/81
|-
| 18
| 873.44
| 224/135
|-
| 19
| 1188.63
| 125/63, 448/225, 486/245
|-
| 20
| 303.82
| 25/21
|-
| 21
| 619.01
| 10/7
|-
| 22
| 934.21
| 12/7
|-
| 23
| 49.40
| 36/35
|-
| 24
| 364.59
| 100/81
|-
| 25
| 679.78
| 40/27
|-
| 26
| 994.97
| '''16/9'''
|-
| 27
| 110.16
| 16/15
|-
| 28
| 425.35
| 32/25
|-
| 29
| 740.54
| 75/49
|-
| 30
| 1055.73
| 90/49
|-
| 31
| 170.93
| 54/49
|-
| 32
| 486.12
| 250/189, 324/245
|-
| 33
| 801.31
| 100/63
|-
| 34
| 1116.50
| 40/21
|-
| 35
| 231.69
| '''8/7'''
|-
| 36
| 546.88
| 48/35
|-
| 37
| 862.07
| 288/175, 400/243
|-
| 38
| 1177.26
| 160/81
|-
| 39
| 292.46
| 32/27
|-
| 40
| 607.65
| 64/45
|}
<nowiki>*</nowiki> In 7-limit CWE tuning, octave reduced

== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]*
! Generator (¢)
! Comments
|-
| 16\61
|
| 314.754
| 61d val, lower bound of 9-odd-limit diamond monotone
|-
|
| 15/14
| 314.930
|
|-
| 21\80
|
| 315.000
|
|-
|
| 9/7
| 315.009
|
|-
|
| 7/5
| 315.118
|
|-
|
| 7/6
| 315.142
|
|-
| 26\99
|
| 315.152
|
|-
|
| 21/20
| 315.163
|
|-
|
| 49/48
| 315.163
|
|-
|
| 36/35
| 315.164
|
|-
|
| 8/7
| 315.176
| 7-odd-limit minimax (error = 1.217 ¢)
|-
|
| 80/63
| 315.183
| 9-odd-limit minimax (error = 1.345 ¢)
|-
|
| 10/9
| 315.200
|
|-
| 57\217
|
| 315.207
|
|-
|
| 4/3
| 315.234
|
|-
|
| 16/15
| 315.249
| 5-odd-limit minimax (error = 0.196 ¢)
|-
| 31\118
|
| 315.254
|
|-
|
| 5/4
| 315.263
|
|-
|
| 25/24
| 315.289
|
|-
|
| 6/5
| 315.641
|
|-
|
| 28/27
| 315.740
|
|-
| 5\19
|
| 315.789
| Upper bound of 9-odd-limit diamond monotone
|}
<nowiki>*</nowiki> Besides the octave

[[Category:Parakleismic| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Hemimean clan]]
[[Category:Ragismic microtemperaments]]
[[Category:Hemimage temperaments]]
[[Category:Canousmic temperaments]]

Sooty fox chords

2026-05-11T09:37:32Z

Xenllium: Created page with "'''Sooty fox chords''' are essentially tempered chords tempered by the sooty fox comma, 343/338. Sooty fox chords are essentially tempered chords for 2.7.13 subgroup, consisted of the '''sooty fox triad''', 1–14/13–8/7 (with steps of 14/13, 14/13, 7/4), and the '''sooty fox tetrad''', 1–14/13–8/7–16/13 (with steps of 14/13, 14/13, 14/13, 13/8). Equal temperaments with sooty fox chords include {{Optimal ET sequence| 9, 10, 11, 19,..."

'''Sooty fox chords''' are [[Dyadic chord|essentially tempered chord]]s tempered by the sooty fox comma, [[343/338]].

Sooty fox chords are essentially tempered chords for 2.7.13 [[subgroup]], consisted of the '''sooty fox triad''', 1–14/13–8/7 (with steps of 14/13, 14/13, 7/4), and the '''sooty fox tetrad''', 1–14/13–8/7–16/13 (with steps of 14/13, 14/13, 14/13, 13/8).

[[Equal temperament]]s with sooty fox chords include {{Optimal ET sequence| 9, 10, 11, 19, 21, 31, and 41}}.

[[Category:13-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Sooty fox]]

Dyadic chord

2026-05-09T10:07:47Z

Xenllium: Jubilismic chords are innate because they cannot be mapped justly with pure octaves

{{Distinguish| Dyad }}

A '''dyadic chord''', also described as a ''dyadically consonant chord'' or ''pairwise consonant chord'', is a [[chord]] each of whose [[interval]]s belongs to a specified set of intervals considered to be [[consonance and dissonance|consonant]]; it is therefore relative to the set of intervals in question.

For example, the [[tetrad]]
* 1 – 6/5 – 7/5 – 8/5

is a dyadic chord in the [[7-odd-limit]] since every interval involved in it is an element of the 7-odd-limit [[tonality diamond]]. Now if we replace 7/5 with 10/7:
* 1 – 6/5 – 10/7 – 8/5

is ''not'' a dyadic chord in the 7-odd-limit. Although each note is 7-odd-limit over the bass, the interval between 10/7 and 6/5 is 25/21, and that between 10/7 and 8/5 is 28/25 – these are not 7-odd-limit.

The significance of dyadic chords and of the paradigm where all interval pairs are examined in the chord has the psychoacoustic basis of timbral fusion and emergence of the [[virtual fundamental]]. In the above examples, it can be shown that the lower harmonics of each note in the first chord blends better than in the second. Meanwhile, the virtual fundamental of the first chord appears 5/1 below the bass, whereas that of the second appears much lower, at 35/1 below the bass as the denominators "fight" each other. For these reasons we tend to find the first chord more consonant than the second.

== Essentially tempered dyadic chord ==
In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in just intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. An ''essentially just'' dyadic chord is a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[4:5:6|1 – 5/4 – 3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0 – 10 – 18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1 – 5/4 – 3/2.

[[File:Essentially tempered chord.png|400px|thumb|right|A more in-depth work-through of the starling 1-6/5-10/7 essentially tempered chord example]]

An '''essentially tempered dyadic chord''', often simply written as '''essentially tempered chord''', or '''dyadically essentially tempered chord''' ('''DET chord'''){{idio}}, is a chord defined in a [[regular temperament]] and relative to a set of consonances such that each interval is within that consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1 – 6/5 – 10/7, when mapped by starling temperament, which makes [[126/125]] [[vanish]], has each of its intervals in the set of [[7-odd-limit]] consonances which is the tempering of the 7-odd-limit diamond by 126/125 (this is because 10/7 is off from 36/25 by 126/125, and therefore 10/7 and 36/25 are tempered together in starling temperament, and since 36/25 = (6/5)², the interval from 6/5 to 10/7 in starling may be heard as a second move by 6/5). However, (10/7)/(6/5) = 25/21 is [[25-odd-limit]], and there is no other 7-odd-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord.

Essentially tempered dyadic chords are a related notion to [[comma pump]]s, and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.

== Innate comma chord ==
An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit. However, instead of specifying the targeted JI ratios as in an [[#Essentially_tempered_dyadic_chord|essentially tempered chord]], an innate comma chord describes only the general chord shape: the [[mapped interval]] representations or the [[interval span|sizes]] of its constituent intervals.

For example, the [[augmented triad]] in 5-limit JI is an innate comma chord below the 25-odd-limit, because it is impossible to tune all three major thirds (in the four-note chord doubling the root up an octave) to [[5/4]] or any other 5-limit interval with odd limit below 25: the innate comma here is 128/125 (41¢). In practice, it might be sung or played justly but with a large odd limit (containing [[wolf interval]]s): for example, 1–5/4–8/5–2, or 1–5/4–25/16–2 (or even 1–5/4–25/16–125/64). Or it might be tempered, e.g. in 12edo as 0¢–400¢–800¢–1200¢. In 7-limit JI, one of the major thirds can be tuned to 9/7, reducing the innate comma to 225/224 (only 8¢). This comma can be distributed among the three thirds, tempering each by only a few cents, which is usually close enough to be acceptable. In 11-limit JI, the augmented chord is not an innate comma chord, because it can be tuned justly as 7:9:11:14, a low enough odd limit to "ring". (However, it is debatable whether this chord qualifies as an augmented triad, because the middle [[11/9]] interval is a neutral third rather than a major third.)

Another example is a [[Jubilismic chords|jubilismic tetrad]], 1–5/4–7/5–7/4–2 (with steps of 5/4, 8/7, 5/4, 8/7), tempered by [[50/49]], which is an innate comma chord in the 7-odd-limit, with impossibility to tune all two ~5/4s and two ~8/7s (or equate ~7/5 with ~10/7) justly with pure octaves. Other jubilismic tetrads are also innate comma chords: 1–6/5–7/5–12/7–2 (with steps of 6/5, 7/6, 6/5, 7/6) and 1–9/7–7/5–9/5–2 (9-odd-limit, with steps of 9/7, 10/9, 9/7, 10/9).

== Anomalous saturated suspension ==
An '''anomalous saturated suspension''' ('''ASS'''), introduced by [[Graham Breed]]<ref>[https://www.webcitation.org/60VBgPSUS ''Anomalous Saturated Suspensions'']</ref>, is a ''q''-odd-limit just dyadic chord to which no pitch ''q''-odd-limit pitch class can be added while keeping it in the ''q''-odd-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5:…:''q'' chord or the 1:1/3:1/5:…:1/''q'' chord. The existence of such chords was discovered by [[Paul Erlich]]<ref>[https://www.webcitation.org/60VCUHe6d ''Anomalous Saturated Suspensions -- Paul Erlich's post'']</ref>. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.

* [[12:15:18:20]]
* [[12:14:18:21]]

For a complete list of ASS chords through the 23-odd-limit see [[List of anomalous saturated suspensions]].

== List of just intonation tetrads ==
* [[Seven limit tetrads]]
* [[Nine limit tetrads]]
* [[1-3-7-11 tetrads]]
* [[Thirteen limit tetrads]]
* [[Fifteen limit tetrads]]
* [[Seventeen limit tetrads]]

== List of essentially tempered dyadic chords ==
Here are some pages on certain essentially tempered dyadic chords, sorted by the lowest odd limit in which they are available. See [[Dyadic chord/Pattern of essentially tempered chords]] for some notable abstract chord patterns.

=== 7-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Archytas chords]] || [[Archytas]] || [[64/63]]
|-
| [[Starling chords]] || [[Starling]] || [[126/125]]
|}

=== 9-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Didymic chords]] || [[Meantone]] || [[81/80]]
|-
| [[Marvel chords]] || [[Marvel]] || [[225/224]]
|-
| [[Sensamagic chords]] || [[Sensamagic]] || [[245/243]]
|}

=== 11-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Mothwellsmic chords]] || [[Mothwellsmic]] || [[99/98]]
|-
| [[Ptolemismic chords]] || [[Ptolemismic]] || [[100/99]]
|-
| [[Biyatismic chords]] || [[Biyatismic]] || [[121/120]]
|-
| [[Valinorsmic chords]] || [[Valinorsmic]] || [[176/175]]
|-
| [[Rastmic chords]] || [[Rastmic]] || [[243/242]]
|-
| [[Frostmic chords]] || [[Frostmic]] || [[245/242]]
|-
| [[Keenanismic chords]] || [[Keenanismic]] || [[385/384]]
|-
| [[Werckismic chords]] || [[Werckismic]] || [[441/440]]
|-
| [[Swetismic chords]] || [[Swetismic]] || [[540/539]]
|-
| [[Pentacircle chords]] || [[Pentacircle]] || [[896/891]]
|-
| [[Undecimal marvel chords]] || [[Marvel]] || 225/224, 385/384
|-
| [[Prodigy chords]] || [[Prodigy]] || 225/224, 441/440
|-
| [[Undecimal sensamagic chords]] || [[Sensamagic]] || 245/243, 385/384
|-
| [[Jove chords]] || [[Jove]] || 243/242, 441/440
|-
| [[Miracle chords]] || [[Miracle]] || 225/224, 243/242, 385/384
|-
| [[Magic chords]] || [[Magic]] || 100/99, 225/224, 245/243
|-
| [[Keemic chords]] || [[Keemic]] || 100/99, 385/384
|-
| [[Orwell tetrad]] || [[Orwellian]] || 99/98, 385/384
|-
| [[Tutonic hexad|Tutonic hexads]] || [[Meantone]] || 81/80, 99/98, 126/125
|-
| [[Baldanders chords|Baldanders hexads]] || [[Andromeda]] || 100/99, 225/224, 245/242
|-
| [[Porcupine heptad]] || [[Porkypine]] || 55/54, 100/99
|}

=== 13-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Buzurgismic chords]] || [[Buzurgismic]] || [[169/168]]
|-
| [[Mynucumic chords]] || [[Mynucumic]] || [[196/195]]
|-
| [[Gassormic chords]] || [[Gassormic]] || [[275/273]]
|-
| [[Marveltwin chords]] || [[Marveltwin]] || [[325/324]]
|-
| [[Sooty fox chords]] || [[Sooty fox]] || [[343/338]]
|-
| [[Ratwolfsmic chords]] || [[Ratwolfsmic]] || [[351/350]]
|-
| [[Major minthmic chords]] || [[Major minthmic]] || [[352/351]]
|-
| [[Minor minthmic chords]] || [[Minor minthmic]] || [[364/363]]
|-
| [[Huntmic chords]] || [[Huntmic]] || [[640/637]]
|-
| [[Squbemic chords]] || [[Squbemic]] || [[729/728]]
|-
| [[Cuthbert chords]] || [[Cuthbert]] || [[847/845]]
|-
| [[Sinbadmic chords]] || [[Sinbadmic]] || [[1001/1000]]
|-
| [[Kestrel chords]] || [[Kestrel]] || [[1188/1183]]
|-
| [[Catadictmic chords]] || [[Catadictmic]] || [[1287/1280]]
|-
| [[Lambeth chords]] || [[Lambeth]] || [[1573/1568]]
|-
| [[Petrmic chords]] || [[Petrmic]] || [[2200/2197]]
|-
| [[Rastmic chords #Rastgross heptad|Rastgross heptad]] || [[Namo]] || 144/143, 243/242
|-
| [[Parapyth chords]] || [[Parapyth]], [[gentle]] || 352/351, 364/363
|-
| [[Hecate hexad]] || [[Hecate]] || 225/224, 325/324, 385/384
|-
| [[Woodpecker octad]] || [[Woodpecker]] || 66/65, 121/120, 126/125
|-
| [[Miraculous decad]] || [[Miraculous]], [[Revelation]] || 105/104, 196/195, 512/507
|}

=== 15-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Island chords]] || [[The Archipelago|Island]] || [[676/675]]
|-
| [[Nicolic chords]] || [[Nicolic]] || [[1575/1573]]
|-
| [[Myhemiwell chords]] || [[Myhemiwell]] || [[3388/3375]]
|-
| [[Battaglia chord]] || [[Marvel]] || 225/224
|-
| [[Tetracot chords]] || [[Tetracot]] || 100/99, 243/242
|-
| [[Orwell tetrad|Orwell ennead]] || [[Guanyin]] || 176/175, 540/539
|}

=== 17-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Augustmic chords]] || [[Augustmic]] || [[154/153]]
|-
| [[Charismic chords]] || [[Charismic]] || [[256/255]]
|-
| [[Prototannismic chords]] || [[Prototannismic]] || [[273/272]]
|-
| [[Ursulismic chords]] || [[Ursulismic]] || [[375/374]]
|-
| [[Seminaiadmic chords]] || [[Seminaiadmic]] || [[442/441]]
|-
| [[Monardismic chords]] || [[Monardismic]] || [[561/560]]
|-
| [[Dakotismic chords]] || [[Dakotismic]] || [[595/594]]
|-
| [[September chords]] || [[September]] || [[715/714]]
|-
| [[Horizmic chords]] || [[Horizmic]] || [[833/832]]
|-
| [[Ainismic chords]] || [[Ainismic]] || [[936/935]]
|-
| [[Twosquare chords]] || [[Twosquare]] || [[1089/1088]]
|-
| [[Quadrantonismic chords]] || [[Quadrantonismic]] || [[1156/1155]]
|-
| [[Cimbrismic chords]] || [[Cimbrismic]] || [[1275/1274]]
|-
| [[Fidesmic chords]] || [[Fidesmic]] || [[2025/2023]]
|-
| [[Heptacircle chords]] || [[Heptacircle]] || [[2431/2430]]
|-
| [[Sextantonismic chords]] || [[Sextantonismic]] || [[2601/2600]]
|}

=== 19-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Photismic chords]] || [[Photismic]] || [[324/323]]
|-
| [[Nutrismic chords]] || [[Nutrismic]] || [[343/342]]
|-
| [[Dudon chords]] || [[Dudon]] || [[361/360]]
|-
| [[Devichromic chords]] || [[Devichromic]] || [[400/399]]
|-
| [[Abnobismic chords]] || [[Abnobismic]] || [[456/455]]
|-
| [[Hedwigmic chords]] || [[Hedwigmic]] || [[476/475]]
|-
| [[Eulalismic chords]] || [[Eulalismic]] || [[495/494]]
|-
| [[Boethius chords]] || [[Boethius]] || [[513/512]]
|-
| [[Kingfisher chords]] || [[Kingfisher]] || [[969/968]]
|-
| [[Bihendrixmic chords]] || [[Bihendrixmic]] || [[1083/1078]]
|-
| [[Eratosthenes chords]] || [[Eratosthenes]] || [[1216/1215]]
|-
| [[Solvejgsmic chords]] || [[Solvejgsmic]] || [[1331/1330]]
|-
| [[Aureusmic chords]] || [[Aureusmic]] || [[1445/1444]]
|-
| [[Pinkanberry chords]] || [[Pinkanberry]] || [[1521/1520]]
|-
| [[Kevolismic chords]] || [[Kevolismic]] || [[1540/1539]]
|-
| [[Ramanujanismic chords]] || [[Ramanujanismic]] || [[1729/1728]]
|-
| [[Blumeyer chords]] || [[Blumeyer]] || [[2432/2431]]
|-
| [[Neovulturismic chords]] || [[Neovulturismic]] || [[2926/2925]]
|-
| [[Neomirkwaismic chords]] || [[Neomirkwaismic]] || [[3136/3135]]
|-
| [[Neosatanismic chords]] || [[Neosatanismic]] || [[4200/4199]]
|-
| [[Langwismic chords]] || [[Langwismic]] || [[6144/6137]]
|}

=== 21-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Slendric pentad]] || [[Gamelismic]] || [[1029/1024]]
|-
| [[Palingenetic chords]] || [[Palingenetic]] || [[1701/1700]]
|-
| [[Xenismic chords]] || [[Xenismic]] || [[2058/2057]]
|-
| [[Ibnsinmic chords]] || [[Ibnsinmic]] || [[2080/2079]]
|-
| [[Heartlandismic chords]] || [[Heartlandismic]] || [[3971/3969]]
|-
| [[Minismic chords]] || [[Minismic]] || [[4096/4095]]
|-
| [[Baladismic chords]] || [[Baladismic]] || [[4914/4913]]
|-
| [[Neogrendelismic chords]] || [[Neogrendelismic]] || [[5985/5984]]
|-
| [[Heartland #Chords|Heartland chords]] || [[Heartland]] || 243/242, 1083/1078
|}

=== 23-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Scanismic chords]] || [[Scanismic]] || [[460/459]]
|-
| [[Pittsburghismic chords]] || [[Pittsburghismic]] || [[484/483]]
|-
| [[Laodicismic chords]] || [[Laodicismic]] || [[507/506]]
|-
| [[Preziosismic chords]] || [[Preziosismic]] || [[529/528]]
|-
| [[Worcester chords]] || [[Worcester]] || [[576/575]]
|-
| [[Harvardismic chords]] || [[Harvardismic]] || [[736/735]]
|-
| [[Squadronismic chords]] || [[Squadronismic]] || [[760/759]]
|-
| [[Lysistratismic chords]] || [[Lysistratismic]] || [[897/896]]
|-
| [[Fragarismic chords]] || [[Fragarismic]] || [[1105/1104]]
|-
| [[Rodessismic chords]] || [[Rodessismic]] || [[1197/1196]]
|-
| [[Triaphonismic chords]] || [[Triaphonismic]] || [[1288/1287]]
|-
| [[Turkismic chords]] || [[Turkismic]] || [[1496/1495]]
|-
| [[Antinousismic chords]] || [[Antinousismic]] || [[1863/1862]]
|-
| [[Artifismic chords]] || [[Artifismic]] || [[2024/2023]]
|-
| [[Cupcake chords]] || [[Cupcake]] || [[2025/2024]]
|-
| [[Guangdongismic chords]] || [[Guangdongismic]] || [[2185/2184]]
|-
| [[Travellismic chords]] || [[Travellismic]] || [[2300/2299]]
|-
| [[Biyativice chords]] || [[Biyativice]] || [[2646/2645]]
|-
| [[Kotkismic chords]] || [[Kotkismic]] || [[2737/2736]]
|-
| [[Vicious chords]] || [[Vicious]] || [[3060/3059]]
|-
| [[Mikkolismic chords]] || [[Mikkolismic]] || [[3381/3380]]
|-
| [[Vicedim chords]] || [[Vicedim]] || [[3520/3519]]
|-
| [[Shoalmic chords]] || [[Shoalmic]] || [[3888/3887]]
|-
| [[Viceaug chords]] || [[Viceaug]] || [[4693/4692]]
|-
| [[Demiquartervice chords]] || [[Demiquartervice]] || [[4761/4760]]
|-
| [[Broadviewsmic chords]] || [[Broadviewsmic]] || [[5083/5082]]
|-
| [[Vicetertismic chords]] || [[Vicetertismic]] || [[Vicetertisma|12168/12167]]
|}

=== 25-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Tunbarsmic chords]] || [[Tunbarsmic]] || [[625/624]]
|-
| [[Nymphismic chords]] || [[Nymphismic]] || [[875/874]]
|-
| [[Noellismic chords]] || [[Noellismic]] || [[1225/1224]]
|-
| [[Trichthonismic chords]] || [[Trichthonismic]] || [[2376/2375]]
|-
| [[Sperasmic chords]] || [[Sperasmic]] || [[2500/2499]]
|-
| [[Lehmerismic chords]] || [[Lehmerismic temperaments|Lehmerismic]] || [[3025/3024]]
|-
| [[Martebismic chords]] || [[Martebismic]] || [[3250/3249]]
|-
| [[Leprechaun chords]] || [[Leprechaun]] || [[4225/4224]]
|-
| [[Neovishmic chords]] || [[Neovishmic]] || [[5776/5775]]
|-
| [[Neonewtismic chords]] || [[Neonewtismic]] || [[6175/6174]]
|}

=== 27-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Ragismic chords]] || [[Ragismic]] || [[4375/4374]]
|-
| [[Hagendorfismic chords]] || [[Hagendorfismic]] || [[8075/8073]]
|}

=== 29-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Vinecute chords]] || [[Vinecute]] || [[4641/4640]]
|-
| [[Large grapevine chords]] || [[Large grapevine]] || [[4901/4900]]
|-
| [[Small grapevine chords]] || [[Small grapevine]] || [[7425/7424]]
|-
| [[Vinous chords]] || [[Vinous]] || [[8671/8670]]
|}

=== 31-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Kibismic chords]] || [[Kibismic]] || [[1024/1023]]
|-
| [[Acronymismic chords]] || [[Acronymismic]] || [[Acronymisma|17577/17576]]
|-
| [[Tricecubismic chords]] || [[Tricecubismic]] || [[Tricecubisma|29792/29791]]
|}

=== 33-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Wizardharry chords]] || [[Wizardharry]] || [[4000/3993]]
|-
| [[Flashmic chords]] || [[The Flashmob|Flashmic]] || [[12376/12375]]
|}

=== 35-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Lummic chords]] || [[Lummic]] || [[1716/1715]]
|}

=== 37-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Bullionismic chords]] || [[Bullionismic]] || [[5292/5291]]
|}

=== 39-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Harmonismic chords]] || [[Harmonismic]] || [[10648/10647]]
|}

=== 55-odd-limit ===
{| class="wikitable"
|-
! Chords !! Associated temperament !! Associated commas
|-
| [[Jacobin chords]] || [[The Jacobins|Jacobin]] || [[6656/6655]]
|}

== List of innate comma chords ==
{| class="wikitable"
|-
! Chords !! Associated Temperament !! Associated Commas
|-
| [[Diminished seventh chord]] || [[Diminished (temperament)|Diminished]] || [[36/35]], [[50/49]]
|-
| [[Jubilismic chords]] || [[Jubilismic]] || [[50/49]]
|-
| [[Augmented triad]] || [[Augmented (temperament)|Augmented]] || [[128/125]]
|}

== List of essentially just dyadic chords ==
As chords that are unambiguous counterparts to common JI chords are not of particular relevance to this page, most of the entries here will be what [[User:Kaiveran|Kaiveran]] calls '''plurichords''', where there are multiple sets of consonances that a given chord can be mapped to. Note that this can still lead to ambiguous tonality in the case of otonal and utonal intervals being identified together.
{| class="wikitable"
|+
! Chords
! Equivalent Mappings
! Associated Commas
|-
| [[Hendrix chord]]
| 8:10:14:19 ~ 12:15:21:28
| [[57/56]]
|-
| [[Rootsubminor triad]]
| 6:7:9 ~ 16:19:24
| 57/56
|-
| [[The Biosphere|Biosphere]] triads
| 6:7:9 ~ 26:30:39 10:13:15 ~ 14:18:21
| [[91/90]]
|-
| [[Rootminor triad]]
| 10:12:15 ~ 16:19:24
| [[96/95]]
|-
| [[Lynchismic chords|Lynchismic plurichords]]
| 12:14:17:20 ~ 1/(20:17:14:12) 6:7:8:10 ~ 1/(20:17:15:12)
| [[120/119]]
|-
| [[Augustmic chords|Augustmic plurichords]]
| 14:17:18:22 ~ 1/(22:18:17:14)
| [[154/153]]
|}

== See also ==
* [[Essential tempering comma]]

== Notes ==

[[Category:Terms]]
[[Category:Regular temperament theory]]
[[Category:Dyadic chords| ]] 
[[Category:Lists of chords]]
<references />

Meantone family

2026-05-09T06:52:49Z

Xenllium: /* Septimal meantone */added reasonable extensions

{{interwiki
| de = Mitteltönig
| en = Meantone family
| es =
| ja =
}}
{{Technical data page}}
The '''meantone family''' is the family of [[rank-2 temperament]]s that [[tempering out|temper out]] the syntonic comma, [[81/80]], and thus can all be seen as [[extension]]s of [[meantone]].

== Meantone ==
{{Main| Meantone }}

Meantone is characterized by an [[2/1|octave]] [[period]], a [[3/2|fifth]] [[generator]], and the relationship that four fifths go to make up a [[5/1|5th harmonic]].

[[Subgroup]]: 2.3.5

[[Comma list]]: 81/80

{{Mapping|legend=1| 1 0 -4 | 0 1 4 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3906{{c}}, ~3/2 = 697.0455{{c}}
: [[error map]]: {{val| +1.391 -3.519 +1.868 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6512{{c}}
: error map: {{val| 0.000 -5.304 +0.291 }}

[[Minimax tuning]]:
* [[5-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Tuning ranges]]:
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 720.000] (4\7 to 3\5)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31, 50, 81, 131b }}

[[Badness]] (Sintel): 0.173

=== Overview to extensions ===
The second comma of the normal comma list defines which [[7-limit]] family member we are looking at.
* Flattertone adds {{monzo| -24 17 0 -1 }}, finding the [[~]][[7/4]] at the double-augmented sixth, for a tuning between 33edo and 26edo.
* Flattone adds {{monzo| -17 9 0 1 }}, finding the ~7/4 at the diminished seventh, for a tuning between 26edo and 19edo.
* Septimal meantone adds [[Harrison's comma|{{monzo| -13 10 0 -1 }}]], finding the ~7/4 at the augmented sixth, for a tuning between 19edo and 12edo.
* Dominant adds [[64/63|{{monzo| 6 -2 0 -1 }}]], finding the ~7/4 at the minor seventh, for a tuning between 12edo and 5edo.
* Sharptone adds [[28/27|{{monzo| 2 -3 0 1 }}]], finding the ~7/4 at the major sixth, for an [[exotemperament]] never exactly well-tuned, and where 5edo is the only [[diamond monotone]] tuning, with a terrible 5-limit part.
Those all have a fifth as generator.
* Injera adds {{monzo| -7 8 0 -2 }} with a half-octave period.
* Mohajira adds {{monzo| -23 11 0 2 }} and splits the fifth in two.
* Godzilla adds [[49/48|{{monzo| -4 -1 0 2 }}]] with an ~[[8/7]] generator, two of which give the [[4/3|fourth]].
* Mothra adds [[1029/1024|{{monzo| -10 1 0 3 }}]] with an ~8/7 generator, three of which give the fifth.
* Liese adds {{monzo| -9 11 0 -3 }} with a ~[[10/7]] generator, three of which give the [[3/1|twelfth]].
* Squares adds {{monzo| -3 9 0 -4 }} with a ~[[9/7]] generator, four of which give the [[8/3|eleventh]].
* Jerome adds {{monzo| 3 7 0 -5 }} and slices the fifth in five.

==== Strong extensions ====
For any meantone generator tuning between 7\12 and 11\19, the augmented sixth is sharper than the diminished seventh and flatter than the minor seventh, befitting an approximation to interval class of 7. This coincides with interpreting the tritone (~9/8)3 as [[7/5]], leading to septimal meantone, a very elegant extension to the 7-limit.

For any tuning flatter than 11\19, the augmented sixth and diminished seventh swap their orders, so the diminished seventh becomes a better approximation to the interval class of 7, resulting in flattone. Likewise, for any tuning sharper than 7\12, the minor seventh is the proper approximation instead, resulting in dominant.

Another way to extend meantone to higher limits involves decomposing the meantone comma into products of smaller commas, or expressing some other comma of interest in terms of the ratio between the meantone comma and another comma. However, this often results in [[weak extension]]s. Another opportunity given by the meantone fifth being flat is that the most obvious ways of dividing it into ''n'' parts leave the part closer to just than usual, because we can allow – and indeed want – more flatwards tempering on the fifth, so may be recommended for this reason.

==== Splitting the meantone fifth into two (243/242) ====
By tempering out [[243/242]] we equate the distance from 9/8 to 10/9 (= [[81/80|S9]]) with the distance between 11/10 to 12/11 (= [[121/120|S11]]), leading to [[mohaha]] which is in some sense thus a trivial tuning of [[rastmic]] (as 81/80 and 121/120 vanish), but an important one, as it leads to the 11/9 being a more in-tune "hemififth" than in non-meantone [[rastmic]] temperaments (which require sharper fifths in good tunings), and it has a natural extension to the full [[11-limit]] by finding [[7/4]] as the semi-diminished seventh, leading to [[mohajira]], which inflates [[64/63]] to equate it with a small quarter-tone, which is characteristic. Mohajira can also be thought of as equating a slightly sharpened [[25/16|(5/4)2]] with [[11/7]], which is also natural as meantone tempering usually has [[5/4]] slightly sharp. There is also the consideration that tempering out [[121/120]] leads to similarly high damage in the 11-limit as tempering [[81/80]] in the 5-limit, because both erase key distinctions of their respective JI subgroups.

==== Splitting the meantone fifth into three (1029/1024) ====
By tempering out [[1029/1024]] we equate the distance from 7/6 to 8/7 (= [[49/48|S7]]) with the distance from 8/7 to 9/8 (= [[64/63|S8]]), so that ([[8/7]])3 is equated with [[3/2]], because of being able to be rewritten as (9/8)(8/7)(7/6) – this observation can be generalized to define the family of [[ultraparticular]] commas. This is an unusually natural extension, with a surprising coincidence: ([[36/35]])/([[64/63]]) = [[81/80]], or using the shorthand notation, S6/S8 = S9. As S6/S8 is already tempered out, it is natural to want [[49/48]] (S7), which is bigger than S8 and smaller than S6 to be equated with both, to avoid inconsistent mappings. This has the surprising consequence of meaning that splitting the meantone fifth into three 8/7's is equivalent to splitting 8/5 into three 7/6's by tempering (8/5)/(7/6)3 = [[1728/1715]] (S6/S7), the orwellisma.

This strategy leads to the 7-limit version of [[mothra]], which is also sometimes called '''cynder'''. Though undecimal mothra is the simplest extension by a small margin, when measured in terms of generators required to reach 11, there is another extension that is perhaps more obvious, by noticing that because we have S6~S7~S8 with S9 tempered out, we can try S8~S10 by tempering out [[176/175]] (S8/S10), which is (11/7)/(5/4)2, taking advantage of 10/9 being tempered sharp in meantone so that we can distinguish 11/10 from it, thus finding 16/11 at 100/99 above the meantone diminished fifth, ([[6/5]])2 = [[36/25]] = ([[3/2]])/([[25/24]]).

==== 31edo as splitting the fifth into two, three and nine ====
[[31edo]] is unique as combining all aforementioned tempering strategies into one elegant [[11-limit]] meantone temperament; it also combines yet more extensions of meantone not discussed here, and it has a very accurate [[5/4]] and [[7/4]] and an even more accurate [[35/32]]. A tempering strategy not mentioned is splitting a flattened [[3/2]] into nine sharpened [[25/24]]'s, resulting in the 5-limit version of [[valentine]] so that 31edo is the unique tuning that combines them. Furthermore, splitting the meantone fifth into two and three in the ways described above leads to meantone + miracle without tempering out 225/224, which interestingly, though a rank-2 temperament, only has 31edo as a [[patent val]] tuning (corresponding to also tempering out 225/224).

Temperaments discussed elsewhere include
* ''[[Plutus]]'' (+15/14) → [[Very low accuracy temperaments #Plutus|Very low accuracy temperaments]]
* [[Godzilla]] (+49/48) → [[Semaphoresmic clan #Godzilla|Semaphoresmic clan]]
* [[Mothra]] (+1029/1024) → [[Gamelismic clan #Mothra|Gamelismic clan]]
* ''[[Mohaha]]'' (+121/120) → [[Rastmic clan #Mohaha|Rastmic clan]]

The rest are considered below.

== Septimal meantone ==
<div style="float:right">[[:de:septimal-mitteltönig|Deutsch]]</div>
{{Main| Meantone #Septimal meantone}}
{{Wikipedia| Septimal meantone temperament }}

In septimal meantone, ten fifths get to the interval class for 7, so that [[7/4]] is an augmented sixth (C–A♯), [[7/6]] is an augmented second (C–D♯), [[7/5]] is an augmented fourth (C–F♯), and [[21/16]] is an augmented third (C–E♯). This mapping is rationalized by the fact that 81/80 factors as ([[126/125]])⋅([[225/224]]), and septimal meantone tempers out both of these commas as well as their difference, [[3136/3125]]. In fact it can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125, 225/224, and 3136/3125.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 126/125

{{Mapping|legend=1| 1 0 -4 -13 | 0 1 4 10 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2358{{c}}, ~3/2 = 697.2122{{c}}
: [[error map]]: {{val| +1.236 -3.507 +2.535 -0.412 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 696.6562{{c}}
: error map: {{val| 0.000 -5.299 +0.311 -2.264 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~3/2 = {{monzo| 0 0 1/4 }} (1/4-comma)
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | -3 0 5/2 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 701.955] (1/3-comma to Pyth.)
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

[[Algebraic generator]]: Cybozem, the real root of 15''x''3 - 10''x''2 - 18, 503.4257 cents. The recurrence converges quickly.

{{Optimal ET sequence|legend=1| 12, 19, 31, 81, 112b, 143b }}

[[Badness]] (Sintel): 0.347

=== 2.3.5.7.23 subgroup ===
There is a reasonable extension to the 2.3.5.7.23 subgroup, which tempers out [[161/160]] and [[162/161]], and maps the [[23/16]] to the diminished fifth (C–G♭).

Subgroup: 2.3.5.7.23

Comma list: 81/80, 126/125, 161/160

Mapping: {{mapping| 1 0 -4 -13 14 | 0 1 4 10 -6 }}

Optimal tunings:
* WE: ~2 = 1201.3627{{c}}, ~3/2 = 697.2263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.5082{{c}}

{{Optimal ET sequence|legend=0| 12, 19, 31, 81, 112bi }}

Badness (Sintel): 0.451

=== Undecimal meantone (huygens) ===
{{Redirect|Huygens|the Dutch mathematician, physicist and astronomer|Wikipedia: Christiaan Huygens}}
{{See also| Huygens vs meanpop }}

Undecimal meantone<ref name="meantone & meanpop 2003">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6048.html#6052 Yahoo! Tuning Group | ''good 11-limit meantones'']</ref> a.k.a. huygens<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref name="meantone & meanpop 2004">[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref> maps the [[11/8]] to the double-augmented third (C–E𝄪). See [[chords of huygens]] for a list of dyadic chords in this temperament.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 | 0 1 4 10 18 }}

Optimal tunings:
* WE: ~2 = 1200.7636{{c}}, ~3/2 = 697.4122{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.0315{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/16 -1/8 0 0 1/16 }}, {{monzo| 9/4 -1/2 0 0 1/4 }}, {{monzo| 21/8 -5/4 0 0 5/8 }}, {{monzo| 25/8 -9/4 0 0 9/8 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [696.774, 700.000] (18\31 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Traverse, the positive real root of ''x''4 + 2''x'' - 13, or 696.9529 cents.

{{Optimal ET sequence|legend=0| 12, 19e, 31, 105, 136b }}

Badness (Sintel): 0.563

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-74-edo.mp3 ''Twinkle canon – 74 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]

==== Grosstone ====
Grosstone, named for tempering out the [[grossma]], is the main extension of interest that extends undecimal meantone to the 13-limit. It maps 13/8 to the double-diminished seventh (C–B♭♭♭). Note also that 11/10 is a double-augmented unison; 12/11~13/12 is a double-diminished third; and 14/13 is a triple-augmented seventh octave reduced. Grosstone is flexible with its tunings; among the good tunings are [[31edo]], [[43edo]], and [[74edo]].

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 | 0 1 4 10 18 -16 }}

Optimal tunings:
* WE: ~2 = 1199.9389{{c}}, ~3/2 = 697.2282{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.2627{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 8/13 0 0 1/26 0 -1/26 }}
: eigenmonzo basis (unchanged-interval basis): 2.13/7

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [696.774, 697.674] (18\31 to 25\43)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

{{Optimal ET sequence|legend=0| 12, 31, 43, 74 }}

Badness (Sintel): 1.07

===== 17-limit =====
This extension maps 17/16 to the minor second (C–D♭), and 19/16 to the minor third (C–E♭), suitable for a system generated by a mildly tempered fifth.

Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 12 | 0 1 4 10 18 -16 -5 }}

Optimal tunings:
* WE: ~2 = 1199.5811{{c}}, ~3/2 = 697.0918{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3303{{c}}

{{Optimal ET sequence|legend=0| 12, 31, 43, 74g }}

Badness (Sintel): 1.06

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 96/95, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 -25 29 12 9 | 0 1 4 10 18 -16 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.2931{{c}}, ~3/2 = 696.9690{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.3736{{c}}

{{Optimal ET sequence|legend=0| 12, 31, 43, 74gh }}

Badness (Sintel): 1.07

==== Fokkertone ====
Fokkertone maps the [[13/8]] to the double-augmented fifth (C–G𝄪). Note that the minor third conflates 13/11 with 6/5, and that 11/10~13/12 is a double-augmented unison; 12/11 is a double-diminished third; and 14/13 is a minor second. 31edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.

This extension used to be known as ''tridecimal meantone'', but was decanonicalized in 2025.

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 105/104

Mapping: {{mapping| 1 0 -4 -13 -25 -20 | 0 1 4 10 18 15 }}

Optimal tunings:
* WE: ~2 = 1200.8149{{c}}, ~3/2 = 697.1155{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.7085{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 9/16 -1/8 0 0 1/16 }}
: unchanged-interval (eigenmonzo) basis: 2.11/9

{{Optimal ET sequence|legend=0| 12f, 19e, 31 }}

Badness (Sintel): 0.746

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 99/98, 105/104, 120/119

Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 | 0 1 4 10 18 15 -5 }}

Optimal tunings:
* WE: ~2 = 1199.5548{{c}}, ~3/2 = 696.7449{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.9823{{c}}

{{Optimal ET sequence|legend=0| 12f, 31 }}

Badness (Sintel): 1.02

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 81/80, 96/95, 99/98, 105/104, 120/119

Mapping: {{mapping| 1 0 -4 -13 -25 -20 12 9 | 0 1 4 10 18 15 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.0408{{c}}, ~3/2 = 696.5824{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.1061{{c}}

{{Optimal ET sequence|legend=0| 12f, 31 }}

Badness (Sintel): 1.10

==== Meridetone ====
Meridetone maps the 13/8 to the quadruple-augmented fourth (C–F𝄪𝄪). 43edo can be recommended as a tuning since it is the only 13-odd-limit diamond monotone tuning.

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 -39 | 0 1 4 10 18 27 }}

Optimal tunings:
* WE: ~2 = 1199.9122{{c}}, ~3/2 = 697.4779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.5241{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/25 -2/25 0 0 0 1/25 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9

{{Optimal ET sequence|legend=0| 12f, 31f, 43 }}

Badness (Sintel): 1.09

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 120/119, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 | 0 1 4 10 18 27 -5 }}

Optimal tunings:
* WE: ~2 = 1199.3793{{c}}, ~3/2 = 697.2833{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6222{{c}}

{{Optimal ET sequence|legend=0| 12f, 43 }}

Badness (Sintel): 1.22

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 96/95, 99/98, 120/119, 126/125

Mapping: {{mapping| 1 0 -4 -13 -25 -39 12 9 | 0 1 4 10 18 27 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.0260{{c}}, ~3/2 = 697.1486{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 697.6887{{c}}

{{Optimal ET sequence|legend=0| 12f, 43 }}

Badness (Sintel): 1.25

==== Hemimeantone ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 169/168

Mapping: {{mapping| 1 0 -4 -13 -25 -5 | 0 2 8 20 36 11 }}

: mapping generators: ~2, ~26/15

Optimal tunings:
* WE: ~2 = 1201.0387{{c}}, ~26/15 = 949.2863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5065{{c}}

{{Optimal ET sequence|legend=0| 19e, 43, 62 }}

Badness (Sintel): 1.30

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 169/168, 221/220

Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 | 0 2 8 20 36 11 33 }}

Optimal tunings:
* WE: ~2 = 1201.0270{{c}}, ~26/15 = 949.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 948.5169{{c}}

{{Optimal ET sequence|legend=0| 19eg, 43, 62 }}

Badness (Sintel): 1.19

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 169/168, 221/220

Mapping: {{mapping| 1 0 -4 -13 -25 -5 -22 -25 | 0 2 8 20 36 11 33 37 }}

Optimal tunings:
* WE: ~2 = 1201.0339{{c}}, ~19/11 = 949.2902{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~19/11 = 948.5111{{c}}

{{Optimal ET sequence|legend=0| 19egh, 43, 62 }}

Badness (Sintel): 1.15

==== Semimeantone ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 126/125, 847/845

Mapping: {{mapping| 2 0 -8 -26 -50 -59 | 0 1 4 10 18 21 }}

: mapping generators: ~55/39, ~3

Optimal tunings:
* WE: ~55/39 = 600.3606{{c}}, ~3/2 = 697.4241{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 697.0545{{c}}

{{Optimal ET sequence|legend=0| 12f, …, 50eff, 62, 136b }}

Badness (Sintel): 1.68

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 126/125, 221/220, 289/288

Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 | 0 1 4 10 18 21 1 }}

Optimal tunings:
* WE: ~17/12 = 600.5426{{c}}, ~3/2 = 697.5571{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9858{{c}}

{{Optimal ET sequence|legend=0| 12f, 50eff, 62, 136bg }}

Badness (Sintel): 1.60

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 126/125, 153/152, 209/208, 221/220

Mapping: {{mapping| 2 0 -8 -26 -50 -59 5 -1 | 0 1 4 10 18 21 1 3 }}

Optimal tunings:
* WE: ~17/12 = 600.5959{{c}}, ~3/2 = 697.5985{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.9638{{c}}

{{Optimal ET sequence|legend=0| 12f, 50eff, 62 }}

Badness (Sintel): 1.47

=== Meanpop ===
{{See also| Huygens vs meanpop }}

Meanpop<ref name="meantone & meanpop 2003"/><ref name="meantone & meanpop 2004"/> maps the 11/8 to the double-diminished fifth (C–G𝄫), and tridecimal meanpop maps the 13/8 to the double-augmented fifth (C–G𝄪), tempering out 144/143 like in grosstone. Note also 11/10 is a double-diminished third; 12/11~13/12, double-augmented unison; and 14/13, minor second.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 385/384

Mapping: {{mapping| 1 0 -4 -13 24 | 0 1 4 10 -13 }}

: mapping generator: ~2, ~3

Optimal tunings:
* WE: ~2 = 1201.3464{{c}}, ~3/2 = 697.2159{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4509{{c}}

Minimax tuning:
* 11-odd-limit: ~3/2 = {{monzo| 0 0 1/4 }}
: projection map: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| -3 0 5/2 0 0 }}, {{monzo| 11 0 -13/4 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3''x''3 + 6''x'' - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112b }}

Badness (Sintel): 0.712

; Music
* [http://soonlabel.com/xenharmonic/archives/607 Scott Joplin's "The Entertainer" tuned into meanpop]{{dead link}}
* [http://micro.soonlabel.com/gene_ward_smith/Others/Meneghin/Claudi-Meneghin-Twinkle-canon-50-edo.mp3 ''Twinkle canon – 50 edo''] by [http://soonlabel.com/xenharmonic/archives/573 Claudi Meneghin]

==== 2.3.5.7.11.23 subgroup ====
It is very reasonable to extend meanpop to the 2.3.5.7.11.23 subgroup. In this temperament, [[21/20]], [[23/22]], [[24/23]], [[25/24]] and [[28/27]] are mapped to the augmented unison.

Subgroup: 2.3.5.7.11.23

Comma list: 81/80, 126/125, 161/160, 231/230

Mapping: {{mapping| 1 0 -4 -13 24 14 | 0 1 4 10 -13 -6 }}

Optimal tunings:
* WE: ~2 = 1201.3759{{c}}, ~3/2 = 697.2251{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4318{{c}}

{{Optimal ET sequence|legend=0| 12e, 19, 31, 81, 112bi }}

Badness (Sintel): 0.615

==== Tridecimal meanpop ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 24 -20 | 0 1 4 10 -13 15 }}

Optimal tunings:
* WE: ~2 = 1201.0765{{c}}, ~3/2 = 696.8361{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2347{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~3/2 = {{monzo| 4/7 0 0 0 -1/28 1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/11

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [694.737, 696.774] (11\19 to 18\31)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955] (1/2-comma to Pyth.)

{{Optimal ET sequence|legend=0| 19, 31, 50, 81 }}

Badness (Sintel): 0.863

===== Meanpoppic =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 144/143, 273/272

Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 | 0 1 4 10 -13 15 26 }}

Optimal tunings:
* WE: ~2 = 1201.0727{{c}}, ~3/2 = 696.8168{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2195{{c}}

{{Optimal ET sequence|legend=0| 19g, 31, 50, 81, 131bd }}

Badness (Sintel): 1.02

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 144/143, 153/152, 273/272

Mapping: {{mapping| 1 0 -4 -13 24 -20 -37 -40 | 0 1 4 10 -13 15 26 28 }}

Optimal tunings:
* WE: ~2 = 1201.0719{{c}}, ~3/2 = 696.8101{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2137{{c}}

{{Optimal ET sequence|legend=0| 19gh, 31, 50, 81 }}

Badness (Sintel): 1.08

===== Meanpoid =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 0 -4 -13 24 -20 12 | 0 1 4 10 -13 15 -5 }}

Optimal tunings:
* WE: ~2 = 1200.2768{{c}}, ~3/2 = 696.5683{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4114{{c}}

{{Optimal ET sequence|legend=0| 19, 31 }}

Badness (Sintel): 1.17

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 96/95, 105/104, 120/119, 126/125

Mapping: {{mapping| 1 0 -4 -13 24 -20 12 9 | 0 1 4 10 -13 15 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.7905{{c}}, ~3/2 = 696.3779{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4973{{c}}

{{Optimal ET sequence|legend=0| 19, 31 }}

Badness (Sintel): 1.25

==== Semimeanpop ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 385/384, 847/845

Mapping: {{mapping| 2 0 -8 -26 48 39 | 0 1 4 10 -13 -10 }}

: mapping generators: ~55/39, ~3

Optimal tunings:
* WE: ~55/39 = 600.6704{{c}}, ~3/2 = 697.2151{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.4341{{c}}

{{Optimal ET sequence|legend=0| 12e, 50, 62, 112b }}

Badness (Sintel): 1.78

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 126/125, 221/220, 273/272, 289/288

Mapping: {{mapping| 2 0 -8 -26 48 39 5 | 0 1 4 10 -13 -10 1 }}

Optimal tunings:
* WE: ~17/12 = 600.7232{{c}}, ~3/2 = 697.2820{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4411{{c}}

{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bg }}

Badness (Sintel): 1.45

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 126/125, 153/152, 209/208, 221/220, 273/272

Mapping: {{mapping| 2 0 -8 -26 48 39 5 -1 | 0 1 4 10 -13 -10 1 3 }}

Optimal tunings:
* WE: ~17/12 = 600.7527{{c}}, ~3/2 = 697.3244{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 696.4525{{c}}

{{Optimal ET sequence|legend=0| 12e, 50, 62, 112bgh }}

Badness (Sintel): 1.28

=== Meanenneadecal ===
Meanenneadecal maps the 11/8 to the augmented fourth (C–F♯), and tridecimal meanenneadecal maps the 13/8 to the double-augmented fifth (C–G𝄪). Note also 11/10 is a major second; 12/11~14/13, minor second; and 13/12, double-augmented unison.

Subgroup: 2.3.5.7.11

Comma list: 45/44, 56/55, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 | 0 1 4 10 6 }}

Optimal tunings:
* WE: ~2 = 1199.6946{{c}}, ~3/2 = 696.0729{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.2083{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [694.737, 700.000] (11\19 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 704.377]

{{Optimal ET sequence|legend=0| 7d, 12, 19, 31e }}

Badness (Sintel): 0.708

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 78/77, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 -20 | 0 1 4 10 6 15 }}

Optimal tunings:
* WE: ~2 = 1199.7931{{c}}, ~3/2 = 696.0258{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1241{{c}}

{{Optimal ET sequence|legend=0| 7df, 12f, 19, 31e }}

Badness (Sintel): 0.875

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 56/55, 78/77, 81/80, 120/119

Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 | 0 1 4 10 6 15 -5 }}

Optimal tunings:
* WE: ~2 = 1198.6665{{c}}, ~3/2 = 695.8010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.4998{{c}}

{{Optimal ET sequence|legend=0| 12f, 19, 31e }}

Badness (Sintel): 1.17

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 56/55, 78/77, 81/80, 96/95, 120/119

Mapping: {{mapping| 1 0 -4 -13 -6 -20 12 9 | 0 1 4 10 6 15 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1198.2880{{c}}, ~3/2 = 695.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.6370{{c}}

{{Optimal ET sequence|legend=0| 12f, 19, 31e }}

Badness (Sintel): 1.23

==== Vincenzo ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 | 0 1 4 10 6 -4 }}

Optimal tunings:
* WE: ~2 = 1202.1684{{c}}, ~3/2 = 696.3160{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.2045{{c}}

{{Optimal ET sequence|legend=0| 7d, 12, 19 }}

Badness (Sintel): 1.02

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 12 | 0 1 4 10 6 -4 -5 }}

Optimal tunings:
* WE: ~2 = 1200.5137{{c}}, ~3/2 = 696.1561{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 695.8771{{c}}

{{Optimal ET sequence|legend=0| 12, 19 }}

Badness (Sintel): 1.30

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80

Mapping: {{mapping| 1 0 -4 -13 -6 10 12 9 | 0 1 4 10 6 -4 -5 -3 }}

Optimal tunings:
* WE: ~2 = 1199.8261{{c}}, ~3/2 = 696.0298{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 696.1262{{c}}

{{Optimal ET sequence|legend=0| 12, 19 }}

Badness (Sintel): 1.36

=== Bimeantone ===
11/8 is mapped to half octave minus the [[128/125|meantone diesis]].

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 245/242

Mapping: {{mapping| 2 0 -8 -26 -31 | 0 1 4 10 12 }}

: mapping generators: ~63/44, ~3

Optimal tunings:
* WE: ~63/44 = 600.7492{{c}}, ~3/2 = 696.8853{{c}}
* CWE: ~63/44 = 600.0000{{c}}, ~3/2 = 696.1908{{c}}

{{Optimal ET sequence|legend=0| 12, 26de, 38d, 50 }}

Badness (Sintel): 1.26

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125, 245/242

Mapping: {{mapping| 2 0 -8 -26 -31 -40 | 0 1 4 10 12 15 }}

Optimal tunings:
* WE: ~55/39 = 600.8309{{c}}, ~3/2 = 696.8000{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 696.0066{{c}}

{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}

Badness (Sintel): 1.19

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 126/125, 189/187, 221/220

Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 | 0 1 4 10 12 15 1 }}

Optimal tunings:
* WE: ~17/12 = 600.9234{{c}}, ~3/2 = 696.8536{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.9317{{c}}

{{Optimal ET sequence|legend=0| 12f, 38df, 50 }}

Badness (Sintel): 1.15

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 126/125, 153/152, 189/187, 221/220

Mapping: {{mapping| 2 0 -8 -26 -31 -40 5 -1 | 0 1 4 10 12 15 1 3 }}

Optimal tunings:
* WE: ~17/12 = 600.9845{{c}}, ~3/2 = 696.8939{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 695.8947{{c}}

{{Optimal ET sequence|legend=0| 12f, 26deff, 38df, 50 }}

Badness (Sintel): 1.08

=== Trimean ===
{{See also| No-sevens subgroup temperaments #Superpine }}

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125, 1344/1331

Mapping: {{mapping| 1 2 4 7 5 | 0 -3 -12 -30 -11 }}

: mapping generators: ~2, ~11/10

Optimal tunings:
* WE: ~2 = 1200.7155{{c}}, ~11/10 = 167.9055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7749{{c}}

{{Optimal ET sequence|legend=0| 7d, 36d, 43, 50, 93 }}

Badness (Sintel): 1.68

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 126/125, 144/143, 364/363

Mapping: {{mapping| 1 2 4 7 5 3 | 0 -3 -12 -30 -11 5 }}

Optimal tunings:
* WE: ~2 = 1200.6104{{c}}, ~11/10 = 167.8749{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7728{{c}}

{{Optimal ET sequence|legend=0| 7d, 43, 50, 93 }}

Badness (Sintel): 1.46

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 126/125, 144/143, 189/187, 221/220

Mapping: {{mapping| 1 2 4 7 5 3 8 | 0 -3 -12 -30 -11 5 -28 }}

Optimal tunings:
* WE: ~2 = 1200.6144{{c}}, ~11/10 = 167.8716{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.7682{{c}}

{{Optimal ET sequence|legend=0| 7dg, 43, 50, 93 }}

Badness (Sintel): 1.28

=== Migration ===
See [[Rastmic clan #Migration|Rastmic clan]].

== Flattone ==
{{Main| Flattone }}

In flattone, 9 fourths get to the interval class for 7, so that [[7/4]] is a diminished seventh (C–B𝄫), [[7/6]] is a diminished third (C–E𝄫), and [[7/5]] is a double-diminished fifth (C–G𝄫). In general, septimal subminor intervals are diminished and septimal supermajor intervals are augmented, which makes it quite easy to learn flattone notation. The fifth in flattone is typically flatter than that of [[19edo]]. Good tunings for flattone include [[45edo]], [[64edo]], and [[71edo]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 525/512

{{Mapping|legend=1| 1 0 -4 17 | 0 1 4 -9 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.6308{{c}}, ~3/2 = 695.8782{{c}}
: [[error map]]: {{val| +3.631 -2.446 -2.801 -2.684 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.7334{{c}}
: error map: {{val| 0.000 -8.222 -11.380 -12.426 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~3/2 = {{monzo| 8/13 0 1/13 -1/13 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 21/13 0 1/13 -1/13 }}, {{monzo| 32/13 0 4/13 -4/13 }}, {{monzo| 32/13 0 -9/13 9/13 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 6/11 2/11 0 -1/11 }}
: [[projection map]]: [{{monzo| 1 0 0 0 }}, {{monzo| 17/11 2/11 0 -1/11 }}, {{monzo| 24/11 8/11 0 -4/11 }}, {{monzo| 34/11 -18/11 0 9/11 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [692.353, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 701.955]

[[Algebraic generator]]: Squarto, the positive root of 8''x''2 - 4''x'' - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.

{{Optimal ET sequence|legend=1| 7, 19, 26, 45 }}

[[Badness]] (Sintel): 0.976

=== 11-limit ===
This can also be considered a no-sevens temperament: [[#Hypnotone|hypnotone]].

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 385/384

Mapping: {{mapping| 1 0 -4 17 -6 | 0 1 4 -9 6 }}

Optimal tuning:
* WE: ~2 = 1202.3247{{c}}, ~3/2 = 694.4688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.1467{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]

{{Optimal ET sequence|legend=0| 7, 19, 26, 45, 71bc, 116bcde }}

Badness (Sintel): 1.12

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 65/64, 78/77, 81/80

Mapping: {{mapping| 1 0 -4 17 -6 10 | 0 1 4 -9 6 -4 }}

Optimal tunings:
* WE: ~2 = 1202.5156{{c}}, ~3/2 = 694.5107{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0538{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [692.308, 694.737] (15\26 to 11\19)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.502, 701.955]

{{Optimal ET sequence|legend=0| 7, 19, 26, 45f, 71bcf, 116bcdef }}

Badness (Sintel): 0.920

=== Ptolemy ===
See [[Rastmic clan #Ptolemy|Rastmic clan]].

== Dominant ==
{{Main| Dominant (temperament) }}
{{See also| Archytas clan }}

The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh (C–Bb). The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure [[3/2]] fifths, and with [[29edo]], [[41edo]], or [[53edo]].

Because dominant entails a near-pure perfect fifth, a small number of generators will not land on an interval close to prime 11. The canonical 11-limit extension identifies 11/8 with the diminished fifth. Domination tempers out 77/75 and identifies 11/8 with the augmented third. Domineering identifies 11/8 with the augmented fourth, which is a very inaccurate mapping; it is however, notable for having the lowest badness among the extensions. Arnold tempers out 33/32 and identifies 11/8 with the perfect fourth. None of them are nearly as good as the weak extension [[neutrominant]], splitting the fifth as well as the chromatic semitone in two like in all [[rastmic clan|rastmic]] temperaments.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 36/35, 64/63

{{Mapping|legend=1| 1 0 -4 6 | 0 1 4 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.3384{{c}}, ~3/2 = 698.8478{{c}}
: [[error map]]: {{val| -4.662 -7.769 +9.077 +14.832 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.1125{{c}}
: error map: {{val| 0.000 -0.842 +18.136 +28.949 }}

[[Tuning ranges]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]] [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 3\5)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [694.786, 715.587]
* 9-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

{{Optimal ET sequence|legend=1| 5, 7, 12, 41cd, 53cdd, 65ccddd }}

[[Badness]] (Sintel): 0.524

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 64/63

Mapping: {{mapping| 1 0 -4 6 13 | 0 1 4 -2 -6 }}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
* 11-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

Optimal tunings:
* WE: ~2 = 1194.0169{{c}}, ~3/2 = 699.7473{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.2672{{c}}

{{Optimal ET sequence|legend=0| 5, 12, 17c, 29cde }}

Badness (Sintel): 0.799

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 36/35, 56/55, 64/63, 66/65

Mapping: {{mapping| 1 0 -4 6 13 18 | 0 1 4 -2 -6 -9 }}

Optimal tunings:
* WE: ~2 = 1193.8055{{c}}, ~3/2 = 700.0042{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 703.8254{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = 705.882 (10\17)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [691.202, 715.587]

{{Optimal ET sequence|legend=0| 12f, 17c, 29cdef }}

Badness (Sintel): 0.996

==== Dominion ====
Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 56/55, 64/63

Mapping: {{mapping| 1 0 -4 6 13 -9 | 0 1 4 -2 -6 8 }}

Optimal tunings:
* WE: ~2 = 1195.0293{{c}}, ~3/2 = 701.9847{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7698{{c}}

{{Optimal ET sequence|legend=0| 5, 12, 17c }}

Badness (Sintel): 1.13

=== Domination ===
Subgroup: 2.3.5.7.11

Comma list: 36/35, 64/63, 77/75

Mapping: {{mapping| 1 0 -4 6 -14 | 0 1 4 -2 11 }}

Optimal tunings:
* WE: ~2 = 1194.8645{{c}}, ~3/2 = 701.9872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.5945{{c}}

{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}

Badness (Sintel): 1.21

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 64/63, 66/65

Mapping: {{mapping| 1 0 -4 6 -14 -9 | 0 1 4 -2 11 8 }}

Optimal tunings:
* WE: ~2 = 1195.1324{{c}}, ~3/2 = 702.6343{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 705.0791{{c}}

{{Optimal ET sequence|legend=0| 5e, 12e, 17c }}

Badness (Sintel): 1.13

=== Domineering ===
Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 64/63

Mapping: {{mapping| 1 0 -4 6 -6 | 0 1 4 -2 6 }}

Optimal tunings:
* WE: ~2 = 1194.7102{{c}}, ~3/2 = 695.6962{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1765{{c}}

{{Optimal ET sequence|legend=0| 5e, 7, 12 }}

Badness (Sintel): 0.727

=== Arnold ===
Subgroup: 2.3.5.7.11

Comma list: 22/21, 33/32, 36/35

Mapping: {{mapping| 1 0 -4 6 5 | 0 1 4 -2 -1 }}

Optimal tunings:
* WE: ~2 = 1199.8507{{c}}, ~3/2 = 698.4045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.4822{{c}}

{{Optimal ET sequence|legend=0| 5, 7, 12e }}

Badness (Sintel): 0.864

=== Neutrominant ===
See [[Rastmic clan #Neutrominant|Rastmic clan]].

== Flattertone ==
In flattertone, 17 fifths get to the interval class for 7, so that [[7/4]] is a double-augmented sixth (C–Ax). The fifth in flattertone is typically at least as flat as [[26edo]]. Here, 26edo and [[33edo|33cd-edo]] are the two primary flattertone tunings. [[1/2-comma meantone]] is also encompassed within flattertone's range. Any flatter than this, the meantone mapping for 5/4 is too inaccurate (it becomes more of a [[16/13]] or [[27/22]]), and [[deeptone]] temperament's mapping is more logical.

Flattertone was named by [[Flora Canou]] in 2024.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 1875/1792

{{Mapping|legend=1| 1 0 -4 -24 | 0 1 4 17 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.4511{{c}}, ~3/2 = 694.3258{{c}}
: [[error map]]: {{val| +4.451 -3.178 -9.011 +3.554 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 692.0479{{c}}
: error map: {{val| 0.000 -9.907 -18.122 -4.012 }}

{{Optimal ET sequence|legend=1| 7d, 19d, 26, 59bcd, 85bccd }}

[[Badness]] (Sintel): 2.43

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1375/1344

Mapping: {{mapping| 1 0 -4 -24 -6 | 0 1 4 17 6 }}

Optimal tunings:
* WE: ~2 = 1203.4653{{c}}, ~3/2 = 693.8144{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 692.0422{{c}}

{{Optimal ET sequence|legend=0| 7d, 19d, 26 }}

Badness (Sintel): 1.53

; Music
* [https://youtu.be/scCuGXnj5IY ''Music in 33EDO (33-Tone Equal Temperament) - Feb 2024''] by [[Budjarn Lambeth]] (2024)

== Sharptone ==
Sharptone is a low-accuracy temperament tempering out [[21/20]] and [[28/27]]. In sharptone, 7/4 is a major sixth, 7/6 a whole tone, and 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val.

However, while 12edo ends up near-optimal, the only valid [[diamond monotone]] tuning for sharptone is [[5edo]]. Anything flat of it has ~12/7 and ~7/4 in the wrong order (and so should be dominant) and anything sharp of it has ~5/4 and ~4/3 in the wrong order (and so should not be meantone).

The 11-limit extension was named by Gene Ward Smith in 2004<ref name="meantone & meanpop 2004"/>.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 21/20, 28/27

{{Mapping|legend=1| 1 0 -4 -2 | 0 1 4 3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1204.2961{{c}}, ~3/2 = 702.6463{{c}}
: [[error map]]: {{val| +4.296 +4.987 +24.271 -56.591 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4928{{c}}
: error map: {{val| 0.000 -0.462 +19.657 -64.347 }}

{{Optimal ET sequence|legend=1| 5, 7d, 12d }}

[[Badness]] (Sintel): 0.629

=== Meanertone ===
Subgroup: 2.3.5.7.11

Comma list: 21/20, 28/27, 33/32

Mapping: {{mapping| 1 0 -4 -2 5 | 0 1 4 3 -1 }}

Optimal tunings:
* WE: ~2 = 1208.5304{{c}}, ~3/2 = 701.5669{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 698.1117{{c}}

{{Optimal ET sequence|legend=0| 5, 7d, 12de }}

Badness (Sintel): 0.832

== Mildtone ==
Mildtone tempers out [[16128/15625]] and finds the interval class of 7 at 22 generators up, as a triple-augmented fifth (C–G#x). [[55edo]] and [[67edo]] are among the possible tunings.

Mildtone was named by [[User: Lucius Chiaraviglio|Lucius Chiaraviglio]] in 2024.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 16128/15625

{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7304{{c}}, ~3/2 = 698.3953{{c}}
: [[error map]]: {{val| -0.270 -3.829 +7.267 -1.434 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.5397{{c}}
: error map: {{val| 0.000 -3.415 +7.845 -0.952 }}

{{Optimal ET sequence|legend=1| 12, 43d, 55, 67 }}

[[Badness]] (Sintel): 2.67

=== 11-limit ===

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 81/80, 176/175, 7058/6875

{{Mapping|legend=1| 1 0 -4 -32 | 0 1 4 22 30}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.816{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.455{{c}}

{{Optimal ET sequence|legend=1| 12, 43de, 55, 67 }}

[[Badness]] (Sintel): 2.15

=== 13-limit ===

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 81/80, 176/175, 196/195, 832/825

{{Mapping|legend=1| 1 0 -4 -32 -44 | 0 1 4 22 30}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.788{{c}}, ~3/2 = 698.355{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.471{{c}}

{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}

[[Badness]] (Sintel): 2.04

=== 17-limit ===

[[Subgroup]]: 2.3.5.7.11.13.17

[[Comma list]]: 81/80, 176/175, 189/197, 196/195, 832/825

{{Mapping|legend=1| 1 0 -4 -32 -44 12| 0 1 4 22 30 -5}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.655{{c}}, ~3/2 = 698.295{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.488{{c}}

{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}

[[Badness]] (Sintel): 1.98

=== 19-limit ===

[[Subgroup]]: 2.3.5.7.11.13.19

[[Comma list]]: 81/80, 96/95, 176/175, 189/187, 196/195, 832/825

{{Mapping|legend=1| 1 0 -4 -32 -44 12 9| 0 1 4 22 30 -5 -3}}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.371{{c}}, ~3/2 = 698.164{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 698.519{{c}}

{{Optimal ET sequence|legend=1| 12f, 55f, 67 }}

[[Badness]] (Sintel): 1.95

{{Todo|unify precision|review}}

== Supermean ==
Supermean tempers out 672/625 and finds the interval class of 7 at 15 generators up, as a double-augmented fifth (C–Gx). As such, it extends [[leapfrog]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 672/625

{{Mapping|legend=1| 1 0 -4 -21 | 0 1 4 15 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.4372{{c}}, ~3/2 = 702.2086{{c}}
: [[error map]]: {{val| -4.563 -4.309 +22.521 -8.319 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5375{{c}}
: error map: {{val| 0.000 +2.583 +31.836 -0.763 }}

{{Optimal ET sequence|legend=1| 5d, 12d, 17c }}

[[Badness]] (Sintel): 3.40

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 132/125

Mapping: {{mapping| 1 0 -4 -21 -14 | 0 1 4 15 11 }}

Optimal tunings:
* WE: ~2 = 1195.7270{{c}}, ~3/2 = 702.5848{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7471{{c}}

{{Optimal ET sequence|legend=0| 5de, 12de, 17c }}

Badness (Sintel): 2.09

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 56/55, 66/65, 81/80

Mapping: {{mapping| 1 0 -4 -21 -14 -9 | 0 1 4 15 11 8 }}

Optimal tunings:
* WE: ~2 = 1196.3958{{c}}, ~3/2 = 702.9766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7940{{c}}

{{Optimal ET sequence|legend=0| 5de, 12de, 17c, 29c }}

Badness (Sintel): 1.67

== Mohajira ==
{{Main| Mohajira }}

Mohajira can be viewed as derived from [[mohaha]] which maps the interval half a [[chromatic semitone|chroma]] flat of the minor seventh to ~7/4 so that 7/4 is mapped to a semidiminished seventh (C–Bdb), although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the [[porwell comma]]. It can be described as {{nowrap| 24 & 31 }}; its ploidacot is dicot. [[31edo]] makes for an excellent mohajira tuning, with generator 9\31. Note that while 24 + 31 = [[55edo]] doesn't apear in the optimal ET sequence, it is a [[patent val]] tuning and recommendable if you prefer a light meantone tempering.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 6144/6125

{{Mapping|legend=1| 1 1 0 6 | 0 2 8 -11 }}

: mapping generators: ~2, ~128/105

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8160{{c}}, ~128/105 = 348.6518{{c}}
: [[error map]]: {{val| +0.816 -3.835 +2.901 +0.900 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 348.4194{{c}}
: error map: {{val| 0.000 -5.116 +1.041 -1.439 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~128/105 = {{monzo| 0 0 1/8 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 6 0 -11/8 0 }}
: [[eigenmonzo basis|Unchanged-interval (eigenmonzo) basis]]: 2.5

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~128/105 = [347.368, 350.000] (11\38 to 7\24)
* 7-odd-limit [[diamond tradeoff]]: ~128/105 = [347.393, 350.978]
* 9-odd-limit diamond tradeoff: ~128/105 = [345.601, 350.978]

[[Algebraic generator]]: Mohabis, real root of 3''x''3 - 3''x''2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.

{{Optimal ET sequence|legend=1| 7, 24, 31 }}

[[Badness]] (Sintel): 1.41

Scales: [[mohaha7]], [[mohaha10]]

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 121/120, 176/175

Mapping: {{mapping| 1 1 0 6 2 | 0 2 8 -11 5 }}

Optimal tunings:
* WE: ~2 = 1201.1562{{c}}, ~11/9 = 348.8124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.4910{{c}}

Minimax tuning:
* 11-odd-limit: ~11/9 = {{monzo| 0 0 1/8 }}
: projection map: [{{Monzo| 1 0 0 0 0 }}, {{monzo| 1 0 1/4 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 6 0 -11/8 0 0 }}, {{monzo| 2 0 5/8 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5

Tuning ranges:
* 11-odd-limit diamond monotone: ~11/9 = [348.387, 350.000] (9\31 to 7\24)
* 11-odd-limit diamond tradeoff: ~11/9 = [344.999, 350.978]

{{Optimal ET sequence|legend=0| 7, 24, 31 }}

Badness (Sintel): 0.862

Scales: [[mohaha7]], [[mohaha10]]

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 105/104, 121/120

Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 8 -11 5 -1 }}

Optimal tunings:
* WE: ~2 = 1200.4256{{c}}, ~11/9 = 348.6819{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.5622{{c}}

{{Optimal ET sequence|legend=0| 7, 24, 31 }}

Badness (Sintel): 0.966

Scales: [[mohaha7]], [[mohaha10]]

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 66/65, 81/80, 105/104, 121/120, 154/153

Mapping: {{mapping| 1 1 0 6 2 4 7 | 0 2 8 -11 5 -1 -10 }}

Optimal tunings:
* WE: ~2 = 1200.0382{{c}}, ~11/9 = 348.7471{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.7360{{c}}

{{Optimal ET sequence|legend=0| 7, 24, 31 }}

Badness (Sintel): 1.05

Scales: [[mohaha7]], [[mohaha10]]

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

Mapping: {{mapping| 1 1 0 6 2 4 7 6 | 0 2 8 -11 5 -1 -10 -6 }}

Optimal tunings:
* WE: ~2 = 1199.7469{{c}}, ~11/9 = 348.7367{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 348.8117{{c}}

{{Optimal ET sequence|legend=0| 7, 24, 31, 55 }}

Badness (Sintel): 1.05

Scales: [[mohaha7]], [[mohaha10]]

== Mohamaq ==
Mohamaq is a lower-accuracy alternative to mohajira that favors tunings sharp of 24edo. It may be described as {{nowrap| 17c & 24 }}; its ploidacot is dicot, the same as mohajira.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 392/375

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

: mapping generators: ~2, ~25/21

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0661{{c}}, ~25/21 = 350.3127{{c}}
: [[error map]]: {{val| -0.934 -2.264 +16.188 -13.827 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/21 = 350.4856{{c}}
: error map: {{val| 0.000 -0.984 +17.571 -12.513 }}

{{Optimal ET sequence|legend=1| 7d, 17c, 24 }}

[[Badness]] (Sintel): 1.97

Scales: [[mohaha7]], [[mohaha10]]

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 77/75, 243/242

Mapping: {{mapping| 1 1 0 -1 2 | 0 2 8 13 5 }}

Optimal tunings:
* WE: ~2 = 1199.1924{{c}}, ~11/9 = 350.3286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.4821{{c}}

{{Optimal ET sequence|legend=0| 7d, 17c, 24 }}

Badness (Sintel): 1.20

Scales: [[mohaha7]], [[mohaha10]]

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 77/75, 243/242

Mapping: {{mapping| 1 1 0 -1 2 4 | 0 2 8 13 5 -1 }}

Optimal tunings:
* WE: ~2 = 1198.5986{{c}}, ~11/9 = 350.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.6459{{c}}

{{Optimal ET sequence|legend=0| 7d, 17c, 24, 41c }}

Badness (Sintel): 1.19

Scales: [[mohaha7]], [[mohaha10]]

== Liese ==
[[:de:Liese|Deutsch]]

Liese splits the [[3/1|perfect twelfth]] into three generators of ~[[10/7]], using the comma [[1029/1000]]. It also tempers out [[686/675]], the senga. It may be described as {{nowrap| 17c & 19 }}; its ploidacot is alpha-tricot. It is a very natural 13-limit tuning, given the generator is so near 13/9. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with mos scales: 7, 9, 11, 13, 15, 17, 19, 36, 55.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 686/675

{{Mapping|legend=1| 1 0 -4 -3 | 0 3 12 11 }}

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.5548{{c}}, ~10/7 = 633.2251{{c}}
: [[error map]]: {{val| +1.555 -2.280 +6.168 -8.015 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 632.5640{{c}}
: error map: {{val| 0.000 -4.263 +4.454 -10.622 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~10/7 = {{monzo| 1/3 0 1/12 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 2/3 0 11/12 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Algebraic generator]]: Radix, the real root of ''x''5 - 2''x''4 + 2''x''3 - 2''x''2 + 2''x'' - 2, also a root of ''x''6 - ''x''5 - 2. The recurrence converges.

{{Optimal ET sequence|legend=1| 17c, 19, 55, 74d }}

[[Badness]] (Sintel): 1.18

=== Liesel ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 81/80, 540/539

Mapping: {{mapping| 1 0 -4 -3 4 | 0 3 12 11 -1 }}

Optimal tunings:
* WE: ~2 = 1198.8507{{c}}, ~10/7 = 632.4668{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 632.9963{{c}}

{{Optimal ET sequence|legend=0| 17c, 19, 36 }}

Badness (Sintel): 1.35

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 81/80, 91/90

Mapping: {{mapping| 1 0 -4 -3 4 0 | 0 3 12 11 -1 7 }}

Optimal tunings:
* WE: ~2 = 1199.4968{{c}}, ~10/7 = 632.7766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.0082{{c}}

{{Optimal ET sequence|legend=0| 17c, 19, 36 }}

Badness (Sintel): 1.13

=== Elisa ===
Subgroup: 2.3.5.7.11

Comma list: 77/75, 81/80, 99/98

Mapping: {{mapping| 1 0 -4 -3 -5 | 0 3 12 11 16 }}

Optimal tunings:
* WE: ~2 = 1201.0489{{c}}, ~10/7 = 633.6147{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1644{{c}}

{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}

Badness (Sintel): 1.37

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 77/75, 81/80, 99/98

Mapping: {{mapping| 1 0 -4 -3 -5 0 | 0 3 12 11 16 7 }}

Optimal tunings:
* WE: ~2 = 1201.4815{{c}}, ~10/7 = 633.7720{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 633.1281{{c}}

{{Optimal ET sequence|legend=0| 17c, 19e, 36e }}

Badness (Sintel): 1.11

=== Lisa ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 343/330

Mapping: {{mapping| 1 0 -4 -3 -6 | 0 3 12 11 18 }}

Optimal tunings:
* WE: ~2 = 1202.6773{{c}}, ~10/7 = 632.7783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.6175{{c}}

{{Optimal ET sequence|legend=0| 17cee, 19 }}

Badness (Sintel): 1.81

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 81/80, 91/88, 147/143

Mapping: {{mapping| 1 0 -4 -3 -6 0 | 0 3 12 11 18 7 }}

Optimal tunings:
* WE: ~2 = 1203.6086{{c}}, ~10/7 = 633.1193{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 631.5346{{c}}

{{Optimal ET sequence|legend=0| 17cee, 19 }}

Badness (Sintel): 1.49

== Superpine ==
{{See also| No-sevens subgroup temperaments #Superpine }}

The superpine temperament is generated by 1/3 of a fourth, represented by [[~]][[35/32]], which resembles [[porcupine]], but it favors flat fifths instead of sharp ones. It may be described as {{nowrap| 36 & 43 }}; its ploidacot is omega-tricot. Unlike in porcupine, the minor third reached by 2 generators up is strongly neutral-flavored and does not represent [[6/5]] – harmonics other than 3 all require the 15-tone mos ([[7L 8s]]) to properly utilize. This temperament has an obvious 11-limit interpretation by treating the generator as [[11/10]] as in porcupine, which makes [[11/8]] high-[[complexity]] like the other harmonics, but in the 13-limit 5 generators up closely approximates [[13/8]]. [[43edo]] is a good tuning especially for the higher-limit extensions.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 1119744/1071875

{{Mapping|legend=1| 1 2 4 1 | 0 -3 -12 13 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3652{{c}}, ~35/32 = 167.1615{{c}}
: [[error map]]: {{val| -0.635 -4.709 +5.209 +3.639 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/32 = 167.2561{{c}}
: error map: {{val| 0.000 -3.723 +6.613 +5.503 }}

{{Optimal ET sequence|legend=1| 7, 36, 43, 79c }}

[[Badness]] (Sintel): 3.46

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 864/847

Mapping: {{mapping| 1 2 4 1 5 | 0 -3 -12 13 -11 }}

Optimal tunings:
* WE: ~2 = 1199.0522{{c}}, ~11/10 = 167.1904{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3382{{c}}

{{Optimal ET sequence|legend=0| 7, 36, 43 }}

Badness (Sintel): 1.90

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 144/143, 176/175

Mapping: {{mapping| 1 2 4 1 5 3 | 0 -3 -12 13 -11 5 }}

Optimal tunings:
* WE: ~2 = 1199.4286{{c}}, ~11/10 = 167.3105{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 167.3958{{c}}

{{Optimal ET sequence|legend=0| 7, 36, 43 }}

Badness (Sintel): 1.52

== Lithium ==
Lithium is named after the 3rd element for having a 3rd-octave period (and also for lithium's molar mass of 6.9 g/mol since 69edo supports it). Its ploidacot is triploid monocot. It supports a [[3L 6s]] scale and thus intuitively can be thought of as "tcherepnin meantone" in that context.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 3125/3087

{{Mapping|legend=1| 3 0 -12 -20 | 0 1 4 6 }}

: mapping generators: ~56/45, ~3

[[Optimal tuning]]s:
* [[WE]]: ~56/45 = 400.6744{{c}}, ~3/2 = 695.8474{{c}} {~15/14 = 105.5015{{c}})
: [[error map]]: {{val| +2.023 -4.084 -2.924 +4.910 }}
* [[CWE]]: ~56/45 = 400.0000{{c}}, ~3/2 = 695.1413{{c}} {~15/14 = 104.8587{{c}})
: error map: {{val| 0.000 -6.814 -5.748 +2.022 }}

{{Optimal ET sequence|legend=1| 12, 33cd, 45, 57 }}

[[Badness]] (Sintel): 1.75

== Squares ==
{{Main| Squares }}

Squares splits the [[6/1|6th harmonic]] into four subminor sixths of [[11/7]]~[[14/9]] (or splits a [[8/3|perfect eleventh]] into four supermajor thirds of [[9/7]]~[[14/11]]), and uses it for a generator. It may be described as {{nowrap| 14c & 17c }}; its ploidacot is beta-tetracot. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8-, 11-, and 14-note mos scales available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 2401/2400

{{Mapping|legend=1| 1 -1 -8 -3 | 0 4 16 9 }}

: mapping generators: ~2, ~14/9

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2488{{c}}, ~14/9 = 774.8640{{c}}
: [[error map]]: {{val| +1.249 -3.748 +1.520 +1.204 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 774.1560{{c}}
: error map: {{val| 0.000 -5.331 +0.183 -1.422 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~9/7 = {{monzo| 1/2 0 -1/16 }}
: [[projection map]]: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3/2 0 9/16 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

[[Algebraic generator]]: Sceptre2, the positive root of 9''x''2 + ''x'' - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.

{{Optimal ET sequence|legend=1| 14c, 17c, 31, 169b, 200b }}

[[Badness]] (Sintel): 1.16

Scales: [[skwares8]], [[skwares11]], [[skwares14]]

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 121/120

Mapping: {{mapping| 1 -1 -8 -3 -3 | 0 4 16 9 10 }}

Optimal tunings:
* WE: ~2 = 1201.6657{{c}}, ~11/7 = 775.1171{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.1754{{c}}

{{Optimal ET sequence|legend=0| 14c, 17c, 31, 130bee, 169beee }}

Badness (Sintel): 0.715

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 81/80, 99/98, 121/120

Mapping: {{mapping| 1 -1 -8 -3 -3 5 | 0 4 16 9 10 -2 }}

Optimal tunings:
* WE: ~2 = 1199.8419{{c}}, ~11/7 = 774.3484{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4422{{c}}

{{Optimal ET sequence|legend=0| 14c, 17c, 31, 79cf }}

Badness (Sintel): 1.05

==== Squad ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 91/90, 99/98

Mapping: {{mapping| 1 -1 -8 -3 -3 -6 | 0 4 16 9 10 15 }}

Optimal tunings:
* WE: ~2 = 1202.0312{{c}}, ~11/7 = 775.5589{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 774.4140{{c}}

{{Optimal ET sequence|legend=0| 14cf, 17c, 31f }}

Badness (Sintel): 1.11

==== Agora ====
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 105/104, 121/120

Mapping: {{mapping| 1 -1 -8 -3 -3 -15 | 0 4 16 9 10 29 }}

Optimal tunings:
* WE: ~2 = 1202.3228{{c}}, ~11/7 = 775.2214{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8617{{c}}

{{Optimal ET sequence|legend=0| 14cf, 31, 45ef, 76e }}

Badness (Sintel): 1.01

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 | 0 4 16 9 10 29 11 }}

Optimal tunings:
* WE: ~2 = 1201.4340{{c}}, ~11/7 = 774.7375{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8955{{c}}

{{Optimal ET sequence|legend=0| 14cf, 31 }}

Badness (Sintel): 1.15

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119

Mapping: {{mapping| 1 -1 -8 -3 -3 -15 -3 -8 | 0 4 16 9 10 29 11 19 }}

Optimal tunings:
* WE: ~2 = 1201.2461{{c}}, ~11/7 = 774.5783{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/7 = 773.8479{{c}}

{{Optimal ET sequence|legend=0| 14cf, 31 }}

Badness (Sintel): 1.15

=== Cuboctahedra ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 1375/1372

Mapping: {{mapping| 1 -1 -8 -3 17 | 0 4 16 9 -21 }}

Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~14/9 = 774.9386{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/9 = 774.0243{{c}}

{{Optimal ET sequence|legend=0| 31, 107b, 138b, 169be, 200be }}

Badness (Sintel): 1.88

== Jerome ==
Jerome is related to [[20ed5|Hieronymus' tuning]]; the Hieronymus generator is 51/20, or 139.316 cents. It may be described as {{nowrap| 17c & 26 }}; its ploidacot is pentacot. While the generator represents both 13/12 and 12/11, the CTE/CWE and Hieronymus generators are close to 13/12 in size.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 17280/16807

{{Mapping|legend=1| 1 1 0 2 | 0 5 20 7 }}

: mapping generators: ~2, ~54/49

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1640{{c}}, ~54/49 = 139.3624{{c}}
: [[error map]]: {{val| +0.164 -4.979 +0.934 +7.039 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~54/49 = 139.3528{{c}}
: error map: {{val| 0.000 -5.191 +0.741 +6.643 }}

{{Optimal ET sequence|legend=1| 17c, 26, 43 }}

[[Badness]] (Sintel): 2.75

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 864/847

Mapping: {{mapping| 1 1 0 2 3 | 0 5 20 7 4 }}

Optimal tunings:
* WE: ~2 = 1201.4436{{c}}, ~12/11 = 139.3714{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~12/11 = 139.4038{{c}}

{{Optimal ET sequence|legend=0| 17c, 26, 43 }}

Badness (Sintel): 1.58

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 81/80, 99/98, 144/143

Mapping: {{mapping| 1 1 0 2 3 3 | 0 5 20 7 4 6 }}

Optimal tunings:
* WE: ~2 = 1199.8860{{c}}, ~13/12 = 139.3737{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3817{{c}}

{{Optimal ET sequence|legend=0| 17c, 26, 43 }}

Badness (Sintel): 1.21

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 81/80, 99/98, 144/143, 189/187

Mapping: {{mapping| 1 1 0 2 3 3 2 | 0 5 20 7 4 6 18 }}

Optimal tunings:
* WE: ~2 = 1199.8346{{c}}, ~13/12 = 139.3431{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3544{{c}}

{{Optimal ET sequence|legend=0| 17cg, 26, 43 }}

Badness (Sintel): 1.06

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143

Mapping: {{mapping| 1 1 0 2 3 3 2 1 | 0 5 20 7 4 6 18 28 }}

Optimal tunings:
* WE: ~2 = 1199.8891{{c}}, ~13/12 = 139.3001{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 139.3080{{c}}

{{Optimal ET sequence|legend=0| 17cgh, 26, 43, 69 }}

Badness (Sintel): 1.11

== Meantritone ==
The meantritone temperament tempers out the [[mirkwai comma]] (16875/16807) and [[trimyna comma]] (50421/50000) in the 7-limit. In this temperament, the 6th harmonic is split into five generators of ~10/7; the ploidacot of this temperament is beta-pentacot. The name ''meantritone'' is a portmanteau of ''meantone'' and ''tritone'', the latter is a generator of this temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 16875/16807

{{Mapping|legend=1| 1 -1 -8 -7 | 0 5 20 19 }}

: mapping generators: ~2, ~10/7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.3832{{c}}, ~10/7 = 619.9478{{c}}
: [[error map]]: {{val| +1.383 -3.599 +1.576 +0.499 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/7 = 619.3176{{c}}
: error map: {{val| 0.000 -5.367 +0.038 -1.791 }}

{{Optimal ET sequence|legend=1| 29cd, 31, 188bcd, 219bbcd }}

[[Badness]] (Sintel): 2.08

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 2541/2500

Mapping: {{mapping| 1 -1 -8 -7 -11 | 0 5 20 19 28 }}

Optimal tunings:
* WE: ~2 = 1201.2054{{c}}, ~10/7 = 619.9752{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/7 = 619.4223{{c}}

{{Optimal ET sequence|legend=0| 29cde, 31 }}

Badness (Sintel): 1.42

== Injera ==
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a ~15/14 semitone difference between a half-octave and a perfect fifth. Injera may be described as {{nowrap| 12 & 26 }}; its ploidacot is diploid monocot. It tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera.

[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3091.html#3091 Origin of the name]

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49, 81/80

{{Mapping|legend=1| 2 0 -8 -7 | 0 1 4 4 }}

: mapping generators: ~7/5, ~3

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 600.6662{{c}}, ~3/2 = 695.1463{{c}} (~21/20 = 94.4801{{c}})
: [[error map]]: {{val| +1.332 -5.476 -5.729 +12.425 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 694.7712{{c}} (~21/20 = 94.7712{{c}})
: error map: {{val| 0.000 -7.184 -7.229 +10.259 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 7-odd-limit [[diamond tradeoff]]: ~3/2 = [688.957, 701.955]
* 9-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

{{Optimal ET sequence|legend=1| 12, 26, 38 }}

[[Badness]] (Sintel): 0.788

; Music
* [https://web.archive.org/web/20201127013520/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Two%20Pairs%20of%20Socks.mp3 ''Two Pairs of Socks''] by [[Igliashon Jones]] – in [[26edo]] tuning

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 81/80

Mapping: {{mapping| 2 0 -8 -7 -12 | 0 1 4 4 6 }}

Optimal tunings:
* WE: ~7/5 = 600.9350{{c}}, ~3/2 = 693.9198{{c}} (~21/20 = 92.9848{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.3539{{c}} (~21/20 = 93.3539{{c}})

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [685.714, 700.000] (8\14 to 7\12)
* 11-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

{{Optimal ET sequence|legend=0| 12, 26 }}

Badness (Sintel): 0.764

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 45/44, 50/49, 78/77, 81/80

Mapping: {{mapping| 2 0 -8 -7 -12 -21 | 0 1 4 4 6 9 }}

Optimal tunings:
* WE: ~7/5 = 600.9982{{c}}, ~3/2 = 693.8249{{c}} (~21/20 = 92.8267{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.0992{{c}} (~21/20 = 93.0992{{c}})

Tuning ranges:
* 13-odd-limit diamond monotone: ~3/2 = 692.308 (15\26)
* 13- and 15-odd-limit diamond tradeoff: ~3/2 = [682.458, 701.955]

{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}

Badness (Sintel): 0.891

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 45/44, 50/49, 78/77, 81/80, 85/84

Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 | 0 1 4 4 6 9 1 }}

Optimal tunings:
* WE: ~7/5 = 601.1757{{c}}, ~3/2 = 693.8441{{c}} (~21/20 = 92.6684{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.8879{{c}} (~21/20 = 92.8879{{c}})

{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}

Badness (Sintel): 0.935

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 45/44, 50/49, 57/56, 78/77, 81/80, 85/84

Mapping: {{mapping| 2 0 -8 -7 -12 -21 5 -1 | 0 1 4 4 6 9 1 3 }}

Optimal tunings:
* WE: ~7/5 = 601.4245{{c}}, ~3/2 = 693.9426{{c}} (~21/20 = 92.5181{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 692.7606{{c}} (~21/20 = 92.7606{{c}})

{{Optimal ET sequence|legend=0| 12f, 14cf, 26 }}

Badness (Sintel): 0.920

==== Enjera ====
Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 40/39, 45/44, 50/49

Mapping: {{mapping| 2 0 -8 -7 -12 -2 | 0 1 4 4 6 3 }}

Optimal tunings:
* WE: ~7/5 = 599.1863{{c}}, ~3/2 = 693.1791{{c}} (~21/20 = 93.9929{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 693.6809{{c}} (~21/20 = 93.6809{{c}})

{{Optimal ET sequence|legend=0| 10cdeef, 12f }}

Badness (Sintel): 1.10

=== Injerous ===
Subgroup: 2.3.5.7.11

Comma list: 33/32, 50/49, 55/54

Mapping: {{mapping| 2 0 -8 -7 10 | 0 1 4 4 -1 }}

Optimal tunings:
* WE: ~7/5 = 603.1682{{c}}, ~3/2 = 694.1945{{c}} (~21/20 = 91.0264{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 691.6107{{c}} (~21/20 = 91.6107{{c}})

{{Optimal ET sequence|legend=0| 12e, 14c, 26e, 40cee }}

Badness (Sintel): 1.28

=== Lahoh ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 81/77

Mapping: {{mapping| 2 0 -8 -7 7 | 0 1 4 4 0 }}

Optimal tunings:
* WE: ~7/5 = 597.3179{{c}}, ~3/2 = 695.8759{{c}} (~21/20 = 98.5581{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 697.8757{{c}} (~21/20 = 97.8757{{c}})

{{Optimal ET sequence|legend=0| 10cd, 12 }}

Badness (Sintel): 1.42

=== Teff ===
{{Main| Teff }}

Teff, found and named by [[Mason Green]], is to injera what mohajira is to meantone; it splits the generator in halves in order to accommodate higher-limit intervals, creating a half-octave quartertone temperament. Its ploidacot is diploid alpha-dicot.

Subgroup: 2.3.5.7.11

Comma list: 50/49, 81/80, 864/847

Mapping: {{mapping| 2 1 -4 -3 8 | 0 2 8 8 -1 }}

: mapping generators: ~7/5, ~16/11

Optimal tunings:
* WE: ~7/5 = 600.2802{{c}}, ~16/11 = 647.7720{{c}} (~33/32 = 47.4918{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5224{{c}} (~33/32 = 47.5224{{c}})

{{Optimal ET sequence|legend=0| 24d, 26, 50d }}

Badness (Sintel): 2.34

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 78/77, 81/80, 144/143

Mapping: {{mapping| 2 1 -4 -3 8 2 | 0 2 8 8 -1 5 }}

Optimal tunings:
* WE: ~7/5 = 600.3037{{c}}, ~16/11 = 647.7954{{c}} (~33/32 = 47.4917{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.5256{{c}} (~33/32 = 47.5256{{c}})

{{Optimal ET sequence|legend=0| 24d, 26, 50d }}

Badness (Sintel): 1.65

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 78/77, 81/80, 85/84, 144/143

Mapping: {{mapping| 2 1 -4 -3 8 2 6 | 0 2 8 8 -1 5 2 }}

Optimal tunings:
* WE: ~7/5 = 600.5123{{c}}, ~16/11 = 647.8970{{c}} (~34/33 = 47.3846{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4314{{c}} (~34/33 = 47.4314{{c}})

{{Optimal ET sequence|legend=0| 24d, 26 }}

Badness (Sintel): 1.50

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 50/49, 57/56, 78/77, 81/80, 85/84, 144/143

Mapping: {{mapping| 2 1 -4 -3 8 2 6 2 | 0 2 8 8 -1 5 2 6 }}

Optimal tunings:
* WE: ~7/5 = 600.6308{{c}}, ~16/11 = 648.0424{{c}} (~34/33 = 47.4116{{c}})
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 647.4715{{c}} (~34/33 = 47.4715{{c}})

{{Optimal ET sequence|legend=0| 24d, 26 }}

Badness (Sintel): 1.41

== Pombe ==
Pombe (named after the African millet beer) is a variant of [[#Teff]] by [[User:Kaiveran|Kaiveran Lugheidh]] that eschews the tempering of 50/49 to attain more accuracy in the 7-limit. Its ploidacot is diploid alpha-dicot, the same as teff. Oddly, the 7th harmonic has a lesser generator distance than in teff (-5 vs +8), but this combined with the fact that other harmonics are in the opposite direction means that the 7-limit diamond is more complex overall.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 300125/294912

{{Mapping|legend=1| 2 1 -4 11 | 0 2 8 -5 }}

: mapping generators: ~735/512, ~35/24

[[Optimal tuning]]s:
* [[WE]]: ~735/512 = 601.0652{{c}}, ~35/24 = 648.9295{{c}} (~36/35 = 47.8642{{c}})
: [[error map]]: {{val| +2.130 -3.031 +0.861 -1.756 }}
* [[CWE]]: ~735/512 = 600.0000{{c}}, ~35/24 = 647.8628{{c}} (~36/35 = 47.8628{{c}})
: error map: {{val| 0.000 -6.229 -3.411 -8.140 }}

{{Optimal ET sequence|legend=1| 24, 26, 50, 126bcd, 176bcdd, 226bbcdd }}

[[Badness]] (Sintel): 2.94

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 245/242, 385/384

Mapping: {{mapping| 2 1 -4 11 8 | 0 2 8 -5 -1 }}

Optimal tunings:
* WE: ~99/70 = 600.7890{{c}}, ~16/11 = 648.7592{{c}} (~36/35 = 47.9701{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.9516{{c}} (~36/35 = 47.9516{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.72

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 245/242

Mapping: {{mapping| 2 1 -4 11 8 2 | 0 2 8 -5 -1 5 }}

Optimal tunings:
* WE: ~99/70 = 600.6971{{c}}, ~16/11 = 648.6029{{c}} (~36/35 = 47.9058{{c}})
* CWE: ~99/70 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.28

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 105/104, 144/143, 245/242, 273/272

Mapping: {{mapping| 2 1 -4 11 8 2 6 | 0 2 8 -5 -1 5 2 }}

Optimal tunings:
* WE: ~17/12 = 600.7610{{c}}, ~16/11 = 648.6638{{c}} (~36/35 = 47.9028{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.8990{{c}} (~36/35 = 47.8990{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.08

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 105/104, 133/132, 144/143, 171/170, 210/209

Mapping: {{mapping| 2 1 -4 11 8 2 6 2 | 0 2 8 -5 -1 5 2 6 }}

Optimal tunings:
* WE: ~17/12 = 600.8048{{c}}, ~16/11 = 648.7494{{c}} (~36/35 = 47.9446{{c}})
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 647.9425{{c}} (~36/35 = 47.9425{{c}})

{{Optimal ET sequence|legend=0| 24, 26, 50 }}

Badness (Sintel): 1.01

== Orphic ==
Orphic has a semi-octave period and four generators plus a period gives the 3rd harmonic; its ploidacot is diploid alpha-tetracot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 5898240/5764801

{{Mapping|legend=1| 2 1 -4 4 | 0 4 16 3 }}

: mapping generators: ~2401/1728, ~343/288

[[Optimal tuning]]s:
* [[WE]]: ~2401/1728 = 600.1767{{c}}, ~343/288 = 324.3015{{c}} (~7/6 = 275.8751{{c}})
: [[error map]]: {{val| +0.353 -4.572 +1.804 +4.785 }}
* [[CWE]]: ~2401/1728 = 600.0000{{c}}, ~343/288 = 324.2285{{c}} (~7/6 = 275.7715{{c}})
: error map: {{val| 0.000 -5.041 +1.342 +3.860 }}

{{Optimal ET sequence|legend=1| 26, 48c, 74 }}

[[Badness]] (Sintel): 6.55

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 73728/73205

Mapping: {{mapping| 2 1 -4 4 8 | 0 4 16 3 -2 }}

Optimal tunings:
* WE: ~363/256 = 600.1011{{c}}, ~77/64 = 324.2923{{c}} (~7/6 = 275.8088{{c}})
* CWE: ~363/256 = 600.0000{{c}}, ~77/64 = 324.2463{{c}} (~7/6 = 275.7537{{c}})

{{Optimal ET sequence|legend=0| 26, 48c, 74 }}

Badness (Sintel): 3.36

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 99/98, 144/143, 2200/2197

Mapping: {{mapping| 2 1 -4 4 8 2 | 0 4 16 3 -2 10 }}

Optimal tunings:
* WE: ~55/39 = 600.0540{{c}}, ~77/64 = 324.2551{{c}} (~7/6 = 275.7989{{c}})
* CWE: ~55/39 = 600.0000{{c}}, ~77/64 = 324.2307{{c}} (~7/6 = 275.7693{{c}})

{{Optimal ET sequence|legend=0| 26, 48c, 74 }}

Badness (Sintel): 2.21

== Cloudtone ==
The cloudtone temperament tempers out the [[cloudy comma]], 16807/16384 and the [[syntonic comma]], 81/80 in the 7-limit. It may be described as {{nowrap| 5 & 50 }}; its ploidacot is pentaploid monocot. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 16807/16384

{{Mapping|legend=1| 5 0 -20 14 | 0 1 4 0 }}

: mapping generators: ~8/7, ~3

[[Optimal tuning]]s:
* [[WE]]: ~8/7 = 240.4267{{c}}, ~3/2 = 696.9566{{c}} (~49/48 = 24.3235{{c}})
: [[error map]]: {{val| +2.133 -2.865 +1.513 -2.852 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~3/2 = 696.1637{{c}} (~49/48 = 23.8373{{c}})
: error map: {{val| 0.000 -5.791 -1.659 -8.826 }}

{{Optimal ET sequence|legend=1| 5, 40c, 45, 50 }}

[[Badness]] (Sintel): 2.59

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384, 2401/2376

Mapping: {{mapping| 5 0 -20 14 41 | 0 1 4 0 -3 }}

Optimal tunings:
* WE: ~8/7 = 240.2740{{c}}, ~3/2 = 697.3317{{c}} (~56/55 = 23.4904{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.6269{{c}} (~56/55 = 23.3731{{c}})

{{Optimal ET sequence|legend=0| 5, 45, 50 }}

Badness (Sintel): 2.33

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 144/143, 2401/2376

Mapping: {{mapping| 5 0 -20 14 41 -21 | 0 1 4 0 -3 5 }}

Optimal tunings:
* WE: ~8/7 = 240.2435{{c}}, ~3/2 = 696.8686{{c}} (~91/90 = 23.8618{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~3/2 = 696.2653{{c}} (~91/90 = 23.7347{{c}})

{{Optimal ET sequence|legend=0| 5, 45f, 50 }}

Badness (Sintel): 2.02

== Subgroup extensions ==
=== Stützel (2.3.5.19) ===
[[Subgroup]]: 2.3.5.19

[[Comma list]]: 81/80, 96/95

{{Mapping|legend=2| 1 0 -4 9 | 0 1 4 -3 }}

{{Mapping|legend=3| 1 0 -4 0 0 0 0 9 | 0 1 4 0 0 0 0 -3 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5513{{c}}, ~3/2 = 697.6058{{c}}
: [[error map]]: {{val| -0.448 -4.798 +4.110 +6.977 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.8222{{c}}
: error map: {{val| 0.000 -4.133 +4.975 +9.020 }}

{{Optimal ET sequence|legend=1| 5, 7, 12, 31, 43, 98h }}

[[Badness]] (Sintel): 0.324

=== Hypnotone ===
Hypnotone is no-sevens [[#Flattone|flattone]].

[[Subgroup]]: 2.3.5.11

[[Comma list]]: 45/44, 81/80

{{Mapping|legend=2| 1 0 -4 -6 | 0 1 4 6 }}

{{Mapping|legend=3| 1 0 -4 0 -6 | 0 1 4 0 6 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.0621{{c}}, ~3/2 = 694.5448{{c}}
: [[error map]]: {{val| +2.062 -5.348 -8.135 +15.951 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 693.9085{{c}}
: error map: {{val| 0.000 -8.047 -10.680 +12.133 }}

{{Optimal ET sequence|legend=1| 7, 12, 19, 26, 45 }}

[[Badness]] (Sintel): 0.326

==== 2.3.5.11.13 subgroup ====
Subgroup: 2.3.5.11.13

Comma list: 45/44, 65/64, 81/80

Subgroup-val mapping: {{mapping| 1 0 -4 -6 10 | 0 1 4 6 -4 }}

Gencom mapping: {{mapping| 1 0 -4 0 -6 10 | 0 1 4 0 6 -4 }}

Optimal tunings:
* WE: ~2 = 1202.6916{{c}}, ~3/2 = 694.4181{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 693.0870{{c}}

{{Optimal ET sequence|legend=0| 7, 12, 19, 26, 45f }}

Badness (Sintel): 0.561

=== Dequarter ===
[[Subgroup]]: 2.3.5.11

[[Comma list]]: 33/32, 55/54

{{Mapping|legend=2| 1 0 -4 5 | 0 1 4 -1 }}

{{Mapping|legend=3| 1 0 -4 0 5 | 0 1 4 0 -1 }}

: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1206.5832{{c}}, ~3/2 = 695.8763{{c}}
: [[error map]]: {{val| +6.583 +0.504 -2.809 -20.862 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 693.1206{{c}}
: error map: {{val| 0.000 -8.834 -13.831 -44.439 }}

{{Optimal ET sequence|legend=1| 5, 7, 19e, 26e }}

[[Badness]] (Sintel): 0.451

==== Dreamtone ====
Subgroup: 2.3.5.11.13

Comma list: 33/32, 55/54, 975/968

Subgroup-val mapping: {{mapping| 1 0 -4 5 21 | 0 1 4 -1 -11 }}

Gencom mapping: {{mapping| 1 0 -4 0 5 21 | 0 1 4 0 -1 -11 }}

Optimal tunings:
* WE: ~2 = 1207.8248{{c}}, ~3/2 = 694.7806{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 690.1826{{c}}

{{Optimal ET sequence|legend=0| 7, 19eff, 26eff, 33ceeff, 40ceeff }}

Badness (Sintel): 1.40

== References ==
<references/>

[[Category:Temperament families]]
[[Category:Meantone family| ]] 
[[Category:Meantone| ]] 
[[Category:Rank 2]]
[[Category:Listen]]

Xenial

2026-05-09T06:00:54Z

Xenllium:

{{Infobox regtemp
| Title = Xenial
| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
| Comma basis = [[126/125]], [[177147/175616]] (7-limit); [[126/125]], [[162/161]], [[169/168]], [[171/170]], [[208/207]], [[221/220]], [[231/230]] (23-limit)
| Edo join 1 = 19 | Edo join 2 = 70
| Mapping = 1; -9 -17 -33 22 -21 26 27 -3
| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]], [[19L 51s]]
| Pergen = (P8, P11/9)
| Odd limit 1 = 7 | Mistuning 1 = 4.60 | Complexity 1 = 51
| Odd limit 2 = 9 | Mistuning 2 = 6.27 | Complexity 2 = 51
| Odd limit 3 = 17 | Mistuning 3 = 8.90 | Complexity 3 = 70
| Odd limit 4 = 23 | Mistuning 4 = 8.96 | Complexity 4 = 70
}}
'''Xenial''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a sharpened minor whole tone of [[~]][[10/9]], so that nine generators reach [[4/3]], 17 reach [[8/5]], 21 reach [[16/13]] and 33 reach [[8/7]] with octave reduction. It is also generated by dividing [[11/1|11th harmonic]] into 22 equal parts, [[17/1|17th harmonic]] into 26 equal parts, or [[19/1|19th harmonic]] into 27 equal parts.

See [[Starling temperaments #Xenial]] for more technical data.

== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.000
| [[1/1]]
|-
| 1
| 188.775
| [[10/9]], [[19/17]], [[28/25]]
|-
| 2
| 377.551
| [[56/45]]
|-
| 3
| 566.326
| [[18/13]], [[32/23]]
|-
| 4
| 755.102
| [[17/11]], [[20/13]]
|-
| 5
| 943.877
| [[19/11]], [[26/15]]
|-
| 6
| 1132.653
| [[23/12]], [[27/14]]
|-
| 7
| 121.428
| [[15/14]]
|-
| 8
| 310.204
| [[6/5]]
|-
| 9
| 498.979
| [[4/3]]
|-
| 10
| 687.755
| [[40/27]]
|-
| 11
| 876.530
|
|-
| 12
| 1065.306
| [[13/7]], [[24/13]]
|-
| 13
| 54.081
| [[26/25]], [[33/32]]
|-
| 14
| 242.857
| [[23/20]]
|-
| 15
| 431.632
| [[9/7]], [[23/18]]
|-
| 16
| 620.408
| [[10/7]]
|-
| 17
| 809.183
| [[8/5]]
|-
| 18
| 997.959
| [[16/9]], [[23/13]]
|-
| 19
| 1186.734
|
|-
| 20
| 175.510
|
|-
| 21
| 364.285
| [[16/13]], [[26/21]]
|-
| 22
| 553.061
| [[11/8]]
|-
| 23
| 741.836
| [[23/15]]
|-
| 24
| 930.612
| [[12/7]]
|-
| 25
| 1119.387
| [[40/21]], [[44/23]], [[48/25]]
|-
| 26
| 108.163
| [[16/15]], [[17/16]]
|-
| 27
| 296.938
| [[19/16]]
|-
| 28
| 485.714
|
|-
| 29
| 674.439
| [[34/23]]
|-
| 30
| 863.265
| [[38/23]], [[23/14]]
|-
| 31
| 1052.040
| [[11/6]], [[46/25]]
|-
| 32
| 40.815
| [[36/35]], [[46/45]], [[50/49]]
|-
| 33
| 229.591
| [[8/7]]
|-
| 34
| 418.366
| [[32/25]]
|}
<nowiki/>* In 23-limit CWE tuning

== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.8535{{c}}
| CWE: ~10/9 = 188.8544{{c}}
| POTE: ~10/9 = 188.8548{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.8295{{c}}
| CWE: ~10/9 = 188.8085{{c}}
| POTE: ~10/9 = 188.8028{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7987{{c}}
| CWE: ~10/9 = 188.7898{{c}}
| POTE: ~10/9 = 188.7875{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 17-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7811{{c}}
| CWE: ~10/9 = 188.7677{{c}}
| POTE: ~10/9 = 188.7655{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 19-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7828{{c}}
| CWE: ~10/9 = 188.7770{{c}}
| POTE: ~10/9 = 188.7762{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7849{{c}}
| CWE: ~10/9 = 188.7755{{c}}
| POTE: ~10/9 = 188.7744{{c}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]
! Generator (¢)
! Comments
|-
|
| 9/5
| 182.4037
|
|-
|
| 13/10
| 186.4465
|
|-
| 5 ⧵ 32
|
| 187.5000
| 32cddefgh val Lower bound of 7-odd-limit diamond monotone
|-
|
| 23/12
| 187.7199
|
|-
|
| 13/9
| 187.7941
|
|-
|
| 23/13
| 188.2081
|
|-
| 8 ⧵ 51
|
| 188.2353
| 51cdh val Lower bound of 9-odd-limit diamond monotone
|-
|
| 23/18
| 188.2910
|
|-
|
| 17/11
| 188.4094
|
|-
|
| 13/12
| 188.4523
|
|-
|
| 15/14
| 188.4918
|
|-
|
| 13/8
| 188.5463
|
|-
| 11 ⧵ 70
|
| 188.5714
| Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 7/5
| 188.5930
|
|-
|
| 17/13
| 188.6048
|
|-
|
| 21/20
| 188.6213
|
|-
|
| 13/11
| 188.6230
| 13-odd-limit minimax
|-
|
| 23/14
| 188.6483
|
|-
|
| 17/16
| 188.6521
|
|-
|
| 23/21
| 188.6537
|
|-
|
| 17/12
| 188.6572
|
|-
|
| 17/9
| 188.6601
|
|-
|
| 3/2
| 188.6717
| 9, 15 and 17-odd-limit minimax
|-
|
| 11/9
| 188.6852
| 11-odd-limit minimax
|-
|
| 19/13
| 188.6872
| 19, 21 and 23-odd-limit minimax
|-
|
| 11/6
| 188.6891
|
|-
|
| 23/15
| 188.6959
|
|-
|
| 11/8
| 188.6963
|
|-
|
| 23/20
| 188.7115
|
|-
|
| 21/17
| 188.7379
|
|-
|
| 19/18
| 188.7467
|
|-
|
| 17/14
| 188.7480
|
|-
|
| 21/11
| 188.7584
|
|-
| 14 ⧵ 89
|
| 188.7640
| 19, 21 and 23-odd-limit diamond monotone (singleton)
|-
|
| 19/12
| 188.7655
|
|-
|
| 11/7
| 188.7726
|
|-
|
| 17/15
| 188.7824
|
|-
|
| 21/16
| 188.7909
|
|-
|
| 21/19
| 188.7932
|
|-
|
| 19/16
| 188.7968
|
|-
|
| 17/10
| 188.8056
|
|-
|
| 19/14
| 188.8115
|
|-
|
| 15/11
| 188.8135
|
|-
|
| 7/4
| 188.8235
|
|-
|
| 11/10
| 188.8463
|
|-
|
| 23/17
| 188.8511
|
|-
|
| 19/15
| 188.8537
|
|-
|
| 7/6
| 188.8804
| 7-odd-limit minimax
|-
|
| 19/10
| 188.8909
|
|-
|
| 15/8
| 188.9127
|
|-
|
| 23/22
| 188.9217
|
|-
|
| 23/19
| 188.9746
|
|-
|
| 9/7
| 189.0056
|
|-
|
| 21/13
| 189.0356
|
|-
|
| 5/4
| 189.0404
| 5-odd-limit minimax
|-
|
| 19/11
| 189.2390
|
|-
|
| 13/7
| 189.3085
|
|-
|
| 5/3
| 189.4552
|
|-
| 3 ⧵ 19
|
| 189.4737
| Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 15/13
| 190.4518
|
|-
|
| 23/16
| 190.5752
|
|-
|
| 19/17
| 192.5576
|
|}

[[Category:Xenial| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]

Xenial

2026-05-09T00:30:32Z

Xenllium:

{{Infobox regtemp
| Title = Xenial
| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
| Comma basis = [[126/125]], [[177147/175616]] (7-limit); [[126/125]], [[162/161]], [[169/168]], [[171/170]], [[208/207]], [[221/220]], [[231/230]] (23-limit)
| Edo join 1 = 19 | Edo join 2 = 70
| Mapping = 1; -9 -17 -33 22 -21 26 27 -3
| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]], [[19L 51s]]
| Pergen = (P8, P11/9)
| Odd limit 1 = 7 | Mistuning 1 = 4.6 | Complexity 1 = 51
| Odd limit 2 = 9 | Mistuning 2 = 6.3 | Complexity 2 = 51
| Odd limit 3 = 17 | Mistuning 3 = 8.9 | Complexity 3 = 70
| Odd limit 4 = 23 | Mistuning 4 = 9.0 | Complexity 4 = 70
}}
'''Xenial''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a sharpened minor whole tone of [[~]][[10/9]], so that nine generators reach [[4/3]], 17 reach [[8/5]], 21 reach [[16/13]] and 33 reach [[8/7]] with octave reduction. It is also generated by dividing [[11/1|11th harmonic]] into 22 equal parts, [[17/1|17th harmonic]] into 26 equal parts, or [[19/1|19th harmonic]] into 27 equal parts.

See [[Starling temperaments #Xenial]] for more technical data.

== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.000
| [[1/1]]
|-
| 1
| 188.775
| [[10/9]], [[19/17]], [[28/25]]
|-
| 2
| 377.551
| [[56/45]]
|-
| 3
| 566.326
| [[18/13]], [[32/23]]
|-
| 4
| 755.102
| [[17/11]], [[20/13]]
|-
| 5
| 943.877
| [[19/11]], [[26/15]]
|-
| 6
| 1132.653
| [[23/12]], [[27/14]]
|-
| 7
| 121.428
| [[15/14]]
|-
| 8
| 310.204
| [[6/5]]
|-
| 9
| 498.979
| [[4/3]]
|-
| 10
| 687.755
| [[40/27]]
|-
| 11
| 876.530
|
|-
| 12
| 1065.306
| [[13/7]], [[24/13]]
|-
| 13
| 54.081
| [[26/25]], [[33/32]]
|-
| 14
| 242.857
| [[23/20]]
|-
| 15
| 431.632
| [[9/7]], [[23/18]]
|-
| 16
| 620.408
| [[10/7]]
|-
| 17
| 809.183
| [[8/5]]
|-
| 18
| 997.959
| [[16/9]], [[23/13]]
|-
| 19
| 1186.734
|
|-
| 20
| 175.510
|
|-
| 21
| 364.285
| [[16/13]], [[26/21]]
|-
| 22
| 553.061
| [[11/8]]
|-
| 23
| 741.836
| [[23/15]]
|-
| 24
| 930.612
| [[12/7]]
|-
| 25
| 1119.387
| [[40/21]], [[44/23]], [[48/25]]
|-
| 26
| 108.163
| [[16/15]], [[17/16]]
|-
| 27
| 296.938
| [[19/16]]
|-
| 28
| 485.714
|
|-
| 29
| 674.439
| [[34/23]]
|-
| 30
| 863.265
| [[38/23]], [[23/14]]
|-
| 31
| 1052.040
| [[11/6]], [[46/25]]
|-
| 32
| 40.815
| [[36/35]], [[46/45]], [[50/49]]
|-
| 33
| 229.591
| [[8/7]]
|-
| 34
| 418.366
| [[32/25]]
|}
<nowiki/>* In 23-limit CWE tuning

== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.8535 {{c}}
| CWE: ~10/9 = 188.8544 {{c}}
| POTE: ~10/9 = 188.8548 {{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7849 {{c}}
| CWE: ~10/9 = 188.7755 {{c}}
| POTE: ~10/9 = 188.7744 {{c}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]
! Generator (¢)
! Comments
|-
|
| 9/5
| 182.404
|
|-
|
| 13/10
| 186.447
|
|-
| 5 ⧵ 32
|
| 187.500
| 32cddefgh val Lower bound of 7-odd-limit diamond monotone
|-
|
| 23/12
| 187.720
|
|-
|
| 13/9
| 187.794
|
|-
|
| 23/13
| 188.208
|
|-
| 8 ⧵ 51
|
| 188.235
| 51cdh val Lower bound of 9-odd-limit diamond monotone
|-
|
| 23/18
| 188.291
|
|-
|
| 17/11
| 188.409
|
|-
|
| 13/12
| 188.452
|
|-
|
| 15/14
| 188.492
|
|-
|
| 13/8
| 188.546
|
|-
| 11 ⧵ 70
|
| 188.571
| Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 7/5
| 188.593
|
|-
|
| 17/13
| 188.605
|
|-
|
| 21/20
| 188.621
|
|-
|
| 13/11
| 188.623
| 13-odd-limit minimax
|-
|
| 23/14
| 188.648
|
|-
|
| 17/16
| 188.652
|
|-
|
| 23/21
| 188.654
|
|-
|
| 17/12
| 188.657
|
|-
|
| 17/9
| 188.660
|
|-
|
| 3/2
| 188.672
| 9, 15 and 17-odd-limit minimax
|-
|
| 11/9
| 188.685
| 11-odd-limit minimax
|-
|
| 19/13
| 188.687
| 19, 21 and 23-odd-limit minimax
|-
|
| 11/6
| 188.689
|
|-
|
| 23/15
| 188.6959
|
|-
|
| 11/8
| 188.6963
|
|-
|
| 23/20
| 188.711
|
|-
|
| 21/17
| 188.738
|
|-
|
| 19/18
| 188.747
|
|-
|
| 17/14
| 188.748
|
|-
|
| 21/11
| 188.758
|
|-
| 14 ⧵ 89
|
| 188.764
| 19, 21 and 23-odd-limit diamond monotone (singleton)
|-
|
| 19/12
| 188.766
|
|-
|
| 11/7
| 188.773
|
|-
|
| 17/15
| 188.782
|
|-
|
| 21/16
| 188.791
|
|-
|
| 21/19
| 188.793
|
|-
|
| 19/16
| 188.797
|
|-
|
| 17/10
| 188.806
|
|-
|
| 19/14
| 188.811
|
|-
|
| 15/11
| 188.814
|
|-
|
| 7/4
| 188.823
|
|-
|
| 11/10
| 188.846
|
|-
|
| 23/17
| 188.851
|
|-
|
| 19/15
| 188.854
|
|-
|
| 7/6
| 188.880
| 7-odd-limit minimax
|-
|
| 19/10
| 188.891
|
|-
|
| 15/8
| 188.913
|
|-
|
| 23/22
| 188.922
|
|-
|
| 23/19
| 188.975
|
|-
|
| 9/7
| 189.006
|
|-
|
| 21/13
| 189.036
|
|-
|
| 5/4
| 189.040
| 5-odd-limit minimax
|-
|
| 19/11
| 189.239
|
|-
|
| 13/7
| 189.308
|
|-
|
| 5/3
| 189.455
|
|-
| 3 ⧵ 19
|
| 189.473
| Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 15/13
| 190.452
|
|-
|
| 23/16
| 190.575
|
|-
|
| 19/17
| 192.558
|
|}

[[Category:Xenial| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]

Xenial

2026-05-04T13:05:28Z

Xenllium:

{{Infobox regtemp
| Title = Xenial
| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
| Comma basis = [[126/125]], [[177147/175616]] (7-limit); [[126/125]], [[162/161]], [[169/168]], [[171/170]], [[208/207]], [[221/220]], [[231/230]] (23-limit)
| Edo join 1 = 19 | Edo join 2 = 70
| Mapping = 1; -9 -17 -33 22 -21 26 27 -3
| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]]
| Pergen = (P8, P11/9)
| Odd limit 1 = 7 | Mistuning 1 = 4.6 | Complexity 1 = 51
}}
'''Xenial''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a sharpened minor whole tone of [[~]][[10/9]], so that nine generators reach [[4/3]], 17 reach [[8/5]], 21 reach [[16/13]] and 33 reach [[8/7]] with octave reduction. It is also generated by dividing [[11/1|11th harmonic]] into 22 equal parts, [[17/1|17th harmonic]] into 26 equal parts, or [[19/1|19th harmonic]] into 27 equal parts.

See [[Starling temperaments #Xenial]] for more technical data.

== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.000
| [[1/1]]
|-
| 1
| 188.775
| [[10/9]], [[19/17]], [[28/25]]
|-
| 2
| 377.551
| [[56/45]]
|-
| 3
| 566.326
| [[18/13]], [[32/23]]
|-
| 4
| 755.102
| [[17/11]], [[20/13]]
|-
| 5
| 943.877
| [[19/11]], [[26/15]]
|-
| 6
| 1132.653
| [[23/12]], [[27/14]]
|-
| 7
| 121.428
| [[15/14]]
|-
| 8
| 310.204
| [[6/5]]
|-
| 9
| 498.979
| [[4/3]]
|-
| 10
| 687.755
| [[40/27]]
|-
| 11
| 876.530
|
|-
| 12
| 1065.306
| [[13/7]], [[24/13]]
|-
| 13
| 54.081
| [[26/25]], [[33/32]]
|-
| 14
| 242.857
| [[23/20]]
|-
| 15
| 431.632
| [[9/7]], [[23/18]]
|-
| 16
| 620.408
| [[10/7]]
|-
| 17
| 809.183
| [[8/5]]
|-
| 18
| 997.959
| [[16/9]], [[23/13]]
|-
| 19
| 1186.734
|
|-
| 20
| 175.510
|
|-
| 21
| 364.285
| [[16/13]], [[26/21]]
|-
| 22
| 553.061
| [[11/8]]
|-
| 23
| 741.836
| [[23/15]]
|-
| 24
| 930.612
| [[12/7]]
|-
| 25
| 1119.387
| [[40/21]], [[44/23]], [[48/25]]
|-
| 26
| 108.163
| [[16/15]], [[17/16]]
|-
| 27
| 296.938
| [[19/16]]
|-
| 28
| 485.714
|
|-
| 29
| 674.439
| [[34/23]]
|-
| 30
| 863.265
| [[38/23]], [[23/14]]
|-
| 31
| 1052.040
| [[11/6]], [[46/25]]
|-
| 32
| 40.815
| [[36/35]], [[46/45]], [[50/49]]
|-
| 33
| 229.591
| [[8/7]]
|-
| 34
| 418.366
| [[32/25]]
|}
<nowiki/>* In 23-limit CWE tuning

== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.8535 {{c}}
| CWE: ~10/9 = 188.8544 {{c}}
| POTE: ~10/9 = 188.8548 {{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7849 {{c}}
| CWE: ~10/9 = 188.7755 {{c}}
| POTE: ~10/9 = 188.7744 {{c}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]
! Generator (¢)
! Comments
|-
|
| 9/5
| 182.404
|
|-
|
| 13/10
| 186.447
|
|-
| 5 ⧵ 32
|
| 187.500
| 32cddefgh val Lower bound of 7-odd-limit diamond monotone
|-
|
| 23/12
| 187.720
|
|-
|
| 13/9
| 187.794
|
|-
|
| 23/13
| 188.208
|
|-
| 8 ⧵ 51
|
| 188.235
| 51cdh val Lower bound of 9-odd-limit diamond monotone
|-
|
| 23/18
| 188.291
|
|-
|
| 17/11
| 188.409
|
|-
|
| 13/12
| 188.452
|
|-
|
| 15/14
| 188.492
|
|-
|
| 13/8
| 188.546
|
|-
| 11 ⧵ 70
|
| 188.571
| Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 7/5
| 188.593
|
|-
|
| 17/13
| 188.605
|
|-
|
| 21/20
| 188.621
|
|-
|
| 13/11
| 188.623
|
|-
|
| 23/14
| 188.648
|
|-
|
| 17/16
| 188.652
|
|-
|
| 23/21
| 188.654
|
|-
|
| 17/12
| 188.657
|
|-
|
| 17/9
| 188.660
|
|-
|
| 3/2
| 188.672
|
|-
|
| 11/9
| 188.685
|
|-
|
| 19/13
| 188.687
|
|-
|
| 11/6
| 188.689
|
|-
|
| 23/15
| 188.6959
|
|-
|
| 11/8
| 188.6963
|
|-
|
| 23/20
| 188.711
|
|-
|
| 21/17
| 188.738
|
|-
|
| 19/18
| 188.747
|
|-
|
| 17/14
| 188.748
|
|-
|
| 21/11
| 188.758
|
|-
| 14 ⧵ 89
|
| 188.764
| 19, 21 and 23-odd-limit diamond monotone (singleton)
|-
|
| 19/12
| 188.766
|
|-
|
| 11/7
| 188.773
|
|-
|
| 17/15
| 188.782
|
|-
|
| 21/16
| 188.791
|
|-
|
| 21/19
| 188.793
|
|-
|
| 19/16
| 188.797
|
|-
|
| 17/10
| 188.806
|
|-
|
| 19/14
| 188.811
|
|-
|
| 15/11
| 188.814
|
|-
|
| 7/4
| 188.823
|
|-
|
| 11/10
| 188.846
|
|-
|
| 23/17
| 188.851
|
|-
|
| 19/15
| 188.854
|
|-
|
| 7/6
| 188.880
|
|-
|
| 19/10
| 188.891
|
|-
|
| 15/8
| 188.913
|
|-
|
| 23/22
| 188.922
|
|-
|
| 23/19
| 188.975
|
|-
|
| 9/7
| 189.006
|
|-
|
| 21/13
| 189.036
|
|-
|
| 5/4
| 189.040
|
|-
|
| 19/11
| 189.239
|
|-
|
| 13/7
| 189.308
|
|-
|
| 5/3
| 189.455
|
|-
| 3 ⧵ 19
|
| 189.473
| Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 15/13
| 190.452
|
|-
|
| 23/16
| 190.575
|
|-
|
| 19/17
| 192.558
|
|}

[[Category:Xenial| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]

Starling temperaments

2026-05-04T12:52:13Z

Xenllium: /* Xenial */

{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.

Temperaments discussed in families and clans are:
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]

Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].

Since {{nowrap|(6/5)3 {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.

== Myna ==
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''

7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). [[11-limit]] myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]). In that sense, it is opposed to [[keemic temperaments]], in particular [[quasitemp]], where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.

In terms of vanishing commas, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.

[[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round cent values may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11- and 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 1728/1715

{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}
: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}

[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}

[[Badness]] (Sintel): 0.684

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}

Optimal tunings:
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}

Badness (Sintel): 0.557

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 196/195

Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}

Optimal tunings:
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}

Badness (Sintel): 0.708

==== Minah ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 126/125, 176/175

Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}

Optimal tunings:
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}

{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}

Badness (Sintel): 1.14

==== Maneh ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 126/125, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}

Optimal tunings:
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}

{{Optimal ET sequence|legend=0| 27eff, 31 }}

Badness (Sintel): 1.23

=== Myno ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 385/384

Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}

Optimal tunings:
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}

{{Optimal ET sequence|legend=0| 27, 31 }}

Badness (Sintel): 1.11

=== Coleto ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 1728/1715

Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}

Optimal tunings:
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}

{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}

Badness (Sintel): 1.61

== Nusecond ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''

Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 70 }}. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.

[[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2430/2401

{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
: mapping generators: ~2, ~49/27

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}
: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}
: error map: {{val| 0.000 -2.655 +3.768 +2.819 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}

[[Badness]] (Sintel): 1.28

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}

Optimal tunings:
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}

Minimax tuning:
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of 15''x''2 - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}

Badness (Sintel): 0.847

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}

{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}

Badness (Sintel): 0.964

== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 117649/116640

{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}
: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}

{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}

[[Badness]] (Sintel): 1.86

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 26411/26244

Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.88

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}

Optimal tunings:
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.47

== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''

Vines may be described as the {{nowrap| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 84035/82944

{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
: [[error map]]: {{val| +0.487 -0.363 +3.036 -4.448 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}
: error map: {{val| 0.000 -0.717 +2.269 -5.552 }}

{{Optimal ET sequence|legend=1| 46, 96d, 142d }}

[[Badness]] (Sintel): 1.98

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2401/2376

Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}

Optimal tunings:
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}

{{Optimal ET sequence|legend=0| 46, 96d, 142d }}

Badness (Sintel): 1.47

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}

Optimal tunings:
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}

{{Optimal ET sequence|legend=0| 46, 96d }}

Badness (Sintel): 1.23

== Xenial ==
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''

Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 177147/175616

{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}

{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}

[[Badness]] (Sintel): 2.13

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 16384/16335

Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}

Optimal tunings:
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.31

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 729/728

Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}

Optimal tunings:
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 1.98

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}

Optimal tunings:
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.06

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}

Optimal tunings:
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 2.03

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 126/125, 162/161, 169/168, 171/170, 208/207, 221/220, 231/230

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}

Optimal tunings:
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 1.93

== Kumonga ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 12288/12005

{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0653{{c}}, ~7/4 = 975.6277{{c}}
: [[error map]]: {{val| -1.935 -1.382 +4.009 +2.932 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 977.1096{{c}}
: error map: {{val| 0.000 +0.470 +7.673 +8.284 }}

{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}

[[Badness]] (Sintel): 2.21

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}

Optimal tunings:
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}

Badness (Sintel): 1.43

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}

Optimal tunings:
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}

Badness (Sintel): 1.19

== Bisemidim ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 118098/117649

{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45

[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}

{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}

[[Badness]] (Sintel): 2.47

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}

Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}

Badness (Sintel): 1.36

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}

Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}

Badness (Sintel): 0.987

== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 19683/19208

{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}

{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}

[[Badness]] (Sintel): 2.53

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}

Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}

{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}

Badness (Sintel): 1.41

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}

Optimal tunings:
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}

{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}

Badness (Sintel): 1.56

== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''

Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.

If we add 385/384 to the list of commas, 48/35 is identified with [[11/8]], and casablanca is revealed as an [[11-limit]] temperament with a very low complexity for [[11/1|11]] and not too high a one for [[7/1|7]]; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit [[meantone]].

Marrakesh, named by [[Herman Miller]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, is a more accurate 11-limit extension where the generator is identified with [[15/11]] as opposed to 11/8 in casablanca.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 589824/588245

{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}
: [[error map]]: {{val| -0.371 -1.087 +3.370 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 542.1684{{c}}
: error map: {{val| 0.000 -0.756 +4.044 -0.152 }}

{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}

[[Badness]] (Sintel): 2.56

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}

Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.22

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}

Optimal tunings:
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.31

=== Marrakesh ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}

Optimal tunings:
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}

Badness (Sintel): 1.34

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}

Optimal tunings:
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}

Badness (Sintel): 1.68

==== Murakuc ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}

Optimal tunings:
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}

{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}

Badness (Sintel): 1.71

== Amigo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2097152/2083725

{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}
: [[error map]]: {{val| -0.565 -0.811 +3.467 -1.206 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.0937{{c}}
: error map: {{val| 0.000 +0.076 +4.780 +0.393 }}

{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}

[[Badness]] (Sintel): 2.81

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 16384/16335

Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}

Optimal tunings:
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}

Badness (Sintel): 1.44

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 176/175, 364/363

Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}

Optimal tunings:
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89 }}

Badness (Sintel): 1.27

== Gilead ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 343/324

{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4516{{c}}, ~5/3 = 879.6394{{c}}
: [[error map]]: {{val| +1.452 +7.542 +2.823 -21.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.7223{{c}}
: error map: {{val| 0.000 +6.545 +0.909 -24.159 }}

{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}

[[Badness]] (Sintel): 2.92

== Supersensi ==
Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 17496/16807

{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}
: [[error map]]: {{val| -0.859 -4.800 +3.337 +6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/270 = 446.5163{{c}}
: error map: {{val| 0.000 -4.210 +4.464 +8.017 }}

{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}

[[Badness]] (Sintel): 3.76

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 864/847

Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}

Optimal tunings:
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}

Badness (Sintel): 1.97

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}

Optimal tunings:
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.46

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}

Optimal tunings:
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.32

== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''

Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.

Cobalt was named by [[Xenllium]] in 2022 after the 27th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 40353607/40310784

{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3

[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 44.4363{{c}}, ~3/2 = 701.1154{{c}}
: [[error map]]: {{val| -0.221 -1.060 +3.307 -1.534 }}
* [[CWE]]: ~36/35 = 44.4444{{c}}, ~3/2 = 701.0414{{c}}
: error map: {{val| 0.000 -0.914 +3.617 -1.118 }}

{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}

[[Badness]] (Sintel): 4.39

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108 }}

Badness (Sintel): 2.58

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}

Optimal tunings:
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}

Badness (Sintel): 2.36

===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445

Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 2.14

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968

Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 1.85

===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968

Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}

Optimal tunings:
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}

Badness (Sintel): 2.40

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083

Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}

Badness (Sintel): 2.08

==== Cobaltite ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 975/968

Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}

{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}

Badness (Sintel): 2.18

== References ==

[[Category:Temperament collections]]
[[Category:Starling temperaments| ]] 
[[Category:Rank 2]]

Starling family

2026-05-04T12:46:45Z

Xenllium:

{{Technical data page}}

The '''starling family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[starling comma]] ([[ratio]]: 126/125, {{monzo|legend=1| 1 2 -3 1 }}), also known as ''septimal semicomma''.

== Starling ==
{{Main| Starling and thrush }}

The head of the starling family is starling, which tempers out 126/125. Starling has a normal generator list of {2, 3, 5}; hence a [[5-limit]] scale can be converted to starling simply by tempering it. One way to do that, and an excellent starling tuning, is given by [[77edo]]. Other possible tunings are [[108edo]] and [[185edo]], and [[135edo]] in the nonpatent 135c val {{val| 135 214 '''314''' 379 }}.

Because no appreciable tuning accuracy is lost by including [[1029/1024]] along with 126/125 in the comma list, which leads to [[valentine]], there is a close relationship between the two. Even if tempering a 5-limit scale, one can assume valentine tempering.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[126/125]]

{{Mapping|legend=1| 1 0 0 -1 | 0 1 0 -2 | 0 0 1 3 }}
: mapping generators: ~2, ~3, ~5

[[Mapping to lattice]]: [{{val| 0 1 0 -2 }}, {{val| 0 1 1 1 }}]

[[Minkowski lattice basis]]:
: ~6/5 length = 1.068, ~5/4 length = 1.206
: angle (~6/5, ~5/4) = 100.364°

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7931{{c}}, ~3/2 = 701.1306{{c}}, ~5/4 = 390.0298{{c}}
: [[error map]]: {{val| -0.207 -1.031 +3.302 -1.618 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.0811{{c}}, ~5/4 = 389.8833{{c}}
: error map: {{val| 0.000 -0.874 +3.570 -1.338 }}

[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: 3 and 7 just, 5 1/3-comma sharp
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 1/3 2/3 0 1/3 | 0 0 0 1 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7

{{Optimal ET sequence|legend=1| 7d, 8d, 12, 19, 27, 31, 46, 58, 77, 135c, 166c }}

[[Badness]] (Sintel): 0.308

[[Projection pair]]: <code>7 125/18</code>

{{Databox|[[Minkowski blocks]]|
* 7: 25/24, 81/80
* 8: 16/15, 648/625
* 9: 27/25, 128/125
* 11: 16/15, 15625/15552
* 12: 128/125, 628/625
* 15: 128/125, 250/243
* 16: 648/625, 3125/3072
* 17: 25/24, 20480/19683
* 19: 81/80, 3125/3072
* 27: 128/125, 78732/78125
* 28: 648/625, 16875/16384
* 31: 81/80, 1990656/1953125
* 34: 15625/15552, 2048/2025
}}

=== Overview to extensions ===
Temperaments discussed elsewhere include
* ''[[Erato]]'' (+81/80) → [[Didymus rank-3 family #Erato|Didymus rank-3 family]]
* ''[[Sensigh]]'' (+245/243) → [[Sengic family #Sensigh|Sengic family]]
* ''[[Oxpecker]]'' (+121/120) → [[Biyatismic clan #Oxpecker|Biyatismic clan]]
* ''[[Cuckoo]]'' (+243/242) → [[Rastmic rank-3 clan #Cuckoo|Rastmic rank-3 clan]]

Considered below are thrush, starnova, thrasher, aplonis, and treecreeper.

== Thrush ==
{{Main| Starling and thrush }}

Thrush is the main extension of starling into the [[11-limit]]. It may be described as 12 & 15 & 31, and is loosely [[associated temperament|associated]] with [[myna]].

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 126/125, 176/175

{{Mapping|legend=1| 1 0 0 -1 -5 | 0 1 0 -2 -2 | 0 0 1 3 5 }}

Mapping to lattice: [{{val| 0 1 1 1 3 }}, {{val| 0 1 0 -2 -2 }}]

Lattice basis:
: ~5/4 length = 0.8576, ~6/5 length = 0.9314
: angle(~5/4, ~6/5) = 74.6239°

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4742{{c}}, ~3/2 = 701.6379{{c}}, ~5/4 = 391.0192{{c}}
: [[error map]]: {{val| -0.526 -0.843 +3.654 -0.621 -1.075 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.6298{{c}}, ~5/4 = 390.8558{{c}}
: error map: {{val| 0.000 -0.325 +4.542 +0.482 -0.298 }}

[[Minimax tuning]]:
* 7- and [[9-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 0 1 0 0 0 }}, {{monzo| 1/3 2/3 0 1/3 0 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| -10/3 4/3 0 5/3 0 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3.7

{{Optimal ET sequence|legend=1| 12, 15, 19e, 27e, 31, 46, 58, 89, 135c, 193c, 224c }}

[[Badness]] (Sintel): 0.424

[[Projection pair]]s: <code>7 125/18</code> <code>11 3125/288</code>

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195

Mapping: {{mapping| 1 0 0 -1 -5 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 5 5 }}

Optimal tunings:
* WE: ~2 = 1199.2727{{c}}, ~3/2 = 702.2858{{c}}, ~5/4 = 391.3567{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4562{{c}}, ~5/4 = 391.2080{{c}}

{{Optimal ET sequence|legend=0| 12f, 19e, 27e, 31, 46, 58, 104c, 135c, 193cf }}

Badness (Sintel): 0.633

=== Bluebird ===
{{Distinguish| Bluebirds }}

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175

Mapping: {{mapping| 1 0 0 -1 -5 9 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -5 }}

Optimal tunings:
* WE: ~2 = 1198.7153{{c}}, ~3/2 = 701.647{{c}}, ~5/4 = 391.9180{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6263{{c}}, ~5/4 = 391.7786{{c}}

{{Optimal ET sequence|legend=0| 12, 15, 27e, 31, 43, 58, 205cceff, 263ccdeefff }}

Badness (Sintel): 0.856

Projection pairs: <code>7 125/18</code> <code>11 3125/288</code> <code>13 41472/3125</code>

=== Nightingale ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 126/125, 176/175

Mapping: {{mapping| 1 0 0 -1 -5 -4 | 0 1 0 -2 -2 -1 | 0 0 1 3 5 4 }}

Optimal tunings:
* WE: ~2 = 1200.1476{{c}}, ~3/2 = 699.2886{{c}}, ~5/4 = 388.2054{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 699.2840{{c}}, ~5/4 = 388.2437{{c}}

{{Optimal ET sequence|legend=0| 12f, 15, 19e, 27eff, 31 }}

Badness (Sintel): 0.783

=== Veery ===
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 176/175

Mapping: {{mapping| 1 0 0 -1 -5 2 | 0 1 0 -2 -2 4 | 0 0 1 3 5 -2 }}

Optimal tunings:
* WE: ~2 = 1199.7998{{c}}, ~3/2 = 704.4117{{c}}, ~5/4 = 391.7571{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.3959{{c}}, ~5/4 = 391.6910{{c}}

{{Optimal ET sequence|legend=0| 12, 15, 19e, 27e, 31f, 46 }}

Badness (Sintel): 0.927

== Starnova ==
Starnova tempers out [[385/384]]. This temperament used to be known as ''undecimal starling'', but was decanonicalized in 2026. The name ''starnova'' was proposed by [[Flora Canou]] after [[Gene Ward Smith]]'s [[Star]] and [[Nova]] scales.

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 126/125, 385/384

{{Mapping|legend=1| 1 0 0 -1 8 | 0 1 0 -2 3 | 0 0 1 3 -4 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.6133{{c}}, ~3/2 = 700.4314{{c}}, ~5/4 = 388.3236{{c}}
: [[error map]]: {{val| +0.613 -0.910 +3.237 -2.878 -1.478 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4020{{c}}, ~5/4 = 388.3631{{c}}
: error map: {{val| 0.000 -1.553 +2.049 -4.541 -3.564 }}

{{Optimal ET sequence|legend=1| 12e, 15, 19, 27, 31, 46, 77, 96d, 127d, 173d, 204de }}

[[Badness]] (Sintel): 0.813

[[Projection pair]]s: <code>7 125/18</code> <code>11 6912/625</code>

Scales: [[Star and Nova]]

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384

Mapping: {{mapping| 1 0 0 -1 8 0 | 0 1 0 -2 3 -5 | 0 0 1 3 -4 5 }}

Optimal tunings:
* WE: ~2 = 1200.5083{{c}}, ~3/2 = 700.7452{{c}}, ~5/4 = 388.4655{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 700.5708{{c}}, ~5/4 = 388.4346{{c}}

{{Optimal ET sequence|legend=0| 15f, 19, 27, 31, 46, 77, 96d }}

Badness (Sintel): 1.07

== Thrasher ==
{{See also| Ptolemismic clan }}

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 56/55, 100/99

{{Mapping|legend=1| 1 0 0 -1 2 | 0 1 0 -2 -2 | 0 0 1 3 2 }}

Mapping to lattice: [{{val| 0 1 0 -2 -2 }}, {{val| 0 1 1 1 0 }}]

Lattice basis:
: ~6/5 length = 0.9089, ~5/4 length = 1.2007
: angle (~6/5, ~5/4) = 98.8447°

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.9183{{c}}, ~3/2 = 706.1793{{c}}, ~5/4 = 394.5223{{c}}
: [[error map]]: {{val| -3.082 +1.143 +2.045 -6.863 +13.041 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 706.2523{{c}}, ~5/4 = 392.7970{{c}}
: error map: {{val| 0.000 +4.297 +6.483 -2.940 +21.771 }}

[[Minimax tuning]]:
* [[11-odd-limit]]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 3/4 0 1/4 -3/8 }}, {{monzo| 1 1/2 0 1/2 -1/4 }}, {{monzo| 0 0 0 1 0 }}, {{monzo| 2 -1/2 0 1/2 1/4 }}]
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7.11/9

{{Optimal ET sequence|legend=1| 7d, 8d, 12, 15, 19, 27e }} *

<nowiki>*</nowiki> [[optimal patent val]]: [[34edo|34]]

[[Badness]] (Sintel): 0.577

Scales: [[Thrasher chromatic]], [[Thrasher diatonic]]

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 100/99

Mapping: {{mapping| 1 0 0 -1 2 2 | 0 1 0 -2 -2 4 | 0 0 1 3 2 -2 }}

Optimal tunings:
* WE: ~2 = 1197.4723{{c}}, ~3/2 = 707.3847{{c}}, ~5/4 = 394.2040{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 707.1440{{c}}, ~5/4 = 392.7909{{c}}

{{Optimal ET sequence|legend=0| 7d, 8d, 12, 15, 19, 27e, 69bceef }} *

<nowiki>*</nowiki> optimal patent val: [[34edo|34]]

Badness (Sintel): 0.820

=== Mockingbird ===
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 100/99

Mapping: {{mapping| 1 0 0 -1 2 3 | 0 1 0 -2 -2 -1 | 0 0 1 3 2 1 }}

Optimal tunings:
* WE: ~2 = 1193.3128{{c}}, ~3/2 = 710.0779{{c}}, ~5/4 = 400.7221{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 711.2922{{c}}, ~5/4 = 398.0867{{c}}

{{Optimal ET sequence|legend=0| 7d, 8d, 12f, 15, 27eff }}

Badness (Sintel): 0.804

=== Catbird ===
Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 100/99

Mapping: {{mapping| 1 0 0 -1 2 0 | 0 1 0 -2 -2 -5 | 0 0 1 3 2 5 }}

Optimal tunings:
* WE: ~2 = 1196.8488{{c}}, ~3/2 = 704.8749{{c}}, ~5/4 = 394.9837{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9205{{c}}, ~5/4 = 393.2290{{c}}

{{Optimal ET sequence|legend=0| 7df, 8d, 12f, 19, 27e, 66cdeeef }}

Badness (Sintel): 0.846

== Aplonis ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 126/125, 540/539

{{Mapping|legend=1| 1 0 0 -1 4 | 0 1 0 -2 7 | 0 0 1 3 -5 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6006{{c}}, ~3/2 = 700.5746{{c}}, ~5/4 = 390.2558{{c}}
: [[error map]]: {{val| -0.399 -1.780 +3.143 -0.406 +1.026 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 700.4342{{c}}, ~5/4 = 389.9796{{c}}
: error map: {{val| 0.000 -1.521 +3.666 +0.244 +1.824 }}

{{Optimal ET sequence|legend=1| 12e, 19, 27e, 31, 58, 89, 197c, 228c }}

[[Badness]] (Sintel): 0.778

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195

Mapping: {{mapping| 1 0 0 -1 4 0 | 0 1 0 -2 7 -5 | 0 0 1 3 -5 5 }}

Optimal tunings:
* WE: ~2 = 1199.0609{{c}}, ~3/2 = 701.2333{{c}}, ~5/4 = 390.8385{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.0203{{c}}, ~5/4 = 390.2413{{c}}

{{Optimal ET sequence|legend=0| 8d, 19, 27e, 31, 50, 58, 166cef, 224ceeff }}

Badness (Sintel): 0.768

== Treecreeper ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 126/125, 1232/1215

{{Mapping|legend=1| 1 0 0 -1 -3 | 0 1 0 -2 7 | 0 0 1 3 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.8957{{c}}, ~3/2 = 704.9055{{c}}, ~5/4 = 392.8725{{c}}
: [[error map]]: {{val| -1.104 +1.846 +4.350 -3.332 -2.725 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 705.0045{{c}}, ~5/4 = 392.3159{{c}}
: error map: {{val| 0.000 +3.050 +6.002 -1.887 -0.918 }}

{{Optimal ET sequence|legend=1| 7d, 12e, 19e, 27e, 39d, 46, 119c }}

[[Badness]] (Sintel): 1.91

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 126/125, 352/351

Mapping: {{mapping| 1 0 0 -1 -3 2 | 0 1 0 -2 7 4 | 0 0 1 3 -2 -2 }}

Optimal tunings:
* WE: ~2 = 1199.7878{{c}}, ~3/2 = 704.6703{{c}}, ~5/4 = 391.3513{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.7011{{c}}, ~5/4 = 391.2846{{c}}

{{Optimal ET sequence|legend=0| 7d, 12e, 19e, 27e, 46 }}

Badness (Sintel): 1.48

[[Category:Temperament families]]
[[Category:Starling family| ]] 
[[Category:Rank 3]]

Xenial

2026-05-04T12:04:52Z

Xenllium:

{{Infobox regtemp
| Title = Xenial
| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
| Comma basis = [[126/125]], [[177147/175616]] (7-limit); [[126/125]], [[162/161]], [[169/168]], [[171/170]], [[221/220]], [[231/230]], [[256/255]] (23-limit)
| Edo join 1 = 19 | Edo join 2 = 70
| Mapping = 1; -9 -17 -33 22 -21 26 27 -3
| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]]
| Pergen = (P8, P11/9)
| Odd limit 1 = 7 | Mistuning 1 = 4.6 | Complexity 1 = 51
}}
'''Xenial''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a sharpened minor whole tone of [[~]][[10/9]], so that nine generators reach [[4/3]], 17 reach [[8/5]], 21 reach [[16/13]] and 33 reach [[8/7]] with octave reduction. It is also generated by dividing [[11/1|11th harmonic]] into 22 equal parts, [[17/1|17th harmonic]] into 26 equal parts, or [[19/1|19th harmonic]] into 27 equal parts.

See [[Starling temperaments #Xenial]] for more technical data.

== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.000
| [[1/1]]
|-
| 1
| 188.775
| [[10/9]], [[19/17]], [[28/25]]
|-
| 2
| 377.551
|
|-
| 3
| 566.326
| [[18/13]], [[32/23]]
|-
| 4
| 755.102
| [[17/11]], [[20/13]]
|-
| 5
| 943.877
| [[19/11]], [[26/15]]
|-
| 6
| 1132.653
| [[23/12]], [[27/14]]
|-
| 7
| 121.428
|
|-
| 8
| 310.204
| [[6/5]]
|-
| 9
| 498.979
| [[4/3]]
|-
| 10
| 687.755
| [[40/27]]
|-
| 11
| 876.530
|
|-
| 12
| 1065.306
| [[13/7]], [[24/13]]
|-
| 13
| 54.081
| [[26/25]], [[33/32]]
|-
| 14
| 242.857
| [[23/20]]
|-
| 15
| 431.632
| [[9/7]], [[23/18]]
|-
| 16
| 620.408
| [[10/7]]
|-
| 17
| 809.183
| [[8/5]]
|-
| 18
| 997.959
| [[16/9]], [[23/13]]
|-
| 19
| 1186.734
|
|-
| 20
| 175.510
|
|-
| 21
| 364.285
| [[16/13]], [[26/21]]
|-
| 22
| 553.061
| [[11/8]]
|-
| 23
| 741.836
| [[23/15]]
|-
| 24
| 930.612
| [[12/7]]
|-
| 25
| 1119.387
| [[40/21]], [[44/23]], [[48/25]]
|-
| 26
| 108.163
| [[16/15]], [[17/16]]
|-
| 27
| 296.938
| [[19/16]]
|-
| 28
| 485.714
|
|-
| 29
| 674.439
| [[34/23]]
|-
| 30
| 863.265
| [[38/23]], [[23/14]]
|-
| 31
| 1052.040
| [[11/6]], [[46/25]]
|-
| 32
| 40.815
|
|-
| 33
| 229.591
| [[8/7]]
|-
| 34
| 418.366
| [[32/25]]
|}
<nowiki/>* In 23-limit CWE tuning

== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.8535 {{c}}
| CWE: ~10/9 = 188.8544 {{c}}
| POTE: ~10/9 = 188.8548 {{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7849 {{c}}
| CWE: ~10/9 = 188.7755 {{c}}
| POTE: ~10/9 = 188.7744 {{c}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]
! Generator (¢)
! Comments
|-
|
| 9/5
| 182.404
|
|-
|
| 13/10
| 186.447
|
|-
| 5 ⧵ 32
|
| 187.500
| 32cddefgh val Lower bound of 7-odd-limit diamond monotone
|-
|
| 23/12
| 187.720
|
|-
|
| 13/9
| 187.794
|
|-
|
| 23/13
| 188.208
|
|-
| 8 ⧵ 51
|
| 188.235
| 51cdh val Lower bound of 9-odd-limit diamond monotone
|-
|
| 23/18
| 188.291
|
|-
|
| 17/11
| 188.409
|
|-
|
| 13/12
| 188.452
|
|-
|
| 15/14
| 188.492
|
|-
|
| 13/8
| 188.546
|
|-
| 11 ⧵ 70
|
| 188.571
| Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 7/5
| 188.593
|
|-
|
| 17/13
| 188.605
|
|-
|
| 21/20
| 188.621
|
|-
|
| 13/11
| 188.623
|
|-
|
| 23/14
| 188.648
|
|-
|
| 17/16
| 188.652
|
|-
|
| 23/21
| 188.654
|
|-
|
| 17/12
| 188.657
|
|-
|
| 17/9
| 188.660
|
|-
|
| 3/2
| 188.672
|
|-
|
| 11/9
| 188.685
|
|-
|
| 19/13
| 188.687
|
|-
|
| 11/6
| 188.689
|
|-
|
| 23/15
| 188.6959
|
|-
|
| 11/8
| 188.6963
|
|-
|
| 23/20
| 188.711
|
|-
|
| 21/17
| 188.738
|
|-
|
| 19/18
| 188.747
|
|-
|
| 17/14
| 188.748
|
|-
|
| 21/11
| 188.758
|
|-
| 14 ⧵ 89
|
| 188.764
| 19, 21 and 23-odd-limit diamond monotone (singleton)
|-
|
| 19/12
| 188.766
|
|-
|
| 11/7
| 188.773
|
|-
|
| 17/15
| 188.782
|
|-
|
| 21/16
| 188.791
|
|-
|
| 21/19
| 188.793
|
|-
|
| 19/16
| 188.797
|
|-
|
| 17/10
| 188.806
|
|-
|
| 19/14
| 188.811
|
|-
|
| 15/11
| 188.814
|
|-
|
| 7/4
| 188.823
|
|-
|
| 11/10
| 188.846
|
|-
|
| 23/17
| 188.851
|
|-
|
| 19/15
| 188.854
|
|-
|
| 7/6
| 188.880
|
|-
|
| 19/10
| 188.891
|
|-
|
| 15/8
| 188.913
|
|-
|
| 23/22
| 188.922
|
|-
|
| 23/19
| 188.975
|
|-
|
| 9/7
| 189.006
|
|-
|
| 21/13
| 189.036
|
|-
|
| 5/4
| 189.040
|
|-
|
| 19/11
| 189.239
|
|-
|
| 13/7
| 189.308
|
|-
|
| 5/3
| 189.455
|
|-
| 3 ⧵ 19
|
| 189.473
| Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 15/13
| 190.452
|
|-
|
| 23/16
| 190.575
|
|-
|
| 19/17
| 192.558
|
|}

[[Category:Xenial| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]

Template:Infobox regtemp/doc

2026-05-04T08:16:20Z

Xenllium:

{{Dochead}}{{substitute|no}}
{{Template rating|rating=ready}}
Infobox for [[regular temperament]]s. Support up to rank 4. As of now, most information has to be entered manually.

== Parameters ==
; Title
: The name of the temperament goes here, and it appears as the title of the infobox.
; Subgroups
: Written here is the [[just intonation subgroup|subgroup(s)]] that the temperament acts on; often a full prime limit.
; Comma basis
: Type the simplest list of commas that defines the temperament in the subgroup(s). Format as:
: <code>comma 1, comma 2, … (subgroup 1); <nowiki> </nowiki>comma 1, comma 2, (subgroup 2); …</code>
; Edo join 1, 2, 3, 4
: The [[equal temperament]]s that reasonably [[support]] the temperament. Despite these parameters being called "edo", they accept [[wart notation]] just fine.
; Mapping
: The number of generators each element of the subgroup is mapped to; see [[Reduced mapping]].
; Generators
: The list of intervals that generates the temperament, in [[ratio]] of [[monzo]] form.
; Generators tuning
: Sizes of the generators in an optimal tuning, in [[cent]]s.
; Optimization method
: The method used to calculate the optimal tuning above. Type the acronym, such as CWE or POTE.
; MOS scales
: For rank-2 temperaments, the sequence of [[mos scale]]s induced by the generator.
; Ploidacot
: For rank-2 temperaments, the [[ploidacot]] signature. Derived from '''Subgroups''', '''Mapping''', and '''Generators tuning''', or overriden here.
; Pergen
: The period and the generator as fractions of simple intervals (most often the octave and a possibly compounded fifth or fourth), if provided.
; Color name
: [[Kite's color notation/Temperament names|Kite's color name]] for the temperament, if provided.
; Odd limit 1, 2, 3, 4
: Reasonable odd limits to use in the temperament.
; Mistuning 1, 2, 3, 4
: [[Minimax|Minimax errors]] of the odd limits.
; Complexity 1, 2, 3, 4
: Smallest numbers of notes for mos scales to contain the odd limits (i.e. greater than the [[Graham complexity|Graham complexities]] of the odd limits).

=== Boilerplate ===
Empty template for copy & paste to avoid copying wrong values from existing temperament pages:

<pre>
{{Infobox regtemp
| Title =
| Subgroups =
| Comma basis =
| Edo join 1 = | Edo join 2 = | Edo join 3 = | Edo join 4 =
| Mapping =
| Generators =
| Generators tuning =
| Optimization method =
| MOS scales =
| Ploidacot =
| Pergen =
| Color name =
| Odd limit 1 = | Mistuning 1 = | Complexity 1 =
| Odd limit 2 = | Mistuning 2 = | Complexity 2 =
| Odd limit 3 = | Mistuning 3 = | Complexity 3 =
| Odd limit 4 = | Mistuning 4 = | Complexity 4 =
}}
</pre>

Template:Infobox regtemp

2026-05-04T08:13:52Z

Xenllium:

<includeonly>{{#invoke: Infobox_regtemp
| infobox_RT
| tempname={{{Title|{{PAGENAME}}}}}
| subgroup={{{Subgroups|}}}
| commas={{{Comma basis|}}}
| edo_first={{{Edo join 1|}}}
| edo_second={{{Edo join 2|}}}
| edo_third={{{Edo join 3|}}}
| edo_fourth={{{Edo join 4|}}}
| mapping={{{Mapping|}}}
| genfrac={{{Generators|}}}
| tuning={{{Generators tuning|}}}
| method={{{Optimization method|}}}
| mosses={{{MOS scales|}}}
| ploidacot={{{Ploidacot|}}}
| pergen={{{Pergen|}}}
| colorname={{{Color name|}}}
| lim1={{{Odd limit 1|}}}
| acc1={{{Mistuning 1|}}}
| comp1={{{Complexity 1|}}}
| lim2={{{Odd limit 2|}}}
| acc2={{{Mistuning 2|}}}
| comp2={{{Complexity 2|}}}
| lim3={{{Odd limit 3|}}}
| acc3={{{Mistuning 3|}}}
| comp3={{{Complexity 3|}}}
| lim4={{{Odd limit 4|}}}
| acc4={{{Mistuning 4|}}}
| comp4={{{Complexity 4|}}}
}}</includeonly><noinclude>
{{Documentation}}

[[Category:Infoboxes]]
</noinclude>

Module:Infobox regtemp

2026-05-04T08:12:19Z

Xenllium:

local p = {}

local infobox = require("Module:Infobox")
local u = require("Module:Utils")
local yesno = require("Module:Yesno")

function p.infobox_RT(frame)

local name = frame.args["tempname"]

local subgroup = frame.args["subgroup"]
local basis = frame.args["commas"]

local et1 = frame.args["edo_first"]
local et2 = frame.args["edo_second"]
local et3 = frame.args["edo_third"]
local et4 = frame.args["edo_fourth"]

local tuning = frame.args["tuning"]
local genfrac = frame.args["genfrac"]
local method = frame.args["method"]
local mapping = frame.args["mapping"]

local mos_override = frame.args["mosses"]
local ploidacot_override = frame.args["ploidacot"]
local colorname = frame.args["colorname"]
local pergen = frame.args["pergen"]

local limit1 = frame.args["lim1"]
local comp1 = frame.args["comp1"]
local acc1 = frame.args["acc1"]
local limit2 = frame.args["lim2"]
local comp2 = frame.args["comp2"]
local acc2 = frame.args["acc2"]
local limit3 = frame.args["lim3"]
local comp3 = frame.args["comp3"]
local acc3 = frame.args["acc3"]
local limit4 = frame.args["lim4"]
local comp4 = frame.args["comp4"]
local acc4 = frame.args["acc4"]
local debugg = frame.args["debug"]

local data = {}

-- process mapping
local rank = p.count_matches(mapping, ";") + 1
local period = string.match(mapping, "(-?%d+);")
local genchain_mapping = {}
for str in mapping:gmatch("(-?%d+)") do
table.insert(genchain_mapping, str)
end

-- interpret subgroup
local subgroup_basis = {}
for str in subgroup:gmatch("(%d+/?%d*)") do
table.insert(subgroup_basis, str)
end

local equave = u.eval_num_arg(subgroup_basis[1])

-- process et join
function process_et_join(et)
local et_eq, et_num = p.warts2ed(et)
local et_eq_letter = p.equave2letter (et_eq)
return "[[" .. et_num .. "ed" .. et_eq_letter .. "|" .. et .. "]]"
end

local et_table = {}
for _, et in ipairs({et1, et2, et3, et4}) do
if u.value_provided(et) then
table.insert(et_table, process_et_join (et))
end
end
local et_join = table.concat(et_table, " & ")

-- process generators
function process_generators(genfrac_item, tuning_item)
return "~" .. genfrac_item .. " = " .. tuning_item .. "{{c}}"
end

local genfrac_table = {}
for str in genfrac:gmatch("(%d+/?%d*)") do
table.insert(genfrac_table, str)
end
local tuning_table = {}
for str in tuning:gmatch("(-?%d+%.?%d*)") do
table.insert(tuning_table, str)
end
local generators_table = {}
for i = 1, math.min(#genfrac_table, #tuning_table) do
table.insert(generators_table,
process_generators (genfrac_table[i], tuning_table[i]))
end
local generators = table.concat (generators_table, ", ")

-- default value for mos
local mos = "n/a"

-- autocalculating ploidacot
if rank == 2 then
local ploid = tonumber(period)
local referent = u.eval_num_arg(subgroup_basis[2])
local cot = tonumber(genchain_mapping[2])
local suffix = "cleft"
if equave == 2 then -- octave-based temp
if referent == 3 then
suffix = "cot"
elseif referent == 5 then
suffix = "seph"
end
elseif equave == 3 then -- twelfth-based temp
if referent == 5 then
referent_candidate = u.eval_num_arg(subgroup_basis[3])
if referent_candidate == 7 then
referent = referent_candidate
cot = tonumber(genchain_mapping[3])
suffix = "gem"
end
end
end

if suffix == "cleft" then
suffix = referent .. suffix
end

local equave_size = 1200 * u.log2(equave)
local referent_size = 1200 * u.log2(referent)
local period_size = equave_size / ploid
local generator_size = tonumber(tuning_table[1])
if cot < 0 then
generator_size = equave_size - generator_size
cot = -cot
end

-- find the shear
local shear
if cot ~= 0 then
shear = (math.floor(generator_size * cot / period_size)
- math.floor(referent_size % equave_size / period_size)) % cot
else
shear = 0
end

-- omega extension
if ploid == 1 and cot > 2 and shear == cot - ploid then
shear = shear - cot
end

-- construct the ploidacot signature
local MAX_GREEK_NUMERAL = 12 -- max number to convert to greek letters/numerals

local ploid_sig
if ploid == 1 then
ploid_sig = "" -- omit the ploid part
elseif ploid <= MAX_GREEK_NUMERAL then
ploid_sig = p.num2greek(tostring(ploid), "ploid") .. "ploid "
else
ploid_sig = tostring (ploid) .. "-ploid "
end

local shear_sig
if shear <= MAX_GREEK_NUMERAL then
shear_sig = p.num2greeklet(tostring(shear))
else
shear_sig = tostring(shear) .. "-sheared "
end

local cot_sig
if cot <= MAX_GREEK_NUMERAL then
cot_sig = p.num2greek(tostring(cot), "cot") .. suffix
else
cot_sig = tostring(cot) .. "-" .. suffix
end

ploidacot = ploid_sig .. shear_sig .. cot_sig
else
ploidacot = "n/a"
end

-- user override
if u.value_provided (mos_override) then
mos = mos_override
end
if u.value_provided (ploidacot_override) then
ploidacot = ploidacot_override
end

-- construct the table
table.insert(data, {
"Subgroups",
subgroup
})

table.insert(data, {
"Comma basis",
basis
})

table.insert(data, {
"Reduced mapping",
"⟨" .. mapping .. "]"
})

table.insert(data, {
"ET join",
et_join
})

table.insert(data, {
"Generators ([[" .. method .. "]])",
generators
})

table.insert(data, {
"MOS scales",
mos
})

table.insert(data, {
"Ploidacot",
ploidacot
})

if u.value_provided (pergen) then
table.insert(data, {
"Pergen",
pergen
})
end

if u.value_provided (colorname) then
table.insert(data, {
"Color name",
colorname
})
end

-- error and stuff
local limit_text
if equave == 2 then
limit_text = "-odd-limit"
elseif equave == 3 then
limit_text = "-throdd-limit"
else
limit_text = "-integer-limit"
end
local string1 = limit1 .. limit_text .. ": " .. acc1 .. "{{c}}"
local string2 = limit1 .. limit_text .. ": " .. comp1 .. " notes"
if u.value_provided (limit2) then
string1 = string1 .. "; " .. limit2 .. limit_text .. ": " .. acc2 .. "{{c}}"
string2 = string2 .. "; " .. limit2 .. limit_text .. ": " .. comp2 .. " notes"
end
if u.value_provided (limit3) then
string1 = string1 .. "; " .. limit3 .. limit_text .. ": " .. acc3 .. "{{c}}"
string2 = string2 .. "; " .. limit3 .. limit_text .. ": " .. comp3 .. " notes"
end
if u.value_provided (limit4) then
string1 = string1 .. "; " .. limit4 .. limit_text .. ": " .. acc4 .. "{{c}}"
string2 = string2 .. "; " .. limit4 .. limit_text .. ": " .. comp4 .. " notes"
end

table.insert(data, {
"Minimax error",
string1
})

table.insert(data, {
"Target scale size",
string2
})

local result = infobox.build(
name,
data
)

return frame:preprocess(debugg == true and "<pre>" .. result .. "</pre>" or result)
end

function p.count_matches(base, pattern)
return select(2, string.gsub(base, pattern, ""))
end

function p.digit2greek(number, purpose)

local greek

if number == "1" then
if purpose == "ploid" then
greek = "ha"
elseif purpose == "cot" then
greek = "mono"
elseif purpose == "decade" then
greek = "deca"
else
greek = "hen"
end
elseif number == "2" then
if purpose == "decade" then
greek = "icosa"
elseif purpose == "unit" then
greek = "do"
else
greek = "di"
end
elseif number == "0" then
if purpose == "unit" then
greek = ""
else
greek = "a"
end
else
if number == "3" then
if purpose == "decade" then
greek = "tria"
else
greek = "tri"
end
elseif number == "4" then
if purpose == "decade" then
greek = "tessera"
else
greek = "tetra"
end
elseif number == "5" then
if purpose == "decade" then
greek = "pente"
else
greek = "penta"
end
elseif number == "6" then
if purpose == "decade" then
greek = "hexe"
else
greek = "hexa"
end
elseif number == "7" then
if purpose == "decade" then
greek = "hebdome"
else
greek = "hepta"
end
elseif number == "8" then
if purpose == "decade" then
greek = "ogdoe"
elseif purpose == "unit" then
greek = "octo"
else
greek = "octa"
end
elseif number == "9" then
if purpose == "decade" then
greek = "enene"
else
greek = "ennea"
end
end
if purpose == "decade" then
greek = greek .. "conta"
end
end

return greek
end

function p.num2greek(number, purpose)

local greek

if string.len(number) == 1 then
greek = p.digit2greek(number, purpose)
elseif string.len(number) == 2 then
local unit = string.char(string.byte(number, 2))
local decade = string.char(string.byte(number, 1))

greek = p.digit2greek(unit, "unit") .. p.digit2greek(decade, "decade")
end
return greek
end

function p.digit2greeklet(number, purpose)

local greek

if number == "1" then
if purpose == "decade" then
greek = "iota-"
elseif purpose == "unit" then
greek = "alpha-"
end
elseif number == "2" then
if purpose == "decade" then
greek = "kappa-"
elseif purpose == "unit" then
greek = "beta-"
end
elseif number == "3" then
if purpose == "decade" then
greek = "lambda-"
elseif purpose == "unit" then
greek = "gamma-"
end
elseif number == "4" then
if purpose == "decade" then
greek = "mu-"
elseif purpose == "unit" then
greek = "delta-"
end
elseif number == "5" then
if purpose == "decade" then
greek = "nu-"
elseif purpose == "unit" then
greek = "epsilon-"
end
elseif number == "6" then
if purpose == "decade" then
greek = "xi-"
elseif purpose == "unit" then
greek = "wau-"
end
elseif number == "7" then
if purpose == "decade" then
greek = "omicron-"
elseif purpose == "unit" then
greek = "zeta-"
end
elseif number == "8" then
if purpose == "decade" then
greek = "pi-"
elseif purpose == "unit" then
greek = "eta-"
end
elseif number == "9" then
if purpose == "decade" then
greek = "qoppa-"
elseif purpose == "unit" then
greek = "theta-"
end
elseif number == "0" then
greek = ""
elseif number == "-1" then
greek = "omega-"
end

return greek
end

function p.num2greeklet(number)

local greek

if string.len(string.match(number, "(%d+)")) == 1 then
greek = p.digit2greeklet(number, "unit")
elseif string.len(string.match(number, "(%d+)")) == 2 then
local unit = string.char(string.byte(number, 2))
local decade = string.char(string.byte(number, 1))

greek = p.digit2greeklet(decade, "decade") .. p.digit2greeklet(unit, "unit")
end
return greek
end

function p.equave2letter(et_eq)
-- converts an equave string to a letter
local et_eq_letter
if et_eq == "2" then
et_eq_letter = "o"
elseif et_eq == "3" then
et_eq_letter = "t"
elseif et_eq == "3/2" then
et_eq_letter = "f"
else
et_eq_letter = et_eq
end
return et_eq_letter
end

function p.warts2ed(warts)
-- converts a wart notation to an equal division
local wart_prefix, et_num = warts:match ("(%a?)(%d+)")
local et_eq = "2"
if wart_prefix == "a" then
et_eq = "2"
elseif wart_prefix == "b" then
et_eq = "3"
elseif wart_prefix == "c" then
et_eq = "5"
elseif wart_prefix == "d" then
et_eq = "7"
elseif wart_prefix == "e" then
et_eq = "11"
elseif wart_prefix == "f" then
et_eq = "13"
elseif wart_prefix == "g" then
et_eq = "17"
elseif wart_prefix == "h" then
et_eq = "19"
elseif wart_prefix == "i" then
et_eq = "23"
elseif wart_prefix == "j" then
et_eq = "29"
elseif wart_prefix == "k" then
et_eq = "31"
end
return et_eq, et_num
end

return p

Xenial

2026-05-04T07:58:33Z

Xenllium:

Xenial

2026-05-03T11:03:34Z

Xenllium:

19edo

2026-05-03T10:44:03Z

Xenllium:

{{interwiki
| de = 19-EDO
| en = 19edo
| es = 19 EDO
| ja = 19平均律
}}
{{Infobox ET}}
{{Wikipedia|19 equal temperament}}
{{ED intro}}

== History ==
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson [[Seigneur Dieu ta pitié]] of 1558. Costeley understood and desired the circulating aspect of this tuning, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577, music theorist Francisco de Salinas proposed [[1/3-comma meantone]], in which the fifth is 694.786{{c}}; the fifth of 19edo is 694.737{{c}}, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as [[50edo|50 equal temperament]] ([http://www.tonalsoft.com/sonic-arts/monzo/woolhouse/essay.htm summary of Woolhouse's essay]).

== Theory ==
19edo is the second edo, after [[12edo]] which is able to approximate [[5-limit]] intervals and chords with tolerable accuracy (unless you count [[15edo]], which has a 18-[[cent]]-sharp fifth). Having an almost just minor third and perfect fifths and major thirds about 7 cents flat, it serves as a good tuning for [[meantone]]. Unlike 12edo, where [[enharmonic]] notes are conflated, 19edo distinguishes them, and differs from [[17edo]] in that its [[diatonic semitone]] is wider than the [[chromatic semitone]], rather than narrower. In fact, it is nearly identical to the enharmonic scale of [[1/3-comma meantone]], and can be considered a closed form thereof.

It is less successful in the [[7-limit]] as it conflates the septimal subminor third ([[7/6]]) with the septimal whole tone ([[8/7]]), but it is still better than 12edo overall.

=== Prime harmonics ===
{{Harmonics in equal|19|columns=12}}

=== As an approximation of other temperaments ===
Besides meantone, 19edo is also suitable for [[magic]]/[[muggles]] temperament, because five of its major thirds are equivalent to one of its twelfths. Its 7-step supermajor third can be used for [[sensi]], whose generator is a very sharp major third, two of which make an approximate 5/3 major sixth.

For all of these there are more optimal tunings: the fifth of 19edo is flatter than the usual for meantone, and [[31edo]] is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; [[41edo]] more closely matches it. It does make for a good tuning for muggles, but in 19edo it is the same as magic. Finally, 19edo can be used as a tuning for sensi, though [[46edo]] provides a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19edo also has the advantage of being excellent for [[negri]], [[keemun]], [[godzilla]], muggles, and [[triton]]/[[liese]]. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their [[mos scale]]s in 19edo offering a great abundance of septimal tetrads. The [[Graham complexity]] of the [[4:5:6:7|7-odd-limit tetrad]] is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles, and 13 for sensi.

=== As a means of extending harmony ===
Because 19edo's 5-limit chords are more blended and concordant than those of 12edo, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. [[William Lynch]] suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non-diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

In addition, [[Joseph Yasser]] talks about the idea of a 12-tone supra-diatonic scale where the 7-tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra-diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of [[Bozuji tuning]], a 21st century tuning based on Gioseffo Zarlino's approach to just intonation. with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

=== Adaptive tuning ===
The no-11's 13-limit is represented relatively well and consistently. 19edo's negri, sensi and godzilla scales have many 13-limit chords. (You can think of the Sensi[8] [[3L 5s]] mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but Sensi[8] gives you additional ratios of 7 and 13.) Its diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3{{c}} off [[23/16]].

Practically 19edo can be used ''adaptively'' on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th, and 13th harmonics are all tuned flat. This is in contrast to 12edo, where this is not possible since the 5th and 7th harmonics are not only much farther from just than they are in 19edo, but fairly sharp already.

Another option would be to use [[octave stretching]], which has similar benefits to adaptive use, but it also works for fixed-pitch Instruments. For more on that see the [[19edo#Octave stretch or compression|section on octave stretch]].

=== Subsets and supersets ===
19edo is the 8th [[prime edo]], following [[17edo]] and preceding [[23edo]]. As such, it does not contain any nontrivial subset edos, though it contains [[19ed4]].

[[38edo]], which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See [[undevigintone]]. [[57edo]] effectively corrects the harmonic 7 to just, although it is [[76edo]] that fits the best. See [[meanmag]].

=== Miscellaneous properties ===
19edo has the flattest possible fifth of any edo that can possibly be [[diamond monotone]] in the [[15-odd-limit]]. Using 11\19 as the fifth, the sharpest possible mapping of [[5/4]] where [[10/9]] is no greater than [[9/8]] is 6\19, so the sharpest possible [[15/8]] is 17\19. Here [[16/15]] is a quarter of [[4/3]] (as in any [[negri]] tuning), so [[15/14]], [[14/13]], and [[13/12]] must all be equated with [[16/15]] to 2\19 for them to be monotone in size. If the fifth were any flatter, 5/4 and 15/8 would have to be flatter, and 16/15 would have to be sharper, so diamond monotone in the 15-odd-limit becomes impossible. 19edo is, in fact, diamond monotone in the 15-odd-limit, and even the [[17-odd-limit]] (see [[Monotonicity limits of small EDOs]]). The sharpest fifth where 15-odd-limit is possible is [[37edo#Miscellaneous properties|22\37]].

== Intervals ==
{| class="wikitable right-1 right-2"
|-
! [[Degree|#]]
! [[Cent]]s
! [[Interval category|Interval categories]]
! Approximated ratios<ref group="note">As a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament.</ref>
|-
| 0
| 0.0
| Unison (prime)
| [[1/1]]
|-
| 1
| 63.2
| Augmented unison
| [[25/24]], [[26/25]], [[27/26]], [[28/27]]
|-
| 2
| 126.3
| Minor second
| [[13/12]], [[14/13]], [[15/14]], [[16/15]]
|-
| 3
| 189.5
| Major second
| [[9/8]], [[10/9]]
|-
| 4
| 252.6
| Augmented second Diminished third
| [[7/6]], [[8/7]], [[15/13]]
|-
| 5
| 315.8
| Minor third
| [[6/5]]
|-
| 6
| 378.9
| Major third
| [[5/4]], [[16/13]], [[56/45]]
|-
| 7
| 442.1
| Augmented third
| [[9/7]], [[13/10]], [[21/16]], [[32/25]]
|-
| 8
| 505.3
| Perfect fourth
| [[4/3]], [[75/56]]
|-
| 9
| 568.4
| Augmented fourth (Small [[tritone]])
| [[7/5]], [[18/13]], [[25/18]]
|-
| 10
| 631.6
| Diminished fifth (Large [[tritone]])
| [[10/7]], [[13/9]], [[36/25]]
|-
| 11
| 694.7
| Perfect fifth
| [[3/2]], [[112/75]]
|-
| 12
| 757.9
| Augmented fifth
| [[14/9]], [[20/13]], [[25/16]], [[32/21]]
|-
| 13
| 821.1
| Minor sixth
| [[8/5]], [[13/8]], [[45/28]]
|-
| 14
| 884.2
| Major sixth
| [[5/3]]
|-
| 15
| 947.4
| Augmented sixth Diminished seventh
| [[7/4]], [[12/7]], [[26/15]]
|-
| 16
| 1010.5
| Minor seventh
| [[9/5]], [[16/9]]
|-
| 17
| 1073.7
| Major seventh
| [[13/7]], [[15/8]], [[24/13]], [[28/15]]
|-
| 18
| 1136.8
| Augmented seventh
| [[25/13]], [[27/14]], [[48/25]], [[52/27]]
|-
| 19
| 1200.0
| Octave
| [[2/1]]
|}
<references group="note"/>

=== Proposed interval names and solfèges ===
{| class="wikitable right-1 right-2 center-3 center-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Table of proposed interval names and solfèges
|-
! #
! Cents
! [[Solfège]]
! colspan="2" | [[SKULO interval names]]
|-
| 0
| 0.0
| Do
| Unison
| P1
|-
| 1
| 63.2
| Di/Ro
| Super unison, subminor second
| S1, sm2
|-
| 2
| 126.3
| Ra
| Minor second
| m2
|-
| 3
| 189.5
| Re
| Major second
| M2
|-
| 4
| 252.6
| Ri/Ma
| Supermajor second, subminor third
| SM2, sm3
|-
| 5
| 315.8
| Me
| Minor third
| m3
|-
| 6
| 378.9
| Mi
| Major third
| M3
|-
| 7
| 442.1
| Mo/Fe
| Supermajor third, sub fourth
| SM3, s4
|-
| 8
| 505.3
| Fa
| Perfect fourth
| P4
|-
| 9
| 568.4
| Fi
| Augmented fourth
| A4
|-
| 10
| 631.6
| Se
| Diminished fifth
| d5
|-
| 11
| 694.7
| So
| Perfect fifth
| P5
|-
| 12
| 757.9
| Si/Lo
| Super fifth, subminor sixth
| S5, sm6
|-
| 13
| 821.1
| Le
| Minor sixth
| m6
|-
| 14
| 884.2
| La
| Major sixth
| M6
|-
| 15
| 947.4
| Li/Ta
| Supermajor sixth, subminor seventh
| SM6, sm7
|-
| 16
| 1010.5
| Te
| Minor seventh
| m7
|-
| 17
| 1073.7
| Ti
| Major seventh
| M7
|-
| 18
| 1136.8
| To/Da
| Supermajor seventh, sub octave
| SM7, s8
|-
| 19
| 1200.0
| Do
| Octave
| P8
|}

=== Interval quality and chord names in color notation ===
Using [[Kite's color notation]], qualities can be loosely associated with colors:

{| class="wikitable" style="text-align: center;"
|-
! Quality
! Color name
! Monzo format
! Examples
|-
| Diminished
| zo
| {{nowrap|(''a'', ''b'', 0, 1)}}
| 7/6, 7/4
|-
| rowspan="2" | Minor
| fourthward wa
| (''a'', ''b''), {{nowrap|''b'' < −1}}
| 32/27, 16/9
|-
| gu
| {{nowrap|(''a'', ''b'', −1)}}
| 6/5, 9/5
|-
| rowspan="2" | Major
| yo
| {{nowrap|(''a'', ''b'', 1)}}
| 5/4, 5/3
|-
| fifthward wa
| (''a'', ''b''), {{nowrap| ''b'' > 1 }}
| 9/8, 27/16
|-
| Augmented
| ru
| {{nowrap|(''a'', ''b'', 0, −1)}}
| 9/7, 12/7
|}

Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of B𝄫 would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A♯ might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished seconds, thirds, sixths and sevenths. Here are the zo, gu, yo and ru triads:

{| class="wikitable center-1 center-2 center-3 center-4"
|-
! Color of the third
! JI chord
! Edosteps
! Notes of C chord
! Written name
! Spoken name
|-
| zo (7-over)
| 6:7:9
| 0–4–11
| C–E𝄫–G
| Cm(♭3) or Cmin(♭3) or C(d3)
| C subminor, C minor flat-three, C dim-three
|-
| gu (5-under)
| 10:12:15
| 0–5–11
| C–E♭–G
| Cm or Cmin
| C minor
|-
| yo (5-over)
| 4:5:6
| 0–6–11
| C–E–G
| C or Cmaj
| C, C major
|-
| ru (7-under)
| 14:18:21
| 0–7–11
| C–E♯–G
| C(♯3) or Cmaj(♯3) or C(A3)
| C supermajor, C major sharp-three, C aug-three
|-
| yo (5-over)
| 4:5:6:7
| 0–6–11–15
| C–E–G–B𝄫
| Ch7 or C,d7 or Cadd(d7)
| C harmonic 7, C (major) add dim-seven
|-
| gu (5-under)
| 1/(12:10:8:7) (1–6/5–3/2–12/7)
| 0–5–11–15
| C–E♭–G–A♯
| Cm♯6 or CmA6 or Cm(add(♯6)) or Cm(add(A6))
| C minor (add) sharp-six, C minor (add) aug-six
|}

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo conflates zo and ru ratios.

For a more complete list, see [[19edo chords #Ups and downs notation]] and [[Kite's ups and downs notation #Chords and chord progressions]].

== Notation ==
=== Standard notation ===
Standard 12edo notation can be used, whether it is staff notation (with five lines), letter [[chain-of-fifths notation]] (with standard accidentals), solfège, or sargam. Note that D# and Eb are two different notes.

Any 19edo note or interval can be [[enharmonic unison|respelled enharmonically]] by adding a double-diminished second to it or subtracting one from it. Adding a dd2 is equivalent to finding the 12edo equivalent with a higher degree, then diminishing it. For example, C# becomes Db, which is diminished to become Dbb.

{| class="wikitable right-1 right-2 center-3 center-4"
|+ style="font-size: 105%;" | Notation of 19edo
|-
! rowspan="2" | [[Degree|#]]
! rowspan="2" | [[Cent]]s
! colspan="2" | [[Chain-of-fifths notation|Standard notation]]
|-
! [[5L 2s|Diatonic interval names]]
! Note names on D
|-
| 0
| 0.0
| '''Perfect unison (P1)'''
| '''D'''
|-
| 1
| 63.2
| Augmented unison (A1) Diminished second (d2)
| D# Ebb
|-
| 2
| 126.3
| Doubly augmented unison (AA1) Minor second (m2)
| Dx Eb
|-
| 3
| 189.5
| '''Major second (M2)''' Doubly diminished third (dd3)
| '''E''' Fbb
|-
| 4
| 252.6
| Augmented second (A2) Diminished third (d3)
| E# Fb
|-
| 5
| 315.8
| Doubly augmented second (AA2) '''Minor third (m3)'''
| Ex '''F'''
|-
| 6
| 378.9
| '''Major third (M3)''' Doubly diminished fourth (dd4)
| '''F#''' Gbb
|-
| 7
| 442.1
| Augmented third (A3) Diminished fourth (d4)
| Fx Gb
|-
| 8
| 505.3
| '''Perfect fourth (P4)'''
| '''G'''
|-
| 9
| 568.4
| Augmented fourth (A4) Doubly diminished fifth (dd5)
| G# Abb
|-
| 10
| 631.6
| Doubly augmented fourth (AA4) Diminished fifth (d5)
| Gx Ab
|-
| 11
| 694.7
| '''Perfect fifth (P5)'''
| '''A'''
|-
| 12
| 757.9
| Augmented fifth (A5) Diminished sixth (d6)
| A# Bbb
|-
| 13
| 821.1
| Doubly augmented fifth (AA5) Minor sixth (m6)
| Ax Bb
|-
| 14
| 884.2
| '''Major sixth (M6)''' Doubly diminished seventh (dd7)
| '''B''' Cbb
|-
| 15
| 947.4
| Augmented sixth (A6) Diminished seventh (d7)
| B# Cb
|-
| 16
| 1010.5
| Doubly augmented sixth (AA6) '''Minor seventh (m7)'''
| Bx '''C'''
|-
| 17
| 1073.7
| Major seventh (M7) Doubly diminished octave (dd8)
| C# Dbb
|-
| 18
| 1136.8
| Augmented seventh (A7) Diminished octave (d8)
| Cx Db
|-
| 19
| 1200.0
| '''Perfect octave (P8)'''
| '''D'''
|}

In 19edo:
* [[Ups and downs notation]] is identical to standard notation;
* Mixed [[sagittal notation]] is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp ([[File:Sagittal sharp.png]]) and sagittal flat ([[File:Sagittal flat.png]]) respectively.

{{Sharpness-sharp1}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[5edo #Sagittal notation|5]], [[12edo #Sagittal notation|12]], and [[26edo #Sagittal notation|26]], and is a subset of the notations for edos [[38edo #Sagittal notation|38]], [[57edo #Sagittal notation|57]], and [[76edo #Sagittal notation|76]].

==== Evo flavor ====
{{Sagittal chart|Evo}}

Because it includes no Sagittal symbols, this Evo Sagittal notation is identical to conventional notation.

==== Revo flavor ====
{{Sagittal chart}}

=== Dodecatonic notation ===
{| class="wikitable right-1 right-2 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Dodecatonic notation of 19edo
|-
! [[Degree|#]]
! [[Cent]]s
! Interval names
|-
| 0
| 0.0
| P1
|-
| 1
| 63.2
| A1, m2
|-
| 2
| 126.3
| M2, m3
|-
| 3
| 189.5
| M3
|-
| 4
| 252.6
| m4, A3
|-
| 5
| 315.8
| M4, m5
|-
| 6
| 378.9
| M5
|-
| 7
| 442.1
| A5, d6
|-
| 8
| 505.3
| P6
|-
| 9
| 568.4
| A6, m7
|-
| 10
| 631.6
| M7, d8
|-
| 11
| 694.7
| P8
|-
| 12
| 757.9
| A8, m9
|-
| 13
| 821.1
| M9, m10
|-
| 14
| 884.2
| M10
|-
| 15
| 947.4
| m11, A10
|-
| 16
| 1010.5
| M11, m12
|-
| 17
| 1073.7
| M12
|-
| 18
| 1136.8
| A12, d13
|-
| 19
| 1200.0
| P13
|}

== Approximation to JI ==
[[File:19ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 19-limit intervals approximated in 19edo]]

=== Interval mappings ===
{{Q-odd-limit intervals|19}}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -30 19 }}
| {{mapping| 19 30 }}
| +2.277
| 2.277
| 3.612
|-
| 2.3.5
| 81/80, 3125/3072
| {{mapping| 19 30 44 }}
| +2.578
| 1.911
| 3.025
|-
| 2.3.5.7
| 49/48, 81/80, 126/125
| {{mapping| 19 30 44 53 }}
| +3.848
| 2.755
| 4.362
|-
| 2.3.5.7.13
| 49/48, 65/64, 81/80, 91/90
| {{mapping| 19 30 44 53 70 }}
| +4.135
| 2.530
| 4.006
|-
| 2.3.5.7.13.23
| 49/48, 65/64, 70/69, 81/80, 91/90
| {{mapping| 19 30 44 53 70 86 }}
| +3.319
| 2.936
| 4.649
|}
* 19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit—''both'' 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next equal temperaments doing better in those subgroups are [[34edo|34]], [[31edo|31]], [[27edo|27e]], [[22edo|22]], and [[26edo|26]], respectively.
* 19et is best in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is [[53edo|53]].

=== Uniform maps ===
{{Uniform map|edo=19}}

=== Commas ===
19et [[tempering out|tempers out]] the following [[comma]]s. (Note: This assumes the [[val]] {{val| 19 30 44 53 66 70 }}.)

{| class="commatable wikitable center-all left-3 right-4 left-6"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cents]]
! [[Color notation/Temperament names|Color name]]
! Name
|-
| 3
| <abbr title="1162261467/1073741824">(20 digits)</abbr>
| {{monzo| -30 19 }}
| 137.14
| Trilawa
| [[19-comma]]
|-
| 5
| [[16875/16384]]
| {{monzo| -14 3 4 }}
| 51.12
| Laquadyo
| Negri comma
|-
| 5
| <abbr title="1594323/1562500">(14 digits)</abbr>
| {{monzo| -2 13 -8}}
| 34.91
| Laquadbigu
| [[Unicorn comma]]
|-
| 5
| [[3125/3072]]
| {{monzo| -10 -1 5 }}
| 29.61
| Laquinyo
| Magic comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Gu
| Syntonic comma
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma
|-
| 5
| [[15625/15552]]
| {{monzo| -6 -5 6 }}
| 8.11
| Tribiyo
| Kleisma
|-
| 5
| <abbr title="1224440064/1220703125">(20 digits)</abbr>
| {{monzo| 8 14 -13 }}
| 5.29
| Thegu
| [[Parakleisma]]
|-
| 5
| <abbr title="19073486328125/19042491875328">(28 digits)</abbr>
| {{monzo| -14 -19 19 }}
| 2.82
| Neyo
| [[Enneadeca]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Laru
| Harrison's comma
|-
| 7
| [[1029/1000]]
| {{monzo| -3 1 -3 3 }}
| 49.49
| Trizogu
| Keega
|-
| 7
| [[525/512]]
| {{monzo| -9 1 2 1 }}
| 43.41
| Lazoyoyo
| Avicennma
|-
| 7
| [[49/48]]
| {{monzo| -4 -1 0 2 }}
| 35.70
| Zozo
| Semaphoresma, slendro diesis
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Laruyo
| Schismean comma
|-
| 7
| [[686/675]]
| {{monzo| 1 -3 -2 3 }}
| 27.99
| Trizo-agugu
| Senga
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotrigu
| Keema
|-
| 7
| [[245/243]]
| {{monzo| 0 -5 1 2 }}
| 14.19
| Zozoyo
| Sensamagic comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Zotrigu
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Ruyoyo
| Marvel comma
|-
| 7
| [[19683/19600]]
| {{monzo| -4 9 -2 -2 }}
| 7.32
| Labirugu
| Cataharry comma
|-
| 7
| [[10976/10935]]
| {{monzo| 5 -7 -1 3 }}
| 6.48
| Satrizo-agu
| Hemimage comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Zozoquingu
| Hemimean comma
|-
| 7
| <abbr title="703125/702464">(12 digits)</abbr>
| {{monzo| -11 2 7 -3 }}
| 1.63
| Latriru-asepyo
| [[Metric comma]]
|-
| 7
| [[4375/4374]]
| {{monzo| -1 -7 4 1 }}
| 0.40
| Zoquadyo
| Ragisma
|-
| 11
| [[45/44]]
| {{monzo| -2 2 1 0 -1 }}
| 38.91
| Luyo
| Undecimal fifth tone
|-
| 11
| [[56/55]]
| {{monzo| 3 0 -1 1 -1 }}
| 31.19
| Luzogu
| Undecimal tritonic comma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[896/891]]
| {{monzo| 7 -4 0 1 -1 }}
| 9.69
| Saluzo
| Pentacircle comma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Lururuyo
| Swetisma
|-
| 13
| [[39/38]]
| {{monzo| -1 1 0 0 0 1 0 -1 }}
| 44.97
| Nutho
| Undevicesimal two-ninth tone
|-
| 13
| [[65/64]]
| {{monzo| -6 0 1 0 0 1 }}
| 26.84
| Thoyo
| Wilsorma
|-
| 13
| [[343/338]]
| {{monzo| -1 0 0 3 0 -2 }}
| 25.42
| Thuthutrizo
|
|-
| 13
| [[91/90]]
| {{monzo| -1 -2 -1 1 0 1 }}
| 19.13
| Thozogu
| Superleap comma, biome comma
|-
| 13
| [[676/675]]
| {{monzo| 2 -3 -2 0 0 2 }}
| 2.56
| Bithogu
| Island comma
|-
| 13
| [[1001/1000]]
| {{monzo| -3 0 -3 1 1 1 }}
| 1.73
| Tholozotrigu
| Fairytale comma, sinbadma
|-
| 23
| [[2187/2116]]
| {{monzo| -2 7 0 0 0 0 0 0 -2 }}
| 57.14
| Labitwethu
| Lipsett comma
|-
| 23
| [[70/69]]
| {{monzo| 1 -1 1 1 0 0 0 0 -1 }}
| 24.91
| Twethuzoyo
| Small vicesimotertial eighth tone
|-
| 23
| 256/253
| {{monzo| 8 0 0 0 -1 0 0 0 -1 }}
| 20.41
| Twethulu
| 253rd subharmonic
|-
| 23
| [[161/160]]
| {{monzo| -5 0 -1 1 0 0 0 0 1 }}
| 10.79
| Twethozogu
| Major kirnbergisma
|-
| 23
| [[208/207]]
| {{monzo| 4 -2 0 0 0 1 0 0 -1 }}
| 8.34
| Twethutho
| Vicetone comma
|-
| 23
| [[529/528]]
| {{monzo| -4 -1 0 0 -1 0 0 0 2 }}
| 3.28
| Bitwetho-alu
| Preziosisma
|-
| 23
| [[576/575]]
| {{monzo| 6 2 -2 0 0 0 0 0 -1 }}
| 3.01
| Twethugugu
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Twethothuluzo
| Triaphonisma
|}
<references group="note" />

=== Linear temperaments ===
* [[List of 19et rank two temperaments by badness]]
* [[List of 19et rank two temperaments by complexity]]
* [[List of edo-distinct 19et rank two temperaments]]
* [[Syntonic–kleismic equivalence continuum]]

Since 19 is prime, all rank-2 temperaments in 19edo have one period per octave (i.e. are linear). Therefore you can make a correspondence between intervals and the linear temperaments they generate.

{| class="wikitable center-1 right-2 center-3"
|-
! Degree
! Cents
! Interval
! Mos scales
! Temperaments
|-
| 1
| 63.16
| A1, d2
|
| [[Unicorn]] / [[Rhinoceros]]
|-
| 2
| 126.32
| m2
| [[1L 8s]], [[9L 1s]]
| [[Negri]]
|-
| 3
| 189.47
| M2
| [[1L 5s]], [[6L 1s]], [[6L 7s]]
| [[Deutone]] [[Xenial]] / [[Sensamagic clan #Xenia|Xenia]] [[Spell]]
|-
| 4
| 252.63
| A2, d3
| [[1L 3s]], [[4L 1s]], [[5L 4s]], [[5L 9s]]
| [[Godzilla]] / [[Helayo]]
|-
| 5
| 315.79
| m3
| [[3L 1s]], [[4L 3s]], [[4L 7s]], [[4L 11s]]
| [[Cata]] / [[keemun]]
|-
| 6
| 378.95
| M3
| [[3L 1s]], [[3L 4s]], [[3L 7s]], [[3L 10s]], [[3L 13s]]
| [[Magic]] / [[muggles]]
|-
| 7
| 442.11
| A3, d4
| [[3L 2s]], [[3L 5s]], [[8L 3s]]
| [[Sensi]]
|-
| 8
| 505.26
| P4
| [[2L 3s]], [[5L 2s]], [[7L 5s]]
| [[Meantone]] / [[flattone]]
|-
| 9
| 568.42
| A4
| [[2L 3s]], [[2L 5s]], [[2L 7s]], [[2L 9s]], [[2L 11s]], [[2L 13s]], [[2L 15s]]
| [[Liese]] [[Triton]] / [[pycnic]]
|}

== Octave stretch or compression ==
19edo is a promising option as a meantone tuning for pianos if split sharps are acceptable, since pianos are frequently tuned with stretched octaves due to the slight [[inharmonicity]] inherent in their strings. It also works well with harpsichords, since many have been, and are, built with split sharps.

Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-[[just]]. For example, if we are using [[49ed6]] or [[30edt]] (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57{{c}}, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-odd-limit [[tonality diamond]]. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. Another possible choice is [[ZPI|65zpi]].

== Scales ==
=== MOS scales ===
{{Main|List of MOS scales in {{PAGENAME}}}}

==== Octave-equivalent mosses ====
* [[Meantone]] pentic, [[2L 3s]] (gen = 11\19): 3 3 5 3 5
* [[Meantone]] diatonic, [[5L 2s]] (gen = 11\19): 3 3 2 3 3 3 2
* [[Meantone]] chromatic, [[7L 5s]] (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
* [[Semaphore]][5], [[4L 1s]] (gen = 4\19): 4 4 3 4 4
* [[Semaphore]][9], [[5L 4s]] (gen = 4\19): 3 1 3 1 3 3 1 3 1
* [[Semaphore]][14], [[5L 9s]] (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
* [[Sensi]][5], [[2L 3s]] (gen = 7\19): 5 2 5 2 5
* [[Sensi]][8], [[3L 5s]] (gen = 7\19): 2 3 2 2 3 2 2 3
* [[Sensi]][11], [[8L 3s]] (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
* [[Negri]][9], [[1L 8s]] (gen = 2\19): 2 2 2 2 3 2 2 2 2
* [[Negri]][10], [[9L 1s]] (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
* [[Kleismic]][7], [[4L 3s]] (gen = 5\19): 1 4 1 4 1 4 4
* [[Kleismic]][11], [[4L 7s]] (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
* [[Kleismic]][15], [[4L 11s]] (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
* [[Magic]][7], [[3L 4s]] (gen = 6\19): 5 1 5 1 5 1 1
* [[Magic]][10], [[3L 7s]] (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
* [[Magic]][13], [[3L 10s]] (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
* [[Magic]][16], [[3L 13s]] (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
* [[Liese]][17], [[2L 15s]] (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

=== Other scales ===
{{Main|19edo modes}}

* Meantone harmonic minor: 3 2 3 3 2 4 2
* Meantone melodic minor: 3 2 3 3 3 3 2 (ascending), 3 2 3 3 2 3 3 (descending)
* Meantone harmonic major: 3 3 2 3 2 4 2
* Chromatic octave species – meantone / [[marvel double harmonic major]] (subset of Negri[9]): 2 4 2 3 2 4 2
* Chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
* Chromatic octave species - [[Sahara]] septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
* [[Marvel hexatonic]] (subset of Negri[9]): 4 2 5 2 4 2
* Enharmonic pentatonic: 2 6 3 2 6
* Enharmonic pentatonic: 6 2 3 6 2
* Enharmonic octave species: 1 1 6 3 1 1 6
* Enharmonic octave species: 6 1 1 3 6 1 1
* Enharmonic octave species: 1 6 1 3 1 6 1
* [[Pinetone #Pinetone octatonic scales|Pinetone major-harmonic octatonic]]: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
* [[Pinetone #Pinetone octatonic scales|Pinetone minor-harmonic octatonic]]: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
* [[Pinetone #Pinetone diminished octatonic|Pinetone diminished octatonic]] / [[Porcusmine]]: 2 3 1 3 2 3 2 3
* [[Pinetone #Pinetone harmonic diminished octatonic|Pinetone harmonic diminished]]: 2 3 1 4 1 3 2 3
* [[Blackville]] / [[SNS ((2/1, 3/2)-5, 16/15)-10|5-limit dipentatonic]] (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
* [[Antipental blues]]: 4 4 1 2 4 4
* [[Semiquartal]] 3|5 b2: 1 3 3 1 3 1 3 3 1
* [[5-odd-limit]] tonality diamond: 5 1 2 3 2 1 5
* [[7-odd-limit]] tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
* [[9-odd-limit]] tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3

== Instruments ==
[[File:Vaisvil-19edo-guitar-IMG00145-1024x768.jpg|512x384px|thumb|none|19 note per octave Ibanez conversion by Brad Smith (Indianapolis)]]
[[File:Bass19.jpg|alt=19edo 5 string Bass 34"-37" scale length|512x384px|thumb|none|19edo bass conversion by Ron Sword]]

== Music ==
{{Main| 19edo/Music }}
{{Catrel| 19edo tracks }}

; [http://micro.soonlabel.com/19-ET/ XA 19-ET Index]
; A number of compositions that were perfomed at the [http://midwestmicrofest.org/concerts.html midwestmicrofest concert in 2007]{{dead link}}

== See also ==
* [[19edo modes]]
* [[19edo chords]]
* [[Strictly proper 19edo scales]]
* [[How to tune a 19edo guitar by ear]]
* [[Primer for 19edo]]
* [[Mason Green's New Common Practice Notation]]
* [[Extraclassical tonality]]
* [[Lumatone mapping for 19edo]]

== Further reading ==
* [[Darreg, Ivor]]. ''[http://www.tonalsoft.com/sonic-arts/darreg/case.htm A Case for Nineteen]''. 1982.
* Darreg, Ivor. ''[http://www.microstick.net/nineteenarticle.htm Nineteen for the Nineties]''{{dead link}}. (Unknown date of publication).
* Howe, Hubert S., Jr. [http://qcpages.qc.edu/%7Ehowe/articles/19-Tone%20Theory.html 19-Tone Theory and Applications]. c. 2004.
* [[Sethares, William A]]. [http://sethares.engr.wisc.edu/tet19/guitarchords19.html Tunings for 19 Tone Equal Tempered Guitar]. 1991.
* [[Sword, Ron]]. ''[http://www.metatonalmusic.com/books.html Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave]''. 2010.
* Yasser, Joseph. ''[https://www.worldcat.org/fr/title/726192994 Theory of Evolving Tonality]''. 1932.

== External links ==
* [http://tonalsoft.com/enc/number/19edo.aspx 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2] on the [[Tonalsoft Encyclopedia]]
* [http://www.n-ism.org/Projects/microtonalism.php Microtonalism] by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
* [http://mtg.redkeylabs.com/index.php?topic=6.0 Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar].
* [[Bostjan Zupancic]]'s [https://sites.google.com/site/bostjanzupancickhereb/home/bostjan/microtones/19edo 19-EDO pages]
* [https://sites.google.com/view/19edoscales Catalog of all 19edo heptatonic scales]

=== References ===
* Bucht, Saku and Huovinen, Erkki, ''Perceived consonance of harmonic intervals in 19-tone equal temperament'', CIM04_proceedings.
* Levy, Kenneth J., ''Costeley's Chromatic Chanson'', Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

[[Category:19-tone scales]]
[[Category:Godzilla]]
[[Category:Golden meantone]]
[[Category:Kleismic]]
[[Category:Meantone]]
[[Category:Magic]]
[[Category:Negri]]
[[Category:Semaphore]]
[[Category:Sensi]]
[[Category:Teentuning]]
[[Category:Historical]]

Xenial

2026-05-03T02:04:40Z

Xenllium: Created page with "{{Infobox regtemp | Title = Xenial | Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23 | Comma basis = 126/125, 177147/175616 (7-limit); 126/125, 162/161, 169/168, 171/170, 221/220, 231/230, 256/255 (23-limit) | Edo join 1 = 19 | Edo join 2 = 70 | Mapping = 1; -9 -17 -33 22 -21 26 27 -3 | Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE | MOS scales = 6L 1s, 6L 7s, 13L 6s, [..."

Oolong

2026-05-03T00:34:47Z

Xenllium:

{{Infobox regtemp
| Title = Oolong
| Subgroups = 2.3.5.7, 2.3.5.7.13
| Comma basis = [[126/125]], [[117649/116640]] (2.3.5.7); [[126/125]], [[196/195]], [[15379/15360]] (2.3.5.7.13)
| Edo join 1 = 27 | Edo join 2 = 50
| Mapping = 1; -17 -18 -20 -5
| Generators = 6/5
| Generators tuning = 311.7
| Optimization method = CWE
| MOS scales = [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]], [[23L 4s]]
| Pergen = (P8, c4P4/17)
| Odd limit 1 = 7 | Mistuning 1 = 5.6 | Complexity 1 = 23
}}

See [[Starling temperaments #Oolong]] for more technical data.

== Interval chain ==
In the following table, prime harmonics are in '''bold'''.

{| class="wikitable center-1 right-2"
! #
! Cents*
! Approximate Ratios
|-
| 0
| 0.0
| 1/1
|-
| 1
| 311.7
| 6/5
|-
| 2
| 623.3
| 10/7
|-
| 3
| 935.0
| 12/7
|-
| 4
| 46.7
| 36/35, 40/39
|-
| 5
| 358.3
| '''16/13'''
|-
| 6
| 670.0
| 96/65
|-
| 7
| 981.7
| 160/91
|-
| 8
| 93.4
| 96/91
|-
| 9
| 405.0
| 91/72
|-
| 10
| 716.7
| 91/60
|-
| 11
| 1028.4
| 49/27, 65/36
|-
| 12
| 140.0
| 13/12
|-
| 13
| 451.7
| 13/10
|-
| 14
| 763.4
| 14/9
|-
| 15
| 1075.0
| 13/7, 28/15, 50/27
|-
| 16
| 186.7
| 10/9, 28/25
|-
| 17
| 498.4
| '''4/3'''
|-
| 18
| 810.0
| '''8/5'''
|-
| 19
| 1121.7
| 40/21, 48/25
|-
| 20
| 233.4
| '''8/7'''
|}
<nowiki>*</nowiki> In 2.3.5.7.13 POTE tuning

== Tuning spectrum ==
{| class="wikitable center-all"
! ET generator
! [[eigenmonzo|eigenmonzo (unchanged-interval]])
! generator (¢)
|-
|
| 7/5
| 308.744
|-
|
| 7/6
| 311.043
|-
| 7\27
|
| 311.111
|-
|
| 10/9
| 311.400
|-
|
| 14/13
| 311.447
|-
|
| 49/48
| 311.491
|-
| 27\104
|
| 311.538
|-
|
| 13/12
| 311.548
|-
|
| 8/7
| 311.559
|-
| 47\181
|
| 311.602
|-
|
| 13/9
| 311.608
|-
|
| 4/3
| 311.650
|-
| 20\77
|
| 311.688
|-
|
| 28/27
| 311.708
|-
|
| 15/13
| 311.742
|-
|
| 16/15
| 311.764
|-
|
| 9/7
| 311.780
|-
|
| 13/10
| 311.863
|-
|
| 5/4
| 311.871
|-
|
| 16/13
| 311.894
|-
| 13\50
|
| 312.000
|-
|
| 15/14
| 312.037
|-
|
| 25/24
| 312.070
|-
| 6\23
|
| 313.043
|-
|
| 6/5
| 315.641
|}

[[Category:Oolong| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]

Starling temperaments

2026-05-03T00:07:02Z

Xenllium:

{{Technical data page}}
This page discusses miscellaneous [[rank-2 temperament]]s tempering out [[126/125]], the starling comma or septimal semicomma.

Temperaments discussed in families and clans are:
* ''[[Pater]]'' (+16/15) → [[Father family #Pater|Father family]]
* ''[[Flattie]]'' (+21/20) → [[Dicot family #Flattie|Dicot family]]
* ''[[Opossum]]'' (+28/27) → [[Trienstonic clan #Opossum|Trienstonic clan]]
* [[Diminished (temperament)|Diminished]] (+36/35) → [[Diminished family #Septimal diminished|Diminished family]]
* [[Keemun]] (+49/48) → [[Kleismic family #Keemun|Kleismic family]]
* [[Augene]] (+64/63) → [[Augmented family #Augene|Augmented family]]
* [[Meantone]] (+81/80) → [[Meantone family #Septimal meantone|Meantone family]]
* [[Mavila]] (+135/128) → [[Pelogic family #Mavila|Pelogic family]]
* [[Sensi]] (+245/243), [[Sensipent family #Sensi|Sensipent family]]
* [[Muggles]] (+525/512) → [[Magic family #Muggles|Magic family]]
* [[Valentine]] (+1029/1024) → [[Gamelismic clan #Valentine|Gamelismic clan]]
* [[Diaschismic]] (+2048/2025) → [[Diaschismic family #Septimal diaschismic|Diaschismic family]]
* [[Wollemia]] (+2240/2187) → [[Tetracot family #Wollemia|Tetracot family]]
* ''[[Unicorn]]'' (+10976/10935) → [[Unicorn family #Unicorn|Unicorn family]]
* ''[[Coblack]]'' (+16807/16384) → [[Trisedodge family #Coblack|Trisedodge family]] / [[Cloudy clan #Coblack|cloudy clan]]
* ''[[Grackle]]'' (+32805/32768) → [[Schismatic family #Grackle|Schismatic family]]
* ''[[Worschmidt]]'' (+33075/32768) → [[Würschmidt family #Worschmidt|Würschmidt family]]
* ''[[Thuja]]'' (+65536/64827) → [[Buzzardsmic clan #Thuja|Buzzardsmic clan]]
* ''[[Passionate]]'' (+131072/127575) → [[Passion family #Passionate|Passion family]]
* ''[[Vishnean]]'' (+540225/524288) → [[Vishnuzmic family #Vishnean|Vishnuzmic family]]
* ''[[Ditonic]]'' (+8751645/8388608) → [[Ditonmic family #Ditonic|Ditonmic family]]
* ''[[Muscogee]]'' (+33756345/33554432) → [[Mabila family #Muscogee|Mabila family]]

Considered below are myna, nusecond, oolong, vines, kumonga, cypress, bisemidim, casablanca, amigo, gilead, supersensi, and cobalt, sorted by increasing [[badness]].

Since {{nowrap|(6/5)3 {{=}} (126/125)⋅(12/7)}}, these temperaments tend to have a relatively small complexity for 6/5. They also possess the [[starling tetrad]], the 6/5–6/5–6/5–7/6 versions of the diminished seventh chord. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before [[12edo]] established itself as the standard tuning, it is actually three stacked minor thirds and an augmented second, contrary to the popular belief that it is four stacked minor thirds.

== Myna ==
{{Main| Myna }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Mynic]].''

7-limit myna is naturally found by establishing a structure of thirds, by making [[7/6]]–[[6/5]]–[[49/40]]–[[5/4]]–[[9/7]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]). [[11-limit]] myna then arises from equating this neutral third to [[11/9]]. Myna's characteristic feature is that the pental thirds are tuned outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds ([[36/35]]). In that sense, it is opposed to [[keemic temperaments]], in particular [[quasitemp]], where the distance between the pental and septimal thirds is the same as the chroma between the pental thirds and different from the septimal dieses.

In terms of vanishing commas, in addition to 126/125, myna adds [[1728/1715]], the orwell comma, and [[2401/2400]], the breedsma. It can also be described as the {{nowrap| 27 & 31 }} temperament, and has a [[ploidacot]] signature of beta-decacot. It has [[~]][[6/5]] as a generator.

[[58edo]] can be used as a tuning, with [[89edo]] being a better one, and fans of round cent values may like [[120edo]]. It is also possible to tune myna with pure fifths by taking 61/10 as the generator. Myna extends naturally but with much increased complexity to the 11- and 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 1728/1715

{{Mapping|legend=1| 1 -1 0 1 | 0 10 9 7 }}
: mapping generators: ~2, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3410{{c}}, ~6/5 = 309.9756{{c}}
: [[error map]]: {{val| -0.659 -1.540 +3.467 +0.344 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~6/5 = 310.0880{{c}}
: error map: {{val| 0.000 -1.075 +4.479 +1.790 }}

[[Minimax tuning]]:
* 7- and [[9-odd-limit]]: ~6/5 = {{monzo| 1/10 1/10 0 0}}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 9/10 9/10 0 0 | 17/10 7/10 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 27, 31, 58, 89, 236cc }}

[[Badness]] (Sintel): 0.684

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 | 0 10 9 7 25 }}

Optimal tunings:
* WE: ~2 = 1199.3441{{c}}, ~6/5 = 309.9748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.0982{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 89, 236cce }}

Badness (Sintel): 0.557

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 196/195

Mapping: {{mapping| 1 -1 0 1 -3 5 | 0 10 9 7 25 -5 }}

Optimal tunings:
* WE: ~2 = 1198.6509{{c}}, ~6/5 = 309.9273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.2218{{c}}

{{Optimal ET sequence|legend=0| 27e, 31, 58, 205cceff, 263ccdeefff }}

Badness (Sintel): 0.708

==== Minah ====
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 91/90, 126/125, 176/175

Mapping: {{mapping| 1 -1 0 1 -3 -2 | 0 10 9 7 25 22 }}

Optimal tunings:
* WE: ~2 = 1199.1929{{c}}, ~6/5 = 310.1724{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.3251{{c}}

{{Optimal ET sequence|legend=0| 27e, 31f, 58f }}

Badness (Sintel): 1.14

==== Maneh ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 126/125, 243/242

Mapping: {{mapping| 1 -1 0 1 -3 -3 | 0 10 9 7 25 26 }}

Optimal tunings:
* WE: ~2 = 1199.9109{{c}}, ~6/5 = 309.7815{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7987{{c}}

{{Optimal ET sequence|legend=0| 27eff, 31 }}

Badness (Sintel): 1.23

=== Myno ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 385/384

Mapping: {{mapping| 1 -1 0 1 5 | 0 10 9 7 -6 }}

Optimal tunings:
* WE: ~2 = 1201.0652{{c}}, ~6/5 = 310.0121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 309.7812{{c}}

{{Optimal ET sequence|legend=0| 27, 31 }}

Badness (Sintel): 1.11

=== Coleto ===
Subgroup: 2.3.5.7.11

Comma list: 56/55, 100/99, 1728/1715

Mapping: {{mapping| 1 -1 0 1 4 | 0 10 9 7 -2 }}

Optimal tunings:
* WE: ~2 = 1196.1024{{c}}, ~6/5 = 309.8434{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 310.6398{{c}}

{{Optimal ET sequence|legend=0| 4, 23bc, 27e }}

Badness (Sintel): 1.61

== Nusecond ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Nusecond]].''

Nusecond tempers out [[2430/2401]] and [[16875/16807]] in addition to 126/125, and may be described as {{nowrap| 31 & 70 }}. It has a neutral second generator of [[49/45]], two of which make up a 6/5 minor third since 2430/2401 is tempered out. Note that in the data below, the generator is its [[octave complement]] since eleven such generators [[octave reduction|octave reduced]] give the [[3/2|perfect fifth]]; its [[ploidacot]] is thus theta-hendecacot.

[[31edo]] can be used as a tuning, or [[132edo]] with a val which is the sum of the [[patent val]]s for 31 and 101. Because 49/45 is flat of 12/11 by only 540/539, nusecond is more naturally thought of as an 11-limit temperament with a combined 12/11 and 11/10 as a generator, tempering out 99/98, 121/120 and 540/539. Because of all the neutral seconds, an exotic Middle Eastern sound comes naturally to nusecond. Mosses of 15, 23, or 31 notes are enough to give fuller effect to the harmony, but the 8-note mos might also be considered from the melodic point of view.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2430/2401

{{Mapping|legend=1| 1 -8 -9 -12 | 0 11 13 17 }}
: mapping generators: ~2, ~49/27

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6138{{c}}, ~49/27 = 1045.0850{{c}}
: [[error map]]: {{val| -0.386 -2.931 +3.267 +2.253 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~49/27 = 1045.3909{{c}}
: error map: {{val| 0.000 -2.655 +3.768 +2.819 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~49/45 = {{monzo| 4/13 0 -1/13 }}
: {{monzo list| 1 0 0 0 | -5/13 0 11/13 0 | 0 0 1 0 | -3/13 0 17/13 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5
* [[9-odd-limit]]: ~49/45 = {{monzo| 3/11 -1/11 }}
: {{monzo list| 1 0 0 0 | 0 1 0 0 | 5/11 13/11 0 0 | 4/11 17/11 0 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.3

{{Optimal ET sequence|legend=1| 8d, 23d, 31, 101, 132c, 163c }}

[[Badness]] (Sintel): 1.28

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 | 0 11 13 17 12 }}

Optimal tunings:
* WE: ~2 = 1200.3420{{c}}, ~11/6 = 1045.6528{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.3816{{c}}

Minimax tuning:
* [[11-odd-limit]]: ~11/6 = {{monzo| 9/10 1/5 0 0 -1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 19/10 11/5 0 0 -11/10 }}, {{monzo| 27/10 13/5 0 0 -13/10 }}, {{monzo| 33/10 17/5 0 0 -17/10 }}, {{monzo| 19/5 12/5 0 0 -6/5 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of 15''x''2 - 10''x'' - 7, or (5 + sqrt (130))/15, at 154.6652 cents. The recurrence converges very quickly.

{{Optimal ET sequence|legend=0| 8d, 23de, 31, 101 }}

Badness (Sintel): 0.847

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 121/120, 126/125

Mapping: {{mapping| 1 -8 -9 -12 -7 -5 | 0 11 13 17 12 10 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~11/6 = 1044.6488{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/6 = 1045.4476{{c}}

{{Optimal ET sequence|legend=0| 8d, 23de, 31 }}

Badness (Sintel): 0.964

== Oolong ==
{{Main| Oolong }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oolong]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 117649/116640

{{Mapping|legend=1| 1 -11 -11 -12 | 0 17 18 20 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9188{{c}}, ~5/3 = 888.2606{{c}}
: [[error map]]: {{val| -0.081 -0.632 +3.269 -2.640 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 888.3163{{c}}
: error map: {{val| 0.000 -0.578 +3.379 -2.500 }}

{{Optimal ET sequence|legend=1| 23d, 27, 50, 77 }}

[[Badness]] (Sintel): 1.86

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 26411/26244

Mapping: {{mapping| 1 -11 -11 -12 -38 | 0 17 18 20 56 }}

Optimal tunings:
* WE: ~2 = 1198.9982{{c}}, ~5/3 = 888.0239{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3941{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.88

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 13013/12960

Mapping: {{mapping| 1 -11 -11 -12 -38 0 | 0 17 18 20 56 5 }}

Optimal tunings:
* WE: ~2 = 1199.5177{{c}}, ~5/3 = 888.0521{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 888.3959{{c}}

{{Optimal ET sequence|legend=0| 27e, 50e, 77, 104c }}

Badness (Sintel): 1.47

== Vines ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Vines]].''

Vines may be described as the {{nowrap| 46 & 50 }} temperament. It has a [[semi-octave]] period and a [[~]][[6/5]] generator. Eight generators minus three periods give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid gamma-octacot. [[96edo]] in the 96d val may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 84035/82944

{{Mapping|legend=1| 2 -1 1 3 | 0 8 7 5 }}
: mapping generators: ~343/240, ~6/5

[[Optimal tuning]]s:
* [[WE]]: ~343/240 = 600.2436{{c}}, ~6/5 = 312.7294{{c}}
: [[error map]]: {{val| +0.487 -0.363 +3.036 -4.448 }}
* [[CWE]]: ~343/240 = 600.0000{{c}}, ~6/5 = 312.6547{{c}}
: error map: {{val| 0.000 -0.717 +2.269 -5.552 }}

{{Optimal ET sequence|legend=1| 46, 96d, 142d }}

[[Badness]] (Sintel): 1.98

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2401/2376

Mapping: {{mapping| 2 -1 1 3 9 | 0 8 7 5 -4 }}

Optimal tunings:
* WE: ~99/70 = 600.2454{{c}}, ~6/5 = 312.7293{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~6/5 = 312.6282{{c}}

{{Optimal ET sequence|legend=0| 46, 96d, 142d }}

Badness (Sintel): 1.47

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 364/363, 385/384

Mapping: {{mapping| 2 -1 1 3 9 10 | 0 8 7 5 -4 -5 }}

Optimal tunings:
* WE: ~55/39 = 600.3065{{c}}, ~6/5 = 312.7240{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~6/5 = 312.5836{{c}}

{{Optimal ET sequence|legend=0| 46, 96d }}

Badness (Sintel): 1.23

== Xenial ==
{{Main| Xenial }}
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Xenial]].''

Named by [[User:Xenllium|Xenllium]] in 2026, xenial may be described as the {{nowrap| 19 & 70 }} temperament, splitting the [[8/3|perfect eleventh]] into nine equal parts, each for ~[[10/9]]. Equivalently, a stack of nine [[9/5]]s is equated with the [[3/2|perfect fifth]] above 7 [[2/1|octave]]s, so the [[ploidacot]] for the temperament is zeta-enneacot, and from this it derives its name.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 177147/175616

{{Mapping|legend=1| 1 -6 -12 -25 | 0 9 17 33 }}
: mapping generators: ~2, ~9/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0095{{c}}, ~9/5 = 1011.1532{{c}}
: [[error map]]: {{val| +0.010 -1.634 +3.176 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~9/5 = 1011.1456{{c}}
: error map: {{val| 0.000 -1.644 +3.162 -1.021 }}

{{Optimal ET sequence|legend=1| 19, 51cd, 70, 89 }}

[[Badness]] (Sintel): 2.13

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 16384/16335

Mapping: {{mapping| 1 -6 -12 -25 22 | 0 9 17 33 -22 }}

Optimal tunings:
* WE: ~2 = 1199.6137{{c}}, ~9/5 = 1010.8717{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.1915{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.31

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 729/728

Mapping: {{mapping| 1 -6 -12 -25 22 -14 | 0 9 17 33 -22 21 }}

Optimal tunings:
* WE: ~2 = 1199.8559{{c}}, ~9/5 = 1011.0911{{c}}
* CWE: ~2 = 1200.000{{c}}, ~9/5 = 1011.2102{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 1.98

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 169/168, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 | 0 9 17 33 -22 21 -26 }}

Optimal tunings:
* WE: ~2 = 1199.6970{{c}}, ~9/5 = 1010.9792{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2323{{c}}

{{Optimal ET sequence|legend=0| 19, 51cd, 70, 89 }}

Badness (Sintel): 2.06

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 169/168, 171/170, 221/220, 256/255, 540/539

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 | 0 9 17 33 -22 21 -26 -27 }}

Optimal tunings:
* WE: ~2 = 1199.7741{{c}}, ~9/5 = 1011.0334{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2230{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 2.03

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 126/125, 162/161, 169/168, 171/170, 221/220, 231/230, 256/255

Mapping: {{mapping| 1 -6 -12 -25 22 -14 26 27 2 | 0 9 17 33 -22 21 -26 -27 3 }}

Optimal tunings:
* WE: ~2 = 1199.6628{{c}}, ~9/5 = 1010.9415{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~9/5 = 1011.2245{{c}}

{{Optimal ET sequence|legend=0| 19, 51cdh, 70, 89 }}

Badness (Sintel): 1.93

== Kumonga ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Kumonga]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 12288/12005

{{Mapping|legend=1| 1 -9 -5 2 | 0 13 9 1 }}
: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.0653{{c}}, ~7/4 = 975.6277{{c}}
: [[error map]]: {{val| -1.935 -1.382 +4.009 +2.932 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 977.1096{{c}}
: error map: {{val| 0.000 +0.470 +7.673 +8.284 }}

{{Optimal ET sequence|legend=1| 16, 27, 43, 70, 167ccdd }}

[[Badness]] (Sintel): 2.21

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 864/847

Mapping: {{mapping| 1 -9 -5 2 -12 | 0 13 9 1 19 }}

Optimal tunings:
* WE: ~2 = 1197.9101{{c}}, ~7/4 = 975.4007{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9964{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e }}

Badness (Sintel): 1.43

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 126/125, 144/143, 176/175

Mapping: {{mapping| 1 -9 -5 2 -12 -2 | 0 13 9 1 19 7 }}

Optimal tunings:
* WE: ~2 = 1198.4987{{c}}, ~7/4 = 975.8162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 976.9677{{c}}

{{Optimal ET sequence|legend=0| 16, 27e, 43, 70e, 113cdee }}

Badness (Sintel): 1.19

== Bisemidim ==
Bisemidim tempers out [[118098/117649]] and may be described as the {{nowrap| 50 & 58 }} temperament. It has a [[semi-octave]] period and a [[~]][[49/45]] generator. Nine generators minus a period give the [[3/2|perfect fifth]], so the [[ploidacot]] for the temperament is diploid alpha-enneacot. [[108edo]] and [[166edo]] in the 166cef val may be recommended as tunings.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 118098/117649

{{Mapping|legend=1| 2 1 2 2 | 0 9 11 15 }}
: mapping generators: ~343/243, ~49/45

[[Optimal tuning]]s:
* [[WE]]: ~343/243 = 599.8915{{c}}, ~49/45 = 144.5293{{c}}
: [[error map]]: {{val| -0.217 -1.299 +3.292 -1.103 }}
* [[CWE]]: ~343/243 = 600.0000{{c}}, ~49/45 = 144.5351{{c}}
: error map: {{val| 0.000 -1.139 +3.572 -0.799 }}

{{Optimal ET sequence|legend=1| 50, 58, 108, 166c, 408ccc }}

[[Badness]] (Sintel): 2.47

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 1344/1331

Mapping: {{mapping| 2 1 2 2 5 | 0 9 11 15 8 }}

Optimal tunings:
* WE: ~99/70 = 599.6360{{c}}, ~12/11 = 144.5388{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~12/11 = 144.5623{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 108, 166ce, 224cee }}

Badness (Sintel): 1.36

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 364/363

Mapping: {{mapping| 2 1 2 2 5 5 | 0 9 11 15 8 10 }}

Optimal tunings:
* WE: ~55/39 = 599.5217{{c}}, ~12/11 = 144.5375{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~12/11 = 144.5698{{c}}

{{Optimal ET sequence|legend=0| 50, 58, 166cef, 224ceeff }}

Badness (Sintel): 0.987

== Cypress ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Cypress]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 19683/19208

{{Mapping|legend=1| 1 -5 -7 -12 | 0 12 17 27 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.1652{{c}}, ~196/135 = 658.2622{{c}}
: [[error map]]: {{val| +0.165 -3.634 +2.988 +2.272 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~196/135 = 658.1814{{c}}
: error map: {{val| 0.000 -3.779 +2.769 +2.071 }}

{{Optimal ET sequence|legend=1| 11cd, 20cd, 31 }}

[[Badness]] (Sintel): 2.53

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 | 0 12 17 27 30 }}

Optimal tunings:
* WE: ~2 = 1200.1117{{c}}, ~22/15 = 658.2892{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2345{{c}}

{{Optimal ET sequence|legend=0| 11cdee, 20cde, 31, 144cd }}

Badness (Sintel): 1.41

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 126/125, 243/242

Mapping: {{mapping| 1 -5 -7 -12 -13 -10 | 0 12 17 27 30 25 }}

Optimal tunings:
* WE: ~2 = 1199.4328{{c}}, ~22/15 = 657.9111{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.1886{{c}}

{{Optimal ET sequence|legend=0| 11cdeef, 20cdef, 31 }}

Badness (Sintel): 1.56

== Casablanca ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Casablanca]].''

Aside from 126/125, casablanca tempers out the no-threes comma [[823543/819200]] and also [[589824/588245]], and may be described as {{nowrap| 31 & 73 }} with a [[ploidacot]] signature of eta-19-cot. 61\135 or 75\166 supply good tunings for the generator, and 20- and 31-note [[mos scale]]s are available.

It may not seem like casablanca has much to offer, but peering under the hood a bit harder suggests otherwise. For one thing, the [[~]][[48/35]] generator is particularly interesting; like [[15/14]] and [[21/20]], it represents an interval between one vertex of a [[hexany]] and the opposite vertex, which makes it particularly simple with regard to the cubic lattice of tetrads.

If we add 385/384 to the list of commas, 48/35 is identified with [[11/8]], and casablanca is revealed as an [[11-limit]] temperament with a very low complexity for [[11/1|11]] and not too high a one for [[7/1|7]]; we might compare 1, 4, 14, 19, the generator steps to 11, 7, 5 and 3 respectively, with 1, 4, 10, 18, the steps to 3, 5, 7 and 11 in 11-limit [[meantone]].

Marrakesh, named by [[Herman Miller]] in 2011<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19166.html#19186 Yahoo! Tuning Group | ''A rose by any other name . . .'']</ref>, is a more accurate 11-limit extension where the generator is identified with [[15/11]] as opposed to 11/8 in casablanca.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 589824/588245

{{Mapping|legend=1| 1 -7 -4 1 | 0 19 14 4 }}
: mapping generators: ~2, ~48/35

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6286{{c}}, ~48/35 = 542.0141{{c}}
: [[error map]]: {{val| -0.371 -1.087 +3.370 -1.141 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~48/35 = 542.1684{{c}}
: error map: {{val| 0.000 -0.756 +4.044 -0.152 }}

{{Optimal ET sequence|legend=1| 11b, 20b, 31, 104c, 135c, 166c }}

[[Badness]] (Sintel): 2.56

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 | 0 19 14 4 1 }}

Optimal tunings:
* WE: ~2 = 1200.6404{{c}}, ~11/8 = 542.3659{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.0945{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.22

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 2420/2401

Mapping: {{mapping| 1 -7 -4 1 3 1 | 0 19 14 4 1 6 }}

Optimal tunings:
* WE: ~2 = 1199.7367{{c}}, ~11/8 = 542.0269{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 542.1392{{c}}

{{Optimal ET sequence|legend=0| 11b, 20b, 31 }}

Badness (Sintel): 2.31

=== Marrakesh ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 | 0 19 14 4 32 }}

Optimal tunings:
* WE: ~2 = 1199.6315{{c}}, ~15/11 = 542.0428{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.1958{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c }}

Badness (Sintel): 1.34

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 14641/14580

Mapping: {{mapping| 1 -7 -4 1 -11 15 | 0 19 14 4 32 -25 }}

Optimal tunings:
* WE: ~2 = 1199.3741{{c}}, ~15/11 = 541.9613{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2361{{c}}

{{Optimal ET sequence|legend=0| 31, 73, 104c, 135c, 239ccf }}

Badness (Sintel): 1.68

==== Murakuc ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 1540/1521

Mapping: {{mapping| 1 -7 -4 1 -11 1 | 0 19 14 4 32 6 }}

Optimal tunings:
* WE: ~2 = 1198.6578{{c}}, ~15/11 = 541.6930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/11 = 542.2577{{c}}

{{Optimal ET sequence|legend=0| 31, 73f, 104cff }}

Badness (Sintel): 1.71

== Amigo ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Magus]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2097152/2083725

{{Mapping|legend=1| 1 -2 2 9 | 0 11 1 -19 }}
: mapping generators: ~2, ~5/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.4354{{c}}, ~5/4 = 390.9104{{c}}
: [[error map]]: {{val| -0.565 -0.811 +3.467 -1.206 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 391.0937{{c}}
: error map: {{val| 0.000 +0.076 +4.780 +0.393 }}

{{Optimal ET sequence|legend=1| 43, 46, 89, 135c, 359cc }}

[[Badness]] (Sintel): 2.81

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 16384/16335

Mapping: {{mapping| 1 -2 2 9 9 | 0 11 1 -19 -17 }}

Optimal tunings:
* WE: ~2 = 1199.5267{{c}}, ~5/4 = 390.9211{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0783{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89, 135c, 224c }}

Badness (Sintel): 1.44

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 176/175, 364/363

Mapping: {{mapping| 1 -2 2 9 9 5 | 0 11 1 -19 -17 -4 }}

Optimal tunings:
* WE: ~2 = 1199.8174{{c}}, ~5/4 = 391.0130{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 391.0737{{c}}

{{Optimal ET sequence|legend=0| 43, 46, 89 }}

Badness (Sintel): 1.27

== Gilead ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 343/324

{{Mapping|legend=1| 1 -5 -5 -6 | 0 9 10 12 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.4516{{c}}, ~5/3 = 879.6394{{c}}
: [[error map]]: {{val| +1.452 +7.542 +2.823 -21.862 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.7223{{c}}
: error map: {{val| 0.000 +6.545 +0.909 -24.159 }}

{{Optimal ET sequence|legend=1| 11cd, 15, 41dd }}

[[Badness]] (Sintel): 2.92

== Supersensi ==
Named by [[Xenllium]] in 2022, supersensi tempers out the no-fives comma [[17496/16807]], and may be described as {{nowrap| 8d & 43 }}. It has a ultramajor third generator, which is sharper than the generator for [[sensi]], hence the name. Its [[ploidacot]] is epsilon-15-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 17496/16807

{{Mapping|legend=1| 1 -4 -4 -5 | 0 15 17 21 }}
: mapping generators: ~2, ~343/270

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1406{{c}}, ~343/270 = 446.2478{{c}}
: [[error map]]: {{val| -0.859 -4.800 +3.337 +6.675 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/270 = 446.5163{{c}}
: error map: {{val| 0.000 -4.210 +4.464 +8.017 }}

{{Optimal ET sequence|legend=1| 8d, …, 35, 43 }}

[[Badness]] (Sintel): 3.76

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 126/125, 864/847

Mapping: {{mapping| 1 -4 -4 -5 -1 | 0 15 17 21 12 }}

Optimal tunings:
* WE: ~2 = 1198.6099{{c}}, ~72/55 = 446.0983{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 446.5381{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35, 43 }}

Badness (Sintel): 1.97

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 99/98, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 | 0 15 17 21 12 18 }}

Optimal tunings:
* WE: ~2 = 1198.9947{{c}}, ~13/10 = 446.2243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5420{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.46

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 78/77, 99/98, 120/119, 126/125, 144/143

Mapping: {{mapping| 1 -4 -4 -5 -1 -3 0 | 0 15 17 21 12 18 11 }}

Optimal tunings:
* WE: ~2 = 1198.7070{{c}}, ~13/10 = 446.1493{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 446.5645{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 35f, 43 }}

Badness (Sintel): 1.32

== Cobalt ==
: ''For the 5-limit version, see [[27th-octave temperaments #Cobalt]].''

Cobalt has a period of 1/27 octave and tempers out 126/125 and 540/539 as in the [[aplonis]] temperament. It may be described as {{nowrap| 27 & 81 }}.

Cobalt was named by [[Xenllium]] in 2022 after the 27th element.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 40353607/40310784

{{Mapping|legend=1| 27 0 20 33 | 0 1 1 1 }}
: mapping generators: ~36/35, ~3

[[Optimal tuning]]s:
* [[WE]]: ~36/35 = 44.4363{{c}}, ~3/2 = 701.1154{{c}}
: [[error map]]: {{val| -0.221 -1.060 +3.307 -1.534 }}
* [[CWE]]: ~36/35 = 44.4444{{c}}, ~3/2 = 701.0414{{c}}
: error map: {{val| 0.000 -0.914 +3.617 -1.118 }}

{{Optimal ET sequence|legend=1| 27, 81, 108, 135c }}

[[Badness]] (Sintel): 4.39

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 | 0 1 1 1 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4418{{c}}, ~3/2 = 699.9594{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.9386{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108 }}

Badness (Sintel): 2.58

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 21609/21296

Mapping: {{mapping| 27 0 20 33 8 100 | 0 1 1 1 2 0 }}

Optimal tunings:
* WE: ~36/35 = 44.4250{{c}}, ~3/2 = 700.5606{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.5524{{c}}

{{Optimal ET sequence|legend=0| 27e, 81, 108, 243ceef }}

Badness (Sintel): 2.36

===== Cobaltous =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 189/187, 196/195, 1452/1445

Mapping: {{mapping| 27 0 20 33 8 100 79 | 0 1 1 1 2 0 2 }}

Optimal tunings:
* WE: ~36/35 = 44.4237{{c}}, ~3/2 = 700.0699{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0569{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 2.14

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 171/170, 189/187, 196/195, 969/968

Mapping: {{mapping| 27 0 20 33 8 100 79 99 | 0 1 1 1 2 0 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4227{{c}}, ~3/2 = 700.0859{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 700.0852{{c}}

{{Optimal ET sequence|legend=0| 27eg, 81, 108g }}

Badness (Sintel): 1.85

===== Cobaltic =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 196/195, 221/220, 12005/11968

Mapping: {{mapping| 27 0 20 33 8 100 -18 | 0 1 1 1 2 0 3 }}

Optimal tunings:
* WE: ~36/35 = 44.4203{{c}}, ~3/2 = 701.2133{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.2530{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ce }}

Badness (Sintel): 2.40

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 126/125, 144/143, 196/195, 210/209, 221/220, 1088/1083

Mapping: {{mapping| 27 0 20 33 8 100 -18 72 | 0 1 1 1 2 0 3 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 701.2519{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 701.3143{{c}}

{{Optimal ET sequence|legend=0| 27eg, 108, 135ceh }}

Badness (Sintel): 2.08

==== Cobaltite ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 169/168, 540/539, 975/968

Mapping: {{mapping| 27 0 20 33 8 57 | 0 1 1 1 2 1 }}

Optimal tunings:
* WE: ~36/35 = 44.4177{{c}}, ~3/2 = 699.5121{{c}}
* CWE: ~36/35 = 44.4444{{c}}, ~3/2 = 699.6606{{c}}

{{Optimal ET sequence|legend=0| 27e, 54bdef, 81f }}

Badness (Sintel): 2.18

== References ==

[[Category:Temperament collections]]
[[Category:Starling temperaments| ]] 
[[Category:Rank 2]]