Xenharmonic Wiki - User contributions [en]

Talk:5120/5103

2026-06-01T20:18:00Z

Lériendil: /* Petition to officialize aberschisma, and change hemifamity to aberschismic */

== "Universal" name for 5120/5103 ==

I've noticed the conversation on the XA Discord about picking a name to replace "hemifamity comma". Suggestions I've seen include ''argent comma'' and ''pele comma''. I'm a bit biased towards ''aberschisma'' since I coined the name, but MidnightBlue pointed out that 6¢ is quite wide to be calling it a schisma, which I've also thought about. Maybe ''aberkleisma'' or even ''pentasept comma''?

: I'm not too invested into this comma, but will add my <strike>schisma</strike> 2c after seeing the Discord convo: ''pele comma'' and ''peleisma'' are fine with me, whereas I'm afraid ''argent comma'', while logical, may be confused with the [[argyria]] (which is more of an ''arg''ument for renaming the latter).

: On a side note, 5120/5103 does function like a kleisma for me, particularly because the ratio of the pental kleisma to it is the [[horwell comma]], which is among the staple commas in my 7-limit analysis of edos incl. 53. Because schismic x kleismic product words are among the best ways to make well-tempered 53-note scales, the pental kleisma is a chroma there, and when horwell tempered, it turns into 5120/5103 and is, among other roles, the scale-chroma between the 81/80 and 64/63 steps.

: Meanwhile, the ratio of 5120/5103 to the pental schisma is the [[garischisma]]. So for fans of the latter, which I'm not, it may act like a schisma instead, but that's less likely because the pental schisma flattens the fifth while the garischisma sharpens it, so if anything, the latter and 5120/5103 would be seen as 'negative schismas', which, btw, brings us to the concept of [[counterpyth]].

: Afaik, counterpyth has never been considered under this name without 5120/5103, whereas [[1216/1215]] works well together with other commas that stack slightly sharp fifths, such as the [[wilschisma]] and the [[symbiotic comma]], and the name ''Eratosthenes' comma'' is good, so I disagree with the assignment of the counterpyth family label to any temp with 1216/1215 in sintel's finder. I.e., to me, 5120/5103 is more related to counterpyth than 1216/1215 is. But I can't be sure of my judgment on this without FloraC's opinion. Either way, I don't mind ''counterpyth comma'' for 5120/5103, its 7-limit rank-3 then called counterpyth like its canonical extension to 2.3.5.7.19 already is.

: --[[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 23:55, 1 June 2025 (UTC)

:: I'm all for ''argent comma''. The similarity with ''argyria'' isn't high enough to worry me. I'm against ''pele comma'' cuz that would set pele as canon which I don't think we should ever do. For the same reason I'd hesitate to call it ''counterpyth comma''. [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 14:17, 2 June 2025 (UTC)

::: OK, then I settle on ''argent comma'' too. That matches my view of argent fifths as a distinct region that's roughly [65\111, 17\29] and sharp of the olympic / garischismic / symbiotic / wilschismic fifths region that's roughly [55\94, 65\111]. --[[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 18:38, 2 June 2025 (UTC)

:::: What about 41edo and 46edo? Those are both notable tunings that temper out the comma and have fifths that fall outside of your *argent* range. -- [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 00:35, 12 June 2025 (UTC)

::::: On my part I include 41edo within the argent range; and there's a case to be made that 46edo and 53edo, while notable, aren't truly representative of the intonation of tempering out 5120/5103. "Argent" strictly speaking refers not to a particular tuning range, anyhow, but to a specific tuning that sets the logarithmic ratio of the perfect fifth to the perfect fourth to be sqrt(2):1, for which one can define bands of tolerance around, but which very closely corresponds to the most accurate tunings that temper out this comma. Perhaps "argentisma" -> argentic, argentismic would be clearer, so as not to imply an RTT interpretation for the term "argent temperament" which is already in use.

::::: Compare this to the intonation of counterpyth, which quite distinctly favors tunings of 3/2 far flatter than the optimum of tempering out 5120/5103 by itself: just 19/15 gives us roughly 1/16-comma hemifamity as opposed to just 15/14, 7/5, or 21/20 which provide 1/5, 1/6, and 1/7-comma tunings. For this reason, I oppose seeing counterpyth as a canonical extension to the 7-limit rank-3 {5120/5103} temperament. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 01:24, 12 June 2025 (UTC)

::::: I did think about making 24\41 the boundary instead, as rank-3 microtemps tend to have flatter fifths than that even if 152fg or 111 support them. My flat end of argent is surely not flatter than 24\41 and not sharper than 41\70. Between those are kwai fifths... that I may consider too damaged indeed on second thought, and so belonging to the "slightly exo" range codenamed argent. [[User:VIxen|VIxen]] ([[User talk:VIxen|talk]]) 20:23, 19 June 2025 (UTC)

:::: I prefer "Saruyo". It's the only name out of all these suggestions that directly indicates 5120/5103. --[[User:TallKite|TallKite]] ([[User talk:TallKite|talk]]) 09:19, 5 June 2025 (UTC)

: I have a dumb idea. Why not call it the *pell comma* after the Pell sequence of numbers, whose convergent ratio gives the approximate ratio between an octave and a perfect fourth for the optimal tuning of the temperament? [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 20:59, 18 September 2025 (UTC)

:: I think referencing the Pell sequence makes way more sense for a member of the family of commas going 50/49, 289/288, 1682/1681, 9801/9800, etc. The only relation of Pell numbers to 5120/5103 is the edo sequence, which seems rather secondary, as much as I'm a promoter of 239edo. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 22:42, 18 September 2025 (UTC)

: I propose the name "interkleisma", since 5120/5103 is the difference between 64/63 and 81/80 (the main formal commas for primes 5 and 7), and is around a kleisma in size.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 03:02, 13 February 2026 (UTC)

:: It's a half kleisma in 270edo (and 311edo if you consider other kleismata such as 1029/1024) tho. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:20, 13 February 2026 (UTC)

::: Your essay on the 13-limit JI space considers 5120/5103~352/351~847/845, 325/324~385/384, 364/363~441/440, 540/539~729/728, and 351/350 as kleismas. Even if it is a half-kleisma in 270edo, the comma is close enough to the rough interval region, and also no single edo should decide the name.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 16:58, 13 February 2026 (UTC)

:::: Oh wow, my bad. I'll change them to hemikleismata. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 21:39, 13 February 2026 (UTC)

Should we do a poll here? The name was basically pre-maturely changed according to a poll on XA Discord. Besides, we need to decide what to do with the temp's name. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 12:51, 22 January 2026 (UTC)

: If enough people want to, then I guess. I like the current name of this comma, and I was thinking of the associated full 7-limit temperament being "argentic" and the 2.3.7/5 subgroup one being "argic". [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 06:12, 27 January 2026 (UTC)

: Why not [[64/63|S8]]/[[81/80|S9]]? Are there many properties of this comma that aren't explained by it being ((8/7)/(9/8)) / ((9/8)/(10/9)) = (64/63) / (81/80) and hence ([[10/7]])/([[9/8]])3? --[[User:Godtone|Godtone]] ([[User talk:Godtone|talk]]) 12:27, 16 February 2026 (UTC)

:: Because "ess eight over ess nine" is too many syllables. [[User:Tristanbay|Tristanbay]] ([[User talk:Tristanbay|talk]]) 05:34, 17 February 2026 (UTC)

: I'm fine calling it saruyoma, y'all sort this out. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 10:04, 6 May 2026 (UTC)

=== Petition to officialize ''aberschisma'', and change ''hemifamity'' to ''aberschismic'' ===
At this point, ''aberschisma'' and ''aberschismic'' seem like the most widely liked and used names in xenharmonic communities. Thereby I request ''aberschisma'' be set as the permanent, main name for 5120/5103, and ''hemifamity'' be officially changed to ''aberschismic''.

Please put your "yes" or "no" and signature below. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:56, 1 June 2026 (UTC)

# '''Yes'''. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 05:56, 1 June 2026 (UTC)
# Yes, sure, why not. --[[User:Eufalesio|Eufalesio]] ([[User talk:Eufalesio|talk]]) 07:48, 1 June 2026 (UTC)
# '''Yes''', I'm only weakly towards that name, but it feels alright, if the community wants it. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 15:40, 1 June 2026 (UTC)
# '''Yes.''' [[User:Inthar|Inthar]] ([[User talk:Inthar|talk]]) 20:16, 1 June 2026 (UTC)
# '''Yes'''. "Argentic" should still refer to the 2.3.7/5 subgroup temperament, however, per the change I made. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 20:18, 1 June 2026 (UTC)

: I was under the impression that "argent" won out in the community? – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 12:30, 1 June 2026 (UTC)
:: In my opinion it's a bit too confusing with the logarithmic argent tuning, even though it is closely related. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 15:40, 1 June 2026 (UTC)
::: Seconded. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 20:18, 1 June 2026 (UTC)

Gammic family

2026-05-24T06:23:50Z

Lériendil: the 7-limit header should still be there because it has subgroup temps under it

{{Technical data page}}
The [[Carlos Gamma]] rank-1 temperament divides 3/2 into 20 equal parts, 11 of which give a 5/4. This is closely related to the rank-2 microtemperament tempering out {{monzo| -29 -11 20 }}, the [[gammic comma]]. This temperament, '''gammic''', takes 11 [[generator]] steps to reach 5/4, and 20 to reach 3/2. The generator in question is 1990656/1953125 = {{monzo| 13 5 -9 }}, which when suitably tempered is very close to 5/171 octaves, which makes for an ideal gammic tuning. As a 5-limit temperament supported by [[171edo]], [[Schismatic family|schismatic]] temperament makes for a natural comparison. Schismic, tempering out {{monzo| -15 8 1 }}, the [[schisma]], is plainly much less complex than gammic, but people seeking the exotic might prefer gammic even so. The 34-note mos is interesting, being a 1L 33s refinement of the [[34edo]] tuning. Of course gammic can be tuned to 34, which makes the two equivalent, and would rather remove the point of Carlos Gamma if used for it.

Because 171 is such a strong [[7-limit]] system, it is natural to extend gammic to the 7-limit. This we may do by adding [[4375/4374]] to the comma list. 96 gammic generators finally reach 7, which is a long way to go compared to the 39 generator steps of pontiac. If someone wants to make the trip, a 103-note mos is possible.

== Gammic ==
[[Subgroup]]: 2.3.5

[[Comma list]]: {{monzo| -29 -11 20 }}

{{Mapping|legend=1| 1 1 2 | 0 20 11 }}

: mapping generators: ~2, ~1990656/1953125

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1990656/1953125 = 35.0964

{{Optimal ET sequence|legend=1| 34, 103, 137, 171, 547, 718, 889, 1607 }}

[[Badness]]: 0.087752

=== 2.3.5.17 subgroup ===
The interval of 3 generators represents one-third of [[6/5]], which is very close to [[17/16]], with the comma between 6/5 and (17/16)3 being [[24576/24565]] = {{S|16/S17}}. This then naturally interprets the generator as [[51/50]] with two generators representing [[25/24]], tempering out [[15625/15606]] = S49×S502.

[[Subgroup]]: 2.3.5.17

[[Comma list]]: 15625/15606, 24576/24565

{{Mapping|legend=1| 1 1 2 4 | 0 20 11 3 }}

: mapping generators: ~2, ~51/50

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~51/50 = 35.1011

{{Optimal ET sequence|legend=1| 34, 103, 137, 171, 376, 547, 2564g, 3111cg, 3658cgg }}

[[Badness]] (Sintel): 0.320

== Septimal gammic ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 6591796875/6576668672

{{Mapping|legend=1| 1 1 2 0 | 0 20 11 96 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~234375/229376 = 35.0904

{{Optimal ET sequence|legend=1| 34d, 171, 205, 1402, 1573, 1744, 1915 }}

[[Badness]]: 0.047362

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

Comma list: 243/242, 4375/4356, 100352/99825

Mapping: {{mapping| 1 1 2 0 2 | 0 20 11 96 50 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.089

{{Optimal ET sequence|legend=1| 34d, 137d, 171 }}

Badness: 0.097061

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

Comma list: 243/242, 364/363, 625/624, 2200/2197

Mapping: {{mapping| 1 1 2 0 2 3 | 0 20 11 96 50 24 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.091

{{Optimal ET sequence|legend=1| 34d, 137d, 171 }}

Badness: 0.047822

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 364/363, 375/374, 595/594, 2200/2197

Mapping: {{mapping| 1 1 2 0 2 3 4 | 0 20 11 96 50 24 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 35.090

{{Optimal ET sequence|legend=1| 34d, 137d, 171 }}

Badness: 0.031466

== Gammy ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 94143178827/91913281250

[[Mapping]]: {{mapping| 1 1 2 1 | 0 20 11 62 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~1990656/1953125 = 34.984

{{Optimal ET sequence|legend=1| 34d, 69d, 103, 240, 343b }}

[[Badness]]: 0.230839

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

Comma list: 225/224, 243/242, 215622/214375

Mapping: {{mapping| 1 1 2 1 2 | 0 20 11 62 50 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.985

{{Optimal ET sequence|legend=1| 34d, 69de, 103, 240, 343be }}

Badness: 0.065326

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

Comma list: 225/224, 243/242, 351/350, 1188/1183

Mapping: {{mapping| 1 1 2 1 2 3 | 0 20 11 62 50 24 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.988

{{Optimal ET sequence|legend=1| 34d, 69de, 103, 240, 343be }}

Badness: 0.033418

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 351/350, 375/374, 1188/1183

Mapping: {{mapping| 1 1 2 1 2 3 4 | 0 20 11 62 50 24 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~45/44 = 34.997

{{Optimal ET sequence|legend=1| 34d, 69de, 103, 137, 240 }}

Badness: 0.025030

== Neptune ==
A more interesting extension is to neptune, which divides an octave plus a gammic generator in half, to get a 10/7 generator. Neptune adds [[2401/2400]] to the gammic comma, and may be described as the 68&171 temperament. The generator chain goes merrily on, stacking one 10/7 over another, until after eighteen generator steps 6/5 (up nine octaves) is reached. Then in succession we get 12/7, the neutral third, 7/4 and 5/4. Two neutral thirds then gives a fifth, and these intervals with their inverses are the full set of septimal consonances. [[171edo]] makes a good tuning, and we can also choose to make any of the consonances besides 7/5 and 10/7 just, including the fifth, which gives a tuning extending [[Carlos Gamma]].

Adding 385/384 or 1375/1372 to the list of commas allows for an extension to the [[11-limit]], where (7/5)3 equates to 11/4.

[[Gene Ward Smith]] once described [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_6001.html neptune as an analog of miracle].

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 48828125/48771072

{{Mapping|legend=1| 1 21 13 13 | 0 -40 -22 -21 }}

: mapping generators: 2, ~7/5

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 582.452

{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 1094, 1265, 1436, 1607, 1778 }}

[[Badness]]: 0.023427

==== 2.3.5.7.17 subgroup ====
Extending 2.3.5.17 gammic via neptune, we find that both 2401/2400 ({{S|49}}) and 2500/2499 (S50) are tempered out; their product, 1225/1224 (S35) is therefore also tempered out.

[[Subgroup]]: 2.3.5.7.17

[[Comma list]]: 1225/1224, 2401/2400, 24576/24565

{{Mapping|legend=1| 1 21 13 13 7 | 0 -40 -22 -21 -6 }}

: mapping generators: ~2, ~7/5

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, 7/5 = 582.450

{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 581, 752, 923, 1094 }}

[[Badness]] (Sintel): 0.404

==== 2.3.5.7.17.31 subgroup ====
Since neptune splits the interval of [[5/3]] into two, we can accurately map each part to [[40/31]]~[[31/24]] by tempering out [[961/960]] (S31). This is especially natural, as combined with tempering out 1225/1224 (S35) and 24576/24565 (S16/S17), we can map (17/16)2 (6 gammic generators) to [[35/31]]. This also gives us its complement with respect to [[5/4]], the interval of 5 gammic generators representing a quarter of a perfect fifth, as [[31/28]].

[[Subgroup]]: 2.3.5.7.17.31

[[Comma list]]: 868/867, 961/960, 1225/1224, 2401/2400

{{Mapping|legend=1| 1 21 13 13 7 20 | 0 -40 -22 -21 -6 -31 }}

: mapping generators: ~2, ~7/5

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, 7/5 = 582.451

{{Optimal ET sequence|legend=1| 35, 68, 103, 171, 752k, 923k, 1094k, 1265kk }}

[[Badness]] (Sintel): 0.393

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

Comma list: 385/384, 1375/1372, 78408/78125

Mapping: {{mapping| 1 21 13 13 2 | 0 -40 -22 -21 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.475

{{Optimal ET sequence|legend=1| 35, 68, 103, 171e, 274e, 445ee }}

Badness: 0.063602

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

Comma list: 385/384, 625/624, 1188/1183, 1375/1372

Mapping: {{mapping| 1 21 13 13 2 27 | 0 -40 -22 -21 3 -48 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.480

{{Optimal ET sequence|legend=1| 35f, 68, 103, 171e, 274e }}

Badness: 0.037156

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 385/384, 561/560, 625/624, 715/714, 1188/1183

Mapping: {{mapping| 1 21 13 13 2 27 7 | 0 -40 -22 -21 3 -48 -6 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.475

{{Optimal ET sequence|legend=1| 35f, 68, 103, 171e, 274e, 445ee }}

Badness: 0.025909

=== Salacia ===
Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 9765625/9732096

Mapping: {{mapping| 1 21 13 13 52 | 0 -40 -22 -21 -100 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.478

{{Optimal ET sequence|legend=1| 68e, 103, 171, 274, 719be, 993bcde, 1267bbcde }}

Badness: 0.069721

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

Comma list: 243/242, 441/440, 625/624, 2200/2197

Mapping: {{mapping| 1 21 13 13 52 27 | 0 -40 -22 -21 -100 -48 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.477

{{Optimal ET sequence|legend=1| 68e, 103, 171, 274, 719be, 993bcde }}

Badness: 0.034977

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 243/242, 375/374, 441/440, 625/624, 2200/2197

Mapping: {{mapping| 1 21 13 13 52 27 7 | 0 -40 -22 -21 -100 -48 -6 }}

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 582.475

{{Optimal ET sequence|legend=1| 68e, 103, 171, 274, 445e, 719be, 1164bcdeef }}

Badness: 0.024577

=== Poseidon ===
Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 9801/9800, 9453125/9437184

Mapping: {{mapping| 2 2 4 5 8 | 0 40 22 21 -37 }}

: mapping generators: ~99/70, ~99/98

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 17.545

{{Optimal ET sequence|legend=1| 68, 206b, 274, 342 }}

Badness: 0.041727

[[Category:Temperament families]]
[[Category:Gammic family| ]] 
[[Category:Gammic| ]] 
[[Category:Rank 2]]

Subgroup temperaments

2026-05-18T16:06:11Z

Lériendil: /* Tridecimal guanyintet */

{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.

For temperaments that omit various prime harmonics, see:
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].

= Composite subgroup temperaments =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.165

==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

: [[gencom]]: [2 9/8; 225/224 325/324 640/637]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

[[Badness]]: 0.00263

=== Joan ===
{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

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

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

=== Genius ===

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

{{Mapping|legend=2| 1 2 4 | 0 3 1 }}

: sval mapping generators: ~2, ~21/16

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

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

=== Tetrameantone ===
{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

=== Tetramagic ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

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

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

{{Todo|inline=1|cleanup}}

<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)
⟨2399.3973, 193.8643]

TE Step Tunings (cents)
⟨25.21211, 47.81337]

TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]

TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]

Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave

Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩

TE Generator Tunings (cents)
⟨2399.4195, 193.8654]

TE Step Tunings (cents)
⟨25.23883, 47.79592]

TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]

TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave

Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)
⟨2400.1097, 193.9498]

TE Step Tunings (cents)
⟨24.18752, 48.52491]

TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave

Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>

==== 4.6.5.7.11.13 ====

<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩

TE Generator Tunings (cents)
⟨2401.2305, 193.5378]

TE Step Tunings (cents)
⟨42.79107, 35.98524]

TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>

==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)
⟨2400.4701, 193.4599]

TE Step Tunings (cents)
⟨43.39350, 35.55764]

TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>

==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩

TE Generator Tunings (cents)
⟨2399.9219, 193.3952]

TE Step Tunings (cents)
⟨44.14256, 35.03670]

TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>

==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

TE Generator Tunings (cents)
⟨2399.3286, 193.5316]

TE Step Tunings (cents)
⟨37.31613, 39.63311]

TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>

== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

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

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

== 8.9.7 subgroup ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

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

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10-3

= Fractional subgroup temperaments =
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

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

{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

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

: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====
{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

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

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

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

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

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

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====
{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====
{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===
{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

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

{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}
: <nowiki/>* wart for 5/3
: † wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

[[Subgroup]]: 2.5/3.13/9

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

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

{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])3 = [[13/10]] and ([[15/14]])3 = [[16/13]], therefore tempering out {S11/S12/S14/S15}. [[40edo]] remains an excellent tuning.

[[Subgroup]]: 2.5.7/3.11/3.13

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})

{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} using subgroup TE 
: <nowiki/>* wart for 7/3

Badness (Sintel): 0.329

==== Laz ====
{{See also | Chromatic pairs #Laz }}

Laz is related to [[avalokita]] as well as to [[winston]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}

{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}

Kryptonite is related to [[krypton]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12

{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}

Kiribati is related to [[nakika]] as well as to [[octacot]].

[[Subgroup]]: 2.9/5.7/3.11/9

[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})

{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20

{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents

=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.

[[Subgroup]]: 2.7/3.11

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3

{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents

== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.

[[Subgroup]]: 2.5.9/7

[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})

{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5

{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents

==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[50/49]]

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

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}

{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830

{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
 based on subgroup TE 

Badness (Sintel): 0.119

==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}

Edson is related to [[pele]] and [[andromeda]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3

{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

{{Optimal ET sequence|legend=1| 12, 17, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

==== Haumea ====
{{See also| Chromatic pairs #Haumea }}

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}

{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: 364/363, 441/440, 1001/1000

{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

[[Subgroup]]: 2.7/5.9/5

[[Comma list]]: [[250047/250000]]

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

{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461

{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents

=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].

[[Subgroup]]: 2.7/5.11/5.13/5

[[Comma list]]: [[847/845]], [[1001/1000]]

{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents

==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95t, 124t }}
: t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents

== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

[[Subgroup]]: 2.11/5.13/5

[[Comma list]]: [[2200/2197]]

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

{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012

{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents

==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168, 540/539, 729/728

{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents

=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].

Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].

[[Subgroup]]: 2.3.7.11/5

[[Comma list]]: 1029/1024, 4000/3993

{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742

{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 1029/1024, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918

{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17

[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19

[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
: <nowiki />* wart for 11/7
: † wart for 13/7

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====
{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

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

{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

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

[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5

=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]

[[Subgroup]]: 2.3.49/5

[[Comma list]]: 2401/2400

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

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

[[Tp tuning #T2 tuning|RMS error]]: ?

== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})

{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

<nowiki/>* wart for 19/7

[[Badness]] (Sintel): 0.082

== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}

=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: [[10976/10935]]

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

[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909

[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

=== Halftone ===
{{Main| Halftone }}

[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: 9604/9375

{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2

[[Comma list]]: 1232/1215, 27783/27500

{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2

[[Comma list]]: 275/273, 1232/1215, 1323/1300

{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2

=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4

[[Comma list]]: 245/243

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

: sval mapping generators: ~3/2, ~9/7

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728

[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]

==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 100/99, 245/243

{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771

[[Optimal ET sequence]]: [[8edf]], [[13edf]]

==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 245/243, 441/440

{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939

[[Optimal ET sequence]]: [[8edf]], [[11edf]]

== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''

[[Subgroup]]: 3/2.5/4.11/8

[[Comma list]]: 121/120

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

: [[gencom]]: [3/2 12/11; 121/120]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29

{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}

== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 87808/85293

{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}

: sval mapping generators: ~3/2, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2

=== Doubleton ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 1352/1323

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

: sval mapping generators: ~26/21, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2

== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11

[[Comma list]]: {{monzo| 11 1 -5 }}

{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}

: [[gencom]]: [5/2 125/88; 341796875/329832448]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2

= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments

[[Category:Subgroup temperaments| ]] 
[[Category:Temperament collections]]
{{Todo| review | cleanup }}

Subgroup temperaments

2026-05-18T16:05:47Z

Lériendil: added tridecimal guanyintet

{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.

For temperaments that omit various prime harmonics, see:
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].

= Composite subgroup temperaments =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.165

==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

: [[gencom]]: [2 9/8; 225/224 325/324 640/637]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

[[Badness]]: 0.00263

=== Joan ===
{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

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

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

=== Genius ===

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

{{Mapping|legend=2| 1 2 4 | 0 3 1 }}

: sval mapping generators: ~2, ~21/16

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

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

=== Tetrameantone ===
{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

=== Tetramagic ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

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

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

{{Todo|inline=1|cleanup}}

<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)
⟨2399.3973, 193.8643]

TE Step Tunings (cents)
⟨25.21211, 47.81337]

TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]

TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]

Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave

Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩

TE Generator Tunings (cents)
⟨2399.4195, 193.8654]

TE Step Tunings (cents)
⟨25.23883, 47.79592]

TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]

TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave

Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)
⟨2400.1097, 193.9498]

TE Step Tunings (cents)
⟨24.18752, 48.52491]

TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave

Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>

==== 4.6.5.7.11.13 ====

<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩

TE Generator Tunings (cents)
⟨2401.2305, 193.5378]

TE Step Tunings (cents)
⟨42.79107, 35.98524]

TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>

==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)
⟨2400.4701, 193.4599]

TE Step Tunings (cents)
⟨43.39350, 35.55764]

TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>

==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩

TE Generator Tunings (cents)
⟨2399.9219, 193.3952]

TE Step Tunings (cents)
⟨44.14256, 35.03670]

TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>

==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

TE Generator Tunings (cents)
⟨2399.3286, 193.5316]

TE Step Tunings (cents)
⟨37.31613, 39.63311]

TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>

== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

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

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

== 8.9.7 subgroup ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

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

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10-3

= Fractional subgroup temperaments =
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

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

{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

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

: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====
{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

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

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

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

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

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

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====
{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====
{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===
{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

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

{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}
: <nowiki/>* wart for 5/3
: † wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

[[Subgroup]]: 2.5/3.13/9

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

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

{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. It is defined by tempering out [[1728/1715]] ({{S|6/S7}}) and [[540/539]] (S12/S14), which imply [[176/175]] (S8/S10) as well as S11/S15 being tempered out. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

{{Mapping|legend=2| 1 0 1 3 | 0 -3 1 -5 }}
: mapping generators: ~2, ~7/6

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.455
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.093

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Tridecimal guanyintet ====
Guanyintet can extend to the 13th harmonic by the equivalences ([[12/11]])3 = [[13/10]] and ([[15/14]])3 = [[16/13]]. [[40edo]] remains an excellent tuning.

[[Subgroup]]: 2.5.7/3.11/3.13

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 0 }}), [[540/539]] ({{monzo| 2 1 -2 -1 0 }}), [[1573/1568]] ({{monzo| -5 0 -2 2 1 }})

{{Mapping|legend=2| 1 0 1 3 1 | 0 -3 1 -5 12 }}
: mapping generators: ~2, ~12/7

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~7/6 = 270.152
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~7/6 = 270.218

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 71, 111, 151, 262c*}} using subgroup TE 
: <nowiki/>* wart for 7/3

Badness (Sintel): 0.329

==== Laz ====
{{See also | Chromatic pairs #Laz }}

Laz is related to [[avalokita]] as well as to [[winston]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}

{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}

Kryptonite is related to [[krypton]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12

{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}

Kiribati is related to [[nakika]] as well as to [[octacot]].

[[Subgroup]]: 2.9/5.7/3.11/9

[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})

{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20

{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents

=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.

[[Subgroup]]: 2.7/3.11

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3

{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents

== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.

[[Subgroup]]: 2.5.9/7

[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})

{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5

{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents

==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[50/49]]

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

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}

{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830

{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
 based on subgroup TE 

Badness (Sintel): 0.119

==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}

Edson is related to [[pele]] and [[andromeda]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3

{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

{{Optimal ET sequence|legend=1| 12, 17, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

==== Haumea ====
{{See also| Chromatic pairs #Haumea }}

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}

{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: 364/363, 441/440, 1001/1000

{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

[[Subgroup]]: 2.7/5.9/5

[[Comma list]]: [[250047/250000]]

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

{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461

{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents

=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].

[[Subgroup]]: 2.7/5.11/5.13/5

[[Comma list]]: [[847/845]], [[1001/1000]]

{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents

==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95t, 124t }}
: t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents

== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

[[Subgroup]]: 2.11/5.13/5

[[Comma list]]: [[2200/2197]]

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

{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012

{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents

==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168, 540/539, 729/728

{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents

=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].

Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].

[[Subgroup]]: 2.3.7.11/5

[[Comma list]]: 1029/1024, 4000/3993

{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742

{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 1029/1024, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918

{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17

[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19

[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
: <nowiki />* wart for 11/7
: † wart for 13/7

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====
{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

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

{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

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

[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5

=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]

[[Subgroup]]: 2.3.49/5

[[Comma list]]: 2401/2400

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

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

[[Tp tuning #T2 tuning|RMS error]]: ?

== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})

{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

<nowiki/>* wart for 19/7

[[Badness]] (Sintel): 0.082

== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}

=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: [[10976/10935]]

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

[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909

[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

=== Halftone ===
{{Main| Halftone }}

[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: 9604/9375

{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2

[[Comma list]]: 1232/1215, 27783/27500

{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2

[[Comma list]]: 275/273, 1232/1215, 1323/1300

{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2

=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4

[[Comma list]]: 245/243

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

: sval mapping generators: ~3/2, ~9/7

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728

[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]

==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 100/99, 245/243

{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771

[[Optimal ET sequence]]: [[8edf]], [[13edf]]

==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 245/243, 441/440

{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939

[[Optimal ET sequence]]: [[8edf]], [[11edf]]

== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''

[[Subgroup]]: 3/2.5/4.11/8

[[Comma list]]: 121/120

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

: [[gencom]]: [3/2 12/11; 121/120]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29

{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}

== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 87808/85293

{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}

: sval mapping generators: ~3/2, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2

=== Doubleton ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 1352/1323

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

: sval mapping generators: ~26/21, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2

== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11

[[Comma list]]: {{monzo| 11 1 -5 }}

{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}

: [[gencom]]: [5/2 125/88; 341796875/329832448]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2

= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments

[[Category:Subgroup temperaments| ]] 
[[Category:Temperament collections]]
{{Todo| review | cleanup }}

Subgroup temperaments

2026-05-18T15:53:44Z

Lériendil: decanonicalized "edson" for hemifamity

{{Technical data page}}
A '''subgroup temperament''' is a regular temperament defined on a [[just intonation subgroup]] that is not a full ''p''-limit group.

For temperaments that omit various prime harmonics, see:
* [[No-thirteens subgroup temperaments]]
* [[No-elevens subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
* [[No-fives subgroup temperaments]]
* [[No-threes subgroup temperaments]]
* [[No-twos subgroup temperaments]] (additionally, [[Catalog of 3.5.7 subgroup rank two temperaments]]).

Below are some temperaments for composite subgroups and fractional subgroups. Obviously, no attempt has been made at completeness; attention is focused on subgroups containing interesting chords. The reader may also want to consult the page on [[Chromatic pairs]].

= Composite subgroup temperaments =
== 2.9.5.7 subgroup ==
See also [[Jubilismic clan #Antikythera|antikythera]] and [[Hemimean clan #Isra|isra]].

=== Commatose ===
Commatose is a [[Dual-fifth temperaments|dual-fifth temperament]] which uses the Pythagorean comma as a generator. It was developed by [[Eliora]] to highlight the near-perfect expression of 9/8 by [[1789edo]], while at the same time the fact that it completely misses 3/2. It is described as the 460 & 1329 temperament. In the 13-limit extension 24 generators are equal to [[~]][[13/9]].

[[Subgroup]]: 2.9.5.7

[[Comma list]]: {{monzo| 28 -2 -19 8 }}, {{monzo| 9 -25 23 6 }}

{{Mapping|legend=2| 1 9 6 13 | 0 -298 -188 -521 }}

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~531441/524288 = 23.4765

{{Optimal ET sequence|legend=1| 460, 869, 1329 }}

[[Badness]]: 0.611

==== 2.9.5.7.11 ====
Subgroup: 2.9.5.7.11

Comma list: {{monzo| -7 7 -3 2 -4 }}, {{monzo| 17 0 -13 1 3 }}, {{monzo| 11 -2 -6 7 -3 }}

Sval mapping: {{mapping| 1 9 6 13 16 | 0 -298 -188 -521 -641 }}

Optimal tuning (CTE): ~2 = 1\1, ~531441/524288 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.165

==== 2.9.5.7.11.13 ====
Subgroup: 2.9.5.7.11.13

Comma list: 123201/123200, 1016064/1015625, 2250423/2249390, 2599051/2598156

Sval mapping: {{mapping| 0 9 6 13 16 10 | -298 -188 -521 -641 -322 }}

Optimal tuning (CTE): ~2 = 1\1, ~3575/3528 = 23.4767

{{Optimal ET sequence|legend=1| 460, 869e, 1329, 1789, 3118 }}

Badness: 0.0564

=== Daemotertiaschis ===
{{See also|Schismatic family#Tertiaschis}}
Daemotertiaschis is produced by taking every other generator of tertiaschis, and the subgroup is chosen so it tempers out exactly the same commas. It is notable due to offering a [[7L 4s|daemotonic 7L 4s]] scale of reasonable hardness, which is notoriously difficult to approximate with simple JI or RTT methods.

Subgroup: 2.9.5.7.33.13.17

Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976

{{Mapping|legend=2|1 1 11 -16 13 -18 20|0 3 -12 26 -11 30 -22}}

Optimal tuning (CTE): ~2 = 1\1, 33/20 = 867.982

[[Support]]ing [[ET]]s: {{Optimal ET sequence|47, 65f, 112, 159, 206, 253}}

=== Baldy ===
{{See also|Schismatic family #Garibaldi}}
{{See also|No-threes subgroup temperaments #Frostburn}}

Baldy results from taking every other generator of the [[garibaldi]] temperament. One of the best extension is 2.9.5.7.13 subgroup with mapping 13/8 to +10 whole tones, as well as the cassandra temperament.

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 225/224, 3125/3087

{{Mapping|legend=2| 1 3 3 4 | 0 1 -4 -7 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.170

{{Optimal ET sequence|legend=1| 6, 29, 35, 41, 47 }}

Related temperament: [[Schismatic family #Garibaldi|Garibaldi]]

==== 2.9.5.7.13 ====
{{See also|Chromatic pairs #Baldy}}

Baldy is every other step of [[garibaldi]], without the mapping of prime 11. It can be described as the 6 & 35 temperament.

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: [[225/224]], [[325/324]], [[640/637]]

{{Mapping|legend=2| 1 0 15 25 -28 | 0 1 -4 -7 10 }}

{{Mapping|legend=3| 1 3/2 3 4 0 2 | 0 1/2 -4 -7 0 10 }}

: [[gencom]]: [2 9/8; 225/224 325/324 640/637]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 204.090

{{Optimal ET sequence|legend=1| 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5999 cents

Related temperament: [[Schismatic family #Garibaldi|Cassandra]]

==== Baldanders ====
Baldanders results from taking every other generator of the andromeda, with mapping 11/8 to -9 whole tones.

Subgroup: 2.9.5.7.11

Comma list: 100/99, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 | 0 1 -4 -7 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.743

{{Optimal ET sequence|legend=1| 6, 23de, 29, 35, 41 }}

Related temperament: [[Schismatic family #Garibaldi|Andromeda]]

===== 2.9.5.7.11.13 =====
Subgroup: 2.9.5.7.11.13

Comma list: 100/99, 144/143, 225/224, 245/242

{{Mapping|legend=2| 1 3 3 4 5 2 | 0 1 -4 -7 -9 10 }}

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.414

{{Optimal ET sequence|legend=1| 6, 23def, 29f, 35, 41, 47 }}

== 2.9.5.11 subgroup ==
=== Glacial ===
{{See also| Chromatic pairs #Glacial }}

[[Subgroup]]: 2.9.5.11.13

[[Comma list]]: 45/44, 65/64, 81/80

{{Mapping|legend=2| 1 0 -4 -6 10 | 0 1 2 3 -2 }}

{{Mapping|legend=3| 1 3/2 2 0 3 4 | 0 1/2 2 0 3 -2 }}

: [[gencom]]: [2 9/8; 45/44 65/64 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 186.151

{{Optimal ET sequence|legend=1| 6, 13, 45be, 58bce, 71bce, 84bce }}

[[Tp tuning #T2 tuning|RMS error]]: 2.887 cents

Music:
* ''[[Thundersnow]]'' - [[Sevish]] (2021)

== 2.9.7 subgroup ==
=== Mabon ===
Derived from a [http://individual.utoronto.ca/kalendis/leap/index.htm#se calendar leap cycle built for the autumn equinox], hence the name. Defined as the 11 & 62 temperament.

Subgroup: 2.9.7

Comma basis: 44957696/43046721

Sval mapping: [{{val|1 1 -3}}, {{val|0 3 8}}]

Optimal tuning (CTE): ~729/448 = 870.792

{{Optimal ET sequence|legend=1|7d, 11, 18d, 29, 40, 62}}, ...

==== 2.9.7.11 subgroup ====
Subgroup: 2.9.7.11

Comma basis: 896/891, 1331/1296

Sval mapping: [{{val|1 1 -3 2}}, {{val|0 3 8 2}}]

Optimal tuning (CTE): ~16/11 = 870.966

{{Optimal ET sequence|legend=1| 7d, 11, 40, 51, 62 }}

== 2.9.7.11 subgroup ==
=== Apparatus ===
[[Subgroup]]: 2.9.7.11

[[Comma list]]: 41503/41472, 322102/321489

{{Mapping|legend=2| 1 5 3 5 | 0 -19 -2 -16 }}

: mapping generators: ~2, ~77/72

{{Mapping|legend=3| 1 5/2 0 3 5 | 0 -19/2 0 -2 -16 }}

: [[gencom]]: [2 77/72; 41503/41472 322102/321489]

[[Optimal tuning]] ([[CTE]]): ~77/72 = 115.5685

{{Optimal ET sequence|legend=1| 10e, 21, 31, 52, 83, 135, 353, 488, 623 }}

[[Badness]]: 0.00263

=== Joan ===
{{See also| Chromatic pairs #Joan }}

Joan is related to [[casablanca]] as well as to [[orwell]].

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 99/98, 9317/9216

{{Mapping|legend=2| 1 0 1 3 | 0 7 4 1 }}

{{Mapping|legend=3| 1 0 0 1 3 | 0 7/2 0 4 1 }}

: [[gencom]]: [2 11/8; 99/98 9317/9216]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~11/8 = 542.672 cents

{{Optimal ET sequence|legend=1| 11, 20, 31, 42, 115bd, 157bd }}

[[Tp tuning #T2 tuning|RMS error]]: 1.424 cents

=== Machine ===
Machine is every other step of [[supra]], most interesting for its scale patterns.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 64/63, 99/98

{{Mapping|legend=2| 1 0 6 13 | 0 1 -1 -3 }}

: sval mapping generators: ~2, ~9

{{Mapping|legend=3| 1 3/2 0 3 4 | 0 1/2 0 -1 -3 }}

: [[gencom]]: [2 8/7; 64/63 99/98]

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~9/8 = 216.9128
* [[POTE]]: ~2 = 1\1, ~9/8 = 214.3843

{{Optimal ET sequence|legend=1| 5, 6, 11, 17, 28 }}

[[Badness]]: 0.00233

=== Penta a.k.a. mechanism ===
Penta or mechanism is the 8 & 11 temperament in the 2.9.7.11 subgroup.

[[Subgroup]]: 2.9.7.11

[[Comma list]]: 896/891, 26411/26244

{{Mapping|legend=2| 1 0 -1 6 | 0 5 6 -4 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 0 5 2 | 0 -5/2 0 -6 4 }}

: [[gencom]]: [2 9/7; 896/891 26411/26244]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~14/9 = 761.3782

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 52 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.4262 cents

[[Badness]]: 0.00439

Scales: [[penta5]], [[penta8]], [[penta11]], [[penta19]]

== 2.9.11 subgroup ==
=== Demon ===
Demon is a temperament which equates 3 [[11/9]] with [[16/9]], or equivalently 3 [[18/11]] with [[9/8]], tempering out [[1331/1296]]. This results in [[11/9]] being tuned flat to a supraminor third, and [[27/22]] being tuned sharp to a submajor third. It was discovered by [[User:CompactStar|CompactStar]] while searching for temperaments assosciated with the [[7L 4s]] ("daemotonic") MOS, known for its lack of representation of simple temperaments. The optimal tuning for demon temperament is near the basic tuning of 7L 4s (13\18), and indeed [[18edo]] supports demon temperament.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[1331/1296]]

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

[[Optimal tuning]] ([[CTE]]): ~[[18/11]] = 870.060

{{Optimal ET sequence|legend=1|4, 7, 11, 18, 29, 76e}}

=== Genius ===

Named after the genius in Roman religion, following the demon (daimon) in Greek mythology.

[[Subgroup]]: 2.9.11

[[Comma list]]: [[131769/131072]]

{{Mapping|legend=2|1 1 4|0 4 -1}}

[[Optimal tuning]] ([[CTE]]): ~[[16/11]] = 650.863

{{Optimal ET sequence|legend=1|9, 11, 24, 59, 83, 142, 225, 367}}[-11], 592[-11], 959[-9, --11], 1326[-9, --11]

== 2.9.15.7 subgroup ==
=== Stacks (a.k.a. 2magic) ===
Stacks, the 11 & 30 temperament in the 2.9.15.7.11.13 subgroup, is every other step of [[magic]].

[[Subgroup]]: 2.9.15.7

[[Comma list]]: 225/224, 245/243

{{Mapping|legend=2| 1 0 2 -1 | 0 5 3 6 }}

: sval mapping generators: ~2, ~14/9

{{Mapping|legend=3| 1 5/2 5/2 5 | 0 -5/2 -1/2 -6 }}

: [[gencom]]: [2 9/7; 225/224 245/243]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~14/9 = 760.704

{{Optimal ET sequence|legend=1| 8, 11, 30, 41, 71, 93, 112c, 134c, 175c }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

==== 2.9.15.7.11 ====
Subgroup: 2.9.15.7.11

Comma list: 100/99, 225/224, 245/243

Sval mapping: {{mapping| 1 0 2 -1 6 | 0 5 3 6 -4 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 | 0 -5/2 -1/2 -6 4 }}

: gencom: [2 9/7; 100/99 225/224 245/243]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.393

Optimal ET sequence: {{Optimal ET sequence| 8, 11, 30, 41, 52, 93, 145, 342bce }}

RMS error: 1.226 cents

==== 2.9.15.7.11.13 ====
Subgroup: 2.9.15.7.11.13

Comma list: 100/99, 105/104, 144/143, 196/195

Sval mapping: {{mapping| 1 0 2 -1 6 -2 | 0 5 3 6 -4 9 }}

Gencom mapping: {{mapping| 1 5/2 5/2 5 2 7 | 0 -5/2 -1/2 -6 4 -9 }}

: gencom: [2 9/7; 100/99 105/104 144/143 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~14/9 = 761.023

Optimal ET sequence: {{Optimal ET sequence| 11, 30, 41, 153cdef, 194cdef, 235cdef }}

RMS error: 1.540 cents

== 2.9.21 subgroup ==
=== A-team ===
A-team is every other step of [[slendric]]; the 2.9.5.21.11 extension below specifically restricts [[mothra]].

[[Subgroup]]: 2.9.21

[[Comma list]]: 1029/1024

{{Mapping|legend=2| 1 2 4 | 0 3 1 }}

: sval mapping generators: ~2, ~21/16

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

: [[gencom]]: [2 21/16; 1029/1024]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~21/16 = 467.375

{{Optimal ET sequence|legend=1| 5, 13, 18, 41, 59, 77, 95 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3202 cents

==== 2.9.5.21 ====
''Lookalike temperament: [[Dual-fifth_temperaments#Dual-3_A-Team|Dual-3 A-Team]]''

Subgroup: 2.9.5.21

[[Comma]] list: 81/80, 1029/1024

Sval mapping: {{mapping| 1 2 0 4 | 0 3 6 1 }}

Mapping generators: ~2, ~21/16

Optimal ([[Lp tuning|POL2]]) generator: 464.3865

{{Optimal ET sequence|legend=1| 13, 18, 31, 44 }}

===== 2.9.5.21.11 =====
Subgroup: 2.9.5.21.11

Comma list: 81/80, 99/98, 385/384

Sval mapping: {{mapping| 1 2 0 4 5 | 0 3 6 1 -4 }}

Gencom mapping: {{mapping| 1 1 0 3 5 | 0 3/2 6 -1/2 -4 }}

: gencom: [2 21/16; 81/80 99/98 385/384]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 463.956

{{Optimal ET sequence|legend=1| 5, 13, 31 }}

==== B-team ====
B-team (23 & 41) is every other step of [[rodan]].

Subgroup: 2.9.15.21.33

Comma list: 245/243, 385/384, 441/440

Sval mapping: {{mapping| 1 2 0 4 7 | 0 3 10 1 -5 }}

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/16 = 468.918

{{Optimal ET sequence|legend=1| 5, 13c, 18, 23, 41, 64, 87, 151 }}

== 4.3.5 subgroup ==
=== Tetrahanson ===
{{Main| Tetrahanson }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 15625/15552

{{Mapping|legend=2| 1 3 3 | 0 -6 -5 }}

: Mapping generators: ~4, ~5/3

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~5/3 = 882.941

[[Support]]ing [[ET]]s: {{EDs|19, 106, 87, 68, 11, 8, 125, 49, 30, 27, 117, 46, 41b, 79|equave=4}}

=== Tetrameantone ===
{{Main| Tetrameantone }}

[[Subgroup]]: 4.3.5

[[Comma list]]: 81/80

{{Mapping|legend=2| 1 1 2 | 0 -1 -4 }}

: Mapping generators: ~4, ~4/3

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~4/3 = 503.761

[[Support]]ing [[ET]]s: {{EDs|5, 9, 14, 19, 24, 43, 62, 81, 100|equave=4}}

=== Tetramagic ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 3125/3072

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

: Mapping generators: ~4, ~5/4

[[Optimal tuning]] ([[POTE]]): 4 = 2400.0, ~5/4 = 380.059

[[Support]]ing [[ET]]s: {{EDs|6, 13, 19, 25, 38, 44, 63, 82|equave=4}}

=== Blacktetra ===

[[Subgroup]]: 4.3.5

[[Comma list]]: 256/243

{{Mapping|legend=2| 5 4 6 | 0 0 -1 }}

: Mapping generators: ~4, ~16/15

[[Optimal tuning]] ([[POTE]]): 1\5ed4 = 480.0, ~16/15 = 80.4062

[[Support]]ing [[ET]]s: {{EDs|5, 10, 15, 20, 25, 30, 55, 85, 115|equave=4}}

== 4.6.5 subgroup ==
=== Meanquad ===
{{Main| Meanquad }}

[[Subgroup]]: 4.6.5

[[Comma list]]: [[81/80]] = {{monzo| -4 4 -1 }}

{{Mapping|legend=2| 1 0 -4| 0 1 4 }}

: mapping generators: ~4, ~6

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 697.214

[[Support]]ing [[ET]]s: *7, *10, *11[-5], *13[+5], *17, *24, *27[+5], *31, *38, *41, *45, *52, *55, *69

<nowiki />* Wart for 4

==== 4.6.5.7 subgroup (tetrominant) ====
[[Subgroup]]: 4.6.5.7

[[Comma list]]: [[36/35]] = {{monzo| 0 2 -1 -1 }}, [[64/63]] = {{monzo| 4 -2 0 -1 }}

{{Mapping|legend=2| 1 0 -4 4 | 0 1 4 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~4 = 2\1, ~3/2 = 699.622

[[Support]]ing [[ET]]s: *7, *10, *17, *24, *27[+5], *31, *38[+7], *41, *44[+5], *55[+7], *58[+5, +7], *65[+5, +7], *75[+5, +7]

<nowiki />* Wart for 4

=== Fourwar ===
The 23-limit version of Fourwar was created first, as an attempt to approximate subgroup 4.6.5.7.11.13.17.19.23 as accurately as possible using 25 to 35 notes per equave. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fourwar is named after the closely related [[hemiwar]] temperament.

{{Todo|inline=1|cleanup}}

<pre>
Reduced Mapping
4 6 5
[ ⟨ 1 0 1 ]
⟨ 0 16 2 ] ⟩

TE Generator Tunings (cents)
⟨2399.3973, 193.8643]

TE Step Tunings (cents)
⟨25.21211, 47.81337]

TE Tuning Map (cents)
⟨2399.397, 3101.829, 2787.126]

TE Mistunings (cents)
⟨-0.603, -0.126, 0.812]

Complexity 1.369085
Adjusted Error 0.692892 cents
TE Error 0.268047 cents/octave

Unison Vector
[8, 1, -8⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7 ====
<pre>
Reduced Mapping
4 6 5 7
[ ⟨ 1 0 1 1 ]
⟨ 0 16 2 5 ] ⟩

TE Generator Tunings (cents)
⟨2399.4195, 193.8654]

TE Step Tunings (cents)
⟨25.23883, 47.79592]

TE Tuning Map (cents)
⟨2399.420, 3101.846, 2787.150, 3368.747]

TE Mistunings (cents)
⟨-0.580, -0.109, 0.837, -0.079]

Complexity 1.192044
Adjusted Error 0.653313 cents
TE Error 0.232715 cents/octave

Unison Vectors
[-2, -1, -2, 4⟩ (2401:2400)
[3, 0, -5, 2⟩ (3136:3125)
[5, 1, -3, -2⟩ (6144:6125)
[8, 1, -8, 0⟩ (393216:390625)

Subsets
q99, q62, q37, q161, q136, q198, q25, q124, q74, q235
</pre>

==== 4.6.5.7.11 ====
<pre>
Reduced Mapping
4 6 5 7 11
[ ⟨ 1 0 1 1 1 ]
⟨ 0 16 2 5 9 ] ⟩

TE Generator Tunings (cents)
⟨2400.1097, 193.9498]

TE Step Tunings (cents)
⟨24.18752, 48.52491]

TE Tuning Map (cents)
⟨2400.110, 3103.196, 2788.009, 3369.859, 4145.658]

TE Mistunings (cents)
⟨0.110, 1.241, 1.696, 1.033, -5.660]

Complexity 1.068792
Adjusted Error 2.926965 cents
TE Error 0.846083 cents/octave

Unison Vectors
[-1, -1, -1, 0, 2⟩ (121:120)
[2, 0, -2, -1, 1⟩ (176:175)
[-3, -1, 1, 1, 1⟩ (385:384)
[-1, 0, 3, -3, 1⟩ (1375:1372)
[-2, -1, -2, 4, 0⟩ (2401:2400)
[1, 0, 1, -4, 2⟩ (2420:2401)

Subsets
q37, q25, q62, q12, q74, q99, q87, q49r, q50r, q124
</pre>

==== 4.6.5.7.11.13 ====

<pre>
Reduced Mapping
4 6 5 7 11 13
[ ⟨ 1 0 1 1 1 0 ]
⟨ 0 16 2 5 9 23 ] ⟩

TE Generator Tunings (cents)
⟨2401.2305, 193.5378]

TE Step Tunings (cents)
⟨42.79107, 35.98524]

TE Tuning Map (cents)
⟨2401.230, 3096.606, 2788.306, 3368.920, 4143.071, 4451.371]

TE Mistunings (cents)
⟨1.230, -5.349, 1.992, 0.094, -8.247, 10.843]

Complexity 1.219191
Adjusted Error 6.699599 cents
TE Error 1.810487 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1⟩ (66:65)
[-1, -1, -1, 0, 2, 0⟩ (121:120)
[1, 2, 0, 0, -1, -1⟩ (144:143)
[2, 0, -2, -1, 1, 0⟩ (176:175)
[-2, 1, 1, 1, 0, -1⟩ (105:104)
[-3, -1, 1, 1, 1, 0⟩ (385:384)
[-3, 0, 0, 1, 2, -1⟩ (847:832)
[1, 3, -1, 0, 0, -2⟩ (864:845)
[-1, 0, 3, -3, 1, 0⟩ (1375:1372)

Subsets
q25, q37f, q12f, q62, q50rf, q13rff, q49rff, q87, q74ff, q24rfff
</pre>

==== 4.6.5.7.11.13.17 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17
[ ⟨ 1 0 1 1 1 0 1 ]
⟨ 0 16 2 5 9 23 13 ] ⟩

TE Generator Tunings (cents)
⟨2400.4701, 193.4599]

TE Step Tunings (cents)
⟨43.39350, 35.55764]

TE Tuning Map (cents)
⟨2400.470, 3095.359, 2787.390, 3367.770, 4141.609, 4449.578, 4915.449]

TE Mistunings (cents)
⟨0.470, -6.596, 1.076, -1.056, -9.709, 9.050, 10.494]

Complexity 1.129881
Adjusted Error 8.082725 cents
TE Error 1.977443 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0⟩ (66:65)
[1, 1, 1, -1, 0, 0, -1⟩ (120:119)
[1, 2, 0, 0, -1, -1, 0⟩ (144:143)
[-2, 1, 1, 1, 0, -1, 0⟩ (105:104)
[-1, 2, 2, 0, 0, -1, -1⟩ (225:221)
[-1, 1, 2, -2, 0, -1, 1⟩ (1275:1274)

Subsets
q25, q12f, q37f, q13rffg, q50rf, q62, q49rffg, q24rfffg, q38rreffg, q74ffg
</pre>

==== 4.6.5.7.11.13.17.19 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19
[ ⟨ 1 0 1 1 1 0 1 1 ]
⟨ 0 16 2 5 9 23 13 14 ] ⟩

TE Generator Tunings (cents)
⟨2399.9219, 193.3952]

TE Step Tunings (cents)
⟨44.14256, 35.03670]

TE Tuning Map (cents)
⟨2399.922, 3094.324, 2786.712, 3366.898, 4140.479, 4448.090, 4914.060, 5107.455]

TE Mistunings (cents)
⟨-0.078, -7.631, 0.399, -1.928, -10.839, 7.562, 9.104, 9.942]

Complexity 1.058472
Adjusted Error 8.712222 cents
TE Error 2.050935 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0⟩ (66:65)
[-1, 0, 0, 1, 1, 0, 0, -1⟩ (77:76)
[2, 1, -1, 0, 0, 0, 0, -1⟩ (96:95)
[1, 1, 1, -1, 0, 0, -1, 0⟩ (120:119)
[0, 1, 1, 1, -1, 0, 0, -1⟩ (210:209)
[0, 0, 1, -2, 1, 0, 1, -1⟩ (935:931)
[2, 0, -3, 1, 0, 0, -1, 1⟩ (2128:2125)

Subsets
q25, q12fh, q37f, q13rffgh, q50rf, q62, q49rffgh, q24rfffghh, q38rreffgh, q74ffgh
</pre>

==== 4.6.5.7.11.13.17.19.23 ====
<pre>
Reduced Mapping
4 6 5 7 11 13 17 19 23
[ ⟨ 1 0 1 1 1 0 1 1 0 ]
⟨ 0 16 2 5 9 23 13 14 28 ] ⟩

TE Generator Tunings (cents)
⟨2399.3286, 193.5316]

TE Step Tunings (cents)
⟨37.31613, 39.63311]

TE Tuning Map (cents)
⟨2399.329, 3096.506, 2786.392, 3366.987, 4141.113, 4451.227, 4915.240, 5108.771, 5418.885]

TE Mistunings (cents)
⟨-0.671, -5.449, 0.078, -1.839, -10.205, 10.699, 10.284, 11.258, -9.389]

Complexity 1.115920
Adjusted Error 9.502017 cents
TE Error 2.100561 cents/octave

Unison Vectors
[0, 1, -1, 0, 1, -1, 0, 0, 0⟩ (66:65)
[1, 0, 0, -1, 0, -1, 0, 0, 1⟩ (92:91)
[0, -1, 1, 0, 0, 0, 0, -1, 1⟩ (115:114)
[1, 1, 1, -1, 0, 0, -1, 0, 0⟩ (120:119)
[2, 0, -2, -1, 1, 0, 0, 0, 0⟩ (176:175)
[-3, -1, 1, 1, 1, 0, 0, 0, 0⟩ (385:384)
[1, 0, -2, 1, 0, 0, 1, -1, 0⟩ (476:475)
[1, 0, 0, -2, 1, 0, -1, 1, 0⟩ (836:833)
[0, 0, 1, -2, 1, 0, 1, -1, 0⟩ (935:931)
[1, -1, 0, 0, 0, 0, -2, 1, 1⟩ (874:867)

Subsets
q25i, q12fhi, q37f, q13rffghii, q62, q50rfii, q49rffghii, q24rfffghhiii, q74ffghi, q38rreffghiii
</pre>

== 4.9.25 subgroup ==
=== Meansquared ===
[[Subgroup]]: 4.9.25

[[Comma list]]: [[6561/6400]]

{{Mapping|legend=2| 1 3 4 | 0 1 4 }}

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~4 = 2\1, ~9/4 = 1394.429

[[Support]]ing [[ET]]s: 12, 7, 19, 5, 31, 26, 17[+25], 43, 9[-25], 33[-25], 45, 29[+25], 8[+25], 22[+25]

== 4.9.49 subgroup ==
=== Archsquared ===
[[Subgroup]]: 4.9.49

[[Comma list]]: 4096/3969

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

Mapping generators: ~4, ~9/64

[[Optimal tuning]] ([[CTE]]): ~9/4 = 1419.190

[[Support]]ing [[ET]]s: 5, 17, 22, 12, 7, 27, 32, 8, 39[+49], 29[+49], 9[+49], 19[+49], 37, 49

== 8.9.7 subgroup ==
=== Sixscared ===
Sixscared is a tuning which still maintains some consonance, while eviscerating the rules of conventional 12-tone harmony. The familiar major, minor and perfect intervals are nowhere to be found, and octaves are far and few between, so the seventh harmonic becomes the backbone of harmony. Approximating the harmonics 7, 8, 9, Sixscared is named for the classic dad joke: "Why was six scared? Because seven ate nine."

[[Subgroup]]: 8.9.7

[[Comma list]]: 64/63

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

: sval mapping generators: ~8, ~9

: [[gencom]]: [8 9/8; 64/63]

[[Optimal tuning]] ([[CTE]]): ~9/8 = 219.1898

[[Optimal ET sequence]]: {{val| 16 17 15 }}, {{val| 33 35 31 }}, {{val| 148 … }}, {{val| 181 … }}, {{val| 214 … }}, {{val| 247 … }}

[[Badness]]: 0.0215 × 10-3

= Fractional subgroup temperaments =
== 2.5/3.… subgroups ==
=== Magicaltet ===
{{See also| Chromatic pairs #Magicaltet }}

Magicaltet is related to [[keemic]], [[superkleismic]], and [[magic]]. The tonic and the first three generator steps make a [[magical seventh chord]], hence the name.

[[Subgroup]]: 2.5/3.7.11

[[Comma list]]: 100/99 ({{monzo| 2 2 0 -1 }}), 385/384 ({{monzo| -7 1 1 1 }})

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

{{Mapping|legend=3| 1 -1/2 1/2 2 4 | 0 1/2 -1/2 3 -2 }}
: [[gencom]]: [2 6/5; 100/99 385/384]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 877.343
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 877.351

{{Optimal ET sequence|legend=1| 4, 7, 11, 15, 26, 67, 93* }}
: <nowiki/>* wart for 5/3

[[Tp tuning #T2 tuning|RMS error]]: 1.206 cents

=== Starlingtet ===
{{See also | Chromatic pairs #Starlingtet }}

Starlingtet, the {{nowrap| 4 & 15 }} temperament in the 2.5/3.7/3 subgroup, is related to [[starling]] as well as to [[myna]]. The tonic and the first three generator steps make a [[starling tetrad]], hence the name.

[[Subgroup]]: 2.5/3.7/3

[[Comma list]]: [[126/125]] ({{monzo| 1 -3 1 }})

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

: mapping generators: ~2, ~5/3

{{Mapping|legend=3| 1 -1 0 1 | 0 4/3 1/3 -5/3 }}
: [[gencom]]: [2 6/5; 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 888.759
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 888.846

{{Optimal ET sequence|legend=1| 4, 15, 19, 23, 27 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.8398 cents

==== Greeley ====
{{See also| Chromatic pairs #Greeley }}

Greeley is related to [[opossum]] as well as to [[nusecond]].

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: 121/120 ({{monzo| -3 -1 0 2 }}), 126/125 ({{monzo| 1 -3 1 }})

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

{{Mapping|legend=3| 1 -5/4 -1/4 3/4 3/4 | 0 9/4 1/4 -15/4 5/4 }}
: [[gencom]]: [2 11/10; 121/120 126/125]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~11/10 = 155.696
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~11/10 = 155.776

{{Optimal ET sequence|legend=1| 8, 15, 23, 54, 77, 100, 131* }}
: <nowiki/>* wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 1.034 cents

==== Skateboard ====
{{See also| Chromatic pairs #Skateboard }}

Skateboard is related to [[thrasher]].

[[Subgroup]]: 2.5/3.7/3.11.13/9

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 91/90 ({{monzo| -1 -1 1 0 1 }}), 100/99 ({{monzo| 2 2 0 -1 }})

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

{{Mapping|legend=3| 1 -3/7 4/7 11/7 4 -6/7 | 0 0 -1 -3 -2 2 }}
: [[gencom]]: [2 6/5; 56/55 91/90 100/99]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 886.158
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 886.158

{{Optimal ET sequence|legend=1| 11, 15, 19, 23, 42d, 65d }}

[[Tp tuning #T2 tuning|RMS error]]: 2.396 cents

=== Gariberttet ===
Gariberttet is the 2.5/3.7/3 [[Subgroup temperament families, relationships, and genes|altergene]] of [[sirius]].

==== Gariberttet (2.5/3.7/3.13/11 subgroup) ====
{{See also | Chromatic pairs #Gariberttet }}

Gariberttet can be described as the {{nowrap| 4 & 29 }} temperament in the 2.5/3.7/3.13/11 subgroup. Extensions to the full 7-, 11-, and 13-limits include [[quasitemp]].

[[Subgroup]]: 2.5/3.7/3.13/11

[[Comma list]]: [[275/273]] ({{monzo| 0 2 -1 -1 }}), [[847/845]] ({{monzo| 0 -1 1 -2 }})

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

{{Mapping|legend=3| 1 0 0 0 0 0 | 0 -8/3 1/3 7/3 -1/2 1/2 }}
: [[gencom]]: [2 13/11; 275/273 847/845]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~13/11 = 293.679

{{Optimal ET sequence|legend=1| 29, 33, 37, 41, 45, 49, 78, 94, 143* }}
: <nowiki/>* wart for 13/11

[[Tp tuning #T2 tuning|RMS error]]: 0.6914 cents

==== Indium ====
{{See also | Chromatic pairs #Indium }}

Indium can be described as the {{nowrap| 8 & 33 }} temperament in the 2.5/3.7/3.11/3 subgroup.

[[Subgroup]]: 2.5/3.7/3.11/3

[[Comma list]]: [[3025/3024]] ({{monzo| -4 2 -1 2 }}), [[3125/3087]] ({{monzo| 0 5 -3 }})

{{Mapping|legend=2| 1 0 0 2 | 0 6 10 -1 }}

{{Mapping|legend=3| 1 -1/2 -1/2 -1/2 3/2 | 0 -15/4 9/4 25/4 -19/4 }}
: [[gencom]]: [2 12/11; 3025/3024 3125/3087]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/11 = 146.978
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/11 = 147.010

{{Optimal ET sequence|legend=1| 8, 33, 41, 49, 204*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.7788 cents

==== Ammon ====
{{See also| Chromatic pairs #Ammon }}

Ammon can be described as the {{nowrap| 8 & 29 }} temperament in the 2.5/3.7/3.11/3.13/3 subgroup. It extends [[tridec]], and is related to [[ammonite]]. It is generated by a semidiminished fourth, hence the old name ''semidim'', which has been rejected since 2025 to avoid confusion with another temperament of the same name.

[[Subgroup]]: 2.5/3.7/3.11/3.13/3

[[Comma list]]: [[121/120]] ({{monzo| -3 -1 0 2 }}), [[169/168]] ({{monzo| -3 0 -1 0 2 }}), [[275/273]] ({{monzo| 0 2 -1 1 -1 }})

{{Mapping|legend=2| 1 3 5 3 4 | 0 -6 -10 -3 -5 }}

{{Mapping|legend=3| 1 -3 0 2 0 1 | 0 24/5 -6/5 -26/5 9/5 -1/5 }}
: [[gencom]]: [2 13/10; 121/120 169/168 275/273]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/10 = 453.121
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/10 = 453.242

{{Optimal ET sequence|legend=1| 8, 29, 37, 45 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.052 cents

=== Sentry ===
{{See also | Chromatic pairs #Sentry }}

Sentry, the {{nowrap| 3 & 5 }} temperament in the 2.5/3.9/7 subgroup, is related to [[sensi]].

[[Subgroup]]: 2.5/3.9/7

[[Comma list]]: [[245/243]] ({{monzo| 0 1 -2 }})

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

{{Mapping|legend=3| 1 0 0 0 | 0 0 2 -1 }}
: [[gencom]]: [2 9/7; 245/243]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]] and [[POTE]]: ~2 = 1200.000, ~9/7 = 440.902

{{Optimal ET sequence|legend=1| 8, 11, 19, 30, 41, 49, 52, 145*, 166†, 197*†, 215†, 264*† }}
: <nowiki/>* wart for 5/3
: † wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.7105 cents

=== Marveltwintri ===
{{See also| Chromatic pairs #Marveltwintri }}

Marveltwintri can be described as the {{nowrap| 3 & 4 }} temperament in the 2.5/3.13/9 subgroup. The tonic and the first two generator steps make a [[marveltwin triad]], hence the name. [[Cata]] is a very natural extension of this temperament to the [[2.3.5.13 subgroup|2.3.5.13-subgroup]].

[[Subgroup]]: 2.5/3.13/9

[[Comma list]]: [[325/324]] ({{monzo| -2 2 1 }})

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

{{Mapping|legend=3| 1 -1/6 5/6 0 0 -1/3 | 0 -1/2 -3/2 0 0 1 }}
: [[gencom]]: [2 6/5; 325/324]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/3 = 882.886
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/3 = 882.861

{{Optimal ET sequence|legend=1| 3, 4, 11, 15, 19, 34, 53, 87, 140 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2444 cents

== 2.….7/3.… subgroups ==
=== Guanyintet ===
{{See also | Chromatic pairs #Guanyintet }}

Guanyintet, the {{nowrap| 4 & 9 }} temperament in the 2.5.7/3.11/3 subgroup, is the main rank-2 chain of [[guanyin]] and a restriction of [[orwell]]. The tonic and the first three generator steps make a [[guanyin tetrad]], hence the name.

[[Subgroup]]: 2.5.7/3.11/3

[[Comma list]]: [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[540/539]] ({{monzo| 2 1 -2 -1 }})

{{Mapping|legend=2| 1 0 2 -2 | 0 3 -1 5 }}
: mapping generators: ~2, ~12/7

{{Mapping|legend=3| 1 -4/3 3 -1/3 5/3 | 0 4/3 -3 7/3 -11/3 }}
: [[gencom]]: [2 7/6; 176/175 540/539]

[[Optimal tuning]]s:
* ([[Tp tuning|subgroup]] [[CTE]]): ~2 = 1200.000, ~12/7 = 929.545
* ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1200.000, ~12/7 = 929.907

{{Optimal ET sequence|legend=1| 9, 22, 31, 40, 191c*, 231c*, 271c*, 311c* }}
: <nowiki/>* wart for 7/3

[[Tp tuning #T2 tuning|RMS error]]: 0.6028 cents

==== Laz ====
{{See also | Chromatic pairs #Laz }}

Laz is related to [[avalokita]] as well as to [[winston]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: [[144/143]] ({{monzo| 4 0 0 -1 -1 }}), [[176/175]] ({{monzo| 4 -2 -1 1 }}), [[196/195]] ({{monzo| 2 -1 2 0 -1 }}

{{Mapping|legend=2| 1 0 2 -2 6 | 0 3 -1 5 -5 }}

{{Mapping|legend=3| 1 -5/4 3 -1/4 7/4 -1/4 | 0 -1/4 -3 3/4 -21/4 19/4 }}
: [[gencom]]: [2 7/6; 144/143 176/175 196/195]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~12/7 = 930.598
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~12/7 = 930.700

{{Optimal ET sequence|legend=1| 9, 31, 40, 49, 156c*†, 205c*† }}
: <nowiki/>* wart for 7/3
: † wart for 11/3

[[Tp tuning #T2 tuning|RMS error]]: 0.8790 cents

=== Kryptonite ===
{{See also| Chromatic pairs #Kryptonite }}

Kryptonite is related to [[krypton]].

[[Subgroup]]: 2.5.7/3.11/3.13/3

[[Comma list]]: 56/55 ({{monzo| 3 -1 1 -1 }}), 78/77 ({{monzo| 1 0 -1 -1 1 }}), 91/90 ({{monzo| -1 -2 1 0 1 }})

{{Mapping|legend=2| 1 2 1 2 2 | 0 3 2 -1 1 }}
: mapping generators: ~2, ~13/12

{{Mapping|legend=3| 1 -5/4 2 -1/4 3/4 3/4 | 0 -1/2 3 3/2 -3/2 1/2 }}
: [[gencom]]: [2 13/12; 56/55 78/77 91/90]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~13/12 = 130.945
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~13/12 = 132.428

{{Optimal ET sequence|legend=1| 1, …, 8, 9 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.545 cents

=== Kiribati ===
{{See also| Chromatic pairs #Kiribati }}

Kiribati is related to [[nakika]] as well as to [[octacot]].

[[Subgroup]]: 2.9/5.7/3.11/9

[[Comma list]]: 100/99 ({{monzo| 2 -2 0 -1 }}), 245/242 ({{monzo| -1 -1 2 -2 }})

{{Mapping|legend=2| 1 1 1 0 | 0 -2 3 4 }}
: mapping generators: ~2, ~21/20

{{Mapping|legend=3| 1 1/10 -4/5 11/10 1/5 | 0 -3/2 -1 3/2 1 }}
: [[gencom]]: [2 21/20; 100/99 245/242]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~21/20 = 87.776
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~21/20 = 87.892

{{Optimal ET sequence|legend=1| 13, 14, 27, 41 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.245 cents

=== Mothwelltri ===
{{See also| Chromatic pairs #Mothwelltri }}

Mothwelltri, the {{nowrap| 1 & 4 }} temperament in the 2.7/3.11 subgroup, is related to [[orwell]]. The tonic and the first two generator steps make a [[mothwellsmic triad]], hence the name.

[[Subgroup]]: 2.7/3.11

[[Comma list]]: [[99/98]] ({{monzo| -1 -2 1 }})

{{Mapping|legend=2| 1 0 1 | 0 1 2 }}
: mapping generators: ~2, ~7/3

{{Mapping|legend=3| 1 -1/2 0 1/2 3 | 0 -1/2 0 1/2 2 }}
: [[gencom]]: [2 7/6; 99/98]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~7/6 = 273.695
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~7/6 = 273.174

{{Optimal ET sequence|legend=1| 4, 9, 13, 22, 79 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.064 cents

== 2.….9/7.… subgroups ==
=== Marveltri ===
{{See also| Chromatic pairs #Marveltri }}

Marveltri, the {{nowrap| 3 & 13 }} temperament in the 2.5.9/7 subgroup, is related to [[marvel]], [[magic]], and the unnamed {{nowrap| 22 & 47 }} temperament. The tonic and the first two generator steps make a [[marvel triad]], hence the name.

[[Subgroup]]: 2.5.9/7

[[Comma list]]: 225/224 ({{monzo| -5 2 1 }})

{{Mapping|legend=2| 1 0 5 | 0 1 -2 }}
: mapping generators: ~2, ~5

{{Mapping|legend=3| 1 2 0 -1 | 0 -4/5 1 2/5 }}
: [[gencom]]: [2 5; 225/224]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 384.208
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 383.638

{{Optimal ET sequence|legend=1| 3, 13, 16, 19, 22, 25, 72, 97, 122, 269c* }}
: <nowiki/>* wart for 9/7

[[Tp tuning #T2 tuning|RMS error]]: 0.4801 cents

==== Sulis ====
Sulis is related to [[minerva]] and [[würschmidt]].

[[Subgroup]]: 2.5.9/7.11/9

[[Comma list]]: 99/98 ({{monzo| -1 0 2 1 }}), 176/175 ({{monzo| 4 -2 1 1 }})

{{Mapping|legend=2| 1 0 5 -9 | 0 1 -2 4 }}]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1200.000, ~5/4 = 386.617
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1200.000, ~5/4 = 386.558

{{Optimal ET sequence|legend=1| 3, …, 22, 25, 28, 31, 59 }}

[[Tp tuning #T2 tuning|RMS error]]: 1.074 cents

== 2.….7/5.… subgroups ==
=== Hydrothermal ===
A tuning whose distinctively sharp (but still consonant) fifth, and flat (but still consonant) octave, lend it a mysterious, heavy atmosphere. The 6-tone (hexatonic) MOS is melodically interesting and flavorful. The 18-tone MOS is a useful 'chromatic' scale for taking subsets of.

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[50/49]]

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

[[Optimal tuning]] (inharmonic [[TE]]): ~1\2 = 590.998, ~[[10/7]]-1\2 = 128.962

[[Support]]ing [[ET]]s: {{EDOs|4, 6, 8, 10, 18, 28, 46, 64, 110}}

=== Argentic ===
Argentic is the 2.3.7/5 subgroup temperament tempering out [[5120/5103]].

[[Subgroup]]: 2.3.7/5

[[Comma list]]: [[5120/5103]] = {{monzo| 10 -6 -1 }}

{{Mapping|legend=2| 1 0 10 | 0 1 -6 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 702.792
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 702.830

{{Optimal ET sequence|legend=1| 12, 29, 41, 70, 321, 391, 461, 531, 601 }}
 based on subgroup TE 

Badness (Sintel): 0.119

==== Edson (2.3.7/5.11/5.13/5 subgroup) ====
{{See also| Chromatic pairs #Edson }}

Edson is related to [[pele]] and [[andromeda]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[196/195]] = {{monzo| 2 -1 2 0 -1 }}, [[352/351]] = {{monzo| 5 -3 0 1 -1 }}, [[364/363]] = {{monzo| 2 -1 1 -2 1 }}

{{Mapping|legend=2| 1 0 10 17 22 | 0 1 -6 -10 -13 }}
: mapping generators: ~2, ~3

{{Mapping|legend=3| 1 1 -5 -1 2 4 | 0 1 29/4 5/4 -11/4 -23/4 }}
: [[gencom]]: [2 3/2; 196/195, 352/351, 364/363]

[[Optimal tuning]]s:
* [[Tp tuning|subgroup]] [[CTE]]: ~2 = 1\1, ~3/2 = 703.4398
* [[Tp tuning|subgroup]] [[POTE]]: ~2 = 1\1, ~3/2 = 703.414

{{Optimal ET sequence|legend=1| 12, 17, 29 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5102 cents

==== Haumea ====
{{See also| Chromatic pairs #Haumea }}

Related temperaments include [[#Bridgetown|bridgetown]], [[namaka]], [[hemigari]], [[#Barbados|barbados]], and [[parizekmic]].

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]], [[847/845]]

{{Mapping|legend=2| 1 0 10 -6 -1 | 0 2 -12 9 3 }}

{{Mapping|legend=3| 1 2 -3/4 -11/4 9/4 5/4 | 0 -2 0 12 -9 -3 }}
: [[gencom]]: [2 15/13; 352/351 676/675 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.491

{{Optimal ET sequence|legend=1| 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2668 cents

=== Historical ===
{{distinguish|Historical temperaments}}
{{distinguish|History (temperament)}}, which is the rank-3 version of this temperament in the full 13-limit.

Historical is essentially an analogue of [[miracle]] that splits [[4/3]] in six rather than [[3/2]]. It tempers out the comma S10/S11 = [[4000/3993]] to set [[11/10]] equal to one-third of 4/3, and S13/S15 = [[676/675]] to equate [[15/13]] to one-half of 4/3, and tempers out S21 = [[441/440]] to split 11/10 into two instances of [[22/21]]~[[21/20]]. [[Sextilifourths]] adds the [[schismic]] mapping of prime 5 (reached by eight fourths) to complete the 13-limit.

[[Subgroup]]: 2.3.7/5.11/5.13/5

[[Comma list]]: 364/363, 441/440, 1001/1000

{{Mapping|legend=2| 1 2 0 1 2 | 0 -6 7 2 -9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~21/20 = 83.016

{{Optimal ET sequence|legend=1| 14, 29, 72, 101, 130, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2562 cents

=== Terrain ===
{{Redirect|Terrain|the scale|Terrain (scale)}}
{{See also| Chromatic pairs #Terrain }}

Terrain, the 6 & 21 temperament in the 2.7/5.9/5 subgroup, is related to [[domain (temperament)|domain]]. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.

[[Subgroup]]: 2.7/5.9/5

[[Comma list]]: [[250047/250000]]

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

{{Mapping|legend=3| 3 10/9 -7/9 2/9 | 0 -2/3 -1/3 2/3 }}
: [[gencom]]: [63/50 10/9; 250047/250000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~63/50 = 1\3, ~10/9 = 182.461

{{Optimal ET sequence|legend=1| 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.00844 cents

=== Tridec ===
{{See also| Chromatic pairs #Tridec }}
{{See also| Non-over-1 temperament #Tridec }}

Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends [[#Petrtri]].

[[Subgroup]]: 2.7/5.11/5.13/5

[[Comma list]]: [[847/845]], [[1001/1000]]

{{Mapping|legend=2| 1 2 0 1 | 0 -4 3 1 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 | 0 0 0 -4 3 1 }}
: [[gencom]]: [2 13/10; 847/845 1001/1000]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.556

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc }}

[[Tp tuning #T2 tuning|RMS error]]: 0.1613 cents

==== Naiadec ====
[[Subgroup]]: 2.7/5.11/5.13/5.17/5

[[Comma list]]: [[170/169]], [[221/220]], [[847/845]]

{{Mapping|legend=2| 1 2 0 1 1 | 0 -4 3 1 2 }}

{{Mapping|legend=3| 1 0 -3/4 5/4 -3/4 1/4 1/4 | 0 0 0 -4 3 1 2 }}
: [[gencom]]: [2 13/10; 170/169 221/220 847/845]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 454.882

{{Optimal ET sequence|legend=1| 5, 8, 21, 29, 95t, 124t }}
: t wart for 17/5

[[Tp tuning #T2 tuning|RMS error]]: 0.7521 cents

== 2.….11/5.… subgroups ==
=== Petrtri ===
{{See also| Chromatic pairs #Petrtri }}
{{See also| 5L 3s/Temperaments #Petrtri }}

Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.

[[Subgroup]]: 2.11/5.13/5

[[Comma list]]: [[2200/2197]]

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

{{Mapping|legend=3| 1 0 -1/3 0 -1/3 2/3 | 0 0 -4/3 0 5/3 -1/3 }}
: [[gencom]]: [2 13/10; 2200/2197]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~13/10 = 455.012

{{Optimal ET sequence|legend=1| 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.0749 cents

==== Bridgetown ====
{{See also| Chromatic pairs #Bridgetown }}

Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to [[#Haumea|haumea]] and [[#Barbados|barbados]].

[[Subgroup]]: 2.3.11/5.13/5

[[Comma list]]: [[352/351]], [[676/675]]

{{Mapping|legend=2| 1 0 -6 -1 | 0 2 9 3 }}

{{Mapping|legend=3| 1 2 -5/3 0 4/3 1/3 | 0 -2 4 0 -5 1 }}
: [[gencom]]: [2 15/13; 352/351 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~2 = 1\1, ~15/13 = 248.399

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2513 cents

=== Hypnosis ===
Related temperaments: [[Swetismic temperaments #Hypnos|hypnos]], [[Alphatricot family #Alphatricot|alphatricot]]

[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 169/168, 540/539, 729/728

{{Mapping|legend=2| 1 0 -3 8 0 | 0 3 11 -13 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~13/9 = 633.518

{{Optimal ET sequence|legend=1| 17, 36, 118f, 125f, 161f, 197f }}

[[Tp tuning #T2 tuning|RMS error]]: 0.5379 cents

=== Trisect ===
Trisect divides every Pythagorean interval into three, and is the much more accurate subgroup restriction of [[Augmented family #Trisected|trisected]].

Extending this temperament to the full [[11-limit|11-]], [[13-limit|13-]], or [[17-limit]] through [[portent]] or [[landscape]] results in the [[weak extension]] known as [[tritikleismic]].

[[Subgroup]]: 2.3.7.11/5

[[Comma list]]: 1029/1024, 4000/3993

{{Mapping|legend=2| 3 0 10 5 | 0 3 -1 -1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.742

{{Optimal ET sequence|legend=1| 15, 21, 36, 123, 159, 195, 231 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13

[[Comma list]]: 1029/1024, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 | 0 3 -1 -1 7 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~44/35 = 1\3, ~13/9 = 633.918

{{Optimal ET sequence|legend=1| 15, 21f, 36, 87, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

==== 2.3.7.11/5.13.17 subgroup ====
[[Subgroup]]: 2.3.7.11/5.13.17

[[Comma list]]: 273/272, 833/832, 1575/1573, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 | 0 3 -1 -1 7 9 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.820

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123, 159 }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== Trisector =====
[[Subgroup]]: 2.3.7.11/5.13.17.19

[[Comma list]]: 210/209, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 | 0 3 -1 -1 7 9 3 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 633.894

{{Optimal ET sequence|legend=1| 15, 21fg, 36, 123h, 159h }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 | 0 3 -1 -1 7 9 3 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~34/27 = 1\3, ~13/9 = 634.038

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

===== 2.3.7.11/5.13.17.19.23.29 subgroup =====
[[Subgroup]]: 2.3.7.11/5.13.17.19.23.29

[[Comma list]]: 210/209, 231/230, 273/272, 286/285, 320/319, 595/594, 2080/2079

{{Mapping|legend=2| 3 0 10 5 0 -2 8 12 13 | 0 3 -1 -1 7 9 3 1 1 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~29/23 = 1\3, ~13/9 = 634.102

{{Optimal ET sequence|legend=1| 15g, 21fg, 36, 87, 123hi }}

[[Tp tuning #T2 tuning|RMS error]]: ???

== 2.….11/7.… subgroups ==
=== Pepperoni ===
{{Main| Parapyth }}
{{See also| Chromatic pairs #Pepperoni }}

Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7 subgroup. It is the single-chain retraction of [[parapyth]]. The [[Peppermint-24|Pepper fifth]], which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.

[[Subgroup]]: 2.3.11/7.13/7

[[Comma list]]: 352/351, 364/363

{{Mapping|legend=2| 1 0 7 12 | 0 1 -4 -7 }}

{{Mapping|legend=3| 1 1 0 -8/3 1/3 7/3 | 0 1 0 11/3 -1/3 -10/3 }}
: [[gencom]]: [2 3/2; 352/351 364/363]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 703.856

{{Optimal ET sequence|legend=1| 5, 7, 12f, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595b*† }}
: <nowiki />* wart for 11/7
: † wart for 13/7

[[Tp tuning #T2 tuning|RMS error]]: 0.3789 cents

== 2.….13/5.… subgroups ==
=== Barbados ===
The [[minimax tuning]] for this makes the generator the cube root of 20/13, or 248.5953 cents. Edos which may be used for it are [[24edo]], [[29edo]], [[53edo]] and [[111edo]], with [[mos scale]]s of size 5, 9, 14, 19, 24 and 29 making for a good variety of scales.

[[Subgroup]]: 2.3.13/5

[[Comma list]]: 676/675 = {{monzo| 2 -3 2 }}

[[Sval]] [[mapping]]: [{{val| 1 0 -1 }}, {{val| 0 2 3 }}]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

; Music
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 ''Desert Island Rain''] in 313edo tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

==== Tobago ====
{{See also| Chromatic pairs #Tobago }}

Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends [[neutral]] and [[barbados]].

[[Subgroup]]: 2.3.11.13/5

[[Comma list]]: [[243/242]], [[676/675]]

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

{{Mapping|legend=3| 2 4 -2 0 9 2 | 0 -2 3/2 0 -5 -3/2 }}
: [[gencom]]: [55/39 15/13; 243/242 676/675]

[[Optimal tuning]] ([[Tp tuning|subgroup]] [[POTE]]): ~55/39 = 1\2, ~15/13 = 249.312

{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}

[[Tp tuning #T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====
[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

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

[[Optimal tuning]]s:
* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
: <nowiki />* wart for 13/5

=== Oceanfront ===
Related temperaments: [[Archytas clan #Superpyth|superpyth]], [[Archytas clan #Ultrapyth|ultrapyth]]

[[Subgroup]]: 2.3.7.13/5

[[Comma list]]: 64/63, 91/90

{{Mapping|legend=2| 1 0 6 -5 | 0 1 -2 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2 = 713.910

{{Optimal ET sequence|legend=1| 5, 22, 27, 32, 37 }}

[[Tp tuning #T2 tuning|RMS error]]: 2.063 cents

Scales: [[Oceanfront scales]]

== 2.….49/5.… subgroups ==
=== Direct breedsmic ===
Related temperament: [[hemithirds]], [[newt]]

[[Subgroup]]: 2.3.49/5

[[Comma list]]: 2401/2400

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

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~49/40 = 350.966

{{Optimal ET sequence|legend=1|7, 10, 17}}

[[Tp tuning #T2 tuning|RMS error]]: ?

== 2.….17/5.… subgroups ==
=== Fiventeen ===
Fiventeen tempers out [[136/135]] ({{monzo| 3 -3 1 }}) in 2.3.17/5. It equates [[17/15]] with [[9/8]], so it implies a [[supersoft]] [[pentic]] [[pentad]] of [[~]]30:34:40:45:51. [[17edo]] makes a good tuning especially for its size, which gives a [[supersoft]] pentic scale corresponding approximately to a just [[20/17]] tuning, although [[80edo]] might be preferred for an approximately just [[51/40]] to optimize plausibility slightly more, and [[97edo]] (= 80 + 17) and [[114edo]] (= 97 + 17) do even better in striking a balance between 80edo's more stable tuning and that having 20/17 more accurate (as in 17edo) is useful because of the more convincing suggestion of the two 15:17:20 chords present in the fiventeen pentad. The same is true of the related rank-3 temperament diatic, for which the [[optimal ET sequence]] is much more characteristic of optimized tunings, finding [[34edo]], then [[80edo]], then [[114edo]] (= 34 + 80) and even [[194edo|194bc-edo]] (= 80 + 114), though because of its focus on primes 5 and 17 it misses 97edo as a tuning, and slightly less optimized though still interesting [[63edo]] and [[143edo]] (= 63 + 80) tunings are found in the optimal ET sequence for fiventeen.

[[Subgroup]]: 2.3.17/5

[[Comma list]]: 136/135 ({{monzo| 3 -3 1 }})

{{Mapping|legend=2| 1 0 -3 | 0 1 3 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1199.2838{{c}}, ~3/2 = 704.4600{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.5286{{c}}

{{Optimal ET sequence|legend=1| 5, 12, 17, 46, 63, 143 }}

== 2.….19/7.… subgroups ==
=== Surprise ===
This temperament was named by [[User:VectorGraphics|Vector]] in 2025, as he was surprised that the temperament of [[57/56]] did not have a name. This is the [[rank-2 temperament|rank-2]] version of the temperament; Vector surmises that the name ''hendrix'' would be more thoughtfully given to the [[rank-3]] version.

[[Subgroup]]: 2.3.19/7

[[Comma list]]: [[57/56]] ({{monzo| -3 1 1 }})

{{Mapping|legend=2| 1 0 3 | 0 1 -1 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[Tp tuning|Subgroup]] [[WE]]: ~2 = 1202.4345{{c}}, ~3/2 = 697.4314{{c}}
* [[Tp tuning|Subgroup]] [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 697.3981{{c}}

{{Optimal ET sequence|legend=1| 5, 7, 12, 19, 31*, 50* }}

<nowiki/>* wart for 19/7

[[Badness]] (Sintel): 0.082

== 3/2.5/2.… subgroups ==
{{Main|Half-prime subgroup}}

=== Hemihemi ===
[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: [[10976/10935]]

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

[[Optimal tuning]] (subgroup [[CTE]]): ~[[3/2]] = 1\[[1edf]], ~[[28/27]] = 60.909

[[Support]]ing [[ET]]s: *23, *12, *11, *35, *34, *10, *13, *47, *9[+5/2], *14[-5/2], *45, *25, *21[+5/2], *8[+5/2]

=== Halftone ===
{{Main| Halftone }}

[[Subgroup]]: 3/2.5/2.7/2

[[Comma list]]: 9604/9375

{{Mapping|legend=2| 1 3 4 | 0 -4 -5 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 128.783

Supporting ETs: *5, *6, *7[+5/2, +7/2], *9[-5/2, --7/2], *11, *16, *17[+5/2], *23[+5/2, +7/2], *21[-7/2], *27, *28[+5/2], *38, *43[-7/2], *49
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2

[[Comma list]]: 1232/1215, 27783/27500

{{Mapping|legend=2| 1 3 4 4 | 0 -4 -5 1 }}
: sval mapping generators: ~3/2, ~15/14

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.186

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2], *21[-7/2], *38[-11/2], *43[-7/2, -11/2], *59[-7/2, -11/2], *70[-7/2, -11/2], *75[--7/2, -11/2]
: <nowiki />* wart for 3/2

==== 3/2.5/2.7/2.11/2.13/2 ====
[[Subgroup]]: 3/2.5/2.7/2.11/2.13/2

[[Comma list]]: 275/273, 1232/1215, 1323/1300

{{Mapping|legend=2| 1 3 4 4 5 | 0 -4 -5 1 -2 }}

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~15/14 = 129.381

[[Support]]ing [[ET]]s: *11, *5, *16, *6, *27[-11/2]
: <nowiki />* wart for 3/2

=== Semiwolf ===
[[Subgroup]]: 3/2.5/2.7/4

[[Comma list]]: 245/243

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

: sval mapping generators: ~3/2, ~9/7

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 262.1728

[[Optimal ET sequence]]: [[3edf]], [[5edf]], [[8edf]]

==== Semilupine ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 100/99, 245/243

{{Mapping|legend=2| 1 1 2 0 | 0 2 -1 4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 264.3771

[[Optimal ET sequence]]: [[8edf]], [[13edf]]

==== Hemilycan ====
[[Subgroup]]: 3/2.5/2.7/4.11/4

[[Comma list]]: 245/243, 441/440

{{Mapping|legend=2| 1 1 2 5 | 0 2 -1 -4 }}

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~7/6 = 261.5939

[[Optimal ET sequence]]: [[8edf]], [[11edf]]

== 3/2.5/4.… subgroups ==
=== Poseidon ===
'''This temperament will be subjected to renaming due to a conflict.'''

[[Subgroup]]: 3/2.5/4.11/8

[[Comma list]]: 121/120

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

: [[gencom]]: [3/2 12/11; 121/120]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~3/2, ~12/11 = 158.29

{{Optimal ET sequence|legend=1|9, 5, 13, 22, 14, 31, 17, 6[+5/4], 23, 40, 35, 21[-5/4], 19[+5/4], 49}}

== Other 3/2-equave subgroups ==
=== Auk ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 87808/85293

{{Mapping|legend=2| 1 0 -8 | 0 1 3 }}

: sval mapping generators: ~3/2, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~3/2 = 1\1edf, ~28/9 = 1950.859

Supporting ETs: *5, *6[+13], *7[-7, -13], *9, *11[+13], *13, *14, *17[-7, -13], *19[+13], *21[-7, -13], *22[-7], *23[+13], *25[-7, -13], *31[-7]
: <nowiki />* wart for 3/2

=== Doubleton ===
[[Subgroup]]: 3/2.7.13

[[Comma list]]: 1352/1323

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

: sval mapping generators: ~26/21, ~7

[[Optimal tuning]] (subgroup [[CTE]]): ~26/21 = 1\2edf, ~28/9 = 1971.772

Supporting ETs: *6, *10, *16, *14[-13], *8[+7], *22, *18[-13], *26, *24[-13], *28[+7], *20[+7], *36[-13], *12[+7, +13], *34[-13]
: <nowiki />* wart for 3/2

== 5/2-equave subgroups ==
=== Hyperion ===
[[Subgroup]]: 5/2.7.11

[[Comma list]]: {{monzo| 11 1 -5 }}

{{Mapping|legend=2| 1 4 3 | 0 -5 -1 }}

: [[gencom]]: [5/2 125/88; 341796875/329832448]

[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~5/2 = 1586.3137, ~125/88 = 593.6668

Supporting ETs: *5[-7], *8, *19[+7], *21[-7], *27[+7], *29[-7], *35[+7], *43[+7], *37[-7], *51[+7, +11], *45[-7], *59[+7, +11]
: <nowiki />* wart for 5/2

= Related temperament collections =
* [[Dual-fifth temperaments]]
* [[Equalizer subgroup]] temperaments
* [[Substitute harmonic]] temperaments

[[Category:Subgroup temperaments| ]] 
[[Category:Temperament collections]]
{{Todo| review | cleanup }}

Euslendric

2026-05-17T16:13:42Z

Lériendil: Changed redirect target from Gamelismic clan#Euslendric to Gamelismic clan#Euslendric (2.3.7.13)

#redirect [[Gamelismic clan #Euslendric (2.3.7.13)]]

[[Category:Rank-2 temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Gamelismic clan]]

Myna

2026-05-14T06:29:52Z

Lériendil: replaced quasitemp with superkleismic, since it's generated by 6/5 like myna and demonstrates the keemic equivalence better; also I believe it to be more known than quasitemp

{{Infobox regtemp
| Title = Myna
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[126/125]], [[1728/1715]] (7-limit); [[126/125]], [[176/175]], [[243/242]] (11-limit)
| Edo join 1 = 27e | Edo join 2 = 31
| Mapping = 1; 10 9 7 25
| Generators = 6/5 | Generators tuning = 310.1 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], …, [[4L 23s]], [[27L 4s]]
| Pergen = (P8, ccP5/10)
| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = 23
| Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = ? | Complexity 2 = 58
}}
'''Myna''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a flattened minor third of [[~]][[6/5]], so that seven generators reach [[7/4]], nine reach [[5/4]] and ten reach [[3/2]]. It can be thought of in terms of a series of equidistances between thirds, that is, making [[8/7]]–[[7/6]]–6/5–[[49/40]]–[[5/4]]–[[9/7]]–[[21/16]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]), or otherwise tuning the pental thirds outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds, 36/35. This is one of two major options for how the thirds are organized in [[edo]]s of medium size – the other one being [[keemic temperaments]], such as [[superkleismic]], where the gap between 6/5 and 5/4 is compressed to equal that between 7/6 and 6/5 instead of widened to equal twice it. In either case, by tempering the septimal dieses together, there is an exact neutral third in between 6/5 and 5/4. [[11-limit]] myna then arises from equating this neutral third to [[11/9]] and 13-limit myna adds the interpretation of [[16/13]] to it as well.

[[27edo|27e-edo]] and [[31edo]] represent natural endpoints of myna's tuning range, and 27 + 31 = [[58edo]] and 58 + 31 = [[89edo]] are very good tunings. In terms of [[commas]], the most characteristic comma that myna [[tempering out|tempers out]] is [[126/125]], the starling comma, so that two generators reach [[10/7]] and four reach the distinctive 36/35~50/49 chroma. Additionally, [[1728/1715]] ([[S-expression|S6/S7]]), the orwellisma, is tempered out to equate 36/35 with 49/48, and so is [[2401/2400]], the breedsma, to equate 49/48 and 50/49 (and find a neutral third at 49/40~60/49). In the 11-limit, [[176/175]], [[243/242]], [[441/440]], and [[540/539]] are tempered out; in the 13-limit, [[144/143]] and [[352/351]] are additionally tempered out.

Note: "myna" is pronounced /'maɪnə/, like {{w|myna|the bird}}, but is also as a pun on "minor".

See [[Starling temperaments #Myna]] 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
| 310.2
| 6/5
|-
| 2
| 620.4
| 10/7
|-
| 3
| 930.7
| 12/7
|-
| 4
| 40.9
| 36/35, 40/39, 45/44, 49/48, 50/49
|-
| 5
| 351.1
| 11/9, '''16/13'''
|-
| 6
| 661.3
| 22/15, 35/24
|-
| 7
| 971.6
| '''7/4'''
|-
| 8
| 81.8
| 21/20, 22/21, 25/24
|-
| 9
| 392.0
| '''5/4'''
|-
| 10
| 702.2
| '''3/2'''
|-
| 11
| 1012.4
| 9/5
|-
| 12
| 122.7
| 14/13, 15/14, 27/25
|-
| 13
| 432.7
| 9/7
|-
| 14
| 743.1
| 20/13
|-
| 15
| 1053.3
| 11/6, 24/13
|-
| 16
| 163.5
| 11/10
|-
| 17
| 473.8
| 21/16
|-
| 18
| 784.0
| 11/7
|-
| 19
| 1094.2
| 15/8
|-
| 20
| 204.4
| 9/8
|-
| 21
| 514.7
| 27/20
|-
| 22
| 824.9
| 21/13
|-
| 23
| 1135.1
| 27/14
|-
| 24
| 245.3
| 15/13
|-
| 25
| 555.5
| '''11/8''', 18/13
|-
| 26
| 865.6
| 33/20
|-
| 27
| 1176.0
| 55/28, 63/32, 77/39, 99/50
|}
<nowiki/>* In 13-limit CWE tuning

== Chords and harmony ==
{{See also| Chords of myna | Chords of tridecimal myna }}

== Scales ==
; Mos scales
* [[Myna7]]
* [[Myna11]]
* [[Myna15]]

; Transversal scales
* [[Myna19trans]]
* [[Myna19trans37]]
* [[Myna23trans]]
* [[Myna23trans37]]
* [[Myna27trans]]
* [[Myna27trans37]]

== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]
! Generator (¢)
! Comments
|-
|
| 7/5
| 308.744
|
|-
|
| 11/9
| 309.482
|
|-
|
| 5/4
| 309.590
|
|-
| 8\31
|
| 309.677
|
|-
|
| 7/4
| 309.832
|
|-
|
| 15/8
| 309.909
|
|-
|
| 15/14
| 309.953
|
|-
|
| 11/6
| 309.958
|
|-
|
| 11/8
| 310.053
|
|-
| 23\89
|
| 310.112
| 89f val
|-
|
| 11/7
| 310.138
|
|-
|
| 3/2
| 310.196
| 5-, 7-, 9- and 11-odd-imit minimax; 5-, 7-, 11- and 13-limit POTT
|-
|
| 11/10
| 310.313
|
|-
|
| 15/13
| 310.323
| 15-odd-limit minimax
|-
| 15\58
|
| 310.345
|
|-
|
| 13/11
| 310.360
| 13-odd-limit minimax
|-
|
| 9/7
| 310.391
|
|-
|
| 13/10
| 310.413
|
|-
|
| 15/11
| 310.508
|
|-
|
| 13/9
| 310.535
|
|-
| 22\85
|
| 310.588
| 85ce val
|-
|
| 9/5
| 310.691
|
|-
|
| 13/7
| 310.692
|
|-
|
| 13/12
| 310.762
|
|-
|
| 7/6
| 311.043
|
|-
| 7\27
|
| 311.111
| 27e val
|-
|
| 13/8
| 311.894
|
|-
|
| 5/3
| 315.641
|
|}

== Music ==
; [[Igliashon Jones]]
* [https://web.archive.org/web/20201129182056/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/89versionof23Myna.mp3 ''Myna Music'']

[[Category:Myna| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]
[[Category:Orwellismic temperaments]]
[[Category:Breedsmic temperaments]]

Ed5/3

2026-05-14T05:54:13Z

Lériendil:

The '''equal division of 5/3''' ('''ed5/3''') is a [[tuning]] obtained by dividing the [[5/3|just major sixth (5/3)]] into a number of [[equal]] steps.

== Properties ==
Division of 5/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed5/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.

5/3 is the most consonant interval in between 3/2 and 2/1, so this suggests it could be useful either as an equivalence, or as just an important structural feature.

[[Joseph Ruhf]] suggested the use of the 6:7:8:(10) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone as a way to evoke 5/3-equivalence. Though it could also be used just as a useful sonority, even without equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224). So, doing this yields 7-, 9-, and 16-note [[mos]] either way, the 16-note mos being 7L 9s. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it (even better than for [[edf]]s as the generator it uses is an excellent fit for heptatonic mos) though it is, technically speaking, micro-[[7L 2s|armotonic]].

If we instead opt to continue using 4:5:6 as the fundamental sonority, then it will take three 3/2 to get to 5/4, resulting in [[Blackcomb]] temperament that tempers out the comma 250/243. This yields mos scales of 4, 5, 6, 11, 16, and 21 notes. Although, it should be noted that doing this will often create a pseudo-octave unlike the 6:7:8 approach.

ED5/3 tuning systems that accurately represent the intervals 5/4 and 4/3 include: [[7ed5/3]] (7.30 cent error), [[9ed5/3]] (6.73 cent error), and [[16ed5/3]] (0.59 cent error).

[[7ed5/3]], [[9ed5/3]], and [[16ed5/3]] are to the [[Ed5/3|division of 5/3]] what [[5edo]], [[7edo]], and [[12edo]] are to the [[EDO|division of 2/1]].

== Individual pages for ed5/3's ==
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…49
| [[0ed5/3|0]]
| [[1ed5/3|1]]
| [[2ed5/3|2]]
| [[3ed5/3|3]]
| [[4ed5/3|4]]
| [[5ed5/3|5]]
| [[6ed5/3|6]]
| [[7ed5/3|7]]
| [[8ed5/3|8]]
| [[9ed5/3|9]]
|-
| [[10ed5/3|10]]
| [[11ed5/3|11]]
| [[12ed5/3|12]]
| [[13ed5/3|13]]
| [[14ed5/3|14]]
| [[15ed5/3|15]]
| [[16ed5/3|16]]
| [[17ed5/3|17]]
| [[18ed5/3|18]]
| [[19ed5/3|19]]
|-
| [[20ed5/3|20]]
| [[21ed5/3|21]]
| [[22ed5/3|22]]
| [[23ed5/3|23]]
| [[24ed5/3|24]]
| [[25ed5/3|25]]
| [[26ed5/3|26]]
| [[27ed5/3|27]]
| [[28ed5/3|28]]
| [[29ed5/3|29]]
|-
| [[30ed5/3|30]]
| [[31ed5/3|31]]
| [[32ed5/3|32]]
| [[33ed5/3|33]]
| [[34ed5/3|34]]
| [[35ed5/3|35]]
| [[36ed5/3|36]]
| [[37ed5/3|37]]
| [[38ed5/3|38]]
| [[39ed5/3|39]]
|-
| [[40ed5/3|40]]
| [[41ed5/3|41]]
| [[42ed5/3|42]]
| [[43ed5/3|43]]
| [[44ed5/3|44]]
| [[45ed5/3|45]]
| [[46ed5/3|46]]
| [[47ed5/3|47]]
| [[48ed5/3|48]]
| [[49ed5/3|49]]
|}

[[Category:Ed5/3's| ]]

[[Category:Edonoi]]
[[Category:Lists of scales]]

{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 5/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.}}

13ed5/3

2026-05-14T04:19:52Z

Lériendil:

{{Stub}}
{{delete|Stub with no additional content beyond tables}}
{{Infobox ET}}
{{ED intro}}

== Intervals ==
{{Interval table}}

== Harmonics ==
{{Harmonics in equal
| steps = 13
| num = 5
| denom = 3
}}
{{Harmonics in equal
| steps = 13
| num = 5
| denom = 3
| start = 12
| collapsed = 1
}}

User talk:Francium/647edo

2026-05-14T04:18:21Z

Lériendil:

== Deletion ==

I vote '''keep''', it has at least one song written in it!
– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 00:18, 13 May 2026 (UTC)

Or maybe move to Francium's userspace. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 00:24, 13 May 2026 (UTC)

: I don't believe that simply having music written in it constitutes a reason to avoid deletion, particularly in cases such as this where it's extremely unclear how exactly the song uses the structures associated with the edo/tuning. 647 might as well just be an arbitrary number. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 04:18, 14 May 2026 (UTC)

Talk:Xenial

2026-05-14T04:04:50Z

Lériendil: Created page with "Hey, Xenllium: just to note a few things, this is another full-on article for a regular temperament that... has barely any explanation of what the temperament actually structurally does and mostly consists of giant tables. The infobox squeezes in 4 different subgroups, half of which aren't even documented in the actual temperament documentation, and with zero explanation for why those specific subgroups are chosen. More or less, it's just another glorified stub that's ma..."

Hey, Xenllium: just to note a few things, this is another full-on article for a regular temperament that... has barely any explanation of what the temperament actually structurally does and mostly consists of giant tables. The infobox squeezes in 4 different subgroups, half of which aren't even documented in the actual temperament documentation, and with zero explanation for why those specific subgroups are chosen. More or less, it's just another glorified stub that's made to look prettier because of the inclusion of the infobox I made. In my opinion and, I'm sure, the opinion of many others, this is the exact kind of thing XenWiki doesn't need. --[[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 04:04, 14 May 2026 (UTC)

Navicular comma

2026-05-14T01:35:30Z

Lériendil:

{{Infobox interval
|Ratio = 24192/24167
|Name = navicular comma
|Color name = 3u31uz1, trithu-aluzo unison
|Comma = yes
}}

'''24192/24167''', the '''navicular comma''' is a no-fives [[13-limit]] comma that is the difference between three [[13/12]]s and one [[14/11]]. It is tempered out in some EDOs such as [[17edo|17]], [[26edo|26]], [[77edo|77]], [[94edo|94]], [[104edo|104]], [[130edo|130]], [[207edo|207]] and [[224edo|224]].

== Etymology ==
The name ''navicular comma'' was given by [[User:Xenllium|Xenllium]] in 2025, for its [[ultraparticular]] property and related temperaments (which?).

[[Category:Commas named for their regular temperament properties]]

Myna

2026-05-14T00:50:01Z

Lériendil:

{{Infobox regtemp
| Title = Myna
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[126/125]], [[1728/1715]] (7-limit); [[126/125]], [[176/175]], [[243/242]] (11-limit)
| Edo join 1 = 27e | Edo join 2 = 31
| Mapping = 1; 10 9 7 25
| Generators = 6/5 | Generators tuning = 310.1 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[4L 3s]], [[4L 7s]], …, [[4L 23s]], [[27L 4s]]
| Pergen = (P8, ccP5/10)
| Odd limit 1 = 7 | Mistuning 1 = ? | Complexity 1 = 23
| Odd limit 2 = (2.3.5.7.11) 21 | Mistuning 2 = ? | Complexity 2 = 58
}}
'''Myna''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a flattened minor third of [[~]][[6/5]], so that seven generators reach [[7/4]], nine reach [[5/4]] and ten reach [[3/2]]. It can be thought of in terms of a series of equidistances between thirds, that is, making [[8/7]]–[[7/6]]–6/5–[[49/40]]–[[5/4]]–[[9/7]]–[[21/16]] all equidistant (the distances between which are [[36/35]], [[49/48]], and [[50/49]]), or otherwise tuning the pental thirds outwards so that the chroma between them ([[25/24]]) is twice the size of the interval between the pental and septimal thirds, 36/35. This is one of two major options for how the thirds are organized in [[edo]]s of medium size – the other one being [[keemic temperaments]], such as [[quasitemp]], where the gap between 6/5 and 5/4 is compressed to equal that between 7/6 and 6/5 instead of widened to equal twice it. In either case, by tempering the septimal dieses together, there is an exact neutral third in between 6/5 and 5/4. [[11-limit]] myna then arises from equating this neutral third to [[11/9]] and 13-limit myna adds the interpretation of [[16/13]] to it as well.

[[27edo|27e-edo]] and [[31edo]] represent natural endpoints of myna's tuning range, and 27 + 31 = [[58edo]] and 58 + 31 = [[89edo]] are very good tunings. In terms of [[commas]], the most characteristic comma that myna [[tempering out|tempers out]] is [[126/125]], the starling comma, so that two generators reach [[10/7]] and four reach the distinctive 36/35~50/49 chroma. Additionally, [[1728/1715]] ([[S-expression|S6/S7]]), the orwellisma, is tempered out to equate 36/35 with 49/48, and so is [[2401/2400]], the breedsma, to equate 49/48 and 50/49 (and find a neutral third at 49/40~60/49). In the 11-limit, [[176/175]], [[243/242]], [[441/440]], and [[540/539]] are tempered out; in the 13-limit, [[144/143]] and [[352/351]] are additionally tempered out.

Note: "myna" is pronounced /'maɪnə/, like {{w|myna|the bird}}, but is also as a pun on "minor".

See [[Starling temperaments #Myna]] 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
| 310.2
| 6/5
|-
| 2
| 620.4
| 10/7
|-
| 3
| 930.7
| 12/7
|-
| 4
| 40.9
| 36/35, 40/39, 45/44, 49/48, 50/49
|-
| 5
| 351.1
| 11/9, '''16/13'''
|-
| 6
| 661.3
| 22/15, 35/24
|-
| 7
| 971.6
| '''7/4'''
|-
| 8
| 81.8
| 21/20, 22/21, 25/24
|-
| 9
| 392.0
| '''5/4'''
|-
| 10
| 702.2
| '''3/2'''
|-
| 11
| 1012.4
| 9/5
|-
| 12
| 122.7
| 14/13, 15/14, 27/25
|-
| 13
| 432.7
| 9/7
|-
| 14
| 743.1
| 20/13
|-
| 15
| 1053.3
| 11/6, 24/13
|-
| 16
| 163.5
| 11/10
|-
| 17
| 473.8
| 21/16
|-
| 18
| 784.0
| 11/7
|-
| 19
| 1094.2
| 15/8
|-
| 20
| 204.4
| 9/8
|-
| 21
| 514.7
| 27/20
|-
| 22
| 824.9
| 21/13
|-
| 23
| 1135.1
| 27/14
|-
| 24
| 245.3
| 15/13
|-
| 25
| 555.5
| '''11/8''', 18/13
|-
| 26
| 865.6
| 33/20
|-
| 27
| 1176.0
| 55/28, 63/32, 77/39, 99/50
|}
<nowiki/>* In 13-limit CWE tuning

== Chords and harmony ==
{{See also| Chords of myna | Chords of tridecimal myna }}

== Scales ==
; Mos scales
* [[Myna7]]
* [[Myna11]]
* [[Myna15]]

; Transversal scales
* [[Myna19trans]]
* [[Myna19trans37]]
* [[Myna23trans]]
* [[Myna23trans37]]
* [[Myna27trans]]
* [[Myna27trans37]]

== Tunings ==
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]
! Generator (¢)
! Comments
|-
|
| 7/5
| 308.744
|
|-
|
| 11/9
| 309.482
|
|-
|
| 5/4
| 309.590
|
|-
| 8\31
|
| 309.677
|
|-
|
| 7/4
| 309.832
|
|-
|
| 15/8
| 309.909
|
|-
|
| 15/14
| 309.953
|
|-
|
| 11/6
| 309.958
|
|-
|
| 11/8
| 310.053
|
|-
| 23\89
|
| 310.112
| 89f val
|-
|
| 11/7
| 310.138
|
|-
|
| 3/2
| 310.196
| 5-, 7-, 9- and 11-odd-imit minimax; 5-, 7-, 11- and 13-limit POTT
|-
|
| 11/10
| 310.313
|
|-
|
| 15/13
| 310.323
| 15-odd-limit minimax
|-
| 15\58
|
| 310.345
|
|-
|
| 13/11
| 310.360
| 13-odd-limit minimax
|-
|
| 9/7
| 310.391
|
|-
|
| 13/10
| 310.413
|
|-
|
| 15/11
| 310.508
|
|-
|
| 13/9
| 310.535
|
|-
| 22\85
|
| 310.588
| 85ce val
|-
|
| 9/5
| 310.691
|
|-
|
| 13/7
| 310.692
|
|-
|
| 13/12
| 310.762
|
|-
|
| 7/6
| 311.043
|
|-
| 7\27
|
| 311.111
| 27e val
|-
|
| 13/8
| 311.894
|
|-
|
| 5/3
| 315.641
|
|}

== Music ==
; [[Igliashon Jones]]
* [https://web.archive.org/web/20201129182056/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Igs/89versionof23Myna.mp3 ''Myna Music'']

[[Category:Myna| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]
[[Category:Orwellismic temperaments]]
[[Category:Breedsmic temperaments]]

85edo

2026-05-14T00:46:58Z

Lériendil: /* Theory */

{{Infobox ET}}
{{ED intro}}

== Theory ==
Since 85 {{=}} 5 × 17, 85edo shares the same 3.9-cent-sharp fifth as [[17edo|17]], [[34edo|34]], and [[68edo|68]]. 3/1 is therefore divisible into 5, and the [[patent val]] correspondingly supports [[magic]], [[tempering out]] [[3125/3072]] in the [[5-limit]] and [[225/224]], [[245/243]], and [[875/864]] in the [[7-limit]]. It tempers out [[100/99]] and [[385/384]] in the [[11-limit]], supporting 11-limit magic, and [[847/845]], [[1188/1183]], and [[1575/1573]] in the 13-limit. It provides the [[optimal patent val]] for the 13-limit 36ce & 49f temperament tempering out 100/99, 540/539, 847/845 and 1575/1573. The 85c val, with a very sharp 5/4 of 395.3{{c}}, supports 7-limit [[myna]].

=== Odd harmonics ===
{{Harmonics in equal|85}}

=== Subsets and supersets ===
85edo contains [[5edo]] and [[17edo]] as subsets. [[255edo]], which triples it, is a notable tuning.

== Interval table ==
{{Interval table}}

== Scales ==
* Amulet{{idiosyncratic}}, (approximated from [[25edo]], subset of [[magic]]): 7 3 7 7 3 7 10 7 7 3 7 10 7

[[Category:Magic]]

== Instruments ==

A [[Lumatone mapping for 85edo]] is available.

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/DbVsxzfKr7I ''microtonal improvisation in 85edo''] (2025)

Valinorsmic clan

2026-05-13T21:55:27Z

Lériendil: /* Valinor */

{{Technical data page}}
The '''valinorsmic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the valinorsma, [[176/175]], which equates [[48/35]] with [[15/11]] (rather than [[11/8]], which is what the keenanisma [[385/384]] does).

For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].

== Valinor ==
''Valinor'' is the 2.5.7.11 temperament defined by tempering out 176/175. When extending to prime 13, two strong extensions are considerable: ''valimar'' and ''valarin''; these intersect in [[Hemimean clan #tridecimal didacus|tridecimal didacus]].

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 176/175

{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
: [[error map]]: {{val| -0.689 +1.849 +1.350 -2.061 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}
: error map: {{val| 0.000 +3.080 +2.694 -1.011 }}

{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}

[[Badness]] (Sintel): 0.0793

=== Valimar ===
This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4s, where two form 11/7, so that four form 16/13. This logic is supported by rank-2s such as tridecimal didacus and the 13-limit [[magus]]/[[amigo]] extensions, as well as [[Orwell extensions|winston]] and [[Valentine extensions|valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 = S8/S10 is (nontrivially) equivalent to (S11*S12)/(S14*S15), and this extension tempers out S11/S14 = [[1573/1568]], and S12/S15 = [[3584/3575]]. It remains well-tuned in ETs with optimal valinorsmic 5/4s (such as [[37edo|37]], [[40edo|40]], and [[43edo|43]]).

[[Subgroup]]: 2.5.7.11.13

[[Comma list]]: 176/175, 1573/1568

{{Mapping|legend=2| 1 0 0 -4 13 | 0 1 0 2 -4 | 0 0 1 1 0 }}
: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.433{{c}}, ~5/4 = 389.226{{c}}, ~7/4 = 971.559{{c}}
: [[error map]]: {{val| -0.567 +1.779 +1.600 -2.440 -0.267 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.466{{c}}, ~7/4 = 971.517{{c}}
: error map: {{val| 0.000 +3.152 +2.691 -0.869 +1.607 }}

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

[[Badness]] (Sintel): 0.336

=== Valarin ===
This extension equates two, ideally slightly flattened, [[8/7]]s to [[13/10]], which is also natural given valinorsmic's tuning tendency towards sharpening [[7/4]]. This extension is supported by rank-2 temperaments including tridecimal didacus, [[Orwell extensions|tridecimal orwell]], [[Valentine extensions|lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].

[[Subgroup]]: 2.5.7.11.13

[[Comma list]]: 176/175, 640/637

{{Mapping|legend=2| 1 0 0 -4 7 | 0 1 0 2 1 | 0 0 1 1 -2 }}
: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.167{{c}}, ~5/4 = 389.474{{c}}, ~7/4 = 972.185{{c}}
: [[error map]]: {{val| -0.833 +1.495 +1.694 -1.851 +0.413 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.189{{c}}, ~7/4 = 972.829{{c}}
: error map: {{val| 0.000 +2.875 +4.003 -0.110 +3.003 }}

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

[[Badness]] (Sintel): 0.227

=== Overview to extensions ===
The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Varda, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have natural extensions to the [[13-limit]] via tempering out both [[351/350]] and [[352/351]].

Discussed elsewhere are:
* [[Zeus]] (+121/120) → [[Biyatismic clan #Zeus|Biyatismic clan]]
* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]
* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[Clio]]'' (+81/80) → [[Didymus rank-3 family #Clio|Didymus rank-3 family]]
* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[Mandos]]'' (+243/242) → [[Rastmic rank-3 clan #Mandos|Rastmic rank-3 clan]]
* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Ulmo]]'' (+2200/2187) → [[Ragismic family #Ulmo|Ragismic family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]

Considered below are manwe and augenic.

== Manwe ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 176/175, 1331/1323

{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
: mapping generators: ~2, ~3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}}
: [[error map]]: {{val| -0.640 +0.100 +1.718 +1.467 -2.401 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5916{{c}}, ~5/4 = 389.3756{{c}}
: error map: {{val| 0.000 +0.637 +3.062 +2.696 -1.045 }}

{{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}

[[Badness]] (Sintel): 0.816

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

Comma list: 176/175, 351/350, 1331/1323

Mapping: {{mapping| 1 0 0 12 8 13 | 0 1 0 3 3 0 | 0 0 1 -6 -4 -4 }}

Optimal tunings:
* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}

{{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }}

Badness (Sintel): 1.07

== Augenic ==
{{Distinguish| Augene }}

Named by [[Xenllium]] in 2026, augenic is closely related to [[augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 56/55, 128/125

{{Mapping|legend=1| 3 0 7 0 2 | 0 1 0 0 0 | 0 0 0 1 1 }}
: mapping generators: ~5/4, ~3, ~7

[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}}
: [[error map]]: {{val| -3.228 -0.039 +6.153 -6.172 +9.184 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.3488{{c}}, ~7/4 = 968.4397{{c}}
: error map: {{val| 0.000 +3.394 +13.686 -0.386 +17.122 }}

{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} *

<nowiki/>* [[optimal patent val]]: [[36edo|36]]

[[Badness]] (Sintel): 0.735

[[Category:Temperament clans]]
[[Category:Valinorsmic clan| ]] 
[[Category:Rank 3]]

Valinorsmic clan

2026-05-13T21:54:22Z

Lériendil: added valimar and valarin following discussion of extensions on xengrove

{{Technical data page}}
The '''valinorsmic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the valinorsma, [[176/175]], which equates [[48/35]] with [[15/11]] (rather than [[11/8]], which is what the keenanisma [[385/384]] does).

For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].

== Valinor ==
''Valinor'' is the 2.5.7.11 temperament defined by tempering out 176/175. When extending to prime 13, two strong extensions are considerable: ''valimar'' and ''valarin''; these intersect in [[tridecimal didacus]].

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 176/175

{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
: [[error map]]: {{val| -0.689 +1.849 +1.350 -2.061 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}
: error map: {{val| 0.000 +3.080 +2.694 -1.011 }}

{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}

[[Badness]] (Sintel): 0.0793

=== Valimar ===
This extension to the no-threes 13-limit is motivated by extending the valinorsmic chain of 5/4s, where two form 11/7, so that four form 16/13. This logic is supported by rank-2s such as tridecimal didacus and the 13-limit [[magus]]/[[amigo]] extensions, as well as [[winston]] and [[valentino]]. This is also natural considering [[S-expression]]s, noting that 176/175 = S8/S10 is (nontrivially) equivalent to (S11*S12)/(S14*S15), and this extension tempers out S11/S14 = [[1573/1568]], and S12/S15 = [[3584/3575]]. It remains well-tuned in ETs with optimal valinorsmic 5/4s (such as [[37edo|37]], [[40edo|40]], and [[43edo|43]]).

[[Subgroup]]: 2.5.7.11.13

[[Comma list]]: 176/175, 1573/1568

{{Mapping|legend=2| 1 0 0 -4 13 | 0 1 0 2 -4 | 0 0 1 1 0 }}
: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.433{{c}}, ~5/4 = 389.226{{c}}, ~7/4 = 971.559{{c}}
: [[error map]]: {{val| -0.567 +1.779 +1.600 -2.440 -0.267 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.466{{c}}, ~7/4 = 971.517{{c}}
: error map: {{val| 0.000 +3.152 +2.691 -0.869 +1.607 }}

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

[[Badness]] (Sintel): 0.336

=== Valarin ===
This extension equates two, ideally slightly flattened, [[8/7]]s to [[13/10]], which is also natural given valinorsmic's tuning tendency towards sharpening [[7/4]]. This extension is supported by rank-2 temperaments including tridecimal didacus, [[tridecimal orwell]], [[lupercalia]] (an extension of [[valentine]]), and [[llywelyn]].

[[Subgroup]]: 2.5.7.11.13

[[Comma list]]: 176/175, 640/637

{{Mapping|legend=2| 1 0 0 -4 7 | 0 1 0 2 1 | 0 0 1 1 -2 }}
: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.167{{c}}, ~5/4 = 389.474{{c}}, ~7/4 = 972.185{{c}}
: [[error map]]: {{val| -0.833 +1.495 +1.694 -1.851 +0.413 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~5/4 = 389.189{{c}}, ~7/4 = 972.829{{c}}
: error map: {{val| 0.000 +2.875 +4.003 -0.110 +3.003 }}

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

[[Badness]] (Sintel): 0.227

=== Overview to extensions ===
The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Varda, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have natural extensions to the [[13-limit]] via tempering out both [[351/350]] and [[352/351]].

Discussed elsewhere are:
* [[Zeus]] (+121/120) → [[Biyatismic clan #Zeus|Biyatismic clan]]
* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]
* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[Clio]]'' (+81/80) → [[Didymus rank-3 family #Clio|Didymus rank-3 family]]
* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[Mandos]]'' (+243/242) → [[Rastmic rank-3 clan #Mandos|Rastmic rank-3 clan]]
* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Ulmo]]'' (+2200/2187) → [[Ragismic family #Ulmo|Ragismic family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]

Considered below are manwe and augenic.

== Manwe ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 176/175, 1331/1323

{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
: mapping generators: ~2, ~3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}}
: [[error map]]: {{val| -0.640 +0.100 +1.718 +1.467 -2.401 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5916{{c}}, ~5/4 = 389.3756{{c}}
: error map: {{val| 0.000 +0.637 +3.062 +2.696 -1.045 }}

{{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}

[[Badness]] (Sintel): 0.816

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

Comma list: 176/175, 351/350, 1331/1323

Mapping: {{mapping| 1 0 0 12 8 13 | 0 1 0 3 3 0 | 0 0 1 -6 -4 -4 }}

Optimal tunings:
* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}

{{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }}

Badness (Sintel): 1.07

== Augenic ==
{{Distinguish| Augene }}

Named by [[Xenllium]] in 2026, augenic is closely related to [[augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 56/55, 128/125

{{Mapping|legend=1| 3 0 7 0 2 | 0 1 0 0 0 | 0 0 0 1 1 }}
: mapping generators: ~5/4, ~3, ~7

[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}}
: [[error map]]: {{val| -3.228 -0.039 +6.153 -6.172 +9.184 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.3488{{c}}, ~7/4 = 968.4397{{c}}
: error map: {{val| 0.000 +3.394 +13.686 -0.386 +17.122 }}

{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} *

<nowiki/>* [[optimal patent val]]: [[36edo|36]]

[[Badness]] (Sintel): 0.735

[[Category:Temperament clans]]
[[Category:Valinorsmic clan| ]] 
[[Category:Rank 3]]

Valinorsmic clan

2026-05-13T21:42:53Z

Lériendil: added description

{{Technical data page}}
The '''valinorsmic clan''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the valinorsma, [[176/175]], which equates a stack of two [[5/4]]s with [[11/7]]; ideally, 5/4 is slightly sharpened to make this equivalence. The valinorsma can also be seen as equating [[35/32]] with [[11/10]] (rather than [[12/11]], which is what the keenanisma [[385/384]] does).

For the rank-4 valinorsmic temperament, see [[Rank-4 temperament #Valinorsmic (176/175)]].

== Valinor ==
''Valinor'' is the 2.5.7.11 temperament defined by tempering out 176/175.

[[Subgroup]]: 2.5.7.11

[[Comma list]]: 176/175

{{Mapping|legend=2| 1 0 0 -4 | 0 1 0 2 | 0 0 1 1 }}
: mapping generators: ~2, ~5, ~7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3113{{c}}, ~5/4 = 389.5404{{c}}, ~7/4 = 971.5534{{c}}
: [[error map]]: {{val| -0.689 +1.849 +1.350 -2.061 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 389.3935{{c}}, ~7/4 = 971.5196{{c}}
: error map: {{val| 0.000 +3.080 +2.694 -1.011 }}

{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}

[[Badness]] (Sintel): 0.0793

=== Overview to extensions ===
The second comma in the comma list determines how we extend the no-3 subgroup temperament to include the harmonic 3. Zeus adds [[121/120]] as well as 385/384, and shares the same lattice structure as no-3 valinorsmic. Varda, adds [[896/891]], slicing the first generator in two with a semi-octave period. Nickel adds [[36/35]] as well as [[45/44]]. Ares adds [[64/63]]. Minerva adds [[99/98]]. Thrush adds [[126/125]]. These slice the last generator in two. Guanyin adds [[540/539]], slicing the first generator in three. Manwe adds [[1331/1323]], slicing the last generator in three. Clio adds [[81/80]], slicing the first generator in four. Lono adds [[5120/5103]], slicing the last generator in six. Mandos adds [[243/242]]. Shrusus adds [[245/243]]. These slice the last generator in five. Ulmo adds [[2200/2187]], slicing the last generator in seven. Finally, draco adds [[19683/19600]], slicing the last generator in nine. Most of these have natural extensions to the [[13-limit]] via tempering out both [[351/350]] and [[352/351]].

Discussed elsewhere are:
* [[Zeus]] (+121/120) → [[Biyatismic clan #Zeus|Biyatismic clan]]
* ''[[Varda]]'' (+896/891) → [[Diaschismic rank-3 family #Varda|Diaschismic rank-3 family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]
* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
* [[Thrush]] (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[Clio]]'' (+81/80) → [[Didymus rank-3 family #Clio|Didymus rank-3 family]]
* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[Mandos]]'' (+243/242) → [[Rastmic rank-3 clan #Mandos|Rastmic rank-3 clan]]
* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Ulmo]]'' (+2200/2187) → [[Ragismic family #Ulmo|Ragismic family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]

Considered below are manwe and augenic.

== Manwe ==
[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 176/175, 1331/1323

{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}
: mapping generators: ~2, ~3, ~5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3600{{c}}, ~3/2 = 702.6947{{c}}, ~5/4 = 389.3118{{c}}
: [[error map]]: {{val| -0.640 +0.100 +1.718 +1.467 -2.401 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.5916{{c}}, ~5/4 = 389.3756{{c}}
: error map: {{val| 0.000 +0.637 +3.062 +2.696 -1.045 }}

{{Optimal ET sequence|legend=1| 15, 28de, 31, 46, 65d, 77, 80, 111, 237cd, 268cd }}

[[Badness]] (Sintel): 0.816

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

Comma list: 176/175, 351/350, 1331/1323

Mapping: {{mapping| 1 0 0 12 8 13 | 0 1 0 3 3 0 | 0 0 1 -6 -4 -4 }}

Optimal tunings:
* WE: ~2 = 1199.4350{{c}}, ~3/2 = 702.5061{{c}}, ~5/4 = 389.2327{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7030{{c}}, ~5/4 = 389.4303{{c}}

{{Optimal ET sequence|legend=0| 31, 46, 65d, 77, 80, 111, 268cd }}

Badness (Sintel): 1.07

== Augenic ==
{{Distinguish| Augene }}

Named by [[Xenllium]] in 2026, augenic is closely related to [[augene]]. It tempers out the [[augmented comma]] but with a free generator for [[7/1|7]], and then extends it to the 11-limit through the identity [[128/125]] = ([[56/55]])⋅(176/175).

[[Subgroup]]: 2.3.5.7.11

[[Comma list]]: 56/55, 128/125

{{Mapping|legend=1| 3 0 7 0 2 | 0 1 0 0 0 | 0 0 0 1 1 }}
: mapping generators: ~5/4, ~3, ~7

[[Optimal tuning]]s:
* [[WE]]: ~5/4 = 398.9239{{c}}, ~3/2 = 705.1447{{c}}, ~7/4 = 969.1106{{c}}
: [[error map]]: {{val| -3.228 -0.039 +6.153 -6.172 +9.184 }}
* [[CWE]]: ~5/4 = 400.0000{{c}}, ~3/2 = 705.3488{{c}}, ~7/4 = 968.4397{{c}}
: error map: {{val| 0.000 +3.394 +13.686 -0.386 +17.122 }}

{{Optimal ET sequence|legend=1| 6, 9, 12, 15, 24, 27e, 51ce, 63cee }} *

<nowiki/>* [[optimal patent val]]: [[36edo|36]]

[[Badness]] (Sintel): 0.735

[[Category:Temperament clans]]
[[Category:Valinorsmic clan| ]] 
[[Category:Rank 3]]

15ed7/3

2026-05-13T21:30:17Z

Lériendil:

{{Stub}}
{{Infobox ET}}
{{ED intro}} The chord [0 5 9]\15ed7/3, closely resembling the 5-limit major triad of [[15edo]], is a close approximation in this tuning system to [[3:4:5]]. A potentially tempered 15ed7/3 chain with octaves as the equivalence results in [[passion]] temperament, with 3:4:5:7 located on the same position that it is in 15ed7/3.

== Intervals ==
{| class="wikitable"
!Degrees
!ed7/3
|-
|1
|97.7914
|-
|2
|195.5828
|-
|3
|293.3742
|-
|4
|391.1656
|-
|5
|488.957
|-
|6
|586.7484
|-
|7
|684.5398
|-
|8
|782.33115
|-
|9
|880.1225
|-
|10
|977.9139
|-
|11
|1075.7053
|-
|12
|1173.4967
|-
|13
|1271.2881
|-
|14
|1369.0795
|-
|15
|1466.8709
|}

== Harmonics ==
{{Harmonics in equal
| steps = 15
| num = 7
| denom = 3
}}
{{Harmonics in equal
| steps = 15
| num = 7
| denom = 3
| start = 12
| collapsed = 1
}}

Ed7/3

2026-05-13T21:27:22Z

Lériendil: removed redlink

Ed7/3

2026-05-13T21:26:48Z

Lériendil: 7/2 shouldn't occur in a 7/3.5/3.4/3 context so this is just wrong RTT

Ed7/3

2026-05-13T21:21:25Z

Lériendil: ed7/3s in the 100s are all deleted now

Ed7/3

2026-05-13T21:21:03Z

Lériendil: removed middletown stuff, archived it on mmtm's userpage

{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 7/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.}}

The '''equal division of 7/3''' ('''ed7/3''') is a [[tuning]] obtained by dividing the [[7/3|septimal minor tenth (7/3)]] in a certain number of [[equal]] steps.

== Applications ==
Division of 7/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed7/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.

The structural utility of 7/3 (or another tenth) is apparent by being the absolute widest range most generally used in popular songs{{citation needed}} (and even the range of a {{w|Dastg%C4%81h-e_M%C4%81hur|dastgah}}{{citation needed}}).

== Chords and harmonies ==
{{main|Pseudo-traditional harmonic functions of enneatonic scale degrees}}
[[:Category:9-tone scales|Enneatonic scale]]s, especially those equivalent at 7/3, can sensibly take [[tetrad]]s as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structurally important as it is. Many, though not all, of these scales have a perceptually important [[Pseudo-octave|pseudo (false) octave]], with various degrees of accuracy.

Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in [[meantone]]. Whereas in meantone it takes four [[3/2]] to get to [[5/1]], here it takes two [[28/15]] to get to [[7/2]] (tempering out the comma [[225/224]]). So, doing this yields 15-, 19-, and 34-note [[mos]] 2/1 apart. While the notes are rather farther apart, the scheme is uncannily similar to meantone. [[Joseph Ruhf]] named this scheme "macrobichromatic".

== Individual pages for ed7/3's ==
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…99
| [[0ed7/3|0]]
| [[1ed7/3|1]]
| [[2ed7/3|2]]
| [[3ed7/3|3]]
| [[4ed7/3|4]]
| [[5ed7/3|5]]
| [[6ed7/3|6]]
| [[7ed7/3|7]]
| [[8ed7/3|8]]
| [[9ed7/3|9]]
|-
| [[10ed7/3|10]]
| [[11ed7/3|11]]
| [[12ed7/3|12]]
| [[13ed7/3|13]]
| [[14ed7/3|14]]
| [[15ed7/3|15]]
| [[16ed7/3|16]]
| [[17ed7/3|17]]
| [[18ed7/3|18]]
| [[19ed7/3|19]]
|-
| [[20ed7/3|20]]
| [[21ed7/3|21]]
| [[22ed7/3|22]]
| [[23ed7/3|23]]
| [[24ed7/3|24]]
| [[25ed7/3|25]]
| [[26ed7/3|26]]
| [[27ed7/3|27]]
| [[28ed7/3|28]]
| [[29ed7/3|29]]
|-
| [[30ed7/3|30]]
| [[31ed7/3|31]]
| [[32ed7/3|32]]
| [[33ed7/3|33]]
| [[34ed7/3|34]]
| [[35ed7/3|35]]
| [[36ed7/3|36]]
| [[37ed7/3|37]]
| [[38ed7/3|38]]
| [[39ed7/3|39]]
|-
| [[40ed7/3|40]]
| [[41ed7/3|41]]
| [[42ed7/3|42]]
| [[43ed7/3|43]]
| [[44ed7/3|44]]
| [[45ed7/3|45]]
| [[46ed7/3|46]]
| [[47ed7/3|47]]
| [[48ed7/3|48]]
| [[49ed7/3|49]]
|-
| [[50ed7/3|50]]
| [[51ed7/3|51]]
| [[52ed7/3|52]]
| [[53ed7/3|53]]
| [[54ed7/3|54]]
| [[55ed7/3|55]]
| [[56ed7/3|56]]
| [[57ed7/3|57]]
| [[58ed7/3|58]]
| [[59ed7/3|59]]
|-
| [[60ed7/3|60]]
| [[61ed7/3|61]]
| [[62ed7/3|62]]
| [[63ed7/3|63]]
| [[64ed7/3|64]]
| [[65ed7/3|65]]
| [[66ed7/3|66]]
| [[67ed7/3|67]]
| [[68ed7/3|68]]
| [[69ed7/3|69]]
|-
| [[70ed7/3|70]]
| [[71ed7/3|71]]
| [[72ed7/3|72]]
| [[73ed7/3|73]]
| [[74ed7/3|74]]
| [[75ed7/3|75]]
| [[76ed7/3|76]]
| [[77ed7/3|77]]
| [[78ed7/3|78]]
| [[79ed7/3|79]]
|-
| [[80ed7/3|80]]
| [[81ed7/3|81]]
| [[82ed7/3|82]]
| [[83ed7/3|83]]
| [[84ed7/3|84]]
| [[85ed7/3|85]]
| [[86ed7/3|86]]
| [[87ed7/3|87]]
| [[88ed7/3|88]]
| [[89ed7/3|89]]
|-
| [[90ed7/3|90]]
| [[91ed7/3|91]]
| [[92ed7/3|92]]
| [[93ed7/3|93]]
| [[94ed7/3|94]]
| [[95ed7/3|95]]
| [[96ed7/3|96]]
| [[97ed7/3|97]]
| [[98ed7/3|98]]
| [[99ed7/3|99]]
|}
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | 100…199
| [[100ed7/3|100]]
| [[101ed7/3|101]]
| [[102ed7/3|102]]
| [[103ed7/3|103]]
| [[104ed7/3|104]]
| [[105ed7/3|105]]
| [[106ed7/3|106]]
| [[107ed7/3|107]]
| [[108ed7/3|108]]
| [[109ed7/3|109]]
|-
| [[110ed7/3|110]]
| [[111ed7/3|111]]
| [[112ed7/3|112]]
| [[113ed7/3|113]]
| [[114ed7/3|114]]
| [[115ed7/3|115]]
| [[116ed7/3|116]]
| [[117ed7/3|117]]
| [[118ed7/3|118]]
| [[119ed7/3|119]]
|-
| [[120ed7/3|120]]
| [[121ed7/3|121]]
| [[122ed7/3|122]]
| [[123ed7/3|123]]
| [[124ed7/3|124]]
| [[125ed7/3|125]]
| [[126ed7/3|126]]
| [[127ed7/3|127]]
| [[128ed7/3|128]]
| [[129ed7/3|129]]
|-
| [[130ed7/3|130]]
| [[131ed7/3|131]]
| [[132ed7/3|132]]
| [[133ed7/3|133]]
| [[134ed7/3|134]]
| [[135ed7/3|135]]
| [[136ed7/3|136]]
| [[137ed7/3|137]]
| [[138ed7/3|138]]
| [[139ed7/3|139]]
|-
| [[140ed7/3|140]]
| [[141ed7/3|141]]
| [[142ed7/3|142]]
| [[143ed7/3|143]]
| [[144ed7/3|144]]
| [[145ed7/3|145]]
| [[146ed7/3|146]]
| [[147ed7/3|147]]
| [[148ed7/3|148]]
| [[149ed7/3|149]]
|-
| [[150ed7/3|150]]
| [[151ed7/3|151]]
| [[152ed7/3|152]]
| [[153ed7/3|153]]
| [[154ed7/3|154]]
| [[155ed7/3|155]]
| [[156ed7/3|156]]
| [[157ed7/3|157]]
| [[158ed7/3|158]]
| [[159ed7/3|159]]
|-
| [[160ed7/3|160]]
| [[161ed7/3|161]]
| [[162ed7/3|162]]
| [[163ed7/3|163]]
| [[164ed7/3|164]]
| [[165ed7/3|165]]
| [[166ed7/3|166]]
| [[167ed7/3|167]]
| [[168ed7/3|168]]
| [[169ed7/3|169]]
|-
| [[170ed7/3|170]]
| [[171ed7/3|171]]
| [[172ed7/3|172]]
| [[173ed7/3|173]]
| [[174ed7/3|174]]
| [[175ed7/3|175]]
| [[176ed7/3|176]]
| [[177ed7/3|177]]
| [[178ed7/3|178]]
| [[179ed7/3|179]]
|-
| [[180ed7/3|180]]
| [[181ed7/3|181]]
| [[182ed7/3|182]]
| [[183ed7/3|183]]
| [[184ed7/3|184]]
| [[185ed7/3|185]]
| [[186ed7/3|186]]
| [[187ed7/3|187]]
| [[188ed7/3|188]]
| [[189ed7/3|189]]
|-
| [[190ed7/3|190]]
| [[191ed7/3|191]]
| [[192ed7/3|192]]
| [[193ed7/3|193]]
| [[194ed7/3|194]]
| [[195ed7/3|195]]
| [[196ed7/3|196]]
| [[197ed7/3|197]]
| [[198ed7/3|198]]
| [[199ed7/3|199]]
|}

[[Category:Ed7/3's| ]]

[[Category:Lists of scales]]

User:Moremajorthanmajor/Ed7/3

2026-05-13T21:20:36Z

Lériendil: restored 2022 version of ed7/3 page

'''Ed7/3''' means '''Division of the septimal minor tenth ([[7/3]]) into n equal parts'''.

== Properties ==
Division of 7/3 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 7:3 (or another tenth) as a base though, is apparent by being the absolute widest range most generally used in popular songs (and even the range of a [https://en.wikipedia.org/wiki/Dastg%C4%81h-e_M%C4%81hur dastgah]) as well as a fairly trivial point to split the difference between the octave and the tritave (which is why I have named the region of intervals between 6 and 7 degrees of 5edo the "Middletown valley", the proper Middletown temperament family being based on an enneatonic scale generated by a third or a fifth optionally with a period of a wolf fourth at most 560 cents wide) and, as is the twelfth, an alternative interval where [[wikipedia:Inversion_(music)#Counterpoint|invertible counterpoint]] has classically occurred. Incidentally [[Pseudo-traditional harmonic functions of enneatonic scale_degrees|enneatonic scales]], especially those equivalent at e. g. 7:3, can sensibly take tetrads as the fundamental complete sonorities of a pseudo-traditional functional harmony due to their seventh degree being as structrally important as it is. Many, though not all, of these scales have a perceptually important pseudo (false) octave, with various degrees of accuracy.

Incidentally, one way to treat 7/3 as an equivalence is the use of the 3:4:5:(7) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to [[5/1]], here it takes two [[28/15]] to get to 7/2 (tempering out the comma 225/224). So, doing this yields 15, 19, and 34 note MOS 2/1 apart. While the notes are rather farther apart, the scheme is uncannily similar to meantone. "Macrobichromatic" might be a practically perfect term for it if it hasn't been named yet.

The branches of the Middletown family are named thus:

* 3&6: Tritetrachordal
* 4&5: Montrose (between 5\4edo and 4\3edo in particular, MOS generated by [pseudo] octaves belong to this branch)
* 2&7: Terra Rubra

The family of interlaced octatonic scale based temperaments in the "Middletown valley" is called Vesuvius (i. e. the volcano east of Naples).

The Middlebury temperament falls in the "Middletown valley", but its enneatonic scales are "generator-remainder".

The temperaments neighboring Middletown proper are named thus:

* 5&6: Rosablanca
* 4&7: Saptimpun (10 1/2)
* 5&7: 8bittone (Old Middetown)

Sort of unsurprisingly, though not so evidently, the pyrite tuning of edXs will turn out to divide a barely mistuned 5:2 of alomst exactly 45\34edo.

== Individual pages for ED7/3s ==
* [[8ed7/3]]
* [[9ed7/3]]
* [[15ed7/3]]
* [[16ed7/3]]
* [[17ed7/3]]
* [[19ed7/3]]
* [[30ed7/3]]
* [[34ed7/3]]
* [[38ed7/3]]
* [[49ed7/3]]
* [[53ed7/3]]
* [[68ed7/3]]
* [[98ed7/3]]
* [[106ed7/3]]

[[Category:Ed7/3| ]]
[[Category:Equal-step tuning]]

Talk:Ed7/3

2026-05-13T20:43:05Z

Lériendil: /* Deletion, again */

Anybody know what this is about? I propose we delete it, or else someone should really break it down and explain what this is. [[User:Keenan Pepper|Keenan Pepper]] ([[User talk:Keenan Pepper|talk]]) 23:35, 20 September 2018 (UTC)

== Request for deletion ==

Since the comment above was never adressed, I will just say we rewrite this current page (I'm willing to take a shot), and delete these:
* [[8edX]]
* [[9edX]]
* [[15edX]]
* [[16edX]]
* [[17edX]]
* [[19edX]]

As they are all nonsensical tables with no practical use. If someone wants to replace them with a simple templated table that generates the intervals (as on the EDO pages) that'd be fine too.

EDIT: signed -- [[User:Sintel|Sintel]] ([[User talk:Sintel|talk]]) 21:16, 19 January 2022 (UTC)

: I do not like all these mentioned tables in any way, and for me personally their value is not evident, at all am against trying to prove hypotheses by handmade tables. But since users have spent time on these tables, I am against simply deleting the pages in question. I would rather move them to the main editor's username space, where they will have time to elaborate them so that they are ready for the article namespace from the other users' point of view as well. I am doing the necessary work for this right now... --[[User:Xenwolf|Xenwolf]] ([[User talk:Xenwolf|talk]]) 13:43, 26 February 2022 (UTC)

== Deletion, again ==
Almost all ed7/3 pages are completely devoid of information, and this page itself is filled with what is clearly nonsense.
Since MMTM has been banned, and apparently nobody cares about any of this, I think we should revisit this.

– [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 00:36, 13 May 2026 (UTC)

: What about ed5/3? —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:27, 13 May 2026 (UTC)

:: Probably same story. Honestly I think we should be somewhat careful in applying [[XW:NG]] retroactively, but the part about:
:::Entries consisting solely of automatically generated content (e.g. interval tables, infoboxes) without context or explanation can be deleted without relocation.
:: Can be enforced without much issues, since we don't lose anything.
:: – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]])

::: I'm in support of deleting most of the ed5/3 and ed7/3 pages based on that guideline. I think this page itself should be considered under a different standard, and we'll have a clearer idea if we clean it up from MMTMisms (like "Middletown Valley" etc.).
::: Regards. -- [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 20:43, 13 May 2026 (UTC)

User:Lériendil/ET harmonic testing page

2026-03-13T00:12:18Z

Lériendil:

{{Infobox Interval
| Ratio = 65712362363534280139543/65536000000000000000000
| Name = deciennealimma
| Comma = yes
}}
==Harmonics==
{{Harmonics in equal|46|5|3|prec=2|columns=15|intervals=prime}}
{{Harmonics in equal|127|6|1|prec=2|columns=15|intervals=prime}}
{{Harmonics in equal|152|7|3|prec=2|columns=15|intervals=odd}}
{{Harmonics in equal|6181|3|1|prec=4|columns=15|intervals=prime}}
{{Harmonics in equal|13822009|2|1|prec=4|columns=11|intervals=prime}}
{{Harmonics in equal|13822009|2|1|prec=4|columns=11|start=12|intervals=prime}}

User:Lériendil/ET harmonic testing page

2026-03-13T00:11:51Z

Lériendil:

User:Lériendil/ET harmonic testing page

2026-03-13T00:11:08Z

Lériendil:

User:Lériendil/ET harmonic testing page

2026-03-12T23:27:33Z

Lériendil:

{{Infobox Interval
| Ratio = 65712362363534280139543/65536000000000000000000
| Name = deciennealimma
| Comma = yes
}}
==Harmonics==
{{Harmonics in equal|46|5|3|prec=2|columns=15|intervals=prime}}
{{Harmonics in equal|127|6|1|prec=2|columns=15|intervals=prime}}
{{Harmonics in equal|152|7|3|prec=2|columns=15|intervals=odd}}
{{Harmonics in equal|6181|3|1|prec=4|columns=15|intervals=prime}}
{{Harmonics in equal|14036269511|2|1|prec=4|columns=11|intervals=prime}}
{{Harmonics in equal|14036269511|2|1|prec=4|columns=11|start=12|intervals=prime}}

Mothra

2026-03-12T04:44:44Z

Lériendil: /* Tuning spectrum */

{{interwiki
| en = Mothra
| de = Slendrisch #Mothra
}}
{{Infobox regtemp
| Title = Mothra
| Subgroups = 2.3.5.7
| Comma basis = [[81/80]], [[1029/1024]]
| Edo join 1 = 26 | Edo join 2 = 31
| Mapping = 1; 3 12 -1
| Generators = 8/7 | Generators tuning = 232.3 | Optimization method = CWE
| MOS scales = [[1L 4s]], [[5L 1s]], [[5L 6s]], …, [[5L 21s]]
| Pergen = (P8, P5/3)
| Odd limit 1 = 7 | Mistuning 1 = 5.4 | Complexity 1 = 31
| Odd limit 2 = (2.3.5.7) 21 | Mistuning 2 = 10.8 | Complexity 2 = 36
}}
'''Mothra''', also known as '''cynder''', is a temperament of the [[7-limit]] that is a strong extension to [[slendric]], which is defined by splitting a perfect fifth representing [[3/2]] into three intervals of [[8/7]], tempering out [[1029/1024]]. The fifth of mothra is flattened to a [[meantone]] fifth, so that it reaches [[5/4]] when stacked four times and [[81/80]] is tempered out, unlike that of the other slendric extension [[rodan]], which is sharpened from just. This has the effect of bringing the generator 8/7 considerably closer to just, and also allowing [[MOS scale]]s of mothra to be more melodically usable than those of other forms of slendric, as the structurally-pervasive small step known as the [[quark]] (the residue between the octave and 5 generators, representing [[49/48]], [[64/63]], and in mothra also [[36/35]]) is larger here. [[EDOs]] that support mothra include [[26edo]], [[31edo]], and [[36edo]], and 31 is a particularly good tuning.

In the [[11-limit]], two extensions are of note: undecimal mothra (26 & 31), which tempers out [[99/98]], [[385/384]] and [[441/440]] to find the 11th harmonic at 8 generators down, and mosura (31 & 36), which tempers out [[176/175]] to find the 11th harmonic at 23 generators up. These two mappings merge at 31edo, which is therefore a uniquely suitable tuning for 11-limit mothra.

In higher limits, one may note that the two-generator interval closely approximates [[17/13]], and that the six-generator interval - the meantone whole tone of [[9/8]][[~]][[10/9]], approximates [[19/17]] - so that the 13:17:19 chord is well-represented; it is worth noting also that this chord is entirely included within the subtemperament obtained from taking every other generator of mothra, which is [[A-team]] (the crawma, [[83521/83486]], is the relevant comma tempered out here). This can be combined with the canonical mapping of 13 for each undecimal extension, which tempers out [[144/143]], to provide a natural route to the [[19-limit]].

For technical data, see [[Gamelismic clan #Mothra]].

== Intervals ==
As a strong extension of slendric, mothra's intervals can be expressed using the same system of extended diatonic interval naming [[Slendric #Interval categories|used for slendric]]. It is particularly convenient to use diatonic conventions for mothra, because its chain of fifths is meantone, and therefore 5/4 is simply read as a major third.

In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.

{| class="wikitable sortable center-1 center-2 right-3"
|-
! rowspan="3" | # !! rowspan="3" | Extended diatonic interval !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
! rowspan="2" | 7-limit intervals !! colspan="2" | Intervals of 11-limit extensions
|-
! Undecimal mothra !! Mosura
|-
| 0
| P1
| 0.0
| '''1/1'''
|
|
|-
| 1
| SM2
| 232.3
| '''8/7'''
| 55/48, 63/55
| 25/22
|-
| 2
| s4
| 464.5
| '''21/16''', 35/27, 64/49
| 55/42, 72/55
| 33/25
|-
| 3
| P5
| 696.8
| '''3/2'''
| 49/33
|
|-
| 4
| SM6
| 929.0
| 12/7
| 55/32, 56/33
|
|-
| 5
| s8
| 1161.3
| 35/18, 63/32, 96/49
| 55/28, 64/33, 108/55
| 88/45
|-
| 6
| M2
| 193.5
| '''9/8''', 10/9
| 49/44, 55/49
|
|-
| 7
| SM3
| 425.8
| 9/7
| 14/11
|
|-
| 8
| s5
| 658.0
| 35/24, 72/49
| '''16/11'''
| 22/15
|-
| 9
| M6
| 890.3
| 5/3, 27/16
|
|
|-
| 10
| SM7
| 1122.5
| 40/21, 27/14
| 21/11
|
|-
| 11
| sM2
| 154.8
| 35/32, 54/49
| 12/11
| 11/10
|-
| 12
| M3
| 387.0
| '''5/4'''
|
| 44/35
|-
| 13
| SA4
| 619.3
| 10/7
| 63/44
|
|-
| 14
| sM6
| 851.5
| 80/49
| 18/11
| 44/27, 33/20
|-
| 15
| M7
| 1083.8
| '''15/8''', 50/27
|
| 66/35
|-
| 16
| SA1
| 116.0
| 15/14
| 35/33
|
|-
| 17
| sM3
| 348.3
| 60/49
| 27/22, 40/33
| 11/9
|-
| 18
| A4
| 580.5
| 25/18, 45/32
|
| 88/63
|-
| 19
| SA5
| 812.8
| 45/28, 100/63
| 35/22
|
|-
| 20
| sM7
| 1045.0
| 90/49
| 20/11
| 11/6
|-
| 21
| A1
| 77.3
| 25/24
|
| 22/21
|-
| 22
| SA2
| 309.5
| 25/21
|
|
|-
| 23
| sA4
| 541.8
|
| 15/11
| '''11/8'''
|-
| 24
| A5
| 774.0
| 25/16
|
| 11/7
|-
| 25
| SA6
| 1006.3
| 25/14
|
| 88/49
|-
| 26
| sA1
| 38.5
| 50/49
| 45/44
| 33/32, 55/54
|}
<nowiki/>* In 7-limit [[CWE tuning]], octave reduced

== 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: ~8/7 = 232.3996{{c}}
| CWE: ~8/7 = 232.2514{{c}}
| POTE: ~8/7 = 232.1933{{c}}
|}

=== Tuning spectrum ===
{{See also| Slendric #Tuning spectrum }}

Vals refer to the appropriate undecimal extension in the edo's range.

{| class="wikitable center-all left-4 left-5"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]*
! Generator (¢)
! Extension
! Comments
|-
| '''[[21edo|4\21]]'''
|
| '''228.571'''
|
| 21c val, '''lower bound of 5-odd-limit diamond monotone'''
|-
|
| [[10/9]]
| 230.401
|
| 1/2-comma meantone fifth
|-
| '''[[26edo|5\26]]'''
|
| '''230.769'''
|
| '''Lower bound of 7- and 9-odd-limit diamond monotone'''
|-
|
| [[8/7]]
| 231.174
|
| Untempered tuning
|-
| [[83edo|16\83]]
|
| 231.325
|
| 83bc val
|-
|
| [[40/21]]
| 231.553
|
|
|-
| [[57edo|11\57]]
|
| 231.579
|
|
|-
|
| [[5/3]]
| 231.595
|
| 1/3-comma meantone fifth
|-
| [[88edo|17\88]]
|
| 231.818
|
|
|-
| [[119edo|23\119]]
|
| 231.933
|
| 119be val
|-
|
| [[25/24]]
| 231.937
|
| 2/7-comma meantone fifth
|-
| [[150edo|29\150]]
|
| 232.000
|
| 150be val
|-
|
| [[19/17]]
| 232.093
|
| As M2
|-
|
| [[10/7]]
| 232.114
|
|
|-
|
| [[19/13]]
| 232.123
|
| As s5
|-
|
| [[5/4]]
| 232.193
|
| 1/4-comma meantone fifth, (7-limit) 5- through 21-odd-limit minimax
|-
|
| [[17/13]]
| 232.214
|
| As s4
|-
| [[31edo|6\31]]
|
| 232.258
| ↑ Undecimal mothra (99/98) ↓ Mosura (176/175)
|
|-
|
| [[15/14]]
| 232.465
|
|
|-
| [[160edo|31\160]]
|
| 232.500
|
| 160be val
|-
|
| [[15/8]]
| 232.551
|
| 1/5-comma meantone fifth
|-
| [[129edo|25\129]]
|
| 232.558
|
|
|-
| [[98edo|19\98]]
|
| 232.653
|
|
|-
| [[67edo|13\67]]
|
| 232.836
|
|
|-
|
| [[96/49]]
| 232.861
|
| 1/5-comma slendric
|-
| [[103edo|20\103]]
|
| 233.010
|
| 103ce val
|-
|
| [[12/7]]
| 233.282
|
| 1/4-comma slendric
|-
| [[36edo|7\36]]
|
| 233.333
|
|
|-
|
| [[3/2]]
| 233.985
|
| 1/3-comma slendric
|-
| '''[[5edo|1\5]]'''
|
| '''240.000'''
|
| 5e val, '''upper bound of 5- to 9-odd-limit diamond monotone'''
|}
<nowiki/>* Besides the octave

== Music ==
[http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil]

[[Category:Mothra| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Gamelismic clan]]
[[Category:Orwellismic temperaments]]

80th-octave temperaments

2026-03-08T22:21:29Z

Lériendil: ]

{{Technical data page}}
{{Infobox fractional-octave|80}}

This page describes 80th-octave temperaments. [[80edo]] is extremely accurate for the [[17/1|17th harmonic]], a relationship which is so far seen in all documented temperaments on this page. In addition, it is the first equal division to be consistent in the [[19-limit]].

Temperaments discussed elsewhere include

* ''Octogintic'', → [[Parkleiness temperaments#Octogintic|Parkleiness temperaments]]

== Tetraicosic ==
Tetraicosic is described as 1600 & 2320, and named after the fact that 4 × 20 = 80.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: {{monzo| -52 17 12 -1 }}, {{monzo| 25 42 -8 -26 }}

[[Mapping]]: [{{val| 80 1 343 -27 }}, {{val| 0 4 -5 8 }}]

Mapping generators: ~31637227888/31381059609, ~7381125/5619712

[[Optimal tuning]] ([[CTE]]): ~7381125/5619712 = 471.7339

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

[[Badness]]: 1.49

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

Comma list: 9801/9800, 928760463360/928426965851, {{monzo| 49 -13 -14 -1 2 }}

Mapping: [{{val| 80 1 343 -27 434 }}, {{val| 0 4 -5 8 -5 }}]

Optimal tuning (CTE): ~2278125/1734656 = 471.7342

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

Badness: 0.178

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

Comma list: 9801/9800, 4100625/4100096, 14236560/14235529, 143327232/143286143

Mapping: [{{val| 80 1 343 -27 434 13 }}, {{val| 0 4 -5 8 -5 9 }}]

Optimal tuning (CTE): ~130/99 = 471.7322

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

Badness: 0.0741

=== 17-limit ===
Subgroup 2.3.5.7.11.13.17

Comma list: 9801/9800, 14400/14399, 373527/373490, 1812608/1812525, 4685824/4685625

Mapping: [{{val| 80 1 343 -27 434 13 327}}, {{val| 0 4 -5 8 -5 9 0}}]

Optimal tuning (CTE): ~130/99 = 471.7...

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 10830/10839, 12636/12635, 23409/23408, 373527/373490, 32133332/32131125

Mapping: [{{val| 80 1 343 -27 434 13 327 -69}}, {{val| 0 4 -5 8 -5 9 0 13}}]

Optimal tuning (CTE): ~67473/51376 = 471.732

{{Optimal ET sequence|legend=1| 720, 1600, 2320, 3920 }}

== Mercury ==
: ''Not to be confused with [[mercury meantone]] and [[mercurial comma|mercurial]].''

Named after the 80th element, defined as the 320 & 2000 temperament.
=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 9801/9800, 67392/67375, 1399680/1399489, {{monzo|14 -8 -8 5 2 -1}}

{{mapping|legend=1| 80 3 161 -23 252 197 | 0 5 1 10 1 4 }}

: mapping generators: ~9295/9216, ~2304/1859

Optimal tuning (CTE): ~2304/1859 = 371.385

{{Optimal ET sequence|legend=1|320, 2000, 4320}}
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 9801/9800, 12376/12375, 67392/67375, 4230144/4229225, 494534656/494515125

{{mapping|legend=1| 80 3 161 -23 252 197 327 | 0 5 1 10 1 4 0 }}

: mapping generators: ~459/455, ~1859/1500

Optimal tuning (CTE): ~2275/1836 = 371.386

{{Optimal ET sequence|legend=1|320, 2000, 4320}}
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 9801/9800, 12376/12375, 67392/67375, 392445/392392, 401408/401375, 1549184/1549125

Mapping: [{{val|80 3 161 -23 252 197 327 117}}, {{val|0 5 1 10 1 4 0 9}}]

: mapping generators: ~459/455, ~2275/1836

Optimal tuning (CTE): ~2275/1836 = 371.386

{{Optimal ET sequence|legend=1|320, 2000, 4320}}

== Octodeca ==
Octodeca can be described as the 80 & 1920 temperament.

Subgroup: 2.3.5.7

Comma list: {{monzo|21 60 -50}}, 184528125/184473632

{{mapping|legend=1|80 2 36 -25|0 5 6 10}}

: mapping generators: ~1071875/1062882 = 1\80, ~7476806640625/6025163444928 = 374.386

Optimal tuning (CTE): ~7476806640625/6025163444928 = 374.386

[[Support]]ing [[ET]]s: {{EDOs|80, 1920, 2000}}
=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 5832/5831, 9801/9800, 89376/89375, 123201/123200, 392445/392392, 653184/653125

{{mapping|legend=1|80 2 36 -25 -127 -321 -327 -240|0 5 6 10 6 -1 0 4}}

: mapping generators: ~12000/12103 = 1\80, ~1625/1309 = 374.386

Optimal tuning (CTE): ~1625/1309 = 374.386

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 5832/5831, 8625/8624, 9801/9800, 10626/10625, 89376/89375, 95013/95000, 123201/123200

{{mapping|legend=1|80 2 36 -25 -127 -321 -327 -240 -287|0 5 6 10 6 -1 0 4 3}}

: mapping generators: ~12000/12103 = 1\80, ~920/741 = 374.387

Optimal tuning (CTE): ~920/741 = 374.387

{{Navbox fractional-octave}}

[[Category:80edo]]

User:Lériendil/ET harmonic testing page

2026-03-08T17:37:16Z

Lériendil:

456/455

2026-03-05T17:59:13Z

Lériendil:

{{Infobox Interval
| Name = abnobisma
| Color name = 19o3urg1, nothurugu 1sn, Nothurugu comma
| Comma = yes
}}
'''456/455''', the '''abnobisma''', is a [[19-limit]] [[Superparticular ratio|superparticular comma]] of about 3.8 [[cent]]s. It is the amount by which a stack consisting of [[7/6]] and [[20/19]] falls short of [[16/13]], and the difference between [[35/24]] = (7/6)([[5/4]]) and [[19/13]]. By tempering it out is defined the '''abnobismic temperament''', which enables the [[abnobismic chords]].

== Etymology ==
The name ''abnobisma'' was named by [[User:Xenllium|Xenllium]] in 2023. It refers to [[Wikipedia:456 Abnoba|456 Abnoba]], the asteroid. Which in turn was named after a Celtic goddess.

== See also ==
* [[Small comma]]
* [[List of superparticular intervals]]

[[Category:Abnobismic]]
[[Category:Commas named after asteroids]]

90/77

2026-03-02T03:25:50Z

Lériendil:

{{Infobox Interval
| Name = swetismic subminor third
| Color name = 1ury2, luruyo 2nd
}}

'''90/77''', the swetismic subminor third, is 540/539 (3.2 cents) sharp of [[7/6]]. It arises in 11-limit scales as the interval between 11/10 and 9/7, 7/6 and 15/11, 11/9 and 10/7, and their inversions.

Additionally, it is [[385/384]] (4.5 cents) flat of [[75/64]], the classic augmented second.

It is very closely approximated in [[40edo]], which contextualizes it further as the product of [[15/14]] and [[12/11]].

[[Category:Third]]
[[Category:Subminor third]]
[[Category:Second]]
[[Category:Augmented second]]
[[Category:Swetismic]]

Talk:Pajara

2026-03-02T03:07:44Z

Lériendil: /* Minimax tunings */

== Minimax tunings ==
Although I'd like to have any excuse for the 7th harmonic to be more in tune, I don't think it makes much sense to call the tuning with eigenmonzo 7/6 the "7-odd-limit minimax".

If I understand correctly, this generator size was chosen because it balances 10/7, which is 17.5 cents flat, with 6/5, which is 17.5 cents sharp, but in pajara 10/7 is always 17.5 cents flat, so there is no point in balancing it. With the same reasoning you can say that the tuning where 5/4 is eigenmonzo, with a fifth of 706.843 cents, is the 7-odd limit minimax, because it balances the two worst intervals – 10/7 with 7/4, which are both, again, 17.5 cents sharp/flat respectively. If you take the term "minimax" literally, i.e. "The tuning in which the maximal error of any consonance is minimal", you get that any tuning between them is also a minimax, because in all of them the maximal error is 17.5 cents – that of 10/7.

I suggest labeling the tuning with eigenminzo 48/35 the 7-odd limit minimax, because that's where 6/5 and 8/7 are balanced. That's also what you get if you take the minimum of the [https://en.wikipedia.org/wiki/Norm_(mathematics)#p-norm p-norm (Wikipedia)] of the errors when p approaches infinity. I haven't checked but some other odd-limit minimax tunings may be changed too. [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 13:56, 16 October 2022 (UTC)

: On the page for [[Minimax tuning]], it says:
: <blockquote>However, this tuning may not be unique, in which case we may break the tie by choosing the tuning, among the set of least maximum error tunings, with the smallest sum of errors squared.</blockquote>
: I'm not sure which tuning would be minimax by this definition, though it seems like it probably wouldn't be the 7/6 eigenmonzo tuning.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 02:01, 11 January 2026 (UTC)

:: I think the limit approach makes more sense. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 08:33, 27 February 2026 (UTC)

::: What do you mean by "the limit approach"? [[User:Roeesi|Roeesi]] ([[User talk:Roeesi|talk]]) 09:18, 1 March 2026 (UTC)

:::: I mean using the ''p''-norm as ''p'' approaches infinity. Alternatively, discarding the bounding intervals and balancing the rest would make sense too. —[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:12, 1 March 2026 (UTC) (last edited [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:14, 1 March 2026 (UTC))

::::: Why are least-squares tunings being called "minimax" here? The most consistent tiebreaker should be second largest error (largest error except for 7/5~10/7). [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 03:07, 2 March 2026 (UTC)

== Unclear/questionable paragraph ==
> Pajara has fundamentally different categories, as a conventional semifourth (~250 cents) is now a neutral interval of some variety.

Different from what? Is this talking about Pajara[10] or Pajara[12]? Afaik pajara doesn't have an exact hemifourth.

> A unique feature of pajara is [1–6/5–3/2–12/7] as an essentially tempered alteration to [1–5/4–3/2–7/4] where both the third and the harmonic seventh are flattened by a chroma.

Isn't it a feature of all jubilismic temperaments? Also "essentially tempered alteration" doesn't make sense to me.

[[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 15:05, 2 August 2025 (UTC)

: Pajara has "~900-1000c" as an interval category. The meaning of "essentially tempered alteration" is clear. Flattening these two intervals by the same amount cannot produce this chord in JI. And be free to rewrite that section in terms of jubilismic temperaments.
: Additionally, you have been previously advised to discuss things on the talk page instead of force-removing them from articles. This does not mean do both at once. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 01:42, 3 August 2025 (UTC)

:: > Pajara has [~900–1000 ¢] as an interval category.

:: I suppose this is about Pajara[10], the 2L 8s scale. It's clearly not the case in Pajara[12].

:: > The meaning of "essentially tempered alteration" is clear.

:: I suppose the term was coined by analogy to essentially tempered chord, in which case, since it doesn't share the basics of essentially tempered chords, I'm afraid it's not a great idea, as Fred related to you on Discord.

:: > You have been previously advised to discuss things on the talk page instead of force-removing them from articles. This does not mean do both at once.

:: In fact, the advised cycle was edit–revert–discuss, becuz we obviously have the right to review your edits. This includes reversion.

:: [[User:FloraC|FloraC]] ([[User talk:FloraC|talk]]) 09:32, 3 August 2025 (UTC)

== Redundant tables ==

There are two tables for the intervals of pajara (canonical 11-limit), just with different optimization. One of them should be removed.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 16:25, 28 September 2025 (UTC)

== Patent vals ==

Planning to revert edit by FloraC that is justified by the entirely opinion-based statement that "patent vals are overrated". To many people, there is no reason to consider non-patent vals.

-- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 21:50, 22 November 2025 (UTC)

: Still, 22edo is the only patent val supporting pajara with 11-odd-limit consistency, so those who don't consider non-patent vals would only use 22edo for a tuning, and not use 32edo or 54edo. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:52, 22 November 2025 (UTC)

: Oops, forgot pajara is defined most simply in the 7-limit and not 11-limit. The only 7-odd-limit consistent EDOs supporting pajara are 10, 12, and 22, and someone who uses only patent vals probably cares about consistency, so wouldn't use anything else anyways. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 21:57, 22 November 2025 (UTC)

:: 32edo and 54edo support pajara in the 7-limit. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 22:20, 22 November 2025 (UTC)

: I have to agree that "being supported by all patent vals that support pajara save 12edo" seems like an arbitrary fact and not a specific advantage. Can you explain why it is useful? – [[User:Sintel|Sintel🎏]] ([[User_talk:Sintel|talk]]) 01:25, 23 November 2025 (UTC)

:: I honestly don't see why everyone sees patent vals as so arbitrary. To me the patent val essentially is the edo, non-patent vals really just exist to define wedgies/ET joins and that's it. -- [[User:VectorGraphics|VectorGraphics]] ([[User talk:VectorGraphics|talk]]) 06:44, 23 November 2025 (UTC)

::: There's a difference between an edo and its patent val; the val describes the corresponding rank-1 temperament. See [[EDO vs ET]]. Sometimes it's better to use the second-best approximation for a prime harmonic; for example, it's much better to use the 34d val than the patent val when using prime [[7/1|7]] in 34edo. In the patent val all of [[7/6]], [[7/5]], [[9/7]], [[14/11]], [[14/13]], [[15/14]], and their [[octave complement]]s are inconsistent, while in the 34d val only [[7/4]] and [[8/7]] are inconsistent. The ratios between harmonics are just as important as the harmonics themselves, and even more so when one realizes that there's much more of them. Also, the 34d val is much more useful than the 32edo or 54edo patent val.--[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 09:14, 25 December 2025 (UTC)

Talk:147edo

2026-03-02T02:44:53Z

Lériendil:

is it worth mentioning that 147edo tempers out 3489660928/3486784401 in the 2.3.13 subgroup? in the way of technical data, there's nothing in the section about its 2.3.13.23 subgroup that isn't also true of shoal. [[User:Squib|Squib]] ([[User talk:Squib|talk]]) 15:25, 1 March 2026 (UTC)

: What does this comma exactly represent? [[User:Lériendil|Lériendil]] ([[User talk:Lériendil|talk]]) 02:44, 2 March 2026 (UTC)

Semicomma family

2026-03-01T17:12:43Z

Lériendil: /* Newspeak */

{{Technical data page}}
The [[5-limit]] parent [[comma]] for the '''semicomma family''' of [[regular temperament|temperaments]] is the [[semicomma]] ({{monzo|legend=1| -21 3 7 }}, [[ratio]]: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.

== Orson ==
'''Orson''', first discovered by [[Erv Wilson]]{{citation needed}}, is the [[5-limit]] temperament [[tempering out]] the semicomma. It has a [[generator]] of [[~]][[75/64]], seven of which give the [[3/1|perfect twelfth]]; its [[ploidacot]] is alpha-heptacot. The generator is sharper than [[7/6]] by [[225/224]] when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

[[Subgroup]]: 2.3.5

[[Comma list]]: 2109375/2097152

{{Mapping|legend=1| 1 0 3 | 0 7 -3 }}
: mapping generators: ~2, ~75/64

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2902{{c}}, ~75/64 = 271.6929{{c}}
: [[error map]]: {{val| +0.290 -0.104 -0.522 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6394{{c}}
: error map: {{val| 0.000 -0.479 -1.232 }}

[[Tuning ranges]]:
* [[5-odd-limit]] [[diamond monotone]]: ~75/64 = [257.143, 276.923] (3\14 to 3\13)
* 5-odd-limit [[diamond tradeoff]]: ~75/64 = [271.229, 271.708] (1/3-comma to 2/7-comma)

{{Optimal ET sequence|legend=1| 22, 31, 53, 190, 243, 296, 645c, 1586bccc }}

[[Badness]] (Sintel): 0.957

=== Overview to extensions ===
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add
* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or
* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), or
* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).

== Orwell ==
{{Main| Orwell }}

So called because 19\84 (as a fraction of the octave) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It is compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the {{nowrap| 22 & 31 }} temperament. It is a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5-, 7- and 11-limit, but it does use its second-closest approximation to 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of its slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out [[2430/2401]] (the nuwell comma), [[1728/1715]] (the orwellisma), [[225/224]] (the marvel comma or septimal kleisma), and [[6144/6125]] (the porwell comma).

The 11-limit version of orwell tempers out [[99/98]], which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1–7/6–11/8–8/5 chord is natural to orwell.

Orwell has [[mos scale]]s of size 9, 13, 22 and 31. The 9-note mos is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13-note mos has those, and of course the 22- and 31-note mos are very well supplied with everything.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 1728/1715

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

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0192{{c}}, ~7/6 = 271.5130{{c}}
: [[error map]]: {{val| +0.019 -1.364 -0.795 +3.297 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/6 = 271.5097{{c}}
: error map: {{val| 0.000 -1.387 -0.843 +3.252 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~7/6 = [266.667, 272.727] (2\9 to 5\22)
* 9-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
* 7-odd-limit [[diamond tradeoff]]: ~7/6 = [266.871, 271.708]
* 9-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

[[Algebraic generator]]: Sabra3, the real root of 12''x3 - 7''x'' - 48.

{{Optimal ET sequence|legend=1| 9, 22, 31, 53, 84, 137, 221d }}

[[Badness]] (Sintel): 0.525

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

Comma list: 99/98, 121/120, 176/175

Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}

Optimal tunings:
* WE: ~2 = 1200.5989{{c}}, ~7/6 = 271.5616{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.4552{{c}}

Minimax tuning:
* 11-odd-limit: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 0 }}, {{monzo| 27/11 0 3/11 -3/11 0 }}, {{monzo| 27/11 0 -8/11 8/11 0 }}, {{monzo| 37/11 0 -2/11 2/11 0 }}]
: Unchanged-interval (eigenmonzo) basis: 2.7/5

Tuning ranges:
* 11-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

{{Optimal ET sequence|legend=0| 9, 22, 31, 53, 84e }}

Badness (Sintel): 0.504

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

Comma list: 99/98, 121/120, 176/175, 275/273

Mapping: {{mapping| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}

Optimal tunings:
* WE: ~2 = 1200.3621{{c}}, ~7/6 = 271.6283{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.5477{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]

{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }}

Badness (Sintel): 0.815

==== Blair ====
Subgroup: 2.3.5.7.11.13

Comma list: 65/64, 78/77, 91/90, 99/98

Mapping: {{mapping| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}

Optimal tunings:
* WE: ~2 = 1201.8031{{c}}, ~7/6 = 271.7083{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.3846{{c}}

{{Optimal ET sequence|legend=0| 9, 22, 31f }}

Badness (Sintel): 0.954

==== Winston ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 99/98, 105/104, 121/120

Mapping: {{mapping| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}

Optimal tunings:
* WE: ~2 = 1200.2846{{c}}, ~7/6 = 271.1524{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.1032{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~7/6 = [270.968, 272.727] (7\31 to 5\22)
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]

{{Optimal ET sequence|legend=0| 9, 22f, 31 }}

Badness (Sintel): 0.824

==== Doublethink ====
Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two [[13/12]]~[[14/13]]'s by tempering out their difference, [[169/168]]. Its ploidacot is alpha-14-cot.

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 176/175

Mapping: {{mapping| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}

Optimal tunings:
* WE: ~2 = 1200.6876{{c}}, ~13/12 = 135.8006{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 135.7410{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~13/12 = [135.484, 136.364] (7\62 to 5\44)
* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]

{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 115ef }}

Badness (Sintel): 1.12

=== Newspeak ===
In newspeak, the simplicity of obtaining ~[[11/8]] by stacking the generator ~[[7/6]] twice (as in basic 11-limit orwell) is sacrificed to gain accuracy for larger equal temperaments (such as [[84edo]] and [[115edo]]), at the cost of much higher complexity: it is reached only after stacking the generator 33 times and octave-reducing. Newspeak intersects with undecimal orwell at [[31edo]].

Subgroup: 2.3.5.7.11

Comma list: 225/224, 441/440, 1728/1715

Mapping: {{mapping| 1 0 3 1 -4 | 0 7 -3 8 33 }}

Optimal tunings:
* WE: ~2 = 1200.2072{{c}}, ~7/6 = 271.3353{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/6 = 271.2952{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~7/6 = [270.968, 271.698] (7\31 to 12\53)
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]

{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }}

Badness (Sintel): 1.04

=== Borwell ===
Subgroup: 2.3.5.7.11

Comma list: 225/224, 243/242, 1728/1715

Mapping: {{mapping| 1 -7 6 -7 -18 | 0 14 -6 16 35 }}
: mapping generators: ~2, ~55/36

Optimal tunings:
* WE: ~2 = 1200.0194{{c}}, ~55/36 = 735.7641{{c}}
* CWE: ~2 = 1200.000{{c}}, ~55/36 = 735.7527{{c}}

{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}

Badness (Sintel): 1.27

== Sabric ==
The sabric temperament tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 53 & 190 }} temperament. It was named by [[Xenllium]] in 2021 for its relation to the Sabra2 tuning (generator: 271.607278 cents).

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2109375/2097152

{{Mapping|legend=1| 1 0 3 -11 | 0 7 -3 61 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3056{{c}}, ~75/64 = 271.6760{{c}}
: [[error map]]: {{val| +0.306 -0.223 -0.425 +0.049 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~75/64 = 271.6110{{c}}
: error map: {{val| 0.000 -0.678 -1.147 -0.558 }}

{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}

[[Badness]] (Sintel): 2.24

== Triwell ==
Triwell tempers out the gamelisma, [[1029/1024]], and the triwellisma, [[235298/234375]]. It may be described as the {{nowrap| 31 & 159 }} temperament. It slices orwell's generator plus two octaves into three generators, and seven generators octave reduced make a ~8/7, which is the generator of [[slendric]]. Its ploidacot is 15-sheared-21-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 235298/234375

{{Mapping|legend=1| 1 -14 9 8 | 0 21 -9 -7 }}
: mapping generators: ~2, ~375/224

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4763{{c}}, ~375/224 = 890.8812{{c}}
: [[error map]]: {{val| +0.476 -0.118 +0.042 -1.184 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~375/224 = 890.5312{{c}}
: error map: {{val| 0.000 -0.799 -1.095 -2.545 }}

{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}

[[Badness]] (Sintel): 2.04

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

Comma list: 385/384, 441/440, 456533/455625

Mapping: {{mapping| 1 -14 9 8 -24 | 0 21 -9 -7 37 }}

Optimal tunings:
* WE: ~2 = 1200.4804{{c}}, ~375/224 = 890.8854{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~375/224 = 890.5344{{c}}

{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}

Badness (Sintel): 0.985

== Quadrawell ==
Quadrawell tempers out [[2401/2400]] and may be described as the {{nowrap| 31 & 212 }} temperament. It has a [[7/4]] generator of about 968 cents, four of which minus three octaves give the original generator of orwell. It can also be viewed as [[2.5.7|2.5.7-subgroup]] [[mothra]] with a different mapping of prime [[3/1|3]]. Its ploidacot is 22-sheared-28-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 2109375/2097152

{{Mapping|legend=1| 1 -21 12 2 | 0 28 -12 1 }}
: mapping generators: ~2, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3006{{c}}, ~7/4 = 968.1489{{c}}
: [[error map]]: {{val| +0.301 -0.098 -0.493 -0.076 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 967.9090{{c}}
: error map: {{val| 0.000 -0.503 -1.222 -0.917 }}

{{Optimal ET sequence|legend=1| 31, 119, 150, 181, 212, 243, 698cd, 941cd }}

[[Badness]] (Sintel): 1.92

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

Comma list: 385/384, 1375/1372, 14641/14580

Mapping: {{mapping| 1 -21 12 2 -28 | 0 28 -12 1 39 }}

Optimal tunings:
* WE: ~2 = 1200.3622{{c}}, ~7/4 = 968.2089{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~7/4 = 967.9206{{c}}

{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}

Badness (Sintel): 1.21

== Rainwell ==
The rainwell temperament tempers out the mirkwai comma, [[16875/16807]], and the rainy comma, [[2100875/2097152]]. It may be described as the {{nowrap| 31 & 265 }} temperament. Its ploidacot is 22-sheared-35-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 16875/16807, 2100875/2097152

{{Mapping|legend=1| 1 -21 12 -3 | 0 35 -15 9 }}

: mapping generators: ~2, ~2625/2048

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2032{{c}}, ~2401/1536 = 774.4577{{c}}
: [[error map]]: {{val| +0.203 -0.204 -0.740 +0.683 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~2401/1536 = 774.3282{{c}}
: error map: {{val| 0.000 -0.469 -1.236 +0.128 }}

{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}

[[Badness]] (Sintel): 3.63

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

Comma list: 540/539, 1375/1372, 2100875/2097152

Mapping: {{mapping| 1 -21 12 -3 -43 | 0 35 -15 9 72 }}

Optimal tunings:
* WE: ~2 = 1200.1915{{c}}, ~2205/1408 = 774.4451{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~2205/1408 = 774.3233{{c}}

{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}

Badness (Sintel): 1.74

== Quinwell ==
The quinwell temperament tempers out the wizma, [[420175/419904]], and may be described as the {{nowrap| 22 & 243 }} temperament. It slices orwell's generator into five quartertones. Its ploidacot is alpha-35-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 420175/419904, 2109375/2097152

{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}
: mapping generators: ~2, ~405/392

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2860{{c}}, ~405/392 = 54.3373{{c}}
: [[error map]]: {{val| +0.286 -0.151 -0.515 +0.084 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~405/392 = 54.3273{{c}}
: error map: {{val| 0.000 -0.501 -1.223 -0.536 }}

{{Optimal ET sequence|legend=1| 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}

[[Badness]] (Sintel): 4.27

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

Comma list: 540/539, 4375/4356, 2109375/2097152

Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}

Optimal tunings:
* WE: ~2 = 1200.0642{{c}}, ~33/32 = 54.3395{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 54.3369{{c}}

{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}

Badness (Sintel): 3.21

=== Quinbetter ===
Subgroup: 2.3.5.7.11

Comma list: 385/384, 24057/24010, 43923/43750

Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}

Optimal tunings:
* WE: ~2 = 1200.0642{{c}}, ~405/392 = 54.3373{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~405/392 = 54.3192{{c}}

{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}

Badness (Sintel): 2.60

[[Category:Temperament families]]
[[Category:Semicomma family| ]] 
[[Category:Rank 2]]
[[Category:Orson]]
[[Category:Orwell]]

Hemimean clan

2026-03-01T04:51:20Z

Lériendil: /* Rectified hebrew */

{{Technical data page}}
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].

The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.

Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.

Didacus has canonical subgroup extensions to primes 11 and 13, at [[#Undecimal didacus|undecimal didacus]]. Other subgroup extensions include rectified hebrew and isra.

Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. Discussed elsewhere are
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Pelogic family #Mohavila|Pelogic family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Bidia]]'' (+2048/2025) → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hexe]]'' (+50/49 or 128/125) → [[Augmented family #Hexe|Augmented family]]
* [[Misty]] (+5120/5103) → [[Misty family #Septimal misty|Misty family]]
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* [[Parakleismic]] (+4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Decistearn]]'' (+118098/117649) → [[Stearnsmic clan #Decistearn|Stearnsmic clan]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]

= 2.5.7 subgroup =
== Didacus ==
{{main|Didacus}}

See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].

[[Subgroup]]: 2.5.7

[[Comma list]]: [[3136/3125]]

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

: sval mapping generators: ~2, ~56/25

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

: [[gencom]]: [2 56/25; 3136/3125]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772

{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents

[[Badness]] (Sintel): 0.091

= Strong extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong extensions
|-
! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*
|-
! Temperament !! 5-limit generator location
|-
| [[#Hemiwürschmidt|Hemiwürschmidt]] || [[Würschmidt family#Würschmidt|Würschmidt]] || +2 || +16 || ↓ [[31edo|31]]
|-
| [[#Hemithirds|Hemithirds]] || [[Luna family#Luna|Luna]] || +1 || -15 || ↑ 31 ↓ [[25edo|25]]
|-
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
<nowiki />* Defined by intersection with other documented extensions

== Hemiwürschmidt ==
''[[#Strong extensions|Return to the map]]''

{{See also| Würschmidt family }}

'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 3136/3125

{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}

Mapping generators: ~2, ~25/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898

{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}

[[Badness]]: 0.020307

=== 2.3.5.7.23 subgroup ===
As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).

Subgroup: 2.3.5.7.23

[[Comma list]]: 576/575, 736/735, 1127/1125

{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}

Mapping generators: ~2, ~25/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901

{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}

Badness (Sintel): 0.304

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

Comma list: 243/242, 441/440, 3136/3125

Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840

{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}

Badness: 0.021069

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

Comma list: 243/242, 351/350, 441/440, 3584/3575

Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829

{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}

Badness: 0.023074

==== Hemithir ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 275/273

Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918

{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}

Badness: 0.031199

=== Hemiwur ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 1375/1372

Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884

{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}

Badness: 0.029270

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

Comma list: 121/120, 176/175, 196/195, 275/273

Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004

{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}

Badness: 0.028432

==== Hemiwar ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 1375/1372

Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698

{{Optimal ET sequence|legend=1| 6f, 31 }}

Badness: 0.044886

=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.

The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 3136/3125

Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}

: mapping generators: ~2, ~147/110

Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404

{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}

Badness: 0.034814

=== Semihemiwür ===
Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3136/3125, 9801/9800

Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}

: mapping generators: ~99/70, ~495/392

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}

Badness: 0.044848

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

Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125

Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}

Badness: 0.023388

===== Semihemiwürat =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625

Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}

Badness: 0.028987

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625

Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}

Badness: 0.021707

===== Semihemiwüram =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224

Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112

{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}

Badness: 0.029718

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224

Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428

{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}

Badness: 0.029545

== Hemithirds ==
''[[#Strong extensions|Return to the map]]''

{{Main| Hemithirds }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 3136/3125

{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244

[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}

[[Badness]]: 0.044284

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

Comma list: 385/384, 441/440, 3136/3125

Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227

Minimax tuning:
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 0 0 -5/9 5/9 }}, {{monzo| 64/27 0 0 2/27 -2/27 }}, {{monzo| 79/27 0 0 5/27 -5/27 }}, {{monzo| 79/27 0 0 -22/27 22/27 }}]
: Eigenmonzos (unchanged-intervals): 2, 11/7

{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}

Badness: 0.019003

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

Comma list: 196/195, 352/351, 385/384, 625/624

Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166

{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}

Badness: 0.021738

== Spell ==
''[[#Strong extensions|Return to the map]]''

{{See also| Magic family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 49/48, 3125/3072

{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927

{{Optimal ET sequence|legend=1| 6, 19, 82dd }}

[[Badness]]: 0.080958

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

Comma list: 49/48, 56/55, 125/121

Mapping: {{mapping| 1 0 2 2 3 | 0 10 2 5 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285

{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}

Badness: 0.059791

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

Comma list: 49/48, 56/55, 78/77, 125/121

Mapping: {{mapping| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928

{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}

Badness: 0.045591

==== Cantrip ====
Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 125/121

Mapping: {{mapping| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360

{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}

Badness: 0.041603

= Weak extensions =

== Semisept ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semisept]].''

The minimal generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 3136/3125

{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}

: mapping generators: ~2, ~75/49

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155

{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}

[[Badness]]: 0.050472

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

Comma list: 176/175, 540/539, 1331/1323

Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125

{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}

Badness: 0.022476

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

Comma list: 176/175, 351/350, 540/539, 1375/1372

Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126

{{Optimal ET sequence|legend=1| 31, 80, 111 }}

Badness: 0.025204

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 640/637, 715/714

Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125

{{Optimal ET sequence|legend=1| 31, 80, 111 }}

Badness: 0.019919

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323

Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116

{{Optimal ET sequence|legend=1| 31, 80, 111 }}

Badness: 0.016301

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459

Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106

{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}

Badness: 0.014957

==== Semishly ====
Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 275/273

Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980

{{Optimal ET sequence|legend=1| 31, 49f, 80f }}

Badness: 0.028408

== Emka ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''

Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 & 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 & 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 84035/82944

{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}

: mapping generators: ~2, ~48/35

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782

{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}

[[Badness]]: 0.144338

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

Comma list: 385/384, 2401/2376, 3136/3125

Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765

{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}

Badness: 0.054744

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

Comma list: 196/195, 364/363, 385/384, 625/624

Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758

{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}

Badness: 0.029741

== Decipentic ==
The generator for the decipentic temperament (43 & 56) is the tenth root of the [[5/1|5th harmonic (5/1)]], 51/10, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 2097152/2083725

{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800

{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}

[[Badness]]: 0.087325

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

Comma list: 441/440, 1344/1331, 3136/3125

Mapping: {{mapping| 1 6 0 -3 3 | 0 -19 10 25 2 }}

Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.061413

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

Comma list: 169/168, 441/440, 832/825, 975/968

Mapping: {{mapping| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.047611

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 256/255, 273/272, 375/374

Mapping: {{mapping| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.031191

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285

Mapping: {{mapping| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.023899

=== Quasijerome ===
Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 15488/15435, 16384/16335

Mapping: {{mapping| 1 6 0 -3 3 | 0 -38 20 50 47 }}

Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403

{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}

Badness: 0.092996

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

Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435

Mapping: {{mapping| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403

{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}

Badness: 0.044328

== Sengagen ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 420175/419904

{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217

{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}

[[Badness]]: 0.057978

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

Comma list: 540/539, 1344/1331, 3136/3125

Mapping: {{mapping| 1 1 2 2 3 | 0 29 16 40 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235

{{Optimal ET sequence|legend=1| 49, 50, 99e }}

Badness: 0.053828

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

Comma list: 351/350, 540/539, 975/968, 1344/1331

Mapping: {{mapping| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181

{{Optimal ET sequence|legend=1| 49, 50, 99e, 149e }}

Badness: 0.053531

==== Sengage ====
Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 364/363, 625/624

Mapping: {{mapping| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234

{{Optimal ET sequence|legend=1| 49f, 50, 99ef }}

Badness: 0.037416

== Mowglic ==
The mowglic temperament (19 & 161) is an extension of the [[Syntonic–kleismic equivalence continuum #Mowgli|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 177147/175000

{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706

{{Optimal ET sequence|legend=1| 19, 123d, 142, 161 }}

[[Badness]]: 0.129915

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

Comma list: 540/539, 3136/3125, 72171/71680

Mapping: {{mapping| 1 0 0 -3 8 | 0 15 22 55 -43 }}

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711

{{Optimal ET sequence|legend=1| 19, 123de, 142, 161 }}

Badness: 0.094032

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

Comma list: 351/350, 540/539, 1701/1690, 3136/3125

Mapping: {{mapping| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705

{{Optimal ET sequence|legend=1| 19, 123def, 142f, 161 }}

Badness: 0.051571

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125

Mapping: {{mapping| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.041918

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.032168

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.026117

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.021398

=== 31-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703

{{Optimal ET sequence|legend=1| 19, 123defgk, 142fk, 161 }}

Badness: 0.019331

== Tremka ==
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 & 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 2125764/2100875

{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173

{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}

[[Badness]]: 0.179925

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

Comma list: 540/539, 3136/3125, 35937/35840

Mapping: {{mapping| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}

Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168

{{Optimal ET sequence|legend=1| 50, 111, 161, 272, 433c }}

Badness: 0.068825

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

Comma list: 351/350, 540/539, 847/845, 3136/3125

Mapping: {{mapping| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}

Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172

{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}

Badness: 0.036070

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088

Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}

Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172

{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}

Badness: 0.022528

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539

Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}

Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170

{{Optimal ET sequence|legend=1| 50, 111, 161, 272h, 433cfh, 705ccdffhh }}

Badness: 0.016900

== Undetrita ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''

The undetrita temperament (111 & 118) tempers out the hemimean comma (3136/3125) and [[scheme comma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 & 118.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 14348907/14336000

{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917

{{Optimal ET sequence|legend=1| 111, 118, 229, 347, 576c }}

[[Badness]]: 0.114188

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

Comma list: 3025/3024, 3136/3125, 8019/8000

Mapping: {{mapping| 1 0 -2 -8 0 | 0 11 30 75 24 }}

Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912

{{Optimal ET sequence|legend=1| 111, 118, 229, 347 }}

Badness: 0.043883

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

Comma list: 352/351, 729/728, 1001/1000, 3025/3024

Mapping: {{mapping| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930

{{Optimal ET sequence|legend=1| 111, 229f }}

Badness: 0.038771

==== Undetritoid ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125

Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}

Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933

{{Optimal ET sequence|legend=1| 111, 229 }}

Badness: 0.042744

= Subgroup extensions =

== Undecimal didacus ==
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])3. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.

Subgroup: 2.5.7.11

Comma list: [[176/175]], [[1375/1372]]

Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}

: sval mapping generators: ~2, ~56/25

Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428

Optimal ET sequence: {{Optimal ET sequence| 6, 19e, 25, 31, 37 }}

RMS error: 0.5567 cents

Badness (Sintel): 0.195

=== Tridecimal didacus ===
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.

Subgroup: 2.5.7.11.13

Comma list: 176/175, 640/637, 1375/1372

Sval mapping: {{mapping| 1 0 -3 -7 13 | 0 2 5 9 -8 }}

: sval mapping generators: ~2, ~56/25

Gencom mapping: {{mapping| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}

: gencom: [2 28/25; 176/175 1375/1372 640/637]

Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594

Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}

Badness (Sintel): 0.324

==== Mediantone ====
Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[~]][[10/9]][[~]][[9/8]].

In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of [[~]][[19/17]][[~]][[28/25]] as ([[~]][[3/2]])2/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.

Subgroup: 2.5.7.11.13.17.19

Comma list: [[176/175]], [[640/637]], [[221/220]], [[476/475]], [[1375/1372]]

Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}

: sval mapping generators: ~2, ~56/25

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927

Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}

<nowiki />* 117d only appears without prime 19

Badness (Sintel): 0.618

==== Roulette ====
{{See also | Chromatic pairs #Roulette }}

Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)2 to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])3 is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.

Subgroup: 2.5.7.11.13.17.19

Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]

Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}

: sval mapping generators: ~2, ~28/25

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259

Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}

Badness (Sintel): 0.676

== Rectified hebrew ==
{{Main| Rectified hebrew }}

Rectified hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three. Notably, it is the no-threes restriction of [[Sycamore family#Septimal sycamore|sycamore]].

Subgroup: 2.5.7.13

Comma list: 3136/3125, 4394/4375

Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}

: sval mapping generators: ~2, ~26/25

Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086

{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}

== Isra ==
Isra (''iss-RAH'') results from taking every other generator of [[septimal meantone]], or from [[didacus]] if the generator is interpreted as 9/8. It is named after the Isrāʾ night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 81/80, 126/125

{{Mapping|legend=2| 1 0 -4 -13 | 0 1 2 5 }}

: sval mapping generators: ~2, ~9

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.9898

{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 56b, 87b }}

=== Tutone ===
Tutone is every other step of [[Meantone vs meanpop|undecimal meantone]], or undecimal [[didacus]] with the generator interpreted as 9/8.

[[Subgroup]]: 2.9.5.7.11

[[Comma list]]: 81/80, 99/98, 126/125

{{Mapping|legend=2| 1 0 -4 -13 -25 | 0 1 2 5 9 }}

{{Mapping|legend=3| 1 3/2 2 2 2 | 0 1/2 2 5 9 }}

: [[gencom]]: [2 9/8; 81/80 99/98 126/125]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 193.937

{{Optimal ET sequence|legend=1| 6, 19e, 25, 31, 68b, 99b }}

[[Badness]]: 0.00536

=== Leantone ===
{{See also| Chromatic pairs #Leantone }}

Leantone is every other step of [[vincenzo]].

[[Subgroup]]: 2.9.5.7.11

[[Comma list]]: 45/44, 56/55, 81/80

{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}

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

: [[gencom]]: [2 9/8; 45/44 56/55 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.500

{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25e, 31e, 56bee, 81beee }}

[[Tp tuning #T2 tuning|RMS error]]: 3.882 cents

=== Deutone ===
{{See also| Chromatic pairs #Deutone }}

Deutone is (also) every other step of [[vincenzo]].

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: 65/64, 81/80, 91/90

{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}

{{Mapping|legend=3| 1 3/2 2 2 0 4 | 0 1/2 2 5 0 -2 }}

: [[gencom]]: [2 9/8; 65/64 81/80 91/90]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 191.059

{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25f, 44df }}

[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents

[[Category:Temperament clans]]
[[Category:Hemimean clan| ]] 
[[Category:Hemimean| ]] 
[[Category:Rank 2]]

Hemimean clan

2026-03-01T04:50:50Z

Lériendil: sycamore

{{Technical data page}}
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].

The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.

Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.

Didacus has canonical subgroup extensions to primes 11 and 13, at [[#Undecimal didacus|undecimal didacus]]. Other subgroup extensions include rectified hebrew and isra.

Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. Discussed elsewhere are
* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]
* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]
* ''[[Mohavila]]'' (+135/128 or 1323/1250) → [[Pelogic family #Mohavila|Pelogic family]]
* ''[[Cohemimabila]]'' (+65536/64827) → [[Mabila family #Cohemimabila|Mabila family]]
* ''[[Sycamore]]'' (+686/675 or 875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]
* ''[[Bidia]]'' (+2048/2025) → [[Diaschismic family #Bidia|Diaschismic family]]
* ''[[Hexe]]'' (+50/49 or 128/125) → [[Augmented family #Hexe|Augmented family]]
* [[Misty]] (+5120/5103) → [[Misty family #Septimal misty|Misty family]]
* ''[[Bischismic]]'' (+32805/32768) → [[Schismatic family #Bischismic|Schismatic family]]
* ''[[Clyde]]'' (+245/243) → [[Kleismic family #Clyde|Kleismic family]]
* [[Parakleismic]] (+4375/4374) → [[Ragismic microtemperaments #Parakleismic|Ragismic microtemperaments]]
* ''[[Arch]]'' (+5250987/5242880) → [[Escapade family #Arch|Escapade family]]
* ''[[Subpental]]'' (+19683/19600) → [[Sensipent family #Sensipent|Sensipent family]]
* ''[[Doubloon]]'' (+33756345/33554432) → [[Vavoom family #Doubloon|Vavoom family]]
* ''[[Decistearn]]'' (+118098/117649) → [[Stearnsmic clan #Decistearn|Stearnsmic clan]]
* ''[[Quintagar]]'' (+33554432/33480783) → [[Quindromeda family #Quintagar|Quindromeda family]]
* ''[[Rubidium]]'' (+4194304/4117715) → [[37th-octave temperaments]]

= 2.5.7 subgroup =
== Didacus ==
{{main|Didacus}}

See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].

[[Subgroup]]: 2.5.7

[[Comma list]]: [[3136/3125]]

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

: sval mapping generators: ~2, ~56/25

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

: [[gencom]]: [2 56/25; 3136/3125]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772

{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents

[[Badness]] (Sintel): 0.091

= Strong extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong extensions
|-
! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*
|-
! Temperament !! 5-limit generator location
|-
| [[#Hemiwürschmidt|Hemiwürschmidt]] || [[Würschmidt family#Würschmidt|Würschmidt]] || +2 || +16 || ↓ [[31edo|31]]
|-
| [[#Hemithirds|Hemithirds]] || [[Luna family#Luna|Luna]] || +1 || -15 || ↑ 31 ↓ [[25edo|25]]
|-
| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25
|}
<nowiki />* Defined by intersection with other documented extensions

== Hemiwürschmidt ==
''[[#Strong extensions|Return to the map]]''

{{See also| Würschmidt family }}

'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 3136/3125

{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}

Mapping generators: ~2, ~25/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898

{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}

[[Badness]]: 0.020307

=== 2.3.5.7.23 subgroup ===
As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).

Subgroup: 2.3.5.7.23

[[Comma list]]: 576/575, 736/735, 1127/1125

{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}

Mapping generators: ~2, ~25/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901

{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}

Badness (Sintel): 0.304

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

Comma list: 243/242, 441/440, 3136/3125

Mapping: {{mapping| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840

{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}

Badness: 0.021069

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

Comma list: 243/242, 351/350, 441/440, 3584/3575

Mapping: {{mapping| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829

{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}

Badness: 0.023074

==== Hemithir ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 275/273

Mapping: {{mapping| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918

{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}

Badness: 0.031199

=== Hemiwur ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 1375/1372

Mapping: {{mapping| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884

{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}

Badness: 0.029270

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

Comma list: 121/120, 176/175, 196/195, 275/273

Mapping: {{mapping| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004

{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}

Badness: 0.028432

==== Hemiwar ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 1375/1372

Mapping: {{mapping| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698

{{Optimal ET sequence|legend=1| 6f, 31 }}

Badness: 0.044886

=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.

The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 3136/3125

Mapping: {{mapping| 1 15 4 7 24 | 0 -32 -4 -10 -49 }}

: mapping generators: ~2, ~147/110

Optimal tuning (POTE): ~2 = 1\1, ~147/110 = 503.0404

{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}

Badness: 0.034814

=== Semihemiwür ===
Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3136/3125, 9801/9800

Mapping: {{mapping| 2 14 6 9 -10 | 0 -16 -2 -5 25 }}

: mapping generators: ~99/70, ~495/392

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9021

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}

Badness: 0.044848

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

Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125

Mapping: {{mapping| 2 14 6 9 -10 25 | 0 -16 -2 -5 25 -26 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9035

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328 }}

Badness: 0.023388

===== Semihemiwürat =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625

Mapping: {{mapping| 2 14 6 9 -10 25 19 | 0 -16 -2 -5 25 -26 -16 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~28/25 = 193.9112

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}

Badness: 0.028987

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625

Mapping: {{mapping| 2 14 6 9 -10 25 19 20 | 0 -16 -2 -5 25 -26 -16 -17 }}

Optimal tuning (POTE): ~17/12 = 1\2, ~19/17 = 193.9145

{{Optimal ET sequence|legend=1| 62e, 68, 130, 198, 328g, 526cfgg }}

Badness: 0.021707

===== Semihemiwüram =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224

Mapping: {{mapping| 2 14 6 9 -10 25 -4 | 0 -16 -2 -5 25 -26 18 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~28/25 = 193.9112

{{Optimal ET sequence|legend=1| 62eg, 68, 130g, 198g }}

Badness: 0.029718

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224

Mapping: {{mapping| 2 14 6 9 -10 25 -4 -3 | 0 -16 -2 -5 25 -26 18 17 }}

Optimal tuning (POTE): ~99/70 = 1\2, ~19/17 = 193.9428

{{Optimal ET sequence|legend=1| 62egh, 68, 130gh, 198gh }}

Badness: 0.029545

== Hemithirds ==
''[[#Strong extensions|Return to the map]]''

{{Main| Hemithirds }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 3136/3125

{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244

[[Minimax tuning]]:
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}

[[Badness]]: 0.044284

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

Comma list: 385/384, 441/440, 3136/3125

Mapping: {{mapping| 1 4 2 2 7 | 0 -15 2 5 -22 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227

Minimax tuning:
* 11-odd-limit: ~28/25 = {{monzo| 5/27 0 0 1/27 -1/27 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 11/9 0 0 -5/9 5/9 }}, {{monzo| 64/27 0 0 2/27 -2/27 }}, {{monzo| 79/27 0 0 5/27 -5/27 }}, {{monzo| 79/27 0 0 -22/27 22/27 }}]
: Eigenmonzos (unchanged-intervals): 2, 11/7

{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}

Badness: 0.019003

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

Comma list: 196/195, 352/351, 385/384, 625/624

Mapping: {{mapping| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166

{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}

Badness: 0.021738

== Spell ==
''[[#Strong extensions|Return to the map]]''

{{See also| Magic family }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 49/48, 3125/3072

{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927

{{Optimal ET sequence|legend=1| 6, 19, 82dd }}

[[Badness]]: 0.080958

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

Comma list: 49/48, 56/55, 125/121

Mapping: {{mapping| 1 0 2 2 3 | 0 10 2 5 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285

{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}

Badness: 0.059791

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

Comma list: 49/48, 56/55, 78/77, 125/121

Mapping: {{mapping| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928

{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}

Badness: 0.045591

==== Cantrip ====
Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 125/121

Mapping: {{mapping| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360

{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}

Badness: 0.041603

= Weak extensions =

== Semisept ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semisept]].''

The minimal generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 3136/3125

{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}

: mapping generators: ~2, ~75/49

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155

{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}

[[Badness]]: 0.050472

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

Comma list: 176/175, 540/539, 1331/1323

Mapping: {{mapping| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125

{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}

Badness: 0.022476

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

Comma list: 176/175, 351/350, 540/539, 1375/1372

Mapping: {{mapping| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}

Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126

{{Optimal ET sequence|legend=1| 31, 80, 111 }}

Badness: 0.025204

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 640/637, 715/714

Mapping: {{mapping| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125

{{Optimal ET sequence|legend=1| 31, 80, 111 }}

Badness: 0.019919

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323

Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116

{{Optimal ET sequence|legend=1| 31, 80, 111 }}

Badness: 0.016301

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459

Mapping: {{mapping| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}

Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106

{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}

Badness: 0.014957

==== Semishly ====
Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 275/273

Mapping: {{mapping| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980

{{Optimal ET sequence|legend=1| 31, 49f, 80f }}

Badness: 0.028408

== Emka ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''

Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 & 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 & 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 84035/82944

{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}

: mapping generators: ~2, ~48/35

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782

{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}

[[Badness]]: 0.144338

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

Comma list: 385/384, 2401/2376, 3136/3125

Mapping: {{mapping| 1 14 6 12 3 | 0 -27 -8 -20 1 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765

{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}

Badness: 0.054744

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

Comma list: 196/195, 364/363, 385/384, 625/624

Mapping: {{mapping| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758

{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}

Badness: 0.029741

== Decipentic ==
The generator for the decipentic temperament (43 & 56) is the tenth root of the [[5/1|5th harmonic (5/1)]], 51/10, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 2097152/2083725

{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800

{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}

[[Badness]]: 0.087325

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

Comma list: 441/440, 1344/1331, 3136/3125

Mapping: {{mapping| 1 6 0 -3 3 | 0 -19 10 25 2 }}

Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.061413

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

Comma list: 169/168, 441/440, 832/825, 975/968

Mapping: {{mapping| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.047611

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 256/255, 273/272, 375/374

Mapping: {{mapping| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.031191

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285

Mapping: {{mapping| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790

{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}

Badness: 0.023899

=== Quasijerome ===
Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 15488/15435, 16384/16335

Mapping: {{mapping| 1 6 0 -3 3 | 0 -38 20 50 47 }}

Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403

{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}

Badness: 0.092996

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

Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435

Mapping: {{mapping| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}

Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403

{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}

Badness: 0.044328

== Sengagen ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 420175/419904

{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217

{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}

[[Badness]]: 0.057978

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

Comma list: 540/539, 1344/1331, 3136/3125

Mapping: {{mapping| 1 1 2 2 3 | 0 29 16 40 23 }}

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235

{{Optimal ET sequence|legend=1| 49, 50, 99e }}

Badness: 0.053828

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

Comma list: 351/350, 540/539, 975/968, 1344/1331

Mapping: {{mapping| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181

{{Optimal ET sequence|legend=1| 49, 50, 99e, 149e }}

Badness: 0.053531

==== Sengage ====
Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 364/363, 625/624

Mapping: {{mapping| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}

Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234

{{Optimal ET sequence|legend=1| 49f, 50, 99ef }}

Badness: 0.037416

== Mowglic ==
The mowglic temperament (19 & 161) is an extension of the [[Syntonic–kleismic equivalence continuum #Mowgli|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 177147/175000

{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706

{{Optimal ET sequence|legend=1| 19, 123d, 142, 161 }}

[[Badness]]: 0.129915

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

Comma list: 540/539, 3136/3125, 72171/71680

Mapping: {{mapping| 1 0 0 -3 8 | 0 15 22 55 -43 }}

Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711

{{Optimal ET sequence|legend=1| 19, 123de, 142, 161 }}

Badness: 0.094032

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

Comma list: 351/350, 540/539, 1701/1690, 3136/3125

Mapping: {{mapping| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705

{{Optimal ET sequence|legend=1| 19, 123def, 142f, 161 }}

Badness: 0.051571

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125

Mapping: {{mapping| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.041918

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.032168

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.026117

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704

{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}

Badness: 0.021398

=== 31-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: {{mapping| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}

Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703

{{Optimal ET sequence|legend=1| 19, 123defgk, 142fk, 161 }}

Badness: 0.019331

== Tremka ==
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 & 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 2125764/2100875

{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173

{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}

[[Badness]]: 0.179925

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

Comma list: 540/539, 3136/3125, 35937/35840

Mapping: {{mapping| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}

Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168

{{Optimal ET sequence|legend=1| 50, 111, 161, 272, 433c }}

Badness: 0.068825

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

Comma list: 351/350, 540/539, 847/845, 3136/3125

Mapping: {{mapping| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}

Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172

{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}

Badness: 0.036070

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088

Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}

Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172

{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}

Badness: 0.022528

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539

Mapping: {{mapping| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}

Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170

{{Optimal ET sequence|legend=1| 50, 111, 161, 272h, 433cfh, 705ccdffhh }}

Badness: 0.016900

== Undetrita ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''

The undetrita temperament (111 & 118) tempers out the hemimean comma (3136/3125) and [[scheme comma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 & 118.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 14348907/14336000

{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917

{{Optimal ET sequence|legend=1| 111, 118, 229, 347, 576c }}

[[Badness]]: 0.114188

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

Comma list: 3025/3024, 3136/3125, 8019/8000

Mapping: {{mapping| 1 0 -2 -8 0 | 0 11 30 75 24 }}

Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912

{{Optimal ET sequence|legend=1| 111, 118, 229, 347 }}

Badness: 0.043883

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

Comma list: 352/351, 729/728, 1001/1000, 3025/3024

Mapping: {{mapping| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930

{{Optimal ET sequence|legend=1| 111, 229f }}

Badness: 0.038771

==== Undetritoid ====
Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125

Mapping: {{mapping| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}

Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933

{{Optimal ET sequence|legend=1| 111, 229 }}

Badness: 0.042744

= Subgroup extensions =

== Undecimal didacus ==
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])3. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.

Subgroup: 2.5.7.11

Comma list: [[176/175]], [[1375/1372]]

Sval mapping: {{mapping| 1 0 -3 -7 | 0 2 5 9 }}

: sval mapping generators: ~2, ~56/25

Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428

Optimal ET sequence: {{Optimal ET sequence| 6, 19e, 25, 31, 37 }}

RMS error: 0.5567 cents

Badness (Sintel): 0.195

=== Tridecimal didacus ===
Tridecimal didacus (formerly ''roulette''; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to [[hemiwur]] or [[grosstone]] with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because [[~]][[8/7]] is already tuned almost 3{{cent}} flat, it makes sense to equate two of it with [[~]][[13/10]] (tempering out the 8{{cent}} [[huntma]]). This mapping of 13 increases the [[badness]] of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.

Subgroup: 2.5.7.11.13

Comma list: 176/175, 640/637, 1375/1372

Sval mapping: {{mapping| 1 0 -3 -7 13 | 0 2 5 9 -8 }}

: sval mapping generators: ~2, ~56/25

Gencom mapping: {{mapping| 1 0 2 2 2 5 | 0 0 2 5 9 -8 }}

: gencom: [2 28/25; 176/175 1375/1372 640/637]

Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594

Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}

Badness (Sintel): 0.324

==== Mediantone ====
Mediantone is named after its whole tone generator serving as the [[mediant]] of [[9/8]] and [[10/9]], namely [[19/17]], in addition to [[28/25]], as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "[[meantone]]", as the context one is most likely to first discover this logic is when the tone also represents [[~]][[10/9]][[~]][[9/8]].

In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of [[~]][[19/17]][[~]][[28/25]] as ([[~]][[3/2]])2/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.

Subgroup: 2.5.7.11.13.17.19

Comma list: [[176/175]], [[640/637]], [[221/220]], [[476/475]], [[1375/1372]]

Sval mapping: {{mapping| 1 0 -3 -7 13 -18 -19 | 0 2 5 9 -8 19 20 }}

: sval mapping generators: ~2, ~56/25

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927

Optimal ET sequence: {{Optimal ET sequence| 6h, 31gh, 37, 80, 117d* }}

<nowiki />* 117d only appears without prime 19

Badness (Sintel): 0.618

==== Roulette ====
{{See also | Chromatic pairs #Roulette }}

Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)2 to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])3 is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.

Subgroup: 2.5.7.11.13.17.19

Comma list: [[170/169]], [[176/175]], [[476/475]], [[640/637]], [[1375/1372]]

Sval mapping: {{mapping| 1 2 2 2 5 7 7 | 0 2 5 9 -8 -18 -17 }}

: sval mapping generators: ~2, ~28/25

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259

Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}

Badness (Sintel): 0.676

== Rectified hebrew ==
{{Main| Rectified hebrew }}

Rectified hebrew (37 & 56) is derived from the [https://individual.utoronto.ca/kalendis/hebrew/rect.htm#353 calendar by the same name]. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out [[4394/4375]] and spliting the generator of didacus in three. Notably, it is the no-threes restriction of [[sycamore]].

Subgroup: 2.5.7.13

Comma list: 3136/3125, 4394/4375

Sval mapping: {{mapping| 1 2 2 3 | 0 6 15 13 }}

: sval mapping generators: ~2, ~26/25

Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086

{{Optimal ET sequence|legend=1| 18, 19, 37, 93, 130 }}

== Isra ==
Isra (''iss-RAH'') results from taking every other generator of [[septimal meantone]], or from [[didacus]] if the generator is interpreted as 9/8. It is named after the Isrāʾ night journey in the Qur'an, because it is similar to [[luna]] (septimal [[hemithirds]], a didacus extension).

[[Subgroup]]: 2.9.5.7

[[Comma list]]: 81/80, 126/125

{{Mapping|legend=2| 1 0 -4 -13 | 0 1 2 5 }}

: sval mapping generators: ~2, ~9

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.9898

{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 56b, 87b }}

=== Tutone ===
Tutone is every other step of [[Meantone vs meanpop|undecimal meantone]], or undecimal [[didacus]] with the generator interpreted as 9/8.

[[Subgroup]]: 2.9.5.7.11

[[Comma list]]: 81/80, 99/98, 126/125

{{Mapping|legend=2| 1 0 -4 -13 -25 | 0 1 2 5 9 }}

{{Mapping|legend=3| 1 3/2 2 2 2 | 0 1/2 2 5 9 }}

: [[gencom]]: [2 9/8; 81/80 99/98 126/125]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 193.937

{{Optimal ET sequence|legend=1| 6, 19e, 25, 31, 68b, 99b }}

[[Badness]]: 0.00536

=== Leantone ===
{{See also| Chromatic pairs #Leantone }}

Leantone is every other step of [[vincenzo]].

[[Subgroup]]: 2.9.5.7.11

[[Comma list]]: 45/44, 56/55, 81/80

{{Mapping|legend=2| 1 0 -4 -13 -6 | 0 1 2 5 3 }}

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

: [[gencom]]: [2 9/8; 45/44 56/55 81/80]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 192.500

{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25e, 31e, 56bee, 81beee }}

[[Tp tuning #T2 tuning|RMS error]]: 3.882 cents

=== Deutone ===
{{See also| Chromatic pairs #Deutone }}

Deutone is (also) every other step of [[vincenzo]].

[[Subgroup]]: 2.9.5.7.13

[[Comma list]]: 65/64, 81/80, 91/90

{{Mapping|legend=2| 1 0 -4 -13 10 | 0 1 2 5 -2 }}

{{Mapping|legend=3| 1 3/2 2 2 0 4 | 0 1/2 2 5 0 -2 }}

: [[gencom]]: [2 9/8; 65/64 81/80 91/90]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/8 = 191.059

{{Optimal ET sequence|legend=1| 6, 7, 13, 19, 25f, 44df }}

[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents

[[Category:Temperament clans]]
[[Category:Hemimean clan| ]] 
[[Category:Hemimean| ]] 
[[Category:Rank 2]]

Valinor

2026-03-01T01:15:32Z

Lériendil: Redirected page to Valinorsmic clan#Valinor

#redirect [[Valinorsmic clan #Valinor]]

User:Lériendil/ET harmonic testing page

2026-02-27T18:29:47Z

Lériendil:

{{Infobox Interval
| Ratio = 65712362363534280139543/65536000000000000000000
| Name = deciennealimma
| Comma = yes
}}
==Harmonics==
{{Harmonics in equal|46|5|3|prec=2|columns=15|intervals=prime}}
{{Harmonics in equal|1086|6|1|prec=2|columns=15|intervals=prime}}
{{Harmonics in equal|152|7|3|prec=2|columns=15|intervals=odd}}
{{Harmonics in equal|6181|3|1|prec=4|columns=15|intervals=prime}}
{{Harmonics in equal|851|2|1|prec=4|columns=15|intervals=prime}}

Gamelismic clan

2026-02-24T18:31:33Z

Lériendil: /* Rodan */

{{Technical data page}}
The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''gamelismic clan''' is the gamelisma, [[1029/1024]], with [[monzo]] {{monzo| -10 1 0 3 }}. For any member of the clan, for the rank-3 [[gamelismic family #Gamelismic|gamelismic temperament]] itself, and for the rank-2 2.3.7 temperament [[slendric]] (a.k.a. gamelic), this means three [[~]][[8/7]] intervals give a fifth, [[3/2]]. In fact, we find that {{nowrap| 3/2 {{=}} (8/7)3⋅(1029/1024) }}. From this it follows that gamelismic temperaments tend to flatten both the fifth and the harmonic seventh, or if they do not, the other of the pair must be flattened even more. [[36edo]] is a good tuning for slendric, though if the full 7-limit is desired, [[72edo]], [[77edo]], or [[118edo]] might be preferred.

== Slendric ==
{{Main| Slendric }}

[[Subgroup]]: 2.3.7

[[Comma list]]: 1029/1024

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

{{Mapping|legend=3| 1 1 0 3 | 0 3 0 -1 }}
: mapping generators: ~2, ~8/7

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4859{{c}}, ~8/7 = 233.7822{{c}}
: [[error map]]: {{val| +0.486 -0.123 -1.151 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~8/7 = 233.7474{{c}}
: error map: {{val| 0.000 -0.713 -2.573 }}

{{Optimal ET sequence|legend=1| 5, 21, 26, 31, 36, 77, 113, 190 }}

[[Badness]] (Sintel): 0.158

=== Overview to extensions ===
==== Full 7-limit extensions ====
To the gamelisma itself we need to add the comma which appears next on the modified [[Normal lists #Normal interval list|normal comma list]] for the full 7-limit. The second comma on the list for mothra is [[81/80]], for rodan [[245/243]], for guiron [[32805/32768]], for gorgo [[36/35]], and for gidorah [[256/245]]. These all use ~8/7 as a generator, though in the case of gidorah that is the same as ~6/5.

Miracle adds [[33075/32768]] and uses the [[secor]], half an ~8/7, as generator. Lemba adds [[525/512]] to the list, and has a half-octave [[period]]. Valentine adds [[6144/6125]] with a generator of ~21/20 and superkleismic adds [[875/864]] with a generator of ~6/5. Unidec adds [[4375/4374]], and has a generator of ~10/9 with a half-octave period. Hemithirds adds [[65625/65536]] with a generator half of a classical major third. Finally, tritikleismic adds [[15625/15552]] and has a generator of 6/5 with a 1/3-octave period.

Full 7-limit temperaments discussed elsewhere are:
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* ''[[Echidnic]]'' (+686/675} → [[Diaschismic family #Echidnic|Diaschismic family]]
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* [[Trismegistus]] (+3125/3072) → [[Magic family #Trismegistus|Magic family]]
* [[Hemithirds]] (+3136/3125) → [[Hemimean clan #Hemithirds|Hemimean clan]]
* ''[[Gamity]]'' (+1071875/1062882) → [[Amity family #Gamity|Amity family]]
* ''[[Tritikleismic]]'' (+15625/15552) → [[Kleismic family #Tritikleismic|Kleismic family]]
* ''[[Heinz]]'' (+78732/78125) → [[Sensipent family #Heinz|Sensipent family]]
* ''[[Triwell]]'' (+235298/234375) → [[Semicomma family #Triwell|Semicomma family]]
* ''[[Gamelstearn]]'' (+118098/117649) → [[Compton family #Gamelstearn|Compton family]]

The rest are considered below.

==== Subgroup extensions ====
No-five subgroup extensions of slendric include radon, a 2.3.7.11-subgroup extension that may be viewed as no-five rodan, considered below, euslendric, a 2.3.7.13-subgroup extension, and baladic, a weak 2.3.7.13.17-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[No-fives subgroup temperaments #Gigapyth|gigapyth]] in the 2.3.7.85 subgroup.

=== Radon ===
Radon is the no-fives version of [[rodan]], equating the diatonic major third to [[14/11]].

Subgroup: 2.3.7.11

Comma list: 896/891, 1029/1024

Subgroup-val mapping: {{mapping| 1 1 3 6 | 0 3 -1 -13 }}

Gencom mapping: {{mapping| 1 1 0 3 6 | 0 3 0 -1 -13 }}

Optimal tunings:
* WE: ~2 = 1199.9708{{c}}, ~8/7 = 234.3748{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.3813{{c}}

{{Optimal ET sequence|legend=0| 5, …, 36, 41, 87, 128 }}

Badness (Sintel): 0.619

== Mothra ==
{{Main| Mothra }}

Mothra tempers out [[81/80]] and finds the prime 5 at a stack of four fifths as does any temperament in the [[meantone family]]. It also tempers out [[1728/1715]], the orwellisma. It can be described as the {{nowrap| 26 & 31 }}. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. However, a pure mos mothra scale is often described as directionless and has limited chord-building potential<ref>[https://www.youtube.com/watch?v=uH3ahBzDSrs 31-EDO Music Theory: Supermajor Hexatonic Scale] by [[Zhea Erose]]</ref>, so something other than a mos may be used as a scale to get the most out of mothra. There are examples of non-mos mothra scales in 31edo [[Strictly proper 7-tone 31edo scales|in the article on strictly proper 7-tone 31edo scales]].

Note that mothra is also called '''cynder''' in the 7-limit, which can be a little confusing sometimes.

Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]])}, taking advantage of the fact that [[81/80]] is a [[semiparticular]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 81/80, 1029/1024

{{Mapping|legend=1| 1 1 0 3 | 0 3 12 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9303{{c}}, ~8/7 = 232.3733{{c}}
: [[error map]]: {{val| +0.930 -3.905 +2.165 +1.592 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.2514{{c}}
: error map: {{val| 0.000 -5.520 +0.703 -1.077 }}

[[Algebraic generator]]: Rabrindanath, largest real root of ''x''8 - 3''x''2 + 1, or 232.0774 cents.

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 0 0 1/12 }}
: {{monzo list| 1 0 0 0 | 1 0 1/4 0 | 0 0 1 0 | 3 0 -1/12 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

{{Optimal ET sequence|legend=1| 5, 21c, 26, 31 }}

[[Badness]] (Sintel): 0.940

=== Undecimal mothra ===
Undecimal mothra is the extension of 7-limit cynder which tempers out 385/384 as is natural in slendric temperaments. It is the simplest extension, supported within a reasonable tuning range (between [[26edo]] and 31edo), and is supported by the patent val of [[5edo]], which implies that it is better behaved as a cluster temperament. It is also notable for being supported by the just tuning of 8/7, and has a restriction to the 2.7.11 subgroup, namely [[amaranthine]], that is a microtemperament.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98, 385/384

Mapping: {{mapping| 1 1 0 3 5 | 0 3 12 -1 -8 }}

Optimal tunings:
* WE: ~2 = 1201.3979{{c}}, ~8/7 = 232.3010{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.0621{{c}}

{{Optimal ET sequence|legend=0| 5, 26, 31, 88, 119be, 150be }}

Badness (Sintel): 0.848

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

Comma list: 81/80, 99/98, 105/104, 144/143

Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 }}

Optimal tunings:
* WE: ~2 = 1201.0985{{c}}, ~8/7 = 232.0231{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.8425{{c}}

{{Optimal ET sequence|legend=0| 5, 26, 31, 57, 88 }}

Badness (Sintel): 0.990

; Music
* ''Prelude for solo piano'' (2014) by [[Chris Vaisvil]] – [https://web.archive.org/web/20201127013310/http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 play] | [https://www.chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ blog] – in Mothra[16], brat 4 tuning

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 99/98, 105/104, 120/119, 144/143

Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 }}

Optimal tunings:
* WE: ~2 = 1200.9734{{c}}, ~8/7 = 231.8960{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.7392{{c}}

{{Optimal ET sequence|legend=0| 5g, 26, 31, 57, 88 }}

Badness (Sintel): 1.00

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 99/98, 105/104, 120/119, 144/143, 153/152

Mapping: {{mapping| 1 1 0 3 5 1 | 0 3 12 -1 -8 14 16 22 }}

Optimal tunings:
* WE: ~2 = 1200.9663{{c}}, ~8/7 = 231.8393{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.6842{{c}}

{{Optimal ET sequence|legend=0| 26, 31, 57 }}

Badness (Sintel): 1.05

=== Mosura ===
The [[S-expression]]-based comma list of mosura suggests it might be the most natural extension of 7-limit cynder to the 11-limit: {[[1728/1715|S6/S7]], [[1029/1024|S7/S8]], ([[81/80|S6/S8 = S9]]), [[176/175|S8/S10]]}.

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175, 540/539

Mapping: {{mapping| 1 1 0 3 -1 | 0 3 12 -1 23 }}

Optimal tunings:
* WE: ~2 = 1200.7675{{c}}, ~8/7 = 232.5673{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.4567{{c}}

{{Optimal ET sequence|legend=0| 5e, 26e, 31, 129 }}

Badness (Sintel): 1.04

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

Comma list: 81/80, 144/143, 176/175, 196/195

Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 }}

Optimal tunings:
* WE: ~2 = 1199.9347{{c}}, ~8/7 = 232.6275{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6392{{c}}

{{Optimal ET sequence|legend=0| 31, 67, 98 }}

Badness (Sintel): 1.52

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 81/80, 144/143, 176/175, 189/187, 196/195

Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 }}

Optimal tunings:
* WE: ~2 = 1199.7124{{c}}, ~8/7 = 232.6376{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.6917{{c}}

{{Optimal ET sequence|legend=0| 31, 67, 98 }}

Badness (Sintel): 1.53

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 81/80, 96/95, 144/143, 153/152, 176/175, 196/195

Mapping: {{mapping| 1 1 0 3 -1 7 | 0 3 12 -1 23 -17 -15 -9 }}

Optimal tunings:
* WE: ~2 = 1199.4885{{c}}, ~8/7 = 232.6310{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 232.7287{{c}}

{{Optimal ET sequence|legend=0| 31, 67, 98h }}

Badness (Sintel): 1.50

=== Cyndra ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80, 1029/1024

Mapping: {{mapping| 1 1 0 3 0 | 0 3 12 -1 18 }}

Optimal tunings:
* WE: ~2 = 1201.1585{{c}}, ~8/7 = 231.5404{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3850{{c}}

{{Optimal ET sequence|legend=0| 5e, 21ce, 26 }}

Badness (Sintel): 1.84

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

Comma list: 45/44, 78/77, 81/80, 640/637

Mapping: {{mapping| 1 1 0 3 0 1 | 0 3 12 -1 18 14 }}

Optimal tunings:
* WE: ~2 = 1201.1152{{c}}, ~8/7 = 231.5079{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 231.3612{{c}}

{{Optimal ET sequence|legend=0| 5e, 21cef, 26 }}

Badness (Sintel): 1.41

== Rodan ==
{{Main| Rodan }}
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Rodan (5-limit)]].''

Rodan tempers out 245/243 and can be described as the {{nowrap| 41 & 46 }} temperament. This temperament is more accurate than mothra and extends neatly to the 13-limit, though the perfect fifth is sharper than ideal for slendric. [[87edo]] is excellent for this, with the 17\87 generator missing the 13-limit CWE tuning by less than a millicent.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 1029/1024

{{Mapping|legend=1| 1 1 -1 3 | 0 3 17 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2146{{c}}, ~8/7 = 234.4587{{c}}
: [[error map]]: {{val| +0.215 +1.636 -0.731 -2.641 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 234.4259{{c}}
: error map: {{val| 0.000 +1.323 -1.073 -3.252 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 2/9 0 1/18 -1/18 }}
: {{monzo list| 1 0 0 0 | 5/3 0 1/6 -1/6 | 25/9 0 17/18 -17/18 | 25/9 0 -1/18 1/18 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5

[[Algebraic generator]]: larger root of 20''x''2 - 36''x'' + 15, or (9 + √6)/10.

{{Optimal ET sequence|legend=1| 41, 87, 128, 215d }}

[[Badness]] (Sintel): 0.939

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

Comma list: 245/243, 385/384, 441/440

Mapping: {{mapping| 1 1 -1 3 6 | 0 3 17 -1 -13 }}

Optimal tunings:
* WE: ~2 = 1200.0553{{c}}, ~8/7 = 234.4695{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4594{{c}}

Minimax tuning:
* 11-odd-limit: ~8/7 = {{monzo| 4/19 2/19 0 0 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 31/19 6/19 0 0 -3/19 }}, {{monzo| 49/19 34/19 0 0 -17/19 }}, {{monzo| 53/19 -2/19 0 0 1/19 }}, {{monzo| 62/19 -26/19 0 0 13/19 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/9

Algebraic generator: positive root of ''x''2 + 16''x'' - 31, or √95 - 8.

{{Optimal ET sequence|legend=0| 41, 87 }}

Badness (Sintel): 0.763

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

Comma list: 196/195, 245/243, 352/351, 364/363

Mapping: {{mapping| 1 1 -1 3 6 8 | 0 3 17 -1 -13 -22 }}

Optimal tunings:
* WE: ~2 = 1199.9868{{c}}, ~8/7 = 234.4796{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.4822{{c}}

Minimax tuning:
* 13- and 15-odd-limit: ~8/7 = {{monzo| 3/14 1/14 0 0 0 -1/28 }}
: unchanged-interval (eigenmonzo) basis: 2.13/9

Algebraic generator: Gatetone, positive root of 4''x''6 - 7''x'' - 1. Recurrence converges slowly.

{{Optimal ET sequence|legend=0| 41, 46, 87 }}

Badness (Sintel): 0.762

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 154/153, 196/195, 245/243, 256/255, 273/272

Mapping: {{mapping| 1 1 -1 3 6 8 8 | 0 3 17 -1 -13 -22 -20 }}

Optimal tunings:
* WE: ~2 = 1199.8331{{c}}, ~8/7 = 234.4919{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.5254{{c}}

Minimax tuning:
* 17-odd-limit: ~8/7 = {{monzo| 3/13 1/13 0 0 0 0 -1/26 }}
: unchanged-interval (eigenmonzo) basis: 2.17/9

{{Optimal ET sequence|legend=0| 41, 46, 87 }}

Badness (Sintel): 0.853

==== Aerodactyl ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 245/243, 385/384, 441/440

Mapping: {{mapping| 1 1 -1 3 6 -1 | 0 3 17 -1 -13 24 }}

Optimal tunings:
* WE: ~2 = 1200.2997{{c}}, ~8/7 = 234.6972{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.6439{{c}}

{{Optimal ET sequence|legend=0| 5, 41f, 46 }}

Badness (Sintel): 1.40

=== Aerodino ===
Subgroup: 2.3.5.7.11

Comma list: 176/175, 245/243, 1029/1024

Mapping: {{mapping| 1 1 -1 3 -3 | 0 3 17 -1 33 }}

Optimal tunings:
* WE: ~2 = 1199.9179{{c}}, ~8/7 = 234.7123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7256{{c}}

{{Optimal ET sequence|legend=0| 5e, 41e, 46 }}

Badness (Sintel): 1.79

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

Comma list: 91/90, 176/175, 245/243, 847/845

Mapping: {{mapping| 1 1 -1 3 -3 -1 | 0 3 17 -1 33 24 }}

Optimal tunings:
* WE: ~2 = 1200.0242{{c}}, ~8/7 = 234.7863{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.7824{{c}}

{{Optimal ET sequence|legend=0| 5e, 41ef, 46 }}

Badness (Sintel): 1.48

=== Varan ===
Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/243, 1029/1024

Mapping: {{mapping| 1 1 -1 3 -2 | 0 3 17 -1 28 }}

Optimal tunings:
* WE: ~2 = 1200.3738{{c}}, ~8/7 = 234.2174{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.1586{{c}}

{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}

Badness (Sintel): 1.49

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

Comma list: 100/99, 105/104, 245/243, 352/351

Mapping: {{mapping| 1 1 -1 3 -2 0 | 0 3 17 -1 28 19 }}

Optimal tunings:
* WE: ~2 = 1200.1389{{c}}, ~8/7 = 234.1162{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 234.0946{{c}}

{{Optimal ET sequence|legend=0| 5e, 36ce, 41 }}

Badness (Sintel): 1.33

== Guiron ==
Guiron tempers out the [[schisma]], and finds the prime 5 at the diminished fourth as does any temperament in the [[schismatic family]]. It can be described as the {{nowrap| 36 & 41 }} temperament. It is more complex than rodan, but the optimal tuning is closer to optimal slendric.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 10976/10935

{{Mapping|legend=1| 1 1 7 3 | 0 3 -24 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3395{{c}}, ~8/7 = 233.9963{{c}}
: [[error map]]: {{val| +0.340 +0.374 +0.151 -1.804 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 233.9239{{c}}
: error map: {{val| 0.000 -0.183 -0.487 -2.750 }}

[[Minimax tuning]]:
* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: {{monzo list| 1 0 0 0 | 15/8 0 -1/8 0 | 0 0 1 0 | 65/24 0 1/24 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5

{{Optimal ET sequence|legend=1| 36, 41, 77, 118, 277d }}

[[Badness]] (Sintel): 1.20

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

Comma list: 385/384, 441/440, 10976/10935

Mapping: {{mapping| 1 1 7 3 -2 | 0 3 -24 -1 28 }}

Optimal tunings:
* WE: ~2 = 1200.3453{{c}}, ~8/7 = 233.9988{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.9312{{c}}

Minimax tuning:
* 11-odd-limit: ~8/7 = {{monzo| 7/24 0 -1/24 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 15/8 0 -1/8 0 0 }}, {{monzo| 0 0 1 0 0 }}, {{monzo| 65/24 0 1/24 0 0 }}, {{monzo| 37/6 0 -7/6 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.5

{{Optimal ET sequence|legend=0| 36e, 41, 77, 118, 159, 277d }}

Badness (Sintel): 0.881

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

Comma list: 196/195, 352/351, 385/384, 729/728

Mapping: {{mapping| 1 1 7 3 -2 0 | 0 3 -24 -1 28 19 }}

Optimal tunings:
* WE: ~2 = 1200.1222{{c}}, ~8/7 = 233.9228{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.8994{{c}}

{{Optimal ET sequence|legend=0| 36e, 41, 77, 118 }}

Badness (Sintel): 1.18

== Gorgo ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #Laconic]].''
{{See also| Llywelynsmic clan }}

Gorgo tempers the generator of ~8/7 together with ~10/9. It can be described as the {{nowrap| 16 & 21 }} temperament.

If we discard the inaccurate mapping of prime 3, we get [[shoe]], so that the large commas of gorgo are explained practically entirely by the inaccurate 3.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 36/35, 1029/1024

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

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.9847{{c}}, ~8/7 = 228.5210{{c}}
: [[error map]]: {{val| +0.985 -15.407 +14.318 +5.607 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 228.4371{{c}}
: error map: {{val| 0.000 -16.644 +12.746 +2.737 }}

{{Optimal ET sequence|legend=1| 5, 11c, 16, 21 }}

[[Badness]] (Sintel): 1.54

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

Comma list: 36/35, 45/44, 1029/1024

Mapping: {{mapping| 1 1 1 3 1 | 0 3 7 -1 13 }}

Optimal tunings:
* WE: ~2 = 1201.3609{{c}}, ~8/7 = 227.6312{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.4955{{c}}

{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}

Badness (Sintel): 1.64

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

Comma list: 27/26, 36/35, 45/44, 507/500

Mapping: {{mapping| 1 1 1 3 1 2 | 0 3 7 -1 13 9 }}

Optimal tunings:
* WE: ~2 = 1201.0996{{c}}, ~8/7 = 227.4378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 227.3327{{c}}

{{Optimal ET sequence|legend=0| 5e, 16, 21, 37b }}

Badness (Sintel): 1.35

=== Spartan ===
Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 1029/1024

Mapping: {{mapping| 1 1 1 3 5 | 0 3 7 -1 -8 }}

Optimal tunings:
* WE: ~2 = 1198.9344{{c}}, ~8/7 = 229.3316{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.5124{{c}}

{{Optimal ET sequence|legend=0| 5, 16e, 21 }}

Badness (Sintel): 2.07

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

Comma list: 27/26, 36/35, 56/55, 507/500

Mapping: {{mapping| 1 1 1 3 5 2 | 0 3 7 -1 -8 9 }}

Optimal tunings:
* WE: ~2 = 1198.3002{{c}}, ~8/7 = 228.7341{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 229.0044{{c}}

{{Optimal ET sequence|legend=0| 5, 16e, 21 }}

Badness (Sintel): 1.95

; Music
* [https://web.archive.org/web/20201127012514/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Herman/gorgo-example.mp3 ''Gorgo Example''] by [[Herman Miller]]

== Gidorah ==
: ''For the 5-limit version, see [[Syntonic–diatonic equivalence continuum #University]].''

Gidorah is a very low-accuracy temperament where the generator of ~8/7 is lumped together with ~6/5. 16c-, 21cc-, and 26ccc-edo are among the possible tunings.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 21/20, 144/125

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

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1192.4932{{c}}, ~8/7 = 229.3187{{c}}
: [[error map]]: {{val| -7.507 -21.506 +57.310 -20.665 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 229.6649{{c}}
: error map: {{val| 0.000 -12.960 +73.016 +1.509 }}

{{Optimal ET sequence|legend=1| 1b, 5 }}

[[Badness]] (Sintel): 1.58

== Oncle ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Oncle]].''

Oncle can be described as the {{nowrap| 31 & 36c }} temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 2430/2401

{{Mapping|legend=1| 1 1 6 3 | 0 3 -19 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.2246{{c}}, ~8/7 = 232.7354{{c}}
: [[error map]]: {{val| +1.225 -2.524 -0.939 +2.112 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 232.4718{{c}}
: error map: {{val| 0.000 -4.539 -3.279 -1.298 }}

{{Optimal ET sequence|legend=1| 31, 98c, 129c, 160bc }}

[[Badness]] (Sintel): 2.24

== Archaeotherium ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Archaeotherium]].''

Archaeotherium can be described as the {{nowrap| 21 & 26 }} temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 405/392, 1029/1024

{{Mapping|legend=1| 1 1 5 3 | 0 3 -14 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.7179{{c}}, ~8/7 = 230.7800{{c}}
: [[error map]]: {{val| +2.718 -6.897 -3.644 +8.548 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 230.1909{{c}}
: error map: {{val| 0.000 -11.382 -8.986 +0.983 }}

{{Optimal ET sequence|legend=1| 21, 26, 47, 73bc }}

[[Badness]] (Sintel): 3.70

== Clyndro ==
Clyndro tempers out [[135/128]] and finds the interval class of 5 at a stack of -3 fifths as does any temperament in the [[mavila family]]. It can be described as the {{nowrap| 11 & 16 }} temperament.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 135/128, 360/343

{{Mapping|legend=1| 1 1 4 3 | 0 3 -9 -1 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1205.6135{{c}}, ~8/7 = 227.5283{{c}}
: [[error map]]: {{val| +5.613 -13.757 -11.614 +20.486 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/7 = 226.3207{{c}}
: error map: {{val| 0.000 -22.993 -23.200 +4.853 }}

{{Optimal ET sequence|legend=1| 5c, 11, 16 }}

[[Badness]] (Sintel): 4.03

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

Comma list: 33/32, 45/44, 352/343

Mapping: {{mapping| 1 1 4 3 4 | 0 3 -9 -1 -3 }}

Optimal tunings:
* WE: ~2 = 1206.2134{{c}}, ~8/7 = 227.6004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 226.2421{{c}}

{{Optimal ET sequence|legend=0| 5c, 11, 16 }}

Badness (Sintel): 2.30

== Miracle ==
{{Main| Miracle }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Ampersand]].''

Miracle is one of the most important entries of this temperament clan. It tempers out [[225/224]], splitting the ~8/7 generator of slendric into 15/14~16/15, and can be described as the {{nowrap| 31 & 41 }} temperament. Its ploidacot is hexacot. It is then extremely natural to equate the neutral third, three generators up, to [[11/9]] and thereby extend miracle to the full [[11-limit]] with essentially no further damage. [[72edo]] makes for an excellent tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 1029/1024

{{Mapping|legend=1| 1 1 3 3 | 0 6 -7 -2 }}
: mapping generator: ~2, ~15/14

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.8209{{c}}, ~15/14 = 116.7550{{c}}
: [[error map]]: {{val| +0.821 -0.604 -1.136 +0.127 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 116.6756{{c}}
: error map: {{val| 0.000 -1.901 -3.043 -2.177 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~15/14 = {{monzo| 2/13 1/13 -1/13 }}
: {{monzo list| 1 0 0 0 | 25/13 6/13 -6/13 0 | 25/13 -7/13 7/13 0 | 35/13 -2/13 2/13 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.5/3
* [[9-odd-limit]]: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
: {{monzo list| 1 0 0 0 | 25/19 12/19 -6/19 0 | 50/19 -14/19 7/19 0 | 55/19 -4/19 2/19 0 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5

[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~15/14 = [114.286, 120.000] (2\21 to 1\10)
* 9-odd-limit diamond monotone: ~15/14 = [116.129, 120.000] (3\31 to 1\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~15/14 = [115.587, 116.993]

[[Algebraic generator]]: Secor59, positive root of 15''x''6 - 8''x''4 - 12

{{Optimal ET sequence|legend=1| 10, 21, 31, 41, 72 }}

[[Badness]] (Sintel): 0.424

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

Comma list: 225/224, 243/242, 385/384

Mapping: {{mapping| 1 1 3 3 2 | 0 6 -7 -2 15 }}

Optimal tunings:
* WE: ~2 = 1200.7626{{c}}, ~15/14 = 116.7069{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.6469{{c}}

Minimax tuning:
* 11-odd-limit: ~15/14 = {{monzo| 1/19 2/19 -1/19 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 25/19 12/19 -6/19 0 0 }}, {{monzo| 50/19 -14/19 7/19 0 0 }}, {{monzo| 55/19 -4/19 2/19 0 0 }}, {{monzo| 53/19 30/19 -15/19 0 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/5

Tuning ranges:
* 11-odd-limit diamond monotone: ~15/14 = [116.129, 117.073] (3\31 to 4\41)
* 11-odd-limit diamond tradeoff: ~15/14 = [115.587, 116.993]

Algebraic generator: Secor59

{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72, 247c, 319bcde, 391bcde, 463bccde }}

Badness (Sintel): 0.353

==== Miraculous ====
Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 144/143, 196/195, 243/242

Mapping: {{mapping| 1 1 3 3 2 4 | 0 6 -7 -2 15 -3 }}

Optimal tunings:
* WE: ~2 = 1200.1267{{c}}, ~15/14 = 116.7596{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7488{{c}}

{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72f }}

Badness (Sintel): 0.771

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 105/104, 120/119, 144/143, 154/153, 170/169

Mapping: {{mapping| 1 1 3 3 2 4 4 | 0 6 -7 -2 15 -3 1 }}

Optimal tunings:
* WE: ~2 = 1199.6759{{c}}, ~15/14 = 116.7378{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7657{{c}}

{{Optimal ET sequence|legend=0| 10, 21e, 31, 41, 72fg }}

Badness (Sintel): 0.870

==== Benediction ====
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 351/350, 385/384

Mapping: {{mapping| 1 1 3 3 2 7 | 0 6 -7 -2 15 -34 }}

Optimal tunings:
* WE: ~2 = 1199.8601{{c}}, ~15/14 = 116.6572{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5688{{c}}

{{Optimal ET sequence|legend=0| 31, 72, 103, 175f }}

Badness (Sintel): 0.649

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 351/350, 375/374

Mapping: {{mapping| 1 1 3 3 2 7 7 | 0 6 -7 -2 15 -34 -30 }}

Optimal tunings:
* WE: ~2 = 1200.8328{{c}}, ~15/14 = 116.6661{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.5774{{c}}

{{Optimal ET sequence|legend=0| 31, 72, 103, 175f, 422bcdefffg }}

Badness (Sintel): 0.639

==== Manna ====
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 325/324, 385/384

Mapping: {{mapping| 1 1 3 3 2 0 | 0 6 -7 -2 15 38 }}

Optimal tunings:
* WE: ~2 = 1200.7564{{c}}, ~15/14 = 116.8129{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7528{{c}}

{{Optimal ET sequence|legend=0| 31f, 41, 72, 185cf, 257cff }}

Badness (Sintel): 0.703

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 325/324, 385/384

Mapping: {{mapping| 1 1 3 3 2 0 0 | 0 6 -7 -2 15 38 42 }}

Optimal tunings:
* WE: ~2 = 1200.7570{{c}}, ~15/14 = 116.8011{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.7408{{c}}

{{Optimal ET sequence|legend=0| 31fg, 41, 72, 185cf, 257cff }}

Badness (Sintel): 0.748

==== Semimiracle ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 243/242, 385/384

Mapping: {{mapping| 2 2 6 6 4 7 | 0 6 -7 -2 15 2 }}
: mapping generators: ~55/39, ~15/14

Optimal tunings:
* WE: ~55/39 = 600.4844{{c}}, ~15/14 = 116.7182{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~15/14 = 116.6413{{c}}

{{Optimal ET sequence|legend=0| 10, 62, 72 }}

Badness (Sintel): 1.02

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 225/224, 243/242, 273/272

Mapping: {{mapping| 2 2 6 6 4 7 7 | 0 6 -7 -2 15 2 6 }}

Optimal tunings:
* WE: ~17/12 = 600.5042{{c}}, ~15/14 = 116.7264{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~15/14 = 116.6485{{c}}

{{Optimal ET sequence|legend=0| 10, 62, 72 }}

Badness (Sintel): 0.822

==== Hemisecordite ====
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 847/845

Mapping: {{mapping| 1 1 3 3 2 2 | 0 12 -14 -4 30 35 }}
: mapping generators: ~2, ~27/26

Optimal tunings:
* WE: ~2 = 1200.6969{{c}}, ~27/26 = 58.3217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2964{{c}}

{{Optimal ET sequence|legend=0| 41, 62, 103, 247c, 350bcde }}

Badness (Sintel): 1.06

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 385/384, 847/845

Mapping: {{mapping| 1 1 3 3 2 2 2 | 0 12 -14 -4 30 35 43 }}

Optimal tunings:
* WE: ~2 = 1200.6557{{c}}, ~27/26 = 58.2932{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/26 = 58.2702{{c}}

{{Optimal ET sequence|legend=0| 41, 62, 103 }}

Badness (Sintel): 1.15

===== Semihemisecordite =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 289/288, 385/384, 847/845

Mapping: {{mapping| 2 2 6 6 4 4 7 | 0 12 -14 -4 30 35 12 }}
: mapping generators: ~17/12, ~27/26

Optimal tunings:
* WE: ~17/12 = 600.3951{{c}}, ~27/26 = 58.3260{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2974{{c}}

{{Optimal ET sequence|legend=0| 62, 144g, 206begg }}

Badness (Sintel): 2.39

====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 209/208, 225/224, 243/242, 289/288, 361/360, 385/384

Mapping: {{mapping| 2 2 6 6 4 4 7 8 | 0 12 -14 -4 30 35 12 5 }}

Optimal tunings:
* WE: ~17/12 = 600.4418{{c}}, ~27/26 = 58.3255{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2928{{c}}

{{Optimal ET sequence|legend=0| 62, 144gh, 206begghh }}

Badness (Sintel): 2.13

====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 209/208, 225/224, 243/242, 289/288, 323/322, 361/360, 385/384

Mapping: {{mapping| 2 2 6 6 4 4 7 8 7 | 0 12 -14 -4 30 35 12 5 21 }}

Optimal tunings:
* WE: ~17/12 = 600.4451{{c}}, ~27/26 = 58.3264{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~27/26 = 58.2942{{c}}

{{Optimal ET sequence|legend=0| 62, 144gh, 206begghhi }}

Badness (Sintel): 1.89

==== Phicordial ====
Subgroup: 2.3.5.7.11.13

Comma list: 225/224, 243/242, 385/384, 2200/2197

Mapping: {{mapping| 1 -11 17 7 -28 3 | 0 18 -21 -6 45 1 }}
: mapping generators: ~2, ~13/8

Optimal tunings:
* WE: ~2 = 1200.7056{{c}}, ~13/8 = 839.3726{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8831{{c}}

{{Optimal ET sequence|legend=0| 103, 216c, 319bcde, 535bccdef }}

Badness (Sintel): 1.37

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 225/224, 243/242, 273/272, 441/440, 2200/2197

Mapping: {{mapping| 1 -11 17 7 -28 3 -5 | 0 18 -21 -6 45 1 13 }}

Optimal tunings:
* WE: ~2 = 1200.5918{{c}}, ~13/8 = 839.2912{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 838.8809{{c}}

{{Optimal ET sequence|legend=0| 103, 216c, 319bcde }}

Badness (Sintel): 1.26

=== Revelation ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 1029/1024

Mapping: {{mapping| 1 1 3 3 5 | 0 6 -7 -2 -16 }}

Optimal tunings:
* WE: ~2 = 1201.3320{{c}}, ~15/14 = 116.4057{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2524{{c}}

{{Optimal ET sequence|legend=0| 10e, 21, 31 }}

Badness (Sintel): 1.09

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

Comma list: 66/65, 99/98, 105/104, 512/507

Mapping: {{mapping| 1 1 3 3 5 4 | 0 6 -7 -2 -16 -3 }}

Optimal tunings:
* WE: ~2 = 1200.6059{{c}}, ~15/14 = 116.3263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 116.2564{{c}}

{{Optimal ET sequence|legend=0| 10e, 21, 31 }}

Badness (Sintel): 1.22

=== Hemimiracle ===
Subgroup: 2.3.5.7.11

Comma list: 225/224, 245/242, 1029/1024

Mapping: {{mapping| 1 1 3 3 4 | 0 12 -14 -4 -11 }}
: mapping generators: ~2, ~33/32

Optimal tunings:
* WE: ~2 = 1200.2902{{c}}, ~33/32 = 58.4217{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4062{{c}}

{{Optimal ET sequence|legend=0| 20, 21, 41 }}

Badness (Sintel): 1.96

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

Comma list: 105/104, 196/195, 245/242, 512/507

Mapping: {{mapping| 1 1 3 3 4 4 | 0 12 -14 -4 -11 -6 }}

Optimal tunings:
* WE: ~2 = 1199.8454{{c}}, ~33/32 = 58.4220{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 58.4305{{c}}

{{Optimal ET sequence|legend=0| 20, 21, 41 }}

Badness (Sintel): 1.78

=== Oracle ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 225/224, 1029/1024

Mapping: {{mapping| 1 -5 10 5 4 | 0 12 -14 -4 -1 }}
: mapping generators: ~2, ~16/11

Optimal tunings:
* WE: ~2 = 1201.2122{{c}}, ~16/11 = 658.9974{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 658.3320{{c}}

{{Optimal ET sequence|legend=0| 11, 20, 31, 82e, 113e, 144ee }}

Badness (Sintel): 1.41

== Hemiseven ==
Unlike miracle which splits 8/7, hemiseven splits ~16/7, an octave above. It can be described as the {{nowrap| 72 & 77 }} temperament; its ploidacot is gamma-hexacot. [[149edo]] is an obvious tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 19683/19600

{{Mapping|legend=1| 1 -2 -15 4 | 0 6 29 -2 }}
: mapping generators: ~2, ~243/160

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5612{{c}}, ~243/160 = 717.0687{{c}}
: [[error map]]: {{val| +0.561 -0.665 +0.260 -0.718 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7478{{c}}
: error map: {{val| 0.000 -1.468 -0.629 -2.321 }}

{{Optimal ET sequence|legend=1| 72, 149, 221, 514bd, 735bcdd }}

[[Badness]] (Sintel): 1.43

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

Comma list: 385/384, 441/440, 19683/19600

Mapping: {{mapping| 1 -2 -15 4 16 | 0 6 29 -2 -21 }}

Optimal tunings:
* WE: ~2 = 1200.6243{{c}}, ~243/160 = 717.0969{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~243/160 = 716.7292{{c}}

{{Optimal ET sequence|legend=0| 72, 149, 221e, 293de }}

Badness (Sintel): 0.941

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

Comma list: 351/350, 385/384, 441/440, 676/675

Mapping: {{mapping| 1 -2 -15 4 16 -19 | 0 6 29 -2 -21 38 }}

Optimal tunings:
* WE: ~2 = 1200.6781{{c}}, ~91/60 = 717.1496{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~91/60 = 716.7520{{c}}

{{Optimal ET sequence|legend=0| 72, 149, 221ef }}

Badness (Sintel): 0.905

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 351/350, 385/384, 441/440, 676/675

Mapping: {{mapping| 1 -2 -15 4 16 -19 -21 | 0 6 29 -2 -21 38 42 }}

Optimal tunings:
* WE: ~2 = 1200.6635{{c}}, ~68/45 = 717.1354{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~68/45 = 716.7472{{c}}

{{Optimal ET sequence|legend=0| 72, 149, 221ef }}

Badness (Sintel): 0.800

== Valentine ==
{{Main| Valentine }}
: ''For the 5-limit version, see [[Syntonic–31 equivalence continuum #Valentine (5-limit)]].''

Valentine tempers out [[126/125]] and [[6144/6125]] as well as 1029/1024. It has a generator of [[~]][[21/20]], three of which make the slendric generator ~8/7. 21/20 can be stripped of its 2 and taken as 3 × 7/5. In this respect it resembles miracle, with a generator of 3 × 5/7, and casablanca, with a generator of 5 × 7/3. These three generators are the simplest in terms of the relationship of tetrads in the [[7-limit symmetrical lattices|lattice of 7-limit tetrads]]. Valentine can be described as the {{nowrap| 31 & 46 }} temperament; its ploidacot is enneacot. [[77edo]], [[108edo]], or [[185edo]] make for excellent tunings, which also happen to be excellent tunings for [[starling]], the rank-3 temperament tempering out 126/125. Hence 7-limit valentine can be used whenever starling is wanted, with the extra tempering out of 1029/1024 having no discernible effect on tuning accuracy. Another tuning for valentine uses (3/2)1/9 as a generator, giving pure 3/2 fifths. Valentine extends naturally to the 11-limit, tempering out 121/120 and 441/440; 46edo has a valentine generator 3\46 which is only 0.0117 cents sharp of the minimax generator, (11/7)1/10.

Valentine has a very straighforward [[S-expression]]-based comma list in the [[11-limit]] add-23 (i.e. the 2.3.5.7.11.23 subgroup) of {([[176/175|S8/S10 = S22 × S23 × S24]], [[121/120|S11]]), [[441/440|S21]], [[484/483|S22]], [[529/528|S23]], [[576/575|S24]]}, so it is the temperament that equalizes the 20::25 segment of the harmonic series.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 1029/1024

{{Mapping|legend=1| 1 1 2 3 | 0 9 5 -3 }}
: mapping generators: ~2, ~21/20

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0749{{c}}, ~21/20 = 77.8687{{c}}
: [[error map]]: {{val| +0.075 -1.062 +3.179 -2.207 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 77.8673{{c}}
: error map: {{val| 0.000 -1.149 +3.023 -2.428 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~21/20 = {{monzo| 1/6 1/12 0 -1/12 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 17/6 5/12 0 -5/12 | 5/2 -1/4 0 1/4 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~21/20 = {{monzo| 1/21 2/21 0 -1/21}}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 47/21 10/21 0 -5/21 | 20/7 -2/7 0 1/7 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

[[Algebraic generator]]: smaller root of ''x''2 - 89''x'' + 92, or (89 - sqrt (7553))/2, at 77.8616 cents.

{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 185 }}

[[Badness]] (Sintel): 0.786

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

Comma list: 121/120, 126/125, 176/175

Mapping: {{mapping| 1 1 2 3 3 | 0 9 5 -3 7 }}

Optimal tunings:
* WE: ~2 = 1200.3890{{c}}, ~22/21 = 77.9065{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9007{{c}}

Minimax tuning:
* 11-odd-limit: ~21/20 = {{monzo| 0 0 0 -1/10 1/10 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 1 0 0 -9/10 9/10 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 3 0 0 3/10 -3/10 }}, {{monzo| 3 0 0 -7/10 7/10 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7

Algebraic generator: positive root of 4''x''3 + 15''x''2 - 21, or else Gontrand2, the smallest positive root of 4''x''7 - 8''x''6 + 5.

{{Optimal ET sequence|legend=0| 15, 31, 46, 77 }}

Badness (Sintel): 0.552

==== Valentino ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 196/195

Mapping: {{mapping| 1 1 2 3 3 5 | 0 9 5 -3 7 -20 }}

Optimal tunings:
* WE: ~2 = 1200.1967{{c}}, ~22/21 = 77.9708{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.9594{{c}}

{{Optimal ET sequence|legend=0| 15f, 31, 46, 77 }}

Badness (Sintel): 0.854

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 126/125, 154/153, 176/175, 196/195

Mapping: {{mapping| 1 1 2 3 3 5 5 | 0 9 5 -3 7 -20 -14 }}

Optimal tunings:
* WE: ~2 = 1200.0404{{c}}, ~22/21 = 78.0055{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.0029{{c}}

{{Optimal ET sequence|legend=0| 15f, 31, 46, 77, 123e }}

Badness (Sintel): 0.854

==== Lupercalia ====
Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 126/125

Mapping: {{mapping| 1 1 2 3 3 3 | 0 9 5 -3 7 11 }}

Optimal tunings:
* WE: ~2 = 1199.9143{{c}}, ~22/21 = 77.7039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 77.7049{{c}}

{{Optimal ET sequence|legend=0| 15, 31 }}

Badness (Sintel): 0.881

==== Dwynwen ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 126/125, 176/175

Mapping: {{mapping| 1 1 2 3 3 2 | 0 9 5 -3 7 26 }}

Optimal tunings:
* WE: ~2 = 1200.1306{{c}}, ~22/21 = 78.2273{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/21 = 78.2241{{c}}

{{Optimal ET sequence|legend=0| 15, 31f, 46 }}

Badness (Sintel): 0.969

==== Semivalentine ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 169/168, 176/175

Mapping: {{mapping| 2 2 4 6 6 7 | 0 9 5 -3 7 3 }}
: mapping generators: ~55/39, ~22/21

Optimal tunings:
* WE: ~55/39 = 600.3497{{c}}, ~22/21 = 77.8845{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~22/21 = 77.8715{{c}}

{{Optimal ET sequence|legend=0| 16, 30, 46, 62, 108ef }}

Badness (Sintel): 1.35

==== Hemivalentine ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 343/338

Mapping: {{mapping| 1 1 2 3 3 4 | 0 18 10 -6 14 -9 }}
: mapping generators: ~2, ~40/39

Optimal tunings:
* WE: ~2 = 1199.6529{{c}}, ~40/39 = 39.0323{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~40/39 = 39.0383{{c}}

{{Optimal ET sequence|legend=0| 30, 31, 61, 92f }}

Badness (Sintel): 1.94

==== Demivalentine ====
Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 126/125, 176/175, 676/675

Mapping: {{mapping| 1 -8 -3 6 -4 -16 | 0 18 10 -6 14 37 }}
: mapping generators: ~2, ~13/9

Optimal tunings:
* WE: ~2 = 1200.3929{{c}}, ~13/9 = 639.1320{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/9 = 638.9325{{c}}

{{Optimal ET sequence|legend=0| 15, 47ef, 62, 77 }}

Badness (Sintel): 1.44

=== Hemivalentino ===
Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 1029/1024

Mapping: {{mapping| 1 1 2 3 2 | 0 18 10 -6 45 }}

Optimal tunings:
* WE: ~2 = 1200.0816{{c}}, ~45/44 = 38.9236{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9228{{c}}

{{Optimal ET sequence|legend=0| 31, 92e, 123, 154, 185 }}

Badness (Sintel): 2.03

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

Comma list: 126/125, 196/195, 243/242, 1029/1024

Mapping: {{mapping| 1 1 2 3 2 5 | 0 18 10 -6 45 -40 }}

Optimal tunings:
* WE: ~2 = 1199.8782{{c}}, ~45/44 = 38.9440{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9472{{c}}

{{Optimal ET sequence|legend=0| 31, 123, 154 }}

Badness (Sintel): 2.39

==== Hemivalentoid ====
Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 243/242, 343/338

Mapping: {{mapping| 1 1 2 3 2 4 | 0 18 10 -6 45 -9 }}

Optimal tunings:
* WE: ~2 = 1199.3614{{c}}, ~45/44 = 38.9721{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~45/44 = 38.9839{{c}}

{{Optimal ET sequence|legend=0| 31, 92ef }}

Badness (Sintel): 2.39

== Superkleismic ==
{{Main| Superkleismic }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Shibboleth]].''

Superkleismic tempers out the keema, [[875/864]], and can be described as the {{nowrap| 15 & 26 }} temperament. It splits the ~7/4 into three ~6/5 generators of around 322 cents. This is noticeably sharper than the [[kleismic]] generator, hence the name.

In the 11-limit, two generator steps can be identified with ~16/11, and in the 13-limit, the same step can be treated as ~13/9. The [[S-expression]]-based comma list of 13-limit superkleismic is {[[875/864|S5/S6]], [[1029/1024|S7/S8]], [[100/99|S10]], [[144/143|S12]], ([[441/440|S21]])}. Through careful observation of the equivalences therein one can derive the mapping of the full 13-limit.

Note that the generator is given as 6/5's octave complement, [[5/3]], in the data that follow, since a stack of 9 such generators octave-reduced is the perfect fifth; the [[ploidacot]] of superkleismic is wau-enneacot.

Superkleismic also sets two intervals of [[21/20]] equal to [[10/9]]; as {{nowrap| 10/9 {{=}} ([[20/19]])⋅([[19/18]]) }}, we can identify 21/20, 20/19, and 19/18 together to add prime 19, tempering out [[361/360]] ({{S|19}}) and [[400/399]] ({{S|20}}). This structure is preserved within the entire superkleismic tuning range between 15edo and 26edo, while extensions for primes 13 and 17 bifurcate and are of higher complexity and lower accuracy.

41edo gives an obvious tuning in all the subgroups.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 1029/1024

{{Mapping|legend=1| 1 -5 -5 5 | 0 9 10 -3 }}
: mapping generators: ~2, ~5/3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.7640{{c}}, ~5/3 = 878.6289{{c}}
: [[error map]]: {{val| +0.764 +1.885 +3.844 -0.893 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 878.1077{{c}}
: error map: {{val| 0.000 +1.014 -5.237 -3.149 }}

{{Optimal ET sequence|legend=1| 11c, 15, 26, 41 }}

[[Badness]] (Sintel): 1.21

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

Comma list: 100/99, 245/242, 385/384

Mapping: {{mapping| 1 -5 -5 5 2 | 0 9 10 -3 2 }}

Optimal tunings:
* WE: ~2 = 1200.1691{{c}}, ~5/3 = 878.2772{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1606{{c}}

{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 179cde, 220cde, 261ccdee }}

Badness (Sintel): 0.848

==== 2.3.5.7.11.19 subgroup ====
Subgroup: 2.3.5.7.11.19

Comma list: 100/99, 133/132, 190/189, 385/384

Mapping: {{mapping| 1 -5 -5 5 2 -6 | 0 9 10 -3 2 14 }}

Optimal tunings:
* WE: ~2 = 1200.2289{{c}}, ~5/3 = 878.3409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.1840{{c}}

{{Optimal ET sequence|legend=0| 11c, 15, 26, 41, 138e }}

Badness (Sintel): 0.692

=== 13-limit ===
Superkleismic in the 13-limit does considerably more damage than in the 11-limit, as indicated by being supported by much fewer [[patent val]]s and having higher Dirichlet badness than its 11-limit counterpart. However, this remains an obvious canonical mapping for prime 13.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 105/104, 144/143, 245/242

Mapping: {{mapping| 1 -5 -5 5 2 -8 | 0 9 10 -3 2 16 }}

Optimal tunings:
* WE: ~2 = 1200.0261{{c}}, ~5/3 = 878.0252{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.0073{{c}}

{{Optimal ET sequence|legend=0| 11cf, 15, 26, 41 }}

Badness (Sintel): 0.887

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 120/119, 144/143, 245/242

Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 | 0 9 10 -3 2 16 22 }}

Optimal tunings:
* WE: ~2 = 1200.0488{{c}}, ~5/3 = 877.8872{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8537{{c}}

{{Optimal ET sequence|legend=0| 11cfg, 15g, 26, 41 }}

Badness (Sintel): 1.01

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 105/104, 120/119, 144/143, 133/132, 190/189

Mapping: {{mapping| 1 -5 -5 5 2 -8 -12 -6 | 0 9 10 -3 2 16 22 14 }}

Optimal tunings:
* WE: ~2 = 1200.2120{{c}}, ~5/3 = 878.0243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 877.8789{{c}}

{{Optimal ET sequence|legend=0| 11cfgh, 15g, 26, 41 }}

Badness (Sintel): 0.964

=== Superana ===
This extension ({{nowrap| 41 & 56 }}) is the counterpart of canonical superkleismic on the other side of 41edo.

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 245/242, 385/384

Mapping: {{mapping| 1 -5 -5 5 2 22 | 0 9 10 -3 2 -25 }}

Optimal tunings:
* WE: ~2 = 1199.8272{{c}}, ~5/3 = 878.1538{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.2795{{c}}

{{Optimal ET sequence|legend=0| 15f, 41, 97, 138e }}

Badness (Sintel): 1.40

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 154/153, 196/195, 245/242, 256/255

Mapping: {{mapping| 1 -5 -5 5 2 22 18 | 0 9 10 -3 2 -25 -19 }}

Optimal tunings:
* WE: ~2 = 1199.5964{{c}}, ~5/3 = 878.0482{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3444{{c}}

{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}

Badness (Sintel): 1.45

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 100/99, 133/132, 154/153, 190/189, 196/195, 256/255

Mapping: {{mapping| 1 -5 -5 5 2 22 18 -6 | 0 9 10 -3 2 -25 -19 14 }}

Optimal tunings:
* WE: ~2 = 1199.6638{{c}}, ~5/3 = 878.1109{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 878.3566{{c}}

{{Optimal ET sequence|legend=0| 15f, 41, 56, 97g }}

Badness (Sintel): 1.36

== Dee leap week ==
{{Main| Dee leap week }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 2460375/2458624

{{Mapping|legend=1| 1 -5 25 5 | 0 9 -31 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4835{{c}}, ~224/135 = 878.2507{{c}}
: [[error map]]: {{val| +0.484 -0.117 +0.004 -1.160 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~224/135 = 877.8926{{c}}
: error map: {{val| 0.000 -0.921 -0.985 -2.504 }}

{{Optimal ET sequence|legend=1| 41, 108, 149, 190 }}

[[Badness]] (Sintel): 2.12

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

Comma list: 385/384, 441/440, 2460375/2458624

Mapping: {{mapping| 1 -5 25 5 -28 | 0 9 -31 -3 43 }}

Optimal tunings:
* WE: ~2 = 1200.4874{{c}}, ~224/135 = 878.2543{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~224/135 = 877.8987{{c}}

{{Optimal ET sequence|legend=0| 41, 108e, 149, 190 }}

Badness (Sintel): 1.35

== Unidec ==
{{Main| Unidec }}

Unidec tempers out the ragisma, [[4375/4374]], and may be described as the {{nowrap| 26 & 46 }} temperament. It has a [[semi-octave]] [[period]] and a generator of ~80/63, two of which minus a period make slendric's generator; its [[ploidacot]] is therefore diploid gamma-hexacot. In the 11-limit, the generator represents [[14/11]]. [[190edo]] makes for an excellent tuning in both the 7-limit and 11-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 4375/4374

{{Mapping|legend=1| 2 -1 -3 7 | 0 6 11 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~1225/864 = 600.2429{{c}}, ~80/63 = 417.0073{{c}}
: [[error map]]: {{val| +0.486 -0.154 +0.038 -1.140 }}
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~80/63 = 416.8688{{c}}
: error map: {{val| 0.000 -0.924 -1.090 -2.503 }}

[[Minimax tuning]]:
* [[7-odd-limit]]: ~10/9 = {{monzo| 3/26 0 -1/13 1/13 }}
: {{monzo list| 1 0 0 0 | 47/26 0 6/13 -6/13 | 71/26 0 11/13 -11/13 | 71/26 0 -2/13 2/13 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* [[9-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
: {{Monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 57/28 11/7 0 -11/14 | 20/7 -2/7 0 1/7 }}
: [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7

{{Optimal ET sequence|legend=1| 26, 46, 72, 118, 190 }}

[[Badness]] (Sintel): 0.972

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

Comma list: 385/384, 441/440, 4375/4374

Mapping: {{mapping| 2 -1 -3 7 9 | 0 6 11 -2 -3 }}

Optimal tunings:
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}

Minimax tuning:
* [[11-odd-limit]]: ~10/9 = {{monzo| 5/28 -1/7 0 1/14 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 10/7 6/7 0 -3/7 0 }}, {{monzo| 57/28 11/7 0 -11/14 0 }}, {{monzo| 20/7 -2/7 0 1/7 0 }}, {{monzo| 99/28 -3/7 0 3/14 0 }}]
: unchanged-interval (eigenmonzo) basis: 2.9/7

{{Optimal ET sequence|legend=0| 26, 46, 72, 118, 190 }}

Badness (Sintel): 0.512

==== Ekadash ====
Subgroup: 2.3.5.7.11.13

Comma list: 385/384, 441/440, 625/624, 729/728

Mapping: {{mapping| 2 -1 -3 7 9 -19 | 0 6 11 -2 -3 38 }}

Optimal tunings:
* WE: ~99/70 = 600.2497{{c}}, ~14/11 = 417.0085{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~14/11 = 416.8543{{c}}

{{Optimal ET sequence|legend=0| 46f, 72, 118, 190, 262df, 452cdef }}

Badness (Sintel): 0.842

==== Hendec ====
Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 385/384

Mapping: {{mapping| 2 -1 -3 7 9 6 | 0 6 11 -2 -3 2 }}

Optimal tunings:
* WE: ~91/64 = 600.3825{{c}}, ~14/11 = 417.0678{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~14/11 = 416.8290{{c}}

{{Optimal ET sequence|legend=0| 26, 46, 72, 190ff }}

Badness (Sintel): 0.732

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 273/272, 325/324, 364/363

Mapping: {{mapping| 2 -1 -3 7 9 6 4 | 0 6 11 -2 -3 2 6 }}

Optimal tunings:
* WE: ~17/12 = 600.3991{{c}}, ~14/11 = 417.0809{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~14/11 = 416.8330{{c}}

{{Optimal ET sequence|legend=0| 26, 46, 72, 190ffg }}

Badness (Sintel): 0.595

== Necromanteion ==
Necromanteion, named by [[Johannes Werpup]] in 2014<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106371.html Yahoo! Tuning Group | ''Temperament ideas: A cuckoo, and two oracles'']</ref> may be described as the {{nowrap| 31 & 51c }} temperament. The generator is a subfifth representing 35/24, four of which minus two octaves make slendric's generator, so its [[ploidacot]] is beta-dodecacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 5103/5000

{{Mapping|legend=1| 1 -5 -7 5 | 0 12 17 -4 }}
: mapping generators: ~2, ~35/24

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2959{{c}}, ~35/24 = 658.3833{{c}}
: [[error map]]: {{val| +0.296 -2.835 +4.130 -0.879 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 658.2313{{c}}
: error map: {{val| 0.000 -3.179 +3.619 -1.751 }}

{{Optimal ET sequence|legend=1| 11c, 20c, 31, 144c, 175c }}

[[Badness]] (Sintel): 2.98

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

Comma list: 176/175, 243/242, 1029/1024

Mapping: {{mapping| 1 -5 -7 5 -13 | 0 12 17 -4 30 }}

Optimal tunings:
* WE: ~2 = 1200.2862{{c}}, ~22/15 = 658.4276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.2805{{c}}

{{Optimal ET sequence|legend=0| 20ce, 31, 113c, 144c }}

Badness (Sintel): 1.77

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

Comma list: 144/143, 176/175, 243/242, 343/338

Mapping: {{mapping| 1 -5 -7 5 -13 7 | 0 12 17 -4 30 -6 }}

Optimal tunings:
* WE: ~2 = 1199.3663{{c}}, ~22/15 = 658.0465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~22/15 = 658.3800{{c}}

{{Optimal ET sequence|legend=0| 20ce, 31, 82cf, 113cf }}

Badness (Sintel): 1.94

== Restles ==
{{See also| Lesser tendoneutralic }}

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 153664/151875

{{Mapping|legend=1| 1 -2 8 4 | 0 12 -19 -4 }}
: mapping generators: ~2. ~315/256

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0322{{c}}, ~315/256 = 358.5581{{c}}
: [[error map]]: {{val| +0.032 +0.678 +1.340 -2.930 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~315/256 = 358.5484{{c}}
: error map: {{val| 0.000 +0.626 +1.267 -3.019 }}

{{Optimal ET sequence|legend=1| 77, 87, 164 }}

[[Badness]] (Sintel): 2.73

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

Comma list: 385/384, 441/440, 153664/151875

Mapping: {{mapping| 1 -2 8 4 -7 | 0 12 -19 -4 35 }}

Optimal tunings:
* WE: ~2 = 1200.1110{{c}}, ~27/22 = 358.6045{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/22 = 358.5720{{c}}

{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}

Badness (Sintel): 1.81

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

Comma list: 196/195, 352/351, 385/384, 676/675

Mapping: {{mapping| 1 -2 8 4 -7 4 | 0 12 -19 -4 35 -1 }}

Optimal tunings:
* WE: ~2 = 1200.0482{{c}}, ~~16/13 = 358.5883{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/13 = 358.5741{{c}}

{{Optimal ET sequence|legend=0| 77, 87, 164, 251d }}

Badness (Sintel): 1.16

== Lagaca ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 11529602/11390625

{{Mapping|legend=1| 2 -4 15 8 | 0 9 -13 -3 }}
: mapping generators: ~3375/2401, ~450/343

[[Optimal tuning]]s:
* [[WE]]: ~3375/2401 = 600.1355{{c}}, ~450/343 = 478.0813{{c}}
: [[error map]]: {{val| +0.271 +0.235 +0.662 -1.986 }}
* [[CWE]]: ~3375/2401 = 600.000{{c}}, ~450/343 = 477.9725{{c}}
: error map: {{val| 0.000 -0.202 +0.043 -2.743 }}

{{Optimal ET sequence|legend=1| 10, 98, 108, 118 }}

[[Badness]] (Sintel): 3.65

== Quartemka ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quartemka]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 1250000/1240029

{{Mapping|legend=1| 1 -17 -26 9 | 0 21 32 -7 }}
: mapping generators: ~2, ~50/27

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.5278{{c}}, ~50/27 = 1062.4614{{c}}
: [[error map]]: {{val| +0.528 +0.762 -1.272 -1.305 }}
* [[CWE]]: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0046{{c}}
: error map: {{val| 0.000 +0.142 -2.167 -2.858 }}

{{Optimal ET sequence|legend=1| 26, 61, 87, 113, 200 }}

[[Badness]] (Sintel): 3.85

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

Comma list: 385/384, 441/440, 800000/793881

Mapping: {{mapping| 1 -17 -26 9 7 | 0 21 32 -7 -4 }}

Optimal tunings:
* WE: ~2 = 1200.3051{{c}}, ~50/27 = 1062.2805{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~50/27 = 1062.0147{{c}}

{{Optimal ET sequence|legend=0| 26, 61, 87, 200, 287d }}

Badness (Sintel): 1.89

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

Comma list: 325/324, 364/363, 385/384, 2200/2197

Mapping: {{mapping| 1 -17 -26 9 7 -14 | 0 21 32 -7 -4 20 }}

Optimal tunings:
* WE: ~2 = 1200.2708{{c}}, ~24/13 = 1062.2496{{c}}
* CWE: ~21 = 1200.0000{{c}}, ~24/13 = 1062.0139{{c}}

{{Optimal ET sequence|legend=0| 26, 61, 87, 200 }}

Badness (Sintel): 1.17

== Tritriple ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Tritriple]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 1959552/1953125

{{Mapping|legend=1| 1 -11 -7 7 | 0 27 20 -9 }}
: mapping generators: ~2, ~864/625

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4239{{c}}, ~864/625 = 559.4921{{c}}
: [[error map]]: {{val| +0.424 -0.331 +0.561 -1.287 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~864/625 = 559.3015{{c}}
: error map: {{val| 0.000 -0.815 -0.284 -2.539 }}

{{Optimal ET sequence|legend=1| 15, …, 88, 103, 118, 221, 339d }}

[[Badness]] (Sintel): 3.00

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

Comma list: 385/384, 441/440, 43923/43750

Mapping: {{mapping| 1 -11 -7 7 -4 | 0 27 20 -9 16 }}

Optimal tunings:
* WE: ~2 = 1200.4953{{c}}, ~242/175 = 559.5243{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~242/175 = 559.3016{{c}}

{{Optimal ET sequence|legend=0| 15, …, 88, 103, 118, 221e, 339de }}

Badness (Sintel): 1.17

== Widefourth ==
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1029/1024, 48828125/48771072

{{Mapping|legend=1| 1 -17 -5 9 | 0 33 13 -11 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4770{{c}}, ~4608/3125 = 676.0584{{c}}
: [[error map]]: {{val| +0.477 -0.137 +0.061 -1.175 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4608/3125 = 675.7954{{c}}
: error map: {{val| 0.000 -0.705 -0.973 -2.576 }}

{{Optimal ET sequence|legend=1| 16, 71, 87, 103, 190 }}

[[Badness]] (Sintel): 3.90

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

Comma list: 385/384, 441/440, 234375/234256

Mapping: {{mapping| 1 16 8 -2 17 | 0 -33 -13 11 -31 }}

Optimal tunings:
* WE: ~2 = 1200.4852{{c}}, ~1250/847 = 676.0634{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~1250/847 = 675.7966{{c}}

{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}

Badness (Sintel): 1.35

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

Comma list: 385/384, 441/440, 625/624, 847/845

Mapping: {{mapping| 1 16 8 -2 17 12 | 0 -33 -13 11 -31 -19 }}

Optimal tunings:
* WE: ~2 = 1200.4217{{c}}, ~77/52 = 676.0286{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~77/52 = 675.7967{{c}}

{{Optimal ET sequence|legend=0| 16, 71, 87, 103, 190 }}

Badness (Sintel): 0.894

== Other subgroup extensions ==
=== Euslendric ===
Forms of slendric in the most optimal range for the 2.3.7 temperament ({{nowrap| 36 & 77 }}) lack an obvious strong mapping of prime 5 or prime 11. However, slendric can extend well to the no-fives no-elevens [[29-limit]] by tempering out [[273/272]], [[343/342]], [[378/377]], [[392/391]], [[513/512]], and [[729/728]], or a comma basis defined in terms of [[S-expression]]s as {S7/S8, S14/S16, S15/S20, S24/S26, S27, S28}. [[113edo]] is an obvious tuning.

Subgroup: 2.3.7.13

Comma list: 729/728, 1029/1024

Subgroup-val mapping: {{mapping| 1 1 3 0 | 0 3 -1 19 }}

Gencom mapping: {{mapping| 1 1 0 3 0 0 | 0 3 0 -1 0 19 }}

Optimal tunings:
* WE: ~2 = 1200.5057{{c}}, ~8/7 = 233.7200{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6534{{c}}

{{Optimal ET sequence|legend=0| 5, 31f, 36, 77, 113, 827bdddff }}

Badness (Sintel): 0.339

==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17

Comma list: 273/272, 729/728, 833/832

Subgroup-val mapping: {{mapping| 1 1 3 0 0 | 0 3 -1 19 21 }}

Gencom mapping: {{mapping| 1 1 0 3 0 0 0 | 0 3 0 -1 0 19 21 }}

Optimal tunings:
* WE: ~2 = 1200.5282{{c}}, ~8/7 = 233.6492{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.5776{{c}}

{{Optimal ET sequence|legend=0| 5g, 31fg, 36, 113, 149 }}

Badness (Sintel): 0.332

==== 2.3.7.13.17.19 subgroup ====
Subgroup: 2.3.7.13.17.19

Comma list: 273/272, 343/342, 513/512, 729/728

Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 | 0 3 -1 19 21 -9 }}

Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 | 0 3 0 -1 0 19 21 -9 }}

Optimal tunings:
* WE: ~2 = 1200.3292{{c}}, ~8/7 = 233.6651{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6106{{c}}

{{Optimal ET sequence|legend=0| 5g, 36, 77, 113, 262df }}

Badness (Sintel): 0.380

==== 2.3.7.13.17.19.23 subgroup ====
Subgroup: 2.3.7.13.17.19.23

Comma list: 273/272, 343/342, 392/391, 513/512, 729/728

Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 | 0 3 -1 19 21 -9 -23 }}

Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 | 0 3 0 -1 0 19 21 -9 -23 }}

Optimal tunings:
* WE: ~2 = 1200.3127{{c}}, ~8/7 = 233.6679{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6091{{c}}

{{Optimal ET sequence|legend=0| 36, 77, 113, 262df }}

Badness (Sintel): 0.474

==== 2.3.7.13.17.19.23.29 subgroup ====
Subgroup: 2.3.7.13.17.19.23.29

Comma list: 273/272, 343/342, 378/377, 392/391, 513/512, 609/608

Subgroup-val mapping: {{mapping| 1 1 3 0 0 6 9 7 | 0 3 -1 19 21 -9 -23 -11 }}

Gencom mapping: {{mapping| 1 1 0 3 0 0 0 6 9 7 | 0 3 0 -1 0 19 21 -9 -23 -11 }}

Optimal tunings:
* WE: ~2 = 1200.2503{{c}}, ~8/7 = 233.6688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~8/7 = 233.6208{{c}}

{{Optimal ET sequence|legend=0| 36, 77, 113 }}

Badness (Sintel): 0.473

=== Baladic ===
Baladic is a 2.3.7.13.17-subgroup temperament that attempts to approximate the Maqam Sikah Baladi scale. It tempers out [[169/168]] ({{S|13}}), which splits [[7/6]] in half ([[13/12]]~[[14/13]]) and one finds that the octave is therefore split in half via the interval [[91/64]], which is then equated to [[17/12]]. 36edo is an excellent baladic tuning.

Subgroup: 2.3.7.13

Comma list: 169/168, 1029/1024

Subgroup-val mapping: {{mapping| 2 2 6 7 | 0 3 -1 1 }}

Gencom mapping: {{mapping| 2 2 0 6 0 7 | 0 3 0 -1 0 1 }}
: mapping generators: ~91/64, ~8/7

Optimal tunings:
* WE: ~91/64 = 600.4315{{c}}, ~8/7 = 233.7724{{c}}
* CWE: ~91/64 = 600.0000{{c}}, ~8/7 = 233.7039{{c}}

{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ff, 226ff, 262dfff }}

Badness (Sintel): 0.434

==== 2.3.7.13.17 subgroup ====
Subgroup: 2.3.7.13.17

Comma list: 169/168, 273/272, 289/288

Subgroup-val mapping: {{mapping| 2 2 6 7 7 | 0 3 -1 1 3 }}

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

Optimal tunings:
* WE: ~17/12 = 600.4436{{c}}, ~8/7 = 233.7883{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 233.7312{{c}}

{{Optimal ET sequence|legend=0| 10, 26, 36, 154f, 190ffg, 226ffg }}

Badness (Sintel): 0.253

== References ==

[[Category:Temperament clans]]
[[Category:Gamelismic clan| ]] 
[[Category:Rank 2]]
[[Category:Listen]]

Amaranthinisma

2026-02-24T16:18:45Z

Lériendil: Lériendil moved page Amaranthinisma to Amaranthine comma

#REDIRECT [[Amaranthine comma]]

Amaranthine comma

2026-02-24T16:18:45Z

Lériendil: Lériendil moved page Amaranthinisma to Amaranthine comma

{{Infobox Interval
| Ratio = 5767168/5764801
| Name = amaranthine comma, convenience comma
| Color name = Loquadbiru comma
| Comma = yes
}}
'''5767168/5764801''', the '''amaranthine comma''', is an [[unnoticeable comma|unnoticeable]] [[11-limit]] [[comma]] between a stack of eight [[8/7]]'s and [[32/11]] (one octave above [[16/11]]), measuring 0.711 cents; it can also be expressed as the difference between [[99/98]] and two [[1029/1024|gamelismas (1029/1024)]]. Since both 99/98 and 1029/1024 are tempered out in undecimal [[mothra]], this comma is also tempered out.

== Etymology ==
The word "amaranthine" means "unfading", and so this comma was named by [[Lériendil]] in 2025, as it is the one portion of the structure of the undecimal [[slendric]] extensions that refuses to fade even at the high precision of a near justly tuned 8/7. It had previously been named the "convenience comma" by [[Scott Dakota]] in 2024, in a reference both to its "convenience" in easily extending septimal harmony and to the 7-11 convenience store (both due to bridging prime 7 to prime 11 and being 0.711 cents), though this name was deemed ambiguous by virtue of coming to refer to several other commas as well.

== Temperaments ==
Tempering out this comma in the full [[11-limit]] results in the rank-4 '''amaranthinesmic temperament''', and tempering it out in the 2.7.11 [[subgroup]] results in the rank-2 '''[[No-threes subgroup temperaments#Amaranthine|amaranthine]] temperament'''.

As its order of 11 is one, any [[7-limit]] temperament can be immediately extended to the 11-limit in theory by tempering out this comma. To make practical sense, however, it requires low complexity and high accuracy of 8/7.

Amaranthine comma

2026-02-24T16:17:21Z

Lériendil: disfavored "-isma"

3136/3125

2026-02-22T06:42:33Z

Lériendil: /* Etymology */

{{Infobox Interval
| Name = hemimean comma, didacus comma
| Color name = zzg53, zozoquingu 3rd, Zozoquingu comma
| Comma = yes
}}
'''3136/3125''', the '''hemimean comma''' or '''didacus comma''', is a [[small comma|small]] [[7-limit]] [[comma]] measuring about 6.1{{cent}}. It is the difference between a stack of five [[5/4|classic major thirds (5/4)]] and a stack of two [[7/4|subminor sevenths (7/4)]]. Perhaps more importantly, it is ([[28/25]])2/([[5/4]]), and in light of the fact that [[28/25]] = ([[7/5]])/([[5/4]]), it is also ([[28/25]])3/([[7/5]]), which means its square is equal to the difference between ([[28/25]])5 and [[7/4]]. The associated temperament has the highly favourable property of putting a number of low complexity 2.5.7 subgroup intervals on a short chain of [[28/25]]'s, itself a 2.5.7 subgroup interval.

In terms of commas, it is the difference between the [[126/125|septimal semicomma (126/125)]] and the [[225/224|septimal kleisma (225/224)]], or between the [[128/125|augmented comma (128/125)]] and the [[50/49|jubilisma (50/49)]]. Examining the latter expression we can observe that this gives us a relatively simple [[S-expression]] of ([[128/125|S4/S5]])/([[50/49|S5/S7]]) which can be rearranged to [[16/15|S4]]*[[49/48|S7]]/[[25/24|S5]]2. Then we can optionally replace S4 with a nontrivial equivalent S-expression, S4 = [[36/35|S6]]*[[49/48|S7]]*[[64/63|S8]] = ([[6/5]])/([[9/8]]); substituting this in and simplifying yields: S6*S72*S8/S52, from which we can obtain an alternative equivalence 3136/3125 = ([[49/45]])/([[25/24]])2, meaning we split [[49/45]] into two [[25/24]]'s in the resulting temperament.

== Temperaments ==
=== Didacus (2.5.7) ===
Tempering out this comma in its minimal prime [[subgroup]] of 2.5.7 leads to [[didacus]] (a variant of [[hemithirds]] without a mapping for 3) with a generator representing [[28/25]]. See [[hemimean clan]] for extensions of didacus.

=== Hemimean (2.3.5.7) ===
Tempering out this comma in the full [[7-limit]] leads to the rank-3 [[hemimean family #Hemimean|hemimean]] temperament, which splits the [[81/80|syntonic comma]] into two equal parts, each representing [[126/125]]~[[225/224]]. See [[hemimean family]] for the family of rank-3 temperaments where it is tempered out.

Note that if we temper 126/125 and/or 225/224 we get [[septimal meantone]].

=== Orion ===
As [[28/25]] is close to [[19/17]] and as the latter is the mediant of [[9/8]] and [[10/9]] (which together make [[5/4]]), it is natural to temper ([[28/25]])/([[19/17]]) = [[476/475]], or equivalently stated, the [[semiparticular]] ([[5/4]])/([[19/17]])2 = [[1445/1444]], which together imply tempering out 3136/3125 and [[2128/2125]], resulting in a rank-3 temperament. The name comes from when it was first proposed on the wiki as part of [[User:Royalmilktea #The Milky Way|The Milky Way realm]].

[[Subgroup]]: 2.5.7.17.19

[[Comma list]]: 476/475, 1445/1444

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

: sval mapping generators: ~2, ~56/25, ~17

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~28/25 = 193.642, ~17/16 = 104.434

{{Optimal ET sequence|legend=1| 12, 18h, 25, 43, 56, 68, 93, 161, 285, 353, 446, 514ch, 799ch }}

[[Badness]]: 0.0150

==== Hemimean orion ====
As tempering either [[256/255|S16]]/[[324/323|S18]] = [[1216/1215]] or [[324/323|S18]]/[[400/399|S20]] = [[1701/1700]] implies the other in the context of orion with the effect of extending to include prime 3 in the subgroup and as this therefore gives us both S16~S18~S20 and S17~S19, it can be considered natural to add these commas, because {S16/S18, S17/S19, S18/S20} implies all the aforementioned commas of orion. However, this is a strong extension of hemimean and weak extension of orion, as we have a ~3/2 generator slicing the second generator of orion into five.

See [[Hemimean family #Hemimean orion]].

==== Semiorion ====
As [[1445/1444]] = [[289/288|S17]]/[[361/360|S19]] we can extend orion to include prime 3 in its subgroup by tempering both [[289/288|S17]] and [[361/360|S19]]. However, note that (because of tempering [[289/288|S17]]) this splits the period in half, representing a [[17/12]]~[[24/17]] half-octave. This has the consequence that the [[17/16]] generator can be described as a [[3/2]] because [[17/16]] up from [[24/17]] is [[3/2]]. As a result, this equates the generators of hemimean orion and orion up to period-equivalence and is a weak extension of both.

See [[Hemimean family #Semiorion]].

== Etymology ==
This comma was first named as ''parahemwuer'' by [[Gene Ward Smith]] in 2005 as a contraction of ''[[parakleismic]]'' and ''[[hemiwürschmidt]]''<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12900.html Yahoo! Tuning Group | ''Seven limit comma names from pairs of temperament names'']</ref>. It is not clear how it later became ''hemimean'', but the root of ''hemimean'' is obvious, being a contraction of ''hemiwürschmidt'' (or ''hemithirds'') and ''meantone''.

The name ''didacus'' seems to be first attested in September 2016 ([https://en.xen.wiki/index.php?title=Subgroup_temperaments&diff=next&oldid=26776 here]), and the name was created by Gene Ward Smith. It is unclear what the origin of this name is; [https://en.wikipedia.org/wiki/Didacus_of_Alcalá St. Didacus] was a Spanish missionary after whom the city of San Diego was named, but there seems to be no relation between this individual and musical temperament.

== Notes ==

[[Category:Hemimean]]
[[Category:Commas named by combining multiple temperament names]]
[[Category:Commas named after individuals]]
[[Category:Commas named after composers]]

Escapade family

2026-02-09T16:13:38Z

Lériendil: /* Escapade */

{{Technical data page}}

<div style="float: right;">
[[File:Escapade.png|alt=Escapade.png|thumb|600x560px|An image of the tuning spectrum of 2.3.5.11 escapade, in terms of the generator; [[Edo]] [[patent val]] tunings are marked with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled.]]
</div>

The '''escapade family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[escapade comma]], {{monzo| 32 -7 -9 }}, of size 9.492 [[cent]]s. The defining feature of this comma is splitting [[5/3]] into sixteen quartertones of which [[5/4]] makes up seven and [[4/3]] makes up nine; therefore [[16/15]] is two generator steps. It most naturally manifests as a [[2.3.5.11 subgroup|2.3.5.11-subgroup]] temperament, tempering out [[4000/3993]] and [[5632/5625]].

Extensions of escapade to incorporate prime 7 (and therefore the full [[11-limit]]) include escapist ({{nowrap| 21 & 22 }}), tempering out [[225/224]] and mapping 7 to −4 generators; escaped ({{nowrap| 22 & 87 }}), tempering out [[245/243]] and mapping 7 to −26 generators; alphaquarter ({{nowrap| 65d & 87 }}), tempering out [[5120/5103]] and mapping 7 to 61 generators; septisuperfourth (a.k.a. biscapade) ({{nowrap| 22 & 86 }}), tempering out [[6144/6125]], splitting the octave in half and mapping 7 to −15 generators; and arch ({{nowrap| 43 & 87 }}), tempering out [[3136/3125]] and splitting the generator into two [[64/63]] intervals; all are considered below.

== Escapade ==
For intervals along the chain of generators in the 2.3.5.11.31 subgroup temperament, out to 22 generators up, see the third column of [[16ed5/3#Intervals]].

=== 5-limit ===
[[Subgroup]]: 2.3.5

[[Comma list]]: 4294967296/4271484375 ({{monzo|32 -7 -9}})

{{Mapping|legend=1| 1 2 2 | 0 -9 7 }}

: mapping generators: ~2, ~16875/16384

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.0000, ~16875/16384 = 55.3052
* [[POTE]]: ~2 = 1200.000, ~16875/16384 = 55.293

{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 152, 217, 586, 803 }}

[[Badness]] (Smith): 0.083778

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.253 || +0.298
|-
| 5/4 || 387.136 || +0.823
|}

=== 2.3.5.11 subgroup ===
Since (an ideally slightly flat) 4/3 is split in three by the interval of 3 generators, it makes sense to equate that interval to [[11/10]] by tempering out 4000/3993, and therefore the generator to {{nowrap|(11/10)/(16/15) {{=}} [[33/32]]}}; this does minimal damage to the temperament. This structure in 2.3.5.11 occurs in all extensions of escapade to include prime 7, and therefore will be considered the fount of all further extensions.

Subgroup: 2.3.5.11

Comma list: 4000/3993 ({{monzo|5 -1 3 -3}}), 5632/5625 ({{monzo|9 -2 -4 1}})

Mapping: {{Mapping| 1 2 2 3 | 0 -9 7 10 }}

Optimal tuning (CTE): ~2 = 1200.0000, ~33/32 = 55.2760

{{Optimal ET sequence|legend=1| 21, 22, 43, 65, 87, 152, 369, 521e }}

Badness: 0.0107

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.516 || +0.561
|-
| 5/4 || 386.932 || +0.618
|-
| 11/8 || 552.760 || +1.442
|}

=== 2.3.5.11.31 subgroup ===
One may note that the generator represents the square root of [[16/15]] and therefore it would be logical to also temper out {{nowrap| S31 {{=}} [[961/960]] }} so that the generator is equated to {{nowrap| [[32/31]] ~ [[31/30]] }} in addition to 33/32.

Subgroup: 2.3.5.11.31

Comma list: 496/495 ({{monzo| 4 -2 -1 -1 1 }}), 961/960 ({{monzo| -6 -1 -1 0 2 }}), 4000/3993 ({{monzo| 5 -1 3 -3 0 }})

Mapping: {{mapping| 1 2 2 3 5 | 0 -9 7 10 -1 }}

Optimal tuning (CTE): ~2 = 1200.000, ~32/31 = 55.276

{{Optimal ET sequence|legend=0| 21, 22, 43, 65, 87, 152, 369, 521e, 673e }}

Badness (Sintel): 0.251

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.518 || +0.563
|-
| 5/4 || 386.931 || +0.617
|-
| 11/8 || 552.758 || +1.440
|-
| 31/16 || 1144.724 || -0.311
|}

= Strong extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to strong full 7- and 11-limit extensions
|-
! rowspan="1" | Extension !! rowspan="1" | Mapping of 7 !! rowspan="1" | Tuning range*
|-
| [[#Escapist|Escapist]] || -4 || ↓ [[65edo|65]]
|-
| [[#Alphaquarter|Alphaquarter]] || +61 || ↑ 65 ↓ [[87edo|87]]
|-
| [[#Escaped|Escaped]] || -26 || ↑ 87
|}
<nowiki/>* Defined as the range in which the extension specified has a better mapping of 7 compared to its neighboring extensions

== Escaped ==
''[[#Strong extensions|Return to the map]]''

This temperament was also known as "sensa" in earlier materials because it tempers out 245/243, 352/351, and 385/384 as a sensamagic temperament. ''Not to be confused with the {{nowrap| 19e & 27 }} temperament (sensi extension).''

Here, [[245/243]] is tempered out so that [[9/7]] is equated to the square root of 5/3 (at 8 generators) present in the temperament. This works best where 5/3 is slightly flat, therefore on the end of the spectrum approaching [[22edo]].

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 65625/65536

{{Mapping|legend=1| 1 2 2 4 | 0 -9 7 -26 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 55.122

{{Optimal ET sequence|legend=1| 22, 65, 87, 196, 283 }}

[[Badness]] (Smith): 0.088746

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

Comma list: 245/243, 385/384, 4000/3993

Mapping: {{mapping| 1 2 2 4 3 | 0 -9 7 -26 10 }}

Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.126

{{Optimal ET sequence|legend=0| 22, 65, 87, 196, 283 }}

Badness (Smith): 0.035844

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 703.831 || +1.876
|-
| 5/4 || 385.909 || -0.405
|-
| 7/4 || 966.624 || -2.202
|-
| 11/8 || 551.299 || -0.019
|}

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

Comma list: 245/243, 352/351, 385/384, 625/624

Mapping: {{mapping| 1 2 2 4 3 2 | 0 -9 7 -26 10 37 }}

Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 55.138

{{Optimal ET sequence|legend=0| 22, 65, 87, 283 }}

Badness (Smith): 0.031366

== Alphaquarter ==
''[[#Strong extensions|Return to the map]]''

Given the slightly sharp ~[[3/2]] in ideal tunings of escapade (between [[65edo]] and [[87edo]]), it does very little damage to temper out [[5120/5103]] to extend it to prime 7; the cost is that the harmonic 7 is exceedingly complex, located all the way at 61 generators up.

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 5120/5103, 29360128/29296875

{{Mapping|legend=1| 1 2 2 0 | 0 -9 7 61 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~16128/15625 = 55.243

{{Optimal ET sequence|legend=1| 65d, 87, 152, 239, 391 }}

[[Badness]] (Smith): 0.116594

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

Comma list: 3025/3024, 4000/3993, 5120/5103

Mapping: {{mapping| 1 2 2 0 3 | 0 -9 7 61 10 }}

Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.243

{{Optimal ET sequence|legend=0| 65d, 87, 152, 239, 391 }}

Badness (Smith): 0.029638

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.918 || +0.963
|-
| 5/4 || 386.620 || +0.306
|-
| 7/4 || 969.113 || +0.287
|-
| 11/8 || 552.314 || +0.996
|}

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

Comma list: 352/351, 625/624, 847/845, 1575/1573

Mapping: {{mapping| 1 2 2 0 3 2 | 0 -9 7 61 10 37 }}

Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.236

{{Optimal ET sequence|legend=0| 65d, 87, 152f, 239f }}

Badness (Smith): 0.025344

== Escapist ==
''[[#Strong extensions|Return to the map]]''

This temperament makes the identification of the 4-generator interval, representing (16/15)2, with [[8/7]] by tempering out [[225/224]] (along with [[12288/12005]]); however, this is somewhat inaccurate as the ~16/15 in escapade is slightly flat, while for a good marvel tuning it needs to be tempered sharpward to equate it with [[15/14]].

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 12288/12005

{{Mapping|legend=1| 1 2 2 3 | 0 -9 7 -4 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~49/48 = 55.327

{{Optimal ET sequence|legend=1| 21, 22, 43, 65d }}

[[Badness]] (Smith): 0.077950

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

Comma list: 99/98, 176/175, 2560/2541

Mapping: {{mapping| 1 2 2 3 3 | 0 -9 7 -4 10 }}

Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 55.354

{{Optimal ET sequence|legend=0| 21, 22, 43, 65d }}

Badness (Smith): 0.036700

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 701.626 || -0.329
|-
| 5/4 || 387.624 || +1.310
|-
| 7/4 || 978.501 || +9.675
|-
| 11/8 || 553.749 || +2.431
|}

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

Comma list: 78/77, 99/98, 176/175, 507/500

Mapping: {{mapping| 1 2 2 3 3 3 | 0 -9 7 -4 10 15 }}

Optimal tuning (POTE): ~2 = 1200.000, ~26/25 = 55.550

{{Optimal ET sequence|legend=0| 21, 22, 43 }}

Badness (Smith): 0.035261

= Weak extensions =
{| class="wikitable center-all"
|+ style="font-size: 105%;" | Map to weak extensions
|-
! rowspan="2" | Extensions !! rowspan="2" | Periods per octave !! colspan="2" | Position of original generator
|-
! Number of generators !! Number of periods
|-
| [[#Septisuperfourth|Septisuperfourth]] || period = 1/2 octave || 1 generator || + 0 periods
|-
| [[#Arch|Arch]] || period = octave || 2 generators || + 0 periods
|}

== Septisuperfourth ==
''[[#Weak extensions|Return to map]]''

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 6144/6125, 118098/117649

{{Mapping|legend=1| 2 4 4 7 | 0 -9 7 -15 }}

: mapping generators: ~343/243, ~16875/16384

[[Optimal tuning]] ([[POTE]]): ~343/243 = 1\2, ~16875/16384 = 55.320

{{Optimal ET sequence|legend=1| 22, 86, 108, 130, 152, 282 }}

[[Badness]] (Smith): 0.059241

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

Comma list: 540/539, 4000/3993, 5632/5625

Mapping: {{mapping| 2 4 4 7 6 | 0 -9 7 -15 10 }}

Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.304

{{Optimal ET sequence|legend=0| 22, 86, 108, 130, 152, 282 }}

Badness (Smith): 0.024619

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.070 || +0.115
|-
| 5/4 || 387.279 || +0.965
|-
| 7/4 || 970.117 || +1.291
|-
| 11/8 || 553.255 || +1.937
|}

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

Comma list: 540/539, 729/728, 1575/1573, 3584/3575

Mapping: {{mapping| 2 4 4 7 6 11 | 0 -9 7 -15 10 -39 }}

Optimal tuning (POTE): ~99/70 = 600.000, ~33/32 = 55.325

{{Optimal ET sequence|legend=0| 22f, 108f, 130, 282 }}

Badness (Smith): 0.022887

==== Septisuperquad ====
This temperament is also known as "biscapade".

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 364/363, 540/539, 4096/4095

Mapping: {{mapping| 2 4 4 7 6 5 | 0 -9 7 -15 10 26 }}

Optimal tuning (POTE): ~55/39 = 600.000, ~33/32 = 55.359

{{Optimal ET sequence|legend=0| 22, 108, 130 }}

Badness (Smith): 0.033038

== Arch ==
''[[#Weak extensions|Return to map]]''

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3136/3125, 5250987/5242880

{{Mapping|legend=1| 1 2 2 2 | 0 -18 14 35 }}

: mapping generators: ~2, ~64/63

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~64/63 = 27.668

{{Optimal ET sequence|legend=1| 43, 87, 130, 217, 347, 824c, 1171c, 1518cd }}

[[Badness]] (Smith): 0.094345

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

Comma list: 441/440, 3136/3125, 4000/3993

Mapping: {{mapping| 1 2 2 2 3 | 0 -18 14 35 20 }}

Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.663

{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 911cde }}

Badness (Smith): 0.036541

{| class="wikitable center-1 center-2 center-3 center-4 mw-collapsible mw-collapsed"
|+ style="font-size: 105%;" | Harmonics
|-
! rowspan="2" | Prime harmonic !! colspan="2" | Tunings
|-
! CTE tuning !! Deviation from just
|-
| 3/2 || 702.178 || +0.223
|-
| 5/4 || 387.195 || +0.881
|-
| 7/4 || 967.987 || -0.839
|-
| 11/8 || 553.135 || +1.817
|}

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

Comma list: 364/363, 441/440, 676/675, 3136/3125

Mapping: {{mapping| 1 2 2 2 3 4 | 0 -18 14 35 20 -13 }}

Optimal tuning (POTE): ~2 = 1200.000, ~64/63 = 27.660

{{Optimal ET sequence|legend=0| 43, 87, 130, 217, 347e, 564e }}

Badness (Smith): 0.019504

[[Category:Temperament families]]
[[Category:Escapade family| ]] 
[[Category:Rank 2]]

Diaschismic family

2026-02-08T16:48:55Z

Lériendil: /* Shrutar */

{{Technical data page}}
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].

== Diaschismic ==
{{Main| Diaschismic }}

The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.

This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.

[[Subgroup]]: 2.3.5

[[Comma list]]: 2048/2025

{{Mapping|legend=1| 2 0 11 | 0 1 -2 }}

: mapping generators: ~45/32, ~3

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4107{{c}}, ~3/2 = 704.2059{{c}}
: [[error map]]: {{val| -1.179 +1.072 +1.150 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}
: error map: {{val| 0.000 +3.003 +3.769 }}

[[Tuning ranges]]:
* [[5-odd-limit]] [[diamond monotone]]: ~3/2 = [600.000 to 720.000] (1\2 to 6\10)
* 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}

[[Badness]] (Sintel): 0.467

=== Overview to extensions ===
==== 7-limit extensions ====
To get the 7-limit extensions, we add another comma:
* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.
* Pajara adds [[50/49]] or [[64/63]] and is a popular and well-known choice.
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.
* Keen adds [[875/864]].

Those all keep the same half-octave period and fifth generator.

Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original generator in four.

==== Subgroup extensions ====
Since the diaschisma factors into ([[256/255]])2([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', documented right below. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.

=== Srutal archagall ===
{{See also | Fiventeen }}

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}

: mapping generators: ~17/12, ~3

Optimal tunings:
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}

{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}

Badness (Sintel): 0.212

== Septimal diaschismic ==
{{Main| Diaschismic }}
{{See also| Srutal vs diaschismic }}

A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] with the 104c [[val]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.

Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 2048/2025

{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4449{{c}}, ~3/2 = 703.0299{{c}}
: [[error map]]: {{val| -1.110 -0.035 +3.740 -1.391 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 705.882] (7\12 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}

[[Badness]] (Sintel): 0.959

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

Comma list: 126/125, 176/175, 896/891

Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}

Optimal tunings:
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 704.348] (7\12 to 27\46)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}

Badness (Sintel): 0.828

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

Comma list: 126/125, 176/175, 196/195, 364/363

Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}

Optimal tunings:
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]

{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}

Badness (Sintel): 0.782

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 136/135, 176/175, 196/195, 256/255

Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}

Optimal tunings:
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}

Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [703.448, 704.348] (34\58 to 27\46)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]

{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}

Badness (Sintel): 0.837

=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
Na"Naa' is a remarkable subgroup temperament of {{nowrap| 46 & 58 }} with a prime harmonic of 23. It is yet to be found why it got this strange name.

Subgroup: 2.3.5.7.11.13.17.23

Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255

Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}

Optimal tunings:
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}

{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}

Badness (Sintel): 0.882

== Pajara ==
{{Main| Pajara }}

Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.

Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 50/49, 64/63

{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}

[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
: [[error map]]: {{val| -2.303 +1.432 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [700.000, 720.000] (7\12 to 6\10)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 715.587]

{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}

[[Badness]] (Sintel): 0.507

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

Comma list: 50/49, 64/63, 99/98

Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}

Optimal tunings:
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [700.000, 709.091] (7\12 to 13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.587]

{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}

Badness (Sintel): 0.673

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

Comma list: 50/49, 64/63, 65/63, 99/98

Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}

Optimal tunings:
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}

{{Optimal ET sequence|legend=0| 10e, 12, 22 }}

Badness (Sintel): 1.14

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 64/63, 65/63, 99/98

Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}

Optimal tunings:
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}

{{Optimal ET sequence|legend=0| 10e, 12, 22 }}

Badness (Sintel): 1.06

==== Pajarina ====
Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 64/63, 78/77, 99/98

Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}

Optimal tunings:
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}

{{Optimal ET sequence|legend=0| 12f, 22, 34d }}

Badness (Sintel): 0.923

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 64/63, 78/77, 85/84, 99/98

Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}

Optimal tunings:
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}

{{Optimal ET sequence|legend=0| 12f, 22, 34d }}

Badness (Sintel): 0.936

==== Pajarita ====
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 64/63, 66/65

Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}

Optimal tunings:
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}

{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}

Badness (Sintel): 0.937

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 64/63, 66/65, 85/84

Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}

Optimal tunings:
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}

{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}

Badness (Sintel): 0.968

=== Pajarous ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 55/54, 64/63

Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}

Optimal tunings:
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = 709.091 (13\22)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 715.803]

{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}

Badness (Sintel): 0.937

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

Comma list: 50/49, 55/54, 64/63, 65/63

Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}

Optimal tunings:
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}

{{Optimal ET sequence|legend=0| 10, 22 }}

Badness (Sintel): 1.04

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}

Optimal tunings:
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}

{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}

Badness (Sintel): 0.930

==== Pajaro ====
Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 50/49, 55/54, 64/63

Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}

Optimal tunings:
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}

{{Optimal ET sequence|legend=0| 10, 22f, 32f }}

Badness (Sintel): 1.13

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 40/39, 50/49, 55/54, 64/63, 85/84

Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}

Optimal tunings:
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}

{{Optimal ET sequence|legend=0| 10, 22f, 32f }}

Badness (Sintel): 1.01

=== Pajaric ===
Subgroup: 2.3.5.7.11

Comma list: 45/44, 50/49, 56/55

Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}

Optimal tunings:
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}

{{Optimal ET sequence|legend=0| 10, 12, 22e }}

Badness (Sintel): 0.787

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

Comma list: 40/39, 45/44, 50/49, 56/55

Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}

Optimal tunings:
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}

{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}

Badness (Sintel): 0.845

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 34/33, 40/39, 45/44, 50/49, 56/55

Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}

Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}

{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}

Badness (Sintel): 0.896

=== Hemipaj ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 121/120

Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}

: mapping generators: ~2, ~16/11

Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}

{{Optimal ET sequence|legend=0| 2, 20, 22 }}

Badness (Sintel): 1.29

=== Hemifourths ===
Subgroup: 2.3.5.7.11

Comma list: 50/49, 64/63, 243/242

Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}

: mapping generators: ~2, ~55/32

Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}

{{Optimal ET sequence|legend=0| 10, 24d, 34d }}

Badness (Sintel): 1.62

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

Comma list: 50/49, 64/63, 78/77, 144/143

Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}

Optimal tunings:
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}

{{Optimal ET sequence|legend=0| 10, 24d, 34d }}

Badness (Sintel): 1.19

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 64/63, 78/77, 85/84, 144/143

Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}

Optimal tunings:
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}

{{Optimal ET sequence|legend=0| 10, 24d, 34d }}

Badness (Sintel): 1.11

== Srutal ==
{{See also| Srutal vs diaschismic }}

Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 4375/4374

{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4046{{c}}, ~3/2 = 704.1150{{c}}
: [[error map]]: {{val| -1.191 +0.969 +1.289 +0.044 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7646{{c}}
: error map: {{val| 0.000 +2.810 +4.157 +2.643 }}

[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [703.448, 705.882] (34\58 to 20\34)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}

[[Badness]] (Sintel): 2.32

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

Comma list: 176/175, 896/891, 1331/1323

Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}

Optimal tunings:
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}

Tuning ranges:
* 11-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]

{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}

Badness (Sintel): 1.17

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

Comma list: 169/168, 176/175, 325/324, 364/363

Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}

Optimal tunings:
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}

Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 13-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]

{{Optimal ET sequence|legend=0| 34d, 46, 80 }}

Badness (Sintel): 1.04

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 169/168, 176/175, 221/220, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}

Optimal tunings:
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}

Tuning ranges:
* 17-odd-limit diamond monotone: ~3/2 = [704.348, 705.882] (27\46 to 20\34)
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]

{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}

Badness (Sintel): 0.947

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 169/168, 176/175, 190/189, 221/220, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}

Optimal tunings:
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}

{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}

Badness (Sintel): 1.04

==== Srutaloo ====
Srutaloo adds 576/575, 736/729 or 208/207, and rhymes with [[skidoo]].

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}

Optimal tunings:
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}

{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}

Badness (Sintel): 0.971

===== 29-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 221/220, 232/231, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}

Optimal tunings:
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}

{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}

Badness (Sintel): 1.10

===== 31-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 136/135, 169/168, 176/175, 190/189, 208/207, 217/216, 221/220, 232/231, 256/255

Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}

Optimal tunings:
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}

{{Optimal ET sequence|legend=0| 46, 80, 126 }}

Badness (Sintel): 1.44

== Keen ==
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 875/864, 2048/2025

{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
: [[error map]]: {{val| -0.679 +4.536 -3.033 -2.591 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}
: error map: {{val| 0.000 +5.574 -1.373 -1.061 }}

{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}

[[Badness]] (Sintel): 2.13

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

Comma list: 100/99, 385/384, 1232/1215

Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}

Optimal tunings:
* WE: ~45/32 = 599.6286{{c}}, ~3/2 = 707.1712{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5984{{c}}

{{Optimal ET sequence|legend=0| 22, 56, 78 }}

Badness (Sintel): 1.50

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

Comma list: 100/99, 105/104, 144/143, 1078/1053

Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}

Optimal tunings:
* WE: ~45/32 = 599.3498{{c}}, ~3/2 = 706.4009{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.1309{{c}}

{{Optimal ET sequence|legend=0| 22f, 34, 56f }}

Badness (Sintel): 1.85

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 100/99, 105/104, 119/117, 144/143, 154/153

Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}

Optimal tunings:
* WE: ~17/12 = 599.4053{{c}}, ~3/2 = 706.4544{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.1243{{c}}

{{Optimal ET sequence|legend=0| 22f, 34, 56f }}

Badness (Sintel): 1.54

==== Keenic ====
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 100/99, 352/351, 385/384

Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}

Optimal tunings:
* WE: ~45/32 = 599.8547{{c}}, ~3/2 = 707.0858{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.2596{{c}}

{{Optimal ET sequence|legend=0| 22, 34, 56 }}

Badness (Sintel): 1.67

===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 100/99, 136/135, 154/153, 256/255

Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}

Optimal tunings:
* WE: ~17/12 = 599.8338{{c}}, ~3/2 = 707.0558{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.2537{{c}}

{{Optimal ET sequence|legend=0| 22, 34, 56 }}

Badness (Sintel): 1.37

== Bidia ==
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 68 }} temperament; its ploidacot is tetraploid monocot. Scales of bidia [[cluster temperament|cluster]] around [[12edo]], with a small residue left behind when three semitones exceed the quarter-octave period. This residue represents [[64/63]], and somewhat peculiarly, [[81/80]] is represented by ''two'' of these intervals.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 3136/3125

{{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}

: mapping generators: ~25/21, ~3

[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.6887{{c}}, ~3/2 = 704.6318{{c}}
: [[error map]]: {{val| -1.245 +1.432 +0.064 +0.854 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 705.5070{{c}}
: error map: {{val| 0.000 +3.552 +2.672 +3.639 }}

{{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}

[[Badness]] (Sintel): 1.43

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

Comma list: 176/175, 896/891, 1375/1372

Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}

Optimal tunings:
* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.2170{{c}}

{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}

Badness (Sintel): 1.33

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

Comma list: 176/175, 325/324, 640/637, 896/891

Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}

Optimal tunings:
* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3241{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}

Badness (Sintel): 1.70

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 256/255, 325/324, 640/637

Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}

Optimal tunings:
* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3496{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}

Badness (Sintel): 1.46

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 136/135, 176/175, 190/189, 256/255, 325/324, 640/637

Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}

Optimal tunings:
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}

Badness (Sintel): 1.25

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 136/135, 176/175, 190/189, 253/252, 256/255, 325/324, 640/637

Mapping: {{mapping| 4 0 22 43 71 -36 10 17 -20 | 0 1 -2 -5 -9 8 1 0 6 }}

Optimal tunings:
* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}

{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}

Badness (Sintel): 1.24

== Shrutar ==
Shrutar adds 245/243 to the commas, and also tempers out [[6144/6125]]. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either ~36/35 or ~35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14(1/7), making 7's just.

By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14(1/7) generator can again be used as tunings.

Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 2048/2025

{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}

: mapping generators: ~45/32, ~35/24

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.5401{{c}}, ~35/24 = 652.3108{{c}}
: [[error map]]: {{val| -0.920 +2.207 +0.304 -1.730 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~35/24 = 652.7736{{c}}
: error map: {{val| 0.000 +3.592 +2.592 +0.589 }}

{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}

[[Badness]] (Sintel): 1.20

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

Comma list: 121/120, 176/175, 245/243

Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}

Optimal tunings:
* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}

{{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}

Badness (Sintel): 0.876

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

Comma list: 121/120, 176/175, 196/195, 245/243

Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}

Optimal tunings:
* WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}

{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}

Badness (Sintel): 1.16

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195

Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}

Optimal tunings:
* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}

{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}

Badness (Sintel): 0.953

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342

Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}

Optimal tunings:
* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}

{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}

Badness (Sintel): 1.07

==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342

Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}

Optimal tunings:
* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}

{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}

Badness (Sintel): 1.03

== Shru ==
Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 392/375, 1323/1280

{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}

: mapping generators: ~45/32, ~10/7

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}

{{Optimal ET sequence|legend=1| 2, 22d, 24 }}

[[Badness]] (Sintel): 3.99

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

Comma list: 56/55, 77/75, 1323/1280

Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}

Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}

{{Optimal ET sequence|legend=0| 2, 22d, 24 }}

Badness (Sintel): 2.10

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

Comma list: 56/55, 77/75, 105/104, 507/500

Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}

Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}

{{Optimal ET sequence|legend=0| 2, 24 }}

Badness (Sintel): 2.12

== Sruti ==
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 19683/19600

{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}

: mapping generators: ~45/32, ~140/81

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.2764{{c}}, ~140/81 = 950.7284{{c}}
: [[error map]]: {{val| -1.447 -0.498 +2.813 +1.497 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}
: error map: {{val| 0.000 +1.690 +6.395 +4.869 }}

{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}

[[Badness]] (Sintel): 2.97

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

Comma list: 176/175, 243/242, 896/891

Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}

Optimal tunings:
* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}

{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}

Badness (Sintel): 1.37

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

Comma list: 144/143, 176/175, 351/350, 676/675

Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}

Optimal tunings:
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}

{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}

Badness (Sintel): 0.983

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 144/143, 170/169, 176/175, 221/220

Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}

Optimal tunings:
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}

{{Optimal ET sequence|legend=0| 24, 34d, 58 }}

Badness (Sintel): 1.05

== Anguirus ==
As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 49/48, 2048/2025

{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}

: mapping generators: ~45/32, ~7/4

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2758{{c}}, ~7/4 = 953.4593{{c}}
: [[error map]]: {{val| +0.552 +4.964 +2.883 -14.264 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}
: error map: {{val| 0.000 +4.083 +1.611 -15.807 }}

{{Optimal ET sequence|legend=1| 10, 24, 34 }}

[[Badness]] (Sintel): 1.97

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

Comma list: 49/48, 56/55, 243/242

Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}

Optimal tunings:
* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}

{{Optimal ET sequence|legend=0| 10, 24, 34 }}

Badness (Sintel): 1.63

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

Comma list: 49/48, 56/55, 91/90, 243/242

Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}

Optimal tunings:
* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}

{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}

Badness (Sintel): 1.27

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 49/48, 56/55, 91/90, 119/117, 154/153

Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}

Optimal tunings:
* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}

{{Optimal ET sequence|legend=0| 10, 24, 34 }}

Badness (Sintel): 1.10

== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.

Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.

The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.

Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 2048/2025

{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}

: mapping generators: ~45/32, ~9/7

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}

{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}

[[Badness]] (Sintel): 1.47

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

Comma list: 176/175, 540/539, 896/891

Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}

Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}

Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7

{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}

Badness (Sintel): 0.859

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

Comma list: 176/175, 351/350, 364/363, 540/539

Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}

Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}

{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}

Badness (Sintel): 0.978

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 176/175, 221/220, 256/255, 540/539

Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}

Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}

{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}

Badness (Sintel): 1.03

== Echidnic ==
Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 686/675, 1029/1024

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

: mapping generators: ~45/32, ~8/7

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}

{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}

[[Badness]] (Sintel): 1.83

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

Comma list: 385/384, 441/440, 686/675

Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}

Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}

{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}

Badness (Sintel): 1.49

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

Comma list: 91/90, 169/168, 385/384, 441/440

Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}

Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}

{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}

Badness (Sintel): 1.19

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 136/135, 154/153, 169/168, 256/255

Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}

Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}

{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}

Badness (Sintel): 0.983

; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)

== Quadrasruta ==
Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2048/2025, 2401/2400

{{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}

: mapping generators: ~45/32, ~21/16

[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4443{{c}}, ~21/16 = 475.7746{{c}}
: [[error map]]: {{val| -1.111 +1.143 +1.377 -0.595 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}
: error map: {{val| 0.000 +3.003 +3.771 +2.456 }}

{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}

[[Badness]] (Sintel): 1.86

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

Comma list: 176/175, 896/891, 2401/2400

Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}

Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}

{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}

Badness (Sintel): 1.62

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

Comma list: 176/175, 196/195, 512/507, 676/675

Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}

Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}

{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}

Badness (Sintel): 1.18

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 170/169, 176/175, 196/195, 256/255

Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}

Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}

{{Optimal ET sequence|legend=0| 10e, 58, 126f }}

Badness (Sintel): 1.21

=== Quadrafourths ===
Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 2048/2025

Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}

Optimal tunings:
* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}

{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}

Badness (Sintel): 1.62

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

Comma list: 144/143, 196/195, 243/242, 676/675

Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}

Optimal tunings:
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}

{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}

Badness (Sintel): 1.11

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 136/135, 144/143, 170/169, 196/195, 221/220

Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}

Optimal tunings:
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}

{{Optimal ET sequence|legend=0| 10, 48c, 58 }}

Badness (Sintel): 1.13

[[Category:Temperament families]]
[[Category:Diaschismic family| ]] 
[[Category:Diaschismic| ]] 
[[Category:Rank 2]]

Buzzardsmic clan

2026-02-08T16:05:29Z

Lériendil: /* Demibuzzard */

{{Technical data page}}
The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''buzzardsmic clan''' is the buzzardsma, [[65536/64827]], with [[monzo]] {{monzo| 16 -3 0 -4 }}, which implies that the tritave, [[3/1]], is divided into four intervals each representing a [[21/16]] subfourth. [[Tempering out]] this comma implies a sharpened [[7/1|7th]] [[harmonic]], and especially a sharpened [[~]]21/16 generator, which approaches the 480{{c}} fourth of [[5edo]].

Extensions of buzzard to incorporate [[prime interval|prime]] [[5/1|5]] along its chain of generators (and therefore the full [[7-limit]]) include septimal buzzard ({{nowrap| 53 & 58 }}), which tempers out [[1728/1715]] (and [[5120/5103]]); subfourth ({{nowrap| 58 & 63 }}), which tempers out [[10976/10935]]; and lemongrass ({{nowrap| 63 & 68 }}), which tempers out [[245/243]]. All are considered below.

Weak extensions include submajor ({{nowrap| 10 & 53 }}), which tempers out [[225/224]] and splits [[32/21]] (the superfifth) in two; and thuja ({{nowrap| 15 & 43 }}), which tempers out [[126/125]] and splits [[21/8]] into three.

Full 7-limit temperaments discussed elsewhere are:
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Cohemimabila]]'' (+3136/3125) → [[Mabila family #Cohemimabila|Mabila family]]

The rest are considered below.

= 2.3.7 subgroup =
== Buzzard ==
{{Main| Buzzard }}

[[Subgroup]]: 2.3.7

[[Comma list]]: 65536/64827

{{Mapping|legend=1| 1 0 4 | 0 4 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2548{{c}}, ~21/16 = 475.5761{{c}}
: [[error map]]: {{val| -0.745 +0.350 +1.465 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.8328{{c}}
: error map: {{val| 0.000 +1.376 +3.676 }}

{{Optimal ET sequence|legend=1| 5, 33, 38, 43, 48, 53, 58 }}

[[Badness]] (Sintel): 0.824

= Strong extensions =
== Septimal buzzard ==
{{Main| Buzzard }}
{{See also| Vulture family }}

Septimal buzzard is not only a naturally motivated extension to 2.3.7 buzzard, but the main extension to [[vulture]] of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~[[21/16]], though buzzard is powerful as a full 13-limit system in its own right. It is most naturally described as {{nowrap| 53 & 58 }} (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, [[111edo]] (111 = 53 + 58) is a great tuning for it. [[Mos scale]]s of 5, 8, 13, 18, 23, 28, 33, 38, 43, 48 or 53 notes are available.

Its 13-limit [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[847/845|S11/S13]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. [[Hemifamity]] leverages it by splitting [[36/35]] into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]]. Then the vanishing of the orwellisma implies [[49/48]], the large septimal diesis, is equated with 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 5120/5103

{{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3061{{c}}, ~21/16 = 475.3611{{c}}
: [[error map]]: {{val| -0.694 -0.511 +0.432 +2.315 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.6144{{c}}
: error map: {{val| 0.000 +0.503 +1.589 +4.331 }}

{{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}

[[Badness]] (Sintel): 1.21

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

Comma list: 176/175, 540/539, 5120/5103

Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }}

Optimal tunings:
* WE: ~2 = 1199.2516{{c}}, ~21/16 = 475.4037{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6806{{c}}

{{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}

Badness (Sintel): 1.14

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

Comma list: 176/175, 351/350, 540/539, 676/675

Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}

Optimal tunings:
* WE: ~2 = 1199.2391{{c}}, ~21/16 = 475.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6760{{c}}

{{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}

Badness (Sintel): 0.779

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 442/441, 540/539

Mapping: {{mapping| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}

Optimal tunings:
* WE: ~2 = 1199.2723{{c}}, ~21/16 = 475.4039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6837{{c}}

{{Optimal ET sequence|legend=0| 53, 58, 111 }}

Badness (Sintel): 0.938

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539

Mapping: {{mapping| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}

Optimal tunings:
* WE: ~2 = 1199.2457{{c}}, ~21/16 = 475.3797{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6690{{c}}

{{Optimal ET sequence|legend=0| 53, 58h, 111 }}

Badness (Sintel): 0.952

=== Buteo ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 2200/2187

Mapping: {{mapping| 1 0 -6 4 9 | 0 4 21 -3 -14 }}

Optimal tunings:
* WE: ~2 = 1200.2867{{c}}, ~21/16 = 475.5498{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4393{{c}}

{{Optimal ET sequence|legend=0| 5, 48, 53 }}

Badness (Sintel): 1.99

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

Comma list: 99/98, 275/273, 385/384, 572/567

Mapping: {{mapping| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}

Optimal tunings:
* WE: ~2 = 1200.3416{{c}}, ~21/16 = 475.5998{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4696{{c}}

{{Optimal ET sequence|legend=0| 5, 48f, 53 }}

Badness (Sintel): 1.65

== Subfourth ==
Subfourth tempers out [[10976/10935]] and may be described as the {{nowrap| 58 & 63 }} temperament, more notable in the higher limits than the lower as it supplies a lot of essentially tempered chords there, including everything from [[parapyth]]. Among the good tunings are [[121edo]] and [[179edo]] using the 179ef val in the 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 65536/64827

{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1804{{c}}, ~21/16 = 475.6659{{c}}
: [[error map]]: {{val| -0.820 +0.709 +0.113 +0.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.0019{{c}}
: error map: {{val| 0.000 +2.052 +1.617 +3.168 }}

{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcdd }}

[[Badness]] (Sintel): 3.56

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

Comma list: 540/539, 896/891, 12005/11979

Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}

Optimal tunings:
* WE: ~2 = 1199.0801{{c}}, ~21/16 = 475.6303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 476.0088{{c}}

{{Optimal ET sequence|legend=0| 58, 121, 179e, 300bdee, 479bcddeee }}

Badness (Sintel): 1.50

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

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}

Optimal tunings:
* WE: ~2 = 1199.0747{{c}}, ~21/16 = 475.6291{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 476.0113{{c}}

{{Optimal ET sequence|legend=0| 58, 121, 179ef, 300bdeef }}

Badness (Sintel): 0.983

== Lemongrass ==
Lemongrass tempers out [[245/243]] and may be described as the {{nowrap| 63 & 68 }} temperament. Characterized by a sharper generator than septimal buzzard, lemongrass compresses the septimal comma so much that the syntonic comma is no longer equated with it but with twice of it, or the large septimal diesis. [[68edo]] itself is a great tuning for this, though [[63edo]] and [[73edo]] are also possible.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 65536/64827

{{Mapping|legend=1| 1 0 17 4 | 0 4 26 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0957{{c}}, ~21/16 = 476.0857{{c}}
: [[error map]]: {{val| -0.904 +2.388 -0.851 -0.700 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.4221{{c}}
: error map: {{val| 0.000 +3.733 +0.660 +1.908 }}

{{Optimal ET sequence|legend=1| 5, …, 63, 68 }}

[[Badness]] (Sintel): 2.90

= Weak extensions =
== Demibuzzard ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum#Demibuzzard]].''
Demibuzzard may be described as the {{nowrap| 10 & 53 }} temperament. It is generated by a submajor third; note that in the data below, the generator is the [[octave complement]], a supraminor sixth, since two of it minus an octave make buzzard's generator of ~21/16. The ploidacot for this temperament is epsilon-octacot.

This temperament naturally comes about from a structure in edos like [[43edo|43-]], [[53edo|53-]], and [[63edo]] where two flattened ~[[13/8]] intervals reach the buzzard generator of ~21/16, two of which produce a semitritave that can here be equated to [[26/15]] – providing a mapping of 5 significantly less complex than the [[vulture]] mapping – and two of those finally reach [[3/1]].

It diverges into two extensions for prime 11: submajor ({{nowrap| 53 & 63 }}) favoring sharp fifths, and interpental ({{nowrap| 43 & 53 }}), favoring flat fifths; the two mappings meet at [[53edo]].

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 51200/50421

{{Mapping|legend=1| 1 -4 10 7 | 0 8 -11 -6 }}
: mapping generators: ~2, ~80/49

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7399{{c}}, ~80/49 = 837.5637{{c}}
: [[error map]]: {{val| -0.260 -0.405 -2.116 +3.971 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 837.7471{{c}}
: error map: {{val| 0.000 +0.022 -1.532 +4.691 }}

{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }}

[[Badness]] (Sintel): 1.53

==== 2.3.5.7.13 subgroup ====
{{See also| Greater tendoneutralic }}

Subgroup: 2.3.5.7.13

Comma list: 169/168, 225/224, 640/637

Mapping: {{mapping| 1 -4 10 7 3 | 0 8 -11 -6 1 }}

Optimal tunings:
* WE: ~2 = 1199.9444{{c}}, ~13/8 = 837.7178{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.7569{{c}}

{{Optimal ET sequence|legend=0| 10, 33, 43, 53 }}

Badness (Sintel): 0.847

=== Submajor ===
Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 6655/6561

Mapping: {{mapping| 1 -4 10 7 -14 | 0 8 -11 -6 25 }}

Optimal tunings:
* WE: ~2 = 1200.0666{{c}}, ~44/27 = 837.9460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/27 = 837.9000{{c}}

{{Optimal ET sequence|legend=0| 10, 43e, 53, 116 }}

Badness (Sintel): 1.67

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

Comma list: 169/168, 225/224, 275/273, 385/384

Mapping: {{mapping| 1 -4 10 7 -14 3 | 0 8 -11 -6 25 1 }}

Optimal tunings:
* WE: ~2 = 1200.1769{{c}}, ~13/8 = 838.0187{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.8965{{c}}

{{Optimal ET sequence|legend=0| 10, 43e, 53, 116 }}

Badness (Sintel): 1.14

=== Interpental ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 51200/50421

Mapping: {{mapping| 1 -4 10 7 23 | 0 8 -11 -6 -28 }}

Optimal tunings:
* WE: ~2 = 1199.9381{{c}}, ~80/49 = 838.5389{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 837.5832{{c}}

{{Optimal ET sequence|legend=0| 43, 53, 96 }}

Badness (Sintel): 1.71

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

Comma list: 99/98, 169/168, 176/175, 640/637

Mapping: {{mapping| 1 -4 10 7 23 3 | 0 8 -11 -6 -28 1 }}

Optimal tunings:
* WE: ~2 = 1200.1048{{c}}, ~13/8 = 837.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.5964{{c}}

{{Optimal ET sequence|legend=0| 43, 53, 96 }}

Badness (Sintel): 1.23

== Thuja ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Thuja]].''

Thuja tempers out 126/125 and may be described as the {{nowrap| 15 & 43 }} temperament. The generator is a somewhat sharp fourth, which may be taken as a ~11/8 in the 11-limit, and three minus an octave make buzzard's generator of ~21/16. The ploidacot for this temperament is epsilon-dodecacot.

Thuja can be extended up to the 29-limit, with a simple and accurate approximation to 29, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 65536/64827

{{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }}
: mapping generators: ~2, ~175/128

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.7356{{c}}, ~175/128 = 558.0168{{c}}
: [[error map]]: {{val| -1.264 -0.696 +3.770 +0.172 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 558.5795{{c}}
: error map: {{val| 0.000 +0.999 +6.584 +3.959 }}

{{Optimal ET sequence|legend=1| 15, 43, 58 }}

[[Badness]] (Sintel): 2.24

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

Comma list: 126/125, 176/175, 1344/1331

Mapping: {{mapping| 1 -4 0 7 3 | 0 12 5 -9 1 }}

Optimal tunings:
* WE: ~2 = 1198.5470{{c}}, ~11/8 = 557.9433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5942{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58 }}

Badness (Sintel): 1.09

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

Comma list: 126/125, 144/143, 176/175, 364/363

Mapping: {{mapping| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}

Optimal tunings:
* WE: ~2 = 1198.5083{{c}}, ~11/8 = 557.8942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5565{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58 }}

Badness (Sintel): 0.944

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 176/175, 221/220, 256/255

Mapping: {{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}

Optimal tunings:
* WE: ~2 = 1198.8533{{c}}, ~11/8 = 557.9750{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4979{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58, 101e, 159cdef }}

Badness (Sintel): 1.14

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220

Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}

Optimal tunings:
* WE: ~2 = 1198.6460{{c}}, ~11/8 = 557.8736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4905{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58h, 101eh }}

Badness (Sintel): 1.15

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}

Optimal tunings:
* WE: ~2 = 1198.4488{{c}}, ~11/8 = 557.7999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5086{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58hi }}

Badness (Sintel): 1.19

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 3 | 0 12 5 -9 1 23 -17 7 -1 4 }}

Optimal tunings:
* WE: ~2 = 1198.5114{{c}}, ~11/8 = 557.8276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5079{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58hi }}

Badness (Sintel): 1.15

== Anthoine ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Anthoine]].''

Anthoine is generated by [[5/4]] and tempers out [[3125/3087]] in addition to the buzzardsma; note that the data below shows the octave complement generator, ~8/5, so that buzzard's generator is found at 5 generators up. It is most notable as the {{nowrap| 25 & 28 }} temperament and as the chain of 5/4's present in 53edo. Its ploidacot is 13-sheared-20-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3125/3087, 65536/64827

{{Mapping|legend=1| 1 -12 3 13 | 0 20 -1 -15 }}
: mapping generators: ~2, ~8/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6282{{c}}, ~8/5 = 814.9050{{c}}
: [[error map]]: {{val| -0.372 +0.605 -2.334 +2.767 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 815.1546{{c}}
: error map: {{val| 0.000 +1.138 -1.468 +3.854 }}

{{Optimal ET sequence|legend=1| 25, 53, 184, 237d }}

[[Badness]] (Sintel): 4.57

[[Category:Temperament clans]]
[[Category:Buzzardsmic clan| ]] 
[[Category:Rank 2]]
[[Category:Listen]]

Buzzardsmic clan

2026-02-08T16:05:19Z

Lériendil: 2.3.5.7.13 submajor -> demibuzzard

{{Technical data page}}
The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''buzzardsmic clan''' is the buzzardsma, [[65536/64827]], with [[monzo]] {{monzo| 16 -3 0 -4 }}, which implies that the tritave, [[3/1]], is divided into four intervals each representing a [[21/16]] subfourth. [[Tempering out]] this comma implies a sharpened [[7/1|7th]] [[harmonic]], and especially a sharpened [[~]]21/16 generator, which approaches the 480{{c}} fourth of [[5edo]].

Extensions of buzzard to incorporate [[prime interval|prime]] [[5/1|5]] along its chain of generators (and therefore the full [[7-limit]]) include septimal buzzard ({{nowrap| 53 & 58 }}), which tempers out [[1728/1715]] (and [[5120/5103]]); subfourth ({{nowrap| 58 & 63 }}), which tempers out [[10976/10935]]; and lemongrass ({{nowrap| 63 & 68 }}), which tempers out [[245/243]]. All are considered below.

Weak extensions include submajor ({{nowrap| 10 & 53 }}), which tempers out [[225/224]] and splits [[32/21]] (the superfifth) in two; and thuja ({{nowrap| 15 & 43 }}), which tempers out [[126/125]] and splits [[21/8]] into three.

Full 7-limit temperaments discussed elsewhere are:
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Cohemimabila]]'' (+3136/3125) → [[Mabila family #Cohemimabila|Mabila family]]

The rest are considered below.

= 2.3.7 subgroup =
== Buzzard ==
{{Main| Buzzard }}

[[Subgroup]]: 2.3.7

[[Comma list]]: 65536/64827

{{Mapping|legend=1| 1 0 4 | 0 4 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2548{{c}}, ~21/16 = 475.5761{{c}}
: [[error map]]: {{val| -0.745 +0.350 +1.465 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.8328{{c}}
: error map: {{val| 0.000 +1.376 +3.676 }}

{{Optimal ET sequence|legend=1| 5, 33, 38, 43, 48, 53, 58 }}

[[Badness]] (Sintel): 0.824

= Strong extensions =
== Septimal buzzard ==
{{Main| Buzzard }}
{{See also| Vulture family }}

Septimal buzzard is not only a naturally motivated extension to 2.3.7 buzzard, but the main extension to [[vulture]] of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~[[21/16]], though buzzard is powerful as a full 13-limit system in its own right. It is most naturally described as {{nowrap| 53 & 58 }} (though [[48edo]] is an interesting higher-damage tuning of it for some purposes). As one might expect, [[111edo]] (111 = 53 + 58) is a great tuning for it. [[Mos scale]]s of 5, 8, 13, 18, 23, 28, 33, 38, 43, 48 or 53 notes are available.

Its 13-limit [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[847/845|S11/S13]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. [[Hemifamity]] leverages it by splitting [[36/35]] into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]]. Then the vanishing of the orwellisma implies [[49/48]], the large septimal diesis, is equated with 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 1728/1715, 5120/5103

{{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3061{{c}}, ~21/16 = 475.3611{{c}}
: [[error map]]: {{val| -0.694 -0.511 +0.432 +2.315 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.6144{{c}}
: error map: {{val| 0.000 +0.503 +1.589 +4.331 }}

{{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}

[[Badness]] (Sintel): 1.21

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

Comma list: 176/175, 540/539, 5120/5103

Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }}

Optimal tunings:
* WE: ~2 = 1199.2516{{c}}, ~21/16 = 475.4037{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6806{{c}}

{{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}

Badness (Sintel): 1.14

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

Comma list: 176/175, 351/350, 540/539, 676/675

Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}

Optimal tunings:
* WE: ~2 = 1199.2391{{c}}, ~21/16 = 475.3956{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6760{{c}}

{{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}

Badness (Sintel): 0.779

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 442/441, 540/539

Mapping: {{mapping| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}

Optimal tunings:
* WE: ~2 = 1199.2723{{c}}, ~21/16 = 475.4039{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6837{{c}}

{{Optimal ET sequence|legend=0| 53, 58, 111 }}

Badness (Sintel): 0.938

==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539

Mapping: {{mapping| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}

Optimal tunings:
* WE: ~2 = 1199.2457{{c}}, ~21/16 = 475.3797{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6690{{c}}

{{Optimal ET sequence|legend=0| 53, 58h, 111 }}

Badness (Sintel): 0.952

=== Buteo ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 2200/2187

Mapping: {{mapping| 1 0 -6 4 9 | 0 4 21 -3 -14 }}

Optimal tunings:
* WE: ~2 = 1200.2867{{c}}, ~21/16 = 475.5498{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4393{{c}}

{{Optimal ET sequence|legend=0| 5, 48, 53 }}

Badness (Sintel): 1.99

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

Comma list: 99/98, 275/273, 385/384, 572/567

Mapping: {{mapping| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}

Optimal tunings:
* WE: ~2 = 1200.3416{{c}}, ~21/16 = 475.5998{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4696{{c}}

{{Optimal ET sequence|legend=0| 5, 48f, 53 }}

Badness (Sintel): 1.65

== Subfourth ==
Subfourth tempers out [[10976/10935]] and may be described as the {{nowrap| 58 & 63 }} temperament, more notable in the higher limits than the lower as it supplies a lot of essentially tempered chords there, including everything from [[parapyth]]. Among the good tunings are [[121edo]] and [[179edo]] using the 179ef val in the 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 65536/64827

{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1804{{c}}, ~21/16 = 475.6659{{c}}
: [[error map]]: {{val| -0.820 +0.709 +0.113 +0.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.0019{{c}}
: error map: {{val| 0.000 +2.052 +1.617 +3.168 }}

{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcdd }}

[[Badness]] (Sintel): 3.56

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

Comma list: 540/539, 896/891, 12005/11979

Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}

Optimal tunings:
* WE: ~2 = 1199.0801{{c}}, ~21/16 = 475.6303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 476.0088{{c}}

{{Optimal ET sequence|legend=0| 58, 121, 179e, 300bdee, 479bcddeee }}

Badness (Sintel): 1.50

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

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}

Optimal tunings:
* WE: ~2 = 1199.0747{{c}}, ~21/16 = 475.6291{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 476.0113{{c}}

{{Optimal ET sequence|legend=0| 58, 121, 179ef, 300bdeef }}

Badness (Sintel): 0.983

== Lemongrass ==
Lemongrass tempers out [[245/243]] and may be described as the {{nowrap| 63 & 68 }} temperament. Characterized by a sharper generator than septimal buzzard, lemongrass compresses the septimal comma so much that the syntonic comma is no longer equated with it but with twice of it, or the large septimal diesis. [[68edo]] itself is a great tuning for this, though [[63edo]] and [[73edo]] are also possible.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 245/243, 65536/64827

{{Mapping|legend=1| 1 0 17 4 | 0 4 26 -3 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0957{{c}}, ~21/16 = 476.0857{{c}}
: [[error map]]: {{val| -0.904 +2.388 -0.851 -0.700 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.4221{{c}}
: error map: {{val| 0.000 +3.733 +0.660 +1.908 }}

{{Optimal ET sequence|legend=1| 5, …, 63, 68 }}

[[Badness]] (Sintel): 2.90

= Weak extensions =
== Demibuzzard ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum#Submajor]].''
Demibuzzard may be described as the {{nowrap| 10 & 53 }} temperament. It is generated by a submajor third; note that in the data below, the generator is the [[octave complement]], a supraminor sixth, since two of it minus an octave make buzzard's generator of ~21/16. The ploidacot for this temperament is epsilon-octacot.

This temperament naturally comes about from a structure in edos like [[43edo|43-]], [[53edo|53-]], and [[63edo]] where two flattened ~[[13/8]] intervals reach the buzzard generator of ~21/16, two of which produce a semitritave that can here be equated to [[26/15]] – providing a mapping of 5 significantly less complex than the [[vulture]] mapping – and two of those finally reach [[3/1]].

It diverges into two extensions for prime 11: submajor ({{nowrap| 53 & 63 }}) favoring sharp fifths, and interpental ({{nowrap| 43 & 53 }}), favoring flat fifths; the two mappings meet at [[53edo]].

=== 7-limit ===
[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 51200/50421

{{Mapping|legend=1| 1 -4 10 7 | 0 8 -11 -6 }}
: mapping generators: ~2, ~80/49

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7399{{c}}, ~80/49 = 837.5637{{c}}
: [[error map]]: {{val| -0.260 -0.405 -2.116 +3.971 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 837.7471{{c}}
: error map: {{val| 0.000 +0.022 -1.532 +4.691 }}

{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }}

[[Badness]] (Sintel): 1.53

==== 2.3.5.7.13 subgroup ====
{{See also| Greater tendoneutralic }}

Subgroup: 2.3.5.7.13

Comma list: 169/168, 225/224, 640/637

Mapping: {{mapping| 1 -4 10 7 3 | 0 8 -11 -6 1 }}

Optimal tunings:
* WE: ~2 = 1199.9444{{c}}, ~13/8 = 837.7178{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.7569{{c}}

{{Optimal ET sequence|legend=0| 10, 33, 43, 53 }}

Badness (Sintel): 0.847

=== Submajor ===
Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 6655/6561

Mapping: {{mapping| 1 -4 10 7 -14 | 0 8 -11 -6 25 }}

Optimal tunings:
* WE: ~2 = 1200.0666{{c}}, ~44/27 = 837.9460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/27 = 837.9000{{c}}

{{Optimal ET sequence|legend=0| 10, 43e, 53, 116 }}

Badness (Sintel): 1.67

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

Comma list: 169/168, 225/224, 275/273, 385/384

Mapping: {{mapping| 1 -4 10 7 -14 3 | 0 8 -11 -6 25 1 }}

Optimal tunings:
* WE: ~2 = 1200.1769{{c}}, ~13/8 = 838.0187{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.8965{{c}}

{{Optimal ET sequence|legend=0| 10, 43e, 53, 116 }}

Badness (Sintel): 1.14

=== Interpental ===
Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 51200/50421

Mapping: {{mapping| 1 -4 10 7 23 | 0 8 -11 -6 -28 }}

Optimal tunings:
* WE: ~2 = 1199.9381{{c}}, ~80/49 = 838.5389{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 837.5832{{c}}

{{Optimal ET sequence|legend=0| 43, 53, 96 }}

Badness (Sintel): 1.71

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

Comma list: 99/98, 169/168, 176/175, 640/637

Mapping: {{mapping| 1 -4 10 7 23 3 | 0 8 -11 -6 -28 1 }}

Optimal tunings:
* WE: ~2 = 1200.1048{{c}}, ~13/8 = 837.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.5964{{c}}

{{Optimal ET sequence|legend=0| 43, 53, 96 }}

Badness (Sintel): 1.23

== Thuja ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Thuja]].''

Thuja tempers out 126/125 and may be described as the {{nowrap| 15 & 43 }} temperament. The generator is a somewhat sharp fourth, which may be taken as a ~11/8 in the 11-limit, and three minus an octave make buzzard's generator of ~21/16. The ploidacot for this temperament is epsilon-dodecacot.

Thuja can be extended up to the 29-limit, with a simple and accurate approximation to 29, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 65536/64827

{{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }}
: mapping generators: ~2, ~175/128

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.7356{{c}}, ~175/128 = 558.0168{{c}}
: [[error map]]: {{val| -1.264 -0.696 +3.770 +0.172 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 558.5795{{c}}
: error map: {{val| 0.000 +0.999 +6.584 +3.959 }}

{{Optimal ET sequence|legend=1| 15, 43, 58 }}

[[Badness]] (Sintel): 2.24

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

Comma list: 126/125, 176/175, 1344/1331

Mapping: {{mapping| 1 -4 0 7 3 | 0 12 5 -9 1 }}

Optimal tunings:
* WE: ~2 = 1198.5470{{c}}, ~11/8 = 557.9433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5942{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58 }}

Badness (Sintel): 1.09

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

Comma list: 126/125, 144/143, 176/175, 364/363

Mapping: {{mapping| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}

Optimal tunings:
* WE: ~2 = 1198.5083{{c}}, ~11/8 = 557.8942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5565{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58 }}

Badness (Sintel): 0.944

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 176/175, 221/220, 256/255

Mapping: {{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}

Optimal tunings:
* WE: ~2 = 1198.8533{{c}}, ~11/8 = 557.9750{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4979{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58, 101e, 159cdef }}

Badness (Sintel): 1.14

=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220

Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}

Optimal tunings:
* WE: ~2 = 1198.6460{{c}}, ~11/8 = 557.8736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4905{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58h, 101eh }}

Badness (Sintel): 1.15

=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}

Optimal tunings:
* WE: ~2 = 1198.4488{{c}}, ~11/8 = 557.7999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5086{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58hi }}

Badness (Sintel): 1.19

=== 29-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 3 | 0 12 5 -9 1 23 -17 7 -1 4 }}

Optimal tunings:
* WE: ~2 = 1198.5114{{c}}, ~11/8 = 557.8276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5079{{c}}

{{Optimal ET sequence|legend=0| 15, 43, 58hi }}

Badness (Sintel): 1.15

== Anthoine ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Anthoine]].''

Anthoine is generated by [[5/4]] and tempers out [[3125/3087]] in addition to the buzzardsma; note that the data below shows the octave complement generator, ~8/5, so that buzzard's generator is found at 5 generators up. It is most notable as the {{nowrap| 25 & 28 }} temperament and as the chain of 5/4's present in 53edo. Its ploidacot is 13-sheared-20-cot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 3125/3087, 65536/64827

{{Mapping|legend=1| 1 -12 3 13 | 0 20 -1 -15 }}
: mapping generators: ~2, ~8/5

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6282{{c}}, ~8/5 = 814.9050{{c}}
: [[error map]]: {{val| -0.372 +0.605 -2.334 +2.767 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 815.1546{{c}}
: error map: {{val| 0.000 +1.138 -1.468 +3.854 }}

{{Optimal ET sequence|legend=1| 25, 53, 184, 237d }}

[[Badness]] (Sintel): 4.57

[[Category:Temperament clans]]
[[Category:Buzzardsmic clan| ]] 
[[Category:Rank 2]]
[[Category:Listen]]

Greater tendoneutralisma

2026-02-08T16:04:40Z

Lériendil:

{{Infobox Interval
| Ratio = 815730721/805306368
| Name = greater tendoneutralisma
| Color name = Laquadbitho comma
| Comma = yes
}}
The '''greater tendoneutralisma''' is a [[small comma]] of the 2.3.13 [[subgroup]] which is the amount by which a stack of eight [[16/13]]'s minus two [[octave]]s falls short of [[4/3]]; that is, it is equal to ([[16/3]])/([[16/13]])8 and so equivalently also to ([[13/3]])/([[16/13]])7.

== Temperaments ==
Although the comma is similar in size to something like 81/80, the corresponding temperament is quite accurate because the error can be split evenly over eight 16/13's, so that the pure-3's tuning (very close to [[53edo]]) has 13 off by only 2.78{{cent}}. A more accurate (lower damage) way of achieving the same (finding 3 by stacking 13's) is by tempering the [[lesser tendoneutralisma]]. Very importantly, both are distinct ways of mapping 2.3.13, so that you cannot combine them unless you want to use the trivial tuning of [[10edo]], so that edos > 10 which have a good 3 and 13 will usually pick between one of these two mappings. A much simpler but relatively much higher error way of mapping 3 for those that prefer sharp fifths is by tempering ([[16/13]])2/([[3/2]]) = [[512/507]].

=== Greater Tendoneutralic ===
Tempering out the greater tendoneutralisma in 2.3.13 leads to the highly notable 10 & 53 temperament, where [[10edo]] is the trivial tuning approximately equal to the pure-13's tuning and [[53edo]] is the tuning practically equal to the pure-3's tuning, although [[43edo]] is an interesting choice for combining this temperament with meantone and [[63edo]] is an interesting choice if you prefer slightly sharp fifths. [[Buzzardsmic clan #Demibuzzard|Demibuzzard]] is an extension of this which maps primes 5 and 7; [[Submajor (temperament)|submajor]] and [[interpental]] map prime 11 and thus the full [[13-limit]].

[[Subgroup]]: 2.3.13

[[Comma list]]: 815730721/805306368

{{Mapping|legend=1| 1 4 4 | 0 -8 -1 }}

[[Optimal tuning]] ([[CTE]]): ~16/13 = 362.248{{cent}}

{{Optimal ET sequence|legend=1| 10, 33, 43, 53, 202, 255f, 308f, 361f }}

[[Badness]] (Sintel): 3.037

== See also ==
* [[Lesser tendoneutralisma]]
* [[512/507]]
* [[Tridecapyth comma]]

[[Category:Commas with unknown etymology]]