Xenharmonic Wiki - User contributions [en]

2.3.5.13 subgroup

2026-07-19T20:59:58Z

Overthink: /* Rank-2 temperaments */ is this word needed

The '''2.3.5.13 subgroup''' (a.k.a. ''yatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1|2]], [[3/1|3]], [[5/1|5]], and [[13/1|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.

It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1|7]] and [[11/1|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[neutral (interval quality)|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].

This subgroup is notable for containing the simplest JI representations of [[interseptimal]] intervals, which are halfway between two interval categories, with [[15/13]] being an ultramajor second/inframinor third, [[13/10]] being an ultramajor third/infrafourth, [[20/13]] being an ultrafifth/inframinor sixth, and [[26/15]] being an ultramajor sixth/inframinor seventh. Importantly, 15/13 is close to half of the [[4/3|perfect fourth]], and 26/15 is close to half of the [[3/1|perfect twelfth]], with two intervals of 15/13 falling short of 4/3 by [[676/675]], the island comma.

== Regular temperaments ==
=== Rank-1 temperaments (edos) ===
The 2.3.5.13 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs| '''7''', 10, 12, 15, '''19''', '''34''', '''53''', 130, 140, 164, 183, 217, 270, 354, 388, 407, '''441''', … }}

=== Rank-2 temperaments ===
The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

[[Tridecaschismic]], which adds prime 13 to schismic via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and especially 53edo. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.

Cata and tridecaschismic unite in 53edo.

Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

For those searching [[very high accuracy temperaments|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.

=== Rank-3 temperaments ===
[[Marveltwin]] offers a very low-complexity approximation to the subgroup, reaching [[16/13]] through ([[10/9]])2, and condensing the subgroup into a 5-limit [[planar temperament]].

{[[Catasma|140625/140608]]}, the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, [[407edo]], [[441edo]], [[494edo]], [[901edo]], and of course [[1342edo]].

[[Category:Just intonation subgroups|#]]
[[Category:Rank-4 temperaments|#]]
[[Category:13-limit|#]]

Decatonic scale

2026-07-16T23:35:13Z

Overthink: add an example

{{Wikipedia}}
A '''decatonic scale''' is a [[scale]] with 10 tones per [[equave]].

A well-known decatonic scale is the 10-note [[2L 8s]] [[mos scale]] of [[pajara]] temperament. Here, [[5/4]] and [[6/5]] are major and minor decatonic fourths, while [[7/6]] and [[8/7]] are major and minor decatonic thirds, respectively. As such, the 5/4 and 7/4 in the harmonic seventh chord [[4:5:6:7]] are major intervals, with the corresponding minor chord being the [[70:84:105:120|1/(7:8:10:12)]] tetrad. There is the symmetrical mos version, and the "[[pentachord|pentachordal]]" [[modmos]] scale, with a step pattern ssLsssLsss, where only the 5-step interval (decatonic sixth) violates the mos criterion by having three qualities (diminished, perfect, and augmented) appear in the scale.

Decatonic mos scales may be found at [[Decatonic MOS]]. Other decatonic scales may be found at [[:Category: 10-tone scales]].

== See also ==
* [[Enneatonic scale]]
* [[Hendecatonic scale]]

[[Category:Scale by size]]
[[Category:Terms]]

{{Stub}}

Semiosiris

2026-07-14T16:27:12Z

Overthink: create redirect

#redirect [[Breedsmic temperaments #Semiosiris]]

Gamelismic clan

2026-07-14T03:28:11Z

Overthink: /* Oracle */ use the correct word here

{{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:
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* [[Lemba]] (+50/49) → [[Jubilismic clan #Lemba|Jubilismic clan]]
* [[Trisected]] (+128/125) → [[Augmented family #Trisected|Augmented family]]
* ''[[Echidnic]]'' (+686/675) → [[Diaschismic family #Echidnic|Diaschismic family]]
* [[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, baladic, a weak 2.3.7.13.17-subgroup extension, and gigapyth, a 2.3.7.85-subgroup extension, considered in [[#Other subgroup extensions]]. Dicussed elsewhere is [[Subgroup temperaments #Trisect|trisect]] in the 2.3.7.11/5 subgroup.

=== Radon ===
{{See also|Chromatic pairs #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

==== 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

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 105/104, 120/119, 144/143, 154/153, 170/169, 210/209

{{Todo|complete temperament data|inline=1}}

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 105/104, 120/119, 144/143, 154/153, 161/160, 170/169, 210/209

{{Todo|complete temperament data|inline=1}}

==== 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

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 243/242, 273/272, 286/285, 375/374

{{Todo|complete temperament data|inline=1}}

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 162/161, 210/209, 225/224, 231/230, 243/242, 273/272, 286/285

{{Todo|complete temperament data|inline=1}}

==== 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

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 243/242, 273/272, 325/324, 343/342

{{Todo|complete temperament data|inline=1}}

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 325/324, 343/342

{{Todo|complete temperament data|inline=1}}

==== 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

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 225/224, 243/242, 273/272

{{Todo|complete temperament data|inline=1}}

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 169/168, 208/207, 210/209, 221/220, 225/224, 243/242, 273/272

{{Todo|complete temperament data|inline=1}}

==== 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

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list:

{{Todo|complete temperament data|inline=1}}

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list:

{{Todo|complete temperament data|inline=1}}

===== 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, 385/384, 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

===== 19-limit =====
Subgroup: 2.3.5.7.11.13.17.19

Comma list: 210/209, 225/224, 243/242, 273/272, 385/384, 2200/2197

{{Todo|complete temperament data|inline=1}}

===== 23-limit =====
Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 210/209, 225/224, 243/242, 273/272, 300/299, 385/384, 1105/1104

{{Todo|complete temperament data|inline=1}}

=== 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 ===
The name is a portmanteau of [[orwell]] and [[miracle]]: Oracle is a weak extension of 7-limit miracle, splitting its ~16/15 generator and an octave into two ~16/11 generators. Additionally, when [[restriction|restricted]] to the 2.15.7.11 subgroup, oracle's generator corresponds to 2 stacked orwell generators.

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. Therefore, its [[ploidacot]] is 6-sheared 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 }}

Restles may be described as the {{nowrap| 77 & 87 }} temperament, and has a [[ploidacot]] signature of wau-dodecacot. It was named by [[Petr Pařízek]] in 2011 for it is some sort of opposite to [[beatles]]<ref name="petr's long post">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.

[[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 ==
Cryptically named by [[Petr Pařízek]] in 2011<ref name="petr's long post"/>, lagaca may be described as the {{nowrap| 10 & 118 }} temperament with a [[ploidacot]] signature of diploid wau-enneacot. The name actually refers to the fact that 12 generator steps in this temperament make ~7/3, where "l", "g", "c" are integers alphabetically converted to letters.

[[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]].''

Quartemka may be described as the {{nowrap| 26 & 61 }} temperament. Its [[ploidacot]] is 18-sheared 21-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is close to 1/4 of the generator for [[emka]]<ref name="petr's long post"/>.

[[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]].''

Tritriple may be described as the {{nowrap| 103 & 118 }} temperament. Its [[ploidacot]] is iota-beta-27-cot. It was named by [[Petr Pařízek]] in 2011 for its generator is 1/9 of the generator for [[slendric]], so that 3×3 generators [[octave reduction|octave reduced]] give slendric's generator, and another ×3 give the [[3/2|perfect fifth]]<ref name="petr's long post"/>.

[[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 (2.3.7.13) ===
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 (2.3.7.13) ===
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

=== Gigapyth (2.3.7.85) ===
Subgroup: 2.3.7.85

Comma list: 1029/1024, 7225/7203

Subgroup-val mapping: {{mapping| 1 -2 4 7 | 0 6 -2 -1 }}

Optimal tunings:
* WE: ~2 = 1200.8295{{c}}, ~128/85 = 717.2597{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~128/85 = 716.7933{{c}}

{{Optimal ET sequence|legend=0| 5, 42*, 47, 52, 57, 62, 67, 72, 149*, 370d***, 519bdd***** }}

<nowiki/>* Wart for 85

== References ==

[[Category:Temperament clans]]
[[Category:Gamelismic clan| ]] 
[[Category:Rank 2]]
[[Category:Listen]]

User:FloraC/Temperament extension issues

2026-07-13T20:31:28Z

Overthink: /* Rank-2 temperaments */ add details

{{Editable user page|Please add ideas and name proposals. }}

A list of issues with temperament extensions.

== Open issues ==
=== Rank-2 temperaments ===
; Alphatrident
: 7-limit → 11-limit
: 11-limit alphatrident should be decanonicalized. The tuning is not very representative of 7-limit alphatrident. Specifically, alphatrident has three 11-limit extensions: 176 & 229, 229 & 282, and 282 & 335d. The first and last are supported by 53 which means they are simpler, but they are not representative of the 7-limit optimum. The middle one is representative of the 7-limit optimum, but is very complex. Therefore, the situation is similar to slendric, where none of the extensions should be canon.
: Proposed solution:

; Diminished
: 7-limit → 11-limit
: 11-limit diminished should be decanonicalized. It flips the optimal tuning from flat to sharp of just.
: Proposed solution:

; Gammic
: 7-limit → 11-limit
: 11-limit gammic should be decanonicalized. It's a low-accuracy extension.
: Proposed solution:

; Grendel
: 11-limit → 13-limit
: 13-limit grendel should be decanonicalized. The tuning is not very representative of 11-limit grendel. Specifically, grendel has three 13-limit extensions: 121 & 152f, 152f & 183, and 183 & 214. The first and last are supported by 31 which means they are simpler, but they are not representative of the 11-limit optimum. The middle one is representative of the 11-limit optimum, but somewhat more complex than the last. Therefore, the situation is similar to slendric, where none of the extensions should be canon.
: Proposed solution:

; Hemiwürschmidt
: 7-limit → 11-limit
: 11-limit hemiwürschmidt should be decanonicalized. The 7-limit optimum is between 99 and 130. The 11-limit extension shifts it to somewhere between 130 and 161, and is a significant drop in accuracy. Hemiwürschmidt is also described as a 2.3.5.7.23-subgroup temperament, which is more natural.
: Proposed solution:

; Keemun
: 11-limit → 13-limit
: 13-limit keemun should be decanonicalized. Kema, which maps prime 13 to +14 steps, is the right extension to use, as in all kleismic temperaments. Its edo join is 15 & 19, while the current canonical 13-limit extension is 15f & 19, and 15 is better than 15f. Kema is additionally supported by patent 34, while the current canonical extension uses the much less accurate 34f. The mapping of prime 13 to -5 generators is inaccurate and in the opposite direction compared to primes 3, 5, and 7, making other intervals of 13 more complex.
: Proposed solution:

; Lunatic
: 7-limit → 11-limit
: 11-limit lunatic should be decanonicalized. Lunatic has at least three extensions: 205 & 323, 323 & 441, and 441 & 559. Perhaps, the canonicity of 441 & 559 could be argued on the basis that the 7-limit optimum is very close to 441; however, compared to the good efficiency and portability of the 7-limit temperament, this 11-limit extension is pretty mid.
: Proposed solution:

; Neptune
: 7-limit → 11-limit
: 11-limit neptune should be decanonicalized. It's a low-accuracy extension. Neptune is also described as a 2.3.5.7.17-subgroup temperament, which is more natural. However, the 2.3.5.7.17.31-subgroup extension is a significant drop in accuracy, so that one should at least be dual-named, like ''triceneptune''.
: Proposed solution:

; Pajara
: 11-limit → 13-limit
: 13-limit pajara should be decanonicalized. It's way too inaccurate. Pajara should be a no-13 17-limit temperament.
: Proposed solution:

; Parakleismic
: 7-limit → 11-limit
: 11-limit parakleismic should be decanonicalized. Extending parakleismic to the 11- and 13-limit is complicated, mainly cuz both 99edo and 118edo have ambiguous interpretations, but paralytic (99e & 118) is the "main" extension here, since septimal parakleismic is really an interpolation between 99 and 118, with 217 giving a good middle ground. Note that 217 is more accurate than both 99 and 118 in the 7-limit, showing that the optimum isn't very near 99. Paralytic uses the 19e val, which is reasonable, and the other extensions zigzag around this segment: 80 & 99e, called paradigmic, 118 & 137d, which is our "undecimal parakleismic" tho the optimum is sharp of 87\118, making it 99 & 118, and finally 61de & 80, called parkleismic, tho the optimum is flat of 59\80, making it 80 & 99.
: Paralytic has four notable 13-limit extensions: 99ef & 118, 99e & 118, 99ef & 118f, and 99e & 118f. Note that 99e & 118 and 99ef & 118f are the more accurate ones, supported by 217, and the other two are less accurate, but use the simpler 19e-form in the 13-limit. The 13-odd-limit Graham complexities of 99e & 118 and 99ef & 118f are 157 and 142, respectively, showing that 99e & 118 isn't simpler than 99ef & 118f; thus 13-limit paralytic needs a distinct name too.
: Paradigmic has two notable 13-limit extensions: 80 & 99e and 80 & 99ef, and like above, 80 & 99e is the simpler 19-form whereas 80 & 99ef is more accurate. 80 & 99ef also supports 13-limit parapyth, which is an extra point. It follows that 13-limit paradigmic needs a distinct name too.
: Proposed solution:

; Secund
: 11-limit → 13-limit
: 13-limit secund should be decanonicalized, for the other extension is obviously way better.
: Proposed solution: delete 13-limit secund. Secundly → 13-limit secund. Status: pending community reactions.

; Tertiaseptal
: 7-limit → 11-limit
: 11-limit tertiaseptal should be decanonicalized. Tertiaseptal has three 11-limit extensions: 109 & 140, 140 & 171, and 171 & 202. Perhaps, the canonicity of 171 & 202 could be argued on the basis that the 7-limit optimum is very close to 171; however, compared to the absolute supremacy of the 7-limit temperament, this 11-limit extension is plain bad. ''Tertia'' and ''tertiaseptia'' are bad-tasted names so we might wanna reconsider these names too.
: Proposed solution:

; Würschmidt
: 5-limit → 7-limit
: 7-limit würschmidt should be decanonicalized. Almost all the notable supporting equal temperaments are dual-7: 34, 65, 96, and 127, and that speaks something about it. Würschmidt is also described as a 2.3.5.23-subgroup temperament, which is more natural.
: Proposed solution:

; Schismatic family
: Various dubious extensions. See [[Talk: Schismatic family #Extensions]]. Pending public reactions.

; Tetracot family
: Various dubious extensions to the 17-limit. See [[Talk: Tetracot family #17- and 19-limit extensions]]. Pending public reactions.

=== Rank-3 temperaments ===
; Agni
: 11-limit → 13-limit
: 13-limit agni should be decanonicalized. There's no reason to favor 31 & 72 over 41 & 72.
: Proposed solution:

; Artemis
: 11-limit → 13-limit
: 13-limit artemis should be decanonicalized, as diana is a very competitive extension. This is a series of awkward cases related to orwell, which doesn't have a simple and accurate 13-limit extension, but since the more accurate one, 31 & 53, was dubbed tridecimal orwell over the simpler 22f & 31, these rank-3 temperaments should not be utterly at odds with that. Related temperaments include guanyin and zeus, both go with the 31 & 53 path.
: Proposed solution:

; Big brother
: 11-limit → 13-limit
: 13-limit big brother should be decanonicalized. See the case with artemis.
: Proposed solution:

== Closed issues ==
=== Rank-2 temperaments ===
; Augmented/augene
: 5-limit → 7-limit
: 7-limit augene should be canonicalized while 13-limit augene should be decanonicalized. Tempering out 126/125 is natural for diaschismic, augmented and diminished, and since both diaschismic and diminished have canonical 7-limit extensions, so should augmented (even tho septimal diaschismic is essentially grandfathered in). Furthermore, the 11-limit extension should remain as canon, for the 11-odd-limit diamond monotone range is the same as 9-odd-limit. The name ''augene'' may remain as an alias for this temperament, but the 13-limit extension should eventually have its name changed cuz that one is much worse.
: Proposed solution: 7-, 11-, and 13-limit augene → augmented (augene). Status: changed.

; Dicot
: 5-limit → 7-limit
: 7-limit dicot should be decanonicalized. It has been clear that it's difficult to discuss these as a single temperament. The 7-limit extension is in fact the best extension of 2.3.7-subgroup {54/49} (in the same way beep is the best extension of bug), known as ''mujannabic'', so it may inherit that name.
: Proposed solution: 7- and 11-limit dicot → mujannabic. Status: changed.

; Ennealimmal
: 7-limit → 11-limit → 13-limit
: 11-limit ennealimmal should be decanonicalized. Further, 13-limit ennealimmal should be decanonicalized with respect to the 11-limit to make way for the other extension to be canon. Compared to the absolute supremacy of the 7-limit temperament, the 11-limit is mid, and the 13-limit is just utterly wrong. This should be obvious from the badness values and the optimal GPV sequences, but also the Graham complexities: 225 for 99e & 171e vs 189 for 99ef & 171ef.
: The names of the ennealimmal extensions demonstrate an absolute devoidness of creativity. The same morpheme combination ''ennealimma'' is re-used again and again, making all the extension names almost undistinguishable. Therefore I believe in rectifying these names we should inject new morphemes into the scene, while keeping the ''ennea-'' part to remind users of their relations. 11-limit ennealimmal (99e & 171e) tempers out the symbiotic comma, so it can be called ''enneabiotic''. Further, since 13-limit 99e & 171e is the worse extension, it can be ''enneabio''. The other high-accuracy 11-limit extension, 99 & 171, tempers out the olympic comma, so it can be called ''ennealympic''. Ennealimnic and ennealiminal were named earlier so I think they can stay.
: ''Semiennealimmal'' is extremely misleading. It tempers out the pine comma so it can be ''ennealimmapine''. This name is a little long but the temp itself isn't very notable anyway.
: Proposed solution: 11-limit ennealimmal, 13-limit ennealimmalis → enneabiotic; 13-, 17-, 19-limit ennealimmal → enneabio; ennealimmia → ennealympic; semiennealimmal → ennealimmapine. Status: changed.

; Hendecatonic
: 7-limit → 11-limit
: 11-limit hendecatonic should be decanonicalized. It's not better than 22 & 99e, known as ''cohendecatonic''.
: Proposed solution: 11-, 13- and 17-limit hendecatonic → hendecaton. Status: changed.

; Ketchup
: 17-limit → 19-limit
: 19-limit ketchup should be decanonicalized. Ketchup should be a no-19 23-limit temperament.
: Proposed solution: replace 19- and 23-limit ketchup with no-19 23-limit. Status: changed.

; Leapday
: 17-limit → 19-limit
: 19-limit leapday should be decanonicalized. Leapday should be a no-19 23-limit temperament.
: Proposed solution: replace 19- and 23-limit leapday and leapling with no-19 23-limit leapday. Status: changed.

; Magus
: 7-limit → 5-limit
: 5-limit magus should be decanonicalized. The 5-limit temperament's optimum is between 43edo and 46edo, which corresponds to the tuning range of amigo (43 & 46), not magus (46 & 49).
: Proposed solution: 5-limit magus → amigo. Status: suspended.

; Misty
: 7-limit → 11-limit
: 11-limit misty should be decanonicalized. Misty is more naturally a 2.3.5.7.17.19-subgroup temperament. Besides, 99 & 111, 99ef & 111, and 87 & 99 are all competitive extensions. Proposing these extensions as murky (87 & 99ef), smoky (87 & 99), hazy (99 & 111), and ashy (99ef & 111).
: Proposed solution: 11-, 13-, 17-, and 19-limit misty → murky, delete mystic (87 & 99e). Status: changed.

; Nessafof
: 7-limit → 11-limit
: 11-limit nessafof should be decanonicalized. Extending nessafof to the 11-limit is complicated since both 84edo and 99edo have ambiguous interpretations (see the case with parakleismic). Currently we have catalogued the two low-complexity 15-form extensions, which makes sense. One is being called ''nessa'' so the other may simply be ''fof''.
: Proposed solution: 11-limit nessafof → fof. Status: changed.

=== Rank-3 temperaments ===
; Aphrodite
: 11-limit → 13-limit
: 13-limit aphrodite should be decanonicalized. It's the worst of the four catalogued extensions. Eros is perhaps the best extension, and based on that, it would make sense to name the other extensions after the Erotes (Eros, Anteros, Himeros, Pothos). However, the other extensions are named after the equivalent deities of Aphrodite in other mythologies, so if we follow that pattern, the temperament in question may be named astarte, after Phoenician goddess Astarte.
: Proposed solution: 13-limit aphrodite → astarte. Status: changed.

; Jubilismic/jubilee
: 7-limit → 11-limit
: Jubilee should be canonicalized. This is obvious. We should spare the name ''jubilee'' for the 2.5.7-subgroup temp, mirroring ''semaphore''.
: Proposed solution: 11-limit jubilee → jubilismic; 2.5.7-subgroup jubilic → jubilee. Status: changed/suspended.

; Laka
: 13-limit → 17-limit
: 17-limit laka should be decanonicalized. Laka should be a no-17 19-limit temperament.
: Proposed solution: delete 17-limit laka. Status: changed.

; Marvel
: 11-limit → 13-limit
: 13-limit marvel should be decanonicalized. Hecate is a very competitive extension, and based on that one could consider naming the temperament in question after Greek deities such as Helios, Hebe, Hermes, or Hephaestus.
: Proposed solution: 13-limit marvel → helios. Status: changed.

; Starling
: 7-limit → 11-limit
: 11-limit starling should be decanonicalized. Thrush is a better extension.
: Proposed solution: undecimal starling → starnova. Status: changed.

59edo

2026-07-12T21:29:18Z

Overthink: /* Intervals */ add these ratios

{{Infobox ET}}
{{ED intro}}

== Theory ==
59edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log25. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].

Using the flat fifth instead of the sharp one allows for the {{nowrap|12 & 35}} temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.

As every other step of [[118edo]], 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the [[50edo|50]] & 59 temperament with a subminor third generator provides an interesting temperament.

=== Odd harmonics ===
{{Harmonics in equal|59|columns=13}}

=== Subsets and supersets ===
59edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]]. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! Steps
! Cents
! Approximate ratios (2.9.5.21.11.39.17-subgroup)
! Ratios of 3, 7, 13 (tending sharp)
! Ratios of 3, 7, 13 (tending flat)
|-
| 0
| 0.0
| [[1/1]]
|
|
|-
| 1
| 20.3
| [[81/80]]
|
|
|-
| 2
| 40.7
| [[40/39]], [[45/44]]
|
|
|-
| 3
| 61.0
| [[27/26]], [[28/27]]
|
|
|-
| 4
| 81.4
| [[21/20]], [[22/21]]
|
|
|-
| 5
| 101.7
| [[17/16]], [[18/17]], [[35/33]]
|
|
|-
| 6
| 122.0
| [[15/14]], [[14/13]]
|
|
|-
| 7
| 142.4
| [[13/12]]
|
|
|-
| 8
| 162.7
| [[11/10]]
|
|
|-
| 9
| 183.1
| [[10/9]]
|
|
|-
| 10
| 203.4
| [[9/8]], [[44/39]]
|
|
|-
| 11
| 223.7
| [[25/22]]
| [[8/7]]
|
|-
| 12
| 244.1
| [[15/13]], [[39/34]]
|
| [[8/7]]
|-
| 13
| 264.4
| [[7/6]], [[64/55]]
|
|
|-
| 14
| 284.7
| [[20/17]], [[33/28]]
|
|
|-
| 15
| 305.1
| [[25/21]]
|
|
|-
| 16
| 325.4
|
|
|
|-
| 17
| 345.8
| [[11/9]], [[39/32]], [[128/105]]
| [[16/13]]
|
|-
| 18
| 366.1
| [[21/17]]
|
| [[16/13]]
|-
| 19
| 386.4
| [[5/4]]
|
|
|-
| 20
| 406.8
| [[81/64]]
|
|
|-
| 21
| 427.1
| [[32/25]], [[50/39]]
|
|
|-
| 22
| 447.5
| [[22/17]], [[35/27]], [[128/99]]
|
|
|-
| 23
| 467.8
| [[21/16]], [[64/49]]
|
|
|-
| 24
| 488.1
| [[45/34]], [[85/64]]
| [[4/3]]
|
|-
| 25
| 508.5
| [[35/26]]
|
| [[4/3]]
|-
| 26
| 528.8
| [[34/25]]
|
|
|-
| 27
| 549.2
| [[11/8]], [[48/35]]
|
|
|-
| 28
| 569.5
| [[25/18]]
|
|
|-
| 29
| 589.8
| [[45/32]], [[128/91]]
|
|
|-
| 30
| 610.2
| [[64/45]], [[91/64]]
|
|
|-
| 31
| 630.5
| [[36/25]]
|
|
|-
| 32
| 650.8
| [[16/11]], [[35/24]]
|
|
|-
| 33
| 671.2
| [[25/17]]
|
|
|-
| 34
| 691.5
| [[52/35]]
|
| [[3/2]]
|-
| 35
| 711.9
| [[68/45]], [[128/85]]
| [[3/2]]
|
|-
| 36
| 732.2
| [[32/21]], [[49/32]]
|
|
|-
| 37
| 752.5
| [[17/11]], [[54/35]], [[99/64]]
|
|
|-
| 38
| 772.9
| [[25/16]], [[39/25]]
|
|
|-
| 39
| 793.2
| [[128/81]]
|
|
|-
| 40
| 813.6
| [[8/5]]
|
|
|-
| 41
| 833.9
| [[34/21]]
|
| [[13/8]]
|-
| 42
| 854.2
| [[18/11]], [[64/39]], [[105/64]]
| [[13/8]]
|
|-
| 43
| 874.6
|
|
|
|-
| 44
| 894.9
| [[42/25]]
|
|
|-
| 45
| 915.3
| [[17/10]], [[56/33]]
|
|
|-
| 46
| 935.6
| [[12/7]], [[55/32]]
|
|
|-
| 47
| 955.9
| [[26/15]], [[68/39]]
|
| [[7/4]]
|-
| 48
| 976.3
| [[44/25]]
| [[7/4]]
|
|-
| 49
| 996.6
| [[16/9]], [[39/22]]
|
|
|-
| 50
| 1016.9
| [[9/5]]
|
|
|-
| 51
| 1037.3
| [[20/11]]
|
|
|-
| 52
| 1057.6
| [[24/13]]
|
|
|-
| 53
| 1078.0
| [[13/7]], [[28/15]]
|
|
|-
| 54
| 1098.3
| [[17/9]], [[32/17]], [[66/35]]
|
|
|-
| 55
| 1118.6
| [[21/11]], [[40/21]]
|
|
|-
| 56
| 1139.0
| [[27/14]], [[52/27]]
|
|
|-
| 57
| 1159.3
| [[39/20]], [[88/45]]
|
|
|-
| 58
| 1179.7
| [[160/81]]
|
|
|-
| 59
| 1200.0
| [[2/1]]
|
|
|}{{Todo|inline=1|complete table}}

== Notation ==

=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].

===== Evo flavor =====
<imagemap>
File:59-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Revo_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

==== Second-best fifth notation ====
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].

===== Evo flavor =====
<imagemap>
File:59b_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59b_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Revo_Sagittal.svg]]
</imagemap>

===== Evo-SZ flavor =====
<imagemap>
File:59b_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.

== Octave stretch or compression ==
59edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

== Scales ==
; [[Porcupine]] scales
* Porcupine[7]: 8 8 8 11 8 8 8
* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')

== Instruments ==
; Lumatone

See [[Lumatone mapping for 59edo]].

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
* [https://www.youtube.com/shorts/unVwXrAWnzI ''icosa - Oliver Buckland (microtonal cover in 59edo)''] (2025)
* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)

; [[Francium]]
* "too powerful if i had social skills" from ''Melancholie'' (2023) – [https://open.spotify.com/track/1J8zDrAstQNKgLnXPjKwdm Spotify] | [https://francium223.bandcamp.com/track/too-powerful-if-i-had-social-skills Bandcamp] | [https://www.youtube.com/watch?v=FyzN0P6icf0 YouTube]
* "Stay Away From The Fog" from ''Void'' (2025) – [https://open.spotify.com/track/6swFGV70cPYwruPrnu3iHX Spotify] | [https://francium223.bandcamp.com/track/stay-away-from-the-fog Bandcamp] | [https://www.youtube.com/watch?v=zVsjM-LRjNo YouTube]

; [[Budjarn Lambeth]]
* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo

[[Category:Porcupine]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]

59edo

2026-07-11T17:25:16Z

Overthink: /* Intervals */ Add these ratios

{{Infobox ET}}
{{ED intro}}

== Theory ==
59edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log25. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].

Using the flat fifth instead of the sharp one allows for the {{nowrap|12 & 35}} temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.

As every other step of [[118edo]], 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the [[50edo|50]] & 59 temperament with a subminor third generator provides an interesting temperament.

=== Odd harmonics ===
{{Harmonics in equal|59|columns=13}}

=== Subsets and supersets ===
59edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]]. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! Steps
! Cents
! Approximate ratios (2.9.5.21.11.39.17-subgroup)
! Ratios of 3, 7, 13 (tending sharp)
! Ratios of 3, 7, 13 (tending flat)
|-
| 0
| 0.0
| [[1/1]]
|
|
|-
| 1
| 20.3
| [[81/80]]
|
|
|-
| 2
| 40.7
| [[40/39]], [[45/44]]
|
|
|-
| 3
| 61.0
| [[27/26]], [[28/27]]
|
|
|-
| 4
| 81.4
| [[21/20]], [[22/21]]
|
|
|-
| 5
| 101.7
| [[17/16]], [[18/17]]
|
|
|-
| 6
| 122.0
| [[15/14]], [[14/13]]
|
|
|-
| 7
| 142.4
| [[13/12]]
|
|
|-
| 8
| 162.7
| [[11/10]]
|
|
|-
| 9
| 183.1
| [[10/9]]
|
|
|-
| 10
| 203.4
| [[9/8]], [[44/39]]
|
|
|-
| 11
| 223.7
| [[25/22]]
| [[8/7]]
|
|-
| 12
| 244.1
| [[15/13]], [[39/34]]
|
| [[8/7]]
|-
| 13
| 264.4
| [[7/6]], [[64/55]]
|
|
|-
| 14
| 284.7
| [[20/17]], [[33/28]]
|
|
|-
| 15
| 305.1
| [[25/21]]
|
|
|-
| 16
| 325.4
|
|
|
|-
| 17
| 345.8
| [[11/9]], [[39/32]], [[128/105]]
| [[16/13]]
|
|-
| 18
| 366.1
| [[21/17]]
|
| [[16/13]]
|-
| 19
| 386.4
| [[5/4]]
|
|
|-
| 20
| 406.8
| [[81/64]]
|
|
|-
| 21
| 427.1
| [[32/25]], [[50/39]]
|
|
|-
| 22
| 447.5
| [[22/17]], [[35/27]], [[128/99]]
|
|
|-
| 23
| 467.8
| [[21/16]], [[64/49]]
|
|
|-
| 24
| 488.1
| [[45/34]], [[85/64]]
| [[4/3]]
|
|-
| 25
| 508.5
| [[35/26]]
|
| [[4/3]]
|-
| 26
| 528.8
| [[34/25]]
|
|
|-
| 27
| 549.2
| [[11/8]]
|
|
|-
| 28
| 569.5
| [[25/18]]
|
|
|-
| 29
| 589.8
| [[45/32]], [[128/91]]
|
|
|-
| 30
| 610.2
| [[64/45]], [[91/64]]
|
|
|-
| 31
| 630.5
| [[36/25]]
|
|
|-
| 32
| 650.8
| [[16/11]]
|
|
|-
| 33
| 671.2
| [[25/17]]
|
|
|-
| 34
| 691.5
| [[52/35]]
|
| [[3/2]]
|-
| 35
| 711.9
| [[68/45]], [[128/85]]
| [[3/2]]
|
|-
| 36
| 732.2
| [[32/21]], [[49/32]]
|
|
|-
| 37
| 752.5
| [[17/11]], [[54/35]], [[99/64]]
|
|
|-
| 38
| 772.9
| [[25/16]], [[39/25]]
|
|
|-
| 39
| 793.2
| [[128/81]]
|
|
|-
| 40
| 813.6
| [[8/5]]
|
|
|-
| 41
| 833.9
| [[34/21]]
|
| [[13/8]]
|-
| 42
| 854.2
| [[18/11]], [[64/39]], [[105/64]]
| [[13/8]]
|
|-
| 43
| 874.6
|
|
|
|-
| 44
| 894.9
| [[42/25]]
|
|
|-
| 45
| 915.3
| [[17/10]], [[56/33]]
|
|
|-
| 46
| 935.6
| [[12/7]], [[55/32]]
|
|
|-
| 47
| 955.9
| [[26/15]], [[68/39]]
|
| [[7/4]]
|-
| 48
| 976.3
| [[44/25]]
| [[7/4]]
|
|-
| 49
| 996.6
| [[16/9]], [[39/22]]
|
|
|-
| 50
| 1016.9
| [[9/5]]
|
|
|-
| 51
| 1037.3
| [[20/11]]
|
|
|-
| 52
| 1057.6
| [[24/13]]
|
|
|-
| 53
| 1078.0
| [[13/7]], [[28/15]]
|
|
|-
| 54
| 1098.3
| [[17/9]], [[32/17]]
|
|
|-
| 55
| 1118.6
| [[21/11]], [[40/21]]
|
|
|-
| 56
| 1139.0
| [[27/14]], [[52/27]]
|
|
|-
| 57
| 1159.3
| [[39/20]], [[88/45]]
|
|
|-
| 58
| 1179.7
| [[160/81]]
|
|
|-
| 59
| 1200.0
| [[2/1]]
|
|
|}{{Todo|inline=1|complete table}}

== Notation ==

=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].

===== Evo flavor =====
<imagemap>
File:59-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Revo_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

==== Second-best fifth notation ====
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].

===== Evo flavor =====
<imagemap>
File:59b_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59b_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Revo_Sagittal.svg]]
</imagemap>

===== Evo-SZ flavor =====
<imagemap>
File:59b_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.

== Octave stretch or compression ==
59edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

== Scales ==
; [[Porcupine]] scales
* Porcupine[7]: 8 8 8 11 8 8 8
* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')

== Instruments ==
; Lumatone

See [[Lumatone mapping for 59edo]].

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
* [https://www.youtube.com/shorts/unVwXrAWnzI ''icosa - Oliver Buckland (microtonal cover in 59edo)''] (2025)
* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)

; [[Francium]]
* "too powerful if i had social skills" from ''Melancholie'' (2023) – [https://open.spotify.com/track/1J8zDrAstQNKgLnXPjKwdm Spotify] | [https://francium223.bandcamp.com/track/too-powerful-if-i-had-social-skills Bandcamp] | [https://www.youtube.com/watch?v=FyzN0P6icf0 YouTube]
* "Stay Away From The Fog" from ''Void'' (2025) – [https://open.spotify.com/track/6swFGV70cPYwruPrnu3iHX Spotify] | [https://francium223.bandcamp.com/track/stay-away-from-the-fog Bandcamp] | [https://www.youtube.com/watch?v=zVsjM-LRjNo YouTube]

; [[Budjarn Lambeth]]
* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo

[[Category:Porcupine]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]

59edo

2026-07-11T17:05:50Z

Overthink: /* Intervals */ Decision made: 13 is dual (plain sets 13/12<14/13 if 3 and 7 are sharp)

{{Infobox ET}}
{{ED intro}}

== Theory ==
59edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log25. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].

Using the flat fifth instead of the sharp one allows for the {{nowrap|12 & 35}} temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.

As every other step of [[118edo]], 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the [[50edo|50]] & 59 temperament with a subminor third generator provides an interesting temperament.

=== Odd harmonics ===
{{Harmonics in equal|59|columns=13}}

=== Subsets and supersets ===
59edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]]. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! Steps
! Cents
! Approximate ratios (2.9.5.21.11.17-subgroup)
! Ratios of 3, 7, 13 (tending sharp)
! Ratios of 3, 7, 13 (tending flat)
|-
| 0
| 0.0
| [[1/1]]
|
|
|-
| 1
| 20.3
| [[81/80]]
|
|
|-
| 2
| 40.7
| [[45/44]]
|
|
|-
| 3
| 61.0
|
|
|
|-
| 4
| 81.4
| [[21/20]], [[22/21]]
|
|
|-
| 5
| 101.7
| [[17/16]], [[18/17]]
|
|
|-
| 6
| 122.0
| [[15/14]]
|
|
|-
| 7
| 142.4
|
|
|
|-
| 8
| 162.7
| [[11/10]]
|
|
|-
| 9
| 183.1
| [[10/9]]
|
|
|-
| 10
| 203.4
| [[9/8]]
|
|
|-
| 11
| 223.7
| [[25/22]]
| [[8/7]]
|
|-
| 12
| 244.1
|
|
| [[8/7]]
|-
| 13
| 264.4
| [[7/6]], [[64/55]]
|
|
|-
| 14
| 284.7
| [[20/17]], [[33/28]]
|
|
|-
| 15
| 305.1
| [[25/21]]
|
|
|-
| 16
| 325.4
|
|
|
|-
| 17
| 345.8
| [[11/9]]
| [[16/13]]
|
|-
| 18
| 366.1
| [[21/17]]
|
| [[16/13]]
|-
| 19
| 386.4
| [[5/4]]
|
|
|-
| 20
| 406.8
| [[81/64]]
|
|
|-
| 21
| 427.1
| [[32/25]]
|
|
|-
| 22
| 447.5
| [[22/17]], [[128/99]]
|
|
|-
| 23
| 467.8
| [[21/16]], [[64/49]]
|
|
|-
| 24
| 488.1
| [[45/34]], [[85/64]]
| [[4/3]]
|
|-
| 25
| 508.5
|
|
| [[4/3]]
|-
| 26
| 528.8
| [[34/25]]
|
|
|-
| 27
| 549.2
| [[11/8]]
|
|
|-
| 28
| 569.5
| [[25/18]]
|
|
|-
| 29
| 589.8
| [[45/32]]
|
|
|-
| 30
| 610.2
| [[64/45]]
|
|
|-
| 31
| 630.5
| [[36/25]]
|
|
|-
| 32
| 650.8
| [[16/11]]
|
|
|-
| 33
| 671.2
| [[25/17]]
|
|
|-
| 34
| 691.5
|
|
| [[3/2]]
|-
| 35
| 711.9
| [[68/45]], [[128/85]]
| [[3/2]]
|
|-
| 36
| 732.2
| [[32/21]], [[49/32]]
|
|
|-
| 37
| 752.5
| [[17/11]], [[99/64]]
|
|
|-
| 38
| 772.9
| [[25/16]]
|
|
|-
| 39
| 793.2
| [[128/81]]
|
|
|-
| 40
| 813.6
| [[8/5]]
|
|
|-
| 41
| 833.9
| [[34/21]]
|
| [[13/8]]
|-
| 42
| 854.2
| [[18/11]]
| [[13/8]]
|
|-
| 43
| 874.6
|
|
|
|-
| 44
| 894.9
| [[42/25]]
|
|
|-
| 45
| 915.3
| [[17/10]], [[56/33]]
|
|
|-
| 46
| 935.6
| [[12/7]], [[55/32]]
|
|
|-
| 47
| 955.9
|
|
| [[7/4]]
|-
| 48
| 976.3
| [[44/25]]
| [[7/4]]
|
|-
| 49
| 996.6
| [[16/9]]
|
|
|-
| 50
| 1016.9
| [[9/5]]
|
|
|-
| 51
| 1037.3
| [[20/11]]
|
|
|-
| 52
| 1057.6
|
|
|
|-
| 53
| 1078.0
| [[28/15]]
|
|
|-
| 54
| 1098.3
| [[17/9]], [[32/17]]
|
|
|-
| 55
| 1118.6
| [[21/11]], [[40/21]]
|
|
|-
| 56
| 1139.0
|
|
|
|-
| 57
| 1159.3
| [[88/45]]
|
|
|-
| 58
| 1179.7
| [[160/81]]
|
|
|-
| 59
| 1200.0
| [[2/1]]
|
|
|}{{Todo|inline=1|complete table}}

== Notation ==

=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].

===== Evo flavor =====
<imagemap>
File:59-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Revo_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

==== Second-best fifth notation ====
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].

===== Evo flavor =====
<imagemap>
File:59b_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59b_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Revo_Sagittal.svg]]
</imagemap>

===== Evo-SZ flavor =====
<imagemap>
File:59b_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.

== Octave stretch or compression ==
59edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

== Scales ==
; [[Porcupine]] scales
* Porcupine[7]: 8 8 8 11 8 8 8
* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')

== Instruments ==
; Lumatone

See [[Lumatone mapping for 59edo]].

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
* [https://www.youtube.com/shorts/unVwXrAWnzI ''icosa - Oliver Buckland (microtonal cover in 59edo)''] (2025)
* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)

; [[Francium]]
* "too powerful if i had social skills" from ''Melancholie'' (2023) – [https://open.spotify.com/track/1J8zDrAstQNKgLnXPjKwdm Spotify] | [https://francium223.bandcamp.com/track/too-powerful-if-i-had-social-skills Bandcamp] | [https://www.youtube.com/watch?v=FyzN0P6icf0 YouTube]
* "Stay Away From The Fog" from ''Void'' (2025) – [https://open.spotify.com/track/6swFGV70cPYwruPrnu3iHX Spotify] | [https://francium223.bandcamp.com/track/stay-away-from-the-fog Bandcamp] | [https://www.youtube.com/watch?v=zVsjM-LRjNo YouTube]

; [[Budjarn Lambeth]]
* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo

[[Category:Porcupine]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]

59edo

2026-07-11T16:51:31Z

Overthink: /* Intervals */ add ratios to first column; start dual columns

{{Infobox ET}}
{{ED intro}}

== Theory ==
59edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log25. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].

Using the flat fifth instead of the sharp one allows for the {{nowrap|12 & 35}} temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.

As every other step of [[118edo]], 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the [[50edo|50]] & 59 temperament with a subminor third generator provides an interesting temperament.

=== Odd harmonics ===
{{Harmonics in equal|59|columns=13}}

=== Subsets and supersets ===
59edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]]. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! Steps
! Cents
! Approximate ratios (2.9.5.21.11.17-subgroup)
!Ratios of 3 and 7 (tending sharp)
!Ratios of 3 and 7 (tending flat)
|-
| 0
| 0.0
| [[1/1]]
|
|
|-
| 1
| 20.3
| [[81/80]]
|
|
|-
| 2
| 40.7
| [[45/44]]
|
|
|-
| 3
| 61.0
|
|
|
|-
| 4
| 81.4
| [[21/20]], [[22/21]]
|
|
|-
| 5
| 101.7
| [[17/16]], [[18/17]]
|
|
|-
| 6
| 122.0
| [[15/14]]
|
|
|-
| 7
| 142.4
|
|
|
|-
| 8
| 162.7
| [[11/10]]
|
|
|-
| 9
| 183.1
| [[10/9]]
|
|
|-
| 10
| 203.4
| [[9/8]]
|
|
|-
| 11
| 223.7
| [[25/22]]
| [[8/7]]
|
|-
| 12
| 244.1
|
|
| [[8/7]]
|-
| 13
| 264.4
| [[7/6]], [[64/55]]
|
|
|-
| 14
| 284.7
| [[20/17]], [[33/28]]
|
|
|-
| 15
| 305.1
| [[25/21]]
|
|
|-
| 16
| 325.4
|
|
|
|-
| 17
| 345.8
| [[11/9]]
|
|
|-
| 18
| 366.1
| [[21/17]]
|
|
|-
| 19
| 386.4
| [[5/4]]
|
|
|-
| 20
| 406.8
| [[81/64]]
|
|
|-
| 21
| 427.1
| [[32/25]]
|
|
|-
| 22
| 447.5
| [[22/17]], [[128/99]]
|
|
|-
| 23
| 467.8
| [[21/16]], [[64/49]]
|
|
|-
| 24
| 488.1
| [[45/34]], [[85/64]]
| [[4/3]]
|
|-
| 25
| 508.5
|
|
| [[4/3]]
|-
| 26
| 528.8
| [[34/25]]
|
|
|-
| 27
| 549.2
| [[11/8]]
|
|
|-
| 28
| 569.5
| [[25/18]]
|
|
|-
| 29
| 589.8
| [[45/32]]
|
|
|-
| 30
| 610.2
| [[64/45]]
|
|
|-
| 31
| 630.5
| [[36/25]]
|
|
|-
| 32
| 650.8
| [[16/11]]
|
|
|-
| 33
| 671.2
| [[25/17]]
|
|
|-
| 34
| 691.5
|
|
| [[3/2]]
|-
| 35
| 711.9
| [[68/45]], [[128/85]]
| [[3/2]]
|
|-
| 36
| 732.2
| [[32/21]], [[49/32]]
|
|
|-
| 37
| 752.5
| [[17/11]], [[99/64]]
|
|
|-
| 38
| 772.9
| [[25/16]]
|
|
|-
| 39
| 793.2
| [[128/81]]
|
|
|-
| 40
| 813.6
| [[8/5]]
|
|
|-
| 41
| 833.9
| [[34/21]]
|
|
|-
| 42
| 854.2
| [[18/11]]
|
|
|-
| 43
| 874.6
|
|
|
|-
| 44
| 894.9
| [[42/25]]
|
|
|-
| 45
| 915.3
| [[17/10]], [[56/33]]
|
|
|-
| 46
| 935.6
| [[12/7]], [[55/32]]
|
|
|-
| 47
| 955.9
|
|
| [[7/4]]
|-
| 48
| 976.3
| [[44/25]]
| [[7/4]]
|
|-
| 49
| 996.6
| [[16/9]]
|
|
|-
| 50
| 1016.9
| [[9/5]]
|
|
|-
| 51
| 1037.3
| [[20/11]]
|
|
|-
| 52
| 1057.6
|
|
|
|-
| 53
| 1078.0
| [[28/15]]
|
|
|-
| 54
| 1098.3
| [[17/9]], [[32/17]]
|
|
|-
| 55
| 1118.6
| [[21/11]], [[40/21]]
|
|
|-
| 56
| 1139.0
|
|
|
|-
| 57
| 1159.3
| [[88/45]]
|
|
|-
| 58
| 1179.7
| [[160/81]]
|
|
|-
| 59
| 1200.0
| [[2/1]]
|
|
|}{{Todo|inline=1|complete table|text=Also figure out what to do about prime 13.}}

== Notation ==

=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].

===== Evo flavor =====
<imagemap>
File:59-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Revo_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

==== Second-best fifth notation ====
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].

===== Evo flavor =====
<imagemap>
File:59b_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59b_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Revo_Sagittal.svg]]
</imagemap>

===== Evo-SZ flavor =====
<imagemap>
File:59b_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.

== Octave stretch or compression ==
59edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

== Scales ==
; [[Porcupine]] scales
* Porcupine[7]: 8 8 8 11 8 8 8
* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')

== Instruments ==
; Lumatone

See [[Lumatone mapping for 59edo]].

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
* [https://www.youtube.com/shorts/unVwXrAWnzI ''icosa - Oliver Buckland (microtonal cover in 59edo)''] (2025)
* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)

; [[Francium]]
* "too powerful if i had social skills" from ''Melancholie'' (2023) – [https://open.spotify.com/track/1J8zDrAstQNKgLnXPjKwdm Spotify] | [https://francium223.bandcamp.com/track/too-powerful-if-i-had-social-skills Bandcamp] | [https://www.youtube.com/watch?v=FyzN0P6icf0 YouTube]
* "Stay Away From The Fog" from ''Void'' (2025) – [https://open.spotify.com/track/6swFGV70cPYwruPrnu3iHX Spotify] | [https://francium223.bandcamp.com/track/stay-away-from-the-fog Bandcamp] | [https://www.youtube.com/watch?v=zVsjM-LRjNo YouTube]

; [[Budjarn Lambeth]]
* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo

[[Category:Porcupine]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]

59edo

2026-07-11T16:01:18Z

Overthink: /* Intervals */ start from a clean slate

{{Infobox ET}}
{{ED intro}}

== Theory ==
59edo's best [[3/2|fifth]] is stretched about 9.91 cents from the just interval, and yet its [[5/4]] is nearly pure (stretched only 0.127{{c}}), as the denominator of a convergent to log25. It is a good [[porcupine]] tuning, giving in fact the [[optimal patent val]] for [[11-limit]] porcupine. This patent val tempers out [[250/243]] in the [[5-limit]], [[64/63]] and [[16875/16807]] in the [[7-limit]], and [[55/54]], [[100/99]] and [[176/175]] in the [[11-limit]].

Using the flat fifth instead of the sharp one allows for the {{nowrap|12 & 35}} temperament, which is a kind of bizarre cousin to [[garibaldi]] with a generator of an approximate 15/14, tuned to the size of a whole tone, rather than a fifth. The flat fifth also acts as a generator for [[flattertone]] temperament in the 59bcd val, a variant of meantone with very flat fifths.

As every other step of [[118edo]], 59edo is an excellent tuning for the 2.9.5.21.11 11-limit [[k*N subgroups|2*59 subgroup]], on which it takes the same tuning and tempers out the same commas. This can be extended to the 19-limit 2*59 subgroup 2.9.5.21.11.39.17.57, for which the [[50edo|50]] & 59 temperament with a subminor third generator provides an interesting temperament.

=== Odd harmonics ===
{{Harmonics in equal|59|columns=13}}

=== Subsets and supersets ===
59edo is the 17th [[prime edo]], following [[53edo]] and before [[61edo]]. As noted above, 118edo is a superset that yields most of the same tuning properties, but it also adds a near-just third harmonic to enable strong full 11-limit tuning.

== Intervals ==
{| class="wikitable center-1 right-2"
|-
! Steps
! Cents
! Approximate ratios (2.5.7/3.9.11.17)
!Ratios of 3 and 7 (tending sharp)
!Ratios of 3 and 7 (tending flat)
|-
| 0
| 0.0
| [[1/1]]
|
|
|-
| 1
| 20.3
|
|
|
|-
| 2
| 40.7
|
|
|
|-
| 3
| 61.0
|
|
|
|-
| 4
| 81.4
|
|
|
|-
| 5
| 101.7
|
|
|
|-
| 6
| 122.0
|
|
|
|-
| 7
| 142.4
|
|
|
|-
| 8
| 162.7
|
|
|
|-
| 9
| 183.1
|
|
|
|-
| 10
| 203.4
|
|
|
|-
| 11
| 223.7
|
|
|
|-
| 12
| 244.1
|
|
|
|-
| 13
| 264.4
|
|
|
|-
| 14
| 284.7
|
|
|
|-
| 15
| 305.1
|
|
|
|-
| 16
| 325.4
|
|
|
|-
| 17
| 345.8
|
|
|
|-
| 18
| 366.1
|
|
|
|-
| 19
| 386.4
|
|
|
|-
| 20
| 406.8
|
|
|
|-
| 21
| 427.1
|
|
|
|-
| 22
| 447.5
|
|
|
|-
| 23
| 467.8
|
|
|
|-
| 24
| 488.1
|
|
|
|-
| 25
| 508.5
|
|
|
|-
| 26
| 528.8
|
|
|
|-
| 27
| 549.2
|
|
|
|-
| 28
| 569.5
|
|
|
|-
| 29
| 589.8
|
|
|
|-
| 30
| 610.2
|
|
|
|-
| 31
| 630.5
|
|
|
|-
| 32
| 650.8
|
|
|
|-
| 33
| 671.2
|
|
|
|-
| 34
| 691.5
|
|
|
|-
| 35
| 711.9
|
|
|
|-
| 36
| 732.2
|
|
|
|-
| 37
| 752.5
|
|
|
|-
| 38
| 772.9
|
|
|
|-
| 39
| 793.2
|
|
|
|-
| 40
| 813.6
|
|
|
|-
| 41
| 833.9
|
|
|
|-
| 42
| 854.2
|
|
|
|-
| 43
| 874.6
|
|
|
|-
| 44
| 894.9
|
|
|
|-
| 45
| 915.3
|
|
|
|-
| 46
| 935.6
|
|
|
|-
| 47
| 955.9
|
|
|
|-
| 48
| 976.3
|
|
|
|-
| 49
| 996.6
|
|
|
|-
| 50
| 1016.9
|
|
|
|-
| 51
| 1037.3
|
|
|
|-
| 52
| 1057.6
|
|
|
|-
| 53
| 1078.0
|
|
|
|-
| 54
| 1098.3
|
|
|
|-
| 55
| 1118.6
|
|
|
|-
| 56
| 1139.0
|
|
|
|-
| 57
| 1159.3
|
|
|
|-
| 58
| 1179.7
|
|
|
|-
| 59
| 1200.0
| [[2/1]]
|
|
|}{{Todo|inline=1|complete table}}

== Notation ==

=== Sagittal notation ===
==== Best fifth notation ====
This notation uses the same sagittal sequence as [[66edo#Sagittal notation|66-EDO]].

===== Evo flavor =====
<imagemap>
File:59-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 743 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 190 106 [[Fractional_3-limit_notation#Bad-fifths_apotome-fraction_notation | apotome-fraction notation]]
rect 190 80 320 106 [[144/143]]
rect 320 80 430 106 [[81/80]]
rect 430 80 570 106 [[1053/1024]]
default [[File:59-EDO_Revo_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this EDO.

==== Second-best fifth notation ====
This notation uses the same sagittal sequence as EDOs [[45edo#Sagittal notation|45]] and [[52edo#Sagittal notation|52]].

===== Evo flavor =====
<imagemap>
File:59b_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 687 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo_Sagittal.svg]]
</imagemap>

===== Revo flavor =====
<imagemap>
File:59b_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 695 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Revo_Sagittal.svg]]
</imagemap>

===== Evo-SZ flavor =====
<imagemap>
File:59b_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 655 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 130 106 [[36/35]]
default [[File:59b_Evo-SZ_Sagittal.svg]]
</imagemap>

Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein–Zimmerman notation.

== Octave stretch or compression ==
59edo’s approximations of 3/1, 7/1 and 11/1 are improved by [[93edt]], a [[Octave stretch|stretched-octave]] version of 59edo. The trade-off is a slightly worse 2/1 and 5/1.

[[ed12|211ed12]] is also a solid stretched-octave option, which improves 59edo's 3/1, doing a little, but not much, damage to most other primes.

If one prefers ''[[Octave shrinking|compressed octaves]]'', then [[ed6|153ed6]] is a viable option. It improves upon 59edo’s 3/1, 7/1 and 13/1 at the cost of a slightly worse 2/1 and 5/1, but substantially worse 11/1.

== Scales ==
; [[Porcupine]] scales
* Porcupine[7]: 8 8 8 11 8 8 8
* Porcupine[15]: 3 5 3 5 3 5 3 5 3 5 3 5 3 5 3
* Porcupine[22]: 3 2 3 3 2 3 3 2 3 3 3 2 3 3 2 3 3 2 3 3 2 3
* [[User:BudjarnLambeth/Antechinus|Antechinus]] (''nonoctave period'')

== Instruments ==
; Lumatone

See [[Lumatone mapping for 59edo]].

== Music ==
; [[Bryan Deister]]
* [https://www.youtube.com/watch?v=-UsnINWSvzo ''Microtonal improvisation in 59edo''] (2025)
* [https://www.youtube.com/shorts/unVwXrAWnzI ''icosa - Oliver Buckland (microtonal cover in 59edo)''] (2025)
* [https://www.youtube.com/shorts/XYr4j6Abwlw ''Le Ciel - Malice Mizer (microtonal cover in 59edo)''] (2026)

; [[Francium]]
* "too powerful if i had social skills" from ''Melancholie'' (2023) – [https://open.spotify.com/track/1J8zDrAstQNKgLnXPjKwdm Spotify] | [https://francium223.bandcamp.com/track/too-powerful-if-i-had-social-skills Bandcamp] | [https://www.youtube.com/watch?v=FyzN0P6icf0 YouTube]
* "Stay Away From The Fog" from ''Void'' (2025) – [https://open.spotify.com/track/6swFGV70cPYwruPrnu3iHX Spotify] | [https://francium223.bandcamp.com/track/stay-away-from-the-fog Bandcamp] | [https://www.youtube.com/watch?v=zVsjM-LRjNo YouTube]

; [[Budjarn Lambeth]]
* [https://youtu.be/YDbqf3g88BE ''The Odd Effects of Breathing the Fairy Dust''] (2026)

; [[Ray Perlner]]
* [https://www.youtube.com/watch?v=JJ4B47S1TUI ''Chinchillian Fugue''] – first mode of the Porcupine[7] scale in 59edo

[[Category:Porcupine]]
[[Category:Listen]]
[[Category:Todo:add rank 2 temperaments table]]

236edo

2026-07-11T15:15:18Z

Overthink: /* Subsets and supersets */ fix error

{{Infobox ET}}
{{ED intro}}

== Theory ==
236edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]]. The 7-limit mapping is worse over that of 118edo in terms of relative error, as it leans on the very sharp side. It tempers out [[6144/6125]] and [[19683/19600]], supporting [[hemischis]]. Using the 236e [[val]] {{val| 236 374 548 663 '''817''' }}, it tempers out [[243/242]], 1375/1372, [[6250/6237]], 14700/14641 and [[16384/16335]].

The 236bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning very close to [[quarter-comma meantone]], although [[205edo]] is even closer. Alternately, sharpening it to 236b gives a fifth that is in the golden [[diaschismic]] sequence.

The 236dghin val (using the second-best mappings of primes 7, 17, 19, 23, and 43) makes for a reasonable flat-tending system as far as the [[43-limit]], and is in fact [[diamond monotone]] in the [[45-odd-limit]]. This works best with a stretched octave.

=== Prime harmonics ===
{{Harmonics in equal|236|columns=11}}
{{Harmonics in equal|236|start=12|columns=11|collapsed=1|title=Approximation of prime harmonics in 236edo (continued)}}

=== Subsets and supersets ===
Since 236 factors into 22 × 59, 236edo has subset edos {{EDOs| 2, 4, 59 and 118 }}. [[472edo]], which doubles it, provides good correction to harmonics 7 and 11.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 6144/6125, 19683/19600, 390625/388962
| {{mapping| 236 374 548 663 }}
| −0.1830
| 0.03883
| 7.64
|}

236edo

2026-07-10T18:10:50Z

Overthink: /* Theory */ found this, actually

{{Infobox ET}}
{{ED intro}}

== Theory ==
236edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]]. The 7-limit mapping is worse over that of 118edo in terms of relative error, as it leans on the very sharp side. It tempers out [[6144/6125]] and [[19683/19600]], supporting [[hemischis]]. Using the 236e [[val]] {{val| 236 374 548 663 '''817''' }}, it tempers out [[243/242]], 1375/1372, [[6250/6237]], 14700/14641 and [[16384/16335]].

The 236bb val (where fifth is flattened by single step, approximately 1/4 comma) gives a tuning very close to [[quarter-comma meantone]], although [[205edo]] is even closer. Alternately, sharpening it to 236b gives a fifth that is in the golden [[diaschismic]] sequence.

The 236dghin val (using the second-best mappings of primes 7, 17, 19, 23, and 43) makes for a reasonable flat-tending system as far as the [[43-limit]], and is in fact [[diamond monotone]] in the [[45-odd-limit]]. This works best with a stretched octave.

=== Prime harmonics ===
{{Harmonics in equal|236|columns=11}}
{{Harmonics in equal|236|start=12|columns=11|collapsed=1|title=Approximation of prime harmonics in 236edo (continued)}}

=== Subsets and supersets ===
Since 236 factors into 22 × 53, 236edo has subset edos {{EDOs| 2, 4, 59 and 118 }}. [[472edo]], which doubles it, provides good correction to harmonics 7 and 11.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 6144/6125, 19683/19600, 390625/388962
| {{mapping| 236 374 548 663 }}
| −0.1830
| 0.03883
| 7.64
|}

15L 6s

2026-07-10T02:14:56Z

Overthink: /* Scale tree */ + temperament interpretation

{{Infobox MOS}}
{{MOS intro}}
== Modes ==
{{MOS modes}}

== Intervals ==
{{MOS intervals}}

== Scale tree ==
{{MOS tuning spectrum
| 3/1 = [[Trisected]]
}}

{{stub}}

Trisected

2026-07-10T02:13:35Z

Overthink: - duplicate link

{{Infobox regtemp
| Title = Trisected
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[128/125]], [[1029/1000]] (7-limit); [[56/55]], [[128/125]], [[1029/1000]] (11-limit); [[56/55]], [[91/90]], [[128/125]], [[1029/1000]] (13-limit)
| Edo join 1 = 15 | Edo join 2 = 36
| Mapping = 3; 3 0 -1 -1 7
| Generators = 10/7
| Generators tuning = 635.0
| Optimization method = CWE
| MOS scales = [[6L 9s]], [[15L 6s]], [[15L 21s]]
| Pergen = (P8/3, P5/3)
| Odd limit 1 = 9 | Mistuning 1 = 16.1 | Complexity 1 = 36
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 17.5 | Complexity 2 = 36
}}
'''Trisected''' is the [[rank-2 temperament]] tempering out [[128/125]], [[1029/1000]], and [[1029/1024]] in the [[7-limit]], making it a member of the [[augmented family]], [[keegic temperaments]], and [[gamelismic clan]]. Since it tempers out 128/125, the [[2/1|octave]] is split into 3 ~[[5/4]]'s, each tuned to 400{{C}} if the octave is pure. Since it tempers out 1029/1024, the [[3/2|perfect fifth]] is split into three intervals of ~[[8/7]]. Since it tempers out 1029/1000, the [[3/1|tritave]] is split into three intervals of [[10/7]]. This means that every [[Pythagorean tuning|Pythagorean]] interval is split into three equal parts.

In the [[11-limit]], the [[4/3|perfect fourth]] is split into three ~[[11/10]]'s, thus tempering out [[4000/3993]]. Additionally, the 1/3-octave period represents [[14/11]], tempering out [[56/55]] and [[176/175]]. The [[13-limit]] extension equates the ~10/7 with [[13/9]], tempering out [[91/90]] and [[2197/2187]].

The 2.3.7.11/5 subgroup [[restriction]], known as [[trisect]], removes the individual mappings for 5 and 11 while still tempering out 1029/1024 and 4000/3993, and is much more accurate.

For technical data, see [[Augmented family #Trisected]].

== Intervals ==
In the following table, odd harmonics 1–21 are in '''bold'''.

{| class="wikitable center-1 right-2 right-4 right-6"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 0.0
| '''1/1'''
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
| 1
| 235.0
| '''8/7'''
| 635.0
| 10/7, '''16/11''', 13/9
| 1035.0
| 20/11
|-
| 2
| 470.0
| '''21/16'''
| 870.0
| 33/20
| 70.0
| 21/20, 33/32
|-
| 3
| 705.0
| '''3/2'''
| 1105.0
| '''15/8''', 21/11
| 305.0
| 6/5
|-
| 4
| 940.0
| 12/7, 26/15
| 140.0
| 12/11, 13/12, 15/14
| 540.0
| 15/11
|-
| 5
| 1175.0
| 63/32
| 375.0
| 26/21
| 775.0
| 52/33
|-
| 6
| 209.9
| 9/8
| 609.9
| 45/32, 63/44
| 1009.9
| 9/5
|-
| 7
| 444.9
| 9/7, 13/10
| 844.9
| 18/11, '''13/8'''
| 44.9
| 36/35
|-
| 8
| 679.9
| 52/35, 72/49
| 1079.9
| 13/7
| 279.9
| 13/11
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced

== Tunings ==
=== Norm-based tunings ===
{{Todo|review}}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/7 = 633.889{{C}}
| CWE: ~10/7 = 634.339{{C}}
| POTE: ~10/7 = 634.476{{C}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/7 = 634.215{{C}}
| CWE: ~10/7 = 634.769{{C}}
| POTE: ~10/7 = 634.893{{C}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/7 = 634.286{{C}}
| CWE: ~10/7 = 634.991{{C}}
| POTE: ~10/7 = 635.144{{C}}
|}

=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~10/7 = 633.282{{C}}
| 7/6
|-
| 9-odd-limit
| ~10/7 = 633.583{{C}}
| 9/7
|-
| 11-odd-limit
| ~10/7 = 633.760{{C}}
| 77/45
|-
| 13-odd-limit
| ~10/7 = 633.962{{C}}
| 13/7
|-
| 15-odd-limit
| ~10/7 = 633.962{{C}}
| 13/7
|-
| 13-limit 21-odd-limit
| ~10/7 = 634.129{{C}}
| 45/44
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| [[7/5]]
| 617.488
|
|-
| [[21edo|11\21]]
|
| 628.571
| Lower bound of 7-odd-limit diamond monotone 21f val
|-
|
| [[15/8]]
| 629.423
|
|-
|
| [[15/14]]
| 629.861
|
|-
|
| [[7/4]]
| 631.174
|
|-
|
| [[7/6]]
| 633.282
| 7-odd-limit minimax
|-
| [[36edo|19\36]]
|
|
| Lower bound of 9- through 13-odd-limit diamond monotone 15-odd-limit diamond monotone (singleton)
|-
|
| [[9/7]]
| 633.583
| 9-odd-limit minimax
|-
|
| [[21/13]]
| 633.949
|
|-
|
| [[13/7]]
| 633.962
| 13- and 15-odd-limit minimax
|-
|
| [[3/2]]
| 633.985
|
|-
|
| [[15/11]]
| 634.238
|
|-
| [[87edo|46\87]]
|
| 634.483
| 87cee val
|-
|
| [[13/8]]
| 634.361
|
|-
|
| [[13/12]]
| 634.643
|
|-
|
| [[11/10]]
| 634.996
|
|-
| [[51edo|27\51]]
|
| 635.294
| 51ce val
|-
|
| [[21/16]]
| 635.390
|
|-
|
| [[11/9]]
| 636.085
|
|-
|
| [[13/11]]
| 636.151
|
|-
|
| [[9/5]]
| 636.266
|
|-
|
| [[13/10]]
| 636.316
|
|-
| [[66edo|35\66]]
|
| 636.364
| 66cef val
|-
|
| [[13/9]]
| 636.618
|
|-
|
| [[11/6]]
| 637.659
|
|-
|
| [[15/13]]
| 638.065
|
|-
|
| [[5/3]]
| 638.547
|
|-
|
| [[21/11]]
| 639.821
|
|-
| [[15edo|8\15]]
|
| 640.000
| Upper bound of 7- through 13-odd-limit diamond monotone
|-
|
| [[21/20]]
| 642.234
|
|}
<nowiki/>* Besides the octave

[[Category:Trisected| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Gamelismic clan]]
[[Category:Keegic temperaments]]

Trisected

2026-07-10T02:12:40Z

Overthink: space

{{Infobox regtemp
| Title = Trisected
| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
| Comma basis = [[128/125]], [[1029/1000]] (7-limit); [[56/55]], [[128/125]], [[1029/1000]] (11-limit); [[56/55]], [[91/90]], [[128/125]], [[1029/1000]] (13-limit)
| Edo join 1 = 15 | Edo join 2 = 36
| Mapping = 3; 3 0 -1 -1 7
| Generators = 10/7
| Generators tuning = 635.0
| Optimization method = CWE
| MOS scales = [[6L 9s]], [[15L 6s]], [[15L 21s]]
| Pergen = (P8/3, P5/3)
| Odd limit 1 = 9 | Mistuning 1 = 16.1 | Complexity 1 = 36
| Odd limit 2 = 13-limit 21 | Mistuning 2 = 17.5 | Complexity 2 = 36
}}
'''Trisected''' is the [[rank-2 temperament]] tempering out [[128/125]], [[1029/1000]], and [[1029/1024]] in the [[7-limit]], making it a member of the [[augmented family]], [[keegic temperaments]], and [[gamelismic clan]]. Since it tempers out 128/125, the [[2/1|octave]] is split into 3 ~[[5/4]]'s, each tuned to 400{{C}} if the octave is pure. Since it tempers out 1029/1024, the [[3/2|perfect fifth]] is split into three intervals of ~[[8/7]]. Since it tempers out [[1029/1000]], the [[3/1|tritave]] is split into three intervals of [[10/7]]. This means that every [[Pythagorean tuning|Pythagorean]] interval is split into three equal parts.

In the [[11-limit]], the [[4/3|perfect fourth]] is split into three ~[[11/10]]'s, thus tempering out [[4000/3993]]. Additionally, the 1/3-octave period represents [[14/11]], tempering out [[56/55]] and [[176/175]]. The [[13-limit]] extension equates the ~10/7 with [[13/9]], tempering out [[91/90]] and [[2197/2187]].

The 2.3.7.11/5 subgroup [[restriction]], known as [[trisect]], removes the individual mappings for 5 and 11 while still tempering out 1029/1024 and 4000/3993, and is much more accurate.

For technical data, see [[Augmented family #Trisected]].

== Intervals ==
In the following table, odd harmonics 1–21 are in '''bold'''.

{| class="wikitable center-1 right-2 right-4 right-6"
|-
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 2
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 0.0
| '''1/1'''
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
| 1
| 235.0
| '''8/7'''
| 635.0
| 10/7, '''16/11''', 13/9
| 1035.0
| 20/11
|-
| 2
| 470.0
| '''21/16'''
| 870.0
| 33/20
| 70.0
| 21/20, 33/32
|-
| 3
| 705.0
| '''3/2'''
| 1105.0
| '''15/8''', 21/11
| 305.0
| 6/5
|-
| 4
| 940.0
| 12/7, 26/15
| 140.0
| 12/11, 13/12, 15/14
| 540.0
| 15/11
|-
| 5
| 1175.0
| 63/32
| 375.0
| 26/21
| 775.0
| 52/33
|-
| 6
| 209.9
| 9/8
| 609.9
| 45/32, 63/44
| 1009.9
| 9/5
|-
| 7
| 444.9
| 9/7, 13/10
| 844.9
| 18/11, '''13/8'''
| 44.9
| 36/35
|-
| 8
| 679.9
| 52/35, 72/49
| 1079.9
| 13/7
| 279.9
| 13/11
|}
<nowiki/>* In 13-limit CWE tuning, octave reduced

== Tunings ==
=== Norm-based tunings ===
{{Todo|review}}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/7 = 633.889{{C}}
| CWE: ~10/7 = 634.339{{C}}
| POTE: ~10/7 = 634.476{{C}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/7 = 634.215{{C}}
| CWE: ~10/7 = 634.769{{C}}
| POTE: ~10/7 = 634.893{{C}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/7 = 634.286{{C}}
| CWE: ~10/7 = 634.991{{C}}
| POTE: ~10/7 = 635.144{{C}}
|}

=== Target tunings ===
{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 7-odd-limit
| ~10/7 = 633.282{{C}}
| 7/6
|-
| 9-odd-limit
| ~10/7 = 633.583{{C}}
| 9/7
|-
| 11-odd-limit
| ~10/7 = 633.760{{C}}
| 77/45
|-
| 13-odd-limit
| ~10/7 = 633.962{{C}}
| 13/7
|-
| 15-odd-limit
| ~10/7 = 633.962{{C}}
| 13/7
|-
| 13-limit 21-odd-limit
| ~10/7 = 634.129{{C}}
| 45/44
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| [[7/5]]
| 617.488
|
|-
| [[21edo|11\21]]
|
| 628.571
| Lower bound of 7-odd-limit diamond monotone 21f val
|-
|
| [[15/8]]
| 629.423
|
|-
|
| [[15/14]]
| 629.861
|
|-
|
| [[7/4]]
| 631.174
|
|-
|
| [[7/6]]
| 633.282
| 7-odd-limit minimax
|-
| [[36edo|19\36]]
|
|
| Lower bound of 9- through 13-odd-limit diamond monotone 15-odd-limit diamond monotone (singleton)
|-
|
| [[9/7]]
| 633.583
| 9-odd-limit minimax
|-
|
| [[21/13]]
| 633.949
|
|-
|
| [[13/7]]
| 633.962
| 13- and 15-odd-limit minimax
|-
|
| [[3/2]]
| 633.985
|
|-
|
| [[15/11]]
| 634.238
|
|-
| [[87edo|46\87]]
|
| 634.483
| 87cee val
|-
|
| [[13/8]]
| 634.361
|
|-
|
| [[13/12]]
| 634.643
|
|-
|
| [[11/10]]
| 634.996
|
|-
| [[51edo|27\51]]
|
| 635.294
| 51ce val
|-
|
| [[21/16]]
| 635.390
|
|-
|
| [[11/9]]
| 636.085
|
|-
|
| [[13/11]]
| 636.151
|
|-
|
| [[9/5]]
| 636.266
|
|-
|
| [[13/10]]
| 636.316
|
|-
| [[66edo|35\66]]
|
| 636.364
| 66cef val
|-
|
| [[13/9]]
| 636.618
|
|-
|
| [[11/6]]
| 637.659
|
|-
|
| [[15/13]]
| 638.065
|
|-
|
| [[5/3]]
| 638.547
|
|-
|
| [[21/11]]
| 639.821
|
|-
| [[15edo|8\15]]
|
| 640.000
| Upper bound of 7- through 13-odd-limit diamond monotone
|-
|
| [[21/20]]
| 642.234
|
|}
<nowiki/>* Besides the octave

[[Category:Trisected| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Gamelismic clan]]
[[Category:Keegic temperaments]]

6L 9s

2026-07-10T02:12:15Z

Overthink: /* Scale tree */ links, + trisected, - exotemp

{{Infobox MOS}}
{{MOS intro}}

== Scale properties ==
{{TAMNAMS use}}

=== Intervals ===
{{MOS intervals}}

=== Generator chain ===
{{MOS genchain}}

=== Modes ===
{{MOS mode degrees}}

== Scale tree ==
{{MOS tuning spectrum
| 4/3 = [[Trisected]] is around here
| 7/3 = [[Oodako]] is around here
| 4/1 = [[Terrain]] is around here
| 9/2 = [[Mirkat]] is around here
}}

{{stub}}

User:Overthink

2026-07-09T19:44:09Z

Overthink: changes.

__NOTOC__
<big>Don't do what my username says. It's a bad habit.</big>

([[User:Overthink/Sandbox|Sandbox]])

== Notes ==
*Proofread before saving edits

== Background ==
I have been playing the piano for a while, though I haven't really been practicing much.

Back in around 2022, I watched the YouTube video [https://www.youtube.com/watch?v=1Hqm0dYKUx4 Why it's impossible to tune a piano] by [https://www.youtube.com/@MinutePhysics minutephysics]. After that, I decided to dig deeper on sound and frequencies, and I also thought about the idea of [[quartertone]]s. I soon found out about Xenharmonic tunings and this wiki.

<youtube>1Hqm0dYKUx4</youtube>

I lost interest in this topic around mid-2023, but I got interested in it again in early 2025. After reading pages for quite a long time, I decided to create an account in September 2025 and contribute to this wiki.

== Facts about me ==
* Likes to use semicolons
* Opinions shift rapidly

''To be expanded''

== What I'm trying to do ==

* None for now

== Subpages ==

{{Special:PrefixIndex/User:Overthink/}}

== External links ==
[https://scratch.mit.edu/projects/1231011747/ Microtonal music player on Scratch] (Haven't really used in a while)

[[Category:User zh-2]]
[[Category:User en-N]]
[[Category:User on Discord]]

Sixtetwoo

2026-07-09T15:07:13Z

Overthink: fix dead links

<pre>
!
Six 7-limit tetrads marvel woo scale with 51 11-limit dyads
! 7-limit transversal [15/14, 7/6, 6/5, 5/4, 9/7, 7/5, 3/2, 8/5, 12/7, 7/4, 15/8, 2]
! Fokblock([225/224, 21/20, 245/192], [6, 8, 5]) <12 19 27 32| Fokker block
! union of tetrads {[-1, 0, -1], [-1, 0, 0], [-1, 0, 1], [0, 0, -1], [0, 0, 0], [0, 0, 1]}
12
!
116.23027
267.51234
316.92773
383.74261
433.15800
584.44007
700.67034
816.90061
933.13088
968.18268
1084.41295
1200.64322
</pre>

Because '''sixtetwoo''' has so many dyads, 12edo music tends to map well to it, though wolf fifths and fourths are likely to turn up. This allows us to get an idea of its capacities by retuning 12edo music.

* [https://web.archive.org/web/20201127012918/http://micro.soonlabel.com/gene_ward_smith/transformers/sixtetwoo/Liszt%20-%20Nuages%20gris%20woo.mp3 Lizst - Nuages gris] 2:10
* [https://web.archive.org/web/20201127014149/http://micro.soonlabel.com/gene_ward_smith/transformers/sixtetwoo/Liszt%20-%20Atonal%20Bagatelle%20woo.mp3 Lizst - Bagatelle sans tonalité] 2:50
* [https://web.archive.org/web/20201127014011/http://micro.soonlabel.com/gene_ward_smith/transformers/sixtetwoo/Delius%20-%20First%20Cuckoo%20woo.mp3 Delius - On Hearing the First Cuckoo in Spring] 6:13
* [https://web.archive.org/web/20201127014622/http://micro.soonlabel.com/gene_ward_smith/transformers/sixtetwoo/Reger%20-%20Clarinet%20Quintet%20Op.%20146%20woo.mp3 Reger - Clarinet Quintet Op. 146] 35:53

[[Category:12-tone scales]]
[[Category:Fokker blocks]]
[[Category:11-limit]]
[[Category:Pages with Scala files]]
[[Category:Listen]]

Miracle

2026-07-09T03:20:20Z

Overthink: slendric

{{Interwiki
| en = Miracle
| de = Miracle
}}
{{Infobox regtemp
| Title = Miracle
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[225/224]], [[1029/1024]] (7-limit); [[225/224]], [[243/242]], [[385/384]] (11-limit)
| Edo join 1 = 31 | Edo join 2 = 41
| Generators = 15/14 | Generators tuning = 116.7 | Optimization method = CTE
| MOS scales = …, [[1L 9s]], [[10L 1s]], [[10L 11s]], [[10L 21s]]
| Mapping = 1; 6 -7 -2 15
| Pergen = (P8, P5/6)
| Odd limit 1 = 9 | Mistuning 1 = 3.32 | Complexity 1 = 21
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 4.86 | Complexity 2 = 31
}}
{{Wikipedia|Miracle temperament}}
'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]] (the [[slendric]] generator), and a stack of seven generators represents [[8/5]]. It is a member of the [[marvel temperaments]], by [[tempering out]] [[225/224]], the [[gamelismic clan]], by tempering out [[1029/1024]], and the [[breedsmic temperaments]], by tempering out [[2401/2400]]. It is naturally a full [[11-limit]] temperament, treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]]. It is supported by the highly notable [[EDO|edos]] [[31edo|31]], [[41edo|41]], and [[72edo|72]], with 72edo being an especially good tuning. (There is an alternative mapping for 11 known as [[revelation]], but there is little reason to use it unless you are using [[31edo]], in which case it is identical to miracle anyway.)

Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.

[[Rastmic clan|Some temperaments]] have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and [[Gamelismic clan|other temperaments]] have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts.

Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]]~[[49/48]]~[[50/49]]~[[55/54]]~[[56/55]]~[[64/63]] all [[tempered]] together.

See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.

== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.

{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 116.6
| 15/14, '''16/15'''
|-
| 2
| 233.3
| '''8/7'''
|-
| 3
| 349.9
| 11/9
|-
| 4
| 466.6
| '''21/16'''
|-
| 5
| 583.2
| 7/5
|-
| 6
| 699.9
| '''3/2'''
|-
| 7
| 816.5
| '''8/5'''
|-
| 8
| 933.2
| 12/7
|-
| 9
| 1049.8
| 11/6
|-
| 10
| 1166.5
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
|-
| 11
| 83.1
| 21/20, 22/21
|-
| 12
| 199.8
| '''9/8'''
|-
| 13
| 316.4
| 6/5
|-
| 14
| 433.1
| 9/7
|-
| 15
| 549.7
| '''11/8'''
|-
| 16
| 666.3
| 22/15
|-
| 17
| 783.0
| 11/7
|-
| 18
| 899.6
| 27/16, 42/25
|-
| 19
| 1016.3
| 9/5
|-
| 20
| 1132.9
| 27/14, 48/25
|-
| 21
| 49.6
| 33/32, 36/35
|-
| 22
| 166.2
| 11/10
|-
| 23
| 282.9
| 33/28
|-
| 24
| 399.5
| 44/35
|-
| 25
| 516.2
| 27/20
|-
| 26
| 632.8
| 36/25
|-
| 27
| 749.5
| 54/35, 77/50
|-
| 28
| 866.1
| 33/20
|-
| 29
| 982.8
| 44/25
|-
| 30
| 1099.4
| 66/35
|-
| 31
| 16.1
| 81/80, 99/98, 121/120
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced

== Chords ==
{{Main| Chords of miracle }}

== Scales ==
{{See also| Miracle 10 MODMOS }}

; Mos scales
* [[Miracle 10|Miracle{{lbrack}}10{{rbrack}}]] – 72edo tuning
* [[Blackjack|Blackjack (miracle{{lbrack}}21{{rbrack}})]] – 72edo tuning
* [[Blackwoo]]
; Transversal scales
* [[Miracle21trans]]
* [[Miracle21trans511]]
* [[Miracle31trans]]
* [[Miracle31trans511]]
; Others
* [[Mir1]] – 6-tone scale, 72edo tuning
* [[Mir2]] – 6-tone scale, 72edo tuning
* [[Miracle 8]] – 8-tone scale, 72edo tuning
* [[Miracle 12]] – 12-tone scale, 72edo tuning
* [[Miracle 12a]] – 12-tone scale, 72edo tuning
* [[Miracle 24hi]] – 24-tone scale, 72edo tuning
* [[Miracle 24lo]] – 24-tone scale, 72edo tuning

== Tunings ==
[[File:Derivation of the secor.png|thumb|600px|right|A diagram taken from George Secor's article "The Miracle Temperament and Decimal Keyboard" which was published in Xenharmonikôn 18 (2006). Highlighting the error band and adding arrows was done for clarity by Douglas Blumeyer on Dave Keenan's request.]]

Displayed on the right is a chart of the tuning spectrum of miracle by how the odd harmonics up to 11 are tuned, showing the minimax generator, i.e. the secor.

=== 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
|-
! Equilateral
| CEE: ~15/14 = 116.5155{{c}}
| CSEE: ~15/14 = 116.5612{{c}}
| POEE: ~15/14 = 116.6465{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.6772{{c}}
| CWE: ~15/14 = 116.6756{{c}}
| POTE: ~15/14 = 116.6752{{c}}
|-
! Benedetti, Wilson
| CBE: ~15/14 = 116.7297{{c}}
| CSBE: ~15/14 = 116.7136{{c}}
| POBE: ~15/14 = 116.6903{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~15/14 = 116.6868{{c}}
| CSEE: ~15/14 = 116.6304{{c}}
| POEE: ~15/14 = 116.5817{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.7112{{c}}
| CWE: ~15/14 = 116.6469{{c}}
| POTE: ~15/14 = 116.6327{{c}}
|-
! Benedetti, Wilson
| CBE: ~15/14 = 116.7355{{c}}
| CSBE: ~15/14 = 116.6768{{c}}
| POBE: ~15/14 = 116.6643{{c}}
|}

=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.588{{c}}
| 5/3
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
| 7-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
| 9-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.721{{c}}
| {{Monzo| 0 117 -44 -19 }}
|-
| 11-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.672{{c}}
| {{Monzo| 0 17 -11 -6 11 }}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
| [[21edo|2\21]]
|
| 114.286
| Lower bound of 7-odd-limit diamond monotone
|-
|
| 7/4
| 115.587
|
|-
|
| 11/9
| 115.803
|
|-
| [[31edo|3\31]]
|
| 116.129
| Lower bound of 9- and 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone
|-
|
| 5/4
| 116.241
|
|-
|
| 21/11
| 116.412
|
|-
|
| 15/11
| 116.441
|
|-
|
| 7/5
| 116.502
|
|-
| [[103edo|10\103]]
|
| 116.505
|
|-
|
| 5/3
| 116.588
| 5- and 7-odd-limit, 11-limit 15- and 21-odd-limit minimax
|-
|
| 11/10
| 116.591
|
|-
|
| 11/6
| 116.596
|
|-
|
| 11/7
| 116.617
|
|-
|
| 7/6
| 116.641
|
|-
| [[72edo|7\72]]
|
| 116.667
|
|-
|
| 9/5
| 116.716
| 9- and 11-odd-limit minimax, Secor's definition of secor
|-
|
| 11/8
| 116.755
|
|-
|
| 21/20
| 116.770
|
|-
|
| 9/7
| 116.792
|
|-
| [[113edo|11\113]]
|
| 116.814
|
|-
|
| 3/2
| 116.993
|
|-
| [[41edo|4\41]]
|
| 117.073
| Upper bound of 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone
|-
|
| 21/16
| 117.695
|
|-
|
| 15/14
| 119.443
|
|-
| [[10edo|1\10]]
|
| 120.000
| Upper bound of 7- and 9-odd-limit diamond monotone
|}
<nowiki/>* Besides the octave

== Music ==
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/realm-of-possibility ''Realm of Possibility''] (2021) – in Miracle[31] with a 116.72-cent generator and 1200.53-cent octave

; [[Joseph Pehrson]]
* ''Blackjack'' (2001) – [https://web.archive.org/web/20201127013023/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 play] | [https://soundclick.com/share.cfm?id=706344 SoundClick] – in [[Blackjack|Blackjack (Miracle{{lbrack}}21{{rbrack}})]]
* ''Blacklight'' (2002) – [https://web.archive.org/web/20201127015033/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 play] | [https://soundclick.com/share.cfm?id=710783 SoundClick] – in Blackjack (Miracle[21])
* ''Black and Jill'' (2003) – in Blackjack (Miracle[21])
** Soprano version – [https://web.archive.org/web/20201127012730/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 play] | [https://soundclick.com/share.cfm?id=2373400 SoundClick]
** [https://soundclick.com/share.cfm?id=9583778 Udderbot version]
* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – in Blackjack (Miracle[21])

; [[Gene Ward Smith]]
* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – in Blackjack (Miracle[21]), 175edo tuning

== External links ==
* [https://x31eq.com/decimal_lattice.htm ''Lattices with Decimal Notation''] by [[Graham Breed]]

[[Category:Miracle| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Gamelismic clan]]

Miracle

2026-07-09T03:18:01Z

Overthink: important comma

{{Interwiki
| en = Miracle
| de = Miracle
}}
{{Infobox regtemp
| Title = Miracle
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[225/224]], [[1029/1024]] (7-limit); [[225/224]], [[243/242]], [[385/384]] (11-limit)
| Edo join 1 = 31 | Edo join 2 = 41
| Generators = 15/14 | Generators tuning = 116.7 | Optimization method = CTE
| MOS scales = …, [[1L 9s]], [[10L 1s]], [[10L 11s]], [[10L 21s]]
| Mapping = 1; 6 -7 -2 15
| Pergen = (P8, P5/6)
| Odd limit 1 = 9 | Mistuning 1 = 3.32 | Complexity 1 = 21
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 4.86 | Complexity 2 = 31
}}
{{Wikipedia|Miracle temperament}}
'''Miracle''' is a [[regular temperament]] discovered by [[George Secor]] in 1974 which splits a tempered [[3/2]] into six [[generator]]s, called ''[[secor]]s'' (after George), that serve as both [[15/14]] and [[16/15]] semitones. A stack of two generators represents [[8/7]], and a stack of seven generators represents [[8/5]]. It is a member of both the [[marvel temperaments]], by [[tempering out]] [[225/224]], the [[gamelismic clan]], by tempering out [[1029/1024]], and the [[breedsmic temperaments]], by tempering out [[2401/2400]]. It is naturally a full [[11-limit]] temperament, treating the neutral third from three generators as [[11/9]], tempering out [[243/242]], [[385/384]], [[441/440]], and [[540/539]]. It is supported by the highly notable [[EDO|edos]] [[31edo|31]], [[41edo|41]], and [[72edo|72]], with 72edo being an especially good tuning. (There is an alternative mapping for 11 known as [[revelation]], but there is little reason to use it unless you are using [[31edo]], in which case it is identical to miracle anyway.)

Miracle is an exceptionally efficient linear temperament. It is quite accurate, with [[TOP]] error only 0.63 [[cent]]s/[[octave]], meaning intervals of the [[11-odd-limit]] [[tonality diamond]] are represented with only one or two cents of error. Yet it is also very low-complexity (efficient), as evidenced by the high density of 11-odd-limit ratios in the [[#Interval chain]]. At least one inversion of every interval in the 11-odd-limit tonality diamond is represented within 22 secors of the starting value.

[[Rastmic clan|Some temperaments]] have 11/9 as a neutral third, meaning it is exactly half of a 3/2 (tempering out 243/242), and [[Gamelismic clan|other temperaments]] have 8/7 as exactly a third of 3/2. Miracle is distinguished by doing both of these things at the same time, so 3/2 is divided into six equal parts.

Miracle can also be thought of as a [[cluster temperament]] with 10 clusters of notes in an octave. The small chroma interval between adjacent notes in each cluster is very versatile, representing [[45/44]]~[[49/48]]~[[50/49]]~[[55/54]]~[[56/55]]~[[64/63]] all [[tempered]] together.

See [[Miracle extensions]] for [[13-limit]] and [[17-limit]] extensions. See [[Gamelismic clan #Miracle]] for technical data.

== Interval chain ==
In the following table, odd harmonics and subharmonics 1–21 are labeled in '''bold'''.

{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 116.6
| 15/14, '''16/15'''
|-
| 2
| 233.3
| '''8/7'''
|-
| 3
| 349.9
| 11/9
|-
| 4
| 466.6
| '''21/16'''
|-
| 5
| 583.2
| 7/5
|-
| 6
| 699.9
| '''3/2'''
|-
| 7
| 816.5
| '''8/5'''
|-
| 8
| 933.2
| 12/7
|-
| 9
| 1049.8
| 11/6
|-
| 10
| 1166.5
| 49/25, 55/28, 63/32, 88/45, 96/49, 108/55
|-
| 11
| 83.1
| 21/20, 22/21
|-
| 12
| 199.8
| '''9/8'''
|-
| 13
| 316.4
| 6/5
|-
| 14
| 433.1
| 9/7
|-
| 15
| 549.7
| '''11/8'''
|-
| 16
| 666.3
| 22/15
|-
| 17
| 783.0
| 11/7
|-
| 18
| 899.6
| 27/16, 42/25
|-
| 19
| 1016.3
| 9/5
|-
| 20
| 1132.9
| 27/14, 48/25
|-
| 21
| 49.6
| 33/32, 36/35
|-
| 22
| 166.2
| 11/10
|-
| 23
| 282.9
| 33/28
|-
| 24
| 399.5
| 44/35
|-
| 25
| 516.2
| 27/20
|-
| 26
| 632.8
| 36/25
|-
| 27
| 749.5
| 54/35, 77/50
|-
| 28
| 866.1
| 33/20
|-
| 29
| 982.8
| 44/25
|-
| 30
| 1099.4
| 66/35
|-
| 31
| 16.1
| 81/80, 99/98, 121/120
|}
<nowiki/>* In 11-limit [[CWE tuning]], octave reduced

== Chords ==
{{Main| Chords of miracle }}

== Scales ==
{{See also| Miracle 10 MODMOS }}

; Mos scales
* [[Miracle 10|Miracle{{lbrack}}10{{rbrack}}]] – 72edo tuning
* [[Blackjack|Blackjack (miracle{{lbrack}}21{{rbrack}})]] – 72edo tuning
* [[Blackwoo]]
; Transversal scales
* [[Miracle21trans]]
* [[Miracle21trans511]]
* [[Miracle31trans]]
* [[Miracle31trans511]]
; Others
* [[Mir1]] – 6-tone scale, 72edo tuning
* [[Mir2]] – 6-tone scale, 72edo tuning
* [[Miracle 8]] – 8-tone scale, 72edo tuning
* [[Miracle 12]] – 12-tone scale, 72edo tuning
* [[Miracle 12a]] – 12-tone scale, 72edo tuning
* [[Miracle 24hi]] – 24-tone scale, 72edo tuning
* [[Miracle 24lo]] – 24-tone scale, 72edo tuning

== Tunings ==
[[File:Derivation of the secor.png|thumb|600px|right|A diagram taken from George Secor's article "The Miracle Temperament and Decimal Keyboard" which was published in Xenharmonikôn 18 (2006). Highlighting the error band and adding arrows was done for clarity by Douglas Blumeyer on Dave Keenan's request.]]

Displayed on the right is a chart of the tuning spectrum of miracle by how the odd harmonics up to 11 are tuned, showing the minimax generator, i.e. the secor.

=== 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
|-
! Equilateral
| CEE: ~15/14 = 116.5155{{c}}
| CSEE: ~15/14 = 116.5612{{c}}
| POEE: ~15/14 = 116.6465{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.6772{{c}}
| CWE: ~15/14 = 116.6756{{c}}
| POTE: ~15/14 = 116.6752{{c}}
|-
! Benedetti, Wilson
| CBE: ~15/14 = 116.7297{{c}}
| CSBE: ~15/14 = 116.7136{{c}}
| POBE: ~15/14 = 116.6903{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~15/14 = 116.6868{{c}}
| CSEE: ~15/14 = 116.6304{{c}}
| POEE: ~15/14 = 116.5817{{c}}
|-
! Tenney
| CTE: ~15/14 = 116.7112{{c}}
| CWE: ~15/14 = 116.6469{{c}}
| POTE: ~15/14 = 116.6327{{c}}
|-
! Benedetti, Wilson
| CBE: ~15/14 = 116.7355{{c}}
| CSBE: ~15/14 = 116.6768{{c}}
| POBE: ~15/14 = 116.6643{{c}}
|}

=== Target tunings ===
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="white-space: nowrap;" | Target tunings
|-
! rowspan="2" | Target
! colspan="2" | Minimax
! colspan="2" | Least squares
|-
! Generator
! Eigenmonzo*
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~16/15 = 116.588{{c}}
| 5/3
| ~16/15 = 116.578{{c}}
| {{Monzo| 0 -19 20 }}
|-
| 7-odd-limit
| ~15/14 = 116.588{{c}}
| 5/3
| ~15/14 = 116.573{{c}}
| {{Monzo| 0 -27 25 5 }}
|-
| 9-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.721{{c}}
| {{Monzo| 0 117 -44 -19 }}
|-
| 11-odd-limit
| ~15/14 = 116.716{{c}}
| 9/5
| ~15/14 = 116.672{{c}}
| {{Monzo| 0 17 -11 -6 11 }}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
| [[21edo|2\21]]
|
| 114.286
| Lower bound of 7-odd-limit diamond monotone
|-
|
| 7/4
| 115.587
|
|-
|
| 11/9
| 115.803
|
|-
| [[31edo|3\31]]
|
| 116.129
| Lower bound of 9- and 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone
|-
|
| 5/4
| 116.241
|
|-
|
| 21/11
| 116.412
|
|-
|
| 15/11
| 116.441
|
|-
|
| 7/5
| 116.502
|
|-
| [[103edo|10\103]]
|
| 116.505
|
|-
|
| 5/3
| 116.588
| 5- and 7-odd-limit, 11-limit 15- and 21-odd-limit minimax
|-
|
| 11/10
| 116.591
|
|-
|
| 11/6
| 116.596
|
|-
|
| 11/7
| 116.617
|
|-
|
| 7/6
| 116.641
|
|-
| [[72edo|7\72]]
|
| 116.667
|
|-
|
| 9/5
| 116.716
| 9- and 11-odd-limit minimax, Secor's definition of secor
|-
|
| 11/8
| 116.755
|
|-
|
| 21/20
| 116.770
|
|-
|
| 9/7
| 116.792
|
|-
| [[113edo|11\113]]
|
| 116.814
|
|-
|
| 3/2
| 116.993
|
|-
| [[41edo|4\41]]
|
| 117.073
| Upper bound of 11-odd-limit, 11-limit 15- and 21-odd-limit diamond monotone
|-
|
| 21/16
| 117.695
|
|-
|
| 15/14
| 119.443
|
|-
| [[10edo|1\10]]
|
| 120.000
| Upper bound of 7- and 9-odd-limit diamond monotone
|}
<nowiki/>* Besides the octave

== Music ==
; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/realm-of-possibility ''Realm of Possibility''] (2021) – in Miracle[31] with a 116.72-cent generator and 1200.53-cent octave

; [[Joseph Pehrson]]
* ''Blackjack'' (2001) – [https://web.archive.org/web/20201127013023/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blackjack.mp3 play] | [https://soundclick.com/share.cfm?id=706344 SoundClick] – in [[Blackjack|Blackjack (Miracle{{lbrack}}21{{rbrack}})]]
* ''Blacklight'' (2002) – [https://web.archive.org/web/20201127015033/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/josephpehrson+blacklight.mp3 play] | [https://soundclick.com/share.cfm?id=710783 SoundClick] – in Blackjack (Miracle[21])
* ''Black and Jill'' (2003) – in Blackjack (Miracle[21])
** Soprano version – [https://web.archive.org/web/20201127012730/http://clones.soonlabel.com/public/micro/gene_ward_smith/Others/Pehrson/blackandjill.mp3 play] | [https://soundclick.com/share.cfm?id=2373400 SoundClick]
** [https://soundclick.com/share.cfm?id=9583778 Udderbot version]
* [https://soundclick.com/share.cfm?id=2623155 ''Inner Voices''] (2005) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=6593353 ''Transpian''] (2006) – in Blackjack (Miracle[21])
* [https://soundclick.com/share.cfm?id=5049231 ''microproj''] (2007) – in Blackjack (Miracle[21])

; [[Gene Ward Smith]]
* ''Rachmaninoff Plays Blackjack'' (archived 2010) – [http://www.archive.org/details/RachmaninoffPlaysBlackjack detail] | [http://www.archive.org/download/RachmaninoffPlaysBlackjack/rachman.mp3 play] – in Blackjack (Miracle[21]), 175edo tuning

== External links ==
* [https://x31eq.com/decimal_lattice.htm ''Lattices with Decimal Notation''] by [[Graham Breed]]

[[Category:Miracle| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Marvel temperaments]]
[[Category:Gamelismic clan]]

Negri

2026-07-09T03:10:08Z

Overthink: add other important commas

{{interwiki
| en = Negri
| de = Negri
| es =
| ja =
}}
{{Infobox regtemp
| Title = Negri
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.13
| Comma basis = [[16875/16384]] (2.3.5); [[49/48]], [[225/224]] (2.3.5.7); [[49/48]], [[65/64]], [[91/90]] (2.3.5.7.13)
| Edo join 1 = 10 | Edo join 2 = 19
| Mapping = 1; -4 3 -2 -3
| Generators = 16/15 | Generators tuning = 125.4 | Optimization method = CWE
| MOS scales = [[1L 8s]], [[9L 1s]], [[10L 9s]]
| Pergen = (P8, P4/4)
| Color name = Laquadyoti
| Odd limit 1 = 7 | Mistuning 1 = 17.8 | Complexity 1 = 8
| Odd limit 2 = 2.3.5.7.13 15 | Mistuning 2 = 17.8 | Complexity 2 = 19
}}
'''Negri''' is a [[regular temperament]] generated by a [[generator]] of approximately 125 [[cent]]s, which can be identified with a tempered [[16/15]], such that:
* Two of them make a tempered [[7/6]]~[[8/7]]~[[15/13]];
* Three of them make a tempered [[5/4]]~[[16/13]];
* Four of them make a tempered [[4/3]].

It is most naturally viewed as a [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament, [[tempering out]] [[49/48]], [[65/64]] and [[91/90]]. This is sometimes called '''negra''', and it is realized consistently in [[19edo]] and [[29edo]]. Other [[edo]]s which may be usable as a negri or negra tuning include [[9edo]], [[10edo]], [[28edo]], [[47edo]], and [[48edo]], all of which are [[consistent]] through (at least) the [[5-odd-limit]], since in the broadest sense, negri is defined as tempering out the [[negri comma]] in the [[5-limit]]. Other important commas that negri tempers out include [[105/104]], [[169/168]], [[196/195]], and [[225/224]]. The 7-limit version can thus be viewed as joining with the [[marvel]] and [[semaphore]] temperament families.

See [[Semaphoresmic clan #Negri]] for technical data. For discussion on the various 11-limit extensions, see [[Negri extensions]].

== Interval chain ==
In the following table, odd harmonics and subharmonics 1–13 are in '''bold'''.

{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.0
| '''1/1'''
|-
| 1
| 125.4
| 13/12, 14/13, 15/14, 16/15
|-
| 2
| 250.7
| 7/6, '''8/7''', 15/13
|-
| 3
| 376.1
| '''5/4''', '''16/13'''
|-
| 4
| 501.4
| '''4/3'''
|-
| 5
| 626.8
| 10/7, 13/9
|-
| 6
| 752.1
| 14/9, 20/13, 32/21
|-
| 7
| 877.5
| 5/3
|-
| 8
| 1002.8
| '''16/9'''
|-
| 9
| 1128.2
| 35/18, 40/21, 52/27
|-
| 10
| 53.5
| 25/24, 28/27, 50/49, 64/63
|}
<nowiki/>* In 2.3.5.7.13-subgroup [[CWE tuning]]

== Scales ==
Negri forms 9-note and 10-note [[mos scale]]s, Negri[9] and Negri[10], at [[1L 8s]] and [[9L 1s]] respectively. In [[19edo]], the negri generator is the diatonic half-step of 2\19, which allows these mosses to be written fairly simply in conventional notation. For example, the ssssLssss mode of 19edo could be written as E F Gb G# A B C Db D# E. This mode is particularly useful as it has identical ssss pentachords (analogous to the [[tetrachord]]s of classical Greek music theory) on the 1/1 and 3/2.

[[File:19edoNegriSymmetricalMajor.mp3|Negri[9], major mode ssssLssss, in 19edo tuning]]

Negri[9], major mode ssssLssss, in 19edo tuning

It is also notable in Negri[9] that a subset of these notes form the E double harmonic major scale, E F G# A B C D# E, which features in a wide variety of world musical traditions. In fact, all modes of Negri[9] and Negri[10] contain at least one mode of the double harmonic scale as a subset.

Another useful mode of Negri[9] is Lssssssss, which in 19edo would be A B C Db D# E F Gb G# A. This has a minor triad (A–C–E) for a tonic chord, which can be extended to a 7-limit utonal tetrad (A–C–E–Gb), as well as 7-limit otonal tetrads on E and F that can function as, respectively, a dominant seventh chord and a German augmented sixth chord. This scale also contains the popular Hungarian minor mode of the double harmonic scale, A B C D# E F G# A.

4 of the 9 modes of Negri[9] are like the Locrian mode of the diatonic major scale in that they do not have a note a perfect 5th above the tonic. These are more difficult to apply conventional music theory to. However even in these modes there are a number of chords built on the tonic that can provide a measure of consonance and stability, such as 13:16:20:24 and 6:7:8.

Negri[10] also has a number of useful features. One of these features is the fact that it makes 4:5:6 and 10:12:15 share the same "shape" of generic intervals in the scale (as in other rank-2 decatonic scales such as [[pajara]] and [[blackwood]] scales; this is because 5/4 and 6/5 get tempered to the same interval in [[10edo]]).

== History and terminology ==
Negri was named by [[Paul Erlich]] in 2001<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_31054.html#31065 Yahoo! Tuning Group | ''The grooviest linear temperaments for 7-limit music'']</ref> after John Negri's 10-out-of-19 maximally even scale<ref>"The Nineteen-Tone System as Ten Plus Nine". [https://interval.xentonic.org/tables-of-contents.html ''Interval, Journal of Music Research and Development''], pp. 11–13 of Volume 5, Number 3 (Winter 1986–1987). John Negri.</ref>. It used to be known by distinct names in the 5- and 7-limit as ''negripent'' and ''negrisept'', respectively (for more information on this, see [[Temperament names #Temperament naming examples]]). It was also earlier known as ''quadrafourths'' and ''tertiathirds''.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_3774#3780 Yahoo! Tuning Group | ''25 best weighted generator steps 5-limit temperaments''] – "I'm calling this tertiathirds (was quadrafourths)." —Dave Keenan</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_41392#41396 Yahoo! Tuning Group | ''! middle-path 7-limit tetradic scales for kalle''] – "Negri [is the new name for quadrafourths]." —Gene Ward Smith</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_12957.html#12970 Yahoo! Tuning Group | ''98 named 7-limit temperaments''] – "[Negri] aka 'tertiathirds', 'negrisept' (MP)" —Herman Miller</ref>

== 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
|-
! Equilateral
| CEE: ~15/14 = 124.602{{c}}
| CSEE: ~15/14 = 125.284{{c}}
| POEE: ~15/14 = 125.468{{c}}
|-
! Tenney
| CTE: ~15/14 = 124.813{{c}}
| CWE: ~15/14 = 125.435{{c}}
| POTE: ~15/14 = 125.608{{c}}
|-
! Benedetti, Wilson
| CBE: ~15/14 = 124.874{{c}}
| CSBE: ~15/14 = 125.429{{c}}
| POBE: ~15/14 = 125.629{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.7.13-subgroup norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~14/13 = 123.471{{c}}
| CSEE: ~14/13 = 124.672{{c}}
| POEE: ~14/13 = 125.528{{c}}
|-
! Tenney
| CTE: ~14/13 = 124.457{{c}}
| CWE: ~14/13 = 125.354{{c}}
| POTE: ~14/13 = 125.567{{c}}
|-
! Benedetti, Wilson
| CBE: ~14/13 = 124.756{{c}}
| CSBE: ~14/13 = 125.428{{c}}
| POBE: ~14/13 = 125.616{{c}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]
! Generator (¢)
! Comments
|-
|
| 15/8
| 111.731
|
|-
|
| 7/4
| 115.587
|
|-
|
| 15/14
| 119.443
|
|-
|
| 13/8
| 119.824
|
|-
| 1\10
|
| 120.000
| Lower bound of 7-, 9-odd-limit, and 2.3.5.7.13-subgroup 13-odd-limit diamond monotone
|-
|
| 7/5
| 123.498
|
|-
|
| 15/13
| 123.871
|
|-
| 3\29
|
| 124.138
|
|-
|
| 13/10
| 124.298
|
|-
|
| 3/2
| 124.511
| 7- and 9-odd-limit minimax
|-
| 5\48
|
| 125.000
| 48df val
|-
|
| 10/9
| 125.673
|
|-
| 2\19
|
| 126.316
| Upper bound of 9-odd-limit and 2.3.5.7.13-subgroup 13-odd-limit diamond monotone
|-
|
| 5/3
| 126.337
| 5-odd-limit minimax
|-
|
| 13/9
| 127.324
|
|-
|
| 9/7
| 127.486
|
|-
| 5\47
|
| 127.660
| 47df val
|-
|
| 13/7
| 128.298
|
|-
| 3\28
|
| 128.571
| 28df val
|-
|
| 5/4
| 128.771
|
|-
| 1\9
|
| 133.333
| Upper bound of 7-odd-limit diamond monotone
|-
|
| 7/6
| 133.435
|
|-
|
| 13/12
| 138.573
|
|}

== See also ==
* [[Intervals of Negri-9]]
* [[Modes of Negri-9]]

== Music ==
; [[Mike Battaglia]]
* [https://youtu.be/bCLZChG6U6c ''Negri Comma Pump'']

; [[Sebastian Dumitrescu]]
* [https://soundcloud.com/sedumitr/la-multi-ani-19_edo ''La Mulți Ani''] ([http://micro.soonlabel.com/gene_ward_smith/Others/Dumitrescu/__La_Mul_i_Ani__negri_10___19edo__by_Sebastian_Dumitrescu.mp3 play]{{dead link}}) – Negri[10] in 19edo tuning

; [[Lillian Hearne]]
* [https://soundcloud.com/lillianhearne/negri-shmegri ''Negri Shmegri''] ([http://micro.soonlabel.com/gene_ward_smith/Others/Hearne/Negri%20Shmegri.mp3 play]{{dead link}}) – Negri[9] symmetric mode in 19edo

; [[Herman Miller]]
* [https://soundcloud.com/morphosyntax-1/without-a-clue ''Without a Clue''] (2024)

; [[Ray Perlner]]
* [https://www.youtube.com/playlist?list=PLkW9S8bpltfy3qYhWKO2vyloaMGiH4JtN ''Negri-9 Modal Fugues''] (YouTube playlist)

== References ==
<references/>

[[Category:Negri| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Semaphoresmic clan]]
[[Category:Marvel temperaments]]
[[Category:Avicennmic temperaments]]

65/64

2026-07-09T03:07:01Z

Overthink: follow-up temperament; sectioning

{{Infobox Interval
| Name = wilsorma
| Color name = 3oy1, thoyo 1sn, Thoyo comma
| Comma = yes
}}
In [[13-limit]] [[just intonation]], '''65/64''', the '''wilsorma''', is a [[superparticular]] interval of around 26.8{{cent}}, nearly a quarter of a semitone or eighth of a tone. 65 is 5 times 13, which means that 65/64 can be treated as a harmonic 13th above a harmonic 5th or vice versa. It is the difference between [[5/4]] and [[16/13]]; [[8/5]] and [[13/8]]; [[13/12]] and [[16/15]]; [[15/8]] and [[24/13]], [[13/10]] and [[32/25]]; [[20/13]] and [[25/16]], and of course, infinitely many other pairs of just intervals. It differs from the septimal comma [[64/63]] by [[4096/4095]] and from the syntonic comma [[81/80]] by [[325/324]].

Tempering it out turns 5/4 and 13/8 into [[octave complement]]s of one another. This is particularly useful in many [[13-limit]] [[magic family]] extensions, as it means they are very simply mapped to plus and minus one generator.

== Notation ==
This interval is the 13th-partial chroma (13-limit formal comma) in [[Ben Johnston's notation]], denoted simply with the number "13", while its reciprocal is denoted as "{{invert|13}}" (a turned "13"). If the base note is C, then [[13/8]] is represented by C–Ab13.

== Temperaments ==
Tempering it out leads to the rank-2 2.5.13 '''wilsormatic''' temperament, which has a generator tuned to abut 370-375 cents that represents both 5/4 and 16/13, or the rank-5 '''wilsormic''' temperament in the 13-limit. However, it is natural to equate [[14/13]] and [[15/14]] with 13/12[[~]]16/15, thus leading to the [[2.3.5.7.13 subgroup|2.3.5.7.13-subgroup]] temperament [[negri]], which splits the perfect fourth [[4/3]] into four equal parts, each being one step of the 12::16 segment of the [[harmonic series]].

== See also ==
* [[Gallery of just intervals]]
* [[64/63]]

[[Category:Commas with unknown etymology]]
{{todo|research|comment=Is it named after Erv Wilson?}}

225edo

2026-07-09T01:34:28Z

Overthink: Undo revision 233541 by Overthink (talk) idk actually

{{Infobox ET}}
{{ED intro}}

== Theory ==
225edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with rather large errors in [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]]. It has three mappings possible for the 7-limit:
* {{val| 225 357 522 632 }} ([[patent val]]),
* {{val| 225 '''356''' 522 '''631''' }} (225bd),
* {{val| 225 357 '''523''' 632 }} (225c).

Using the patent val, it tempers out 20000/19683 and 2109375/2097152 in the 5-limit; 3125/3087, 10976/10935, and 589824/588245 in the 7-limit.

Using the 225bd val, it tempers out 78732/78125 (sensipent comma) and {{monzo| -52 27 4 }} in the 5-limit; 225/224, 177147/175000, and 40353607/40000000 in the 7-limit.

Using the 225c val, it tempers out 131072000/129140163 (rodan comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 2401/2400, 4375/4374, and 2097152/2066715 in the 7-limit.

=== Odd harmonics ===
{{Harmonics in equal|225}}

== Scales ==
* [[Bossier scales]]

[[Category:Bossier]]

225edo

2026-07-09T01:31:51Z

Overthink: Delete (the edo is only vaguely relevant to the scales)

{{Delete|[[XW:NG]]}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
225edo is in[[consistent]] to the [[5-odd-limit]] and higher limits, with rather large errors in [[harmonic]]s [[3/1|3]], [[5/1|5]], [[7/1|7]], [[11/1|11]], and [[13/1|13]]. It has three mappings possible for the 7-limit:
* {{val| 225 357 522 632 }} ([[patent val]]),
* {{val| 225 '''356''' 522 '''631''' }} (225bd),
* {{val| 225 357 '''523''' 632 }} (225c).

Using the patent val, it tempers out 20000/19683 and 2109375/2097152 in the 5-limit; 3125/3087, 10976/10935, and 589824/588245 in the 7-limit.

Using the 225bd val, it tempers out 78732/78125 (sensipent comma) and {{monzo| -52 27 4 }} in the 5-limit; 225/224, 177147/175000, and 40353607/40000000 in the 7-limit.

Using the 225c val, it tempers out 131072000/129140163 (rodan comma) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; 2401/2400, 4375/4374, and 2097152/2066715 in the 7-limit.

=== Odd harmonics ===
{{Harmonics in equal|225}}

== Scales ==
* [[Bossier scales]]

[[Category:Bossier]]

226edo

2026-07-09T01:29:14Z

Overthink: nominate for deletion

{{Delete|[[XW:NG]]}}
{{Infobox ET}}
{{ED intro}}

== Theory ==
226edo is closely related to [[113edo]], but its mapping of [[harmonic]] [[5/1|5]] is sharp instead of flat. Unlike 113, 226 is only [[consistent]] to the [[5-odd-limit]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[1029/1024]] and [[19683/19600]] in the [[7-limit]]; [[243/242]], [[9801/9800]] and notably the [[quartisma]] in the [[11-limit]]; and [[364/363]] and [[729/728]] in the [[13-limit]].

=== Odd harmonics ===
{{Harmonics in equal|226}}

=== Subsets and supersets ===
226 factors into 2 × 113, with [[2edo]] and [[113edo]] as its subset edos. [[904edo]], which quadruples it, gives a good correction to the harmonic 7.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{monzo| 17 1 -8 }}, {{monzo| -32 29 -6 }}
| {{mapping| 226 358 525 }}
| +0.0386
| 0.5044
| 9.50
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperaments
|-
| 1
| 73\226
| 387.61
| 5/4
| [[Würschmidt]] (5-limit)
|-
| 1
| 91\226
| 483.19
| 320/243
| [[Hemiseven]] (7-limit)
|-
| 2
| 23\226
| 122.12
| 15/14
| [[Lagaca]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

97edt

2026-07-09T01:25:15Z

Overthink: formatting

{{Infobox ET}}
'''[[Edt|Division of the third harmonic]] into 97 equal parts''' (97EDT) is related to [[61edo|61 edo]], but with the 3/1 rather than the 2/1 being just; in fact, 97edt is close to 61.2002edo, meaning that its step is very accurately represented by 5 steps of [[306edo]]. The octave is about 3.9252 cents compressed and the step size is about 19.6078 cents.

Lookalikes: [[61edo]], [[141ed5]], [[158ed6]]

== Intervals ==
{{Interval table}}

== Harmonics ==
{{Harmonics in equal
| steps = 97
| num = 3
| denom = 1
| intervals = prime
}}
{{Harmonics in equal
| steps = 97
| num = 3
| denom = 1
| start = 12
| collapsed = 1
| intervals = prime
}}

== Music ==
===[[Francium]]===
* [https://www.youtube.com/watch?v=EYHumK-fsw8 Imagination] (2023)

153edt

2026-07-09T01:13:12Z

Overthink: ED intro

{{Infobox ET}}
{{ED intro}}

== Theory ==
153edt is notable for being the denominator of a convergent to log3(7/3), after [[9edt]], [[13edt]] and [[35edt]], and the last before [[3401edt]], and therefore has an extremely accurate approximation to [[7/3]], a mere 0.0036 cents flat. In fact, 153edt demonstrates 11-strong 7-3 [[telicity]], due to the next term in the continued fraction expansion being large (note how much larger 3401 is than 153), although 3401edt in fact surpasses it, demonstrating 16-strong 7-3 telicity.

In the no-twos [[7-limit]], 153edt supports [[canopus]] temperament, which gives it a rather accurate approximation of the 5th harmonic; and it additionally is accurate in the [[11-limit]], tempering out the comma [[387420489/386683451]] in the 3.7.11 subgroup. Harmonics 19 and 29 are also notably good.

However, 153edt's approximation of [[2/1]] is close to maximally bad, meaning that it is as far from an octave-equivalent tuning that an [[EDT]] of this size can be (though by this point, it is only 6 or so cents off).

== Harmonics ==
{{Harmonics in equal|153|3|1|intervals = prime|columns = 9}}
{{Harmonics in equal|153|3|1|start = 12|collapsed = 1|intervals = odd}}

230edt

2026-07-09T00:51:33Z

Overthink: nominate for deletion

{{Delete|[[XW:NG]]}}
{{Infobox ET}}
{{ED intro}}

230edt is related to [[145edo]], but with the [[3/1|perfect twelfth]] instead of the [[2/1|octave]] tuned just. It is [[consistent]] to the [[integer limit|13-integer-limit]]. In comparison, 145edo is only consistent to the 12-integer-limit.

=== Harmonics ===
{{Harmonics in equal|230|3|1|intervals=integer|columns=11}}
{{Harmonics in equal|230|3|1|intervals=integer|columns=12|start=12|collapsed=1|title=Approximation of harmonics in 230edt (continued)}}

=== Subsets and supersets ===
Since 230 factors into primes as {{nowrap| 2 × 5 × 23 }}, 230edt contains subset edts {{EDs|equave=t| 2, 5, 10, 23, 46, and 115 }}.

User:Francium/239edt

2026-07-09T00:48:01Z

Overthink: Overthink moved page 239edt to User:Francium/239edt: XW:NG

{{Stub}}
{{Infobox ET}}
{{ED intro}}

== Harmonics ==
{{Harmonics in equal
| steps = 239
| num = 3
| denom = 1
}}
{{Harmonics in equal
| steps = 239
| num = 3
| denom = 1
| start = 12
| collapsed = 1
}}

== Music ==
; [[Francium]]
* "Don't Treat Me Like Potato" from ''Don't'' (2025) – [https://open.spotify.com/track/2E16vtSM7e1zW8PFjD2qbj Spotify] | [https://francium223.bandcamp.com/track/dont-treat-me-like-potato Bandcamp] | [https://www.youtube.com/watch?v=f-PeHvwgcy8 YouTube]

EDT

2026-07-09T00:46:27Z

Overthink: /* Individual pages for EDTs */ remove unneeded links

__FORCETOC__
{{Interwiki
| en = EDT
| de = Edt
}}
The '''equal division of the tritave''' or '''twelfth''' ('''EDT''') or '''3rd harmonic''' ('''ED3''') is a [[tuning]] obtained by dividing the [[3/1|3rd harmonic]] in a certain number of [[equal]] steps.

== Introduction to tritave equivalence ==
Western music generally revolves around the principle of [[octave equivalence]]: notes an octave apart are often perceived in western music as being the same ''chroma'' but differing in pitch height. As the octave corresponds to a 2/1 frequency ratio, it has been proposed that the next-simplest after the octave, the 3/1, can also be used to evoke a sense of chroma equivalence. This interval corresponds to a perfect twelfth in the diatonic scale, but when used to refer to an equivalence interval it is often called the "[[tritave]]".

It has been argued that pitches a tritave apart can never truly be heard as equivalent in all of the ways that octaves are, with some claiming that the [http://www.mmk.ei.tum.de/persons/ter/top/octequiv.html tonotopic representation of the mammalian auditory system]{{dead link}} is inherently biased towards octave-equivalence. With proper context, experience, and training, however, at least some people find that they can experience some degree of tritave equivalence, especially when using timbres whose overtones consist of primarily or only odd harmonics such as clarinets, square waves, or triangle waves. While is not known whether odd harmonics actually facilitate the ability to hear in tritave-equivalence, it is known that musically valuable organizations of pitch can arise through the equal division of non-octave intervals, regardless of whether the period is perceived as being truly chroma-equivalent, and as such the multitude of equal divisions of the tritave are rich and ripe for exploration.

The [[Bohlen–Pierce scale]], most commonly consisting of 13 equal divisions of the tritave (although a justly-intoned version exists as well), seems to have been the first such arrangement to be seriously studied and made into music. The BP scale was independently discovered by Heinz Bohlen, John Pierce and Kees Van Prooijen. Bohlen found it while looking for triads with equal-difference tones, Prooijen uncovered it while searching for equally-tempered scales with accurate higher harmonics, and Pierce stumbled upon it trying to find consonant chords other than 4:5:6. Though they all started with different goals in mind, each of them amazingly ended up at the same destination.

== As generator chains for temperaments ==
There are other uses, or conceptualizations, of tritave-based tunings. Purely intuitive use of these myriad, assuredly xenharmonic structures comes to mind (see "EDO" versus "equal temperament"). Another intent might be to find or define temperaments (such as Magic, Hanson, etc.), or to provide exact formulae for stretching/compressing what would musically be used as an "ordinary" octave of ~2:1. (And given the stable nature of octave-based systems, some aesthetic overlap even in the most tritave-equivalent of music, would be forseeable.) For instance, the Bernhard-Stopper (19edt) temperament, might for instance be found useful in tuning pianoforti, being equivalent to 12edo, except for a 1.2 cents sharp octave which is relevant to inharmonicity.

Below is a large list of EDTs; additionally, some equal divisions of the tritave are known by alternate names or have special interest:

* [[3edt]] (Liese generator)
* [[4edt]] (Vulture generator)
* [[5edt]] (Tritave counterpart of Magic)
* [[6edt]] (Tritave counterpart of Hanson)
* [[7edt]] (Tritave counterpart of Orwell)
* [[8edt]] (Tritave counterpart of Vulture)
* [[11edt]] "Euler Temperament"
* [[BP|"Bohlen–Pierce" or "BP"]]
* [[15edt]] (Mowgli generator)
* [[19edt|"Bernhard Stopper"]]
* [[39edt]] Triple Bohlen–Pierce (Erlich)

== Individual pages for EDTs ==
{| class="wikitable center-all mw-collapsible"
|+ style="font-size: 105%; white-space: nowrap;" | 0…99
|-
| [[0edt|0]] || [[1edt|1]] || [[2edt|2]] || [[3edt|3]] || [[4edt|4]] || [[5edt|5]] || [[6edt|6]] || [[7edt|7]] || [[8edt|8]] || [[9edt|9]]
|-
| [[10edt|10]] || [[11edt|11]] || [[12edt|12]] || [[13edt|13]] || [[14edt|14]] || [[15edt|15]] || [[16edt|16]] || [[17edt|17]] || [[18edt|18]] || [[19edt|19]]
|-
| [[20edt|20]] || [[21edt|21]] || [[22edt|22]] || [[23edt|23]] || [[24edt|24]] || [[25edt|25]] || [[26edt|26]] || [[27edt|27]] || [[28edt|28]] || [[29edt|29]]
|-
| [[30edt|30]] || [[31edt|31]] || [[32edt|32]] || [[33edt|33]] || [[34edt|34]] || [[35edt|35]] || [[36edt|36]] || [[37edt|37]] || [[38edt|38]] || [[39edt|39]]
|-
| [[40edt|40]] || [[41edt|41]] || [[42edt|42]] || [[43edt|43]] || [[44edt|44]] || [[45edt|45]] || [[46edt|46]] || [[47edt|47]] || [[48edt|48]] || [[49edt|49]]
|-
| [[50edt|50]] || [[51edt|51]] || [[52edt|52]] || [[53edt|53]] || [[54edt|54]] || [[55edt|55]] || [[56edt|56]] || [[57edt|57]] || [[58edt|58]] || [[59edt|59]]
|-
| [[60edt|60]] || [[61edt|61]] || [[62edt|62]] || [[63edt|63]] || [[64edt|64]] || [[65edt|65]] || [[66edt|66]] || [[67edt|67]] || [[68edt|68]] || [[69edt|69]]
|-
| [[70edt|70]] || [[71edt|71]] || [[72edt|72]] || [[73edt|73]] || [[74edt|74]] || [[75edt|75]] || [[76edt|76]] || [[77edt|77]] || [[78edt|78]] || [[79edt|79]]
|-
| [[80edt|80]] || [[81edt|81]] || [[82edt|82]] || [[83edt|83]] || [[84edt|84]] || [[85edt|85]] || [[86edt|86]] || [[87edt|87]] || [[88edt|88]] || [[89edt|89]]
|-
| [[90edt|90]] || [[91edt|91]] || [[92edt|92]] || [[93edt|93]] || [[94edt|94]] || [[95edt|95]] || [[96edt|96]] || [[97edt|97]] || [[98edt|98]] || [[99edt|99]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 100…199
|-
| [[100edt|100]] || [[101edt|101]] || [[102edt|102]] || [[103edt|103]] || [[104edt|104]] || [[105edt|105]] || [[106edt|106]] || [[107edt|107]] || [[108edt|108]] || [[109edt|109]]
|-
| [[110edt|110]] || [[111edt|111]] || [[112edt|112]] || [[113edt|113]] || [[114edt|114]] || [[115edt|115]] || [[116edt|116]] || [[117edt|117]] || [[118edt|118]] || [[119edt|119]]
|-
| [[120edt|120]] || [[121edt|121]] || [[122edt|122]] || [[123edt|123]] || [[124edt|124]] || [[125edt|125]] || [[126edt|126]] || [[127edt|127]] || [[128edt|128]] || [[129edt|129]]
|-
| [[130edt|130]] || [[131edt|131]] || [[132edt|132]] || [[133edt|133]] || [[134edt|134]] || [[135edt|135]] || [[136edt|136]] || [[137edt|137]] || [[138edt|138]] || [[139edt|139]]
|-
| [[140edt|140]] || [[141edt|141]] || [[142edt|142]] || [[143edt|143]] || [[144edt|144]] || [[145edt|145]] || [[146edt|146]] || [[147edt|147]] || [[148edt|148]] || [[149edt|149]]
|-
| [[150edt|150]] || [[151edt|151]] || [[152edt|152]] || [[153edt|153]] || [[154edt|154]] || [[155edt|155]] || [[156edt|156]] || [[157edt|157]] || [[158edt|158]] || [[159edt|159]]
|-
| [[160edt|160]] || [[161edt|161]] || [[162edt|162]] || [[163edt|163]] || [[164edt|164]] || [[165edt|165]] || [[166edt|166]] || [[167edt|167]] || [[168edt|168]] || [[169edt|169]]
|-
| [[170edt|170]] || [[171edt|171]] || [[172edt|172]] || [[173edt|173]] || [[174edt|174]] || [[175edt|175]] || [[176edt|176]] || [[177edt|177]] || [[178edt|178]] || [[179edt|179]]
|-
| [[180edt|180]] || [[181edt|181]] || [[182edt|182]] || [[183edt|183]] || [[184edt|184]] || [[185edt|185]] || [[186edt|186]] || [[187edt|187]] || [[188edt|188]] || [[189edt|189]]
|-
| [[190edt|190]] || [[191edt|191]] || [[192edt|192]] || [[193edt|193]] || [[194edt|194]] || [[195edt|195]] || [[196edt|196]] || [[197edt|197]] || [[198edt|198]] || [[199edt|199]]
|}

{| class="wikitable center-all mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 200…299
|-
| [[200edt|200]] || [[201edt|201]] || [[202edt|202]] || [[203edt|203]] || [[204edt|204]] || [[205edt|205]] || [[206edt|206]] || [[207edt|207]] || [[208edt|208]] || [[209edt|209]]
|-
| [[210edt|210]] || [[211edt|211]] || [[212edt|212]] || [[213edt|213]] || [[214edt|214]] || [[215edt|215]] || [[216edt|216]] || [[217edt|217]] || [[218edt|218]] || [[219edt|219]]
|-
| [[220edt|220]] || [[221edt|221]] || [[222edt|222]] || [[223edt|223]] || [[224edt|224]] || [[225edt|225]] || [[226edt|226]] || [[227edt|227]] || [[228edt|228]] || [[229edt|229]]
|-
| [[230edt|230]] || [[231edt|231]] || [[232edt|232]] || [[233edt|233]] || [[234edt|234]] || [[235edt|235]] || [[236edt|236]] || [[237edt|237]] || [[238edt|238]] || [[239edt|239]]
|-
| [[240edt|240]] || [[241edt|241]] || [[242edt|242]] || [[243edt|243]] || [[244edt|244]] || [[245edt|245]] || [[246edt|246]] || [[247edt|247]] || [[248edt|248]] || [[249edt|249]]
|-
| [[250edt|250]] || [[251edt|251]] || [[252edt|252]] || [[253edt|253]] || [[254edt|254]] || [[255edt|255]] || [[256edt|256]] || [[257edt|257]] || [[258edt|258]] || [[259edt|259]]
|-
| [[260edt|260]] || [[261edt|261]] || [[262edt|262]] || [[263edt|263]] || [[264edt|264]] || [[265edt|265]] || [[266edt|266]] || [[267edt|267]] || [[268edt|268]] || [[269edt|269]]
|-
| [[270edt|270]] || [[271edt|271]] || [[272edt|272]] || [[273edt|273]] || [[274edt|274]] || [[275edt|275]] || [[276edt|276]] || [[277edt|277]] || [[278edt|278]] || [[279edt|279]]
|-
| [[280edt|280]] || [[281edt|281]] || [[282edt|282]] || [[283edt|283]] || [[284edt|284]] || [[285edt|285]] || [[286edt|286]] || [[287edt|287]] || [[288edt|288]] || [[289edt|289]]
|-
| [[290edt|290]] || [[291edt|291]] || [[292edt|292]] || [[293edt|293]] || [[294edt|294]] || [[295edt|295]] || [[296edt|296]] || [[297edt|297]] || [[298edt|298]] || [[299edt|299]]
|}

; 300 and beyond
* [[314edt|314]], [[316edt|316]], [[336edt|336]], [[372edt|372]], [[415edt|415]], [[428edt|428]], [[499edt|499]], [[527edt|527]], [[613edt|613]], [[729edt|729]], [[800edt|800]], [[953edt|953]], [[1213edt|1213]], [[1342edt|1342]], [[3401edt|3401]], [[6181edt|6181]], [[27208edt|27208]]

* A [[list of tritave reduced harmonics]] for easy comparison of JI and temperaments in tritave-based systems.
* Also may be found convenient: [http://www.nonoctave.com/tuning/twelfth.html Nonoctave.com | Tuning | Equal Divisions of the Twelfth]

== EDT-EDO correspondences ==
It is useful to consider EDTs that both ''closely'' and ''poorly'' approximate EDOs. The former are usable as stretches and compressions of EDOs with strong flat or sharp tendencies, while the latter allow for no-twos harmony without the distraction of octaves appearing. It is possible to define "dual-octave" EDTs similar to dual-fifth EDOs, as those whose closest approximation of 2 is more than 1/3 of a step off (so in other words, they have a better closest approximation of the 4th harmonic than the 2nd).

Otherwise, one can speak of EDTs that correspond to a diatonic [[val]] (i.e. the EDT's size is some EDO added to an approximation of [[3/2]] in that EDO that is a [[5L 2s|diatonic]] generator), which is equivalent to the EDT's approximation of [[2/1]] generating the {{mos scalesig|8L 3s<3/1>|link=1}} scale against the tritave, therefore being between 5\8edt and 7\11edt.

EDTs with this property include {{EDTs| 19, 27, 30, 35, 38, 41, 43, 46, 49, 51, 52, 54, 57, 59, 60, 62, 63, 65, 67, 68, 70, 71, 73 to 76, 78, 79, 81 to 87, and all greater than 88.}}

EDTs ''without'' a diatonic val are 1 to {{EDTs| 7, 9, 10, 12 to 15, 17, 18, 20, 21, 23, 25, 26, 28, 29, 31, 34, 36, 37, 39, 42, 45, 47, 50, 53, 58, 61, and 69.}}

Borderline cases (i.e. EDTs corresponding to a heptatonic or pentatonic fifth) are {{EDTs| 8, 11, 16, 22, 24, 32, 33, 40, 44, 48, 55, 56, 64, 66, 72, 77, 80, and 88.}}

Correspondences are explained in more detail in the table below.

==== Multiples of 13EDT which approximate EDO ====
On the topic of multiples of 13EDT, 26 (double) and 39 (triple) offer very good harmonic approximations, the former of the 8th, 13th and 17th partials, and the latter of the 11th and 13th. However, quadruple through sextuple, ie. 52, 65 and 78EDT, also exist offering good approximations of the octave. 52EDT is very nearly [[33edo]] and 78EDT is very nearly [[49edo]], while 65EDT is practically identical to [[41edo]].

=== Table of correspondences ===
{| class="wikitable mw-collapsible"
|+ style="font-size: 105%; white-space: nowrap;" | EDT–EDO correspondences
|-
! EDT
! EDO
! Comments
|-
| [[8edt]]
| [[5edo]]
| 8edt is equivalent to 5edo with ~11 cent octave compression. Equivalently, 5edo is 8edt with ~18 cent stretched tritaves. [[Patent_val|Patent vals]] match through the 13-limit.
|-
| [[9edt]]
|
| rowspan="2" | Neither 9edt nor 10edt is equivalent to [[6edo]].
|-
| [[10edt]]
|
|-
| [[11edt]]
| [[7edo]]
| 11edt is equivalent to 7edo with ~10 cent stretched octaves. Patent vals differ in the 7-limit, but neither can really be said to represent the 7th harmonic with a straight face.
|-
| [[12edt]]
|
| 12edt entirely misses 2/1, falling halfway between 7 and 8 edos.
|-
| [[13edt]]
|
| The equal-tempered BP scale cannot be considered equivalent to [[8edo]].
|-
| [[14edt]]
| [[9edo]]
| There is a lot of mismatch between the pure-octave and pure-tritave tunings, but the patent vals match through the 13-limit. Great for stretched-octave pelog!
|-
| [[15edt]]
|
| 15edt entirely misses 2/1, falling halfway between 9 and 10 edos.
|-
| [[16edt]]
| [[10edo]]
| Similar situation to 8edt~5edo. Patent vals match through the 17-limit.
|-
| [[17edt]]
|
| rowspan="2" | Neither 17edt nor 18edt is equivalent to [[11edo]].
|-
| [[18edt]]
|
|-
| [[19edt]]
| [[12edo]]
| 19edt is 12edo with ~1.2 cent octave stretch. Patent vals match through the 31-limit, with the exception of 11.
|-
| [[20edt]]
|
| rowspan="2" | Neither 20edt nor 21edt is equivalent to [[13edo]].
|-
| [[21edt]]
|
|-
| [[22edt]]
| [[14edo]]
| Similar situation to 11edt~7edo, but the equivalence is rough. Patent vals match through the 11-limit, with the exception of 5 (which neither represents well).
|-
| [[23edt]]
|
| 23edt entirely misses 2/1, falling halfway between 14 and 15 edos.
|-
| [[24edt]]
| [[15edo]]
| This is only a rough correspondence, as the (8n)edt ~ (5n)edo sequence begins to break down. Patent vals match through the 13-limit, with the exception of 7.
|-
| [[25edt]]
| [[16edo]]
| Also only a rough correspondence; 25edt corresponds to 16edo with ~17 cent octave stretch, and patent vals match through the 5-limit.
|-
| [[26edt]]
|
| Double BP scale entirely misses 2/1, falling halfway between 16 and 17 edos.
|-
| [[27edt]]
| [[17edo]]
| 27edt is 17edo with ~2.5 cent compressed octaves. With the exception of 5 (which neither represents well), patent vals match through the 13-limit.
|-
| [[28edt]]
|
| rowspan="2" | Neither 28edt nor 29edt is equivalent to [[18edo]].
|-
| [[29edt]]
|
|-
| [[30edt]]
| [[19edo]]
| 30edt is 19edo with ~4.6 cent stretched octaves. Patent vals match through the 7-limit.
|-
| [[31edt]]
|
| 31edt entirely misses 2/1, falling halfway between 19 and 20 edos.
|-
| [[32edt]]
|
| 32edt cannot be considered equivalent to [[20edo]].
|-
| [[33edt]]
|
| 33edt cannot be considered equivalent to [[21edo]].
|-
| [[34edt]]
|
| 34edt entirely misses 2/1, falling halfway between 21 and 22 edos.
|-
| [[35edt]]
| [[22edo]]
| 35edt is 22edo with ~4.5 cent compressed octaves. Patent vals match through the 11-limit.
|-
| [[36edt]]
|
| rowspan="2" | Neither 36edt nor 37edt is equivalent to [[23edo]], although step of 36edt is close to step recommended for [[23edo and octave stretching]].
|-
| [[37edt]]
|
|-
| [[38edt]]
| [[24edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 19-limit.
|-
| [[39edt]]
|
| rowspan="2" | Neither 39edt (Triple BP scale) nor 40edt is equivalent to [[25edo]].
|-
| [[40edt]]
|
|-
| [[41edt]]
| [[26edo]]
| 41edt is 26edo with ~6.1 cent stretched octaves. Patent vals match through the 7-limit.
|-
| [[42edt]]
|
| 42edt falls exactly halfway between 26 and 27 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\53 of an octave.
|-
| [[43edt]]
| [[27edo]]
| 43edt is 27edo with ~5.7 cent compressed octaves. Patent vals match through the 7-limit.
|-
| [[44edt]]
|
| rowspan="2" | Neither 44edt nor 45edt is equivalent to [[28edo]].
|-
| [[45edt]]
|
|-
| [[46edt]]
| [[29edo]]
| 46edt is 29edo with ~0.94 cent compressed octaves. Patent vals match through the 89-limit.
|-
| [[47edt]]
|
| rowspan="2" | Neither 47edt nor 48edt is equivalent to [[30edo]].
|-
| [[48edt]]
|
|-
| [[49edt]]
| [[31edo]]
| 49edt is 31edo with ~3.3 cent stretched octaves. Patent vals match through the 11-limit.
|-
| [[50edt]]
|
| 50edt falls exactly halfway between 31 and 32 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\63 of an octave.
|-
| [[51edt]]
| [[32edo]]
| 51edt is 32edo with ~6.6 cent octave compression. Patent vals match through the 11-limit, with the exception of 5.
|-
| [[52edt]]
| [[33edo]]
| 52edt is 33edo with ~7 cent octave stretch (rough correspondence). Patent vals differ in the 5-limit.
|-
| [[53edt]]
|
| 53edt falls exactly halfway between 33 and 34 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\67 of an octave.
|-
| [[54edt]]
| [[34edo]]
| Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 17-limit, with the exception of 7.
|-
| [[55edt]]
|
| rowspan="2" | Neither 55edt nor 56edt is equivalent to [[35edo]].
|-
| [[56edt]]
|
|-
| [[57edt]]
| [[36edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 31-limit, with the exception of 11.
|-
| [[58edt]]
|
| 58edt falls exactly halfway between 36 and 37 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it resembles the scale with generator 2\73 of an octave.
|-
| [[59edt]]
| [[37edo]]
| 59edt is 37edo with ~7.2 cent octave compression (rough correspondence). Patent vals match through the 5-limit.
|-
| [[60edt]]
| [[38edo]]
| Same ~4.6 cent octave stretch as 30edt~19edo. Patent vals match through the 5-limit.
|-
| [[61edt]]
|
| 61edt falls exactly halfway between 38 and 39 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\77 of an octave.
|-
| [[62edt]]
| [[39edo]]
| 62edt is 39edo with ~3.6 cent compressed octaves. Patent vals match through the 5-limit.
|-
| [[63edt]]
| [[40edo]]
| 63edt is 40edo with ~7.6 cent stretched octaves (rough correspondence). Patent vals match through the 11-limit, with the exception of 5.
|-
| [[64edt]]
|
| 64edt falls exactly halfway between 40 and 41 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it resembles the scale with generator 2\81 of an octave.
|-
| [[65edt]]
| [[41edo]]
| 65edt is 41edo with ~0.31 cent compressed octaves. Patent vals match through the 19-limit.
|-
| [[66edt]]
|
| 66edt falls exactly halfway between 41 and 42 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it resembles the scale with generator 2\83 of an octave.
|-
| [[67edt]]
| [[42edo]]
| 67edt is 42edo with ~7.3 cent compressed octaves (rough correspondence). Patent vals match through the 5-limit, though the 5s cannot be said to match with a straight face.
|-
| [[68edt]]
| [[43edo]]
| 68edt is 43edo with ~2.7 cent stretched octaves. Patent vals match through the 5-limit.
|-
| [[69edt]]
|
| 69edt falls exactly halfway between 43 and 44 edos. It entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\87 of an octave.
|-
| [[70edt]]
| [[44edo]]
| Same ~4.5 cent octave compression as 35edt~22edo. Patent vals match through the 7-limit, with the exception of 5, though the 7s cannot be said to match with a straight face.
|-
| [[71edt]]
| [[45edo]]
| 71edt is is 45edo with ~4.5 cent stretched octaves (rough correspondence). Patent vals match through the 7-limit, with the exception of 5.
|-
| [[72edt]]
|
| ''72edt falls exactly halfway between 45 and 46 edos. It is the last edt which entirely misses 2/1, but nails the "double octave" 4/1, so it strongly resembles the scale with generator 2\91 of an octave.''
|-
| [[73edt]]
| [[46edo]]
| 73edt is 46edo with ~1.5 cent compressed octaves. Patent vals match through the 17-limit.
|-
| [[74edt]]
|
| rowspan="2" | Neither 74edt nor 75edt is equivalent to [[47edo]].
|-
| [[75edt]]
|
|-
| [[76edt]]
| [[48edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 11-limit.
|-
| [[77edt]]
|
| ''77edt falls exactly halfway between 48 and 49 edos, so it strongly resembles the scale with generator 2\97 of an octave, but technically does not entirely miss 2/1 due to having a step of ~24.7¢.''
|-
| [[78edt]]
| [[49edo]]
| 78edt is 49edo with ~5.2 cent compressed octaves (rough correspondence). Patent vals match through the 11-limit, though the 77s rather than either the 7s or 11s individually can be said to match with a straight face.
|-
| [[79edt]]
| [[50edo]]
| 79edt is 50edo with ~3.8 cent stretched octaves. Patent vals match through the 7-limit.
|-
| [[80edt]]
|
| 80edt falls exactly halfway between 50 and 51 edos, so it strongly resembles the scale with generator 2\101 of an octave, but technically does not entirely miss 2/1 due to having a step of ~23.8¢.
|-
| [[81edt]]
| [[51edo]]
| Same ~2.5 cent octave compression as 27edt~17edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[82edt]]
| [[52edo]]
| Same ~6.1 cent octave stretch as 41edt~26edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[83edt]]
|
| 83edt falls exactly halfway between 52 and 53 edos. so it resembles the scale with generator 2\105 of an octave, but technically does not entirely miss 2/1 due to having a step of ~22.9¢.
|-
| [[84edt]]
| [[53edo]]
| 84edt is 53edo with ~0.04 cent stretched octaves. Patent vals match through the 61-limit.
|-
| [[85edt]]
|
| 85edt falls exactly halfway between 53 and 54 edos, so it resembles the scale with generator 2\107 of an octave, but technically does not entirely miss 2/1 due to having a step of ~22.4¢.
|-
| [[86edt]]
| [[54edo]]
| Same ~5.7 cent octave compression as 43edt~27edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[87edt]]
| [[55edo]]
| 87edt is 55edo with ~2.4 cent stretched octaves. Patent vals match through the 11-limit, with the exception of 5.
|-
| [[88edt]]
|
| 88edt falls exactly halfway between 55 and 56 edos, so it strongly resembles the scale with generator 2\111 of an octave, but technically does not entirely miss 2/1 due to having a step of ~21.6¢.
|-
| [[89edt]]
| [[56edo]]
| 89edt is 56edo with ~3.3 cent compressed octaves. Patent vals match through the 5-limit.
|-
| [[90edt]]
| [[57edo]]
| Same ~4.6 cent octave stretch as 30edt~19edo (rough correspondence). Patent vals match through the 5-limit.
|-
| [[91edt]]
|
| 91edt falls exactly halfway between 57 and 58 edos, so it strongly resembles the scale with generator 2\115 of an octave, but technically does not entirely miss 2/1 due to having a step of ~20.9¢.
|-
| [[92edt]]
| [[58edo]]
| Same ~0.94 cent octave compression as 46edt~29edo. Patent vals match through the 17-limit.
|-
| [[93edt]]
|
| rowspan="2" | Neither 93edt nor 94edt is equivalent to [[59edo]].
|-
| [[94edt]]
|
|-
| [[95edt]]
| [[60edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 7-limit.
|-
| [[96edt]]
|
| 96edt falls exactly halfway between 60 and 61 edos, so it strongly resembles the scale with generator 2\121 of an octave, but technically does not entirely miss 2/1 due to having a step of ~19.8¢.
|-
| [[97edt]]
| [[61edo]]
| 97edt is 61edo with ~3.9 cent compressed octaves (rough correspondence). Patent vals match through the 5-limit.
|-
| [[98edt]]
| [[62edo]]
| Same ~3.3 cent octave compression as 49edt~31edo. Patent vals match through the 23-limit.
|-
| [[99edt]]
|
| 99edt falls exactly halfway between 62 and 63 edos, so it strongly resembles the scale with generator 2\125 of an octave, but technically does not entirely miss 2/1 due to having a step of ~19.2¢.
|-
| [[100edt]]
| [[63edo]]
| 100edt is 63edo with ~1.8 cent compressed octaves. Patent vals match through the 23-limit.
|-
| [[101edt]]
|
| rowspan="2" | Neither 101edt nor 102edt is equivalent to [[64edo]].
|-
| [[102edt]]
|
|-
| [[103edt]]
| [[65edo]]
| 103edt is 65edo with ~2.4 cent stretched octaves. Patent vals match through the 181-limit, with the exception of 13.
|-
| [[104edt]]
|
| 104edt falls exactly halfway between 65 and 66 edos, so it resembles the scale with generator 2\131 of an octave, but technically does not entirely miss 2/1 due to having a step of ~18.3¢.
|-
| [[105edt]]
| [[66edo]]
| Same ~4.5 cent octave compression as 35edt~22edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[106edt]]
| [[67edo]]
| 106edt is 67edo with ~2.2 cent stretched octaves, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[107edt]]
|
| 107edt falls exactly halfway between 67 and 68 edos, so it strongly resembles the scale with generator 2\135 of an octave, but technically does not entirely miss 2/1 due to having a step of ~17.8¢.
|-
| [[108edt]]
| [[68edo]]
| Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 7-limit.
|-
| [[109edt]]
| [[69edo]]
| 109edt is 69edo with ~4 cent stretched octaves (rough correspondence). Patent vals match through the 5-limit.
|-
| [[110edt]]
|
| 110edt falls exactly halfway between 69 and 70 edos, so it resembles the scale with generator 2\139 of an octave, but technically does not entirely miss 2/1 due to having a step of ~17.3¢.
|-
| [[111edt]]
| [[70edo]]
| 111edt is 70edo with ~0.57 cent compressed octaves. Patent vals match through the 67-limit.
|-
| [[112edt]]
|
| rowspan="2" | Neither 112edt nor 113edt is equivalent to 71edo.
|-
| [[113edt]]
|
|-
| [[114edt]]
| [[72edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 19-limit.
|-
| [[115edt]]
|
| 115edt falls exactly halfway between 72 and 73 edos, so it strongly resembles the scale with generator 2\145 of an octave, but technically does not entirely miss 2/1 due to having a step of ~16.6¢.
|-
| [[116edt]]
| [[73edo]]
| 116edt is 73edo with ~3.1 cent compressed octaves. Patent vals match through the 11-limit, though products of of any two of 5, 7 and 11 rather than 5, 7 and 11 themselves can be said to match with a straight face.
|-
| [[117edt]]
| [[74edo]]
| 117edt is 74edo with ~2.95 cent stretched octaves, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[118edt]]
|
| 118edt falls exactly halfway between 74 and 75 edos, so it strongly resembles the scale with generator 2\149 of an octave, but technically does not entirely miss 2/1 due to having a step of ~16.1¢.
|-
| [[119edt]]
| [[75edo]]
| 119edt is 75edo with ~1.3 cent compressed octaves. Patent vals match through the 19-limit, with the exception of 11.
|-
| [[120edt]]
| [[76edo]]
| Same ~4.6 cent octave stretch as 30edt~19edo (rough correspondence). Patent vals match through the 7-limit, though the 7s cannot be said to match with a straight face.
|-
| [[121edt]]
|
| 121edt falls exactly halfway between 76 and 77 edos, so it resembles the scale with generator 2\153 of an octave, but technically does not entirely miss 2/1 due to having a step of ~15.7¢.
|-
| [[122edt]]
| [[77edo]]
| 122edt is 77edo with ~0.41 cent stretched octaves. Patent vals match through the 37-limit.
|-
| [[123edt]]
|
| Same ~6.1 cent octave stretch as 41edt~26edo, but actually more strongly resembles the scale with generator 2\155 of an octave.
|-
| [[124edt]]
| [[78edo]]
| Same ~3.6 cent octave compression as 62edt~39edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[125edt]]
| [[79edo]]
| 125edt is 79edo with ~2 cent stretched octaves. Patent vals match through the 13-limit, with the exception of 7.
|-
| [[126edt]]
|
| 126edt falls exactly halfway between 79 and 80 edos, so it strongly resembles the scale with generator 2\159 of an octave, but technically does not entirely miss 2/1 due to having a step of ~15.1¢.
|-
| [[127edt]]
| [[80edo]]
| 127edt is 80edo with ~1.9 cent compressed octaves. Patent vals match through the 11-limit.
|-
| [[128edt]]
| [[81edo]]
| 128edt is 81edo with ~3.6 cent stretched octaves. Patent vals match through the 7-limit, though the 5s cannot be said to match with a straight face.
|-
| [[129edt]]
|
| Same ~5.7 cent octave compression as 43edt~27edo, but actually more strongly resembles the scale with generator 2\163 of an octave.
|-
| [[130edt]]
| [[82edo]]
| Same ~0.31 cent octave compression as 65edt~41edo. Patent vals match through the 43-limit, with the exception of 13.
|-
| [[131edt]]
|
| 131edt falls exactly halfway between 82 and 83 edos, so it resembles the scale with generator 2\165 of an octave, but technically does not entirely miss 2/1 due to having a step of ~14.5¢.
|-
| [[132edt]]
| [[83edo]]
| 132edt is 83edo with ~4.1 cent compressed octaves (rough correspondence). Patent vals match through the 5-limit.
|-
| [[133edt]]
| [[84edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo. Patent vals match through the 7-limit.
|-
| [[134edt]]
|
| 134edt falls exactly halfway between 84 and 85 edos, so it strongly resembles the scale with generator 2\169 of an octave, but technically does not entirely miss 2/1 due to having a step of ~14.2¢.
|-
| [[135edt]]
| [[85edo]]
| Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 7-limit, with the exception of 5.
|-
| [[136edt]]
| [[86edo]]
| Same ~2.7 cent octave stretch as 68edt~43edo.
|-
| [[137edt]]
|
| 137edt falls exactly halfway between 86 and 87 edos, so it strongly resembles the scale with generator 2\173 of an octave, but technically does not entirely miss 2/1 due to having a step of ~13.9¢.
|-
| [[138edt]]
| [[87edo]]
| Same ~0.94 cent octave compression as 46edt~29edo.
|-
| [[139edt]]
|
| ''139edt is 88edo with a ~4.1 cent stretched octave, but also 175ed4 with a ~5.55 cent compressed 4/1.''
|-
| [[140edt]]
|
| ''Same ~4.5 cent octave compression as 35edt~22edo, but actually equally strongly resembles the scale with generator 2\177 of an octave.''
|-
| [[141edt]]
| [[89edo]]
| 141edt is 89edo with ~0.52 cent stretched octaves.
|-
| [[142edt]]
|
| 142edt falls exactly halfway between 89 and 90 edos, so it strongly resembles the scale with generator 2\179 of an octave, but technically does not entirely miss 2/1 due to having a step of ~13.4¢.
|-
| [[143edt]]
| [[90edo]]
| 143edt is 90edo with ~3 cent compressed octaves.
|-
| [[144edt]]
| [[91edo]]
| 144edt is 91edo with ~1.9 cent stretched octaves.
|-
| [[145edt]]
|
| 145edt falls exactly halfway between 91 and 92 edos, so it strongly resembles the scale with generator 2\183 of an octave, but technically does not entirely miss 2/1 due to having a step of ~13.1¢.
|-
| [[146edt]]
| [[92edo]]
| Same ~1.5 cent octave compression as 73edt~46edo.
|-
| [[147edt]]
| [[93edo]]
| Same ~3.3 cent octave compression as 49edt~31edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[148edt]]
|
| 148edt falls exactly halfway between 93 and 94 edos, so it strongly resembles the scale with generator 2\187 of an octave, but technically does not entirely miss 2/1 due to having a step of ~12.85¢.
|-
| [[149edt]]
| [[94edo]]
| 149edt is 94edo with ~0.11 cent compressed octaves.
|-
| [[150edt]]
|
| ''Same ~4.6 cent octave stretch as 30edt~19edo, but actually equally strongly resembles the scale with generator 2\189 of an octave.''
|-
| [[151edt]]
|
| ''151edt is 95edo with a ~3.4 cent compressed octave, but also 191ed4 with a ~5.7 cent stretched 4/1.''
|-
| [[152edt]]
| [[96edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo.
|-
| [[153edt]]
|
| 153edt falls exactly halfway between 96 and 97 edos, so it strongly resembles the scale with generator 2\193 of an octave, but technically does not entirely miss 2/1 due to having a step of ~12.4¢.
|-
| [[154edt]]
| [[97edo]]
| 154edt is 97edo with ~2 cent compressed octaves.
|-
| [[155edt]]
| [[98edo]]
| 155edt is 98edo with ~2.5 cent stretched octaves.
|-
| [[156edt]]
|
| Same ~5.2 cent octave stretch as 78edt~49edo, but actually equally strongly resembles the scale with generator 2\197 of an octave.
|-
| [[157edt]]
| [[99edo]]
| 157edt is 99edo with ~0.68 cent compressed octaves.
|-
| [[158edt]]
|
| ''Same ~3.6 cent octave stretch as 79edt~50edo, but actually equally strongly resembles the scale with generator 2\199 of an octave.''
|-
| [[159edt]]
|
| ''159edt is 100edo with a ~3.8 cent compressed octave, but also 201ed4 with a ~4.4 cent stretched 4/1.''
|-
| [[160edt]]
| [[101edo]]
| 160edt is 101edo with ~0.61 cent stretched octaves.
|-
| [[161edt]]
|
| 161edt falls exactly halfway between 101 and 102 edos, so it strongly resembles the scale with generator 2\203 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.8¢.
|-
| [[162edt]]
| [[102edo]]
| Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 5-limit.
|-
| [[163edt]]
| [[103edo]]
| 163edt is 103edo with ~1.85 cent stretched octaves.
|-
| [[164edt]]
|
| ''Same ~6.1 cent octave stretch as 41edt~26edo, but actually more strongly resembles the scale with generator 2\207 of an octave.''
|-
| [[165edt]]
| [[104edo]]
| 165edt is 104edo with ~1.2 cent compressed octaves.
|-
| [[166edt]]
| [[105edo]]
| 166edt is 105edo with ~3 cent stretched octaves.
|-
| [[167edt]]
|
| 167edt falls exactly halfway between 105 and 106 edos, so it strongly resembles the scale with generator 2\211 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.4¢.
|-
| [[168edt]]
| [[106edo]]
| Same ~0.04 cent octave stretch as 84edt~53edo.
|-
| [[169edt]]
|
| 169edt falls exactly halfway between 106 and 107 edos, so it strongly resembles the scale with generator 2\213 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.25¢.
|-
| [[170edt]]
| [[107edo]]
| 170edt is 107edo with ~2.9 cent compressed octaves.
|-
| [[171edt]]
| [[108edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo.
|-
| [[172edt]]
|
| 172edt falls exactly halfway between 108 and 109 edos, so it strongly resembles the scale with generator 2\217 of an octave, but technically does not entirely miss 2/1 due to having a step of ~11.1¢.
|-
| [[173edt]]
| [[109edo]]
| 173edt is 109edo with ~1.7 cent compressed octaves.
|-
| [[174edt]]
| [[110edo]]
| 174edt is 110edo with ~2.4 cent stretched octaves.
|-
| [[175edt]]
|
| ''Same ~4.5 cent octave compression as 35edt~22edo, but actually more strongly resembles the scale with generator 2\221 of an octave.''
|-
| [[176edt]]
| [[111edo]]
| 176edt is 111edo with ~0.47 cent compressed octaves.
|-
| [[177edt]]
|
| ''177edt is 112edo with a ~3.5 cent compressed octave, but also 223ed4 with a ~3.75 cent compressed 4/1.''
|-
| [[178edt]]
|
| ''Same ~3.3 cent octave stretch as 89edt~56edo, but actually more strongly resembles the scale with generator 2\225 of an octave.''
|-
| [[179edt]]
| [[113edo]]
| 179edt is 113edo with ~0.68 cent stretched octaves.
|-
| [[180edt]]
|
| ''Same ~4.6 cent octave stretch as 30edt~19edo, but actually more strongly resembles the scale with generator 2\227 of an octave.''
|-
| [[181edt]]
| [[114edo]]
| 181edt is 114edo with ~2.1 cent compressed octaves.
|-
| [[182edt]]
| [[115edo]]
| 182edt is 115edo with ~1.8 cent stretched octaves.
|-
| [[183edt]]
|
| 183edt falls exactly halfway between 115 and 116 edos, so it strongly resembles the scale with generator 2\231 of an octave, but technically does not entirely miss 2/1 due to having a step of ~10.4¢.
|-
| [[184edt]]
| [[116edo]]
| Same ~0.94 cent octave compression as 46edt~29edo.
|-
| [[185edt]]
| [[117edo]]
| 185edt is 117edo with a ~2.9 cent stretched octave.
|-
| [[186edt]]
|
| ''Same ~3.6 cent octave compression as 62edt~39edo, but actually more strongly resembles the scale with generator 2\235 of an octave.''
|-
| [[187edt]]
| [[118edo]]
| 187edt is 118edo with ~0.16 cent stretched octaves.
|-
| [[188edt]]
|
| 188edt is 119edo with ~3.9 cent stretched octaves, but also 237ed4 with an ~2.3 cent compressed 4/1.
|-
| [[189edt]]
| [[119edo]]
| Same ~2.5 cent octave compression as 27edt~17edo, but there is a lot of mismatch between the pure-octave and pure-tritave tunings. Patent vals differ in the 5-limit.
|-
| [[190edt]]
| [[120edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo.
|-
| [[191edt]]
|
| 191edt falls exactly halfway between 120 and 121 edos, so it strongly resembles the scale with generator 2\241 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.96¢.
|-
| [[192edt]]
| [[121edo]]
| 192edt is 121edo with ~1.4 cent compressed octaves.
|-
| [[193edt]]
| [[122edo]]
| 193edt is 122edo with ~2.3 cent stretched octaves.
|-
| [[194edt]]
|
| ''Same ~3.9 cent octave compression as 97edt~61edo, but actually more strongly resembles the scale with generator 2\245 of an octave.''
|-
| [[195edt]]
| [[123edo]]
| Same ~0.31 cent octave compression as 65edt~41edo.
|-
| [[196edt]]
|
| ''Same ~3.3 cent octave compression as 49edt~31edo, but actually more strongly resembles the scale with generator 2\247 of an octave.''
|-
| [[197edt]]
|
| ''197edt is 124edo with a ~2.9 cent compressed octave, but also 249ed4 with a ~4 cent stretched 4/1.''
|-
| [[198edt]]
| [[125edo]]
| 198edt is 125edo with ~0.73 cent stretched octaves.
|-
| [[199edt]]
|
| 199edt falls exactly halfway between 125 and 126 edos, so it strongly resembles the scale with generator 2\251 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.56¢.
|-
| [[200edt]]
| [[126edo]]
| Same ~1.8 cent octave compression as 100edt~63edo.
|-
| [[201edt]]
| [[127edo]]
| 201edt is 127edo with ~1.7 cent compressed octave.
|-
| [[202edt]]
|
| 202edt falls exactly halfway between 127 and 128 edos, so it strongly resembles the scale with generator 2\255 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.42¢.
|-
| [[203edt]]
| [[128edo]]
| 203edt is 128edo with ~0.74 cent compressed octaves.
|-
| [[204edt]]
|
| ''Same ~2.7 cent octave stretch as 68edt~43edo, but actually more strongly resembles the scale with generator 2\257 of an octave.''
|-
| [[205edt]]
|
| ''205edt is 129edo with a ~3.2 cent compressed octave, but also 259ed4 with a ~3 cent stretched 4/1.''
|-
| [[206edt]]
| [[130edo]]
| Same ~2.4 cent octave stretch as 103edt~65edo.
|-
| [[207edt]]
|
| 207edt falls exactly halfway between 130 and 131 edos, so it strongly resembles the scale with generator 2\261 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.19¢.
|-
| [[208edt]]
| [[131edo]]
| 208edt is 131edo with ~2.1 cent compressed octaves.
|-
| [[209edt]]
| [[132edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo.
|-
| [[210edt]]
|
| 210edt falls exactly halfway between 132 and 133 edos, so it strongly resembles the scale with generator 2\265 of an octave, but technically does not entirely miss 2/1 due to having a step of ~9.06¢.
|-
| [[211edt]]
| [[133edo]]
| 211edt is 133edo with ~1.1 cent compressed octaves.
|-
| [[212edt]]
| [[134edo]]
|Same ~2.2 cent octave stretch as 106edt~67edo, but patent vals surprisingly actually match through the 7-limit, though the 7s nevertheless cannot be said to match with a straight face.
|-
| [[213edt]]
|
| 213edt falls exactly halfway between 134 and 135 edos, so it strongly resembles the scale with generator 2\269 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.92¢
|-
| [[214edt]]
| [[135edo]]
| 214edt is 135edo with ~0.17 cent compressed octaves.
|-
| [[215edt]]
|
| 215edt falls exactly halfway between 135 and 136 edos, so it strongly resembles the scale with generator 2\271 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.85¢.
|-
| [[216edt]]
| [[136edo]]
| Same ~2.5 cent octave compression as 27edt~17edo. Patent vals match through the 5-limit.
|-
| [[217edt]]
| [[137edo]]
| 217edt is 137edo with ~0.77 cent stretched octaves.
|-
| [[218edt]]
|
| 218edt falls exactly halfway between 137 and 138 edos, so it strongly resembles the scale with generator 2\275 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.725¢.
|-
| [[219edt]]
| [[138edo]]
|Same ~1.5 cent octave compression as 73edt~46edo.
|-
| [[220edt]]
| [[139edo]]
| 220edt is 139edo with ~1.6 cent stretched octaves
|-
| [[221edt]]
|
| 221edt falls exactly halfway between 139 and 140 edos, so it strongly resembles the scale with generator 2\279 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.61¢.
|-
| [[222edt]]
| [[140edo]]
| Same ~0.57 cent octave compression as 111edt~70edo.
|-
| [[223edt]]
|
| rowspan="2" | Neither 223edt nor 224edt is equivalent to 141edo.
|-
| [[224edt]]
|-
| [[225edt]]
| [[142edo]]
| 225edt is 142edo with ~0.345 cent stretched octaves.
|-
| [[226edt]]
|
| 226edt falls exactly halfway between 142 and 143 edos, so it strongly resembles the scale with generator 2\285 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.42¢.
|-
| [[227edt]]
| [[143edo]]
| 227edt is 143edo with ~1.85 cent compressed octaves.
|-
| [[228edt]]
| [[144edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo.
|-
| [[229edt]]
|
| 229edt falls exactly halfway between 144 and 145 edos, so it strongly resembles the scale with generator 2\279 of an octave, but technically does not entirely miss 2/1 due to having a step of ~8.305¢.
|-
| [[230edt]]
| [[145edo]]
| Same ~0.94 cent octave compression as 46edt~29edo.
|-
| [[231edt]]
|
| rowspan="2" | Neither 231edt nor 232edt is equivalent to 146edo.
|-
| [[232edt]]
|-
| [[233edt]]
| [[147edo]]
| 233edt is 147edo with ~.05 cent compressed octaves.
|-
| [[234edt]]
|
| ''Same ~2.95 cent octave stretch as 117edt~74edo, but actually more strongly resembles the scale with generator 2\295 of an octave.''
|-
| [[235edt]]
|
| ''235edt is 148edo with a ~2.2 cent compressed octave, but also 297ed4 with a ~3.75 cent stretched 4/1.''
|-
| [[236edt]]
| [[149edo]]
| 236edt is 149edo with a ~0.81 cent stretched octave
|-
| [[237edt]]
|
| ''Same ~3.8 cent octave stretch as 79edt~50edo, but actually more strongly resembles the scale with generator 2/299 of an octave.''
|-
| [[238edt]]
| [[150edo]]
| Same ~1.9 cent octave compression as 119edt~75edo, but actually do not start matching patent vals until 11.
|-
| [[239edt]]
| [[151edo]]
| 239edt is 151edo with ~1.65 cent stretched octaves.
|-
| [[240edt]]
|
| 240edt falls exactly halfway between 151 and 152 edos, so it strongly resembles the scale with generator 2\303 of an octave, but technically does not entirely miss 2/1 due to having a step of ~7.925¢.
|-
| [[241edt]]
| [[152edo]]
| 241edt is 152edo with ~0.43 cent compressed octaves.
|-
| [[242edt]]
|
| rowspan="2" | Neither 242edt nor 243edt is equivalent to 153edo.
|-
| [[243edt]]
|-
| [[244edt]]
| [[154edo|''154edo'']]
| ''Same ~0.41 cent octave stretch as 122edt~77edo, but actually do not start matching patent vals until 7.''
|-
| [[245edt]]
|
| 245edt falls exactly halfway between 154 and 155 edos, so it strongly resembles the scale with generator 2\309 of an octave, but technically does not entirely miss 2/1 due to having a step of ~7.76¢.
|-
| [[246edt]]
| [[155edo]]
| 246edt is 155edo with ~1.6 cent compressed octaves.
|-
| [[247edt]]
| [[156edo]]
| Same ~1.2 cent octave stretch as 19edt~12edo.
|-
| [[248edt]]
|
| 248edt falls exactly halfway between 156 and 157 edos, so it strongly resembles the scale with generator 2\313 of an octave, but technically does not entirely miss 2/1 due to having a step of ~7.67¢.
|-
| [[249edt]]
| [[157edo]]
| 249edt is 157edo with ~.775 cent compressed octaves.
|-
| [[250edt]]
|
| ''Same ~2 cent octave stretch as 125edt~79edo, but actually more strongly resembles the scale with generator 2\315 of an octave.''
|-
| [[251edt]]
|
| ''251edt is 158edo with a ~2.8 cent compressed octave, but also 317ed4 with a ~2.1 cent stretched 4/1.''
|-
| [[252edt]]
| [[159edo]]
| Same ~0.04 cent octave stretch as 84edt~53edo.
|}

== See also ==
* [[Consistency levels of small EDTs]]
* [[Relative errors of small EDTs]]
* [[List of tritave reduced harmonics]]
* [[List of no-twos chords in JI]]
* Heinz Bohlen's work: [http://www.huygens-fokker.org/bpsite/otherscales.html ''The Bohlen-Pierce Site: Other Unusual Scales'']

[[Category:Edt| ]] 
[[Category:Tritave]]
[[Category:Acronyms]]

6076/6075

2026-07-09T00:45:11Z

Overthink: category

{{Infobox Interval
| Name = large ricegrain comma
| Comma = yes
}}
'''6076/6075''', the '''large ricegrain comma''', is a [[31-limit]] [[superparticular]] [[comma]] measuring about 0.3{{cent}}. It is the amount by which a stack of two [[15/14]]'s exceeds [[31/27]].

It is also the difference between the following superparticular pairs:
* [[217/216]] and [[225/224]]
* [[406/405]] and [[435/434]]
* [[496/495]] and [[540/539]]
* [[651/650]] and [[729/728]]
* [[784/783]] and [[900/899]]
* [[1054/1053]] and [[1275/1274]]
* [[1216/1215]] and [[1520/1519]]
* [[1519/1518]] and [[2025/2024]]
* [[2401/2400]] and [[3969/3968]]
* [[2676/2975]] and [[5832/5831]]
* [[3136/3135]] and [[6480/6479]]
* [[3751/3750]] and [[9801/9800]]
* [[4375/4374]] and [[15625/15624]]

[[Category:Commas with unknown etymology]]

{{Stub}}

91edo

2026-07-09T00:43:05Z

Overthink: /* Theory */ remove "second highest in a series of four edos", as this info has been removed from the 89edo, 90edo, and 92edo pages

{{Infobox ET}}
{{ED intro}}

== Theory ==
The [[harmonic]]s [[3/1|3]], [[5/1|5]] and [[7/1|7]] for 91edo are on the flat side, making this a mostly flat system. It [[tempering out|tempers out]] [[15625/15552]] in the 5-limit, [[225/224]] and [[4375/4374]] in the 7-limit, [[245/242]], [[385/384]] in the 11-limit, and [[105/104]], [[144/143]], [[196/195]] in the 13-limit. It provides the [[optimal patent val]] for 11- and 13-limit [[septimin]] temperament, and the 13-limit rank-3 [[tripod]] temperament, as well as the 11-limit rank-4 temperament tempering out 245/242 and the 13-limit rank-5 temperament tempering out 105/104, or rank-4 tempering out 105/104 and 144/143, or else 105/104 and 196/195 and hence 225/224 also.

Using the 91c val, it is audibly indistinguishable from a closed system of [[1/7-comma meantone]], with a 5th only 0.018 cents sharper. The chromatic semitone in this scale corresponds to 135/128, the [[eigenmonzo]] (unchanged interval) of [[1/7-comma meantone]]. Being 7 steps, what is also remarkable is that in this instance the chromatic semitone is equal to one step of [[13edo]]. Since 135/128 is also equal to 1/13 of the octave, the 91c [[val]] tempers out the [[aluminium comma]] in the 5-limit.

It also tempers out the {{monzo| -11 26 -13 }}, the tridecatonic comma, which assigns [[10/9]] to 2/13 of the octave, and it supports [[trideci]] in the 7-limit, tempering out 4375/4374 and 83349/81920. It supports a variant of [[semaphore]] temperament which tempers out the {{monzo| -42 23 2 }} comma in the 2.3.7 [[subgroup]], and is generated by a 19\91 generator. It also tempers out the [[quartisma]] ({{monzo| 24 -6 0 1 -5 }}) in the 11-limit, and as a corollary it is a tuning for the [[quartkeenlig]] temperament, which can also act as a [[23edo and octave stretching|stretched]] [[23edo]]. In the 13-limit, it supports [[vidar]] and gives a reasonable tuning for its size.

=== Odd harmonics ===
{{Harmonics in equal|91}}

=== Subsets and supersets ===
91 is the smallest composite number whose composite character is not immediately evident in the decimal system; it is, in fact, the product of 7 and 13. As such, 91edo contains [[7edo]] and [[13edo]] as subsets.

=== Miscellany ===
The [[concoctic scale]] for 91edo is 27 steps, where two concoctic neutral thirds make a sharp fifth of 54\91, representing 3/2 in the 91b val.

== Intervals ==
{{Interval table}}

== Notation ==
=== Ups and downs notation ===
91edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
{{Sharpness-sharp7a}}

Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used:
{{Sharpness-sharp7}}

=== Eliora's notation ===
[[User:Eliora|Eliora]], who believes the diatonic way of naming intervals in 91edo is not useful due to the fact that other temperaments and techniques for 91edo are more prominent, proposes a way of naming that merges the factors 7 and 13—7 equidistant notes are named do, re, mi, and 13 are named by some other virtue. The proposition is to use Old Slavic letter names, since no one uses them for naming or in mathematics. The {{nowrap|7 + 13}} naming convention can be called a duality notation. Intervals can be named through Latin ones for the 7-note scale, and Greek ones for the 13-note.

{| class="wikitable center-1 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 91edo
|-
! #
! Eliora's naming system
! Eliora's notation
! Associated ratio
|-
| 0
| unison perfect prime perfect prota
| A Az (А)
| [[1/1]]
|-
| 1
| major prime major prota
| A# Az#
| [[1728/1715]]
|-
| 2
| augmented prota
| Az##
|
|-
| 3
| biaugmented prota
| Az###
|
|-
| 4
| bidiminished deiteria
| Buki♭♭♭
| [[33/32]]
|-
|5
| diminished deiteria
| Buki♭♭
|
|-
| 6
| minor deiteria
| Buki♭
|
|-
| 7
| neutral deiteria
| Buki (Б)
| [[135/128]]
|-
| 8
| major deiteria
| Buki#
|
|-
| 9
| augmented deiteria
| Buki##
|
|-
| 10
| biaugmented deiteria
| Buki###
|
|-
| 11
| bidiminished tritia
| Vedi♭♭♭
| [[13/12]], [[12/11]]
|-
| 12
| diminished tritia
| Vedi♭♭
|
|-
| 13
| neutral secunde minor tritia
| B Vedi♭
| [[11/10]]
|-
| 14
| neural tritia
| Vedi (В)
| [[10/9]]
|-
| 15
| major tritia
| Vedi#
| [[9/8]]
|-
| 16
| augmented tritia
| Vedi##
|
|-
| 17
| biaugmented tritia
| Vedi###
|
|-
| 18
| bidiminished tesseria
| Glagol♭♭♭
| [[8/7]]
|-
| 19
| diminished tesseria
| Glagol♭♭
|
|-
| 20
| minor tesseria
| Glagol♭
| [[7/6]]
|-
| 21
| neutral tesseria
| Glagol (Г)
|
|-
| 22
| major tesseria
| Glagol#
| [[13/11]]
|-
| 23
| augmented tesseria
| Glagol##
|
|-
| 24
| biaugmented tesseria
| Glagol###
| [[6/5]]
|-
| 25
| bidiminished pemptia
| Dobro♭♭♭
|
|-
| 26
| neutral tertie diminished pemptia
| C Dobro♭♭
| [[11/9]]
|-
| 27
| major tertie minor pemptia
| C# Dobro♭
| [[16/13]], 27/22
|-
| 28
| neutral pemptia
| Dobro (Д)
|
|-
| 29
| major pemptia
| Dobro#
| [[5/4]]
|-
| 30
| augmented pemptia
| Dobro##
|
|-
| 31
| biaugmented pemptia
| Dobro###
|
|-
| 32
| bidiminished hektia
| Yest♭♭♭
| [[14/11]]
|-
| 33
| diminished hektia
| Yest♭♭
| [[9/7]]
|-
| 34
| minor hektia
| Yest♭
|
|-
| 35
| neutral hektia
| Yest (Е)
|
|-
| 36
| major hektia
| Yest#
|
|-
| 37
| augmented hektia
| Yest##
|
|-
| 38
| biaugmented hektia
| Yest###
| [[4/3]]
|-
| 39
| neutral quarte bidiminished hebdomia
| D Zhivete♭♭♭
|
|-
| 40
| diminished hebdomia
| Zhivete♭♭
|
|-
| 41
| minor hebdomia
| Zhivete♭
|
|-
| 42
| neutral hebdomia
| Zhivete (Ж)
| [[11/8]]
|-
| 43
| major hebdomia
| Zhivete#
|
|-
| 44
| augmented hebdomia
| Zhivete##
| [[7/5]]
|-
| 45
| biaugmented hebdomia
| Zhivete###
|
|-
| 46
| bidiminished ogdonia
| Dzelo♭♭♭
|
|-
| 47
| diminished ogdonia
| Dzelo♭♭
| [[10/7]]
|-
| 48
| minor ogdonia
| Dzelo♭
|
|-
| 49
| neutral ogdonia
| Dzelo (Ѕ)
|
|-
| 52
| neutral quinte
| E
| 121/81
|-
| 53
| major quinte
| E#
| [[3/2]]
|-
| 54
| augmented quinte diminished ennatia
| E## Zemle♭♭
| [[256/169]]
|-
| 55
| minor ennatia
| Zemle♭
|
|-
| 56
| neutral ennatia
| Zemle (З)
|
|-
| 63
| neutral decatia
| Izhe (И)
|
|-
| 64
| major decatia minor sexte
| Izhe# F♭
|
|-
| 65
| neutral sexte
| F
|
|-
| 70
| neutral hendecatia
| Jerve (Ђ)
|
|-
| 77
| neutral dodecatia
| Kako (К)
|
|-
| 78
| neutral septime
| G
|
|-
| 84
| neutral decatotritia
| Ludi (Л)
|
|-
| 91
| perfect octave perfect decatotetartia
| A Az (А)
| [[2/1]] exact
|}

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -144 91 }}
| {{Mapping| 91 144 }}
| +0.963
| 0.964
| 7.31
|-
| 2.3.5
| 15625/15552, 43046721/41943040
| {{Mapping| 91 144 211 }}
| +1.202
| 0.857
| 6.49
|-
| 2.3.5.7
| 225/224, 4375/4374, 50421/50000
| {{Mapping| 91 144 211 255 }}
| +1.453
| 0.860
| 6.51
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 2\91
| 26.37
| 49/48
| [[Sfourth]]
|-
| 1
| 4\91
| 52.75
| 33/32
| [[Quartkeenlig]] (91f)
|-
| 1
| 11\91
| 145.05
| 49/45
| [[Swetneus]] (91ef)
|-
| 1
| 19\91
| 250.55
| 1240029/1048576
| ''[[Semaphore]] variant'' (24 & 91)**
|-
| 1
| 20\91
| 263.74
| 7/6
| [[Septimin]] (91)
|-
| 1
| 24\91
| 316.48
| 6/5
| [[Catakleismic]] (91f)
|-
| 1
| 33\91
| 435.16
| 9/7
| [[Supermajor (temperament)|Supermajor]]
|-
| 1
| 34\91
| 448.35
| 35/27
| [[Semidimfourth]]
|-
| 1
| 38\91
| 501.10
| 4/3
| [[Python]]
|-
| 1
| 44\91
| 580.22
| 7/5
| [[Tritonic]]
|-
| 7
| 38\91 (1\91)
| 501.10 (13.19)
| 4/3 (81/80)
| [[Absurdity]]
|-
| 13
| 38\91 (1\91)
| 501.10 (13.19)
| 4/3 (265/252)
| [[Trideci]] (91) [[Aluminium]] (91c)
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

<nowiki/>** Derived from scales in the Scales section, official name not decided upon yet.

== Scales ==
{{See also| 5- to 10-tone scales in 91edo }}

* [[Semaphore5]]: 19 15 19 19 19
* [[Semaphore9]]: 15 4 15 4 15 4 15 15 4
* [[Semaphore14]]: 4 11 4 4 11 4 4 11 4 11 4 4 11 4
* NaiveMajor[7]: 13 16 10 13 16 13 10
* NaiveMinor[7]: 13 10 16 13 10 13 16
* Septimin[9]: 11 9 11 9 11 9 11 9 11
* SeptiminHijaz[9]: 5 15 11 9 11 9 5 15 11
* Meantone[12]: 878787887878
* [[Meantone43 in 91edo]]
* [[Meantone55 in 91edo]]
* NaiveOrwell[13]: 5795797597579
* ArabicNaiveOrwell[13]: 1 11 9 5 1 15 7 5 9 7 1 11 9
* HungarianNaiveSurorwell[13]: 7 7 8 6 11 5 5 7 10 4 4 13 4
* Quartkeenlig[23]: 44444444444444444444443
* ConcocticSubset[7]: 17 10 17 10 17 10 17
* ConcocticMaqamSikah: 10 17 17 10 10 17 10

== Instruments ==
A [[Lumatone mapping for 91edo]] is available.

== Music ==
=== Modern renderings ===
; {{W|Maretu}}
* [https://www.youtube.com/shorts/7RDvArkSJrk ''Aishite ita no ni''] (2023) – microtonal cover in 91edo by [[Bryan Deister]] (2026)

=== 21st century ===
; [[Mercury Amalgam]]
* ''Sadness - Nope'' (2022) – [https://mercuryamalgam.bandcamp.com/track/sadness-nope-the-molecular-agoge-pt-2 Bandcamp] | [https://www.youtube.com/watch?v=_5WS7AGZxm4 YouTube]

; [[Bryan Deister]]
* [https://www.youtube.com/shorts/HaYUAg30298 ''microtonal improvisation in 91edo''] (2025)
* [https://www.youtube.com/shorts/z6PeEocYMV8 ''improv 91edo''] (2025)

; [[Chris Vaisvil]]
* ''DPRK ISON CHASE'' (2014) – [http://chrisvaisvil.com/dprk-ison-chase-12-of-91-edo-ambient/ blog] | [https://www.youtube.com/watch?v=StCR6hcm5tM YouTube]

[[Category:Animist]]
[[Category:Frostmic]]
[[Category:Listen]]
[[Category:Quartismic]]
[[Category:Septimin]]
[[Category:Tripod]]

92edo

2026-07-09T00:41:07Z

Overthink: /* Theory */ remove this since it's been removed from the 89edo and 90edo pages

{{Infobox ET}}
{{ED intro}}

== Theory ==
The equal temperament is [[contorted]] through the 17-limit, with the same tuning and [[comma]]s as [[46edo]], and hence attracts little interest. That said, the approximation to the [[19/1|19th harmonic]] is much improved. Like 46, the [[patent fifth]] (54\92) is about 2.4{{c}} sharp. The alternate fifth 53\92 is a very flat fifth, flatter even than that of [[26edo]] and only 0.102{{c}} sharp of [[1/2-comma meantone]]; the 92bcccd val [[support]]s [[flattone]], while the 92bcccdd val supports [[Meantone_family#Flattertone|flattertone]].

=== Odd harmonics ===
{{Harmonics in equal|92}}

=== Subsets and supersets ===
Since 92 factors into 22 × 23, 92edo has subset edos {{EDOs| 2, 4, 23, and 46 }}.

== Intervals ==
{{Interval table}}

== Instruments ==

A [[Lumatone mapping for 92edo]] is available.

== Music==
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/5XFOALAzLiA ''microtonal improvisation in 92edo''] (2025)
* [https://www.youtube.com/watch?v=qWAinBYHwtE ''92edo waltz''] (2025)

50edo

2026-07-07T19:33:50Z

Overthink: /* Theory */ grammar

{{Infobox ET}}
{{ED intro}}

== Theory ==
As an equal temperament, 50et [[tempering out|tempers out]] [[81/80]] in the [[5-limit]], making it a [[meantone]] system, and in that capacity has historically drawn some notice; it is a somewhat sharp approximation of [[2/7-comma meantone]] (and is almost exactly 5/18-comma meantone). In [http://lit.gfax.ch/Harmonics%202nd%20Edition%20%28Robert%20Smith%29.pdf "Harmonics or the Philosophy of Musical Sounds"] (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, {{w|W. S. B. Woolhouse}} noted it was fairly close to the [[Target_tunings|least squares]] tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher-limit point of view. While [[31edo]] extends meantone with a [[7/4]] which is nearly pure, 50 has a flat 7/4 but both [[11/8]] and [[13/8]] are nearly pure. It is also the highest edo where the mapping of [[9/8]] and [[10/9]] to the same interval is [[consistent]], with two stacked fifths falling almost exactly 3/7-syntonic-comma sharp of 10/9 and 4/7-comma flat of 9/8. It is also almost consistent to the no-21s [[25-odd-limit]], only barely missing consistent mappings of [[11/9]] and [[18/11]].

50edo is also quite strong in the realm of tertian harmony for a meantone system, as the errors on [[7/6]], [[6/5]], [[5/4]], and [[9/7]] are all balanced to be roughly half as flat as the fifth, meaning that this set of thirds taken as a whole is minimally out-of-tune given the damage induced by meantone. Though it fails to approximate [[11/9]] well by virtue of not having a perfect hemififth, it inherits the excellent [[16/13]] from [[10edo]] and additionally has a 1.2{{c}} flat [[13/11]], providing even more qualities of roughly just thirds alongside their more complex [[fifth complement]]s.

=== Odd harmonics ===
{{Harmonics in equal|50|intervals=odd|columns=11}}
{{Harmonics in equal|50|intervals=odd|columns=12|start=12|collapsed=true|title=Approximation of odd harmonics in 50edo (continued)}}

=== As a tuning of other temperaments ===
50et tempers out [[126/125]], [[225/224]] and [[3136/3125]] in the [[7-limit]], indicating it [[support]]s septimal meantone; [[245/242]], [[385/384]] and [[540/539]] in the [[11-limit]] and [[105/104]], [[144/143]] and 196/195 in the [[13-limit]], and can be used for even higher limits. Aside from meantone and its extension [[meanpop]], it can be used to advantage for the [[coblack]] temperament (15 & 50), and provides the optimal patent val for 11- and 13-limit [[Meantone family #Bimeantone|bimeantone]]. It is also the unique equal temperament tempering out both 81/80 and the [[vishnuzma]], {{monzo| 23 6 -14 }}, so that in 50edo seven chromatic semitones stack to a perfect fourth. By comparison, this gives a perfect fifth in 12edo, a doubly diminished fifth in 31edo, and a diminished fourth in 19edo.

=== Relations ===
The 50edo system is related to [[7edo]], [[12edo]], [[19edo]], [[31edo]] as the next approximation to the "[[Golden meantone|Golden Tone System]]" ([[Das Goldene Tonsystem]]) of [[Thorvald Kornerup]] (and similarly as the next step from 31edo in [[Joseph Yasser]]'s "[https://books.google.com.au/books/about/A_theory_of_evolving_tonality.html?id=-XUsAAAAMAAJ&redir_esc=y A Theory of Evolving Tonality]").

== Intervals ==
{| class="wikitable center-all right-2 left-3"
|-
! #
! Cents
! Ratios<ref group="note">{{sg|50edo|limit=13-limit}}</ref>
! colspan="3" | [[Ups and downs notation]]
([[Enharmonic unisons in ups and downs notation|EUs]]: v3A1 and vvd2)
|-
| 0
| 0
| [[1/1]]
| Perfect 1sn
| P1
| D
|-
| 1
| 24
| ''[[45/44]]'', [[49/48]], [[56/55]], [[65/64]], [[66/65]], [[78/77]], [[91/90]], [[99/98]], [[100/99]], [[121/120]], ''[[169/168]]''
| Up 1sn
| ^1
| ^D
|-
| 2
| 48
| ''[[27/26]]'', [[33/32]], [[36/35]], ''[[50/49]]'', ''[[55/54]]'', ''[[64/63]]''
| Dim 2nd, Downaug 1sn
| d2, vA1
| Ebb, vD#
|-
| 3
| 72
| ''[[21/20]]'', [[25/24]], [[26/25]], [[28/27]]
| Aug 1sn, Updim 2nd
| A1, ^d2
| D#, ^Ebb
|-
| 4
| 96
| ''[[22/21]]''
| Downminor 2nd
| vm2
| vEb
|-
| 5
| 120
| [[16/15]], [[15/14]], [[14/13]]
| Minor 2nd
| m2
| Eb
|-
| 6
| 144
| [[13/12]], [[12/11]]
| Upminor 2nd
| ^m2
| ^Eb
|-
| 7
| 168
| [[11/10]]
| Downmajor 2nd
| vM2
| vE
|-
| 8
| 192
| [[9/8]], [[10/9]]
| Major 2nd
| M2
| E
|-
| 9
| 216
| [[25/22]]
| Upmajor 2nd
| ^M2
| ^E
|-
| 10
| 240
| [[8/7]], [[15/13]]
| Downaug 2nd, Dim 3rd
| vA2, d3
| vE#, Fb
|-
| 11
| 264
| [[7/6]]
| Updim 3rd, Aug 2nd
| ^d3, A2
| ^Fb, E#
|-
| 12
| 288
| [[13/11]]
| Downminor 3rd
| vm3
| vF
|-
| 13
| 312
| [[6/5]]
| Minor 3rd
| m3
| F
|-
| 14
| 336
| ''[[27/22]]'', [[39/32]], [[40/33]], ''[[49/40]]''
| Upminor 3rd
| ^m3
| ^F
|-
| 15
| 360
| [[16/13]], ''[[11/9]]''
| Downmajor 3rd
| vM3
| vF#
|-
| 16
| 384
| [[5/4]]
| Major 3rd
| M3
| F#
|-
| 17
| 408
| [[14/11]]
| Upmajor 3rd
| ^M3
| ^F#
|-
| 18
| 432
| [[9/7]]
| Downaug 3rd, Dim 4th
| vA3, d4
| vFx, Gb
|-
| 19
| 456
| [[13/10]]
| Updim 4th, Aug 3rd
| A3, ^d4
| ^Gb, Fx
|-
| 20
| 480
| [[33/25]], ''[[55/42]]'', ''[[64/49]]''
| Down 4th
| v4
| vG
|-
| 21
| 504
| [[4/3]]
| Perfect 4th
| P4
| G
|-
| 22
| 528
| [[15/11]]
| Up 4th
| ^4
| ^G
|-
| 23
| 552
| [[11/8]], [[18/13]]
| Downaug 4th
| vA4
| vG#
|-
| 24
| 576
| [[7/5]]
| Aug 4th
| A4
| G#
|-
| 25
| 600
| ''[[63/44]]'', ''[[88/63]]'', [[78/55]], [[55/39]]
| Upaug 4th, Downdim 5th
| ^A4, vd5
| ^G#, vAb
|-
| 26
| 624
| [[10/7]]
| Dim 5th
| d5
| Ab
|-
| 27
| 648
| [[16/11]], [[13/9]]
| Updim 5th
| ^d5
| ^Ab
|-
| 28
| 672
| [[22/15]]
| Down 5th
| v5
| vA
|-
| 29
| 696
| [[3/2]]
| Perfect 5th
| P5
| A
|-
| 30
| 720
| [[50/33]], ''[[84/55]]'', ''[[49/32]]''
| Up 5th
| ^5
| ^A
|-
| 31
| 744
| [[20/13]]
| Downaug 5th, Dim 6th
| vA5, d6
| vA#, Bbb
|-
| 32
| 768
| [[14/9]]
| Updim 6th, Aug 5th
| ^d6, A5
| ^Bbb, A#
|-
| 33
| 792
| [[11/7]]
| Downminor 6th
| vm6
| vBb
|-
| 34
| 816
| [[8/5]]
| Minor 6th
| m6
| Bb
|-
| 35
| 840
| [[13/8]], ''[[18/11]]''
| Upminor 6th
| ^m6
| ^Bb
|-
| 36
| 864
| ''[[44/27]]'', [[64/39]], [[33/20]], ''[[80/49]]''
| Downmajor 6th
| vM6
| vB
|-
| 37
| 888
| [[5/3]]
| Major 6th
| M6
| B
|-
| 38
| 912
| [[22/13]]
| Upmajor 6th
| ^M6
| ^B
|-
| 39
| 936
| [[12/7]]
| Downaug 6th, Dim 7th
| vA6, d7
| vB#, Cb
|-
| 40
| 960
| [[7/4]]
| Updim 7th, Aug 6th
| ^d7, A6
| ^Cb, B#
|-
| 41
| 984
| [[44/25]]
| Downminor 7th
| vm7
| vC
|-
| 42
| 1008
| [[16/9]], [[9/5]]
| Minor 7th
| m7
| C
|-
| 43
| 1032
| [[20/11]]
| Upminor 7th
| ^m7
| ^C
|-
| 44
| 1056
| [[24/13]], [[11/6]]
| Downmajor 7th
| vM7
| vC#
|-
| 45
| 1080
| [[15/8]], [[28/15]], [[13/7]]
| Major 7th
| M7
| C#
|-
| 46
| 1104
| ''[[21/11]]''
| Upmajor 7th
| ^M7
| ^C#
|-
| 47
| 1128
| ''[[40/21]]'', [[48/25]], [[25/13]], [[27/14]]
| Downaug 7th, Dim 8ve
| vA7, d8
| vCx, Db
|-
| 48
| 1152
| ''[[52/27]]'', [[64/33]], [[35/18]], ''[[49/25]]'', ''[[108/55]]'', ''[[63/32]]''
| Updim 8ve, Aug 7th
| ^d8, A7
| ^Db, Cx
|-
| 49
| 1176
| ''[[88/45]]'', [[96/49]], [[55/28]], [[128/65]], [[65/33]], [[77/39]], [[180/91]], [[196/99]], [[99/50]], [[240/121]], ''[[336/169]]''
| Down 8ve
| v8
| vD
|-
| 50
| 1200
| [[2/1]]
| Perfect 8ve
| P8
| D
|}
<references group="note" />

== Notation ==
=== Stein–Zimmermann–Gould notation ===
50edo can be notated with [[Stein–Zimmermann–Gould notation]]:
{{Sharpness-sharp3-szg}}

Here, a sharp raises by three steps, and a flat lowers by three steps, so arrows can be used to fill in the gap. If the arrows are taken to have their own layer of enharmonic spellings, some notes may be best spelled with double arrows.

=== Kite's ups and downs notation ===
Spoken as up, downsharp, sharp, upsharp, etc. Note that downsharp can be respelled as dup (double-up), and upflat as dud.
{{Ups and downs sharpness}}

=== Sagittal notation ===
This notation uses the same sagittal sequence as edos [[57edo #Sagittal notation|57]], [[64edo #Sagittal notation|64]], and [[71edo #Second-best fifth notation|71b]].

==== Evo flavor ====
<imagemap>
File:50-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 599 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:50-EDO_Evo_Sagittal.svg]]
</imagemap>

==== Revo flavor ====
<imagemap>
File:50-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 300 0 583 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 160 106 [[1053/1024]]
default [[File:50-EDO_Revo_Sagittal.svg]]
</imagemap>

In the diagrams above, a sagittal symbol followed by an equals sign (=) means that the following comma is the symbol's [[Sagittal notation#Primary comma|primary comma]] (the comma it ''exactly'' represents in JI), while an approximately equals sign (≈) means it is a secondary comma (a comma it ''approximately'' represents in JI). In both cases the symbol exactly represents the tempered version of the comma in this edo.

== Approximation to JI ==
[[File:50ed2.svg|250px|thumb|right|alt=alt : Your browser has no SVG support.|Selected 29-limit intervals approximated in 50edo]]

=== 15-odd-limit interval mappings ===
{{Q-odd-limit intervals|50|15}}

== Regular temperament properties ==
=== Temperament measures ===
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -79 50 }}
| {{mapping| 50 79 }}
| +1.88
| 1.88
| 7.83
|-
| 2.3.5
| 81/80, {{monzo| -27 -2 13 }}
| {{mapping| 50 79 116 }}
| +1.58
| 1.59
| 6.62
|-
| 2.3.5.7
| 81/80, 126/125, 84035/82944
| {{mapping| 50 79 116 140 }}
| +1.98
| 1.54
| 6.39
|-
| 2.3.5.7.11
| 81/80, 126/125, 245/242, 385/384
| {{mapping| 50 79 116 140 173 }}
| +1.54
| 1.63
| 6.76
|-
| 2.3.5.7.11.13
| 81/80, 105/104, 126/125, 144/143, 245/242
| {{mapping| 50 79 116 140 173 185 }}
| +1.31
| 1.57
| 6.54
|}

=== Commas ===
50et [[tempering out|tempers out]] the following [[comma]]s. This assumes the [[val]] {{val| 50 79 116 140 173 185 204 212 226 }}, comma values in cents rounded to 2 decimal places. This list is not all-inclusive, and is based on the interval table from Scala version 2.2.

{| class="commatable wikitable center-all left-3 right-4 left-5"
|-
! [[Harmonic limit|Prime limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Monzo]]
! [[Cent]]s
! Name
|-
| 3
| <abbr title="717897987691852588770249/604462909807314587353088">(20 digits)</abbr>
| {{monzo| -79 50 }}
| 297.75
| 50-comma
|-
| 5
| [[81/80]]
| {{monzo| -4 4 -1 }}
| 21.51
| Syntonic comma
|-
| 5
| <abbr title="1220703125/1207959552">(20 digits)</abbr>
| {{monzo| -27 -2 13 }}
| 18.17
| [[Ditonma]]
|-
| 5
| [[6115295232/6103515625|(20 digits)]]
| {{monzo| 23 6 -14 }}
| 3.34
| [[Vishnuzma]]
|-
| 7
| [[59049/57344]]
| {{monzo| -13 10 0 -1 }}
| 50.72
| Harrison's comma
|-
| 7
| [[16807/16384]]
| {{monzo| -14 0 0 5}}
| 44.13
| Cloudy comma
|-
| 7
| [[3645/3584]]
| {{monzo| -9 6 1 -1 }}
| 29.22
| Schismean comma
|-
| 7
| [[126/125]]
| {{monzo| 1 2 -3 1 }}
| 13.79
| Starling comma
|-
| 7
| [[225/224]]
| {{monzo| -5 2 2 -1 }}
| 7.71
| Marvel comma
|-
| 7
| [[3136/3125]]
| {{monzo| 6 0 -5 2 }}
| 6.08
| Hemimean comma
|-
| 7
| <abbr title="578509309952/576650390625">(24 digits)</abbr>
| {{monzo| 11 -10 -10 10 }}
| 5.57
| [[Linus comma]]
|-
| 7
| [[703125/702464|(12 digits)]]
| {{monzo| -11 2 7 -3 }}
| 1.63
| [[Meter]]
|-
| 7
| <abbr title="420175/419904">(12 digits)</abbr>
| {{monzo| -6 -8 2 5 }}
| 1.12
| [[Wizma]]
|-
| 11
| [[245/242]]
| {{monzo| -1 0 1 2 -2 }}
| 21.33
| Frostma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Keenanisma
|-
| 11
| [[540/539]]
| {{monzo| 2 3 1 -2 -1 }}
| 3.21
| Swetisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| Wizardharry comma
|-
| 11
| [[9801/9800]]
| {{monzo| -3 4 -2 -2 2 }}
| 0.18
| Kalisma
|-
| 13
| [[105/104]]
| {{monzo| -3 1 1 1 0 -1 }}
| 16.57
| Animist comma
|-
| 13
| [[144/143]]
| {{monzo| 4 2 0 0 -1 -1 }}
| 12.06
| Grossma
|-
| 13
| [[196/195]]
| {{monzo| 2 -1 -1 2 0 -1 }}
| 8.86
| Mynucuma
|-
| 13
| [[1188/1183]]
| {{monzo| 2 3 0 -1 1 -2 }}
| 7.30
| Kestrel comma
|-
| 13
| [[31213/31104]]
| {{monzo| -7 -5 0 4 0 1 }}
| 6.06
| Praveensma
|-
| 13
| [[364/363]]
| {{monzo| 2 -1 0 1 -2 1 }}
| 4.76
| Minor minthma
|-
| 13
| [[2200/2197]]
| {{monzo| 3 0 2 0 1 -3 }}
| 2.36
| Petrma
|-
| 17
| [[170/169]]
| {{monzo| 1 0 1 0 0 -2 1 }}
| 10.21
| Major naiadma
|-
| 17
| [[221/220]]
| {{monzo| -2 0 -1 0 -1 1 1 }}
| 7.85
| Minor naiadma
|-
| 17
| [[289/288]]
| {{monzo| -5 -2 0 0 0 0 2 }}
| 6.00
| Semitonisma
|-
| 17
| [[375/374]]
| {{monzo| -1 1 3 0 -1 0 -1 }}
| 4.62
| Ursulisma
|-
| 19
| [[153/152]]
| {{monzo| -3 2 0 0 0 0 1 -1 }}
| 11.35
| Ganassisma
|-
| 19
| [[171/170]]
| {{monzo| -1 2 -1 0 0 0 -1 1}}
| 10.15
| Malcolmisma
|-
| 19
| [[210/209]]
| {{monzo| 1 1 1 1 -1 0 0 1}}
| 8.26
| Spleen comma
|-
| 19
| [[324/323]]
| {{monzo| 2 4 0 0 0 0 -1 -1 }}
| 5.35
| Photisma
|-
| 19
| [[361/360]]
| {{monzo| -3 -2 -1 0 0 0 0 2 }}
| 4.80
| Go comma
|-
| 19
| [[495/494]]
| {{monzo| -1 2 1 0 1 -1 0 -1 }}
| 3.50
| Eulalisma
|-
| 23
| [[507/506]]
| 2.3.11.13.23 {{monzo| -1 1 -1 2 -1 }}
| 3.42
| Laodicisma
|-
| 23
| [[529/528]]
| 2.3.11.23 {{monzo| -4 -1 -1 2 }}
| 3.28
| Preziosisma
|-
| 23
| [[576/575]]
| 2.3.5.23 {{monzo| 6 2 -2 -1 }}
| 3.01
| Worcester comma
|-
| 23
| [[1288/1287]]
| {{monzo| 3 -2 0 1 -1 -1 0 0 1 }}
| 1.34
| Triaphonisma
|}
<references group="note" />

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperament
|-
| 1
| 1\50
| 24.0
| 686/675
| [[Sengagen]]
|-
| 1
| 9\50
| 216.0
| 17/15
| [[Tremka]]
|-
| 1
| 11\50
| 264.0
| 7/6
| [[Septimin]]
|-
| 1
| 13\50
| 312.0
| 6/5
| [[Oolong]]
|-
| 1
| 17\50
| 408.0
| 325/256
| [[Coditone]]
|-
| 1
| 19\50
| 456.0
| 125/96
| [[Qak]]
|-
| 1
| 21\50
| 504.0
| 4/3
| [[Meantone]] / [[meanpop]]
|-
| 1
| 23\50
| 552.0
| 11/8
| [[Emka]]
|-
| 2
| 2\50
| 48.0
| 36/35
| [[Pombe]]
|-
| 2
| 3\50
| 72.0
| 25/24
| [[Vishnu]] / [[vishnean]]
|-
| 2
| 6\50
| 144.0
| 12/11
| [[Bisemidim]]
|-
| 2
| 9\50
| 216.0
| 17/15
| [[Wizard]] / [[lizard]] / [[gizzard]]
|-
| 2
| 12\50
| 288.0
| 13/11
| [[Vines]]
|-
| 2
| 21\50 (4\50)
| 504.0 (96.0)
| 4/3 (35/33)
| [[Bimeantone]]
|-
| 5
| 21\50 (1\50)
| 504.0 (24.0)
| 4/3 (49/48)
| [[Cloudtone]]
|-
| 5
| 23\50 (3\50)
| 552.0 (72.0)
| 11/8 (21/20)
| [[Coblack]]
|-
| 10
| 7\50 (3\50)
| 168.0 (72.0)
| 54/49 (25/24)
| [[Decavish]]
|-
| 10
| 21\50 (1\50)
| 504.0 (24.0)
| 4/3 (78/77)
| [[Decic]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==
50edo's [[prime]]s 3, 5, 7, 17, 19, and 23 are all tuned flat and its 11 and 13 have close to no error, so 50edo can benefit from slight [[octave stretching]]. Some slightly stretched-octave tunings of 50edo include (least to most stretch): [[equal tuning|166ed10]], [[ed5|116ed5]], [[zpi|238zpi]] and [[ed12|179ed12]].

== Instruments ==
; Lumatone

See [[Lumatone mapping for 50edo]].

; Piano

A [[:Category:Piano|piano]] playing with a 50edo ensemble may wish to use the tuning [[116ed5]]. This tuning is almost exactly the same as 50edo, but with octaves [[octave stretch|stretched]] by 1 cent. Because pianos usually use stretched octaves, this tuning will sit better with the [[timbre]] of the piano, while still being close enough that it sounds perfectly in-tune with the other instruments tuned to 50edo.

== Music ==
=== Modern renderings ===
; {{W|Johann Sebastian Bach}}
* [https://www.youtube.com/watch?v=RnYqc0NKMLM "Ricercar a 3" from ''The Musical Offering'', BWV 1079] (1747) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=e6fMO-sue4Y "Contrapunctus 4" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024)
* [https://www.youtube.com/watch?v=M3wQu4UF1pg "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742–1749) – rendered by Claudi Meneghin (2024, organ sound rendering)
* [https://www.youtube.com/watch?v=qjb9DDM32Ic "Contrapunctus 11" from ''The Art of Fugue'', BWV 1080] (1742-1749) — rendered by Claudi Meneghin (2025, harpsichord sound rendering)

; {{W|David Belasco}}
* [https://www.youtube.com/shorts/WcExL9W2Gyc ''The Prettiest Little Song Of All''] (1908) - microtonal cover in 50edo by [[Bryan Deister]] (2025)

; {{W|Nicolaus Bruhns}}
* [https://www.youtube.com/watch?v=yrM50pvmD5c ''Prelude in E Minor "The Great"''] – rendered by Claudi Meneghin (2023)

; {{W|John Bull (composer)|John Bull}}
* [https://www.youtube.com/watch?v=6RewllRJ5rU ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2026)

; {{w|Frédéric Chopin}}
* [https://www.youtube.com/shorts/7Bisk0I2H4o ''Prelude Op. 28, No. 7 in A major''] (1839), arranged for fortepiano, tuned into 50-edo – rendered by [[Claudi Meneghin]] (2025)

; {{W|Louis Couperin}}
* [https://www.youtube.com/shorts/NSzakO66Roc ''«La Piémontoise»''] (1658?) – rendered by Claudi Meneghin (2026)

; {{W|Gabriel Fauré}}
* [https://www.youtube.com/watch?v=7djfrUlw2ck ''Pavane'', op. 50] (1887) – arranged for harpsichord and rendered by Claudi Meneghin (2020)

; {{W|Toby Fox}}
* [https://www.youtube.com/shorts/ynz5XvJOHiE ''Piano that may not be played that well''] via ''{{W|Deltarune}} Chapters 3 + 4'' (2025) (microtonal cover in 50edo) by [[Bryan Deister]] (2025)

; {{W|Iyowa}}
* [https://www.youtube.com/shorts/L6jF5_HEGkM ''Heat Abnormal''] (2024) - microtonal cover in 50edo by [[Bryan Deister]] (2025)

; {{W|Akira Kamiya}}
* [https://www.youtube.com/watch?v=5UnPAhRqmb4 ''funfunfun ta yo''] (2007) – rendered by MortisTheneRd (2024)

; {{W|Laufey_(singer)|Laufey}}
* [https://www.youtube.com/shorts/J34qt45jZW4 ''Snow White''] (2025) - microtonal cover in 50edo by [[Bryan Deister]] (2025)

; {{W|Wolfgang Amadeus Mozart}}
* [https://www.youtube.com/watch?v=YK_kFs4PL2g&list=WL&index=347 ''Gigue KV 574 («Leipziger Gigue»)''] (1789) – rendered by Claudi Meneghin (2026)

; {{W|Akiko Shikata}}
* [https://www.youtube.com/shorts/eXGcC52YMh8 ''Mother''] via ''{{W|Umineko_When_They_Cry|Umineko no Naku Koro ni}}'' (2007) - microtonal cover in 50edo by [[Bryan Deister]] (2026)

=== 21st century===
; [[Bryan Deister]]
* [https://www.youtube.com/shorts/zCsc5n6dr_I ''microtonal improv in 50edo''] (2024)
* [https://www.youtube.com/shorts/dAyMY-14yZo ''50edo improv''] (2025-10-13)
* [https://www.youtube.com/shorts/DIiLORDPfUI ''50edo improv''] (2026-05-25)
* [https://www.youtube.com/watch?v=LqzPpl01WXc ''Fantasy in 50edo''] (2026)

; [[Francium]]
* [https://www.youtube.com/watch?v=pH6E35hwUnM ''On My Way To Somewhere''] (2023)

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=Zh2jWoIXAf8 ''La Petite Poule Grise - Fugue''] (2014, uploaded 2019)
* [https://www.youtube.com/shorts/IhKVro5YEcA ''Canon on «Twinkle Twinkle Little Star» in 50-edo, for Organ''] (≤2014, restored and re-hosted 2025)
* [https://www.youtube.com/watch?v=TRXy0FJOKIA ''Fugue on the Dragnet theme''] (2014)
* [https://www.youtube.com/watch?v=wcTVED9zFrU ''Blue Fugue for Organ''] (2018)
* [https://www.youtube.com/watch?v=28x3vqw9kDI ''Happy Birthday Canon'', 6-in-1 Canon in 50edo] (2019)
* [https://www.youtube.com/watch?v=szUpO3FAOes ''Fantasia Catalana''] (2020)
* [https://www.youtube.com/watch?v=38UMa3oWSIE ''Preludi Nocturn i Fuga sobre la Lluna la Pruna''] (2020)
* [https://www.youtube.com/watch?v=C4EkNEu4EeU ''Canon at the Semitone on The Mother's Malison Theme'', for Organ] (2022)
* [https://www.youtube.com/watch?v=FyDKSjS9Qtg ''Fugue on an Original Theme'', for Baroque Ensemble] (2023) ([https://www.youtube.com/watch?v=TXwlLV2TCsw for Organ])
* [https://www.youtube.com/watch?v=2nD_7Ot8-0A ''Catalan Fugue (La Santa Espina)''] (2023)
* [https://www.youtube.com/watch?v=TBxDmpM9Xa8 ''Canon in C='' for Baroque Wind Ensemble] (2023)
* [https://www.youtube.com/watch?v=sIr394fGEEg ''Fantasia Catalana'', for Baroque Ensemble] (2023)
* ''Chord Progression: The Octave Divided into Five Parts in 50 edo'' (intended for demonstrating chord progressions, as the title indicates, but actually works as a short composition)
** [https://www.youtube.com/shorts/7_kROtWc4Sw <nowiki>organ rendition</nowiki>] (2023)
** [https://www.youtube.com/shorts/qsuM1sA2-A0 <nowiki>harpsichord rendition</nowiki>] (2024)
* [https://www.youtube.com/shorts/2x5atFuN6WA ''Baroque Blues - 4-Part Fugue for Baroque Consort''] (2026)
* ''Fugue on the French Lullaby «La Petite Poule Grise»'' (2026)
** [https://www.youtube.com/watch?v=RrsZ-bzf1mE Baroque ensemble rendition]
** [https://www.youtube.com/watch?v=wgsWzn_vfIM organ rendition]

; [[Cam Taylor]]
* [https://soundcloud.com/camtaylor-1/sets/the-late-little-xmas-album ''the late little xmas album''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/harpsichord-meantone ''Harpsichord meantone improvisation 1 in 50EDO''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/long-improvisation-2-in-50edo ''Long improvisation 2 in 50EDO''] (2014)
* [https://soundcloud.com/camtaylor-1/chord-sequence-for-difference ''Chord sequence for Difference tones in 50EDO''] (2014)
* [https://soundcloud.com/camtaylor-1/enharmonic-modulations-in ''Enharmonic Modulations in 50EDO''] (2014)
* [https://soundcloud.com/cam-taylor-2-1/harmonic-clusters-on-50edo-harpsichord-bosanquet-axis-through-pianoteq ''Harmonic Clusters on 50EDO Harpsichord''] (2014)
* [https://soundcloud.com/camtaylor-1/fragment-in-fifty ''Fragment in Fifty''] (2014)

== Additional reading ==
* [http://www.archive.org/details/harmonicsorphilo00smit Robert Smith's book online]
* [http://www.music.ed.ac.uk/russell/conference/robertsmithkirckman.html More information about Robert Smith's temperament]{{Dead link}}
* [https://www.dropbox.com/sh/4x81rzpkot32qzk/MQ3cJljjkh 50EDO Theory - Intervals, Chords and Scales in 50EDO by Cam Taylor]{{Dead link}}
* [http://iamcamtaylor.wordpress.com/ iamcamtaylor - Blog on 50EDO and extended meantone theory by Cam Taylor]

[[Category:50edo]]
[[Category:Equal divisions of the octave|##]] 
[[Category:Golden meantone]]
[[Category:Historical]]
[[Category:Listen]]
[[Category:Meantone]]
[[Category:Meanpop]]

328edo

2026-07-07T19:23:10Z

Overthink: /* Theory */ important fact

{{Infobox ET}}
{{ED intro}}

== Theory ==
328edo is [[enfactoring|enfactored]] in the [[5-limit]], with the same tuning as [[164edo]], but the approximation of higher [[harmonic]]s are much improved. Like 164edo, it inherits the [[3/2|perfect fifth]] from [[41edo]]. It has a sharp tendency, with harmonics 3 through 17 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[2401/2400]], [[3136/3125]], and [[6144/6125]] in the 7-limit, [[9801/9800]], [[16384/16335]] and [[19712/19683]] in the 11-limit, [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]] in the 13-limit, [[936/935]], [[1156/1155]] and [[2601/2600]] in the 17-limit, so that it [[support]]s [[würschmidt]] and [[hemiwürschmidt]], and provides the [[optimal patent val]] for 7-limit hemiwürschmidt, 11- and 13-limit [[semihemiwür]], and 13-limit [[semiporwell]].

=== Prime harmonics ===
{{Harmonics in equal|328|intervals=prime|columns=11}}

=== Subsets and supersets ===
Since 328 factors into {{Factorisation|328}}, 328edo has subset edos {{EDOs| 2, 4, 8, 41, 82, and 164 }}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5.7
| 2401/2400, 3136/3125, 589824/588245
| {{mapping| 328 520 762 921 }}
| −0.298
| 0.229
| 6.27
|-
| 2.3.5.7.11
| 2401/2400, 3136/3125, 9801/9800, 19712/19683
| {{mapping| 328 520 762 921 1135 }}
| −0.303
| 0.205
| 5.61
|-
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3136/3125, 10648/10647
| {{mapping| 328 520 762 921 1135 1214 }}
| −0.295
| 0.188
| 5.15
|-
| 2.3.5.7.11.13.17
| 676/675, 936/935, 1001/1000, 1156/1155, 1716/1715, 3136/3125
| {{mapping| 328 520 762 921 1135 1214 1341 }}
| −0.293
| 0.174
| 4.77
|}

=== Rank-2 temperaments ===
Note: 5-limit temperaments supported by 164et are not listed.

{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods per 8ve
! Generator*
! Cents*
! Associated ratio*
! Temperaments
|-
| 1
| 53\328
| 193.90
| 28/25
| [[Hemiwürschmidt]]
|-
| 1
| 117\328
| 428.05
| 2800/2187
| [[Osiris]]
|-
| 2
| 17\328
| 62.20
| 28/27
| [[Eagle]]
|-
| 2
| 111\328 (53\328)
| 406.10 (193.90)
| 495/392 (28/25)
| [[Semihemiwürschmidt]]
|-
| 8
| 136\328 (13\328)
| 497.56 (47.56)
| 4/3 (36/35)
| [[Twilight]]
|-
| 41
| 49\328 (1\328)
| 179.27 (3.66)
| 567/512 (352/351)
| [[Hemicountercomp]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

[[Category:Hemiwürschmidt]]
[[Category:Semiporwell]]

User talk:Eufalesio/Ultimate

2026-07-07T04:13:10Z

Overthink: /* Potential error */ new section

== Potential error ==

In the "Quick definition" section, you mention the [[nexus comma|tribilo comma]] being nullified. However, the corresponding temperament splits the octave in three, while not all Ultimate edos are divisible by 3 (217 and 311 aren't), meaning it can't possibly be nullified by the temperament. --[[User:Overthink|Overthink]] ([[User talk:Overthink|talk]]) 04:13, 7 July 2026 (UTC)

Xenial

2026-07-07T03:26:16Z

Overthink: make infobox less wide

{{Infobox regtemp
| Title = Xenial
| Subgroups = 2.3.5.7, 2.3.5.7.13, 2.3.5.7.13.23, 2.3.5.7.11.13.17.19.23
| Comma basis = [[126/125]], [[177147/175616]] (7-limit); [[126/125]], [[162/161]], [[169/168]], [[171/170]], [[208/207]], [[221/220]], [[231/230]] (23-limit)
| Edo join 1 = 19 | Edo join 2 = 70
| Mapping = 1; -9 -17 -33 22 -21 26 27 -3
| Generators = 10/9 | Generators tuning = 188.8 | Optimization method = CWE
| MOS scales = [[6L 1s]], [[6L 7s]], [[13L 6s]], [[19L 13s]], [[19L 32s]], [[19L 51s]]
| Pergen = (P8, P11/9)
| Odd limit 1 = 7 | Mistuning 1 = 4.60 | Complexity 1 = 51
| Odd limit 2 = 9 | Mistuning 2 = 6.27 | Complexity 2 = 51
| Odd limit 3 = 17 | Mistuning 3 = 8.90 | Complexity 3 = 70
| Odd limit 4 = 23 | Mistuning 4 = 8.96 | Complexity 4 = 70
}}
'''Xenial''' is a [[rank-2]] [[regular temperament|temperament]] that is [[generator|generated]] by a sharpened minor whole tone of [[~]][[10/9]], so that nine generators reach [[4/3]], 17 reach [[8/5]], 21 reach [[16/13]] and 33 reach [[8/7]] with octave reduction. It is also generated by dividing [[11/1|11th harmonic]] into 22 equal parts, [[17/1|17th harmonic]] into 26 equal parts, or [[19/1|19th harmonic]] into 27 equal parts.

See [[Starling temperaments #Xenial]] for more technical data.

== Interval chain ==
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.000
| [[1/1]]
|-
| 1
| 188.775
| [[10/9]], [[19/17]], [[28/25]]
|-
| 2
| 377.551
| [[56/45]]
|-
| 3
| 566.326
| [[18/13]], [[32/23]]
|-
| 4
| 755.102
| [[17/11]], [[20/13]]
|-
| 5
| 943.877
| [[19/11]], [[26/15]]
|-
| 6
| 1132.653
| [[23/12]], [[27/14]]
|-
| 7
| 121.428
| [[15/14]]
|-
| 8
| 310.204
| [[6/5]]
|-
| 9
| 498.979
| [[4/3]]
|-
| 10
| 687.755
| [[40/27]]
|-
| 11
| 876.530
|
|-
| 12
| 1065.306
| [[13/7]], [[24/13]]
|-
| 13
| 54.081
| [[26/25]], [[33/32]]
|-
| 14
| 242.857
| [[23/20]]
|-
| 15
| 431.632
| [[9/7]], [[23/18]]
|-
| 16
| 620.408
| [[10/7]]
|-
| 17
| 809.183
| [[8/5]]
|-
| 18
| 997.959
| [[16/9]], [[23/13]]
|-
| 19
| 1186.734
|
|-
| 20
| 175.510
|
|-
| 21
| 364.285
| [[16/13]], [[26/21]]
|-
| 22
| 553.061
| [[11/8]]
|-
| 23
| 741.836
| [[23/15]]
|-
| 24
| 930.612
| [[12/7]]
|-
| 25
| 1119.387
| [[40/21]], [[44/23]], [[48/25]]
|-
| 26
| 108.163
| [[16/15]], [[17/16]]
|-
| 27
| 296.938
| [[19/16]]
|-
| 28
| 485.714
|
|-
| 29
| 674.439
| [[34/23]]
|-
| 30
| 863.265
| [[38/23]], [[23/14]]
|-
| 31
| 1052.040
| [[11/6]], [[46/25]]
|-
| 32
| 40.815
| [[36/35]], [[46/45]], [[50/49]]
|-
| 33
| 229.591
| [[8/7]]
|-
| 34
| 418.366
| [[32/25]]
|}
<nowiki/>* In 23-limit CWE tuning

== Tunings ==
=== Norm-based tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.8535{{c}}
| CWE: ~10/9 = 188.8544{{c}}
| POTE: ~10/9 = 188.8548{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.8295{{c}}
| CWE: ~10/9 = 188.8085{{c}}
| POTE: ~10/9 = 188.8028{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 13-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7987{{c}}
| CWE: ~10/9 = 188.7898{{c}}
| POTE: ~10/9 = 188.7875{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 17-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7811{{c}}
| CWE: ~10/9 = 188.7677{{c}}
| POTE: ~10/9 = 188.7655{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 19-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7828{{c}}
| CWE: ~10/9 = 188.7770{{c}}
| POTE: ~10/9 = 188.7762{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 23-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~10/9 = 188.7849{{c}}
| CWE: ~10/9 = 188.7755{{c}}
| POTE: ~10/9 = 188.7744{{c}}
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]
! Generator (¢)
! Comments
|-
|
| 9/5
| 182.4037
|
|-
|
| 13/10
| 186.4465
|
|-
| 5 ⧵ 32
|
| 187.5000
| 32cddefgh val Lower bound of 7-odd-limit diamond monotone
|-
|
| 23/12
| 187.7199
|
|-
|
| 13/9
| 187.7941
|
|-
|
| 23/13
| 188.2081
|
|-
| 8 ⧵ 51
|
| 188.2353
| 51cdh val Lower bound of 9-odd-limit diamond monotone
|-
|
| 23/18
| 188.2910
|
|-
|
| 17/11
| 188.4094
|
|-
|
| 13/12
| 188.4523
|
|-
|
| 15/14
| 188.4918
|
|-
|
| 13/8
| 188.5463
|
|-
| 11 ⧵ 70
|
| 188.5714
| Lower bound of 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 7/5
| 188.5930
|
|-
|
| 17/13
| 188.6048
|
|-
|
| 21/20
| 188.6213
|
|-
|
| 13/11
| 188.6230
| 13-odd-limit minimax
|-
|
| 23/14
| 188.6483
|
|-
|
| 17/16
| 188.6521
|
|-
|
| 23/21
| 188.6537
|
|-
|
| 17/12
| 188.6572
|
|-
|
| 17/9
| 188.6601
|
|-
|
| 3/2
| 188.6717
| 9, 15 and 17-odd-limit minimax
|-
|
| 11/9
| 188.6852
| 11-odd-limit minimax
|-
|
| 19/13
| 188.6872
| 19, 21 and 23-odd-limit minimax
|-
|
| 11/6
| 188.6891
|
|-
|
| 23/15
| 188.6959
|
|-
|
| 11/8
| 188.6963
|
|-
|
| 23/20
| 188.7115
|
|-
|
| 21/17
| 188.7379
|
|-
|
| 19/18
| 188.7467
|
|-
|
| 17/14
| 188.7480
|
|-
|
| 21/11
| 188.7584
|
|-
| 14 ⧵ 89
|
| 188.7640
| 19, 21 and 23-odd-limit diamond monotone (singleton)
|-
|
| 19/12
| 188.7655
|
|-
|
| 11/7
| 188.7726
|
|-
|
| 17/15
| 188.7824
|
|-
|
| 21/16
| 188.7909
|
|-
|
| 21/19
| 188.7932
|
|-
|
| 19/16
| 188.7968
|
|-
|
| 17/10
| 188.8056
|
|-
|
| 19/14
| 188.8115
|
|-
|
| 15/11
| 188.8135
|
|-
|
| 7/4
| 188.8235
|
|-
|
| 11/10
| 188.8463
|
|-
|
| 23/17
| 188.8511
|
|-
|
| 19/15
| 188.8537
|
|-
|
| 7/6
| 188.8804
| 7-odd-limit minimax
|-
|
| 19/10
| 188.8909
|
|-
|
| 15/8
| 188.9127
|
|-
|
| 23/22
| 188.9217
|
|-
|
| 23/19
| 188.9746
|
|-
|
| 9/7
| 189.0056
|
|-
|
| 21/13
| 189.0356
|
|-
|
| 5/4
| 189.0404
| 5-odd-limit minimax
|-
|
| 19/11
| 189.2390
|
|-
|
| 13/7
| 189.3085
|
|-
|
| 5/3
| 189.4552
|
|-
| 3 ⧵ 19
|
| 189.4737
| Upper bound of 7, 9, 11, 13, 15 and 17-odd-limit diamond monotone
|-
|
| 15/13
| 190.4518
|
|-
|
| 23/16
| 190.5752
|
|-
|
| 19/17
| 192.5576
|
|}

[[Category:Xenial| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Starling temperaments]]

Cassaschismic

2026-07-06T02:42:21Z

Overthink: /* External links */ link text; name creator

{{Infobox regtemp
| Title = Cassaschismic
| Subgroups = 2.3.5.7.11, 2.3.5.7.11.13, 2.3.5.7.11.13.19
| Comma basis = [[19712/19683]], [[41503/41472]] (11-limit); [[2080/2079]], [[4096/4095]], [[19712/19683]] (13-limit); [[1216/1215]], [[1540/1539]], [[1729/1728]], [[2080/2079]] (2.3.5.7.11.13.19)
| Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 270
| Mapping = 1; 1 0 -14 23 12 5; 0 1 0 0 -1 1
| Generators = 3/2; 5/4 | Generators tuning = 702.2307; 386.3245
| Optimization method = CWE
| Pergen = (P8, P5, ^1)
| Color name = Salozo & Sasaru + Ya Salozo & Sasaru (& Sathoyo (& Sanogu))
| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
}}
'''Cassaschismic''' is a [[rank-3 temperament]] that expands [[gary]]'s [[chain of fifths]] into the full [[11-limit]] by adding an independent [[generator]] for the [[5/1|5th]] [[harmonic]]. It is therefore a member of the [[garischismic family]] and [[olympic clan]].

The generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. By moving the generators around, it can also be taken to be a ~4.5{{c}} generic aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], the [[352/351|minor minthma]] and many other important commas around that size. [[Tempering out]] this aberschisma results in [[cassandra]], so cassaschismic is a rank-3 [[detemperament]] of it, modifying its mapping by ±1 aberschisma to reach primes 5, 13, and 19.

Other rank-2 temperaments of cassaschismic include [[cotoneum]], [[gariwizmic]], [[newt]], [[satin]], [[vulture]], [[paramity]] and [[heptacot]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.

{{Databox|Generators needed to reach the aberschisma|
* Newt (41 & 270): -41 hemififths;
* Cotoneum (41 & 217): -41 fifths, equating it with the 41-comma;
* Gariwizmic (94 & 270): +53 fifths (mercator comma) - 1/2 pythagorean comma;
* Vulture (53 & 217): -41 1/4-fifths;
* Satin (94 & 217): -94 1/3-fourths;
* Paramity (53 & 311): -53 1/5-elevenths;
* Heptacot (12e & 311): 12 1/7-fifths.
}}

Cassaschismic is [[support]]ed by notable [[equal temperament]]s such as {{EDOs| 217, 270, 311, and 364 }}, where the aberschisma step is well represented by one edostep. It is also trivially supported by edos of cassandra, these being [[41edo|41]], [[53edo|53]], [[94edo|94]]. [[12edo]] supports it trivially through the 12e [[val]], where both the comma step and the aberschisma step are tempered out. It can be used in any of those forms.

See [[Garischismic family #Cassaschismic]] for technical data.

== Interval lattice ==
Here is a quick compressed cheat sheet of octave-reduced intervals. This is a simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to 8 fifths - 1 aberschisma step, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 aberschisma step. There are no octave reduced primes or prime inverses with positive fifth step and aberschisma step.

{| class="wikitable center-1 right-2 right-4"
! rowspan="2" | #
! colspan="2" | Aberschisma offset -1
! colspan="2" | Aberschisma offset 0
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 1195.83
| 351/176
| 0.00
| '''1/1'''
|-
| 1
| 698.06
| 256/171
| 702.23
| '''3/2'''
|-
| 2
| 200.29
| 64/57
| 204.46
| '''9/8'''
|-
| 3
| 902.52
| '''32/19'''
| 906.69
| 27/16
|-
| 4
| 404.75
| 24/19
| 408.92
| 19/15
|-
| 5
| 1106.98
| 36/19
| 1111.15
| 19/10
|-
| 6
| 609.21
| 27/19
| 613.38
| 57/40
|-
| 7
| 111.44
| '''16/15'''
| 115.62
| 77/72
|-
| 8
| 813.68
| '''8/5'''
| 817.85
| 77/48
|-
| 9
| 315.91
| 6/5
| 320.08
| 77/64
|-
| 10
| 1018.14
| 9/5
| 1022.31
| 65/36
|-
| 11
| 520.37
| 27/20
| 524.54
| 65/48
|-
| 12
| 22.60
| 81/80
| 26.77
| 64/63
|-
| 13
| 724.83
| 38/25
| 729.00
| '''32/21'''
|-
| 14
| 227.06
| 57/50
| 231.23
| '''8/7'''
|-
| 15
| 929.29
| 77/45
| 933.46
| 12/7
|-
| 16
| 431.52
| 77/60
| 435.69
| 9/7
|-
| 17
| 1133.75
| 52/27
| 1137.92
| 27/14
|-
| 18
| 635.98
| 13/9
| 640.15
| 81/56
|-
| 19
| 138.21
| 13/12
| 142.38
| 88/81
|-
| 20
| 840.44
| '''13/8'''
| 844.61
| 44/27
|-
| 21
| 342.67
| 39/32
| 346.85
| 11/9
|-
| 22
| 1044.91
| 64/35
| 1049.08
| 11/6
|-
| 23
| 547.14
| 48/35
| 551.31
| '''11/8'''
|-
| 24
| 49.37
| 36/35
| 53.54
| 33/32
|}
<nowiki>*</nowiki> In 2.3.5.7.11.13.19-subgroup CWE tuning, octave reduced

== Notation ==
Cassaschismic is easily notated with [[chain-of-fifths notation]] with two extra pairs of accidentals: one for the comma step, and the other for the aberschisma step. It can therefore be seen as an addition to the cassandra chain of fifths, which itself can be seen as an addition to the 12edo chain of fifths, providing a layered-precision system of notation that ranges from rough (12), to moderately accurate (41, 53, 94), to highly accurate (217, 270, 311, …).

As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step.

{| class="wikitable center-all"
|+Nomenclature of selected intervals
! Ratio
! Example on C
|-
| 3/2
| C–G (perfect fifth)
|-
| 5/4
| C–^↓E (upsubmajor third)
|-
| 7/4
| C–↓Bb (subminor seventh)
|-
| 11/8
| C–↑↑F (hyperfourth)
|-
| 13/8
| C–v↑↑Ab (downhyperminor sixth)
|-
| 19/16
| C–^Eb (upminor third)
|}

== See also ==
* [[User:Eufalesio/Ultimate]] – An opinion-based derivation of cassaschismic

== External links ==
* [https://www.desmos.com/calculator/pbyqpjgrrn Ultimate mapgraph] – A [https://www.desmos.com/calculator Desmos] graph by [[Eufalesio]] showing cassaschismic edos up to 311, [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is the [[patent val]] prime 11, which is not used in cassaschismic. The blue dots indicate going up and down by Pythagorean commas in the 12L 29s scale, and the orange dots indicate the leftover edosteps. The jump from 94 to 270 is due to 135edo being next in line for cassandra; since halving it results in 270edo, it is used instead, and also to showcase the use of aberschismas to reach primes 5, 13, and 19.

[[Category:Cassaschismic| ]] 
[[Category:Rank-3 temperaments]]
[[Category:Microtemperaments]]
[[Category:Garischismic family]]
[[Category:Olympic clan]]

Shallowtone

2026-07-06T02:37:33Z

Overthink: This category feels fitting enough (no 9-odd-limit monotone and high error)

{{Infobox regtemp
| Title = Shallowtone
| Subgroups = 2.3.5, 2.3.5.7
| Comma basis = [[295245/262144]] (5-limit) [[36/35]], [[295245/262144]] (7-limit)
| Mapping = 1; 1 -10 12
| Edo join 1 = 7 | Edo join 2 = 30b
| Generators = 3/2
| Generators tuning = 681.2
| Optimization method = CWE
| MOS scales = [[2L 3s]], [[2L 5s]], [[7L 2s]]
| Odd limit 1 = 5 | Mistuning 1 = 18.7 | Complexity 1 = 16
| Odd limit 2 = 9 | Mistuning 2 = 45.3 | Complexity 2 = 23
}}
'''Shallowtone''' is a [[regular temperament|temperament]] where the [[shallowtone comma]] is [[tempering out|tempered out]]. It is generated by a flattened [[3/2|perfect fifth]], typically sharper than in [[mavila]] but flatter than in [[7edo]], and [[5/4]] is reached by minus ten fifths [[octave reduction|octave-reduced]], which is an augmented third (C–E𝄪) in melodic antidiatonic notation and a diminished third (C–E𝄫) in harmonic antidiatonic notation. This gives it a high [[error]], although not as much as mavila.

The only reasonable [[extension]] to the [[7-limit]] tempers out [[36/35]]. Additionally, there is a weak extension [[Greenwoodmic temperaments #Semishallowtone|semishallowtone]] with a half-octave period, tempering out [[405/392]].

The name was coined by [[User:CompactStar|CompactStar]] in 2024. It refers to how it is an opposite to [[deeptone]] (which is on the other side of 7edo), as well as because it is just below the diatonic "surface".

See [[Mint temperaments #Shallowtone]] for technical data.

== Interval chain ==
In the following table, prime harmonics are labeled in '''bold'''.

{| class="wikitable"
|-
! rowspan="2" | #
! rowspan="2" | Cents*
! colspan="2" | Approximate ratios
|-
! 5-limit
! 13-limit extension
|-
| 0
| 0.000
| [[1/1]]
|
|-
| 1
| 681.801
| '''[[3/2]]'''
| [[13/9]]
|-
| 2
| 163.602
| [[9/8]]
| [[11/10]], [[13/12]], [[35/32]], [[81/70]], [[8192/8019]]
|-
| 3
| 845.403
| [[27/16]]
| '''[[13/8]]''', [[117/70]], [[4096/2673]]
|-
| 4
| 327.204
| [[81/64]]
| [[39/32]], [[99/80]], [[1024/891]]
|-
| 5
| 1009.0511
| [[243/128]]
| [[117/64]], [[512/297]]
|-
| 6
| 490.806
| [[512/405]], [[729/512]]
| [[128/99]]
|-
| 7
| 1172.607
| [[256/135]]
| [[64/33]]
|-
| 8
| 654.408
| [[64/45]]
| '''[[16/11]]''', [[112/81]]
|-
| 9
| 136.209
| [[16/15]]
| [[12/11]], [[28/27]]
|-
| 10
| 818.010
| '''[[8/5]]'''
| [[14/9]], [[18/11]], [[64/39]]
|-
| 11
| 299.811
| [[6/5]]
| [[7/6]], [[27/22]], [[63/52]]
|-
| 12
| 981.612
| [[9/5]]
| '''[[7/4]]''', [[26/15]], [[81/44]]
|-
| 13
| 463.413
| [[27/20]]
| [[13/10]], [[21/16]], [[243/176]]
|-
| 14
| 1145.214
| [[81/40]]
| [[39/20]], [[63/32]], [[729/352]]
|-
| 15
| 627.015
| [[243/160]]
| [[117/80]], [[189/128]]
|}
<nowiki/>* In 5-limit CTE tuning, octave reduced

[[Category:Shallowtone| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Mint temperaments]]
[[Category:Exotemperaments]]

Very low accuracy temperaments

2026-07-06T02:32:21Z

Overthink: /* Oxygen */ add equivalence to show damage; In some tunings the 6/5 might be okay, so "either of those intervals" feels like suboptimal wording

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

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

[[Subgroup]]: 2.3.5

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

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

: mapping generators: ~3/2, ~5

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

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

[[Badness]] (Sintel): 0.508

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

=== Septimal antitonic ===
Subgroup: 2.3.5.7

Comma list: 9/8, 15/14

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

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

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

Badness (Sintel): 0.490

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

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

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

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

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

Badness (Sintel): 0.721

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

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

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

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

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

Badness (Sintel): 0.628

=== Antaeus ===
Subgroup: 2.3.5.7

Comma list: 9/8, 35/32

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

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

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

Badness (Sintel): 0.950

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

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

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

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

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

Badness (Sintel): 1.12

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

[[Subgroup]]: 2.3.5

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

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

: mapping generators: ~4/3, ~5

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

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

[[Badness]] (Sintel): 1.83

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

[[Subgroup]]: 2.3.5

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

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

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

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

[[Badness]] (Sintel): 0.392

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

Subgroup: 2.3.5.7

Comma list: 10/9, 15/14

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

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

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

Badness (Sintel): 0.606

=== Antonym ===
Subgroup: 2.3.5.7

Comma list: 7/6, 10/9

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

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

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

Badness (Sintel): 0.568

=== Antony ===
Subgroup: 2.3.5.7

Comma list: 8/7, 10/9

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

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

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

Badness (Sintel): 0.720

=== Brutus ===
Subgroup: 2.3.5.7

Comma list: 10/9, 28/25

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

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

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

Badness (Sintel): 1.35

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

Subgroup: 2.3.5.7

Comma list: 10/9, 35/32

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

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

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

Badness (Sintel): 1.30

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

Subgroup: 2.3.5.7

Comma list: 10/9, 49/45

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

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

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

Badness (Sintel): 1.43

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

Subgroup: 2.3.5.7

Comma list: 10/9, 49/48

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

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

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

Badness (Sintel): 1.49

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

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~5/4, ~7

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

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

[[Badness]] (Sintel): 0.726

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

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

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

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

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

Badness (Sintel): 0.846

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

[[Badness]] (Sintel): 1.16

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

[[Subgroup]]: 2.3.5.7

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

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

: Mapping generators: ~6/5, ~7

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

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

[[Badness]] (Sintel): 1.22

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

[[Badness]] (Sintel): 1.29

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

[[Subgroup]]: 2.3.5.7

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

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

: mapping generators: ~2, ~7/4

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

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

[[Badness]] (Sintel): 0.942

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

[[Badness]] (Sintel): 1.11

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

[[Badness]] (Sintel): 1.08

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

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

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

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

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

Badness (Sintel): 0.887

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

[[Badness]] (Sintel): 1.48

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

[[Badness]] (Sintel): 1.14

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

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

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

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

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

Badness (Sintel): 1.08

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

[[Subgroup]]: 2.3.5.7

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

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

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

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

[[Badness]] (Sintel): 1.95

== Oxygen ==
{{See also| Porcupine family }}
Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7, and conflating 10/9 with 7/6. While this means it does not represent 7-limit JI with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block|Fokker blocks]].

[[Subgroup]]: 2.3.5.7

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

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

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

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

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

[[Badness]] (Sintel): 1.52

2023/2000

2026-07-02T04:25:01Z

Overthink: /* Pentagoth */ fix incorrect generator ratio

{{Infobox Interval
| Ratio = 2023/2000
| Name = pentagoth comma
| Color name = 17oozg33 sosozotrigu 3rd Sosozotrigu comma
| Comma = yes
}}
'''2023/2000''', the '''pentagoth comma''', is the interval between a stack of two [[20/17]] minor thirds and one [[7/5]] tritone.

[[Tempering out]] this comma alone in the 2.5.7.17 subgroup leads to the '''pentagoth''' temperament.

== Temperaments ==
Note: groundfault uses extensions that identify the ~20/17~13/11 with ~19/16 and tempering out 209/208; however, this leads to considerable damage to prime 19.
=== Pentagoth ===
[[Subgroup]]: 2.5.7.17

[[Comma list]]: 2023/2000

{{Mapping|legend=1| 1 2 2 4 | 0 1 1 1 | 0 0 2 -1 }}
: mapping generators: ~2, ~5/4, ~20/17

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.566{{c}}, ~5/4 = 387.475{{c}}, ~20/17 = 289.545{{c}}
: [[error map]]: {{val| +0.566 2.294 -1.127 -4.760}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 388.049{{c}}, ~20/17 = 289.369{{c}}
: error map: {{val| 0.000 1.735 -2.038 -6.275 }}

[[Badness]] (Sintel): 0.509

==== 2.5.7.13/11.17.23 ====
[[Subgroup]]: 2.5.7.13/11.17.23

[[Comma list]]: 221/220, 161/160, 1309/1300

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

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~5/4 = 387.513, ~[[13/11]] = 288.786

[[Badness]] (Sintel): 0.191

===== No-3 no-19 23-limit =====
[[Subgroup]]: 2.5.7.11.13.17.23

[[Comma list]]: 176/175, 221/220, 161/160, 1309/1300

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

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~5/4 = 390.453, ~[[13/11]] = 288.844

[[Badness]] (Sintel): 0.497

=== Pentagoth clan ===
Rank-2 temperaments supporting pentagoth
==== Vengeance ====
{{Main| Vengeance }}
A lower-error replica of mavila, with the fifth being ~[[25/17]] instead of ~[[3/2]].

[[Subgroup]]: 2.5.7.17

[[Comma list]]: 2023/2000, 78608/78125

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

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[34/25]] = 527.718

{{Optimal ET sequence|legend=1|7g, 9, 25, 34, 93, 127, 288, 415}}

[[Badness]] (Sintel): 0.720

==== Sidewalk ====
Sidewalk, 2.5.7.17[21 & 25], tempers out 823543/800000 in the 2.5.7 subgroup.

[[Subgroup]]: 2.5.7.17

[[Comma list]]: [[343/340]], 2023/2000

{{Mapping|legend=2|1 4 4 6|0 -7 -5 -8}}

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[20/17]] = 287.377

[[Badness]] (Sintel): 0.447

===== 2.5.7.13/11.17.23 =====
[[Subgroup]]: 2.5.7.13/11.17.23

[[Comma list]]: 221/220, 161/160, 1309/1300, 343/340

{{Mapping|legend=2|1 4 4 0 6 5|0 -7 -5 1 -8 -2}}

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[13/11]] = 287.412

[[Badness]] (Sintel): 0.256
===== No-3 no-19 23-limit =====
[[Subgroup]]: 2.5.7.11.13.17.23

[[Comma list]]: 176/175, 221/220, 161/160, 1309/1300, 343/340

{{Mapping|legend=2|1 4 4 8 8 6 5|0 -7 -5 -19 -18 -8 -2}}

[[Optimal tuning]] ([[CWE]]): ~2 = 1\1, ~[[13/11]] = 286.821

[[Badness]] (Sintel): 0.671

== Etymology ==
As of June 6, 2026, pentagoth is the name given to the rank-3 temperament 2.5.7.17[2023/2000] by [[groundfault]]. It originally referred to a rank-2 temperament now called [[vengeance]], which as a 2.5.7.17 temperament tempers out this comma.

[[Category:Pentagoth]]
[[Category:Rank 2]]

Orwell

2026-06-30T16:18:26Z

Overthink: 9-odd-limit instead of 7

{{interwiki
| en = Orwell
| de = Orwell
| es =
| ja =
}}
{{Infobox regtemp
| Title = Orwell
| Subgroups = 2.3.5.7, 2.3.5.7.11
| Comma basis = [[225/224]], [[1728/1715]] (7-limit); [[99/98]], [[121/120]], [[176/175]] (11-limit)
| Edo join 1 = 22 | Edo join 2 = 31
| Mapping = 1; 7 -3 8 2
| Generators = 7/6 | Generators tuning = 271.5 | Optimization method = CWE
| MOS scales = [[4L 1s]], [[4L 5s]], [[9L 4s]], [[9L 13s]]
| Pergen = (P8, cP5/7)
| Odd limit 1 = 9 | Mistuning 1 = 5.35 | Complexity 1 = 13
| Odd limit 2 = 11-limit 21 | Mistuning 2 = 9.32 | Complexity 2 = 22
}}
[[File:Orwell generator in 31.jpg|thumb|Martin Aurell's diagram showing Orwell[9] generated in 31 tone equal temperament.]]

'''Orwell''' – so named because 19 steps of [[84edo]], i.e. 19\84, is a possible generator – is an excellent [[7-limit]] [[regular temperament|temperament]] and an amazing [[11-limit]] temperament because of the simplicity of [[harmonic]] [[11/1|11]].

In orwell, [[8/5]] is divided into three equal steps, each of which represent [[7/6]], so that [[1728/1715]] ({{S|6/S7}}) is tempered out. This means that the [[5/1|5th harmonic (5/1)]] is divided into three equal steps that represent [[~]][[12/7]]. After two 8/5's (six generators), [[9/7]] is found by [[tempering out]] the marvel comma, [[225/224]], and thus the [[3/1|just perfect twelfth (3/1)]] is divided into 7 equal steps.

In the 11-limit, two generators are equated to [[15/11]] and [[11/8]] (meaning [[99/98]] and [[121/120]] are tempered out). This means that three stacked generators makes the [[orwell tetrad]] 1–7/6–11/8–8/5, a chord in which every interval is a (tempered) 11-odd-limit consonance. Other such chords in undecimal orwell are the [[keenanismic chords]] and the [[swetismic chords]]. A far more complicated mapping of 11 at 33 generators, tempering out [[441/440]] instead, is also possible and is known as [[newspeak]] temperament; these two mappings unite on 31edo.

Compatible [[equal temperaments]] include [[22edo]], [[31edo]], [[53edo]], and [[84edo]] (though in 84edo, 11-limit orwell uses the 84e [[val]]). Orwell is in better tune in lower limits than higher ones; the [[optimal patent val]] is [[296edo]] in the 5-limit, [[137edo]] in the 7-limit, and [[53edo]] in the 11-limit.

See [[Semicomma family #Orwell]] for technical details. See [[Orwell extensions]] for details about 13-limit extensions.

== Theory ==
=== Interval chain ===
Odd harmonics 1–21 and their inverses are in '''bold'''.
{| class="wikitable center-1 right-2"
|-
! #
! Cents*
! Approximate ratios
|-
| 0
| 0.00
| '''1/1'''
|-
| 1
| 271.46
| 7/6
|-
| 2
| 542.91
| '''11/8''', 15/11
|-
| 3
| 814.37
| '''8/5'''
|-
| 4
| 1085.82
| '''15/8''', 28/15
|-
| 5
| 157.28
| 11/10, 12/11, 35/32
|-
| 6
| 428.73
| 9/7, 14/11, 32/25
|-
| 7
| 700.19
| '''3/2'''
|-
| 8
| 971.64
| '''7/4'''
|-
| 9
| 43.10
| 33/32, 36/35, 49/48
|-
| 10
| 314.55
| 6/5
|-
| 11
| 586.01
| 7/5
|-
| 12
| 857.46
| 18/11
|-
| 13
| 1128.92
| 21/11, 27/14, 48/25
|-
| 14
| 200.37
| '''9/8''', 28/25
|-
| 15
| 471.83
| '''21/16'''
|-
| 16
| 743.28
| 49/32, 54/35
|-
| 17
| 1014.74
| 9/5
|-
| 18
| 86.19
| 21/20
|-
| 19
| 357.65
| 27/22, 49/40
|-
| 20
| 629.10
| 36/25
|-
| 21
| 900.56
| 27/16, 42/25
|-
| 22
| 1172.01
| 63/32
|}
<nowiki/>* In 11-limit CWE tuning, octave reduced

=== Chords and harmony ===
{{See also| Chords of orwell | Functional harmony in rank-2 temperaments }}

The fundamental otonal consonance of orwell, voiced in a roughly {{w|tertian harmony|tertian}} manner, is 4:5:6:7:9:11. In terms of generator steps this is 0–(−3)–7–8–14–2, only available in a 22-tone mos. However, some subsets of this chord are way simpler, such as 8:11:12:14, which is 1–11/8–3/2–7/4 (0–2–7–8).

The generator, ~7/6, is a septimal interval, so chords could instead be built around it as 1–7/6–3/2 (0–1–7), 1–7/4–3 (0–8–7), or tetrads such as 1–7/6–3/2–7/4 (0–1–7–8).

To 1–7/6–3/2–7/4 we may add 11/8, or to 1–11/8–3/2–7/4 we may add 7/6, to form an essentially tempered pentad, 1–7/6–11/8–3/2–7/4 (0–1–2–7–8). Its inverse is 1–12/11–9/7–3/2–12/7 (0–5–6–7–(−1)), which can serve as a minor counterpart. This is similar, but also in clear contrast to the 1–5/4–3/2 (0–4–1) and 1–6/5–3/2 (0–(−3)–1) chords of [[meantone]]. Two approaches to functional harmony thus arise.

First, we can treat the septimal chords above as the basis of harmony, but swapping the roles of 3 and 7 according to their temperamental complexities (number of generator steps). Thus a "dominant" chord is either 7/6 or 12/7 over tonic; a "subdominant" chord is either 7/6 or 12/7 under tonic. This leads to an approach closely adherent to mos scales. The 9-tone mos contains a tonic and a "dominant" triad. The 13-tone mos is good for encapsulating tonic, "pre-dominant", and "dominant" functions, triads to pentads alike.

Second, we can treat the same chords as the basis of harmony, and keeping the role of the [[chain of fifths]] as the spine on which the functions are defined. This means dominant is still 3/2 over tonic, for example. A consequence is we must step out of the logic of mos scales, as they are often too restrictive without the many fifths to stack. This is essentially working in JI, but using the commas tempered out in some way to lock into the identity of the temperament.

== Scales ==
{{Main| Orwell scales }}

=== Mos scales ===
* [[Orwell5]]

; 9-tone scales (sLsLsLsLs, proper)
* [[Orwell9]] – 84edo tuning
* [[Orwell9-12]] – 7-limit POTE tuning, mapped to 12-tones

[[file:OrwellNonatonicPOTE.mp3]] in POTE tuning

[[file:OrwellNonatonic22edo.mp3]] in 22edo

[[file:OrwellNonatonic53edo.mp3]] in 53edo

{| class="wikitable center-all"
|-
! Small ("minor") interval
| 114.29
| 228.59
| 385.72
| 500.02
| 657.15
| 771.44
| 928.57
| 1042.87
|-
! JI intervals represented
| 15/14~16/15
| 8/7
| 5/4
| 4/3
| 16/11
| 14/9~11/7
| 12/7
| 11/6
|-
! Large ("major") interval
| 157.13
| 271.43
| 428.56
| 542.85
| 699.98
| 814.28
| 971.41
| 1085.71
|-
! JI intervals represented
| 12/11~11/10
| 7/6
| 14/11~9/7
| 11/8
| 3/2
| 8/5
| 7/4
| 15/8
|}

; 13-tone scales (LsLLsLLLsLLsL, improper)
* [[Orwell13]] – 84edo tuning
* [[Orwellwoo13]] – [6 5/2] unchanged-interval (eigenmonzo) tuning

{| class="wikitable center-all"
|-
! Small ("minor") interval
| 42.84
| 157.13
| 271.43
| 314.26
| 428.56
| 542.85
| 585.69
| 699.98
| 814.28
| 857
| 971.41
| 1085.71
|-
! JI intervals represented
|
| 12/11~11/10
| 7/6
| 6/5
| 14/11~9/7
| 11/8
| 7/5
| 3/2
| 8/5
| 18/11
| 7/4
| 15/8
|-
! Large ("major") interval
| 114.29
| 228.59
| 342.88
| 385.72
| 500.02
| 614.31
| 657.15
| 771.44
| 885.74
| 928.57
| 1042.87
| 1157.16
|-
! JI intervals represented
| 15/14~16/15
| 8/7
| 11/9
| 5/4
| 4/3
| 10/7
| 16/11
| 14/9~11/7
| 5/3
| 12/7
| 11/6
|
|}

; 22-tone scales
* [[Orwell22]]
* [[Orwellwoo22]] – [6 5/2] unchanged-interval (eigenmonzo) tuning

=== Transversal scales ===
* [[Orwell13trans]]
* [[Orwell13trans57]]
* [[Orwell22trans]]
* [[Orwell22trans57]]
* [[Orwell31trans]]
* [[Orwell31trans57]]

=== Others ===
* [[Orwell-graham]] – 13-tone modmos in 53edo tuning
* [[Orwell13-modmos-containing-minerva12]] – 13-tone modmos in POTE tuning
* [[Minerva12-orwell-tempered]] – Minerva[12] tempered to orwell

== 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
|-
! Equilateral
| CEE: ~7/6 = 271.3553{{c}}
| CSEE: ~7/6 = 271.3339{{c}}
| POEE: ~7/6 = 271.3727{{c}}
|-
! Tenney
| CTE: ~7/6 = 271.5130{{c}}
| CWE: ~7/6 = 271.5097{{c}}
| POTE: ~7/6 = 271.5087{{c}}
|-
! Benedetti, Wilson
| CBE: ~7/6 = 271.5725{{c}}
| CSBE: ~7/6 = 271.5741{{c}}
| POBE: ~7/6 = 271.5576{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Equilateral
| CEE: ~7/6 = 271.4920{{c}}
| CSEE: ~7/6 = 271.3038{{c}}
| POEE: ~7/6 = 271.1665{{c}}
|-
! Tenney
| CTE: ~7/6 = 271.5597{{c}}
| CWE: ~7/6 = 271.4552{{c}}
| POTE: ~7/6 = 271.4261{{c}}
|-
! Benedetti, Wilson
| CBE: ~7/6 = 271.5915{{c}}
| CSBE: ~7/6 = 271.5302{{c}}
| POBE: ~7/6 = 271.5174{{c}}
|}

{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | DR and equal-beating tunings
|-
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
| 3:4:5 (+1 +1) || ~7/6 = 272.890{{c}} || ''f''10 − 8''f''3 + 8 = 0 || 1–3–5 equal-beating tuning
|-
| 4:5:6 (+1 +1) || ~7/6 = 271.508{{c}} || ''f''10 + 2''f''3 - 8 = 0 || 1–3–5 equal-beating tuning
|}

=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo generator
! [[Eigenmonzo|Unchanged interval (eigenmonzo)]]*
! Generator (¢)
! Comments
|-
| 2\9
|
| 266.667
| Lower bound of 7-odd-limit diamond monotone
|-
|
| 7/6
| 266.871
|
|-
|
| 15/11
| 268.475
|
|-
|
| 11/7
| 269.585
|
|-
|
| 11/6
| 270.127
|
|-
|
| 15/14
| 270.139
|
|-
|
| 49/48
| 270.633
|
|-
|
| 21/11
| 270.728
|
|-
| 7\31
|
| 270.968
| Lower bound of 9- and 11-odd-limit diamond monotone
|-
|
| 11/9
| 271.049
|
|-
|
| 7/4
| 271.103
|
|-
|
| 7/5
| 271.137
| 7- and 11-odd-limit minimax
|-
|
| 5/4
| 271.229
|
|-
|
| 21/20
| 271.359
|
|-
|
| 21/16
| 271.385
|
|-
| 19\84
|
| 271.429
| 84e val
|-
|
| 25/24
| 271.487
|
|-
|
| 64/63
| 271.488
|
|-
|
| 5/3
| 271.564
| 5-odd-limit minimax
|-
|
| 9/5
| 271.623
| 9-odd-limit minimax
|-
|
| 81/80
| 271.661
|
|-
| 12\53
|
| 271.698
|
|-
|
| 3/2
| 271.708
|
|-
| 17\75
|
| 272.000
|
|-
|
| 15/8
| 272.067
|
|-
|
| 36/35
| 272.086
|
|-
|
| 9/7
| 272.514
|
|-
| 5\22
|
| 272.727
| Upper bound of 7-, 9- and 11-odd-limit diamond monotone
|-
|
| 11/10
| 273.001
|
|-
|
| 11/8
| 275.659
|
|}
<nowiki/>* Besides the octave

== Non-octave settings ==
=== Watcher ===
By switching the roles of the period and generator, we end up with a nonoctave temperament that is to orwell what [[angel]] and [[devadoot]] are to [[meantone]] and [[magic]], respectively. There is an interesting mos with 7 notes per period; if this is derived as a subset of [[84edt]] (which has 12 notes per period, and is almost identical to 53edo), the resulting mos has the same structure as the 12edo diatonic scale, only compressed so that the period is ~272 cents rather than an octave. Thus, a piano keyboard for this mos would look exactly the same as a typical keyboard, only what looks like an octave would not be one anymore. This temperament could be called [[Wikipedia: Watcher (angel)|watcher]], a reference to a class of angels whose very name carries Orwellian connotations. The 12-integer-limit otonality (1::12) and utonality (1/(1::12)) both have complexity 4. If we consider these to be the fundamental consonances, then using the 7-note-per-period mos, there are exactly 3 of each type per period, which again is analogous to the diatonic scale. While angel and devadoot do not perform well past the 10-integer-limit, watcher handles the 12-integer-limit with ease. Straight-fretted watcher guitars could be built as long as the strings were all tuned to period-equivalent notes.

== Rank-3 temperaments ==
Following is a list of rank-3, or planar temperaments that are supported by orwell temperament.

{| class="wikitable"
|-
! colspan="2" | Rank-3 temperament
! colspan="4" | Among others, rank-3 temperament is also supported by…
|-
! 7-limit
! 11-limit Extension
! 9tet
! 22tet
! 31tet
! 53tet
|-
| [[Marvel]]
|
| Negri, septimin, august, amavil, enneaportent
| Magic, pajara, wizard, porky
| Meantone, miracle, tritonic, slender, würschmidt
| Garibaldi, catakleismic
|-
|
| Marvel
| Negri, septimin, enneaportent
| Magic, pajarous, wizard
| Meanpop, miracle, tritoni, slender
| Garibaldi, catakleismic
|-
|
| Minerva
| Negric, august, amavil
| Telepathy, pajara
| Meantone, revelation, würschmidt
| Cataclysmic
|-
|
| Artemis*
| Wilsec
| Divination, hemipaj, porky
| Migration, oracle, tritonic
|
|-
| [[Porwell]]
|
| Triforce, armodue, twothirdtonic
| Porcupine, astrology, shrutar, hendecatonic, septisuperfourth
| Hemiwürschmidt, valentine, mohajira, grendel
| Amity, hemischis, hemikleismic
|-
|
| Zeus
| Triforce, armodue, twothirdtonic
| Porcupine, astrology, shrutar, hendecatonic
| Hemiwur, valentine, mohajira
| Hitchcock, hemikleismic
|-
|
| Jupiter
|
| Septisuperfourth
| Hemiwürschmidt, grendel
| Amity, hemischis
|-
| [[Orwellismic]]
|
| Beep, secund, infraorwell, niner
| Superpyth, doublewide, echidna
| Myna, mothra, sentinel, semisept
| Quartonic, buzzard
|-
|
| Orwellian
| Pentoid, secund
| Suprapyth, doublewide
| Myno, mothra, sentinel
|
|-
|
| Guanyin
| Infraorwell, niner
| Superpyth, fleetwood, echidna
| Myna, mosura, semisept
| Quartonic, buzzard
|-
| [[Nuwell]]
|
| Progression, superpelog
| Quasisuper, hedgehog
| Squares, nusecond
| Alphatrimot, hamity
|-
|
| Big brother
| Progression, superpelog
| Quasisupra, hedgehog
| Squares, nusecond
| Alphatrimot, hamity
|-
| [[Horwell]]
|
|
| Bisupermajor, escaped, fifthplus
| Hemithirds, worschmidt, tertiaseptal
| Countercata, pontiac
|-
|
| Zelda
|
| Bisupermajor, sensa
| Hemithirds, worschmidt, tertia
| Countercata
|}
<nowiki/>* [[Weak extension]] (one or more generators from the parent temperament are split)

== Music ==
; [[Tarkan Grood]]
* ''Mountain Villiage'' (2013) – [https://web.archive.org/web/20201127012514/http://micro.soonlabel.com/gene_ward_smith/Others/Grood/Mountain_Village_TarkanGrood.mp3 play] | [https://soundcloud.com/tarkan-grood/mountain-village-tarkangrood SoundCloud] – in Orwell[9]

; [[Andrew Heathwaite]]
* ''[[Earwig]]'' (2012) – [https://web.archive.org/web/20201127015238/http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/earwig.mp3 play] – in 31edo tuning
* [[Technical Notes for Newbeams #Elf Dine on Ho Ho|''Elf Dine on Ho Ho'']] (2012) – [https://web.archive.org/web/20201127015137/http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2004%20Hypnocloudsmack%201.mp3 play] – in 53edo tuning
* [[Technical Notes for Newbeams #Spun|''Spun'']] (2012) – [https://web.archive.org/web/20201112021340/http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/Newbeams/Andrew%20Heathwaite%20-%20Newbeams%20-%2008%20Spun.mp3 play] – in Orwell[13]
* [https://web.archive.org/web/20201127013436/http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/andrewheathwaite+onedropofrain.mp3 ''one drop of rain'']
* [https://web.archive.org/web/20201127014501/http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/andrewheathwaite+ivecomewithabucketofroses.mp3 ''i've come with a bucket of roses'']
* [https://web.archive.org/web/20201127014830/http://micro.soonlabel.com/gene_ward_smith/Others/Heathwaite/andrewheathwaite+myownhouse.mp3 ''my own house'']

; [[Peter Kosmorsky]]
* ''Tunicata and Fugue'' – [http://www.archive.org/details/TunicataAndFugue details] | [http://archive.org/download/TunicataAndFugue/TunicataAndFugueVer2.mp3 play]

; [[Löis Lancaster]] ([[Roncevaux]])
* ''Schizo Blue'' – [https://web.archive.org/web/20201127012220/http://micro.soonlabel.com/gene_ward_smith/Others/Roncevaux/Schizo_Blue__22_EDO_Orwell__first_mix_by_Roncevaux_on_SoundCloud___Hear_the_world_s_sounds.mp3 play] | [https://soundcloud.com/lois-lancaster/schizo-blue-22-edo-orwell SoundCloud]{{dead link}}
* ''Sejaliscos'' (2013) – [https://web.archive.org/web/20201127012431/http://micro.soonlabel.com/gene_ward_smith/Others/Roncevaux/Sejaliscos_by_Roncevaux_on_SoundCloud___Hear_the_world_s_sounds.mp3 play] | [https://soundcloud.com/lois-lancaster/sejaliscos SoundCloud] – in Orwell[9], 22edo tuning

; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=zWrOiih7raY ''Orwell Canon 3 in 1 upon a Ground for Baroque Oboe, Viola, Clarinet, and Viola da Gamba''] (2024)
* [https://www.youtube.com/shorts/g7C2OrFd-nk ''Orwell Micro Trio, for Organ (Just: 7 Orwells = 1 Twelfth)''] (2025) — in open-ended Orwell tuning, but with the generator adjusted to be extremely close to 12\53, at 271.71{{c}}

; [[Herman Miller]]
* ''[https://soundcloud.com/morphosyntax-1/a-hidden-world A Hidden World]'' (2022) – in Orwell[31]
* ''[https://soundcloud.com/morphosyntax-1/zurg-tuun-vantu-war-is-peace Zurğ tuun vantu]'' (2024) – in Orwell[13], with a generator of 271.5{{c}} and a period of 1199.5{{c}}

; [[Sevish]]
* "[[Droplet]]", from ''[[Rhythm and Xen]]'' (2015) – [https://sevish.bandcamp.com/track/droplet Bandcamp] | [https://soundcloud.com/sevish/droplet?in=sevish/sets/rhythm-and-xen SoundCloud] | [https://www.youtube.com/watch?v=xVZy9GUeMqY YouTube] – drum and bass in Orwell[9], 53edo tuning

; [[Gene Ward Smith]]
* ''Trio in Orwell'' (archived 2010) – [http://www.archive.org/details/TrioInOrwell details] | [http://www.archive.org/download/TrioInOrwell/TrioInOrwell.mp3 play] – in Orwell[9], 53edo tuning
* [https://web.archive.org/web/20201112015404/http://micro.soonlabel.com/gene_ward_smith/transformers/swing-orwell9.mp3 ''Swing in Orwell-9'']

; [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127014716/http://micro.soonlabel.com/orwell/daily20100721-gpo-owellian-cameras.mp3 ''Orwellian Cameras'']

== Keyboards ==
{{See also| Orwell on an isomorphic keyboard | Lumatone mapping for orwell }}

To play interactive versions of these keyboards, check out [https://github.com/vsicurella/SuperVirtualKeyboard Vito Sicurella's plugin], which works with REAPER:

[[File:Orwell_13.png|alt=Orwell_13.png|1023x292px|Orwell_13.png]]

[[File:Orwell_22.png|alt=Orwell_22.png|1023x292px|Orwell_22.png]]

[[File:orwell13_axis49.png|alt=orwell13_axis49.png|orwell13_axis49.png]]

[[Category:Orwell| ]] 
[[Category:Rank-2 temperaments]]
[[Category:Semicomma family]]
[[Category:Marvel temperaments]]
[[Category:Orwellismic temperaments]]
[[Category:Listen]]

Bird's eye view of temperaments by accuracy

2026-06-30T16:15:15Z

Overthink: /* Orwell */ link

== Introduction ==
This page serves to document temperaments people deem to be highly valuable, and organizing them broadly by accuracy preference, and then approximately by subgroup focus, that is, what sort of harmonies, broadly speaking, the temperament is targetting.

Under each accuracy and subgroup focus is found an incomplete list of temperaments, organized ''approximately'' by complexity (how many notes per octave are required). The complexity given is the note count per octave (or for no-2's, per tritave), with the set of odds used for deriving the complexity given. Sometimes two complexities are given and the average is taken for the purpose of ranking.

== Edit guides ==

=== Do <big>not</big> add a temperament if you do not deem it unusually/uniquely valuable for making music. ===
Temperaments here should only be ones that one or more people seriously consider to be "cream of the crop". Therefore, when adding a temperament, ''make sure to include a description of how it works'', ideally one that motivates it as to why someone might want to use it.

=== The ''accuracy'' classification of a temperament is the maximum error allowed on (almost*) all intervals. ===
<nowiki>*</nowiki>If there are intervals in the corresponding (thr)odd-limit that violate this bound slightly even in an optimized tuning, they should be noted as "'''Bound-violating intervals:'''" under the header for easy comprehension and consideration.

This should not be abused to attempt to reclassify temperaments. There should be ideally zero, at most 2 intervals (more in more complex subgroups). The main exception that justifies more than two bound-violating interval pairs is when a single odd is responsible for all of them (e.g. odd 7 in [[#Buzzard]]). Therefore, most temperaments in a category are more accurate than the bound suggests.

The bounds are:

* [[Bird's eye view of temperaments by accuracy#Microtemperaments (<1c)|Microtemperament]] (~ <1c)
* [[Bird's eye view of temperaments by accuracy#High accuracy (<4c)|High accuracy]] (1 ~ 4c)
* [[Bird's eye view of temperaments by accuracy#Medium accuracy|Medium accuracy]] (4 ~ 7c)
* [[Bird's eye view of temperaments by accuracy#Low accuracy (<12c)|Low accuracy]] (7 ~ 12c)
* [[Bird's eye view of temperaments by accuracy#Very low accuracy (12~18c)|Very low accuracy]] (12 ~ 18c)
* [[Bird's eye view of temperaments by accuracy#Exotemperament|Exotemperament]] (~ >18c)

The two name system is for two reasons: to account for people's varying preferences and terminology for accuracies and to make the system of categories symmetrical, with [[exotemperament]]s and [[microtemperament]]s as the extrema.

The cent errors are a result of a set of compromises between people of different preference, and being given in cents, are somewhat arbitrary. The exotemperament bound is very high because it is not meaningful to pretend that the bound is even remotely precise. The "<~18c" bound was chosen to allow using [[5edo]] and [[7edo]] as the circle of [[~]][[3/2]]'s to barely qualify as not being exotemperaments, due to both being the extrema of [[5L 2s|diatonic]].

=== Style guide ===
The convention for this page in contrast to most Xenharmonic Wiki pages, is that if you should link rank 2 temperaments '''on this page,''' unless you are merely discussing competing extensions or don't intend to log that temperament separately. That way, the page can be fairly self-contained to avoid intimidating or confusing someone using this page as a reference.

The obvious exception is the title referring to the main entry. The purpose of using #Temperament here is to indicate that clicking on the link will send you to the section in '''this''' page, if it exists. If not, feel free to add it.

== Generator tunings ==
"Generator tunings" on this page are given in the format '''a\b''', meaning ''a'' steps of ''b'' [[edo]]; the frequency ratio 2''a''/''b''.

You can input 44\111 in [[Scale Workshop]] by hovering over "New scale" on the left, then pressing "Rank-2 temperament", then entering 44\111 in the "Generator" input field.

You will then get a list of '''MOS sizes''' to pick from before pressing '''OK''' to generate the scale. The importance of this is that picking from this list gives you a [[MOS scale]] which has only up to two sizes of minimum intervals. The UI shows an incomplete list of MOSes, so you can still add your own amount of notes by modifying the '''Scale size'''.

In some cases, the [[octave]] is split or tempered, in which case you will instead see pairs.

* Taking [[Bird's eye view of temperaments by accuracy#Diaschismic, srutal archagall|#Diaschismic]] as a split example, it is (3\34, 1\2). This means that 3\34 goes in the "Generator" input field and 1\2 goes in the "Period" input field. You must then change '''Number of periods''', to 2 in this case. This will create a series of [[Multi-MOS]] scales that repeat more than once per octave. Notice also that "Generators up/down from 1/1" goes up in pairs. This because you are adding a generator per period, and in diaschismic there are two periods, thus two generators. In some pages like [[Sintel]]'s [https://sintel.pythonanywhere.com/ temperament calculator], this is not displayed. For instance, in diaschismic, the calculator says you need -2 fifths to reach ~5, so to use this in In Scale Workshop you multiply this number by the number of periods to get your mapped in interval: input 4 down generators.

* Taking [[#Godzilla]] as a tempering example, it is (4\30<3>, 19\30<3>). This means that 4\30<3> goes in the "Generator" input field and 19\30<3> goes in the "Period" input field. This reflects tempering the octave sharp so that 3/1 is just in order to get various intervals of 3, 5, 7 and 13 more in tune as they are all tuned flat in [[19edo]]; very significantly so in the case of 7 and 13.

Note that the generator tunings are listed in order of increasing accuracy, with the least accurate being the leftmost; the least accurate tuning is not required to satisfy the maximum cent error, so that a tone-efficient example is included, though the example should be unambiguously representative of the full scope of its harmony within reason. All tunings after it should thus be increasingly accurate so that it satisfies the tuning bounds indicated by the accuracy classification of the temperament (up to any indicated exceptions for the temperament).

'''Once you have generated your MOS scale in Scale Workshop, you are ready to select a tuning file format from the right side; check which one is needed by your DAW/synthesizer.'''

If you're having trouble, don't be afraid to ask in the [[Links#Discord_server|Xenharmonic Alliance Discord server]]! Remember to be persistent and patient.

== Note counts ==
Wherever you see "Note count(s):" on this page, those are the note counts needed to find all of those harmonics relative to the same note, so that you can build an otonal/harmonic series chord out of all of them.

For example, for [[#Septimal meantone]] we see:

Note count: 11 for {3, 5, 7, 9, 15, 21, 25} ([[7L 5s]], [[12L 7s]])

This tells us that making a 1:3:5:7:9:15:21:25 chord requires using 11 notes, and the bracketed [[7L 5s]], [[12L 7s]] tell us that these are potential MOS scales of interest for the purpose of getting these harmonies.

Because most temperaments on this page have an octave or an ''n''th of an octave (1\''n'') as the period, it suffices in those cases to describe the set of odd harmonics that they target, and how many notes it takes to do so according to their mappings. Because different note counts may allow for different amounts of odd harmonics to be targeted, there is sometimes multiple note counts, to help the reader decide between them.

== Explanation of subgroup focuses ==
For temperaments with prime 2, complexity is judged by the odd-limit of the temperament's subgroup, potentially plus some composite odds that the temperament can reasonably be said to target if it doesn't hurt complexity a lot. For temperaments without prime 2, we use the analogous concept of throdd-limit, where the [[equave]] is the [[tritave]] ([[3/1]]).

Each accuracy category is split into the following subgroup focuses, which are enumerated here so as to explain what is meant by them:
=== 5-limit focus ===
The main purpose of the temperament is [[5-limit]] harmony, and may admit one or two "sporadic primes" > 11 if they don't damage the 5-limit more than it already is.
=== 7-limit focus ===
The main purpose of the temperament is [[7-limit]] harmony, and may admit one or two "sporadic primes" > 13 if they don't damage the 7-limit more than it already is.
=== 11-limit focus ===
The main purpose of the temperament is [[11-limit]] harmony, and may admit one or two "sporadic primes" > 17 if they don't damage the 7-limit more than it already is.
=== ~17-limit focus ===
The main purpose of the temperament is approximately [[17-limit]] harmony, potentially minus one prime (hence the ~). The omitted prime could be prime 17, so pure [[13-limit]] temperaments are documented under this category. However, such a temperament may not omit primes 2, 3 or 5, due to [[#No-2's focus|no-2's focus]], [[#No-3's focus|no-3's focus]] and [[#No-5's focus|no-5's focus]] categories. If such a temperament admits a "sporadic prime" that is mapped somewhat simply relative to the temperament's lower-limit complexity, then it should instead be classified under [[#Higher-limit focus|higher-limit focus]].
=== Higher-limit focus ===
The main purpose of the temperament is subgroup harmonies of the [[19-limit]], [[23-limit]], [[29-limit]], [[31-limit]], etc. Subgroup temperaments with multiple primes > 17 should usually go here, unless one of those primes is very complex to reach relative to the temperament's complexity and only included because it's essentially "free" in the sense of not damaging the temperament.
=== No-2's focus ===
All no-2's temperaments go under this category, which includes all [[tritave]] temperaments. If prime 3 is not present, it must be clearly noted as "(no-3's)" so that those looking for no-2's and no-3's temperaments can find them easily. Even harmonics are allowed as long as they don't implicate the existence of prime 2 in the subgroup, but judging their complexity becomes more difficult as a result, so all such temperaments must be clearly noted as "(with even harmonics)".
=== No-3's focus ===
All temperaments with prime 2 but no prime 3 go under this category.
=== No-5's focus ===
All temperaments with primes 2 and 3 but no prime 5 go under this category.
== '''Microtemperaments (<1c)''' ==
These temperaments essentially serve as ways of simultaneously simplifying and imparting new structure onto [[JI]] with minimal to unnoticeable tuning damage.
=== 5-limit focus ===
==== [[Schismic]] ====
[[#Note counts|Note count]]: 12 for {3, 5, 9, 15, 27, 45(, 81)} ([[5L 7s]] or [[12L 5s]])

[[#Generator tunings|Generator tunings]]: (24\41,) 31\53, 69\118, 100\171, 131\224

Schismic is an extremely accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])4, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]].

In schismic, (9/8)6 overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated.

Many extensions to other primes exist, but most are not accurate enough to be microtemperaments, except for the extension to prime 41 by tempering out [[1025/1024]] = ([[41/32]])/([[32/25]]). However, as it is common to want to extend schismic, we will note common extensions here:

* [[#Garibaldi]] finds [[~]][[8/7]] as [[9/8]] * [[81/80]] by tempering out [[5120/5103]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth.

* [[Schismatic family#Tridecaschismic (2.3.5.13)|Tridecaschismic]] (which is the 2.3.5.13 version of [[#Cassandra]]) finds 13/4 as (9/8)10 and demands an approximately Pythagorean tuning.

* [[#Nestoria]] equates [[~]][[19/16]] with [[32/27]] and [[~]][[19/15]] with [[81/64]].
* [[#Pontiac]] equates (27/25)2 with 7/6, tempering out [[4375/4374]]. Extremely accurate, but more complex, finding 7/4 as a major sixth + 3 pyth-syntonic commas.

=== 7-limit focus ===

==== [[Ennealimmal]] ====
[[Bird's eye view of temperaments by accuracy#Note counts|Note counts]]:

45 for {3, 5, 7, 9} ([[27L 18s]])

72 for {3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63} ([[27L 45s]])

[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: (3\72, 1\9), (4\99, 1\9), (7\171, 1\9), (11\270, 1\9)

Ennealimmal has a 1/9-octave period representing [[27/25]], and two of them represent [[7/6]], tempering out [[4375/4374]]. It is generated by a [[~]][[36/35]] quartertone, with 3/2 being mapped to 6 periods minus 2 generators, 5/4 mapped to 4 periods minus 3 generators, and 7/4 mapped to 8 periods minus 2 generators. It finds a neutral third representing [[49/40]]~[[60/49]] at 3 periods minus 1 generator, tempering out [[2401/2400]].

This temperament therefore tempers out the two smallest superparticular ratios in the [[7-limit]], 2401/2400 and 4375/4374. It is very accurate, with errors of around 0.2 cents in optimized tunings. However, due to the high note count, one may prefer lower accuracy temperaments.

=== 11-limit focus ===
=== ~17-limit focus ===

==== [[Gariwizmic]] ====
[[Bird's eye view of temperaments by accuracy#Note counts|Note counts]]:

* 82: see [[#Gary]]
* 176 for {3, 5, 7, 9, 11, 13, 15, 21, 25}

Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2)

Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a [[#Gary]] fifth. As its name implies, it also tempers out the [[wizma]].

Tempering out the kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (-27 fifths - 1 period). This structure is practically identical to that of [[cassaschismic]], only that the minicomma is not an independent generator and is instead found in the deep in the diploid chain of fifths.

It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It is very complex mapping-wise despite its great accuracy, but it can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 (and 19) which is still reachable within the rank-2 structure. It isn't a true generator; were it independent, the temperament would be [[cassaschismic]].

It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]].

==== [[Decoid]] ====
Note counts: TBA

Bound-violating intervals: [[15/13]]

Generator tunings: (103\130, 1\10), (111\140, 1\10), (214\270, 1\10)

Decoid has a 1/10-octave period representing [[15/14]], 7 of them being [[13/8]] as in 10edo. The generator can be a [[26/15]] semitritave, tempering out [[676/675]]. Thanks to ''relatively'' good approximation of 10edo of the 2.3.5.7.13, these primes require little change, only 2, -3, -1, and 0 generators respectively. Prime 11 is a bit more complex, at -8 generators. This also tempers out [[2080/2079]], [[4096/4095]] and [[1716/1715]].

The main selling point of decoid is 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its rank-2 layout] which has a a lot of important intervals close together, namely, all the primes except 11 are close to horizontal, and the fact that it is supported too by 270edo, making it an incredible 13-limit temperament. Prime 19 can also be reached by tempering [[1216/1215]], and is thus reached by 7 generators.

=== Higher-limit focus ===
=== No-2's focus ===
=== No-3's focus ===
=== No-5's focus ===

==== [[Gary]] ====
Note counts:

* 41 for {3, 7, 9, 11, 21, 27, 33, 77, 81, 99} ([[12L 29s]])

Generator tunings: 24\41, 55\94, 79\135, 498\851

Gary is an extremely good 2.3.7.11 temperament generated by a slightly sharp fifth which maps the pythagorean comma to 64/63, and two of those to 33/32, tempering out [[19712/19683]] and [[131072/130977]], making it an [[Olympic clan|olympic]] and [[Symbiotic comma|symbiotic]] temperament. It can be seen as the no-5, no-13 restriction of [[#Cassandra]]. The lack of 5 and 13 makes these two mappings be optimal, reaching errors well below a cent.

It also provides an astonighingly accurate approximation to [[19/15]], which then mapped to the major third tempers out 1540/1539 '''and''' 1216/1215, whose subgroup is 2.3.7.11.19/5, which has no independent prime 19 or 5; only the specific compound interval of dividing the two.

41edo is the simplest and easiest to use, [[135edo]] provides an essentially perfect tuning with manageable fineness, to which no other tuning is as good until [[851edo]] makes scene.

Prime 31 can also be easily included by tempering out [[1024/1023]], essentially equating [[64/63]] with [[63/62]] and [[33/32]] with [[32/31]]. It accrues more error here than the rest of primes, wanting to tune the fifth a tenth of a cent sharper to get everything within subcent values, which makes everything ever so slightly worsely tuned but still within microtemperament range.

There are multiple ways of incorporating prime 5 and 13:

* [[#Cassandra]] is the simplest, tempering out the [[225/224|marvel]] and [[325/324|marveltwin]] commas. High accuracy.
* [[#Cotoneum]] is a good one, though with high complexity, tempering out the [[quince comma]] and the [[minisma]]. Very high accuracy.
* [[#Gariwizmic]] is one of the best, also with high complexity, tempering out the [[1716/1715|lumma]] and the minisma. Extremely high accuracy.

== '''High accuracy (1~4c)''' ==
The bound is the approximate [[JND|melodic JND (Just-Noticeable-Difference)]], though note that this doesn't mean that damage/mistuning is ''imperceptible'' in these temperaments as the harmonic JND can often be significantly smaller, depending largely on context, timbre and who is listening/who you ask.
=== 5-limit focus ===
==== [[Cata]] ====
[[#Note counts|Note count]]: 15 for {3, 5, 9, 13, 15, 25} ([[4L 7s]])

[[#Generator tunings|Generator tunings]]: 9\34, 14\53, 23\87, 37\140

Cata is a very efficient 5-limit and 2.3.5.13-subgroup temperament with a generator of a very slightly sharpened [[6/5]], two of which make [[13/9]] and thus three of which make [[26/15]] which is made into half of [[3/1]] so that its octave complement of [[15/13]] is half of [[4/3]]. It is amazing for its combination of accuracy and simplicity, because making six [[~]][[6/5]] generators equal to a fourth or fifth (up to octave-reduction) is the simplest equivalence possible without incurring a lot of damage. Its 7-note scale of [[4L 3s]] is usable, and its interpretation is accurately {[[25/24]], [[6/5]], [[5/4]], [[36/25]][[~]][[13/9]], [[3/2]], [[26/15]], [[2/1]]} so that it is at the simplest structural level well-supplied with plausible harmony, as this structure will persist and be duplicated in every superset/derived MOS scale, such as the likely more useful 15-note one, whose tuning range is at broadest in the [[15edo]] to [[19edo]] range, corresponding to the small step being at least half the size of the large step so that it has [[Rothenberg propriety]] (for those that care about this property).

Cata admits an elegant extension to prime 7 called [[Catakleismic]], at the cost of some accuracy, a higher complexity and a smaller valid tuning range.

This extension can be observed based on an [[S-expression]]-based comma list of: {[[169/168|S13]], [[225/224|S15 = S25*S26*S27]], [[325/324|S10/S12 = S25*S26]](, [[625/624|S25]], [[676/675|S26 = S13/S15]], [[729/728|S27]])}, which is notable as making use of the record prime gap between 23 and 29 for an opportune no-11's 13-limit tempering opportunity [[~]][[28/27]][[~]][[27/26]][[~]][[26/25]][[~]][[25/24]], which as shown, implies tempering many notable commas, the most accurate of which is the [[ragisma]] (S25/S27), corresponding here to having an interval [[~]][[14/13]][[~]][[27/25]][[~]][[13/12]], and (arguably) the most interesting of which is making use of the exceptional numerical coincidence that [[676/675|S13/S15 = S26]]. The tuning range for catakleismic is approximately [[53edo]] to [[72edo]] - which are both reasonable tunings for it, with 53edo more accurate on the full subgroup and 72edo more accurate in the [[7-limit]].

==== [[Sensipent]] ====
[[#Note counts|Note counts]]:
* 10 for {3, 5, 31} ([[8L 3s]])
* 19 for adding {9, 15, 25} ([[8L 11s]])

[[#Generator tunings|Generator tunings]]: 17\46, 24\65, 55\149, 199\539

Sensipent is an accurate 2.3.5.31 temperament with a generator of [[~]][[31/24]][[~]][[40/31]], where the two interpretations of the generator differ by [[961/960|S31 = (31/30)/(32/31)]], which is the best extension of 5-limit sensipent as its generator serves as half of 40/24 = [[5/3]], so that the generator is the midpoint of [[4/3]] and [[5/4]], whose difference is [[16/15]], hence the relevance of making [[~]][[32/31]][[~]][[31/30]].

Sensipent finds [[6/1]] (the fifth plus two octaves) at 7 generators.

It admits a number of extensions of varying accuracy:
* the most accurate is [[#Sendai]] which finds primes 23 and 29
* the second most accurate is [[#Sensible]], which finds primes 11, 17 and 23
* the simplest but least accurate is [[Sensipent family#Sensor|Sensor]] (commonly just called "sensi"), which interprets it as a full 17-limit temperament, for which the best tuning is [[46edo]].

==== [[Würschmidt]] ====
[[#Note counts|Note counts]]:
* 10 for {3, 5, 15, 25, 125} ([[3L 7s]])
* 18 for adding {9, 23, 45, 75, 115} ([[3L 16s]])
* 22 (or 23) for adding {11, 55(, 69)} ([[3L 19s]] or [[3L 22s]])

[[#Generator tunings|Generator tunings]]: 21\65, 32\99, 53\164

Würschmidt (sometimes written wurschmidt or wuerschmidt for convenience) is a temperament with an approximately 1{{cent}} sharp [[5/4]] as the generator, so that [[6/1]] is reached as (5/4)8. The rationale for this is that (5/4)3 falls short of the octave by [[128/125]], and this is approximately half of [[25/24]], so that if we flatten (5/4)2 = [[25/16]] by [[128/125]] twice we get [[~]][[3/2]]. Therefore, in an optimized tuning, we can expect the fifth to be slightly flat, so that [[25/24]] is sharpened so that it makes sense to equate with a slightly flattened [[~]][[24/23]] by tempering out their difference, [[576/575|S24]], which is favourable as finding interpretations of a variety of intervals that are otherwise given somewhat questionable 5-limit interpretations, [[Würschmidt#Interval chain|as documented in its interval chain]]. Because of dividing 6/1 into eight, it admits a neutral third at 4 generators so that an extension to prime 11 is also natural by tempering out [[243/242|S9/S11 = (12/8)/(11/9)2 = (3/2)/(11/9)2]] so that [[~]][[11/9]][[~]][[27/22]] is the neutral third.

Würschmidt can be seen as something like a [[cluster temperament]] with 3 main clusters, and with [[~]][[128/125]][[~]][[46/45]] as the interval separating intervals in a given cluster. A notable extension to prime 7 is [[#Hemiwürschmidt]] by splitting the generator into two [[~]][[28/25]]'s, which is thus the result of combining würschmidt with [[#Didacus]].

=== 7-limit focus ===
==== [[Garibaldi]] ====
[[#Note counts|Note count]]: 18 for {3, 5, 7, 9, 15, 21, 27, 35, 45} ([[12L 5s]] or [[12L 17s]])

Bound-violating intervals: [[7/5]], [[21/20]], [[15/14]] (all derived from contrasting odd 7 (sharp) and 5 (flat))

[[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94

Garibaldi is arguably the best way to bestow prime 7 upon [[#Schismic]] effectively, at the cost of some accuracy. It uses a slightly sharper fifth that tunes the 5-limit worse, making it no longer a microtemperament. This is done by interpreting (9/8)3 as [[~]][[10/7]] by tempering out S8/S9 = [[5120/5103]] so that 8/7 and 10/9 are equidistant from 9/8, corresponding to equating S8 = [[64/63]] and S9 = [[81/80]] respectively. This results in a conveniently general tempered comma-sized interval that also represents the [[Pythagorean comma]], which is equal to (9/8)6 / (2/1). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has a (barely) inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 41 = [[135edo]] and 94 + 53 = [[147edo]] also support it but with yet more inconsistencies due to the finer gamut, so it's worth checking the "Prime harmonics" tables to see if you're okay with the errors.

Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about prime 11 or prime 13:
* For prime 11, [[41edo]] is better, as it finds [[~]][[11/9]] as half of the fifth and as a comma above [[~]][[6/5]] (as in [[cassandra]]) or a comma below [[~]][[5/4]] (as in [[andromeda]]), corresponding to [[tetracot]] in 2.3.5.11 (by tempering out (S9/S11 = [[243/242]],) S10 = [[100/99]] and S10/(S9/S11) = [[2200/2187]] respectively) which splits the halved fifth into two small major seconds of [[~]][[11/10]][[~]][[10/9]] around 175.6 cents. However, there is significant damage to 15/13 and 13/10.
* For primes 5 and 13, [[53edo]] is better, as it finds [[interseptimal interval]]s distinctly from adjacent [[septimal]] intervals so that [[~]][[15/13]] is half of a practically-just [[4/3]] (tempering out [[676/675|S13/S15]]) and is (resultantly) found as a comma above [[~]][[8/7]] or a comma below [[~]][[7/6]], which reflects to (3/2)/(15/13) = [[~]][[13/10]] being made the midpoint of [[~]][[21/16]] and [[~]][[9/7]] respectively. It also makes [[~]][[16/13]] a comma below [[~]][[5/4]] (by tempering out ((5/4)/(16/13))/(81/80) = 325/324). This corresponds to a number of temperaments; the most relevant of which for [[#Schismic]] is the very accurate extension to prime 13 called [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]], corresponding to reaching 13/4 through (9/8)10 (tempering out the [[tridecapyth comma]]) and also corresponding to tempering out [[325/324]] = S25*S26 = S10/S12 as mentioned. However, there is significant damage to 14/11. (Also, 53edo's fifth is flatter so better tuned for schismic/for the 5-limit, as implicitly aforementioned.)

Both support [[cassandra]], a 13-limit extension which finds [[~]][[16/13]] as a comma below [[~]][[5/4]] and equates (3/2)/(16/13) = [[39/32]] with [[11/9]]. (This means that in 41edo, we have a single neutral third at the cost of damage to prime 13, while in 53edo we have two neutral thirds at the cost of damage to prime 11, hence 41 + 53 = [[94edo]] is a lot more characteristic of cassandra's tuning.)

=== 11-limit focus ===
==== [[Miracle]] ====
[[#Note counts|Note count]]: 23 for {3, 5, 7, 9, 11, 15, 21} ([[10L 11s]] or [[10L 21s]])

[[#Generator tunings|Generator tunings]]: 3\31, 4\41, 7\72

Bound-violating intervals: 16/15 (none in 11-odd-limit)

Miracle is an elegant temperament that splits [[3/2]] into six equal parts that can be derived as the most natural and efficient way of doing so through [[S-expression]]s by splitting 3/2 into two by tempering out {{nowrap| [[243/242|S9/S11 {{=}} (12/8)/(11/9)2 {{=}} (3/2)/(11/9)2]] }}, splitting 3/2 into three by tempering out [[1029/1024|S7/S8 = (9/6)/(8/7)3 = (3/2)/(8/7)3]] and then splitting the [[~]][[8/7]] in two by tempering out [[225/224|S15 = (15/14)/(16/15)]] so that [[15/14]] and [[16/15]] are equated. [[72edo]] is a very good tuning of miracle, though [[31edo]] and [[41edo]] may be preferred for smaller note counts and for the various things they support, EG [[#Meantone]] for 31edo and [[#Garibaldi]] for 41edo.

=== ~17-limit focus ===

==== [[Cassandra]] ====
See [[#Garibaldi]].

==== [[Buzzard]] ====
[[#Note counts|Note counts]]:
* 31 for {3, 5, ''7'', 9, 13, 15, (19*,) ''21'', 27, ''35''(, 81)} ([[5L 28s]])
* 47 for {3, 5, ''7'', 9, 11, 13, 15, 19*, ''21'', 27, 33, ''35'', 39(, 81)} (5L 43s (minimum) or 53L 5s)

Bound-violating intervals: [[7/4]], [[11/7]] and various intervals made with compound intervals of 7 (corresponding odds italicized), which are the simplest (see [[#2.3.7 Buzzard]])

[[#Generator tunings|Generator tunings]]: 21\53, 23\58, 44\111

Buzzard is generated by a sharp ~21/16 or near-just ~25/19, tuned strictly in the 475 to 476 [[cent]] range; two of these make a hemitwelfth (sqrt(3) interval) of ~26/15, which means four makes ~3/1, and combined with finding ~13/11 as ~32/27, this is enough to determine the no-17's [[19-limit]] mapping.

Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[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]]. Tempering out S8/S9 leverages this by splitting [[36/35|S6]] 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]], while tempering out S6/S7 implies that [[49/48|S7]] 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 at least the full 13-limit. Because of the sharp 3 and 5 (in an optimized tuning), [[~]][[16/15]] is tuned quite flat so that a very natural extension to prime 17 exists by equating it with a sharp [[~]][[17/16]] (tempering out [[256/255|S16]]).

[[53edo]] and [[58edo]] are good tunings for the 13-limit; though 58edo is more accurate, 53edo supports a variety of structures that might be preferred over the ones 58edo supports due to including many contextually-usable harmonies of 19 so that the gen can (also) more accurately be interpreted as ~25/19, such as [[#Cata]] (and its best extension to prime 7, [[#Catakleismic]], as well as another more complex extension to prime 7 called [[countercata]]), [[#Schismic]] and especially [[marvel]] (so that it supports [[#Garibaldi]]); by contrast, 58edo may be preferred for supporting [[#Echidna]] and [[#Diaschismic]]. [[111edo]] is a very elegant tuning for higher limits, combining it with [[#Sensible]] and [[#Sendai]] (two extensions of [[#Sensipent]]). ([[48edo]] (gen exactly 475{{cent}}) supports a ''significantly'' more damaged version of the 7-limit (and arguably no-11's 13-limit**) part of buzzard that uses different mappings for primes 11 and 19.)

<nowiki>*</nowiki> Notably, the mapping for 19 simplifies extremely in 48edo and 53edo, via ~19/16 = ~32/27 (at the cost of damage); its canonic mapping (following the microtemperament [[vulture]]) requires 41 generators, so only really makes sense to consider if you are using [[111edo]] for buzzard (or opportunistically in 53edo***), as there is no ~19/16 in 58edo. Specifically, 53edo attempts to equate 13/11 with 19/16 by using 32/27 to contextually suggest/pun both of them, and is likely about as optimal for that purpose as it can be. In other words, buzzard naturally distinguishes ~19/16 and ~13/11 in most tunings, explaining the mapping complexity of ~19/16 which is canonically reached distinctly from the simpler ~13/11.

(<nowiki>**</nowiki> If we use the 48f val, so that ~13/8 is mapped as 825{{cent}} instead of 850{{cent}}, then we preserve all the mappings of buzzard in the no-11's 13-limit; using the accurate 11 inherited from 24edo requires using a different mapping for 11, but as a result, ~13/11 is at 275{{cent}}, while if we used the [[patent val]] (which in 2.3.11.13 is the same as [[24edo]]) it would be mapped to the more dubious 300{{cent}} (though both are not great). In other words, in a strange sense, 48edo preserves the "spirit" of no-17's 19-limit buzzard of distinguishing ~13/11 from ~19/16, though this isn't a recommendable tuning unless you are interested in other things 48edo supports that are unrelated to buzzard. The main reason for considering this non-patent flat mapping of 13, then, is that the interseptimals 15/13 and 13/10 (and their octave-complements) are quite accurate in 24edo, via error-cancellation with the flat 5.)

(<nowiki>***</nowiki> 53edo is uniquely the tuning that uses both the canonic mapping at 41 gens and the simplified higher-damage mapping at -12 gens = ~32/27, as 41 + 12 = 53.)

=== Higher-limit focus ===
==== [[Sendai]] ====
{{ See also | Sensipent#Sendai interval table }}
[[#Note counts|Note counts]]:
* 13 for {3, 5, 23, 31, 69, 115} ([[8L 11s]])
* 29 for adding {9, 15, 25, 29, 87, 145} ([[19L 8s]])

[[#Generator tunings|Generator tunings]]: 24\65, 31\84, 55\149

Sendai is an accuracy-focused extension of [[#Sensipent]] to primes 23 and 29. If one is fine with lowering the accuracy but increasing the number of interpretations of harmony, it can merge meaningfully with [[#Sensible]], giving access to primes 11 and 17, and this has the benefit that combining them does not force an [[edo]] (or more generally a rank 1) tuning, though if one wants to use an edo/rank 1 tuning, the obvious choice is [[65edo]] which gets you prime 19 too (though that could be added as a more complex extension of either).

=== No-2's focus ===
=== No-3's focus ===
=== No-5's focus ===
== '''Medium accuracy (4~7c)''' ==
Many temperaments that people consider theoretically tend to fall into this category, due to its balance of simplicity and accuracy and due to the common usage of [[meantone]] temperaments, though plenty of simple temperaments exist that are even more accurate, documented in higher-accuracy categories.
=== 5-limit focus ===
==== [[Meantone]] ====
[[#Note counts|Note count]]: 5 for {1, 3, 5}

[[#Generator tunings|Generator tunings]]: 11\19, 18\31, 29\50

Meantone is an incredibly efficient temperament for targetting [[5-odd-limit]] harmony whose characteristic is flattening [[3/2]] (the generator) by a few cents. Perhaps unsurprisingly, it was historically the most commonly used temperament. It does this by sacrificing a distinction between [[9/8]] and [[10/9]] so that two "tones" makes [[5/4]], hence its name.

==== [[Srutal archagall|Diaschismic, Srutal archagall]] ====
[[#Note counts|Note counts]]:
* 10 for {3, 5, 9, 15, 17} ([[2L 8s]])
* 12 for adding {25} ([[10L 2s]])

[[#Generator tunings|Generator tunings]]: (3\34, 1\2), (4\46, 1\2), (7\80, 1\2)

Srutal archagall is the natural extension of [[Diaschismic family#Srutal aka diaschismic|5-limit diaschismic]] to prime 17 by interpreting the generator as a near-just [[17/16]] and the period as [[~]][[24/17]][[~]][[17/12]].

It is notable as preserving a lot of intuitions of 12edo like [[9/8]] as two semitones, [[6/5]] as three semitones, [[5/4]] as a semitone less than [[4/3]] which is itself a semitone less than half an octave. It essentially "doubles up" on familiar categories by: two minor seconds, [[~]][[25/24]] and [[~]][[18/17]][[~]][[17/16]][[~]][[16/15]], two major seconds, [[~]][[10/9]] and [[~]][[9/8]][[~]][[17/15]], two minor thirds, [[~]][[20/17]] and [[~]][[6/5]], and two major thirds, [[~]][[5/4]] and [[~]][[51/40]][[~]][[32/25]]. Furthermore, major and minor intervals are separated by a semitone of [[17/16]], though interestingly this results in alternating the tuning of the otherwise-familiar category, e.g. [[20/17]] = ([[5/4]])/([[17/16]])). The distance between these pairs of familiar intervals is an exaggerated syntonic comma ([[81/80]]) which is also equal to a flattened diesis ([[128/125]]), making it useful as a more accurate alternative to meantone. Notably the intervals of 5 require using the period offset to reach, so that the minor third reached by the circle of fifths is actually [[20/17]].

For abundant options, one might prefer a 22-note MOS over a 12-note one, so that srutal archagall can be seen as a detempering of [[22edo]], but the 12-note MOS is likely the easiest and most intuitive to approach for a beginner. [[34edo]] is a good tuning for optimizing the 2.3.5.17 subgroup.

=== 7-limit focus ===
==== [[Septimal meantone]] ====
[[#Note counts|Note count]]: 11 for {3, 5, 7, 9, 15, 21, 25} ([[7L 5s]], [[12L 7s]])

Bound-violating intervals: [[9/8]], [[10/9]], [[36/25]] (none if odd 9 is omitted)

[[#Generator tunings|Generator tunings]]: 18\31

Septimal meantone is an extension of [[#meantone]] that finds 8/7 as the diminished third and 7/6 as the augmented second, so that using an augmented sixth with a major triad forms a harmonic seventh chord. Though 12 notes is about sufficient for achieving its harmony, often one wants to use a 19-note MOS ([[12L 7s]]) for more freedom and availability.

It has two main extensions to prime 11, both similarly complex, discussed in [[meantone vs meanpop]], though the one called [[undecimal meantone]] is arguably more elegant as being the merge of septimal meantone and the no-3's 11-limit temperament [[#Didacus]], which can be seen as every other gen of undecimal meantone. An alternative extension that splits the generator in half is by interpreting [[~]][[11/9]] as half of the meantone fifth, by tempering out [[243/242|S9/S11 = (12/8)/(11/9)2 = (3/2)/(11/9)2]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. This leads to [[#Migration]] if you accept the septimal meantone mapping of 7 (which becomes double as complex), or [[mohaha]] if you interpret it as no-7's.

==== [[Mothra]] ====
[[#Note counts|Note count]]: 15 for {3, 5, 7, 9, 21, 35, 49} ([[5L 11s]], or [[5L 16s]] for odd 15)

Bound-violating intervals: [[9/8]], [[10/9]] (none in [[7-odd-limit]] or if 9 is omitted)

[[#Generator tunings|Generator tunings]]: 5\26, 6\31

Mothra makes a near-just [[8/7]] equal to a third of a meantone fifth ([[~]][[3/2]]) and is most notable for being a surprisingly elegant extension of meantone to the 7-limit (though it splits the generator), as tempering out 81/80 makes S6 = [[36/35]] (the distance between [[6/5]] and [[7/6]]) and S8 = [[64/63]] (the distance between [[8/7]] and [[9/8]]) equivalent, so it seems natural to want to equate S6 = S7 = S8, where S7 = [[49/48]] (the distance between 7/6 and 8/7), so that 9/8, 8/7, 7/6, 6/5 are made equidistant. As a result, not only is 8/7 a third of 3/2, but also, because of tempering out [[1728/1715|S6/S7]], we have that 7/6 is a third of [[8/5]]. Combining it with [[#Septimal meantone]] (among other things) results in [[31edo]], where it is quite accurate, while combining it with the less accurate [[#Flattone]] results in [[26edo]], where it is quite damaged.

The only real drawback of mothra is that because it splits the meantone fifth in three, it takes 12 generators to reach prime 5.

==== [[Magic]] ====
[[#Note counts|Note count]]: 13 for {3, 5, 7, 9, 15, 25} ([[3L 10s]], [[3L 13s]], or [[3L 16s]])

[[#Generator tunings|Generator tunings]]: 6\19, 7\22, 13\41

Magic is generated by a major third of around 380 cents, five of which make a perfect twelft ([[3/1]]). This results in the magic comma, [[3125/3072]], being tempered out. Since the major third is tuned quite flat, it makes sense to equate two of them to 14/9, tempering out [[225/224]] and mapping 7/4 to +12 generators, which also tempers out [[245/243]]. While magic has slightly higher complexity and error than [[#Septimal meantone]], it doesn't temper out [[81/80]] and therefore can distinguish [[9/8]] and [[10/9]], and is in fact one of the simplest temperaments capable of mapping the [[9-odd-limit]] distinctly. The canonical extension to the 11- and 13-limits tempers out 100/99 and 105/104, but that increases complexity and lowers accuracy.

==== [[Orwell]] ====

[[#Note counts|Note count]]: 22 for {3, 5, 7, 9, 15, 21, 25, 35, 45, 75} ([[9L 13s]])

[[#Generator tunings|Generator tunings]]: 5\22, 7\31, 12\53, 19\84

Orwell is a tempermant generated by a slightly sharp subminor third representing [[7/6]], seven of which reach [[6/1]]. Its name stems from George Orwell's book 1984, as a good generator is [[84edo|19\84]]. Three subminor thirds reach [[8/5]], the 5th subharmonic. This temperament is supported by notable edos [[22edo|22]], [[31edo|31]], and [[53edo|53]], as well as the less known though still notable [[84edo]]. This temperament represents the 7-limit with good accuracy and relatively low complexity. However, the mapping for 11 is very simple, so this temperament really comes into its own in the 11-limit (see [[#Undecimal Orwell]]). Important commas tempered out by orwell include [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[65625/65536]], and [[2109375/2097152]].

=== 11-limit focus ===
=== ~17-limit focus ===

==== [[Echidna]] ====
[[#Note counts|Note counts]]:
* 24 for {3, 5, 7, 11, 15, 17, 33} ([[14L 8s]])
* 26 for adding {9} (14L 8s or 22L 14s)
* 46 for adding {13} (22L 14s)

[[#Generator tunings|Generator tunings]]: (8\58, 1\2), (11\80, 1\2)

Echidna has a generator of [[11/10]] or equivalently [[9/7]] because 11/10 * 9/7 = [[99/70]] is its period of half an octave, and can be seen as splitting the fourth of [[srutal archagall]] into three [[11/10]]'s by tempering out [[4000/3993|S10/S11]] = (12/9)/(11/10)3 = (4/3)/(11/10)3 so that [[12/11]] and [[10/9]] are made equidistant from 11/10. It can be seen as a high-accuracy version of [[hedgehog]] and as a mild detempering of [[22edo]] that achieves an accurate and distinctly consistent [[11-odd-limit]]. In fact, of the three smallest edos that are distinctly consistent in the 11-odd-limit, which are [[58edo]], [[72edo]] and [[80edo]], echidna is supported by the smallest and third-smallest (so 72edo is in a sense the odd one out, being the one that ''doesn't'' support echidna). The smallest edo consistent in the 11-odd-limit, 22edo, is in fact a trivial tuning of echidna, where the generator is conflated with 12/11 and 10/9. 58edo and 80edo are both interesting options, being the merge of echidna and a variety of other notable temperaments, so depending on preference and tuning needs, though 80edo is the more optimal tuning for it (especially in the full 17-limit).

Echidna is notable as achieving no-13's [[17-limit]] harmony with accuracy in a surprisingly small number of notes. [[13/8]] can be found too but is the most complex, being found at (11/10)16 plus a half-octave period, octave-reduced. However, as primes 5 and 11 are also found in the same direction, intervals of 13 are common even in the 22-note MOS, so the 36-note MOS is more useful than might be suspected, despite not finding every odd from the same position.

==== [[Catakleismic]] ====
[[#Note counts|Note counts]]:
* 23 for {3, 5, 7, 9, 13, 15, 25, 27, 39, 45, 65, 75(, 125)} ([[15L 4s]] or 19L 15s)
* 29 for adding {21, 35, 63} (19L 15s)

[[#Generator tunings|Generator tunings]]: 14\53, 19\72

As catakleismic is largely just a certain extension of cata to prime 7, see [[#Cata]]. It admits a number of possible extensions to prime 11 depending on user preference and the tuning used, hence its listing here as a ~13-limit temperament.

==== [[Diaschismic]] ====
[[#Note counts|Note count]]: 36 for {3, 5, 7, 9, 11, 13, 15, 17, 21, 25, 33, 35, 39, 51} (12L 22s)

[[#Generator tunings|Generator tunings]]: (27\46, 1\2), (34\58, 1\2), (61\104, 1\2)

Diaschismic is an extension of [[#Srutal archagall]] to the full [[17-limit]] of similar complexity to [[#Srutal]] (with which it merges in [[46edo]] so that they're complimentary) but which damages the 2.3.5.17 subgroup slightly more. Familiarizing oneself with the structure of srutal archagall is recommendable, even if the ideal tunings differ slightly, as navigation will be similar.

=== Higher-limit focus ===
==== [[Sensible]] ====
{{ See also | Sensipent#Sensible interval table }}

[[#Note counts|Note counts]]:
* 26 for {3, 5, 11, 17, 23, 31, 33, 51, 55, 69, 85, 99(, 115)} ([[19L 8s]])
* 35 for adding {9, 15, 25, 93(, 155)} ([[19L 8s]] or 19L 27s)

[[#Generator tunings|Generator tunings]]: 24\65, 41\111

Sensible is a fairly accurate (though quite complex) subgroup interpretation of sensi (or extension of [[#Sensipent]]) which is especially accurate in the [[CEE tuning]] (443.115{{cent}}) if avoiding composite odds and intervals of 23 so that under such restrictions it classifies as [[#High accuracy]], meaning it's one of the most accurate temperaments in this category, especially if you use error-cancellations of sharp harmonics to your advantage to construct complex harmonic series chords in order to justify the generally-higher errors of composite harmonics. [[46edo]] is a somewhat reasonable but trivial tuning of it, [[65edo]] is somewhat better, and [[111edo]] is even better, but it works especially well in a more optimized tuning, hence its significance as a rank 2 temperament. The 27-note MOS, though not achieving all of its odd harmonics from a single note, is sufficient, because the missing odds {9, 15, 25, 93(, 155)} also do reasonably occur within the span of the 27-note MOS. Therefore, sensible can be seen as a detempering of [[27edo]] to a more accurate rank 2 temperament on a mostly-unrelated subgroup.

==== [[Srutal]] ====
[[#Note counts|Note count]]: 36 for {3, 5, 7, 9, 11, 13, 15, 17, 23, 33, 35, 51} (12L 22s)

Bound-violating intervals: [[13/9]]

[[#Generator tunings|Generator tunings]]: (4\46, 1\2), (7\80, 1\2)

Srutal is an at least no-19's [[23-limit]] temperament, being an extension of [[#Srutal archagall]] to the full [[17-limit]] and finding [[23/16]] as an augmented fourth, that is, as a tritone of ([[~]][[9/8]])3. Therefore, familiarizing oneself with srutal archagall is recommendable as the structures and tunings are nearly identical (and their tunings merge meaningfully in [[80edo]]), with the main difference being number of notes and breadth of harmonies targetted. Srutal can find more primes than just those in the no-19's 23-limit but they are more complex so more likely to be used opportunistically in a 34-note MOS, so that this temperament can be seen as a detemperament of [[34edo]]. [[80edo]] is a good tuning for it, though [[46edo]] deals well enough with the no-19's 23-limit part (potentially add-31) at the cost of a variety of distinctions. An alternative extension of srutal archagall to just the full 17-limit which is damages the 2.3.5.17 subgroup slightly more is [[#Diaschismic]].

=== No-2's focus ===

==== [[BPS]] ====
[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]:
* 7 for {5, 7, 25, 35, 49} ([[4L 5s (3/1-equivalent)|4L 5s<3/1>]])
Bound-violating intervals: [[35/27]], [[49/27]] and/or [[25/9]] (none if those higher harmonics are omitted)

[[#Generator tunings|Generator tuning]]: 10\13edt

BPS is arguably the most important temperament of the nonoctave [[3.5.7 subgroup]]. This temperament has a tritave period, and a generator of [[~]][[9/7]]. The tritave-reduced 7th harmonic, [[7/3]], is found at -1 generators, and the tritave reduced 5th harmonic, [[5/3]], is found at +2 generators, tempering out [[245/243]]. It is as simple as a good temperament in its subgroup can be, covering the entire no-evens 7-throdd-limit tonality diamond in 7 notes, with no redundant or missing notes, and any simpler temperament would have to equate simple consonances and have very low accuracy. Its accuracy is quite good, with a no-evens 7-throdd-limit minimax error of 4.73 cents. An excellent scale to explore this temperament is the 9-note mos, or lambda scale, which can be considered the 3.5.7 analog of the diatonic scale.

=== No-3's focus ===
==== [[Didacus]] ====
[[#Note counts|Note counts]]:
* 10 for {5, 7, 11, 25, 35, 55} ([[6L 7s]])
* 15 for {5, 7, 11, 25, 35, 49, 55, 77} ([[6L 13s]])

Bound-violating intervals: [[35/22]], [[49/44]]

[[#Generator tunings|Generator tunings]]: 5\31, 6\37, 11\68

Didacus is a uniquely efficient no-3's focus temperament, mapping the no-3's 11-limit with ease and at the cost of accuracy admitting an extension to the no-3's 19-limit called [[#Mediantone]].

Its generator is [[~]][[28/25]] slightly flattened; two make [[~]][[5/4]], three make [[~]][[7/5]], four make a slightly flattened [[~]][[11/7]] (which is thus a sharp [[~]][[25/16]]), thus five make [[~]][[7/4]] and nine make [[~]][[11/4]] so that seven make [[~]][[11/5]].

[[37edo]] is a good tuning for its size and is practically equivalent to the pure-11's tuning, though for less damage on 5 and 7, [[68edo]] is a good tuning that much better targets composite harmonies, so might be advised as the first edo tuning to try.

==== [[Mediantone]] ====
[[#Note counts|Note count]]: 29 for {5, 7, 11, 13, 17, 19, 25, 55, 65, 91} ([[6L 25s]])

Bound-violating intervals: [[25/17]], [[28/17]], [[55/34]] (all intervals of 17; note odd 35 can be added but is high error)

[[#Generator tunings|Generator tunings]]: 13\80

Though not quite as accurate as [[#Didacus]], mediantone is notable for extending didacus to the full no-3's 19-limit. It does well in an optimized tuning, though if one wants to use an edo tuning, the main tuning of interest is [[80edo]], which finds primes 3 and 23 as well.

=== No-5's focus ===
==== 2.3.7 [[Buzzard]] ====
[[#Note counts|Note counts]]:
* 8 for {3, 7, 21} ([[5L 3s]])
* 12 for {3, 7, 9, 21} ([[5L 8s]])

[[#Generator tunings|Generator tunings]]: 19\48, 21\53, 23\58, 44\111

The 2.3.7 part of [[#Buzzard]] is not as accurate as everything else in the 13-limit; specifically, its interval of 7 barely violates the 4{{cent}} bound, however it makes up for it by being much simpler (mapping-wise) so that it is interesting as a 2.3.7-subgroup temperament that splits [[3/1]] into four equal parts, each representing a sharp [[~]][[21/16]], which defines it in the 2.3.7 subgroup. [[53edo]], [[58edo]] and [[111edo]] are good tunings.

== '''Low accuracy (7~12c)''' ==
Low accuracy temperaments in small prime limits are commonly considered due to their simplicity. As a result, "higher-limit focus" tends to not be focused on at this accuracy, as the error involved on intervals beyond the [[17-limit]] is potentially too much depending on the context and who you ask, though again such temperaments are commonly relevant as targets for detempering.
=== 5-limit focus ===
=== 7-limit focus ===
==== [[Superpyth]] ====
[[#Note counts|Note counts]]:
* 4 for {3, 7} ([[2L 3s]])
* 12 for {3, 5, 7, 9} ([[5L 7s]] and [[5L 12s]])
Bound-violating intervals: [[9/8]] (and [[8/7]] in flatter tunings like 22edo)

[[#Generator tunings|Generator tunings]]: 13\22, 16\27, 29\49

Superpyth is the natural "opposite" of [[#Septimal meantone]] in a surprisingly large number of surprisingly exact senses; the main ones of note are that the [[3/2|fifth]] is mistuned in opposite directions (sharp in superpyth), and that superpyth makes prime 7 most immediately accessible on the chain of fifths with prime 5 requiring more complex movements (when measured in number of fifths), while septimal meantone does the opposite. Superpyth makes the major third [[~]][[9/7]], the minor third [[~]][[7/6]], the major second a blend between a sharp [[~]][[9/8]] and a flat [[~]][[8/7]] and the minor second [[~]][[28/27]], which is tuned very flat so that it becomes a quarter-tone in any good tuning of superpyth. Superpyth finds [[~]][[5/4]] as the augmented second and [[~]][[6/5]] correspondingly as the diminished fourth.

Superpyth is a common choice for a beginner, with [[22edo]] and [[27edo]] having different advantages and 22edo the most explored by far, though the number of unique advantages and opportunities in 27edo make it formidable as a competitor. [[22edo]] is approximately the pure-[[9/7]]'s tuning while [[27edo]] is approximately the pure-[[7/6]]'s tuning. For 5 more notes, 27edo has the advantage of not equating [[7/5]] and [[10/7]] and having a more accurate [[8/7]] and [[7/4]]. A more optimized edo tuning is [[49edo]] but that comes at the cost of a lot of notes if you aren't merely looking to take a [[MOS]] scale subset of it.

=== 11-limit focus ===
==== [[Mohaha]] ====
[[#Note counts|Note counts]]:
* 9 for {3, 5, 9, 11(, 33, 35)} ([[7L 3s]], note odd 35 comes from the [[mohajira]] mapping of 7 specifically)
* 20 or 23 for adding {7, 15, 21(, 35)} ([[7L 10s]] and [[7L 17s]])

[[#Generator tunings|Generator tunings]]: 9\31, 16\55

Mohaha is a 2.3.5.11 (no-7's [[11-limit]]) "hemi-meantone" temperament that splits [[#Meantone]]'s fifth into two [[~]][[11/9]]'s by tempering out [[243/242|S9/S11 = (12/8)/(11/9)2 = (3/2)/(11/9)2]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. It has two main extensions to the full 11-limit; if you accept the [[#Septimal meantone]] mapping of 7 you get [[migration]], but maybe more natural is if you instead equate the flat [[~]][[33/32]] interval with S6 = [[36/35]] = ([[6/5]])/([[7/6]]), which results in [[mohajira]], which finds 7 at a negative number of gens so that composite harmonics of 7 are simpler to find (as primes 3, 5 and 11 are all found at a positive number of gens). Because of this, mohajira is usually the preferred extension as it is more note-efficient, but both extensions merge in [[31edo]], which is a good tuning for both.

==== [[Orwell|Undecimal Orwell]] ====
[[#Note counts|Note count]]: 22 for {3, 5, 7, 9, 11, 15, 21, 25, 33, 35, 45, 55, 75, 77} ([[9L 13s]])

[[#Generator tunings|Generator tunings]]: 5\22, 7\31, 12\53, 19\84

Orwell is a tempermant generated by a slightly sharp subminor third representing [[7/6]], seven of which reach [[6/1]]. Three subminor thirds reach [[8/5]], and two reach [[11/8]]. This temperament is arguably one of the simplest 11-limit temperaments with decent accuracy, and is supported by the highly notable edos {{edos|22, 31, and 53.}} This temperament is more accurate in lower limits, very nearly being a microtemperament in the 5-limit (5-odd-limit minimax error 1.006{{c}}!), and quite good accuracy in the 7-limit, with the least accurate prime being 11, so this temperament can only barely be considered "low accuracy". However, the mapping for 11 is very simple, so this temperament really comes into its own in the 11-limit. Important commas tempered out by orwell include [[99/98]], [[121/120]], [[176/175]], [[225/224]], [[385/384]], [[540/539]], and [[1728/1715]].

=== ~17-limit focus ===
=== Higher-limit focus ===
Temperaments in the higher-limit focus category imparting more than 7 cents of damage tend not to be considered, but are most common as implicitly being the targets of detempering of various JI scales.
=== No-2's focus ===
=== No-3's focus ===
=== No-5's focus ===

== '''Very low accuracy (12~18c)''' ==
Very low accuracy temperaments are of interest to people wanting simple scales and who are fine with high damage. As a result, they tend not to have "higher-limit focus", as the error involved on intervals beyond the [[17-limit]] is too much. A variety of people consider this category to largely or even entirely be composed of exotemperaments, while others argue for various entries in this category being reasonable to consider harmonically based on the temperability of the simplest [[LCJI]] intervals.
=== 5-limit focus ===
=== 7-limit focus ===
==== [[Godzilla]] ====
[[#Note counts|Note count]]: 9 for {3, 5, 7, 9(, 21)} ([[5L 4s]])

[[#Generator tunings|Generator tuning]]: (4\30<3>, 19\30<3>)

Godzilla is a very coarse temperament, where we use context to suggest ~4:5:6:7(:8). It admits a natural extension to prime 13 based on interpreting its semifourth of [[~]][[8/7]][[~]][[7/6]] much more accurately as [[15/13]], though if you specifically want 15/13 as the semifourth, there is much more accurate temperaments available that don't require interpreting it inaccurately as ~8/7~7/6, such as [[immunity]], or if you don't need a semifourth as the generator, [[#Cata]]. Nonetheless, insofar as it makes sense, it's notable for providing a usefully-small 9-note scale for the entire [[9-odd-limit]] (insofar as it is capable of approximating its sound with context). Due to its inaccuracy, it is recommended to use a sharp octave-tempering for this temperament, such as [[30edt]] instead of [[19edo]], in which case you also improve various intervals of 13 as well. Doing this means that you can use voicing across octaves to improve the accuracy and hence psychoacoustic convincingness of godzilla.

==== [[Negri]] ====
[[#Note counts|Note count]]: 9 for {3, 5, 7(, 13, 15)} ([[9L 1s]])

[[#Generator tunings|Generator tuning]]: (2\30<3>, 19\30<3>)

Like [[#Godzilla]], negri is a temperament that trades accuracy for simplicity. It divides the 4/3 into four equal semitone-like steps that resemble steps of 10edo, so that it can be seen as halving the generator of godzilla, which it uses to find prime 5 more quickly. As a result, odd 9 is found at 8 generators, so you can't find odd 9 from the same position as odd 5 unless you use at least 12 notes, which corresponds to the 19-note MOS scale of [[10L 9s]], at which point (like godzilla) you may as well use [[19edo]] with octave-tempering (e.g. [[30edt]]), though [[29edo]] also supports a stranger tuning of it if you want to try a subset of that with sharp-octave-tempering. Negri can also be interpreted as a temperament of the 2.3.5.13 subgroup, for which an interesting tuning is [[48edo]] by using the 48f val also used by its strange rendition of [[#Buzzard]], and as a 5-limit temperament, it is supported by yet more tunings (generally: ones using a flat mapping for 5/4 and 3/2 which split 4/3 into four of the implied ~16/15).

==== [[Augene]] ====
[[#Note counts|Note counts]]:
* 6 for {3, 5(, 15)} ([[3L 3s]])
* 12 for adding {7} ([[3L 9s]])
* 15 for adding {9} {[[12L 3s]])

[[#Generator tunings|Generator tunings]]: (9\15, 1\3), (16\27, 1\3)

Augene is the result of using 1\3 as an approximation of ~5/4 so that the period is a third of 2/1, and then using a sharp 3/2 or equivalently a flat 6/5 as the generator. Because the approximation of 5/4 is so sharp, any good tuning of augene will have a noticeably sharp ~3/2 so that ~6/5 becomes more in-tune (which is also important as it's ''also'' the generator, being equal to ~3/2 minus a period). Because the amount required to get ~6/5 reasonably in-tune is quite significant, it makes sense to lean into this and take advantage of it by tempering out the difference between the very sharp ~9/8 and a flat ~8/7, so that the minor third becomes ~7/6 as in [[#Superpyth]]. Augene merges with superpyth in [[27edo]], which is a recommendable tuning for also being the smallest edo to have all intervals of the [[7-odd-limit]] tuned distinctly (not equated), in which case you get harmonies of the no-11's [[13-odd-limit]] too. It's unclear whether a psychoacoustically optimal tuning of augene would have the fifth sharper or flatter than the 27edo tuning; if you think having the fifth more in-tune is preferable, you could try the [[39edo]] tuning {{nowrap| (23\39, 1\3) }}, which uses a very sharp mapping for ~7/4 so that 7 is barely sharper than 5; by some metrics this tuning is more optimal than 27edo. If you want 6/5 more in-tune, you could try the [[42edo]] tuning {{nowrap| (25\42, 1\3) }}, which may be preferred for having a potentially more convincing approximation of ~4:5:6 than all the other options discussed, as well as for being distinctly consistent in the 7-odd-limit like 27edo. Some even prefer the [[15edo]] tuning, as though it damages the 7-limit even more than augene does so that the 7-odd-limit is no longer tuned distinctly (because of ~8/7 = 1\5 = ~7/6), it includes an approximation of ~11/8, and can be thought of as xenmelodically/structurally interesting for a version of augene where the 5edo fifth is the generator and the 3edo major third is the period.

==== [[Pajara]] ====
[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]: 10 for {3, 5, 7, 9} ([[2L 8s]])

[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: 13\22, 20\34, 33\56

This temperament was discovered by [[Paul Erlich]], and its scales are known for having properties similar to diatonic in the 5-limit. This temperament tempers out [[50/49]], setting [[7/5]] and [[10/7]] to the half-octave. It is generated by a fifth, or alternatively a semitone that is the fifth minus the half-octave. It tempers out [[64/63]] and [[2048/2025]], meaning harmonics 5 is mapped to a half-octave minus two semitones, and harmonic 7 is an octave minus two semitones. This temperament is best used with a decatonic interval classification, so 3/2 is a Perfect 7th10, 5/4 is a Major 4th10, and 7/4 is a Major 9th10. The [[4:5:6:7]] otonal tetrad is written P1-M4-P7-M9. Changing both major intervals to minor ones gives us the [[70:84:105:120|1/(7:8:10:12)]] utonal tetrad. Since 50/49 is tempered out, [[25/24]] and [[49/48]] are equated, both to the augmented unison of the decatonic scale. It works just like in diatonic, where changing the major third of the [[4:5:6]] triad to a minor one gives the [[10:12:15|1/(4:5:6)]] triad. While this temperament has poor accuracy overall, since 12edo's accuracy is accepted by many, this temperament's accuracy can still be considered reasonable.

=== 11-limit focus ===

==== [[Porcupine]] ====
[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]: 15 for {3, 5, 7, 9, 11, 15} ([[7L 8s]])

[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: 3\22, 5\37, 8\59

This temperament is generated by a submajor second of around 163 cents representing all of [[10/9]], [[11/10]], and [[12/11]]. It maps prime 3 to -3 generators, and prime 5 to -5 generators, tempering out [[250/243]]. This means two 10/9s are equated to 6/5, and three 10/9s are equated to 4/3. The 10/9 generator is then naturally equated to 11/10 and 12/11, mapping prime 11 to -5 generators. An important high-damage equivalence of this mapping is 11/9~6/5 via tempering of [[55/54]], a comma of over 31 cents. The 2.3.5.11 version of this temperament is arguably one of the simplest temperaments in this subgroup with acceptable accuracy, even if barely so. Even if one doesn't accept its accuracy, it is still useful as a mapping to organise intervals. The canonical mapping for prime 7 finds it at +6 generators, tempering out [[64/63]], which makes sense as the fifth is already very sharp. In that regard, it is notable in the full 11-limit as well, and is one of the best ways to analyse the 11-limit of [[22edo]].
=== ~17-limit focus ===
==== [[Flattone]] ====
[[#Note counts|Note count]]: 14 for {3, 5, 7, 9, 11, 13}

[[#Generator tunings|Generator tunings]]: (11\30<3>, 19\30<3>), (15\93<12>, 26\93<12>)

Flattone is a low-accuracy [[11-limit|11-]] or [[13-limit]] temperament. It is an alternative extension of [[#Meantone]] of interest because it maps [[7/4]] to the arguably more intuitive diminished seventh and [[11/8]] to the similarly simple augmented fourth (aka tritone). If one maps the 13-limit, the best way is by equating a sharpened [[~]][[16/13]] with the already-very-flat [[~]][[5/4]], continuing the strong flat tendency. It tunes meantone much flatter than usual so that the whole tone is much closer to [[10/9]] than it is to [[9/8]], and is maybe most notable as being supported by [[26edo]], the smallest edo consistent in the [[13-odd-limit]]. Maybe surprisingly, it is one of the most accurate temperaments in this accuracy category; its most off primes are 5 and 13, which are the only ones to meaningfully transgress the 12{{cent}} bound, along with odd 9 being tuned very flat which has the benefit of causing the tuning of [[6/5]] to be relatively accurate.

=== Higher-limit focus ===
Temperaments in the higher-limit focus category imparting more than 12 cents of damage are rare, but are most common as implicitly being the targets of detempering of various JI scales.
=== No-2's focus ===
=== No-3's focus ===
=== No-5's focus ===

Xenharmonic Wiki:I'm a new editor, what is everything I can do to help?

2026-06-28T02:03:03Z

Overthink: /* Transfer a page from riters.com */ site no longer availabe

There are many possible things you can do. This is a sampling of some of them. Pick and choose whichever sound the most interesting to you.

== Small tasks ==
=== Add examples ===
Add examples to one of the pages in [[:Category:Todo:add examples]].

=== Write a missing introduction ===
Write an introduction for one of the pages in [[:Category:Todo:intro]].

=== Finish a list or table ===
Complete one of the incomplete lists or tables from [[:Category:Todo:complete list]] or [[:Category:Todo:complete table]] respectively.

=== Resurrect a dead link ===
Go to one of the pages in [[:Category:Pages containing dead links]].

Try to find a live version of the link's target online. Perhaps [https://web.archive.org/ Wayback Machine] might have one.

If you can find a live version, then update the link to point to that.

If you can't find one, '''do not delete''' the link. Instead, post on the talk page and list every place you've looked so far, so that future searchers don't have to tread the same ground.

== Intermediate tasks ==
=== Rate a piece of music ===
Add your ratings of xenharmonic pieces of music to [[List of xenharmonic music by community ratings]]. Carefully follow the instructions on that page to add them.

=== Answer a question ===
If you know the answers to any questions on [[FAQ]] or any other page in [[:Category:Todo:answer questions]], then edit the page to add your answers.

=== Add an audio example ===
Almost every page on the wiki is in need of audio examples, so that readers can hear what the concepts actually sound like.

If you can add such audio examples, please do so.

=== Add an illustration ===
Add an illustration to one of the pages in [[:Category:Todo:add illustration]].

=== Create a choose-an-EDO flow chart ===
Create a flow chart to help readers choose an [[EDO]] to make music with. Link your flow chart in [[Edo recommendation hub page]].

=== Add a rank 2 temperaments table ===
Add a rank 2 temperaments table to one of the pages in [[:Category:Todo:add rank 2 temperaments table]].

Check out some edo pages (e.g. [[26edo]], [[27edo]], [[28edo]]) to get an idea of what rank 2 temperaments tables look like.

=== Finish a section ===
Complete one of the incomplete page subsections from [[:Category:Todo:complete section]].

== Large tasks ==
=== Recommend the EDOs you like ===
Create a page where you recommend a list of [[edo]]s to beginners.

Follow the standardised structure shown on the page [[Edo recommendation hub page]] to make it easy for readers to compare with the other lists.

When your page is done, link it on [[Edo recommendation hub page]].

=== Become the resident expert on x ===
Choose an article about a tuning or scale from [[:Category:Stubs]] or [[:Category:Todo:expand]].

Make some music with that tuning, and document your process every step along your way:
* What subset scales you tried that didn't work.
** What ones did.
* What instruments or synth settings sounded bad.
** What ones sounded good.
* Literally every little detail, document it all.

By the time you're done, you will probably know more about that tuning than anybody else in the world ever has. You will now be more qualified to write its page than anyone else.

Go ahead and expand the stub or todo:expand page with those notes you gathered as a guide, and add a link to your musical example(s) too.

Now you have taken the page from almost empty, to one of the best written, most complete on the wiki!

== For special skillsets ==

=== Mathematics ===
==== Correct the mathematics on a page ====
If you have some experience with mathematics, go to one of the pages in [[:Category:Todo:correct maths]] and make sure all the mathematics written on the page is correct. If it's not, correct it.

==== Make a page more accessible ====
Write a simplified version of one of the pages listed in either [[:Category:Inaccessible pages]] or [[:Category:Todo:reduce mathslang]].

Make it fully understandable to non-mathematicians.

Post it as a new, separate page on the wiki.

=== Journalism, modern history or social science ===
==== Fact check ====
Do any of the following, whichever ones interest you:
* Check facts for pages in [[:Category:Todo:confirm]] and [[:Category:Todo:research]]
* Update time-sensitive facts (e.g. whether someone is or was active, whethee they are studying or did study at x, etc.) for pages in [[:Category:Todo:update]]

==== Find sources ====
Do any of the following, whichever ones interest you:
* Find and add etymology for pages in [[:Category:Todo:add etymology]]
* Find and add sources for pages in [[:Category:Todo:add source]] and [[:Category:Pages with unsourced statements]]
* Find how a page in [[:Category:Todo:explain its xenharmonic value]] relates to microtonality or musical tuning, and edit the page to explain how

=== Lua ===
==== Write documentation for a template or module ====
If you have some experience with Lua, we need your help to write documentation for the templates or modules in [[:Category:Todo:add documentation]]

=== Ethnomusicology ===
==== Improve a page ====
The following pages are in need of drastic improvement by experts in the field, please improve them:
* [[African music]]
* [[Arabic, Turkish, Persian music]]
* [[Georgian]] music
* [[Indian music]]
* [[Indonesian]] music
** [[Gamelan]]
** [[Pelog]]
** [[Slendro]]
* [[Pre-Columbian South American music]]

==== Create a page ====
The following pages are needed, but have not yet been created. If you are an expert in the field, please create them.

Note that given the nature of this wiki, the pages should focus on musical ''tuning'' first and foremost:
* Separate pages for some different musical traditions within Africa
* Separate pages for Arabic music, Turkish music and Iranian music
* A page for Thai music
* A page for Chinese music (ancient & modern)
* Pages for non-Western composers and musicians who use microtuning

=== Music history ===
==== Improve a page ====
The following articles are in need of drastic improvement by experts in the field, please improve them:
* [[Historical temperaments]]
** And all the examples listed therein
* [[Ancient Greek music]]
** [[Teleic scales]]
** [[Tetrachord]]

==== Create a page ====
The following pages are needed, but have not yet been created. If you are an expert in the field, please create them.

Note that given the nature of this wiki, the pages should focus on musical ''tuning'' first and foremost:
* A page for Byzantine music
* A page for Medieval European music
* All the red links on the page [[Historical temperaments]]
* Pages for historical composers who used microtuning

=== Any non-English language ===
==== Contribute ====
If the language you know already has a Xen Wiki interwiki - contribute to it! Write pages for it.

==== Initiate ====
If it does not yet have an interwiki, start writing pages in your language in the main English Xen Wiki and eventually they will be moved into their own site.

=== Large EDOs ===
==== Curate a table ====
Go to one of the pages in [[:Category:Todo:Replace auto-generated table of intervals with manually curated table]].

Create a manual, annotated version of the table using a wikitable.

Google how to do make wikitables if you're not sure how, or use the visual editor. [https://excel2wiki.toolforge.org/index.php Excel2Wiki] may also prove helpful.

=== MOS scales ===
==== Write an introductory guide ====
Write a thorough introduction to MOS scales from the point of view of a musician who knows nothing about them and just wants to know how to make music with them and how different ones sound and feel.

=== Comma pumps ===
==== Add a comma pump to a comma page ====
Go to any page in [[:Category:Commas]] and its subcategories. Edit the page and describe a comma pump that uses the comma. Follow the style of the [[Frameshift comma]] page.

== Want more to do? ==
Check out [[Xenharmonic Wiki:Things to do]] and [[Wikifuture]].

[[Category:Xenharmonic Wiki]]

Cohemiripple

2026-06-28T01:55:42Z

Overthink: nominate for deletion

{{Delete|temperament deleted}}
#REDIRECT [[Ripple family#Cohemiripple]]

[[Category:Cohemiripple]]
[[Category:Ripple family]]
[[Category:Sensamagic clan]]
[[Category:Temperaments]]

Ripple family

2026-06-28T01:54:19Z

Overthink: impractically complex given low accuracy

{{Technical data page}}
The '''ripple family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[ripple comma]] ([[ratio]]: 6561/6250, {{monzo|legend=1| -1 8 -5 }}), which equates a stack of five [[27/25]]'s with [[4/3]].

== Ripple ==
The generator of ripple is a semitone representing 27/25, five of which give 4/3, and eight of which give 8/5. The [[ploidacot]] of ripple is omega-pentacot. This means that 27/25 is severely flattened, so that the characteristic damage is a strongly flat-tempered fourth reached at 5 semitones. Interestingly, in optimal tunings, the major third of ~5/4 does not tend to be damaged much sharpwards as one might expect from the equivalence, and is in practice sometimes even flat, so that prime 3 takes on practically the whole damage of the 5-limit equivalence, for which it has the advantage of being the simplest so still having a good chance at psychoacoustic viability. As a result though, the mapping of ~9/8 is often very flat, so that ripple can in practice be thought of as a [[dual-fifth temperament]] unless you use tunings close to [[12edo]].

Reasonable [[patent val]] tunings not appearing in the optimal ET sequence are [[35edo]] and [[47edo]].

[[Subgroup]]: 2.3.5

[[Comma list]]: 6561/6250

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

: mapping generators: ~2, ~27/25

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.2636{{c}}, ~27/25 = 100.8602{{c}}
: [[error map]]: {{val| +0.264 -5.729 +7.596 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 100.7982{{c}}
: [[error map]]: {{val| 0.000 -5.946 +7.300 }}

[[Tuning ranges]]:
* [[5-odd-limit]] [[diamond monotone]]: [92.308, 109.091] (1\13 to 1\11)
* 5-odd-limit [[diamond tradeoff]]: [99.609, 105.214]

{{Optimal ET sequence|legend=1| 11c, 12, 71b, 83b }}

[[Badness]] (Sintel): 3.26

=== Overview to extensions ===
The second comma of the comma list defines which 7-limit family member we are looking at:
* Septimal ripple adds [[126/125]];
* Rip adds [[36/35]];

Both use the same nominal generator as ripple.

For weak extensions, we have hemiripple and cohemiripple. Hemiripple adds [[49/48]], spliting the semitone generator in two. Cohemiripple adds [[245/243]], spliting the [[octave complement]] of the semitone generator in two.

== Septimal ripple ==
{{See also| Dual-fifth temperaments }}

Septimal ripple interprets the generator as a very flat ~15/14, so that 3 and 5 are flat and 7 is sharp; of these, 3 is the most damaged, but is also the simplest, so is still viable as an approximation. Due to the sharp 7 and flatter 3, ~21/16 can be fairly in-tune, acting as the alternate fourth in a dual-fourth interpretation, so that the inconsistent but more accurate ~16/9 is reached as ~(21/16)⋅(4/3) = ~7/4, though this assumes you are putting the most damage on 3 as to get larger primes more in tune. This has another advantage, specific to the 11-limit: this accurate but inconsistent ~9/8 (which is usually just to slightly sharp) can find the neutral third ~11/9 with reasonable accuracy.

''If you are looking for the former canonical extension, see [[#Rip]].''

[[Subgroup]]: 2.3.5.7

[[Comma list]]: [[126/125]], [[405/392]]

{{Mapping|legend=1| 1 2 3 4 | 0 -5 -8 -14 }}

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1201.7546{{c}}, ~15/14 = 102.1309{{c}}
: error map: {{val| +1.755 -9.100 +1.903 +8.360 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 101.7772{{c}}
: error map: {{val| 0.000 -10.841 -0.531 +6.294 }}

{{Optimal ET sequence|legend=1| 11cd, 12, 35, 47 }}

[[Badness]] (Sintel): 1.52

=== 11-limit ===
A notable [[patent val]] tuning of 11-limit ripple not appearing in the optimal ET sequence is [[47edo]].

Subgroup: 2.3.5.7.11

Comma list: [[45/44]], [[99/98]], [[126/125]]

Mapping: {{mapping| 1 2 3 4 5 | 0 -5 -8 -14 -18 }}

Optimal tunings:
* WE: ~2 = 1202.5973{{c}}, ~15/14 = 102.7900{{c}}
: error map: {{val| +2.597 -10.710 -0.842 +2.504 +11.449 }}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 102.2972{{c}}
: error map: {{val| 0.000 -13.441 -4.691 -0.986 +7.333 }}

{{Optimal ET sequence|legend=0| 11cdee, 12, 23de, 35 }}

Badness (Sintel): 1.33

== Rip ==
Formerly known as ''septimal ripple'', but de-canonized in favour of canonizing a significantly more accurate extension of similar efficiency so that [[#Ripple]] admits nontrivial edo tunings of interest. The reason for de-canonization is not coming close to preserving the damage level of 5-limit ripple to the 7-limit or even of this 7-limit damage level to the 11-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 36/35, 2560/2401

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

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1195.0347{{c}}, ~21/20 = 99.0710{{c}}
: error map: {{val| -4.965 -7.240 +6.223 +18.136 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 100.1093{{c}}
: error map: {{val| 0.000 -2.501 +12.812 +30.956 }}

{{Optimal ET sequence|legend=1| 11c, 12 }}

[[Badness]] (Sintel): 1.51

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

Comma list: 36/35, 80/77, 126/121

Mapping: {{mapping| 1 2 3 3 4 | 0 -5 -8 -2 -6 }}

Optimal tunings:
* WE: ~2 = 1192.7877{{c}}, ~21/20 = 98.7876{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 100.3202{{c}}

{{Optimal ET sequence|legend=0| 11c, 12 }}

Badness (Sintel): 1.28

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

Comma list: 36/35, 40/39, 66/65, 147/143

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -5 -8 -2 -6 -3 }}

Optimal tunings:
* WE: ~2 = 1189.8521{{c}}, ~21/20 = 97.7384{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 99.7618{{c}}

{{Optimal ET sequence|legend=0| 11c, 12f, 37ccddeeeeffff }}

Badness (Sintel): 1.31

== Hemiripple ==
Hemiripple tempers out 49/48 and splits the semitone generator in two for ~36/35. Its ploidacot is omega-decacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 49/48, 6561/6250

{{Mapping|legend=1| 1 2 3 3 | 0 -10 -16 -5 }}

: mapping generators: ~2, ~36/35

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.5561{{c}}, ~36/35 = 50.9765{{c}}
: error map: {{val| +3.556 -4.608 +8.730 -13.040 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 50.5928{{c}}
: error map: {{val| 0.000 -7.883 +4.201 -21.790 }}

{{Optimal ET sequence|legend=1| 23d, 24, 47d }}

[[Badness]] (Sintel): 4.43

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

Comma list: 49/48, 121/120, 567/550

Mapping: {{mapping| 1 2 3 3 4 | 0 -10 -16 -5 -13 }}

Optimal tunings:
* WE: ~2 = 1203.5344{{c}}, ~36/35 = 50.9757{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5870{{c}}

{{Optimal ET sequence|legend=0| 23de, 24, 47de }}

Badness (Sintel): 2.21

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

Comma list: 49/48, 66/65, 121/120, 351/350

Mapping: {{mapping| 1 2 3 3 4 4 | 0 -10 -16 -5 -13 -7 }}

Optimal tunings:
* WE: ~2 = 1202.0936{{c}}, ~36/35 = 50.7232{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 50.5048{{c}}

{{Optimal ET sequence|legend=0| 23de, 24 }}

Badness (Sintel): 1.93

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

Xenharmonic Wiki:Deleted temperament entries

2026-06-28T01:54:14Z

Overthink: impractically complex given low accuracy

This page collects deleted entries of [[regular temperament]]s from the Xenharmonic Wiki due to not meeting our [[Xenharmonic Wiki: Notability guidelines|notability guidelines]], although other materials may still reference them. We recommend to refer to these temperaments with an ET [[join]] (e. g. "11-limit 12 & 85") or as a [[restriction]] of a different temperament (e. g. "no-7's cassandra").

{| class="wikitable" style="margin: auto;"
|+
! Name
! Subgroup
! Mapping
! Reason for deletion
|-
| Meanplop
| 13-limit
| {{Mapping| 1 0 -4 -13 24 10 | 0 1 4 10 -13 -4 }}
| Low accuracy is not worth medium complexity; poor-tasted name
|-
| Oviminor
| 5- to 7-limit
| {{Mapping| 1 50 51 147 | 0 184 185 548 }}
| Pointless microtemperament: more complex than egads (which shares the same generator) at lower accuracy
|-
| Maqamschismic
| 2.3.5.11(.13)
| {{Mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }}
| Unenlightening restriction of cassandra to remove a lower-complexity, accurate prime
|-
| Quintapole
| 7- to 11-limit
| {{Mapping| 1 2 1 1 0 | 0 5 -16 -22 -42 }}
| Medium accuracy is not worth the high complexity; confusing name (could lead to confusion with similarly named "quintupole")
|-
| Quintapole > galeleic
| 13- to 19-limit
| {{Mapping| 1 2 1 1 0 -1 5 4 | 0 5 -16 -22 -42 -57 11 -3 }}
| Medium accuracy is not worth the high complexity; not even supported by any patent vals
|-
| Quintapole > catagali
| 13- to 19-limit
| {{Mapping| 1 2 1 1 0 0 5 4 | 0 5 -16 -22 -42 -45 11 -3 }}
| Medium accuracy is not worth the high complexity; not even supported by any patent vals
|-
| Quintapole > quintain
| 11- to 13-limit
| {{Mapping| 1 2 1 1 1 0 | 0 5 -16 -22 -30 -45 }}
| Medium accuracy is not worth the high complexity
|-
| Bixby
| 5-limit
| {{Mapping| 1 2 0 | 0 0 1 }}
| Extreme exotemperament: too coarse even for detempering
|-
| Archon
| 5-limit
| {{Mapping| 1 0 2 | 0 1 0 }}
| Extreme exotemperament: too coarse even for detempering
|-
| Seesaw
| 2.3.5(.11)
| {{Mapping| 1 0 1 2 | 0 1 1 1 }}
| Extreme exotemperament: too coarse even for detempering
|-
| Seesaw > heavy windmill
| 7- to 11-limit
| {{Mapping| 1 0 1 0 | 0 1 1 2 }}
| Extreme exotemperament: too coarse even for detempering
|-
| Seesaw > light windmill
| 7- to 11-limit
| {{Mapping| 1 0 1 3 | 0 1 1 0 }}
| Extreme exotemperament: too coarse even for detempering
|-
| Sixseven
| 2.3.7(.13)
| {{Mapping| 1 0 1 2 | 0 1 1 1 }}
| Extreme exotemperament: too coarse even for detempering
|-
| Sixseven > heaven
| 2.3.7.11(.13)
| {{Mapping| 1 0 1 0 2 | 0 1 1 2 1 }}
| Extreme exotemperament: too coarse even for detempering
|-
| Dog
| 2.3.19
| {{Mapping| 1 0 -2 | 0 1 4 }}
| unenlightening restriction of armodue/armodog, lack of unique value
|-
| Sept a.k.a. mujannab
| 7- to 13-limit
| {{Mapping| 7 11 0 20 8 26 | 0 0 1 0 1 0 }}
| Pointless exotemperament: no reason to leave prime 5 free in 7et rather than prime 7; its former name was also confusable with ''mujannabic''
|-
| Counterorson
| 7-limit
| {{Mapping| 1 0 -21 85 | 0 7 103 -363 }}
| Pointless microtemperament: high badness at unusable complexity
|-
| Quectismic > nanoquectismic
| 7-limit
| {{Mapping| 1 0 -554 109 | 0 1 351 -67 }}
| Pointless microtemperament: high badness at unusable complexity
|-
| Quectismic > conanoquectismic
| 7-limit
| {{Mapping| 1 0 -554 -945 | 0 1 351 598 }}
| Pointless microtemperament: extremely high badness at unusable complexity
|-
| White dwarf
| 2.3.7.11.101
| {{Mapping| 1 25 2 -3 131 | 0 29 -1 -8 154 }}
| Extension of [[chrysanthemum]] to an extremely esoteric subgroup with extremely high complexity for its extension
|-
| Countermiracle > countermanna
| 13- to 17-limit
| {{Mapping| 1 -21 -4 1 -39 -102 -98 | 0 25 7 2 47 117 113 }}
| Medium accuracy is not worth the high complexity
|-
| Countermiracle > counterbenediction
| 13- to 17-limit
| {{Mapping| 1 -21 -4 1 -39 29 57 61 | 0 25 7 2 47 -59 -63 }}
| Medium accuracy is not worth the high complexity
|-
| Countermiracle > counterrevelation
| 11- to 17-limit
| {{Mapping| 1 -21 -4 1 -11 29 33 | 0 25 7 2 16 -28 -32 }}
| Medium accuracy is not worth the high complexity; obvious 11/9 candidate at 3 gens is not treated as such
|-
| Cohemiripple
| 7- to 13-limit
| {{Mapping| 1 -3 -5 -5 -8 -5 | 0 10 16 17 25 19 }}
| Impractical complexity for its accuracy; 25 generators for prime 11 (despite one generator being 550{{C}}), while 24edo is about as accurate as possible
|}

[[Category:Xenharmonic Wiki guidelines]]

Cassaschismic

2026-06-27T16:25:56Z

Overthink: /* External links */ en dash

{{Infobox regtemp
| Title = Cassaschismic
| Subgroups = 2.3.5.7.11, 2.3.5.7.11.13, 2.3.5.7.11.13.19
| Comma basis = [[19712/19683]], [[41503/41472]] (11-limit); [[2080/2079]], [[4096/4095]], [[19712/19683]] (13-limit); [[1216/1215]], [[1540/1539]], [[1729/1728]], [[2080/2079]] (2.3.5.7.11.13.19)
| Edo join 1 = 41 | Edo join 2 = 53 | Edo join 3 = 270
| Mapping = 1; 1 0 -14 23 12 5; 0 1 0 0 -1 1
| Generators = 3/2; 5/4 | Generators tuning = 702.2307; 386.3245
| Optimization method = CWE
| Pergen = (P8, P5, ^1)
| Color name = Salozo & Sasaru + Ya Salozo & Sasaru (& Sathoyo (& Sanogu))
| Odd limit 1 = 11 | Mistuning 1 = 0.588 | Complexity 1 = ?
}}
'''Cassaschismic''' is a [[rank-3 temperament]] that expands [[gary]]'s [[chain of fifths]] into the full [[11-limit]] by adding an independent [[generator]] for the [[5/1|5th]] [[harmonic]]. It is therefore a member of the [[garischismic family]] and [[olympic clan]].

The generator for 5 can be used for [[13/1|13]] and [[19/1|19]]. By moving the generators around, it can also be taken to be a ~4.5{{c}} generic aberschisma, which represents the [[schisma]], the [[aberschisma]], the [[undevicesimal schisma]], the [[352/351|minor minthma]] and many other important commas around that size. [[Tempering out]] this aberschisma results in [[cassandra]], so cassaschismic is a rank-3 [[detemperament]] of it, modifying its mapping by ±1 aberschisma to reach primes 5, 13, and 19.

Other rank-2 temperaments of cassaschismic include [[cotoneum]], [[gariwizmic]], [[newt]], [[satin]], [[vulture]], [[paramity]] and [[heptacot]]; these temperaments, instead of tempering out the aberschisma, find it deep in the generator chain.

{{Databox|Generators needed to reach the aberschisma|
* Newt (41 & 270): -41 hemififths;
* Cotoneum (41 & 217): -41 fifths, equating it with the 41-comma;
* Gariwizmic (94 & 270): +53 fifths (mercator comma) - 1/2 pythagorean comma;
* Vulture (53 & 217): -41 1/4-fifths;
* Satin (94 & 217): -94 1/3-fourths;
* Paramity (53 & 311): -53 1/5-elevenths;
* Heptacot (12e & 311): 12 1/7-fifths.
}}

Cassaschismic is [[support]]ed by notable [[equal temperament]]s such as {{EDOs| 217, 270, 311, and 364 }}, where the aberschisma step is well represented by one edostep. It is also trivially supported by edos of cassandra, these being [[41edo|41]], [[53edo|53]], [[94edo|94]]. [[12edo]] supports it trivially through the 12e [[val]], where both the comma step and the aberschisma step are tempered out. It can be used in any of those forms.

See [[Garischismic family #Cassaschismic]] for technical data.

== Interval lattice ==
Here is a quick compressed cheat sheet of octave-reduced intervals. This is a simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to 8 fifths - 1 aberschisma step, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 aberschisma step. There are no octave reduced primes or prime inverses with positive fifth step and aberschisma step.

{| class="wikitable center-1 right-2 right-4"
! rowspan="2" | #
! colspan="2" | Aberschisma offset -1
! colspan="2" | Aberschisma offset 0
|-
! Cents*
! Approx. ratios
! Cents*
! Approx. ratios
|-
| 0
| 1195.83
| 351/176
| 0.00
| '''1/1'''
|-
| 1
| 698.06
| 256/171
| 702.23
| '''3/2'''
|-
| 2
| 200.29
| 64/57
| 204.46
| '''9/8'''
|-
| 3
| 902.52
| '''32/19'''
| 906.69
| 27/16
|-
| 4
| 404.75
| 24/19
| 408.92
| 19/15
|-
| 5
| 1106.98
| 36/19
| 1111.15
| 19/10
|-
| 6
| 609.21
| 27/19
| 613.38
| 57/40
|-
| 7
| 111.44
| '''16/15'''
| 115.62
| 77/72
|-
| 8
| 813.68
| '''8/5'''
| 817.85
| 77/48
|-
| 9
| 315.91
| 6/5
| 320.08
| 77/64
|-
| 10
| 1018.14
| 9/5
| 1022.31
| 65/36
|-
| 11
| 520.37
| 27/20
| 524.54
| 65/48
|-
| 12
| 22.60
| 81/80
| 26.77
| 64/63
|-
| 13
| 724.83
| 38/25
| 729.00
| '''32/21'''
|-
| 14
| 227.06
| 57/50
| 231.23
| '''8/7'''
|-
| 15
| 929.29
| 77/45
| 933.46
| 12/7
|-
| 16
| 431.52
| 77/60
| 435.69
| 9/7
|-
| 17
| 1133.75
| 52/27
| 1137.92
| 27/14
|-
| 18
| 635.98
| 13/9
| 640.15
| 81/56
|-
| 19
| 138.21
| 13/12
| 142.38
| 88/81
|-
| 20
| 840.44
| '''13/8'''
| 844.61
| 44/27
|-
| 21
| 342.67
| 39/32
| 346.85
| 11/9
|-
| 22
| 1044.91
| 64/35
| 1049.08
| 11/6
|-
| 23
| 547.14
| 48/35
| 551.31
| '''11/8'''
|-
| 24
| 49.37
| 36/35
| 53.54
| 33/32
|}
<nowiki>*</nowiki> In 2.3.5.7.11.13.19-subgroup CWE tuning, octave reduced

== Notation ==
Cassaschismic is easily notated with [[chain-of-fifths notation]] with two extra pairs of accidentals: one for the comma step, and the other for the aberschisma step. It can therefore be seen as an addition to the cassandra chain of fifths, which itself can be seen as an addition to the 12edo chain of fifths, providing a layered-precision system of notation that ranges from rough (12), to moderately accurate (41, 53, 94), to highly accurate (217, 270, 311, …).

As an example, we can use up and down arrows with shafts (↑/↓) for the comma step, and arrows without shafts (^/v) for the aberschisma step.

{| class="wikitable center-all"
|+Nomenclature of selected intervals
! Ratio
! Example on C
|-
| 3/2
| C–G (perfect fifth)
|-
| 5/4
| C–^↓E (upsubmajor third)
|-
| 7/4
| C–↓Bb (subminor seventh)
|-
| 11/8
| C–↑↑F (hyperfourth)
|-
| 13/8
| C–v↑↑Ab (downhyperminor sixth)
|-
| 19/16
| C–^Eb (upminor third)
|}

== See also ==
* [[User:Eufalesio/Ultimate]] – An opinion-based derivation of cassaschismic

== External links ==
* [https://www.desmos.com/calculator/pbyqpjgrrn] – A [https://www.desmos.com/calculator Desmos] graph showing cassaschismic edos up to 311, [[8afdo|harmonic mode 8]] (green), and [[5L 7s]] 6|5 (red). The purple line on 12 is the [[patent val]] prime 11, which is not used in cassaschismic. The blue dots indicate going up and down by Pythagorean commas in the 12L 29s scale, and the orange dots indicate the leftover edosteps. The jump from 94 to 270 is due to 135edo being next in line for cassandra; since halving it results in 270edo, it is used instead, and also to showcase the use of aberschismas to reach primes 5, 13, and 19.

[[Category:Cassaschismic| ]] 
[[Category:Rank-3 temperaments]]
[[Category:Microtemperaments]]
[[Category:Garischismic family]]
[[Category:Olympic clan]]

Würschmidt family

2026-06-27T16:16:28Z

Overthink: /* 2.3.5.23 subgroup */ note further extensions

{{Technical data page}}
The [[5-limit]] parent comma for the '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt. The [[generator]] is a classic major third, and to get to the interval class of fifths requires eight of these. In fact, (5/4)8 × 393216/390625 = 6.

Similar to [[meantone]], würschmidt implies that 3/2 will be tempered flat and/or 5/4 will be tempered sharp, and therefore 6/5 will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]], which notably is the point where it is combined with [[schismic]]/[[nestoria]] and [[gravity]]/[[larry]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]].

Another tuning solution is to sharpen the major third by 1/8 of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]].

[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.

== Würschmidt ==
{{Main| Würschmidt }}

[[Subgroup]]: 2.3.5

[[Comma list]]: 393216/390625

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

: mapping generators: ~2, ~5/4

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.734
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.799

{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}

[[Badness]] (Smith): 0.040603

=== Overview to extensions ===
==== 7-limit extensions ====
The 7-limit extensions can be obtained by adding another comma. Septimal würschmidt adds [[225/224]], worschmidt adds [[126/125]], whirrschmidt adds [[4375/4374]]. These all use the same generator as 5-limit würschmidt.

Hemiwürschmidt adds [[3136/3125]] and splits the generator in two. This temperament is the best extension available for würschmidt despite its complexity. The details can be found in [[Hemimean clan #Hemiwürschmidt|Hemimean clan]].

==== Subgroup extensions ====
Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to [[11/9]], that is by tempering out [[5632/5625]] in addition to [[243/242]]; furthermore, like practically any 5-limit temperament with this accuracy level of [[3/2]] available, extensions to prime 19 exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo and [[nestoria]]).

However, as discussed in the main article, the "free" higher prime for würschmidt outside the 5-limit is in fact 23, via tempering out S24 = [[576/575]] and S462 × S47 = [[12167/12150]]. Therefore, the below discusses the 2.3.5.23 and 2.3.5.11.23 extensions.

=== 2.3.5.23 subgroup ===
Extensions to harmonics [[47/1|47]] and [[49/1|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].

Subgroup: 2.3.5.23

Comma list: 576/575, 12167/12150

Sval mapping: {{mapping| 1 -1 2 0 | 0 8 1 14 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.734
* POTE: ~2 = 1\1, ~5/4 = 387.805

Optimal ET sequence: {{optimal ET sequence| 3, …, 28i, 31, 34, 65, 99, 164 }}

Badness (Smith): 0.00530

==== 2.3.5.11.23 subgroup ====
Subgroup: 2.3.5.11.23

Comma list: 243/242, 276/275, 529/528

Sval mapping: {{mapping| 1 -1 2 -3 0 | 0 8 1 20 14 }}

Optimal tuning:
* CTE: ~2 = 1\1, ~5/4 = 387.652
* POTE: ~2 = 1\1, ~5/4 = 387.690

Optimal ET sequence: {{optimal ET sequence| 31, 34, 65 }}

Badness (Smith): 0.00660

== Septimal würschmidt ==
Würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.

2-würschmidt, the temperament with all the same commas as würschmidt but a generator of twice the size, is equivalent to [[skwares]] as a 2.3.7.11 subgroup temperament.

The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]], but specifically, the 34d [[val]] where we accept 17edo's mapping of ~7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by equating them as S8 = S9 = S10 = S11, hence its S-expression-based comma list is {{nowrap| {[[5120/5103|S8/S9]], [[8019/8000|S9/S10]], [[4000/3993|S10/S11]]} }}, which may be expressed in shortened form as {{nowrap| {S8/9/10/11} }}*. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7 no matter what mapping you use (with the sharp 7 being used for this temperament); even so, it's fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, 65d + 31 = [[96edo]] is also a reasonable choice, as it has the advantage of being [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.

(<nowiki>*</nowiki> The advantage of this form is we can easily see that all of the [[semiparticular]] commas expected are implied as well as any other commas expressible as the difference between two square superparticular commas by reading them off as ratios like 8/10 (S8/S10) and 9/11 (S9/S11).)

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 8748/8575

{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.379
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.383

{{Optimal ET sequence|legend=1| 31, 96, 127 }}

[[Badness]] (Smith): 0.050776

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

Comma list: 99/98, 176/175, 243/242

Mapping: {{mapping| 1 -1 2 -3 -3 | 0 8 1 18 20 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.441
* POTE: ~2 = 1\1, ~5/4 = 387.447

Optimal ET sequence: {{optimal ET sequence| 31, 65d, 96, 127 }}

Badness (Smith): 0.024413

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

Comma list: 99/98, 144/143, 176/175, 275/273

Mapping: {{mapping| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.469
* POTE: ~2 = 1\1, ~5/4 = 387.626

Optimal ET sequence: {{optimal ET sequence| 31, 65d }}

Badness (Smith): 0.023593

==== Worseschmidt ====
Subgroup: 2.3.5.7.11.13

Commas: 66/65, 99/98, 105/104, 243/242

Mapping: {{mapping| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.179
* POTE: ~2 = 1\1, ~5/4 = 387.099

Optimal ET sequence: {{optimal ET sequence| 3def, 28def, 31 }}

Badness (Smith): 0.034382

== Worschmidt ==
Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo]], [[34edo]], or [[127edo]] as a tuning. If 127 is used, note that the val is {{val| 127 201 295 '''356''' }} (127d) and not {{val| 127 201 295 '''357''' }} as with würschmidt. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 33075/32768

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

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.406
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.392

{{Optimal ET sequence|legend=1| 31, 96d, 127d }}

[[Badness]] (Smith): 0.064614

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

Comma list: 126/125, 243/242, 385/384

Mapping: {{mapping| 1 -1 2 7 -3 | 0 8 1 -13 20 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.472
* POTE: ~2 = 1\1, ~5/4 = 387.407

Optimal ET sequence: {{optimal ET sequence| 31, 65, 96d, 127d }}

Badness (Smith): 0.033436

== Whirrschmidt ==
[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with 7 mapped to the 52nd generator step.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 393216/390625

{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }}

[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1\1, ~5/4 = 387.853
* [[POTE]]: ~2 = 1\1, ~5/4 = 387.881

{{Optimal ET sequence|legend=1| 34d, 65, 99 }}

[[Badness]] (Smith): 0.086334

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

Comma list: 243/242, 896/891, 4375/4356

Mapping: {{mapping| 1 -1 2 -14 -3 | 0 8 1 52 20 }}

Optimal tunings:
* CTE: ~2 = 1\1, ~5/4 = 387.829
* POTE: ~2 = 1\1, ~5/4 = 387.882

Optimal ET sequence: {{optimal ET sequence| 34d, 65, 99e }}

Badness (Smith): 0.058325

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

Microtemperament

2026-06-22T05:16:15Z

Overthink: add this sentence

A '''microtemperament''' is a [[regular temperament]] in which all of the intervals considered to be consonances of the system are so closely approximated as to be effectively [[just]].

There is no agreed on value for the threshold to '''microtempering''', and here we will use less than a cent in error as our line of demarcation, which some people would call a ''nanotemperament''. Using this definition and the tonality diamond as our consonance set, we find that the equal temperaments {{EDOs|118, 171, 270 …}} are [[5-limit]] microtemperaments; that {{EDOs|171, 270, 441, 612, 643 …}} are [[7-limit]] microtemperaments; {{EDOs|764, 836, 1084, 1106 …}} are [[11-limit]] microtemperaments, and that {{EDOs|1171, 1178, 1186 …}} are [[13-limit]] microtemperaments.

Higher-rank microtemperaments are also useful as simplifications of just intonation. Putting 118 and 171 together, we see that the 5-limit [[schismatic]] temperament, which tempers out the [[schisma]] (32805/32768), is a microtemperament when appropriately tuned. Putting 171 and 270 together, we see that [[ennealimmal]] temperament, which tempers out the [[breedsma]] (2401/2400) and the [[ragisma]] (4375/4374), is also a microtemperament. A list of microtemperaments can be found at [[:Category: Microtemperaments]].

The opposite of a microtemperament is an [[exotemperament]], which is very inaccurate and tempers out large and/or high-damage commas, such as [[16/15]], [[25/24]], or [[28/27]].

== Practical notes ==
Tuning microtemperaments can be challenging due to the sheer number of pitch materials, but one may take advantage of MIDI channels. See [[Tuning per channel]].

== See also ==
* [[Unnoticeable comma]]
* [[Very high accuracy temperaments]]

[[Category:Regular temperament theory]]
[[Category:Method]]