Ragismic microtemperaments: Difference between revisions

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

* ''[[Alphatrillium]]'' (+{{monzo| 40 -22 -1 -1 }}) → [[Alphatricot family #Trillium|Alphatricot family]]

* [[Modus]] (+64/63) → [[Tetracot family #Modus|Tetracot family]]

* ''[[Vulture]]'' (+33554432/33480783) → [[Vulture family #Septimal vulture|Vulture family]]

* ''[[Unlit]]'' (+{{monzo| 41 -20 -4 }}) → [[Undim family #Unlit|Undim family]]

* ''[[Quindro]]'' (+{{monzo| 56 -28 -5 }}) → [[Quindromeda family #Quindro|Quindromeda family]]

* [[Catakleismic]] (+225/224) → [[Kleismic family #Catakleismic|Kleismic family]]

* ''[[Crepuscular]]'' (+50/49) → [[Fifive family #Crepuscular|Fifive family]]

* ''[[Quartonic]]'' (+1728/1715 or 4000/3969) → [[Quartonic family]]

* [[Parakleismic]] (+3136/3125) → [[Parakleismic family #Septimal parakleismic|Parakleismic family]]

* ''[[Vishnu]]'' (+29360128/29296875) → [[Vishnu family #Septimal vishnu|Vishnu family]]

* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Nuwell temperaments #Maja|Nuwell temperaments]]

* ''[[Mitonic]]'' (+2100875/2097152) → [[Minortone family #Mitonic|Minortone family]]

* [[Ennealimmal]] (+2401/2400) → [[Septiennealimmal clan #Ennealimmal|Septiennealimmal clan]]

* [[Octoid]] (+16875/16807) → [[Stearnsmic clan #Octoid|Stearnsmic clan]]

* ''[[Ludizemlic]]'' (+{{monzo|-223 47 -11 62}}) → [[37th-octave temperaments #Ludizemlic|37th-octave temperaments]]

 

The generator for supermajor temperament is a supermajor third, [[9/7]], tuned about 0.002 cents flat. Note that in the data that follow, the generator is given as its [[octave complement]]. 37 of these give 3/2²², 46 give 5/2²⁷, and 75 give 7/2⁴⁵. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: [[1106edo]] or [[1277edo]] can be used as tunings, leading to accuracy even greater than that of [[ennealimmal]]. The 80-note generator chain is presumably the place to start, and if that is not enough notes for you, there is always the 171-note generator chain.

{{Mapping|legend=1| 1 -22 -27 -45 | 0 37 46 75 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0067{{c}}, ~14/9 = 764.9222{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~14/9 = 764.9181{{c}}

: error map: {{val| 0.000 +0.013 -0.083 +0.029 }}

{{Optimal ET sequence|legend=1| 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}

[[Badness]] (Sintel): 0.274

{{Mapping|legend=0| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 164.9180{{c}}

{{Optimal ET sequence|legend=0| 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee }}

Badness (Sintel): 0.422

 

Enneadecal tempers out the [[enneadeca]], {{monzo| -14 -19 19 }}, and as a consequence has a period of 1/19 octave. This is because the enneadeca is the amount by which nineteen [[6/5|just minor thirds]] fall short of an octave. If to this we add 4375/4374 we get the 7-limit temperament we are considering here, but note should be taken of the fact that it makes for a reasonable 5-limit microtemperament also, where the generator can be ~25/24, ~27/25, ~10/9, ~5/4 or ~3/2. To this we may add possible 7-limit generators such as ~225/224, ~15/14 or ~9/7. Since enneadecal tempers out [[703125/702464]], the amount by which 81/80 falls short of three stacked 225/224, we can equate the 225/224 generator with (81/80)^1/3. This is the interval needed to adjust the 1/3-comma meantone flat fifths and major thirds of [[19edo]] up to just ones.  

[[171edo]] is a good tuning for either the 5- or 7-limit, and [[494edo]] shows how to extend the temperament to the 11- or 13-limit, where it is accurate but very complex. Fans of near-perfect fifths may want to use [[665edo]] for a tuning.

[[Optimal tuning]]s:

* [[WE]]: ~28/27 = 63.1599{{c}}, ~3/2 = 701.9027{{c}} (~225/224 = 7.1437{{c}})

: [[error map]]: {{val| +0.038 -0.014 -0.134 +0.080 }}

* [[CWE]]: ~28/27 = 63.1579{{c}}, ~3/2 = 701.9002{{c}} (~225/224 = 7.1634{{c}})

[[Badness]] (Sintel): 0.277

{{Mapping|legend=0| 19 0 14 -37 126 | 0 1 1 3 -2 }}

Optimal tunings:

* WE: ~28/27 = 63.1431{{c}}, ~3/2 = 702.1956{{c}} (~225/224 = 7.6216{{c}})

* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.3164{{c}} (~225/224 = 7.5795{{c}})

{{Optimal ET sequence|legend=0| 19, 133d, 152, 323e, 475de, 627de }}

Badness (Sintel): 1.45

{{Mapping|legend=0| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}

Optimal tunings:

* WE: ~28/27 = 63.1406{{c}}, ~3/2 = 702.0192{{c}} (~225/224 = 7.4730{{c}})

* CWE: ~28/27 = 63.1579{{c}}, ~3/2 = 702.1539{{c}} (~225/224 = 7.4171{{c}})

{{Optimal ET sequence|legend=0| 19, 133df, 152f, 323ef }}

Badness (Sintel): 1.39

{{Mapping|legend=0| 38 0 28 -74 11 | 0 1 1 3 2 }}

Optimal tunings:

* WE: ~55/54 = 31.5800{{c}}, ~3/2 = 701.9053{{c}} (~243/242 = 7.1448{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9034{{c}} (~243/242 = 7.1666{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 836, 1178, 2014, 3192ce, 5206ce }}

Badness (Sintel): 0.330

{{Mapping|legend=0| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}

Optimal tunings:

* WE: ~55/54 = 31.5785{{c}}, ~3/2 = 701.9995{{c}} (~243/242 = 7.2727{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 702.0053{{c}} (~243/242 = 7.2685{{c}})

{{Optimal ET sequence|legend=0| 152f, 342f, 494 }}

Badness (Sintel): 0.859

{{Mapping|legend=0| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}

Optimal tunings:

* WE: ~55/54 = 31.5784{{c}}, ~3/2 = 701.9736{{c}} (~243/242 = 7.2493{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~3/2 = 701.9855{{c}} (~243/242 = 7.2487{{c}})

{{Optimal ET sequence|legend=0| 152, 342, 494, 1330, 1824, 2318d }}

Badness (Sintel): 1.26

{{Mapping|legend=0| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}

: mapping generators: ~55/54, ~429/250

Optimal tunings:

* WE: ~55/54 = 31.5799{{c}}, ~429/250 = 935.1824{{c}} (~144/143 = 12.2152{{c}})

* CWE: ~55/54 = 31.5789{{c}}, ~429/250 = 935.1617{{c}} (~144/143 = 12.2067{{c}})

{{Optimal ET sequence|legend=0| 190, 304d, 494, 684, 1178, 2850, 4028ce }}

Badness (Sintel): 0.607

{{Mapping|legend=0| 19 3 17 -28 82 92 159 78 | 0 10 10 30 -6 -8 -30 1 }}

 

* CWE: ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.2439{{c}}

{{Optimal ET sequence|legend=0| 855, 988, 1843 }}

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimi]].''

The generator of semidimi is a semi-diminished fourth interval tuned between 162/125 and 35/27. It tempers out 5-limit {{monzo| -12 -73 55 }} and 7-limit 3955078125/3954653486, as well as 4375/4374.

{{Mapping|legend=1| 1 -19 -25 -32 | 0 55 73 93 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0018{{c}}, ~35/27 = 449.1277{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}

{{Optimal ET sequence|legend=1| 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419 }}

[[Badness]] (Sintel): 0.382

The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}), and may be described as the {{nowrap| 217 & 224 }} temperament.  

Early in the design of the [[Sagittal]] notation system, [[George Secor|Secor]] and [[Dave Keenan|Keenan]] found that an economical JI notation system could be defined, which divided the apotome (Pythagorean sharp or flat) into 21 almost-equal divisions. This required only 10 microtonal accidentals, although a few others were added for convenience in alternative spellings. This is called the Athenian symbol set (which includes the Spartan set). Its symbols are defined to exactly notate many common 11-limit ratios and the 17th harmonic, and to approximate within ±0.4{{c}} many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of brahmagupta that has pure octaves and pure fifths, which can also be described as a 17-limit extension having a 1/7-octave period (171.4286{{c}}) and 1/21-apotome generator (5.4136{{c}}).

[[Comma list]]: 4375/4374, {{monzo| 46 -14 -3 -6 }}

[[Optimal tuning]]s:

* [[WE]]: ~1157625/1048576 = 171.4275{{c}}, ~27/20 = 519.7125{{c}}

{{Optimal ET sequence|legend=1| 7, …, 217, 224, 441, 1106, 1547 }}

[[Badness]] (Sintel): 0.737

{{Mapping|legend=0| 7 2 -8 53 3 | 0 3 8 -11 7 }}

Optimal tunings:

* WE: ~243/220 = 171.4208{{c}}, ~27/20 = 519.6807{{c}}

* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7034{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665 }}

Badness (Sintel): 1.73

{{Mapping|legend=0| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}

Optimal tunings:

* WE: ~243/220 = 171.4197{{c}}, ~27/20 = 519.6789{{c}}

* CWE: ~243/220 = 171.4286{{c}}, ~27/20 = 519.7052{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1106e }}

Badness (Sintel): 0.956

 

Abigail tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit, and may be described as the {{nowrap| 46 & 224 }} temperament, with a [[ploidacot]] signature of diploid wau-hendecacot. It extends into a very strong 11- and 13-limit temperament. [[494edo]], [[764edo]] and [[1258edo]] are among the possible tunings.  

Abigail was named by [[Gene Ward Smith]] in 2010 after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930 Yahoo! Tuning Group | ''11-limit rank 2 using only wedgies''] "I propose Abigail as a name, on the grounds 313/1798 is an excellent generator, and Abigail Fillmore, wife of Millard, was born on 3-13-1798 at least as Americans recon things." —Gene Ward Smith</ref>

{{Mapping|legend=1| 2 -4 -11 18 | 0 11 24 -19 }}

[[Optimal tuning]]s:  

* [[WE]]: ~46305/32768 = 599.9699{{c}}, ~1536/1225 = 391.0818{{c}}

: [[error map]]: {{val| -0.060 +0.065 -0.021 +0.079 }}

* [[CWE]]: ~46305/32768 = 600.0000{{c}}, ~1536/1225 = 391.1007{{c}}

: error map: {{val| 0.000 +0.152 +0.102 +0.262 }}

{{Optimal ET sequence|legend=1| 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd }}

[[Badness]] (Sintel): 0.936

{{Mapping|legend=0| 2 -4 -11 18 18 | 0 11 24 -19 -17 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~1536/1225 = 391.0992{{c}}

{{Optimal ET sequence|legend=0| 46, 132, 178, 224, 270, 494, 764 }}

Badness (Sintel): 0.425

{{Mapping|legend=0| 2 -4 -11 18 18 25 | 0 11 24 -19 -17 -27 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~351/280 = 391.0969{{c}}

{{Optimal ET sequence|legend=0| 46, 178, 224, 270, 494, 764, 1258 }}

Badness (Sintel): 0.366

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Gamera]].''

{{Mapping|legend=1| 1 -17 -30 2 | 0 23 40 1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.8483{{c}}, ~7/4 = 969.5415{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~7/4 = 969.6608{{c}}

[[Badness]] (Sintel): 0.953

{{Mapping|legend=0| 2 -11 -20 5 10 | 0 23 40 1 -5 }}

: mapping generators: ~99/70, ~99/80

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~99/80 = 369.6610{{c}}

{{Optimal ET sequence|legend=0| 26, 172c, 198, 224, 422, 646, 1068d }}

Badness (Sintel): 1.35

{{Mapping|legend=0| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~26/21 = 369.6603{{c}}

{{Optimal ET sequence|legend=0| 26, 172cf, 198, 224, 422, 646f, 1068df }}

Badness (Sintel): 0.844

{{Mapping|legend=0| 1 -40 -70 1 -77 | 0 46 80 2 89 }}

: mapping generators: ~2, ~144/77

Optimal tunings:

* WE: ~2 = 1199.8845{{c}}, ~144/77 = 1084.7314{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8345{{c}}

{{Optimal ET sequence|legend=0| 73, 125, 198, 323, 521 }}

Badness (Sintel): 2.59

{{Mapping|legend=0| 1 -40 -70 1 -77 -131 | 0 46 80 2 89 149 }}

Optimal tunings:

* WE: ~2 = 1199.8726{{c}}, ~144/77 = 1084.7220{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~144/77 = 1084.8359{{c}}

{{Optimal ET sequence|legend=0| 73f, 125f, 198, 323, 521 }}

Badness (Sintel): 1.82

Crazy tempers out the [[kwazy comma]] in the 5-limit, and adds the ragisma to extend it to the 7-limit. It can be described as the {{nowrap| 118 & 494 }} temperament, with a [[ploidacot]] of diploid alpha-octacot. [[1106edo]] gives a strong tuning.

 

Crazy was named by [[Flora Canou]] in 2025 by removing the mutation from ''kwazy'', the name for the 5-limit microtemperament.  

* [[WE]]: ~332150625/234881024 = 600.0019{{c}}, ~1125/1024 = 162.7479{{c}}

: [[error map]]: {{val| +0.004 +0.030 -0.042 -0.014 }}

* [[CWE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7474{{c}}

: error map: {{val| 0.000 +0.024 -0.051 -0.022 }}

[[Badness]] (Sintel): 0.998

{{Mapping|legend=0| 2 1 6 -15 -8 | 0 8 -5 76 55 }}

* WE: ~99/70 = 600.0047{{c}}, ~1125/1024 = 162.7493{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~1125/1024 = 162.7481{{c}}

Badness (Sintel): 0.562

[[Comma list]]: 4375/4374, {{monzo| 41 -4 2 -14 }}

{{Mapping|legend=1| 2 -8 -15 6 | 0 29 51 -1 }}

[[Optimal tuning]]s:  

* [[WE]]: ~7411887/5242880 = 599.9927{{c}}, ~8/7 = 231.1012{{c}}

: [[error map]]: {{val| -0.015 +0.037 -0.045 +0.029 }}

* [[CWE]]: ~7411887/5242880 = 600.0000{{c}}, ~8/7 = 231.1037{{c}}

{{Optimal ET sequence|legend=1| 26, …, 244, 270, 836, 1106, 1376, 2482 }}

[[Badness]] (Sintel): 1.02

{{Mapping|legend=0| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}

Optimal tunings:

* WE: ~99/70 = 600.0025{{c}}, ~8/7 = 231.1039{{c}}

* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1030{{c}}

{{Optimal ET sequence|legend=0| 26, 244, 270, 566, 836, 1106 }}

Badness (Sintel): 0.535

{{Mapping|legend=0| 2 -8 -15 6 10 -3 | 0 29 51 -1 -8 27 }}

 

Optimal tunings:  

* WE: ~99/70 = 600.0192{{c}}, ~8/7 = 231.1102{{c}}

 

 

 

 

 

 

 

{{Mapping|legend=1| 17 0 26 -87 | 0 2 1 10 }}

 

* [[WE]]: ~25/24 = 70.5880{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9962{{c}}

* [[CWE]]: ~25/24 = 70.5882{{c}}, ~{{monzo| 24 -5 -9 2 }} = 950.9990{{c}}

: error map: {{val| 0.000 +0.043 -0.021 -0.013 }}

 

{{Optimal ET sequence|legend=1| 289, 323, 612, 935, 1547, 3706, 5253 }}

 

 

 

{{Mapping|legend=0| 17 0 26 -87 207 | 0 2 1 10 -11 }}

* WE: ~25/24 = 70.5905{{c}}, ~693/400 = 951.0054{{c}}

* CWE: ~25/24 = 70.5882{{c}}, ~693/400 = 950.9754{{c}}

 

Badness (Sintel): 2.11

: ''For the 5-limit version, see [[Very high accuracy temperaments #Senior]].

Aside from the ragisma, the seniority temperament tempers out the [[wadisma]], 201768035/201326592, and may be described as {{nowrap| 26 & 145 }}. It is so named because the [[senior comma]] ({{monzo| -17 62 -35 }}) is tempered out.

{{Mapping|legend=1| 1 -24 -43 5 | 0 35 62 -3 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0745{{c}}, ~5120/3087 = 877.2500{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5120/3087 = 877.1965{{c}}

: error map: {{val| 0.000 -0.077 -0.130 -0.415 }}

{{Optimal ET sequence|legend=1| 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d }}

[[Badness]] (Sintel): 1.14

Senator (26 & 145) extends seniority by tempering out [[441/440]] and [[65536/65219]], and can be extended to the 13- and 17-limit immediately by adding [[364/363]] and [[595/594]] to the comma list in this order.

{{Mapping|legend=0| 1 -24 -43 5 2 | 0 35 62 -3 2 }}

Optimal tunings:

* WE: ~2 = 1199.7665{{c}}, ~128/77 = 877.0367{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~128/77 = 877.2051{{c}}

{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e }}

Badness (Sintel): 3.05

{{Mapping|legend=0| 1 -24 -43 5 2 -27 | 0 35 62 -3 2 42 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2038{{c}}

{{Optimal ET sequence|legend=0| 26, 119cf, 145, 171, 316ef }}

Badness (Sintel): 1.85

{{Mapping|legend=0| 1 -24 -43 5 2 -27 -31 | 0 35 62 -3 2 42 48 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~108/65 = 877.2039{{c}}

{{Optimal ET sequence|legend=0| 26, 119cfg, 145, 171, 316ef }}

Badness (Sintel): 1.35

: ''For the 5-limit version, see [[Very high accuracy temperaments #Monzismic]].  

Monzismic tempers out the [[monzisma]], {{monzo| 54 -37 2 }}, and in the 7-limit, the [[nanisma]], {{monzo| 109 -67 0 -1 }}, as well as the ragisma, [[4375/4374]]. It may be described as the {{nowrap| 53 & 612 }} temperament, with a [[ploidacot]] signature of alpha-dicot. A notable tuning not appearing on the optimal ET sequence is [[665edo]], which is nearly equivalent to the pure-3's tuning.

{{Mapping|legend=1| 1 0 -27 109 | 0 2 37 -134 }}

: mapping generators: ~2, ~{{monzo| 28 -11 -3 -1 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1200.0128{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9895{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 28 -11 -3 -1 }} = 950.9793{{c}}

: error map: {{val| 0.000 +0.004 -0.080 -0.050 }}

{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd }}

[[Badness]] (Sintel): 1.18

{{Mapping|legend=0| 1 0 -27 109 -159 | 0 2 37 -134 205 }}

Optimal tunings:  

* WE: ~2 = 1200.0347{{c}}, ~400/231 = 951.0082{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9807{{c}}

{{Optimal ET sequence|legend=0| 53, 559, 612, 3619de, 4231de, …, 6067ddee }}

Badness (Sintel): 1.89

{{Mapping|legend=0| 1 0 -27 109 -159 -70 | 0 2 37 -134 205 93 }}

Optimal tunings:  

* WE: ~2 = 1200.0036{{c}}, ~400/231 = 950.9829{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~400/231 = 950.9801{{c}}

{{Optimal ET sequence|legend=0| 53, 559, 612 }}

Badness (Sintel): 2.22

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semidimfourth]].''

The semidimfourth temperament is featured by a semidiminished fourth inverval which is [[128/125]] above the pythagorean major third [[81/64]]. In the 7-limit, this temperament tempers out the ragisma and the triwellisma, [[235298/234375]].

{{Mapping|legend=1| 1 -10 -13 -17 | 0 31 41 53 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9936{{c}}, ~35/27 = 448.4533{{c}}

{{Optimal ET sequence|legend=1| 8d, …, 91, 99, 289, 388, 875 }}

[[Badness]] (Sintel): 1.40

{{Mapping|legend=0| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4546{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 190, 388 }}

Badness (Sintel): 1.95

{{Mapping|legend=0| 2 -20 -26 -34 -17 -21 | 0 31 41 53 32 38 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~35/27 = 448.4549{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 190, 198, 388 }}

Badness (Sintel): 1.28

{{Mapping|legend=1| 1 -22 -22 -65 | 0 32 33 92 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.9305{{c}}, ~5/3 = 884.3923{{c}}

: [[error map]]: {{val| -0.070 +0.126 +0.160 -0.221 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.4423{{c}}

: error map: {{val| 0.000 +0.198 +0.282 -0.136 }}

[[Badness]] (Sintel): 1.42

{{Mapping|legend=0| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}

Optimal tunings:

* WE: ~2 = 1199.9698{{c}}, ~5/3 = 884.4193{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4414{{c}}

{{Optimal ET sequence|legend=0| 19, 251, 270, 829, 1099, 1369, 1639 }}

Badness (Sintel): 1.22

{{Mapping|legend=0| 1 -22 -22 -65 58 -56 | 0 32 33 92 -74 81 }}

Optimal tunings:

* WE: ~2 = 1199.9939{{c}}, ~5/3 = 884.4384{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4429{{c}}

{{Optimal ET sequence|legend=0| 19, 251, 270 }}

Badness (Sintel): 1.11

{{Mapping|legend=0| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}

Optimal tunings:

* WE: ~2 = 1199.8877{{c}}, ~5/3 = 884.3639{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4457{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc }}

Badness (Sintel): 1.41

{{Mapping|legend=0| 1 -22 -22 -65 -141 -56 | 0 32 33 92 196 81 }}

Optimal tunings:

* WE: ~2 = 1199.9285{{c}}, ~5/3 = 884.3937{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/3 = 884.4458{{c}}

{{Optimal ET sequence|legend=0| 19e, …, 251e, 270, 1331c }}

Badness (Sintel): 1.08

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Quasithird]].''

Quasithird may be described as the {{nowrap| 224 & 388 }} temperament, featured by a major third interval which is 1600000/1594323 ([[amity comma]]) or 5120/5103 ([[5120/5103|hemifamity comma]]) below the just major third [[5/4]] as a generator, five of which give a fifth with octave reduction. This temperament has a period of a quarter octave, which allows it to temper out the ragisma and {{monzo| -60 29 0 5 }}. Its [[ploidacot]] is tetraploid delta-pentacot.  

[[Optimal tuning]]s:

* [[WE]]: ~65536/55125 = 300.0052{{c}}, ~5103/4096 = 380.3949{{c}}

[[Badness]] (Sintel): 1.56

{{Mapping|legend=0| 4 0 -11 48 43 | 0 5 16 -29 -23 }}

Optimal tunings:

* WE: ~65536/51125 = 300.0073{{c}}, ~5103/4096 = 380.3963{{c}} (or ~22/21 = 80.3890{{c}})

* CWE: ~65536/51125 = 300.0000{{c}}, ~5103/4096 = 380.3868{{c}} (or ~22/21 = 80.3868{{c}})

{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee }}

Badness (Sintel): 0.698

{{Mapping|legend=0| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}

Optimal tunings:

* WE: ~65536/51125 = 299.9985{{c}}, ~81/65 = 380.3833{{c}} (or ~22/21 = 80.3848{{c}})

* CWE: ~65536/51125 = 300.0000{{c}}, ~81/65 = 380.3852{{c}} (or ~22/21 = 80.3852{{c}})

{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836 }}

Badness (Sintel): 1.22

[[Comma list]]: 4375/4374, 823543/819200

{{Mapping|legend=1| 1 2 3 3 | 0 -30 -49 -14 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.2169{{c}}, ~1728/1715 = 16.6160{{c}}

: [[error map]]: {{val| +0.217 +0.000 +0.155 -0.799 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~1728/1715 = 16.6083{{c}}

: error map: {{val| 0.000 -0.205 -0.122 -1.343 }}

[[Badness]] (Sintel): 2.02

Comma list: 441/440, 4000/3993, 4375/4374

{{Mapping|legend=0| 1 2 3 3 4 | 0 -30 -49 -14 -39 }}

* WE: ~2 = 1200.1286{{c}}, ~100/99 = 16.6147{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6101{{c}}

{{Optimal ET sequence|legend=0| 72, 217, 289 }}

Badness (Sintel): 1.02

Comma list: 364/363, 441/440, 676/675, 4375/4374

{{Mapping|legend=0| 1 2 3 3 4 5 | 0 -30 -49 -14 -39 -94 }}

Optimal tunings:

* WE: ~2 = 1200.0554{{c}}, ~100/99 = 16.6028{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6011{{c}}

{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}

Badness (Sintel): 0.986

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 676/675, 1156/1155

Optimal tunings:  

* WE: ~2 = 1200.0647{{c}}, ~100/99 = 16.6025{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.6004{{c}}

{{Optimal ET sequence|legend=0| 72, 145, 217, 289 }}

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 343/342, 364/363, 441/440, 476/475, 595/594, 676/675

{{Mapping|legend=0| 1 2 3 3 4 5 5 4 | 0 -30 -49 -14 -39 -94 -66 18 }}

Optimal tunings:  

* WE: ~2 = 1199.9287{{c}}, ~100/99 = 16.5930{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 16.5948{{c}}

{{Optimal ET sequence|legend=0| 72, 145, 217 }}

Badness (Sintel): 0.924

== Deca ==

: ''For 5-limit version, see [[10th-octave temperaments#Neon]].''