Ragismic microtemperaments: Difference between revisions

This is a collection of [[rank-2 temperament|rank-2]] [[regular temperament|temperaments]] [[tempering out]] the ragisma, [[4375/4374]] ({{monzo| -1 -7 4 1 }}). The ragisma is the smallest [[7-limit]] [[superparticular ratio]].  

Since {{nowrap|(10/9)⁴ {{=}} (4375/4374)⋅(32/21) }}, the minor tone 10/9 tends to be an interval of relatively low [[complexity]] in temperaments tempering out the ragisma, though when looking at [[microtemperament]]s the word "relatively" should be emphasized. Even so mitonic uses it as a generator, which ennealimmal and enneadecal can do also, and amity reaches it in three generators. We also have {{nowrap| 7/6 {{=}} (4375/4374)⋅(27/25)² }}, so 27/25 also tends to relatively low complexity, with the same caveat about "relatively"; however 27/25 is the period for ennealimmal.

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

* ''[[Maja]]'' (+2430/2401 or 3125/3087) → [[Maja family #Septimal maja|Maja family]]

* [[Amity]] (+5120/5103) → [[Amity family #Septimal amity|Amity family]]

* [[Pontiac]] (+32805/32768) → [[Schismatic family #Pontiac|Schismatic family]]

* ''[[Zarvo]]'' (+33075/32768) → [[Gravity family #Zarvo|Gravity family]]

* ''[[Whirrschmidt]]'' (+393216/390625) → [[Würschmidt family #Whirrschmidt|Würschmidt family]]

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

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

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

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

* ''[[Vacuum]]'' (+{{monzo| -68 18 17 }}) → [[Vavoom family #Vacuum|Vavoom family]]

* ''[[Chlorine]]'' (+{{monzo| -52 -17 34}}) → [[17th-octave temperaments #Chlorine|17th-octave temperaments]]

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 52734375/52706752

 

 

 

 

 

 

Comma list: 3025/3024, 4375/4374, 35156250/35153041

 

 

 

 

: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Enneadecal (5-limit)]].''

 

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

 

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 703125/702464

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

: mapping generators: ~28/27, ~3

[[Optimal tuning]]s:

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

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

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

: error map: {{val| 0.000 -0.055 -0.203 +0.033 }}

{{Optimal ET sequence|legend=1| 19, …, 152, 171, 665, 836, 1007, 2185, 3192c }}

[[Badness]] (Sintel): 0.277

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 16384/16335

Mapping: {{mapping| 19 0 14 -37 126 | 0 1 1 3 -2 }}

Optimal tunings:  

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

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

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

Badness (Sintel): 1.45

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 2205/2197

Mapping: {{mapping| 19 0 14 -37 126 -20 | 0 1 1 3 -2 3 }}

Optimal tunings:  

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

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

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

Badness (Sintel): 1.39

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 234375/234256

: mapping generators: ~55/54, ~3

Optimal tunings:  

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

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

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

Badness (Sintel): 0.330

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 234375/234256

Mapping: {{mapping| 38 0 28 -74 11 -281 | 0 1 1 3 2 7 }}

Optimal tunings:  

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4096/4095, 4375/4374, 31250/31213

Mapping: {{mapping| 38 0 28 -74 11 502 | 0 1 1 3 2 -6 }}

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 78125/78078

: 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

Named after the 19th element, potassium, and after an archaic variant of the element's name to resolve a name conflict. [[19/16]] can be used as a generator. Since it is enfactored in the 17-limit and lower, it makes no sense to name it for the lower subgroups.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 2500/2499, 3250/3249, 4225/4224, 4375/4374, 11016/11011, 57375/57344

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

Optimal tunings:  

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

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

Badness (Sintel): 3.15

== Semidimi ==

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

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 3955078125/3954653486

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

[[Optimal tuning]]s:  

: error map: {{val| 0.000 +0.030 -0.043 -0.015 }}

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

[[Badness]] (Sintel): 0.382

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

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

[[Subgroup]]: 2.3.5.7

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

{{Mapping|legend=1| 7 2 -8 53 | 0 3 8 -11 }}

: mapping generators: ~1157625/1048576, ~27/20

[[Optimal tuning]]s:  

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

: [[error map]]: {{val| -0.007 +0.037 -0.034 -0.004 }}

* [[CWE]]: ~1157625/1048576 = 171.4286{{c}}, ~27/20 = 519.7156{{c}}

: error map: {{val| 0.000 +0.049 -0.018 +0.017 }}

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

[[Badness]] (Sintel): 0.737

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 4375/4374, 131072/130977

Mapping: {{mapping| 7 2 -8 53 3 | 0 3 8 -11 7 }}

Optimal tunings:  

* 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

 

Comma list: 1575/1573, 2080/2079, 4096/4095, 4375/4374

 

 

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

 

 

 

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

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2147483648/2144153025

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

: mapping generators: ~46305/32768, ~1536/1225

[[Optimal tuning]]s:  

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

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

: 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

 

 

 

 

 

Badness (Sintel): 0.425

 

 

 

 

 

 

 

 

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

[[Optimal tuning]]s:  

{{Optimal ET sequence|legend=1| 26, 73, 99, 224, 323, 422, 745d }}

[[Badness]] (Sintel): 0.953

Subgroup: 2.3.5.7.11

Mapping: {{mapping| 2 -11 -20 5 10 | 0 23 40 1 -5 }}

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: {{mapping| 2 -11 -20 5 10 -8 | 0 23 40 1 -5 25 }}

 

 

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

Badness (Sintel): 0.844

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 14641/14580, 15488/15435

Mapping: {{mapping| 1 -40 -70 1 -77 | 0 46 80 2 89 }}

: mapping generators: ~2, ~144/77

 

 

 

 

 

 

 

 

 

 

 

 

 

[[Comma list]]: 4375/4374, {{monzo| -53 10 16 }}

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

[[Optimal tuning]]s:  

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

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

{{Optimal ET sequence|legend=1| 118, 376, 494, 612, 1106, 1718 }}

[[Badness]] (Sintel): 0.998

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2791309312/2790703125

Mapping: {{mapping| 2 1 6 -15 -8 | 0 8 -5 76 55 }}

Optimal tunings:

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

{{Optimal ET sequence|legend=0| 118, 376, 494, 612, 1106, 2824, 3930e }}

Badness (Sintel): 0.562

== Orga ==

Orga may be described as the {{nowrap| 26 & 270 }} temperament, and [[1106edo]] gives a strong tuning.  

[[Subgroup]]: 2.3.5.7

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

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

* [[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

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 5767168/5764801

Mapping: {{mapping| 2 -8 -15 6 10 | 0 29 51 -1 -8 }}

Optimal tunings:  

* 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

 

Comma list: 1716/1715, 2080/2079, 3025/3024, 15379/15360

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Comma list: 441/440, 4375/4374, 65536/65219

 

 

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

 

 

 

 

 

 

 

 

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 2200/2197

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

Optimal tunings:  

* 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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -55 30 2 1 }}

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

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

[[Optimal tuning]]s:  

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

* [[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

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 184549376/184528125

Mapping: {{mapping| 1 0 -27 109 -159 | 0 2 37 -134 205 }}

Optimal tunings:  

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

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

Badness (Sintel): 1.89

==== 13-limit ====

Comma list: 2200/2197, 4096/4095, 4375/4374, 40656/40625

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

Optimal tunings:  

{{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]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 235298/234375

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

: mapping generators: ~2, ~35/27

[[Optimal tuning]]s:

: [[error map]]: {{val| -0.007 +0.160 +0.353 -0.694 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/27 = 448.4555{{c}}

: error map: {{val| 0.000 +0.165 +0.361 -0.685 }}

[[Badness]] (Sintel): 1.40

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 235298/234375

Mapping: {{mapping| 2 -20 -26 -34 -17 | 0 31 41 53 32 }}

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

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 3025/3024, 4375/4374

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

* WE: ~99/70 = 600.0034{{c}}, ~35/27 = 448.4573{{c}}

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

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

Badness (Sintel): 1.28

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 2202927104/2197265625

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

[[Optimal tuning]]s:  

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

{{Optimal ET sequence|legend=1| 19, …, 251, 270, 2449c, 2719c, 2989bc }}

[[Badness]] (Sintel): 1.42

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 172032/171875

Mapping: {{mapping| 1 -22 -22 -65 58 | 0 32 33 92 -74 }}

Optimal tunings:  

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 10985/10976

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

Optimal tunings:  

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

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

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

Badness (Sintel): 1.11

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 117649/117612

Mapping: {{mapping| 1 -22 -22 -65 -141 | 0 32 33 92 196 }}

Optimal tunings:  

* 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

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1716/1715, 4225/4224, 4375/4374

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

* 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.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -60 29 0 5 }}

{{Mapping|legend=1| 4 0 -11 48 | 0 5 16 -29 }}

[[Optimal tuning]]s:  

: [[error map]]: {{val| +0.021 +0.020 -0.052 -0.031 }}

: error map: {{val| 0.000 -0.013 -0.100 -0.089 }}

{{Optimal ET sequence|legend=1| 60d, 164, 224, 388, 612, 1448, 2060 }}

[[Badness]] (Sintel): 1.56

Subgroup: 2.3.5.7.11

Mapping: {{mapping| 4 0 -11 48 43 | 0 5 16 -29 -23 }}

 

 

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

 

 

 

 

Mapping: {{mapping| 4 0 -11 48 43 11 | 0 5 16 -29 -23 3 }}

 

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

Badness (Sintel): 1.22

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

Deca has a period of 1/10 octave and tempers out the [[neon comma]] ({{monzo| 21 60 -50 }}) in the 5-limit, the [[linus comma]] ({{monzo| 11 -10 -10 10 }}) and {{monzo| 12 -3 -14 9 }} (165288374272/164794921875) in the 7-limit. It may be described as the {{nowrap| 80 & 190 }} temperament, and has a [[ploidacot]] of decaploid wau-pentacot.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 165288374272/164794921875

{{Mapping|legend=1| 10 4 9 2 | 0 5 6 11 }}

[[Optimal tuning]]s:  

: error map: {{val| 0.000 +0.136 +0.195 -0.226 }}

{{Optimal ET sequence|legend=1| 80, 190, 270, 1270, 1540, 1810, 2080 }}

[[Badness]] (Sintel): 2.04

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 391314/390625

Mapping: {{mapping| 10 4 9 2 18 | 0 5 6 11 7 }}

Optimal tunings:  

* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4189{{c}} (77/75 = 44.4189{{c}})

{{Optimal ET sequence|legend=0| 80, 190, 270, 1000, 1270, 1540e, 1810e }}

Badness (Sintel): 0.804

Subgroup: 2.3.5.7.11.13

Comma list: 1001/1000, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 10 4 9 2 18 37 | 0 5 6 11 7 0 }}

Optimal tunings:  

* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4048{{c}} (~40/39 = 44.4048{{c}})

{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}

Badness (Sintel): 0.695

Subgroup: 2.3.5.7.11.13.19

Comma list: 1001/1000, 1521/1520, 3025/3024, 4225/4224, 4375/4374

Mapping: {{mapping| 10 4 9 2 18 37 33 | 0 5 6 11 7 0 4 }}

* WE: ~15/14 = 120.0045{{c}}, ~33/28 = 284.4140{{c}} (~39/38 = 44.4050{{c}})

* CWE: ~15/14 = 120.0000{{c}}, ~33/28 = 284.4075{{c}} (~39/38 = 44.4075{{c}})

{{Optimal ET sequence|legend=0| 80, 190, 270, 730, 1000 }}

Badness (Sintel): 0.556

Keenanose, the {{nowrap| 270 & 1889 }} temperament, was named by [[Eliora]] in 2022 for the fact that it uses [[385/384]], the keenanisma, as the generator.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -56 1 -8 26 }}

{{Mapping|legend=1| 1 2 3 3 | 0 -112 -183 -52 }}

: mapping generators: ~2, ~{{monzo| 21 3 1 -10 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1200.0068{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4467{{c}}

: [[error map]]: {{val| +0.007 +0.031 -0.035 -0.032 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4466{{c}}

{{Optimal ET sequence|legend=1| 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd }}

[[Badness]] (Sintel): 2.17

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 117649/117612, 67110351/67108864