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.

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

* ''[[Hystrix]]'' (+36/35) → [[Porcupine family #Hystrix|Porcupine family]]

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

* [[Unidec]] (+1029/1024) → [[Gamelismic clan #Unidec|Gamelismic clan]]

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

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

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

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

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

Considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, chlorine, octoid, seniority, monzismic, semidimfourth, acrokleismic, quasithird, quincy, deca, keenanose, counterkleismic, sfourth, aluminium, ragitritonic, quatracot, trideci, moulin, and palladium.  

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

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~2, ~14/9

[[Optimal tuning]]s:  

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

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

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

[[Badness]] (Sintel): 0.274

Subgroup: 2.3.5.7.11

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

Mapping: {{mapping| 2 -7 -8 -15 -6 | 0 37 46 75 47 }}

* WE: ~99/70 = 600.0103{{c}}, ~11/10 = 164.9205{{c}}

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

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

Badness (Sintel): 0.422

: ''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}})

{{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:  

* 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 }}

* 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: {{mapping| 38 0 28 -74 11 | 0 1 1 3 2 }}

: mapping generators: ~55/54, ~3

Optimal tunings:  

* 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:  

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

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

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

Badness (Sintel): 0.859

==== Hemienneadec ====

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:  

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

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

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

Badness (Sintel): 1.26

Subgroup: 2.3.5.7.11.13

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

Mapping: {{mapping| 38 1 29 -71 13 111 | 0 2 2 6 4 1 }}

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

Optimal tunings:  

* 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 }}

 

 

 

 

 

 

 

 

 

 

 

== Brahmagupta ==

 

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}}).

 

 

 

 

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

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

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

 

 

 

 

 

 

* 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 }}

 

 

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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

Badness (Sintel): 2.59

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 4375/4374, 14641/14580

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

Optimal tunings:  

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

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

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

Badness (Sintel): 1.82

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

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

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

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~332150625/234881024, ~1125/1024

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

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

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

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

[[Badness]] (Sintel): 0.998

=== 11-limit ===

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:

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

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

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

Badness (Sintel): 0.562

== Orga ==

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

[[Subgroup]]: 2.3.5.7

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

: mapping generators: ~7411887/5242880, ~8/7

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

* [[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

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:  

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

* CWE: ~99/70 = 600.0000{{c}}, ~8/7 = 231.1033{{c}}

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

Badness (Sintel): 0.899

: ''For the 5-limit version, see [[17th-octave temperaments #Chlorine]].''

Chlorine (named after the 17th element) tempers out the [[septendecima]] in the 5-limit, and {{monzo| -49 4 22 -3 }} as well as the ragisma in the 7-limit. It has a 1/17-octave period, and can be described as {{nowrap| 289 & 323 }} temperament. Not only the semitwelfth, but also the ~5/4 can be used as a generator.  

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, {{monzo| -49 4 22 -3 }}

{{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 }}

[[Badness]] (Sintel): 1.05

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 41503/41472, 1879453125/1879048192

Mapping: {{mapping| 17 0 26 -87 207 | 0 2 1 10 -11 }}

Optimal tunings:  

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

{{Optimal ET sequence|legend=0| 289, 323, 612, 3349de, 3961de, …, 5797ddee }}

Badness (Sintel): 2.11

: {{Main| Octoid }}

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

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 ([[4375/4374|ragisma]]) and 16875/16807 ([[16875/16807|mirkwai comma]]). In the 11-limit, it tempers out [[540/539]], [[1375/1372]], and [[6250/6237]]. In this temperament, one period gives ~[[12/11]], two give ~[[25/21]], three give ~[[35/27]], and four give [[99/70]]~[[140/99]].

The [[11-limit]] is the last place where all the extensions of octoid shown here agree in the mappings of primes. [[80edo]] is an alternative tuning for octoid in the 11-limit; though [[72edo]] does better for minimizing the average damage on the [[11-odd-limit]], 80edo damages prime 7 in favor of practically-just [[17/16]]'s, [[11/10]]'s and [[9/7]]'s. In higher limits, the mapping supported by 80edo is octopus – not octoid – as 80edo does not temper out [[324/323]], [[375/374]], [[495/494]], [[625/624]], [[715/714]] or [[729/728]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 16875/16807

{{Mapping|legend=1| 8 1 3 3 | 0 3 4 5 }}

[[Optimal tuning]]s:  

* [[WE]]: ~49/45 = 150.0003{{c}}, ~7/5 = 583.9416{{c}}

* [[CWE]]: ~49/45 = 150.0000{{c}}, ~7/5 = 583.9411{{c}}

[[Tuning ranges]]:  

* 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=1| 8d, …, 72, 152, 224 }}

[[Badness]] (Sintel): 1.08

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 4000/3993

Mapping: {{mapping| 8 1 3 3 16 | 0 3 4 5 3 }}

* WE: ~12/11 = 149.9932{{c}}, ~7/5 = 583.9356{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9477{{c}}

* 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224, 824d }}

Badness (Sintel): 0.466

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 1375/1372

Mapping: {{mapping| 8 1 3 3 16 -21 | 0 3 4 5 3 13 }}

Optimal tunings:  

* WE: ~12/11 = 150.0005{{c}}, ~7/5 = 583.9066{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9052{{c}}

{{Optimal ET sequence|legend=0| 72, 152f, 224 }}

Badness (Sintel): 0.631

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 375/374, 540/539, 625/624, 715/714, 729/728

Mapping: {{mapping| 8 1 3 3 16 -21 -14 | 0 3 4 5 3 13 12 }}

Optimal tunings:  

* WE: ~12/11 = 150.0064{{c}}, ~7/5 = 583.8666{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.8489{{c}}

{{Optimal ET sequence|legend=0| 72, 152fg, 224, 296, 520g }}

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 375/374, 400/399, 495/494, 540/539, 715/714

Mapping: {{mapping| 8 1 3 3 16 -21 -14 34 | 0 3 4 5 3 13 12 0 }}

Optimal tunings:  

* WE: ~12/11 = 149.9785{{c}}, ~7/5 = 583.8482{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9138{{c}}

{{Optimal ET sequence|legend=0| 72, 152fg, 224 }}

Badness (Sintel): 0.975

A reasonable alternative tuning of octopus not shown here which works well for 23-limit harmony (and beyond) is [[80edo]], which has a strong sharp tendency that can be thought of as matching the sharpness of mapping [[19/16]] to 1\4 = 300{{c}}.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 | 0 3 4 5 3 4 }}

Optimal tunings:  

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9583{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224f }}

Badness (Sintel): 0.896

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 540/539

Mapping: {{mapping| 8 1 3 3 16 14 21 | 0 3 4 5 3 4 3 }}

* WE: ~12/11 = 150.0528{{c}}, ~7/5 = 584.0161{{c}}

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 583.9166{{c}}

{{Optimal ET sequence|legend=0| 8d, …, 72, 152, 224fg, 296ffg }}

Badness (Sintel): 0.795

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 286/285, 289/288, 325/324, 400/399

Mapping: {{mapping| 8 1 3 3 16 14 21 34 | 0 3 4 5 3 4 3 0 }}

Optimal tunings:  

* CWE: ~12/11 = 150.0000{{c}}, ~7/5 = 584.0712{{c}}

{{Optimal ET sequence|legend=0| 8d, 72, 152 }}

Badness (Sintel): 0.993

Scales: [[Octoid72]], [[Octoid80]]

Hexadecoid (80 & 144) has a period of 1/16 octave and tempers out 4225/4224.

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 1375/1372, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 | 0 3 4 5 3 -1 }}

Optimal tunings:  

* CWE: ~448/429 = 75.0000{{c}}, ~7/5 = 583.9709{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224 }}

Badness (Sintel): 1.27

===== 17-limit =====

Subgroup: 2.3.5.7.11.13.17

Comma list: 540/539, 715/714, 936/935, 4000/3993, 4225/4224

Mapping: {{mapping| 16 2 6 6 32 67 81 | 0 3 4 5 3 -1 -2 }}

Optimal tunings:  

{{Optimal ET sequence|legend=0| 80, 144, 224, 528dg }}

Badness (Sintel): 1.46

===== 19-limit =====

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 400/399, 540/539, 715/714, 936/935, 1331/1330, 1445/1444

Mapping: {{mapping| 16 2 6 6 32 67 81 68 | 0 3 4 5 3 -1 -2 0 }}

Optimal tunings:  

* WE: ~117/112 = 74.9865{{c}}, ~7/5 = 583.9642{{c}}

* CWE: ~117/112 = 75.0000{{c}}, ~7/5 = 584.0803{{c}}

{{Optimal ET sequence|legend=0| 80, 144, 224, 304dh, 528dghh }}

Badness (Sintel): 1.44

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

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

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 201768035/201326592

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

: mapping generators: ~2, ~5120/3087

[[Optimal tuning]]s:  

* [[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.

Subgroup: 2.3.5.7.11

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

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

Optimal tunings:  

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

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

Badness (Sintel): 3.05

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 2200/2197, 4375/4374

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

Optimal tunings:  

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

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

Badness (Sintel): 1.85

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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

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

Optimal tunings:  

* WE: ~2 = 1199.7195{{c}}, ~108/65 = 877.0018{{c}}

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

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

Badness (Sintel): 1.35

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

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

: [[error map]]: {{val| +0.013 +0.024 -0.049 -0.019 }}

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

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

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

[[Badness]] (Sintel): 1.18

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 }}

* 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

Subgroup: 2.3.5.7.11.13

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

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

Optimal tunings:  

* 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]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 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}}

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

[[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 }}

: mapping generators: ~99/70, ~35/27

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