Ragismic microtemperaments: Difference between revisions

Microtemperaments considered below, sorted by [[badness]], are supermajor, enneadecal, semidimi, brahmagupta, abigail, gamera, crazy, orga, seniority, monzismic, semidimfourth, acrokleismic, quasithird, deca, keenanose, aluminium, quatracot, moulin, and palladium. Some near-microtemperaments are appended as octoid, parakleismic, counterkleismic, quincy, sfourth, and trideci. Discussed elsewhere are:

* ''[[Rhinoceros]]'' (+49/48) → [[Unicorn family #Rhinoceros|Unicorn family]]

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

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

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

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

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

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

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

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

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

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

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

* ''[[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. 37 of these give (2¹⁵)/3, 46 give (2¹⁹)/5, and 75 give (2³⁰)/7. This is clearly quite a complex temperament; it makes up for it, to the extent it does, with extreme accuracy: 1106 or 1277 can be used as tunings, leading to accuracy even greater than that of ennealimmal. The 80-note mos is presumably the place to start, and if that is not enough notes for you, there is always the 171-note mos.

{{Mapping|legend=1| 1 15 19 30 | 0 -37 -46 -75 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~9/7 = 435.082{{c}}

{{Optimal ET sequence|legend=1| 11, 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214 }}

Mapping: {{mapping| 2 30 38 60 41 | 0 -37 -46 -75 -47 }}

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~9/7 = 435.082{{c}}

{{Optimal ET sequence|legend=0| 80, 342, 764, 1106, 1448, 2554, 4002f, 6556cf }}

Enneadecal temperament 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 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.

: Mapping generators: ~28/27, ~3

[[Optimal tuning]] ([[CTE]]): ~28/27 = 63.1579{{c}}, ~3/2 = 701.9275{{c}} (~225/224 = 7.1907{{c}})

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

Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~3/2 = 702.1483{{c}} (~225/224 = 7.4115{{c}})

Badness (Sintel): 1.446

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

Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~3/2 = 701.9258{{c}} (~225/224 = 7.1890{{c}})

Badness (Sintel): 1.386

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

Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9351{{c}} (~225/224 = 7.1983{{c}})

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

Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9955{{c}} (~225/224 = 7.2587{{c}})

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

Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~3/2 = 701.9812{{c}} (~225/224 = 7.2444{{c}})

Badness (Sintel): 1.256

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

Optimal tuning (CTE): ~55/54 = 31.5789{{c}}, ~429/250 = 935.1789{{c}} (~144/143 = 12.1895{{c}})

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

Optimal tuning (CTE): ~28/27 = 63.1579{{c}}, ~6545/5928 = 171.244{{c}}

Badness (Sintel): 3.153

The generator of semidimi temperament 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 36 48 61 | 0 -55 -73 -93 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~35/27 = 449.1270{{c}}

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

The brahmagupta temperament has a period of 1/7 octave, tempering out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}).  

Early in the design of the [[Sagittal]] notation system, Secor and 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 ¢ many common 13-limit ratios. If the divisions were made exactly equal, this would be the specific tuning of Brahmagupta temperament 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 ¢) and 1/21-apotome generator (5.4136 ¢).

[[Comma list]]: 4375/4374, 70368744177664/70338939985125

[[Optimal tuning]] ([[POTE]]): ~1157625/1048576 = 171.429{{c}}, ~27/20 = 519.716{{c}}

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

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

Optimal tuning (POTE): ~243/220 = 171.429{{c}}, ~27/20 = 519.704{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1771ee }}

Badness (Sintel): 1.725

Mapping: {{mapping| 7 2 -8 53 3 35 | 0 3 8 -11 7 -3 }}

Optimal tuning (POTE): ~243/220 = 171.429{{c}}, ~27/20 = 519.706{{c}}

{{Optimal ET sequence|legend=0| 7, 217, 224, 441, 665, 1771eef }}

: ''For the 5-limit versino, see [[Miscellaneous 5-limit temperaments #Abigail]].''

Abigail temperament tempers out the [[pessoalisma]] in addition to the ragisma in the 7-limit. It was named by Gene Ward Smith after the birthday of First Lady Abigail Fillmore.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_17927.html#17930]: "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."</ref>

{{Mapping|legend=1| 2 7 13 -1 | 0 -11 -24 19 }}

: Mapping generators: ~46305/32768, ~27/20

[[Optimal tuning]] ([[POTE]]): ~46305/32768 = 600.000{{c}}, ~6912/6125 = 208.899{{c}}

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

Mapping: {{mapping| 2 7 13 -1 1 | 0 -11 -24 19 17 }}

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~1155/1024 = 208.901{{c}}

Mapping: {{mapping| 2 7 13 -1 1 -2 | 0 -11 -24 19 17 27 }}

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~44/39 = 208.903{{c}}

{{Mapping|legend=1| 1 6 10 3 | 0 -23 -40 -1 }}

: mapping generators: ~2, ~8/7

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~8/7 = 230.336{{c}}

Mapping: {{mapping| 2 12 20 6 5 | 0 -23 -40 -1 5 }}

: mapping generators: ~99/70, ~8/7

Optimal tuning (POTE): ~99/70 = 600.0000{{c}}, ~8/7 = 230.3370{{c}}

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

Badness (Sintel): 1.354

Mapping: {{mapping| 2 12 20 6 5 17 | 0 -23 -40 -1 5 -25 }}

Optimal tuning (POTE): ~99/70 = 600.0000{{c}}, ~8/7 = 230.3373{{c}}

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

Mapping: {{mapping| 1 6 10 3 12 | 0 -46 -80 -2 -89 }}

: mapping generators: ~2, ~77/72

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.1642{{c}}

{{Optimal ET sequence|legend=1| 73, 125, 198, 323, 521 }}

Badness (Sintel): 2.589

Mapping: {{mapping| 1 6 10 3 12 18 | 0 -46 -80 -2 -89 -149 }}

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~77/72 = 115.1628{{c}}

Badness (Sintel): 1.821

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. [[1106edo]] is a strong tuning.  

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

* [[CTE]]: ~332150625/234881024 = 600.0000{{c}}, ~1125/1024 = 162.7475{{c}}

* [[error map]]: {{val| 0.0000 +0.0253 -0.0514 -0.0133 }}

* error map: {{val| 0.0000 +0.0244 -0.0508 -0.0218 }}

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

* CTE: ~99/70 = 162.7485{{c}}, ~1125/1024 = 162.7485{{c}}

* CWE: ~99/70 = 162.7485{{c}}, ~1125/1024 = 162.7481{{c}}

[[Comma list]]: 4375/4374, 54975581388800/54936068900769

{{Mapping|legend=1| 2 21 36 5 | 0 -29 -51 1 }}

: Mapping generators: ~7411887/5242880, ~1310720/1058841

[[Optimal tuning]] ([[POTE]]): ~7411887/5242880 = 600.000{{c}}, ~8/7 = 231.104{{c}}

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

[[Badness]] (Sintel): 1.018

Mapping: {{mapping| 2 21 36 5 2 | 0 -29 -51 1 8 }}

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~8/7 = 231.103{{c}}

Mapping: {{mapping| 2 21 36 5 2 24 | 0 -29 -51 1 8 -27 }}

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~8/7 = 231.103{{c}}

Aside from the ragisma, the seniority temperament (26 & 145) tempers out the wadisma, 201768035/201326592. It is so named because the senior comma ({{monzo| -17 62 -35 }}, quadla-sepquingu) is tempered out.

{{Mapping|legend=1| 1 11 19 2 | 0 -35 -62 3 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~3087/2560 = 322.804{{c}}

{{Optimal ET sequence|legend=1| 26, 145, 171, 1513d, 1684d, 1855d, 2026d, 2197d, 2368d, 2539d, 2710d }}

[[Badness]] (Sintel): 1.136

The senator temperament (26 & 145) is an 11-limit extension of the seniority, which tempers out 441/440 and 65536/65219. It 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: {{mapping| 1 11 19 2 4 | 0 -35 -62 3 -2 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}

{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316e, 487ee }}

Badness (Sintel): 3.049

Mapping: {{mapping| 1 11 19 2 4 15 | 0 -35 -62 3 -2 -42 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}

{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }}

Badness (Sintel): 1.845

Mapping: {{mapping| 1 11 19 2 4 15 17 | 0 -35 -62 3 -2 -42 -48 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~77/64 = 322.793{{c}}

{{Optimal ET sequence|legend=0| 26, 119c, 145, 171, 316ef, 487eef }}

Badness (Sintel): 1.353

The monzismic temperament (53 & 612) 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]]. 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 2 10 -25 | 0 -2 -37 134 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| -27 11 3 1 }} = 249.0207{{c}}

{{Optimal ET sequence|legend=1| 53, …, 559, 612, 1277, 1889 }}

[[Badness]] (Sintel): 1.179

Mapping: {{mapping| 1 2 10 -25 46 | 0 -2 -37 134 -205 }}

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0193{{c}}

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

Badness (Sintel): 1.887

Mapping: {{mapping| 1 2 10 -25 46 23 | 0 -2 -37 134 -205 -93 }}

Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~231/200 = 249.0199{{c}}

Badness (Sintel): 2.222

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]]: {{mapping| 1 21 28 36 | 0 -31 -41 -53 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~35/27 = 448.456{{c}}

{{Optimal ET sequence|legend=1| 8d, 91, 99, 289, 388, 875, 1263d, 1651d }}

[[Badness]] (Sintel): 1.398

Mapping: {{mapping| 2 11 15 19 15 | 0 -31 -41 -53 -32 }}

Optimal tuning (POTE): ~99/70 = 600.000{{c}}, ~12/11 = 151.547{{c}}

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

Badness (Sintel): 1.955

Mapping: {{mapping| 2 11 15 19 15 17 | 0 -31 -41 -53 -32 -38 }}

Optimal tuning (POTE): ~99/70 = 1200.000{{c}}, ~12/11 = 151.545{{c}}

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

Badness (Sintel): 1.279

{{Mapping|legend=1| 1 10 11 27 | 0 -32 -33 -92 }}

: mapping generators: ~2, ~6/5

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 315.557{{c}}

[[Badness]] (Sintel): 1.422

Mapping: {{mapping| 1 10 11 27 -16 | 0 -32 -33 -92 74 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.558{{c}}

Badness (Sintel): 1.219

Mapping: {{mapping| 1 10 11 27 -16 25 | 0 -32 -33 -92 74 -81 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.557{{c}}

Badness (Sintel): 1.108

Mapping: {{mapping| 1 10 11 27 55 | 0 -32 -33 -92 -196 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.553{{c}}

{{Optimal ET sequence|legend=0| 19e, 251e, 270, 1061e, 1331c, 1601c, 1871bc, 4012bcde }}

Badness (Sintel): 1.407

Mapping: {{mapping| 1 10 11 27 55 25 | 0 -32 -33 -92 -196 -81 }}

Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.554{{c}}

{{Optimal ET sequence|legend=0| 19e, 251e, 270, 1331c, 1601c, 1871bcf, 2141bcf }}

Badness (Sintel): 1.076

The quasithird temperament is 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 to temper out the [[4375/4374|ragisma]] and {{monzo|-60 29 0 5}}.

 

 

 

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

 

[[Optimal tuning]] ([[POTE]]): ~51200000/43046721 = 300.000{{c}}, ~1594323/1280000 = 380.395{{c}}

 

[[Badness]] (Sintel): 2.335

[[Optimal tuning]] ([[POTE]]): ~65536/55125 = 300.000{{c}}, ~5103/4096 = 380.388 {{c}}

[[Badness]] (Sintel): 1.564

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

Optimal tuning (POTE): ~65536/51125 = 300.000{{c}}, ~5103/4096 = 380.387{{c}} (or ~22/21 = 80.387{{c}})

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

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

Optimal tuning (POTE): ~65536/51125 = 300.000{{c}}, ~81/65 = 380.385{{c}} (or ~22/21 = 80.385{{c}})

{{Optimal ET sequence|legend=0| 60d, 164, 224, 388, 612, 836, 1448f, 2284f }}

Badness (Sintel): 1.219

 

 

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

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

: mapping generators: ~15/14, ~6/5

[[Optimal tuning]] ([[POTE]]): ~15/14 = 120.000{{c}}, ~6/5 = 315.577{{c}}

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

[[Badness]] (Sintel): 2.041

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

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

Optimal tuning (POTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.582{{c}}

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

Badness (Sintel): 0.804

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 tuning (POTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.602{{c}} (~40/39 = 44.398{{c}})

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

Badness (Sintel): 0.695

=== 2.3.5.7.11.13.19 subgroup ===

Subgroup: 2.3.5.7.11.13.19

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

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

Optimal tuning (CTE): ~15/14 = 120.000{{c}}, ~6/5 = 315.581{{c}} (~39/38 = 44.419{{c}})

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

Badness (Sintel): 0.556

== Keenanose ==

Keenanose is named for the fact that it uses [[385/384]], the keenanisma, as the generator.

 

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

 

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

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~{{monzo| 21 3 1 -10 }} = 4.4465{{c}}

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

Subgroup: 2.3.5.7.11

 

 

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~385/384 = 4.4465{{c}}

 

{{Optimal ET sequence|legend=0| 270, 1349, 1619, 1889, 2159, 11065, 13224 }}

 

Badness (Sintel): 1.020

 

=== 13-limit ===

 

Comma list: 4225/4224, 4375/4374, 6656/6655, 117649/117612

 

Mapping: {{mapping| 1 2 3 3 3 3 | 0 -112 -183 -52 124 189 }}

 

Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~385/384 = 4.4466{{c}}

 

Badness (Sintel): 0.879

 

: ''For the 5-limit version, see [[13th-octave temperaments #Aluminium]].''

 

Aluminium is named after the 13th element, and tempers out the {{monzo| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.

[[Comma list]]: 4375/4374, {{monzo| 92 -39 -13 }}

[[Mapping]]: {{mapping| 13 0 92 -355 | 0 1 -3 19 }}

: Mapping generators: ~135/128, ~3

[[Optimal tuning]] ([[CTE]]): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0024{{c}}

{{Optimal ET sequence|legend=1| 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b }}

[[Badness]] (Sintel): 3.201

Comma list: 4375/4374, 234375/234256, 2097152/2096325

Mapping: {{mapping| 13 0 92 -355 148 | 0 1 -3 19 -5 }}

Optimal tuning (CTE): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0042{{c}}

{{Optimal ET sequence|legend=0| 494, 1053, 1547, 3588e, 5135e }}

Badness (Sintel): 1.393

Comma list: 4096/4095, 4375/4374, 6656/6655, 78125/78078

Mapping: {{mapping| 13 0 92 -355 148 419 | 0 1 -3 19 -5 -18 }}

Optimal tuning (CTE): ~135/128 = 92.3077{{c}}, ~3/2 = 702.0099{{c}}

{{Optimal ET sequence|legend=0| 494, 1547, 2041, 4576def }}

Badness (Sintel): 1.180

== Countritonic ==

: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Countritonic (5-limit)]].''

Countritonic (''co-un-tritonic'') can be described as the {{nowrap| 53 & 422 }} temperament, generated by an octave-reduced 91st harmonic or subharmonic in the 13-limit.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 68719476736/68356598625

{{Mapping|legend=1| 1 6 19 -33 | 0 -9 -34 73 }}

[[Optimal tuning]] (CTE): ~2 = 1200.0000{{c}}, ~45927/32768 = 588.6216{{c}}

 

 

=== 11-limit ===

 

Comma list: 4375/4374, 5632/5625, 2621440/2614689