Ragismic microtemperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

This is a collection of rank-2 temperaments tempering out the ragisma, 4375/4374 ([-1 -7 4 1⟩). The ragisma is the smallest 7-limit superparticular ratio.

Since (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 microtemperaments 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 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.

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

• Hystrix (+36/35) → Porcupine family
• Rhinoceros (+49/48) → Unicorn family
• Crepuscular (+50/49) → Fifive family
• Modus (+64/63) → Tetracot family
• Flattone (+81/80) → Meantone family
• Sensi (+126/125 or 245/243) → Sensipent family
• Catakleismic (+225/224) → Kleismic family
• Unidec (+1029/1024) → Gamelismic clan
• Quartonic (+1728/1715 or 4000/3969) → Quartonic family
• Srutal (+2048/2025) → Diaschismic family
• Ennealimmal (+2401/2400) → Septiennealimmal clan
• Maja (+2430/2401 or 3125/3087) → Maja family
• Amity (+5120/5103) → Amity family
• Pontiac (+32805/32768) → Schismatic family
• Zarvo (+33075/32768) → Gravity family
• Whirrschmidt (+393216/390625) → Würschmidt family
• Mitonic (+2100875/2097152) → Minortonic family
• Vishnu (+29360128/29296875) → Vishnuzmic family
• Vulture (+33554432/33480783) → Vulture family
• Alphatrillium (+[40 -22 -1 -1⟩) → Alphatricot family
• Vacuum (+[-68 18 17⟩) → Vavoom family
• Unlit (+[41 -20 -4⟩) → Undim family
• Chlorine (+[-52 -17 34⟩) → 17th-octave temperaments
• Quindro (+[56 -28 -5⟩) → Quindromeda family
• Dzelic (+[-223 47 -11 62⟩) → 37th-octave temperaments

Supermajor

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: [⟨1 -22 -27 -45], ⟨0 37 46 75]]

mapping generators: ~2, ~14/9

Optimal tunings:

• WE: ~2 = 1200.0067 ¢, ~14/9 = 764.9222 ¢

error map: ⟨+0.007 +0.019 -0.074 +0.037]

• CWE: ~2 = 1200.0000 ¢, ~14/9 = 764.9181 ¢

error map: ⟨0.000 +0.013 -0.083 +0.029]

Optimal ET sequence: 80, 171, 764, 935, 1106, 1277, 3660, 4937, 6214

Badness (Sintel): 0.274

Semisupermajor

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -7 -8 -15 -6], ⟨0 37 46 75 47]]

mapping generators: ~99/70, ~11/10

Optimal tunings:

• WE: ~99/70 = 600.0103 ¢, ~11/10 = 164.9205 ¢
• CWE: ~99/70 = 600.0000 ¢, ~11/10 = 164.9180 ¢

Optimal ET sequence: 80, 262d, 342, 764, 1106, 1448, 2554, 4002e, 6556cee

Badness (Sintel): 0.422

Enneadecal

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

Enneadecal tempers out the enneadeca, [-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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 703125/702464

Mapping: [⟨19 0 14 -37], ⟨0 1 1 3]]

mapping generators: ~28/27, ~3

Optimal tunings:

• WE: ~28/27 = 63.1599 ¢, ~3/2 = 701.9027 ¢ (~225/224 = 7.1437 ¢)

error map: ⟨+0.038 -0.014 -0.134 +0.080]

• CWE: ~28/27 = 63.1579 ¢, ~3/2 = 701.9002 ¢ (~225/224 = 7.1634 ¢)

error map: ⟨0.000 -0.055 -0.203 +0.033]

Optimal ET sequence: 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: [⟨19 0 14 -37 126], ⟨0 1 1 3 -2]]

Optimal tunings:

• WE: ~28/27 = 63.1431 ¢, ~3/2 = 702.1956 ¢ (~225/224 = 7.6216 ¢)
• CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.3164 ¢ (~225/224 = 7.5795 ¢)

Optimal ET sequence: 19, 133d, 152, 323e, 475de, 627de

Badness (Sintel): 1.45

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨19 0 14 -37 126 -20], ⟨0 1 1 3 -2 3]]

Optimal tunings:

• WE: ~28/27 = 63.1406 ¢, ~3/2 = 702.0192 ¢ (~225/224 = 7.4730 ¢)
• CWE: ~28/27 = 63.1579 ¢, ~3/2 = 702.1539 ¢ (~225/224 = 7.4171 ¢)

Optimal ET sequence: 19, 133df, 152f, 323ef

Badness (Sintel): 1.39

Hemienneadecal

Subgroup: 2.3.5.7.11

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

Mapping: [⟨38 0 28 -74 11], ⟨0 1 1 3 2]]

mapping generators: ~55/54, ~3

Optimal tunings:

• WE: ~55/54 = 31.5800 ¢, ~3/2 = 701.9053 ¢ (~243/242 = 7.1448 ¢)
• CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9034 ¢ (~243/242 = 7.1666 ¢)

Optimal ET sequence: 152, 342, 836, 1178, 2014, 3192ce, 5206ce

Badness (Sintel): 0.330

Hemienneadecalis

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨38 0 28 -74 11 -281], ⟨0 1 1 3 2 7]]

Optimal tunings:

• WE: ~55/54 = 31.5785 ¢, ~3/2 = 701.9995 ¢ (~243/242 = 7.2727 ¢)
• CWE: ~55/54 = 31.5789 ¢, ~3/2 = 702.0053 ¢ (~243/242 = 7.2685 ¢)

Optimal ET sequence: 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: [⟨38 0 28 -74 11 502], ⟨0 1 1 3 2 -6]]

Optimal tunings:

• WE: ~55/54 = 31.5784 ¢, ~3/2 = 701.9736 ¢ (~243/242 = 7.2493 ¢)
• CWE: ~55/54 = 31.5789 ¢, ~3/2 = 701.9855 ¢ (~243/242 = 7.2487 ¢)

Optimal ET sequence: 152, 342, 494, 1330, 1824, 2318d

Badness (Sintel): 1.26

Semihemienneadecal

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨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 ¢, ~429/250 = 935.1824 ¢ (~144/143 = 12.2152 ¢)
• CWE: ~55/54 = 31.5789 ¢, ~429/250 = 935.1617 ¢ (~144/143 = 12.2067 ¢)

Optimal ET sequence: 190, 304d, 494, 684, 1178, 2850, 4028ce

Badness (Sintel): 0.607

Kalium

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: [⟨19 3 17 -28 82 92 159 78], ⟨0 10 10 30 -6 -8 -30 1]]

Optimal tunings:

• WE: ~28/27 = 63.1582 ¢, ~6545/5928 = 171.2448 ¢
• CWE: ~28/27 = 63.1579 ¢, ~6545/5928 = 171.2439 ¢

Optimal ET sequence: 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 [-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: [⟨1 -19 -25 -32], ⟨0 55 73 93]]

mapping generators: ~2, ~35/27

Optimal tunings:

• WE: ~2 = 1200.0018 ¢, ~35/27 = 449.1277 ¢

error map: ⟨+0.002 +0.031 -0.040 -0.012]

• CWE: ~2 = 1200.0000 ¢, ~35/27 = 449.1270 ¢

error map: ⟨0.000 +0.030 -0.043 -0.015]

Optimal ET sequence: 8d, …, 171, 863, 1034, 1205, 1376, 1547, 1718, 4983, 6701, 8419

Badness (Sintel): 0.382

Brahmagupta

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

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 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 ¢).

Subgroup: 2.3.5.7

Comma list: 4375/4374, [46 -14 -3 -6⟩

Mapping: [⟨7 2 -8 53], ⟨0 3 8 -11]]

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

Optimal tunings:

• WE: ~1157625/1048576 = 171.4275 ¢, ~27/20 = 519.7125 ¢

error map: ⟨-0.007 +0.037 -0.034 -0.004]

• CWE: ~1157625/1048576 = 171.4286 ¢, ~27/20 = 519.7156 ¢

error map: ⟨0.000 +0.049 -0.018 +0.017]

Optimal ET sequence: 7, …, 217, 224, 441, 1106, 1547

Badness (Sintel): 0.737

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨7 2 -8 53 3], ⟨0 3 8 -11 7]]

Optimal tunings:

• WE: ~243/220 = 171.4208 ¢, ~27/20 = 519.6807 ¢
• CWE: ~243/220 = 171.4286 ¢, ~27/20 = 519.7034 ¢

Optimal ET sequence: 7, 217, 224, 441, 665

Badness (Sintel): 1.73

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨7 2 -8 53 3 35], ⟨0 3 8 -11 7 -3]]

Optimal tunings:

• WE: ~243/220 = 171.4197 ¢, ~27/20 = 519.6789 ¢
• CWE: ~243/220 = 171.4286 ¢, ~27/20 = 519.7052 ¢

Optimal ET sequence: 7, 217, 224, 441, 665, 1106e

Badness (Sintel): 0.956

Abigail

For the 5-limit version, see Miscellaneous 5-limit temperaments #Abigail.

Abigail tempers out the pessoalisma in addition to the ragisma in the 7-limit, and may be described as the 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.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2147483648/2144153025

Mapping: [⟨2 -4 -11 18], ⟨0 11 24 -19]]

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

Optimal tunings:

• WE: ~46305/32768 = 599.9699 ¢, ~1536/1225 = 391.0818 ¢

error map: ⟨-0.060 +0.065 -0.021 +0.079]

• CWE: ~46305/32768 = 600.0000 ¢, ~1536/1225 = 391.1007 ¢

error map: ⟨0.000 +0.152 +0.102 +0.262]

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764, 1034, 1798, 6428bcdd, 8226bbcddd

Badness (Sintel): 0.936

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 131072/130977

Mapping: [⟨2 -4 -11 18 18], ⟨0 11 24 -19 -17]]

Optimal tunings:

• WE: ~99/70 = 599.9782 ¢, ~1536/1225 = 391.0852 ¢
• CWE: ~99/70 = 600.0000 ¢, ~1536/1225 = 391.0992 ¢

Optimal ET sequence: 46, 132, 178, 224, 270, 494, 764

Badness (Sintel): 0.425

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 3025/3024, 4096/4095

Mapping: [⟨2 -4 -11 18 18 25], ⟨0 11 24 -19 -17 -27]]

Optimal tunings:

• WE: ~99/70 = 599.9862 ¢, ~351/280 = 391.0879 ¢
• CWE: ~99/70 = 600.0000 ¢, ~351/280 = 391.0969 ¢

Optimal ET sequence: 46, 178, 224, 270, 494, 764, 1258

Badness (Sintel): 0.366

Gamera

For the 5-limit version, see Miscellaneous 5-limit temperaments #Gamera.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 589824/588245

Mapping: [⟨1 -17 -30 2], ⟨0 23 40 1]]

mapping generators: ~2, ~7/4

Optimal tunings:

• WE: ~2 = 1199.8483 ¢, ~7/4 = 969.5415 ¢
• CWE: ~2 = 1200.0000 ¢, ~7/4 = 969.6608 ¢

Optimal ET sequence: 26, 73, 99, 224, 323, 422, 745d

Badness (Sintel): 0.953

Hemigamera

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 589824/588245

Mapping: [⟨2 -11 -20 5 10], ⟨0 23 40 1 -5]]

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

Optimal tunings:

• WE: ~99/70 = 599.9323 ¢, ~99/80 = 369.6212 ¢
• CWE: ~99/70 = 600.0000 ¢, ~99/80 = 369.6610 ¢

Optimal ET sequence: 26, 172c, 198, 224, 422, 646, 1068d

Badness (Sintel): 1.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 2080/2079, 2200/2197, 3025/3024

Mapping: [⟨2 -11 -20 5 10 -8], ⟨0 23 40 1 -5 25]]

Optimal tunings:

• WE: ~99/70 = 599.9207 ¢, ~26/21 = 369.6139 ¢
• CWE: ~99/70 = 600.0000 ¢, ~26/21 = 369.6603 ¢

Optimal ET sequence: 26, 172cf, 198, 224, 422, 646f, 1068df

Badness (Sintel): 0.844

Semigamera

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -40 -70 1 -77], ⟨0 46 80 2 89]]

mapping generators: ~2, ~144/77

Optimal tunings:

• WE: ~2 = 1199.8845 ¢, ~144/77 = 1084.7314 ¢
• CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.8345 ¢

Optimal ET sequence: 73, 125, 198, 323, 521

Badness (Sintel): 2.59

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -40 -70 1 -77 -131], ⟨0 46 80 2 89 149]]

Optimal tunings:

• WE: ~2 = 1199.8726 ¢, ~144/77 = 1084.7220 ¢
• CWE: ~2 = 1200.0000 ¢, ~144/77 = 1084.8359 ¢

Optimal ET sequence: 73f, 125f, 198, 323, 521

Badness (Sintel): 1.82

Crazy

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 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, [-53 10 16⟩

Mapping: [⟨2 1 6 -15], ⟨0 8 -5 76]]

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

Optimal tunings:

• WE: ~332150625/234881024 = 600.0019 ¢, ~1125/1024 = 162.7479 ¢

error map: ⟨+0.004 +0.030 -0.042 -0.014]

• CWE: ~332150625/234881024 = 600.0000 ¢, ~1125/1024 = 162.7474 ¢

error map: ⟨0.000 +0.024 -0.051 -0.022]

Optimal ET sequence: 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: [⟨2 1 6 -15 -8], ⟨0 8 -5 76 55]]

Optimal tunings:

• WE: ~99/70 = 600.0047 ¢, ~1125/1024 = 162.7493 ¢
• CWE: ~99/70 = 600.0000 ¢, ~1125/1024 = 162.7481 ¢

Optimal ET sequence: 118, 376, 494, 612, 1106, 2824, 3930e

Badness (Sintel): 0.562

Orga

Orga may be described as the 26 & 270 temperament, and 1106edo gives a strong tuning.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [41 -4 2 -14⟩

Mapping: [⟨2 -8 -15 6], ⟨0 29 51 -1]]

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

Optimal tunings:

• WE: ~7411887/5242880 = 599.9927 ¢, ~8/7 = 231.1012 ¢

error map: ⟨-0.015 +0.037 -0.045 +0.029]

• CWE: ~7411887/5242880 = 600.0000 ¢, ~8/7 = 231.1037 ¢

error map: ⟨0.000 +0.053 -0.023 +0.070]

Optimal ET sequence: 26, …, 244, 270, 836, 1106, 1376, 2482

Badness (Sintel): 1.02

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -8 -15 6 10], ⟨0 29 51 -1 -8]]

Optimal tunings:

• WE: ~99/70 = 600.0025 ¢, ~8/7 = 231.1039 ¢
• CWE: ~99/70 = 600.0000 ¢, ~8/7 = 231.1030 ¢

Optimal ET sequence: 26, 244, 270, 566, 836, 1106

Badness (Sintel): 0.535

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -8 -15 6 10 -3], ⟨0 29 51 -1 -8 27]]

Optimal tunings:

• WE: ~99/70 = 600.0192 ¢, ~8/7 = 231.1102 ¢
• CWE: ~99/70 = 600.0000 ¢, ~8/7 = 231.1033 ¢

Optimal ET sequence: 26, 244, 270, 566, 836f, 1106f

Badness (Sintel): 0.899

Seniority

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 26 & 145. It is so named because the senior comma ([-17 62 -35⟩) is tempered out.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 201768035/201326592

Mapping: [⟨1 -24 -43 5], ⟨0 35 62 -3]]

mapping generators: ~2, ~5120/3087

Optimal tunings:

• WE: ~2 = 1200.0745 ¢, ~5120/3087 = 877.2500 ¢

error map: ⟨+0.075 +0.008 -0.016 -0.203]

• CWE: ~2 = 1200.0000 ¢, ~5120/3087 = 877.1965 ¢

error map: ⟨0.000 -0.077 -0.130 -0.415]

Optimal ET sequence: 26, 119c, 145, 171, 1513d, 1684d, …, 2539d, 2710d

Badness (Sintel): 1.14

Senator

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: [⟨1 -24 -43 5 2], ⟨0 35 62 -3 2]]

Optimal tunings:

• WE: ~2 = 1199.7665 ¢, ~128/77 = 877.0367 ¢
• CWE: ~2 = 1200.0000 ¢, ~128/77 = 877.2051 ¢

Optimal ET sequence: 26, 119c, 145, 171, 316e

Badness (Sintel): 3.05

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -24 -43 5 2 -27], ⟨0 35 62 -3 2 42]]

Optimal tunings:

• WE: ~2 = 1199.7136 ¢, ~108/65 = 877.9974 ¢
• CWE: ~2 = 1200.0000 ¢, ~108/65 = 877.2038 ¢

Optimal ET sequence: 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: [⟨1 -24 -43 5 2 -27 -31], ⟨0 35 62 -3 2 42 48]]

Optimal tunings:

• WE: ~2 = 1199.7195 ¢, ~108/65 = 877.0018 ¢
• CWE: ~2 = 1200.0000 ¢, ~108/65 = 877.2039 ¢

Optimal ET sequence: 26, 119cfg, 145, 171, 316ef

Badness (Sintel): 1.35

Monzismic

For the 5-limit version, see Very high accuracy temperaments #Monzismic.

Monzismic tempers out the monzisma, [54 -37 2⟩, and in the 7-limit, the nanisma, [109 -67 0 -1⟩, as well as the ragisma, 4375/4374. It may be described as the 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, [-55 30 2 1⟩

Mapping: [⟨1 0 -27 109], ⟨0 2 37 -134]]

mapping generators: ~2, ~[28 -11 -3 -1⟩

Optimal tunings:

• WE: ~2 = 1200.0128 ¢, ~[28 -11 -3 -1⟩ = 950.9895 ¢

error map: ⟨+0.013 +0.024 -0.049 -0.019]

• CWE: ~2 = 1200.0000 ¢, ~[28 -11 -3 -1⟩ = 950.9793 ¢

error map: ⟨0.000 +0.004 -0.080 -0.050]

Optimal ET sequence: 53, …, 559, 612, 1277, 1889, 10722c, 12611cd, 14500cd, 16389ccd

Badness (Sintel): 1.18

Monzism

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 0 -27 109 -159], ⟨0 2 37 -134 205]]

Optimal tunings:

• WE: ~2 = 1200.0347 ¢, ~400/231 = 951.0082 ¢
• CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.9807 ¢

Optimal ET sequence: 53, 559, 612, 3619de, 4231de, …, 6067ddee

Badness (Sintel): 1.89

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 0 -27 109 -159 -70], ⟨0 2 37 -134 205 93]]

Optimal tunings:

• WE: ~2 = 1200.0036 ¢, ~400/231 = 950.9829 ¢
• CWE: ~2 = 1200.0000 ¢, ~400/231 = 950.9801 ¢

Optimal ET sequence: 53, 559, 612

Badness (Sintel): 2.22

Semidimfourth

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: [⟨1 -10 -13 -17], ⟨0 31 41 53]]

mapping generators: ~2, ~35/27

Optimal tunings:

• WE: ~2 = 1199.9936 ¢, ~35/27 = 448.4533 ¢

error map: ⟨-0.007 +0.160 +0.353 -0.694]

• CWE: ~2 = 1200.0000 ¢, ~35/27 = 448.4555 ¢

error map: ⟨0.000 +0.165 +0.361 -0.685]

Optimal ET sequence: 8d, …, 91, 99, 289, 388, 875

Badness (Sintel): 1.40

Neusec

Subgroup: 2.3.5.7.11

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

Mapping: [⟨2 -20 -26 -34 -17], ⟨0 31 41 53 32]]

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

Optimal tunings:

• WE: ~99/70 = 600.0381 ¢, ~35/27 = 448.4812 ¢
• CWE: ~99/70 = 600.0000 ¢, ~35/27 = 448.4546 ¢

Optimal ET sequence: 8d, …, 190, 388

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨2 -20 -26 -34 -17 -21], ⟨0 31 41 53 32 38]]

Optimal tunings:

• WE: ~99/70 = 600.0034 ¢, ~35/27 = 448.4573 ¢
• CWE: ~99/70 = 600.0000 ¢, ~35/27 = 448.4549 ¢

Optimal ET sequence: 8d, …, 190, 198, 388

Badness (Sintel): 1.28

Acrokleismic

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2202927104/2197265625

Mapping: [⟨1 -22 -22 -65], ⟨0 32 33 92]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1199.9305 ¢, ~5/3 = 884.3923 ¢

error map: ⟨-0.070 +0.126 +0.160 -0.221]

• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4423 ¢

error map: ⟨0.000 +0.198 +0.282 -0.136]

Optimal ET sequence: 19, …, 251, 270, 2449c, 2719c, 2989bc

Badness (Sintel): 1.42

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -22 -22 -65 58], ⟨0 32 33 92 -74]]

Optimal tunings:

• WE: ~2 = 1199.9698 ¢, ~5/3 = 884.4193 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4414 ¢

Optimal ET sequence: 19, 251, 270, 829, 1099, 1369, 1639

Badness (Sintel): 1.22

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -22 -22 -65 58 -56], ⟨0 32 33 92 -74 81]]

Optimal tunings:

• WE: ~2 = 1199.9939 ¢, ~5/3 = 884.4384 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4429 ¢

Optimal ET sequence: 19, 251, 270

Badness (Sintel): 1.11

Counteracro

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -22 -22 -65 -141], ⟨0 32 33 92 196]]

Optimal tunings:

• WE: ~2 = 1199.8877 ¢, ~5/3 = 884.3639 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4457 ¢

Optimal ET sequence: 19e, …, 251e, 270, 1061e, 1331c, 1601c, 1871bc

Badness (Sintel): 1.41

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -22 -22 -65 -141 -56], ⟨0 32 33 92 196 81]]

Optimal tunings:

• WE: ~2 = 1199.9285 ¢, ~5/3 = 884.3937 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.4458 ¢

Optimal ET sequence: 19e, …, 251e, 270, 1331c

Badness (Sintel): 1.08

Quasithird

For the 5-limit version, see Miscellaneous 5-limit temperaments #Quasithird.

Quasithird may be described as the 224 & 388 temperament, featured by a major third interval which is 1600000/1594323 (amity comma) or 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 [-60 29 0 5⟩. Its ploidacot is tetraploid delta-pentacot.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-60 29 0 5⟩

Mapping: [⟨4 0 -11 48], ⟨0 5 16 -29]]

mapping generators: ~65536/55125, ~5103/4096

Optimal tunings:

• WE: ~65536/55125 = 300.0052 ¢, ~5103/4096 = 380.3949 ¢

error map: ⟨+0.021 +0.020 -0.052 -0.031]

• CWE: ~65536/55125 = 300.0000 ¢, ~5103/4096 = 380.3884 ¢

error map: ⟨0.000 -0.013 -0.100 -0.089]

Optimal ET sequence: 60d, 164, 224, 388, 612, 1448, 2060

Badness (Sintel): 1.56

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 4296700485/4294967296

Mapping: [⟨4 0 -11 48 43], ⟨0 5 16 -29 -23]]

Optimal tunings:

• WE: ~65536/51125 = 300.0073 ¢, ~5103/4096 = 380.3963 ¢ (or ~22/21 = 80.3890 ¢)
• CWE: ~65536/51125 = 300.0000 ¢, ~5103/4096 = 380.3868 ¢ (or ~22/21 = 80.3868 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836, 1448, 6404cee, 7852cee

Badness (Sintel): 0.698

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2200/2197, 3025/3024, 4096/4095, 4375/4374

Mapping: [⟨4 0 -11 48 43 11], ⟨0 5 16 -29 -23 3]]

Optimal tunings:

• WE: ~65536/51125 = 299.9985 ¢, ~81/65 = 380.3833 ¢ (or ~22/21 = 80.3848 ¢)
• CWE: ~65536/51125 = 300.0000 ¢, ~81/65 = 380.3852 ¢ (or ~22/21 = 80.3852 ¢)

Optimal ET sequence: 60d, 164, 224, 388, 612, 836

Badness (Sintel): 1.22

Deca

For 5-limit version, see 10th-octave temperaments #Neon.

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

Subgroup: 2.3.5.7

Comma list: 4375/4374, 165288374272/164794921875

Mapping: [⟨10 4 9 2], ⟨0 5 6 11]]

mapping generators: ~15/14, ~460992/390625

Optimal tunings:

• WE: ~15/14 = 119.9966 ¢, ~460992/390625 = 284.4150 ¢ (5625/5488 = 44.4219 ¢)

error map: ⟨-0.034 +0.106 +0.145 -0.268]

• CWE: ~15/14 = 120.0000 ¢, ~460992/390625 = 284.4182 ¢ (5625/5488 = 44.4182 ¢)

error map: ⟨0.000 +0.136 +0.195 -0.226]

Optimal ET sequence: 80, 190, 270, 1270, 1540, 1810, 2080

Badness (Sintel): 2.04

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨10 4 9 2 18], ⟨0 5 6 11 7]]

Optimal tunings:

• WE: ~15/14 = 120.0004 ¢, ~33/28 = 284.4193 ¢ (77/75 = 44.4185 ¢)
• CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4189 ¢ (77/75 = 44.4189 ¢)

Optimal ET sequence: 80, 190, 270, 1000, 1270, 1540e, 1810e

Badness (Sintel): 0.804

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨10 4 9 2 18 37], ⟨0 5 6 11 7 0]]

Optimal tunings:

• WE: ~15/14 = 120.0067 ¢, ~33/28 = 284.4139 ¢ (~40/39 = 44.4006 ¢)
• CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4048 ¢ (~40/39 = 44.4048 ¢)

Optimal ET sequence: 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, 1521/1520, 3025/3024, 4225/4224, 4375/4374

Mapping: [⟨10 4 9 2 18 37 33], ⟨0 5 6 11 7 0 4]]

Optimal tunings:

• WE: ~15/14 = 120.0045 ¢, ~33/28 = 284.4140 ¢ (~39/38 = 44.4050 ¢)
• CWE: ~15/14 = 120.0000 ¢, ~33/28 = 284.4075 ¢ (~39/38 = 44.4075 ¢)

Optimal ET sequence: 80, 190, 270, 730, 1000

Badness (Sintel): 0.556

Keenanose

Keenanose, the 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, [-56 1 -8 26⟩

Mapping: [⟨1 2 3 3], ⟨0 -112 -183 -52]]

mapping generators: ~2, ~[21 3 1 -10⟩

Optimal tunings:

• WE: ~2 = 1200.0068 ¢, ~[21 3 1 -10⟩ = 4.4467 ¢

error map: ⟨+0.007 +0.031 -0.035 -0.032]

• CWE: ~2 = 1200.0000 ¢, ~[21 3 1 -10⟩ = 4.4466 ¢

error map: ⟨0.000 +0.025 -0.043 -0.050]

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 2159, 4048, 18081cd

Badness (Sintel): 2.17

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 3], ⟨0 -112 -183 -52 124]]

Optimal tunings:

• WE: ~2 = 1199.9970 ¢, ~385/384 = 4.4465 ¢
• CWE: ~2 = 1200.0000 ¢, ~385/384 = 4.4465 ¢

Optimal ET sequence: 270, 1349, 1619, 1889, 2159, 11065, 13224

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 3 3], ⟨0 -112 -183 -52 124 189]]

Optimal tunings:

• WE: ~2 = 1200.0065 ¢, ~385/384 = 4.4467 ¢
• CWE: ~2 = 1200.0000 ¢, ~385/384 = 4.4467 ¢

Optimal ET sequence: 270, 1079, 1349, 1619, 1889, 4048

Badness (Sintel): 0.879

Aluminium

For the 5-limit version, see 13th-octave temperaments #Aluminium.

Aluminium was named by Eliora in 2023 after the 13th element. It tempers out the [92 -39 -13⟩ comma, which sets 135/128 interval to be equal to 1/13th of the octave.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [92 -39 -13⟩

Mapping: [⟨13 0 92 -355], ⟨0 1 -3 19]]

Mapping generators: ~135/128, ~3

Optimal tunings:

• WE: ~135/128 = 92.3072 ¢, ~3/2 = 701.9995 ¢

error map: ⟨-0.006 +0.038 -0.030 -0.013]

• CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0030 ¢

error map: ⟨0.000 +0.048 -0.015 +0.001]

Optimal ET sequence: 494, 1053, 1547, 8788, 10335, 11882, 13429b, 14976b

Badness (Sintel): 3.20

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨13 0 92 -355 148], ⟨0 1 -3 19 -5]]

Optimal tunings:

• WE: ~135/128 = 92.3062 ¢, ~3/2 = 701.9946 ¢
• CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0056 ¢

Optimal ET sequence: 494, 1053, 1547, 3588e, 5135e

Badness (Sintel): 1.39

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨13 0 92 -355 148 419], ⟨0 1 -3 19 -5 -18]]

Optimal tunings:

• WE: ~135/128 = 92.3055 ¢, ~3/2 = 701.9928 ¢
• CWE: ~135/128 = 92.3077 ¢, ~3/2 = 702.0098 ¢

Optimal ET sequence: 494, 1547, 2041, 4576def

Badness (Sintel): 1.18

Ragitritonic

For the 5-limit version, see Schismic–Mercator equivalence continuum #Countritonic.

Ragitritonic may be described as the 53 & 369 temperament, splitting the 24th harmonic into nine tritone generators; its ploidacot is thus delta-enneacot. 422edo makes for a strong tuning.

Ragitritonic was named by Flora Canou in 2026 as a contraction of ragismic and tritonic.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 68719476736/68356598625

Mapping: [⟨1 -3 -15 40], ⟨0 9 34 -73]]

mapping generators: ~2, ~65536/45927

Optimal tunings:

• WE: ~2 = 1199.8189 ¢, ~65536/45927 = 611.2850 ¢

error map: ⟨-0.181 +0.153 +0.094 +0.123]

• CWE: ~2 = 1200.0000 ¢, ~65536/45927 = 611.3775 ¢

error map: ⟨0.000 +0.443 +0.522 +0.615]

Optimal ET sequence: 53, 210d, 263, 316, 369, 422, 791, 1213cd, 2004bcdd

Badness (Sintel): 3.37

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -3 -15 40 -75], ⟨0 9 34 -73 154]]

Optimal tunings:

• WE: ~2 = 1199.8147 ¢, ~768/539 = 611.2822 ¢
• CWE: ~2 = 1200.0000 ¢, ~768/539 = 611.3762 ¢

Optimal ET sequence: 53, 316e, 369, 422, 791e, 1213cde

Badness (Sintel): 2.34

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 2200/2197, 4375/4374, 5632/5625

Mapping: [⟨1 -3 -15 40 -75 -34], ⟨0 9 34 -73 154 74]]

Optimal tunings:

• WE: ~2 = 1199.7916 ¢, ~91/64 = 611.2698 ¢
• CWE: ~2 = 1200.0000 ¢, ~91/64 = 611.3754 ¢

Optimal ET sequence: 53, 316ef, 369f, 422, 1213cdeff, 1635bcdefff

Badness (Sintel): 1.51

Quatracot

See also: Stratosphere

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-32 5 14 -3⟩

Mapping: [⟨2 -6 -1 -36], ⟨0 13 8 59]]

mapping generators: ~2278125/1605632, ~7168/5625

Optimal tunings:

• WE: ~2278125/1605632 = 600.0888 ¢, ~7168/5625 = 423.2574 ¢

error map: ⟨+0.178 -0.141 -0.343 +0.165]

• CWE: ~2278125/1605632 = 600.0000 ¢, ~7168/5625 = 423.1986 ¢

error map: ⟨0.000 -0.374 -0.725 -0.111]

Optimal ET sequence: 34d, 156d, 190, 224, 414, 638, 1052c, 1690bcc

Badness (Sintel): 4.45

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 1265625/1261568

Mapping: [⟨2 -6 -1 -36 -22], ⟨0 13 8 59 41]]

Optimal tunings:

• WE: ~99/70 = 600.0847 ¢, ~225/176 = 423.2536 ¢
• CWE: ~99/70 = 600.0000 ¢, ~225/176 = 423.1977 ¢

Optimal ET sequence: 34d, 156de, 190, 224, 414, 638, 1052c

Badness (Sintel): 1.36

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1575/1573, 2200/2197

Mapping: [⟨2 -6 -1 -36 -22 -6], ⟨0 13 8 59 41 19]]

Optimal tunings:

• WE: ~99/70 = 600.0571 ¢, ~143/112 = 423.2366 ¢
• CWE: ~99/70 = 600.0000 ¢, ~143/112 = 423.1987 ¢

Optimal ET sequence: 34d, 156de, 190, 224, 414, 638

Badness (Sintel): 0.936

Moulin

Moulin can be described as the 494 & 1619 temperament. It has a generator of ~22/13, and it was named by Eliora in 2022 after the Law & Order: Special Victims Unit episode Season 22, Episode 13. "Trick-Rolled At The Moulin". However, the functional generator is ~13/11, and 73 of them octave reduced reach the perfect fifth. Since 11/8 is within 23 generators, the 25-tone generator chain (4L 21s) of this temperament contains the 8:11:13 triad.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-88 2 45 -7⟩

Mapping: [⟨1 -16 -9 -75], ⟨0 73 47 323]]

mapping generators: ~2, ~3796875/3211264

Optimal tunings:

• WE: ~2 = 1200.0272 ¢, ~3796875/3211264 = 289.0675 ¢

error map: ⟨+0.027 +0.007 -0.084 +0.013]

• CWE: ~2 = 1200.0000 ¢, ~3796875/3211264 = 289.0675 ¢

error map: ⟨0.000 -0.029 -0.142 -0.029]

Optimal ET sequence: 494, 1125, 1619, 8589cc, 10208cc

Badness (Sintel): 5.93

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 759375/758912, 100663296/100656875

Mapping: [⟨1 -16 -9 -75 9], ⟨0 73 47 323 -23]]

Optimal tunings:

• WE: ~2 = 1200.0043 ¢, ~605/512 = 289.0687 ¢
• CWE: ~2 = 1200.0000 ¢, ~605/512 = 289.0677 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 2.24

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 4225/4224, 4375/4374, 6656/6655, 78125/78078

Mapping: [⟨1 -16 -9 -75 9 9], ⟨0 73 47 323 -23 -22]]

Optimal tunings:

• WE: ~2 = 1200.0043 ¢, ~13/11 = 289.0687 ¢
• CWE: ~2 = 1200.0000 ¢, ~13/11 = 289.0677 ¢

Optimal ET sequence: 494, 1125, 1619, 2113

Badness (Sintel): 1.12

Palladium

For the 5-limit version, see 46th-octave temperaments #Palladium.

The name of the palladium temperament comes from palladium, the 46th element. Palladium has a period of 1/46 octave. It tempers out the 46-9/5-comma, [-39 92 -46⟩, by which 46 minor whole tones (10/9) fall short of seven octaves. This temperament can be described as 46 & 414 temperament, which tempers out [-51 8 2 12⟩ as well as the ragisma.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-51 8 2 12⟩

Mapping: [⟨46 0 -39 202], ⟨0 1 2 -1]]

mapping generators: ~83349/81920, ~3

Optimal tunings:

• WE: ~83349/81920 = 26.0910 ¢, ~3/2 = 701.7155 ¢

error map: ⟨+0.185 -0.055 -0.061 +0.349]

• CWE: ~83349/81920 = 26.0870 ¢, ~3/2 = 701.6491 ¢

error map: ⟨0.000 -0.306 -0.407 -0.910]

Optimal ET sequence: 46, …, 368, 414, 460, 874d

Badness (Sintel): 7.81

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 134775333/134217728

Mapping: [⟨46 0 -39 202 232], ⟨0 1 2 -1 -1]]

Optimal tunings:

• WE: ~8192/8085 = 26.0912 ¢, ~3/2 = 701.7082 ¢
• CWE: ~8192/8085 = 26.0870 ¢, ~3/2 = 701.6173 ¢

Optimal ET sequence: 46, …, 368, 414, 460, 874de

Badness (Sintel): 2.44

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 3025/3024, 4225/4224, 4375/4374, 26411/26364

Mapping: [⟨46 0 -39 202 232 316], ⟨0 1 2 -1 -1 -2]]

Optimal tunings:

• WE: ~65/64 = 26.0906 ¢, ~3/2 = 701.7411 ¢
• CWE: ~65/64 = 26.0870 ¢, ~3/2 = 701.6465 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334dde

Badness (Sintel): 1.68

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 833/832, 1089/1088, 1225/1224, 1701/1700, 4225/4224

Mapping: [⟨46 0 -39 202 232 316 188], ⟨0 1 2 -1 -1 -2 0]]

Optimal tunings:

• WE: ~65/64 = 26.0906 ¢, ~3/2 = 701.7399 ¢
• CWE: ~65/64 = 26.0870 ¢, ~3/2 = 701.6464 ¢

Optimal ET sequence: 46, 368, 414, 460, 874de, 1334ddeg

Badness (Sintel): 1.14

Counterorson

Counterorson tempers out the [147 -103 7⟩ comma in the 5-limit. It uses a generator that reaches the 3rd harmonic in 7 steps, but unlike the semicomma family, 5th harmonic is 103 generators up and not 3 generators down. The two mappings converge on 53edo.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [154 -54 -21 -7⟩

Mapping: [⟨1 0 -21 85], ⟨0 7 103 -363]]

mapping generators: ~2, ~[66 -23 -9 -3⟩

Optimal tunings:

• WE: ~2 = 1200.0040 ¢, ~[66 -23 -9 -3⟩ = 271.7122 ¢

error map: ⟨+0.004 -0.303 -0.041 -0.015]

• CWE: ~2 = 1200.0000 ¢, ~[66 -23 -9 -3⟩ = 271.7113 ¢

error map: ⟨0.000 +0.024 -0.051 -0.025]

Optimal ET sequence: 53, …, 1612, 1665, 1718

Badness (Sintel): 7.92

Oviminor

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Oviminor (5-limit).

Oviminor was named by Eliora in 2022 after the facts that it takes 184 minor thirds of 6/5 to reach the interval class of 4/3, the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past egads, though it is less accurate.

Subgroup: 2.3.5.7

Comma list: 4375/4374, [-100 53 48 -34⟩

Mapping: [⟨1 -134 -134 -401], ⟨0 184 185 548]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1200.0193 ¢, ~5/3 = 884.2638 ¢

error map: ⟨+0.019 +0.010 -0.085 +0.032]

• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.2497 ¢

error map: ⟨0.000 -0.011 -0.120 +0.008]

Optimal ET sequence: 19, …, 1600, 1619, 4838, 6457c

Badness (Sintel): 14.7

Octoid

Main article: Octoid

For the 5-limit version, see 8th-octave temperaments #Octoid.

The octoid temperament has a period of 1/8 octave and tempers out 4375/4374 (ragisma) and 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: [⟨8 1 3 3], ⟨0 3 4 5]]

mapping generators: ~49/45, ~7/5

Optimal tunings:

• WE: ~49/45 = 150.0003 ¢, ~7/5 = 583.9416 ¢

error map: ⟨+0.002 -0.130 -0.547 +0.883]

• CWE: ~49/45 = 150.0000 ¢, ~7/5 = 583.9411 ¢

error map: ⟨0.000 -0.132 -0.549 +0.880]

Tuning ranges:

• 7-odd-limit diamond monotone: ~7/5 = [578.571, 600.000] (27\56 to 4\8)
• 9-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64 to 43\88)
• 7-odd-limit diamond tradeoff: ~7/5 = [582.512, 584.359]
• 9-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨8 1 3 3 16], ⟨0 3 4 5 3]]

Optimal tunings:

• WE: ~12/11 = 149.9932 ¢, ~7/5 = 583.9356 ¢
• CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9477 ¢

Tuning ranges:

• 11-odd-limit diamond monotone: ~7/5 = [581.250, 586.364] (31\64, 43\88)
• 11-odd-limit diamond tradeoff: ~7/5 = [582.512, 585.084]

Optimal ET sequence: 8d, …, 72, 152, 224, 824d

Badness (Sintel): 0.466

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨8 1 3 3 16 -21], ⟨0 3 4 5 3 13]]

Optimal tunings:

• WE: ~12/11 = 150.0005 ¢, ~7/5 = 583.9066 ¢
• CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9052 ¢

Optimal ET sequence: 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: [⟨8 1 3 3 16 -21 -14], ⟨0 3 4 5 3 13 12]]

Optimal tunings:

• WE: ~12/11 = 150.0064 ¢, ~7/5 = 583.8666 ¢
• CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.8489 ¢

Optimal ET sequence: 72, 152fg, 224, 296, 520g

Badness (Sintel): 0.729

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨8 1 3 3 16 -21 -14 34], ⟨0 3 4 5 3 13 12 0]]

Optimal tunings:

• WE: ~12/11 = 149.9785 ¢, ~7/5 = 583.8482 ¢
• CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9138 ¢

Optimal ET sequence: 72, 152fg, 224

Badness (Sintel): 0.975

Octopus

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 ¢.

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨8 1 3 3 16 14], ⟨0 3 4 5 3 4]]

Optimal tunings:

• WE: ~12/11 = 150.0313 ¢, ~7/5 = 584.0134 ¢
• CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9583 ¢

Optimal ET sequence: 8d, …, 72, 152, 224f

Badness (Sintel): 0.896

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨8 1 3 3 16 14 21], ⟨0 3 4 5 3 4 3]]

Optimal tunings:

• WE: ~12/11 = 150.0528 ¢, ~7/5 = 584.0161 ¢
• CWE: ~12/11 = 150.0000 ¢, ~7/5 = 583.9166 ¢

Optimal ET sequence: 8d, …, 72, 152, 224fg, 296ffg

Badness (Sintel): 0.795

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨8 1 3 3 16 14 21 34], ⟨0 3 4 5 3 4 3 0]]

Optimal tunings:

• WE: ~12/11 = 150.0049 ¢, ~7/5 = 584.0833 ¢
• CWE: ~12/11 = 150.0000 ¢, ~7/5 = 584.0712 ¢

Optimal ET sequence: 8d, 72, 152

Badness (Sintel): 0.993

Scales: Octoid72, Octoid80

Hexadecoid

See also: 16th-octave temperaments

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: [⟨16 2 6 6 32 67], ⟨0 3 4 5 3 -1]]

mapping generators: ~448/429, ~7/5

Optimal tunings:

• WE: ~448/429 = 74.9943 ¢, ~7/5 = 583.9408 ¢
• CWE: ~448/429 = 75.0000 ¢, ~7/5 = 583.9709 ¢

Optimal ET sequence: 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: [⟨16 2 6 6 32 67 81], ⟨0 3 4 5 3 -1 -2]]

Optimal tunings:

• WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9626 ¢
• CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0463 ¢

Optimal ET sequence: 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: [⟨16 2 6 6 32 67 81 68], ⟨0 3 4 5 3 -1 -2 0]]

Optimal tunings:

• WE: ~117/112 = 74.9865 ¢, ~7/5 = 583.9642 ¢
• CWE: ~117/112 = 75.0000 ¢, ~7/5 = 584.0803 ¢

Optimal ET sequence: 80, 144, 224, 304dh, 528dghh

Badness (Sintel): 1.44

Parakleismic

Main article: Parakleismic

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Parakleismic (5-limit).

In the 5-limit, parakleismic is an undoubted microtemperament, tempering out the parakleisma, [8 14 -13⟩, with the 118edo tuning giving errors well under a cent. It has a generator a very slightly (half a cent or less) flat 6/5, 13 of which give 32/3, and 14 give 64/5. While 118 no longer has better than a cent of accuracy in the 7-limit, it is a decent temperament there nonetheless, and this allows an extension adding 3136/3125 and 4375/4374, for which 99edo, 118edo, and especially 217edo are accurate tunings.

Parakleismic does not extend easily to the 11- or 13-limit. Possible 11-limit extensions include undecimal parakleismic (99 & 118), paralytic (99e & 118), parkleismic (80 & 99), and paradigmic (80 & 99e).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23], ⟨0 13 14 35]]

mapping generators: ~2, ~5/3

Optimal tunings:

• WE: ~2 = 1199.7820 ¢, ~5/3 = 884.6581 ¢

error map: ⟨-0.218 +0.344 +0.644 -0.779]

• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8088 ¢

error map: ⟨0.000 +0.560 +1.010 -0.516]

Optimal ET sequence: 19, 61d, 80, 99, 217, 316, 415

Badness (Sintel): 0.694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 30], ⟨0 13 14 35 -36]]

Optimal tunings:

• WE: ~2 = 1200.3296 ¢, ~5/3 = 884.9921 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7519 ¢

Optimal ET sequence: 19, 99, 118

Badness (Sintel): 1.64

Paralytic

Paralytic (99e & 118) tempers out 441/440, 5632/5625, and 19712/19683. In 13-limit, 118 & 217 tempers out 1001/1000, 1575/1573, and 3584/3575.

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57], ⟨0 13 14 35 82]]

Optimal tunings:

• WE: ~2 = 1199.9940 ¢, ~5/3 = 884.7757 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7800 ¢

Optimal ET sequence: 19e, …, 99e, 118, 217, 335

Badness (Sintel): 1.19

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 441/440, 1001/1000, 3136/3125, 4375/4374

Mapping: [⟨1 -8 -8 -23 -57 59], ⟨0 13 14 35 82 -75]]

Optimal tunings:

• WE: ~2 = 1199.9218 ¢, ~5/3 = 884.7285 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7858 ¢

Optimal ET sequence: 99e, 118, 217

Badness (Sintel): 1.85

Paraklein

Paraklein (19e & 118) is another 13-limit extension of paralytic, which equates 13/11 with 32/27, 14/13 with 15/14, 25/24 with 26/25, and 27/26 with 28/27.

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 625/624, 729/728

Mapping: [⟨1 -8 -8 -23 -57 -28], ⟨0 13 14 35 82 43]]

Optimal tunings:

• WE: ~2 = 1199.8239 ¢, ~5/3 = 884.6449 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.7709 ¢

Optimal ET sequence: 19e, …, 99ef, 118

Badness (Sintel): 1.55

Parkleismic

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1375/1372, 2200/2187

Mapping: [⟨1 -8 -8 -23 -43], ⟨0 13 14 35 63]]

Optimal tunings:

• WE: ~2 = 1199.1848 ¢, ~5/3 = 884.3386 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9158 ¢

Optimal ET sequence: 19e, 61de, 80, 179, 259cd

Badness (Sintel): 1.85

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 1375/1372

Mapping: [⟨1 -8 -8 -23 -43 -14], ⟨0 13 14 35 63 24]]

Optimal tunings:

• WE: ~2 = 1199.5318 ¢, ~5/3 = 884.5800 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9118 ¢

Optimal ET sequence: 19e, 61de, 80, 179

Badness (Sintel): 1.51

Paradigmic

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 3136/3125

Mapping: [⟨1 -8 -8 -23 16], ⟨0 13 14 35 -17]]

Optimal tunings:

• WE: ~2 = 1199.0616 ¢, ~5/3 = 884.2124 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.8877 ¢

Optimal ET sequence: 19, 61d, 80, 99e, 179e, 457bcddeeee

Badness (Sintel): 1.38

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 540/539, 832/825

Mapping: [⟨1 -8 -8 -23 16 -14], ⟨0 13 14 35 -17 24]]

Optimal tunings:

• WE: ~2 = 1199.2683 ¢, ~5/3 = 884.3805 ¢
• CWE: ~2 = 1200.0000 ¢, ~5/3 = 884.9061 ¢

Optimal ET sequence: 19, 61d, 80, 99e

Badness (Sintel): 1.48

Semiparakleismic

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4], ⟨0 13 14 35 23]]

mapping generators: ~99/70, ~33/28

Optimal tunings:

• WE: ~99/70 = 599.9270 ¢, ~33/28 = 284.7841 ¢
• CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8119 ¢

Optimal ET sequence: 80, 118, 198, 316, 514c

Badness (Sintel): 1.13

Semiparamint

This extension was named semiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 1001/1000, 3025/3024, 4375/4374

Mapping: [⟨2 -3 -2 -11 -4 15], ⟨0 13 14 35 23 -16]]

Optimal tunings:

• WE: ~99/70 = 599.8253 ¢, ~33/28 = 284.7608 ¢
• CWE: ~99/70 = 600.0000 ¢, ~33/28 = 284.8366 ¢

Optimal ET sequence: 80, 118, 198

Badness (Sintel): 1.40

Semiparawolf

This extension was named gentsemiparakleismic in the earlier materials.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 3136/3125

Mapping: [⟨2 -3 -2 -11 -4 -4], ⟨0 13 14 35 23 24]]

Optimal tunings:

• WE: ~99/70 = 600.0569 ¢, ~13/11 = 284.8431 ¢
• CWE: ~99/70 = 600.0000 ¢, ~13/11 = 284.8216 ¢

Optimal ET sequence: 80, 118f, 198f

Badness (Sintel): 1.67

Counterkleismic

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Counterhanson.

In the 5-limit, the counterhanson temperament tempers out the counterhanson (quinquinyo) comma, [-20 -24 25⟩, the amount by which six major dieses ((648/625)⁶) fall short of the classic major third (5/4). It can be described as 19 & 224 temperament, tempering out the ragisma and 158203125/157351936 (laquadru-atritriyo comma). It was named by analogy to catakleismic and parakleismic)

Subgroup: 2.3.5.7

Comma list: 4375/4374, 158203125/157351936

Mapping: [⟨1 -5 -4 -18], ⟨0 25 24 79]]

mapping generators: ~2, ~6/5

Optimal tunings:

• WE: ~2 = 1200.1778 ¢, ~6/5 = 316.1065 ¢

error map: ⟨+0.178 -0.181 -0.469 +0.388]

• CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0631 ¢

error map: ⟨0.000 -0.377 -0.799 +0.161]

Optimal ET sequence: 19, …, 205, 224, 243, 467

Badness (Sintel): 2.29

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 4375/4374, 2097152/2096325

Mapping: [⟨1 -5 -4 -18 19], ⟨0 25 24 79 -59]]

Optimal tunings:

• WE: ~2 = 1199.9944 ¢, ~6/5 = 316.0690 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0705 ¢

Optimal ET sequence: 19, 205, 224

Badness (Sintel): 2.35

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 540/539, 625/624, 729/728, 10985/10976

Mapping: [⟨1 -5 -4 -18 19 -15], ⟨0 25 24 79 -59 71]]

Optimal tunings:

• WE: ~2 = 1199.9827 ¢, ~6/5 = 316.0650 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0695 ¢

Optimal ET sequence: 19, 205, 224

Badness (Sintel): 1.40

Counterlytic

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 4375/4374, 496125/495616

Mapping: [⟨1 -5 -4 -18 -40], ⟨0 25 24 79 165]]

Optimal tunings:

• WE: ~2 = 1200.1247 ¢, ~6/5 = 316.0976 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0660 ¢

Optimal ET sequence: 19e, 205e, 224, 467e, 691, 915c

Badness (Sintel): 2.16

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 625/624, 729/728, 1375/1372, 10985/10976

Mapping: [⟨1 -5 -4 -18 -40 -15], ⟨0 25 24 79 165 71]]

Optimal tunings:

• WE: ~2 = 1200.0987 ¢, ~6/5 = 316.0908 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 316.0658 ¢

Optimal ET sequence: 19e, 205e, 224, 467e, 691, 915c

Badness (Sintel): 1.23

Quincy

Subgroup: 2.3.5.7

Comma list: 4375/4374, 823543/819200

Mapping: [⟨1 2 3 3], ⟨0 -30 -49 -14]]

mapping generators: ~2, ~1728/1715

Optimal tunings:

• WE: ~2 = 1200.2169 ¢, ~1728/1715 = 16.6160 ¢

error map: ⟨+0.217 +0.000 +0.155 -0.799]

• CWE: ~2 = 1200.0000 ¢, ~1728/1715 = 16.6083 ¢

error map: ⟨0.000 -0.205 -0.122 -1.343]

Optimal ET sequence: 72, 217, 289, 650d, 939dd

Badness (Sintel): 2.02

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 2 3 3 4], ⟨0 -30 -49 -14 -39]]

Optimal tunings:

• WE: ~2 = 1200.1286 ¢, ~100/99 = 16.6147 ¢
• CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6101 ¢

Optimal ET sequence: 72, 217, 289

Badness (Sintel): 1.02

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 2 3 3 4 5], ⟨0 -30 -49 -14 -39 -94]]

Optimal tunings:

• WE: ~2 = 1200.0554 ¢, ~100/99 = 16.6028 ¢
• CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6011 ¢

Optimal ET sequence: 72, 145, 217, 289

Badness (Sintel): 0.986

17-limit

Subgroup: 2.3.5.7.11.13.17

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

Mapping: [⟨1 2 3 3 4 5 5], ⟨0 -30 -49 -14 -39 -94 -66]]

Optimal tunings:

• WE: ~2 = 1200.0647 ¢, ~100/99 = 16.6025 ¢
• CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.6004 ¢

Optimal ET sequence: 72, 145, 217, 289

Badness (Sintel): 0.751

19-limit

Subgroup: 2.3.5.7.11.13.17.19

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

Mapping: [⟨1 2 3 3 4 5 5 4], ⟨0 -30 -49 -14 -39 -94 -66 18]]

Optimal tunings:

• WE: ~2 = 1199.9287 ¢, ~100/99 = 16.5930 ¢
• CWE: ~2 = 1200.0000 ¢, ~100/99 = 16.5948 ¢

Optimal ET sequence: 72, 145, 217

Badness (Sintel): 0.924

Sfourth

For the 5-limit version, see Miscellaneous 5-limit temperaments #Sfourth.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 64827/64000

Mapping: [⟨1 2 3 3], ⟨0 -19 -31 -9]]

mapping generators: ~2, ~49/48

Optimal tunings:

• WE: ~2 = 1200.8332 ¢, ~49/48 = 26.3053 ¢

error map: ⟨+0.833 -0.090 +0.721 -3.074]

• CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2590 ¢

error map: ⟨0.000 -0.876 -0.343 -5.157]

Optimal ET sequence: 45, 46, 91, 137d

Badness (Sintel): 3.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 4375/4374

Mapping: [⟨1 2 3 3 4], ⟨0 -19 -31 -9 -25]]

Optimal tunings:

• WE: ~2 = 1201.1486 ¢, ~49/48 = 26.3112 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2461 ¢

Optimal ET sequence: 45e, 46, 91e, 137de

Badness (Sintel): 1.78

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 169/168, 325/324, 441/440

Mapping: [⟨1 2 3 3 4 4], ⟨0 -19 -31 -9 -25 -14]]

Optimal tunings:

• WE: ~2 = 1201.4956 ¢, ~49/48 = 26.3423 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2614 ¢

Optimal ET sequence: 45ef, 46, 91ef, 137def, 228ddeeefff

Badness (Sintel): 1.37

Sfour

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 4375/4374

Mapping: [⟨1 2 3 3 3], ⟨0 -19 -31 -9 21]]

Optimal tunings:

• WE: ~2 = 1200.4402 ¢, ~49/48 = 26.2557 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2403 ¢

Optimal ET sequence: 45, 46, 91, 137d, 183d

Badness (Sintel): 2.53

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 4375/4374

Mapping: [⟨1 2 3 3 3 3], ⟨0 -19 -31 -9 21 32]]

Optimal tunings:

• WE: ~2 = 1200.3796 ¢, ~49/48 = 26.2473 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 26.2372 ¢

Optimal ET sequence: 45, 46, 91, 137d, 183d

Badness (Sintel): 2.14

Trideci

For the 5-limit version, see 13th-octave temperaments #Tridecatonic.

The trideci temperament (26 & 65) has a period of 1/13 octave and tempers out 245/242 and 385/384 in the 11-limit. It tempers out the same 5-limit comma as the tridecatonic temperament, but with the ragisma (4375/4374) rather than the octagar (4000/3969) tempered out. The name trideci comes from tridecim (Latin for "thirteen").

Subgroup: 2.3.5.7

Comma list: 4375/4374, 83349/81920

Mapping: [⟨13 0 -11 57], ⟨0 1 2 -1]]

mapping generators: ~256/245, ~3

Optimal tunings:

• WE: ~256/245 = 92.4141 ¢, ~3/2 = 699.9466 ¢

error map: ⟨+1.383 -0.626 -0.210 -2.554]

• CWE: ~256/245 = 92.3077 ¢, ~3/2 = 699.4521 ¢

error map: ⟨0.000 -2.503 -2.794 -6.740]

Optimal ET sequence: 26, 65, 91

Badness (Sintel): 4.67

11-limit

Subgroup: 2.3.5.7.11

Comma list: 245/242, 385/384, 4375/4374

Mapping: [⟨13 0 -11 57 45], ⟨0 1 2 -1 0]]

Optimal tunings:

• WE: ~22/21 = 92.3729 ¢, ~3/2 = 700.1118 ¢
• CWE: ~22/21 = 92.3077 ¢, ~3/2 = 699.7703 ¢

Optimal ET sequence: 26, 65, 91

Badness (Sintel): 2.80

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 245/242, 325/324, 385/384

Mapping: [⟨13 0 -11 57 45 48], ⟨0 1 2 -1 0 0]]

Optimal tunings:

• WE: ~22/21 = 92.4003 ¢, ~3/2 = 699.9983 ¢
• CWE: ~22/21 = 92.3077 ¢, ~3/2 = 699.4772 ¢

Optimal ET sequence: 26, 65f, 91f

Badness (Sintel): 2.16