Octagar temperaments

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 octagar temperaments, which temper out the octagar comma (monzo: [5 -4 3 -2⟩, ratio: 4000/3969).

Temperaments discussed elsewhere are:

• Garibaldi (+225/224) → Schismatic family
• Injera (+50/49 or 81/80) → Meantone family
• Augene (+64/63 or 126/125) → Augmented family
• Parakangaroo (+28/27) → Trienstonic clan
• Sida (+25/24) → Dicot family
• Bidia (+2048/2025 or 3136/3125) → Diaschismic family
• Rip (+36/35) → Ripple family
• Nautilus (+49/48) → Porcupine family
• Roman (+525/512) → Wesley family
• Octacot (+245/243) → Tetracot family
• Superkleismic (+875/864 or 1029/1024) → Gamelismic clan
• Hamity (+2430/2401) → Amity family
• Hocum (+3125/3072) → Magic family
• Quartonic (+1728/1715) → Quartonic family
• Hemikleismic (+6144/6125) → Kleismic family
• Trisedodge (+110592/109375) → Trisedodge family

Considered below are pluto, slithy, quintupole, tridecatonic, slendi and dhaivatic, in the order of increasing badness.

Pluto

Not to be confused with Plutus.

Pluto, named by Gene Ward Smith in 2010, can be described as the 41 & 80 temperament. It is generated by a flattened ~10/7 tritone, seven of which give the interval class of 3, and the ploidacot for this temperament is gamma-heptacot. 59\121 is about perfect as a tuning.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 10976/10935

Mapping: [⟨1 -2 -11 -10], ⟨0 7 26 25]]

mapping generators: ~2, ~10/7

Optimal tunings:

• WE: ~2 = 1199.4720 ¢, ~10/7 = 614.5826 ¢

error map: ⟨-0.528 +1.179 -1.358 +1.020]

• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8282 ¢

error map: ⟨0.000 +1.843 -0.780 +1.880]

Optimal ET sequence: 39d, 41, 80, 121, 283d, 404bd

Badness (Sintel): 1.46

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 1375/1372

Mapping: [⟨1 -2 -11 -10 5], ⟨0 7 26 25 -3]]

Optimal tunings:

• WE: ~2 = 1200.1835 ¢, ~10/7 = 614.4677 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8616 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 0.987

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 364/363, 540/539

Mapping: [⟨1 -2 -11 -10 5 16], ⟨0 7 26 25 -3 -24]]

Optimal tunings:

• WE: ~2 = 1199.2469 ¢, ~10/7 = 614.4908 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8684 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 1.06

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 325/324, 352/351, 364/363, 540/539

Mapping: [⟨1 -2 -11 -10 5 16 21], ⟨0 7 26 25 -3 -24 -33]]

Optimal tunings:

• WE: ~2 = 1199.2009 ¢, ~10/7 = 614.4746 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8813 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 1.09

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 256/255, 325/324, 352/351, 361/360, 595/594

Mapping: [⟨1 -2 -11 -10 5 16 21 -6], ⟨0 7 26 25 -3 -24 -33 20]]

Optimal tunings:

• WE: ~2 = 1199.2833 ¢, ~10/7 = 614.5237 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8876 ¢

Optimal ET sequence: 39d, 41, 80, 121

Badness (Sintel): 1.07

Orcus

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 275/273, 896/891

Mapping: [⟨1 -2 -11 -10 5 -5], ⟨0 7 26 25 -3 17]]

Optimal tunings:

• WE: ~2 = 1198.7372 ¢, ~10/7 = 614.2418 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.8502 ¢

Optimal ET sequence: 39df, 41, 80f, 121ff

Badness (Sintel): 1.38

Plutino

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 10976/10935

Mapping: [⟨1 -2 -11 -10 -16], ⟨0 7 26 25 38]]

Optimal tunings:

• WE: ~2 = 1199.7793 ¢, ~10/7 = 614.6040 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.7074 ¢

Optimal ET sequence: 39dee, 41

Badness (Sintel): 1.92

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 196/195, 245/242, 729/728

Mapping: [⟨1 5 15 15 22 12], ⟨0 7 26 25 38 17]]

Optimal tunings:

• WE: ~2 = 1199.0328 ¢, ~10/7 = 614.2729 ¢
• CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.7277 ¢

Optimal ET sequence: 39deef, 41

Badness (Sintel): 1.66

Slithy

Slithy tempers out 420175/419904, the wizma, and may be described as the 27 & 94 temperament with a ploidacot signature of 17-sheared 19-cot. 94edo, 121edo and 215edo can all be used as tunings.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 420175/419904

Mapping: [⟨1 -16 -31 -12], ⟨0 19 36 16]]

mapping generators: ~2, ~40/21

Optimal tunings:

• WE: ~2 = 1199.5464 ¢, ~40/21 = 1110.2981 ¢

error map: ⟨-0.454 +0.996 -1.521 +1.387]

• CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.7026 ¢

error map: ⟨0.000 +1.394 -1.021 +2.415]

Optimal ET sequence: 27, 67c, 94, 121, 215, 336d

Badness (Sintel): 2.72

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 2200/2187, 4375/4356

Mapping: [⟨1 -16 -31 -12 -53], ⟨0 19 36 16 61]]

Optimal tunings:

• WE: ~2 = 1199.5884 ¢, ~40/21 = 1110.3276 ¢
• CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.6969 ¢

Optimal ET sequence: 27e, 67ce, 94, 121, 215, 336de

Badness (Sintel): 1.52

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 352/351, 540/539, 1575/1573

Mapping: [⟨1 -16 -31 -12 -53 0], ⟨0 19 36 16 61 4]]

Optimal tunings:

• WE: ~2 = 1199.5032 ¢, ~40/21 = 1110.2545 ¢
• CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.7037 ¢

Optimal ET sequence: 27e, 67ce, 94, 121, 215, 336def

Badness (Sintel): 1.18

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 352/351, 375/374, 540/539, 595/594

Mapping: [⟨1 -16 -31 -12 -53 0 -57], ⟨0 19 36 16 61 4 66]]

Optimal tunings:

• WE: ~2 = 1199.5051 ¢, ~40/21 = 1110.2555 ¢
• CWE: ~2 = 1200.0000 ¢, ~40/21 = 1110.7033 ¢

Optimal ET sequence: 27eg, 67ceg, 94, 121, 215, 336defg

Badness (Sintel): 1.01

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 325/324, 352/351, 361/360, 375/374, 595/594

Mapping: [⟨1 -16 -31 -12 -53 0 -57 -30], ⟨0 19 36 16 61 4 66 37]]

Optimal tunings:

• WE: ~2 = 1199.6184 ¢, ~19/10 = 1110.3583 ¢
• CWE: ~2 = 1200.0000 ¢, ~19/10 = 1110.7036 ¢

Optimal ET sequence: 27eg, 67cegh, 94, 121, 215

Badness (Sintel): 0.998

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 190/189, 300/299, 323/322, 325/324, 352/351, 361/360, 484/483

Mapping: [⟨1 -16 -31 -12 -53 0 -57 -30 -76], ⟨0 19 36 16 61 4 66 37 87]]

Optimal tunings:

• WE: ~2 = 1199.6651 ¢, ~19/10 = 1110.3928 ¢
• CWE: ~2 = 1200.0000 ¢, ~19/10 = 1110.6965 ¢

Optimal ET sequence: 27egi, 94, 121i, 215

Badness (Sintel): 1.07

Quintupole

Main article: Quintupole

For the 5-limit version, see Schismic–commatic equivalence continuum #Quintaleap.

Quintupole may be described as the 12 & 121 temperament with a ploidacot signature of omega-pentacot. It was named by Xenllium in 2021; the word quintupole means "five poles", but also a play on the words quintuple and polypyth, so named because the generator is 1/5 of a perfect fourth of about 495.8 cents, which can serve as a tuning for polypyth.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 458752/455625

Mapping: [⟨1 2 1 0], ⟨0 -5 16 34]]

mapping generators: ~2, ~135/128

Optimal tunings:

• WE: ~2 = 1199.3243 ¢, ~135/128 = 99.1196 ¢

error map: ⟨-0.676 +1.095 -1.075 +1.241]

• CWE: ~2 = 1200.0000 ¢, ~135/128 = 99.1615 ¢

error map: ⟨0.000 +2.237 +0.271 +2.666]

Optimal ET sequence: 12, 85d, 97, 109, 121

Badness (Sintel): 2.82

11-limit

Subgroup: 2.3.5.7.11

Comma list: 896/891, 1375/1372, 4375/4356

Mapping: [⟨1 2 1 0 -1], ⟨0 -5 16 34 54]]

Optimal tunings:

• WE: ~2 = 1199.3424 ¢, ~35/33 = 99.1015 ¢
• CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.1436 ¢

Optimal ET sequence: 12, 97e, 109, 121, 351bde

Badness (Sintel): 1.87

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 625/624, 2704/2695

Mapping: [⟨1 2 1 0 -1 -2], ⟨0 -5 16 34 54 69]]

Optimal tunings:

• WE: ~2 = 1199.3223 ¢, ~35/33 = 99.1087 ¢
• CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.1536 ¢

Optimal ET sequence: 12f, 97ef, 109, 121

Badness (Sintel): 1.59

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 352/351, 364/363, 375/374, 442/441

Mapping: [⟨1 2 1 0 -1 -2 5], ⟨0 -5 16 34 54 69 -11]]

Optimal tunings:

• WE: ~2 = 1199.2234 ¢, ~18/17 = 99.1076 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.1629 ¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.46

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374

Mapping: [⟨1 2 1 0 -1 -2 5 4], ⟨0 -5 16 34 54 69 -11 3]]

Optimal tunings:

• WE: ~2 = 1199.3678 ¢, ~18/17 = 99.1115 ¢
• CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.1576 ¢

Optimal ET sequence: 12f, 109, 121

Badness (Sintel): 1.45

Tridecatonic

For the 5-limit version, see Miscellaneous 5-limit temperaments #Tridecatonic

Tridecatonic has a period of 1/13 octave, and may be described as 26 & 39. It tempers out devil's tridecalimma, [-11 26 -13⟩ in the 5-limit. Its ploidacot is 13-ploid monocot.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 8748/8575

Mapping: [⟨13 0 -11 16], ⟨0 1 2 1]]

mapping generators: ~21/20, ~3

Optimal tunings:

• WE: ~21/20 = 92.2197 ¢, ~3/2 = 701.2915 ¢

error map: ⟨-1.144 -1.807 -0.435 +6.837]

• CWE: ~21/20 = 92.3077 ¢, ~3/2 = 701.2352 ¢

error map: ⟨0.000 -0.720 +0.772 +9.332]

Optimal ET sequence: 26, 39d, 65d

Badness (Sintel): 3.52

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 1944/1925, 2560/2541

Mapping: [⟨13 0 -11 16 45], ⟨0 1 2 1 0]]

Optimal tunings:

• WE: ~21/20 = 92.2333 ¢, ~3/2 = 701.1371 ¢
• CWE: ~21/20 = 92.3077 ¢, ~3/2 = 701.2751 ¢

Optimal ET sequence: 26, 39d, 65d

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 275/273, 1040/1029

Mapping: [⟨13 0 -11 16 45 7], ⟨0 1 2 1 0 2]]

Optimal tunings:

• WE: ~21/20 = 92.2248 ¢, ~3/2 = 700.7272 ¢
• CWE: ~21/20 = 92.3077 ¢, ~3/2 = 700.8621 ¢

Optimal ET sequence: 26, 39df, 65d

Badness (Sintel): 1.56

Slendi

Slendi may be described as the 41 & 42 temperament. It has a generator of 49/48, seventeen of which give 4/3. Its ploidacot is omega-17-cot.

The name was likely derived from slendro diesis, one of the names for the interval 49/48.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 84035/82944

Mapping: [⟨1 2 3 3], ⟨0 -17 -28 -8]]

mapping generators: ~2, ~49/48

Optimal tunings:

• WE: ~2 = 1200.2515 ¢, ~49/48 = 29.1870 ¢

error map: ⟨+0.252 +2.368 -2.796 -1.568]

• CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1715 ¢

error map: ⟨0.000 +2.129 -3.116 -2.198]

Optimal ET sequence: 1c, 40c, 41, 206

Badness (Sintel): 3.62

11-limit

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 46656/46585

Mapping: [⟨1 2 3 3 4], ⟨0 -17 -28 -8 -22]]

Optimal tunings:

• WE: ~2 = 1199.8027 ¢, ~49/48 = 29.1751 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1875 ¢

Optimal ET sequence: 1ce, 40c, 41

Badness (Sintel): 2.09

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 100/99, 144/143, 245/242, 343/338

Mapping: [⟨1 2 3 3 4 4], ⟨0 -17 -28 -8 -22 -12]]

Optimal tunings:

• WE: ~2 = 1199.0915 ¢, ~49/48 = 29.1033 ¢
• CWE: ~2 = 1200.0000 ¢, ~49/48 = 29.1576 ¢

Optimal ET sequence: 1ce, 41

Badness (Sintel): 1.80

Dhaivatic

Named by Xenllium in 2023, dhaivatic tempers out the rainy comma and may be described as the 15 & 94 temperament. Its ploidacot is epsilon-21-cot.

Subgroup: 2.3.5.7

Comma list: 4000/3969, 2100875/2097152

Mapping: [⟨1 -4 -3 6], ⟨0 21 20 -12]]

mapping generators: ~2, ~6/5

Optimal tunings:

• WE: ~2 = 1199.9902 ¢, ~6/5 = 319.1955 ¢

error map: ⟨-0.010 +1.190 -2.374 +0.769]

• CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.1980 ¢

error map: ⟨0.000 +1.202 -2.354 +0.798]

Optimal ET sequence: 15, 64c, 79, 94, 203, 297c

Badness (Sintel): 4.26

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 4000/3969

Mapping: [⟨1 -4 -3 6 0], ⟨0 21 20 -12 13]]

Optimal tunings:

• WE: ~2 = 1200.0983 ¢, ~6/5 = 319.2229 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.1985 ¢

Optimal ET sequence: 15, 64c, 79, 94, 203, 297c

Badness (Sintel): 1.69

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 275/273, 325/324, 385/384, 1331/1323

Mapping: [⟨1 -4 -3 6 0 -8], ⟨0 21 20 -12 13 44]]

Optimal tunings:

• WE: ~2 = 1200.0732 ¢, ~6/5 = 319.1840 ¢
• CWE: ~2 = 1200.0000 ¢, ~6/5 = 319.1658 ¢

Optimal ET sequence: 15, 79, 94

Badness (Sintel): 1.55