Canopic clan
- 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.
The mirkwai clan of temperaments tempers out the mirkwai comma (monzo: [0 3 4 -5⟩, ratio: 16875/16807), a no-twos comma.
Canopus
Subgroup: 3.5.7
Comma list: 16875/16807
Subgroup-val mapping: [⟨1 3 3], ⟨0 -5 -4]]
- mapping generators: ~3, ~7/5
Optimal tuning (POTE): ~3 = 1901.9550 ¢, ~7/5 = 583.9584 ¢
Optimal ET sequence: b13, b62, b75, b88, b101, b114, b355, b469, b583, b697
Overview to extensions
The full 7-limit extensions' relation to canopus is clearer if the mapping is normalized in terms of 3.5.7.2. In fact, the strong extensions are nusecond and octoid. These temperaments are distributed into different temperament collection pages.
- Nusecond (+126/125) → Starling temperaments
- Octoid (+4375/4374) → Ragismic microtemperaments
The others are weak extensions. Mirkat tempers out 19683/19600, splitting the generator in two with a semitwelfth period. Sqrtphi tempers out 15625/15552, splitting the period in six. Semisept tempers out 1728/1715 and 3136/3125, splitting the generator in six. Miracle tempers out 225/224. Pluto tempers out 4000/3969. These split the generator in five. Kwai tempers out 5120/5103, splitting the generator in ten. Quanharuk tempers out 32805/32768, splitting the generator in three with a 1/5-twelfth period. Grendel tempers out 6144/6125, splitting the generator in eleven. Finally, eris tempers out 65625/65536, splitting the generator in sixteen.
Members of the clan discussed elsewhere are:
- Kwai (+5120/5103) → Hemifamity temperaments
- Octokaidecal (+28/27 or 50/49) → Trienstonic clan
- Meantritone (+81/80) → Meantone family
- Quanharuk (+32805/32768) → Schismatic family
- Miracle (+225/224) → Gamelismic clan
- Pluto (+4000/3969) → Octagar temperaments
- Bohpier (+245/243) → Sensamagic clan
- Semisept (+1728/1715 or 3136/3125) → Hemimean clan
- Grendel (+6144/6125) → Porwell temperaments
- Quinmage (+3125/3072) → Magic family
- Familia (+1600000/1594323) → Amity family
- Sqrtphi (+15625/15552) → Kleismic family
- Rainwell (+2100875/2097152) → Semicomma family
- Quintiquart (+390625000/387420489) → Quartonic family
For no-twos extensions, see No-twos subgroup temperaments #Canopus.
Considered below are mirkat, eris, subsemifourth, septendesemi, gaster, subsedia, hemiseptisix, browser, and grazer.
Mirkat
Subgroup: 2.3.5.7
Comma list: 16875/16807, 19683/19600
Mapping: [⟨3 2 1 2], ⟨0 6 13 14]]
Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~10/9 = 183.539 ¢
Optimal ET sequence: 39d, 72, 111, 183, 255
Badness (Smith): 0.059376
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 8019/8000
Mapping: [⟨3 2 1 2 9], ⟨0 6 13 14 3]]
Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~10/9 = 183.528 ¢
Optimal ET sequence: 39d, 72, 111, 183, 255
Badness (Smith): 0.022126
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 1375/1372
Mapping: [⟨3 2 1 2 9 1], ⟨0 6 13 14 3 22]]
Optimal tuning (POTE): ~63/50 = 400.000 ¢, ~10/9 = 183.577 ¢
Optimal ET sequence: 39df, 72, 111, 183
Badness (Smith): 0.018632
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 442/441, 540/539, 561/560, 715/714
Mapping: [⟨3 2 1 2 9 1 4], ⟨0 6 13 14 3 22 18]]
Optimal tuning (POTE): ~34/27 = 400.000 ¢, ~10/9 = 183.579 ¢
Optimal ET sequence: 39dfg, 72, 111, 183
Badness (Smith): 0.011775
Eris
The 2.5.7 subgroup restriction of this temperament is exodia.
Subgroup: 2.3.5.7
Comma list: 16875/16807, 65625/65536
Mapping: [⟨1 10 0 6], ⟨0 -29 8 -11]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~60/49 = 348.216 ¢
Optimal ET sequence: 31, 131, 162, 193, 224, 1823cd, 2271cd
Badness (Smith): 0.074719
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 65625/65536
Mapping: [⟨1 10 0 6 20], ⟨0 -29 8 -11 -57]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/9 = 348.219 ¢
Optimal ET sequence: 31, 193, 224, 703, 927d, 1151cd
Badness (Smith): 0.027621
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 1375/1372, 4096/4095
Mapping: [⟨1 10 0 6 20 -14], ⟨0 -29 8 -11 -57 61]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/9 = 348.213 ¢
Optimal ET sequence: 31, 193, 224
Badness (Smith): 0.025137
Subsemifourth
Subgroup: 2.3.5.7
Comma list: 16875/16807, 26873856/26796875
Mapping: [⟨1 39 27 45], ⟨0 -47 -31 -53]]
- mapping generators: ~2, ~125/72
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~144/125 = 244.719 ¢
Optimal ET sequence: 49, 103, 152, 255, 407
Badness (Smith): 0.135173
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 234375/234256
Mapping: [⟨1 39 27 45 56], ⟨0 -47 -31 -53 -66]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~121/105 = 244.719 ¢
Optimal ET sequence: 49, 103, 152, 255, 407, 966d
Badness (Smith): 0.034276
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 847/845, 1375/1372, 1575/1573
Mapping: [⟨1 39 27 45 56 65], ⟨0 -47 -31 -53 -66 -77]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/13 = 244.714 ¢
Optimal ET sequence: 49f, 103, 152f, 255, 407f, 662df
Badness (Smith): 0.028387
Septendesemi
The name septendesemi means a septendecimal semitone (17/16). The septendesemi temperament (80 & 103) tempers out the mirkwai comma and 1959552/1953125 (parkleiness comma, zotritrigu) in the 7-limit. 183edo provides an excellent tuning for 7-, 11-, 13-, and 17-limit septendesemi.
Subgroup: 2.3.5.7
Comma list: 16875/16807, 1959552/1953125
Mapping: [⟨1 39 37 53], ⟨0 -41 -38 -55]]
- mapping generators: ~2, ~648/343
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~343/324 = 104.916 ¢
Optimal ET sequence: 80, 103, 183
Badness (Smith): 0.146795
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 43923/43750
Mapping: [⟨1 39 37 53 50], ⟨0 -41 -38 -55 -51]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 104.916 ¢
Optimal ET sequence: 80, 103, 183
Badness (Smith): 0.041554
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 1375/1372, 4225/4224
Mapping: [⟨1 39 37 53 50 11], ⟨0 -41 -38 -55 -51 -8]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~35/33 = 104.908 ¢
Optimal ET sequence: 80, 103, 183, 469f, 652def
Badness (Smith): 0.027908
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 540/539, 561/560, 715/714, 4225/4224
Mapping: [⟨1 39 37 53 50 11 5], ⟨0 -41 -38 -55 -51 -8 -1]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~17/16 = 104.909 ¢
Optimal ET sequence: 80, 103, 183, 469f, 652def
Badness (Smith): 0.020128
Gaster
- For the 5-limit version, see Very high accuracy temperaments #Gaster.
The gaster temperament (111 & 113) tempers out [-70 72 -19⟩ (quadbila-negu) in the 5-limit; mirkwai comma (16875/16807) and scheme comma (14348907/14336000) in the 7-limit. The word "gaster" means abdomen or stomach, but also a restructuring of the words "gassormic tritone", which is a generator of this temperament. This temperament is sufficient to obtain high prime limit harmonics like a stomach, so that patent vals 111, 113 and 224 support it even in the 41-limit.
Subgroup: 2.3.5.7
Comma list: 16875/16807, 14348907/14336000
Mapping: [⟨1 11 38 37], ⟨0 -19 -72 -69]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~800/567 = 594.641 ¢
Badness (Smith): 0.154521
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 14348907/14336000
Mapping: [⟨1 11 38 37 -1], ⟨0 -19 -72 -69 9]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~512/363 = 594.639 ¢
Optimal ET sequence: 111, 224, 783d, 1007d, 1231dd
Badness (Smith): 0.054060
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1375/1372, 2200/2197
Mapping: [⟨1 11 38 37 -1 26], ⟨0 -19 -72 -69 9 -45]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/39 = 594.639 ¢
Optimal ET sequence: 111, 224, 783df, 1007df, 1231ddf
Badness (Smith): 0.024882
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 715/714, 729/728, 936/935, 2200/2197
Mapping: [⟨1 11 38 37 -1 26 14], ⟨0 -19 -72 -69 9 -45 -20]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.636 ¢
Optimal ET sequence: 111, 224, 559dgg
Badness (Smith): 0.021436
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 324/323, 400/399, 495/494, 540/539, 715/714, 1445/1444
Mapping: [⟨1 11 38 37 -1 26 14 32], ⟨0 -19 -72 -69 9 -45 -20 -56]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.636 ¢
Badness (Smith): 0.018370
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
Mapping: [⟨1 11 38 37 -1 26 14 32 7], ⟨0 -19 -72 -69 9 -45 -20 -56 -5]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.641 ¢
Badness (Smith): 0.017619
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 290/289, 324/323, 400/399, 460/459, 495/494, 529/528, 540/539, 715/714
Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.646 ¢
Optimal ET sequence: 111, 113, 224
Badness (Smith): 0.016815
31-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 715/714
Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.644 ¢
Optimal ET sequence: 111, 113, 224
Badness (Smith): 0.014790
37-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 540/539, 667/666, 715/714
Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0 -27], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.644 ¢
Optimal ET sequence: 111, 113, 224
Badness (Smith): 0.014377
41-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Comma list: 290/289, 324/323, 400/399, 435/434, 460/459, 495/494, 528/527, 533/532, 540/539, 575/574, 667/666
Mapping: [⟨1 11 38 37 -1 26 14 32 7 -11 0 -27 45], ⟨0 -19 -72 -69 9 -45 -20 -56 -5 32 10 65 -80]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~24/17 = 594.643 ¢
Optimal ET sequence: 111, 113, 224
Badness (Smith): 0.012858
Subsedia
The generator for subsedia (10 & 111) is 0.5 cents flat of 15/14-wide semitone and tempers out the mirkwai comma and buzzardsma. In this temperament, three generators makes ~16/13, five of them equals ~24/17, twelve of them equals ~16/7, sixteen of them equals ~3/1, and 45 of them equals ~22/1.
Subgroup: 2.3.5.7
Comma list: 16875/16807, 65536/64827
Mapping: [⟨1 0 5 4], ⟨0 16 -27 -12]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.965 ¢
Optimal ET sequence: 10, 101, 111, 121, 232d
Badness (Smith): 0.157658
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 65536/64827
Mapping: [⟨1 0 5 4 -1], ⟨0 16 -27 -12 45]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.968 ¢
Optimal ET sequence: 10, 101, 111, 121, 232d
Badness (Smith): 0.066838
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 540/539, 676/675, 1375/1372
Mapping: [⟨1 0 5 4 -1 4], ⟨0 16 -27 -12 45 -3]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.968 ¢
Optimal ET sequence: 10, 101, 111, 121, 232d
Badness (Smith): 0.031635
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 256/255, 352/351, 442/441, 540/539, 715/714
Mapping: [⟨1 0 5 4 -1 4 3], ⟨0 16 -27 -12 45 -3 11]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.968 ¢
Optimal ET sequence: 10, 101, 111, 121, 232dg
Badness (Smith): 0.019707
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714
Mapping: [⟨1 0 5 4 -1 4 3 10], ⟨0 16 -27 -12 45 -3 11 -58]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~15/14 = 118.964 ¢
Optimal ET sequence: 10, 101h, 111, 121, 232dg
Badness (Smith): 0.017935
Hemiseptisix
The name hemiseptisix means a half of septimal major sixth (12/7). The hemiseptisix temperament (103 & 121) tempers out the mirkwai comma and 95703125/95551488 (pontiqak comma, lazozotritriyo) in the 7-limit. 224edo provides an excellent tuning for 7-, 11-, and 13-limit hemiseptisix.
Subgroup: 2.3.5.7
Comma list: 16875/16807, 95703125/95551488
Mapping: [⟨1 34 17 34], ⟨0 -53 -24 -51]]
- mapping generators: ~2, ~75/49
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~98/75 = 466.071 ¢
Optimal ET sequence: 103, 121, 224
Badness (Smith): 0.162826
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 2734375/2725888
Mapping: [⟨1 34 17 34 53], ⟨0 -53 -24 -51 -81]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/42 = 466.070 ¢
Optimal ET sequence: 103, 121, 224
Badness (Smith): 0.043381
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 1375/1372, 2200/2197
Mapping: [⟨1 34 17 34 53 30], ⟨0 -53 -24 -51 -81 -43]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~55/42 = 466.071 ¢
Optimal ET sequence: 103, 121, 224
Badness (Smith): 0.021127
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 540/539, 625/624, 715/714, 2200/2197
Mapping: [⟨1 34 17 34 53 30 31], ⟨0 -53 -24 -51 -81 -43 -44]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~17/13 = 466.074 ¢
Optimal ET sequence: 103, 121, 224
Badness (Smith): 0.018611
Browser
This can also be considered a non-over-1 temperament, with considerable scope for harmony in the 2.5/3.7/3.11/3.13/3.17/3 subgroup with MOSes of 8, 15, 23, and 31 notes despite no harmonics from the root. It can be considered a detemperament of 8et, with a generator very slightly sharp of 1\8.
Subgroup: 2.3.5.7
Comma list: 16875/16807, 78732/78125
Mapping: [⟨1 6 8 10], ⟨0 -35 -45 -57]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/45 = 151.399 ¢
Optimal ET sequence: 103, 111, 214
Badness (Smith): 0.180645
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 78732/78125
Mapping: [⟨1 6 8 10 8], ⟨0 -35 -45 -57 -36]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.405 ¢
Badness (Smith): 0.057634
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 847/845, 1375/1372
Mapping: [⟨1 6 8 10 8 9], ⟨0 -35 -45 -57 -36 -42]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.403 ¢
Optimal ET sequence: 103, 111, 214
Badness (Smith): 0.028822
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 540/539, 561/560, 715/714, 847/845
Mapping: [⟨1 6 8 10 8 9 8], ⟨0 -35 -45 -57 -36 -42 -31]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.397 ¢
Optimal ET sequence: 103, 111, 214
Badness (Smith): 0.020384
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 324/323, 351/350, 456/455, 495/494, 540/539, 715/714
Mapping: [⟨1 6 8 10 8 9 8 18], ⟨0 -35 -45 -57 -36 -42 -31 -109]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 151.396 ¢
Optimal ET sequence: 103h, 111, 214
Badness (Smith): 0.017570
Grazer
Subgroup: 2.3.5.7
Comma list: 16875/16807, 1071875/1062882
Mapping: [⟨1 34 47 58], ⟨0 -37 -51 -63]]
- mapping generators: ~2, ~90/49
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/45 = 148.719 ¢
Optimal ET sequence: 113, 121, 234
Badness (Smith): 0.217166
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 218750/216513
Mapping: [⟨1 34 47 58 35], ⟨0 -37 -51 -63 -36]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.729 ¢
Optimal ET sequence: 113, 121, 234, 355e, 589cee
Badness (Smith): 0.076062
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 364/363, 540/539, 2200/2197
Mapping: [⟨1 34 47 58 35 44], ⟨0 -37 -51 -63 -36 -46]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.729 ¢
Optimal ET sequence: 113, 121, 234, 355e, 589cee
Badness (Smith): 0.036248
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 325/324, 364/363, 540/539, 595/594, 2000/1989
Mapping: [⟨1 34 47 58 35 44 33], ⟨0 -37 -51 -63 -36 -46 -33]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.735 ¢
Optimal ET sequence: 113, 121, 234g, 355eg
Badness (Smith): 0.025410
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 325/324, 364/363, 400/399, 540/539, 595/594, 665/663
Mapping: [⟨1 34 47 58 35 44 33 6], ⟨0 -37 -51 -63 -36 -46 -33 -2]]
Optimal tuning (POTE): ~2 = 1200.000 ¢, ~12/11 = 148.727 ¢
Optimal ET sequence: 113, 121, 234g, 355eg, 589ceegg
Badness (Smith): 0.022574