Mint 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 low complexity, high error, temperaments which temper out the septimal quarter-tone, 36/35. 36 is both a square and a triangular number, and this helps make 36/35 a septimal interval of considerable significance. These temperaments equate 6/5 with 7/6, 5/4 with 9/7, and 7/4 with 9/5, so minor and major thirds and sixths are intervals of 5 and 7 at the same time.
Temperaments discussed elsewhere include
- Father (+16/15) → Father family
- Dominant (+64/63) → Meantone family
- Armodue (+135/128) → Mavila family
- Mujannabic (+25/24) → Dicot family
- Beep (+21/20) → Bug family
- August (+128/125) → Augmented family
- Gorgo (+1029/1024) → Gamelismic clan
- Hystrix (+160/147) → Porcupine family
- Diminished (+50/49) → Diminished family
- Smate (+2048/1875) → Smate family
- Darkstone (+1875/1792) → Magic family
- Rip (+2560/2401) → Ripple family
- Whitewood (+2187/2048) → Whitewood family
Penta
- For the 5-limit version, see Syntonic–diatonic equivalence continuum #University.
Subgroup: 2.3.5.7
Comma list: 28/25, 36/35
Mapping: [⟨1 1 2 2], ⟨0 3 2 4]]
- mapping generators: ~2, ~7/6
- WE: ~2 = 1186.8093 ¢, ~7/6 = 237.3390 ¢
- error map: ⟨-13.191 -3.129 +61.983 -45.851]
- CWE: ~2 = 1200.0000 ¢, ~7/6 = 231.845 ¢
- error map: ⟨0.000 +6.291 +85.850 -24.498]
Optimal ET sequence: 1bd, …, 4bcd, 5
Badness (Sintel): 1.19
Progression
- Not to be confused with Progress.
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Lafayette.
Named by Gene Ward Smith in 2011[1], progression can be described as the 8d & 9 temperament. It has a generator that is a somewhat flat neutral second, three make 5/4, five make 3/2, and seven make 7/4, with a ploidacot signature of pentacot. 17edo is an obvious tuning for it.
Subgroup: 2.3.5.7
Comma list: 36/35, 392/375
Mapping: [⟨1 1 2 2], ⟨0 5 3 7]]
- mapping generators: ~2, ~15/14
- WE: ~2 = 1193.9544 ¢, ~15/14 = 140.2169 ¢
- error map: ⟨-6.046 -6.916 +22.246 +0.601]
- CWE: ~2 = 1200.000 ¢, ~15/14 = 139.8991 ¢
- error map: ⟨0.000 -2.460 +33.384 +10.468]
Optimal ET sequence: 8d, 9, 17c
Badness (Sintel): 1.22
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 77/75
Mapping: [⟨1 1 2 2 3], ⟨0 5 3 7 4]]
Optimal tunings:
- WE: ~2 = 1194.7089 ¢, ~12/11 = 140.1262 ¢
- CWE: ~2 = 1200.0000 ¢, ~12/11 = 139.8776 ¢
Optimal ET sequence: 8d, 9, 17c
Badness (Sintel): 0.861
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 56/55, 66/65
Mapping: [⟨1 1 2 2 3 3], ⟨0 5 3 7 4 6]]
Optimal tunings:
- WE: ~2 = 1195.3694 ¢, ~13/12 = 140.2080 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.9749 ¢
Optimal ET sequence: 8d, 9, 17c
Badness (Sintel): 0.750
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 26/25, 36/35, 51/50, 56/55, 66/65
Mapping: [⟨1 1 2 2 3 3 4], ⟨0 5 3 7 4 6 1]]
Optimal tunings:
- WE: ~2 = 1193.8350 ¢, ~13/12 = 140.6779 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.5885 ¢
Optimal ET sequence: 8d, 9, 17cg
Badness (Sintel): 0.853
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 26/25, 36/35, 51/50, 56/55, 57/55, 66/65
Mapping: [⟨1 1 2 2 3 3 4 4], ⟨0 5 3 7 4 6 1 2]]
Optimal tunings:
- WE: ~2 = 1196.1446 ¢, ~13/12 = 140.0276 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.0301 ¢
Optimal ET sequence: 8d, 9, 17cg
Badness (Sintel): 1.02
Subklei
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Delorean.
Subklei is likely named for its flatter minor third generator than kleismic. It tempers out 1029/1000 as well as 2401/2400 and can be described as 17c & 21. Its ploidacot is delta-hexacot. Note that in the data below, the generator is the 5/3~12/7 major sixth, so that six generators minus four octaves give the perfect fifth.
Subgroup: 2.3.5.7
Comma list: 36/35, 1029/1000
Mapping: [⟨1 -3 -3 -1], ⟨0 6 7 5]]
- mapping generators: ~2, ~5/3
- WE: ~2 = 1197.5285 ¢, ~5/3 = 915.8950 ¢
- error map: ⟨-2.471 -11.171 +18.366 +3.121]
- CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.4228 ¢
- error map: ⟨0.000 -9.418 +21.646 +8.288]
Optimal ET sequence: 4, 13cd, 17c, 21, 38c
Badness (Sintel): 1.55
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 77/75, 352/343
Mapping: [⟨1 -3 -3 -1 -8], ⟨0 6 7 5 15]]
Optimal tunings:
- WE: ~2 = 1196.3345 ¢, ~5/3 = 913.9469 ¢
- CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.2970 ¢
Optimal ET sequence: 4e, …, 13cdee, 17c
Badness (Sintel): 1.48
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 66/65, 352/343
Mapping: [⟨1 3 4 4 7 7], ⟨0 6 7 5 15 14]]
Optimal tunings:
- WE: ~2 = 1196.0784 ¢, ~5/3 = 914.1466 ¢
- CWE: ~2 = 1200.0000 ¢, ~5/3 = 916.6712 ¢
Optimal ET sequence: 4ef, …, 13cdeef, 17c
Badness (Sintel): 1.34
Subkla
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 1029/1000
Mapping: [⟨1 -3 -3 -1 5], ⟨0 6 7 5 -2]]
Optimal tunings:
- WE: ~2 = 1196.3809 ¢, ~5/3 = 913.4153 ¢
- CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.8804 ¢
Optimal ET sequence: 4, 13cd, 17c, 38ce
Badness (Sintel): 1.56
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 56/55, 66/65, 640/637
Mapping: [⟨1 -3 -3 -1 5 6], ⟨0 6 7 5 -2 -3]]
Optimal tunings:
- WE: ~2 = 1196.5477 ¢, ~5/3 = 913.4822 ¢
- CWE: ~2 = 1200.0000 ¢, ~5/3 = 915.9416 ¢
Optimal ET sequence: 4, 17c, 38ce
Badness (Sintel): 1.52
Naian
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Naian.
Named by Xenllium in 2026, naian may be described as 8d & 21 with a ploidacot signature of epsilon-enneacot.
Subgroup: 2.3.5.7
Comma list: 36/35, 9604/9375
Mapping: [⟨1 -4 -2 -4], ⟨0 9 7 11]]
- mapping generators: ~2, ~75/49
- WE: ~2 = 1195.8807 ¢, ~75/49 = 741.8522 ¢
- error map: ⟨-4.119 -8.808 +14.890 +8.026]
- CWE: ~2 = 1200.0000 ¢, ~75/49 = 743.9885 ¢
- error map: ⟨0.000 -6.059 +21.606 +15.047]
Optimal ET sequence: 8d, 21, 29cd
Badness (Sintel): 2.89
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 2541/2500
Mapping: [⟨1 -4 -2 -4 1], ⟨0 9 7 11 4]]
Optimal tunings:
- WE: ~2 = 1194.3937 ¢, ~75/49 = 741.2117 ¢
- CWE: ~2 = 1200.0000 ¢, ~75/49 = 744.2018 ¢
Optimal ET sequence: 8d, 21, 29cde
Badness (Sintel): 1.95
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 56/55, 66/65, 507/500
Mapping: [⟨1 -4 -2 -4 1 0], ⟨0 9 7 11 4 6]]
Optimal tunings:
- WE: ~2 = 1193.8564 ¢, ~20/13 = 740.9635 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2703 ¢
Optimal ET sequence: 8d, 21, 29cdef
Badness (Sintel): 1.55
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 36/35, 51/50, 56/55, 66/65, 170/169
Mapping: [⟨1 -4 -2 -4 1 0 1], ⟨0 9 7 11 4 6 5]]
Optimal tunings:
- WE: ~2 = 1194.2330 ¢, ~20/13 = 741.1242 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.2679 ¢
Optimal ET sequence: 8d, 21, 29cdef
Badness (Sintel): 1.43
Slurpee
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Slurpee.
Slurpee may be described as the 16 & 17c temperament. It has a generator that is a somewhat flat semitone of ~21/20, three make 8/7, seven make 4/3, and eleven make 8/5, with a ploidacot signature of omega-heptacot.
Subgroup: 2.3.5.7
Comma list: 36/35, 51200/50421
Mapping: [⟨1 2 3 3], ⟨0 -7 -11 -3]]
- mapping generators: ~2, ~21/20
- WE: ~2 = 1197.9281 ¢, ~21/20 = 72.1780 ¢
- error map: ⟨-2.072 -11.345 +13.512 +8.424]
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4941 ¢
- error map: ⟨0.000 -9.414 +16.251 +13.692]
Optimal ET sequence: 16, 17c, 33
Badness (Sintel): 2.91
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 121/120, 352/343
Mapping: [⟨1 2 3 3 4], ⟨0 -7 -11 -3 -9]]
Optimal tunings:
- WE: ~2 = 1198.4220 ¢, ~21/20 = 72.2015 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4470 ¢
Optimal ET sequence: 16, 17c, 33
Badness (Sintel): 1.67
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 66/65, 143/140, 352/343
Mapping: [⟨1 2 3 3 4 4], ⟨0 -7 -11 -3 -9 -5]]
Optimal tunings:
- WE: ~2 = 1199.0238 ¢, ~21/20 = 72.3506 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 72.4956 ¢
Optimal ET sequence: 16, 17c, 33
Badness (Sintel): 1.37
Shallowtone
- For the 5-limit version, see Syntonic–chromatic equivalence continuum #Shallowtone (5-limit).
Subgroup: 2.3.5.7
Comma list: 36/35, 295245/262144
Mapping: [⟨1 0 18 -16], ⟨0 1 -10 12]]
- mapping generators: ~2, ~3
- WE: ~2 = 1202.4397 ¢, ~3/2 = 682.6219 ¢
- error map: ⟨+2.440 -16.893 +6.986 +12.877]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2447 ¢
- error map: ⟨0.000 -20.710 +1.239 +6.110]
Optimal ET sequence: 7, 30b, 37b
Badness (Sintel): 7.79
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 72171/65536
Mapping: [⟨1 0 18 -16 16], ⟨0 1 -10 12 -8]]
Optimal tunings:
- WE: ~2 = 1202.0285 ¢, ~3/2 = 682.4922 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.3267 ¢
Optimal ET sequence: 7, 30b, 37b
Badness (Sintel): 4.29
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 36/35, 45/44, 16731/16384
Mapping: [⟨1 0 18 -16 16 -1], ⟨0 1 -10 12 -8 3]]
Optimal tunings:
- WE: ~2 = 1201.4928 ¢, ~3/2 = 682.1188 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 681.2669 ¢
Optimal ET sequence: 7, 30b, 37b
Badness (Sintel): 3.19