Xenharmonic Wiki

Mint temperaments

Revision as of 07:05, 10 March 2026 by FloraC (talk | contribs) (Units & misc. cleanup. - redundant category)
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

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~7/6 = 231.845 ¢
error map: ⟨0.000 -6.421 +77.376 -41.447]
  • POTE: ~2 = 1200.000 ¢, ~7/6 = 239.977 ¢
error map: ⟨0.000 +17.976 +93.640 -8.918]

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.

Progression, named by Gene Ward Smith in 2011[1] can be described as the 8d & 9 temperament, and 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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~15/14 = 138.611 ¢
error map: ⟨0.000 -8.900 +29.519 +1.451]
  • POTE: ~2 = 1200.000 ¢, ~15/14 = 140.927 ¢
error map: ⟨0.000 +2.680 +36.467 +17.663]

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:

  • CTE: ~2 = 1200.000 ¢, ~12/11 = 138.556 ¢
  • POTE: ~2 = 1200.000 ¢, ~12/11 = 140.747 ¢

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:

  • CTE: ~2 = 1200.000 ¢, ~13/12 = 138.741 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/12 = 140.751 ¢

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:

  • CTE: ~2 = 1200.000 ¢, ~13/12 = 138.649 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/12 = 141.404 ¢

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:

  • CTE: ~2 = 1200.000 ¢, ~13/12 = 138.750 ¢
  • POTE: ~2 = 1200.000 ¢, ~13/12 = 140.479 ¢

Optimal ET sequence: 8d, 9, 17cg

Badness (Sintel): 1.02

Subklei

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

Subgroup: 2.3.5.7

Comma list: 36/35, 1029/1000

Mapping: [⟨1 3 4 4], ⟨0 -6 -7 -5]]

mapping generators: ~2, ~7/6

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~7/6 = 284.986 ¢
error map: ⟨0.000 -11.870 +18.785 +6.245]
  • POTE: ~2 = 1200.000 ¢, ~7/6 = 284.219 ¢
error map: ⟨0.000 -7.268 +24.154 +10.080]

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 4 4 7], ⟨0 -6 -7 -5 -15]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~7/6 = 284.265 ¢
  • POTE: ~2 = 1200.000 ¢, ~7/6 = 283.253 ¢

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:

  • CTE: ~2 = 1200.000 ¢, ~7/6 = 283.918 ¢
  • POTE: ~2 = 1200.000 ¢, ~7/6 = 282.856 ¢

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 4 4 3], ⟨0 -6 -7 -5 2]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~7/6 = 284.870 ¢
  • POTE: ~2 = 1200.000 ¢, ~7/6 = 283.822 ¢

Optimal ET sequence: 4, 17c, 21, 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 4 4 3 3], ⟨0 -6 -7 -5 2 3]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~7/6 = 284.758 ¢
  • POTE: ~2 = 1200.000 ¢, ~7/6 = 283.882 ¢

Optimal ET sequence: 4, 17c, 38ce

Badness (Sintel): 1.52

Naian

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

Subgroup: 2.3.5.7

Comma list: 36/35, 9604/9375

Mapping: [⟨1 5 5 7], ⟨0 -9 -7 -11]]

mapping generators: ~2, ~98/75

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~98/75 = 456.494 ¢
error map: ⟨0.000, -10.401, +18.228, +9.740]
  • CWE: ~2 = 1200.000 ¢, ~98/75 = 456.012 ¢
error map: ⟨0.000, -6.059, +21.606, +15.047]

Optimal ET sequence: 8d, 13b, 21

Badness (Sintel): 2.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 56/55, 2541/2500

Mapping: [⟨1 5 5 7 5], ⟨0 -9 -7 -11 -4]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~77/60 = 456.625 ¢
  • CWE: ~2 = 1200.000 ¢, ~77/60 = 455.798 ¢

Optimal ET sequence: 8d, 13b, 21

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 5 5 7 5 6], ⟨0 -9 -7 -11 -4 -6]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~13/10 = 456.767 ¢
  • CWE: ~2 = 1200.000 ¢, ~13/10 = 455.730 ¢

Optimal ET sequence: 8d, 13b, 21

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 5 5 7 5 6 6], ⟨0 -9 -7 -11 -4 -6 -5]]

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~13/10 = 456.821 ¢
  • CWE: ~2 = 1200.000 ¢, ~13/10 = 455.732 ¢

Optimal ET sequence: 8d, 13b, 21

Badness (Sintel): 1.43

Slurpee

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

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~21/20 = 72.776 ¢
error map: ⟨0.000 -11.385 +13.153 +12.847]
  • POTE: ~2 = 1200.000 ¢, ~21/20 = 72.303 ¢
error map: ⟨0.000 -8.075 +18.355 +14.265]

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:

  • CTE: ~2 = 1200.000 ¢, ~21/20 = 72.681 ¢
  • POTE: ~2 = 1200.000 ¢, ~21/20 = 72.297 ¢

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:

  • CTE: ~2 = 1200.000 ¢, ~21/20 = 72.653 ¢
  • POTE: ~2 = 1200.000 ¢, ~21/20 = 72.409 ¢

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

Optimal tunings:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 681.278 ¢
error map: ⟨0.000 -20.677 +0.906 +6.510]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 681.245 ¢
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:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 681.254 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 681.327 ¢

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:

  • CTE: ~2 = 1200.000 ¢, ~3/2 = 681.237 ¢
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 681.267 ¢

Optimal ET sequence: 7, 30b, 37b

Badness (Sintel): 3.19

References

  1. ↑ Yahoo! Tuning Group | Progression temperament
Retrieved from "https://en.xen.wiki/index.php?title=Mint_temperaments&oldid=225498"