Minortone family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings. Improve intro to each entry
Hemidomain -> semidomain; 13-limit ore -> orefield (see talk)
 
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Mitonic takes advantage of the fact that [[32/21]] is only a [[ragisma]] shy of (10/9)<sup>4</sup>, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While [[559edo|559]] or [[730edo|730]] are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. Generator chains of 46 or 79 notes can be used for this temperament.
Mitonic takes advantage of the fact that [[32/21]] is only a [[ragisma]] shy of (10/9)<sup>4</sup>, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While [[559edo|559]] or [[730edo|730]] are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. Generator chains of 46 or 79 notes can be used for this temperament.


Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & 125). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators. In the 17-limit, both mineral and ore temper out 833/832, 1225/1224, 1701/1700, and 4096/4095 (2.3.5.7.13.17 commas). The word ''mineral'' is related to ''mine'' (an excavation from which ore or solid minerals are taken) and ''miner'' (a person who works in a mine, also as a pun on ''minor'').
Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & 171e). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators. The latter further branches to goldmine (46 & 125f), which equates [[16/13]] with a stack of two [[10/9]]'s and tempers together [[13/12]] and [[14/13]], and orefield (46 & 125), which uses the same mapping for 13 as mineral. All have the same mapping for 17, at -6 generators. The word ''mineral'' is related to ''mine'' and ''miner'', a pun on ''minor''.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness (Sintel): 1.77
Badness (Sintel): 1.77


==== 13-limit ====
==== Goldmine ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 385/384, 1331/1323, 3267/3250
Comma list: 169/168, 325/324, 385/384, 1331/1323


Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}
Mapping: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}


{{Optimal ET sequence|legend=0| 46, 125, 171e }}
{{Optimal ET sequence|legend=0| 46, 79, 125f, 171ef }}


Badness (Sintel): 1.91
Badness (Sintel): 1.62


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445
Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323


Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}
Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}


{{Optimal ET sequence|legend=0| 46, 125, 171e }}
{{Optimal ET sequence|legend=0| 46, 125f, 171ef }}


Badness (Sintel): 1.45
Badness (Sintel): 1.40
 
==== Goldmine ====
The goldmine temperament (46 & 79) is another 13-limit extension of ore, equating [[13/12]] with [[14/13]] and [[16/13]] with two [[10/9]]s.


==== Orefield ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 385/384, 1331/1323
Comma list: 352/351, 385/384, 1331/1323, 3267/3250


Mapping: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}
Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}


{{Optimal ET sequence|legend=0| 46, 79, 125f, 171ef }}
{{Optimal ET sequence|legend=0| 46, 125, 171e }}


Badness (Sintel): 1.62
Badness (Sintel): 1.91


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323
Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445


Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}
Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}


{{Optimal ET sequence|legend=0| 46, 125f, 171ef }}
{{Optimal ET sequence|legend=0| 46, 125, 171e }}


Badness (Sintel): 1.40
Badness (Sintel): 1.45


=== Seminar ===
=== Seminar ===
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[[Badness]] (Sintel): 0.354
[[Badness]] (Sintel): 0.354


=== Hemidomain ===
=== Semidomain ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Latest revision as of 12:53, 20 June 2026

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 minortone family of temperaments tempers out the minortone comma (monzo: [-16 35 -17⟩).

Minortone

The head of this family is the 5-limit minortone temperament, a super-accurate microtemperament with generator a minor tone of 10/9, seventeen of which make the 6th harmonic, and another eighteen generators make the interval class of the 5th harmonic. Its ploidacot is beta-17-cot. The generator should be tuned about 1/16 of a cent flat, with 61/17 being 0.06423 cents flat and 401/35 being 0.06234 cents flat. 171-, 559- and 730edo are among the possible edo tunings.

Subgroup: 2.3.5

Comma list: [-16 35 -17⟩

Mapping: [⟨1 -1 -3], ⟨0 17 35]]

mapping generators: ~2, ~10/9

Optimal tunings:

  • WE: ~2 = 1200.0036 ¢, ~10/9 = 182.4670 ¢
error map: ⟨+0.004 -0.020 +0.020]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4666 ¢
error map: ⟨0.000 -0.023 +0.017]

Optimal ET sequence: 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091, 25859b

Badness (Sintel): 0.698

Mitonic

Mitonic takes advantage of the fact that 32/21 is only a ragisma shy of (10/9)4, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in 171edo. Generator chains of 46 or 79 notes can be used for this temperament.

Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & 171e). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps 11/8 to three generators. The latter further branches to goldmine (46 & 125f), which equates 16/13 with a stack of two 10/9's and tempers together 13/12 and 14/13, and orefield (46 & 125), which uses the same mapping for 13 as mineral. All have the same mapping for 17, at -6 generators. The word mineral is related to mine and miner, a pun on minor.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2100875/2097152

Mapping: [⟨1 -1 -3 6], ⟨0 17 35 -21]]

Optimal tunings:

  • WE: ~2 = 1200.0963 ¢, ~10/9 = 182.4727 ¢
error map: ⟨+0.096 -0.015 -0.057 -0.176]
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4592 ¢
error map: ⟨0.000 -0.149 -0.243 -0.468]

Optimal ET sequence: 46, 125, 171, 1927d, 2098d, …, 3637bcdd

Badness (Sintel): 0.637

Mineral

Subgroup: 2.3.5.7.11

Comma list: 441/440, 4375/4374, 16384/16335

Mapping: [⟨1 -1 -3 6 10], ⟨0 17 35 -21 -43]]

Optimal tuning:

  • WE: ~2 = 1200.8759 ¢, ~10/9 = 182.4631 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4818 ¢

Optimal ET sequence: 46, 125e, 171, 217

Badness (Sintel): 1.95

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 364/363, 441/440, 3584/3575, 4375/4374

Mapping: [⟨1 -1 -3 6 10 11], ⟨0 17 35 -21 -43 -48]]

Optimal tunings:

  • WE: ~2 = 1199.8808 ¢, ~10/9 = 182.4633 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4816 ¢

Optimal ET sequence: 46, 125e, 171, 217

Badness (Sintel): 1.37

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 364/363, 441/440, 595/594, 1156/1155, 3584/3575

Mapping: [⟨1 -1 -3 6 10 11 5], ⟨0 17 35 -21 -43 -48 -6]]

Optimal tunings:

  • WE: ~2 = 1199.8923 ¢, ~10/9 = 182.4650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4816 ¢

Optimal ET sequence: 46, 125e, 171, 217

Badness (Sintel): 1.01

Ore

Subgroup: 2.3.5.7.11

Comma list: 385/384, 1331/1323, 4375/4374

Mapping: [⟨1 -1 -3 6 3], ⟨0 17 35 -21 3]]

Optimal tunings:

  • WE: ~2 = 1199.3408 ¢, ~10/9 = 182.5004 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4518 ¢

Optimal ET sequence: 46, 125, 171e

Badness (Sintel): 1.77

Goldmine

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 1331/1323

Mapping: [⟨1 -1 -3 6 3 4], ⟨0 17 35 -21 3 -2]]

Optimal tunings:

  • WE: ~2 = 1200.5647 ¢, ~10/9 = 182.5224 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4409 ¢

Optimal ET sequence: 46, 79, 125f, 171ef

Badness (Sintel): 1.62

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323

Mapping: [⟨1 -1 -3 6 3 4 5], ⟨0 17 35 -21 3 -2 -6]]

Optimal tunings:

  • WE: ~2 = 1200.4506 ¢, ~10/9 = 182.5124 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4465 ¢

Optimal ET sequence: 46, 125f, 171ef

Badness (Sintel): 1.40

Orefield

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 385/384, 1331/1323, 3267/3250

Mapping: [⟨1 -1 -3 6 3 11], ⟨0 17 35 -21 3 -48]]

Optimal tunings:

  • WE: ~2 = 1200.1737 ¢, ~10/9 = 182.4966 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4705 ¢

Optimal ET sequence: 46, 125, 171e

Badness (Sintel): 1.91

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445

Mapping: [⟨1 -1 -3 6 3 11 5], ⟨0 17 35 -21 3 -48 -6]]

Optimal tunings:

  • WE: ~2 = 1200.1403 ¢, ~10/9 = 182.4924 ¢
  • CWE: ~2 = 1200.0000 ¢, ~10/9 = 182.4712 ¢

Optimal ET sequence: 46, 125, 171e

Badness (Sintel): 1.45

Seminar

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2100875/2097152

Mapping: [⟨2 -2 -6 12 13], ⟨0 17 35 -21 -20]]

Optimal tunings:

  • WE: ~99/70 = 600.0535 ¢, ~10/9 = 182.4735 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~10/9 = 182.4578 ¢

Optimal ET sequence: 46, 250, 296, 342, 3466cddee, 3808bcddeee, …, 5176bccdddeeee

Badness (Sintel): 0.886

Domain

Domain adds the landscape comma, 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament terrain.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 645700815/645657712

Mapping: [⟨3 -3 -9 -8], ⟨0 17 35 36]]

mapping generators: ~63/50, ~10/9

Optimal tunings:

  • WE: ~63/50 = 400.0013 ¢, ~10/9 = 182.4672 ¢
error map: ⟨+0.004 -0.016 +0.028 -0.016]
  • CWE: ~63/50 = 400.0000 ¢, ~10/9 = 182.4668 ¢
error map: ⟨0.000 -0.020 +0.024 -0.021]

Optimal ET sequence: 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038

Badness (Sintel): 0.354

Semidomain

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 151263/151250, 1771561/1771470

Mapping: [⟨6 -6 -18 -16 -13], ⟨0 17 35 36 37]]

mapping generators: ~55/49, ~10/9

Optimal tunings:

  • WE: ~55/49 = 200.0009 ¢, ~10/9 = 182.4669 ¢
  • CWE: ~55/49 = 200.0000 ¢, ~10/9 = 182.4675 ¢

Optimal ET sequence: 342, 1164, 1506, 1848, 4038, 5886, 9924

Badness (Sintel): 0.360

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