Minortone family: Difference between revisions

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This tempers out the minortone comma, |-16 35 -17>. The head of the family is minortonic temperament, with generator a minor tone.
{{Technical data page}}
The '''minortone family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[minortone comma]] ({{monzo|legend=1| -16 35 -17 }}).  


[[Comma|Comma]]: |-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 [[6/1|6th harmonic]], and another eighteen generators make the interval class of the [[5/1|5th harmonic]]. Its [[ploidacot]] is beta-17-cot. The generator should be tuned about 1/16 of a cent flat, with 6<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. [[171edo|171-]], [[559edo|559-]] and [[730edo]] are among the possible edo tunings.


[[POTE_tuning|POTE generator]]: ~10/9 = 182.466
[[Subgroup]]: 2.3.5


[[Map|Map]]: [&lt;1 16 32|, &lt;0 -17 -35|]
[[Comma list]]: {{monzo| -16 35 -17 }}


[[EDO|EDO]]s: [[46edo|46]], [[125edo|125]], [[171edo|171]], [[388edo|388]], 559, 730, 1289, 2019, 2749, 4768, 16323, 21091
{{Mapping|legend=1| 1 -1 -3 | 0 17 35 }}
__FORCETOC__
: mapping generators: ~2, ~10/9
= Mitonic =
{{see also|Ragismic microtemperaments #Mitonic}}


As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, |-16 35 -17&gt;. Flipping that gives the 5-limit wedgie &lt;&lt;17 35 16||, which tells us that 10/9 can be taken as the generator, with 17 of them giving a 6, 18 of them a 20/3, and 35 of them giving a 40. The generator should be tuned about 1/16 of a cent flat, with 6^(1/17) being 0.06423 cents flat and 40^(1/35) being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0036{{c}}, ~10/9 = 182.4670{{c}}
: [[error map]]: {{val| +0.004 -0.020 +0.020 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 182.4666{{c}}
: error map: {{val| 0.000 -0.023 +0.017 }}


However, as noted before, 32/21 is only a ragisma shy of (10/9)^4, and so a 7-limit interpretation, 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|171edo]]. The wedgie is now &lt;&lt;17 35 -21 16 -81 -147||, with 21 10/9 generators giving a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.
{{Optimal ET sequence|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091, 25859b }}


[[Comma|Comma]]s: 4375/4374, 2100875/2097152
[[Badness]] (Sintel): 0.698


[[POTE_generator|POTE generator]]: ~10/9 = 182.458
== Mitonic ==
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.


[[Map|Map]]: [&lt;1 16 32 -15|, &lt;0 -17 -35 21|]
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''.


[[EDO|EDOs]]: {{EDOs|7, 20c, 33c, 46, 125, 171}}
[[Subgroup]]: 2.3.5.7


[[Badness|Badness]]: 0.0252
[[Comma list]]: 4375/4374, 2100875/2097152


=Domain=
{{Mapping|legend=1| 1 -1 -3 6 | 0 17 35 -21 }}
Domain temperament 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 [[Chromatic_pairs#Terrain|terrain]].


[[Comma|Comma]]s: 250047/250000, 645700815/645657712
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0963{{c}}, ~10/9 = 182.4727{{c}}
: [[error map]]: {{val| +0.096 -0.015 -0.057 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 182.4592{{c}}
: error map: {{val| 0.000 -0.149 -0.243 -0.468 }}


[[POTE_generator|POTE generator]]: ~10/9 = 182.467
{{Optimal ET sequence|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}


[[Map|Map]]: [&lt;3 14 26 28|, &lt;0 -17 -35 -36|]
[[Badness]] (Sintel): 0.637


[[EDO|EDO]]S: 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038
=== Mineral ===
Subgroup: 2.3.5.7.11


[[Badness|Badness]]: 0.0140
Comma list: 441/440, 4375/4374, 16384/16335


[[Category:Theory]]
Mapping: {{mapping| 1 -1 -3 6 10 | 0 17 35 -21 -43 }}
[[Category:Temperament family]]
[[Category:Minortonic]]


[[Category:Todo:intro]]
Optimal tuning:
* WE: ~2 = 1200.8759{{c}}, ~10/9 = 182.4631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4818{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -3 6 10 11 | 0 17 35 -21 -43 -48 }}
 
Optimal tunings:
* WE: ~2 = 1199.8808{{c}}, ~10/9 = 182.4633{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4816{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -3 6 10 11 5 | 0 17 35 -21 -43 -48 -6 }}
 
Optimal tunings:
* WE: ~2 = 1199.8923{{c}}, ~10/9 = 182.4650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4816{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}
 
Badness (Sintel): 1.01
 
=== Ore ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 4375/4374
 
Mapping: {{mapping| 1 -1 -3 6 3 | 0 17 35 -21 3 }}
 
Optimal tunings:
* WE: ~2 = 1199.3408{{c}}, ~10/9 = 182.5004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4518{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}
 
Optimal tunings:
* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}
 
Optimal tunings:
* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}
 
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125, 171e }}
 
Badness (Sintel): 1.45
 
=== Seminar ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 2100875/2097152
 
Mapping: {{mapping| 2 -2 -6 12 13 | 0 17 35 -21 -20 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0535{{c}}, ~10/9 = 182.4735{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 182.4578{{c}}
 
{{Optimal ET sequence|legend=0| 46, 250, 296, 342, 3466cddee, 3808bcddeee, …, 5176bccdddeeee }}
 
Badness (Sintel): 0.886
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17
 
Comma list: 1089/1088, 1225/1224, 1701/1700, 262395/262144
 
Mapping: {{mapping| 2 -2 -6 12 13 4 | 0 17 35 -21 -20 6 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0371{{c}}, ~10/9 = 182.4699{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 182.4589{{c}}
 
{{Optimal ET sequence|legend=0| 46, 250, 296, 342, 1414de }}
 
Badness (Sintel): 0.831
 
== 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|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
: mapping generators: ~63/50, ~10/9
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0013{{c}}, ~10/9 = 182.4672{{c}}
: [[error map]]: {{val| +0.004 -0.016 +0.028 -0.016 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 182.4668{{c}}
: error map: {{val| 0.000 -0.020 +0.024 -0.021 }}
 
{{Optimal ET sequence|legend=1| 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: {{mapping| 6 -6 -18 -16 -13 | 0 17 35 36 37 }}
: mapping generators: ~55/49, ~10/9
 
Optimal tunings:
* WE: ~55/49 = 200.0009{{c}}, ~10/9 = 182.4669{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~10/9 = 182.4675{{c}}
 
{{Optimal ET sequence|legend=0| 342, 1164, 1506, 1848, 4038, 5886, 9924 }}
 
Badness (Sintel): 0.360
 
[[Category:Minortone family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Catalogs of rank-2 temperaments]]

Latest revision as of 12:12, 14 August 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

2.3.5.7.11.17 subgroup

Subgroup: 2.3.5.7.11.17

Comma list: 1089/1088, 1225/1224, 1701/1700, 262395/262144

Mapping: [⟨2 -2 -6 12 13 4], ⟨0 17 35 -21 -20 6]]

Optimal tunings:

  • WE: ~99/70 = 600.0371 ¢, ~10/9 = 182.4699 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~10/9 = 182.4589 ¢

Optimal ET sequence: 46, 250, 296, 342, 1414de

Badness (Sintel): 0.831

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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