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Minortone family: Difference between revisions

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== Minortone ==
== Minortone ==
The head of this family is the 5-limit minortone temperament, with generator a minor tone.
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.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
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{{Mapping|legend=1| 1 -1 -3 | 0 17 35 }}
{{Mapping|legend=1| 1 -1 -3 | 0 17 35 }}
: mapping generators: ~2, ~10/9


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10/9 = 182.466{{c}}
[[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 }}


{{Optimal ET sequence|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091 }}
{{Optimal ET sequence|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091, 25859b }}


[[Badness]] (Smith): 0.029765
[[Badness]] (Sintel): 0.698


== Mitonic ==
== Mitonic ==
As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo| -16 35 -17 }}. 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<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.
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.


However, as noted before, 32/21 is only a ragisma shy of (10/9)<sup>4</sup>, 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]]. 21 generators gives a ~64/7. [[Mos scale]]s of size 20, 33, 46 or 79 notes can be used for mitonic.
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'').


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 -1 -3 6 | 0 17 35 -21 }}
{{Mapping|legend=1| 1 -1 -3 6 | 0 17 35 -21 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~10/9 = 182.458{{c}}
[[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 }}


{{Optimal ET sequence|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}
{{Optimal ET sequence|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}


[[Badness]] (Smith): 0.025184
[[Badness]] (Sintel): 0.637


=== Mineral ===
=== Mineral ===
Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: ''mineral'' (46 &amp; 171) and ''ore'' (46 &amp; 125). The mineral temperament tempers out 441/440 and 16384/16335 in the 11-limit. 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").
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 43: Line 50:
Mapping: {{mapping| 1 -1 -3 6 10 | 0 17 35 -21 -43 }}
Mapping: {{mapping| 1 -1 -3 6 10 | 0 17 35 -21 -43 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.482{{c}}
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, 605ee, 822dee }}
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}


Badness (Smith): 0.059060
Badness (Sintel): 1.95


==== 13-limit ====
==== 13-limit ====
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Mapping: {{mapping| 1 -1 -3 6 10 11 | 0 17 35 -21 -43 -48 }}
Mapping: {{mapping| 1 -1 -3 6 10 11 | 0 17 35 -21 -43 -48 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.481{{c}}
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, 605ee, 822dee }}
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}


Badness (Smith): 0.033140
Badness (Sintel): 1.37


==== 17-limit ====
==== 17-limit ====
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Mapping: {{mapping| 1 -1 -3 6 10 11 5 | 0 17 35 -21 -43 -48 -6 }}
Mapping: {{mapping| 1 -1 -3 6 10 11 5 | 0 17 35 -21 -43 -48 -6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.481{{c}}
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, 605ee, 822dee }}
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}


Badness (Smith): 0.019792
Badness (Sintel): 1.01


=== Ore ===
=== Ore ===
The ore temperament tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Mapping: {{mapping| 1 -1 -3 6 3 | 0 17 35 -21 3 }}
Mapping: {{mapping| 1 -1 -3 6 3 | 0 17 35 -21 3 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.449{{c}}
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=1| 46, 125, 171e }}
{{Optimal ET sequence|legend=0| 46, 125, 171e }}


Badness (Smith): 0.053662
Badness (Sintel): 1.77


==== 13-limit ====
==== 13-limit ====
Line 97: Line 110:
Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}
Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.470{{c}}
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, 388ee }}
{{Optimal ET sequence|legend=0| 46, 125, 171e }}


Badness (Smith): 0.046170
Badness (Sintel): 1.91


===== 17-limit =====
===== 17-limit =====
Line 110: Line 125:
Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}
Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.471{{c}}
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, 388ee }}
{{Optimal ET sequence|legend=0| 46, 125, 171e }}


Badness (Smith): 0.028423
Badness (Sintel): 1.45


==== Goldmine ====
==== Goldmine ====
Line 125: Line 142:
Mapping: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}
Mapping: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.437{{c}}
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, 296eff }}
{{Optimal ET sequence|legend=0| 46, 79, 125f, 171ef }}


Badness (Smith): 0.039302
Badness (Sintel): 1.62


===== 17-limit =====
===== 17-limit =====
Line 138: Line 157:
Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}
Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.444{{c}}
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 }}
{{Optimal ET sequence|legend=0| 46, 125f, 171ef }}


Badness (Smith): 0.027440
Badness (Sintel): 1.40


=== Seminar ===
=== Seminar ===
Line 151: Line 172:
Mapping: {{mapping| 2 -2 -6 12 13 | 0 17 35 -21 -20 }}
Mapping: {{mapping| 2 -2 -6 12 13 | 0 17 35 -21 -20 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~10/9 = 182.457{{c}}
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, 204c, 250, 296, 342 }}
{{Optimal ET sequence|legend=0| 46, 250, 296, 342, 3466cddee, 3808bcddeee, …, 5176bccdddeeee }}


Badness (Smith): 0.026808
Badness (Sintel): 0.886


== Domain ==
== Domain ==
{{See also| Landscape microtemperaments #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]].
 
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 [[Subgroup temperaments #Terrain|terrain]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
: mapping generators: ~63/50, ~10/9


[[Optimal tuning]] ([[POTE]]): ~63/50 = 400.000{{c}}, ~10/9 = 182.467{{c}}
[[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 }}
{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038 }}


[[Badness]] (Smith): 0.013979
[[Badness]] (Sintel): 0.354


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


Comma list: 9801/9800, 250047/250000, 14348907/14348180
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


Mapping: {{mapping| 6 11 17 20 24 | 0 -17 -35 -36 -37 }}
Optimal tunings:  
: mapping generators: ~55/49 = 200.000{{c}}, ~100/99 = 17.533{{c}}
* WE: ~55/49 = 200.0009{{c}}, ~10/9 = 182.4669{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~10/9 = 182.4675{{c}}


Optimal tuning (CTE): ~100/99 = 17.533
{{Optimal ET sequence|legend=0| 342, 1164, 1506, 1848, 4038, 5886, 9924 }}


{{Optimal ET sequence|legend=0| 342, 480, 822, 1164, 1506, 1848, … }}
Badness (Sintel): 0.360


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Minortone family| ]] <!-- main article -->
[[Category:Minortone family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Minortone_family"