Mint family: Difference between revisions

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The '''mint family''' of rank-3 temperaments tempers out [[36/35]], identifying both [[7/6]] with [[6/5]] and [[5/4]] with [[9/7]].
{{Technical data page}}
The '''mint family''' of [[rank-3 temperament]]s tempers out [[36/35]], identifying both [[7/6]] with [[6/5]] and [[5/4]] with [[9/7]].


== Mint  ==
== Mint  ==
 
[[Subgroup]]: 2.3.5.7
Subgroup: 2.3.5.7


[[Comma list]]: 36/35
[[Comma list]]: 36/35


[[Mapping]]: [{{val| 1 0 0 2 }}, {{val| 0 1 0 2 }}, {{val| 0 0 1 -1 }}]
{{Mapping|legend=1| 1 0 0 2 | 0 1 0 2 | 0 0 1 -1 }}


[[POTE generator]]s: ~3/2 = 698.6351, ~5/4 = 404.2809
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 698.6351, ~5/4 = 404.2809


{{Val list|legend=1| 4, 5, 7, 8d, 9, 12 }}
{{Optimal ET sequence|legend=1| 4, 5, 7, 8d, 9, 12 }}


[[Badness]]: 0.0637 × 10<sup>-3</sup>
[[Badness]]: 0.0637 × 10<sup>-3</sup>


== Nickel  ==
== Nickel  ==
 
[[Subgroup]]: 2.3.5.7.11
Subgroup: 2.3.5.7.11


[[Comma list]]: 36/35, [[45/44]]
[[Comma list]]: 36/35, [[45/44]]


[[Mapping]]: [{{val| 1 0 0 2 -2 }}, {{val| 0 1 0 2 2 }}, {{val| 0 0 1 -1 1 }}]
{{Mapping|legend=1| 1 0 0 2 -2 | 0 1 0 2 2 | 0 0 1 -1 1 }}


[[POTE generator]]s: ~3/2 = 688.8370, ~5/4 = 391.8801
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 688.8370, ~5/4 = 391.8801


{{Val list|legend=1| 7, 9, 12, 21, 28 }}
{{Optimal ET sequence|legend=1| 7, 9, 12, 21, 28 }}


[[Badness]]: 0.463 × 10<sup>-3</sup>
[[Badness]]: 0.463 × 10<sup>-3</sup>


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


Comma list: [[27/26]], 36/35, 45/44
Comma list: [[27/26]], 36/35, 45/44


Mapping: [{{val| 1 0 0 2 -2 -1 }}, {{val| 0 1 0 2 2 3 }}, {{val| 0 0 1 -1 1 0 }}]
Mapping: {{mapping| 1 0 0 2 -2 -1 | 0 1 0 2 2 3 | 0 0 1 -1 1 0 }}


POTE generators: ~3/2 = 685.0048, ~5/4 = 390.2563
Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 685.0048, ~5/4 = 390.2563


{{Val list|legend=1| 7, 9, 12f, 16, 21, 28 }}
{{Optimal ET sequence|legend=1| 7, 9, 12f, 16, 21, 28 }}


Badness: 0.647 × 10<sup>-3</sup>
Badness: 0.647 × 10<sup>-3</sup>


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Mint]]
[[Category:Mint family| ]] <!-- main article -->
[[Category:Mint| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Latest revision as of 00:41, 24 June 2025

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 mint family of rank-3 temperaments tempers out 36/35, identifying both 7/6 with 6/5 and 5/4 with 9/7.

Mint

Subgroup: 2.3.5.7

Comma list: 36/35

Mapping: [⟨1 0 0 2], ⟨0 1 0 2], ⟨0 0 1 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 698.6351, ~5/4 = 404.2809

Optimal ET sequence: 4, 5, 7, 8d, 9, 12

Badness: 0.0637 × 10-3

Nickel

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44

Mapping: [⟨1 0 0 2 -2], ⟨0 1 0 2 2], ⟨0 0 1 -1 1]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 688.8370, ~5/4 = 391.8801

Optimal ET sequence: 7, 9, 12, 21, 28

Badness: 0.463 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44

Mapping: [⟨1 0 0 2 -2 -1], ⟨0 1 0 2 2 3], ⟨0 0 1 -1 1 0]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 685.0048, ~5/4 = 390.2563

Optimal ET sequence: 7, 9, 12f, 16, 21, 28

Badness: 0.647 × 10-3

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