Didymus rank three family: Difference between revisions

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The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth.
The '''didymus rank-3 family''' are rank-3 temperaments tempering out the didymus comma, [[81/80]]. If nothing else is tempered out we have a [[7-limit]] [[planar temperament]], with an [[11-limit]] comma we get an 11-limit temperament, and so forth.


Most of these temperaments were named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100647.html#100648 ''Yahoo! Tuning Group | Higher-rank meantone extensions (muse temperaments?)'']</ref>.  
Most of these temperaments were named by [[Gene Ward Smith]] in 2010<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_100647.html#100648 Yahoo! Tuning Group | ''Higher-rank meantone extensions (muse temperaments?)'']</ref>.  


Temperaments discussed elsewhere include:
Temperaments discussed elsewhere include:
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{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }}
{{Mapping|legend=1| 1 0 -4 0 | 0 1 4 0 | 0 0 0 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.2387, ~7/4 = 964.9090
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.2387{{c}}, ~7/4 = 964.9090{{c}}


{{Optimal ET sequence|legend=1| 12, 19, 31, 81 }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 81 }}


[[Badness]]: 0.095 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.095 × 10<sup>-3</sup>


== Euterpe ==
== Euterpe ==
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{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }}
{{Mapping|legend=1| 1 0 -4 0 1 | 0 1 4 0 -2 | 0 0 0 1 2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.1982{{c}}, ~7/4 = 968.4280{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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{{Optimal ET sequence|legend=1| 12, 17c, 19e, 26, 31, 88 }}
{{Optimal ET sequence|legend=1| 12, 17c, 19e, 26, 31, 88 }}


[[Badness]]: 0.536 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.536 × 10<sup>-3</sup>


== Calliope  ==
== Calliope  ==
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{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 -4 0 -6 | 0 1 4 0 6 | 0 0 0 1 0 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.1982, ~7/4 = 968.4280
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.1982{{c}}, ~7/4 = 968.4280{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}
{{Optimal ET sequence|legend=1| 7d, 12, 19, 26, 45 }}


[[Badness]]: 0.530 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.530 × 10<sup>-3</sup>


== Erato ==
== Erato ==
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{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }}
{{Mapping|legend=1| 1 0 -4 -13 0 | 0 1 4 10 0 | 0 0 0 0 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.4949, ~11/8 = 547.0252
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.4949{{c}}, ~11/8 = 547.0252{{c}}


{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81 }}
{{Optimal ET sequence|legend=1| 12, 19, 31, 50, 81 }}


[[Badness]]: 0.558 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.558 × 10<sup>-3</sup>


=== 13-limit  ===
=== 13-limit  ===
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Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }}
Mapping: {{mapping| 1 0 -4 -13 0 -20 | 0 1 4 10 0 15 | 0 0 0 0 1 0 }}


Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 695.9883, ~11/8 = 545.6817
Optimal tuning (POTE): ~2 = 1200.0000{{c}}, ~3/2 = 695.9883{{c}}, ~11/8 = 545.6817{{c}}


{{Optimal ET sequence|legend=1| 12f, 19, 31, 50, 81 }}
{{Optimal ET sequence|legend=0| 12f, 19, 31, 50, 81 }}


== Clio  ==
== Clio  ==
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{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }}
{{Mapping|legend=1| 1 0 -4 0 -12 | 0 1 4 0 8 | 0 0 0 1 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 697.2502, ~7/4 = 968.6295
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 697.2502{{c}}, ~7/4 = 968.6295{{c}}


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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{{Optimal ET sequence|legend=1| 7, 12, 19e, 24, 31, 105, 129 }}
{{Optimal ET sequence|legend=1| 7, 12, 19e, 24, 31, 105, 129 }}


[[Badness]]: 0.738 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.738 × 10<sup>-3</sup>


== Polyhymnia ==
== Polyhymnia ==


[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11
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{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }}
{{Mapping|legend=1| 1 0 -4 0 11 | 0 1 4 0 -3 | 0 0 0 1 -1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.2305, ~7/4 = 964.8695
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.2305{{c}}, ~7/4 = 964.8695{{c}}


{{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31 }}
{{Optimal ET sequence|legend=1| 7, 12e, 19, 24, 26, 31 }}
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Scales: [[polyhymnia12]]
Scales: [[polyhymnia12]]


== Thalia ==
== Thalia ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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{{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }}
{{Mapping|legend=1| 1 0 -4 0 5 | 0 1 4 0 -1 | 0 0 0 1 0 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 692.0796, ~7/4 = 950.2565
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 692.0796{{c}}, ~7/4 = 950.2565{{c}}


{{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e }}
{{Optimal ET sequence|legend=1| 5, 7, 12e, 14c, 19e }}


== Melpomene ==
== Melpomene ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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{{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }}
{{Mapping|legend=1| 1 0 -4 0 7 | 0 1 4 0 -4 | 0 0 0 1 1 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 699.2230, ~7/4 = 964.2363
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 699.2230{{c}}, ~7/4 = 964.2363{{c}}


{{Optimal ET sequence|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }}
{{Optimal ET sequence|legend=1| 7d, 12, 17c, 19, 24, 31e, 36 }}


 
== Terpsichore ==
== Terpsichore ==
[[Subgroup]]: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


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{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}
{{Mapping|legend=1| 1 0 -4 0 -2 | 0 1 4 0 7 | 0 0 0 1 -2 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 696.2358, ~7/4 = 964.0006
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 696.2358{{c}}, ~7/4 = 964.0006{{c}}


{{Optimal ET sequence|legend=1| 14c, 17c, 19, 31, 81, 112b }}
{{Optimal ET sequence|legend=1| 14c, 17c, 19, 31, 81, 112b }}


[[Badness]]: 0.850 × 10<sup>-3</sup>
[[Badness]] (Smith): 0.850 × 10<sup>-3</sup>


[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11
[[Complexity spectrum]]: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11

Latest revision as of 12:48, 26 August 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 didymus rank-3 family are rank-3 temperaments tempering out the didymus comma, 81/80. If nothing else is tempered out we have a 7-limit planar temperament, with an 11-limit comma we get an 11-limit temperament, and so forth.

Most of these temperaments were named by Gene Ward Smith in 2010[1].

Temperaments discussed elsewhere include:

Didymus

Subgroup: 2.3.5.7

Comma list: 81/80

Mapping: [⟨1 0 -4 0], ⟨0 1 4 0], ⟨0 0 0 1]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 696.2387 ¢, ~7/4 = 964.9090 ¢

Optimal ET sequence: 12, 19, 31, 81

Badness (Smith): 0.095 × 10-3

Euterpe

Subgroup: 2.3.5.7.11

Comma list: 81/80, 99/98

Mapping: [⟨1 0 -4 0 1], ⟨0 1 4 0 -2], ⟨0 0 0 1 2]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 696.1982 ¢, ~7/4 = 968.4280 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-1 0 -1/2 2 0⟩]
Eigenmonzos (unchanged-intervals): 2, 5, 7

Optimal ET sequence: 12, 17c, 19e, 26, 31, 88

Badness (Smith): 0.536 × 10-3

Calliope

Subgroup: 2.3.5.7.11

Comma list: 45/44, 81/80

Mapping: [⟨1 0 -4 0 -6], ⟨0 1 4 0 6], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 696.1982 ¢, ~7/4 = 968.4280 ¢

Minimax tuning:

[[1 0 0 0 0⟩, [1 0 0 0 1/6⟩, [0 0 0 0 2/3⟩, [1 -1 0 1 1/6⟩, [0 0 0 0 1⟩]
Unchanged-interval (eigenmonzo) basis: 2.7/3.11

Optimal ET sequence: 7d, 12, 19, 26, 45

Badness (Smith): 0.530 × 10-3

Erato

Subgroup: 2.3.5.7.11

Comma list: 81/80, 126/125

Mapping: [⟨1 0 -4 -13 0], ⟨0 1 4 10 0], ⟨0 0 0 0 1]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 696.4949 ¢, ~11/8 = 547.0252 ¢

Optimal ET sequence: 12, 19, 31, 50, 81

Badness (Smith): 0.558 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 81/80, 105/104, 126/125

Mapping: [⟨1 0 -4 -13 0 -20], ⟨0 1 4 10 0 15], ⟨0 0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 695.9883 ¢, ~11/8 = 545.6817 ¢

Optimal ET sequence: 12f, 19, 31, 50, 81

Clio

Subgroup: 2.3.5.7.11

Comma list: 81/80, 176/175

Mapping: [⟨1 0 -4 0 -12], ⟨0 1 4 0 8], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 697.2502 ¢, ~7/4 = 968.6295 ¢

Minimax tuning:

[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [0 0 0 1 0⟩, [-4 0 2 1 0⟩]
Eigenmonzos (unchanged-intervals): 2, 5, 7

Optimal ET sequence: 7, 12, 19e, 24, 31, 105, 129

Badness (Smith): 0.738 × 10-3

Polyhymnia

Subgroup: 2.3.5.7.11

Comma list: 81/80, 385/384

Mapping: [⟨1 0 -4 0 11], ⟨0 1 4 0 -3], ⟨0 0 0 1 -1]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 696.2305 ¢, ~7/4 = 964.8695 ¢

Optimal ET sequence: 7, 12e, 19, 24, 26, 31

Scales: polyhymnia12

Thalia

Subgroup: 2.3.5.7.11

Comma list: 33/32, 55/54

Mapping: [⟨1 0 -4 0 5], ⟨0 1 4 0 -1], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 692.0796 ¢, ~7/4 = 950.2565 ¢

Optimal ET sequence: 5, 7, 12e, 14c, 19e

Melpomene

Subgroup: 2.3.5.7.11

Comma list: 81/80, 56/55

Mapping: [⟨1 0 -4 0 7], ⟨0 1 4 0 -4], ⟨0 0 0 1 1]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 699.2230 ¢, ~7/4 = 964.2363 ¢

Optimal ET sequence: 7d, 12, 17c, 19, 24, 31e, 36

Terpsichore

Subgroup: 2.3.5.7.11

Comma list: 81/80, 540/539

Mapping: [⟨1 0 -4 0 -2], ⟨0 1 4 0 7], ⟨0 0 0 1 -2]]

Optimal tuning (POTE): ~2 = 1200.0000 ¢, ~3/2 = 696.2358 ¢, ~7/4 = 964.0006 ¢

Optimal ET sequence: 14c, 17c, 19, 31, 81, 112b

Badness (Smith): 0.850 × 10-3

Complexity spectrum: 4/3, 10/9, 9/8, 6/5, 9/7, 7/5, 7/6, 5/4, 8/7, 11/9, 12/11, 11/8, 11/10, 14/11

References

  1. ↑ Yahoo! Tuning Group | Higher-rank meantone extensions (muse temperaments?)
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