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239edo excels as an [[11-limit]] system with a sharp tendency: [[prime harmonic]]s 3 through 11 are all tuned sharp. The accuracy of [[12/11]] is particularly notable, as 30\239 represents a [[convergent]] to this interval, which forms a barely [[consistent circle]], with the closing error being about 45% of a step.
239edo excels as an [[11-limit]] system with a sharp tendency: [[prime harmonic]]s 3 through 11 are all tuned sharp. The accuracy of [[12/11]] is particularly notable, as 30\239 represents a [[convergent]] to this interval, which forms a barely [[consistent circle]], with the closing error being about 45% of a step.


239 is a convergent to the [[argent temperament]], where the perfect fourth and fifth are in the logarithmic ratio of 1:√2, and with a fifth in this range, tempers out [[5120/5103]] with great accuracy. This implies that [[81/80]] and [[64/63]] are equated (to 5 steps), that three wholetones ([[9/8]]) stack to [[10/7]], and that the [[2187/2048|apotome]], the [[256/243|limma]], and the [[Pythagorean comma]] are equated with [[15/14]] (24 steps), [[21/20]] (17 steps), and [[50/49]] (7 steps) respectively.  
239 is a convergent to the [[argent tuning]], where the perfect fourth and fifth are in the logarithmic ratio of 1:√2, and with a fifth in this range, tempers out [[5120/5103]] with great accuracy. This implies that [[81/80]] and [[64/63]] are equated (to 5 steps), that three wholetones ([[9/8]]) stack to [[10/7]], and that the [[2187/2048|apotome]], the [[256/243|limma]], and the [[Pythagorean comma]] are equated with [[15/14]] (24 steps), [[21/20]] (17 steps), and [[50/49]] (7 steps) respectively.  


Another notable feature of 239edo is that many of its [[5-limit]] intervals are mapped to composite numbers of steps: [[3/2]] to 140, [[4/3]] to 99, [[5/3]] to 176, [[5/4]] to 77, [[6/5]] to 63, and [[8/5]] to 162. Not only that, but plenty of these form relatively simple logarithmic ratios with each other, as described by [[Don Page comma]]s. In particular: [[gammic]], where 9 units form 6/5, 11 form 5/4, and 20 form 3/2; [[quartonic]], where 7 units form 6/5, 11 form 4/3, and 18 form 8/5; and [[escapade]], where 7 units form 5/4, 9 form 4/3, and 16 form 5/3, are all supported by 239edo with generators 7\239, 9\239, and 11\239 respectively. 239edo is describable as the unique intersection of any two of these.
Another notable feature of 239edo is that many of its [[5-limit]] intervals are mapped to composite numbers of steps: [[3/2]] to 140, [[4/3]] to 99, [[5/3]] to 176, [[5/4]] to 77, [[6/5]] to 63, and [[8/5]] to 162. Not only that, but plenty of these form relatively simple logarithmic ratios with each other, as described by [[Don Page comma]]s. In particular: [[gammic]], where 9 units form 6/5, 11 form 5/4, and 20 form 3/2; [[quartonic]], where 7 units form 6/5, 11 form 4/3, and 18 form 8/5; and [[escapade]], where 7 units form 5/4, 9 form 4/3, and 16 form 5/3, are all supported by 239edo with generators 7\239, 9\239, and 11\239 respectively. 239edo is describable as the unique intersection of any two of these.
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== Intervals ==
== Intervals ==
{{Interval table}}
{{Main|Table of 239edo intervals}}


== Regular temperament properties ==
== Regular temperament properties ==
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 379 -239 }}
| {{Monzo| 379 -239 }}
| {{mapping| 239 379 }}
| {{Mapping| 239 379 }}
| −0.307
| −0.307
| 0.307
| 0.307
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 3 -18 11}}, {{monzo| 32 -7 -9 }}
| {{Monzo| 3 -18 11 }}, {{monzo| 32 -7 -9 }}
| {{mapping| 239 379 555 }}
| {{Mapping| 239 379 555 }}
| −0.247
| −0.247
| 0.265
| 0.265
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 5120/5103, 29360128/29296875
| 2401/2400, 5120/5103, 29360128/29296875
| {{mapping| 239 379 555 671 }}
| {{Mapping| 239 379 555 671 }}
| −0.204
| −0.204
| 0.241
| 0.241
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4000/3993, 5120/5103
| 2401/2400, 3025/3024, 4000/3993, 5120/5103
| {{mapping| 239 379 555 671 827 }}
| {{Mapping| 239 379 555 671 827 }}
| −0.220
| −0.220
| 0.218
| 0.218
| 4.34
| 4.34
|-
| 2.3.5.7.11.17
| 595/594, 1156/1155, 2058/2057, 2401/2400, 5120/5103
| {{Mapping| 239 379 555 671 827 977 }}
| −0.203
| 0.203
| 4.03
|- style="border-top: double;"
| 2.3.5.7.11.13
| 352/351, 625/624, 847/845, 1575/1573, 2401/2400
| {{Mapping| 239 379 555 671 827 885 }} (239f)
| −0.318
| 0.296
| 5.89
|-
| 2.3.5.7.11.13.17
| 352/351, 595/594, 625/624, 833/832, 1156/1155, 1575/1573
| {{Mapping| 239 379 555 671 827 885 977 }} (239f)
| −0.290
| 0.282
| 5.63
|}
|}


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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 55.23
| 55.23
| 33/32
| 33/32
| [[Escapade]] / [[alphaquarter]]
| [[Escapade]] / [[alphaquarter]] (239f)
|-
|-
| 1
| 1
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| 14/11
| 14/11
| [[Unthirds]] (239f)
| [[Unthirds]] (239f)
|-
| 1
| 103\239
| 517.15
| 66/49
| [[Cutefourths]] (239f)
|-
|-
| 1
| 1
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| [[Neptune]] (7-limit)
| [[Neptune]] (7-limit)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Music ==
== Music ==
Retrieved from "https://en.xen.wiki/w/239edo"