239edo: Difference between revisions

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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.

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.

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 edo 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.
-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.
+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.