Logarithmic approximants: Difference between revisions

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As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to 21/20 (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to 15/14 (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to 50/49 (34.976 cents).

If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (~~3120~~/~~3103~~) is the bimodular comma formed from 10/7 and 9/8

If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (5120/5103) is the bimodular comma formed from 10/7 and 9/8

By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (''r'' = 2√2''a''+''b'', where'' a'' and ''b'' are integers) are transcendental, with the exception of octave multiples (''a'' = 0). The frequency ratio of the tempered perfect eleventh (8/3 = 2.6666...) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2√2 = 2.665144...

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 As a consequence of these relationships the tempered diatonic semitone (85.281 cents) is close to <u>21/20</u> (84.467 cents), the tempered chromatic semitone (120.606 cents) is close to <u>15/14</u> (119.443 cents), and the tempered Pythagorean comma (35.325 cents) is close to <u>50/49</u> (34.976 cents).
-If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (<u>3120/3103</u>) is the bimodular comma formed from <u>10/7</u> and <u>9/8</u>
+If these 7-limit intervals are considered to be tempered to their 3-limit counterparts argent is an example of hemifamity temperament. Hemifamity (5120/5103) is the bimodular comma formed from <u>10/7</u> and <u>9/8</u>
 By the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem Gelfond-Schneider theorem] the frequency ratios of all argent intervals (''r'' = 2√2''a''+''b'', where'' a'' and ''b'' are integers) are transcendental, with the exception of octave multiples (''a'' = 0). The frequency ratio of the tempered perfect eleventh (<u>8/3</u> = <u>2.6666...</u>) is the [http://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant Gelfond-Schneider constant] or Hilbert number, 2√2 = 2.665144...