Logarithmic approximants: Difference between revisions

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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...

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...

==Golden temperaments==

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 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...
+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<sup>√2</sup> = 2.665144...
 ==Golden temperaments==