29edo: Difference between revisions

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== Theory ==

29 is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]]—a [[Parapyth|Parapythagorean]] tuning instead of a meantone system.

29 is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] — a [[Parapyth|Parapythagorean]] tuning instead of a meantone system.

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29edo could be thought of as the "twin" of [[12edo]] in the 5-limit, since 5-limit intervals in 12edo and 29edo are tuned with almost exactly the same absolute errors, but in opposite directions. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).

29edo could be thought of as the "twin" of [[12edo]] in the 5-limit, since 5-limit intervals in 12edo and 29edo are tuned with almost exactly the same absolute errors, but in opposite directions. There are other ways in which they are counterparts, like how 12 tempers out [[50/49]] but not [[49/48]]; 29 does the opposite. Each also supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).

A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone.

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 == Theory ==
-is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]]&mdash;a [[Parapyth|Parapythagorean]] tuning instead of a meantone system.
+is the lowest edo which approximates the [[3/2]] just fifth more accurately than [[12edo]]: 3/2 = 701.955… cents; 17 degrees of 29edo = 703.448… cents. Since the fifth is sharp instead of flat, 29edo is a [[Erv Wilson's Linear Notations|positive temperament]] &mdash; a [[Parapyth|Parapythagorean]] tuning instead of a meantone system.
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 | [[File:29edoSuperpythDiatonic.mp3]] [[:File:29edoSuperpythDiatonic.mp3|[File info]]]
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 |}
-edo could be thought of as the "twin" of [[12edo]] in the 5-limit, since 5-limit intervals in 12edo and 29edo are tuned with almost exactly the same absolute errors, but in opposite directions. There are other ways in which they are counterparts (12 tempers out 50:49 but not 49:48; 29 does the opposite). Each supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).
+edo could be thought of as the "twin" of [[12edo]] in the 5-limit, since 5-limit intervals in 12edo and 29edo are tuned with almost exactly the same absolute errors, but in opposite directions. There are other ways in which they are counterparts, like how 12 tempers out [[50/49]] but not [[49/48]]; 29 does the opposite. Each also supports a particularly good tonal framework (meantone[7] and nautilus[14], respectively).
 A more coincidental similarity is that just as the 12-tone scale is also a 1/2-tone scale (the whole tone being divided into 2 semitones), the 29-tone temperament may also be called 2/9-tone. This is because it has two different sizes of whole tone (4 and 5 steps wide, respectively). So the step size of 29edo may be called a 2/9-tone, just as 24edo's step size is called a quarter tone.