18edf: Difference between revisions

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== Theory ==

18edf corresponds to [[31edo]] with an [[octave stretching]] of about 9 [[cent]]s. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor [[7/6]] (+6{{cent}}), minor [[6/5]] (-3{{cent}}), neutral [[11/9]] (+4{{cent}}), major [[5/4]] (+4{{cent}}), and supermajor [[9/7]] (-6{{cent}}). These intervals may be used to form a variety of [[triad]]s and [[tetrad]]s in close harmony along with the tuning's pure fifth.

In comparison, [[20edf]] (and [[Carlos Gamma]]) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds.

=== Regular temperaments ===

18edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].

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{{Harmonics in equal|18|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edf (continued)}}

=== Subsets and supersets ===

Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18edf has subset edfs {{EDs|equave=f| 2, 3, 6, and 9 }}.

== Intervals ==

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== See also ==

* [[31edo]] – relative edo

* [[49edt]] – relative ~~edft~~

* [[49edt]] – relative edt

* [[72ed5]] – relative ed5

* [[80ed6]] – relative ed6

* [[87ed7]] – relative ed7

* [[107ed11]] – relative ed11

* [[111ed12]] – relative ed12

* [[138ed22]] – relative ed22

* [[204ed96]] – close to the zeta-optimized tuning for 31edo

* [[39cET]]

[[Category:31edo]]

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 == Theory ==
+edf corresponds to [[31edo]] with an [[octave stretching]] of about 9 [[cent]]s. Consequently, it does not provide 31edo's good approximations of most low harmonics, but it provides good approximations to many simple ratios in the thirds region: subminor [[7/6]] (+6{{cent}}), minor [[6/5]] (-3{{cent}}), neutral [[11/9]] (+4{{cent}}), major [[5/4]] (+4{{cent}}), and supermajor [[9/7]] (-6{{cent}}). These intervals may be used to form a variety of [[triad]]s and [[tetrad]]s in close harmony along with the tuning's pure fifth.
+In comparison, [[20edf]] (and [[Carlos Gamma]]) offers more accurate pental (minor and major) and undecimal (neutral) thirds, but less accurate septimal (subminor and supermajor) thirds.
+=== Regular temperaments ===
 edf is related to the [[regular temperament]] which [[tempering out|tempers out]] 2401/2400 and 8589934592/8544921875 in the [[7-limit]]; with 5632/5625, 46656/46585, and 166698/166375 in the [[11-limit]], which is supported by [[31edo]], [[369edo]], [[400edo]], [[431edo]], and [[462edo]].
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 {{Harmonics in equal|18|3|2|intervals=integer|columns=11}}
 {{Harmonics in equal|18|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 18edf (continued)}}
+=== Subsets and supersets ===
+Since 18 factors into primes as {{nowrap| 2 × 3<sup>2</sup> }}, 18edf has subset edfs {{EDs|equave=f| 2, 3, 6, and 9 }}.
 == Intervals ==
@@ Line 199: / Line 207: @@
 == See also ==
 * [[31edo]] – relative edo
-* [[49edt]] – relative edft
+* [[49edt]] – relative edt
 * [[72ed5]] – relative ed5
 * [[80ed6]] – relative ed6
 * [[87ed7]] – relative ed7
+* [[107ed11]] – relative ed11
 * [[111ed12]] – relative ed12
+* [[138ed22]] – relative ed22
+* [[204ed96]] – close to the zeta-optimized tuning for 31edo
 * [[39cET]]
+[[Category:31edo]]