18edf - Revision history

Hstraub: Link to German

2026-01-02T15:16:26Z

Link to German

2025-08-01T05:43:40Z

Add comparison with 20edf

← Older revision		Revision as of 05:43, 1 August 2025
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	== Theory ==		== 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.		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 ===		=== Regular temperaments ===

2025-08-01T05:36:56Z

Remove red link

← Older revision		Revision as of 05:36, 1 August 2025
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	== Theory ==		== 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 ~~[[third]]s~~ 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.		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.

	=== Regular temperaments ===		=== Regular temperaments ===

2025-08-01T05:35:50Z

Add a bit of human-readable and musically applicable theory

← Older revision		Revision as of 05:35, 1 August 2025
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	== Theory ==		== 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 [[third]]s 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.

			=== 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]].		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]].

2025-07-16T15:00:26Z

See also

← Older revision		Revision as of 15:00, 16 July 2025
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	* [[80ed6]] – relative ed6		* [[80ed6]] – relative ed6
	* [[87ed7]] – relative ed7		* [[87ed7]] – relative ed7
			* [[107ed11]] – relative ed11
	* [[111ed12]] – relative ed12		* [[111ed12]] – relative ed12
			* [[138ed22]] – relative ed22
	* [[204ed96]] – close to the zeta-optimized tuning for 31edo		* [[204ed96]] – close to the zeta-optimized tuning for 31edo
	* [[39cET]]		* [[39cET]]

	[[Category:31edo]]		[[Category:31edo]]

2025-06-10T13:30:35Z

See also

2025-06-10T13:30:29Z

See also

← Older revision		Revision as of 13:30, 10 June 2025
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	* [[87ed7]] – relative ed7		* [[87ed7]] – relative ed7
	* [[111ed12]] – relative ed12		* [[111ed12]] – relative ed12
			* [[204ed96]] – close to the zeta-optimized tuning for 31edo
	* [[39cET]]		* [[39cET]]

	[[Category:31edo]]		[[Category:31edo]]

2025-06-10T13:28:34Z

See also

2025-03-24T12:50:28Z

Theory: +subsets and supersets

← Older revision		Revision as of 12:50, 24 March 2025
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	{{Harmonics in equal\|18\|3\|2\|intervals=integer\|columns=11}}		{{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)}}		{{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 ==		== Intervals ==

2025-03-24T12:49:04Z