Dual-fifth tuning - Revision history

Overthink: /* Stretched or compressed edos */ correct maths

2026-05-01T19:12:32Z

Stretched or compressed edos: correct maths

← Older revision		Revision as of 19:12, 1 May 2026
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	Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:		Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:

			{{Todo\|correct maths\|inline=1\|text=This list gives half the unstretched step size for each edo. The correct values should be half of the step size after stretching, where the 9/4 reached by stacking both fifths is just.}}

	'''>25{{c}} error'''		'''>25{{c}} error'''

Overthink: /* Stretched or compressed edos */ not "worst"

2026-05-01T19:09:53Z

Stretched or compressed edos: not "worst"

← Older revision		Revision as of 19:09, 1 May 2026
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	=== Stretched or compressed edos ===		=== Stretched or compressed edos ===
	[[Octave stretching]] or [[octave shrinking]] can be used to equally share the error between an edo's best and ~~worst~~ fifths (i.e. bring them both to exactly 50% [[relative error]]).		[[Octave stretching]] or [[octave shrinking]] can be used to equally share the error between an edo's best and second-best fifths (i.e. bring them both to exactly 50% [[relative error]]).

	Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:		Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:

BudjarnLambeth: /* Stretched or compressed edos */

2025-09-19T08:56:07Z

Stretched or compressed edos

← Older revision		Revision as of 08:56, 19 September 2025
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	Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:		Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:

			'''>25{{c}} error'''
	* [[16edo]]: 37.50{{c}}		* [[16edo]]: 37.50{{c}}
	* [[18edo]]: 33.33{{c}}		* [[18edo]]: 33.33{{c}}
	* [[20edo]]: 30.00{{c}}		* [[20edo]]: 30.00{{c}}
	* [[23edo]]: 26.09{{c}}		* [[23edo]]: 26.09{{c}}

			'''15-25{{c}} error'''
	* [[25edo]]: 24.00{{c}}		* [[25edo]]: 24.00{{c}}
	* [[28edo]]: 21.43{{c}}		* [[28edo]]: 21.43{{c}}
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	* [[37edo]]: 16.22{{c}}		* [[37edo]]: 16.22{{c}}
	* [[40edo]]: 15.00{{c}}		* [[40edo]]: 15.00{{c}}

			'''<15{{c}} error'''
	* [[42edo]]: 14.29{{c}}		* [[42edo]]: 14.29{{c}}
	* [[47edo]]: 12.77{{c}}		* [[47edo]]: 12.77{{c}}

BudjarnLambeth: /* Stretched or compressed edos */

2025-09-19T08:38:21Z

Stretched or compressed edos

← Older revision		Revision as of 08:38, 19 September 2025
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	* [[28edo]]: 21.43{{c}}		* [[28edo]]: 21.43{{c}}
	* [[30edo]]: 20.00{{c}}		* [[30edo]]: 20.00{{c}}
	* [[33edo]]: 18.18{{c}}
	* [[35edo]]: 17.14{{c}}		* [[35edo]]: 17.14{{c}}
	* [[37edo]]: 16.22{{c}}		* [[37edo]]: 16.22{{c}}

BudjarnLambeth: /* Stretched or compressed edos */

2025-09-19T08:37:08Z

Stretched or compressed edos

← Older revision		Revision as of 08:37, 19 September 2025
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	* [[28edo]]: 21.43{{c}}		* [[28edo]]: 21.43{{c}}
	* [[30edo]]: 20.00{{c}}		* [[30edo]]: 20.00{{c}}
	* [[32edo]]: 18.75{{c}}
	* [[33edo]]: 18.18{{c}}		* [[33edo]]: 18.18{{c}}
	* [[35edo]]: 17.14{{c}}		* [[35edo]]: 17.14{{c}}

BudjarnLambeth: /* Dual-fifth edos */ Discuss impure octaves

2025-09-19T08:29:17Z

Dual-fifth edos: Discuss impure octaves

← Older revision		Revision as of 08:29, 19 September 2025
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	We may, heuristically, define dual-fifth edos as those whose [[relative error]] of the third harmonic is greater than 1/3. In that case 1/3 of all edos will be dual-fifth and the other 2/3 will be plain-fifth.		We may, heuristically, define dual-fifth edos as those whose [[relative error]] of the third harmonic is greater than 1/3. In that case 1/3 of all edos will be dual-fifth and the other 2/3 will be plain-fifth.

			=== Stretched or compressed edos ===
			[[Octave stretching]] or [[octave shrinking]] can be used to equally share the error between an edo's best and worst fifths (i.e. bring them both to exactly 50% [[relative error]]).

			Here is the amount of error the sharp and flat fifth will both exhibit in some possible dual-fifth edos after they have been stretched/compressed in this way:
			* [[16edo]]: 37.50{{c}}
			* [[18edo]]: 33.33{{c}}
			* [[20edo]]: 30.00{{c}}
			* [[23edo]]: 26.09{{c}}
			* [[25edo]]: 24.00{{c}}
			* [[28edo]]: 21.43{{c}}
			* [[30edo]]: 20.00{{c}}
			* [[32edo]]: 18.75{{c}}
			* [[33edo]]: 18.18{{c}}
			* [[35edo]]: 17.14{{c}}
			* [[37edo]]: 16.22{{c}}
			* [[40edo]]: 15.00{{c}}
			* [[42edo]]: 14.29{{c}}
			* [[47edo]]: 12.77{{c}}
			* [[49edo]]: 12.25{{c}}
			* [[52edo]]: 11.54{{c}}
			* [[54edo]]: 11.11{{c}}
			* [[57edo]]: 10.53{{c}}
			* [[59edo]]: 10.17{{c}}
			* [[64edo]]: 9.38{{c}}
			* [[66edo]]: 9.09{{c}}
			* [[69edo]]: 8.70{{c}}
			* [[71edo]]: 8.45{{c}}

			You can calculate this for any edo by subtracting the smaller fifth from the bigger fifth, then dividing the result by two (i.e., by dividing the size of one edo-step by two).

	== Dual-fifth temperaments ==		== Dual-fifth temperaments ==

BudjarnLambeth: /* Dual-fifth edos */ Reword a little

2025-09-19T07:38:27Z

Dual-fifth edos: Reword a little

← Older revision		Revision as of 07:38, 19 September 2025
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	[[35edo]] is the [[equal temperament]] which can be said to most authentically represent the concept of "dual-fifth", since its fifths of 20\35 and 21\35 correspond to the bounds of the tuning range for the [[diatonic]] scale where the term ''fifth'' in the standard Western practice originates from. 35edo is the largest [[edo]] without a diatonic scale, and it is therefore the smallest whose sharp and flat fifth can be equally treated as being approximants of five staff positions of the diatonic scale.		[[35edo]] is the [[equal temperament]] which can be said to most authentically represent the concept of "dual-fifth", since its fifths of 20\35 and 21\35 correspond to the bounds of the tuning range for the [[diatonic]] scale where the term ''fifth'' in the standard Western practice originates from. 35edo is the largest [[edo]] without a diatonic scale, and it is therefore the smallest whose sharp and flat fifth can be equally treated as being approximants of five staff positions of the diatonic scale.

	~~Although edos like~~ [[18edo]]~~, [[23edo]]~~ and ~~[[25edo]] have been extensively studied as dual-~~fifth, ~~their corresponding dual-~~fifth ~~intervals that~~ are ~~also often considered as~~ [[~~2L 5s\|antidiatonic~~]] ~~generators or subminor sixths~~, and ~~not every musical approach treats them as approximants of 3/2 or intervals playing the role of the~~ fifth.		Perhaps a more familiar dual-fifth system to many is [[18edo]]. It is the first system that has intervals that are close enough to 3/2 that they can be regarded as sharp and flat fifth, but also far enough to sound different. Its sharp fifth and flat fifth are almost equally off from just: it has a 733.3¢ sharp fifth 31.4¢ sharp from pure [[3/2]], and a 666.7¢ flat fifth is 35.3¢ flat.

	~~Perhaps a more familiar dual-fifth system to many is~~ [[18edo]]~~. It is the first system that has intervals that are close enough to 3/2 that they can be regarded~~ as ~~sharp and flat~~ fifth, ~~but also far enough to sound different. Its sharp fifth and flat~~ fifth are ~~almost equally off from just: it has a 733.3¢ sharp fifth 31.4¢ sharp from pure~~ [[~~3/2~~]], and ~~a 666.7¢ flat~~ fifth ~~is 35.3¢ flat~~.		Although edos like [[18edo]], [[23edo]] and [[25edo]] have been studied as dual-fifth, their corresponding dual-fifth intervals that are also often considered as [[2L 5s\|antidiatonic]] generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the fifth.

	For a list of edos which could be considered dual-fifth, see:		For a list of edos which could be considered dual-fifth, see:

Inthar at 11:32, 21 September 2024

2024-09-21T11:32:42Z

← Older revision		Revision as of 11:32, 21 September 2024
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	For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 (3<sup>-</sup>), 53\91 (3), and 54\91 (3<sup>+</sup>). Thus, if used this way they do not represent distinct dimensions, but rather correspond to 3 × 3<sup>-</sup> × 3<sup>+</sup> = 27/1.		For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 (3<sup>-</sup>), 53\91 (3), and 54\91 (3<sup>+</sup>). Thus, if used this way they do not represent distinct dimensions, but rather correspond to 3 × 3<sup>-</sup> × 3<sup>+</sup> = 27/1.

	=== Music ===		== Music ==
	*[[File:Sixix Fugue.mp3\|256x256px]] A fugue in [[18edo]] as a dual-fifth tuning (WIP)		*[[File:Sixix Fugue.mp3\|256x256px]] A fugue in [[18edo]] as a dual-fifth tuning (WIP)
	[[Category:Dual-fifth\| ]] <!-- main article -->		[[Category:Dual-fifth\| ]] <!-- main article -->
	[[Category:Terms]]		[[Category:Terms]]

Inthar: /* Multiple-fifth temperaments */

2024-09-20T04:03:49Z

Multiple-fifth temperaments

← Older revision		Revision as of 04:03, 20 September 2024
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	For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 (3<sup>-</sup>), 53\91 (3), and 54\91 (3<sup>+</sup>). Thus, if used this way they do not represent distinct dimensions, but rather correspond to 3 × 3<sup>-</sup> × 3<sup>+</sup> = 27/1.		For example, [[91edo]] has 3 usable fifths with their own functions - 52\91 (3<sup>-</sup>), 53\91 (3), and 54\91 (3<sup>+</sup>). Thus, if used this way they do not represent distinct dimensions, but rather correspond to 3 × 3<sup>-</sup> × 3<sup>+</sup> = 27/1.

			=== Music ===
			*[[File:Sixix Fugue.mp3\|256x256px]] A fugue in [[18edo]] as a dual-fifth tuning (WIP)
	[[Category:Dual-fifth\| ]] <!-- main article -->		[[Category:Dual-fifth\| ]] <!-- main article -->
	[[Category:Terms]]		[[Category:Terms]]

CompactStar at 19:52, 11 March 2024

2024-03-11T19:52:31Z

← Older revision		Revision as of 19:52, 11 March 2024
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	== Dual-fifth scales ==		== Dual-fifth scales ==
	[[Sixix]][7] can be regarded as the scale which is the most authentic representation of the "dual-fifth" phenomenon via its [[mode]]s, since it features both sharp and flat fifth on different modes, and the interval in this case occupies 5 staff positions. For example, in [[25edo]], sixix can take form of 4 3 4 3 4 3 4, where five staff positions occupy 18\25 (sharp fifth), but if the mode is 3 4 3 4 3 4 4, then five staff positions are equal to 17\25 (flat fifth).		[[Sixix]][7] can be regarded as the scale which is the most authentic representation of the "dual-fifth" phenomenon via its [[mode]]s, since it features both sharp and flat fifth on different modes, and the interval in this case occupies 5 staff positions. For example, in [[25edo]], sixix can take form of 4 3 4 3 4 3 4, where five staff positions occupy 18\25 (sharp fifth), but if the mode is 3 4 3 4 3 4 4, then five staff positions are equal to 17\25 (flat fifth). However, it should be noted that in better tunings of sixix, the flat fifth is Mavila-like in quality while the sharp fifth is a comparatively accurate [[superpyth]] diatonic fifth.

	== Dual-fifth edos ==		== Dual-fifth edos ==
	[[35edo]] is the [[equal temperament]] which can be said to most authentically represent the concept of "dual-fifth", since its fifths of 20\35 and 21\35 correspond to the bounds of the tuning range for the [[diatonic]] scale where the term ''fifth'' in the standard Western practice originates from. 35edo is the largest [[edo]] without a diatonic scale, and it is therefore the smallest whose sharp and flat fifth can be equally treated as being approximants of five staff positions of the diatonic scale.		[[35edo]] is the [[equal temperament]] which can be said to most authentically represent the concept of "dual-fifth", since its fifths of 20\35 and 21\35 correspond to the bounds of the tuning range for the [[diatonic]] scale where the term ''fifth'' in the standard Western practice originates from. 35edo is the largest [[edo]] without a diatonic scale, and it is therefore the smallest whose sharp and flat fifth can be equally treated as being approximants of five staff positions of the diatonic scale.

	Although edos like [[18edo]], [[23edo]] and [[25edo]] have been extensively studied as dual-fifth, their corresponding dual-fifth intervals that are also often considered as [[2L 5s\|~~mavila~~]] generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the fifth.		Although edos like [[18edo]], [[23edo]] and [[25edo]] have been extensively studied as dual-fifth, their corresponding dual-fifth intervals that are also often considered as [[2L 5s\|antidiatonic]] generators or subminor sixths, and not every musical approach treats them as approximants of 3/2 or intervals playing the role of the fifth.

	Perhaps a more familiar dual-fifth system to many is [[18edo]]. It is the first system that has intervals that are close enough to 3/2 that they can be regarded as sharp and flat fifth, but also far enough to sound different. Its sharp fifth and flat fifth are almost equally off from just: it has a 733.3¢ sharp fifth 31.4¢ sharp from pure [[3/2]], and a 666.7¢ flat fifth is 35.3¢ flat.		Perhaps a more familiar dual-fifth system to many is [[18edo]]. It is the first system that has intervals that are close enough to 3/2 that they can be regarded as sharp and flat fifth, but also far enough to sound different. Its sharp fifth and flat fifth are almost equally off from just: it has a 733.3¢ sharp fifth 31.4¢ sharp from pure [[3/2]], and a 666.7¢ flat fifth is 35.3¢ flat.