7edo - Revision history

Xenllium: /* See also */

2026-02-26T09:45:53Z

Contribution at 01:45, 28 October 2025

2025-10-28T01:45:44Z

← Older revision		Revision as of 01:45, 28 October 2025
Line 494:		Line 494:
	== Ear training ==		== Ear training ==
	7edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].		7edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].

			== See also ==
			* [https://en.xen.wiki/w/Equal-step_tuning#Alpha-beta-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions]

	== Notes ==		== Notes ==

BudjarnLambeth: /* Octave stretch or compression */

2025-09-23T02:20:40Z

Octave stretch or compression

@@ Line 483: / Line 483: @@
 == Octave stretch or compression ==
-[[Stretched and compressed tuning|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.
+[[Stretched and compressed tuning|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[zpi|15zpi]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for full [[11-limit]] harmonies, then these are good choices to make that easier.
 == Instruments ==

FloraC: /* Octave stretch */ unify section titles

2025-08-25T13:41:03Z

Octave stretch: unify section titles

← Older revision		Revision as of 13:41, 25 August 2025
Line 482:		Line 482:
	3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.		3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.

	== Octave stretch ==		== Octave stretch or compression ==
	[[Stretched and compressed tuning\|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128\|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.		[[Stretched and compressed tuning\|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128\|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.

FloraC: /* Octave stretch */ fixes & unify the precision

2025-08-23T12:05:41Z

Octave stretch: fixes & unify the precision

FloraC: /* Octave stretch */ restore the summary

2025-08-23T11:37:04Z

Octave stretch: restore the summary

← Older revision		Revision as of 11:37, 23 August 2025
Line 483:		Line 483:

	== Octave stretch ==		== Octave stretch ==
	~~What follows~~ is ~~a comparison of stretched~~-~~octave 7edo tunings~~.		[[Stretched and compressed tuning\|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128\|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.

	; 7edo		; 7edo

BudjarnLambeth: /* Octave stretch */

2025-08-21T23:45:12Z

Octave stretch

BudjarnLambeth: /* Octave stretch */

2025-08-21T21:57:31Z

Octave stretch

← Older revision		Revision as of 21:57, 21 August 2025
Line 499:		Line 499:
	; [[18ed6]]		; [[18ed6]]
	* Step size: 172.331{{c}}, octave size: 1206.3{{c}}		* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
	Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 14. This approximates all harmonics up to 16 within 48.7{{c}}. The tuning 18ed6 does this.		Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 13. This approximates all harmonics up to 16 within 48.7{{c}}. The tuning 18ed6 does this.
	{{Harmonics in equal\|18\|6\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6}}		{{Harmonics in equal\|18\|6\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6}}
	{{Harmonics in equal\|18\|6\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6 (continued)}}		{{Harmonics in equal\|18\|6\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6 (continued)}}

BudjarnLambeth: /* Temperaments */ octave stretch section

2025-08-21T21:56:32Z

Temperaments: octave stretch section

← Older revision		Revision as of 21:56, 21 August 2025
Line 481:		Line 481:

	3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.		3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1L''n''s [[pentatonic]] mos. This is in part because 7edo is close to low-complexity JI for its size, and is the second edo with a good fifth for its size (after [[5edo]]), the fifth serving as a generator for the edo's meantone and mavila interpertations.

			== Octave stretch ==
			What follows is a comparison of stretched-octave 7edo tunings.

			; 7edo
			* Step size: 171.429{{c}}, octave size: 1200.0{{c}}
			Pure-octaves 7edo approximates all harmonics up to 16 within NNN{{c}}.
			{{Harmonics in equal\|7\|2\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7edo}}
			{{Harmonics in equal\|7\|2\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7edo (continued)}}

			; [[WE\|7et, 2.3.11.13 WE]]
			* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
			Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 75.0{{c}}. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
			{{Harmonics in cet\|171.993\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
			{{Harmonics in cet\|171.993\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}

			; [[18ed6]]
			* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
			Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 14. This approximates all harmonics up to 16 within 48.7{{c}}. The tuning 18ed6 does this.
			{{Harmonics in equal\|18\|6\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6}}
			{{Harmonics in equal\|18\|6\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6 (continued)}}

			; [[WE\|7et, 2.3.5.11.13 WE]]
			* Step size: 172.390{{c}}, octave size: 1206.7{{c}}
			Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. This approximates all harmonics up to 16 within 85.7{{c}}. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
			{{Harmonics in cet\|172.390\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}
			{{Harmonics in cet\|172.390\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}

			; [[zpi\|15zpi]]
			* Step size: 172.495{{c}}, octave size: 1207.5{{c}}
			Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 84.0{{c}}. The tuning 15zpi does this.
			{{Harmonics in cet\|172.495\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 15zpi}}
			{{Harmonics in cet\|172.495\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 15zpi (continued)}}

			; [[11edt]]
			* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
			Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. This approximates all harmonics up to 16 within 83.6{{c}}. The tuning 11edt does this.
			{{Harmonics in equal\|11\|3\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 11edt}}
			{{Harmonics in equal\|11\|3\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 11edt (continued)}}

	== Instruments ==		== Instruments ==

BudjarnLambeth: /* Octave stretch */

2025-08-21T21:55:52Z

Octave stretch

← Older revision		Revision as of 21:55, 21 August 2025
Line 40:		Line 40:

	It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>		It has often been stated that 7edo approximates tunings used in [[Thai]] classical music, though this is a myth unsupported by [[empirical]] studies of the instruments.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>

	~~=== Octave stretch ===~~
	[[Stretched and compressed tuning\|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128\|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.

	~~The stretched 7edo tuning [[zpi\|15zpi]] can also be used to improve 7edo's approximation of JI in a similar way.~~

	=== Subsets and supersets ===		=== Subsets and supersets ===

← Older revision		Revision as of 09:45, 26 February 2026
Line 496:		Line 496:

	== See also ==		== See also ==
	* [~~https://en.xen.wiki/w/Equal-step_tuning#~~Alpha-beta~~-gamma_family_of_equal_divisions Alpha-beta-gamma_family_of_equal_divisions~~]		* [[Alpha, beta, and gamma family of equal divisions]]

	== Notes ==		== Notes ==

← Older revision		Revision as of 12:05, 23 August 2025
Line 486:		Line 486:

	; 7edo		; 7edo
	* Step size: 171.429{{c}}, octave size: 1200.0{{c}}		* Step size: 171.429{{c}}, octave size: 1200.000{{c}}
	Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It ~~doesn't~~ approximate 7.		Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it is arguable whether it approximates 5 – if it does it does so poorly. It does not approximate 7.
	{{Harmonics in equal\|7\|2\|1\|columns=11\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 7edo}}		{{Harmonics in equal\|7\|2\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 7edo}}
	{{Harmonics in equal\|7\|2\|1\|columns=12\|start=12\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 7edo (continued)}}		{{Harmonics in equal\|7\|2\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 7edo (continued)}}

	; [[WE\|7et, 2.3.11.13 WE]]		; [[WE\|7et, 2.3.11.13 WE]]
	* Step size: 171.993{{c}}, octave size: ~~1204~~.0{{c}}		* Step size: 171.993{{c}}, octave size: 1203.948{{c}}
	Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.		Stretching the octave of 7edo by around 3.9{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
	{{Harmonics in cet\|171.~~993~~\|columns=11\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}		{{Harmonics in cet\|171.992645\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
	{{Harmonics in cet\|171.~~993~~\|columns=12\|start=12\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}		{{Harmonics in cet\|171.992645\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}

	; [[18ed6]]		; [[18ed6]]
	* Step size: 172.331{{c}}, octave size: 1206.3{{c}}		* Step size: 172.331{{c}}, octave size: 1206.316{{c}}
	Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 13. The tuning 18ed6 does this.		Stretching the octave of 7edo by around 6.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 18ed6 does this.
	{{Harmonics in equal\|18\|6\|1\|columns=11\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 18ed6}}		{{Harmonics in equal\|18\|6\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 18ed6}}
	{{Harmonics in equal\|18\|6\|1\|columns=12\|start=12\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 18ed6 (continued)}}		{{Harmonics in equal\|18\|6\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 18ed6 (continued)}}

	; [[WE\|7et, 2.3.5.11.13 WE]]		; [[WE\|7et, 2.3.5.11.13 WE]]
	* Step size: 172.390{{c}}, octave size: 1206.7{{c}}		* Step size: 172.390{{c}}, octave size: 1206.728{{c}}
	Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.		Stretching the octave of 7edo by around 7.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
	{{Harmonics in cet\|172.~~390~~\|columns=11\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}		{{Harmonics in cet\|172.389769\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}
	{{Harmonics in cet\|172.~~390~~\|columns=12\|start=12\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}		{{Harmonics in cet\|172.389769\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}

	; [[zpi\|15zpi]]		; [[zpi\|15zpi]]
	* Step size: 172.495{{c}}, octave size: 1207.5{{c}}		* Step size: 172.495{{c}}, octave size: 1207.471{{c}}
	Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this.		Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this.
	{{Harmonics in cet\|172.~~495~~\|columns=11\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 15zpi}}		{{Harmonics in cet\|172.495886\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 15zpi}}
	{{Harmonics in cet\|172.~~495~~\|columns=12\|start=12\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 15zpi (continued)}}		{{Harmonics in cet\|172.495886\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 15zpi (continued)}}

	; [[11edt]]		; [[11edt]]
	* Step size: 172.905{{c}}, octave size: 1210.3{{c}}		* Step size: 172.905{{c}}, octave size: 1210.335{{c}}
	Stretching the octave of 7edo by around ~~NNN~~{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this.		Stretching the octave of 7edo by around 10.3{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this.
	{{Harmonics in equal\|11\|3\|1\|columns=11\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 11edt}}		{{Harmonics in equal\|11\|3\|1\|intervals=integer\|columns=11\|collapsed=true\|title=Approximation of harmonics in 11edt}}
	{{Harmonics in equal\|11\|3\|1\|columns=12\|start=12\|collapsed=true~~\|intervals=integer~~\|title=Approximation of harmonics in 11edt (continued)}}		{{Harmonics in equal\|11\|3\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 11edt (continued)}}

	== Instruments ==		== Instruments ==

← Older revision		Revision as of 23:45, 21 August 2025
Line 487:		Line 487:
	; 7edo		; 7edo
	* Step size: 171.429{{c}}, octave size: 1200.0{{c}}		* Step size: 171.429{{c}}, octave size: 1200.0{{c}}
	Pure-octaves 7edo approximates ~~all~~ harmonics ~~up to 16 within NNN{{c}}~~.		Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It doesn't approximate 7.
	{{Harmonics in equal\|7\|2\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7edo}}		{{Harmonics in equal\|7\|2\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7edo}}
	{{Harmonics in equal\|7\|2\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7edo (continued)}}		{{Harmonics in equal\|7\|2\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7edo (continued)}}
Line 493:		Line 493:
	; [[WE\|7et, 2.3.11.13 WE]]		; [[WE\|7et, 2.3.11.13 WE]]
	* Step size: 171.993{{c}}, octave size: 1204.0{{c}}		* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
	Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13~~. This approximates all harmonics up to 16 within 75.0{{c}}~~. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.		Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
	{{Harmonics in cet\|171.993\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}		{{Harmonics in cet\|171.993\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
	{{Harmonics in cet\|171.993\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}		{{Harmonics in cet\|171.993\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}
Line 499:		Line 499:
	; [[18ed6]]		; [[18ed6]]
	* Step size: 172.331{{c}}, octave size: 1206.3{{c}}		* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
	Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 13~~. This approximates all harmonics up to 16 within 48.7{{c}}~~. The tuning 18ed6 does this.		Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 13. The tuning 18ed6 does this.
	{{Harmonics in equal\|18\|6\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6}}		{{Harmonics in equal\|18\|6\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6}}
	{{Harmonics in equal\|18\|6\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6 (continued)}}		{{Harmonics in equal\|18\|6\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 18ed6 (continued)}}
Line 505:		Line 505:
	; [[WE\|7et, 2.3.5.11.13 WE]]		; [[WE\|7et, 2.3.5.11.13 WE]]
	* Step size: 172.390{{c}}, octave size: 1206.7{{c}}		* Step size: 172.390{{c}}, octave size: 1206.7{{c}}
	Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13~~. This approximates all harmonics up to 16 within 85.7{{c}}~~. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.		Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
	{{Harmonics in cet\|172.390\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}		{{Harmonics in cet\|172.390\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}
	{{Harmonics in cet\|172.390\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}		{{Harmonics in cet\|172.390\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}
Line 511:		Line 511:
	; [[zpi\|15zpi]]		; [[zpi\|15zpi]]
	* Step size: 172.495{{c}}, octave size: 1207.5{{c}}		* Step size: 172.495{{c}}, octave size: 1207.5{{c}}
	Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13~~. This approximates all harmonics up to 16 within 84.0{{c}}~~. The tuning 15zpi does this.		Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this.
	{{Harmonics in cet\|172.495\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 15zpi}}		{{Harmonics in cet\|172.495\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 15zpi}}
	{{Harmonics in cet\|172.495\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 15zpi (continued)}}		{{Harmonics in cet\|172.495\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 15zpi (continued)}}
Line 517:		Line 517:
	; [[11edt]]		; [[11edt]]
	* Step size: 172.905{{c}}, octave size: 1210.3{{c}}		* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
	Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13~~. This approximates all harmonics up to 16 within 83.6{{c}}~~. The tuning 11edt does this.		Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this.
	{{Harmonics in equal\|11\|3\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 11edt}}		{{Harmonics in equal\|11\|3\|1\|columns=11\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 11edt}}
	{{Harmonics in equal\|11\|3\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 11edt (continued)}}		{{Harmonics in equal\|11\|3\|1\|columns=12\|start=12\|collapsed=true\|intervals=integer\|title=Approximation of harmonics in 11edt (continued)}}