57ed6 - Revision history

Overthink: /* Theory */ note non-integer edo

2026-01-02T03:11:00Z

Theory: note non-integer edo

← Older revision		Revision as of 03:11, 2 January 2026
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	== Theory ==		== Theory ==
	57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), which is milder and more suited for the 11-limit. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1\|3]] and [[7/1\|7]], but the [[5/1\|5]] and [[11/1\|11]] become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.		57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}, corresponding to about 22.050610edo. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), which is milder and more suited for the 11-limit. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1\|3]] and [[7/1\|7]], but the [[5/1\|5]] and [[11/1\|11]] become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.

	=== Harmonics ===		=== Harmonics ===

Overthink: /* Theory */ more relevant

2025-12-24T10:05:25Z

Theory: more relevant

← Older revision		Revision as of 10:05, 24 December 2025
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	== Theory ==		== Theory ==
	57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), which is milder and more suited for ~~widely voiced harmonies~~. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1\|3]] and [[7/1\|7]], but the [[5/1\|5]] and [[11/1\|11]] become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.		57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), which is milder and more suited for the 11-limit. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1\|3]] and [[7/1\|7]], but the [[5/1\|5]] and [[11/1\|11]] become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.

	=== Harmonics ===		=== Harmonics ===

Overthink: doesn't look that close to optimal

2025-12-24T00:43:31Z

doesn't look that close to optimal

← Older revision		Revision as of 00:43, 24 December 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), ~~making 57ed6 close to optimal~~ for ~~22edo~~. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1\|3]] and [[7/1\|7]], but the [[5/1\|5]] and [[11/1\|11]] become worse. Compared to 22edo, it ~~pushes~~ some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.		57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), which is milder and more suited for widely voiced harmonies. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1\|3]] and [[7/1\|7]], but the [[5/1\|5]] and [[11/1\|11]] become worse. Compared to 22edo, it brings some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.

	=== Harmonics ===		=== Harmonics ===

Overthink: expanded theory a bit

2025-10-31T03:07:02Z

expanded theory a bit

← Older revision		Revision as of 03:07, 31 October 2025
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	== Theory ==		== Theory ==
	57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 close to optimal for 22edo.		57ed6 is closely related to [[22edo]], but with the 6th harmonic rather than the [[2/1\|octave]] being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.75{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 close to optimal for 22edo. Like 22edo, it is consistent to the 12-[[integer-limit]]. It corrects harmonics [[3/1\|3]] and [[7/1\|7]], but the [[5/1\|5]] and [[11/1\|11]] become worse. Compared to 22edo, it pushes some intervals that are more out of tune in 22edo closer to just, such as 3/2, 6/5, and 7/4.

	=== Harmonics ===		=== Harmonics ===

FloraC: /* Theory */

2025-05-27T09:18:49Z

Theory

← Older revision		Revision as of 09:18, 27 May 2025
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	=== Harmonics ===		=== Harmonics ===
	{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=11}}		{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=11}}
	{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=11\|start=12\|collapsed=true\|title=Approximation of harmonics in 57ed6 (continued)}}		{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 57ed6 (continued)}}

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

FloraC: +subsets and supersets; +see also

2025-05-27T09:12:44Z

+subsets and supersets; +see also

← Older revision		Revision as of 09:12, 27 May 2025
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	{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=11}}		{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=11}}
	{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=11\|start=12\|collapsed=true\|title=Approximation of harmonics in 57ed6 (continued)}}		{{Harmonics in equal\|57\|6\|1\|intervals=integer\|columns=11\|start=12\|collapsed=true\|title=Approximation of harmonics in 57ed6 (continued)}}

			=== Subsets and supersets ===
			Since 57 factors into primes as {{nowrap\| 3 × 19 }}, 57ed6 contains subset ed6's [[3ed6]] and [[19ed6]].

	== Intervals ==		== Intervals ==
	{{Interval table}}		{{Interval table}}

			== See also ==
			* [[22edo]] – relative edo
			* [[35edt]] – relative edt
			* [[79ed12]] – relative ed12

FloraC: Cleanup

2025-05-27T09:10:38Z

Cleanup

@@ Line 2: / Line 2: @@
 {{ED intro}}
-ed6 is very nearly identical to [[22edo|22 EDO]], but with the [[6/1]] rather than the 2/1 being just, which results in octaves being [[stretched and compressed tuning|compressed]] by about 2.754{{c}}. The local [[The_Riemann_zeta_function_and_tuning#Optimal_octave_stretch|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 very close to optimal for 22edo.
+== Theory ==
 == Intervals ==
 {{Interval table}}

ArrowHead294 at 16:44, 15 January 2025

2025-01-15T16:44:58Z

← Older revision		Revision as of 16:44, 15 January 2025
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	{{ED intro}}		{{ED intro}}

	57ed6 is very nearly identical to [[22edo\|22 EDO]], but with the [[6/1]] rather than the 2/1 being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.754{{~~cent~~}}. The local [[The_Riemann_zeta_function_and_tuning#Optimal_octave_stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 very close to optimal for 22edo.		57ed6 is very nearly identical to [[22edo\|22 EDO]], but with the [[6/1]] rather than the 2/1 being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.754{{c}}. The local [[The_Riemann_zeta_function_and_tuning#Optimal_octave_stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 very close to optimal for 22edo.

	== Intervals ==		== Intervals ==

ArrowHead294 at 16:40, 15 January 2025

2025-01-15T16:40:51Z

← Older revision		Revision as of 16:40, 15 January 2025
Line 2:		Line 2:
	{{ED intro}}		{{ED intro}}

	57ed6 is very nearly identical to [[22edo\|22 EDO]], but with the [[6/1]] rather than the 2/1 being just~~. The octave is about 2.754{{cent}}~~ [[stretched and compressed tuning\|compressed]] ~~and the step size is~~ about 54.~~4203~~{{c}}. The local [[The_Riemann_zeta_function_and_tuning#Optimal_octave_stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 very close to optimal for 22edo.		57ed6 is very nearly identical to [[22edo\|22 EDO]], but with the [[6/1]] rather than the 2/1 being just, which results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.754{{cent}}. The local [[The_Riemann_zeta_function_and_tuning#Optimal_octave_stretch\|zeta peak]] around 22 is located at 22.025147, which has a step size of 54.483{{c}} and an octave of 1198.63{{c}} (which is compressed by 1.37{{c}}), making 57ed6 very close to optimal for 22edo.

	== Intervals ==		== Intervals ==

ArrowHead294 at 16:39, 15 January 2025

2025-01-15T16:39:08Z

← Older revision		Revision as of 16:39, 15 January 2025
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	}}		}}

	~~[[Category:22edo]]~~
	[[Category:Edonoi]]		[[Category:Edonoi]]

	{{stub}}		{{stub}}