328ed1536 - Revision history

Overthink: /* Theory */ typo

2025-12-07T05:17:51Z

Theory: typo

← Older revision		Revision as of 05:17, 7 December 2025
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	== Theory ==		== Theory ==
	The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. While 31edo's approximation of the 13th harmonic is of limited accuracy, [[31edo#Regular_temperament_properties\|its optimal 13-limit octave stretch]] is 0.502314{{c}}; 328ed1536 is very close, with a difference of only {{sfrac\|1\|79}}{{c}}.		The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.49{{c}}. While 31edo's approximation of the 13th harmonic is of limited accuracy, [[31edo#Regular_temperament_properties\|its optimal 13-limit octave stretch]] is 0.502314{{c}}; 328ed1536 is very close, with a difference of only {{sfrac\|1\|79}}{{c}}.

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

FloraC: Text replacement - "{{Infobox et}}↵{{ED intro}}" to "{{Infobox ET}} {{ED intro}}"

2025-06-26T12:00:21Z

Text replacement - "{{Infobox et}}↵{{ED intro}}" to "{{Infobox ET}} {{ED intro}}"

← Older revision		Revision as of 12:00, 26 June 2025
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	{{Infobox et}}		{{Infobox ET}}
	{{ED intro}}		{{ED intro}}

FloraC: Text replacement - "{{mathematical interest}}" to "{{Mathematical interest}}"

2025-06-26T11:57:52Z

Text replacement - "{{mathematical interest}}" to "{{Mathematical interest}}"

← Older revision		Revision as of 11:57, 26 June 2025
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	{{~~mathematical~~ interest}}		{{Mathematical interest}}
	{{Infobox et}}		{{Infobox et}}
	{{ED intro}}		{{ED intro}}

FloraC: Text replacement - "{{infobox et}}↵{{ED intro}}" to "{{Infobox et}} {{ED intro}}"

2025-06-26T11:57:26Z

Text replacement - "{{infobox et}}↵{{ED intro}}" to "{{Infobox et}} {{ED intro}}"

← Older revision		Revision as of 11:57, 26 June 2025
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	{{mathematical interest}}		{{mathematical interest}}
	{{~~infobox~~ et}}		{{Infobox et}}
	{{ED intro}}		{{ED intro}}

ArrowHead294: /* Theory */

2025-06-18T13:23:30Z

Theory

← Older revision		Revision as of 13:23, 18 June 2025
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	== Theory ==		== Theory ==
	The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. While 31edo's ~~approximations~~ the 13th harmonic is of limited accuracy, [[31edo#Regular_temperament_properties\|its optimal 13-limit octave stretch]] is 0.502314{{c}}; 328ed1536 is very close, with a difference of only {{sfrac\|1\|79}}{{c}}.		The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. While 31edo's approximation of the 13th harmonic is of limited accuracy, [[31edo#Regular_temperament_properties\|its optimal 13-limit octave stretch]] is 0.502314{{c}}; 328ed1536 is very close, with a difference of only {{sfrac\|1\|79}}{{c}}.

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

ArrowHead294 at 13:22, 18 June 2025

2025-06-18T13:22:54Z

← Older revision		Revision as of 13:22, 18 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. While 31edo's ~~13-limit accuracy~~ is limited, ~~the~~ [[31edo#Regular_temperament_properties\|its optimal ~~octave stretch [[~~13-limit]] is 0.502314{{c}}; 328ed1536 is very close, with a difference of only {{sfrac\|1\|79}}{{c}}.		The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. While 31edo's approximations the 13th harmonic is of limited accuracy, [[31edo#Regular_temperament_properties\|its optimal 13-limit octave stretch]] is 0.502314{{c}}; 328ed1536 is very close, with a difference of only {{sfrac\|1\|79}}{{c}}.

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

ArrowHead294: /* Theory */

2025-06-18T13:21:42Z

Theory

← Older revision		Revision as of 13:21, 18 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. ~~The~~ [[31edo#Regular_temperament_properties\|optimal octave stretch ~~for 31edo]] in the~~ [[13-limit]] is 0.502314{{c}}~~, making~~ 328ed1536 very close ~~to optimal for 31edo~~.		The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|stretched]] by about 0.79{{c}}. While 31edo's 13-limit accuracy is limited, the [[31edo#Regular_temperament_properties\|its optimal octave stretch [[13-limit]] is 0.502314{{c}}; 328ed1536 is very close, with a difference of only {{sfrac\|1\|79}}{{c}}.

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

ArrowHead294: Closest ED to 13-limit optimality for 31edo

2025-06-18T13:19:15Z

Closest ED to 13-limit optimality for 31edo

New page

{{mathematical interest}}
{{infobox et}}
{{ED intro}}

== Theory ==
The 1536th harmonic is impossibly wide for a useful equivalence, so 328ed1536 is better thought of as a stretched version of [[31edo]]. Indeed, tuning the 1536/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning|stretched]] by about 0.79{{c}}. The [[31edo#Regular_temperament_properties|optimal octave stretch for 31edo]] in the [[13-limit]] is 0.502314{{c}}, making 328ed1536 very close to optimal for 31edo.

=== Harmonics ===
{{Harmonics in equal|328|1536|1|intervals=integer|columns=11}}
{{Harmonics in equal|328|1536|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 328ed1536 (continued)}}

== See also ==
* [[18edf]] – relative edf
* [[31edo]] – relative edo
* [[49edt]] – relative edt
* [[72ed5]] – relative ed5
* [[80ed6]] – relative ed6
* [[111ed12]] – relative ed12
* [[204ed96]] – close to the zeta-optimized tuning for 31edo

[[Category:31edo]]