76ed80 - Revision history

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

2025-06-26T11:57:50Z

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

2025-06-18T17:32:08Z

Theory

← Older revision		Revision as of 17:32, 18 June 2025
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	== Theory ==		== Theory ==
	The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}; 76ed80 ~~has an octave only about {{sfrac\|1\|6}}{{c}} wider than just and is therefore~~ extremely close to optimal for 12edo.		The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}, making 76ed80 extremely close to optimal for 12edo.

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

2025-06-18T17:30:47Z

← Older revision		Revision as of 17:30, 18 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}; 76ed80's octave ~~is off from the zeta peak by~~ only {{~~frac~~\|1\|6}}{{c}} and is therefore extremely close to optimal for 12edo.		The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}; 76ed80 has an octave only about {{sfrac\|1\|6}}{{c}} wider than just and is therefore extremely close to optimal for 12edo.

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

2025-06-18T17:28:24Z

Theory

← Older revision		Revision as of 17:28, 18 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}} ~~(which~~ is ~~compressed~~ by ~~2.31~~{{c}}~~), making 76ed80~~ extremely close to optimal for 12edo.		The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}}; 76ed80's octave is off from the zeta peak by only {{frac\|1\|6}}{{c}} and is therefore extremely close to optimal for 12edo.

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

2025-06-18T15:47:29Z

Theory

← Older revision		Revision as of 15:47, 18 June 2025
Line 4:		Line 4:

	== Theory ==		== Theory ==
	The 80th harmonic ~~would~~ be ~~extremely wide for an~~ equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}} (which is compressed by 2.31{{c}}), making 76ed80 extremely close to optimal for 12edo.		The 80th harmonic is too wide to be a useful equivalence, so 76ed80 is better thought of as a compressed version of the ubiquitous [[12edo]]. Indeed, tuning the 80/1 ratio just instead of 2/1 results in octaves being [[stretched and compressed tuning\|compressed]] by about 2.16{{c}}. The local [[The Riemann zeta function and tuning #Optimal octave stretch\|zeta peak]] around 12 is located at 12.023183, which has a step size of 99.807{{c}} and an octave of 1197.686{{c}} (which is compressed by 2.31{{c}}), making 76ed80 extremely close to optimal for 12edo.

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

2025-06-18T13:27:48Z

← Older revision		Revision as of 13:27, 18 June 2025
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	[[Category:12edo]]		[[Category:12edo]]
			[[Category:Zeta-optimized tunings]]

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