User:BudjarnLambeth/116zpi - Revision history

BudjarnLambeth at 03:18, 16 August 2025

2025-08-16T03:18:01Z

← Older revision		Revision as of 03:18, 16 August 2025
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	All instruments listed under [[29edo#Instruments]] also work for 116zpi.		All instruments listed under [[29edo#Instruments]] also work for 116zpi.

	[[Category:Zeta peak indexes]] [[Category:29edo~~]] [[Category:Edonoi~~]]		[[Category:Zeta peak indexes]] [[Category:29edo]]

BudjarnLambeth at 03:10, 16 August 2025

2025-08-16T03:10:06Z

← Older revision		Revision as of 03:10, 16 August 2025
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	For [[prime]]s up to 29:		For [[prime]]s up to 29:

	~~435zpi~~ approximates these with less than 7 cents error (<17% relative error):		116zpi approximates these with less than 7 cents error (<17% relative error):
	* 2, 3, 11, 13, 19, 23		* 2, 3, 11, 13, 19, 23

BudjarnLambeth at 03:06, 16 August 2025

2025-08-16T03:06:28Z

← Older revision		Revision as of 03:06, 16 August 2025
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	{{Editable user page}}		{{Editable user page}}

	'''116zpi''', the 116th [[zeta peak index]], is a [[Octave stretching\|stretched-octaves]] version of [[29edo]]. It can be thought of as '''~~80ed1202~~.49c''' or as '''41.465cet'''.		'''116zpi''', the 116th [[zeta peak index]], is a [[Octave stretching\|stretched-octaves]] version of [[29edo]]. It can be thought of as '''29ed1202.49c''' or as '''41.465cet'''.

	It has a step size of 41.465 [[cent]]s, and the [[octave]] ([[2/1]]) is tuned slightly impurely to ~~80ed1202~~.49 cents.		It has a step size of 41.465 [[cent]]s, and the [[octave]] ([[2/1]]) is tuned slightly impurely to 1202.49 cents.

	29edo tunes almost all simple [[harmonic]]s slightly flat by roughly the same amount, so 116zpi is one possible way of correcting for this.		29edo tunes almost all simple [[harmonic]]s slightly flat by roughly the same amount, so 116zpi is one possible way of correcting for this.

BudjarnLambeth: /* Prime harmonics */

2025-08-16T02:57:38Z

Prime harmonics

← Older revision		Revision as of 02:57, 16 August 2025
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	* 29		* 29

	This makes it usable as a full [[29-limit]] tuning, or as a more accurate 2.3.5.11.13 [[subgroup]] tuning.		This makes it usable as a full [[23-limit]] tuning, or as a more accurate 2.3.5.11.13 [[subgroup]] tuning.

	==Scales==		==Scales==

BudjarnLambeth: /* Prime harmonics */

2025-08-16T02:57:28Z

Prime harmonics

← Older revision		Revision as of 02:57, 16 August 2025
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	==Prime harmonics==		==Prime harmonics==
	{{Harmonics in cet\|41.465\|columns=~~7\|intervals=prime\|title=Approximation of prime harmonics in 116zpi}}~~		{{Harmonics in cet\|41.465\|columns=10\|intervals=prime\|title=Approximation of prime harmonics in 116zpi}}
	~~{{Harmonics in cet\|41.465\|columns=7\|start=8~~\|intervals=prime\|title=Approximation of prime harmonics in 116zpi}}

	For [[prime]]s up to 29:		For [[prime]]s up to 29:

BudjarnLambeth: Created page with "{{Editable user page}} '''116zpi''', the 116th zeta peak index, is a stretched-octaves version of 29edo. It can be thought of as '''80ed1202.49c''' or as '''41.465cet'''. It has a step size of 41.465 cents, and the octave (2/1) is tuned slightly impurely to 80ed1202.49 cents. 29edo tunes almost all simple harmonics slightly flat by roughly the same amount, so 116zpi is one possible way of correcting for this. ==Prime har..."

2025-08-16T02:57:00Z

Created page with "{{Editable user page}} '''116zpi''', the 116th zeta peak index, is a stretched-octaves version of 29edo. It can be thought of as '''80ed1202.49c''' or as '''41.465cet'''. It has a step size of 41.465 cents, and the octave (2/1) is tuned slightly impurely to 80ed1202.49 cents. 29edo tunes almost all simple harmonics slightly flat by roughly the same amount, so 116zpi is one possible way of correcting for this. ==Prime har..."

New page

{{Editable user page}}

'''116zpi''', the 116th [[zeta peak index]], is a [[Octave stretching|stretched-octaves]] version of [[29edo]]. It can be thought of as '''80ed1202.49c''' or as '''41.465cet'''.

It has a step size of 41.465 [[cent]]s, and the [[octave]] ([[2/1]]) is tuned slightly impurely to 80ed1202.49 cents.

29edo tunes almost all simple [[harmonic]]s slightly flat by roughly the same amount, so 116zpi is one possible way of correcting for this.

==Prime harmonics==
{{Harmonics in cet|41.465|columns=7|intervals=prime|title=Approximation of prime harmonics in 116zpi}}
{{Harmonics in cet|41.465|columns=7|start=8|intervals=prime|title=Approximation of prime harmonics in 116zpi}}

For [[prime]]s up to 29:

435zpi approximates these with less than 7 cents error (<17% relative error):
* 2, 3, 11, 13, 19, 23

...these with 7-14 cents error (17-34% relative error):
* 5, 7, 17

...and these with more than 14 cents error (>34% relative error):
* 29

This makes it usable as a full [[29-limit]] tuning, or as a more accurate 2.3.5.11.13 [[subgroup]] tuning.

==Scales==
Any [[29edo#Scales|scales from 29edo]] should also be useable here.

==Instruments==
All instruments listed under [[29edo#Instruments]] also work for 116zpi.

[[Category:Zeta peak indexes]] [[Category:29edo]] [[Category:Edonoi]]