118th-octave temperaments - Revision history

ArrowHead294 at 03:43, 14 February 2025

2025-02-14T03:43:35Z

← Older revision		Revision as of 03:43, 14 February 2025
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	{{~~Fractional~~-octave ~~navigation~~\|118}}		{{Infobox fractional-octave\|118}}
	[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].		[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].

BudjarnLambeth: navbox

2024-12-05T07:43:48Z

navbox

← Older revision		Revision as of 07:43, 5 December 2024
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	{{Optimal ET sequence\|legend=1\| 354, ..., 3422, 3776 }}		{{Optimal ET sequence\|legend=1\| 354, ..., 3422, 3776 }}

			{{Navbox fractional-octave}}

	[[Category:118edo]]		[[Category:118edo]]
	~~[[Category:Temperament collections]]~~

Eliora: /* Parakleischis */ rename centenniamajor to peithoian by namer's decision + the fact that the "-major" ending in centenniamajor can potentially be confusing as the temp has no intrinsic "major" quality.

2024-08-27T15:57:35Z

Parakleischis: rename centenniamajor to peithoian by namer's decision + the fact that the "-major" ending in centenniamajor can potentially be confusing as the temp has no intrinsic "major" quality.

← Older revision		Revision as of 15:57, 27 August 2024
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	Badness: 0.049316		Badness: 0.049316

	=== ~~Centenniamajor~~ ===		=== Peithoian ===
	~~Centenniamajor~~ is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of ~~centenniamajor~~ temperament is to expand on that. Named after the ~~fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) =~~ 118.		Peithoian is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of peithoian temperament is to expand on that. Named after the minor planet [[wikipedia:118 Peitho\|118 Peitho]].

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

Eliora: /* Oganesson */ oops

2024-03-21T13:00:08Z

Oganesson: oops

← Older revision		Revision as of 13:00, 21 March 2024
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	[[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068		[[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068

	{{Optimal ET sequence\|legend=1\| 354, 2360, 2714, 3068, ~~3442~~, 3776, 7198cd, 10974bccdd }}		{{Optimal ET sequence\|legend=1\| 354, 2360, 2714, 3068, 3422, 3776, 7198cd, 10974bccdd }}

	[[Badness]]: 2.66		[[Badness]]: 2.66
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	Optimal tuning (CTE): ~1953125/1354752 = 634.0085		Optimal tuning (CTE): ~1953125/1354752 = 634.0085

	{{Optimal ET sequence\|legend=1\| 354, 3068e, ~~3442~~, 3776, 11682ccdde }}		{{Optimal ET sequence\|legend=1\| 354, 3068e, 3422, 3776, 11682ccdde }}

	Badness: 0.568		Badness: 0.568

Eliora at 17:51, 17 September 2023

2023-09-17T17:51:54Z

← Older revision		Revision as of 17:51, 17 September 2023
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			{{Fractional-octave navigation\|118}}
	[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].		[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].

Eliora: 19-limit oganesson

2023-06-10T14:14:36Z

19-limit oganesson

← Older revision		Revision as of 14:14, 10 June 2023
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	[[Badness]]: 0.145166		[[Badness]]: 0.145166

	==== 11-limit ====		=== 11-limit ===
	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

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	Badness: 0.357		Badness: 0.357

	===== 13-limit =====		==== 13-limit ====
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

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	[[Badness]]: 2.66		[[Badness]]: 2.66

	==== 11-limit ====		=== 11-limit ===
	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

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	Badness: 0.568		Badness: 0.568

	==== 13-limit ====		=== 13-limit ===
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

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	Badness: 0.172		Badness: 0.172

	==== 17-limit ====		=== 17-limit ===
	Subgroup: 2.3.5.7.11.13.17		Subgroup: 2.3.5.7.11.13.17

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	{{Optimal ET sequence\|legend=1\| 354, 3068e, 3422, 3776 }}		{{Optimal ET sequence\|legend=1\| 354, 3068e, 3422, 3776 }}

	~~Badness~~: 0.~~105~~		=== 19-limit ===
			The closest superparticular to one step of 118edo is [[171/170]], so 19-limit extension for oganesson is prescribed.

			Subgroup: 2.3.5.7.11.13.17.19

			Comma list: 4096/4095, 6175/6174, 9801/9800, 14365/14364, 28900/28899, 3438981/3437500

			Mapping: [{{val\| 118 0 274 643 1094 499 607 1000}}, {{val\| 0 3 0 -5 -11 -1 2 -8}}]

			: mapping generators: ~171/170, ~238/165

			Optimal tuning (CTE): ~238/165 = 634.006

			{{Optimal ET sequence\|legend=1\| 354, ..., 3422, 3776 }}

	[[Category:118edo]]		[[Category:118edo]]
	[[Category:Temperament collections]]		[[Category:Temperament collections]]

Eliora at 13:59, 10 June 2023

2023-06-10T13:59:21Z

← Older revision		Revision as of 13:59, 10 June 2023
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	Badness: 0.105		Badness: 0.105

			[[Category:118edo]]
			[[Category:Temperament collections]]

Eliora at 13:35, 10 June 2023

2023-06-10T13:35:58Z

← Older revision		Revision as of 13:35, 10 June 2023
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	[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets.		[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets. Furthermore, one step of 118edo is in direct proximity to essential tempering commas [[169/168]] and [[170/169]].

	=== Parakleischis ===		== Parakleischis ==
	118edo and its ~~multiples are members of both~~ [[parakleismic]] and [[Schismatic family\|schismic]], and ~~from this it derives its name~~.		118edo's is an excellent 5-limit system and its comma basis constitutes the [[parakleismic]] and [[Schismatic family\|schismic]] temperaments together. Parakleischis retains the 5-limit mapping from 118edo and leaves other harmonics as independent generators.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	[[Mapping]]: [{{val\| 118 187 274 0 }}, {{val\| 0 0 0 1 }}]		[[Mapping]]: [{{val\| 118 187 274 0 }}, {{val\| 0 0 0 1 }}]

	~~Mapping~~ generators: ~15625/15552, ~7		: mapping generators: ~15625/15552, ~7

	[[Optimal tuning]] ([[~~POTE~~]]): ~7/4 = 968.7235		[[Optimal tuning]] ([[CTE]]): ~7/4 = 968.7235

	{{Optimal ET sequence\|legend=1\| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}		{{Optimal ET sequence\|legend=1\| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}
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	Mapping: [{{val\| 118 187 274 0 77 }}, {{val\| 0 0 0 1 1 }}]		Mapping: [{{val\| 118 187 274 0 77 }}, {{val\| 0 0 0 1 1 }}]

	Optimal tuning (~~POTE~~): ~7/4 = 968.5117		: mapping generators: ~176/176, ~7

			Optimal tuning (CTE): ~7/4 = 968.5117

	{{Optimal ET sequence\|legend=1\| 118, 354, 472 }}		{{Optimal ET sequence\|legend=1\| 118, 354, 472 }}
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	Badness: 0.049316		Badness: 0.049316

	==== Centenniamajor ====		=== Centenniamajor ===
	Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.		Centenniamajor is an extension of parakleischis which retains the 5-limit mapping of 118edo and provides the correction for 13th harmonic. 13-limit is the first prime limit that 118edo does not tune consistently, and the goal of centenniamajor temperament is to expand on that. Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11
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	Badness: 0.122		Badness: 0.122

	=== Oganesson ===		== Oganesson ==
	Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.		Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.

Eliora: there's enough for its own page

2023-06-10T13:29:29Z

there's enough for its own page

New page

[[118edo]] is the 17th [[zeta peak edo]], and it is accurate for harmonics 3 and 5, so various 118th-octave temperaments naturally occur through temperament merging of its supersets.

=== Parakleischis ===
118edo and its multiples are members of both [[parakleismic]] and [[Schismatic family|schismic]], and from this it derives its name.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 32805/32768, 1224440064/1220703125

[[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]

Mapping generators: ~15625/15552, ~7

[[Optimal tuning]] ([[POTE]]): ~7/4 = 968.7235

{{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}

[[Badness]]: 0.145166

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 32805/32768, 137781/137500

Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]

Optimal tuning (POTE): ~7/4 = 968.5117

{{Optimal ET sequence|legend=1| 118, 354, 472 }}

Badness: 0.049316

==== Centenniamajor ====
Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.

Subgroup: 2.3.5.7.11

Comma list: 32805/32768, 151263/151250, 1224440064/1220703125

Mapping: [{{val| 118 187 274 0 -420 }}, {{val| 0 0 0 2 5 }}]

Mapping generators: ~15625/15552, ~405504/153125

Optimal tuning (CTE): ~202752/153125 = 484.4837

{{Optimal ET sequence|legend=1| 354, 944e, 1298 }}

Badness: 0.357

===== 13-limit =====
Subgroup: 2.3.5.7.11.13

Comma list: 1716/1715, 32805/32768, 34398/34375, 384912/384475

Mapping: [{{val| 118 187 274 0 -420 271 }}, {{val| 0 0 0 2 5 1 }}]

Optimal tuning (CTE): ~8125/6144 = 484.4867

{{Optimal ET sequence|legend=1| 354, 944e, 1298 }}

Badness: 0.122

=== Oganesson ===
Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.

In the 7-limit and 11-limit, the period corresponds to [[bronzisma]].

[[Subgroup]]: 2.3.5.7

[[Comma list]]: {{monzo| 30 10 -27 6 }}, {{monzo| 77 -20 -5 -12 }}

[[Mapping]]: [{{val| 118 0 274 643 }}, {{val| 0 3 0 -5 }}]

Mapping generators: ~2097152/2083725, ~1953125/1354752

[[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068

{{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3442, 3776, 7198cd, 10974bccdd }}

[[Badness]]: 2.66

==== 11-limit ====
Subgroup: 2.3.5.7.11

Comma list: 9801/9800, {{monzo| 13 -1 4 -16 7 }}, {{monzo| 55 -7 -15 -2 -1 }}

Mapping: [{{val| 118 0 274 643 1094 }}, {{val| 0 3 0 -5 -11 }}]

Optimal tuning (CTE): ~1953125/1354752 = 634.0085

{{Optimal ET sequence|legend=1| 354, 3068e, 3442, 3776, 11682ccdde }}

Badness: 0.568

==== 13-limit ====
Subgroup: 2.3.5.7.11.13

Comma list: 4096/4095, 9801/9800, 537403776/537109375, 453874312332/453857421875

Mapping: [{{val| 118 0 274 643 1094 499}}, {{val| 0 3 0 -5 -11 -1}}]

Mapping generators: ~169/168, ~1124864/779625

Optimal tuning (CTE): ~1124864/779625 = 634.0087

{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}

Badness: 0.172

==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17

Comma list: 4096/4095, 9801/9800, 34391/34375, 361250/361179, 562432/562275

Mapping: [{{val| 118 0 274 643 1094 499 607 }}, {{val| 0 3 0 -5 -11 -1 2 }}]

Mapping generators: ~170/169, ~238/165

Optimal tuning (CTE): ~238/165 = 634.0080

{{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}

Badness: 0.105