13th-octave temperaments - Revision history

Eliora: per discussion on discord: for the same logic as neon should belong in 10th-octave, aluminium similarly should be placed here. besides it adds to this page as opposed to just listing "discussed elsewhere".

2026-08-06T21:37:36Z

per discussion on discord: for the same logic as neon should belong in 10th-octave, aluminium similarly should be placed here. besides it adds to this page as opposed to just listing "discussed elsewhere".

@@ Line 3: / Line 3: @@
 A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.
-Temperaments discussed elsewhere are [[ragismic microtemperaments #Aluminium|aluminium]], [[octagar temperaments #Tridecatonic|tridecatonic]], [[ragismic microtemperaments #Trideci|trideci]], and [[orwellismic temperaments #Triskaidekic|triskaidekic]].
+Furthermore, one step of 13edo is very close to [[135/128]], a notable 5-limit interval, in fact it is a semiconvergent to the said representation. Hence, considered below is aluminium.
 {{Navbox fractional-octave}}

FloraC: - duplicate data (moved to misc)

2026-07-21T03:52:05Z

- duplicate data (moved to misc)

@@ Line 3: / Line 3: @@
 A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.
-== Aluminium ==
+Temperaments discussed elsewhere are [[ragismic microtemperaments #Aluminium|aluminium]], [[octagar temperaments #Tridecatonic|tridecatonic]], [[ragismic microtemperaments #Trideci|trideci]], and [[orwellismic temperaments #Triskaidekic|triskaidekic]].
 {{Navbox fractional-octave}}

FloraC: Switch to Sintel's badness, WE & CWE tunings

2026-07-21T03:46:32Z

Switch to Sintel's badness, WE & CWE tunings

← Older revision		Revision as of 03:46, 21 July 2026
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	== Aluminium ==		== Aluminium ==
	: ''~~See also:~~ [[Ragismic microtemperaments #Aluminium]]''		: ''For extensions, see [[Ragismic microtemperaments #Aluminium]].''

	Aluminium ~~is named after the 13th element, and~~ tempers out ~~the~~ {{monzo\| 92 -39 -13 }} ~~comma which~~ sets [[135/128]] ~~interval to be equal~~ to 1/~~13th~~ of ~~the~~ octave.		Aluminium tempers out {{monzo\| 92 -39 -13 }} in the 5-limit, and sets [[135/128]] to 1/13 of an [[octave]].

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5
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	[[Comma list]]: {{monzo\| 92 -39 -13 }}		[[Comma list]]: {{monzo\| 92 -39 -13 }}

	~~[[Mapping]]:~~ {{~~mapping~~\| 13 0 92 \| 0 1 -3 }}		{{Mapping\|legend=1\| 13 0 92 \| 0 1 -3 }}
			: mapping generators: ~135/128, ~3

	: ~~Mapping generators~~: ~135/128, ~3		[[Optimal tuning]]s:
			* [[WE]]: ~135/128 = 92.3068{{c}}, ~3/2 = 701.9791{{c}}
	[[~~Optimal tuning~~]] ([[~~CTE~~]]): ~135/128 = ~~1\13~~, ~3/2 = 701.9897		: [[error map]]: {{val\| -0.012 +0.012 +0.009 }}
			* [[CWE]]: ~135/128 = 92.3077{{c}}, ~3/2 = 701.9897{{c}}
			: error map: {{val\| 0.000 +0.032 +0.033 }}

	{{Optimal ET sequence\|legend=1\| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}		{{Optimal ET sequence\|legend=1\| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}

	[[Badness]] (Sintel): 2.~~893~~		[[Badness]] (Sintel): 2.89

	== Tridecatonic ==		== Tridecatonic ==
	: ''~~See also:~~ [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[~~26th-octave~~ temperaments]]''		: ''For extensions, see [[Octagar temperaments #Tridecatonic]], [[Ragismic microtemperaments #Trideci]], and [[Varunismic temperaments #Bosonic]].''

	''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 ~~steps of 13edo~~.		Tridecatonic tempers out the [[devil's tridecalimma]], the comma which associates the [[10/9]] minor tone with [[13edo\|2\13]].

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

	[[Comma list]]: ~~2541865828329/2500000000000~~		[[Comma list]]: {{monzo\| -11 26 -13 }}

	~~[[Mapping]]: [~~{{~~Val~~\| 13 ~~21 31 }}, {{Val~~\| 0 -1 -2 }}]		{{Mapping\|legend=1\| 13 0 -11 \| 0 1 2 }}
			: mapping generators: ~531441/500000, ~3

	: ~~Mapping generators~~: ~531441/500000, ~3		[[Optimal tuning]]s:
			* [[WE]]: ~531441/500000 = 92.3166{{c}}, ~3/2 = 701.2353{{c}}
	[[~~Optimal tuning~~]] ([[~~CTE~~]]): ~531441/500000 = ~~1\13~~, ~3/2 = 701.2353		: [[error map]]: {{val\| +0.116 -0.688 +0.737 }}
			* [[CWE]]: ~531441/500000 = 92.3077{{c}}, ~3/2 = 701.2353{{c}}
			: error map: {{val\| 0.000 -0.797 +0.617 }}

	{{Optimal ET sequence\|legend=1\| 26, 39, 65, 286, 351, 416, 481, 546 }}		{{Optimal ET sequence\|legend=1\| 26, 39, 65, 286, 351, 416, 481, 546 }}

	[[Badness]] (Sintel): 10.~~263~~		[[Badness]] (Sintel): 10.3

	== Triskaidekic ==		== Triskaidekic ==
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	{{Mapping\|legend=1\| 13 0 30 \| 0 1 0 }}		{{Mapping\|legend=1\| 13 0 30 \| 0 1 0 }}

	: Mapping generators: ~128/125, ~3		: Mapping generators: ~128/125, ~3

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~128/125 = 92.589, ~3/2 = 698.263		* [[WE]]: ~128/125 = 92.589{{c}}, ~3/2 = 698.263{{c}}
	* [[CWE]]: ~128/125 = 92.308, ~3/2 = 698.068		: [[error map]]: {{val\| +3.656 -0.036 -8.647 }}
			* [[CWE]]: ~128/125 = 92.308{{c}}, ~3/2 = 698.068{{c}}
			: error map: {{val\| 0.000 -3.887 -17.083 }}

	{{Optimal ET sequence\|legend=1\| 13, 26, 91cc }}		{{Optimal ET sequence\|legend=1\| 13, 26, 91cc }}

	[[Badness]] (Sintel): 52.~~583~~		[[Badness]] (Sintel): 52.6

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

Xenllium at 22:32, 21 January 2026

2026-01-21T22:32:12Z

← Older revision		Revision as of 22:32, 21 January 2026
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	[[Badness]] (Sintel): 10.263		[[Badness]] (Sintel): 10.263

			== Triskaidekic ==
			: ''For extensions, see [[Orwellismic temperaments #Triskaidekic]].''

			[[Subgroup]]: 2.3.5

			[[Comma list]]: 1220703125/1073741824

			{{Mapping\|legend=1\| 13 0 30 \| 0 1 0 }}

			: Mapping generators: ~128/125, ~3

			[[Optimal tuning]]s:
			* [[WE]]: ~128/125 = 92.589, ~3/2 = 698.263
			* [[CWE]]: ~128/125 = 92.308, ~3/2 = 698.068

			{{Optimal ET sequence\|legend=1\| 13, 26, 91cc }}

			[[Badness]] (Sintel): 52.583

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

Xenllium at 01:15, 17 January 2026

2026-01-17T01:15:34Z

← Older revision		Revision as of 01:15, 17 January 2026
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	A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.		A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.

	~~Discussed elsewhere are~~:		== Aluminium ==
			: ''See also: [[Ragismic microtemperaments #Aluminium]]''

	* ''[[~~Aluminium~~]]~~'', →~~ [[~~Ragismic microtemperaments#Aluminium\|Ragismic microtemperaments~~]]~~, which is notable as~~ [[135/128]] ~~is a~~ [[~~semiconvergent~~]] to 1\13 ~~as well as a 5-limit interval~~.		Aluminium is named after the 13th element, and tempers out the {{monzo\| 92 -39 -13 }} comma which sets [[135/128]] interval to be equal to 1/13th of the octave.

			[[Subgroup]]: 2.3.5

			[[Comma list]]: {{monzo\| 92 -39 -13 }}

			[[Mapping]]: {{mapping\| 13 0 92 \| 0 1 -3 }}

			: Mapping generators: ~135/128, ~3

			[[Optimal tuning]] ([[CTE]]): ~135/128 = 1\13, ~3/2 = 701.9897

			{{Optimal ET sequence\|legend=1\| 65, 299, 364, 429, 494, 559, 1053, 1612, 5889, 7501, 9113, 10725, 23062bc, 33787bcc, 44512bbcc }}

			[[Badness]] (Sintel): 2.893

	== Tridecatonic ==		== Tridecatonic ==
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	''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.		''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.

	Subgroup: 2.3.5		[[Subgroup]]: 2.3.5

	[[Comma]]: 2541865828329/2500000000000		[[Comma list]]: 2541865828329/2500000000000

	[[Mapping]]: [{{Val\| 13 21 31 }}, {{Val\| 0 -1 -2 }}]		[[Mapping]]: [{{Val\| 13 21 31 }}, {{Val\| 0 -1 -2 }}]

	[[~~POTE generator~~]]: ~3/2 = 701.~~0837~~		: Mapping generators: ~531441/500000, ~3

			[[Optimal tuning]] ([[CTE]]): ~531441/500000 = 1\13, ~3/2 = 701.2353

	{{Optimal ET sequence\|legend=1\| 26, 39, 65, 286, 351, 416, 481, 546 }}		{{Optimal ET sequence\|legend=1\| 26, 39, 65, 286, 351, 416, 481, 546 }}

	[[Badness]]: 0.~~437494~~		[[Badness]] (Sintel): 10.263

	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}

Xenllium at 13:41, 13 April 2025

2025-04-13T13:41:30Z

← Older revision		Revision as of 13:41, 13 April 2025
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			{{Technical data page}}
	{{Infobox fractional-octave\|13}}		{{Infobox fractional-octave\|13}}
	Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.		A 13th-octave temperament can be described by temperament merging of edos whose greatest common divisor is 13. Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.

	Discussed elsewhere are:		Discussed elsewhere are:

Xenllium at 12:10, 13 April 2025

2025-04-13T12:10:07Z

← Older revision		Revision as of 12:10, 13 April 2025
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	{{Infobox fractional-octave\|13}}		{{Infobox fractional-octave\|13}}
			Although [[13edo]] itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 13 are.

	Discussed elsewhere are:		Discussed elsewhere are:

ArrowHead294 at 03:59, 14 February 2025

2025-02-14T03:59:07Z

← Older revision		Revision as of 03:59, 14 February 2025
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	{{~~Fractional~~-octave ~~navigation~~\|13}}		{{Infobox fractional-octave\|13}}

	Discussed elsewhere are:		Discussed elsewhere are:

BudjarnLambeth: navbox

2024-12-05T07:17:45Z

navbox

← Older revision		Revision as of 07:17, 5 December 2024
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	[[Badness]]: 0.437494		[[Badness]]: 0.437494

			{{Navbox fractional-octave}}

Eliora: /* Tridecatonic */

2024-02-21T13:08:02Z

Tridecatonic

← Older revision		Revision as of 13:08, 21 February 2024
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	''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.		''Tridecatonic'' tempers out the devil's tridecalimma, the comma which associates the [[10/9]] minor tone with 2 steps of 13edo.

	Subgroup: 2.3.5		Subgroup: 2.3.5