Septiennealimmal clan - Revision history

FloraC: Change several names, as planned

2026-05-19T08:42:14Z

Change several names, as planned

← Older revision		Revision as of 08:42, 19 May 2026
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	[[Badness]] (Sintel): 0.0914		[[Badness]] (Sintel): 0.0914

	=== ~~11-limit~~ ===		=== Enneabiotic ===
	~~The undecimal ennealimmal temperament can be described as~~ {{nowrap\| 99e & 171e }}~~, which~~ tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma).		Enneabiotic ({{nowrap\| 99e & 171e }}) tempers out [[5632/5625]] (vishdel comma) and [[19712/19683]] (symbiotic comma). It is catalogued as ''undecimal ennealimmal'' in [[Graham Breed]]'s [https://x31eq.com/temper/ Temperament Finder].

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11
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	==== 13-limit ====		==== 13-limit ====
			Subgroup: 2.3.5.7.11.13

			Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625

			Mapping: {{mapping\| 9 1 1 12 -75 -106 \| 0 2 3 2 16 21 }}

			Optimal tunings:
			* WE: ~27/25 = 133.3215{{c}}, ~5/3 = 884.4027{{c}} (~36/35 = 48.8479{{c}})
			* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4745{{c}} (~36/35 = 48.8589{{c}})

			{{Optimal ET sequence\|legend=0\| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}

			Badness (Sintel): 0.912

			==== Enneabio ====
	Subgroup: 2.3.5.7.11.13		Subgroup: 2.3.5.7.11.13

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	Badness (Sintel): 1.25		Badness (Sintel): 1.25

	===~~= Ennealimmalis =~~===		=== Ennealympic ===
	~~Subgroup: 2.3.5.7.11.13~~		Ennealympic ({{nowrap\| 99 & 171 }}, formerly ''ennealimmia'') is an alternative extension which tempers out [[131072/130977]] (olympia).

	~~Comma list: 2080/2079, 2401/2400, 4375/4374, 5632/5625~~

	~~Mapping:~~ {{~~mapping~~\| ~~9 1 1 12 -75 -106 \| 0 2 3 2 16 21 }}~~

	~~Optimal tunings:~~
	* WE: ~27/25 = 133.3215{{c}}, ~~~5/3 = 884.4027{{c}} (~36/35 = 48.8479{{c}}~~)
	* CWE: ~27/25 = 133.3333{{c}}, ~5/3 = 884.4745{{c}} (~36/35 = 48.8589{{c}})

	~~{{Optimal ET sequence\|legend=0\| 99ef, 171ef, 270, 639, 909, 1179, 2088bce }}~~

	~~Badness (Sintel): 0.912~~

	~~=== Ennealimmia ===~~
	~~The ennealimmia temperament~~ is an alternative extension ~~and can be described as {{nowrap\| 99 & 171 }},~~ which tempers out [[131072/130977]] (olympia).

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11
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	=== Ennealimnic ===		=== Ennealimnic ===
	{{Distinguish\| Ennealimmic }}		{{Distinguish\| Ennealimmic }}
			{{See also\| Chords of ennealimnic }}

	Ennealimnic ({{nowrap\| 72 & 171 }}) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.		Ennealimnic ({{nowrap\| 72 & 171 }}) equates 11/9 with 27/22, 49/40, and 60/49 as a neutral third interval.
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	Badness (Sintel): 0.673		Badness (Sintel): 0.673

	~~See also: [[Chords of ennealimnic]]~~

	==== 13-limit ====		==== 13-limit ====
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	* CWE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0386{{c}}		* CWE: ~27/26 = 66.6667{{c}}, ~26/15 = 951.0386{{c}}

	{{Optimal ET sequence\|legend=1\| 72, 198g, 270 }}		{{Optimal ET sequence\|legend=0\| 72, 198g, 270 }}

	Badness (Sintel): 0.812		Badness (Sintel): 0.812
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	Badness (Sintel): 0.927		Badness (Sintel): 0.927

	=== ~~Semiennealimmal~~ ===		=== Ennealimmapine ===
	~~Semiennealimmal~~ tempers out [[4000/3993]], and uses a ~140/121 semifourth generator. ~~Notably, however~~, ~~two generator steps do not reach~~ ~4/3, ~~despite that~~ the ~~name may suggest so. In fact~~, ~~it splits~~ the ~~generator~~ of ~~ennealimmal into three~~.		Ennealimmapine (formerly ''semiennealimmal'') tempers out [[4000/3993]], and uses a ~140/121 semifourth generator, six of which and 1/3 octave give the 3rd harmonic. Perhaps a better generator is the [[secor]], ~77/72, six of which give the perfect fifth, or the [[ptolemisma]], six of which and 1/3 octave give the perfect fourth.

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

FloraC: Move septiennealic to the bottom as it skips two primes

2026-04-30T12:31:09Z

Move septiennealic to the bottom as it skips two primes

@@ Line 29: / Line 29: @@
 [[Badness]] (Sintel): 0.191
 === Ennea ===
 Subgroup: 2.3.7.11
@@ Line 804: / Line 783: @@
 Badness (Sintel): 1.49
 [[Category:Septiennealimmal clan| ]] <!-- main article -->
 [[Category:Temperament clans]]

FloraC: - undecipherable nonsense

2026-03-20T11:21:16Z

- undecipherable nonsense

← Older revision		Revision as of 11:21, 20 March 2026
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	=== Hemiennealimmal ===		=== Hemiennealimmal ===
	Hemiennealimmal ({{nowrap\| 72 & 198 }}) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, [[3025/3024]], 4375/4374, and [[9801/9800]]. ~~Tempering out 9801/9800 leads to an octave split into two equal parts. Notably, every one of these commas is part of one or more known infinite comma families; see directly below.~~		Hemiennealimmal ({{nowrap\| 72 & 198 }}) has a period of 1/18 octave and tempers out the four smallest superparticular commas of the 11-limit JI, 2401/2400, [[3025/3024]], 4375/4374, and [[9801/9800]]. Its [[S-expression]]-based comma list is {([[3025/3024\|S22/S24 = S55 = S25/S27 × S99]]), [[4375/4374\|S25/S27]], [[2401/2400\|S49]], [[9801/9800\|S33/S35 = S99]]}. Tempering out 9801/9800 leads to an octave split into two equal parts.

	Its [[S-expression]]-based comma list is {([[3025/3024\|S22/S24 = S55 = S25/S27 × S99]]), [[4375/4374\|S25/S27]], [[2401/2400\|S49]], [[9801/9800\|S33/S35 = S99]]}.

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

FloraC: Note source of names

2026-03-20T11:18:25Z

Note source of names

← Older revision		Revision as of 11:18, 20 March 2026
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	== Undecentic ==		== Undecentic ==
	~~Undecentic~~ ({{nowrap\| 99 & 198 }}) has a period of 1/99 octave.		{{Distinguish\| Undecental }}

			Named by [[Xenllium]] in 2021, undecentic ({{nowrap\| 99 & 198 }}) has a period of 1/99 octave.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	== Schisennealimmal ==		== Schisennealimmal ==
	Schisennealimmal ({{nowrap\| 171 & 342 }}) has a period of 1/171 octave. [[171edo]] and its multiples are members of both [[schismic]] and ennealimmal~~, and from this it derives its name~~.		Schisennealimmal ({{nowrap\| 171 & 342 }}) has a period of 1/171 octave. It was named by [[Xenllium]] in 2021 for the fact that [[171edo]] and its multiples are members of both [[schismic]] and ennealimmal.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11
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	== Lunennealimmal ==		== Lunennealimmal ==
	Lunennealimmal ({{nowrap\| 441 & 882 }}) has has a period of 1/441 octave. [[441edo]] and its multiples are members of both [[luna]] and ennealimmal~~, and from this it derives its name~~.		Lunennealimmal ({{nowrap\| 441 & 882 }}) has has a period of 1/441 octave. It was named by [[Xenllium]] in 2021 for the fact that [[441edo]] and its multiples are members of both [[luna]] and ennealimmal.

	[[Subgroup]]: 2.3.5.7.11		[[Subgroup]]: 2.3.5.7.11

FloraC: Switch to Sintel's badness, WE & CWE tunings (5/5)

2026-03-20T11:11:27Z

Switch to Sintel's badness, WE & CWE tunings (5/5)

Show changes

FloraC: Switch to Sintel's badness, WE & CWE tunings (4/)

2026-03-20T10:51:09Z

Switch to Sintel's badness, WE & CWE tunings (4/)

Show changes

FloraC: Switch to Sintel's badness, WE & CWE tunings (3/)

2026-03-20T09:28:28Z

Switch to Sintel's badness, WE & CWE tunings (3/)

Show changes

FloraC: Switch to Sintel's badness, WE & CWE tunings (2/)

2026-03-20T08:53:12Z

Switch to Sintel's badness, WE & CWE tunings (2/)

Show changes

FloraC: Switch to Sintel's badness, WE & CWE tunings (1/)

2026-03-20T08:09:44Z

Switch to Sintel's badness, WE & CWE tunings (1/)

← Older revision		Revision as of 08:09, 20 March 2026
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	== No-five septiennealimmal ==		== No-five septiennealimmal ==
	This rank-2 temperament is of interest to anyone who wants a different generator for the ennealimmal-like structure because it represents the part of ennealimmal supported by non-ennealimmal equal temperaments of interest that do well in the 2.3.7 subgroup, such as [[36edo]], which adds the [[1029/1024\|gamelisma]], or [[63edo]], which in the 7-limit can be used for [[magic]] and in higher limits for [[parapyth]] among other things.		This rank-2 temperament simply equates a stack of nine [[7/6]] subminor thirds with two octaves. It is of interest to anyone who wants a different generator for the ennealimmal-like structure because it represents the part of ennealimmal supported by non-ennealimmal equal temperaments of interest that do well in the [[2.3.7 subgroup]], such as [[36edo]], which adds the [[1029/1024\|gamelisma]], or [[63edo]], which in the 7-limit can be used for [[magic]] and in higher limits for [[parapyth]] among other things.

	[[Subgroup]]: 2.3.7		[[Subgroup]]: 2.3.7
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[~~CTE~~]]: ~2592/2401 = 133.~~3333~~{{c}}, ~3/2 = ~~702~~.~~0044~~{{c}}		* [[WE]]: ~2592/2401 = 133.3357{{c}}, ~3/2 = 701.9772{{c}}
	: [[error map]]: {{val\| 0.~~0000~~ +0.~~0494~~ -0.~~1549~~ }}		: [[error map]]: {{val\| +0.021 +0.043 -0.135 }}
	* [[~~POTE~~]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.~~9649~~{{c}}		* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.9833{{c}}
	: error map: {{val\| 0.0000 +0.~~0099~~ -0.~~1944~~ }}		: error map: {{val\| 0.0000 +0.028 -0.176 }}

	{{Optimal ET sequence\|legend=1\| 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd }}		{{Optimal ET sequence\|legend=1\| 27, 36, 99, 135, 171, 306, 4419d, 4725d, …, 8397dd, 8703dd }}

			[[Badness]] (Sintel): 0.191

	=== Septiennealic ===		=== Septiennealic ===
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	A notable tuning of septiennealic not appearing in the optimal ET sequence is [[63edo]]. If we include a somewhat more complex mapping for 11 via {{nowrap\| 36e & 63 }}, it will become the optimal patent val and largest in the sequence.		A notable tuning of septiennealic not appearing in the optimal ET sequence is [[63edo]]. If we include a somewhat more complex mapping for 11 via {{nowrap\| 36e & 63 }}, it will become the optimal patent val and largest in the sequence.

	[[Subgroup]]: 2.3.7.13		Subgroup: 2.3.7.13

	[[Comma list]]: [[169/168]], [[31213/31104]]		Comma list: 169/168, 31213/31104

	{{~~Mapping\|legend=2~~\| 9 0 11 19 \| 0 1 1 1 }}		Subgroup-val mapping: {{mapping\| 9 0 11 19 \| 0 1 1 1 }}

	[[Optimal ~~tuning]] ([[CTE]])~~: ~13/12 = 133.~~333~~{{c}}, ~3/2 = 702.~~638~~{{c}}		Optimal tunings:
			* WE: ~13/12 = 133.3847{{c}}, ~3/2 = 701.9342{{c}}
			* CWE: ~13/12 = 133.3333{{c}}, ~3/2 = 702.0763{{c}}

	{{Optimal ET sequence\|legend=1\| 27, 36, 99, 135f, 171f }}		{{Optimal ET sequence\|legend=0\| 27, 36, 99, 135f, 171f }}

	[[Badness~~]]:~~		Badness (Sintel): 0.540
	* Smith: 0.0134
	* Dirichlet: 0.540

	=== Ennea ===		=== Ennea ===
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	Comma list: 41503/41472, 43923/43904		Comma list: 41503/41472, 43923/43904

	~~Mapping~~: {{mapping\| 9 0 11 24 \| 0 2 2 1 }}		Subgroup-val mapping: {{mapping\| 9 0 11 24 \| 0 2 2 1 }}
			: mapping generators: ~121/112, ~343/198

	Optimal tunings:		Optimal tunings:
	* ~~CTE~~: ~121/112 = 133.~~3333~~{{c}}, 343/198 = 951.~~0143~~{{c}} (~99/98 = 17.~~6810~~{{c}})		* WE: ~121/112 = 133.3392{{c}}, 343/198 = 951.0013{{c}} (~99/98 = 17.6266{{c}})
	* ~~POTE~~: ~121/112 = 133.3333{{c}}, 343/198 = 950.~~9591~~{{c}} (~99/98 = 17.~~6258~~{{c}})		* CWE: ~121/112 = 133.3333{{c}}, 343/198 = 950.9799{{c}} (~99/98 = 17.6466{{c}})

			{{Optimal ET sequence\|legend=0\| 63, 72, 135, 342, 477, 1089, 1566 }}

	~~{{Optimal ET sequence\|legend=~~0~~\| 54, 63, 72, 135, 342, 477, 1089, 1566 }}~~		Badness (Sintel): 0.161

	== Ennealimmal ==		== Ennealimmal ==

FloraC: Units & misc. cleanup. + category

2026-03-20T07:54:45Z

Units & misc. cleanup. + category

Show changes