Very low accuracy temperaments - Revision history

FloraC: /* Quint */ review

2026-07-23T10:51:10Z

Quint: review

← Older revision		Revision as of 10:51, 23 July 2026
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	== Quint ==		== Quint ==
	Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is ~~way more~~ accurately approximated ~~than 5~~ by 5edo, this temperament provides little improvement ~~to 5edo's 7-limit tuning~~, so ~~in what way this temperament~~ is ~~useful remains unexplained~~. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.		Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is already quite accurately approximated by 5edo for its size, this temperament provides little improvement, so it is only provided below for bookkeeping purposes. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	{{Mapping\|legend=1\| 5 8 12 0 \| 0 0 0 1 }}		{{Mapping\|legend=1\| 5 8 12 0 \| 0 0 0 1 }}

	: Mapping generators: ~6/5, ~7		: Mapping generators: ~6/5, ~7

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) ~~<!-- Please review, last digit may be wrong -->~~		* [[WE]]: ~6/5 = 236.416{{c}}, ~7/4 = 1002.701{{c}} (~28/27 = 57.038{{c}})
	: [[error map]]: {{val\| -17.921 -10.628 +50.676 -1.966 }}		: [[error map]]: {{val\| -17.921 -10.628 +50.676 -1.966 }}
	* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)		* [[CWE]]: ~6/5 = 240.000{{c}}, ~7/4 = 1005.135{{c}} (~28/27 = 45.135{{c}})
	: error map: {{val\| 0.000 +18.045 +93.686 +36.309 }}		: error map: {{val\| 0.000 +18.045 +93.686 +36.309 }}

	{{Optimal ET sequence\|legend=1\| 5, 15ccd }}		{{Optimal ET sequence\|legend=1\| 5, 15ccd, 20cccd }}

	[[Badness]] (Sintel): 1.22		[[Badness]] (Sintel): 1.22

FloraC: Restore quint. It's present in the 7-limit catalog page, so good just for bookkeeping reasons

2026-07-23T10:39:13Z

Restore quint. It's present in the 7-limit catalog page, so good just for bookkeeping reasons

← Older revision		Revision as of 10:39, 23 July 2026
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	[[Badness]] (Sintel): 1.16		[[Badness]] (Sintel): 1.16

			== Quint ==
			Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

			[[Subgroup]]: 2.3.5.7

			[[Comma list]]: 16/15, 27/25

			{{Mapping\|legend=1\| 5 8 12 0 \| 0 0 0 1 }}

			: Mapping generators: ~6/5, ~7

			[[Optimal tuning]]s:
			* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) <!-- Please review, last digit may be wrong -->
			: [[error map]]: {{val\| -17.921 -10.628 +50.676 -1.966 }}
			* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
			: error map: {{val\| 0.000 +18.045 +93.686 +36.309 }}

			{{Optimal ET sequence\|legend=1\| 5, 15ccd }}

			[[Badness]] (Sintel): 1.22

	== Geryon ==		== Geryon ==

Hkm: /* Quint */

2026-07-23T06:38:35Z

Quint

← Older revision		Revision as of 06:38, 23 July 2026
Line 311:		Line 311:

	[[Badness]] (Sintel): 1.16		[[Badness]] (Sintel): 1.16

	~~== Quint ==~~
	Quint preserves the 5-limit mapping of 5edo, and harmonic 7 is mapped to an independent generator. As harmonic 7 is way more accurately approximated than 5 by 5edo, this temperament provides little improvement to 5edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 2.3.7-subgroup structure of 5edo and give prime 5 an independent generator instead, which is exactly what [[blackwood]] does.

	~~[[Subgroup]]: 2.3.5.7~~

	~~[[Comma list]]: 16/15, 27/25~~

	~~{{Mapping\|legend=1\| 5 8 12 0 \| 0 0 0 1 }}~~

	~~: Mapping generators: ~6/5, ~7~~

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~6/5 = 236.416, ~7/4 = 1002.701 (~28/27 = 57.038) <!-- Please review, last digit may be wrong -->
	~~: [[error map]]: {{val\| -17.921 -10.628 +50.676 -1.966 }}~~
	* [[CWE]]: ~6/5 = 240.000, ~7/4 = 1005.135 (~28/27 = 45.135)
	~~: error map: {{val\| 0.000 +18.045 +93.686 +36.309 }}~~

	~~{{Optimal ET sequence\|legend=1\| 5, 15ccd }}~~

	~~[[Badness]] (Sintel): 1.22~~

	== Geryon ==		== Geryon ==

Overthink: /* Oxygen */ add equivalence to show damage; In some tunings the 6/5 might be okay, so "either of those intervals" feels like suboptimal wording

2026-07-06T02:32:21Z

Oxygen: add equivalence to show damage; In some tunings the 6/5 might be okay, so "either of those intervals" feels like suboptimal wording

← Older revision		Revision as of 02:32, 6 July 2026
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	== Oxygen ==		== Oxygen ==
	{{See also\| Porcupine family }}		{{See also\| Porcupine family }}
	Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7. While this means it does not represent ~~either of those intervals~~ with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block\|Fokker blocks]].		Oxygen extends porcupine into the 7-limit by conflating 6/5 with 8/7, and conflating 10/9 with 7/6. While this means it does not represent 7-limit JI with any real accuracy, it is still of interest because its comma basis suggests potential utility to construct [[fokker block\|Fokker blocks]].

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	{{Optimal ET sequence\|legend=1\| 1c, …, 6bcd, 7d }}*		{{Optimal ET sequence\|legend=1\| 1c, …, 6bcd, 7d }}*

	<nowiki/>* [[~~Optimal~~ patent val]]: [[8edo\|8]]		<nowiki/>* [[optimal patent val]]: [[8edo\|8]]

	[[Badness]] (Sintel): 1.52		[[Badness]] (Sintel): 1.52

FloraC: Delete involution and devolution

2026-07-02T08:14:00Z

Delete involution and devolution

@@ Line 479: / Line 479: @@
 Badness (Sintel): 1.08
 == Codex ==
@@ Line 575: / Line 518: @@
 {{Optimal ET sequence|legend=1| 1c, …, 6bcd, 7d }}*
-<nowiki/>*[[Optimal patent val]]: [[8edo|8]]
+<nowiki/>* [[Optimal patent val]]: [[8edo|8]]
 [[Badness]] (Sintel): 1.52

Hkm: /* Sept */

2026-06-16T03:58:38Z

Sept

← Older revision		Revision as of 03:58, 16 June 2026
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	[[Badness]] (Sintel): 1.22		[[Badness]] (Sintel): 1.22

	~~== Sept ==~~
	~~{{See also\| Whitewood family }}~~

	Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what [[jamesbond]] does.

	~~This temperament used to be known as ''mujannab''.~~

	~~[[Subgroup]]: 2.3.5.7~~

	~~[[Comma list]]: 54/49, 64/63~~

	~~{{Mapping\|legend=1\| 7 11 0 20 \| 0 0 1 0 }}~~

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~9/8 = 170.823{{c}}, ~5/4 = 393.792{{c}} (~15/14 = 52.145{{c}})
	~~: [[error map]]: {{val\| -4.236 -22.898 -0.994 +47.642 }}~~
	* [[CWE]]: ~9/8 = 171.429{{c}}, ~5/4 = 392.719{{c}} (~15/14 = 49.862{{c}})
	~~: error map: {{val\| 0.000 -16.241 +6.406 +59.746 }}~~

	~~{{Optimal ET sequence\|legend=1\| 7, 14d }}~~

	~~[[Badness]] (Sintel): 2.68~~

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

	~~Comma list: 45/44, 54/49, 64/63~~

	~~Mapping: {{mapping\| 7 11 0 20 8 \| 0 0 1 0 1 }}~~

	~~Optimal tunings:~~
	* WE: ~11/10 = 170.817{{c}}, ~5/4 = 393.252{{c}} (~33/32 = 51.619{{c}})
	* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 391.840{{c}} (~33/32 = 48.983{{c}})

	~~{{Optimal ET sequence\|legend=0\| 7, 14de }}~~

	~~Badness (Sintel): 2.02~~

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

	~~Comma list: 27/26, 45/44, 52/49, 64/63~~

	~~Mapping: {{mapping\| 7 11 0 20 8 26 \| 0 0 1 0 1 0 }}~~

	~~Optimal tunings:~~
	* WE: ~11/10 = 170.795{{c}}, ~5/4 = 393.611{{c}} (~33/32 = 52.021{{c}})
	* CWE: ~11/10 = 171.429{{c}}, ~5/4 = 392.725{{c}} (~33/32 = 49.868{{c}})

	~~{{Optimal ET sequence\|legend=0\| 7, 14de }}~~

	~~Badness (Sintel): 1.77~~

	== Geryon ==		== Geryon ==

Overthink: /* Oviminor */ That's not exactly "very low" accuracy...

2026-06-16T03:13:36Z

Oviminor: That's not exactly "very low" accuracy...

← Older revision		Revision as of 03:13, 16 June 2026
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	[[Badness]] (Sintel): 1.52		[[Badness]] (Sintel): 1.52

	~~== Oviminor ==~~
	~~: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Oviminor (5-limit)]].''~~

	Oviminor was named by [[Eliora]] in 2022 after the facts that it takes 184 minor thirds of [[6/5]] to reach the interval class of [[4/3]], the Roman consul was Eggius in the year 184 AD, and the Latin word for egg is ovum, and with prefix ovi-. It sets a new record of complexity for a chain of nineteen 6/5's past [[egads]], though it is less accurate.

	~~[[Subgroup]]: 2.3.5.7~~

	~~[[Comma list]]: 4375/4374, {{monzo\| -100 53 48 -34 }}~~

	~~{{Mapping\|legend=1\| 1 -134 -134 -401 \| 0 184 185 548 }}~~
	~~: mapping generators: ~2, ~5/3~~

	~~[[Optimal tuning]]s:~~
	* [[WE]]: ~2 = 1200.0193{{c}}, ~5/3 = 884.2638{{c}}
	~~: [[error map]]: {{val\| +0.019 +0.010 -0.085 +0.032 }}~~
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/3 = 884.2497{{c}}
	~~: error map: {{val\| 0.000 -0.011 -0.120 +0.008 }}~~

	~~{{Optimal ET sequence\|legend=1\| 19, …, 1600, 1619, 4838, 6457c }}~~

	~~[[Badness]] (Sintel): 14.7~~

Hkm at 02:29, 16 June 2026

2026-06-16T02:29:38Z

Show changes

Xenllium at 09:10, 13 June 2026

2026-06-13T09:10:51Z

@@ Line 334: / Line 334: @@
 == Sept ==
-{{See also| Apotome family }}
+{{See also| Whitewood family }}
 Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what [[jamesbond]] does.
@@ Line 772: / Line 772: @@
 == Sixseven ==
 Sixseven tempers out the [[7/6|septimal minor third (7/6)]], equating the [[6/1|sixth]] and [[7/1|seventh harmonic]]s. It was named by [[User:Xenllium|Xenllium]] in 2026.
@@ Line 807: / Line 806: @@
 Badness (Sintel): 0.422
 [[Category:Temperament collections]]

Xenllium at 05:39, 13 June 2026

2026-06-13T05:39:55Z

@@ Line 334: / Line 334: @@
 == Sept ==
 Sept preserves the 2.3.7-subgroup of mapping of 7edo, and harmonic 5 is mapped to an independent generator. As harmonic 5 is way more accurately approximated than 7 by 7edo, this temperament provides little improvement to 7edo's 7-limit tuning, so in what way this temperament is useful remains unexplained. It would make much more sense to, for example, preserve the 5-limit structure of 7edo and give prime 7 an independent generator instead, which is exactly what [[jamesbond]] does.
@@ Line 768: / Line 770: @@
 Badness (Sintel): 0.681
 [[Category:Temperament collections]]