Catalog of rank-4 temperaments - Revision history

Eufalesio: Added azazoth the temperament

2026-10-23T22:07:21Z

Added azazoth the temperament

@@ Line 1,305: / Line 1,305: @@
 [[Badness]] (Sintel): 0.456
 [[Category:Temperament collections]]
 [[Category:Catalogs of rank-4 temperaments]]

Eufalesio: Put 11-comma in title

2026-10-20T16:31:08Z

Put 11-comma in title

← Older revision		Revision as of 16:31, 20 October 2026
Line 1,183:		Line 1,183:
	Badness (Sintel): 1.69		Badness (Sintel): 1.69

	== Microorganismic ==		== Microorganismic (67110351/67108864) ==
	It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]]. It has three valid 13-limit extensions, each of them basically just as accurate, but only koch is a strong extension.		It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]]. It has three valid 13-limit extensions, each of them basically just as accurate, but only koch is a strong extension.

Eufalesio: Actually my bad, there are 3 valid extensions to microorganismic so here they are

2026-10-20T15:31:55Z

Actually my bad, there are 3 valid extensions to microorganismic so here they are

@@ Line 1,184: / Line 1,184: @@
 == Microorganismic ==
-It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]].
+It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]]. It has three valid 13-limit extensions, each of them basically just as accurate, but only koch is a strong extension.
 [[Subgroup]]: 2.3.5.7.11
@@ Line 1,208: / Line 1,208: @@
 Notable temperaments not appearing in the optimal ET sequence are {{Edos|8269, 8539, 16808}}.
-=== 13-limit ===
+=== Leeuwenhoek ===
 [[Subgroup]]: 2.3.5.7.11.13
@@ Line 1,227: / Line 1,269: @@
 [[Optimal ET sequence]]: {{Edos|11c, 15, 38f, 53, 121e, 149e, 202, 217, 270, 783f, 836, 1038, 1106, 1308, 1578, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808…}}
-[[Badness]] (Sintel): 0.387
+[[Badness]] (Sintel): 0.912
 == Quartismic (117440512/117406179) ==

Eufalesio: Microorganismic and proposed canon extension to 13-limit

2026-10-20T14:25:12Z

Microorganismic and proposed canon extension to 13-limit

← Older revision		Revision as of 14:25, 20 October 2026
Line 1,182:		Line 1,182:

	Badness (Sintel): 1.69		Badness (Sintel): 1.69

			== Microorganismic ==
			It splits the septischisma in 3 parts and the pythrabian comma in 5 parts, each of these parts being a tempered [[olympia]]~[[argyria]].

			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: [[67110351/67108864]]

			[[Mapping]]: {{Mapping\|1 0 0 7 7\|0 1 0 1 -2\|0 0 1 0 0\|0 0 0 3 -5}}

			[[Optimal tuning\|Optimal tunings]]:

			* [[WE]]: ~2 = 1200.001014{{C}}, ~3 = 1901.953889{{C}}, ~5 = 2786.311685{{C}}, ~539/512 = 88.956151{{C}}

			: [[error map]]: ⟨+0.001014 −0.000098 0.000000 −0.001537 −0.001401]

			* [[CWE]]: ~2 = 1200.000000{{C}}, ~3 = 1901.953627{{C}}, ~5 = 2786.311919{{C}}, ~539/512 = 88.955923{{C}}

			: error map: ⟨0.000000 −0.001374 −0.001795 −0.004512 −0.004808]

			[[Optimal ET sequence]]: {{Edos\|11c, 12e, 15, 26, 38, 41, 53, 68, 94, 161, 176, 202, 217, 255, 270, 566, 742, 836, 1106, 1578, 1848, 2684, 3737, 4007, 4843, 5585, 6421, 6691, 12276, 14960…}}

			[[Badness]] (Sintel): 0.387

			Notable temperaments not appearing in the optimal ET sequence are {{Edos\|8269, 8539, 16808}}.

			=== 13-limit ===
			[[Subgroup]]: 2.3.5.7.11.13

			[[Comma list]]: [[67110351/67108864]], [[1990656/1990625]]

			[[Mapping]]: {{Mapping\|1 0 0 7 7 11\|0 1 0 1 -2 3\|0 0 1 0 0 -5\|0 0 0 3 -5 -6}}

			[[Optimal tuning\|Optimal tunings]]:

			* [[WE]]: ~2 = 1200.001014{{C}}, ~3 = 1901.953889{{C}}, ~5 = 2786.311685{{C}}, ~539/512 = 88.956151{{C}}

			: [[error map]]: {{Val\|+0.000612 −0.002463 +0.004690 −0.001614 −0.003969 +0.002382}}

			* [[CWE]]: ~2 = 1200.000000{{C}}, ~3 = 1901.953627{{C}}, ~5 = 2786.311919{{C}}, ~539/512 = 88.955923{{C}}

			: error map: {{Val\|0.000000 −0.003186 +0.003468 −0.003438 −0.005995 +0.000566}}

			[[Optimal ET sequence]]: {{Edos\|11c, 15, 38f, 53, 121e, 149e, 202, 217, 270, 783f, 836, 1038, 1106, 1308, 1578, 2414, 2631, 2684, 5315, 5585, 8269, 8539, 14124, 16808…}}

			[[Badness]] (Sintel): 0.387

	== Quartismic (117440512/117406179) ==		== Quartismic (117440512/117406179) ==

FloraC: /* Keenanismic (385/384) */ - duplicates

2026-09-30T13:32:07Z

Keenanismic (385/384): - duplicates

@@ Line 167: / Line 167: @@
 == Keenanismic (385/384) ==
 {{Main| Keenanismic family }}
 === Commas 351/350, 385/384 ===
@@ Line 246: / Line 184: @@
 [[Badness]] (Sintel): 0.781
 === Commas 364/363, 385/384 ===

FloraC: /* Vishdelismic (5632/5625) */ + intro

2026-09-29T13:32:11Z

Vishdelismic (5632/5625): + intro

← Older revision		Revision as of 13:32, 29 September 2026
Line 633:		Line 633:

	== Vishdelismic (5632/5625) ==		== Vishdelismic (5632/5625) ==
			Vishdelismic tempers out 5632/5625, the [[vishdelisma]], and adds to [[vishdelic]] an independent generator for prime 7. From 5632/5625 = ([[176/175]])/([[225/224]]), vishdelismic has the feature of making [[14/9]], [[25/16]], and [[11/7]] equidistant.

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

FloraC: Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

2026-09-24T12:33:40Z

Text replacement - "Mapping: {{mapping| " to "{{Mapping|legend=0| "

← Older revision		Revision as of 12:33, 24 September 2026
Line 407:		Line 407:
	Comma list: 273/272, 441/440, 561/560		Comma list: 273/272, 441/440, 561/560

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 0 -3 11 7 \| 0 1 0 0 2 -4 -3 \| 0 0 1 0 -1 2 2 \| 0 0 0 1 2 -2 -1 }}		{{Mapping\|legend=0\| 1 0 0 0 -3 11 7 \| 0 1 0 0 2 -4 -3 \| 0 0 1 0 -1 2 2 \| 0 0 0 1 2 -2 -1 }}

	Optimal tunings:		Optimal tunings:
Line 911:		Line 911:
	Comma list: 1216/1215, 1540/1539, 2080/2079		Comma list: 1216/1215, 1540/1539, 2080/2079

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 0 -8 -13 -6 \| 0 1 0 0 9 12 5 \| 0 0 1 0 0 -1 1 \| 0 0 0 1 -1 0 0 }}		{{Mapping\|legend=0\| 1 0 0 0 -8 -13 -6 \| 0 1 0 0 9 12 5 \| 0 0 1 0 0 -1 1 \| 0 0 0 1 -1 0 0 }}

	Optimal tunings:		Optimal tunings:
Line 1,022:		Line 1,022:
	Comma list: 2080/2079, 3510/3509, 4096/4095		Comma list: 2080/2079, 3510/3509, 4096/4095

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 0 17 12 -21 \| 0 1 0 0 -5 -2 11 \| 0 0 1 0 0 -1 0 \| 0 0 0 1 -2 -1 3 }}		{{Mapping\|legend=0\| 1 0 0 0 17 12 -21 \| 0 1 0 0 -5 -2 11 \| 0 0 1 0 0 -1 0 \| 0 0 0 1 -2 -1 3 }}

	Optimal tunings:		Optimal tunings:
Line 1,041:		Line 1,041:
	Comma list: 2080/2079, 3250/3249, 4096/4095		Comma list: 2080/2079, 3250/3249, 4096/4095

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 1 15 11 -13 6 \| 0 1 0 0 -5 -2 5 -2 \| 0 0 1 0 0 -1 2 1 \| 0 0 0 2 -4 -2 5 -1 }}		{{Mapping\|legend=0\| 1 0 0 1 15 11 -13 6 \| 0 1 0 0 -5 -2 5 -2 \| 0 0 1 0 0 -1 2 1 \| 0 0 0 2 -4 -2 5 -1 }}

	Optimal tunings:		Optimal tunings:
Line 1,058:		Line 1,058:
	Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095		Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 1 15 11 -13 6 -18 \| 0 1 0 0 -5 -2 5 -2 -11 \| 0 0 1 0 0 -1 2 1 0 \| 0 0 0 2 -4 -2 5 -1 6 }}		{{Mapping\|legend=0\| 1 0 0 1 15 11 -13 6 -18 \| 0 1 0 0 -5 -2 5 -2 -11 \| 0 0 1 0 0 -1 2 1 0 \| 0 0 0 2 -4 -2 5 -1 6 }}

	Optimal tunings:		Optimal tunings:
Line 1,077:		Line 1,077:
	Comma list: 2080/2079, 4096/4095, 35000/34969		Comma list: 2080/2079, 4096/4095, 35000/34969

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 1 15 11 -13 \| 0 1 0 0 -5 -2 5 \| 0 0 1 0 0 -1 2 \| 0 0 0 2 -4 -2 5 }}		{{Mapping\|legend=0\| 1 0 0 1 15 11 -13 \| 0 1 0 0 -5 -2 5 \| 0 0 1 0 0 -1 2 \| 0 0 0 2 -4 -2 5 }}
	: mapping generators: ~2, ~3, ~5, ~187/100		: mapping generators: ~2, ~3, ~5, ~187/100

Line 1,095:		Line 1,095:
	Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199		Comma list: 2080/2079, 3250/3249, 4096/4095, 4200/4199

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 1 15 11 -13 6 \| 0 1 0 0 -5 -2 5 -2 \| 0 0 1 0 0 -1 2 1 \| 0 0 0 2 -4 -2 5 -1 }}		{{Mapping\|legend=0\| 1 0 0 1 15 11 -13 6 \| 0 1 0 0 -5 -2 5 -2 \| 0 0 1 0 0 -1 2 1 \| 0 0 0 2 -4 -2 5 -1 }}

	Optimal tunings:		Optimal tunings:
Line 1,112:		Line 1,112:
	Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095, 4200/4199		Comma list: 2080/2079, 3250/3249, 3510/3509, 4096/4095, 4200/4199

	~~Mapping:~~ {{~~mapping~~\| 1 0 0 1 15 11 -13 6 -18 \| 0 1 0 0 -5 -2 5 -2 -11 \| 0 0 1 0 0 -1 2 1 0 \| 0 0 0 2 -4 -2 5 -1 6 }}		{{Mapping\|legend=0\| 1 0 0 1 15 11 -13 6 -18 \| 0 1 0 0 -5 -2 5 -2 -11 \| 0 0 1 0 0 -1 2 1 0 \| 0 0 0 2 -4 -2 5 -1 6 }}

	Optimal tunings:		Optimal tunings:
Line 1,202:		Line 1,202:
	Comma list: 352/351, 715/714, 155771/155520		Comma list: 352/351, 715/714, 155771/155520

	~~Mapping:~~ {{~~mapping~~\| 1 0 -6 0 1 6 0 \| 0 1 3 0 2 -1 3 \| 0 0 5 0 -1 -1 3 \| 0 0 0 1 0 0 -1 }}		{{Mapping\|legend=0\| 1 0 -6 0 1 6 0 \| 0 1 3 0 2 -1 3 \| 0 0 5 0 -1 -1 3 \| 0 0 0 1 0 0 -1 }}

	Optimal tunings:		Optimal tunings:
Line 1,217:		Line 1,217:
	Comma list: 352/351, 715/714, 1463/1458, 2057/2052		Comma list: 352/351, 715/714, 1463/1458, 2057/2052

	~~Mapping:~~ {{~~mapping~~\| 1 0 -6 0 1 6 0 0 \| 0 1 3 0 2 -1 3 4 \| 0 0 5 0 -1 -1 3 1 \| 0 0 0 1 0 0 -1 -1 }}		{{Mapping\|legend=0\| 1 0 -6 0 1 6 0 0 \| 0 1 3 0 2 -1 3 4 \| 0 0 5 0 -1 -1 3 1 \| 0 0 0 1 0 0 -1 -1 }}

	Optimal tunings:		Optimal tunings:
Line 1,250:		Line 1,250:
	Comma list: 676/675, 6656/6655		Comma list: 676/675, 6656/6655

	~~Mapping:~~ {{~~mapping~~\| 3 0 0 0 8 -3 \| 0 2 0 0 1 3 \| 0 0 1 0 0 1 \| 0 0 0 1 0 0 }}		{{Mapping\|legend=0\| 3 0 0 0 8 -3 \| 0 2 0 0 1 3 \| 0 0 1 0 0 1 \| 0 0 0 1 0 0 }}

	Optimal tunings:		Optimal tunings:

FloraC: /* Nexus (1771561/1769472) */ fix generator preimage

2026-09-17T08:21:09Z

Nexus (1771561/1769472): fix generator preimage

← Older revision		Revision as of 08:21, 17 September 2026
Line 1,236:		Line 1,236:

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~121/96 = 400.0154{{c}}, ~3/2 = 950.9883{{c}}, ~5/4 = 386.2216{{c}}, ~7/4 = 968.7338{{c}}		* [[WE]]: ~121/96 = 400.0154{{c}}, ~1331/768 = 950.9883{{c}}, ~5/4 = 386.2216{{c}}, ~7/4 = 968.7338{{c}}
	: [[error map]]: {{val\| +0.046 +0.022 -0.000 -0.000 -0.207 }}		: [[error map]]: {{val\| +0.046 +0.022 -0.000 -0.000 -0.207 }}
	* [[CWE]]: ~121/96 = 400.0000{{c}}, ~3/2 = 950.9685{{c}}, ~5/4 = 386.2485{{c}}, ~7/4 = 968.7470{{c}}		* [[CWE]]: ~121/96 = 400.0000{{c}}, ~1331/768 = 950.9685{{c}}, ~5/4 = 386.2485{{c}}, ~7/4 = 968.7470{{c}}
	: error map: {{val\| 0.000 -0.018 -0.065 -0.079 -0.349 }}		: error map: {{val\| 0.000 -0.018 -0.065 -0.079 -0.349 }}

Line 1,253:		Line 1,253:

	Optimal tunings:		Optimal tunings:
	* WE: ~121/96 = 400.0003{{c}}, ~3/2 = 951.0845{{c}}, ~5/4 = 386.5832{{c}}, ~7/4 = 968.8238{{c}}		* WE: ~121/96 = 400.0003{{c}}, ~1331/768 = 951.0845{{c}}, ~5/4 = 386.5832{{c}}, ~7/4 = 968.8238{{c}}
	* CWE: ~121/96 = 400.0000{{c}}, ~3/2 = 951.0840{{c}}, ~5/4 = 386.5837{{c}}, ~7/4 = 968.8241{{c}}		* CWE: ~121/96 = 400.0000{{c}}, ~1331/768 = 951.0840{{c}}, ~5/4 = 386.5837{{c}}, ~7/4 = 968.8241{{c}}

	{{Optimal ET sequence\|legend=0\| 9, 15, 24, 39c/df, 48c, 57f, 63, 72, 87, 111, 159, 183, 198, 270 }}		{{Optimal ET sequence\|legend=0\| 9, 15, 24, 39c/df, 48c, 57f, 63, 72, 87, 111, 159, 183, 198, 270 }}

FloraC: Move nexus here

2026-09-17T08:04:05Z

Move nexus here

← Older revision		Revision as of 08:04, 17 September 2026
Line 1,226:		Line 1,226:

	Badness (Sintel): 2.63		Badness (Sintel): 2.63

			== Nexus (1771561/1769472) ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: 1771561/1769472

			{{Mapping\|legend=1\| 3 0 0 0 8 \| 0 2 0 0 1 \| 0 0 1 0 0 \| 0 0 0 1 0 }}
			: mapping generators: ~121/96, ~1331/768, ~5, ~7

			[[Optimal tuning]]s:
			* [[WE]]: ~121/96 = 400.0154{{c}}, ~3/2 = 950.9883{{c}}, ~5/4 = 386.2216{{c}}, ~7/4 = 968.7338{{c}}
			: [[error map]]: {{val\| +0.046 +0.022 -0.000 -0.000 -0.207 }}
			* [[CWE]]: ~121/96 = 400.0000{{c}}, ~3/2 = 950.9685{{c}}, ~5/4 = 386.2485{{c}}, ~7/4 = 968.7470{{c}}
			: error map: {{val\| 0.000 -0.018 -0.065 -0.079 -0.349 }}

			{{Optimal ET sequence\|legend=1\| 9, 15, 24, 33cd, 39(d), 48, 57, 63, 72, 159, 183, 198, 255, 270, 342, 612, 954, 1566, 2091e, 2361e, 2433e, 2703e, 3045e, 3315ee, 3657ee }}

			[[Badness]] (Sintel): 10.7

			=== Commas 676/675, 6656/6655 ===
			Subgroup: 2.3.5.7.11.13

			Comma list: 676/675, 6656/6655

			Mapping: {{mapping\| 3 0 0 0 8 -3 \| 0 2 0 0 1 3 \| 0 0 1 0 0 1 \| 0 0 0 1 0 0 }}

			Optimal tunings:
			* WE: ~121/96 = 400.0003{{c}}, ~3/2 = 951.0845{{c}}, ~5/4 = 386.5832{{c}}, ~7/4 = 968.8238{{c}}
			* CWE: ~121/96 = 400.0000{{c}}, ~3/2 = 951.0840{{c}}, ~5/4 = 386.5837{{c}}, ~7/4 = 968.8241{{c}}

			{{Optimal ET sequence\|legend=0\| 9, 15, 24, 39c/df, 48c, 57f, 63, 72, 87, 111, 159, 183, 198, 270 }}

			Badness (Sintel): 1.69

	== Quartismic (117440512/117406179) ==		== Quartismic (117440512/117406179) ==

FloraC: + vishdelismic (5632/5625)

2026-09-05T12:32:21Z

+ vishdelismic (5632/5625)

← Older revision		Revision as of 12:32, 5 September 2026
Line 631:		Line 631:

	[[Badness]] (Sintel): 0.272		[[Badness]] (Sintel): 0.272

			== Vishdelismic (5632/5625) ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: 5632/5625

			{{Mapping\|legend=1\| 1 0 0 0 2 \| 0 1 0 0 0 \| 0 0 1 0 -3 \| 0 0 0 1 3 }}
			: mapping generators: ~2, ~3, ~5, ~7

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.8976{{c}}, ~3/2 = 702.1145{{c}}, ~5/4 = 386.7638{{c}}, ~7/4 = 969.0306{{c}}
			: [[error map]]: {{val\| -0.102 +0.057 +0.245 -0.000 -0.136 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.1309{{c}}, ~5/4 = 386.7770{{c}}, ~7/4 = 969.0087{{c}}
			: error map: {{val\| 0.000 +0.176 +0.463 +0.183 +0.052 }}

			{{Optimal ET sequence\|legend=1\| 12, 19e, 22, 31, 53, 84e, 87, 99e, 118, 130, 152, 239, 270, 670, 822, 940, 1092, 1362c, 2032c, 2302c }}

			[[Badness]] (Sintel): 0.804

	== Trimitone (8019/8000) ==		== Trimitone (8019/8000) ==