Catalog of rank-4 temperaments - Revision history

FloraC: + elysic

2026-06-05T05:05:29Z

+ elysic

← Older revision		Revision as of 05:05, 5 June 2026
Line 898:		Line 898:

	[[Badness]] (Sintel): 0.770		[[Badness]] (Sintel): 0.770

			== Elysic (117649/117612) ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: [[117649/117612]]

			{{Mapping\|legend=1\| 1 0 0 0 -1 \| 0 2 0 0 -5 \| 0 0 1 0 0 \| 0 0 0 1 3 }}
			: Mapping generators: ~2, ~343/198, ~5, ~7

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.0027{{c}}, ~343/198 = 950.9861{{c}}, ~5/4 = 386.3082{{c}}, ~7/4 = 968.7558{{c}}
			: [[error map]]: {{val\| +0.003 +0.017 -0.000 -0.065 +0.033 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~343/198 = 950.9850{{c}}, ~5/4 = 386.3105{{c}}, ~7/4 = 968.7571{{c}}
			: error map: {{val\| 0.000 +0.015 -0.003 -0.069 +0.028 }}

			{{Optimal ET sequence\|legend=1\| 49, 58, 72, 130, 193, 198, 212, 270, 342, 612, 954, 1084, 1354, 1547, 1817, 2159, 3243, 3513, 5060, 5402, 5672, 8915d, 11074d }}

			[[Badness]] (Sintel): 0.901

	== Olympic (131072/130977) ==		== Olympic (131072/130977) ==

FloraC: /* Olympic (131072/130977) */ + intro

2026-05-29T09:01:50Z

Olympic (131072/130977): + intro

← Older revision		Revision as of 09:01, 29 May 2026
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	== Olympic (131072/130977) ==		== Olympic (131072/130977) ==
			Olympic is highly accurate yet easy to conceptualize and notate on the staff. It equates [[33/32]] with ([[64/63]])<sup>2</sup>, and [[36/35]] with [[1053/1024]], reducing the four [[formal comma]]s required for the [[13-limit]] to two. Therefore it can be notated in [[chain-of-fifths notation]] with two additional set of accidentals, one for the septimal comma step, and the other for the [[aberschisma]] step. It can be viewed as the [[detemperament]] of [[cassaschismic]] that decouples the Pythagorean and septimal commas.

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

FloraC: Move alphaxenic here

2026-05-15T11:08:42Z

Move alphaxenic here

← Older revision		Revision as of 11:08, 15 May 2026
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	[[Badness]] (Sintel): 0.292		[[Badness]] (Sintel): 0.292

			== Alphaxenic (131769/131072) ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: 131769/131072

			{{Mapping\|legend=1\| 2 1 0 0 8 \| 0 2 0 0 -1 \| 0 0 1 0 0 \| 0 0 0 1 0 }}
			: mapping generators: ~363/256, ~16/11, ~5, ~7

			[[Optimal tuning]]s:
			* [[WE]]: ~363/256 = 600.1590{{c}}, ~16/11 = 650.8508{{c}}, ~5/4 = 385.6773{{c}}, ~7/4 = 968.1894{{c}}
			: [[error map]]: {{val\| +0.318 -0.094 -0.000 -0.000 -0.897 }}
			* [[CWE]]: ~363/256 = 600.0000{{c}}, ~16/11 = 650.7321{{c}}, ~5/4 = 385.7900{{c}}, ~7/4 = 968.1927{{c}}
			: error map: {{val\| 0.000 -0.491 -0.524 -0.633 -2.050 }}

			{{Optimal ET sequence\|legend=1\| 22, 46, 68, 72, 118, 190, 212, 284, 330e, 402de, 498e, 544e, 616ee, 806dee, 828cee, 946cdeee, 1018cdeee }}

			[[Badness]] (Sintel): 23.0

	== Seascape (160083/160000) ==		== Seascape (160083/160000) ==

FloraC: Sorting. + alt extension of trimitone

2026-04-22T13:06:57Z

Sorting. + alt extension of trimitone

@@ Line 470: / Line 470: @@
 [[Badness]] (Sintel): 0.454
 === Commas 540/539, 625/624 ===
@@ Line 523: / Line 505: @@
 [[Badness]] (Sintel): 0.802
 == Pentacircle (896/891) ==
@@ Line 638: / Line 638: @@
 {{Mapping|legend=1| 1 0 0 0 6 -3 | 0 1 0 0 -6 6 | 0 0 1 0 3 0 | 0 0 0 1 0 -1 }}
 [[Optimal tuning]]s:
@@ Line 649: / Line 648: @@
 [[Badness]] (Sintel): 0.875
 == Kalismic (9801/9800) ==

FloraC: Move quartismic here

2026-04-17T15:01:10Z

Move quartismic here

← Older revision		Revision as of 15:01, 17 April 2026
Line 1,004:		Line 1,004:

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

			== Quartismic (117440512/117406179) ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: 117440512/117406179

			{{Mapping\|legend=1\| 1 0 0 1 5 \| 0 1 0 1 -1 \| 0 0 1 0 0 \| 0 0 0 5 1 }}
			: mapping generators: ~2, ~3, ~5, ~33/32

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9875{{c}}, ~3/2 = 701.9753{{c}}, ~5/4 = 386.3387{{c}}, ~33/32 = 53.3743{{c}}
			: [[error map]]: {{val\| -0.012 +0.008 -0.000 -0.004 +0.031 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9802{{c}}, ~5/4 = 386.3343{{c}}, ~33/32 = 53.3730{{c}}
			: error map: {{val\| 0.000 +0.025 +0.021 +0.019 +0.075 }}

			{{Optimal ET sequence\|legend=1\| 21, 22, 43, 46, 65d, 68, 89, 111, 159, 202, 224, 270, 494, 742, 764, 966, 1236, 1506, 2159, 2653, 3125, 3395, 7060, 7554, 10949e, 14614e, 15850ee, 22168bdee, 23404bcdee, 26799bcdeee, 34353bcdeeee }}

			[[Badness]] (Sintel): 5.00

			=== Tridecimal quartismic ===
			[[Subgroup]]: 2.3.5.7.11.13

			[[Comma list]]: 6656/6655, 123201/123200

			{{Mapping\|legend=1\| 1 0 0 1 5 6 \| 0 1 0 1 -1 -3 \| 0 0 1 0 0 1 \| 0 0 0 5 1 3 }}

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9884{{c}}, ~3/2 = 701.9674{{c}}, ~5/4 = 386.3432{{c}}, ~33/32 = 53.3763{{c}}
			: [[error map]]: {{val\| -0.012 +0.001 +0.006 -0.000 +0.045 -0.016 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9728{{c}}, ~5/4 = 386.3385{{c}}, ~33/32 = 53.3748{{c}}
			: error map: {{val\| 0.000 +0.018 +0.025 +0.021 +0.084 +0.017 }}

			{{Optimal ET sequence\|legend=1\| 22, 43f, 46, 65d, 89f, 111, 159, 224, 270, 494, 764, 1012, 1236, 1506, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816dee }}

			[[Badness]] (Sintel): 0.456

	[[Category:Temperaments]]		[[Category:Temperaments]]

FloraC: + argyric and chrysic

2026-03-09T16:32:41Z

+ argyric and chrysic

← Older revision		Revision as of 16:32, 9 March 2026
Line 846:		Line 846:

	Badness (Sintel): 0.375		Badness (Sintel): 0.375

			== Argyric (41503/41472) ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: [[41503/41472]]

			{{Mapping\|legend=1\| 1 0 0 1 3 \| 0 1 0 0 2 \| 0 0 1 0 0 \| 0 0 0 2 -3 }}
			: Mapping generators: ~2, ~3, ~5, ~144/77

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.0485{{c}}, ~3/2 = 701.9606{{c}}, ~5/4 = 386.2167{{c}}, ~144/77 = 1084.3250{{c}}
			: [[error map]]: {{val\| +0.049 +0.054 -0.000 -0.127 -0.129 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9739{{c}}, ~5/4 = 386.2460{{c}}, ~144/77 = 1084.2953{{c}}
			: error map: {{val\| 0.000 +0.019 -0.068 -0.235 -0.256 }}

			{{Optimal ET sequence\|legend=1\| 19, 22, 31, 41, 53, 72, 166, 167, 176, 198, 239, 270, 342, 612, 954, 1566, 3101de, 3443de, 4055de, 4397cdee, 4667dee, 5009cddee }}

			[[Badness]] (Sintel): 0.777

			== Chrysic (43923/43904) ==
			[[Subgroup]]: 2.3.5.7.11

			[[Comma list]]: [[43923/43904]]

			{{Mapping\|legend=1\| 1 0 0 3 4 \| 0 1 0 3 2 \| 0 0 1 0 0 \| 0 0 0 -4 -3 }}
			: Mapping generators: ~2, ~3, ~5, ~33/14

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.0167{{c}}, ~3/2 = 701.9323{{c}}, ~5/4 = 386.2803{{c}}, ~33/28 = 284.2370{{c}}
			: [[error map]]: {{val\| +0.017 -0.006 -0.000 +0.056 -0.114 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9341{{c}}, ~5/4 = 386.2927{{c}}, ~33/28 = 284.2352{{c}}
			: error map: {{val\| 0.000 -0.021 -0.021 +0.036 -0.155 }}

			{{Optimal ET sequence\|legend=1\| 17c, 26, 29, 34d, 43, 46, 72, 118, 152, 224, 270, 342, 612, 836, 1106, 1448, 2554, 2715, 2985, 3327e, 3479, 3821, 4091e, 4433e, 5539e }}

			[[Badness]] (Sintel): 0.770

	== Olympic (131072/130977) ==		== Olympic (131072/130977) ==

FloraC: /* Semiporwellismic (16384/16335) */ fix tuning

2026-03-09T16:16:58Z

Semiporwellismic (16384/16335): fix tuning

← Older revision		Revision as of 16:16, 9 March 2026
Line 790:		Line 790:
	* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}		* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}
	: [[error map]]: {{val\| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}		: [[error map]]: {{val\| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.~~4546~~{{c}}, ~84/65 = 442.~~3626~~{{c}}, ~7/4 = ~~968~~.~~9590~~{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.3927{{c}}, ~84/65 = 442.4436{{c}}, ~7/4 = 969.1894{{c}}
	: error map: {{val\| 0.000 +0.~~500~~ +0.~~866~~ +0.~~133~~ +1.~~410~~ +1.~~344~~ }}		: error map: {{val\| 0.000 +0.438 +0.966 +0.364 +1.453 +1.331 }}

	{{Optimal ET sequence\|legend=1\| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}		{{Optimal ET sequence\|legend=1\| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}

FloraC: /* Semiporwellismic (16384/16335) */ + tridecimal

2026-03-09T15:55:17Z

Semiporwellismic (16384/16335): + tridecimal

← Older revision		Revision as of 15:55, 9 March 2026
Line 779:		Line 779:

	[[Badness]] (Sintel): 4.00		[[Badness]] (Sintel): 4.00

			=== Tridecimal semiporwellismic ===
			[[Subgroup]]: 2.3.5.7.11.13

			[[Comma list]]: 2080/2079, 3584/3575

			{{Mapping\|legend=1\| 1 0 0 0 7 2 \| 0 1 1 0 -2 0 \| 0 0 2 0 -1 -3 \| 0 0 0 1 0 1 }}

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.7102{{c}}, ~3/2 = 702.2488{{c}}, ~84/65 = 442.3476{{c}}, ~7/4 = 969.1583{{c}}
			: [[error map]]: {{val\| -0.290 +0.004 +0.341 -0.247 +0.388 +0.429 }}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.4546{{c}}, ~84/65 = 442.3626{{c}}, ~7/4 = 968.9590{{c}}
			: error map: {{val\| 0.000 +0.500 +0.866 +0.133 +1.410 +1.344 }}

			{{Optimal ET sequence\|legend=1\| 19, 22f, 27, 38e, 41, 46, 65d, 68, 84, 87, 111, 130, 198, 282, 328, 369f, 697cef, 1546bccdeeeffff, 1676bccdeeeffff }}

			[[Badness]] (Sintel): 1.13

	== Symbiotic (19712/19683) ==		== Symbiotic (19712/19683) ==

FloraC: - tridecimal semicanousmic (not a very natural extension, feel free to revert if anyone thinks it's worth documenting)

2026-03-07T19:41:59Z

- tridecimal semicanousmic (not a very natural extension, feel free to revert if anyone thinks it's worth documenting)

← Older revision		Revision as of 19:41, 7 March 2026
Line 761:		Line 761:

	[[Badness]] (Sintel): 6.40		[[Badness]] (Sintel): 6.40

	~~=== Tridecimal semicanousmic ===~~
	~~Subgroup: 2.3.5.7.11.13~~

	~~Comma list: 2080/2079, 14641/14580~~

	~~Mapping: {{mapping\| 1 0 -2 0 0 -3 \| 0 1 2 0 2 3 \| 0 0 4 0 1 -3 \| 0 0 0 1 0 1 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1199.9764{{c}}, ~3/2 = 702.4793{{c}}, ~11/9 = 345.3532{{c}}, ~7/4 = 968.9908{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.4828{{c}}, ~11/9 = 345.3583{{c}}, ~7/4 = 968.9886{{c}}

	~~{{Optimal ET sequence\|legend=0\| 87, 111, 181, 198, 323, 410 }}~~

	~~Badness (Sintel): 4.50~~

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

	~~Comma list: 715/714, 1089/1088, 14641/14580~~

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

	~~Optimal tunings:~~
	* WE: ~2 = 1200.0035{{c}}, ~3/2 = 702.4119{{c}}, ~11/9 = 345.3672{{c}}, ~7/4 = 969.0257{{c}}
	* CWE: ~2 = 1200.00.0{{c}}, ~3/2 = 702.4110{{c}}, ~11/9 = 345.3664{{c}}, ~7/4 = 969.0262{{c}}

	~~{{Optimal ET sequence\|legend=0\| 87, 94, 111, 181, 198g, 212g, 292, 323, 410 }}~~

	~~Badness (Sintel): 2.77~~

	~~==== 19-limit ====~~
	~~Subgroup: 2.3.5.7.11.13.17.19~~

	~~Comma list: 715/714, 1089/1088, 1216/1215, 1445/1444~~

	~~Mapping: {{mapping\| 1 0 -2 0 0 -3 -6 -8 \| 0 1 2 0 2 3 6 7 \| 0 0 4 0 1 -3 2 4 \| 0 0 0 1 0 1 0 0 }}~~

	~~Optimal tunings:~~
	* WE: ~2 = 1200.0386{{c}}, ~3/2 = 702.3639{{c}}, ~18/11 = 345.3639{{c}}, ~7/4 = 969.0046{{c}}
	* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3508{{c}}, ~18/11 = 345.3556{{c}}, ~7/4 = 969.0100{{c}}

	~~{{Optimal ET sequence\|legend=0\| 87, 94, 111, 181, 205, 212gh, 292h, 299, 323, 410, 622ef }}~~

	~~Badness (Sintel): 2.13~~

	== Semiporwellismic (16384/16335) ==		== Semiporwellismic (16384/16335) ==

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

2026-03-06T16:26:22Z

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

Show changes