Catalog of 3.5.7 subgroup rank two temperaments

Todo: merge articles Merge into No-twos subgroup temperaments

Table

name	comma 3.5.7 |>	comma n/d	comma c	comma name	a&b (3)	a&b (2)	complexity	damage	mapping	TE gen1	TE gen2	per/3	MOS
*Izar	| -11 -7 12 >	13841287201/13839609375	0.210		b271 & b1071	171p & 676cd	11.675	0.003	[< 271 397 480 ], < 1071 1569 1897 ]>	1901.9584	877.2825	1	2*,3,5,7,9,11*,13*,24,37,50,63,76,...258*,271*
Canopus	| 3 4 -5 >	16875/16807	6.990	small BP diesis, mirkwai	b13 & b101	8d & 64cd	4.866	0.249	[< 13 19 23 ], < 101 148 179 ]>	1901.7828	583.9055	1	2,3*,4,7,10*,13*,23,36,49,62,75,88,101,114*
*Procyon	| -8 -3 7 >	823543/820125	7.200		b13 & b144	8d & 91p	6.810	0.183	[< 13 19 23 ], < 144 211 255 ]>	1902.198	145.4124	1	...13*,14,27,40,53,66,79,92,105,118,131,144,157*
*Gacrux	| 6 8 -10 >	284765625/282475249	13.981
BPS (Bohlen-Pierce-Stearns)	| -5 1 2 >	245/243	14.191	minor BP diesis, sensamagic	b13 & b17	8d & 11cd	2.749	0.895	[< 13 19 23 ], < 17 25 30 ]>	1903.7424	440.902	1	2,3,4*,5,9,13*,17,30,43,56,69,82*,95*
Sirius	| -2 5 -3 >	3125/3087	21.181	major BP diesis, gariboh	b13 & b19	8d & 12p	4.021	0.913	[< 13 19 23 ], < 19 28 34 ]>	1902.4482	293.7397	1	2,3,4,5,6*,7,13*,19,32,45,58,71,84,97,110,123,136*
*Betelgeuse	| -13 -2 9 >	40353607/39858075	21.391		b13 & b148	8d & 93bccd	8.754	0.423	[< 13 19 23 ], < 148 217 262 ]>	1902.5991	732.4139	1	2*,3*,5*,8,13*,...226*
*Miaplacidus	| 1 9 -8 >	5859375/5764801	28.171		b13 & b94d	8d & 59ccdd	7.788	0.627	[< 13 19 23 ], < 94 138 167 ]>	1901.88	585.8431	1	2,3*,4,7,10,13*,16,29,42,55,68,81,94,...224*
Arcturus	| -7 6 -1 >	15625/15309	35.371	great BP diesis	b13 & b2	8d & 1cdd	4.823	1.271	[< 13 19 23 ], < 2 3 4 ]>	1903.8685	878.9253	1	2*,3,5,7,9,11,13*
*Pollux	| -18 -1 11 >	1977326743/1937102445	35.581		b13 & b139	8d & 88dd	10.697	0.576	[< 13 19 23 ], < 139 204 246 ]>	1902.8625	438.6389	1	2,3,4*,5,9*,13
*Castor	| -1 14 -11 >	6103515625/5931980229	49.352		b13 & b113d	8d & 71ccddd	11.259	0.760	[< 13 19 23 ], < 113 166 201 ]>	1902.0166	438.5094	1	2,3,4*,5,9*,13
*Muphrid	| -12 7 1 >	546875/531441	49.562		b13 & b2d	8d & 1c	6.599	1.302	[< 13 19 23 ], < 2 3 3 ]>	1904.3123	879.0503	1	2*,3,5,7,9,11,13*
*Rigel	| -9 11 -4 >	48828125/47258883	56.552		b13 & b21d	8d & 5d	8.844	1.108	[< 13 19 23 ], < 21 31 38 ]>	1903.2571	731.7183	1	2*,3*,5*,8*,13*
*Mirzam	| -17 8 3 >	133984375/129140163	63.752		b13 & b32d	8d & 20c	9.347	1.182	[< 13 19 23 ], < 32 47 56 ]>	1904.3391	293.1889	1	2,3,4,5,6*,7,13*
	| 15 -9 -1 >	14348907/13671875	83.676		b17 & b32	11cd & 20cd	8.228	1.762	[< 17 25 30 ], < 32 47 57 ]>	1898.8326	891.5018	1	2*,3,5,7,9,11,13,15*,17*,32,49*,66,115,164*
*Capella	| 10 -8 1 >	413343/390625	97.866	small link	b17 & b19	12p & 11cd	6.452	2.629	[< 17 25 30 ], < 19 28 34 ]>	1897.8856	895.415	1	2*,3,5,7,9,11,13,15,17*,19,36*,53*,89,142*,195*
	| 13 -4 -4 >	1594323/1500625	104.857		b4 & b32	3bcdd & 20cd	7.126	2.550	[< 4 6 7 ], < 32 47 57 ]>	474.1446	55.0036	4	...32*,36*,68*,104*,172*,276*
	| 16 0 -9 >	43046721/40353607	111.847		b9 & b36	6bcd & 23d	8.773	2.209	[< 9 13 16 ], < 36 53 64 ]>	210.9385	44.1126	9	18,27,36*,45*,81,126,171*,216*,387*
Toliman	| 5 -7 3 >	83349/78125	112.057	minor link	b6 & b17	4p & 11cd	5.649	3.437	[< 6 9 11 ], < 17 25 30 ]>	1898.4178	333.298	1	2,3,4,5*,6*,11,17*,23,40,57*,74,131*
Vega	| 8 -3 -2 >	6561/6125	119.047	major link	b4 & b15	3bcdd & 10bccdd	4.377	4.713	[< 4 6 7 ], < 15 22 27 ]>	1892.0204	494.8658	1	2,3*,4*,7,11,15,19*,23*,42*,65*
*Achernar	| 0 -6 5 >	16807/15625	126.247		b7 & b17	4bcd & 11cd	4.847	4.514	[< 7 10 12 ], < 17 25 30 ]>	1901.955	559.351	1	2,3*,4,7*,10,17*
%	| 3 -2 0 >	27/25	133.238	large limma, BP small semitone	b4 & b6	3bcdd & 4p	1.629	14.176	[< 4 6 7 ], < 6 9 11 ]>	939.6121	549.9897	2	4*,6,10*,14,24*,34,58,82*
	| 6 2 -5 >	18225/16807	140.228	minimal BP chroma	b9 & b10	6bcd & 6ccdd	4.885	4.974	[< 9 13 16 ], < 10 15 18 ]>	1895.1156	201.762	1	...9*,10,19*,28*,47*
%	| -2 -1 2 >	49/45	147.428	BP minor semitone, swetismic neutral second	b4 & b7	3bcdd & 4bcd	1.925	13.274	[< 4 6 7 ], < 7 10 12 ]>	1917.7266	516.1077	1	2,3*,4*,7,11*,15*,26*
	| 1 3 -3 >	375/343	154.418	BP major semitone, minor BP chroma	b3 & b6	2cd & 4p	2.941	9.098	[< 3 4 5 ], < 6 9 11 ]>	632.9248	234.1367	3	6*,9*,15,24*,33,57*,81*
	| -7 0 4 >	2401/2187	161.619		b4 & b40cd	3bcdd & 25p	3.868	7.240	[< 4 6 7 ], < 40 58 70 ]>	478.34	41.8631	4	...44*,48,92*,136,228*,320*
	| -4 4 -1 >	625/567	168.609	BP great semitone, major BP chroma	b2 & b7	1cdd & 4bcd	3.195	9.146	[< 2 3 4 ], < 7 10 12 ]>	1914.567	1069.8876	1	2*,3,5,7*,9*,16,25,34*
	| -9 5 1 >	21875/19683	182.799	maximal BP chroma	b9 & b11	6bcd & 7d	4.970	6.373	[< 9 13 16 ], < 11 16 19 ]>	1913.8948	858.1831	1	2*,3,5,7,9*,11,20*,29*
%	| 9 0 -5 >	19683/16807	273.465		b5 & b10	3d & 6ccdd	4.904	9.663	[< 5 7 9 ], < 10 15 18 ]>	377.3036	145.1888	5	10*,15*,25*,40,65*

names: * = proposed

names: % = not shown on PTS plots

MOS: * = maximally even

complexity = ln(n*d)/4

damage = 1200 *log(n/d) / log(n*d)

(complexity and damage are calculated as in the "Middle Path" tables)

Graphs

Complexity vs. damage plot. z<1 corresponds to the "Middle Path" inclusion criterion.

Projective tuning space diagrams

Temperaments with smaller commas, labeled by name

Temperaments with smaller commas, labeled by comma

Temperaments with larger commas, labeled by name

Temperaments with larger commas, labeled by comma

Both sets, labeled by name

Both sets, labeled by comma

11.675