Map of rank-2 temperaments

This is intended to be a map of all interesting rank-2 temperaments that are compatible with octave equivalence. The only rank-2 temperaments not appearing here should be ones like Bohlen-Pierce that completely lack octaves.

Please make sure each fraction of an octave is always the mediant of the ones directly above and below.

One period per octave

Since this is the largest subset, it has its own page: Map of linear temperaments.

Two periods per octave

Generator												Cents	Comments
0\2												0
											1\26	46.154
										1\24		50.000	Shrutar
											2\46	52.174	Shrutar
									1\22			54.545	Shrutar
								1\20				60.000
							1\18					66.667
								2\34				70.588	Vishnu
											7\118	71.186	Vishnu
										5\84		71.429
									3\50			72.000
						1\16						75.000
							2\30					80.000
								3\44				81.818
									4\58			82.759	Harry
										5\72		83.333	Harry
					1\14							85.714
						2\26						92.308	Injera
							3\38					94.737	Injera
				1\12								100.000	Srutal/pajara/injera
							4\46					104.348	Srutal/pajara/diaschismic
						3\34						105.882	Srutal/pajara/diaschismic
					2\22							109.091	Srutal/pajara
			1\10									120.000
				2\18								133.333	Octokaidecal
					3\26							138.462
						4\34						141.176	Fifive
		1\8										150.000
					4\30							160.000
				3\22								163.636	Hedgehog/echidna
						8\58						165.517	Hedgehog/echidna
					5\36							166.667	Hedgehog/echidna
			2\14									171.429
				3\20								180.000
						7\46						182.609	Unidec/hendec
							11\72					183.333	Unidec/hendec
					4\26							184.615
						5\32						187.500
							6\38					189.474
								7\44				190.909
									8\50			192.000
										9\56		192.857
											10\62	193.544
	1\6											200.000
											11\64	206.250
										10\58		206.897
									9\52			207.692
								8\46				208.696
							7\40					210.000
						6\34						211.765
					5\28							214.286
						9\50						216.000
							13\72					216.667	Antikythera/astrology/wizard
				4\22								218.182	Antikythera/astrology
			3\16									225.000
					8\42							228.571
				5\26								230.769	Lemba
					7\36							233.333	Lemba
						9\46						234.783	Echidnic
		2\10										240.000	Decimal
					7\34							247.059	Decimal
				5\24								250.000	Decimal
			3\14									257.143
				4\18								266.667
					5\22							272.727	Doublewide
							11\48					275.000	Doublewide
						6\26						276.923	Doublewide
							7\30					280.000
								8/34				282.353
									9/38			284.2105
										10/42		285.714
											11\46	286.9565
1\4												300.000

Three periods per octave

Generator									Cents	Comments
0\3									0
								1\30	40.000
							1\27		44.444	Semiaug
								2\51	47.059	Semiaug
						1\24			50.000	Semiaug
					1\21				57.143
				1\18					66.667
			1\15						80.000
								6\87	82.759	Tritikleismic
							5\72		83.333	Tritikleismic
						4\57			84.2105
					3\42				85.714
				2\27					88.889	Augmented/augene
		1\12							100.000	Augmented/augene/august
				3\33					109.091	Augmented/august
			2\21						114.286
				3\30					120.000
					4\39				123.077
						5\48			125.000
							6\57		126.316
								7\66	127.273
	1\9								133.333
								8\69	139.134
							7\60		140.000
						6\51			141.176
					5\42				142.857
				4\33					145.4545
			3\24						150.000	Triforce
		2\15							160.000
			3/21						171.429
				4/27					177.778
					5/33				181.818
						6\39			184.615
							7\45		186.667
								8\51	118.235
1\6									200.000

Four periods per octave

Generator																		Cents	Comments
0\4																		0
																	1\76	15.78947
																1\72		16.6	Quadritikleismic
																	2\140	17.14286
															1\68			17.64706
														1\64				18.75
													1\60					20
												1\56						21.42857
											1\52							23.06792
										1\48								25
									1\44									27.27
								1\40										30
							1\36											33.3
						1\32												37.5
					1\28													42.85714
				1\24														50
			1\20															60
		1\16																75	Diminished
			2\28															85.714285
				3\40														90
					4\52													92.30769
						5\64												93.75
							6\76											94.73684
								7\88										95.45
									8\100									96
										9\112								96.42857
											10\124							96.77419
												11\136						97.08852
													12\148					97.297
														13\160				97.5
															14\172			97.67442
																15\184		97.82609
																	16\196	97.95918
	1\12																	100	Diminished
																	17\200	102
																16\188		102.12766
															15\176			102.27
														14\164				102.43902
													13\152					102.63158
												12\140						102.85714
											11\128							103.125
										10\116								103.448275
									9\104									103.84615
								8\92										104.34783
							7\80											105
						6\68												105.88235	Bidia
					5\56													107.14286
				4\44														109.09
			3\32															112.5
		2\20																120
			3\28															128.57143
				4\36														133.3
					5\44													136.36
						6\52												138.46154
							7\60											140
								8\68										141.17646
									9\76									142.10526
										10\84								142.85714
											11\92							143.47826
												12\100						144
													13\108					144.4
														14\116				144.82759
															15\124			145.16129
																16\132		145.45
																	17\140	145.714285
1\8																		150

Five periods per octave

• Blackwood/blacksmith - The prime 3, and in blacksmith also 7, is represented using 5edo. The generator gets you to all intervals of 5.
• Elderthing - generator of phi. Two generators up to 3, two down to 7, other primes are more complex. (One generator up or one down are ambiguous 13.)

Six periods per octave

• Hexe - The 2.5.7 subgroup is represented using 6edo, and the generator gets you to 4/3 and 3/2. Makes little sense not to additionally temper down to 12edo.

Seven periods per octave

• Whitewood - Analogue of blackwood. The prime 3 is represented using 7edo, the generator is used for 5.
• Jamesbond/septimal - The 5-limit (and in septimal the prime 11) is represented using 7edo, and the generator is only used for intervals of 7.
• Sevond - 10/9 is tempered to be exactly 1\7 of an octave. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.
• Absurdity - A complex temperament (perhaps "absurdly" so).

Eight periods per octave

• Octoid - 16-cent generator, sub-cent accuracy.

Nine periods per octave

• Ennealimmal - The generator is 49.02 cents, and don't forget the ".02" because it really is that accurate.

Twelve periods per octave

See also: Pythagorean family

Temperaments in this family are interesting because they can be thought of as 12edo with microtonal alterations.

• Compton - 3-limit as in 12edo; intervals of 5 are off by one generator. In the 7-limit (sometimes called waage), intervals of 7 are off by two generators. In the 11-limit, intervals of 11 are off by 3 generators. Thinking of 72edo might make this more concrete.
• Catler - 5-limit as in 12edo; intervals of 7 are off by one generator.
• Atomic - Does not temper out the schisma, so 3/2 is one schisma sharp of its 12edo value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the Pythagorean comma is twelve schismas, and hence 81/80, the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.