Tree of rank two temperaments

The temperament tree

Using the normal comma list it is possible to define a tree of temperaments of a given rank. Below is given the top level of a tree for rank two temperaments in the form of a 5-limit monzo, followed by a link to a 7-limit node of the tree. This in turn may be followed, via the hyperlinks, to further nodes in higher prime limits. Temperaments are denoted via names and the normal comma list. The number at the end of the row is a logflat badness measure. The branches near the top tend to favor the 5-limit part of the temperament in terms of complexity, while those near the bottom tend to favor higher limits after the links are followed to those limits.

The 5-limit top branches

Monzo	Ratio	Temperament name	Badness
[-15 8 1⟩	32805 / 32768	helmholtz	4.25910
[-4 4 -1⟩	81 / 80	meantone	7.38134
[-3 -1 2⟩	25 / 24	dicot	13.02798
[-6 -5 6⟩	15625 / 15552	hanson	13.23353
[4 -1 -1⟩	16 / 15	father	14.88434
[1 -27 18⟩	7629394531250 / 7625597484987	ennealimmic	17.19058
[11 -4 -2⟩	2048 / 2025	diaschismic	19.91485
[38 -2 -15⟩	274877906944 / 274658203125	luna	20.57600
[9 -13 5⟩	1600000 / 1594323	amity	21.95959
[7 0 -3⟩	128 / 125	augmented	22.31538
[-16 35 -17⟩	50031545098999707 / 50000000000000000	minortone	29.76534
[1 -5 3⟩	250 / 243	porcupine	30.77769
[23 6 -14⟩	6115295232 / 6103515625	vishnuzmic	31.18117
[0 3 -2⟩	27 / 25	bug	32.80131
[2 9 -7⟩	78732 / 78125	sensipent	35.22030
[-10 -1 5⟩	3125 / 3072	magic	39.16293
[-7 3 1⟩	135 / 128	mavila	39.55645
[17 1 -8⟩	393216 / 390625	würschmidt	40.60312
[-21 3 7⟩	2109375 / 2097152	orson	40.80736
[24 -21 4⟩	10485760000 / 10460353203	vulture	41.43092
[8 14 -13⟩	1224440064 / 1220703125	parakleismic	43.27862
[39 -29 3⟩	68719476736000 / 68630377364883	tricot	46.09293
[3 4 -4⟩	648 / 625	dimipent	47.23052
[-14 -19 19⟩	19073486328125 / 19042491875328	enneadecal	47.84488
[5 -9 4⟩	20000 / 19683	tetracot	48.51756
[8 -5 0⟩	256 / 243	limmic (blackwood)	63.76017
[12 -6 -1⟩	4096 / 3645	uncle	72.65308
[-52 -17 34⟩		chlorine	77.072
[32 -7 -9⟩	4294967296 / 4271484375	escapade	83.77811
[-14 3 4⟩	16875 / 16384	negri	86.85590
[-29 -11 20⟩	95367431640625 / 95105071448064	gammic	87.75217
[-13 17 -6⟩	129140163 / 128000000	graviton	93.18426
[-19 12 0⟩	531441 / 524288	compton	94.49449
[26 -12 -3⟩	67108864 / 66430125	misty	106.54043
[13 5 -9⟩	1990656 / 1953125	valensixthtone	122.76461
[5 -6 2⟩	800 / 729	okai	122.84790
[12 -9 1⟩	20480 / 19683	superpyth	135.14075
[47 -15 -10⟩	140737488355328 / 140126044921875	deco (qintosec)	139.19066
[-2 13 -8⟩	1594323 / 1562500	unicorn	150.48658
[-11 7 0⟩	2187 / 2048	apotomic (whitewood)	154.65113
[-4 7 -3⟩	2187 / 2000	laconic	161.79907
[-44 -3 21⟩	476837158203125 / 474989023199232	unit (mutt)	162.46707
[-25 7 6⟩	34171875 / 33554432	ampersand	165.75484
[20 -17 3⟩	131072000 / 129140163	roda	168.26415
[-18 7 3⟩	273375 / 262144	stump	200.60049
[-9 -6 8⟩	390625 / 373248	doublewide	226.75870
[-5 -10 9⟩	1953125 / 1889568	shibboleth	227.55270
[28 -3 -10⟩	268435456 / 263671875	amavil	232.48148
[-28 25 -5⟩	847288609443 / 838860800000	pental	240.04961
[9 9 -10⟩	10077696 / 9765625	mynic	249.96513
[19 10 -15⟩	30958682112 / 30517578125	trisedodge	252.72417
[25 -48 22⟩		abigail	254.51011
[-13 -46 37⟩		supermajor	263.66615
[10 -40 23⟩	12207031250000000000 / 12157665459056928801	gamera	272.56201
[-5 -32 24⟩	59604644775390625 / 59296646043258912	octoid	285.05756
[-59 5 22⟩	579357147216796875 / 576460752303423488	tertiaseptal	297.54417
[25 15 -21⟩	481469424205824 / 476837158203125	nessafof	332.79289
[-17 2 6⟩	140625 / 131072	lemba	334.71996
[19 -9 -2⟩	524288 / 492075	beatles	358.54157
[35 -25 2⟩	858993459200 / 847288609443	hemififths	372.84807
[10 23 -20⟩	96402615118848 / 95367431640625	countermeantone	373.47690
[-29 11 5⟩	553584375 / 536870912	tritonic	378.52735
[-31 2 12⟩	2197265625 / 2147483648	wizard	386.36650
[5 13 -11⟩	51018336 / 48828125	nusecond	466.49271
[31 -21 1⟩	10737418240 / 10460353203	leapday	523.18249
[30 6 -17⟩	782757789696 / 762939453125	semisept	630.57587
[-35 6 11⟩	35595703125 / 34359738368	septimin	670.94722
[-20 39 -18⟩	4052555153018976267 / 4000000000000000000	mirkat	751.58057
[-46 10 13⟩	72081298828125 / 70368744177664	slender	760.56781
[0 -19 13⟩	1220703125 / 1162261467	bohpier	860.53382
[65 -41 0⟩		41-3 node	934.30950
[46 -29 0⟩	70368744177664 / 68630377364883	mystery	1020.55630
[93 -3 -38⟩		quasiorwell	1303.61642
[72 0 -31⟩		31-5 node	1402.24568
[22 14 -19⟩	20061226008576 / 19073486328125	casablanca	1423.73592
[-56 9 18⟩	75084686279296875 / 72057594037927936	ennea	2019.43349
[104 -70 3⟩		satin	2853.04905
[-49 31 0⟩	617673396283947 / 562949953421312	31-3 node	4309.84584