Tree of rank two temperaments

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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, wedgies, and the normal comma list. The wedgies, for those with the proper software, can be used to do all of the basic regular temperament operations, and their presence allows them to be searched for using the "Search Wiki" function. By putting together the commas in the normal list, Graham Breed's temperament finder gives an alternative to wedgies for finding the properties of the temperament. The number at the end of the row is a logflat badness measure (1000 times wedgie badness.) 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