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, 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 |