User:2^67-1/Blackwood-dicot-semaphore equivalence continuum
The blackwood-dicot-semaphore equivalence continuum is a continuum of 7-limit rank-3 temperaments describing the set of all 7-limit rank-3 temperaments supported by 10edo. Any rank-2 temperament supported by 10edo can thus be represented by a line between two points in this continuum.
All temperaments in the continuum satisfy (25/24)^{p}(49/48)^{q} ~ 256/243, equating a stack of dicot commas (25/24) and semaphore commas (49/48) with the blackwood comma (256/243).
The blackwood comma is the characteristic 3-limit comma tempered out in 10edo.
User:Godtone notes the following JIP's, each one corresponding to a 1D continua contained therein, for which increasingly efficient approximations generally represents increasingly efficient 7-limit temperaments:
- log_{2}(256/243) / log_{2}(25/24) = 1.2766647429 ; this is the JIP of p=1, q=0 (equiv. to p=-1, q=0)
- log_{2}(256/243) / log_{2}(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)
- log_{2}(256/243) / log_{2}(1225/1152) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)
- log_{2}(256/243) / log_{2}(50/49) = 2.5796543166 ; this is the JIP of p=1, q=-1 (equiv. to p=-1, q=1)
- log_{2}(256/243) / log_{2}(60025/55296) = 0.6350918818 ; this is the JIP of p=1, q=2 (equiv. to p=-1, q=-2)
- log_{2}(256/243) / log_{2}(2401/2400) = 125.1044589 ; this is the JIP of p=1, q=-2 (equiv. to p=-1, q=2)
- log_{2}(256/243) / log_{2}(30625/27648) = 0.5096257724 ; this is the JIP of p=2, q=1 (equiv. to p=-2, q=-1)
- log_{2}(256/243) / log_{2}(625/588) = 0.8540148427 ; this is the JIP of p=2, q=-1 (equiv. to p=-2, q=1)
Importantly, each JIP corresponds to a rational, so that, for example, (p, q) = (1, -2) is equivalent to (p, q) = (2, -4) and to (p, q) = (-1, 2) but not to (1, 2). Note that all these JIPs lie on the JIL (just intonation line).
Also note that continua separated by 2401/2400 are meaningfully different, but due to the efficiency of 2401/2400, one may want to examine the continuum of all 7-limit temperaments supported by 10edo for which 2401/2400 is tempered.
p | q | Temperament | Comma | _{Added bysomeone else?} | |
---|---|---|---|---|---|
Ratio | Monzo | ||||
0 | -1 | No-fives trienstonic (squared) | 28/27 (reduced), 784/729 (unreduced) | [2 -3 0 1⟩ (reduced), [4 -6 0 2⟩ (unreduced) | n |
0 | 0 | Blackwood | 256/243 | [8 -5 0 0⟩ | n |
0 | 1 | Archytas | 64/63 (reduced), 4096/3969 (unreduced) | [6 -2 0 -1⟩ (reduced), [12 -4 0 -2⟩ (unreduced) | n |
0 | 1.5 = 3/2 | 2.3.7 5 & 97d | 268435456/257298363 | [28 -7 0 -6⟩ | Y |
0 | 1.6 = 5/3 | 2.3.7 5 & 70 | 4194304/4084101 (reduced), 17592186044416/16679880978201 (unreduced) | [22 -5 0 -5⟩ (reduced), [44 -10 0 -10⟩ (unreduced) | n |
0 | 2 | Buzzard | 65536/64827 | [16 -3 0 -4⟩ | n |
0 | 2.3 = 7/3 | Septiness | 67108864/66706983 (reduced), 4503599627370496/4449821580962289 (unreduced) | [26 -4 0 -7⟩ (reduced), [52 -8 0 -14⟩ (unreduced) | n |
0 | 2.5 = 5/2 | Slendrismic | 68719476736/68641485507 | [36 -5 0 -10⟩ | Y |
0 | 2.571428 = 18/7 | 2.3.7 5 & 472 | (39 digits) | [36 -5 0 -10⟩ | n |
0 | 2.6 = 13/5 | 2.3.7 5 & 171 | 70632088586703/70368744177664 (reduced), 4988891938119860195948410209/4951760157141521099596496896 (unreduced) | [-46 6 0 13⟩ (reduced) [-92 12 0 26⟩ (unreduced) | Y |
0 | 2.6 = 8/3 | 2.3.7 5 & 212 | 72680419155717387/72057594037927936 | [-56 7 0 16⟩ | n |
0 | 3 | Slendric (squared) | 1029/1024 (reduced), 1058841/1048576 (unreduced) | [-10 1 0 3⟩ (reduced), [-20 2 0 6⟩ (unreduced) | n |
0 | 4 | 5 & 81 | 17294403/16777216 | [-24 1 0 8⟩ | Y |
0 | 5 | Cloudy (squared) | 16807/16384 (reduced), 282475249/268435456 (unreduced) | [-14 0 0 5⟩ (reduced), [-28 0 0 10⟩ (unreduced) | Y |
0 | ∞ | Semaphore | 49/48 | [-4 -1 0 2⟩ | n |
1 | 0 | Srutal | 2048/2025 | [11 -4 -2 0⟩ | n |
1.25 = 5/4 | 0 | Quintosec | 140737488355328/140126044921875 | [47 -15 -10 0⟩ | n |
1.285714 = 9/7 | 0 | 2.3.5 Lagaca | 9696448624912261962890625/9671406556917033397649408 | [-83 26 18 0⟩ | Y |
1.3 = 13/10 | 0 | 2.3.5 10 & 171 | (36 digits) | [-119 37 26 0⟩ | n |
1.3 = 4/3 | 0 | Submajor | 69198046875/68719476736 | [-36 11 8 0⟩ | Y |
1.5 = 3/2 | 0 | 2.3.5 Miracle/Ampersand | 34171875/33554432 | [-25 7 6 0⟩ | Y |
2 | 0 | Negri | 16875/16384 | [-14 3 4 0⟩ | n |
∞ | 0 | Dicot | 25/24 | [-3 -1 2 0⟩ | n |
1 | 0.5 = 1/2 | 10 & 53 & 130 | 67108864/66976875 | [26 -7 -4 -2⟩ | n |
1.2 = 6/5 | 0.6 = 3/5 | p=2, q=1 Miracle projection | 35162859375/34359738368 (reduced), 1236426679426025390625/1180591620717411303424 (unreduced) | [-35 8 6 3⟩ (reduced), [-70 16 12 6⟩ (unreduced) | Y |
2 | 1 | Avicennmic | 525/512 (reduced), 275625/262144 (unreduced) | [-9 1 2 1⟩ (reduced), [-18 2 4 2⟩ (unreduced) | Y |
1 | 1 | Mirwomo | 33075/32768 | [-15 3 2 2⟩ | n |
0.857142 = 6/7 | 0.857142 = 6/7 | 10 & 77 & 308 | (30 digits) | [-98 23 12 12⟩ | Y |
0.83 = 5/6 | 0.83 = 5/6 | 80d & 130 & 140 (decoid detemp.) | (25 digits) | [83 -20 -10 -10⟩ | Y |
0.8 = 4/5 | 0.8 = 4/5 | 10 & 53 & 275 | 295147905179352825856/290807555001001171875 | [68 -17 -8 -8⟩ | Y |
0.75 = 3/4 | 0.75 = 3/4 | (10 or 36 or 46 or 56) & 118d & 220 | 9007199254740992/8792367498140625 | [53 -14 -6 -6⟩ | Y |
0.6 = 2/3 | 0.6 = 2/3 | 39 & 58 & 68 | 274877906944/265831216875 | [38 -11 -4 -4⟩ | Y |
0.5 = 1/2 | 0.5 = 1/2 | p=1, q=1 Pajara projection | 8388608/8037225 | [23 -8 -2 -2⟩ | Y |
1 | ∞ | Jubilic | 50/49 | [1 0 2 -2⟩ | n |
1 | -1 | 10 & 14 & 27 | 6272/6075 | [7 -5 -2 2⟩ | n |
2 | -2 | 10 & 19 & 58 | 153664/151875 | [6 -5 -4 4⟩ | n |
2.5 = 5/2 | -2.5 = -5/2 | Linus | 578509309952/576650390625 | [11 -10 -10 10⟩ | n |
2.6 = 13/5 | -2.6 = -13/5 | 10 & 171 & 20cd | (31 digits) | [-27 25 26 -26⟩ | n |
2.6 = 8/3 | -2.6 = -8/3 | 10 & 111 & 91 | 2189469451904296875/2177953337809371136 | [-16 15 16 -16⟩ | n |
3 | -3 | 10 & 41 & 133d | 3796875/3764768 | [-5 5 6 -6⟩ | n |
∞ | 1 | Jubilic | 50/49 | [1 0 2 -2⟩ | n |
∞ | 2 | Breedsmic | 2401/2400 | [-5 -1 -2 4⟩ | n |