User:2^67-1/Blackwood-dicot-semaphore equivalence continuum: Difference between revisions

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Fixed a small error
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m since 5/3 is present in p=0, q=1, we should include 3/2 for completeness. ive expressed it as 5 & 67d as 67d supports septimal meantone so is potentially notable as a very sharp tuning thereof
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| 0 || 1 || [[Archytas]] (squared) || [[64/63|4096/3969]] || {{monzo| 12 -4 0 -2 }} || n
| 0 || 1 || [[Archytas]] (squared) || [[64/63|4096/3969]] || {{monzo| 12 -4 0 -2 }} || n
|-
| 0 || 1.5 = 3/2 || 2.3.7 5 & 97d || 268435456/257298363 || {{monzo| 28 -7 -6 }} || Y
|-
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| 0 || 1.{{overline|6}} = 5/3 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=5+70&submit_edo=submit&commas= 5 & 70] (squared) || 17592186044416/16679880978201 || {{monzo| -56 7 0 16 }} || n
| 0 || 1.{{overline|6}} = 5/3 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=5+70&submit_edo=submit&commas= 5 & 70] (squared) || 17592186044416/16679880978201 || {{monzo| -56 7 0 16 }} || n

Revision as of 18:37, 27 July 2024

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:

  • log2(256/243) / log2(25/24) = 1.2766647429 ; this is the JIP of p=1, q=0 (equiv. to p=-1, q=0)
  • log2(256/243) / log2(49/48) = 2.5275365063 ; this is the JIP of p=0, q=1 (equiv. to p=0, q=-1)
  • log2(256/243) / log2(1225/1152) = 0.8482245109 ; this is the JIP of p=1, q=1 (equiv. to p=-1, q=-1)
  • log2(256/243) / log2(50/49) = 2.5796543166 ; this is the JIP of p=1, q=-1 (equiv. to p=-1, q=1)
  • log2(256/243) / log2(60025/55296) = 0.6350918818 ; this is the JIP of p=1, q=2 (equiv. to p=-1, q=-2)
  • log2(256/243) / log2(2401/2400) = 125.1044589 ; this is the JIP of p=1, q=-2 (equiv. to p=-1, q=2)
  • log2(256/243) / log2(30625/27648) = 0.5096257724 ; this is the JIP of p=2, q=1 (equiv. to p=-2, q=-1)
  • log2(256/243) / log2(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.

Selected temperaments with integer p and q
p q Temperament Comma Added by
someone else?
Ratio Monzo
0 -1 No-fives trienstonic (squared) 784/729 [2 -3 0 1⟩ n
0 0 Blackwood 256/243 [8 -5 0 0⟩ n
0 1 Archytas (squared) 4096/3969 [12 -4 0 -2⟩ n
0 1.5 = 3/2 2.3.7 5 & 97d 268435456/257298363 [28 -7 -6⟩ Y
0 1.6 = 5/3 2.3.7 5 & 70 (squared) 17592186044416/16679880978201 [-56 7 0 16⟩ n
0 2 Buzzard 65536/64827 [16 -3 0 -4⟩ n
0 2.3 = 7/3 Septiness (squared) 4503599627370496/4449821580962289 [52 -8 0 -14⟩ n
0 2.5 = 5/2 Slendrismic 68719476736/68641485507 [36 -5 0 -10⟩ Y
0 2.6 = 13/5 2.3.7 5 & 171 (28 digits) [-92 12 0 26⟩ Y
0 2.6 = 8/3 2.3.7 5 & 212 72680419155717387/72057594037927936 [-56 7 0 16⟩ n
0 3 Slendric (squared) 1058841/1048576 [-20 2 0 6⟩ n
0 4 5 & 81 17294403/16777216 [-24 1 0 8⟩ Y
0 5 Cloudy (squared) 282475249/268435456 [-28 0 0 10⟩ 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⟩ 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⟩ Y
1.5 = 3/2 0 2.3.5 Miracle 34171875/33554432 [-25 7 6⟩ 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=1,q=2 Miracle projection (squared) 1236426679426025390625/11805916207174113034249 [-35 8 6 3⟩ Y
2 1 Avicennmic 275625/262144 [-18 2 4 2⟩ 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
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