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

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! Monzo
! Monzo
|-
|-
| 0 || -1 || [[Trienstonic clan #No-fives trienstonic|No-fives trienstonic]] (squared) || [[28/27|784/729]] || {{monzo| 2 -3 0 1 }} || n
| 0 || -1 || [[Trienstonic clan #No-fives trienstonic|No-fives trienstonic]] (squared) || [[28/27]] (reduced), 784/729 (unreduced) || {{monzo| 2 -3 0 1 }} (reduced), {{monzo| 4 -6 0 2 }} (unreduced) || n
|-
|-
| 0 || 0 || [[Blackwood]] || [[256/243]] || {{monzo| 8 -5 0 0 }} || n
| 0 || 0 || [[Blackwood]] || [[256/243]] || {{monzo| 8 -5 0 0 }} || n
|-
|-
| 0 || 1 || [[Archytas]] (squared) || [[64/63|4096/3969]] || {{monzo| 12 -4 0 -2 }} || n
| 0 || 1 || [[Archytas]] || [[64/63]] (reduced), 4096/3969 (unreduced) || {{monzo| 6 -2 0 -1 }} (reduced), {{monzo| 12 -4 0 -2 }} (unreduced) || 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
| 0 || 1.5 = 3/2 || 2.3.7 5 & 97d || 268435456/257298363 || {{monzo| 28 -7 0 -6 }} || Y
|-
| 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] || 4194304/4084101 (reduced), 17592186044416/16679880978201 (unreduced) || {{monzo| 22 -5 0 -5 }} (reduced), {{monzo| 44 -10 0 -10}} (unreduced) || n
|-
|-
| 0 || 2 || [[Buzzard]] || [[65536/64827]] || {{monzo| 16 -3 0 -4 }} || n
| 0 || 2 || [[Buzzard]] || [[65536/64827]] || {{monzo| 16 -3 0 -4 }} || n
|-
|-
| 0 || 2.{{overline|3}} = 7/3 || [[Septiness]] (squared) || [[67108864/66706983|4503599627370496/4449821580962289]] || {{monzo| 52 -8 0 -14 }} || n
| 0 || 2.{{overline|3}} = 7/3 || [[Septiness]] || [[67108864/66706983]] (reduced), 4503599627370496/4449821580962289 (unreduced) || {{monzo| 26 -4 0 -7 }} (reduced), {{monzo| 52 -8 0 -14 }} (unreduced) || n
|-
|-
| 0 || 2.5 = 5/2 || [[Slendric schisma|Slendrismic]] || [[Slendric schisma|68719476736/68641485507]] || {{monzo| 36 -5 0 -10 }} || Y
| 0 || 2.5 = 5/2 || [[Slendric schisma|Slendrismic]] || [[Slendric schisma|68719476736/68641485507]] || {{monzo| 36 -5 0 -10 }} || Y
|-
|-
| 0 || 2.6 = 13/5 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=&commas=4988891938119860195948410209%2F4951760157141521099596496896&submit_comma=submit 5 & 171] || [[4988891938119860195948410209/4951760157141521099596496896|(28 digits)]] || {{monzo| -92 12 0 26 }} || Y
| 0 || 2.{{overline|571428}} = 18/7 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=&commas=%5B-128+17+36%3E&submit_comma=submit 5 & 472] || [[342444953666443524469058979480764340963/340282366920938463463374607431768211456|(39 digits)]] || {{monzo| 36 -5 0 -10 }} || n
|-
| 0 || 2.6 = 13/5 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=&commas=4988891938119860195948410209%2F4951760157141521099596496896&submit_comma=submit 5 & 171] || 70632088586703/70368744177664 (reduced), 4988891938119860195948410209/4951760157141521099596496896 (unreduced) || {{monzo| -46 6 0 13}} (reduced) {{monzo| -92 12 0 26 }} (unreduced) || Y
|-
|-
| 0 || 2.{{overline|6}} = 8/3 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=&commas=72680419155717387%2F72057594037927936&submit_comma=submit 5 & 212] || 72680419155717387/72057594037927936 || {{monzo| -56 7 0 16 }} || n
| 0 || 2.{{overline|6}} = 8/3 || 2.3.7 [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=on&weights=weil&target=&edos=&commas=72680419155717387%2F72057594037927936&submit_comma=submit 5 & 212] || 72680419155717387/72057594037927936 || {{monzo| -56 7 0 16 }} || n
|-
|-
| 0 || 3 || [[Slendric]] (squared) || [[1029/1024|1058841/1048576]] || {{monzo| -20 2 0 6 }} || n
| 0 || 3 || [[Slendric]] (squared) || [[1029/1024]] (reduced), 1058841/1048576 (unreduced) || {{monzo| -10 1 0 3 }} (reduced), {{monzo| -20 2 0 6 }} (unreduced)|| n
|-
|-
| 0 || 4 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=&commas=17294403%2F16777216&submit_comma=submit 5 & 81] || 17294403/16777216 || {{monzo| -24 1 0 8 }} || Y
| 0 || 4 || [https://sintel.pythonanywhere.com/result?subgroup=2.3.7&reduce=off&weights=weil&target=&edos=&commas=17294403%2F16777216&submit_comma=submit 5 & 81] || 17294403/16777216 || {{monzo| -24 1 0 8 }} || Y
|-
|-
| 0 || 5 || [[Cloudy]] (squared) || [[16807/16384|282475249/268435456]] || {{monzo| -28 0 0 10 }} || Y
| 0 || 5 || [[Cloudy]] (squared) || [[16807/16384]] (reduced), 282475249/268435456 (unreduced) || {{monzo| -14 0 0 5 }} (reduced), {{monzo| -28 0 0 10 }} (unreduced) || Y
|-
|-
| 0 || ∞ || [[Semaphore]] || [[49/48]] || {{monzo| -4 -1 0 2 }} || n
| 0 || ∞ || [[Semaphore]] || [[49/48]] || {{monzo| -4 -1 0 2 }} || n
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| 1.25 = 5/4 || 0 || [[Quintosec]] || 140737488355328/140126044921875 || {{monzo| 47 -15 -10 0 }} || n
| 1.25 = 5/4 || 0 || [[Quintosec]] || 140737488355328/140126044921875 || {{monzo| 47 -15 -10 0 }} || n
|-
|-
| 1.{{overline|285714}} = 9/7 || 0 || 2.3.5 [[Lagaca]] || 9696448624912261962890625/9671406556917033397649408 || {{monzo| -83 26 18 }} || Y
| 1.{{overline|285714}} = 9/7 || 0 || 2.3.5 [[Lagaca]] || 9696448624912261962890625/9671406556917033397649408 || {{monzo| -83 26 18 0 }} || Y
|-
|-
| 1.3 = 13/10 || 0 || 2.3.5 [https://sintel.pythonanywhere.com/result?subgroup=5&reduce=on&weights=weil&target=&edos=&commas=%5B119+-37+-26%3E&submit_comma=submit 10 & 171] || [[670975306467707996070384979248046875/664613997892457936451903530140172288|(36 digits)]] || {{monzo| -119 37 26 0 }} || n
| 1.3 = 13/10 || 0 || 2.3.5 [https://sintel.pythonanywhere.com/result?subgroup=5&reduce=on&weights=weil&target=&edos=&commas=%5B119+-37+-26%3E&submit_comma=submit 10 & 171] || [[670975306467707996070384979248046875/664613997892457936451903530140172288|(36 digits)]] || {{monzo| -119 37 26 0 }} || n
|-
|-
| 1.{{overline|3}} = 4/3 || 0 || [[Marvel temperaments#Submajor|Submajor]] || 69198046875/68719476736 || {{monzo| -36 11 8 }} || Y
| 1.{{overline|3}} = 4/3 || 0 || [[Marvel temperaments#Submajor|Submajor]] || 69198046875/68719476736 || {{monzo| -36 11 8 0 }} || Y
|-
|-
| 1.5 = 3/2 || 0 || 2.3.5 [[Miracle]] || [[34171875/33554432]] || {{monzo| -25 7 6 }} || Y
| 1.5 = 3/2 || 0 || 2.3.5 [[Miracle]]/[[Ampersand]] || [[34171875/33554432]] || {{monzo| -25 7 6 0 }} || Y
|-
|-
| 2 || 0 || [[Negri]] || [[16875/16384]] || {{monzo| -14 3 4 0 }} || n
| 2 || 0 || [[Negri]] || [[16875/16384]] || {{monzo| -14 3 4 0 }} || n
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| 1 || 0.5 = 1/2 || [https://sintel.pythonanywhere.com/result?subgroup=7&reduce=on&weights=weil&target=&edos=&commas=67108864%2F66976875&submit_comma=submit 10 & 53 & 130] || 67108864/66976875 || {{monzo| 26 -7 -4 -2 }} || n
| 1 || 0.5 = 1/2 || [https://sintel.pythonanywhere.com/result?subgroup=7&reduce=on&weights=weil&target=&edos=&commas=67108864%2F66976875&submit_comma=submit 10 & 53 & 130] || 67108864/66976875 || {{monzo| 26 -7 -4 -2 }} || n
|-
|-
| 1.2 = 6/5 || 0.6 = 3/5 || p=1,q=2 Miracle projection (squared) || 1236426679426025390625/11805916207174113034249 || {{monzo| -35 8 6 3 }} || Y
| 1.2 = 6/5 || 0.6 = 3/5 || p=2, q=1 Miracle projection || 35162859375/34359738368 (reduced), 1236426679426025390625/1180591620717411303424 (unreduced) || {{monzo|-35 8 6 3 }} (reduced), {{monzo| -70 16 12 6 }} (unreduced) || Y
|-
|-
| 2 || 1 || [[Avicennmic temperaments|Avicennmic]] || [[525/512|275625/262144]] || {{monzo| -18 2 4 2 }} || Y
| 2 || 1 || [[Avicennmic temperaments|Avicennmic]] || [[525/512]] (reduced), 275625/262144 (unreduced) || {{monzo| -9 1 2 1 }} (reduced), {{monzo| -18 2 4 2 }} (unreduced) || Y
|-
|-
| 1 || 1 || [[Mirwomo temperaments|Mirwomo]] || [[33075/32768]] || {{monzo| -15 3 2 2 }} || n
| 1 || 1 || [[Mirwomo temperaments|Mirwomo]] || [[33075/32768]] || {{monzo| -15 3 2 2 }} || n

Latest revision as of 22:10, 31 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) 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
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