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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
84edo shares the [[3/2|perfect fifth]] with [[12edo]], [[tempering out]] the [[Pythagorean comma]] in its [[patent val]]. In the [[5-limit]] it tempers out the [[sensipent comma]]; in the [[7-limit]] [[225/224]], [[1728/1715]], [[2430/2401]], [[6144/6125]], [[support]]ing [[orwell]], [[compton]], and [[sensei]]. In the [[13-limit]] it is the [[optimal patent val]] for the rank-5 temperament tempering out [[144/143]]. | |||
84edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two | 84edo is where the orwell temperament takes its name from, since the generator of [[7/6]] is equal to 19 steps of the edo, referencing the [[Wikipedia: Nineteen Eighty-Four|book 1984]]. Orwell in 84edo comes in two varieties—the 84e val {{val| 84 133 195 236 '''290''' }}, supporting the original orwell, and its [[patent val]] {{val| 84 133 195 236 '''291''' }} supporting [[newspeak]]. 84edo orwell offers [[mos scale]]s of size 9, 13, 22, and 31, of which the 31-note scale is the [[maximal evenness]] scale. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 17: | Line 15: | ||
== Intervals == | == Intervals == | ||
{| class="wikitable" | {| class="wikitable center-1 right-2" | ||
|- | |||
! # | ! # | ||
! Cents | ! Cents | ||
| Line 24: | Line 23: | ||
|- | |- | ||
| 0 | | 0 | ||
| 0. | | 0.0 | ||
| 1/1 | | [[1/1]] | ||
| Perfect 1sn | | Perfect 1sn | ||
| P1 | | P1 | ||
| Line 31: | Line 30: | ||
|- | |- | ||
| 1 | | 1 | ||
| 14. | | 14.3 | ||
| ''81/80'', 105/104, 126/125, 169/168, 196/195 | | ''[[81/80]]'', [[105/104]], [[126/125]], [[169/168]], [[196/195]] | ||
| Up 1sn | | Up 1sn | ||
| ^1 | | ^1 | ||
| Line 38: | Line 37: | ||
|- | |- | ||
| 2 | | 2 | ||
| 28. | | 28.6 | ||
| 50/49, 64/63, 65/64, ''91/90'' | | [[50/49]], [[64/63]], [[65/64]], ''[[91/90]]'' | ||
| Dup 1sn | | Dup 1sn | ||
| ^^1 | | ^^1 | ||
| Line 45: | Line 44: | ||
|- | |- | ||
| 3 | | 3 | ||
| 42. | | 42.9 | ||
| 36/35, 40/39, 46/45, 49/48 | | [[36/35]], [[40/39]], [[46/45]], [[49/48]] | ||
| Trup 1sn | | Trup 1sn | ||
| ^^^1 | | ^^^1 | ||
| Line 52: | Line 51: | ||
|- | |- | ||
| 4 | | 4 | ||
| 57. | | 57.1 | ||
| ''27/26'' | | ''[[27/26]]'' | ||
| Trudminor 2nd | | Trudminor 2nd | ||
| vvvm2 | | vvvm2 | ||
| Line 59: | Line 58: | ||
|- | |- | ||
| 5 | | 5 | ||
| 71. | | 71.4 | ||
| 24/23, 25/24, 26/25, ''28/27'' | | [[24/23]], [[25/24]], [[26/25]], ''[[28/27]]'' | ||
| Dudminor 2nd | | Dudminor 2nd | ||
| vvm2 | | vvm2 | ||
| Line 66: | Line 65: | ||
|- | |- | ||
| 6 | | 6 | ||
| 85. | | 85.7 | ||
| 20/19, 21/20 | | [[20/19]], [[21/20]] | ||
| Downminor 2nd | | Downminor 2nd | ||
| vm2 | | vm2 | ||
| Line 73: | Line 72: | ||
|- | |- | ||
| 7 | | 7 | ||
| 100. | | 100.0 | ||
| 19/18 | | [[19/18]] | ||
| Minor 2nd | | Minor 2nd | ||
| m2 | | m2 | ||
| Line 80: | Line 79: | ||
|- | |- | ||
| 8 | | 8 | ||
| 114. | | 114.3 | ||
| 15/14, 16/15 | | [[15/14]], [[16/15]] | ||
| Upminor 2nd | | Upminor 2nd | ||
| ^m2 | | ^m2 | ||
| Line 87: | Line 86: | ||
|- | |- | ||
| 9 | | 9 | ||
| 128. | | 128.6 | ||
| 14/13 | | [[14/13]] | ||
| Dupminor 2nd | | Dupminor 2nd | ||
| ^^m2 | | ^^m2 | ||
| Line 94: | Line 93: | ||
|- | |- | ||
| 10 | | 10 | ||
| 142. | | 142.9 | ||
| 13/12 | | [[13/12]] | ||
| Trupminor 2nd | | Trupminor 2nd | ||
| ^^^m2 | | ^^^m2 | ||
| Line 101: | Line 100: | ||
|- | |- | ||
| 11 | | 11 | ||
| 157. | | 157.1 | ||
| 23/21 | | [[23/21]] | ||
| Trudmajor 2nd | | Trudmajor 2nd | ||
| vvvM2 | | vvvM2 | ||
| Line 108: | Line 107: | ||
|- | |- | ||
| 12 | | 12 | ||
| 171. | | 171.4 | ||
| 21/19 | | [[21/19]] | ||
| Dudmajor 2nd | | Dudmajor 2nd | ||
| vvM2 | | vvM2 | ||
| Line 115: | Line 114: | ||
|- | |- | ||
| 13 | | 13 | ||
| 185. | | 185.7 | ||
| 10/9 | | [[10/9]] | ||
| Downmajor 2nd | | Downmajor 2nd | ||
| vM2 | | vM2 | ||
| Line 122: | Line 121: | ||
|- | |- | ||
| 14 | | 14 | ||
| 200. | | 200.0 | ||
| 9/8 | | [[9/8]] | ||
| Major 2nd | | Major 2nd | ||
| M2 | | M2 | ||
| Line 129: | Line 128: | ||
|- | |- | ||
| 15 | | 15 | ||
| 214. | | 214.3 | ||
| 26/23 | | [[26/23]] | ||
| Upmajor 2nd | | Upmajor 2nd | ||
| ^M2 | | ^M2 | ||
| Line 136: | Line 135: | ||
|- | |- | ||
| 16 | | 16 | ||
| 228. | | 228.6 | ||
| 8/7 | | [[8/7]] | ||
| Dupmajor 2nd | | Dupmajor 2nd | ||
| ^^M2 | | ^^M2 | ||
| Line 143: | Line 142: | ||
|- | |- | ||
| 17 | | 17 | ||
| 242. | | 242.9 | ||
| 15/13, 23/20 | | [[15/13]], [[23/20]] | ||
| Trupmajor 2nd | | Trupmajor 2nd | ||
| ^^^M2 | | ^^^M2 | ||
| Line 150: | Line 149: | ||
|- | |- | ||
| 18 | | 18 | ||
| 257. | | 257.1 | ||
| 52/45 | | [[52/45]] | ||
| Trudminor 3rd | | Trudminor 3rd | ||
| vvvm3 | | vvvm3 | ||
| Line 157: | Line 156: | ||
|- | |- | ||
| 19 | | 19 | ||
| 271. | | 271.4 | ||
| 7/6 | | [[7/6]] | ||
| Dudminor 3rd | | Dudminor 3rd | ||
| vvm2 | | vvm2 | ||
| Line 164: | Line 163: | ||
|- | |- | ||
| 20 | | 20 | ||
| 285. | | 285.7 | ||
| 45/38, 46/39 | | [[45/38]], [[46/39]] | ||
| Downminor 3rd | | Downminor 3rd | ||
| vm3 | | vm3 | ||
| Line 171: | Line 170: | ||
|- | |- | ||
| 21 | | 21 | ||
| 300. | | 300.0 | ||
| 19/16, 25/21, 32/27 | | [[19/16]], [[25/21]], [[32/27]] | ||
| Minor 3rd | | Minor 3rd | ||
| m3 | | m3 | ||
| Line 178: | Line 177: | ||
|- | |- | ||
| 22 | | 22 | ||
| 314. | | 314.3 | ||
| 6/5 | | [[6/5]] | ||
| Upminor 3rd | | Upminor 3rd | ||
| ^m3 | | ^m3 | ||
| Line 185: | Line 184: | ||
|- | |- | ||
| 23 | | 23 | ||
| 328. | | 328.6 | ||
| 23/19 | | [[23/19]] | ||
| Dupminor 3rd | | Dupminor 3rd | ||
| ^^m3 | | ^^m3 | ||
| Line 192: | Line 191: | ||
|- | |- | ||
| 24 | | 24 | ||
| 342. | | 342.9 | ||
| 28/23, 39/32 | | [[28/23]], [[39/32]] | ||
| Trupminor 3rd | | Trupminor 3rd | ||
| ^^^m3 | | ^^^m3 | ||
| Line 199: | Line 198: | ||
|- | |- | ||
| 25 | | 25 | ||
| 357. | | 357.1 | ||
| 16/13 | | [[16/13]] | ||
| Trudmajor 3rd | | Trudmajor 3rd | ||
| vvvM3 | | vvvM3 | ||
| Line 206: | Line 205: | ||
|- | |- | ||
| 26 | | 26 | ||
| 371. | | 371.4 | ||
| 26/21 | | [[26/21]] | ||
| Dudmajor 3rd | | Dudmajor 3rd | ||
| vvM3 | | vvM3 | ||
| Line 213: | Line 212: | ||
|- | |- | ||
| 27 | | 27 | ||
| 385. | | 385.7 | ||
| 5/4 | | [[5/4]] | ||
| Downmajor 3rd | | Downmajor 3rd | ||
| vM3 | | vM3 | ||
| Line 220: | Line 219: | ||
|- | |- | ||
| 28 | | 28 | ||
| 400. | | 400.0 | ||
| 24/19 | | [[24/19]] | ||
| Major 3rd | | Major 3rd | ||
| M3 | | M3 | ||
| Line 227: | Line 226: | ||
|- | |- | ||
| 29 | | 29 | ||
| 414. | | 414.3 | ||
| 19/15 | | [[19/15]] | ||
| Upmajor 3rd | | Upmajor 3rd | ||
| ^M3 | | ^M3 | ||
| Line 234: | Line 233: | ||
|- | |- | ||
| 30 | | 30 | ||
| 428. | | 428.6 | ||
| 9/7, 23/18, 32/25 | | [[9/7]], [[23/18]], [[32/25]] | ||
| Dupmajor 3rd | | Dupmajor 3rd | ||
| ^^M3 | | ^^M3 | ||
| Line 241: | Line 240: | ||
|- | |- | ||
| 31 | | 31 | ||
| 442. | | 442.9 | ||
| 84/65 | | [[84/65]] | ||
| Trupmajor 3rd | | Trupmajor 3rd | ||
| ^^^M3 | | ^^^M3 | ||
| Line 248: | Line 247: | ||
|- | |- | ||
| 32 | | 32 | ||
| 457. | | 457.1 | ||
| 13/10, 30/23 | | [[13/10]], [[30/23]] | ||
| Trud 4th | | Trud 4th | ||
| vvv4 | | vvv4 | ||
| Line 255: | Line 254: | ||
|- | |- | ||
| 33 | | 33 | ||
| 471. | | 471.4 | ||
| 21/16 | | [[21/16]] | ||
| Dud 4th | | Dud 4th | ||
| vv4 | | vv4 | ||
| Line 262: | Line 261: | ||
|- | |- | ||
| 34 | | 34 | ||
| 485. | | 485.7 | ||
| 65/49 | | [[65/49]] | ||
| Down 4th | | Down 4th | ||
| v4 | | v4 | ||
| Line 269: | Line 268: | ||
|- | |- | ||
| 35 | | 35 | ||
| 500. | | 500.0 | ||
| 4/3 | | [[4/3]] | ||
| Perfect 4th | | Perfect 4th | ||
| P4 | | P4 | ||
| Line 276: | Line 275: | ||
|- | |- | ||
| 36 | | 36 | ||
| 514. | | 514.3 | ||
| 27/20 | | [[27/20]] | ||
| Up 4th | | Up 4th | ||
| ^4 | | ^4 | ||
| Line 283: | Line 282: | ||
|- | |- | ||
| 37 | | 37 | ||
| 528. | | 528.6 | ||
| 19/14 | | [[19/14]] | ||
| Dup 4th | | Dup 4th | ||
| ^^4 | | ^^4 | ||
| Line 290: | Line 289: | ||
|- | |- | ||
| 38 | | 38 | ||
| 542. | | 542.9 | ||
| 26/19 | | [[26/19]] | ||
| Trup 4th | | Trup 4th | ||
| ^^^4 | | ^^^4 | ||
| Line 297: | Line 296: | ||
|- | |- | ||
| 39 | | 39 | ||
| 557. | | 557.1 | ||
| 18/13 | | [[18/13]] | ||
| Trudaug 4th | | Trudaug 4th | ||
| vvvA4 | | vvvA4 | ||
| Line 304: | Line 303: | ||
|- | |- | ||
| 40 | | 40 | ||
| 571. | | 571.4 | ||
| 25/18, 32/23 | | [[25/18]], [[32/23]] | ||
| Dudaug 4th | | Dudaug 4th | ||
| vvA4 | | vvA4 | ||
| Line 311: | Line 310: | ||
|- | |- | ||
| 41 | | 41 | ||
| 585. | | 585.7 | ||
| 7/5 | | [[7/5]] | ||
| Downaug 4th | | Downaug 4th | ||
| vA4 | | vA4 | ||
| Line 318: | Line 317: | ||
|- | |- | ||
| 42 | | 42 | ||
| 600. | | 600.0 | ||
| 27/19, 38/27 | | [[27/19]], [[38/27]] | ||
| Aug 4th, Dim 5th | | Aug 4th, Dim 5th | ||
| A4, d5 | | A4, d5 | ||
| Line 325: | Line 324: | ||
|- | |- | ||
| 43 | | 43 | ||
| 614. | | 614.3 | ||
| 10/7 | | [[10/7]] | ||
| Updim 5th | | Updim 5th | ||
| ^d5 | | ^d5 | ||
| Line 332: | Line 331: | ||
|- | |- | ||
| 44 | | 44 | ||
| 628. | | 628.6 | ||
| 23/16, 36/25 | | [[23/16]], [[36/25]] | ||
| Dupdim 5th | | Dupdim 5th | ||
| ^^d5 | | ^^d5 | ||
| Line 339: | Line 338: | ||
|- | |- | ||
| 45 | | 45 | ||
| 642. | | 642.9 | ||
| 13/9 | | [[13/9]] | ||
| Trupdim 5th | | Trupdim 5th | ||
| ^^^d5 | | ^^^d5 | ||
| Line 346: | Line 345: | ||
|- | |- | ||
| 46 | | 46 | ||
| 657. | | 657.1 | ||
| 19/13 | | [[19/13]] | ||
| Trud 5th | | Trud 5th | ||
| vvv5 | | vvv5 | ||
| Line 353: | Line 352: | ||
|- | |- | ||
| 47 | | 47 | ||
| 671. | | 671.4 | ||
| 28/19 | | [[28/19]] | ||
| Dud 5th | | Dud 5th | ||
| vv5 | | vv5 | ||
| Line 360: | Line 359: | ||
|- | |- | ||
| 48 | | 48 | ||
| 685. | | 685.7 | ||
| 40/27 | | [[40/27]] | ||
| Down 5th | | Down 5th | ||
| v5 | | v5 | ||
| Line 367: | Line 366: | ||
|- | |- | ||
| 49 | | 49 | ||
| 700. | | 700.0 | ||
| 3/2 | | [[3/2]] | ||
| Perfect 5th | | Perfect 5th | ||
| P5 | | P5 | ||
| Line 374: | Line 373: | ||
|- | |- | ||
| 50 | | 50 | ||
| 714. | | 714.3 | ||
| 98/65 | | [[98/65]] | ||
| Up 5th | | Up 5th | ||
| ^5 | | ^5 | ||
| Line 381: | Line 380: | ||
|- | |- | ||
| 51 | | 51 | ||
| 728. | | 728.6 | ||
| 32/21 | | [[32/21]] | ||
| Dup 5th | | Dup 5th | ||
| ^^5 | | ^^5 | ||
| Line 388: | Line 387: | ||
|- | |- | ||
| 52 | | 52 | ||
| 742. | | 742.9 | ||
| 20/13, 23/15 | | [[20/13]], [[23/15]] | ||
| Trup 5th | | Trup 5th | ||
| ^^^5 | | ^^^5 | ||
| Line 395: | Line 394: | ||
|- | |- | ||
| 53 | | 53 | ||
| 757. | | 757.1 | ||
| 65/42 | | [[65/42]] | ||
| Trudminor 6th | | Trudminor 6th | ||
| vvvm6 | | vvvm6 | ||
| Line 402: | Line 401: | ||
|- | |- | ||
| 54 | | 54 | ||
| 771. | | 771.4 | ||
| 14/9, 25/16, 36/23 | | [[14/9]], [[25/16]], [[36/23]] | ||
| Dudminor 6th | | Dudminor 6th | ||
| vvm6 | | vvm6 | ||
| Line 409: | Line 408: | ||
|- | |- | ||
| 55 | | 55 | ||
| 785. | | 785.7 | ||
| 30/19 | | [[30/19]] | ||
| Downminor 6th | | Downminor 6th | ||
| vm6 | | vm6 | ||
| Line 416: | Line 415: | ||
|- | |- | ||
| 56 | | 56 | ||
| 800. | | 800.0 | ||
| 19/12 | | [[19/12]] | ||
| Minor 6th | | Minor 6th | ||
| m6 | | m6 | ||
| Line 423: | Line 422: | ||
|- | |- | ||
| 57 | | 57 | ||
| 814. | | 814.3 | ||
| 8/5 | | [[8/5]] | ||
| Upminor 6th | | Upminor 6th | ||
| ^m6 | | ^m6 | ||
| Line 430: | Line 429: | ||
|- | |- | ||
| 58 | | 58 | ||
| 828. | | 828.6 | ||
| 21/13 | | [[21/13]] | ||
| Dupminor 6th | | Dupminor 6th | ||
| ^^m6 | | ^^m6 | ||
| Line 437: | Line 436: | ||
|- | |- | ||
| 59 | | 59 | ||
| 842. | | 842.9 | ||
| 13/8 | | [[13/8]] | ||
| Trupminor 6th | | Trupminor 6th | ||
| ^^^m6 | | ^^^m6 | ||
| Line 444: | Line 443: | ||
|- | |- | ||
| 60 | | 60 | ||
| 857. | | 857.1 | ||
| 23/14, 64/39 | | [[23/14]], [[64/39]] | ||
| Trudmajor 6th | | Trudmajor 6th | ||
| vvvM6 | | vvvM6 | ||
| Line 451: | Line 450: | ||
|- | |- | ||
| 61 | | 61 | ||
| 871. | | 871.4 | ||
| 38/23 | | [[38/23]] | ||
| Dudmajor 6th | | Dudmajor 6th | ||
| vvM6 | | vvM6 | ||
| Line 458: | Line 457: | ||
|- | |- | ||
| 62 | | 62 | ||
| 885. | | 885.7 | ||
| 5/3 | | [[5/3]] | ||
| Downmajor 6th | | Downmajor 6th | ||
| vM6 | | vM6 | ||
| Line 465: | Line 464: | ||
|- | |- | ||
| 63 | | 63 | ||
| 900. | | 900.0 | ||
| 32/19, 27/16, 42/25 | | [[32/19]], [[27/16]], [[42/25]] | ||
| Major 6th | | Major 6th | ||
| M6 | | M6 | ||
| Line 472: | Line 471: | ||
|- | |- | ||
| 64 | | 64 | ||
| 914. | | 914.3 | ||
| 39/23, 76/45 | | [[39/23]], [[76/45]] | ||
| Upmajor 6th | | Upmajor 6th | ||
| ^M6 | | ^M6 | ||
| Line 479: | Line 478: | ||
|- | |- | ||
| 65 | | 65 | ||
| 928. | | 928.6 | ||
| 12/7 | | [[12/7]] | ||
| Dupmajor 6th | | Dupmajor 6th | ||
| ^^M6 | | ^^M6 | ||
| Line 486: | Line 485: | ||
|- | |- | ||
| 66 | | 66 | ||
| 942. | | 942.9 | ||
| 45/26 | | [[45/26]] | ||
| Trupmajor 6th | | Trupmajor 6th | ||
| ^^^M6 | | ^^^M6 | ||
| Line 493: | Line 492: | ||
|- | |- | ||
| 67 | | 67 | ||
| 957. | | 957.1 | ||
| 26/15, 40/23 | | [[26/15]], [[40/23]] | ||
| Trudminor 7th | | Trudminor 7th | ||
| vvvm7 | | vvvm7 | ||
| Line 500: | Line 499: | ||
|- | |- | ||
| 68 | | 68 | ||
| 971. | | 971.4 | ||
| 7/4 | | [[7/4]] | ||
| Dudminor 7th | | Dudminor 7th | ||
| vvm7 | | vvm7 | ||
| Line 507: | Line 506: | ||
|- | |- | ||
| 69 | | 69 | ||
| 985. | | 985.7 | ||
| 23/13 | | [[23/13]] | ||
| Downminor 7th | | Downminor 7th | ||
| vm7 | | vm7 | ||
| Line 514: | Line 513: | ||
|- | |- | ||
| 70 | | 70 | ||
| 1000. | | 1000.0 | ||
| 16/9 | | [[16/9]] | ||
| Minor 7th | | Minor 7th | ||
| m7 | | m7 | ||
| Line 521: | Line 520: | ||
|- | |- | ||
| 71 | | 71 | ||
| 1014. | | 1014.3 | ||
| 9/5 | | [[9/5]] | ||
| Upminor 7th | | Upminor 7th | ||
| ^m7 | | ^m7 | ||
| Line 528: | Line 527: | ||
|- | |- | ||
| 72 | | 72 | ||
| 1028. | | 1028.6 | ||
| 38/21 | | [[38/21]] | ||
| Dupminor 7th | | Dupminor 7th | ||
| ^^m7 | | ^^m7 | ||
| Line 535: | Line 534: | ||
|- | |- | ||
| 73 | | 73 | ||
| 1042. | | 1042.9 | ||
| 42/23 | | [[42/23]] | ||
| Trupminor 7th | | Trupminor 7th | ||
| ^^^m7 | | ^^^m7 | ||
| Line 542: | Line 541: | ||
|- | |- | ||
| 74 | | 74 | ||
| 1057. | | 1057.1 | ||
| 24/13 | | [[24/13]] | ||
| Trudmajor 7th | | Trudmajor 7th | ||
| vvvM7 | | vvvM7 | ||
| Line 549: | Line 548: | ||
|- | |- | ||
| 75 | | 75 | ||
| 1071. | | 1071.4 | ||
| 13/7 | | [[13/7]] | ||
| Dudmajor 7th | | Dudmajor 7th | ||
| vvM7 | | vvM7 | ||
| Line 556: | Line 555: | ||
|- | |- | ||
| 76 | | 76 | ||
| 1085. | | 1085.7 | ||
| 15/8, 28/15 | | [[15/8]], [[28/15]] | ||
| Downmajor 7th | | Downmajor 7th | ||
| vM7 | | vM7 | ||
| Line 563: | Line 562: | ||
|- | |- | ||
| 77 | | 77 | ||
| 1100. | | 1100.0 | ||
| 36/19 | | [[36/19]] | ||
| Major 7th | | Major 7th | ||
| M7 | | M7 | ||
| Line 570: | Line 569: | ||
|- | |- | ||
| 78 | | 78 | ||
| 1114. | | 1114.3 | ||
| 19/10, 40/21 | | [[19/10]], [[40/21]] | ||
| Upmajor 7th | | Upmajor 7th | ||
| ^M7 | | ^M7 | ||
| Line 577: | Line 576: | ||
|- | |- | ||
| 79 | | 79 | ||
| 1128. | | 1128.6 | ||
| 23/12, 25/13, ''27/14'', 48/25 | | [[23/12]], [[25/13]], ''[[27/14]]'', [[48/25]] | ||
| Dupmajor 7th | | Dupmajor 7th | ||
| ^^M7 | | ^^M7 | ||
| Line 584: | Line 583: | ||
|- | |- | ||
| 80 | | 80 | ||
| 1142. | | 1142.9 | ||
| ''52/27'' | | ''[[52/27]]'' | ||
| Trupmajor 7th | | Trupmajor 7th | ||
| ^^^M7 | | ^^^M7 | ||
| Line 591: | Line 590: | ||
|- | |- | ||
| 81 | | 81 | ||
| 1157. | | 1157.1 | ||
| 35/18, 39/20, 96/49 | | [[35/18]], [[39/20]], [[96/49]] | ||
| Trud 8ve | | Trud 8ve | ||
| vvv8 | | vvv8 | ||
| Line 598: | Line 597: | ||
|- | |- | ||
| 82 | | 82 | ||
| 1171. | | 1171.4 | ||
| 45/23, 49/25, 63/32, 128/65, ''180/91'' | | [[45/23]], [[49/25]], [[63/32]], [[128/65]], ''[[180/91]]'' | ||
| Dud 8ve | | Dud 8ve | ||
| vv8 | | vv8 | ||
| Line 605: | Line 604: | ||
|- | |- | ||
| 83 | | 83 | ||
| 1185. | | 1185.7 | ||
| 125/63, ''160/81'', 195/98, 336/169 | | [[125/63]], ''[[160/81]]'', [[195/98]], [[336/169]] | ||
| Down 8ve | | Down 8ve | ||
| v8 | | v8 | ||
| Line 612: | Line 611: | ||
|- | |- | ||
| 84 | | 84 | ||
| 1200. | | 1200.0 | ||
| 2/1 | | [[2/1]] | ||
| Perfect 8ve | | Perfect 8ve | ||
| P8 | | P8 | ||
| D | | D | ||
|} | |} | ||
<nowiki />* | <nowiki/>* As a 2.3.5.7.13.19.23-subgroup temperament | ||
== Notation == | == Notation == | ||
=== Ups and downs notation === | |||
84edo can be notated using [[ups and downs notation|ups and downs]]. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc. | |||
{{Ups and downs sharpness|84}} | |||
Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used: | |||
{{Sharpness-sharp7|84}} | |||
=== 4L 5s (gramitonic) notation === | === 4L 5s (gramitonic) notation === | ||
This notation is based on Orwell[9]. Notes are denoted as {{nowrap|LsLsLsLss {{=}} JKLMNOPQRJ}}, and raising and lowering by a chroma ({{nowrap|L − s}}), 3 steps in this instance, is denoted by & ("amp") and @ ("at"). | |||
{| class="wikitable center-1 right-2 center-3" | {| class="wikitable center-1 right-2 center-3" | ||
|- | |- | ||
! | ! # | ||
! Cents | ! Cents | ||
! Note | ! Note | ||
! Name | ! Name | ||
! Associated | ! Associated Ratio | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 740: | Line 747: | ||
| 2/1 | | 2/1 | ||
|} | |} | ||
== Approximation to JI == | |||
=== 15-odd-limit intervals === | |||
{{Q-odd-limit intervals|84}} | |||
=== Higher-limit JI === | |||
84edo has fairly good approximation to higher [[prime harmonic]]s such as [[13/1|13]], [[19/1|19]], [[23/1|23]], [[29/1|29]], [[31/1|31]], 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is [[consistent]] in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~0.13{{cent}} (off by ~7.3{{cent}}), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result). | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| 78732/78125, 531441/524288 | | 78732/78125, 531441/524288 | ||
| {{ | | {{Mapping| 84 133 195 }} | ||
| +0.498 | | +0.498 | ||
| 0.531 | | 0.531 | ||
| Line 753: | Line 776: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 225/224, 1728/1715, 78732/78125 | | 225/224, 1728/1715, 78732/78125 | ||
| {{ | | {{Mapping| 84 133 195 236 }} | ||
| +0.141 | | +0.141 | ||
| 0.769 | | 0.769 | ||
| Line 760: | Line 783: | ||
| 2.3.5.7.13 | | 2.3.5.7.13 | ||
| 225/224, 351/350, 640/637, 1701/1690 | | 225/224, 351/350, 640/637, 1701/1690 | ||
| {{ | | {{Mapping| 84 133 195 236 311 }} | ||
| | | −0.013 | ||
| 0.754 | | 0.754 | ||
| 5.28 | | 5.28 | ||
| Line 767: | Line 790: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 225/224, 441/440, 1344/1331, 1728/1715 | | 225/224, 441/440, 1344/1331, 1728/1715 | ||
| {{ | | {{Mapping| 84 133 195 236 291 }} (84) | ||
| | | −0.225 | ||
| 1.003 | | 1.003 | ||
| 7.02 | | 7.02 | ||
| Line 774: | Line 797: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 144/143, 225/224, 351/350, 441/440, 975/968 | | 144/143, 225/224, 351/350, 441/440, 975/968 | ||
| {{ | | {{Mapping| 84 133 195 236 291 311 }} (84) | ||
| | | −0.292 | ||
| 0.928 | | 0.928 | ||
| 6.50 | | 6.50 | ||
| Line 781: | Line 804: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 99/98, 121/120, 176/175, 78732/78125 | | 99/98, 121/120, 176/175, 78732/78125 | ||
| {{ | | {{Mapping| 84 133 195 236 290 }} (84e) | ||
| +0.601 | | +0.601 | ||
| 1.151 | | 1.151 | ||
| Line 788: | Line 811: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 99/98, 121/120, 176/175, 275/273, 1701/1690 | | 99/98, 121/120, 176/175, 275/273, 1701/1690 | ||
| {{ | | {{Mapping| 84 133 195 236 290 311 }} (84e) | ||
| +0.396 | | +0.396 | ||
| 1.146 | | 1.146 | ||
| 8.02 | | 8.02 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br>per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br>ratio* | |||
! Temperament | |||
|- | |- | ||
| 1 | | 1 | ||
| 19\84 | | 19\84 | ||
| 271. | | 271.4 | ||
| 7/6 | | 7/6 | ||
| [[Orwell]] (84e) / [[newspeak]] (84) | | [[Orwell]] (84e) / [[newspeak]] (84) | ||
| Line 805: | Line 835: | ||
| 1 | | 1 | ||
| 25\84 | | 25\84 | ||
| 357. | | 357.1 | ||
| 768/625 | | 768/625 | ||
| [[Dodifo]] | | [[Dodifo]] | ||
| Line 811: | Line 841: | ||
| 1 | | 1 | ||
| 27\84 | | 27\84 | ||
| 385. | | 385.7 | ||
| 5/4 | | 5/4 | ||
| [[Mutt]] | | [[Mutt]] | ||
| Line 817: | Line 847: | ||
| 1 | | 1 | ||
| 31\84 | | 31\84 | ||
| 442. | | 442.9 | ||
| 125 | | 162/125 | ||
| [[Sensei]] | | [[Sensei]] | ||
|- | |- | ||
| 1 | | 1 | ||
| 41\84 | | 41\84 | ||
| 585. | | 585.7 | ||
| 7/5 | | 7/5 | ||
| [[Merman]] | | [[Merman]] | ||
| Line 829: | Line 859: | ||
| 2 | | 2 | ||
| 5\84 | | 5\84 | ||
| 71. | | 71.4 | ||
| 25/24 | | 25/24 | ||
| [[Narayana]] | | [[Narayana]] | ||
| Line 835: | Line 865: | ||
| 2 | | 2 | ||
| 11\84 | | 11\84 | ||
| 157. | | 157.1 | ||
| 35/32 | | 35/32 | ||
| [[Bison]] | | [[Bison]] | ||
| Line 841: | Line 871: | ||
| 2 | | 2 | ||
| 13\84 | | 13\84 | ||
| 185. | | 185.7 | ||
| 10/9 | | 10/9 | ||
| [[Secant]] | | [[Secant]] | ||
| Line 847: | Line 877: | ||
| 3 | | 3 | ||
| 11\84 | | 11\84 | ||
| 157. | | 157.1 | ||
| 35/32 | | 35/32 | ||
| [[Nessafof]] | | [[Nessafof]] | ||
|- | |||
| 6 | |||
| 5\84 | |||
| 71.4 | |||
| 25/24 | |||
| [[Trivish]] | |||
|- | |- | ||
| 7 | | 7 | ||
| 5\84 | | 5\84 | ||
| | | 14.3 | ||
| | | 81/80 | ||
| [[Absurdity]] | | [[Absurdity]] | ||
|- | |- | ||
| 12 | | 12 | ||
| | | 1\84 | ||
| | | 14.3 | ||
| | | 126/125 | ||
| [[Compton]] | | [[Compton]] | ||
|- | |||
| 12 | |||
| 2\84 | |||
| 28.6 | |||
| 64/63 | |||
| [[Catler]] (84c) | |||
|- | |- | ||
| 21 | | 21 | ||
| | | 1\84) | ||
| | | 14.3 | ||
| | | 126/125 | ||
| [[Akjayland]] | | [[Akjayland]] | ||
|- | |- | ||
| 28 | | 28 | ||
| | | 1\84 | ||
| | | 14.3 | ||
| | | 105/104 | ||
| [[Oquatonic]] | | [[Oquatonic]] | ||
|} | |||
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | |||
== Scales == | == Scales == | ||
| Line 882: | Line 924: | ||
* [[Orwell]] | * [[Orwell]] | ||
** Orwell[9] | ** Orwell[9] ([[4L 5s]]) – 11 8 11 8 11 8 11 8 8 | ||
** Orwell[13] | ** Orwell[13] ([[9L 4s]]) – 8 8 8 3 8 8 3 8 8 3 8 8 3 | ||
** Orwell[22] | ** Orwell[22] ([[13L 9s]]) | ||
** Orwell[31] | ** Orwell[31] ([[22L 9s]]) | ||
=== Other === | === Other === | ||
* [[5- to 10-tone scales in 84edo]] | * [[5- to 10-tone scales in 84edo]] | ||
* [[Maeve Gutierrez|Gutierrez Moonglade scale]] | |||
== Instruments == | |||
If you have a precise enough tuner and stable enough instruments, 84edo can be played using 7 instruments tuned a 14th of a tone apart. | |||
You could also try the [[Lumatone mapping for 84edo]] | |||
== Music == | == Music == | ||
| Line 896: | Line 944: | ||
* ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson & Denyer recording (YouTube)] | * ''Two5'' for tenor trombone and piano (1991) [https://youtu.be/YOtQZIqfY1w Fulkerson & Denyer recording (YouTube)] | ||
* ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar & Snijders recording (YouTube)] | * ''Two6'' for violin and piano (1992) [https://youtu.be/XkX37zH6AbU Haar & Snijders recording (YouTube)] | ||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/shorts/Sqkxrmwggr0 ''microtonal improvisation in 84edo''] (2025) | |||
* [https://www.youtube.com/shorts/Qu6UIA2NmmQ ''84edo groove''] (2026) | |||
; [[Eliora]] | ; [[Eliora]] | ||
Latest revision as of 15:15, 20 June 2026
| ← 83edo | 84edo | 85edo → |
84 equal divisions of the octave (abbreviated 84edo or 84ed2), also called 84-tone equal temperament (84tet) or 84 equal temperament (84et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 84 equal parts of about 14.3 ¢ each. Each step represents a frequency ratio of 21/84, or the 84th root of 2.
Theory
84edo shares the perfect fifth with 12edo, tempering out the Pythagorean comma in its patent val. In the 5-limit it tempers out the sensipent comma; in the 7-limit 225/224, 1728/1715, 2430/2401, 6144/6125, supporting orwell, compton, and sensei. In the 13-limit it is the optimal patent val for the rank-5 temperament tempering out 144/143.
84edo is where the orwell temperament takes its name from, since the generator of 7/6 is equal to 19 steps of the edo, referencing the book 1984. Orwell in 84edo comes in two varieties—the 84e val ⟨84 133 195 236 290], supporting the original orwell, and its patent val ⟨84 133 195 236 291] supporting newspeak. 84edo orwell offers mos scales of size 9, 13, 22, and 31, of which the 31-note scale is the maximal evenness scale.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -1.96 | -0.60 | +2.60 | +5.82 | +2.33 | -4.96 | +2.49 | +0.30 | -1.01 | -2.18 | +5.80 |
| Relative (%) | +0.0 | -13.7 | -4.2 | +18.2 | +40.8 | +16.3 | -34.7 | +17.4 | +2.1 | -7.0 | -15.2 | +40.6 | |
| Steps (reduced) |
84 (0) |
133 (49) |
195 (27) |
236 (68) |
291 (39) |
311 (59) |
343 (7) |
357 (21) |
380 (44) |
408 (72) |
416 (80) |
438 (18) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.49 | +2.77 | +5.92 | -2.08 | -2.03 | -2.60 | +6.41 | +6.02 | +0.78 | +6.89 | +7.10 | +0.55 |
| Relative (%) | -3.4 | +19.4 | +41.5 | -14.5 | -14.2 | -18.2 | +44.9 | +42.1 | +5.5 | +48.2 | +49.7 | +3.8 | |
| Steps (reduced) |
450 (30) |
456 (36) |
467 (47) |
481 (61) |
494 (74) |
498 (78) |
510 (6) |
517 (13) |
520 (16) |
530 (26) |
536 (32) |
544 (40) | |
Subsets and supersets
84 is a largely composite number. Since 84 factors as 22 × 3 × 7, 84edo has subset edos 2, 3, 4, 6, 7, 12, 14, 21, 28, 42. Being a small multiple of 28, it tempers out the oquatonic comma, which maps 5/4 to 9\28.
Intervals
| # | Cents | Approximate ratios* | Ups and downs notation | ||
|---|---|---|---|---|---|
| 0 | 0.0 | 1/1 | Perfect 1sn | P1 | D |
| 1 | 14.3 | 81/80, 105/104, 126/125, 169/168, 196/195 | Up 1sn | ^1 | ^D |
| 2 | 28.6 | 50/49, 64/63, 65/64, 91/90 | Dup 1sn | ^^1 | ^^D |
| 3 | 42.9 | 36/35, 40/39, 46/45, 49/48 | Trup 1sn | ^^^1 | ^^^D |
| 4 | 57.1 | 27/26 | Trudminor 2nd | vvvm2 | vvvEb |
| 5 | 71.4 | 24/23, 25/24, 26/25, 28/27 | Dudminor 2nd | vvm2 | vvEb |
| 6 | 85.7 | 20/19, 21/20 | Downminor 2nd | vm2 | vEb |
| 7 | 100.0 | 19/18 | Minor 2nd | m2 | Eb |
| 8 | 114.3 | 15/14, 16/15 | Upminor 2nd | ^m2 | ^Eb |
| 9 | 128.6 | 14/13 | Dupminor 2nd | ^^m2 | ^^Eb |
| 10 | 142.9 | 13/12 | Trupminor 2nd | ^^^m2 | ^^^Eb |
| 11 | 157.1 | 23/21 | Trudmajor 2nd | vvvM2 | vvvE |
| 12 | 171.4 | 21/19 | Dudmajor 2nd | vvM2 | vvE |
| 13 | 185.7 | 10/9 | Downmajor 2nd | vM2 | vE |
| 14 | 200.0 | 9/8 | Major 2nd | M2 | E |
| 15 | 214.3 | 26/23 | Upmajor 2nd | ^M2 | ^E |
| 16 | 228.6 | 8/7 | Dupmajor 2nd | ^^M2 | ^^E |
| 17 | 242.9 | 15/13, 23/20 | Trupmajor 2nd | ^^^M2 | ^^^E |
| 18 | 257.1 | 52/45 | Trudminor 3rd | vvvm3 | vvvF |
| 19 | 271.4 | 7/6 | Dudminor 3rd | vvm2 | vvF |
| 20 | 285.7 | 45/38, 46/39 | Downminor 3rd | vm3 | vF |
| 21 | 300.0 | 19/16, 25/21, 32/27 | Minor 3rd | m3 | F |
| 22 | 314.3 | 6/5 | Upminor 3rd | ^m3 | ^F |
| 23 | 328.6 | 23/19 | Dupminor 3rd | ^^m3 | ^^F |
| 24 | 342.9 | 28/23, 39/32 | Trupminor 3rd | ^^^m3 | ^^^F |
| 25 | 357.1 | 16/13 | Trudmajor 3rd | vvvM3 | vvvF# |
| 26 | 371.4 | 26/21 | Dudmajor 3rd | vvM3 | vvF# |
| 27 | 385.7 | 5/4 | Downmajor 3rd | vM3 | vF# |
| 28 | 400.0 | 24/19 | Major 3rd | M3 | F# |
| 29 | 414.3 | 19/15 | Upmajor 3rd | ^M3 | ^F# |
| 30 | 428.6 | 9/7, 23/18, 32/25 | Dupmajor 3rd | ^^M3 | ^^F# |
| 31 | 442.9 | 84/65 | Trupmajor 3rd | ^^^M3 | ^^^F# |
| 32 | 457.1 | 13/10, 30/23 | Trud 4th | vvv4 | vvvG |
| 33 | 471.4 | 21/16 | Dud 4th | vv4 | vvG |
| 34 | 485.7 | 65/49 | Down 4th | v4 | vG |
| 35 | 500.0 | 4/3 | Perfect 4th | P4 | G |
| 36 | 514.3 | 27/20 | Up 4th | ^4 | ^G |
| 37 | 528.6 | 19/14 | Dup 4th | ^^4 | ^^G |
| 38 | 542.9 | 26/19 | Trup 4th | ^^^4 | ^^^G |
| 39 | 557.1 | 18/13 | Trudaug 4th | vvvA4 | vvvG# |
| 40 | 571.4 | 25/18, 32/23 | Dudaug 4th | vvA4 | vvG# |
| 41 | 585.7 | 7/5 | Downaug 4th | vA4 | vG# |
| 42 | 600.0 | 27/19, 38/27 | Aug 4th, Dim 5th | A4, d5 | G#, Ab |
| 43 | 614.3 | 10/7 | Updim 5th | ^d5 | ^Ab |
| 44 | 628.6 | 23/16, 36/25 | Dupdim 5th | ^^d5 | ^^Ab |
| 45 | 642.9 | 13/9 | Trupdim 5th | ^^^d5 | ^^^Ab |
| 46 | 657.1 | 19/13 | Trud 5th | vvv5 | vvvA |
| 47 | 671.4 | 28/19 | Dud 5th | vv5 | vvA |
| 48 | 685.7 | 40/27 | Down 5th | v5 | vA |
| 49 | 700.0 | 3/2 | Perfect 5th | P5 | A |
| 50 | 714.3 | 98/65 | Up 5th | ^5 | ^A |
| 51 | 728.6 | 32/21 | Dup 5th | ^^5 | ^^A |
| 52 | 742.9 | 20/13, 23/15 | Trup 5th | ^^^5 | ^^^A |
| 53 | 757.1 | 65/42 | Trudminor 6th | vvvm6 | vvvBb |
| 54 | 771.4 | 14/9, 25/16, 36/23 | Dudminor 6th | vvm6 | vvBb |
| 55 | 785.7 | 30/19 | Downminor 6th | vm6 | vBb |
| 56 | 800.0 | 19/12 | Minor 6th | m6 | Bb |
| 57 | 814.3 | 8/5 | Upminor 6th | ^m6 | ^Bb |
| 58 | 828.6 | 21/13 | Dupminor 6th | ^^m6 | ^^Bb |
| 59 | 842.9 | 13/8 | Trupminor 6th | ^^^m6 | ^^^Bb |
| 60 | 857.1 | 23/14, 64/39 | Trudmajor 6th | vvvM6 | vvvB |
| 61 | 871.4 | 38/23 | Dudmajor 6th | vvM6 | vvB |
| 62 | 885.7 | 5/3 | Downmajor 6th | vM6 | vB |
| 63 | 900.0 | 32/19, 27/16, 42/25 | Major 6th | M6 | B |
| 64 | 914.3 | 39/23, 76/45 | Upmajor 6th | ^M6 | ^B |
| 65 | 928.6 | 12/7 | Dupmajor 6th | ^^M6 | ^^B |
| 66 | 942.9 | 45/26 | Trupmajor 6th | ^^^M6 | ^^^B |
| 67 | 957.1 | 26/15, 40/23 | Trudminor 7th | vvvm7 | vvvC |
| 68 | 971.4 | 7/4 | Dudminor 7th | vvm7 | vvC |
| 69 | 985.7 | 23/13 | Downminor 7th | vm7 | vC |
| 70 | 1000.0 | 16/9 | Minor 7th | m7 | C |
| 71 | 1014.3 | 9/5 | Upminor 7th | ^m7 | ^C |
| 72 | 1028.6 | 38/21 | Dupminor 7th | ^^m7 | ^^C |
| 73 | 1042.9 | 42/23 | Trupminor 7th | ^^^m7 | ^^^C |
| 74 | 1057.1 | 24/13 | Trudmajor 7th | vvvM7 | vvvC# |
| 75 | 1071.4 | 13/7 | Dudmajor 7th | vvM7 | vvC# |
| 76 | 1085.7 | 15/8, 28/15 | Downmajor 7th | vM7 | vC# |
| 77 | 1100.0 | 36/19 | Major 7th | M7 | C# |
| 78 | 1114.3 | 19/10, 40/21 | Upmajor 7th | ^M7 | ^C# |
| 79 | 1128.6 | 23/12, 25/13, 27/14, 48/25 | Dupmajor 7th | ^^M7 | ^^C# |
| 80 | 1142.9 | 52/27 | Trupmajor 7th | ^^^M7 | ^^^C# |
| 81 | 1157.1 | 35/18, 39/20, 96/49 | Trud 8ve | vvv8 | vvvD |
| 82 | 1171.4 | 45/23, 49/25, 63/32, 128/65, 180/91 | Dud 8ve | vv8 | vvD |
| 83 | 1185.7 | 125/63, 160/81, 195/98, 336/169 | Down 8ve | v8 | vD |
| 84 | 1200.0 | 2/1 | Perfect 8ve | P8 | D |
* As a 2.3.5.7.13.19.23-subgroup temperament
Notation
Ups and downs notation
84edo can be notated using ups and downs. Trup is equivalent to quudsharp, trudsharp is equivalent to quup, etc.
| Semitones | 0 | 1⁄7 | 2⁄7 | 3⁄7 | 4⁄7 | 5⁄7 | 6⁄7 | 1 | 1 1⁄7 | 1 2⁄7 | 1 3⁄7 | 1 4⁄7 | 1 5⁄7 | 1 6⁄7 | 2 | 2 1⁄7 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol | 
|
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| Flat symbol | |
|
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Alternatively, sharps and flats with arrows borrowed from Helmholtz–Ellis notation can be used:
| Semitones | 0 | 1⁄7 | 2⁄7 | 3⁄7 | 4⁄7 | 5⁄7 | 6⁄7 | 1 | 1+1⁄7 | 1+2⁄7 | 1+3⁄7 | 1+4⁄7 | 1+5⁄7 | 1+6⁄7 | 2 | 1+1⁄7 | 1+2⁄7 | 1+3⁄7 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Sharp symbol |
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| Flat symbol |  |
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4L 5s (gramitonic) notation
This notation is based on Orwell[9]. Notes are denoted as LsLsLsLss = JKLMNOPQRJ, and raising and lowering by a chroma (L − s), 3 steps in this instance, is denoted by & ("amp") and @ ("at").
| # | Cents | Note | Name | Associated Ratio |
|---|---|---|---|---|
| 0 | 0.0 | J | Perfect 0-gramstep | 1/1 |
| 8 | 114.3 | K@ | Minor 1-gramstep | 15/14~16/15 |
| 11 | 157.1 | K | Major 1-gramstep | 11/10~12/11 |
| 16 | 228.6 | L@ | Diminished 2-gramstep | 8/7 |
| 19 | 271.4 | L | Perfect 2-gramstep | 7/6 |
| 27 | 385.7 | M@ | Minor 3-gramstep | 5/4 |
| 30 | 428.6 | M | Major 3-gramstep | 9/7 |
| 35 | 500.0 | N@ | Minor 4-gramstep | 4/3 |
| 38 | 542.9 | N | Major 4-gramstep | 11/8~15/11 |
| 46 | 657.1 | O@ | Minor 5-gramstep | 16/11~22/15 |
| 49 | 700.0 | O | Major 5-gramstep | 3/2 |
| 54 | 771.4 | P@ | Minor 6-gramstep | 14/9 |
| 57 | 814.3 | P | Major 6-gramstep | 8/5 |
| 65 | 928.6 | Q@ | Perfect 7-gramstep | 12/7 |
| 68 | 971.4 | Q | Augmented 7-gramstep | 7/4 |
| 73 | 1042.9 | R@ | Minor 8-gramstep | 11/6~20/11 |
| 76 | 1085.7 | R | Major 8-gramstep | 15/8~28/15 |
| 84 | 1200.0 | J | Perfect 9-gramstep | 2/1 |
Approximation to JI
15-odd-limit intervals
The following tables show how 15-odd-limit intervals are represented in 84edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 13/7, 14/13 | 0.273 | 1.9 |
| 5/4, 8/5 | 0.599 | 4.2 |
| 5/3, 6/5 | 1.356 | 9.5 |
| 3/2, 4/3 | 1.955 | 13.7 |
| 13/8, 16/13 | 2.329 | 16.3 |
| 15/8, 16/15 | 2.554 | 17.9 |
| 7/4, 8/7 | 2.603 | 18.2 |
| 13/10, 20/13 | 2.929 | 20.5 |
| 7/5, 10/7 | 3.202 | 22.4 |
| 11/7, 14/11 | 3.222 | 22.6 |
| 9/5, 10/9 | 3.311 | 23.2 |
| 13/11, 22/13 | 3.495 | 24.5 |
| 9/8, 16/9 | 3.910 | 27.4 |
| 13/12, 24/13 | 4.284 | 30.0 |
| 11/9, 18/11 | 4.551 | 31.9 |
| 7/6, 12/7 | 4.558 | 31.9 |
| 15/13, 26/15 | 4.884 | 34.2 |
| 15/14, 28/15 | 5.157 | 36.1 |
| 11/8, 16/11 | 5.825 | 40.8 |
| 15/11, 22/15 | 5.906 | 41.3 |
| 13/9, 18/13 | 6.239 | 43.7 |
| 11/10, 20/11 | 6.424 | 45.0 |
| 11/6, 12/11 | 6.506 | 45.5 |
| 9/7, 14/9 | 6.513 | 45.6 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 13/7, 14/13 | 0.273 | 1.9 |
| 5/4, 8/5 | 0.599 | 4.2 |
| 5/3, 6/5 | 1.356 | 9.5 |
| 3/2, 4/3 | 1.955 | 13.7 |
| 13/8, 16/13 | 2.329 | 16.3 |
| 15/8, 16/15 | 2.554 | 17.9 |
| 7/4, 8/7 | 2.603 | 18.2 |
| 13/10, 20/13 | 2.929 | 20.5 |
| 7/5, 10/7 | 3.202 | 22.4 |
| 11/7, 14/11 | 3.222 | 22.6 |
| 9/5, 10/9 | 3.311 | 23.2 |
| 13/11, 22/13 | 3.495 | 24.5 |
| 9/8, 16/9 | 3.910 | 27.4 |
| 13/12, 24/13 | 4.284 | 30.0 |
| 7/6, 12/7 | 4.558 | 31.9 |
| 15/13, 26/15 | 4.884 | 34.2 |
| 15/14, 28/15 | 5.157 | 36.1 |
| 11/8, 16/11 | 5.825 | 40.8 |
| 13/9, 18/13 | 6.239 | 43.7 |
| 11/10, 20/11 | 6.424 | 45.0 |
| 9/7, 14/9 | 6.513 | 45.6 |
| 11/6, 12/11 | 7.780 | 54.5 |
| 15/11, 22/15 | 8.379 | 58.7 |
| 11/9, 18/11 | 9.735 | 68.1 |
Higher-limit JI
84edo has fairly good approximation to higher prime harmonics such as 13, 19, 23, 29, 31, 41, 43, 53, 59, 61, 73 and 89, so that it is for its size very performant for much of the 61-limit, with more off primes usually being sharp so that they can cancel opportunistically with other sharp harmonics. In fact, it is consistent in the no-11 no-17 no-27 no-37 no-47 no-49 no-51 no-55 65-odd-limit excepting only 1 inconsistent pair, 45/43 and 86/45, which are inconsistent by ~0.13 ¢ (off by ~7.3 ¢), offering a truly vast inventory of harmony to draw from that has mostly been unexplored. This is especially true because its approximation powers do not end there: prime 11, due to its simplicity (and thus lesser tuning fidelity), is certainly usable (just causes some inconsistencies), and there are higher primes that are reasonably in-tune too when supported by context. The only missing primes are thus 17, 37, 47, 67, 71, 79 and 83, which except for 17 are all about 6 cents sharp, similar to the sharpness of prime 11, so that it somewhat makes up for these omissions by having a very accurate 22:37:47:67:71:79:83 chord, to which various additions are possible (though usually increasing the error as a result).
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | 78732/78125, 531441/524288 | [⟨84 133 195]] | +0.498 | 0.531 | 3.72 |
| 2.3.5.7 | 225/224, 1728/1715, 78732/78125 | [⟨84 133 195 236]] | +0.141 | 0.769 | 5.39 |
| 2.3.5.7.13 | 225/224, 351/350, 640/637, 1701/1690 | [⟨84 133 195 236 311]] | −0.013 | 0.754 | 5.28 |
| 2.3.5.7.11 | 225/224, 441/440, 1344/1331, 1728/1715 | [⟨84 133 195 236 291]] (84) | −0.225 | 1.003 | 7.02 |
| 2.3.5.7.11.13 | 144/143, 225/224, 351/350, 441/440, 975/968 | [⟨84 133 195 236 291 311]] (84) | −0.292 | 0.928 | 6.50 |
| 2.3.5.7.11 | 99/98, 121/120, 176/175, 78732/78125 | [⟨84 133 195 236 290]] (84e) | +0.601 | 1.151 | 8.05 |
| 2.3.5.7.11.13 | 99/98, 121/120, 176/175, 275/273, 1701/1690 | [⟨84 133 195 236 290 311]] (84e) | +0.396 | 1.146 | 8.02 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 19\84 | 271.4 | 7/6 | Orwell (84e) / newspeak (84) |
| 1 | 25\84 | 357.1 | 768/625 | Dodifo |
| 1 | 27\84 | 385.7 | 5/4 | Mutt |
| 1 | 31\84 | 442.9 | 162/125 | Sensei |
| 1 | 41\84 | 585.7 | 7/5 | Merman |
| 2 | 5\84 | 71.4 | 25/24 | Narayana |
| 2 | 11\84 | 157.1 | 35/32 | Bison |
| 2 | 13\84 | 185.7 | 10/9 | Secant |
| 3 | 11\84 | 157.1 | 35/32 | Nessafof |
| 6 | 5\84 | 71.4 | 25/24 | Trivish |
| 7 | 5\84 | 14.3 | 81/80 | Absurdity |
| 12 | 1\84 | 14.3 | 126/125 | Compton |
| 12 | 2\84 | 28.6 | 64/63 | Catler (84c) |
| 21 | 1\84) | 14.3 | 126/125 | Akjayland |
| 28 | 1\84 | 14.3 | 105/104 | Oquatonic |
Scales
MOS
Brightest mode is listed.
Other
Instruments
If you have a precise enough tuner and stable enough instruments, 84edo can be played using 7 instruments tuned a 14th of a tone apart.
You could also try the Lumatone mapping for 84edo
Music
- Ten for chamber ensemble (1991) Ives Ensemble recording (YouTube) [dead link]
- Two4 for violin and piano or shō (1991) Harr & Miyata recording (YouTube)
- Two5 for tenor trombone and piano (1991) Fulkerson & Denyer recording (YouTube)
- Two6 for violin and piano (1992) Haar & Snijders recording (YouTube)
- microtonal improvisation in 84edo (2025)
- 84edo groove (2026)
- Requiem in Gb 1/7 Orwell (2023)
- Undiminished (2023)