84edo: Difference between revisions
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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. | 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 27: | Line 24: | ||
| 0 | | 0 | ||
| 0.0 | | 0.0 | ||
| 1/1 | | [[1/1]] | ||
| Perfect 1sn | | Perfect 1sn | ||
| P1 | | P1 | ||
| Line 34: | Line 31: | ||
| 1 | | 1 | ||
| 14.3 | | 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 41: | Line 38: | ||
| 2 | | 2 | ||
| 28.6 | | 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 48: | Line 45: | ||
| 3 | | 3 | ||
| 42.9 | | 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 55: | Line 52: | ||
| 4 | | 4 | ||
| 57.1 | | 57.1 | ||
| ''27/26'' | | ''[[27/26]]'' | ||
| Trudminor 2nd | | Trudminor 2nd | ||
| vvvm2 | | vvvm2 | ||
| Line 62: | Line 59: | ||
| 5 | | 5 | ||
| 71.4 | | 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 69: | Line 66: | ||
| 6 | | 6 | ||
| 85.7 | | 85.7 | ||
| 20/19, 21/20 | | [[20/19]], [[21/20]] | ||
| Downminor 2nd | | Downminor 2nd | ||
| vm2 | | vm2 | ||
| Line 76: | Line 73: | ||
| 7 | | 7 | ||
| 100.0 | | 100.0 | ||
| 19/18 | | [[19/18]] | ||
| Minor 2nd | | Minor 2nd | ||
| m2 | | m2 | ||
| Line 83: | Line 80: | ||
| 8 | | 8 | ||
| 114.3 | | 114.3 | ||
| 15/14, 16/15 | | [[15/14]], [[16/15]] | ||
| Upminor 2nd | | Upminor 2nd | ||
| ^m2 | | ^m2 | ||
| Line 90: | Line 87: | ||
| 9 | | 9 | ||
| 128.6 | | 128.6 | ||
| 14/13 | | [[14/13]] | ||
| Dupminor 2nd | | Dupminor 2nd | ||
| ^^m2 | | ^^m2 | ||
| Line 97: | Line 94: | ||
| 10 | | 10 | ||
| 142.9 | | 142.9 | ||
| 13/12 | | [[13/12]] | ||
| Trupminor 2nd | | Trupminor 2nd | ||
| ^^^m2 | | ^^^m2 | ||
| Line 104: | Line 101: | ||
| 11 | | 11 | ||
| 157.1 | | 157.1 | ||
| 23/21 | | [[23/21]] | ||
| Trudmajor 2nd | | Trudmajor 2nd | ||
| vvvM2 | | vvvM2 | ||
| Line 111: | Line 108: | ||
| 12 | | 12 | ||
| 171.4 | | 171.4 | ||
| 21/19 | | [[21/19]] | ||
| Dudmajor 2nd | | Dudmajor 2nd | ||
| vvM2 | | vvM2 | ||
| Line 118: | Line 115: | ||
| 13 | | 13 | ||
| 185.7 | | 185.7 | ||
| 10/9 | | [[10/9]] | ||
| Downmajor 2nd | | Downmajor 2nd | ||
| vM2 | | vM2 | ||
| Line 125: | Line 122: | ||
| 14 | | 14 | ||
| 200.0 | | 200.0 | ||
| 9/8 | | [[9/8]] | ||
| Major 2nd | | Major 2nd | ||
| M2 | | M2 | ||
| Line 132: | Line 129: | ||
| 15 | | 15 | ||
| 214.3 | | 214.3 | ||
| 26/23 | | [[26/23]] | ||
| Upmajor 2nd | | Upmajor 2nd | ||
| ^M2 | | ^M2 | ||
| Line 139: | Line 136: | ||
| 16 | | 16 | ||
| 228.6 | | 228.6 | ||
| 8/7 | | [[8/7]] | ||
| Dupmajor 2nd | | Dupmajor 2nd | ||
| ^^M2 | | ^^M2 | ||
| Line 146: | Line 143: | ||
| 17 | | 17 | ||
| 242.9 | | 242.9 | ||
| 15/13, 23/20 | | [[15/13]], [[23/20]] | ||
| Trupmajor 2nd | | Trupmajor 2nd | ||
| ^^^M2 | | ^^^M2 | ||
| Line 153: | Line 150: | ||
| 18 | | 18 | ||
| 257.1 | | 257.1 | ||
| 52/45 | | [[52/45]] | ||
| Trudminor 3rd | | Trudminor 3rd | ||
| vvvm3 | | vvvm3 | ||
| Line 160: | Line 157: | ||
| 19 | | 19 | ||
| 271.4 | | 271.4 | ||
| 7/6 | | [[7/6]] | ||
| Dudminor 3rd | | Dudminor 3rd | ||
| vvm2 | | vvm2 | ||
| Line 167: | Line 164: | ||
| 20 | | 20 | ||
| 285.7 | | 285.7 | ||
| 45/38, 46/39 | | [[45/38]], [[46/39]] | ||
| Downminor 3rd | | Downminor 3rd | ||
| vm3 | | vm3 | ||
| Line 174: | Line 171: | ||
| 21 | | 21 | ||
| 300.0 | | 300.0 | ||
| 19/16, 25/21, 32/27 | | [[19/16]], [[25/21]], [[32/27]] | ||
| Minor 3rd | | Minor 3rd | ||
| m3 | | m3 | ||
| Line 181: | Line 178: | ||
| 22 | | 22 | ||
| 314.3 | | 314.3 | ||
| 6/5 | | [[6/5]] | ||
| Upminor 3rd | | Upminor 3rd | ||
| ^m3 | | ^m3 | ||
| Line 188: | Line 185: | ||
| 23 | | 23 | ||
| 328.6 | | 328.6 | ||
| 23/19 | | [[23/19]] | ||
| Dupminor 3rd | | Dupminor 3rd | ||
| ^^m3 | | ^^m3 | ||
| Line 195: | Line 192: | ||
| 24 | | 24 | ||
| 342.9 | | 342.9 | ||
| 28/23, 39/32 | | [[28/23]], [[39/32]] | ||
| Trupminor 3rd | | Trupminor 3rd | ||
| ^^^m3 | | ^^^m3 | ||
| Line 202: | Line 199: | ||
| 25 | | 25 | ||
| 357.1 | | 357.1 | ||
| 16/13 | | [[16/13]] | ||
| Trudmajor 3rd | | Trudmajor 3rd | ||
| vvvM3 | | vvvM3 | ||
| Line 209: | Line 206: | ||
| 26 | | 26 | ||
| 371.4 | | 371.4 | ||
| 26/21 | | [[26/21]] | ||
| Dudmajor 3rd | | Dudmajor 3rd | ||
| vvM3 | | vvM3 | ||
| Line 216: | Line 213: | ||
| 27 | | 27 | ||
| 385.7 | | 385.7 | ||
| 5/4 | | [[5/4]] | ||
| Downmajor 3rd | | Downmajor 3rd | ||
| vM3 | | vM3 | ||
| Line 223: | Line 220: | ||
| 28 | | 28 | ||
| 400.0 | | 400.0 | ||
| 24/19 | | [[24/19]] | ||
| Major 3rd | | Major 3rd | ||
| M3 | | M3 | ||
| Line 230: | Line 227: | ||
| 29 | | 29 | ||
| 414.3 | | 414.3 | ||
| 19/15 | | [[19/15]] | ||
| Upmajor 3rd | | Upmajor 3rd | ||
| ^M3 | | ^M3 | ||
| Line 237: | Line 234: | ||
| 30 | | 30 | ||
| 428.6 | | 428.6 | ||
| 9/7, 23/18, 32/25 | | [[9/7]], [[23/18]], [[32/25]] | ||
| Dupmajor 3rd | | Dupmajor 3rd | ||
| ^^M3 | | ^^M3 | ||
| Line 244: | Line 241: | ||
| 31 | | 31 | ||
| 442.9 | | 442.9 | ||
| 84/65 | | [[84/65]] | ||
| Trupmajor 3rd | | Trupmajor 3rd | ||
| ^^^M3 | | ^^^M3 | ||
| Line 251: | Line 248: | ||
| 32 | | 32 | ||
| 457.1 | | 457.1 | ||
| 13/10, 30/23 | | [[13/10]], [[30/23]] | ||
| Trud 4th | | Trud 4th | ||
| vvv4 | | vvv4 | ||
| Line 258: | Line 255: | ||
| 33 | | 33 | ||
| 471.4 | | 471.4 | ||
| 21/16 | | [[21/16]] | ||
| Dud 4th | | Dud 4th | ||
| vv4 | | vv4 | ||
| Line 265: | Line 262: | ||
| 34 | | 34 | ||
| 485.7 | | 485.7 | ||
| 65/49 | | [[65/49]] | ||
| Down 4th | | Down 4th | ||
| v4 | | v4 | ||
| Line 272: | Line 269: | ||
| 35 | | 35 | ||
| 500.0 | | 500.0 | ||
| 4/3 | | [[4/3]] | ||
| Perfect 4th | | Perfect 4th | ||
| P4 | | P4 | ||
| Line 279: | Line 276: | ||
| 36 | | 36 | ||
| 514.3 | | 514.3 | ||
| 27/20 | | [[27/20]] | ||
| Up 4th | | Up 4th | ||
| ^4 | | ^4 | ||
| Line 286: | Line 283: | ||
| 37 | | 37 | ||
| 528.6 | | 528.6 | ||
| 19/14 | | [[19/14]] | ||
| Dup 4th | | Dup 4th | ||
| ^^4 | | ^^4 | ||
| Line 293: | Line 290: | ||
| 38 | | 38 | ||
| 542.9 | | 542.9 | ||
| 26/19 | | [[26/19]] | ||
| Trup 4th | | Trup 4th | ||
| ^^^4 | | ^^^4 | ||
| Line 300: | Line 297: | ||
| 39 | | 39 | ||
| 557.1 | | 557.1 | ||
| 18/13 | | [[18/13]] | ||
| Trudaug 4th | | Trudaug 4th | ||
| vvvA4 | | vvvA4 | ||
| Line 307: | Line 304: | ||
| 40 | | 40 | ||
| 571.4 | | 571.4 | ||
| 25/18, 32/23 | | [[25/18]], [[32/23]] | ||
| Dudaug 4th | | Dudaug 4th | ||
| vvA4 | | vvA4 | ||
| Line 314: | Line 311: | ||
| 41 | | 41 | ||
| 585.7 | | 585.7 | ||
| 7/5 | | [[7/5]] | ||
| Downaug 4th | | Downaug 4th | ||
| vA4 | | vA4 | ||
| Line 321: | Line 318: | ||
| 42 | | 42 | ||
| 600.0 | | 600.0 | ||
| 27/19, 38/27 | | [[27/19]], [[38/27]] | ||
| Aug 4th, Dim 5th | | Aug 4th, Dim 5th | ||
| A4, d5 | | A4, d5 | ||
| Line 328: | Line 325: | ||
| 43 | | 43 | ||
| 614.3 | | 614.3 | ||
| 10/7 | | [[10/7]] | ||
| Updim 5th | | Updim 5th | ||
| ^d5 | | ^d5 | ||
| Line 335: | Line 332: | ||
| 44 | | 44 | ||
| 628.6 | | 628.6 | ||
| 23/16, 36/25 | | [[23/16]], [[36/25]] | ||
| Dupdim 5th | | Dupdim 5th | ||
| ^^d5 | | ^^d5 | ||
| Line 342: | Line 339: | ||
| 45 | | 45 | ||
| 642.9 | | 642.9 | ||
| 13/9 | | [[13/9]] | ||
| Trupdim 5th | | Trupdim 5th | ||
| ^^^d5 | | ^^^d5 | ||
| Line 349: | Line 346: | ||
| 46 | | 46 | ||
| 657.1 | | 657.1 | ||
| 19/13 | | [[19/13]] | ||
| Trud 5th | | Trud 5th | ||
| vvv5 | | vvv5 | ||
| Line 356: | Line 353: | ||
| 47 | | 47 | ||
| 671.4 | | 671.4 | ||
| 28/19 | | [[28/19]] | ||
| Dud 5th | | Dud 5th | ||
| vv5 | | vv5 | ||
| Line 363: | Line 360: | ||
| 48 | | 48 | ||
| 685.7 | | 685.7 | ||
| 40/27 | | [[40/27]] | ||
| Down 5th | | Down 5th | ||
| v5 | | v5 | ||
| Line 370: | Line 367: | ||
| 49 | | 49 | ||
| 700.0 | | 700.0 | ||
| 3/2 | | [[3/2]] | ||
| Perfect 5th | | Perfect 5th | ||
| P5 | | P5 | ||
| Line 377: | Line 374: | ||
| 50 | | 50 | ||
| 714.3 | | 714.3 | ||
| 98/65 | | [[98/65]] | ||
| Up 5th | | Up 5th | ||
| ^5 | | ^5 | ||
| Line 384: | Line 381: | ||
| 51 | | 51 | ||
| 728.6 | | 728.6 | ||
| 32/21 | | [[32/21]] | ||
| Dup 5th | | Dup 5th | ||
| ^^5 | | ^^5 | ||
| Line 391: | Line 388: | ||
| 52 | | 52 | ||
| 742.9 | | 742.9 | ||
| 20/13, 23/15 | | [[20/13]], [[23/15]] | ||
| Trup 5th | | Trup 5th | ||
| ^^^5 | | ^^^5 | ||
| Line 398: | Line 395: | ||
| 53 | | 53 | ||
| 757.1 | | 757.1 | ||
| 65/42 | | [[65/42]] | ||
| Trudminor 6th | | Trudminor 6th | ||
| vvvm6 | | vvvm6 | ||
| Line 405: | Line 402: | ||
| 54 | | 54 | ||
| 771.4 | | 771.4 | ||
| 14/9, 25/16, 36/23 | | [[14/9]], [[25/16]], [[36/23]] | ||
| Dudminor 6th | | Dudminor 6th | ||
| vvm6 | | vvm6 | ||
| Line 412: | Line 409: | ||
| 55 | | 55 | ||
| 785.7 | | 785.7 | ||
| 30/19 | | [[30/19]] | ||
| Downminor 6th | | Downminor 6th | ||
| vm6 | | vm6 | ||
| Line 419: | Line 416: | ||
| 56 | | 56 | ||
| 800.0 | | 800.0 | ||
| 19/12 | | [[19/12]] | ||
| Minor 6th | | Minor 6th | ||
| m6 | | m6 | ||
| Line 426: | Line 423: | ||
| 57 | | 57 | ||
| 814.3 | | 814.3 | ||
| 8/5 | | [[8/5]] | ||
| Upminor 6th | | Upminor 6th | ||
| ^m6 | | ^m6 | ||
| Line 433: | Line 430: | ||
| 58 | | 58 | ||
| 828.6 | | 828.6 | ||
| 21/13 | | [[21/13]] | ||
| Dupminor 6th | | Dupminor 6th | ||
| ^^m6 | | ^^m6 | ||
| Line 440: | Line 437: | ||
| 59 | | 59 | ||
| 842.9 | | 842.9 | ||
| 13/8 | | [[13/8]] | ||
| Trupminor 6th | | Trupminor 6th | ||
| ^^^m6 | | ^^^m6 | ||
| Line 447: | Line 444: | ||
| 60 | | 60 | ||
| 857.1 | | 857.1 | ||
| 23/14, 64/39 | | [[23/14]], [[64/39]] | ||
| Trudmajor 6th | | Trudmajor 6th | ||
| vvvM6 | | vvvM6 | ||
| Line 454: | Line 451: | ||
| 61 | | 61 | ||
| 871.4 | | 871.4 | ||
| 38/23 | | [[38/23]] | ||
| Dudmajor 6th | | Dudmajor 6th | ||
| vvM6 | | vvM6 | ||
| Line 461: | Line 458: | ||
| 62 | | 62 | ||
| 885.7 | | 885.7 | ||
| 5/3 | | [[5/3]] | ||
| Downmajor 6th | | Downmajor 6th | ||
| vM6 | | vM6 | ||
| Line 468: | Line 465: | ||
| 63 | | 63 | ||
| 900.0 | | 900.0 | ||
| 32/19, 27/16, 42/25 | | [[32/19]], [[27/16]], [[42/25]] | ||
| Major 6th | | Major 6th | ||
| M6 | | M6 | ||
| Line 475: | Line 472: | ||
| 64 | | 64 | ||
| 914.3 | | 914.3 | ||
| 39/23, 76/45 | | [[39/23]], [[76/45]] | ||
| Upmajor 6th | | Upmajor 6th | ||
| ^M6 | | ^M6 | ||
| Line 482: | Line 479: | ||
| 65 | | 65 | ||
| 928.6 | | 928.6 | ||
| 12/7 | | [[12/7]] | ||
| Dupmajor 6th | | Dupmajor 6th | ||
| ^^M6 | | ^^M6 | ||
| Line 489: | Line 486: | ||
| 66 | | 66 | ||
| 942.9 | | 942.9 | ||
| 45/26 | | [[45/26]] | ||
| Trupmajor 6th | | Trupmajor 6th | ||
| ^^^M6 | | ^^^M6 | ||
| Line 496: | Line 493: | ||
| 67 | | 67 | ||
| 957.1 | | 957.1 | ||
| 26/15, 40/23 | | [[26/15]], [[40/23]] | ||
| Trudminor 7th | | Trudminor 7th | ||
| vvvm7 | | vvvm7 | ||
| Line 503: | Line 500: | ||
| 68 | | 68 | ||
| 971.4 | | 971.4 | ||
| 7/4 | | [[7/4]] | ||
| Dudminor 7th | | Dudminor 7th | ||
| vvm7 | | vvm7 | ||
| Line 510: | Line 507: | ||
| 69 | | 69 | ||
| 985.7 | | 985.7 | ||
| 23/13 | | [[23/13]] | ||
| Downminor 7th | | Downminor 7th | ||
| vm7 | | vm7 | ||
| Line 517: | Line 514: | ||
| 70 | | 70 | ||
| 1000.0 | | 1000.0 | ||
| 16/9 | | [[16/9]] | ||
| Minor 7th | | Minor 7th | ||
| m7 | | m7 | ||
| Line 524: | Line 521: | ||
| 71 | | 71 | ||
| 1014.3 | | 1014.3 | ||
| 9/5 | | [[9/5]] | ||
| Upminor 7th | | Upminor 7th | ||
| ^m7 | | ^m7 | ||
| Line 531: | Line 528: | ||
| 72 | | 72 | ||
| 1028.6 | | 1028.6 | ||
| 38/21 | | [[38/21]] | ||
| Dupminor 7th | | Dupminor 7th | ||
| ^^m7 | | ^^m7 | ||
| Line 538: | Line 535: | ||
| 73 | | 73 | ||
| 1042.9 | | 1042.9 | ||
| 42/23 | | [[42/23]] | ||
| Trupminor 7th | | Trupminor 7th | ||
| ^^^m7 | | ^^^m7 | ||
| Line 545: | Line 542: | ||
| 74 | | 74 | ||
| 1057.1 | | 1057.1 | ||
| 24/13 | | [[24/13]] | ||
| Trudmajor 7th | | Trudmajor 7th | ||
| vvvM7 | | vvvM7 | ||
| Line 552: | Line 549: | ||
| 75 | | 75 | ||
| 1071.4 | | 1071.4 | ||
| 13/7 | | [[13/7]] | ||
| Dudmajor 7th | | Dudmajor 7th | ||
| vvM7 | | vvM7 | ||
| Line 559: | Line 556: | ||
| 76 | | 76 | ||
| 1085.7 | | 1085.7 | ||
| 15/8, 28/15 | | [[15/8]], [[28/15]] | ||
| Downmajor 7th | | Downmajor 7th | ||
| vM7 | | vM7 | ||
| Line 566: | Line 563: | ||
| 77 | | 77 | ||
| 1100.0 | | 1100.0 | ||
| 36/19 | | [[36/19]] | ||
| Major 7th | | Major 7th | ||
| M7 | | M7 | ||
| Line 573: | Line 570: | ||
| 78 | | 78 | ||
| 1114.3 | | 1114.3 | ||
| 19/10, 40/21 | | [[19/10]], [[40/21]] | ||
| Upmajor 7th | | Upmajor 7th | ||
| ^M7 | | ^M7 | ||
| Line 580: | Line 577: | ||
| 79 | | 79 | ||
| 1128.6 | | 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 587: | Line 584: | ||
| 80 | | 80 | ||
| 1142.9 | | 1142.9 | ||
| ''52/27'' | | ''[[52/27]]'' | ||
| Trupmajor 7th | | Trupmajor 7th | ||
| ^^^M7 | | ^^^M7 | ||
| Line 594: | Line 591: | ||
| 81 | | 81 | ||
| 1157.1 | | 1157.1 | ||
| 35/18, 39/20, 96/49 | | [[35/18]], [[39/20]], [[96/49]] | ||
| Trud 8ve | | Trud 8ve | ||
| vvv8 | | vvv8 | ||
| Line 601: | Line 598: | ||
| 82 | | 82 | ||
| 1171.4 | | 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 608: | Line 605: | ||
| 83 | | 83 | ||
| 1185.7 | | 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 615: | Line 612: | ||
| 84 | | 84 | ||
| 1200.0 | | 1200.0 | ||
| 2/1 | | [[2/1]] | ||
| Perfect 8ve | | Perfect 8ve | ||
| P8 | | P8 | ||
| Line 623: | Line 620: | ||
== Notation == | == Notation == | ||
=== Ups and downs 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. | 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: | Alternatively, sharps and flats with arrows borrowed from [[Helmholtz–Ellis notation]] can be used: | ||
| Line 754: | Line 749: | ||
== Approximation to JI == | == 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). | |||
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}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 782: | Line 769: | ||
| 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 789: | 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 796: | 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 | | −0.013 | ||
| 0.754 | | 0.754 | ||
| Line 803: | 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 | | −0.225 | ||
| 1.003 | | 1.003 | ||
| Line 810: | 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 | | −0.292 | ||
| 0.928 | | 0.928 | ||
| Line 817: | 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 824: | 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 | ||
| Line 842: | Line 829: | ||
| 1 | | 1 | ||
| 19\84 | | 19\84 | ||
| 271. | | 271.4 | ||
| 7/6 | | 7/6 | ||
| [[Orwell]] (84e) / [[newspeak]] (84) | | [[Orwell]] (84e) / [[newspeak]] (84) | ||
| Line 848: | Line 835: | ||
| 1 | | 1 | ||
| 25\84 | | 25\84 | ||
| 357. | | 357.1 | ||
| 768/625 | | 768/625 | ||
| [[Dodifo]] | | [[Dodifo]] | ||
| Line 854: | Line 841: | ||
| 1 | | 1 | ||
| 27\84 | | 27\84 | ||
| 385. | | 385.7 | ||
| 5/4 | | 5/4 | ||
| [[Mutt]] | | [[Mutt]] | ||
| Line 860: | 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 872: | Line 859: | ||
| 2 | | 2 | ||
| 5\84 | | 5\84 | ||
| 71. | | 71.4 | ||
| 25/24 | | 25/24 | ||
| [[Narayana]] | | [[Narayana]] | ||
| Line 878: | Line 865: | ||
| 2 | | 2 | ||
| 11\84 | | 11\84 | ||
| 157. | | 157.1 | ||
| 35/32 | | 35/32 | ||
| [[Bison]] | | [[Bison]] | ||
| Line 884: | Line 871: | ||
| 2 | | 2 | ||
| 13\84 | | 13\84 | ||
| 185. | | 185.7 | ||
| 10/9 | | 10/9 | ||
| [[Secant]] | | [[Secant]] | ||
| Line 890: | 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/>* [[ | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
== Scales == | == Scales == | ||
| Line 932: | Line 931: | ||
=== Other === | === Other === | ||
* [[5- to 10-tone scales in 84edo]] | * [[5- to 10-tone scales in 84edo]] | ||
* [[Maeve Gutierrez|Gutierrez Moonglade scale]] | |||
== Instruments == | == Instruments == | ||
| Line 944: | 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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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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 |
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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)