53edo: Difference between revisions
Jump to navigation
Jump to search
m The ratio of supermajor chord was wrong! |
Removed pions column. These pages especially should reflect the knowledge and use of the community, not an individual. |
||
| Line 20: | Line 20: | ||
! | solfege | ! | solfege | ||
! | cents | ! | cents | ||
! | approximate ratios | ! | approximate ratios | ||
! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]] | ! colspan="3" | [[Ups_and_Downs_Notation|ups and downs]] [[Ups_and_Downs_Notation|notation]] | ||
| Line 28: | Line 27: | ||
| style="text-align:center;" | do | | style="text-align:center;" | do | ||
| style="text-align:center;" | 0.00 | | style="text-align:center;" | 0.00 | ||
| style="text-align:center;" | 1/1 | | style="text-align:center;" | 1/1 | ||
| style="text-align:center;" | P1 | | style="text-align:center;" | P1 | ||
| Line 38: | Line 36: | ||
| style="text-align:center;" | di | | style="text-align:center;" | di | ||
| style="text-align:center;" | 22.64 | | style="text-align:center;" | 22.64 | ||
| style="text-align:center;" | 81/80, 64/63, 50/49 | | style="text-align:center;" | 81/80, 64/63, 50/49 | ||
| style="text-align:center;" | ^1 | | style="text-align:center;" | ^1 | ||
| Line 48: | Line 45: | ||
| style="text-align:center;" | daw | | style="text-align:center;" | daw | ||
| style="text-align:center;" | 45.28 | | style="text-align:center;" | 45.28 | ||
| style="text-align:center;" | 49/48, 36/35, 33/32, 128/125 | | style="text-align:center;" | 49/48, 36/35, 33/32, 128/125 | ||
| style="text-align:center;" | ^^1, | | style="text-align:center;" | ^^1, | ||
| Line 64: | Line 60: | ||
| style="text-align:center;" | ro | | style="text-align:center;" | ro | ||
| style="text-align:center;" | 67.925 | | style="text-align:center;" | 67.925 | ||
| style="text-align:center;" | 27/26, 26/25, 25/24, 22/21 | | style="text-align:center;" | 27/26, 26/25, 25/24, 22/21 | ||
| style="text-align:center;" | vm2 | | style="text-align:center;" | vm2 | ||
| Line 74: | Line 69: | ||
| style="text-align:center;" | rih | | style="text-align:center;" | rih | ||
| style="text-align:center;" | 90.57 | | style="text-align:center;" | 90.57 | ||
| style="text-align:center;" | 21/20, 256/243 | | style="text-align:center;" | 21/20, 256/243 | ||
| style="text-align:center;" | m2 | | style="text-align:center;" | m2 | ||
| Line 84: | Line 78: | ||
| style="text-align:center;" | ra | | style="text-align:center;" | ra | ||
| style="text-align:center;" | 113.21 | | style="text-align:center;" | 113.21 | ||
| style="text-align:center;" | 16/15, 15/14 | | style="text-align:center;" | 16/15, 15/14 | ||
| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| Line 94: | Line 87: | ||
| style="text-align:center;" | ru | | style="text-align:center;" | ru | ||
| style="text-align:center;" | 135.85 | | style="text-align:center;" | 135.85 | ||
| style="text-align:center;" | 14/13, 13/12, 27/25 | | style="text-align:center;" | 14/13, 13/12, 27/25 | ||
| style="text-align:center;" | v~2 | | style="text-align:center;" | v~2 | ||
| Line 104: | Line 96: | ||
| style="text-align:center;" | ruh | | style="text-align:center;" | ruh | ||
| style="text-align:center;" | 158.49 | | style="text-align:center;" | 158.49 | ||
| style="text-align:center;" | 12/11, 11/10, 800/729 | | style="text-align:center;" | 12/11, 11/10, 800/729 | ||
| style="text-align:center;" | ^~2 | | style="text-align:center;" | ^~2 | ||
| Line 114: | Line 105: | ||
| style="text-align:center;" | reh | | style="text-align:center;" | reh | ||
| style="text-align:center;" | 181.13 | | style="text-align:center;" | 181.13 | ||
| style="text-align:center;" | 10/9 | | style="text-align:center;" | 10/9 | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| Line 124: | Line 114: | ||
| style="text-align:center;" | re | | style="text-align:center;" | re | ||
| style="text-align:center;" | 203.77 | | style="text-align:center;" | 203.77 | ||
| style="text-align:center;" | 9/8 | | style="text-align:center;" | 9/8 | ||
| style="text-align:center;" | M2 | | style="text-align:center;" | M2 | ||
| Line 134: | Line 123: | ||
| style="text-align:center;" | ri | | style="text-align:center;" | ri | ||
| style="text-align:center;" | 226.42 | | style="text-align:center;" | 226.42 | ||
| style="text-align:center;" | 8/7, 256/225 | | style="text-align:center;" | 8/7, 256/225 | ||
| style="text-align:center;" | ^M2 | | style="text-align:center;" | ^M2 | ||
| Line 144: | Line 132: | ||
| style="text-align:center;" | raw | | style="text-align:center;" | raw | ||
| style="text-align:center;" | 249.06 | | style="text-align:center;" | 249.06 | ||
| style="text-align:center;" | 15/13, 144/125 | | style="text-align:center;" | 15/13, 144/125 | ||
| style="text-align:center;" | ^^M2, | | style="text-align:center;" | ^^M2, | ||
| Line 160: | Line 147: | ||
| style="text-align:center;" | ma | | style="text-align:center;" | ma | ||
| style="text-align:center;" | 271.70 | | style="text-align:center;" | 271.70 | ||
| style="text-align:center;" | 7/6, 75/64 | | style="text-align:center;" | 7/6, 75/64 | ||
| style="text-align:center;" | vm3 | | style="text-align:center;" | vm3 | ||
| Line 170: | Line 156: | ||
| style="text-align:center;" | meh | | style="text-align:center;" | meh | ||
| style="text-align:center;" | 294.34 | | style="text-align:center;" | 294.34 | ||
| style="text-align:center;" | 13/11, 32/27 | | style="text-align:center;" | 13/11, 32/27 | ||
| style="text-align:center;" | m3 | | style="text-align:center;" | m3 | ||
| Line 180: | Line 165: | ||
| style="text-align:center;" | me | | style="text-align:center;" | me | ||
| style="text-align:center;" | 316.98 | | style="text-align:center;" | 316.98 | ||
| style="text-align:center;" | 6/5 | | style="text-align:center;" | 6/5 | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| Line 190: | Line 174: | ||
| style="text-align:center;" | mu | | style="text-align:center;" | mu | ||
| style="text-align:center;" | 339.62 | | style="text-align:center;" | 339.62 | ||
| style="text-align:center;" | 11/9, 243/200 | | style="text-align:center;" | 11/9, 243/200 | ||
| style="text-align:center;" | v~3 | | style="text-align:center;" | v~3 | ||
| Line 200: | Line 183: | ||
| style="text-align:center;" | muh | | style="text-align:center;" | muh | ||
| style="text-align:center;" | 362.26 | | style="text-align:center;" | 362.26 | ||
| style="text-align:center;" | 16/13, 100/81 | | style="text-align:center;" | 16/13, 100/81 | ||
| style="text-align:center;" | ^~3 | | style="text-align:center;" | ^~3 | ||
| Line 210: | Line 192: | ||
| style="text-align:center;" | mi | | style="text-align:center;" | mi | ||
| style="text-align:center;" | 384.91 | | style="text-align:center;" | 384.91 | ||
| style="text-align:center;" | 5/4 | | style="text-align:center;" | 5/4 | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| Line 220: | Line 201: | ||
| style="text-align:center;" | maa | | style="text-align:center;" | maa | ||
| style="text-align:center;" | 407.55 | | style="text-align:center;" | 407.55 | ||
| style="text-align:center;" | 81/64 | | style="text-align:center;" | 81/64 | ||
| style="text-align:center;" | M3 | | style="text-align:center;" | M3 | ||
| Line 230: | Line 210: | ||
| style="text-align:center;" | mo | | style="text-align:center;" | mo | ||
| style="text-align:center;" | 430.19 | | style="text-align:center;" | 430.19 | ||
| style="text-align:center;" | 9/7, 14/11 | | style="text-align:center;" | 9/7, 14/11 | ||
| style="text-align:center;" | ^M3 | | style="text-align:center;" | ^M3 | ||
| Line 240: | Line 219: | ||
| style="text-align:center;" | maw | | style="text-align:center;" | maw | ||
| style="text-align:center;" | 452.83 | | style="text-align:center;" | 452.83 | ||
| style="text-align:center;" | 13/10, 125/96 | | style="text-align:center;" | 13/10, 125/96 | ||
| style="text-align:center;" | ^^M3, | | style="text-align:center;" | ^^M3, | ||
| Line 256: | Line 234: | ||
| style="text-align:center;" | fe | | style="text-align:center;" | fe | ||
| style="text-align:center;" | 475.47 | | style="text-align:center;" | 475.47 | ||
| style="text-align:center;" | 21/16, 675/512, 320/243 | | style="text-align:center;" | 21/16, 675/512, 320/243 | ||
| style="text-align:center;" | v4 | | style="text-align:center;" | v4 | ||
| Line 266: | Line 243: | ||
| style="text-align:center;" | fa | | style="text-align:center;" | fa | ||
| style="text-align:center;" | 498.11 | | style="text-align:center;" | 498.11 | ||
| style="text-align:center;" | 4/3 | | style="text-align:center;" | 4/3 | ||
| style="text-align:center;" | P4 | | style="text-align:center;" | P4 | ||
| Line 276: | Line 252: | ||
| style="text-align:center;" | fih | | style="text-align:center;" | fih | ||
| style="text-align:center;" | 520.75 | | style="text-align:center;" | 520.75 | ||
| style="text-align:center;" | 27/20 | | style="text-align:center;" | 27/20 | ||
| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| Line 286: | Line 261: | ||
| style="text-align:center;" | fu | | style="text-align:center;" | fu | ||
| style="text-align:center;" | 543.40 | | style="text-align:center;" | 543.40 | ||
| style="text-align:center;" | 11/8, 15/11 | | style="text-align:center;" | 11/8, 15/11 | ||
| style="text-align:center;" | v~4 | | style="text-align:center;" | v~4 | ||
| Line 296: | Line 270: | ||
| style="text-align:center;" | fuh | | style="text-align:center;" | fuh | ||
| style="text-align:center;" | 566.04 | | style="text-align:center;" | 566.04 | ||
| style="text-align:center;" | 18/13 | | style="text-align:center;" | 18/13 | ||
| style="text-align:center;" | ^~4, | | style="text-align:center;" | ^~4, | ||
| Line 312: | Line 285: | ||
| style="text-align:center;" | fi | | style="text-align:center;" | fi | ||
| style="text-align:center;" | 588.68 | | style="text-align:center;" | 588.68 | ||
| style="text-align:center;" | 7/5, 45/32 | | style="text-align:center;" | 7/5, 45/32 | ||
| style="text-align:center;" | vA4, | | style="text-align:center;" | vA4, | ||
| Line 328: | Line 300: | ||
| style="text-align:center;" | se | | style="text-align:center;" | se | ||
| style="text-align:center;" | 611.32 | | style="text-align:center;" | 611.32 | ||
| style="text-align:center;" | 10/7, 64/45 | | style="text-align:center;" | 10/7, 64/45 | ||
| style="text-align:center;" | A4, | | style="text-align:center;" | A4, | ||
| Line 344: | Line 315: | ||
| style="text-align:center;" | suh | | style="text-align:center;" | suh | ||
| style="text-align:center;" | 633.96 | | style="text-align:center;" | 633.96 | ||
| style="text-align:center;" | 13/9 | | style="text-align:center;" | 13/9 | ||
| style="text-align:center;" | ^A4, | | style="text-align:center;" | ^A4, | ||
| Line 360: | Line 330: | ||
| style="text-align:center;" | su | | style="text-align:center;" | su | ||
| style="text-align:center;" | 656.60 | | style="text-align:center;" | 656.60 | ||
| style="text-align:center;" | 16/11, 22/15 | | style="text-align:center;" | 16/11, 22/15 | ||
| style="text-align:center;" | ^~5 | | style="text-align:center;" | ^~5 | ||
| Line 370: | Line 339: | ||
| style="text-align:center;" | sih | | style="text-align:center;" | sih | ||
| style="text-align:center;" | 679.25 | | style="text-align:center;" | 679.25 | ||
| style="text-align:center;" | 40/27 | | style="text-align:center;" | 40/27 | ||
| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| Line 380: | Line 348: | ||
| style="text-align:center;" | sol | | style="text-align:center;" | sol | ||
| style="text-align:center;" | 701.89 | | style="text-align:center;" | 701.89 | ||
| style="text-align:center;" | 3/2 | | style="text-align:center;" | 3/2 | ||
| style="text-align:center;" | P5 | | style="text-align:center;" | P5 | ||
| Line 390: | Line 357: | ||
| style="text-align:center;" | si | | style="text-align:center;" | si | ||
| style="text-align:center;" | 724.53 | | style="text-align:center;" | 724.53 | ||
| style="text-align:center;" | 32/21, 243/160, 1024/675 | | style="text-align:center;" | 32/21, 243/160, 1024/675 | ||
| style="text-align:center;" | ^5 | | style="text-align:center;" | ^5 | ||
| Line 400: | Line 366: | ||
| style="text-align:center;" | saw | | style="text-align:center;" | saw | ||
| style="text-align:center;" | 747.17 | | style="text-align:center;" | 747.17 | ||
| style="text-align:center;" | 20/13, 192/125 | | style="text-align:center;" | 20/13, 192/125 | ||
| style="text-align:center;" | ^^5, | | style="text-align:center;" | ^^5, | ||
| Line 416: | Line 381: | ||
| style="text-align:center;" | lo | | style="text-align:center;" | lo | ||
| style="text-align:center;" | 769.81 | | style="text-align:center;" | 769.81 | ||
| style="text-align:center;" | 14/9, 25/16, 11/7 | | style="text-align:center;" | 14/9, 25/16, 11/7 | ||
| style="text-align:center;" | vm6 | | style="text-align:center;" | vm6 | ||
| Line 426: | Line 390: | ||
| style="text-align:center;" | leh | | style="text-align:center;" | leh | ||
| style="text-align:center;" | 792.45 | | style="text-align:center;" | 792.45 | ||
| style="text-align:center;" | 128/81 | | style="text-align:center;" | 128/81 | ||
| style="text-align:center;" | m6 | | style="text-align:center;" | m6 | ||
| Line 436: | Line 399: | ||
| style="text-align:center;" | le | | style="text-align:center;" | le | ||
| style="text-align:center;" | 815.09 | | style="text-align:center;" | 815.09 | ||
| style="text-align:center;" | 8/5 | | style="text-align:center;" | 8/5 | ||
| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| Line 446: | Line 408: | ||
| style="text-align:center;" | lu | | style="text-align:center;" | lu | ||
| style="text-align:center;" | 837.74 | | style="text-align:center;" | 837.74 | ||
| style="text-align:center;" | 13/8, 81/50 | | style="text-align:center;" | 13/8, 81/50 | ||
| style="text-align:center;" | v~6 | | style="text-align:center;" | v~6 | ||
| Line 456: | Line 417: | ||
| style="text-align:center;" | luh | | style="text-align:center;" | luh | ||
| style="text-align:center;" | 860.38 | | style="text-align:center;" | 860.38 | ||
| style="text-align:center;" | 18/11, 400/243 | | style="text-align:center;" | 18/11, 400/243 | ||
| style="text-align:center;" | ^~6 | | style="text-align:center;" | ^~6 | ||
| Line 466: | Line 426: | ||
| style="text-align:center;" | la | | style="text-align:center;" | la | ||
| style="text-align:center;" | 883.02 | | style="text-align:center;" | 883.02 | ||
| style="text-align:center;" | 5/3 | | style="text-align:center;" | 5/3 | ||
| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| Line 476: | Line 435: | ||
| style="text-align:center;" | laa | | style="text-align:center;" | laa | ||
| style="text-align:center;" | 905.66 | | style="text-align:center;" | 905.66 | ||
| style="text-align:center;" | 22/13, 27/16 | | style="text-align:center;" | 22/13, 27/16 | ||
| style="text-align:center;" | M6 | | style="text-align:center;" | M6 | ||
| Line 486: | Line 444: | ||
| style="text-align:center;" | lo | | style="text-align:center;" | lo | ||
| style="text-align:center;" | 928.30 | | style="text-align:center;" | 928.30 | ||
| style="text-align:center;" | 12/7 | | style="text-align:center;" | 12/7 | ||
| style="text-align:center;" | ^M6 | | style="text-align:center;" | ^M6 | ||
| Line 496: | Line 453: | ||
| style="text-align:center;" | law | | style="text-align:center;" | law | ||
| style="text-align:center;" | 950.94 | | style="text-align:center;" | 950.94 | ||
| style="text-align:center;" | 26/15, 125/72 | | style="text-align:center;" | 26/15, 125/72 | ||
| style="text-align:center;" | ^^M6 | | style="text-align:center;" | ^^M6 | ||
| Line 512: | Line 468: | ||
| style="text-align:center;" | ta | | style="text-align:center;" | ta | ||
| style="text-align:center;" | 973.58 | | style="text-align:center;" | 973.58 | ||
| style="text-align:center;" | 7/4 | | style="text-align:center;" | 7/4 | ||
| style="text-align:center;" | vm7 | | style="text-align:center;" | vm7 | ||
| Line 522: | Line 477: | ||
| style="text-align:center;" | teh | | style="text-align:center;" | teh | ||
| style="text-align:center;" | 996.23 | | style="text-align:center;" | 996.23 | ||
| style="text-align:center;" | 16/9 | | style="text-align:center;" | 16/9 | ||
| style="text-align:center;" | m7 | | style="text-align:center;" | m7 | ||
| Line 532: | Line 486: | ||
| style="text-align:center;" | te | | style="text-align:center;" | te | ||
| style="text-align:center;" | 1018.87 | | style="text-align:center;" | 1018.87 | ||
| style="text-align:center;" | 9/5 | | style="text-align:center;" | 9/5 | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| Line 542: | Line 495: | ||
| style="text-align:center;" | tu | | style="text-align:center;" | tu | ||
| style="text-align:center;" | 1041.51 | | style="text-align:center;" | 1041.51 | ||
| style="text-align:center;" | 11/6, 20/11, 729/400 | | style="text-align:center;" | 11/6, 20/11, 729/400 | ||
| style="text-align:center;" | v~7 | | style="text-align:center;" | v~7 | ||
| Line 552: | Line 504: | ||
| style="text-align:center;" | tuh | | style="text-align:center;" | tuh | ||
| style="text-align:center;" | 1064.15 | | style="text-align:center;" | 1064.15 | ||
| style="text-align:center;" | 13/7, 24/13, 50/27 | | style="text-align:center;" | 13/7, 24/13, 50/27 | ||
| style="text-align:center;" | ^~7 | | style="text-align:center;" | ^~7 | ||
| Line 562: | Line 513: | ||
| style="text-align:center;" | ti | | style="text-align:center;" | ti | ||
| style="text-align:center;" | 1086.79 | | style="text-align:center;" | 1086.79 | ||
| style="text-align:center;" | 15/8 | | style="text-align:center;" | 15/8 | ||
| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| Line 572: | Line 522: | ||
| style="text-align:center;" | tih | | style="text-align:center;" | tih | ||
| style="text-align:center;" | 1109.43 | | style="text-align:center;" | 1109.43 | ||
| style="text-align:center;" | 40/21, 243/128 | | style="text-align:center;" | 40/21, 243/128 | ||
| style="text-align:center;" | M7 | | style="text-align:center;" | M7 | ||
| Line 582: | Line 531: | ||
| style="text-align:center;" | to | | style="text-align:center;" | to | ||
| style="text-align:center;" | 1132.08 | | style="text-align:center;" | 1132.08 | ||
| style="text-align:center;" | 48/25, 27/14 | | style="text-align:center;" | 48/25, 27/14 | ||
| style="text-align:center;" | ^M7 | | style="text-align:center;" | ^M7 | ||
| Line 592: | Line 540: | ||
| style="text-align:center;" | taw | | style="text-align:center;" | taw | ||
| style="text-align:center;" | 1154.72 | | style="text-align:center;" | 1154.72 | ||
| style="text-align:center;" | 125/64 | | style="text-align:center;" | 125/64 | ||
| style="text-align:center;" | ^^M7, | | style="text-align:center;" | ^^M7, | ||
| Line 608: | Line 555: | ||
| style="text-align:center;" | da | | style="text-align:center;" | da | ||
| style="text-align:center;" | 1177.36 | | style="text-align:center;" | 1177.36 | ||
| style="text-align:center;" | 160/81 | | style="text-align:center;" | 160/81 | ||
| style="text-align:center;" | v8 | | style="text-align:center;" | v8 | ||
| Line 618: | Line 564: | ||
| style="text-align:center;" | do | | style="text-align:center;" | do | ||
| style="text-align:center;" | 1200 | | style="text-align:center;" | 1200 | ||
| style="text-align:center;" | 2/1 | | style="text-align:center;" | 2/1 | ||
| style="text-align:center;" | P8 | | style="text-align:center;" | P8 | ||
Revision as of 16:04, 3 May 2020
Theory
The famous 53 equal division divides the octave into 53 equal comma-sized parts of 22.642 cents each. It is notable as a 5-limit system, a fact apparently first noted by Isaac Newton, tempering out the schisma, 32805/32768, the kleisma, 15625/15552, the amity comma, 1600000/1594323 and the semicomma, 2109375/2097152. In the 7-limit it tempers out 225/224, 1728/1715 and 3125/3087, the marvel comma, the gariboh, and the orwell comma. In the 11-limit, it tempers out 99/98 and 121/120, and is the optimal patent val for Big Brother temperament, which tempers out both, as well as 11-limit orwell temperament, which also tempers out the 11-limit comma 176/175. In the 13-limit, it tempers out 169/168 and 245/243, and gives the optimal patent val for athene temperament. It is the eighth zeta integral edo and the 16th prime edo, following 47edo and coming before 59edo.
53EDO has also found a certain dissemination as an EDO tuning for Arabic/Turkish/Persian music.
It can also be treated as a no-elevens, no-seventeens tuning, on which it is consistent all the way up to the 21-limit.
Intervals
| degree | solfege | cents | approximate ratios | ups and downs notation | generator for | ||
|---|---|---|---|---|---|---|---|
| 0 | do | 0.00 | 1/1 | P1 | unison | D | |
| 1 | di | 22.64 | 81/80, 64/63, 50/49 | ^1 | up unison | ^D | |
| 2 | daw | 45.28 | 49/48, 36/35, 33/32, 128/125 | ^^1,
vvm2 |
double-up unison,
double-down minor 2nd |
^^D,
vvEb |
Quartonic |
| 3 | ro | 67.925 | 27/26, 26/25, 25/24, 22/21 | vm2 | downminor 2nd | vEb | |
| 4 | rih | 90.57 | 21/20, 256/243 | m2 | minor 2nd | Eb | |
| 5 | ra | 113.21 | 16/15, 15/14 | ^m2 | upminor 2nd | ^Eb | |
| 6 | ru | 135.85 | 14/13, 13/12, 27/25 | v~2 | downmid 2nd | ^^Eb | |
| 7 | ruh | 158.49 | 12/11, 11/10, 800/729 | ^~2 | upmid 2nd | vvE | Hemikleismic |
| 8 | reh | 181.13 | 10/9 | vM2 | downmajor 2nd | vE | |
| 9 | re | 203.77 | 9/8 | M2 | major 2nd | E | |
| 10 | ri | 226.42 | 8/7, 256/225 | ^M2 | upmajor 2nd | ^E | |
| 11 | raw | 249.06 | 15/13, 144/125 | ^^M2,
vvm3 |
double-up major 2nd,
double-down minor 3rd |
^^E,
vvF |
Hemischis |
| 12 | ma | 271.70 | 7/6, 75/64 | vm3 | downminor 3rd | vF | Orwell |
| 13 | meh | 294.34 | 13/11, 32/27 | m3 | minor 3rd | F | |
| 14 | me | 316.98 | 6/5 | ^m3 | upminor 3rd | ^F | Hanson/Catakleismic |
| 15 | mu | 339.62 | 11/9, 243/200 | v~3 | downmid 3rd | ^^F | Amity/Hitchcock |
| 16 | muh | 362.26 | 16/13, 100/81 | ^~3 | upmid 3rd | vvF# | |
| 17 | mi | 384.91 | 5/4 | vM3 | downmajor 3rd | vF# | |
| 18 | maa | 407.55 | 81/64 | M3 | major 3rd | F# | |
| 19 | mo | 430.19 | 9/7, 14/11 | ^M3 | upmajor 3rd | ^F# | Hamity |
| 20 | maw | 452.83 | 13/10, 125/96 | ^^M3,
vv4 |
double-up major 3rd,
double-down 4th |
^^F#,
vvG |
|
| 21 | fe | 475.47 | 21/16, 675/512, 320/243 | v4 | down 4th | vG | Vulture/Buzzard |
| 22 | fa | 498.11 | 4/3 | P4 | perfect 4th | G | |
| 23 | fih | 520.75 | 27/20 | ^4 | up 4th | ^G | |
| 24 | fu | 543.40 | 11/8, 15/11 | v~4 | downmid 4th | ^^G | |
| 25 | fuh | 566.04 | 18/13 | ^~4,
vd5 |
upmid 4th,
downdim 5th |
vvG#,
vAb |
Tricot |
| 26 | fi | 588.68 | 7/5, 45/32 | vA4,
d5 |
downaug 4th,
dim 5th |
vG#,
Ab |
|
| 27 | se | 611.32 | 10/7, 64/45 | A4,
^d5 |
aug 4th,
updim 5th |
G#,
^Ab |
|
| 28 | suh | 633.96 | 13/9 | ^A4,
v~5 |
upaug 4th,
downmid 5th |
^G#,
^^Ab |
|
| 29 | su | 656.60 | 16/11, 22/15 | ^~5 | upmid 5th | vvA | |
| 30 | sih | 679.25 | 40/27 | v5 | down 5th | vA | |
| 31 | sol | 701.89 | 3/2 | P5 | perfect 5th | A | Helmholtz/Garibaldi |
| 32 | si | 724.53 | 32/21, 243/160, 1024/675 | ^5 | up 5th | ^A | |
| 33 | saw | 747.17 | 20/13, 192/125 | ^^5,
vvm6 |
double-up 5th,
double-down minor 6th |
^^A,
vvBb |
|
| 34 | lo | 769.81 | 14/9, 25/16, 11/7 | vm6 | downminor 6th | vBb | |
| 35 | leh | 792.45 | 128/81 | m6 | minor 6th | Bb | |
| 36 | le | 815.09 | 8/5 | ^m6 | upminor 6th | ^Bb | |
| 37 | lu | 837.74 | 13/8, 81/50 | v~6 | downmid 6th | ^^Bb | |
| 38 | luh | 860.38 | 18/11, 400/243 | ^~6 | upmid 6th | vvB | |
| 39 | la | 883.02 | 5/3 | vM6 | downmajor 6th | vB | |
| 40 | laa | 905.66 | 22/13, 27/16 | M6 | major 6th | B | |
| 41 | lo | 928.30 | 12/7 | ^M6 | upmajor 6th | ^B | |
| 42 | law | 950.94 | 26/15, 125/72 | ^^M6
vvm7 |
double-up major 6th,
double-down minor 7th |
^^B,
vvC |
|
| 43 | ta | 973.58 | 7/4 | vm7 | downminor 7th | vC | |
| 44 | teh | 996.23 | 16/9 | m7 | minor 7th | C | |
| 45 | te | 1018.87 | 9/5 | ^m7 | upminor 7th | ^C | |
| 46 | tu | 1041.51 | 11/6, 20/11, 729/400 | v~7 | downmid 7th | ^^C | |
| 47 | tuh | 1064.15 | 13/7, 24/13, 50/27 | ^~7 | upmid 7th | vvC# | |
| 48 | ti | 1086.79 | 15/8 | vM7 | downmajor 7th | vC# | |
| 49 | tih | 1109.43 | 40/21, 243/128 | M7 | major 7th | C# | |
| 50 | to | 1132.08 | 48/25, 27/14 | ^M7 | upmajor 7th | ^C# | |
| 51 | taw | 1154.72 | 125/64 | ^^M7,
vv8 |
double-up major 7th,
double-down 8ve |
^^C#,
vvD |
|
| 52 | da | 1177.36 | 160/81 | v8 | down 8ve | vD | |
| 53 | do | 1200 | 2/1 | P8 | perfect 8ve | D | |
Combining ups and downs notation with color notation, qualities can be loosely associated with colors:
| quality | color | monzo format | examples |
|---|---|---|---|
| downminor | zo | (a, b, 0, 1) | 7/6, 7/4 |
| minor | fourthward wa | (a, b) with b < -1 | 32/27, 16/9 |
| upminor | gu | (a, b, -1) | 6/5, 9/5 |
| downmid | ilo | (a, b, 0, 0, 1) | 11/9, 11/6 |
| upmid | lu | (a, b, 0, 0, -1) | 12/11, 18/11 |
| downmajor | yo | (a, b, 1) | 5/4, 5/3 |
| major | fifthward wa | (a, b) with b > 1 | 9/8, 27/16 |
| upmajor | ru | (a, b, 0, -1) | 9/7, 12/7 |
All 53edo chords can be named using ups and downs. An up, down or mid after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13). Alterations are always enclosed in parentheses, additions never are.
Here are the zo, gu, ilo, lu, yo and ru triads:
| color of the 3rd | JI chord | notes as edosteps | notes of C chord | written name | spoken name |
|---|---|---|---|---|---|
| zo | 6:7:9 | 0-12-31 | C vEb G | Cvm | C downminor |
| gu | 10:12:15 | 0-14-31 | C ^Eb G | C^m | C upminor |
| ilo | 18:22:27 | 0-15-31 | C ^^Eb G | Cv~ | C downmid |
| lu | 22:27:33 | 0-16-31 | C vvE G | C^~ | C upmid |
| yo | 4:5:6 | 0-17-31 | C vE G | Cv | C downmajor or C down |
| ru | 14:18:21 | 0-19-31 | C ^E G | C^ | C upmajor or C up |
For a more complete list, see Ups and Downs Notation - Chords and Chord Progressions.
Relationship to 12-edo
Whereas 12-edo has a circle of twelve 5ths, 53-edo has a spiral of twelve 5ths (since 31\53 is on the 7\12 kite in the scale tree). This shows 53-edo in a 12-edo-friendly format. Excellent for introducing 53-edo to musicians unfamiliar with microtonal music. The two innermost and two outermost intervals on the spiral are duplicates.
Just Approximation
53edo provides excellent approximations for the classic 5-limit just chords and scales, such as the Ptolemy-Zarlino "just major" scale.
| interval | ratio | size | difference |
|---|---|---|---|
| perfect fifth | 3/2 | 31 | −0.07 cents |
| major third | 5/4 | 17 | −1.40 cents |
| minor third | 6/5 | 14 | +1.34 cents |
| major tone | 9/8 | 9 | −0.14 cents |
| minor tone | 10/9 | 8 | −1.27 cents |
| diat. semitone | 16/15 | 5 | +1.48 cents |
One notable property of 53EDO is that it offers good approximations for both pure and pythagorean major thirds.
The perfect fifth is almost perfectly equal to the just interval 3/2, with only a 0.07 cent difference! 53EDO is practically equal to an extended Pythagorean. The 14- and 17- degree intervals are also very close to 6/5 and 5/4 respectively, and so 5-limit tuning can also be closely approximated. In addition, the 43-degree interval is only 4.8 cents away from the just ratio 7/4, so 53EDO can also be used for 7-limit harmony, tempering out the septimal kleisma, 225/224.
Selected just intervals by error
| prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | |
|---|---|---|---|---|---|---|
| error | 0.0¢ | -0.07¢ | -1.4¢ | +4.8¢ | -7.9¢ | -2.8¢ |
| fifthspan | 0 | +1 | -8 | -14 | +23 | +20 |
The following table shows how some prominent just intervals are represented in 53edo (ordered by absolute error). Octave-reduced prime harmonics are bolded.
| Interval, complement | Error (abs., in cents) |
| 4/3, 3/2 | 0.068 |
| 9/8, 16/9 | 0.136 |
| 10/9, 9/5 | 1.272 |
| 15/13, 26/15 | 1.316 |
| 6/5, 5/3 | 1.340 |
| 13/10, 20/13 | 1.384 |
| 5/4, 8/5 | 1.408 |
| 16/15, 15/8 | 1.476 |
| 18/13, 13/9 | 2.655 |
| 13/12, 24/13 | 2.724 |
| 16/13, 13/8 | 2.792 |
| 8/7, 7/4 | 4.759 |
| 7/6, 12/7 | 4.827 |
| 9/7, 14/9 | 4.895 |
| 13/11, 22/13 | 5.130 |
| 7/5, 10/7 | 6.167 |
| 15/14, 28/15 | 6.235 |
| 15/11, 22/15 | 6.445 |
| 11/10, 20/11 | 6.514 |
| 14/13, 13/7 | 7.551 |
| 11/9, 18/11 | 7.785 |
| 12/11, 11/6 | 7.854 |
| 11/8, 16/11 | 7.922 |
| 14/11, 11/7 | 9.961 |
Linear temperaments
List of edo-distinct 53et rank two temperaments
Music
Bach WTC1 Prelude 1 in 53 by Bach and Mykhaylo Khramov
Bach WTC1 Fugue 1 in 53 by Bach and Mykhaylo Khramov
Whisper Song in 53EDO play by Prent Rodgers
Trio in Orwell play by Gene Ward Smith
Desert Prayer by Aaron Krister Johnson
Whisper Song in 53 EDO by Prent Rodgers
Elf Dine on Ho Ho play and Spun play by Andrew Heathwaite
The Fallen of Kleismic15play by Chris Vaisvil
mothers by Cam Taylor