43edo: Difference between revisions
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=43 tone equal temperament= | =43 tone equal temperament= | ||
'''43edo''' divides the [[Octave|octave]] into 43 [[Equal|equal]] parts of 27.907 [[cent|cent]]s each. It is strongly associated with meantone temperament, particularly 1/5 comma meantone, being a good tuning system in the 5, 7, 11, and 13-limit. The version of 11-limit meantone is the one tempering out 99/98, 176/175 and 441/440 sometimes called Huygens. 43-equal has the first good 13-limit meantone available as an equal division of the octave. The baroque, | '''43edo''' divides the [[Octave|octave]] into 43 [[Equal|equal]] parts of 27.907 [[cent|cent]]s each. It is strongly associated with meantone temperament, particularly 1/5 comma meantone, being a good tuning system in the 5, 7, 11, and 13-limit. The version of 11-limit meantone is the one tempering out 99/98, 176/175 and 441/440 sometimes called Huygens. 43-equal has the first good 13-limit meantone available as an equal division of the octave. The baroque, French, ironically hearing and speech impaired acoustician [http://en.wikipedia.org/wiki/Joseph_Sauveur Joseph Sauveur] based his system on 43 equal tones to the octave, calling them "merides". Further information: http://tonalsoft.com/enc/m/meride.aspx | ||
In the 13-limit, we get two versions of meantone equivalent in 43et, one, [[Meantone_family#Septimal meantone-Unidecimal meantone aka Huygens-Meridetone|meridetone]], tempering out 78/77, the other, [[Meantone_family#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone|grosstone]], 144/143. Meridetone has generator mapping <0 1 4 10 18 27|, and grosstone <0 1 4 10 18 -16|; 43 supplies the optimal patent val for meridetone. | In the 13-limit, we get two versions of meantone equivalent in 43et, one, [[Meantone_family#Septimal meantone-Unidecimal meantone aka Huygens-Meridetone|meridetone]], tempering out 78/77, the other, [[Meantone_family#Septimal meantone-Unidecimal meantone aka Huygens-Grosstone|grosstone]], 144/143. Meridetone has generator mapping <0 1 4 10 18 27|, and grosstone <0 1 4 10 18 -16|; 43 supplies the optimal patent val for meridetone. | ||
| Line 17: | Line 17: | ||
| style="text-align:center;" | Degrees | | style="text-align:center;" | Degrees | ||
| style="text-align:center;" | Cents value | | style="text-align:center;" | Cents value | ||
|pions | |||
|7mus | |||
| | Approximate 13-limit Ratios | | | Approximate 13-limit Ratios | ||
| colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]] | | colspan="3" style="text-align:center;" | [[Ups_and_Downs_Notation|ups and downs notation]] | ||
|- | |- | ||
| style="text-align:center;" | 0 | | style="text-align:center;" | 0 | ||
| style="text-align:center;" | 0 | | colspan="3" style="text-align:center;" | 0 | ||
| | 1/1 | | | 1/1 | ||
| style="text-align:center;" | P1 | | style="text-align:center;" | P1 | ||
| Line 29: | Line 31: | ||
| style="text-align:center;" | 1 | | style="text-align:center;" | 1 | ||
| style="text-align:center;" | 27.907 | | style="text-align:center;" | 27.907 | ||
|29.581 | |||
|35.721 (23.B89<sub>16</sub>) | |||
| | | | | | ||
| style="text-align:center;" | ^1, d2 | | style="text-align:center;" | ^1, d2 | ||
| Line 36: | Line 40: | ||
| style="text-align:center;" | 2 | | style="text-align:center;" | 2 | ||
| style="text-align:center;" | 55.814 | | style="text-align:center;" | 55.814 | ||
|59.163 | |||
|71.442 (47.612<sub>16</sub>) | |||
| | | | | | ||
| style="text-align:center;" | vA1, ^d2 | | style="text-align:center;" | vA1, ^d2 | ||
| Line 43: | Line 49: | ||
| style="text-align:center;" | 3 | | style="text-align:center;" | 3 | ||
| style="text-align:center;" | 83.721 | | style="text-align:center;" | 83.721 | ||
|88.744 | |||
|107.163 (6B.29B<sub>16</sub>) | |||
| | | | | | ||
| style="text-align:center;" | vm2 | | style="text-align:center;" | vm2 | ||
| Line 50: | Line 58: | ||
| style="text-align:center;" | 4 | | style="text-align:center;" | 4 | ||
| style="text-align:center;" | 111.628 | | style="text-align:center;" | 111.628 | ||
|118.326 | |||
|142.884 (8E.E24<sub>16</sub>) | |||
| | 17/16, 16/15, 15/14 | | | 17/16, 16/15, 15/14 | ||
| style="text-align:center;" | m2 | | style="text-align:center;" | m2 | ||
| Line 57: | Line 67: | ||
| style="text-align:center;" | 5 | | style="text-align:center;" | 5 | ||
| style="text-align:center;" | 139.535 | | style="text-align:center;" | 139.535 | ||
|147.907 | |||
|178.605 (B2.9AD<sub>16</sub>) | |||
| | 12/11, 13/12, 14/13 | | | 12/11, 13/12, 14/13 | ||
| style="text-align:center;" | ^m2 | | style="text-align:center;" | ^m2 | ||
| Line 64: | Line 76: | ||
| style="text-align:center;" | 6 | | style="text-align:center;" | 6 | ||
| style="text-align:center;" | 167.442 | | style="text-align:center;" | 167.442 | ||
|177.488 | |||
|214.326 (D6.536<sub>16</sub>) | |||
| | 11/10 | | | 11/10 | ||
| style="text-align:center;" | vM2 | | style="text-align:center;" | vM2 | ||
| Line 71: | Line 85: | ||
| style="text-align:center;" | 7 | | style="text-align:center;" | 7 | ||
| style="text-align:center;" | 195.349 | | style="text-align:center;" | 195.349 | ||
|207.07 | |||
|250.0465 (FA.0BE8<sub>16</sub>) | |||
| | 9/8, 10/9 | | | 9/8, 10/9 | ||
| style="text-align:center;" | M2 | | style="text-align:center;" | M2 | ||
| Line 78: | Line 94: | ||
| style="text-align:center;" | 8 | | style="text-align:center;" | 8 | ||
| style="text-align:center;" | 223.256 | | style="text-align:center;" | 223.256 | ||
|236.651 | |||
|285.767 (11D.C47<sub>16</sub>) | |||
| | 8/7 | | | 8/7 | ||
| style="text-align:center;" | ^M2 | | style="text-align:center;" | ^M2 | ||
| Line 85: | Line 103: | ||
| style="text-align:center;" | 9 | | style="text-align:center;" | 9 | ||
| style="text-align:center;" | 251.163 | | style="text-align:center;" | 251.163 | ||
|266.233 | |||
|321.488 (141.7D08<sub>16</sub>) | |||
| | 15/13 | | | 15/13 | ||
| style="text-align:center;" | vA2, ^d3 | | style="text-align:center;" | vA2, ^d3 | ||
| Line 92: | Line 112: | ||
| style="text-align:center;" | 10 | | style="text-align:center;" | 10 | ||
| style="text-align:center;" | 279.07 | | style="text-align:center;" | 279.07 | ||
|295.814 | |||
|357.209 (165.359<sub>16</sub>) | |||
| | 7/6, 13/11 | | | 7/6, 13/11 | ||
| style="text-align:center;" | vm3 | | style="text-align:center;" | vm3 | ||
| Line 99: | Line 121: | ||
| style="text-align:center;" | 11 | | style="text-align:center;" | 11 | ||
| style="text-align:center;" | 306.977 | | style="text-align:center;" | 306.977 | ||
|325.395 | |||
|392.93 (188.EE2<sub>16</sub>) | |||
| | 6/5 | | | 6/5 | ||
| style="text-align:center;" | m3 | | style="text-align:center;" | m3 | ||
| Line 106: | Line 130: | ||
| style="text-align:center;" | 12 | | style="text-align:center;" | 12 | ||
| style="text-align:center;" | 334.884 | | style="text-align:center;" | 334.884 | ||
|354.977 | |||
|428.651 (1AC.A6B<sub>16</sub>) | |||
| | 17/14, 39/32 | | | 17/14, 39/32 | ||
| style="text-align:center;" | ^m3 | | style="text-align:center;" | ^m3 | ||
| Line 113: | Line 139: | ||
| style="text-align:center;" | 13 | | style="text-align:center;" | 13 | ||
| style="text-align:center;" | 362.791 | | style="text-align:center;" | 362.791 | ||
|384.558 | |||
|464.372 (1D0.6F4<sub>16</sub>) | |||
| | 11/9, 16/13 | | | 11/9, 16/13 | ||
| style="text-align:center;" | vM3 | | style="text-align:center;" | vM3 | ||
| Line 120: | Line 148: | ||
| style="text-align:center;" | 14 | | style="text-align:center;" | 14 | ||
| style="text-align:center;" | 390.698 | | style="text-align:center;" | 390.698 | ||
|414.1395 | |||
|500.093 (1F4.2FD<sub>16</sub>) | |||
| | 5/4 | | | 5/4 | ||
| style="text-align:center;" | M3 | | style="text-align:center;" | M3 | ||
| Line 127: | Line 157: | ||
| style="text-align:center;" | 15 | | style="text-align:center;" | 15 | ||
| style="text-align:center;" | 418.605 | | style="text-align:center;" | 418.605 | ||
|443.721 | |||
|535.814 (217.D06<sub>16</sub>) | |||
| | 9/7, 14/11 | | | 9/7, 14/11 | ||
| style="text-align:center;" | ^M3 | | style="text-align:center;" | ^M3 | ||
| Line 134: | Line 166: | ||
| style="text-align:center;" | 16 | | style="text-align:center;" | 16 | ||
| style="text-align:center;" | 446.512 | | style="text-align:center;" | 446.512 | ||
|473.302 | |||
|571.535 (23B.88F<sub>16</sub>) | |||
| | 13/10 | | | 13/10 | ||
| style="text-align:center;" | vA3, ^d4 | | style="text-align:center;" | vA3, ^d4 | ||
| Line 141: | Line 175: | ||
| style="text-align:center;" | 17 | | style="text-align:center;" | 17 | ||
| style="text-align:center;" | 474.419 | | style="text-align:center;" | 474.419 | ||
|502.884 | |||
|607.256 (25F.418<sub>16</sub>) | |||
| | 21/16 | | | 21/16 | ||
| style="text-align:center;" | v4 | | style="text-align:center;" | v4 | ||
| Line 148: | Line 184: | ||
| style="text-align:center;" | 18 | | style="text-align:center;" | 18 | ||
| style="text-align:center;" | 502.326 | | style="text-align:center;" | 502.326 | ||
|532.465 | |||
|642.977 (282.FA1<sub>16</sub>) | |||
| | 4/3 | | | 4/3 | ||
| style="text-align:center;" | P4 | | style="text-align:center;" | P4 | ||
| Line 155: | Line 193: | ||
| style="text-align:center;" | 19 | | style="text-align:center;" | 19 | ||
| style="text-align:center;" | 530.233 | | style="text-align:center;" | 530.233 | ||
|562.0465 | |||
|678.698 (2A5.B2A<sub>16</sub>) | |||
| | 15/11 | | | 15/11 | ||
| style="text-align:center;" | ^4 | | style="text-align:center;" | ^4 | ||
| Line 161: | Line 201: | ||
|- | |- | ||
| style="text-align:center;" | 20 | | style="text-align:center;" | 20 | ||
| style="text-align:center;" | 558. | | style="text-align:center;" | 558.1395 | ||
|591.628 | |||
|714.419 (2C8.6FD<sub>16</sub>) | |||
| | 11/8, 18/13 | | | 11/8, 18/13 | ||
| style="text-align:center;" | vA4 | | style="text-align:center;" | vA4 | ||
| Line 168: | Line 210: | ||
|- | |- | ||
| style="text-align:center;" | 21 | | style="text-align:center;" | 21 | ||
| style="text-align:center;" | 586. | | style="text-align:center;" | 586.0465 | ||
|621.209 | |||
|750.1395 (2EE.5318<sub>16</sub>) | |||
| | 7/5 | | | 7/5 | ||
| style="text-align:center;" | A4, vd5 | | style="text-align:center;" | A4, vd5 | ||
| Line 175: | Line 219: | ||
|- | |- | ||
| style="text-align:center;" | 22 | | style="text-align:center;" | 22 | ||
| style="text-align:center;" | 613. | | style="text-align:center;" | 613.9535 | ||
|650.791 | |||
|785.8605 (311.ACE8<sub>16</sub>) | |||
| | 10/7 | | | 10/7 | ||
| style="text-align:center;" | ^A4, d5 | | style="text-align:center;" | ^A4, d5 | ||
| Line 182: | Line 228: | ||
|- | |- | ||
| style="text-align:center;" | 23 | | style="text-align:center;" | 23 | ||
| style="text-align:center;" | 641. | | style="text-align:center;" | 641.8605 | ||
|680.372 | |||
|821.581 (335,A03<sub>16</sub>) | |||
| | 16/11, 13/9 | | | 16/11, 13/9 | ||
| style="text-align:center;" | ^d5 | | style="text-align:center;" | ^d5 | ||
| Line 190: | Line 238: | ||
| style="text-align:center;" | 24 | | style="text-align:center;" | 24 | ||
| style="text-align:center;" | 669.767 | | style="text-align:center;" | 669.767 | ||
|709.9535 | |||
|857.302 (359.4D6<sub>16</sub>) | |||
| | 22/15 | | | 22/15 | ||
| style="text-align:center;" | v5 | | style="text-align:center;" | v5 | ||
| Line 197: | Line 247: | ||
| style="text-align:center;" | 25 | | style="text-align:center;" | 25 | ||
| style="text-align:center;" | 697.674 | | style="text-align:center;" | 697.674 | ||
|739.535 | |||
|893.023 (37D.05F<sub>16</sub>) | |||
| | 3/2 | | | 3/2 | ||
| style="text-align:center;" | P5 | | style="text-align:center;" | P5 | ||
| Line 204: | Line 256: | ||
| style="text-align:center;" | 26 | | style="text-align:center;" | 26 | ||
| style="text-align:center;" | 725.581 | | style="text-align:center;" | 725.581 | ||
|769.116 | |||
|928.744 (3A0.BE8<sub>16</sub>) | |||
| | 32/21 | | | 32/21 | ||
| style="text-align:center;" | ^5 | | style="text-align:center;" | ^5 | ||
| Line 211: | Line 265: | ||
| style="text-align:center;" | 27 | | style="text-align:center;" | 27 | ||
| style="text-align:center;" | 753.488 | | style="text-align:center;" | 753.488 | ||
|798.698 | |||
|964.465 (3D4.771<sub>16</sub>) | |||
| | 20/13 | | | 20/13 | ||
| style="text-align:center;" | vA5, ^d6 | | style="text-align:center;" | vA5, ^d6 | ||
| Line 218: | Line 274: | ||
| style="text-align:center;" | 28 | | style="text-align:center;" | 28 | ||
| style="text-align:center;" | 781.395 | | style="text-align:center;" | 781.395 | ||
|828.279 | |||
|1000.186 (3E8.2FA<sub>16</sub>) | |||
| | 14/9, 11/7 | | | 14/9, 11/7 | ||
| style="text-align:center;" | vm6 | | style="text-align:center;" | vm6 | ||
| Line 225: | Line 283: | ||
| style="text-align:center;" | 29 | | style="text-align:center;" | 29 | ||
| style="text-align:center;" | 809.302 | | style="text-align:center;" | 809.302 | ||
|857.8605 | |||
|1035.917 (40B.D02<sub>16</sub>) | |||
| | 8/5 | | | 8/5 | ||
| style="text-align:center;" | m6 | | style="text-align:center;" | m6 | ||
| Line 232: | Line 292: | ||
| style="text-align:center;" | 30 | | style="text-align:center;" | 30 | ||
| style="text-align:center;" | 837.209 | | style="text-align:center;" | 837.209 | ||
|887.442 | |||
|1071.628 (42F.90C<sub>16</sub>) | |||
| | 18/11, 13/8 | | | 18/11, 13/8 | ||
| style="text-align:center;" | ^m6 | | style="text-align:center;" | ^m6 | ||
| Line 239: | Line 301: | ||
| style="text-align:center;" | 31 | | style="text-align:center;" | 31 | ||
| style="text-align:center;" | 865.116 | | style="text-align:center;" | 865.116 | ||
|917.023 | |||
|1107.349 (453.595<sub>16</sub>) | |||
| |28\17, 64\39 | | |28\17, 64\39 | ||
| style="text-align:center;" | vM6 | | style="text-align:center;" | vM6 | ||
| Line 246: | Line 310: | ||
| style="text-align:center;" | 32 | | style="text-align:center;" | 32 | ||
| style="text-align:center;" | 893.023 | | style="text-align:center;" | 893.023 | ||
|946.605 | |||
|1143.07 (477.11E<sub>16</sub>) | |||
| | 5/3 | | | 5/3 | ||
| style="text-align:center;" | M6 | | style="text-align:center;" | M6 | ||
| Line 253: | Line 319: | ||
| style="text-align:center;" | 33 | | style="text-align:center;" | 33 | ||
| style="text-align:center;" | 920.93 | | style="text-align:center;" | 920.93 | ||
| | 12/7 | |976.186 | ||
|1178.791 (49A.DB7<sub>16</sub>) | |||
| | 12/7, 22/13 | |||
| style="text-align:center;" | ^M6 | | style="text-align:center;" | ^M6 | ||
| style="text-align:center;" | upmajor 6th | | style="text-align:center;" | upmajor 6th | ||
| Line 260: | Line 328: | ||
| style="text-align:center;" | 34 | | style="text-align:center;" | 34 | ||
| style="text-align:center;" | 948.837 | | style="text-align:center;" | 948.837 | ||
|1005.767 | |||
|1214.512 (4BF.82F8<sub>16</sub>) | |||
| | 26/15 | | | 26/15 | ||
| style="text-align:center;" | vA6, ^d7 | | style="text-align:center;" | vA6, ^d7 | ||
| Line 267: | Line 337: | ||
| style="text-align:center;" | 35 | | style="text-align:center;" | 35 | ||
| style="text-align:center;" | 976.744 | | style="text-align:center;" | 976.744 | ||
|1035.349 | |||
|1250.233 (4E2.3B9<sub>16</sub>) | |||
| | 7/4 | | | 7/4 | ||
| style="text-align:center;" | vm7 | | style="text-align:center;" | vm7 | ||
| Line 274: | Line 346: | ||
| style="text-align:center;" | 36 | | style="text-align:center;" | 36 | ||
| style="text-align:center;" | 1004.651 | | style="text-align:center;" | 1004.651 | ||
|1064.93 | |||
|1285.9535 (505.F418<sub>16</sub>) | |||
| | 16/9, 9/5 | | | 16/9, 9/5 | ||
| style="text-align:center;" | m7 | | style="text-align:center;" | m7 | ||
| Line 281: | Line 355: | ||
| style="text-align:center;" | 37 | | style="text-align:center;" | 37 | ||
| style="text-align:center;" | 1032.558 | | style="text-align:center;" | 1032.558 | ||
|1094.512 | |||
|1321.674 (529.ACA<sub>16</sub>) | |||
| | 20/11 | | | 20/11 | ||
| style="text-align:center;" | ^m7 | | style="text-align:center;" | ^m7 | ||
| Line 288: | Line 364: | ||
| style="text-align:center;" | 38 | | style="text-align:center;" | 38 | ||
| style="text-align:center;" | 1060.465 | | style="text-align:center;" | 1060.465 | ||
|1124.093 | |||
|1357.395 (54D.652<sub>16</sub>) | |||
| | 11/6, 24/13, 13/7 | | | 11/6, 24/13, 13/7 | ||
| style="text-align:center;" | vM7 | | style="text-align:center;" | vM7 | ||
| Line 295: | Line 373: | ||
| style="text-align:center;" | 39 | | style="text-align:center;" | 39 | ||
| style="text-align:center;" | 1088.372 | | style="text-align:center;" | 1088.372 | ||
|1153.674 | |||
|1393.116 (571.1DC<sub>16</sub>) | |||
| | 15/8, 28/15 | | | 15/8, 28/15 | ||
| style="text-align:center;" | M7 | | style="text-align:center;" | M7 | ||
| Line 302: | Line 382: | ||
| style="text-align:center;" | 40 | | style="text-align:center;" | 40 | ||
| style="text-align:center;" | 1116.279 | | style="text-align:center;" | 1116.279 | ||
|1183.256 | |||
|1428.847 (594.D65<sub>16</sub>) | |||
| | | | | | ||
| style="text-align:center;" | ^M7 | | style="text-align:center;" | ^M7 | ||
| Line 309: | Line 391: | ||
| style="text-align:center;" | 41 | | style="text-align:center;" | 41 | ||
| style="text-align:center;" | 1144.186 | | style="text-align:center;" | 1144.186 | ||
|1212.837 | |||
|1464.488 (5B8.BEE<sub>16</sub>) | |||
| | | | | | ||
| style="text-align:center;" | vA7, ^d8 | | style="text-align:center;" | vA7, ^d8 | ||
| Line 316: | Line 400: | ||
| style="text-align:center;" | 42 | | style="text-align:center;" | 42 | ||
| style="text-align:center;" | 1172.093 | | style="text-align:center;" | 1172.093 | ||
|1242.419 | |||
|1500.279 (5DC.476<sub>16</sub>) | |||
| | | | | | ||
| style="text-align:center;" | A7, v8 | | style="text-align:center;" | A7, v8 | ||
| Line 323: | Line 409: | ||
| style="text-align:center;" | 43 | | style="text-align:center;" | 43 | ||
| style="text-align:center;" | 1200 | | style="text-align:center;" | 1200 | ||
|1272 | |||
|1536 (600<sub>16</sub>) | |||
| | 2/1 | | | 2/1 | ||
| style="text-align:center;" | P8 | | style="text-align:center;" | P8 | ||