Mina
The mina is a unit of interval size which has been proposed by George Secor and Dave Keenan, and which is defined as 1/2460 of an octave, the step size of 2460edo. 2460 is divisible by both 12 and 41, two important systems, and it's been suggested that degrees and minutes can be used to express values in it, so that for instance 3/2, which is 1439 minas, could be denoted by 23°59', meaning very slightly flat of the 24\41 41edo fifths. This works out since 41 * 60 = 2460; an octave is therefore expressed as if it were an angle of 41 degrees.
Other popular systems that can be represented exactly in whole numbers of minas include 10edo and 15edo. Moreover a cent is exactly 2.05 minas, and a mem, 1\205 octaves, is exactly 12 minas.
The following table lists some intervals which may be represented exactly in minas and in degrees and minutes, with the sizes listed in both cents and minas and expressed as degrees and minutes.
| interval | size in cents |
size in minas |
size as degrees and minutes |
|---|---|---|---|
| 1\2460 | 0.488 | 1 | 1' |
| 1\205 | 5.835 | 12 | 12' |
| 1\41 | 29.268 | 60 | 1° |
| 1\15 | 80 | 164 | 2°44' |
| 1\12 | 100 | 205 | 3°25' |
| 1\10 | 120 | 246 | 4°6' |
| 1\6 | 200 | 410 | 6°50' |
| 1\5 | 240 | 492 | 8°12' |
| 1\4 | 300 | 615 | 10°15' |
| 1\3 | 400 | 820 | 13°40' |
| 2\5 | 480 | 984 | 16°24' |
| 5\12 | 500 | 1025 | 17°5' |
| 1\2 | 600 | 1230 | 20°30' |
| 7\12 | 700 | 1435 | 23°55' |
| 3\5 | 720 | 1476 | 24°36' |
| 2\3 | 800 | 1640 | 27°20' |
| 3\4 | 900 | 1845 | 30°45' |
| 4\5 | 960 | 1960 | 32°48' |
| 5\6 | 1000 | 2050 | 34°10 |
| 11\12 | 1100 | 2255 | 37°35 |
| 2/1 | 1200 | 2460 | 41° |
Another notable feature of the mina is the accuracy and breadth of its approximations to just intervals. Accordingly it is hardly necessary to express intervals in non-integer values of mina, something that arguably cannot be said of cents. 2460edo is uniquely consistent through to the 27-limit, which is not very remarkable in itself (388edo is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-limit intervals. It is also a zeta peak edo and has a lower 19-limit relative error than any edo until 3395, and a lower 23-limit relative error than any until 8269. Also it has a lower 23-limit TE logflat badness than any smaller edo and less than any until 16808.
Below the intervals of the 27-limit tonality diamond are tabulated, with the sizes listed in both cents and minas and expressed as degrees and minutes (rounded to the nearest minute). The value in minas, rounded to the nearest integer, can be found by applying the 23-limit patent val ⟨2460 3899 5712 6906 8510 9103 10055 10450 11128] for 2460edo; this will not work for 1200edo and cents.
| interval ratio |
size in cent |
size in mina |
size as degrees and minutes |
|---|---|---|---|
| 1 | 0.000 | 0.000 | 0° |
| 28/27 | 62.961 | 129.070 | 2°9' |
| 27/26 | 65.337 | 133.942 | 2°14' |
| 26/25 | 67.900 | 139.195 | 2°19' |
| 25/24 | 70.672 | 144.878 | 2°25' |
| 24/23 | 73.681 | 151.045 | 2°31' |
| 23/22 | 76.956 | 157.761 | 2°38' |
| 22/21 | 80.537 | 165.101 | 2°45' |
| 21/20 | 84.467 | 173.158 | 2°53' |
| 20/19 | 88.801 | 182.041 | 3°2' |
| 19/18 | 93.603 | 191.886 | 3°12' |
| 18/17 | 98.955 | 202.857 | 3°23' |
| 17/16 | 104.955 | 215.159 | 3°35' |
| 16/15 | 111.731 | 229.049 | 3°49' |
| 15/14 | 119.443 | 244.858 | 4°5' |
| 14/13 | 128.298 | 263.011 | 4°23' |
| 27/25 | 133.238 | 273.137 | 4°33' |
| 13/12 | 138.573 | 284.074 | 4°44' |
| 25/23 | 144.353 | 295.924 | 4°56' |
| 12/11 | 150.637 | 308.806 | 5°9' |
| 23/21 | 157.493 | 322.862 | 5°23' |
| 11/10 | 165.004 | 338.259 | 5°38' |
| 21/19 | 173.268 | 355.199 | 5°55' |
| 10/9 | 182.404 | 373.928 | 6°14' |
| 19/17 | 192.558 | 394.743 | 6°35' |
| 28/25 | 196.198 | 402.207 | 6°42' |
| 9/8 | 203.910 | 418.016 | 6°58' |
| 26/23 | 212.253 | 435.119 | 7°15' |
| 17/15 | 216.687 | 444.208 | 7°24' |
| 25/22 | 221.309 | 453.684 | 7°34' |
| 8/7 | 231.174 | 473.907 | 7°54' |
| 23/20 | 241.961 | 496.019 | 8°16' |
| 15/13 | 247.741 | 507.869 | 8°28' |
| 22/19 | 253.805 | 520.300 | 8°40' |
| 7/6 | 266.871 | 547.085 | 9°7' |
| 27/23 | 277.591 | 569.061 | 9°29' |
| 20/17 | 281.358 | 576.785 | 9°37' |
| 13/11 | 289.210 | 592.880 | 9°53' |
| 32/27 | 294.135 | 602.977 | 10°3' |
| 19/16 | 297.513 | 609.902 | 10°10' |
| 25/21 | 301.847 | 618.785 | 10°19' |
| 6/5 | 315.641 | 647.065 | 10°47' |
| 23/19 | 330.761 | 678.061 | 11°18' |
| 17/14 | 336.130 | 689.065 | 11°29' |
| 28/23 | 340.552 | 698.131 | 11°38' |
| 11/9 | 347.408 | 712.186 | 11°52' |
| 27/22 | 354.547 | 726.821 | 12°7' |
| 16/13 | 359.472 | 736.918 | 12°17' |
| 21/17 | 365.825 | 749.942 | 12°30' |
| 26/21 | 369.747 | 757.981 | 12°38' |
| 5/4 | 386.314 | 791.943 | 13°12' |
| 34/27 | 399.090 | 818.135 | 13°38' |
| 24/19 | 404.442 | 829.106 | 13°49' |
| 19/15 | 409.244 | 838.951 | 13°59' |
| 14/11 | 417.508 | 855.891 | 14°15' |
| 23/18 | 424.364 | 869.947 | 14°30' |
| 32/25 | 427.373 | 876.114 | 14°36' |
| 9/7 | 435.084 | 891.922 | 14°52' |
| 22/17 | 446.363 | 915.043 | 15°15' |
| 13/10 | 454.214 | 931.139 | 15°31' |
| 30/23 | 459.994 | 942.988 | 15°43' |
| 17/13 | 464.428 | 952.077 | 15°52' |
| 21/16 | 470.781 | 965.101 | 16°5' |
| 25/19 | 475.114 | 973.985 | 16°14' |
| 4/3 | 498.045 | 1020.992 | 17°1' |
| 27/20 | 519.551 | 1065.080 | 17°45' |
| 23/17 | 523.319 | 1072.804 | 17°53' |
| 19/14 | 528.687 | 1083.809 | 18°4' |
| 34/25 | 532.328 | 1091.272 | 18°11' |
| 15/11 | 536.951 | 1100.749 | 18°21' |
| 26/19 | 543.015 | 1113.180 | 18°33' |
| 11/8 | 551.318 | 1130.202 | 18°50' |
| 18/13 | 563.382 | 1154.934 | 19°15' |
| 25/18 | 568.717 | 1165.871 | 19°26' |
| 32/23 | 571.726 | 1172.038 | 19°32' |
| 7/5 | 582.512 | 1194.150 | 19°54' |
| 38/27 | 591.648 | 1212.878 | 20°13' |
| 24/17 | 597.000 | 1223.849 | 20°24' |
| 17/12 | 603.000 | 1236.151 | 20°36' |
| 27/19 | 608.352 | 1247.122 | 20°47' |
| 10/7 | 617.488 | 1265.850 | 21°6' |
| 23/16 | 628.274 | 1287.962 | 21°28' |
| 36/25 | 631.283 | 1294.129 | 21°34' |
| 13/9 | 636.618 | 1305.066 | 21°45' |
| 16/11 | 648.682 | 1329.798 | 22°10' |
| 19/13 | 656.985 | 1346.820 | 22°27' |
| 22/15 | 663.049 | 1359.251 | 22°39' |
| 25/17 | 667.672 | 1368.728 | 22°49' |
| 28/19 | 671.313 | 1376.191 | 22°56' |
| 34/23 | 676.681 | 1387.196 | 23°7' |
| 40/27 | 680.449 | 1394.920 | 23°15' |
| 3/2 | 701.955 | 1439.008 | 23°59' |
| 38/25 | 724.886 | 1486.015 | 24°46' |
| 32/21 | 729.219 | 1494.899 | 24°55' |
| 26/17 | 735.572 | 1507.923 | 25°8' |
| 23/15 | 740.006 | 1517.012 | 25°17' |
| 20/13 | 745.786 | 1528.861 | 25°29' |
| 17/11 | 753.637 | 1544.957 | 25°45' |
| 14/9 | 764.916 | 1568.078 | 26°8' |
| 25/16 | 772.627 | 1583.886 | 26°24' |
| 36/23 | 775.636 | 1590.053 | 26°30' |
| 11/7 | 782.492 | 1604.109 | 26°44' |
| 30/19 | 790.756 | 1621.049 | 27°1' |
| 19/12 | 795.558 | 1630.894 | 27°11' |
| 27/17 | 800.910 | 1641.865 | 27°22' |
| 8/5 | 813.686 | 1668.057 | 27°48' |
| 21/13 | 830.253 | 1702.019 | 28°22' |
| 34/21 | 834.175 | 1710.058 | 28°30' |
| 13/8 | 840.528 | 1723.082 | 28°43' |
| 44/27 | 845.453 | 1733.179 | 28°53' |
| 18/11 | 852.592 | 1747.814 | 29°8' |
| 23/14 | 859.448 | 1761.869 | 29°22' |
| 28/17 | 863.870 | 1770.935 | 29°31' |
| 38/23 | 869.239 | 1781.939 | 29°42' |
| 5/3 | 884.359 | 1812.935 | 30°13' |
| 42/25 | 898.153 | 1841.215 | 30°41' |
| 32/19 | 902.487 | 1850.098 | 30°50' |
| 27/16 | 905.865 | 1857.023 | 30°57' |
| 22/13 | 910.790 | 1867.120 | 31°7' |
| 17/10 | 918.642 | 1883.215 | 31°23' |
| 46/27 | 922.409 | 1890.939 | 31°31' |
| 12/7 | 933.129 | 1912.915 | 31°53' |
| 19/11 | 946.195 | 1939.700 | 32°20' |
| 26/15 | 952.259 | 1952.131 | 32°32' |
| 40/23 | 958.039 | 1963.981 | 32°44' |
| 7/4 | 968.826 | 1986.093 | 33°6' |
| 44/25 | 978.691 | 2006.316 | 33°26' |
| 30/17 | 983.313 | 2015.792 | 33°36' |
| 23/13 | 987.747 | 2024.881 | 33°45' |
| 16/9 | 996.090 | 2041.984 | 34°2' |
| 25/14 | 1003.802 | 2057.793 | 34°18' |
| 34/19 | 1007.442 | 2065.257 | 34°25' |
| 9/5 | 1017.596 | 2086.072 | 34°46' |
| 38/21 | 1026.732 | 2104.801 | 35°4' |
| 20/11 | 1034.996 | 2121.741 | 35°22' |
| 42/23 | 1042.507 | 2137.138 | 35°37' |
| 11/6 | 1049.363 | 2151.194 | 35°51' |
| 46/25 | 1055.647 | 2164.076 | 36°4' |
| 24/13 | 1061.427 | 2175.926 | 36°16' |
| 50/27 | 1066.762 | 2186.863 | 36°27' |
| 13/7 | 1071.702 | 2196.989 | 36°37' |
| 28/15 | 1080.557 | 2215.142 | 36°55' |
| 15/8 | 1088.269 | 2230.951 | 37°11' |
| 32/17 | 1095.045 | 2244.841 | 37°23' |
| 17/9 | 1101.045 | 2257.143 | 37°37' |
| 36/19 | 1106.397 | 2268.114 | 37°48' |
| 19/10 | 1111.199 | 2277.959 | 37°58' |
| 40/21 | 1115.533 | 2286.842 | 38°7' |
| 21/11 | 1119.463 | 2294.899 | 38°15' |
| 44/23 | 1123.044 | 2302.239 | 38°22' |
| 23/12 | 1126.319 | 2308.955 | 38°29' |
| 48/25 | 1129.328 | 2315.122 | 38°35' |
| 25/13 | 1132.100 | 2320.805 | 38°41' |
| 52/27 | 1134.663 | 2326.058 | 38°46' |
| 27/14 | 1137.039 | 2330.930 | 38°51' |
| 2 | 1200.000 | 2460.000 | 41° |