Mina

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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\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
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°