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°

External links

• mina on Tonalsoft Encyclopedia