Ed15/4

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The equal division of 15/4 (ed15/4) is a tuning obtained by dividing the classic major fourteenth (15/4) into a number of equal steps.

Properties

Division of 15/4 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed15/4 scales have a perceptually important false octave, with various degrees of accuracy.

Ed15/4s are in the region where they may experience structural beating with the interval 4/1.

Notable ed15/4s

17ed15/4

  • 2.3.7.13.19 all within 18 cents
  • Stretched 9edo
  • Much better 3.7.13.19 than 9edo
  • Much worse 2.5.11 than 9edo
Approximation of prime harmonics in 17ed15/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +11.4 -17.5 +40.4 -3.7 +21.4 +1.4 -59.2 +17.4 -44.1 -41.6 -22.5
Relative (%) +8.5 -13.0 +30.0 -2.8 +15.9 +1.0 -44.0 +13.0 -32.8 -30.9 -16.7
Steps
(reduced) 		9
(9) 		14
(14) 		21
(4) 		25
(8) 		31
(14) 		33
(16) 		36
(2) 		38
(4) 		40
(6) 		43
(9) 		44
(10)
Approximation of prime harmonics in 9edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -35.3 +13.7 -35.5 -18.0 -40.5 +28.4 -30.8 +38.4 +37.1 +55.0
Relative (%) +0.0 -26.5 +10.3 -26.6 -13.5 -30.4 +21.3 -23.1 +28.8 +27.8 +41.2
Steps
(reduced) 		9
(0) 		14
(5) 		21
(3) 		25
(7) 		31
(4) 		33
(6) 		37
(1) 		38
(2) 		41
(5) 		44
(8) 		45
(0)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 134.6 12/11, 13/12, 14/13, 15/14
2 269.2 7/6, 22/19
3 403.8 5/4, 14/11, 19/15, 24/19
4 538.4 11/8, 15/11, 19/14, 23/17
5 673 19/13, 22/15
6 807.6 8/5, 19/12
7 942.2 12/7, 19/11
8 1076.8 13/7, 15/8, 24/13
9 1211.4 2/1
10 1346 11/5, 13/6, 24/11
11 1480.6 7/3, 19/8
12 1615.2 23/9
13 1749.9 11/4
14 1884.5 3/1
15 2019.1 16/5
16 2153.7 7/2
17 2288.3 15/4

43ed15/4

  • 3.5.7.11.17.19.23.31 all within 20 cents
  • Stretched 36edt
  • Virtually identical to 78ed11
  • Much better 11.19.23.31 than 36edt
  • Much worse 2.3.5.7.13 than 36edt
Approximation of prime harmonics in 43ed15/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +24.0 +13.8 -19.1 -16.2 -0.5 -23.6 -9.1 +11.2 -0.3 +24.1 +15.1
Relative (%) +45.0 +25.9 -35.9 -30.5 -0.9 -44.4 -17.1 +21.0 -0.5 +45.4 +28.4
Steps
(reduced) 		23
(23) 		36
(36) 		52
(9) 		63
(20) 		78
(35) 		83
(40) 		92
(6) 		96
(10) 		102
(16) 		110
(24) 		112
(26)
Approximation of prime harmonics in 36edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +15.1 +0.0 +13.8 +12.4 +22.4 -2.6 +8.4 -25.6 +13.4 -18.0 +25.0
Relative (%) +28.7 +0.0 +26.1 +23.5 +42.4 -5.0 +16.0 -48.5 +25.4 -34.2 +47.3
Steps
(reduced) 		23
(23) 		36
(0) 		53
(17) 		64
(28) 		79
(7) 		84
(12) 		93
(21) 		96
(24) 		103
(31) 		110
(2) 		113
(5)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 53.2 30/29
2 106.4
3 159.6 11/10, 23/21
4 212.9 17/15, 26/23
5 266.1
6 319.3
7 372.5 26/21
8 425.7
9 478.9 29/22
10 532.2 15/11, 19/14
11 585.4 7/5
12 638.6 29/20
13 691.8
14 745 23/15
15 798.2
16 851.4
17 904.7
18 957.9 26/15
19 1011.1
20 1064.3
21 1117.5 19/10, 21/11
22 1170.7
23 1224
24 1277.2 23/11
25 1330.4
26 1383.6 20/9
27 1436.8 23/10
28 1490 26/11
29 1543.3 22/9
30 1596.5
31 1649.7 13/5
32 1702.9
33 1756.1
34 1809.3
35 1862.5
36 1915.8
37 1969 28/9
38 2022.2 29/9
39 2075.4
40 2128.6
41 2181.8
42 2235.1
43 2288.3

47ed15/4

  • 2.3.5.7.11.13.17.19.29.31 all within 18 cents
  • Stretched 25edo
  • Compressed 39edt
  • Much better 3.11.13.29 than 25edo
  • Much worse 2.5.23 than 25edo
  • Much better 2.17.19.29 than 39edt
  • Much worse 3.7.11.13.23 than 39edt
Approximation of prime harmonics in 47ed15/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +17.2 -3.2 -11.2 -9.5 -13.0 -10.0 +12.4 +14.6 -24.1 +12.8 -5.3
Relative (%) +35.3 -6.5 -23.0 -19.4 -26.6 -20.6 +25.4 +29.9 -49.4 +26.3 -10.8
Steps
(reduced) 		25
(25) 		39
(39) 		57
(10) 		69
(22) 		85
(38) 		91
(44) 		101
(7) 		105
(11) 		111
(17) 		120
(26) 		122
(28)
Approximation of prime harmonics in 25edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +18.0 -2.3 -8.8 -23.3 +23.5 -9.0 -9.5 -4.3 -21.6 +7.0
Relative (%) +0.0 +37.6 -4.8 -18.4 -48.6 +48.9 -18.7 -19.8 -8.9 -45.0 +14.5
Steps
(reduced) 		25
(0) 		40
(15) 		58
(8) 		70
(20) 		86
(11) 		93
(18) 		102
(2) 		106
(6) 		113
(13) 		121
(21) 		124
(24)
Approximation of prime harmonics in 39edt
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +19.2 +0.0 -6.5 -3.8 -6.0 -2.6 +20.6 +23.1 -15.0 +22.6 +4.7
Relative (%) +39.4 +0.0 -13.4 -7.9 -12.4 -5.4 +42.3 +47.4 -30.8 +46.3 +9.6
Steps
(reduced) 		25
(25) 		39
(0) 		57
(18) 		69
(30) 		85
(7) 		91
(13) 		101
(23) 		105
(27) 		111
(33) 		120
(3) 		122
(5)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 48.7
2 97.4 18/17, 19/18
3 146.1 25/23
4 194.7 19/17, 29/26
5 243.4 15/13, 31/27
6 292.1 13/11
7 340.8 17/14
8 389.5
9 438.2 9/7
10 486.9
11 535.6 15/11
12 584.2 7/5
13 632.9 13/9
14 681.6
15 730.3 26/17, 29/19
16 779 11/7
17 827.7 21/13, 29/18
18 876.4
19 925 29/17
20 973.7
21 1022.4 9/5
22 1071.1 13/7
23 1119.8 21/11
24 1168.5
25 1217.2
26 1265.9 27/13, 29/14
27 1314.5 15/7
28 1363.2 11/5
29 1411.9
30 1460.6
31 1509.3 31/13
32 1558 27/11
33 1606.7
34 1655.3 13/5
35 1704
36 1752.7
37 1801.4 17/6
38 1850.1
39 1898.8 3/1
40 1947.5
41 1996.1 19/6
42 2044.8
43 2093.5
44 2142.2 31/9
45 2190.9
46 2239.6
47 2288.3

55ed15/4

  • 2.3.5.7.11.13.17.29.31 all within 12 cents
  • Stretched 29edo
  • Much better 5.7.11.17.31 than 29edo
  • Much worse 3.19.23 than 29edo
Approximation of prime harmonics in 55ed15/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +6.5 +11.9 +1.2 +1.2 +9.2 +11.2 +4.4 +19.9 -19.6 -4.9 +4.5
Relative (%) +15.7 +28.5 +2.9 +2.8 +22.0 +26.9 +10.6 +47.8 -47.2 -11.8 +10.7
Steps
(reduced) 		29
(29) 		46
(46) 		67
(12) 		81
(26) 		100
(45) 		107
(52) 		118
(8) 		123
(13) 		130
(20) 		140
(30) 		143
(33)
Approximation of prime harmonics in 29edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +1.5 -13.9 -17.1 -13.4 -12.9 +19.2 -7.9 -7.6 +4.9 +13.6
Relative (%) +0.0 +3.6 -33.6 -41.3 -32.4 -31.3 +46.4 -19.0 -18.3 +11.9 +32.8
Steps
(reduced) 		29
(0) 		46
(17) 		67
(9) 		81
(23) 		100
(13) 		107
(20) 		119
(3) 		123
(7) 		131
(15) 		141
(25) 		144
(28)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 41.6
2 83.2 21/20, 22/21
3 124.8 14/13
4 166.4 11/10
5 208 9/8
6 249.6 15/13, 22/19
7 291.2 13/11, 32/27
8 332.8 17/14
9 374.4 26/21, 31/25
10 416 14/11, 33/26
11 457.7 13/10
12 499.3 4/3
13 540.9 15/11, 26/19
14 582.5 7/5
15 624.1
16 665.7 22/15, 25/17
17 707.3
18 748.9 17/11, 20/13
19 790.5 30/19
20 832.1 21/13
21 873.7
22 915.3 17/10, 22/13
23 956.9 26/15, 33/19
24 998.5 16/9
25 1040.1 31/17
26 1081.7 28/15
27 1123.3 21/11
28 1164.9
29 1206.5
30 1248.1
31 1289.8 19/9
32 1331.4 28/13
33 1373 31/14
34 1414.6
35 1456.2
36 1497.8 19/8
37 1539.4 17/7
38 1581
39 1622.6
40 1664.2
41 1705.8
42 1747.4 11/4
43 1789 31/11
44 1830.6
45 1872.2
46 1913.8
47 1955.4 31/10
48 1997 19/6
49 2038.6 13/4
50 2080.2 10/3
51 2121.8 17/5
52 2163.5
53 2205.1 25/7
54 2246.7 11/3
55 2288.3 15/4

57ed15/4

  • 2.3.5.7.11.13.17.19.23.29.31 all within 17 cents
  • Stretched 30edo
  • Much better 3.7.17.19.31 than 30edo
  • Much worse 11.13 than 30edo
Approximation of prime harmonics in 57ed15/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +4.4 -15.1 -16.3 +3.4 -16.4 +15.6 -7.3 +0.9 -8.7 -8.5 -3.6
Relative (%) +10.8 -37.7 -40.6 +8.4 -40.8 +38.8 -18.1 +2.3 -21.6 -21.3 -8.9
Steps
(reduced) 		30
(30) 		47
(47) 		69
(12) 		84
(27) 		103
(46) 		111
(54) 		122
(8) 		127
(13) 		135
(21) 		145
(31) 		148
(34)
Approximation of prime harmonics in 30edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +18.0 +13.7 -8.8 +8.7 -0.5 +15.0 -17.5 +11.7 +10.4 +15.0
Relative (%) +0.0 +45.1 +34.2 -22.1 +21.7 -1.3 +37.6 -43.8 +29.3 +26.1 +37.4
Steps
(reduced) 		30
(0) 		48
(18) 		70
(10) 		84
(24) 		104
(14) 		111
(21) 		123
(3) 		127
(7) 		136
(16) 		146
(26) 		149
(29)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 40.1
2 80.3 21/20, 22/21, 23/22
3 120.4
4 160.6 11/10, 23/21, 34/31
5 200.7
6 240.9 23/20
7 281 20/17
8 321.2
9 361.3 16/13, 21/17
10 401.5 24/19, 29/23
11 441.6 22/17, 31/24
12 481.7 29/22, 33/25
13 521.9 23/17
14 562 29/21
15 602.2 17/12
16 642.3 29/20
17 682.5
18 722.6
19 762.8 31/20
20 802.9
21 843 13/8, 31/19
22 883.2 5/3
23 923.3 17/10, 29/17
24 963.5
25 1003.6 34/19
26 1043.8 31/17
27 1083.9
28 1124.1 21/11, 23/12
29 1164.2
30 1204.4 2/1
31 1244.5
32 1284.6 21/10
33 1324.8 28/13
34 1364.9 11/5
35 1405.1
36 1445.2 23/10
37 1485.4
38 1525.5 29/12
39 1565.7
40 1605.8
41 1645.9 31/12
42 1686.1
43 1726.2 19/7
44 1766.4 25/9
45 1806.5 17/6
46 1846.7 29/10
47 1886.8
48 1927
49 1967.1
50 2007.3
51 2047.4
52 2087.5 10/3
53 2127.7
54 2167.8 7/2
55 2208
56 2248.1 11/3
57 2288.3
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