Ed13/4

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The equal division of 13/4 (ed13/4) is a tuning obtained by dividing the tridecimal neutral thirteenth (13/4) into a number of equal steps.

Properties

Division of 13/4 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed13/4 scales have a perceptually important false octave, with various degrees of accuracy.

Equivalence aside, the structural significance of thirteenths like 13/4 is apparent by being the the top of the upper structure of jazz voicings, as well as a fairly trivial point to split the difference between the tritave and the double octave.

Ed13/4s are in the region where they may experience structural beating with the interval 3/1.

Notable ed13/4s

22ed13/4

  • 2.5.13.17.19.29.31 all within 14 cents
  • Stretched 13edo
  • Much better 5.19.31 than 13edo
  • Much worse 11.23 than 13edo
Approximation of prime harmonics in 22ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +5.8 +45.8 -3.8 -29.8 +22.5 +11.5 +10.9 +3.8 +44.0 +13.8 -9.0
Relative (%) +6.2 +49.4 -4.1 -32.1 +24.2 +12.4 +11.7 +4.1 +47.5 +14.8 -9.7
Steps
(reduced) 		13
(13) 		21
(21) 		30
(8) 		36
(14) 		45
(1) 		48
(4) 		53
(9) 		55
(11) 		59
(15) 		63
(19) 		64
(20)
Approximation of prime harmonics in 13edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +36.5 -17.1 -45.7 +2.5 -9.8 -12.6 -20.6 +17.9 -14.2 -37.3
Relative (%) +0.0 +39.5 -18.5 -49.6 +2.7 -10.6 -13.7 -22.3 +19.4 -15.4 -40.5
Steps
(reduced) 		13
(0) 		21
(8) 		30
(4) 		36
(10) 		45
(6) 		48
(9) 		53
(1) 		55
(3) 		59
(7) 		63
(11) 		64
(12)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 92.8 17/16, 20/19, 21/20, 22/21
2 185.5 19/17, 21/19
3 278.3 13/11, 20/17
4 371 16/13, 21/17
5 463.8 13/10, 17/13, 21/16, 25/19
6 556.5 11/8
7 649.3 16/11, 19/13, 22/15
8 742 17/11, 20/13, 23/15
9 834.8 13/8, 21/13
10 927.5 17/10
11 1020.3
12 1113 19/10, 21/11, 23/12
13 1205.8 2/1
14 1298.5 17/8, 21/10
15 1391.3
16 1484 19/8
17 1576.8 5/2
18 1669.5 21/8
19 1762.3 11/4
20 1855
21 1947.8
22 2040.5 13/4

27ed13/4

  • 2.3.5.11.17.23.29 all within 14 cents
  • Stretched 16edo
  • Much better 3.11.17.23.29 than 16edo
  • Much worse 7.19 than 16edo
Approximation of prime harmonics in 27ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +9.2 -12.6 +10.0 +32.1 +5.3 +18.4 +7.4 -34.0 +13.1 -10.3 +25.4
Relative (%) +12.2 -16.6 +13.2 +42.4 +7.0 +24.4 +9.8 -45.0 +17.4 -13.6 +33.6
Steps
(reduced) 		16
(16) 		25
(25) 		37
(10) 		45
(18) 		55
(1) 		59
(5) 		65
(11) 		67
(13) 		72
(18) 		77
(23) 		79
(25)
Approximation of prime harmonics in 16edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -27.0 -11.3 +6.2 -26.3 -15.5 -30.0 +2.5 -28.3 +20.4 -20.0
Relative (%) +0.0 -35.9 -15.1 +8.2 -35.1 -20.7 -39.9 +3.3 -37.7 +27.2 -26.7
Steps
(reduced) 		16
(0) 		25
(9) 		37
(5) 		45
(13) 		55
(7) 		59
(11) 		65
(1) 		68
(4) 		72
(8) 		78
(14) 		79
(15)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 75.6 21/20, 22/21, 23/22, 24/23, 25/24, 26/25
2 151.2 12/11, 23/21, 25/23
3 226.7 8/7, 17/15, 25/22
4 302.3 25/21
5 377.9 5/4, 26/21
6 453.5 13/10, 17/13, 22/17
7 529 15/11, 23/17
8 604.6 17/12, 24/17
9 680.2
10 755.8 17/11, 20/13
11 831.3 13/8, 21/13
12 906.9 22/13
13 982.5 23/13
14 1058.1 11/6, 24/13
15 1133.6 23/12, 25/13
16 1209.2 2/1
17 1284.8 19/9, 21/10, 23/11
18 1360.4 11/5, 24/11
19 1435.9 16/7, 23/10
20 1511.5 12/5
21 1587.1 5/2
22 1662.7 13/5, 21/8
23 1738.2
24 1813.8 17/6, 20/7
25 1889.4
26 1965 25/8
27 2040.5 13/4

29ed13/4

  • 2.3.7.11.13.23.29 all within 11 cents
  • Compressed 17edo
  • Much better 7.11.29 than 17edo
  • Much worse 19.31 than 17edo
Approximation of prime harmonics in 29ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -3.8 -2.2 +28.2 +8.6 +0.1 -7.7 +20.5 -31.4 -10.3 +10.6 -34.5
Relative (%) -5.4 -3.1 +40.1 +12.2 +0.1 -10.9 +29.1 -44.6 -14.7 +15.0 -49.1
Steps
(reduced) 		17
(17) 		27
(27) 		40
(11) 		48
(19) 		59
(1) 		63
(5) 		70
(12) 		72
(14) 		77
(19) 		83
(25) 		84
(26)
Approximation of prime harmonics in 17edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +3.9 -33.4 +19.4 +13.4 +6.5 -34.4 -15.2 +7.0 +29.2 -15.6
Relative (%) +0.0 +5.6 -47.3 +27.5 +19.0 +9.3 -48.7 -21.5 +9.9 +41.4 -22.1
Steps
(reduced) 		17
(0) 		27
(10) 		39
(5) 		48
(14) 		59
(8) 		63
(12) 		69
(1) 		72
(4) 		77
(9) 		83
(15) 		84
(16)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 70.4 22/21, 23/22, 24/23, 27/26
2 140.7 12/11, 13/12
3 211.1 9/8, 17/15, 26/23
4 281.5 13/11, 20/17, 27/23
5 351.8 11/9, 16/13, 27/22
6 422.2 14/11, 23/18
7 492.5 4/3
8 562.9 18/13
9 633.3 13/9, 23/16
10 703.6 3/2
11 774 11/7, 14/9
12 844.4 13/8, 18/11
13 914.7 17/10, 22/13, 27/16
14 985.1 23/13
15 1055.4 11/6, 24/13
16 1125.8 21/11, 23/12
17 1196.2 2/1
18 1266.5 23/11, 27/13
19 1336.9 13/6
20 1407.3 9/4
21 1477.6
22 1548 22/9, 27/11
23 1618.3 23/9
24 1688.7 8/3
25 1759.1 11/4
26 1829.4 23/8, 26/9
27 1899.8 3/1
28 1970.2
29 2040.5 13/4

31ed13/4

  • 2.3.7.11 all within 16 cents
  • Compressed 18edo
  • Much better 3.7.11 than 17edo
  • Much worse 2.5.31 than 17edo
Approximation of prime harmonics in 31ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -15.2 +6.9 -21.7 -11.8 -4.4 -30.4 +31.8 -29.1 -30.7 +28.7 -20.9
Relative (%) -23.1 +10.5 -33.0 -18.0 -6.7 -46.1 +48.3 -44.2 -46.7 +43.6 -31.8
Steps
(reduced) 		18
(18) 		29
(29) 		42
(11) 		51
(20) 		63
(1) 		67
(5) 		75
(13) 		77
(15) 		82
(20) 		89
(27) 		90
(28)
Approximation of prime harmonics in 18edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +31.4 +13.7 +31.2 -18.0 +26.1 +28.4 -30.8 -28.3 -29.6 -11.7
Relative (%) +0.0 +47.1 +20.5 +46.8 -27.0 +39.2 +42.6 -46.3 -42.4 -44.4 -17.6
Steps
(reduced) 		18
(0) 		29
(11) 		42
(6) 		51
(15) 		62
(8) 		67
(13) 		74
(2) 		76
(4) 		81
(9) 		87
(15) 		89
(17)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 65.8 24/23, 25/24, 26/25
2 131.6 13/12, 14/13
3 197.5
4 263.3 7/6, 22/19
5 329.1 23/19
6 394.9 5/4, 24/19
7 460.8 13/10
8 526.6 19/14
9 592.4
10 658.2 19/13, 22/15
11 724.1
12 789.9 11/7, 19/12
13 855.7 18/11, 23/14
14 921.5
15 987.4 23/13
16 1053.2 11/6, 24/13
17 1119 19/10, 21/11, 23/12
18 1184.8
19 1250.6
20 1316.5 15/7
21 1382.3
22 1448.1 23/10
23 1513.9 12/5
24 1579.8 5/2
25 1645.6 13/5
26 1711.4
27 1777.2 14/5
28 1843.1
29 1908.9 3/1
30 1974.7 22/7, 25/8
31 2040.5 13/4

32ed13/4

  • 2.3.7.11.17.19.23.31 all within 15 cents
  • Stretched 19edo
  • Much better 7.11.17.19 than 19edo
  • Much worse 2.5 than 19edo
Approximation of prime harmonics in 32ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +11.6 +11.0 +19.4 +10.8 -6.5 +23.1 +5.1 +3.8 -8.1 -26.8 -14.8
Relative (%) +18.1 +17.3 +30.4 +16.9 -10.2 +36.3 +7.9 +6.0 -12.7 -42.1 -23.1
Steps
(reduced) 		19
(19) 		30
(30) 		44
(12) 		53
(21) 		65
(1) 		70
(6) 		77
(13) 		80
(16) 		85
(21) 		91
(27) 		93
(29)
Approximation of prime harmonics in 19edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -7.2 -7.4 -21.5 +17.1 -19.5 +21.4 +18.3 +3.3 -19.1 -8.2
Relative (%) +0.0 -11.4 -11.7 -34.0 +27.1 -30.8 +33.8 +28.9 +5.2 -30.2 -13.0
Steps
(reduced) 		19
(0) 		30
(11) 		44
(6) 		53
(15) 		66
(9) 		70
(13) 		78
(2) 		81
(5) 		86
(10) 		92
(16) 		94
(18)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 63.8 25/24, 26/25, 27/26, 28/27
2 127.5 14/13, 15/14, 27/25
3 191.3 10/9, 19/17, 28/25
4 255.1 15/13, 22/19
5 318.8 6/5
6 382.6 5/4
7 446.4 13/10, 22/17
8 510.1
9 573.9 7/5, 25/18
10 637.7 13/9
11 701.4 3/2
12 765.2 14/9, 25/16
13 829 21/13
14 892.7 5/3
15 956.5 26/15
16 1020.3 9/5
17 1084 15/8, 28/15
18 1147.8
19 1211.6
20 1275.3 21/10, 23/11, 25/12
21 1339.1 13/6
22 1402.9 9/4
23 1466.6 7/3
24 1530.4 17/7
25 1594.2 5/2
26 1657.9 13/5
27 1721.7 19/7, 27/10
28 1785.5 14/5
29 1849.2
30 1913
31 1976.8 22/7, 25/8
32 2040.5 13/4

33ed13/4

  • 3.5.11.13.23.31 all within 15 cents
  • Compressed 45ed5
  • Much better 3.13.23 than 45ed5
  • Much worse 17.29 than 45ed5
Approximation of prime harmonics in 33ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -25.2 +14.9 -3.8 -29.8 -8.4 +11.5 -20.1 -27.1 +13.1 -17.2 -9.0
Relative (%) -40.7 +24.1 -6.1 -48.2 -13.6 +18.7 -32.4 -43.8 +21.2 -27.8 -14.5
Steps
(reduced) 		19
(19) 		31
(31) 		45
(12) 		54
(21) 		67
(1) 		72
(6) 		79
(13) 		82
(16) 		88
(22) 		94
(28) 		96
(30)
Approximation of prime harmonics in 45ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -23.6 +17.5 +0.0 -25.2 -2.8 +17.6 -13.4 -20.2 +20.5 -9.3 -0.9
Relative (%) -38.0 +28.3 +0.0 -40.8 -4.5 +28.4 -21.7 -32.7 +33.1 -15.0 -1.5
Steps
(reduced) 		19
(19) 		31
(31) 		45
(0) 		54
(9) 		67
(22) 		72
(27) 		79
(34) 		82
(37) 		88
(43) 		94
(4) 		96
(6)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 61.8 26/25
2 123.7
3 185.5 19/17
4 247.3 15/13, 22/19
5 309.2 6/5, 25/21
6 371 21/17, 26/21
7 432.8 23/18
8 494.7
9 556.5 18/13
10 618.3 10/7
11 680.2
12 742 23/15, 26/17
13 803.8 19/12
14 865.7
15 927.5 12/7, 17/10
16 989.3 23/13
17 1051.2 11/6
18 1113 19/10, 21/11
19 1174.8
20 1236.7
21 1298.5
22 1360.4 11/5
23 1422.2 25/11
24 1484 26/11
25 1545.9
26 1607.7
27 1669.5
28 1731.4 19/7
29 1793.2
30 1855
31 1916.9
32 1978.7 22/7
33 2040.5

35ed13/4

  • 5.7.11.13.17.23.29.31 all within 13 cents
  • Stretched 48ed5
  • Much better 11.13.17.23.29.31 than 48ed5
  • Much worse 3.5.7.19 than 48ed5
Approximation of prime harmonics in 35ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +24.3 +22.0 +12.1 +12.6 -12.0 -9.7 -7.7 -25.3 -6.3 +0.5 +1.6
Relative (%) +41.7 +37.7 +20.8 +21.6 -20.5 -16.6 -13.2 -43.5 -10.8 +0.9 +2.8
Steps
(reduced) 		21
(21) 		33
(33) 		48
(13) 		58
(23) 		71
(1) 		76
(6) 		84
(14) 		87
(17) 		93
(23) 		100
(30) 		102
(32)
Approximation of prime harmonics in 48ed5
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +19.0 +13.6 +0.0 -2.0 +28.2 -28.9 -28.9 +10.7 +28.3 -24.8 -24.1
Relative (%) +32.8 +23.5 +0.0 -3.5 +48.5 -49.7 -49.8 +18.5 +48.7 -42.6 -41.5
Steps
(reduced) 		21
(21) 		33
(33) 		48
(0) 		58
(10) 		72
(24) 		76
(28) 		84
(36) 		88
(40) 		94
(46) 		100
(4) 		102
(6)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 58.3 28/27
2 116.6 15/14
3 174.9 10/9
4 233.2
5 291.5 13/11
6 349.8
7 408.1
8 466.4 17/13
9 524.7 23/17, 27/20
10 583 7/5
11 641.3
12 699.6 3/2
13 757.9 14/9, 17/11
14 816.2
15 874.5
16 932.8
17 991.1 23/13
18 1049.4
19 1107.7
20 1166
21 1224.3
22 1282.6 21/10, 23/11
23 1340.9
24 1399.2 9/4
25 1457.5
26 1515.8
27 1574.1
28 1632.4
29 1690.7
30 1749
31 1807.3
32 1865.6
33 1923.9
34 1982.2 22/7
35 2040.5

43ed13/4

  • 2.3.5.7.29.31 all within 14 cents
  • Compressed 25edo
  • Much better 3.7.29 than 48ed5
  • Much worse 2.5.17.19.23 than 48ed5
Approximation of prime harmonics in 43ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -13.6 -3.8 +13.5 +0.4 -22.8 +20.2 -17.2 -19.9 -18.5 +7.3 -13.3
Relative (%) -28.8 -8.0 +28.4 +0.9 -48.1 +42.5 -36.2 -42.0 -39.0 +15.3 -28.0
Steps
(reduced) 		25
(25) 		40
(40) 		59
(16) 		71
(28) 		87
(1) 		94
(8) 		103
(17) 		107
(21) 		114
(28) 		123
(37) 		125
(39)
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)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 47.5
2 94.9 18/17, 19/18, 20/19
3 142.4
4 189.8 19/17, 29/26
5 237.3 23/20
6 284.7 20/17
7 332.2 17/14, 23/19
8 379.6
9 427.1 23/18
10 474.5
11 522 23/17, 27/20
12 569.4
13 616.9 10/7
14 664.4
15 711.8
16 759.3 14/9, 17/11
17 806.7 27/17
18 854.2 18/11, 23/14
19 901.6
20 949.1 19/11, 26/15
21 996.5
22 1044 11/6
23 1091.4
24 1138.9 27/14, 29/15
25 1186.4
26 1233.8
27 1281.3 21/10, 23/11
28 1328.7
29 1376.2 20/9
30 1423.6
31 1471.1 7/3
32 1518.5
33 1566
34 1613.4 28/11
35 1660.9
36 1708.3
37 1755.8 11/4
38 1803.3 17/6
39 1850.7
40 1898.2 3/1
41 1945.6
42 1993.1 19/6
43 2040.5

46ed13/4

  • 2.3.5.7.13.19.31 all within 9 cents
  • Compressed 27edo
  • Much better 3.5.7.19.31 than 27edo
  • Much worse 23.29 than 27edo
Approximation of prime harmonics in 46ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -2.3 +5.5 +8.3 +2.5 +18.5 -4.6 +18.9 +3.8 -16.4 -18.5 -0.9
Relative (%) -5.2 +12.4 +18.8 +5.6 +41.6 -10.4 +42.7 +8.6 -37.1 -41.7 -2.0
Steps
(reduced) 		27
(27) 		43
(43) 		63
(17) 		76
(30) 		94
(2) 		100
(8) 		111
(19) 		115
(23) 		122
(30) 		131
(39) 		134
(42)
Approximation of prime harmonics in 27edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 +9.2 +13.7 +9.0 -18.0 +3.9 -16.1 +13.6 -6.1 -7.4 +10.5
Relative (%) +0.0 +20.6 +30.8 +20.1 -40.5 +8.8 -36.1 +30.6 -13.6 -16.5 +23.7
Steps
(reduced) 		27
(0) 		43
(16) 		63
(9) 		76
(22) 		93
(12) 		100
(19) 		110
(2) 		115
(7) 		122
(14) 		131
(23) 		134
(26)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 47.5
2 94.9 18/17, 19/18, 20/19
3 142.4
4 189.8 19/17, 29/26
5 237.3 23/20
6 284.7 20/17
7 332.2 17/14, 23/19
8 379.6
9 427.1 23/18
10 474.5
11 522 23/17, 27/20
12 569.4
13 616.9 10/7
14 664.4
15 711.8
16 759.3 14/9, 17/11
17 806.7 27/17
18 854.2 18/11, 23/14
19 901.6
20 949.1 19/11, 26/15
21 996.5
22 1044 11/6
23 1091.4
24 1138.9 27/14, 29/15
25 1186.4
26 1233.8
27 1281.3 21/10, 23/11
28 1328.7
29 1376.2 20/9
30 1423.6
31 1471.1 7/3
32 1518.5
33 1566
34 1613.4 28/11
35 1660.9
36 1708.3
37 1755.8 11/4
38 1803.3 17/6
39 1850.7
40 1898.2 3/1
41 1945.6
42 1993.1 19/6
43 2040.5

47ed13/4

  • 2.3.5.7.11.13.17.19.23.29.31 all within 18 cents
  • Stretched 28edo
  • Much better 3.13.17.23.31 than 28edo
  • Much worse 2.11.19.29 than 28edo
Approximation of prime harmonics in 47ed13/4
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +15.6 +8.3 -7.7 +17.6 +16.6 -12.1 +1.0 -17.9 -1.3 -11.9 +2.9
Relative (%) +36.0 +19.2 -17.8 +40.5 +38.2 -28.0 +2.3 -41.2 -3.1 -27.4 +6.6
Steps
(reduced) 		28
(28) 		44
(44) 		64
(17) 		78
(31) 		96
(2) 		102
(8) 		113
(19) 		117
(23) 		125
(31) 		134
(40) 		137
(43)
Approximation of prime harmonics in 28edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -16.2 -0.6 +16.9 +5.8 +16.6 -19.2 +2.5 +14.6 -1.0 +12.1
Relative (%) +0.0 -37.9 -1.4 +39.4 +13.6 +38.8 -44.9 +5.8 +34.0 -2.3 +28.3
Steps
(reduced) 		28
(0) 		44
(16) 		65
(9) 		79
(23) 		97
(13) 		104
(20) 		114
(2) 		119
(7) 		127
(15) 		136
(24) 		139
(27)
Intervals
Steps Cents Approximate ratios
0 0 1/1
1 47.5
2 94.9 18/17, 19/18, 20/19
3 142.4
4 189.8 19/17, 29/26
5 237.3 23/20
6 284.7 20/17
7 332.2 17/14, 23/19
8 379.6
9 427.1 23/18
10 474.5
11 522 23/17, 27/20
12 569.4
13 616.9 10/7
14 664.4
15 711.8
16 759.3 14/9, 17/11
17 806.7 27/17
18 854.2 18/11, 23/14
19 901.6
20 949.1 19/11, 26/15
21 996.5
22 1044 11/6
23 1091.4
24 1138.9 27/14, 29/15
25 1186.4
26 1233.8
27 1281.3 21/10, 23/11
28 1328.7
29 1376.2 20/9
30 1423.6
31 1471.1 7/3
32 1518.5
33 1566
34 1613.4 28/11
35 1660.9
36 1708.3
37 1755.8 11/4
38 1803.3 17/6
39 1850.7
40 1898.2 3/1
41 1945.6
42 1993.1 19/6
43 2040.5
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