Ed17/5

The equal division of 17/5 (ed17/5) is a tuning obtained by dividing the septendecimal major thirteenth (17/5) into a number of equal steps.

Properties

Division of 17/5 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed17/5 scales have a perceptually important false octave, with various degrees of accuracy.

Equivalence aside, the structural significance of thirteenths like 17/5 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.

Ed17/5s are in the region where they may experience structural beating with the interval 3/1.

Notable ed17/5s

22ed17/5

• 5.7.11.13.17.19 all within 11 cents
• Stretched 29ed5
• Much better 11.13 than 29ed5
• Much worse 3.31 than 29ed5

Approximation of prime harmonics in 22ed17/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-44.4	+24.1	+6.4	+1.7	-10.3	-10.6	+6.4	+6.5	-35.4	+44.8	+25.7
	Relative (%)	-46.1	+25.0	+6.7	+1.8	-10.7	-11.0	+6.7	+6.7	-36.7	+46.6	+26.7
Steps (reduced)		12 (12)	20 (20)	29 (7)	35 (13)	43 (21)	46 (2)	51 (7)	53 (9)	56 (12)	61 (17)	62 (18)

Approximation of prime harmonics in 29ed5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-47.0	+19.6	+0.0	-6.0	-19.9	-20.9	-4.9	-5.3	-47.8	+31.3	+11.9
	Relative (%)	-49.0	+20.4	+0.0	-6.3	-20.7	-21.7	-5.1	-5.5	-49.8	+32.6	+12.4
Steps (reduced)		12 (12)	20 (20)	29 (0)	35 (6)	43 (14)	46 (17)	51 (22)	53 (24)	56 (27)	61 (3)	62 (4)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	96.3	18/17, 19/18
2	192.6	19/17
3	288.9	13/11, 25/21
4	385.2
5	481.5	25/19
6	577.8	7/5, 25/18
7	674.1	25/17
8	770.4	11/7
9	866.7	18/11, 23/14
10	963
11	1059.3	11/6, 13/7
12	1155.6
13	1251.9
14	1348.2	13/6
15	1444.5	23/10
16	1540.8	17/7
17	1637.1	18/7
18	1733.4	19/7
19	1829.7
20	1926
21	2022.3
22	2118.6	17/5

28ed17/5

• 2.3.5.11.17.29 all within 14 cents
• Stretched 16edo
• Much better 3.11.17.23.29 than 16edo
• Much worse 2.7.13.19.31 than 16edo

Approximation of prime harmonics in 28ed17/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+10.7	-10.3	+13.3	+36.1	+10.3	+23.8	+13.3	-27.9	+19.7	-3.3	+32.6
	Relative (%)	+14.1	-13.6	+17.6	+47.8	+13.6	+31.4	+17.6	-36.9	+26.0	-4.4	+43.0
Steps (reduced)		16 (16)	25 (25)	37 (9)	45 (17)	55 (27)	59 (3)	65 (9)	67 (11)	72 (16)	77 (21)	79 (23)

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	75.7	21/20, 22/21, 23/22, 24/23, 25/24, 26/25
2	151.3	12/11, 23/21, 25/23
3	227	8/7, 17/15, 25/22
4	302.7	25/21
5	378.3	5/4, 26/21
6	454	13/10, 17/13, 22/17
7	529.7	15/11, 23/17
8	605.3	17/12, 24/17, 27/19
9	681
10	756.7	17/11, 20/13
11	832.3	13/8, 21/13
12	908	17/10, 22/13
13	983.7	23/13
14	1059.3	11/6, 24/13
15	1135	23/12, 25/13
16	1210.7	2/1
17	1286.3	19/9, 21/10, 23/11
18	1362	11/5
19	1437.6	16/7, 23/10
20	1513.3	12/5
21	1589	5/2
22	1664.6	13/5, 21/8
23	1740.3
24	1816	20/7
25	1891.6	3/1
26	1967.3	25/8
27	2043	13/4
28	2118.6	17/5

29ed17/5

• 3.5.7.11.13.17.19.29 all within 16 cents
• Compressed 26edt (double Bohlen-Pierce)
• Much better 11.13.19.29 than 26edt
• Much worse 17.23.31 than 26edt

Approximation of prime harmonics in 29ed17/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-31.1	-2.5	-10.2	-8.2	+12.9	+15.9	-10.2	+16.4	-22.1	+15.0	-27.5
	Relative (%)	-42.6	-3.4	-13.9	-11.3	+17.7	+21.8	-13.9	+22.5	-30.2	+20.5	-37.6
Steps (reduced)		16 (16)	26 (26)	38 (9)	46 (17)	57 (28)	61 (3)	67 (9)	70 (12)	74 (16)	80 (22)	81 (23)

Approximation of prime harmonics in 26edt

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-29.6	+0.0	-6.5	-3.8	+18.4	+21.8	-3.8	+23.1	-15.0	+22.6	-19.7
	Relative (%)	-40.4	+0.0	-8.9	-5.2	+25.1	+29.7	-5.1	+31.6	-20.5	+30.9	-26.9
Steps (reduced)		16 (16)	26 (0)	38 (12)	46 (20)	57 (5)	61 (9)	67 (15)	70 (18)	74 (22)	80 (2)	81 (3)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	73.1	22/21, 23/22, 26/25, 27/26
2	146.1	25/23
3	219.2	17/15, 25/22, 26/23
4	292.2	13/11, 25/21
5	365.3	21/17, 26/21
6	438.3	9/7, 22/17
7	511.4
8	584.5	7/5
9	657.5	19/13, 22/15, 25/17
10	730.6	23/15, 26/17
11	803.6	27/17
12	876.7	5/3
13	949.7	19/11, 26/15
14	1022.8	9/5
15	1095.8	17/9
16	1168.9
17	1242
18	1315	15/7
19	1388.1
20	1461.1	7/3
21	1534.2	17/7
22	1607.2
23	1680.3
24	1753.4
25	1826.4	26/9
26	1899.5	3/1
27	1972.5	22/7
28	2045.6
29	2118.6	17/5

32ed17/5

• 2.3.5.7.11.13.17.19.23.29.31 all within 20 cents
• Compressed 18edo
• Much better 3.5.7.13.17.19.23.29 than 18edo
• Much worse 2 than 18edo

Approximation of prime harmonics in 32ed17/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-8.3	+18.1	-5.6	+7.8	+19.8	-4.6	-5.6	+0.5	+0.7	-3.3	+13.6
	Relative (%)	-12.5	+27.3	-8.5	+11.7	+29.8	-7.0	-8.5	+0.7	+1.1	-5.0	+20.6
Steps (reduced)		18 (18)	29 (29)	42 (10)	51 (19)	63 (31)	67 (3)	74 (10)	77 (13)	82 (18)	88 (24)	90 (26)

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	66.2	24/23, 25/24, 26/25
2	132.4	13/12, 14/13
3	198.6	19/17, 28/25
4	264.8	7/6
5	331	17/14, 23/19
6	397.2	24/19
7	463.5	13/10, 17/13
8	529.7	15/11, 19/14, 23/17
9	595.9	17/12, 24/17
10	662.1	19/13, 22/15, 25/17, 28/19
11	728.3	26/17
12	794.5	19/12
13	860.7	18/11, 23/14, 28/17
14	926.9	12/7, 17/10
15	993.1	23/13
16	1059.3	24/13
17	1125.5	21/11, 23/12, 25/13
18	1191.7	2/1
19	1257.9
20	1324.2	15/7, 28/13
21	1390.4
22	1456.6
23	1522.8	12/5
24	1589	5/2
25	1655.2	13/5
26	1721.4	19/7
27	1787.6	14/5
28	1853.8
29	1920
30	1986.2	19/6, 22/7
31	2052.4	23/7
32	2118.6	17/5

41ed17/5

• 2.3.5.7.11.13.17.19.23.29.31 all within 19 cents
• 2.3.5.7.13.17.23.29.31 all within 12 cents
• Compressed 23edo
• Much better 3.5.7.11.29 than 23edo
• Much worse 2.19 than 23edo

Approximation of prime harmonics in 41ed17/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-11.5	+10.0	+4.1	-10.0	-17.4	+3.5	+4.1	+18.2	-2.5	+9.6	-2.5
	Relative (%)	-22.2	+19.3	+7.9	-19.4	-33.6	+6.7	+7.9	+35.3	-4.8	+18.6	-4.8
Steps (reduced)		23 (23)	37 (37)	54 (13)	65 (24)	80 (39)	86 (4)	95 (13)	99 (17)	105 (23)	113 (31)	115 (33)

Approximation of prime harmonics in 23edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	-23.7	-21.1	+22.5	+22.6	-5.7	-0.6	+15.5	-2.2	+13.9	+2.8
	Relative (%)	+0.0	-45.4	-40.4	+43.1	+43.3	-11.0	-1.2	+29.8	-4.2	+26.6	+5.3
Steps (reduced)		23 (0)	36 (13)	53 (7)	65 (19)	80 (11)	85 (16)	94 (2)	98 (6)	104 (12)	112 (20)	114 (22)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	51.7	30/29
2	103.3	18/17
3	155	12/11, 23/21
4	206.7	26/23
5	258.4	29/25
6	310	6/5
7	361.7	21/17
8	413.4	14/11, 19/15
9	465.1	17/13, 30/23
10	516.7	23/17
11	568.4	18/13, 25/18
12	620.1	10/7
13	671.8	25/17
14	723.4
15	775.1
16	826.8	21/13, 29/18
17	878.5	5/3
18	930.1	12/7, 29/17
19	981.8	23/13, 30/17
20	1033.5	20/11
21	1085.2
22	1136.8	25/13, 29/15
23	1188.5
24	1240.2
25	1291.9	19/9
26	1343.5	13/6
27	1395.2	29/13
28	1446.9	23/10, 30/13
29	1498.6
30	1550.2
31	1601.9
32	1653.6	13/5
33	1705.2
34	1756.9	11/4
35	1808.6	17/6
36	1860.3
37	1911.9
38	1963.6
39	2015.3
40	2067
41	2118.6	17/5

42ed17/5

• 2.3.5.7.11.13.17.19.31 all within 15 cents
• Stretched 24edo
• Much better 7.13 than 24edo
• Much worse 2.3.11.17 than 24edo
• The appeal of this tuning is that it brings 7/1 within 15 cents without pushing any prime below 23/1 outside 15 cents

Approximation of prime harmonics in 42ed17/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+10.7	+14.9	-11.9	+10.9	-14.9	-1.5	-11.9	-2.7	+19.7	+21.9	+7.3
	Relative (%)	+21.1	+29.6	-23.6	+21.6	-29.6	-2.9	-23.6	-5.3	+39.0	+43.4	+14.5
Steps (reduced)		24 (24)	38 (38)	55 (13)	67 (25)	82 (40)	88 (4)	97 (13)	101 (17)	108 (24)	116 (32)	118 (34)

Approximation of prime harmonics in 24edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.0	-2.0	+13.7	-18.8	-1.3	+9.5	-5.0	+2.5	+21.7	+20.4	+5.0
	Relative (%)	+0.0	-3.9	+27.4	-37.7	-2.6	+18.9	-9.9	+5.0	+43.5	+40.8	+9.9
Steps (reduced)		24 (0)	38 (14)	56 (8)	67 (19)	83 (11)	89 (17)	98 (2)	102 (6)	109 (13)	117 (21)	119 (23)

Intervals

Steps	Cents	Approximate ratios
0	0	1/1
1	50.4
2	100.9
3	151.3	23/21
4	201.8	9/8
5	252.2	15/13, 22/19
6	302.7
7	353.1
8	403.6	19/15, 29/23
9	454	13/10, 30/23
10	504.4	4/3
11	554.9	29/21
12	605.3
13	655.8	19/13
14	706.2	3/2
15	756.7	17/11
16	807.1
17	857.5	23/14
18	908	22/13, 27/16
19	958.4	26/15
20	1008.9
21	1059.3
22	1109.8	19/10
23	1160.2
24	1210.7
25	1261.1	29/14
26	1311.5
27	1362	11/5
28	1412.4
29	1462.9	7/3
30	1513.3
31	1563.8
32	1614.2
33	1664.6	21/8
34	1715.1
35	1765.5
36	1816	20/7
37	1866.4
38	1916.9
39	1967.3	28/9
40	2017.8
41	2068.2
42	2118.6	17/5

48ed17/5

• 2.3.5.7.11.17.23.29.31 all within 15 cents
• Compressed 27edo
• Much better 3.5.11.17 than 27edo
• Much worse 2.7.13.19 than 27edo
• The appeal of this tuning is that it brings 11/1 within 15 cents without pushing any prime below 19/1 outside 15 cents

Approximation of prime harmonics in 48ed17/5

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	-8.3	-4.0	-5.6	-14.3	-2.3	+17.4	-5.6	-21.6	+0.7	-3.3	+13.6
	Relative (%)	-18.7	-9.1	-12.7	-32.4	-5.2	+39.5	-12.7	-48.9	+1.7	-7.5	+30.9
Steps (reduced)		27 (27)	43 (43)	63 (15)	76 (28)	94 (46)	101 (5)	111 (15)	115 (19)	123 (27)	132 (36)	135 (39)

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	44.1
2	88.3	19/18, 20/19, 21/20
3	132.4	27/25
4	176.6	21/19
5	220.7	17/15, 25/22
6	264.8	7/6
7	309	31/26
8	353.1	11/9, 27/22
9	397.2	29/23
10	441.4	22/17
11	485.5
12	529.7	19/14
13	573.8	25/18
14	617.9	10/7
15	662.1	22/15, 25/17
16	706.2	3/2
17	750.4	17/11
18	794.5	19/12, 30/19
19	838.6
20	882.8	5/3
21	926.9	29/17
22	971	7/4
23	1015.2	9/5
24	1059.3
25	1103.5	17/9
26	1147.6
27	1191.7
28	1235.9
29	1280	21/10, 23/11
30	1324.2	15/7
31	1368.3	11/5
32	1412.4
33	1456.6
34	1500.7	19/8, 31/13
35	1544.8	22/9
36	1589	5/2
37	1633.1	18/7
38	1677.3	29/11
39	1721.4	27/10
40	1765.5	25/9
41	1809.7
42	1853.8
43	1897.9	3/1
44	1942.1
45	1986.2	22/7
46	2030.4	29/9
47	2074.5
48	2118.6	17/5