# Ed257/128

An equal division of reduced harmonic 257 (ed257/128) is an equal-step tuning in which the 4ve-reduced 257th harmonic (257/128) is justly tuned and is divided in a given number of equal steps. 257/128 is very close to the octave, 2/1, but it is slightly sharper. This makes it suitable as an alternative to edos whose consonances are too flat, such as 7edo.

## 7ed257/128

### Harmonics

Approximation of harmonics in 7ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 -5.6 +13.5 -28.0 +1.1 +79.0 +20.2 -11.3 -21.3 -13.9 +7.9
relative (%) +4 -3 +8 -16 +1 +46 +12 -7 -12 -8 +5
Steps
(reduced)
7
(0)
11
(4)
14
(0)
16
(2)
18
(4)
20
(6)
21
(0)
22
(1)
23
(2)
24
(3)
25
(4)

7edo, 16ed5, 22ed9 for comparison:

Approximation of harmonics in 7edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -16.2 +0.0 -43.5 -16.2 +59.7 +0.0 -32.5 -43.5 -37.0 -16.2
relative (%) +0 -9 +0 -25 -9 +35 +0 -19 -25 -22 -9
Steps
(reduced)
7
(0)
11
(4)
14
(0)
16
(2)
18
(4)
20
(6)
21
(0)
22
(1)
23
(2)
24
(3)
25
(4)
Approximation of harmonics in 16ed5
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +19.0 +13.6 +38.0 +0.0 +32.6 -60.1 +57.0 +27.3 +19.0 +28.2 +51.7
relative (%) +11 +8 +22 +0 +19 -34 +33 +16 +11 +16 +30
Steps
(reduced)
7
(7)
11
(11)
14
(14)
16
(0)
18
(2)
19
(3)
21
(5)
22
(6)
23
(7)
24
(8)
25
(9)
Approximation of harmonics in 22ed9
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
relative (%) +6 +0 +12 -11 +6 -48 +18 +0 -5 -1 +12
Steps
(reduced)
7
(7)
11
(11)
14
(14)
16
(16)
18
(18)
19
(19)
21
(21)
22
(0)
23
(1)
24
(2)
25
(3)

• 172.393
• 344.786
• 517.178
• 689.571
• 861.964
• 1034.357
• 1206.749

## 9ed257/128

### Harmonics

Approximation of harmonics in 9ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 -24.8 +13.5 +29.4 -18.0 -16.7 +20.2 -49.6 +36.2 +5.3 -11.3
relative (%) +5 -18 +10 +22 -13 -12 +15 -37 +27 +4 -8
Steps
(reduced)
9
(0)
14
(5)
18
(0)
21
(3)
23
(5)
25
(7)
27
(0)
28
(1)
30
(3)
31
(4)
32
(5)

9edo for comparison:

Approximation of harmonics in 9edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -35.3 +0.0 +13.7 -35.3 -35.5 +0.0 +62.8 +13.7 -18.0 -35.3
relative (%) +0 -26 +0 +10 -26 -27 +0 +47 +10 -13 -26
Steps
(reduced)
9
(0)
14
(5)
18
(0)
21
(3)
23
(5)
25
(7)
27
(0)
29
(2)
30
(3)
31
(4)
32
(5)

• 134.083
• 268.167
• 402.25
• 536.333
• 670.416
• 804.5
• 938.583
• 1072.666
• 1206.749

## 14ed257/128

### Harmonics

Approximation of harmonics in 14ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 -5.6 +13.5 -28.0 +1.1 -7.2 +20.2 -11.3 -21.3 -13.9 +7.9
relative (%) +8 -7 +16 -33 +1 -8 +23 -13 -25 -16 +9
Steps
(reduced)
14
(0)
22
(8)
28
(0)
32
(4)
36
(8)
39
(11)
42
(0)
44
(2)
46
(4)
48
(6)
50
(8)

14edo for comparison:

Approximation of harmonics in 14edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -16.2 +0.0 +42.3 -16.2 -26.0 +0.0 -32.5 +42.3 -37.0 -16.2
relative (%) +0 -19 +0 +49 -19 -30 +0 -38 +49 -43 -19
Steps
(reduced)
14
(0)
22
(8)
28
(0)
33
(5)
36
(8)
39
(11)
42
(0)
44
(2)
47
(5)
48
(6)
50
(8)

• 86.196
• 172.393
• 258.589
• 344.786
• 430.982
• 517.178
• 603.375
• 689.571
• 775.768
• 861.964
• 948.16
• 1034.357
• 1120.553
• 1206.749

## 16ed257/128

### Harmonics

Approximation of harmonics in 16ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 -16.4 +13.5 +4.3 -9.7 +25.2 +20.2 -32.8 +11.0 -3.1 -2.9
relative (%) +9 -22 +18 +6 -13 +33 +27 -44 +15 -4 -4
Steps
(reduced)
16
(0)
25
(9)
32
(0)
37
(5)
41
(9)
45
(13)
48
(0)
50
(2)
53
(5)
55
(7)
57
(9)

16edo for comparison:

Approximation of harmonics in 16edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -27.0 +0.0 -11.3 -27.0 +6.2 +0.0 +21.1 -11.3 -26.3 -27.0
relative (%) +0 -36 +0 -15 -36 +8 +0 +28 -15 -35 -36
Steps
(reduced)
16
(0)
25
(9)
32
(0)
37
(5)
41
(9)
45
(13)
48
(0)
51
(3)
53
(5)
55
(7)
57
(9)

• 75.422
• 150.844
• 226.266
• 301.687
• 377.109
• 452.531
• 527.953
• 603.375
• 678.797
• 754.218
• 829.64
• 905.062
• 980.484
• 1055.906
• 1131.328
• 1206.749

## 19ed257/128

### Harmonics

Approximation of harmonics in 19ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 +3.4 +13.5 +8.3 +10.2 -2.6 +20.2 +6.9 +15.0 -23.0 +16.9
relative (%) +11 +5 +21 +13 +16 -4 +32 +11 +24 -36 +27
Steps
(reduced)
19
(0)
30
(11)
38
(0)
44
(6)
49
(11)
53
(15)
57
(0)
60
(3)
63
(6)
65
(8)
68
(11)

19edo for comparison:

Approximation of harmonics in 19edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 -21.5 +0.0 -14.4 -7.4 +17.1 -7.2
relative (%) +0 -11 +0 -12 -11 -34 +0 -23 -12 +27 -11
Steps
(reduced)
19
(0)
30
(11)
38
(0)
44
(6)
49
(11)
53
(15)
57
(0)
60
(3)
63
(6)
66
(9)
68
(11)

• 63.513
• 127.026
• 190.539
• 254.053
• 317.566
• 381.079
• 444.592
• 508.105
• 571.618
• 635.131
• 698.644
• 762.158
• 825.671
• 889.184
• 952.697
• 1016.21
• 1079.723
• 1143.236
• 1206.749

## 33ed257/128

### Harmonics

Approximation of harmonics in 33ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 -0.4 +13.5 -7.1 +6.3 -4.6 -16.3 -0.8 -0.4 +17.5 +13.1
relative (%) +18 -1 +37 -20 +17 -12 -45 -2 -1 +48 +36
Steps
(reduced)
33
(0)
52
(19)
66
(0)
76
(10)
85
(19)
92
(26)
98
(32)
104
(5)
109
(10)
114
(15)
118
(19)

33edo for comparison:

Approximation of harmonics in 33edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -11.0 +0.0 +13.7 -11.0 +13.0 +0.0 +14.3 +13.7 -5.9 -11.0
relative (%) +0 -30 +0 +38 -30 +36 +0 +39 +38 -16 -30
Steps
(reduced)
33
(0)
52
(19)
66
(0)
77
(11)
85
(19)
93
(27)
99
(0)
105
(6)
110
(11)
114
(15)
118
(19)

• 36.568
• 73.136
• 109.704
• 146.273
• 182.841
• 219.409
• 255.977
• 292.545
• 329.113
• 365.682
• 402.25
• 438.818
• 475.386
• 511.954
• 548.522
• 585.09
• 621.659
• 658.227
• 694.795
• 731.363
• 767.931
• 804.499
• 841.067
• 877.636
• 914.204
• 950.772
• 987.34
• 1023.908
• 1060.476
• 1097.045
• 1133.613
• 1170.181
• 1206.749

### Regular temperament properties

This is an excellent tuning for dreamtone temperament, much better than standard 33edo. It is almost exactly the TE tuning.

## 38ed257/128

### Harmonics

Approximation of harmonics in 38ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 +3.4 +13.5 +8.3 +10.2 -2.6 -11.5 +6.9 +15.0 +8.8 -14.8
relative (%) +21 +11 +43 +26 +32 -8 -36 +22 +47 +28 -47
Steps
(reduced)
38
(0)
60
(22)
76
(0)
88
(12)
98
(22)
106
(30)
113
(37)
120
(6)
126
(12)
131
(17)
135
(21)

38edo for comparison:

Approximation of harmonics in 38edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -7.2 +0.0 -7.4 -7.2 +10.1 +0.0 -14.4 -7.4 -14.5 -7.2
relative (%) +0 -23 +0 -23 -23 +32 +0 -46 -23 -46 -23
Steps
(reduced)
38
(0)
60
(22)
76
(0)
88
(12)
98
(22)
107
(31)
114
(0)
120
(6)
126
(12)
131
(17)
136
(22)

## 45ed257/128

### Harmonics

Approximation of harmonics in 45ed257/128
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +6.7 +2.0 -13.3 +2.6 +8.8 +10.1 -6.6 +4.1 +9.4 +5.3 -11.3
relative (%) +25 +8 -50 +10 +33 +38 -24 +15 +35 +20 -42
Steps
(reduced)
45
(0)
71
(26)
89
(44)
104
(14)
116
(26)
126
(36)
134
(44)
142
(7)
149
(14)
155
(20)
160
(25)

45edo for comparison:

Approximation of harmonics in 45edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error absolute (¢) +0.0 -8.6 +0.0 -13.0 -8.6 -8.8 +0.0 +9.4 -13.0 +8.7 -8.6
relative (%) +0 -32 +0 -49 -32 -33 +0 +35 -49 +33 -32
Steps
(reduced)
45
(0)
71
(26)
90
(0)
104
(14)
116
(26)
126
(36)
135
(0)
143
(8)
149
(14)
156
(21)
161
(26)