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.
Ed257/128s really only make sense for that purpose with 65 or fewer tones per pseudo-octave. With more tones than that, the relative error on 2/1 becomes unacceptably high and it makes more sense to switch to a different tuning like a zpi or ed513/256.
Ed257/128s are the complementary opposite of ed255/128s.
7ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | -5.6 | -28.0 | +79.0 | -13.9 | +41.7 | -78.0 | +74.3 | -84.1 | +31.8 | -83.7 |
Relative (%) | +3.9 | -3.3 | -16.3 | +45.8 | -8.1 | +24.2 | -45.2 | +43.1 | -48.8 | +18.4 | -48.5 | |
Steps (reduced) |
7 (0) |
11 (4) |
16 (2) |
20 (6) |
24 (3) |
26 (5) |
28 (0) |
30 (2) |
31 (3) |
34 (6) |
34 (6) |
7edo, 16ed5, 22ed9 for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.0 | -16.2 | -43.5 | +59.7 | -37.0 | +16.6 | +66.5 | +45.3 | +57.4 | -1.0 | +55.0 |
Relative (%) | +0.0 | -9.5 | -25.3 | +34.9 | -21.6 | +9.7 | +38.8 | +26.5 | +33.5 | -0.6 | +32.1 | |
Steps (reduced) |
7 (0) |
11 (4) |
16 (2) |
20 (6) |
24 (3) |
26 (5) |
29 (1) |
30 (2) |
32 (4) |
34 (6) |
35 (0) |
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +19.0 | +13.6 | +0.0 | -60.1 | +28.2 | -86.9 | -28.9 | -47.3 | -29.8 | -82.8 | -24.1 |
Relative (%) | +10.9 | +7.8 | +0.0 | -34.5 | +16.2 | -49.9 | -16.6 | -27.2 | -17.1 | -47.5 | -13.8 | |
Steps (reduced) |
7 (7) |
11 (11) |
16 (0) |
19 (3) |
24 (8) |
25 (9) |
28 (12) |
29 (13) |
31 (15) |
33 (1) |
34 (2) |
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +10.3 | +0.0 | -19.8 | -83.6 | -1.6 | +55.0 | -63.6 | -83.3 | -68.2 | +49.2 | -66.3 |
Relative (%) | +6.0 | +0.0 | -11.5 | -48.4 | -0.9 | +31.8 | -36.8 | -48.2 | -39.5 | +28.5 | -38.3 | |
Steps (reduced) |
7 (7) |
11 (11) |
16 (16) |
19 (19) |
24 (2) |
26 (4) |
28 (6) |
29 (7) |
31 (9) |
34 (12) |
34 (12) |
Intervals
- 172.393
- 344.786
- 517.178
- 689.571
- 861.964
- 1034.357
- 1206.749
9ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | -24.8 | +29.4 | -16.7 | +5.3 | -15.8 | +56.1 | -2.3 | -64.9 | -64.0 | -45.4 |
Relative (%) | +5.0 | -18.5 | +22.0 | -12.5 | +3.9 | -11.8 | +41.9 | -1.8 | -48.4 | -47.7 | -33.8 | |
Steps (reduced) |
9 (0) |
14 (5) |
21 (3) |
25 (7) |
31 (4) |
33 (6) |
37 (1) |
38 (2) |
40 (4) |
43 (7) |
44 (8) |
9edo for comparison:
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
- 134.083
- 268.167
- 402.25
- 536.333
- 670.416
- 804.5
- 938.583
- 1072.666
- 1206.749
14ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | -5.6 | -28.0 | -7.2 | -13.9 | +41.7 | +8.2 | -11.9 | +2.1 | +31.8 | +2.5 |
Relative (%) | +7.8 | -6.5 | -32.5 | -8.3 | -16.1 | +48.4 | +9.6 | -13.8 | +2.4 | +36.9 | +2.9 | |
Steps (reduced) |
14 (0) |
22 (8) |
32 (4) |
39 (11) |
48 (6) |
52 (10) |
57 (1) |
59 (3) |
63 (7) |
68 (12) |
69 (13) |
14edo for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.0 | -16.2 | +42.3 | -26.0 | -37.0 | +16.6 | -19.2 | -40.4 | -28.3 | -1.0 | -30.7 |
Relative (%) | +0.0 | -18.9 | +49.3 | -30.3 | -43.2 | +19.4 | -22.4 | -47.1 | -33.0 | -1.2 | -35.9 | |
Steps (reduced) |
14 (0) |
22 (8) |
33 (5) |
39 (11) |
48 (6) |
52 (10) |
57 (1) |
59 (3) |
63 (7) |
68 (12) |
69 (13) |
Intervals
- 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
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | -16.4 | +4.3 | +25.2 | -3.1 | +9.4 | -2.5 | +31.2 | +2.1 | -22.1 | +13.3 |
Relative (%) | +8.9 | -21.8 | +5.7 | +33.4 | -4.1 | +12.4 | -3.4 | +41.3 | +2.8 | -29.3 | +17.6 | |
Steps (reduced) |
16 (0) |
25 (9) |
37 (5) |
45 (13) |
55 (7) |
59 (11) |
65 (1) |
68 (4) |
72 (8) |
77 (13) |
79 (15) |
16edo for comparison:
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
- 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
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | +3.4 | +8.3 | -2.6 | -23.0 | +5.4 | -14.4 | -16.5 | -29.7 | +13.6 | +25.2 |
Relative (%) | +10.6 | +5.4 | +13.0 | -4.1 | -36.2 | +8.5 | -22.7 | -25.9 | -46.7 | +21.5 | +39.7 | |
Steps (reduced) |
19 (0) |
30 (11) |
44 (6) |
53 (15) |
65 (8) |
70 (13) |
77 (1) |
80 (4) |
85 (9) |
92 (16) |
94 (18) |
19edo for comparison:
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
- 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
This is an excellent tuning for dreamtone temperament, much better than standard 33edo. It is almost exactly the TE tuning.
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | -0.4 | -7.1 | -4.6 | +17.5 | -15.8 | -4.8 | -14.5 | -16.2 | -15.2 | +15.6 |
Relative (%) | +18.5 | -1.1 | -19.5 | -12.5 | +47.7 | -43.2 | -13.2 | -39.8 | -44.3 | -41.7 | +42.6 | |
Steps (reduced) |
33 (0) |
52 (19) |
76 (10) |
92 (26) |
114 (15) |
121 (22) |
134 (2) |
139 (7) |
148 (16) |
159 (27) |
163 (31) |
33edo for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.0 | -11.0 | +13.7 | +13.0 | -5.9 | -4.2 | +4.1 | -6.6 | -10.1 | -11.4 | -17.8 |
Relative (%) | +0.0 | -30.4 | +37.6 | +35.7 | -16.1 | -11.5 | +11.4 | -18.2 | -27.8 | -31.3 | -48.8 | |
Steps (reduced) |
33 (0) |
52 (19) |
77 (11) |
93 (27) |
114 (15) |
122 (23) |
135 (3) |
140 (8) |
149 (17) |
160 (28) |
163 (31) |
Intervals
- 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
38ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | +3.4 | +8.3 | -2.6 | +8.8 | +5.4 | -14.4 | +15.3 | +2.1 | +13.6 | -6.6 |
Relative (%) | +21.3 | +10.8 | +26.0 | -8.3 | +27.7 | +17.0 | -45.5 | +48.2 | +6.6 | +42.9 | -20.7 | |
Steps (reduced) |
38 (0) |
60 (22) |
88 (12) |
106 (30) |
131 (17) |
140 (26) |
154 (2) |
161 (9) |
171 (19) |
184 (32) |
187 (35) |
38edo for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.0 | -7.2 | -7.4 | +10.1 | -14.5 | +12.1 | -10.2 | -13.3 | +3.3 | +12.5 | -8.2 |
Relative (%) | +0.0 | -22.9 | -23.3 | +32.1 | -45.8 | +38.3 | -32.4 | -42.1 | +10.5 | +39.7 | -25.9 | |
Steps (reduced) |
38 (0) |
60 (22) |
88 (12) |
107 (31) |
131 (17) |
141 (27) |
155 (3) |
161 (9) |
172 (20) |
185 (33) |
188 (36) |
42ed257/128
See 42ed257/128.
45ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.7 | +2.0 | +2.6 | +10.1 | +5.3 | +11.0 | +2.5 | -2.3 | -11.3 | -10.4 | +8.3 |
Relative (%) | +25.2 | +7.6 | +9.8 | +37.6 | +19.6 | +41.2 | +9.3 | -8.8 | -42.2 | -38.6 | +30.8 | |
Steps (reduced) |
45 (0) |
71 (26) |
104 (14) |
126 (36) |
155 (20) |
166 (31) |
183 (3) |
190 (10) |
202 (22) |
217 (37) |
222 (42) |
45edo for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.0 | -8.6 | -13.0 | -8.8 | +8.7 | +12.8 | +1.7 | -4.2 | +11.7 | +10.4 | +1.6 |
Relative (%) | +0.0 | -32.3 | -48.7 | -33.1 | +32.6 | +48.0 | +6.4 | -15.7 | +44.0 | +39.1 | +6.1 | |
Steps (reduced) |
45 (0) |
71 (26) |
104 (14) |
126 (36) |
156 (21) |
167 (32) |
184 (4) |
191 (11) |
204 (24) |
219 (39) |
223 (43) |
54ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.75 | -2.44 | +7.09 | +5.60 | +5.26 | +6.57 | -10.92 | -2.35 | +2.10 | +3.05 | -0.68 |
Relative (%) | +30.2 | -10.9 | +31.7 | +25.1 | +23.6 | +29.4 | -48.8 | -10.5 | +9.4 | +13.6 | -3.0 | |
Steps (reduced) |
54 (0) |
85 (31) |
125 (17) |
151 (43) |
186 (24) |
199 (37) |
219 (3) |
228 (12) |
243 (27) |
261 (45) |
266 (50) |
45edo for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.00 | +9.16 | -8.54 | +8.95 | +4.24 | +3.92 | +6.16 | -8.62 | -6.05 | -7.35 | +10.52 |
Relative (%) | +0.0 | +41.2 | -38.4 | +40.3 | +19.1 | +17.6 | +27.7 | -38.8 | -27.2 | -33.1 | +47.3 | |
Steps (reduced) |
54 (0) |
86 (32) |
125 (17) |
152 (44) |
187 (25) |
200 (38) |
221 (5) |
229 (13) |
244 (28) |
262 (46) |
268 (52) |
59ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.75 | +0.21 | -4.65 | +5.98 | +0.72 | -2.14 | +3.86 | -4.62 | -8.13 | -0.36 | +6.90 |
Relative (%) | +33.0 | +1.0 | -22.8 | +29.2 | +3.5 | -10.5 | +18.9 | -22.6 | -39.7 | -1.8 | +33.7 | |
Steps (reduced) |
59 (0) |
93 (34) |
136 (18) |
165 (47) |
203 (26) |
217 (40) |
240 (4) |
249 (13) |
265 (29) |
285 (49) |
291 (55) |
59edo for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.00 | +9.91 | +0.13 | +7.45 | -2.17 | -6.63 | -3.26 | +7.57 | +2.23 | +7.71 | -6.05 |
Relative (%) | +0.0 | +48.7 | +0.6 | +36.6 | -10.6 | -32.6 | -16.0 | +37.2 | +11.0 | +37.9 | -29.8 | |
Steps (reduced) |
59 (0) |
94 (35) |
137 (19) |
166 (48) |
204 (27) |
218 (41) |
241 (5) |
251 (15) |
267 (31) |
287 (51) |
292 (56) |
64ed257/128
Harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +6.75 | +2.45 | +4.29 | +6.30 | -3.12 | +9.36 | -2.54 | -6.54 | +2.10 | -3.24 | -5.57 |
Relative (%) | +35.8 | +13.0 | +22.8 | +33.4 | -16.5 | +49.6 | -13.4 | -34.7 | +11.1 | -17.2 | -29.5 | |
Steps (reduced) |
64 (0) |
101 (37) |
148 (20) |
179 (51) |
220 (28) |
236 (44) |
260 (4) |
270 (14) |
288 (32) |
309 (53) |
315 (59) |
64edo for comparison:
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | Absolute (¢) | +0.00 | -8.21 | +7.44 | +6.17 | -7.57 | +3.22 | +7.54 | +2.49 | +9.23 | +1.67 | -1.29 |
Relative (%) | +0.0 | -43.8 | +39.7 | +32.9 | -40.4 | +17.2 | +40.2 | +13.3 | +49.2 | +8.9 | -6.9 | |
Steps (reduced) |
64 (0) |
101 (37) |
149 (21) |
180 (52) |
221 (29) |
237 (45) |
262 (6) |
272 (16) |
290 (34) |
311 (55) |
317 (61) |