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If you have a 12edo guitar, or other fretted string instrument, and you want to play in an EDO that is numerically near 12 (e.g. 11edo or 13edo), then rather than redoing the whole fretboard, you might be tempted simply to move the bridge. If you move the bridge so that the 13th fret is now precisely 2/1, the frets will play precisely 13edo, right?

...well, actually, no. The frets form a geometric series of lengths that converges at a specific point, which is where the bridge ought to be. (That's what an EDO is - a geometric sequence of frequencies, corresponding to a geometric sequence of string lengths.) If you move the bridge, the new string lengths no longer form a mathematically correct geometric sequence. However, depending on what range of the fretboard you want to be usable, and what accuracy you desire, a moving-the-bridge solution may be possible.

Derivation of the resulting scale


Let the EDO number of the original instrument be N (so very often N=12). Let the original scale length of the instrument (distance from bridge to nut) be 1. In other words we're measuring all lengths relative to the original scale length. Then the playable string lengths of the unmodified instrument are



If the bridge is moved so that the new scale length is x, this adds (x-1) to all string lengths, so the new string lengths are simply



The frequencies are inversely proportional to the string lengths. If we plug in i=0 to the above formula, we get x, so the frequency ratios relative to the open string are



Converting those frequency ratios into cents in the usual way (taking the log to base 2 and multiplying by 1200) gives the new scale in cents.

Examples for converting a 12edo instrument


9edo


The naive way to position the bridge for 9edo would be to make the 9th fret play an exact 2/1. However, this causes a rather large amount of error in some lower frets (as well as, of course, the higher ones above fret 9):

Fret number
Cents
Deviation from 9edo
0 (nut)
0.000
0.000
1
124.191
-9.142
2
250.196
-16.471
3
378.179
-21.821
4
508.328
-25.005
5
640.854
-25.813
6
775.993
-24.007
7
914.017
-19.317
8
1055.234
-11.433
9
1200.000
0.000
10
1348.726
15.392
11
1501.890
35.223
12
1660.056
60.056
13
1823.890
90.557

Can the error be reduced? Yes, at the expense of having a smaller usable range of fretboard. For example, let's say we want to limit the maximum error to 10 cents. How many frets can we use at this level of accuracy?

Fret number
Cents
Deviation from 9edo
0 (nut)
0.000
0.000
1
127.785
-5.548
2
257.717
-8.950
3
390.004
-9.996
4
524.880
-8.453
5
662.614
-4.053
6
803.509
3.509
7
947.916
14.583
8
1096.241
29.574
9
1248.955
48.955

The answer is six frets. (It's impossible to make the seventh fret more accurate without the error of the third fret exceeding 10 cents.) Perhaps surprisingly, this level of accuracy for the first six frets is only achievable by making the 9th fret at least 1249 cents, rather than 2/1.

10edo


Pure 2/1:

Fret number
Cents
Deviation from 10edo
0 (nut)
0.000
0.000
1
114.426
-5.574
2
229.842
-10.158
3
346.326
-13.674
4
463.963
-16.037
5
582.847
-17.153
6
703.082
-16.918
7
824.780
-15.220
8
948.070
-11.930
9
1073.091
-6.909
10
1200.000
0.000
11
1328.973
8.973
12
1460.208
20.208
13
1593.926
33.926
14
1730.381
50.381
15
1869.859
69.859

Max error limited to 10 cents (only the first 9 frets may be used):

Fret number
Cents
Deviation from 10edo
0 (nut)
0.000
0.000
1
115.766
-4.234
2
232.628
-7.372
3
350.674
-9.326
4
470.000
-10.000
5
590.714
-9.286
6
712.933
-7.067
7
836.788
-3.212
8
962.426
2.426
9
1090.010
10.010
10
1219.720
19.720
11
1351.764
31.764
12
1486.373
46.373
13
1623.811
63.811

11edo


Pure 2/1:

Fret number
Cents
Deviation from 11edo
0 (nut)
0.000
0.000
1
106.521
-2.570
2
213.456
-4.726
3
320.834
-6.439
4
428.685
-7.678
5
537.043
-8.412
6
645.941
-8.604
7
755.419
-8.218
8
865.517
-7.210
9
976.280
-5.538
10
1087.757
-3.152
11
1200.000
0.000
12
1313.066
3.975
13
1427.017
8.836
14
1541.922
14.649
15
1657.854
21.491
16
1774.895
29.441
17
1893.135
38.589
18
2012.670
49.034
19
2133.611
60.884

Super-accurate, error limited to 1 cent (only the first 4 frets may be used):

Fret number
Cents
Deviation from 11edo
0 (nut)
0.000
0.000
1
108.322
-0.769
2
217.183
-0.998
3
326.621
-0.652
4
436.676
0.313
5
547.394
1.939
6
658.821
4.276

13edo


Pure 2/1:

Fret number
Cents
Deviation from 13edo
0 (nut)
0.000
0.000
1
94.538
2.230
2
188.770
4.154
3
282.679
5.756
4
376.250
7.019
5
469.465
7.926
6
562.305
8.458
7
654.751
8.597
8
746.784
8.322
9
838.383
7.613
10
929.526
6.449
11
1020.193
4.808
12
1110.358
2.666
13
1200.000
0.000
14
1289.093
-3.215
15
1377.612
-7.004
16
1465.530
-11.393
17
1552.822
-16.409
18
1639.458
-22.080
19
1725.412
-28.434
20
1810.654
-35.500
21
1895.155
-43.307
22
1978.883
-51.886
23
2061.810
-61.267

Super-accurate, error limited to 1 cent (only the first 5 frets may be used):

Fret number
Cents
Deviation from 13edo
0 (nut)
0.000
0.000
1
93.001
0.693
2
185.616
1.000
3
277.826
0.903
4
369.611
0.380
5
460.950
-0.588
6
551.821
-2.025
7
642.202
-3.952

14edo


Pure 2/1:

Fret number
Cents
Deviation from 14edo
0 (nut)
0.000
0.000
1
89.903
4.189
2
179.269
7.841
3
268.074
10.931
4
356.292
13.435
5
443.896
15.324
6
530.859
16.573
7
617.153
17.153
8
702.749
17.035
9
787.619
16.190
10
871.731
14.588
11
955.057
12.200
12
1037.564
8.993
13
1119.223
4.937
14
1200.000
0.000
15
1279.864
-5.850
16
1358.784
-12.645
17
1436.727
-20.416
18
1513.660
-29.197
19
1589.553
-39.019
20
1664.373
-49.913
21
1738.090
-61.910

Max error limited to 10 cents (only the first 12 frets may be used):

Fret number
Cents
Deviation from 14edo
0 (nut)
0.000
0.000
1
88.925
3.210
2
177.267
5.839
3
265.002
7.859
4
352.101
9.244
5
438.538
9.966
6
524.283
9.997
7
609.307
9.307
8
693.580
7.866
9
777.073
5.645
10
859.755
2.612
11
941.593
-1.264
12
1022.558
-6.014
13
1102.616
-11.670
14
1181.736
-18.264
15
1259.886
-25.829
16
1337.033
-34.395
17
1413.146
-43.996
18
1488.194
-54.663
19
1562.144
-66.427

15edo


Pure 2/1:

Fret number
Cents
Deviation from 15edo
0 (nut)
0.000
0.000
1
85.927
5.927
2
171.140
11.140
3
255.611
15.611
4
339.310
19.310
5
422.204
22.204
6
504.264
24.264
7
585.458
25.458
8
665.754
25.754
9
745.120
25.120
10
823.524
23.524
11
900.935
20.935
12
977.319
17.319
13
1052.645
12.645
14
1126.883
6.883
15
1200.000
0.000
16
1271.967
-8.033
17
1342.754
-17.246
18
1412.333
-27.667
19
1480.675
-39.325
20
1547.754
-52.246
21
1613.546
-66.454

Max error limited to 10 cents (only the first 11 frets may be used):

Fret number
Cents
Deviation from 15edo
0 (nut)
0.000
0.000
1
83.670
3.670
2
166.535
6.535
3
248.564
8.564
4
329.727
9.727
5
409.990
9.990
6
489.322
9.322
7
567.691
7.691
8
645.065
5.065
9
721.412
1.412
10
796.700
-3.300
11
870.897
-9.103
12
943.974
-16.026
13
1015.899
-24.101
14
1086.643
-33.357
15
1156.178
-43.822
16
1224.475
-55.525
17
1291.509
-68.491

16edo


Pure 2/1:

Fret number
Cents
Deviation from 16edo
0 (nut)
0.000
0.000
1
82.483
7.483
2
164.117
14.117
3
244.868
19.868
4
324.706
24.706
5
403.598
28.598
6
481.511
31.511
7
558.415
33.415
8
634.278
34.278
9
709.066
34.066
10
782.751
32.751
11
855.300
30.300
12
926.683
26.683
13
996.873
21.873
14
1065.840
15.840
15
1133.558
8.558
16
1200.000
0.000
17
1265.142
-9.858
18
1328.962
-21.038
19
1391.438
-33.562
20
1452.550
-47.450
21
1512.281
-62.719

Max error limited to 10 cents (only the first 10 frets may be used):

Fret number
Cents
Deviation from 16edo
0 (nut)
0.000
0.000
1
78.993
3.993
2
157.011
7.011
3
234.022
9.022
4
309.995
9.995
5
384.898
9.898
6
458.700
8.700
7
531.369
6.369
8
602.877
2.877
9
673.193
-1.807
10
742.290
-7.710
11
810.140
-14.860
12
876.717
-23.283
13
941.997
-33.003
14
1005.957
-44.043
15
1068.574
-56.426
16
1129.830
-70.170

Bohlen-Pierce


As an example of a non-octave temperament, let's try to approximate the equal-tempered Bohlen-Pierce scale (13ed3) by moving the bridge of a 12edo guitar.

The naive method of making the 13th fret an exact 3/1 would be terrible:

Fret number
Cents
Deviation from 13ed3
0 (nut)
0.000
0.000
1
127.235
-19.072
2
256.565
-36.050
3
388.191
-50.732
4
522.340
-62.891
5
659.270
-72.269
6
799.275
-78.572
7
942.691
-81.462
8
1089.910
-80.552
9
1241.382
-75.387
10
1397.638
-65.439
11
1559.303
-50.081
12
1727.124
-28.568
13
1902.000
0.000
14
2085.028
36.720
15
2277.566
82.951
16
2481.323
140.400

Even though the error of the 13th fret is 0, the error of frets 7 and 8 is about 80 cents, which is unacceptable.

A much more satisfactory solution is to minimize the maximum error over the first 6 frets:

Fret number
Cents
Deviation from 13ed3
0 (nut)
0.000
0.000
1
138.788
-7.520
2
280.853
-11.763
3
426.563
-12.360
4
576.345
-8.886
5
730.697
-0.841
6
890.206
12.360
7
1055.569
31.415
8
1227.620
57.158
9
1407.376
90.606

Or even just the first 5, for better accuracy:

Fret number
Cents
Deviation from 13e3
0 (nut)
0.000
0.000
1
140.285
-6.023
2
284.014
-8.601
3
431.581
-7.343
4
583.443
-1.787
5
740.140
8.601
6
902.305
24.459
7
1070.696
46.543
8
1246.229
75.768

If the open strings are tuned in 9/7s or 7/5s, this makes a perfectly playable BP guitar with fret error under 9 cents. The only restriction is that you should never play frets 6 or above, because they're increasingly out of tune.

Advanced tricks


By adjusting the nut in addition to the bridge, it may be possible to make a surprisingly large portion of the fretboard usable with good accuracy. This is because there is another degree of freedom to tweak. For decreasing the EDO number you would want to either raise the nut or move it away from the first fret, and for increasing the EDO number you would want to either lower it or move it towards the first fret.