Moving the bridge hack

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

$2^{-i/N} \text{ for } i = 0, 1, 2, 3\dots$

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

$2^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots$

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

$\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots$

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 (13edt) 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.