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

[math]\displaystyle{ 2^{-i/N} \text{ for } i = 0, 1, 2, 3\dots }[/math]

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

[math]\displaystyle{ 2^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots }[/math]

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

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

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.

Each fret will raise the pitch by a different amount, as these charts show

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.

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.

See also

Tuning A Kite Guitar To 31edo or 62edo