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Moving the bridge hack: Difference between revisions

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If you have a [[12edo|12edo]] guitar, or other fretted [[string_instruments|string instrument]], and you want to play in an EDO that is numerically near 12 (e.g. [[11edo|11edo]] or [[13edo|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?
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.
...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 ==
== Derivation of the resulting scale ==
 
Let the EDO number of the original instrument be ''N'' (so very often {{nowrap|''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
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>2^{-i/N} \text{ for } i = 0, 1, 2, 3\dots</math>
<math>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
If the bridge is moved so that the new scale length is ''x'', this adds ({{nowrap|''x'' − 1}}) to all string lengths, so the new string lengths are simply


<math>2^{-i/N} + x - 1 \text{ for } i = 0, 1, 2, 3\dots</math>
<math>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
The frequencies are inversely proportional to the string lengths. If we plug in {{nowrap|''i'' {{=}} 0}} to the above formula, we get ''x'', so the frequency ratios relative to the open string are


<math>\frac{x}{2^{-i/N} + x - 1} \text{ for } i = 0, 1, 2, 3\dots</math>
<math>\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.
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
[[File:Bridge-sliding chart 1.jpg|none|thumb]]
[[File:Bridge-sliding chart 2.jpg|none|thumb]]
[[File:Bridge-sliding chart 3.jpg|none|thumb]]


== Examples for converting a 12edo instrument ==
== Examples for converting a 12edo instrument ==
Line 25: Line 29:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 9edo
! Deviation from 9edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 35: Line 39:
| 1
| 1
| 124.191
| 124.191
| -9.142
| −9.142
|-
|-
| 2
| 2
| 250.196
| 250.196
| -16.471
| −16.471
|-
|-
| 3
| 3
| 378.179
| 378.179
| -21.821
| −21.821
|-
|-
| 4
| 4
| 508.328
| 508.328
| -25.005
| −25.005
|-
|-
| 5
| 5
| 640.854
| 640.854
| -25.813
| −25.813
|-
|-
| 6
| 6
| 775.993
| 775.993
| -24.007
| −24.007
|-
|-
| 7
| 7
| 914.017
| 914.017
| -19.317
| −19.317
|-
|-
| 8
| 8
| 1055.234
| 1055.234
| -11.433
| −11.433
|-
|-
| 9
| 9
Line 90: Line 94:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 9edo
! Deviation from 9edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 100: Line 104:
| 1
| 1
| 127.785
| 127.785
| -5.548
| −5.548
|-
|-
| 2
| 2
| 257.717
| 257.717
| -8.950
| −8.950
|-
|-
| 3
| 3
| 390.004
| 390.004
| -9.996
| −9.996
|-
|-
| 4
| 4
| 524.880
| 524.880
| -8.453
| −8.453
|-
|-
| 5
| 5
| 662.614
| 662.614
| -4.053
| −4.053
|-
|-
| 6
| 6
Line 142: Line 146:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 10edo
! Deviation from 10edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 152: Line 156:
| 1
| 1
| 114.426
| 114.426
| -5.574
| −5.574
|-
|-
| 2
| 2
| 229.842
| 229.842
| -10.158
| −10.158
|-
|-
| 3
| 3
| 346.326
| 346.326
| -13.674
| −13.674
|-
|-
| 4
| 4
| 463.963
| 463.963
| -16.037
| −16.037
|-
|-
| 5
| 5
| 582.847
| 582.847
| -17.153
| −17.153
|-
|-
| 6
| 6
| 703.082
| 703.082
| -16.918
| −16.918
|-
|-
| 7
| 7
| 824.780
| 824.780
| -15.220
| −15.220
|-
|-
| 8
| 8
| 948.070
| 948.070
| -11.930
| −11.930
|-
|-
| 9
| 9
| 1073.091
| 1073.091
| -6.909
| −6.909
|-
|-
| 10
| 10
Line 215: Line 219:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 10edo
! Deviation from 10edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 225: Line 229:
| 1
| 1
| 115.766
| 115.766
| -4.234
| −4.234
|-
|-
| 2
| 2
| 232.628
| 232.628
| -7.372
| −7.372
|-
|-
| 3
| 3
| 350.674
| 350.674
| -9.326
| −9.326
|-
|-
| 4
| 4
| 470.000
| 470.000
| -10.000
| −10.000
|-
|-
| 5
| 5
| 590.714
| 590.714
| -9.286
| −9.286
|-
|-
| 6
| 6
| 712.933
| 712.933
| -7.067
| −7.067
|-
|-
| 7
| 7
| 836.788
| 836.788
| -3.212
| −3.212
|-
|-
| 8
| 8
Line 281: Line 285:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 11edo
! Deviation from 11edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 291: Line 295:
| 1
| 1
| 106.521
| 106.521
| -2.570
| −2.570
|-
|-
| 2
| 2
| 213.456
| 213.456
| -4.726
| −4.726
|-
|-
| 3
| 3
| 320.834
| 320.834
| -6.439
| −6.439
|-
|-
| 4
| 4
| 428.685
| 428.685
| -7.678
| −7.678
|-
|-
| 5
| 5
| 537.043
| 537.043
| -8.412
| −8.412
|-
|-
| 6
| 6
| 645.941
| 645.941
| -8.604
| −8.604
|-
|-
| 7
| 7
| 755.419
| 755.419
| -8.218
| −8.218
|-
|-
| 8
| 8
| 865.517
| 865.517
| -7.210
| −7.210
|-
|-
| 9
| 9
| 976.280
| 976.280
| -5.538
| −5.538
|-
|-
| 10
| 10
| 1087.757
| 1087.757
| -3.152
| −3.152
|-
|-
| 11
| 11
Line 370: Line 374:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 11edo
! Deviation from 11edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 380: Line 384:
| 1
| 1
| 108.322
| 108.322
| -0.769
| −0.769
|-
|-
| 2
| 2
| 217.183
| 217.183
| -0.998
| −0.998
|-
|-
| 3
| 3
| 326.621
| 326.621
| -0.652
| −0.652
|-
|-
| 4
| 4
Line 408: Line 412:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 13edo
! Deviation from 13edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 470: Line 474:
| 14
| 14
| 1289.093
| 1289.093
| -3.215
| −3.215
|-
|-
| 15
| 15
| 1377.612
| 1377.612
| -7.004
| −7.004
|-
|-
| 16
| 16
| 1465.530
| 1465.530
| -11.393
| −11.393
|-
|-
| 17
| 17
| 1552.822
| 1552.822
| -16.409
| −16.409
|-
|-
| 18
| 18
| 1639.458
| 1639.458
| -22.080
| −22.080
|-
|-
| 19
| 19
| 1725.412
| 1725.412
| -28.434
| −28.434
|-
|-
| 20
| 20
| 1810.654
| 1810.654
| -35.500
| −35.500
|-
|-
| 21
| 21
| 1895.155
| 1895.155
| -43.307
| −43.307
|-
|-
| 22
| 22
| 1978.883
| 1978.883
| -51.886
| −51.886
|-
|-
| 23
| 23
| 2061.810
| 2061.810
| -61.267
| −61.267
|}
|}


Line 513: Line 517:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 13edo
! Deviation from 13edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 539: Line 543:
| 5
| 5
| 460.950
| 460.950
| -0.588
| −0.588
|-
|-
| 6
| 6
| 551.821
| 551.821
| -2.025
| −2.025
|-
|-
| 7
| 7
| 642.202
| 642.202
| -3.952
| −3.952
|}
|}


Line 555: Line 559:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 14edo
! Deviation from 14edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 621: Line 625:
| 15
| 15
| 1279.864
| 1279.864
| -5.850
| −5.850
|-
|-
| 16
| 16
| 1358.784
| 1358.784
| -12.645
| −12.645
|-
|-
| 17
| 17
| 1436.727
| 1436.727
| -20.416
| −20.416
|-
|-
| 18
| 18
| 1513.660
| 1513.660
| -29.197
| −29.197
|-
|-
| 19
| 19
| 1589.553
| 1589.553
| -39.019
| −39.019
|-
|-
| 20
| 20
| 1664.373
| 1664.373
| -49.913
| −49.913
|-
|-
| 21
| 21
| 1738.090
| 1738.090
| -61.910
| −61.910
|}
|}


Line 652: Line 656:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 14edo
! Deviation from 14edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 702: Line 706:
| 11
| 11
| 941.593
| 941.593
| -1.264
| −1.264
|-
|-
| 12
| 12
| 1022.558
| 1022.558
| -6.014
| −6.014
|-
|-
| 13
| 13
| 1102.616
| 1102.616
| -11.670
| −11.670
|-
|-
| 14
| 14
| 1181.736
| 1181.736
| -18.264
| −18.264
|-
|-
| 15
| 15
| 1259.886
| 1259.886
| -25.829
| −25.829
|-
|-
| 16
| 16
| 1337.033
| 1337.033
| -34.395
| −34.395
|-
|-
| 17
| 17
| 1413.146
| 1413.146
| -43.996
| −43.996
|-
|-
| 18
| 18
| 1488.194
| 1488.194
| -54.663
| −54.663
|-
|-
| 19
| 19
| 1562.144
| 1562.144
| -66.427
| −66.427
|}
|}


Line 742: Line 746:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 15edo
! Deviation from 15edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 812: Line 816:
| 16
| 16
| 1271.967
| 1271.967
| -8.033
| −8.033
|-
|-
| 17
| 17
| 1342.754
| 1342.754
| -17.246
| −17.246
|-
|-
| 18
| 18
| 1412.333
| 1412.333
| -27.667
| −27.667
|-
|-
| 19
| 19
| 1480.675
| 1480.675
| -39.325
| −39.325
|-
|-
| 20
| 20
| 1547.754
| 1547.754
| -52.246
| −52.246
|-
|-
| 21
| 21
| 1613.546
| 1613.546
| -66.454
| −66.454
|}
|}


Line 839: Line 843:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 15edo
! Deviation from 15edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 885: Line 889:
| 10
| 10
| 796.700
| 796.700
| -3.300
| −3.300
|-
|-
| 11
| 11
| 870.897
| 870.897
| -9.103
| −9.103
|-
|-
| 12
| 12
| 943.974
| 943.974
| -16.026
| −16.026
|-
|-
| 13
| 13
| 1015.899
| 1015.899
| -24.101
| −24.101
|-
|-
| 14
| 14
| 1086.643
| 1086.643
| -33.357
| −33.357
|-
|-
| 15
| 15
| 1156.178
| 1156.178
| -43.822
| −43.822
|-
|-
| 16
| 16
| 1224.475
| 1224.475
| -55.525
| −55.525
|-
|-
| 17
| 17
| 1291.509
| 1291.509
| -68.491
| −68.491
|}
|}


Line 921: Line 925:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 16edo
! Deviation from 16edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 995: Line 999:
| 17
| 17
| 1265.142
| 1265.142
| -9.858
| −9.858
|-
|-
| 18
| 18
| 1328.962
| 1328.962
| -21.038
| −21.038
|-
|-
| 19
| 19
| 1391.438
| 1391.438
| -33.562
| −33.562
|-
|-
| 20
| 20
| 1452.550
| 1452.550
| -47.450
| −47.450
|-
|-
| 21
| 21
| 1512.281
| 1512.281
| -62.719
| −62.719
|}
|}


Line 1,018: Line 1,022:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 16edo
! Deviation from 16edo
|-
|-
| 0 (nut)
| 0 (nut)
Line 1,060: Line 1,064:
| 9
| 9
| 673.193
| 673.193
| -1.807
| −1.807
|-
|-
| 10
| 10
| 742.290
| 742.290
| -7.710
| −7.710
|-
|-
| 11
| 11
| 810.140
| 810.140
| -14.860
| −14.860
|-
|-
| 12
| 12
| 876.717
| 876.717
| -23.283
| −23.283
|-
|-
| 13
| 13
| 941.997
| 941.997
| -33.003
| −33.003
|-
|-
| 14
| 14
| 1005.957
| 1005.957
| -44.043
| −44.043
|-
|-
| 15
| 15
| 1068.574
| 1068.574
| -56.426
| −56.426
|-
|-
| 16
| 16
| 1129.830
| 1129.830
| -70.170
| −70.170
|}
|}


=== Bohlen-Pierce ===
=== 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.
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:
The naive method of making the 13th fret an exact 3/1 would be terrible:
Line 1,098: Line 1,102:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 13ed3
! Deviation from 13ed3
|-
|-
| 0 (nut)
| 0 (nut)
Line 1,108: Line 1,112:
| 1
| 1
| 127.235
| 127.235
| -19.072
| −19.072
|-
|-
| 2
| 2
| 256.565
| 256.565
| -36.050
| −36.050
|-
|-
| 3
| 3
| 388.191
| 388.191
| -50.732
| −50.732
|-
|-
| 4
| 4
| 522.340
| 522.340
| -62.891
| −62.891
|-
|-
| 5
| 5
| 659.270
| 659.270
| -72.269
| −72.269
|-
|-
| 6
| 6
| 799.275
| 799.275
| -78.572
| −78.572
|-
|-
| 7
| 7
| 942.691
| 942.691
| -81.462
| −81.462
|-
|-
| 8
| 8
| 1089.910
| 1089.910
| -80.552
| −80.552
|-
|-
| 9
| 9
| 1241.382
| 1241.382
| -75.387
| −75.387
|-
|-
| 10
| 10
| 1397.638
| 1397.638
| -65.439
| −65.439
|-
|-
| 11
| 11
| 1559.303
| 1559.303
| -50.081
| −50.081
|-
|-
| 12
| 12
| 1727.124
| 1727.124
| -28.568
| −28.568
|-
|-
| 13
| 13
Line 1,177: Line 1,181:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 13ed3
! Deviation from 13ed3
|-
|-
| 0 (nut)
| 0 (nut)
Line 1,187: Line 1,191:
| 1
| 1
| 138.788
| 138.788
| -7.520
| −7.520
|-
|-
| 2
| 2
| 280.853
| 280.853
| -11.763
| −11.763
|-
|-
| 3
| 3
| 426.563
| 426.563
| -12.360
| −12.360
|-
|-
| 4
| 4
| 576.345
| 576.345
| -8.886
| −8.886
|-
|-
| 5
| 5
| 730.697
| 730.697
| -0.841
| −0.841
|-
|-
| 6
| 6
Line 1,226: Line 1,230:
{| class="wikitable right-2 right-3"
{| class="wikitable right-2 right-3"
|-
|-
! | Fret number
! Fret number
! | Cents
! Cents
! | Deviation from 13e3
! Deviation from 13e3
|-
|-
| 0 (nut)
| 0 (nut)
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| 1
| 1
| 140.285
| 140.285
| -6.023
| −6.023
|-
|-
| 2
| 2
| 284.014
| 284.014
| -8.601
| −8.601
|-
|-
| 3
| 3
| 431.581
| 431.581
| -7.343
| −7.343
|-
|-
| 4
| 4
| 583.443
| 583.443
| -1.787
| −1.787
|-
|-
| 5
| 5
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== Advanced tricks ==
== 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.
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]]


[[Category:Guitar]]
[[Category:Guitar]]
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