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Moving the bridge hack: Difference between revisions

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== Derivation of the resulting scale ==
== 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
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


Line 30: 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 40: 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 95: 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 105: 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 147: 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 157: 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 220: 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 230: 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 286: 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 296: 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 375: 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 385: 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 413: 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 475: 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 518: 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 544: 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 560: 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 626: 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 657: 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 707: 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 747: 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 817: 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 844: 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 890: 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 926: 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 1,000: 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 1,023: 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 1,065: 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 1,103: 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 1,113: 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 1,182: 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 1,192: 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 1,231: 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)
Line 1,241: Line 1,240:
| 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
Retrieved from "https://en.xen.wiki/w/Moving_the_bridge_hack"