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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>
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{| 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)
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| 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
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| −21.821
|-
|-
| 4
| 4
| 508.328
| 508.328
| -25.005
| −25.005
|-
|-
| 5
| 5
| 640.854
| 640.854
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| −25.813
|-
|-
| 6
| 6
| 775.993
| 775.993
| -24.007
| −24.007
|-
|-
| 7
| 7
| 914.017
| 914.017
| -19.317
| −19.317
|-
|-
| 8
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| 1055.234
| 1055.234
| -11.433
| −11.433
|-
|-
| 9
| 9
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{| 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)
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| 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
|-
|-
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| 5
| 662.614
| 662.614
| -4.053
| −4.053
|-
|-
| 6
| 6
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{| 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)
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| 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
|-
|-
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| 6
| 703.082
| 703.082
| -16.918
| −16.918
|-
|-
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| 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
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{| 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)
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| 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
|-
|-
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| 712.933
| 712.933
| -7.067
| −7.067
|-
|-
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| 836.788
| 836.788
| -3.212
| −3.212
|-
|-
| 8
| 8
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{| 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)
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| 1
| 1
| 106.521
| 106.521
| -2.570
| −2.570
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|-
| 2
| 2
| 213.456
| 213.456
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| −4.726
|-
|-
| 3
| 3
| 320.834
| 320.834
| -6.439
| −6.439
|-
|-
| 4
| 4
| 428.685
| 428.685
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| −7.678
|-
|-
| 5
| 5
| 537.043
| 537.043
| -8.412
| −8.412
|-
|-
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| 6
| 645.941
| 645.941
| -8.604
| −8.604
|-
|-
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| 7
| 755.419
| 755.419
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| −8.218
|-
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| 865.517
| 865.517
| -7.210
| −7.210
|-
|-
| 9
| 9
| 976.280
| 976.280
| -5.538
| −5.538
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|-
| 10
| 10
| 1087.757
| 1087.757
| -3.152
| −3.152
|-
|-
| 11
| 11
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{| 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)
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| 1
| 1
| 108.322
| 108.322
| -0.769
| −0.769
|-
|-
| 2
| 2
| 217.183
| 217.183
| -0.998
| −0.998
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|-
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| 3
| 326.621
| 326.621
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| −0.652
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|-
| 4
| 4
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{| 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)
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| 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
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| −16.409
|-
|-
| 18
| 18
| 1639.458
| 1639.458
| -22.080
| −22.080
|-
|-
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| 19
| 1725.412
| 1725.412
| -28.434
| −28.434
|-
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| 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
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|-
| 23
| 23
| 2061.810
| 2061.810
| -61.267
| −61.267
|}
|}


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{| 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)
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| 5
| 5
| 460.950
| 460.950
| -0.588
| −0.588
|-
|-
| 6
| 6
| 551.821
| 551.821
| -2.025
| −2.025
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|-
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| 7
| 642.202
| 642.202
| -3.952
| −3.952
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|}


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{| 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)
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| 15
| 15
| 1279.864
| 1279.864
| -5.850
| −5.850
|-
|-
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| 1358.784
| 1358.784
| -12.645
| −12.645
|-
|-
| 17
| 17
| 1436.727
| 1436.727
| -20.416
| −20.416
|-
|-
| 18
| 18
| 1513.660
| 1513.660
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| −29.197
|-
|-
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| 19
| 1589.553
| 1589.553
| -39.019
| −39.019
|-
|-
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| 20
| 1664.373
| 1664.373
| -49.913
| −49.913
|-
|-
| 21
| 21
| 1738.090
| 1738.090
| -61.910
| −61.910
|}
|}


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{| 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)
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| 11
| 11
| 941.593
| 941.593
| -1.264
| −1.264
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| 12
| 12
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| 1022.558
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| −6.014
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| 1102.616
| 1102.616
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| −11.670
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|-
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| 1181.736
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| −18.264
|-
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| 1259.886
| 1259.886
| -25.829
| −25.829
|-
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| 1337.033
| 1337.033
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| −34.395
|-
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| 1413.146
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| −43.996
|-
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| 1488.194
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|-
|-
| 19
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| 1562.144
| 1562.144
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| −66.427
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|}


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{| 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)
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| 16
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| 1271.967
| 1271.967
| -8.033
| −8.033
|-
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| 1342.754
| -17.246
| −17.246
|-
|-
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| 18
| 1412.333
| 1412.333
| -27.667
| −27.667
|-
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| 1480.675
| 1480.675
| -39.325
| −39.325
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| 1547.754
| 1547.754
| -52.246
| −52.246
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| 1613.546
| 1613.546
| -66.454
| −66.454
|}
|}


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{| 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)
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| 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
|-
|-
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| 15
| 1156.178
| 1156.178
| -43.822
| −43.822
|-
|-
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| 16
| 1224.475
| 1224.475
| -55.525
| −55.525
|-
|-
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| 1291.509
| 1291.509
| -68.491
| −68.491
|}
|}


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{| 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)
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| 17
| 17
| 1265.142
| 1265.142
| -9.858
| −9.858
|-
|-
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| 18
| 1328.962
| 1328.962
| -21.038
| −21.038
|-
|-
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| 19
| 1391.438
| 1391.438
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| −33.562
|-
|-
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| 20
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| 1452.550
| -47.450
| −47.450
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|-
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| 1512.281
| 1512.281
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| −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
|-
|-
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| 941.997
| 941.997
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| −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
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{| 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
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{| 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
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