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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? |
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| ...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. |
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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 {{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 | |
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| <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> |
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| 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 |
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| <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> |
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| 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 |
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| <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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| 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. |
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| =Examples for converting a 12edo instrument=
| | Each fret will raise the pitch by a different amount, as these charts show |
| | | [[File:Bridge-sliding chart 1.jpg|none|thumb]] |
| ==9edo==
| | [[File:Bridge-sliding chart 2.jpg|none|thumb]] |
| | [[File:Bridge-sliding chart 3.jpg|none|thumb]] |
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| | == 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): | | 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): |
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| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 9edo | | ! Deviation from 9edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 124.191
| | | 124.191 |
| | style="text-align:right;" | -9.142 | | | −9.142 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 250.196
| | | 250.196 |
| | style="text-align:right;" | -16.471 | | | −16.471 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 378.179
| | | 378.179 |
| | style="text-align:right;" | -21.821 | | | −21.821 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 508.328
| | | 508.328 |
| | style="text-align:right;" | -25.005 | | | −25.005 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 640.854
| | | 640.854 |
| | style="text-align:right;" | -25.813 | | | −25.813 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 775.993
| | | 775.993 |
| | style="text-align:right;" | -24.007 | | | −24.007 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 914.017
| | | 914.017 |
| | style="text-align:right;" | -19.317 | | | −19.317 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 1055.234
| | | 1055.234 |
| | style="text-align:right;" | -11.433 | | | −11.433 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 1200.000
| | | 1200.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 1348.726
| | | 1348.726 |
| | style="text-align:right;" | 15.392
| | | 15.392 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 1501.890
| | | 1501.890 |
| | style="text-align:right;" | 35.223
| | | 35.223 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1660.056
| | | 1660.056 |
| | style="text-align:right;" | 60.056
| | | 60.056 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1823.890
| | | 1823.890 |
| | style="text-align:right;" | 90.557
| | | 90.557 |
| |} | | |} |
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| 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? | | 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? |
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| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 9edo | | ! Deviation from 9edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 127.785
| | | 127.785 |
| | style="text-align:right;" | -5.548 | | | −5.548 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 257.717
| | | 257.717 |
| | style="text-align:right;" | -8.950 | | | −8.950 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 390.004
| | | 390.004 |
| | style="text-align:right;" | -9.996 | | | −9.996 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 524.880
| | | 524.880 |
| | style="text-align:right;" | -8.453 | | | −8.453 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 662.614
| | | 662.614 |
| | style="text-align:right;" | -4.053 | | | −4.053 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 803.509
| | | 803.509 |
| | style="text-align:right;" | 3.509
| | | 3.509 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 947.916
| | | 947.916 |
| | style="text-align:right;" | 14.583
| | | 14.583 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 1096.241
| | | 1096.241 |
| | style="text-align:right;" | 29.574
| | | 29.574 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 1248.955
| | | 1248.955 |
| | style="text-align:right;" | 48.955
| | | 48.955 |
| |} | | |} |
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| 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. | | 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. |
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| ==10edo== | | === 10edo === |
| | |
| Pure 2/1: | | Pure 2/1: |
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| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 10edo | | ! Deviation from 10edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 114.426
| | | 114.426 |
| | style="text-align:right;" | -5.574 | | | −5.574 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 229.842
| | | 229.842 |
| | style="text-align:right;" | -10.158 | | | −10.158 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 346.326
| | | 346.326 |
| | style="text-align:right;" | -13.674 | | | −13.674 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 463.963
| | | 463.963 |
| | style="text-align:right;" | -16.037 | | | −16.037 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 582.847
| | | 582.847 |
| | style="text-align:right;" | -17.153 | | | −17.153 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 703.082
| | | 703.082 |
| | style="text-align:right;" | -16.918 | | | −16.918 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 824.780
| | | 824.780 |
| | style="text-align:right;" | -15.220 | | | −15.220 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 948.070
| | | 948.070 |
| | style="text-align:right;" | -11.930 | | | −11.930 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 1073.091
| | | 1073.091 |
| | style="text-align:right;" | -6.909 | | | −6.909 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 1200.000
| | | 1200.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 1328.973
| | | 1328.973 |
| | style="text-align:right;" | 8.973
| | | 8.973 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1460.208
| | | 1460.208 |
| | style="text-align:right;" | 20.208
| | | 20.208 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1593.926
| | | 1593.926 |
| | style="text-align:right;" | 33.926
| | | 33.926 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1730.381
| | | 1730.381 |
| | style="text-align:right;" | 50.381
| | | 50.381 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1869.859
| | | 1869.859 |
| | style="text-align:right;" | 69.859
| | | 69.859 |
| |} | | |} |
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| Max error limited to 10 cents (only the first 9 frets may be used): | | Max error limited to 10 cents (only the first 9 frets may be used): |
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| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 10edo | | ! Deviation from 10edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 115.766
| | | 115.766 |
| | style="text-align:right;" | -4.234 | | | −4.234 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 232.628
| | | 232.628 |
| | style="text-align:right;" | -7.372 | | | −7.372 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 350.674
| | | 350.674 |
| | style="text-align:right;" | -9.326 | | | −9.326 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 470.000
| | | 470.000 |
| | style="text-align:right;" | -10.000 | | | −10.000 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 590.714
| | | 590.714 |
| | style="text-align:right;" | -9.286 | | | −9.286 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 712.933
| | | 712.933 |
| | style="text-align:right;" | -7.067 | | | −7.067 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 836.788
| | | 836.788 |
| | style="text-align:right;" | -3.212 | | | −3.212 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 962.426
| | | 962.426 |
| | style="text-align:right;" | 2.426
| | | 2.426 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 1090.010
| | | 1090.010 |
| | style="text-align:right;" | 10.010
| | | 10.010 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 1219.720
| | | 1219.720 |
| | style="text-align:right;" | 19.720
| | | 19.720 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 1351.764
| | | 1351.764 |
| | style="text-align:right;" | 31.764
| | | 31.764 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1486.373
| | | 1486.373 |
| | style="text-align:right;" | 46.373
| | | 46.373 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1623.811
| | | 1623.811 |
| | style="text-align:right;" | 63.811
| | | 63.811 |
| |} | | |} |
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| ==11edo== | | === 11edo === |
| | |
| Pure 2/1: | | Pure 2/1: |
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| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 11edo | | ! Deviation from 11edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 106.521
| | | 106.521 |
| | style="text-align:right;" | -2.570 | | | −2.570 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 213.456
| | | 213.456 |
| | style="text-align:right;" | -4.726 | | | −4.726 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 320.834
| | | 320.834 |
| | style="text-align:right;" | -6.439 | | | −6.439 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 428.685
| | | 428.685 |
| | style="text-align:right;" | -7.678 | | | −7.678 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 537.043
| | | 537.043 |
| | style="text-align:right;" | -8.412 | | | −8.412 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 645.941
| | | 645.941 |
| | style="text-align:right;" | -8.604 | | | −8.604 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 755.419
| | | 755.419 |
| | style="text-align:right;" | -8.218 | | | −8.218 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 865.517
| | | 865.517 |
| | style="text-align:right;" | -7.210 | | | −7.210 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 976.280
| | | 976.280 |
| | style="text-align:right;" | -5.538 | | | −5.538 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 1087.757
| | | 1087.757 |
| | style="text-align:right;" | -3.152 | | | −3.152 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 1200.000
| | | 1200.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1313.066
| | | 1313.066 |
| | style="text-align:right;" | 3.975
| | | 3.975 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1427.017
| | | 1427.017 |
| | style="text-align:right;" | 8.836
| | | 8.836 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1541.922
| | | 1541.922 |
| | style="text-align:right;" | 14.649
| | | 14.649 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1657.854
| | | 1657.854 |
| | style="text-align:right;" | 21.491
| | | 21.491 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1774.895
| | | 1774.895 |
| | style="text-align:right;" | 29.441
| | | 29.441 |
| |- | | |- |
| | | 17
| | | 17 |
| | style="text-align:right;" | 1893.135
| | | 1893.135 |
| | style="text-align:right;" | 38.589
| | | 38.589 |
| |- | | |- |
| | | 18
| | | 18 |
| | style="text-align:right;" | 2012.670
| | | 2012.670 |
| | style="text-align:right;" | 49.034
| | | 49.034 |
| |- | | |- |
| | | 19
| | | 19 |
| | style="text-align:right;" | 2133.611
| | | 2133.611 |
| | style="text-align:right;" | 60.884
| | | 60.884 |
| |} | | |} |
|
| |
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| Super-accurate, error limited to 1 cent (only the first 4 frets may be used): | | Super-accurate, error limited to 1 cent (only the first 4 frets may be used): |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 11edo | | ! Deviation from 11edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 108.322
| | | 108.322 |
| | style="text-align:right;" | -0.769 | | | −0.769 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 217.183
| | | 217.183 |
| | style="text-align:right;" | -0.998 | | | −0.998 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 326.621
| | | 326.621 |
| | style="text-align:right;" | -0.652 | | | −0.652 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 436.676
| | | 436.676 |
| | style="text-align:right;" | 0.313
| | | 0.313 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 547.394
| | | 547.394 |
| | style="text-align:right;" | 1.939
| | | 1.939 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 658.821
| | | 658.821 |
| | style="text-align:right;" | 4.276
| | | 4.276 |
| |} | | |} |
|
| |
|
| ==13edo== | | === 13edo === |
| | |
| Pure 2/1: | | Pure 2/1: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 13edo | | ! Deviation from 13edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 94.538
| | | 94.538 |
| | style="text-align:right;" | 2.230
| | | 2.230 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 188.770
| | | 188.770 |
| | style="text-align:right;" | 4.154
| | | 4.154 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 282.679
| | | 282.679 |
| | style="text-align:right;" | 5.756
| | | 5.756 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 376.250
| | | 376.250 |
| | style="text-align:right;" | 7.019
| | | 7.019 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 469.465
| | | 469.465 |
| | style="text-align:right;" | 7.926
| | | 7.926 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 562.305
| | | 562.305 |
| | style="text-align:right;" | 8.458
| | | 8.458 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 654.751
| | | 654.751 |
| | style="text-align:right;" | 8.597
| | | 8.597 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 746.784
| | | 746.784 |
| | style="text-align:right;" | 8.322
| | | 8.322 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 838.383
| | | 838.383 |
| | style="text-align:right;" | 7.613
| | | 7.613 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 929.526
| | | 929.526 |
| | style="text-align:right;" | 6.449
| | | 6.449 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 1020.193
| | | 1020.193 |
| | style="text-align:right;" | 4.808
| | | 4.808 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1110.358
| | | 1110.358 |
| | style="text-align:right;" | 2.666
| | | 2.666 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1200.000
| | | 1200.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1289.093
| | | 1289.093 |
| | style="text-align:right;" | -3.215 | | | −3.215 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1377.612
| | | 1377.612 |
| | style="text-align:right;" | -7.004 | | | −7.004 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1465.530
| | | 1465.530 |
| | style="text-align:right;" | -11.393 | | | −11.393 |
| |- | | |- |
| | | 17
| | | 17 |
| | style="text-align:right;" | 1552.822
| | | 1552.822 |
| | style="text-align:right;" | -16.409 | | | −16.409 |
| |- | | |- |
| | | 18
| | | 18 |
| | style="text-align:right;" | 1639.458
| | | 1639.458 |
| | style="text-align:right;" | -22.080 | | | −22.080 |
| |- | | |- |
| | | 19
| | | 19 |
| | style="text-align:right;" | 1725.412
| | | 1725.412 |
| | style="text-align:right;" | -28.434 | | | −28.434 |
| |- | | |- |
| | | 20
| | | 20 |
| | style="text-align:right;" | 1810.654
| | | 1810.654 |
| | style="text-align:right;" | -35.500 | | | −35.500 |
| |- | | |- |
| | | 21
| | | 21 |
| | style="text-align:right;" | 1895.155
| | | 1895.155 |
| | style="text-align:right;" | -43.307 | | | −43.307 |
| |- | | |- |
| | | 22
| | | 22 |
| | style="text-align:right;" | 1978.883
| | | 1978.883 |
| | style="text-align:right;" | -51.886 | | | −51.886 |
| |- | | |- |
| | | 23
| | | 23 |
| | style="text-align:right;" | 2061.810
| | | 2061.810 |
| | style="text-align:right;" | -61.267 | | | −61.267 |
| |} | | |} |
|
| |
|
| Super-accurate, error limited to 1 cent (only the first 5 frets may be used): | | Super-accurate, error limited to 1 cent (only the first 5 frets may be used): |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 13edo | | ! Deviation from 13edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 93.001
| | | 93.001 |
| | style="text-align:right;" | 0.693
| | | 0.693 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 185.616
| | | 185.616 |
| | style="text-align:right;" | 1.000
| | | 1.000 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 277.826
| | | 277.826 |
| | style="text-align:right;" | 0.903
| | | 0.903 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 369.611
| | | 369.611 |
| | style="text-align:right;" | 0.380
| | | 0.380 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 460.950
| | | 460.950 |
| | style="text-align:right;" | -0.588 | | | −0.588 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 551.821
| | | 551.821 |
| | style="text-align:right;" | -2.025 | | | −2.025 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 642.202
| | | 642.202 |
| | style="text-align:right;" | -3.952 | | | −3.952 |
| |} | | |} |
|
| |
|
| ==14edo== | | === 14edo === |
| | |
| Pure 2/1: | | Pure 2/1: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 14edo | | ! Deviation from 14edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 89.903
| | | 89.903 |
| | style="text-align:right;" | 4.189
| | | 4.189 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 179.269
| | | 179.269 |
| | style="text-align:right;" | 7.841
| | | 7.841 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 268.074
| | | 268.074 |
| | style="text-align:right;" | 10.931
| | | 10.931 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 356.292
| | | 356.292 |
| | style="text-align:right;" | 13.435
| | | 13.435 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 443.896
| | | 443.896 |
| | style="text-align:right;" | 15.324
| | | 15.324 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 530.859
| | | 530.859 |
| | style="text-align:right;" | 16.573
| | | 16.573 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 617.153
| | | 617.153 |
| | style="text-align:right;" | 17.153
| | | 17.153 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 702.749
| | | 702.749 |
| | style="text-align:right;" | 17.035
| | | 17.035 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 787.619
| | | 787.619 |
| | style="text-align:right;" | 16.190
| | | 16.190 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 871.731
| | | 871.731 |
| | style="text-align:right;" | 14.588
| | | 14.588 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 955.057
| | | 955.057 |
| | style="text-align:right;" | 12.200
| | | 12.200 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1037.564
| | | 1037.564 |
| | style="text-align:right;" | 8.993
| | | 8.993 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1119.223
| | | 1119.223 |
| | style="text-align:right;" | 4.937
| | | 4.937 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1200.000
| | | 1200.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1279.864
| | | 1279.864 |
| | style="text-align:right;" | -5.850 | | | −5.850 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1358.784
| | | 1358.784 |
| | style="text-align:right;" | -12.645 | | | −12.645 |
| |- | | |- |
| | | 17
| | | 17 |
| | style="text-align:right;" | 1436.727
| | | 1436.727 |
| | style="text-align:right;" | -20.416 | | | −20.416 |
| |- | | |- |
| | | 18
| | | 18 |
| | style="text-align:right;" | 1513.660
| | | 1513.660 |
| | style="text-align:right;" | -29.197 | | | −29.197 |
| |- | | |- |
| | | 19
| | | 19 |
| | style="text-align:right;" | 1589.553
| | | 1589.553 |
| | style="text-align:right;" | -39.019 | | | −39.019 |
| |- | | |- |
| | | 20
| | | 20 |
| | style="text-align:right;" | 1664.373
| | | 1664.373 |
| | style="text-align:right;" | -49.913 | | | −49.913 |
| |- | | |- |
| | | 21
| | | 21 |
| | style="text-align:right;" | 1738.090
| | | 1738.090 |
| | style="text-align:right;" | -61.910 | | | −61.910 |
| |} | | |} |
|
| |
|
| Max error limited to 10 cents (only the first 12 frets may be used): | | Max error limited to 10 cents (only the first 12 frets may be used): |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 14edo | | ! Deviation from 14edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 88.925
| | | 88.925 |
| | style="text-align:right;" | 3.210
| | | 3.210 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 177.267
| | | 177.267 |
| | style="text-align:right;" | 5.839
| | | 5.839 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 265.002
| | | 265.002 |
| | style="text-align:right;" | 7.859
| | | 7.859 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 352.101
| | | 352.101 |
| | style="text-align:right;" | 9.244
| | | 9.244 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 438.538
| | | 438.538 |
| | style="text-align:right;" | 9.966
| | | 9.966 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 524.283
| | | 524.283 |
| | style="text-align:right;" | 9.997
| | | 9.997 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 609.307
| | | 609.307 |
| | style="text-align:right;" | 9.307
| | | 9.307 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 693.580
| | | 693.580 |
| | style="text-align:right;" | 7.866
| | | 7.866 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 777.073
| | | 777.073 |
| | style="text-align:right;" | 5.645
| | | 5.645 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 859.755
| | | 859.755 |
| | style="text-align:right;" | 2.612
| | | 2.612 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 941.593
| | | 941.593 |
| | style="text-align:right;" | -1.264 | | | −1.264 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1022.558
| | | 1022.558 |
| | style="text-align:right;" | -6.014 | | | −6.014 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1102.616
| | | 1102.616 |
| | style="text-align:right;" | -11.670 | | | −11.670 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1181.736
| | | 1181.736 |
| | style="text-align:right;" | -18.264 | | | −18.264 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1259.886
| | | 1259.886 |
| | style="text-align:right;" | -25.829 | | | −25.829 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1337.033
| | | 1337.033 |
| | style="text-align:right;" | -34.395 | | | −34.395 |
| |- | | |- |
| | | 17
| | | 17 |
| | style="text-align:right;" | 1413.146
| | | 1413.146 |
| | style="text-align:right;" | -43.996 | | | −43.996 |
| |- | | |- |
| | | 18
| | | 18 |
| | style="text-align:right;" | 1488.194
| | | 1488.194 |
| | style="text-align:right;" | -54.663 | | | −54.663 |
| |- | | |- |
| | | 19
| | | 19 |
| | style="text-align:right;" | 1562.144
| | | 1562.144 |
| | style="text-align:right;" | -66.427 | | | −66.427 |
| |} | | |} |
|
| |
|
| ==15edo== | | === 15edo === |
| | |
| Pure 2/1: | | Pure 2/1: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 15edo | | ! Deviation from 15edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 85.927
| | | 85.927 |
| | style="text-align:right;" | 5.927
| | | 5.927 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 171.140
| | | 171.140 |
| | style="text-align:right;" | 11.140
| | | 11.140 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 255.611
| | | 255.611 |
| | style="text-align:right;" | 15.611
| | | 15.611 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 339.310
| | | 339.310 |
| | style="text-align:right;" | 19.310
| | | 19.310 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 422.204
| | | 422.204 |
| | style="text-align:right;" | 22.204
| | | 22.204 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 504.264
| | | 504.264 |
| | style="text-align:right;" | 24.264
| | | 24.264 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 585.458
| | | 585.458 |
| | style="text-align:right;" | 25.458
| | | 25.458 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 665.754
| | | 665.754 |
| | style="text-align:right;" | 25.754
| | | 25.754 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 745.120
| | | 745.120 |
| | style="text-align:right;" | 25.120
| | | 25.120 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 823.524
| | | 823.524 |
| | style="text-align:right;" | 23.524
| | | 23.524 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 900.935
| | | 900.935 |
| | style="text-align:right;" | 20.935
| | | 20.935 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 977.319
| | | 977.319 |
| | style="text-align:right;" | 17.319
| | | 17.319 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1052.645
| | | 1052.645 |
| | style="text-align:right;" | 12.645
| | | 12.645 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1126.883
| | | 1126.883 |
| | style="text-align:right;" | 6.883
| | | 6.883 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1200.000
| | | 1200.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1271.967
| | | 1271.967 |
| | style="text-align:right;" | -8.033 | | | −8.033 |
| |- | | |- |
| | | 17
| | | 17 |
| | style="text-align:right;" | 1342.754
| | | 1342.754 |
| | style="text-align:right;" | -17.246 | | | −17.246 |
| |- | | |- |
| | | 18
| | | 18 |
| | style="text-align:right;" | 1412.333
| | | 1412.333 |
| | style="text-align:right;" | -27.667 | | | −27.667 |
| |- | | |- |
| | | 19
| | | 19 |
| | style="text-align:right;" | 1480.675
| | | 1480.675 |
| | style="text-align:right;" | -39.325 | | | −39.325 |
| |- | | |- |
| | | 20
| | | 20 |
| | style="text-align:right;" | 1547.754
| | | 1547.754 |
| | style="text-align:right;" | -52.246 | | | −52.246 |
| |- | | |- |
| | | 21
| | | 21 |
| | style="text-align:right;" | 1613.546
| | | 1613.546 |
| | style="text-align:right;" | -66.454 | | | −66.454 |
| |} | | |} |
|
| |
|
| Max error limited to 10 cents (only the first 11 frets may be used): | | Max error limited to 10 cents (only the first 11 frets may be used): |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 15edo | | ! Deviation from 15edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 83.670
| | | 83.670 |
| | style="text-align:right;" | 3.670
| | | 3.670 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 166.535
| | | 166.535 |
| | style="text-align:right;" | 6.535
| | | 6.535 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 248.564
| | | 248.564 |
| | style="text-align:right;" | 8.564
| | | 8.564 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 329.727
| | | 329.727 |
| | style="text-align:right;" | 9.727
| | | 9.727 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 409.990
| | | 409.990 |
| | style="text-align:right;" | 9.990
| | | 9.990 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 489.322
| | | 489.322 |
| | style="text-align:right;" | 9.322
| | | 9.322 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 567.691
| | | 567.691 |
| | style="text-align:right;" | 7.691
| | | 7.691 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 645.065
| | | 645.065 |
| | style="text-align:right;" | 5.065
| | | 5.065 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 721.412
| | | 721.412 |
| | style="text-align:right;" | 1.412
| | | 1.412 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 796.700
| | | 796.700 |
| | style="text-align:right;" | -3.300 | | | −3.300 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 870.897
| | | 870.897 |
| | style="text-align:right;" | -9.103 | | | −9.103 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 943.974
| | | 943.974 |
| | style="text-align:right;" | -16.026 | | | −16.026 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1015.899
| | | 1015.899 |
| | style="text-align:right;" | -24.101 | | | −24.101 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1086.643
| | | 1086.643 |
| | style="text-align:right;" | -33.357 | | | −33.357 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1156.178
| | | 1156.178 |
| | style="text-align:right;" | -43.822 | | | −43.822 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1224.475
| | | 1224.475 |
| | style="text-align:right;" | -55.525 | | | −55.525 |
| |- | | |- |
| | | 17
| | | 17 |
| | style="text-align:right;" | 1291.509
| | | 1291.509 |
| | style="text-align:right;" | -68.491 | | | −68.491 |
| |} | | |} |
|
| |
|
| ==16edo== | | === 16edo === |
| | |
| Pure 2/1: | | Pure 2/1: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 16edo | | ! Deviation from 16edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 82.483
| | | 82.483 |
| | style="text-align:right;" | 7.483
| | | 7.483 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 164.117
| | | 164.117 |
| | style="text-align:right;" | 14.117
| | | 14.117 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 244.868
| | | 244.868 |
| | style="text-align:right;" | 19.868
| | | 19.868 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 324.706
| | | 324.706 |
| | style="text-align:right;" | 24.706
| | | 24.706 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 403.598
| | | 403.598 |
| | style="text-align:right;" | 28.598
| | | 28.598 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 481.511
| | | 481.511 |
| | style="text-align:right;" | 31.511
| | | 31.511 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 558.415
| | | 558.415 |
| | style="text-align:right;" | 33.415
| | | 33.415 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 634.278
| | | 634.278 |
| | style="text-align:right;" | 34.278
| | | 34.278 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 709.066
| | | 709.066 |
| | style="text-align:right;" | 34.066
| | | 34.066 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 782.751
| | | 782.751 |
| | style="text-align:right;" | 32.751
| | | 32.751 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 855.300
| | | 855.300 |
| | style="text-align:right;" | 30.300
| | | 30.300 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 926.683
| | | 926.683 |
| | style="text-align:right;" | 26.683
| | | 26.683 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 996.873
| | | 996.873 |
| | style="text-align:right;" | 21.873
| | | 21.873 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1065.840
| | | 1065.840 |
| | style="text-align:right;" | 15.840
| | | 15.840 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1133.558
| | | 1133.558 |
| | style="text-align:right;" | 8.558
| | | 8.558 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1200.000
| | | 1200.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 17
| | | 17 |
| | style="text-align:right;" | 1265.142
| | | 1265.142 |
| | style="text-align:right;" | -9.858 | | | −9.858 |
| |- | | |- |
| | | 18
| | | 18 |
| | style="text-align:right;" | 1328.962
| | | 1328.962 |
| | style="text-align:right;" | -21.038 | | | −21.038 |
| |- | | |- |
| | | 19
| | | 19 |
| | style="text-align:right;" | 1391.438
| | | 1391.438 |
| | style="text-align:right;" | -33.562 | | | −33.562 |
| |- | | |- |
| | | 20
| | | 20 |
| | style="text-align:right;" | 1452.550
| | | 1452.550 |
| | style="text-align:right;" | -47.450 | | | −47.450 |
| |- | | |- |
| | | 21
| | | 21 |
| | style="text-align:right;" | 1512.281
| | | 1512.281 |
| | style="text-align:right;" | -62.719 | | | −62.719 |
| |} | | |} |
|
| |
|
| Max error limited to 10 cents (only the first 10 frets may be used): | | Max error limited to 10 cents (only the first 10 frets may be used): |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 16edo | | ! Deviation from 16edo |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 78.993
| | | 78.993 |
| | style="text-align:right;" | 3.993
| | | 3.993 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 157.011
| | | 157.011 |
| | style="text-align:right;" | 7.011
| | | 7.011 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 234.022
| | | 234.022 |
| | style="text-align:right;" | 9.022
| | | 9.022 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 309.995
| | | 309.995 |
| | style="text-align:right;" | 9.995
| | | 9.995 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 384.898
| | | 384.898 |
| | style="text-align:right;" | 9.898
| | | 9.898 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 458.700
| | | 458.700 |
| | style="text-align:right;" | 8.700
| | | 8.700 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 531.369
| | | 531.369 |
| | style="text-align:right;" | 6.369
| | | 6.369 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 602.877
| | | 602.877 |
| | style="text-align:right;" | 2.877
| | | 2.877 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 673.193
| | | 673.193 |
| | style="text-align:right;" | -1.807 | | | −1.807 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 742.290
| | | 742.290 |
| | style="text-align:right;" | -7.710 | | | −7.710 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 810.140
| | | 810.140 |
| | style="text-align:right;" | -14.860 | | | −14.860 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 876.717
| | | 876.717 |
| | style="text-align:right;" | -23.283 | | | −23.283 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 941.997
| | | 941.997 |
| | style="text-align:right;" | -33.003 | | | −33.003 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 1005.957
| | | 1005.957 |
| | style="text-align:right;" | -44.043 | | | −44.043 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 1068.574
| | | 1068.574 |
| | style="text-align:right;" | -56.426 | | | −56.426 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 1129.830
| | | 1129.830 |
| | style="text-align:right;" | -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|Bohlen-Pierce]] scale (13ed3) 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: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 13ed3 | | ! Deviation from 13ed3 |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 127.235
| | | 127.235 |
| | style="text-align:right;" | -19.072 | | | −19.072 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 256.565
| | | 256.565 |
| | style="text-align:right;" | -36.050 | | | −36.050 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 388.191
| | | 388.191 |
| | style="text-align:right;" | -50.732 | | | −50.732 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 522.340
| | | 522.340 |
| | style="text-align:right;" | -62.891 | | | −62.891 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 659.270
| | | 659.270 |
| | style="text-align:right;" | -72.269 | | | −72.269 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 799.275
| | | 799.275 |
| | style="text-align:right;" | -78.572 | | | −78.572 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 942.691
| | | 942.691 |
| | style="text-align:right;" | -81.462 | | | −81.462 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 1089.910
| | | 1089.910 |
| | style="text-align:right;" | -80.552 | | | −80.552 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 1241.382
| | | 1241.382 |
| | style="text-align:right;" | -75.387 | | | −75.387 |
| |- | | |- |
| | | 10
| | | 10 |
| | style="text-align:right;" | 1397.638
| | | 1397.638 |
| | style="text-align:right;" | -65.439 | | | −65.439 |
| |- | | |- |
| | | 11
| | | 11 |
| | style="text-align:right;" | 1559.303
| | | 1559.303 |
| | style="text-align:right;" | -50.081 | | | −50.081 |
| |- | | |- |
| | | 12
| | | 12 |
| | style="text-align:right;" | 1727.124
| | | 1727.124 |
| | style="text-align:right;" | -28.568 | | | −28.568 |
| |- | | |- |
| | | 13
| | | 13 |
| | style="text-align:right;" | 1902.000
| | | 1902.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 14
| | | 14 |
| | style="text-align:right;" | 2085.028
| | | 2085.028 |
| | style="text-align:right;" | 36.720
| | | 36.720 |
| |- | | |- |
| | | 15
| | | 15 |
| | style="text-align:right;" | 2277.566
| | | 2277.566 |
| | style="text-align:right;" | 82.951
| | | 82.951 |
| |- | | |- |
| | | 16
| | | 16 |
| | style="text-align:right;" | 2481.323
| | | 2481.323 |
| | style="text-align:right;" | 140.400
| | | 140.400 |
| |} | | |} |
|
| |
|
| Line 1,184: |
Line 1,179: |
| A much more satisfactory solution is to minimize the maximum error over the first 6 frets: | | A much more satisfactory solution is to minimize the maximum error over the first 6 frets: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 13ed3 | | ! Deviation from 13ed3 |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 138.788
| | | 138.788 |
| | style="text-align:right;" | -7.520 | | | −7.520 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 280.853
| | | 280.853 |
| | style="text-align:right;" | -11.763 | | | −11.763 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 426.563
| | | 426.563 |
| | style="text-align:right;" | -12.360 | | | −12.360 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 576.345
| | | 576.345 |
| | style="text-align:right;" | -8.886 | | | −8.886 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 730.697
| | | 730.697 |
| | style="text-align:right;" | -0.841 | | | −0.841 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 890.206
| | | 890.206 |
| | style="text-align:right;" | 12.360
| | | 12.360 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 1055.569
| | | 1055.569 |
| | style="text-align:right;" | 31.415
| | | 31.415 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 1227.620
| | | 1227.620 |
| | style="text-align:right;" | 57.158
| | | 57.158 |
| |- | | |- |
| | | 9
| | | 9 |
| | style="text-align:right;" | 1407.376
| | | 1407.376 |
| | style="text-align:right;" | 90.606
| | | 90.606 |
| |} | | |} |
|
| |
|
| Or even just the first 5, for better accuracy: | | Or even just the first 5, for better accuracy: |
|
| |
|
| {| class="wikitable" | | {| class="wikitable right-2 right-3" |
| |- | | |- |
| ! | Fret number | | ! Fret number |
| ! | Cents | | ! Cents |
| ! | Deviation from 13e3 | | ! Deviation from 13e3 |
| |- | | |- |
| | | 0 (nut)
| | | 0 (nut) |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| | style="text-align:right;" | 0.000
| | | 0.000 |
| |- | | |- |
| | | 1
| | | 1 |
| | style="text-align:right;" | 140.285
| | | 140.285 |
| | style="text-align:right;" | -6.023 | | | −6.023 |
| |- | | |- |
| | | 2
| | | 2 |
| | style="text-align:right;" | 284.014
| | | 284.014 |
| | style="text-align:right;" | -8.601 | | | −8.601 |
| |- | | |- |
| | | 3
| | | 3 |
| | style="text-align:right;" | 431.581
| | | 431.581 |
| | style="text-align:right;" | -7.343 | | | −7.343 |
| |- | | |- |
| | | 4
| | | 4 |
| | style="text-align:right;" | 583.443
| | | 583.443 |
| | style="text-align:right;" | -1.787 | | | −1.787 |
| |- | | |- |
| | | 5
| | | 5 |
| | style="text-align:right;" | 740.140
| | | 740.140 |
| | style="text-align:right;" | 8.601
| | | 8.601 |
| |- | | |- |
| | | 6
| | | 6 |
| | style="text-align:right;" | 902.305
| | | 902.305 |
| | style="text-align:right;" | 24.459
| | | 24.459 |
| |- | | |- |
| | | 7
| | | 7 |
| | style="text-align:right;" | 1070.696
| | | 1070.696 |
| | style="text-align:right;" | 46.543
| | | 46.543 |
| |- | | |- |
| | | 8
| | | 8 |
| | style="text-align:right;" | 1246.229
| | | 1246.229 |
| | style="text-align:right;" | 75.768
| | | 75.768 |
| |} | | |} |
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| 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. | | 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. |
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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. |
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| | == See also == |
| | [[Tuning A Kite Guitar To 31edo or 62edo]] |
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| 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.
| | [[Category:Guitar]] |