Equal Temperaments and their Approximations of 13 Odd-Limit ratios |
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These tables show all the notes in each equal temperament up to 36 tone and compare each scale degree to the nearest Just intonation ratio. Where 2 ratios have similar distances and the further is more consonant, both ratios are shown for comparison.
An average distance from JI ratios is calculated, then compared to the number of notes required to get that degree of precision. The results are shown as Average Distance and roughness/complexity. Lower r/c numbers indicate more consonance. Since dissonances are plentiful in all these systems, lower r/c numbers could indicate more useful tuning systems.
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, (More tables will be added), Reference scaleAverage distance = 8.75
Roughness/Complexity = 4.37
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 600.00 | 7/5 10/7 | 582.51 617.49 | 17.49 17.49 |
Average Distance = 9.13
Roughness/Complexity = 3.04
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 400.00 | 5/4 | 386.31 | 13.69 |
| 2 | 800.00 | 8/5 | 813.69 | 13.69 |
Average Distance = 9.77
Roughness/Complexity = 2.44
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 300.00 | 13/11 | 289.21 | 10.79 |
| 2 | 600.00 | 7/5 10/7 | 582.51 617.49 | 17.49 17.49 |
| 3 | 900.00 | 22/13 | 910.79 | 10.79 |
Average Distance = 10.75
Roughness/Complexity = 2.15
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 240.00 | 8/7 | 231.17 | 8.83 |
| 2 | 480.00 | 4/3 | 498.04 | 18.04 |
| 3 | 720.00 | 3/2 | 701.96 | 18.04 |
| 4 | 960.00 | 7/4 | 968.83 | 8.83 |
Average Distance = 8.78
Roughness/Complexity = 1.46
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 200.00 | 9/8 | 203.91 | 3.91 |
| 2 | 400.00 | 5/4 | 386.31 | 13.69 |
| 3 | 600.00 | 7/5 10/7 | 582.51 617.49 | 17.49 17.49 |
| 4 | 800.00 | 8/5 | 813.69 | 13.69 |
| 5 | 1000.00 | 16/9 | 996.09 | 3.91 |
Average Distance = 7.78
Roughness/Complexity = 1.11
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 171.43 | 11/10 | 165.00 | 6.43 |
| 2 | 342.86 | 11/9 | 347.41 | 4.55 |
| 3 | 514.29 | 4/3 | 498.04 | 16.25 |
| 4 | 685.71 | 3/2 | 701.96 | 16.25 |
| 5 | 857.14 | 18/11 | 852.59 | 4.55 |
| 6 | 1028.57 | 20/11 | 1035.00 | 6.43 |
Average Distance = 6.10
Roughness/Complexity = 0.76
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 150.00 | 12/11 | 150.64 | 0.64 |
| 2 | 300.00 | 13/11 | 289.21 | 10.79 |
| 3 | 450.00 | 13/10 | 454.21 | 4.21 |
| 4 | 600.00 | 7/5 10/7 | 582.51 617.49 | 17.49 17.49 |
| 5 | 750.00 | 20/13 | 745.79 | 4.21 |
| 6 | 900.00 | 22/13 | 910.79 | 10.79 |
| 7 | 1050.00 | 11/6 | 1049.36 | 0.64 |
Average Distance = 8.20
Roughness/Complexity = 0.91
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 133.33 | 14/13 13/12 | 128.30 138.57 | 5.03 5.24 |
| 2 | 266.67 | 7/6 | 266.87 | 0.20 |
| 3 | 400.00 | 5/4 | 386.31 | 13.69 |
| 4 | 533.33 | 11/8 | 551.32 | 17.99 |
| 5 | 666.67 | 16/11 | 648.68 | 17.99 |
| 6 | 800.00 | 8/5 | 813.69 | 13.69 |
| 7 | 933.33 | 12/7 | 933.13 | 0.20 |
| 8 | 1066.67 | 24/13 13/7 | 1061.43 1071.70 | 5.24 5.03 |
Average Distance = 7.35
Roughness/Complexity = 0.74
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 120.00 | 15/14 | 119.44 | 0.56 |
| 2 | 240.00 | 8/7 | 231.17 | 8.83 |
| 3 | 360.00 | 16/13 | 359.47 | 0.53 |
| 4 | 480.00 | 4/3 | 498.04 | 18.04 |
| 5 | 600.00 | 7/5 10/7 | 582.51 617.49 | 17.49 17.49 |
| 6 | 720.00 | 3/2 | 701.96 | 18.04 |
| 7 | 840.00 | 13/8 | 840.53 | 0.53 |
| 8 | 960.00 | 7/4 | 968.83 | 8.83 |
| 9 | 1080.00 | 28/15 | 1080.56 | 0.56 |
Average Distance = 9.27
Roughness/Complexity = 0.84
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 109.09 | 14/13 | 128.30 | 19.21 |
| 2 | 218.18 | 8/7 | 231.17 | 12.99 |
| 3 | 327.27 | 6/5 | 315.64 | 11.63 |
| 4 | 436.36 | 9/7 | 435.08 | 1.28 |
| 5 | 545.45 | 11/8 | 551.32 | 5.87 |
| 6 | 654.54 | 16/11 | 648.68 | 5.86 |
| 7 | 763.63 | 14/9 | 764.92 | 1.29 |
| 8 | 872.72 | 5/3 | 884.36 | 11.64 |
| 9 | 981.81 | 7/4 | 968.83 | 12.98 |
| 10 | 1090.90 | 13/7 | 1071.70 | 19.20 |
Average Distance = 11.23
Roughness/Complexity = 0.94
| Scale Degree | Nearest Ratio | Difference | ||
| Fret | Cents | Ratio | Cents | |
| 0 | 0.00 | 1/1 | 0.00 | 0.00 |
| 1 | 100.00 | 14/13 | 128.30 | 28.30 |
| 2 | 200.00 | 9/8 | 203.91 | 3.91 |
| 3 | 300.00 | 13/11 | 289.21 | 10.79 |
| 4 | 400.00 | 5/4 | 386.31 | 13.69 |
| 5 | 500.00 | 4/3 | 498.04 | 1.96 |
| 6 | 600.00 | 7/5 10/7 | 582.51 617.49 | 17.49 17.49 |
| 7 | 700.00 | 3/2 | 701.96 | 1.96 |
| 8 | 800.00 | 8/5 | 813.69 | 13.69 |
| 9 | 900.00 | 22/13 | 910.79 | 10.79 |
| 10 | 1000.00 | 16/9 | 996.09 | 3.91 |
| 11 | 1100.00 | 13/7 | 1071.70 | 28.30 |
This is the list of reference intervals for comparison with equal temperaments.
| Ratio | Cents | Note |
| 1/1 | 0.00 | C +0 |
| 14/13 | 128.30 | C# +28 |
| 13/12 | 138.57 | C# +39 |
| 12/11 | 150.64 | D -49 |
| 11/10 | 165.00 | D -35 |
| 10/9 | 182.40 | D -18 |
| 9/8 | 203.91 | D +4 |
| 8/7 | 231.17 | D +31 |
| 7/6 | 266.87 | D# -33 |
| 13/11 | 289.21 | D# -11 |
| 6/5 | 315.64 | D# +16 |
| 11/9 | 347.41 | D# +47 |
| 16/13 | 359.47 | E -41 |
| 5/4 | 386.31 | E -14 |
| 14/11 | 417.51 | E +18 |
| 9/7 | 435.08 | E +35 |
| 13/10 | 454.21 | F -46 |
| 4/3 | 498.04 | F -2 |
| 11/8 | 551.32 | F# -49 |
| 18/13 | 563.38 | F# -37 |
| 7/5 | 582.51 | F# -17 |
| 10/7 | 617.49 | F# +17 |
| 13/9 | 636.62 | F# +37 |
| 16/11 | 648.68 | F# +49 |
| 3/2 | 701.96 | G +2 |
| 20/13 | 745.79 | G +46 |
| 14/9 | 764.92 | G# -35 |
| 11/7 | 782.49 | G# -18 |
| 8/5 | 813.69 | G# +14 |
| 13/8 | 840.53 | G# +41 |
| 18/11 | 852.59 | A -47 |
| 5/3 | 884.36 | A -16 |
| 22/13 | 910.79 | A +11 |
| 12/7 | 933.13 | A +33 |
| 7/4 | 968.83 | A# -31 |
| 16/9 | 996.09 | A# -4 |
| 9/5 | 1017.60 | A# +18 |
| 20/11 | 1035.00 | A# +35 |
| 11/6 | 1049.36 | A# +49 |
| 24/13 | 1061.43 | B -39 |
| 13/7 | 1071.70 | B -28 |
| 2/0 | 0.00 | C +0 |