Comparison To Other Scales
Because a distance of 31 steps in this scale is almost precisely equal to a just perfect fifth, for practical purposes this scale can be considered a form of Pythagorean tuning that has been extended to 53 tones. As such the intervals available can have the same properties as any Pythagorean tuning, such as fifths that are (practically) pure, major thirds that are wide from just (about 81/64 opposed to the purer 5/4), and minor thirds that are conversely narrow (32/27 compared to 6/5).
However, 53-TET contains additional intervals that are very close to just intonation. For instance, the interval of 17 steps is also a major third, but only 1.4 cents narrower than the very pure just interval 5/4. 53-TET is very good as an approximation to any interval in 5-limit just intonation.
| interval name | size (steps) | size (cents) | just ratio | just (cents) | error |
| perfect fifth | 31 | 701.89 | 3:2 | 701.96 | −0.07 |
| diatonic tritone | 26 | 588.68 | 45:32 | 590.22 | −1.54 |
| septimal tritone | 26 | 588.68 | 7:5 | 582.51 | +6.17 |
| classic tritone | 25 | 566.04 | 25:18 | 568.72 | −2.68 |
| undecimal tritone | 24 | 543.40 | 11:8 | 551.32 | −7.92 |
| double diminished fifth | 24 | 543.40 | 512:375 | 539.10 | +4.30 |
| undecimal augmented fourth | 24 | 543.40 | 15:11 | 536.95 | +6.45 |
| acute fourth | 23 | 520.76 | 27:20 | 519.55 | +1.21 |
| perfect fourth | 22 | 498.11 | 4:3 | 498.04 | +0.07 |
| grave fourth | 21 | 475.47 | 320:243 | 476.54 | −1.07 |
| septimal narrow fourth | 21 | 475.47 | 21:16 | 470.78 | +4.69 |
| classic augmented third | 20 | 452.83 | 125:96 | 456.99 | −4.16 |
| tridecimal augmented third | 20 | 452.83 | 13:10 | 454.21 | −1.38 |
| septimal major third | 19 | 430.19 | 9:7 | 435.08 | −4.90 |
| classic diminished fourth | 19 | 430.19 | 32:25 | 427.37 | +2.82 |
| ditone | 18 | 407.54 | 81:64 | 407.82 | −0.28 |
| just major third | 17 | 384.91 | 5:4 | 386.31 | −1.40 |
| grave major third | 16 | 362.26 | 100:81 | 364.80 | −2.54 |
| acute minor third | 15 | 339.62 | 243:200 | 337.15 | +2.47 |
| just minor third | 14 | 316.98 | 6:5 | 315.64 | +1.34 |
| semiditone | 13 | 294.34 | 32:27 | 294.13 | +0.21 |
| classic augmented second | 12 | 271.70 | 75:64 | 274.58 | −2.88 |
| septimal minor third | 12 | 271.70 | 7:6 | 266.87 | +4.83 |
| classic diminished third | 11 | 249.06 | 144:125 | 244.97 | +4.09 |
| septimal whole tone | 10 | 226.41 | 8:7 | 231.17 | −4.76 |
| diminished third | 10 | 226.41 | 256:225 | 223.46 | +2.95 |
| whole tone, major tone | 9 | 203.77 | 9:8 | 203.91 | −0.14 |
| whole tone, minor tone | 8 | 181.13 | 10:9 | 182.40 | −1.27 |
| neutral second, grave whole tone | 7 | 158.49 | 800:729 | 160.90 | −2.41 |
| neutral second, large limma | 6 | 135.85 | 27:25 | 133.24 | +2.61 |
| Pythagorean major semitone | 5 | 113.21 | 2187:2048 | 113.69 | −0.48 |
| just diatonic semitone | 5 | 113.21 | 16:15 | 111.73 | +1.48 |
| major limma | 4 | 90.57 | 135:128 | 92.18 | −1.61 |
| Pythagorean minor semitone | 4 | 90.57 | 256:243 | 90.22 | +0.34 |
| just chromatic semitone | 3 | 67.92 | 25:24 | 70.67 | −2.75 |
| just diesis | 2 | 45.28 | 128:125 | 41.06 | +4.22 |
| syntonic comma | 1 | 22.64 | 81:80 | 21.51 | +1.14 |
Read more about this topic: 53 Equal Temperament
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