53 Equal Temperament - Comparison To Other Scales

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

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