Interval Size
Here are the sizes of some common intervals:
| interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
| harmonic seventh | 25 | 967.74 | Play | 7:4 | 968.83 | Play | −1.09 |
| perfect fifth | 18 | 696.77 | Play | 3:2 | 701.96 | Play | −5.19 |
| greater septimal tritone | 16 | 619.35 | 10:7 | 617.49 | +1.87 | ||
| lesser septimal tritone | 15 | 580.65 | Play | 7:5 | 582.51 | Play | −1.86 |
| undecimal tritone, 11th harmonic | 14 | 541.94 | Play | 11:8 | 551.32 | Play | −9.38 |
| perfect fourth | 13 | 503.23 | Play | 4:3 | 498.04 | Play | +5.19 |
| septimal narrow fourth | 12 | 464.52 | Play | 21:16 | 470.78 | play | −6.26 |
| tridecimal major third | 12 | 464.52 | Play | 13:10 | 454.21 | Play | +10.31 |
| septimal major third | 11 | 425.81 | Play | 9:7 | 435.08 | Play | −9.27 |
| undecimal major third | 11 | 425.81 | Play | 14:11 | 417.51 | Play | +8.30 |
| major third | 10 | 387.10 | Play | 5:4 | 386.31 | Play | +0.79 |
| tridecimal neutral third | 9 | 348.39 | Play | 16:13 | 359.47 | play | −11.09 |
| undecimal neutral third | 9 | 348.39 | Play | 11:9 | 347.41 | Play | +0.98 |
| minor third | 8 | 309.68 | Play | 6:5 | 315.64 | Play | −5.96 |
| septimal minor third | 7 | 270.97 | Play | 7:6 | 266.87 | Play | +4.10 |
| septimal whole tone | 6 | 232.26 | Play | 8:7 | 231.17 | Play | +1.09 |
| whole tone, major tone | 5 | 193.55 | Play | 9:8 | 203.91 | Play | −10.36 |
| whole tone, minor tone | 5 | 193.55 | Play | 10:9 | 182.40 | Play | +11.15 |
| greater undecimal neutral second | 4 | 154.84 | Play | 11:10 | 165.00 | −10.16 | |
| lesser undecimal neutral second | 4 | 154.84 | Play | 12:11 | 150.64 | Play | +4.20 |
| septimal diatonic semitone | 3 | 116.13 | Play | 15:14 | 119.44 | Play | −3.31 |
| diatonic semitone, just | 3 | 116.13 | Play | 16:15 | 111.73 | Play | +4.40 |
| septimal chromatic semitone | 3 | 77.42 | Play | 21:20 | 84.47 | Play | −7.05 |
| chromatic semitone, just | 2 | 77.42 | Play | 25:24 | 70.67 | Play | +6.75 |
| lesser diesis | 1 | 38.71 | Play | 128:125 | 41.06 | Play | −2.35 |
| undecimal diesis | 1 | 38.71 | Play | 45:44 | 38.91 | Play | −0.20 |
| septimal diesis | 1 | 38.71 | Play | 49:48 | 35.70 | Play | +3.01 |
The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, which have no approximate fits in 12 equal temperament and only poor fits in 19 equal temperament. The composer Joel Mandelbaum (born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.
This tuning can be considered a meantone temperament. It has the necessary property that a chain of its four fifths is equivalent to its major third (the syntonic comma 81:80 is tempered out), which also means that it contains a "meantone" that falls between the sizes of 10:9 and 9:8 as the combination of one of each of its chromatic and diatonic semitones.
Read more about this topic: 31 Equal Temperament
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