Interval Size in Equal Temperament
Here are the sizes of some common intervals in a 24-note equally tempered scale, with the interval names proposed by Alois Hába (neutral third, etc.) and Ivan Wyschnegradsky (major fourth, etc.):
interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
---|---|---|---|---|---|---|---|
octave | 24 | 1200 | play | 2:1 | 1200.00 | play | 0.00 |
semidiminished octave | 23 | 1150 | play | 2:1 | 1200.00 | play | −50.00 |
supermajor seventh | 23 | 1150 | play | 35:18 | 1151.23 | −1.23 | |
major seventh | 22 | 1100 | play | 15:8 | 1088.27 | play | +11.73 |
neutral seventh | 21 | 1050 | play | 11:6 | 1049.36 | play | +0.64 |
minor seventh | 20 | 1000 | play | 16:9 | 996.09 | play | +3.91 |
supermajor sixth/subminor seventh | 19 | 950 | play | 7:4 | 968.83 | play | −18.83 |
major sixth | 18 | 900 | play | 5:3 | 884.36 | play | +15.64 |
neutral sixth | 17 | 850 | play | 18:11 | 852.59 | play | −2.59 |
minor sixth | 16 | 800 | play | 8:5 | 813.69 | play | −13.69 |
subminor sixth | 15 | 750 | play | 14:9 | 764.92 | play | −14.92 |
perfect fifth | 14 | 700 | play | 3:2 | 701.95 | play | −1.95 |
minor fifth | 13 | 650 | play | 16:11 | 648.68 | play | +1.32 |
lesser septimal tritone | 12 | 600 | play | 7:5 | 582.51 | play | +17.49 |
major fourth | 11 | 550 | play | 11:8 | 551.32 | play | −1.32 |
perfect fourth | 10 | 500 | play | 4:3 | 498.05 | play | +1.95 |
tridecimal major third | 9 | 450 | play | 13:10 | 454.21 | play | −4.21 |
septimal major third | 9 | 450 | play | 9:7 | 435.08 | play | +14.92 |
major third | 8 | 400 | play | 5:4 | 386.31 | play | +13.69 |
undecimal neutral third | 7 | 350 | play | 11:9 | 347.41 | play | +2.59 |
minor third | 6 | 300 | play | 6:5 | 315.64 | play | −15.64 |
septimal minor third | 5 | 250 | play | 7:6 | 266.88 | play | −16.88 |
tridecimal minor third | 5 | 250 | play | 15:13 | 247.74 | play | +2.26 |
septimal whole tone | 5 | 250 | play | 8:7 | 231.17 | play | +18.83 |
whole tone, major tone | 4 | 200 | play | 9:8 | 203.91 | play | −3.91 |
whole tone, minor tone | 4 | 200 | 10:9 | 182.40 | +17.60 | ||
neutral second, greater undecimal | 3 | 150 | play | 11:10 | 165.00 | play | −15.00 |
neutral second, lesser undecimal | 3 | 150 | play | 12:11 | 150.64 | play | −0.64 |
15:14 semitone | 2 | 100 | play | 15:14 | 119.44 | −19.44 | |
diatonic semitone, just | 2 | 100 | play | 16:15 | 111.73 | play | −11.73 |
21:20 semitone | 2 | 100 | play | 21:20 | 84.47 | play | +15.53 |
28:27 semitone | 1 | 50 | play | 28:27 | 62.96 | play | −12.96 |
septimal quarter tone | 1 | 50 | play | 36:35 | 48.77 | play | +1.23 |
Moving from 12-TET to 24-TET allows the better approximation of a number of intervals. Intervals matched particularly closely include the neutral second, neutral third, and (11:8) ratio, or the 11th harmonic. The septimal minor third and septimal major third are approximated rather poorly; the (13:10) and (15:13) ratios, involving the 13th harmonic, are matched very closely. Overall, 24-TET can be viewed as matching the 11th harmonic more closely than the 7th.
Read more about this topic: Quarter Tone
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