Quarter Tone - Interval Size in Equal Temperament

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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