Intervals · Exhibit
Hear every interval
in the octave.
Pick an interval, play it on any root, and weigh pure-ratio tuning against equal steps.
Perfect fifth
Choose an interval, root, and tuning, then play the pair together and one note at a time. Listen for its characteristic width and for any beating when pure ratio and equal steps differ. The frequency relationship sets both the sound and the displayed string length; comparing the two tunings reveals the small compromise behind a reusable keyboard.
Play both forms and follow the lower note as it rises one octave above the upper. Listen for the same two pitch names to return with a different interval between them. Moving one note reverses which frequency is the reference, so the two ratios multiply to a 2:1 octave. Inversion shows how one pitch pair can carry two complementary distances.
3:2 × 4:3 = 2:1, the octave the two sizes fill between them.
This 3:2 tuning sounds 1.96 cents above written E4. An octave spans 1,200 cents. The notehead leaves that tuning gap to your ear.
Reading the labels
How the names work
The number counts letter names from the lower note, with both ends included. C up to E includes C, D and E, so the interval is a third. The quality word names its size. Pure ratio tunes the major third here to 5:4 and the minor third to 6:5. Unisons, fourths, fifths and octaves take the word perfect. The tritone splits the octave in half; its interval code is A4 for augmented fourth, or d5 for diminished fifth.
Pure ratio and Equal steps name both systems on the switch and the notes rail. Pure ratio uses whole-number interval ratios (just intonation). Equal steps split the octave into twelve equal parts (12-TET), and each part is a semitone. Cents split the octave into 1,200 equal parts, so one semitone measures exactly 100. The difference readings above use that ruler. Two cents is a fiftieth of a semitone and slips past most ears. A pure major third sits 13.69 cents under its keyboard neighbour. The Hear pure ratio, then equal steps button turns that gap into audible beating.
You opened Intervals from Small numbers.
Return to Small numbers →