Say the sound "iowwoi" to yourself slowly once -- like a kid making a cartoon spring sound: boioioioing... :0) silently if you prefer. You might also recognize this sound from the movies: westerns use it to identify a rural, 'hick' character. The instrument used is called -- guess what? -- a 'Jew's harp'.
Notice the motions of your lips and tongue as you make this sound. The distance of the tongue from the front of the mouth is what creates the different vowel sounds.
What's happening in your mouth is called phase shift. Rock musicians do something similar with an electronic phase shifter. The signal from the instrument is split in two, then one half is delayed by a tiny interval, which is varied from zero -- in phase -- to x milliseconds behind the other half.
The human mouth accomplishes this by splitting the "signal" -- the voice -- at the aperture of the lips. One fraction of the voice comes straight out, unimpeded, while the other is delayed by the fraction of a second it takes to echo off the teeth and lips and rebound from the tongue. The greater the distance of the tongue from the front of the mouth, the longer the delay.
At certain precise points in the range of vowels between "ee" and "oh", the echo reinforces a harmonic overtone of the initial note. When a singer focuses on one of these precise points you hear, if you are listening, a higher note, quite distinct from the singer's voice.