Ever try writing (the old-fashioned way, with a pen or pencil) on a moving train? If so, you know how extraneous movement interferes with fine motor coordination. You might end up with a page of barely decipherable chicken-scratching.
Like the muscles in our hands that allow us to write clearly in a stationary environment, our vocal muscles—the tiny muscles that move and stretch our vocal cords in various subtle ways in singing—require a stable environment in order to respond precisely and delicately. A turbulent environment throws them off, adding unwanted random "noise" to our sound, like the unwanted random jittering our pen makes on the paper when the train rattles along the tracks.
So how do you provide your vocal muscles with the stable environment they need to accomplish their delicate task well? You maintain stability by not unnecessarily altering the shape of your internal resonance space when singing. Does that sound complicated? It's not, really: the configuration of your resonance space is largely determined by the vowel you sing. If you maintain a consistent vowel, you stabilize the environment in which your vocal muscles are working.
For example, if you are singing a two-note interval on [ah], you must make sure not to change the [ah] when moving to the second pitch. Often, we alter our vowels so automatically, that we are not even able to perceive the change. It was months after my mentor Cornelius Reid first identified my habit of singing [ah-ah-ah-UH] as I went up the scale before I was able to note the difference myself. I remember thinking (incorrectly, as it turned out) that he must have been imagining things; it sure sounded like [ah-ah-ah-ah] to me!
The effect of modifying our vowels is amplified for larger intervals. You can try it for yourself by singing [ah] on a lowish note followed by the same note an octave higher. If you change the vowel (for example, to [uh] or [oo] as in book or [a] as in cat), you will jostle the vocal muscles unnecessarily, interfering with their ability to respond with precision and ease. If you maintain a consistent [ah], your musculature, as if by magic, will have the freedom to create just the right conditions for the second note.
It may take some time before this process works efficiently. A lot depends on the current state of coordination in your vocal musculature and your ability to hear accurately what sounds you are producing. Until things are working at their peak, it is very helpful to have a teacher's ear to analyze what is going on.
One of my primary goals in voice teaching is to lead my students to an ability to accurately discern the quality of their vowels so they eventually are able to maintain a stable resonance environment in their singing.