Consider what makes up harmony. Harmony is a combination of notes that are sung or played at the same time. Harmonic notes make up chords. Basic chords follow the following form: 1--3--5, or do--mi--so. One, or "do," is the first note of the chord, or the root note. Three, or "mi", is four half steps up from do. Five, or "so," is three more half steps up from mi, or seven half steps up from do. A half step, by the way, is the distance from one note to next closest note, as can be seen on the keys of a piano, for instance.
Play middle C on the keyboard. Sing a syllable such as "la" in tune with this pitch. Using the key of C, this will be our root note. You could begin with any singable note, but the key of C will be the example used in this article.
Sing the next harmonic note in the C chord: E. The E note will be four half steps to the right of middle C. Play the note on the keyboard first, and then match the pitch with your voice.
Sing the "so" note of the C chord. So far, C has been do (or 1), and E has been mi (or 3). "So" (or 5) will be three half steps to the right of E on the keyboard. Play it and vocalize a syllable that matches this pitch.
Extend your learning. If you don't already know how to read music, consider learning either from a teacher or from self-study. In traditional music, such as hymns, the notes are usually written in four harmonic parts: soprano, alto, tenor and bass. The soprano note usually is the top note on the music staff. This note also usually corresponds with the melody--that is, the familiar tune that most people sing. Try singing the next note down; this is the alto note. The tenor note is the top note on the bass clef of the written music. It usually is a little lower than the alto note. Finally, try to sing the bass note if you're able to reach it vocally. This is the bottom note and deepest tone of the four. Practice with other people and with recorded music as much as possible to hone your harmony skills!