Play the main melody on the piano or sing it. Note the key of the melody. For example, is the song in C major, or is it in a minor key? Knowing the key of the melody will help you narrow down the notes you can use for the bass chords and make you aware of any sharps and flats.
Consider the first phrase of "Twinkle, Twinkle Little Star," played in C major (with no sharps or flats), as an example. The first phrase starts with the following notes at an even tempo:
C C G G A A G---do do so so la la so
Each bass chords you create to accompany a note in the melody should include the note in the melody. That is, to accompany C in the treble or higher clef (range of sound), the bass chord should contain C to sound good.
Make one chord for each change in pitch: C, G, A, and G. The other notes are repeats.
Focus on major chords because the example tune is in a major key (has a "happy" melody rather than a minor, or "sad," melody). Use the standard triad (chords made up of three notes) as the foundation. According to the major triad rule, a major triad is made of a major 3rd interval and a dominant 5th interval. This means that the triad is made of three notes that are each the 3rd note up from the previous, where the last note in the triad is five steps from the base note. The formation should look like this: 1 3 5 (*1* >2> *3*, *3* >4> *5*).
Go back to the main melody, keeping the preceding rule in mind, and construct the first chord to accompany the first sound in the tune. In the example, it is C. For C, which is also the tonic (the first note) of the C major scale, you can create a tonic chord (a chord starting with the tonic of a scale) by applying the major tonic triad rule: Use the tonic as the base (the lowest note) and build up using major intervals of a 3rd. Specifically, match the base, the lowest note in the chord, with the note in the melody; in this case, it is C. Next, play the third note from the base; in this case, it is E, at an interval of 3 up from C. Add the next third note from E, which is G, at an interval of 5 up from C. Your major triad chord should be C E G, played simultaneously with the C note in the melody.
Find the next pitch in the phrase: G, which is the dominant (fifth note) of the C major scale. You can use the same chord you created in the previous step, that is, C E G. You can choose any of these notes to use as a new base; for example, use E G C or G C E instead of the original C E G. As long as you stay within those three notes, the sound will be consonant and pleasant to the ear. The tonic triad (consisting of C, E, and G) is the most solid, basic chord to use with the tonic (the first note of the major scale) and the dominant (the fifth note of the major scale) before going into more advanced music theory for variations in sound.
Find the next pitch in the phrase: A, which is the submediant (sixth note) of the C major scale. This is where it gets a little tricky. Here, you can use a subdominant chord (built from the fourth chord in the major triad). The subdominant, or fourth note, in the C major scale is F. (C D E *F*; or do re mi *FA*).
Remember the major triad rule and build a major triad from F. The first 3rd interval from F is A; this matches the note in the melody, which is another reason you are using this chord. The next 3rd up from A is C, taking you back to the tonic. Your subdominant chord should look like this: F A C. It will have the following sound: fa la do. Play it together with the A in the melody to hear it. It should be pleasant to the ear.
Find the last note in the melody phrase: G, for which you have already constructed the C E G triad. Use this. Again, you can vary the order but not the notes. You should now play the whole melody again and try your chords with the melody.