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If the tune of Twinkle, twinkle little star is played on the piano and then some chords are added, this is harmony, not counterpoint. Instead, if the tune of Twinkle, twinkle little star was played while another melody was played, this would be considered counterpoint.
A different way of playing it would be to start the tune with the right hand. Then, in the second bar (measure), as the fifth note is played, the left hand starts to play the tune an octave lower. This works well for a time, but in the fifth bar (on the word “Up” in the right hand part) it starts to get dissonant (sounding unpleasant), so changes need to made to the left hand to make it sound nicer. This way of writing with a particular number of parts (in this case: two) is called "contrapuntal music".
In that example the left hand imitated the right hand at first. This is called imitation.
If the second part had continued to imitate all the way through the piece it would have been a canon. But “Twinkle, twinkle” does not work well as a canon. One famous canon is by Thomas Tallis. A canon that can be repeated is called a round. This is all contrapuntal music.
Counterpoint does not have to have imitation, although it often does. The important thing is that each part (i.e. each voice) is equally important. It is not one part singing the tune and the rest just accompanying.
Counterpoint does not have to be one note against one note. There can be two or more notes in one part against one in the other e.g. crotchets (quarter notes) in one part and quavers (eighth notes) in another. There is a whole system for this called “species”.
Counterpoint can be varied by inverting it, i.e. putting the top part at the bottom. When music is written so that the parts can be swapped round it is called "invertible counterpoint".
The word “counterpoint” comes from the Latin “punctus contra punctum” meaning “point against point”. The word “point” meant “note”. Several hundred years ago composers found how to write contrapuntal music. They often took a main tune (called a “Cantus Firmus”) and then added one or two or more parts to it. The more parts there were the harder it was to compose because it all had to fit so that it sounded good. Music for several voices written in this way is called polyphonic music. Polyphony was used in all church music in the Renaissance. The greatest composer of polyphony was Giovanni da Palestrina (1525-1594). Students learning the art of composition today still learn counterpoint by taking Palestrina’s music as their model.