Wheras pianos respond to velocity (hit the key faster to make it louder) and organs respond to displacement (position of the key), hydraulophones respond to absement.

Absement is the time-integral of displacement

The word "absement" derives from the words "absence" and "displacement". I will attempt to explain the concept of absement by way of the following simple example:

Consider a 5-hour train ride that takes you 500 miles directly away from your home, in a straight line, to another destination where you stay for 5 hours and then return. Suppose you want to stay wirelessly glogged into your home computer at a "roaming" communications cost of $1/mile/hour. For simplicity, assume a linear long-distance rate, i.e. $1/hour when you're 1 mile away, $2/hour when you're 2 miles away, $3.14/hour when you're 3.14 miles away, etc.. The total cost of your online communications is $5000, since the absement (time-integral of displacement) is 5000 mile hours (1250 mile hours on the way to your destination, plus 500 miles * 5 hours stay = 2500 mile hours, plus 1250 mile hours of absement during the return trip).

The middle plot shows Displacement. The first 5 hours are spent in the train going at velocity 100mph (miles per hour) away from home. The area under this triangular part is 1/2 five times 500 mile hours, which is 1/2 times 2500 mile hours, i.e. 1250 mile hours. The next 5 hours are spent at your destination, 500miles from your home, where you pay $500/hour for 5 hours, for a cost of $2500. Staying online during your return trip costs you another $1250.

Your total cumulative running cost is the area under the middle plot up to a particular point in time. This integral is called absement and is shown on the top plot.

Each of the three plots is the time-derivative of the plot above it:

Area under the graph of displacement and time

See some recent discussion on "area under a distance vs time graph".

Musical instruments that respond to absement

Absement-responsive musical instruments: Gallery of hydraulophone pictures

Hydraulophone in Experimental Musical Instruments publication

See and hear a musical composition for hydraulophone and orchestra:

Here's a paper on teaching absement to children, published March 13, 2008, by Harbourfront Centre in Toronto: