The term "mapping" refers to making a map of correspondences between a source domain and some other "target" range. (Think of the game where you are given words in one category and are challenged to try to find an appropriate correspondence in another category, as in "Kitten is to cat as puppy is to ...".) The simplest kind of numerical mapping is called "linear mapping". That's when a one-to-one correspondence is drawn from every value in a source range X to a value that holds an exactly comparable position in a target range Y.
There are many objects available for delaying events (i.e., for scheduling events to happen at a specific future moment). For timing and scheduling events, the most common object is the metro object (send bang periodically at a specified time interval), which can be used to trigger events, start/stop entire processes, or trigger a counter to step through a table or a coll or any sort of sequence of things.
Here are three ways of generating MIDI notes. Admittedly they don't result in very interesting music, but they show ways how numbers can be converted for usage as pitch information.
One of the most intriguing features of the transport object is that there can be multiple independent transports in action at the same time. If a transport object has not had its name attribute set, it refers to the one global transport. However, you can create an independent transport by giving the transport object a name (i.e., by setting its name attribute to some name of your choice).
The timepoint object sends out a bang when the transport reaches a specified time position. This can be useful for causing something to happen—or for starting an entire process—at a particular instant during the transport's progress. A timepoint might, for example, even trigger a new time position value to be sent to the transport object itself, thus causing the transport to leap to a different time.
The translate object converts a message from one type of time unit to another. It uses the tempo and the time signature of the transport to do that calculation. If the tempo or the time signature changes, the result of the calculation would be different, so translate always resends its output whenever the transport receives a change to one of those values.
This patch does some of the same things as the "GlobalTransport" patch in the Extras menu, and shows what is likely going on behind the scenes in that patch. The toggle labeled "Start/Stop" starts the transport and immediately turns on the metro to begin triggering time reports. The button labeled "Rewind" sends a time position of bar 1, beat 1, 0 ticks to the transport to reset its time.
Timing objects such as metro normally operate with their time interval specified in milliseconds, and they are controlled by the Max scheduler, which is always running.
Onscreen drawing of lines, shapes, text, and stored images is done using drawing commands derived from Apple's QuickDraw/Quartz systems for creating 2D graphics. The lcd object understands many such commands, and can serve as a 2D drawing surface (and/or animation stage) within a Max patch. Note that almost everything that can be drawn in the lcd object can also be drawn into a Jitter matrix with the jit.lcd object.
How would you go about playing a melody (or arpeggio) that has a sinusoidal shape? Using a steadily increasing number, you can step repeatedly through a sine function stored in a lookup table, and then scale and offset the result to be in the desired pitch range.