Syncopation is the synchronization of notes with a steady rhythm. The notes are to be spread across a beat in time-equal intervals. For example, a series of eighth notes is played on the beat and offbeat, evenly spaced apart from one another. And, when playing triplets, three notes are to be played in time equal intervals, spread across one beat.

Theoretically, the concept of syncopation is very simple; play a set number of notes while simultaneously tapping your foot to a steady rhythm. In the case of quarter notes, or any rhythm that occurs in an even number of notes, i.e. eighth-notes (2 per beat), sixteenth-notes (4 per beat), thirty-second notes (8 per beat), rhythm can be kept more easily because they beats and off-beats are both synchronized with the notes- in contrast to odd numbers, triplets for example, where the notes are synchronized with the beats, but not with the off-beats.

Example;

A rhythm of 120 beats per minute= 1 beat per 500milliseconds
The beat occurs at 0 ms, and the offbeat occurs at 250 milliseconds

Eighth notes will fall directly on the beat and off beat, 0ms and 250ms, respectively.
Sixteenth notes occur at 0ms, 125ms, 250ms and 375ms

Triplets occur, approximately, at 0ms, 333ms, and 667ms

As you can see, odd numbers do not synchronize with the off-beats. When soloing, you will encounter many rapid series of notes that are in contrast to their supporting rhythm. When playing these series, you must ignore the offbeat because of the potentially disorienting polyrhythm that occurs when notes are out of sync. I would suggest paying very close attention to the beat, while placing less priority on the offbeat, until you become very comfortable with polyrhythm.

A solo may include a series of 11 notes, played with hammer-ons and pull-offs that all lie on one solid beat. When approaching confusing rhythms such as these, I would suggest practicing playing the notes in even intervals, before attempting to play them within the context of the song. Tap your foot, and simultaneously play the series of 11 notes, and then repeat slowly. You will find that your hands develop a sort of robotic instinctive movement (muscle memory). It is this robotic movement that you must develop for oddly-syncopated solos. The movement of your fingers must become a single movement, in other words, like pushing a button, your hands will move automatically in time equal intervals. You will target the beginning of this sequence with the beat, and then initiate the instinctive movement.

When playing even numbers of notes, you must develop the same quick movement technique. But, in the case of even numbers of notes, you do not have to avoid the off-beats. I would suggest taking advantage of the additional guidance from the off-beats.

Example: If a rhythm section is playing a steady sixteenth-note rhythm, and the lead melody is playing twelve notes per beat, then the lead will synchronize with the rhythm on its beat and offbeat.

The rhythms are as follows

16th note supporting rhythm: 0ms (beat), 31.25ms, 62.5ms, 93.75ms, 125ms (2nd point of synchronization), 156.25ms, 187.5ms, 218.75ms, 250ms (offbeat), 281.25ms, 312.5ms, 343.75, 375ms (3rd point of synchronization), 406.25ms, 437.5ms, and 468.75ms

12 note lead rhythm: 0ms (beat), 41.67, 83.3ms, 125ms (2nd point of synchronization), 166.67, 208.33ms, 250ms (offbeat), 291.67ms, 333ms, and 375ms (3rd point of synchronization), 416.67ms, and 458.33ms

As you can see there are points of intersection at 0ms (beat), 125ms, 250ms (offbeat), and 375ms. The notes in each seat that do not occur simultaneously are out of synchronization- or in other words, in polyrhythm.

Targets are starting points for note sequences, it is a term that I have coined to describe the starting point in time/rhythm when you push of the button to unleash your hands and automatically play an exhaustively rehearsed complex series of notes in rapid succession while almost entirely ignoring, but without losing, the accompanying rhythm. The concept of using targets when playing rapid note flurries is essential. Sometimes you will initiate rapid movements, starting on the off-beats. Whence you have developed some elaborate hammer-on and pull-off patterns, I would suggest targeting them on the beats, off-beats, between the beat and off-beat, and between the offbeat and the next beat. The goal is to play the notes, starting on a particular targeted point, without losing your place from polyrhythmic distractions.

The examples demonstrated in this article should give some clue as to how much of the rhythmic landscape will be out of sync when using some types of melodies/rhythms. I would suggest developing the rapid, instinctive movements in your hands, and then practice by targeting them on particular beats/off-beats.