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Arpeggiators are fairly common in dance music, particularly trance. In this example you can see he is pressing just one key and letting the keyboard do the rest for him, and it changes up and down.
How does this tie in with the theoretical side of music or chords to pick complimentary chords?
The machine just follows rules.
Just looking at the video, I guess the rules are these:
Follow a pattern (like a drum machine) for the tempo and rhythm of the notes.
Follow a pattern based on a setting as to whether the arpeggio goes up and repeats, goes down and repeats, or goes up and down.
Change the arpeggio only at the beginning of each measure.
At the beginning of the measure:
4a. if only one key is pressed, arpeggiate a major chord with that note as the root.
4b. If two notes a major third apart are played, arpeggiate a major chord with the lower note as the root.
4c. If two notes a minor third apart are played, arpeggiate a minor chord with the lower note as the root.
4d. If more than two notes are played, arpeggiate that chord.
Again, I'm just guessing, there can be more complex rules. The system could allow you to set the key, and then build a chord only on notes within the key. In this way, in the key of C, playing an A would result in an A minor chord.
An even more complex system could have a database of examples and use a Markov-chain or Bayesian method. These are a strictly mathematical technique, and (surprisingly) can be much easier and more effective than programming musical theory into the machine. Markov chains were used in the 1950's to program computers to automatically compose music.
Those methods work something like this: Suppose the key is C, the last time the chord changed was three measures ago, the current chord is G7, and the user presses E. In the past, under those circumstances, if the new chord had an E, 90% of the time it was C major, 8% it was A minor, 2% it was C# minor. For predictability, pick C major every time; for automatic composition, pick C major with 90% probability, A minor with 8% or C# minor with 2%.
Most likely the Roland arpeggiator does NOT do that, but if you like to program and have a system that has a MIDI keyboard for input and can play MIDI output you can try writing programs to do this.
Postscript: You may find it surprising that pure math can work as well as or better than music theory for doing something like this, but this just shows that there was a good reason that in ancient and Medieval times Music was considered to be a branch of Mathematics.
Arpeggiators repeat a rhythmic pattern. One not can be programmed to trigger a chord that gets repeated rhythmically. OR (more commonly), arpeggiators repeat individual notes of a played and sustained chord that is being triggered in sequence. So, if you are holding a C major triad down on the keyboard, the arpeggiator will run through all the notes in that chord, one at a time (not play the chord as a chord), sync'd to a tempo. So it will play the C, then E then G. The order in which the notes are played can be defined, as can the rhythm, as can the range - meaning the arpeggiator can go up multiple octaves or down multiple octaves, triggering the various C's, E's and G's within that range (say you select 2 octaves, then you'll have a low c, as well as a higher c being triggered). The order that the notes get triggered can also be defined. So it may not go C, E, G, C, E G. It could go E, C, G, C, E, G or G, E, C, G, E, C. Etc. Again, these notes can be played with triplet quarter note values, as eigth notes, as swung eighth notes, as dotted quarter notes, as half notes or any combo of anything note length value you can think of.
An arpeggio in classical music refers to playing the notes that make up a chord, separately and in sequence. A BAZILLION compositions employ arpeggios. Think Chopin, or Beetoven Piano music. The left hand outlines chords while a melody is played in the right hand. Arpeggiators automate this fundamental methodology of opening, or prolonging chordal/harmonic functions.
Modern synthesizers have taken that traditional arpeggio and expanded on it - by using gating and more. Where chords essentially can be triggered (common in trance for example) and repeat certain rhythmic patterns - this technically isn't an arpeggio, but the audio is triggered in the same way that arpeggiators traditionally function, so it falls under "arpeggiator" type plug-ins.
