Question is self-explanatory. What's the difference? I don't see it explained at all online, much less clearly and simply.
A broken chord is a form of arpeggio where a chord's notes are played in a certain order to form a pattern. A classic example of this is the "estudio brilliante" by Tarrega. What guitarists call "picking patterns" is in fact just whatever variation of broken chord the guitarist is fond of.
An arpeggio is also a chord played one note at a time, but here the notes are played in a strict order a-c-e over how many octaves. If you want an example of this effect then you can listen to Eddie Van Halen's guitar playing. Arpeggiating was kind of his thing.
A rolled chord is an effect that mimics the strumming of string instruments. It is one of the ways you can get a piano to sound guitar-like. The Spanish composers of the Romantic era were especially fond of this effect as it was a sound familiar in Spain. This effect just plays each note slightly after each other, but the chords are still notated as a single sound. The succession of notes is very quick and when done it just gives an effect. Each note is still part of the collective.
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arpeggio (the chord itself), arpeggiato (the manner of playing using rolled chords)
accordo arpeggiato (Italian m.), gebrochener Akkord (German m.), accord brisé (French m.), acorde harpejado (Spanish m.), acorde arpeggiado (Spanish m.) an arpeggiated chord where the notes are played, not simultaneously, but one after the other as in an Alberti bass or two notes may be immediately followed by another two
So, going by this, there is no difference.
In simple terms, in a rolled chord, the three (or more) notes are played in quick succession. They will typically all be played in the space of one beat, unless it's a really long rolled chord with many notes.
A broken chord is played as three (or more) distinct notes, one after the other, in time with the music.
It's matter of degree. The rolled chord delays (or anticipates) the attack of a chord, but the whole thing is one item, not rolling it would just be a different gesture. The timing of the 'roll' is not strictly notated. The broken chord has precise rhythmic notation and is perceived as separate notes.
But they also have many similarities.
A broken chord is one that is arpeggiated, and similarly, a rolled chord is also a form of arpeggio. The distinguishing points between the two are generally the method and speed of the technique. Broken chords are the individual notes of a chord played individually in any number of patterns, whereas a rolled chord plays the notes in rapid succession usually accomplished by rolling the wrist on either keyboards or guitar, and the succession of notes is either low to high in order, or less often high to low in order. A rolled chord generally counts as one beat, because it is played so rapidly.