: What is the meaning of the dotted lines in Hanon exercise 33? In The Virtuoso Pianist in 60 exercises exercise 33 (page 44 of the pdf) there are dotted lines with the numeral 8 above and below joining three measures together

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?t means it's supposed to be played an octave higher than indicated. You may also see this as 8va or 8a and 15ma for two octaves. Respectively, 8vb and 15mb is for 1 and 2 octaves lower than indicated.

People use this when they try to avoid adding many lines above the existing ones.

You can also see it on Wikipedia.

8va means all' ottava = at the octave.

Moreover, on some sheets you might see that after the 8va they write loco which means you are supposed to play it at the octave written.