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Transposition moves a selection of notes higher or lower on the staff. MuseScore supports several kinds of transposition.
Chromatic transposition moves notes up or down in semitone increments. From the main menu choose Notes → Transpose.... You can also transpose a selection of notes using the arrow keys (↑ or ↓).
Scalar transposition (also known as diatonic transposition) moves notes up or down the current scale according to the key signature. You can move a single note by dragging it up or down. You can move a selection of multiple notes with Ctrl + click and drag.
Certain instruments such as B-flat trumpet or alto sax are known as transposing instruments. These instruments sound lower or higher than their written pitch. MuseScore has built-in support for transposing instruments.
The Concert Pitch button and Notes → Concert Pitch from the main menu lets you switch between concert pitch and transposing pitch. Concert pitch helps composers and arrangers because it displays every instrument in the same key, so the notes on the staff match their sounding pitches. When concert pitch is turned off, the notes on some instrument staves may not match their sounding pitches, but they are ready for an instrumentalist to play from. If you use concert pitch during your session, remember to turn off concert pitch before printing the parts.
Instrument transpositions are already set up in MuseScore. However if you want a rare instrument or transposition that is not available in MuseScore you may need to edit the instrument transposition manually. Right-click on an empty part of the instrument staff and choose Staff Properties.... At the bottom of the Staff Properties window you can select the interval of transposition, any octave shifts, and whether the interval is "Up" (sounds higher than written) or "Down" (sounds lower than written).