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From this workbook extract it seems that transposing instruments are always written higher than they sound, is this always true?
EXTRACT: If you compose music for any transposing instrument you have to remember the following.
This is not true in general. For example clarinets, cornets, and sopranino saxes, all in E flat, are written a third lower than sounding pitch.
Note: to avoid any confusion, "standard size" clarinets and cornets are both B flat instruments written a second higher than sounding pitch, but the smaller, higher pitched E flat instruments are quite common in wind bands, etc.
Another example, no longer manufactured, was the Piccolo in D flat (written a minor 9th below sounding pitch) - they were made in that key for use in military bands where key signatures with lots of flats (e.g. Ab, Db, Gb major) are easier to play in tune on brass instruments in Bb and Eb than keys with sharps.
This is not always the case. Most of the time but not always the transposing descends but there are some exceptions. THIS link has a really comprehensive list of orchestral instruments and how they transpose.
THIS website tells us there is a Clarinet in Eb that also has a minor third ascending transposition and I also remember having to learn about a Trumpet in E that also goes up a Major Third.
A simple counterexample is the Piccolo. Written an octave lower than it sounds. Db Piccolos are pretty rare, but I believe they are written a minor ninth lower than they sound.
Glockenspiels are similar, but they are written two octaves lower than they sound. There are other examples.
