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In order to write, you will either need staff paper and a writing implement, such as a pencil or pen, or some kind of music writing software. Writing by hand is problematic because, in order to hear what you’ve played, you have to be able to play an instrument or you need to find someone else who can. With music notation software, you’re able to hear what you’ve written immediately, using the computer playback.  Some examples of music writing software include: Sibelius First, Finale and MagicScore Maestro.  For free music writing software, try MuseScore and Lilypond. If you play or have access to an instrument, you can try playing your melodies using your instrument to hear how they sound before writing them down. Using the outline you created earlier, construct each individual movement. While writing, you’ll need to be constantly making decisions regarding different musical techniques and styles within your piece. Some of those techniques include:  Dynamics Rhythms Intervals Harmonies Theme development Counterpoint Voice leading Orchestration Traditionally, symphonies are written for the following instrumental types: strings (violin, viola, cello, and bass) woodwinds (2 flutes, 2 oboes, 2 clarinets, and 2 bassoons)  brass (2 French horn, 2 trumpets with a small orchestra) and percussion (2 tympani, triangle, and cymbal) for a small orchestra. A medium size is when you can add a piccolo, English horn, bass clarinet, contra bassoon to the woodwinds and double the number of French horns and add 3 trombone plus a tuba to the brass as well as extra percussion instruments. If you can go larger, go for it.  You don't however, have to stick with traditional instrumentation. If the instruments are available and there are players to play them, feel free to use other instruments in your symphony either adding them or using them like different colors (eg. replace one traditional instrument with one non-traditional one). For example, the inventor of the saxophone family designed his instruments for the orchestra (built in the keys of B♭ and E♭) as well as the military band. You could also include an instrument that's been revived like one of the recorders, the viola da gamba or the oboe da caccia for example. Some symphonies also have vocal accompaniment. Try giving the same melodies to different instruments at different times to create a motif within the piece.  Pay attention to layers and texture. There should never be any more than five layers happening at one time.
Gather your materials. Start writing. Assign melodies to different instruments.