What makes a wind instrument sound louder or softer?
So obviously we can say "if you want to play louder, blow harder." However, is the connection really as simple as "more air means more volume," or is there a secondary effect which can permit louder sounds without having to expend more air.
I'm particularly thinking about how to play long fortissimo phrases where my instincts are to cut back on the loudness to complete the phrase, or long pianissimo phrases where so little air is used that I feel like I can't get enough oxygen unless I explicitly exhale before every breath (which there's often no time to do)
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More air does not mean more volume. Sound waves move by alternating between pressure differences and air part velocities. The actually moved air is only important with regard to how much it contributes to the original vibration. Take a look at an accordion: it generates sound regardless of whether you push or pull, so it's not how much air goes "out".
It's how much effect the air has on the original sound. With recorders and flutes, that's quite related to the amount of airflow (though one can trick with a more focused air stream, easier to do with flutes than recorders): while the resonating chamber is an important part of the tone production, the sound originates at the labium and the flowback from the air column is only part of the source.
With reed instruments, the pressure difference against which the reed resonates is important. That is formed by embouchure shape and strength as well as static pressure. The actual amount of air flow is comparatively small, particularly with double reeds.
So one key to achieving loud volumes is efficiency. Reducing waste/side air is important.
For reed instruments efficiency also plays a large role. Covering more of the reed with your lips damps the vibrations and thus lowers the volume, sometimes to the extent that you have to blow harder in soft passages. Conversely, allowing the reed to vibrate freely (like in a bagpipe) makes full use of the energy and heightens the volume.
Furthermore, less stiff reeds require less energy to play loudly (provided that they are still stiff enough to vibrate at all). So thinner reeds and/or reeds made from softer materials are louder in proportion to the energy put into blowing the air through them. However, stiffness also increases the maximum possible energy input.
Blowing harder is typically not the best idea, since it suggests to use muscles in the mouth region and increased tension there - these are tiring quickly. It helps to focus on increased support originating in the diaphragm instead.
There are various ways to influence the energy contained in the breathing stream: one can try more air, faster air flow or more focused air flow (works well on woodwinds at least).
Your idea of fortissimo deserves some comments:
fortissimo nearly never means full power throughout. Typically one starts with that, but fades away and possibly adds a crescendo towards the end. Details depend on the piece and the conductor.
In an ensemble there are various additional options: just half of the instrument group may play, alternating before suffocation; it may also be possible (if much louder instruments are around you and playing in the same time) to simply drop a bar for breathing.
For the most part, yes, volume is determined by the amount of air. You're not going to be able to play loudly with a tiny amount of air, or quietly with a lot.
However, there is an efficiency factor. Part of this is the player, and part is the equipment. If you can play in a proper relaxed manner that doesn't choke the vibration in the mouth, and use equipment that's in good shape and works well together (primary example: reed/mouthpiece combination for single reeds) so it responds easily, you'll be able to make more sound with less effort.
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