Woodwind v brass instruments - what is the defining characteristic?
The names of these classes of instruments suggest that the material from which they are made is significant but this is not so. Flutes are frequently made of metal and saxophones always are yet they are classified as woodwind.
(Note that I play the clarinet and saxophone but no brass instrument so my knowledge of those will be weaker.)
So, what is the defining characteristic?
The form of the mouthpiece. As far as I know, all brass instruments have similar mouthpieces. Woodwind instruments have at least three distinct forms: flute / piccolo, single reed (e.g. clarinet and saxophone), double reed (e.g. oboe and bassoon).
How notes between harmonics are obtained. Brass instruments adjust the length of the tube with valves or slides. Woodwind instruments open or close holes in the tube with fingers or keys.
Whether the fundamental note is used: rarely on brass instruments but routinely with woodwind.
As far as I know, these properties go together. For example, I don't know of any commonly used instrument with a brass style mouthpiece but woodwind style holes in the tube for pitch control. It has been tried long ago but it was not successful: Keyed trumpet (Wikipedia).
The use of the fundamental could be significant. If it is not used then the valves just to fill in a gap of a fifth between the second and third harmonic. If the fundamental were used then an octave or more (clarinet) would be required and this might be hard to achieve with valves.
It might be hard to add valves to a wooden instruments though, of course, not all woodwind instruments are made of wood.
It has always been a puzzle to me that the holes in the tube technique produces a poor effect with brass instruments but satisfactory with woodwind instruments.
Edit: I forgot the recorder in point 1.
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Musical instruments are classified by the way the sound is produced. The material is immaterial and brass and woodwind instruments can both be made from metal, plastic or wood.
Woodwind instruments are those where you blow across an opening (flute), or use a single reed (sax, clarinet), or double reed (oboe, bassoon). I think pipe organs are in this class too.
Brass instruments are those where you use your lips to create the vibration, like blowing a raspberry.
Plucked string instruments are another class: guitar, harpsichord.
Percussion is where you make the sound by hitting something with something else: snare drum, piano (striking strings with hammers).
I play saxophone (a woodwind instrument made from brass) and didgeridoo (a brass instrument made from wood). I've offered the didge to some brass ensembles but this never seems to get taken up for some reason.
You might be interested to read up on the Hornbostel-Sachs classification system for musical instruments. Specifically, under this system, wind instruments are categorized as follows:
Edge-blown aerophones or flutes (421)
The player makes a ribbon-shaped flow of air with his/her lips (421.1), or his/her breath is directed through a duct against an edge (421.2).
Reed aerophones (422)
The player's breath is directed against a lamella or pair of lamellae which periodically interrupt the airflow and cause the air to be set in motion.
Trumpets (423)
The player's vibrating lips set the air in motion.
Flutes are in category 421; "woodwinds" other than flutes are in category 422. All "brass" instruments are in category 423.
Within each of these subcategories, there are many, many, many sub-subcategories. Within category 422, there are subcategories for double reeds (422.1) and single reeds (422.2). This system therefore categorizes bassoons and clarinets as more closely related to each other than to flutes.
Concerning the use of keys on brass instruments: category 423.2 is "chromatic trumpets", for which the pitch of the instrument can be altered mechanically (as opposed to natural trumpets.) Within this category, we have
423.21 Keyed trumpets – Ophicleide.
423.22 Slide trumpets – Trombone.
423.23 Valved trumpets – French horn, euphonium, baritone horn, trumpet, and tuba.
423.231 Conical bore
423.232 Semi-conical bore
423.233 Cylindrical bore
"Keyed trumpets" do exist, and this attribute is (in this system) secondary to the more fundamental attribute of how the sound is produced. In addition to the "keyed trumpet" mentioned in the OP, the ophiclede, cornett, serpent, and Vladimir horn are all keyed trumpets under this system. In some sense, all of these could be considered "brass instruments" even though the last three are typically made mostly of wood.
To answer part of your question, concerning why keyed brass instruments don't work out:
The "puckered lips" which produce the note in a brass instrument create a sound pressure wave which is very nearly a square wave. By comparison, a flute produces close to a sine wave, and clarinet/sax a sort of triangle wave. Now, what the keys/holes in a woodwind do is mess up the impedance at their location along the bore, basically disallowing a peak in the standing wave pattern. Unfortunately, when this is done to a standing square-wave, (following Fourier decomposition analysis) the waveform gets badly distorted because some of the sine frequencies which make up the square wave are completely destroyed but others aren't.
In comparison, changing the length of the bore (without any holes/keys) simply changes the fundamental standing wave frequency. Any type of source waveform is "happy" to operate in this situation - consider a slide whistle, for example.
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