In simple fingering if you uncover more holes, the effective tube length shortens, raising the pitch. But in cross fingerings, does covering and uncovering alternating holes give you a longer or shorter tube length? Why is the pattern of which holes are covered seem random and different between instruments?
Think of a straw for drinking juice. A shorter straw makes a higher sound because the air has less room to vibrate and the sound waves are closer together.
Some musical instruments like flutes or recorders make different notes by changing the tube length that the air goes through. You can make the tube shorter by covering some holes with your fingers, which makes a higher note.
Cross fingerings are when you cover and uncover different holes to change the tube length. The fingering may look random and different for each instrument because they have different hole sizes and locations. Sometimes, you may use different fingerings for special effects or decorations in the music.
For the most part, cross fingerings will lower the pitch compared to the fingering which does not use the cross fingering. For example, on saxophone, G is played by pressing down the left thumb and the three fingers in the left hand, and Gb is played by adding the second finger in the right hand (skipping the first finger, making it a cross fingering).
The pattern of fingerings may seem “random” between different instruments because different instruments use different fingering/key systems and because of bore shape. The oboe and saxophone, for example, have a conical bore, meaning that an additional octave can be reached with the same fingerings simply by adding one additional key. The saxophone G mentioned previously is played T123 and an octave above is play octave key + T123. Clarinet, however, has a cylindrical bore, which means that the same key raises the pitch by a 12th rather than an octave. So, T123 on a clarinet is C and register key + T123 is a G a 12th above. I can go into more detail on this but wasn’t sure it would remain ELI5.
Any other seeming “randomness” has to do with the layout of the keywork; if you examine a keyed wind instrument you will see that many of the keys actually open or close tone holes quite far away from where the key is actually pressed. Certain fingerings may seem counterintuitive based on the locations of the keys you’re pressing, but often if you look at which tone holes are actually being manipulated, the fingerings may make more sense. Different instruments use different fingering systems based on the physical limitations of the instrument; there are actually multiple different key systems being used on clarinets today. Look up Boehm system and Oehler system clarinets. These two systems make compromises in different areas in an attempt to deal with the limitations of the fundamental shape of the instrument.