Do most professional musicians that can sight read have the ability to see what they hear? That is, do years of sight reading train the brain to be able to do the reverse? Sight reading is see->hear but wondering if it also generally works in reverse, hear->see(mentally) or is that a different skill set that one learns.
A well-trained musician can do this, yes. But my experience suggests that it's a relatively rare skill.
Although the "music as language" metaphor is overdone and often inexact, it works in this case. If you're fluent in a language, you can not only write it, read it, and speak it, but you can easily convert between written notation and sound. If you're reading this, you can reproduce this written English into sound. If you heard this sentence spoken aloud, you could transcribe it into written notation.
In my opinion, this level of fluency in music should be expected of high-level musicians. It is certainly possible, and it does not require absolute pitch, only proper instruction.
Years of sight reading certainly helps, but so do other, more specific forms of pedagogy. Error-detection work is especially helpful for this task, as is general score study (which helps your brain map sounds to written notation).
It's not that rare a skill for those who pass Musicianship classes in conservatories. It's called music dictation, and it is usually a prerequisite for getting a Music degree.
How well and how much you can do...that's another matter. It would be very hard to take full dictation of, say, even a string quartet at one hearing, even though you might be expected to sight-read the same string quartet score all at once. Most musicians can sight read at a much higher level than that at which they can take dictation.
One of the most durable legends of Mozart's musical genius is based around music dictation. Legend has it that the Allegri Miserere was the exclusive property of the Sistine Choir in Rome, so nobody but them had the music. Mozart is supposed have heard the multi-part work once, gone home, and written it all out from memory. There's significant doubt as to whether the piece was that exclusive--but Mozart had an amazing memory and ability to visualize the aural, that's for sure.
"Music dictation", is not one skill but rather a set of related skills, and each can be developed by systematic exercices and drills, most of the times building on other typical musical activities required in "practical musicianship" examinations, such as sight-reading and quick study (see the second family of exercices below). It also gives basic composition and arrangement skills to musical students. Another use is to transcribe folk music into western music notation.
Hearing music once or several times then playing or singing it.
Insert writing it before playing or singing at it. Start with simple and short music say one bar and straight time in a simple scale, beginning each exercice with a tonal and time reference, on material significantly easier than what the student can usually sing or play. Repeat and use gradually longer and more complex examples. Quite easy to do if you have a friend or a teacher to practice with. You can use sight-reading courses and tests as support material. It gives a lot more life to music notation as a strong equivalent to letters and punctuation for ordinary language.
A kind of exercice useful for music dictation comes directly from sight-reading: instead of immediate sight-reading (you play an instrument or sing while looking at a score you have never seen before), you practice delayed sight-reading, playing, singing or writing small then larger and larger number of bars from a score after looking at it and removing the score from your eyes. Two typical strategies for students are
- sounding the music repeatedly in their head while they have the score in front of them to remember it before they can write, play or sing it.
- memorising it visually it as a score in their head while they look at it, so that they can "read it" in their mind while they write, play or sing it.
Again, starting with very simple music on a single staff, as soon as possible during musical training and increasing complexity gradually, you can go quite far, and it improves your classical sight-reading skills, notably to quickly memorise last bars of a page before page turning, and finding patterns in music. My advice is to mainly use well-known works of the repertoire of your instrument for this activity, once you have used up sight-reading exercice books. Also have instrument player students practice the singing versions as well, first by having them sing the extract first at sight, before they try to write it or play it.
When I started practicing this often as a student, I had sometimes vivid dreams of seeing music scores while hearing music as the page wrote "itself" in sync with the sound.
If you want to prepare students to orchestra direction, insert at some point the requirement to write the extract in another key or for a transposing instrument (such as clarinet or english horn).
On the historical side, Bach is said (see Forkel) to have had all his students work "on the table" and pure keyboard scales and finger exercices for a very long time before making them study and play musical works, only inserting small improvised works of his own (such as the two-parts inventions) as composition prototypes and sweeteners. And several musicians, such as Bartok were noted for their ability to notate what they heard (he travelled extensively in Balkans and North-Africa to record folk music tunes and traditions).