"Who is not aware of all the demands made on the Clavier [i.e.., keyboard]? It is not enough that the player of the clavier should satisfy the expectations we rightly have of any instrumentalist, that is, that he be able to perform a piece composed for his instrument according to the rules of good execution. He must, besides, extemporize in a variety of manners; work up a given composition on the spot, observing the strictest rules of harmony and melody; play with equal ease in all keys, transpose unhesitatingly and faultlessly, play anything whatsoever at sight, whether composed specially for his instrument or not; he must have a total command of the science of thoroughbass and be able to execute it in a variety of ways, often contradictory; now in many parts, now in few, now in strict harmony, now elegantly, from bass parts that may be insufficiently or over abundantly furnished with figures, or not at all, or incorrectly....And who can count all the other requirements?"
---C. P. E. Bach "Essay on the Correct Manner of Playing the Clavier" (1753)
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Absolute Pitch--see perfect pitch.
Discipline--The most important musical gift there is because all the others grow from it. An ability to consistently and doggedly work to improve, usually in tiny increments, in all areas of musical talent. In the minds of many people this skill is unnecessary, because you "either have it or you don't." Legends about great composers like Mozart just being able to write down music he heard in his head without having to do any work conspire with the human tendency, whenever confronted with the results of long years of hard work to just see the results and assume away the hard work. Discipline does not imply not having any fun; on the contrary, the more fun the process is, the more likely we are to stick to it. But it means not giving up when we don't get results that we can see immediately. Inexperienced musicians are less able to see progress in stages, so it helps to have a mentor who can encourage such progress and help to develop good habits. Recently I told a student that Mozart spent at least five hours every morning composing music and watched her jaw drop. He also, for those of you scoring at home, sometimes made numerous sketches of his pieces, and even spent months or years working on some of his finest. Not that some things didn't come easily to him, but things get easier when you work your tail off. Trust me.
Endlessly bragging about musical talent --not really much of a skill, but it does get results. If you tell people how great you are long enough, some of them will believe you. The most well-developed of abilities for some musicians; it is sometimes considered an ample substitute for everything else.
Flying by the seat of your pants--A sort of catch-all musical skill which can only be acquired from the conservatory of life. No degrees are awarded at such an institution because you will never graduate. Each situation is different. Sometimes challenges are caused by working with non-musicians who are unable to communicate what they'd like you to do, trusting you to just "figure it out." General experience helps, as does patience. Being unable to adequately prepare also leads to such a condition. A sense of humor helps in this area. A good one.
Hearing music in your head--There are really two different musical gifts here. One is the ability to hear the music of other composers; the other is to hear original music that you might like to write down. The first can come about in two ways: one is to hear a composition perhaps a number of times and aurally memorize it, which can be a handy way to pass the time when you are mowing the lawn (they didn't have walkmen or Ipods when I was a kid so I just listened to records I already knew!) Another is to be able to "hear" a piece of music just by looking at the notes, a talent that is not as magical as many non-musicians believe. Whenever I see something about Beethoven's Ninth Symphony, the one he wrote when he was totally deaf, it seems people consider the most amazing thing about that was that he could write a piece without being able to physically hear the notes. This is not such a miracle to me. I think the piece itself is the miracle! After years of training, connecting notes to sounds, any number of musicians can "hear" what they have written without needing to actually have it played. On the other hand, having original music streaming through your head is probably a good sign that you should try composing. The musicologist Eduard Hanslick (a friend of Brahms) said that music is not composed as a result of an inner feeling, but rather from an inner singing. Taming such internal melody by way of a conscious compositional process can be a bit tricky, which is where the years of study come in. So do years of self-criticism; as Brahms put it "the hardest thing about composing is letting the extra notes fall from the table." In other words, not every idea we have is golden. Or worth putting on paper. Much of what we compose, particularly at first, is largely imitative, stemming, at least in part, if not a very large part, from the kind of music we have been exposed to the most. Which is why, if you are trying to become original, it is better not to isolate yourself from the music of others, but rather to listen to all sorts of music "until your ears fall off." However, being able to hear entire concerts filled with original music is also a handy way to pass time in Geometry class (sorry, Mr. Bonar.)
Improvising--We all deal with language in three ways. We read it, we write it, and we speak it. Improvising means being able to speak it: to create all or part of a musical composition on the spot. Jazz pianists frequently improvise solos during a piece; church organists will sometimes improvise during a worship service. The skill is a must whenever music is needed in such quantities and frequency that it can't all be learned and practiced in time, but it also allows a musician to express himself creatively in whatever medium he has chosen. The famous composers of old were all inveterate improvisers; Beethoven spent hours at his piano making stuff up, and, if he liked what he heard, it might serve as the springboard for a written composition later. Although there are a few pianists out there doing it, improvising outside of jazz solos and the occasional (mostly European) worship service is rarely part of a musician's makeup. It requires knowing the musical process so well that you can make a musical statement with no time to reflect on potential alternatives. Anyone who uses a lead sheet, where only the melody is written out, usually with chord names, has to engage in some improvisation, though there is a big difference between this and creating an entire musical composition without such a guideline. Today there are sometimes competitions for organists in improvisation (pretty hard to judge objectively, no doubt).
Mozart used to entertain his audience by taking a popular tune of the day and improvising a complete composition around it; sometimes musicians would ask the audience to submit themes (short melodies) for improvisation, picking one out of a hat, for instance, to prove that they hadn't had time to consider it beforehand.
Memorization--see "how do you memorize all that music?"
Perfect Pitch--Many musicians consider this the holy grail of musical talents, and they will look at you funny if you have it, as though you are a member of a different species. Basically, PP is the ability to tell the name of a note just by hearing it. The ability seems to be more common among pianists, who don't really need it, the other instrumentalists point out sourly. This is because, unless you are trying to tune an instrument, perfect pitch is largely a useless skill, and can even be an impediment to things like transposition (see below) since one's sense of a pitch is so strongly bound to a particular note. Cashing in on the aura of mystery that surrounds this talent, a showman like Yanni once complained that his 800 zillion-piece orchestra needed to have music in front of them unlike his godlike self (who could play from memory of course) because they didn't have perfect pitch, which is a lot like saying that the reason you need a flashlight when the power goes out is because you can't fly. Being able to establish a strong connection between the name, or look, of a note, and the sound of a note, is important, and allows you to do things like hear the music without having to play it, or compose without using a piano. In tandem with other, equally mysterious, skills, it can be quite useful. Otherwise it is pretty much a circus act.
Perfect pitch can come in handy if you are traveling with a choir that sings a capella and there is no pitch pipe. ("Michael, give us an F.")
Playing by ear--listening to a piece of music and then being able to play it without having a written score in front of you. As with sightreading (see below) a quick recognition of musical patterns is necessary, but the ear alone has to sort out what is being heard, understand it, and reproduce it. Persons who play by ear frequently do not read music and sometimes flatter themselves that they are superior to those who do (hey, the reverse is also true). My experience has been that strengthening the ear's ability to connect with the notes also helps to play written music, to understand what I am playing, and to memorize written music. At the conservatory, classes in ear-training generally include exercises in dictation, which means that the professor will play a short melody or chord progression on the piano, and the students, seated at their desks, have to write down what they heard without recourse to a musical instrument. As a former teacher of this, I can tell you it is the part of the class that makes the students most anxious. Persons who play by ear most tend to be grounded in more popular styles of music where the harmonic patterns are simpler, and the chord progressions more predictable. For a nice challenge, try learning a Beethoven sonata by ear! (I did once, in high school) Since music has to be heard to be "alive" it is important to establish a connection between sounds and their recognition that doesn't depend on what the eye sees. Many musicians (sadly) have not developed this skill.
Growing up, my parents also found there is an economic benefit to playing by ear. They would call out a tune, I would play it; no cash need be tendered for written music! (And hence they could afford to send me to college; but perhaps I exaggerate)
Sight-reading--being able to sit down with piece of music you've never seen before and perform it adequately. This means being able to decode musical signs in a hurry; having such a thorough acquaintance with notes and rhythms, dynamics and articulation signs as well as other written instructions means you can, basically, read music the same way you read a book, the same way you are reading this. This is astounding to some people who have pointed out to me that on a piano, at least, there are so many things going on at the same time that, while reading a single note makes sense, being able to deal with a whole herd at once is just magical. But good reading means you are able to recognize not just individual letters, but entire words and phrases instantly. It is the same way with sight-reading. If you've seen a pattern enough times, that stack of four notes in my left hand becomes immediately recognizable as a D major chord; I've got plenty of brain cells left over to deal with what is in the right hand, and besides, after years of piano playing, my brain pretty much operates in stereo anyway. While sight-reading is not my strongest ability, doing it a lot certainly helps to build the skill, just like anything else. While a graduate assistant at school I had plenty of opportunities to sight-read. I was an accompanist for singers, who often bring you the music at the last possible minute. Often, for auditions, the accompanist is sight-reading for the singer; this also happens in voice classes, and sometimes even for recitals (not recommended) depending on how organized the singer is! Sight-reading in front of world-famous singer types is a nice way to get the brain going in the morning.
Classical musical training tends to develop sight-reading to an extent well beyond what other styles do, since it depends on being able to realize the written intentions of the composer as carefully as possible, and departure from the written notes is not considered a good idea. Even additions to the score, such as tempo modifications, or dynamic changes, can make for a long argument. Conservatory musicians tend to be able to sight-read pretty well, although they often can't do things such as playing by ear or improvising, activities for which no written notes are available to be followed.
Transposing--(changing key) Being able to play music which was originally in one key in another key. This is essentially a non-written activity, since, even if you are looking at notes when you transpose, you are playing different ones. It requires a thorough knowledge of the patterns of sharps and flats which are particular to each key so that by preserving the pattern of the original (major or minor, for example) you are playing something that sounds just like the same piece; only the pitches are all a step or two lower, higher, etc. Singers will often ask for a piece to be transposed because, in the original key, it is too high/low for them to sing. They will love you for it.