Although these are two completely different things, they do indeed have a one-to-one relationship. The definition of a musical note is, 'a single pitch that is heard for a given amount of time'. Traditional music notation revolves around this definition.
When music is written, each note will be represented individually - both those notes that are heard sequentially as well as those notes that are all heard at once. Each of these individually represented notes has a single pitch or frequency that lasts for a small amount of time.
During the length of a note, whether it is long or short, there may be a change in its loudness or even in its tone or quality, but it will still be considered the same note. A single note in a single part is sung or played by only one type of voice or instrument.
However, there are a couple of clarifications that are required:
- There are a few percussion instruments that have certain notes written for them with what you would call indeterminate lengths because a note played on then cannot be sustained for a particular length of time, the drum being the most obvious example.
- There are some types of instruments that have the ability to change its pitch very gradually during its length. The perfect example would be the Hawaiian guitar, or any one of those stringed instruments without frets, like the violin.
- There are some instruments that can play many notes at one single time. Examples include the keyboard instruments and many of the stringed instruments, like the harp. These notes could be a chord, or perhaps notes of different parts that move independently. In such cases, each note will be written with specific lengths.
Notation is the way of writing down music and has developed and evolved over the years. Just like stories, many of the different types of early music were passed down from generation to generation without really being notated, which is why they evolved over the years. Notation is required for precision and consistency.
Notation was created and developed on the same lines with music theory, because it is impossible for us to record the notes that are being used if there are no names for the notes, or no way of identifying the relationship between the notes. Hence, when the concepts of keys and scales began to take shape, musical notes were also being named.
The Romans and Greeks used non-graphical notations that used letters of their respective alphabets to symbolize notes. From this came the letters A to G that we use today. They are sometimes called the 'Boethian notation' after the Roman writer and statesman Boethius, allegedly the first person to document the use of the Roman letters as names for the notes.
Another method for naming the notes was introduced circa 1000 AD by the wise monk Guido d'Arezzo. This method has survived the passage of time and is known as tonic sol-fa till today. The most important thing about this musical development is that at that time it used six notes that are used in the major scale even today.
Italy and France and other such European countries are now using these tonic sol-fa names as names for the notes, instead of the alphabetical letters, but this change has only taken place in the last two hundred years or so.
The early system of notation that used alphabetical letters was essentially the origin of some of the symbols that are in use today. In earlier times, the B flat was a actually a different note and a lower-case, rounded 'B' was used at that time to represent it.
A more gothic, squarer lower-case 'B' was used as the B natural. However, the musical notation today is a lot more precise than the older notations used.
Over the years, there have been many adaptations and additions in music notations, and many new methods, signs, and complications have been added. Those of the older order that were considered to be useful for today's music have stuck on, while those that were more cumbersome and complicated have long been forgotten.