Is it possible to have a piece which can't be notated using the musical notes we currently have?
If you mean can one compose a melody using notes that are not defined in any recognized form of musical notation - then the answer is certainly!
And if you compose a musical work that uses tones that are not defined in say 12 Tone Equal Temperament or other common tuning or notation system, there is no universally accepted way to transcribe those tones into a written representation that others could follow.
According to Wikipedia Article on Microtonal Music
Microtonal music or microtonality is the use in music of microtones—intervals smaller than a semitone, which are also called "microintervals". It may also be extended to include any music using intervals not found in the customary Western tuning of twelve equal intervals per octave.
The advent of technology of the type that is used to create Electronic Dance Music (EDM) is capable of instructing (through computer code) a MIDI capable instrument to play an infinite number of notes that lie between any two notes that are in common use. There is a limit to how many different notes our ears can distinguish as unique sounds however. But it's safe to say there might be ten or more distinguishable unique tones between any two common notes.
Also many instruments can easily play the notes between the ones that are commonly used. Any fretless stringed instrument such as a violin or fretless bass, a trombone (through slide position), woodwinds like clarinet and saxophone via embouchure, or even a fretted guitar through the use of string bending can play notes that we have no way to notate using standard notation systems that are universally understood.
Before computers and the ability to record music, transcribing a musical work on paper into a recognizable form of written musical notation was the best way a composer could communicate his musical creation to others.
Today it is very simple to memorialize a musical composition through a digital recording which can then be disseminated to the entire world with a few clicks of a mouse on your computer. This fact, along with the technology that facilitate the creation of electronic music, is giving momentum to a trend towards composing microtonal music. And given that it's easier to record it than transcribe it, and given that a computer can convert the recording to code, it is unlikely that we will ever see a new system of notation for music that uses notes that are not already defined in existing music notation systems.
It could be argued that any digital recording of the melody in question is a notation of that particular performance:
- It's something that can be followed to reproduce the original performance.
- It can be written down (any digital recording is made of bits that can be encoded as ink on paper, if need be).
- Anyone with the right training and equipment can use that notation to recreate the melody.
- The music can be 'reinterpreted' by, say, playing it at a higher speed, adding special effects, or any of the thousands of other ways DJs remix recordings.
A particularly picky person might claim that digital encoding of a recording doesn't contain all of the sounds from the original, due to finite bandwidth, but that's true of old-fashioned classical musical notation too: "Largo" or "piano" are very approximate terms.
It's actually quite difficult to come up with a distinguishing feature that separates a digital recording from classical musical notation.
Finally, note that "classical notation" is very flexible in that composers often make up new symbols and markings, which are usually described in a foreword for the score. Sometimes these markings become popular enough that they become part of standard classical notation - which is how the classical notation that we use came about in the first place. You can see some particularly wacky examples here.
Sure. Here are some examples of scores by John Cage, Steve Reich, Brian Eno, Krzysztof Penderecki, and Cornelius Cardew - all recognized "mainstream" 20th century composers.
The melodies of Turkish classical, Arabic, and Hindustani have all come to terms with various means of notating their microtonal and non-Western tones and intervals, the Turkish Makams being, probably, one of the best of such systems. Still, all notation is just a guideline. Instruments that play glissandos by sliding on strings or by any other means obviously smoothly cross between discrete tones, and although rough means of representing these transits exist in all systems of notation, listen to a master playing the indian sarangi and tell me if that can be notated. The Turkish Ney or Arabic Nay not only play non-Western tones and glissandos but can be under- and overblown to render various transients and harmonic spectrums in conjunction with nameable notes much like Hendrix did with feedback. As others have pointed out, "melodies" if recognizable as such should be subject to some form of notation. "Music," however, is a more broadly understood concept than melody, and almost any kind of sound can become an element of music. Instead of notation, some kinds of "music" might only be communicable as live performances, recordings, or verbal descriptions.
Certainly current musical notation is incapable of handling note durations that aren't rational multiples of each other. For instance if you had a melody where one of the long tones was pi times the duration of one of the short tones then there is no way to explicitly notate this (it is possible to get as close an approximation as is desired though).
I don't have any examples, but there are melodies (possibly stretching the definition of "music" a bit) composed mostly or solely of "effects", like "a crashing glass", "a gunshot", "a ringing telephone", "a low moan", "the sound of a cat" etc. You could adapt standard notation for it, but a great part of your music would be in the "legend" (which effect corresponds to which "percussion instrument").
Also, in such melodies, you could use digital distortion of the "effects", like pitch shift (for extra effects, make the pitch go down until it reaches below 20 Hz), reverse playback, creative amplitude modulation, etc.
If you make it sufficiently complex, only a digital recording (or a live performance) can represent it.
Not really, because notation systems have always evolved to accommodate the music composers are writing. Baroque composers, particularly French ones, came up with all kinds of weird notation, some of it unique to the composer.
An example: La Sylva by Forqueray (pg. 37 of this document), written in the 18th century. The information about what the odd notation means is in the preface.
Similarly, if you compose music and then come up with additional notation in our current system to help indicate your intent, you have participated in that evolution.
I'm not sure about "can't" be notated, but if you take the question to be "is there music for which a notation system does not exist, then certainly. The answer is yes, and not just obscure music, or "world" music, or even high-brow music. In this answer I'm going to use the example of Elvis Presley, believe it or not.
Other answers have pointed out that hindustani music, turkish music and persian music for example all follow different divisions of the octave which can't be notated in western notation: listen to singers from any of these traditions and you will hear melodies that you can't find on a piano. That's because, by western divisions, they are "microtonal", that is to say they use note distinctions smaller than a semitone.
However, there are examples which are much closer to home. Blues music at its roots contains melodies which are fundamentally microtonal. Often this is described in the sense that the blues adds "bent notes" or "blue notes" to the traditional western palette. However, often these "blue notes" are so fundamental to the melody itself, that it's not so much that the music is based around a western scale with additional "bent notes", and more that the western scale is completely absent. People obviously attempt to analyse it using familiar terms, but when projecting the notes A, Bb, B, C ... etc. onto the music, the result is something which, with its "nearest available note" resolution, loses the melody altogether.
Having said that, blues music, which originated as in unaccompanied work songs and field hollers, was then played on western instruments. Mostly these instruments can play outside of the 12 note scale, but have to be wrangled into doing so (like the guitar and the harmonica), and so this becomes increasingly influential on how blues music develops (along with the other musical traditions surrounding the blues of course).
Blues, and later jazz music runs the whole gamut from music completely unnotatable in western notation, through music with a non-microtonal skeleton over which more microtonal melodies are sung, to music which has microtonal inflections in the melodies here and there, but those melodies are very music within the 12-note western scale.
For example, this louis armstrong tune plays around with microtonal bent notes here and there, but is fundamentally "in key". You could absolutely write this down, leave the bent notes to largely to performance practice, and give it to 10 different players who had never heard the record and end up with something decent played every time.
Smokestack Lightnin' - Howling wolf. This is much more microtonal, but still an argument could be make that this is still fundamentally notateable. You could put some annotations about bent notes here and there and argue you have fairly represented the original.
However, because of the pervasive influence the blues had on western music, there are some pretty "mainstream" melodies that are pretty damn microtonal; Listen to Jailhouse Rock with honest ears, and you will find its melody completely outside the traditional western scale. Some people will write that off as somehow not a "serious" way to look at the music as if it's just stylistic variation and ornamentation on a melody which is really made up of "normal" notes. But anyone who sings jailhouse rock to themselves, or covers it, sings a melody with pretty much the same shape as the original; a shape which falls completely outside of western notation. Notably it spends most of it's time somewhere between a second and a third (as in between D and E when playing in C), with a couple of different distinctive (and unnotatable) patterns in that space. You can probably sing "jailhouse rock" from memory, but you couldn't play what you just sang on a piano.