It's known that on a piano, no note is perfectly in-tune because of equal temperament tuning. However, I'm confused why we need this special tuning. Can't we just tighten each string on the piano so that they are each perfectly in tune, eliminating problems? (I'm sorry if this question is trivial...)
The simplest answer is that if we use the Just Intonation (most consonant) frequency ratios, things don't line up.
If we go up from A 440 by major thirds, we would have the notes A, then C#, then F, then A again, one octave higher. So the second A should have a frequency of 880 Hz.
But if we repeatedly multiply 440 by the JI ratio of 5/4 for a major third, we get:
- A - 440
- C# - 550
- F - 687.5
- A - 859.375???
One way to make it work is to not have all the thirds have the same frequency ratio, so some thirds would sound better and others worse. The other compromise is to make all the thirds equally wrong so that we can play any third we want and it sounds about the same. The second compromise is called Equal Temperment. As we can see from mdtracy's chart, that would look like this:
- A - 440
- C# - 554.37
- F - 698.46
- A - 880
Note that the problem occurs with all intervals (except octaves) not just the thirds. I merely used them as an example of how things don't line up naturally. The gap between what a frequency should be to equal the next octave and what it is if you generate the notes using just ratios is called the Pythagorean comma, since Pythagoras wrote down most of the first theories of music theory and notes in the western world.
Also note that on the piano specifically, there are other tuning problems, the main one being that the stiffness of the strings makes notes sound funny. If we tune the highest and lowest strings to the exact frequencies, then they will sound wrong to our ears, so we have to deliberately tune them "off" to make them sound right. This is called stretch tuning.
Also, in order to make the piano sound louder and more interesting, each "string" is actually two or three strings, except the lowest notes which are one string. For the two and three string notes, the different strings are not tuned exactly the same but just very slightly differently from each other to create a fuller sound. The same tuning trick is used on 12 string guitars, mandolins, and similar instruments.
If perfect tuning was possible then the ratios of the harmonic series (the notes a bugle can play using a fixed tube length) would be pure multiples across the piano keyboard.
Octave Up - multiply the frequency by 2 Octave + 5th - multiply by 3 2 Octaves - Multiply by 4 2 Octaves + 3rd - Multiply by 5
Notice the 5th (15th interval) is tuned a bit below 3 multiples while the major 3rds are tuned a bit above the pure 5.
A good piano tuner pushes these frequencies a bit further because the strings tend to sound somewhat out of tune at these listed frequencies due to lack of alignment with each string's overtones (muliple frequncies produced by the striking of a single string). Ideally the overtones are tweaked a bit to make the chords being played sound better. This is typically called a "stretch tuning" with the upper octaves being tuned a little "sharp".
Some of these ideas have found their way to suggested guitar tunings that detune the 6 strings from the expected pure frequencies to product better sounding chords.
There are two common tuning systems in Western music, which are equal temperament and just-intonation. Just intonation is based around simple frequency ratios, and is therefore more pleasing due to having perfect harmonies. So why wouldn't we use this? It would make sense that if we had a system that supports perfect harmonies, we should use it.
Unfortunately, those perfect ratios are centered around one note. So while harmonies may sound more in-tune in one key, they vary wildly and sound borderline unusable in some keys.
The advantage of equal-temperament is that compositions can be rewritten and played in any key, and sound relatively unchanged. So while I may not be able to play a piano piece in just-intonation in C#, I can play it in equal temperament and have it sound more in-tune.
I just tried some of the different "temperments" (various tunings developed over the years) and the "Pure Major" sounds amazing playing C to F to G and back to C but aweful playing in the other keys with C# - F# - G# - C# being the worst: like an upright left outdoors in the rain for a year. Orchestra players always try to play the 3rd note of a chord a bit flat and the fifths a bit sharp to make the chords sound better. It seems that digital piano's might have the capability to consider groups of notes in context and drop the 3rds and raise the 5ths to create pure tunings in any key and use the well tempered defaults when the context is unclear or ambiguous. Maybe there's already one out there in the market.