The alto clef is used in sheet music for the viola and other instruments whose range dwells somewhere between the treble clef and the bass clef.

It's also called the C clef because the center line is middle C. Musically speaking, if you combined the bottom two lines of the treble clef and the top two of the bass clef, and added a fifth line between them, you'd have the alto clef.

In music notation, the alto clef looks something like this:


II |---------
II |  *  \   
II |------|--
II |  |__/   
II |-{ __----
II |  |  \   
II |------|--
II |  *__/   
II |---------
Learning the alto clef after the treble clef, or after the bass clef, or both, is painfully difficult. Here's the deal:
---------------- In treble clef, this is a B
---------------- 
-------O-------- In alto clef, this is a C.
----------------
---------------- In bass clef, this is a D.
It challenges you to read in a completely different way that is just nearly the same: like if someone told you that all the english letter sounds had shifted, so an "M" sounded like an "L", a "B" like an "A". You know the alphabet, you know how to read words, but would this be an easy transition? Of course not!

Truly, the only way to learn the alto clef is to play in it a lot. Forget mnemonics, don't use the tricks that are only acceptable when you need to know the alto clef for a one song stint (see the end of this writeup). Simply remember that the middle line is middle C, and play through every piece of music in the alto clef on your instrument of choice as possible. If you're learning the alto clef for theory purposes and not for playing a specific instrument, get music written in alto clef (ie, viola parts of orchestral works) and just like you did when you were little, write the letter name of each note underneath the staff. If you've got one at your disposal, try to play the melody lines on a piano.

The most common setback when trying to learn the alto clef is that you will start playing by interval: you figure out the starting note of the piece of music the right way, but thereafter, instead of connecting positions to notes, you figure out the next note based on the interval away from the note you're playing. This is confusing on paper, but nearly everyone does it when learning a new clef. For example:

--------------------------

------------O-------------
                 O   
--O-----------------------

-------O------------------

--------------------------
The first note is middle C: you know this. The second note is an A, but you don't think of it this way. You think "Oh, that's a third below C. Therefore, it's an A!" The next note is an E, which you'll undoubtedly arrive at by concluding that it's a fifth higher than A. Finally, the last note is a D, which can easily be identified because it's a second below E. This seems like a foolproof method at first; however, it will run you into troubles. If you ever find yourself playing music that has a jump from a low note to a high note of an interval larger than an octave, and you do not possess a degree in theory, you are probably screwed. No matter what you may say, you simply do not know what note is a thirteenth above A fast enough to play by interval. (It's an F, by the way)

That said, there are situations where certain tricks are neccessary and forgivable. A friend of mine who plays violin was once forced to fill in for a missing violist, since the two are technically similar and the part was fairly simple. Of course, she didn't know how to read the alto clef. After being told that she could easily read it because each note is one above what it would be on the treble clef (which is, by the way, not an easy way to think about playing in any clef, it just makes everything more confusing), someone taught her this trick, world renowned, for your every day violinist filling in as a violist:

Playing the viola in alto clef is the same as playing the violin in third position

How does this work? Quite simply, actually. Consider the following:

------------------- This is a C in alto clef and a B in treble clef.
------------------- On the viola, in first position,
--------O---------- this would be played with the 3rd finger on the 2nd string.
------------------- On the violin, in third position, 
------------------- this would be played with the 3rd finger on the 2nd string.

--------O---------- This is a G in alto clef and an F in treble clef.
------------------- On the viola, in first position,
------------------- this would be played with the 3rd finger on the 3rd string.
------------------- On the violin, in third position,
------------------- this would be played with the 3rd finger on the 3rd string.
No joke. This works for every note in the first position range on the viola except for the lowest, the open C. Certain problems will arise (most notably with key signature), but they can usually be worked through. This does work, but please, do not rely on it as the extent of your knowledge of the alto clef: You will be looked at oddly when you cannot name notes without your violin handy.
The focus of this writeup is obviously the viola, the instrument that most commonly uses the alto clef, and the violin, the instrument whose players most often need to learn the alto clef in addition to the treble clef. So there.

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