Cannot identify notation

[jiatu]

There's a notation that I do not know what it means.  It may be a trill, but it's vertical and not horizontal, plus it's below the note and not above, below middle C which is the first note for a triplet.   I've not seen it before and could not find it online.  Any help much appreciated.


Link to photo of what I'm talking about:
https://s1149.photobucket.com/user/Jiatu/media/Askmenow2_zpsxy0sojce.jpg.html

[timothy42b]

In that chord, the three lower notes are of quarter note time value and the upper note is an eighth note.

The long vertical line is just the note. 

The clue is the quarter note rest under the triplet.

[jiatu]

Many thanks. 

[jiatu]

https://i1149.photobucket.com/albums/o598/Jiatu/Askmenow2_zpsxy0sojce.jpg

Above is a link for an image of a measure from "Ask Me Now" found in T. Monk Easy Piano Solos.

In this measure, there is a quarter note rest. Why it is even in the measure I have no idea. If someone could explain, I would deeply appreciate it.

[quantum]

https://i1149.photobucket.com/albums/o598/Jiatu/Askmenow2_zpsxy0sojce.jpg

Above is a link for an image of a measure from "Ask Me Now" found in T. Monk Easy Piano Solos.

In this measure, there is a quarter note rest. Why it is even in the measure I have no idea. If someone could explain, I would deeply appreciate it.

Notice how there are notes with stems up, and there are notes with stems down, all with unnatural stem direction.  In this measure there are two voices (stems up and stems down) - that is two independent lines where you need to account for the rhythmic values.  It is like taking taking notes written on two staves but compressing them to fit on a single staff, the direction of the stems giving an indication to which voice one is referring to.  

The quarter rest belongs to the lower voice.  So in the lower voice you would have rhythmic values of: quarter, quarter, half.  

[timothy42b]

In this measure, there is a quarter note rest. Why it is even in the measure I have no idea. If someone could explain, I would deeply appreciate it.

Quantum's explanation is correct but I'll add to it.

In the bottom stave you have two half note chords.  There is no confusion what beat to put the second half note on.  The first half note is beats 1 and 2, and the second is beats 3 and 4.

In the top stave, the half note chord is in the physical position of beat 3, but physical positions don't necessarily determine the beat, though they are very convenient if well done.  The quarter rest is there to make it absolutely clear that there is a quarter note chord, a quarter note rest, and the half note chord must start exactly on beat 3. 

Most people would play it that way anyway, but it is good practice to make it clear.

Handbell music is rife with this kind of thing.  (I direct a handbell choir.)  Every piece has at least one example where the ringers have to ask me what beat a note is actually intended to be on, because the chords have multiple notes that often vary in time value.  A good editor always leaves a clue. 

[jiatu]

Many thanks for all the replies.  Very very helpful!