There is a “piccolo” (a wind instrument), there is a “dill pickle” (a pickled cucumber) and then there is a “dill piccolo—the wind instrument that plays only sour notes.” The alleged musical instrument has been shown on an image.
 
“When music is the topic: ‘That sour note just then must have emanated from the dill piccolo’” was printed in The Sunday Star (Washington, DC) on August 11, 1912. “If a piccolo player hits a a sour note is it a dill piccolo?” was printed in The Sentinel (Franklin, NE) on August 9, 1934.
 
Peter Schickele (who often performs the works of the fictional “P. D. Q. Bach”), according to his Wikipedia page, “invented the ‘dill piccolo’ for playing sour notes.” “The Dill Piccolo! @joshgroban The wind instrument that plays only sour notes” (with a picture of the instrument) was posted on Twitter by SuzanneLovesJGMusic on September 3, 2016.
   
   
Wikipedia: Piccolo
The piccolo /ˈpɪkəloʊ/ (Italian pronunciation: [ˈpikkolo]; Italian for “compact”, but named ottavino in Italy) is a half-size flute, and a member of the woodwind family of musical instruments. The modern piccolo has most of the same fingerings as its larger sibling, the standard transverse flute, but the sound it produces is an octave higher than written. This gave rise to the name ottavino (Italian for “little octave”), which the instrument is called in the scores of Italian composers. It is also called flauto piccolo or flautino.
   
Wikipedia: Pickled cucumber
Kosher dill
A “kosher” dill pickle is not necessarily kosher in the sense that it has been prepared in accordance with Jewish dietary law. Rather, it is a pickle made in the traditional manner of Jewish New York City pickle makers, with generous addition of garlic and dill to a natural salt brine.
 
In New York terminology, a “full-sour” kosher dill is one that has fully fermented, while a “half-sour”, given a shorter stay in the brine, is still crisp and bright green. Elsewhere, these pickles may sometimes be termed “old” and “new” dills.
         
Newspapers.com 
11 August 1912, The Sunday Star (Washington, DC), “The Grim Tragedy of Being a Comedian” by Raymond Hitchcock, Sunday Magazine, pg. 16, col. 3:
When music is the topic: “That sour note just then must have emanated from the dill piccolo.”
   
Newspapers.com
22 February 1929, The Daily Times (Davenport, IA), pg. 3, col. 3:
“PLEASURE BOUND,” a new revue, opened this week in N. Y. Here is a sample joke from it:
   
I love cornet-on-the-cob.
Can you play the dill piccolo?
 
Newspapers.com
13 January 1932, Des Moines (IA) Tribune, “‘Crazy Quilt’ Jokes Old Friends of Show-Goers” by Walter Graham, pg. 1, col. 3:
There was reference to somebody’s ability of the “fruit cello” and the “dill piccolo” in “A Night in Spain.”
 
Newspapers.com
9 August 1934, The Sentinel (Franklin, NE), “The Street Sweeper,” pg. 5, col. 3:
If a piccolo player hits a a sour note is it a dill piccolo?
 
Newspapers.com
17 July 1971, Nashville (TN) Tennessean, “Laugh-In’s Mod Mod World,” comics sec., pg. 19, col. 5:
IF H. J. HEINZ MERGED…
...WITH CONN INSTRMENTS…
...THEY’D MAKE DILL PICCOLOS
 
Google Books
The Definitive Biography of P. D. Q. Bach, 1807-1742?
By Peter Schickele
New York, NY: Random House
1976
Pg. 198:
... the legend cannot be verified, but we do know that Enfant wrote poetry and took lessons on the dill piccolo from Thomas Pollex, and in a letter to his parents in Rien, dated the 7th of November 1806, he says that he has written the text for a ...
 
Twitter
Steve Hayman
@shayman
The official flute of @Argonotes is the Dill Piccolo.  Which only plays sour notes. #10ConsecutivePiccoloThemedTweets
1:30 PM · May 2, 2012·Twitterrific for Mac
 
Twitter
Hugh Sung
@musicallifeshow
Meet PDQ Bach, inventor of the dill piccolo for playing sour notes (groan) http://bit.ly/22lDBMo #composer
10:37 AM · May 27, 2016·TweetDeck
 
Twitter
SuzanneLovesJGMusic
@suzannede4
The Dill Piccolo!
@joshgroban The wind instrument that plays only sour notes.
6:44 PM · Sep 3, 2016·Twitter for iPhone
 
Twitter
Band Jokes 101
@bandjokes101
Tag a [Dill] Piccolo player who only plays sour notes! Face with tears of joy http://ift.tt/2c2mFsK
11:03 PM · Sep 8, 2016·IFTTT
 
Twitter
Hiring devs! Slickdeals.net (11m monthly uniques)
@strantheman
Dill Piccolo for sale in my local Craigslist music section. Only plays sour notes. #random https://lasvegas.craigslist.org/msg/5773229662.html
4:53 AM · Sep 9, 2016·Twitter for Android
 
Twitter
Janet Horvath
@playinglesshurt
#jokeoftheday — unusual instruments of the orchestra : the Dill Piccolo… the instrument that plays only sour notes #music #piccolo #instruments
8:51 AM · Sep 3, 2018·Twitter for iPhone
     
Facebook
Peter Schickele
January 25, 2019 ·
Came across the Dill Piccolo today while rummaging around.
 
Twitter
Hat
@meltedbutter
Replying to @itsginnydi
Good.  Leaving out the dill piccolo flute saves your listeners from inevitable sour notes.  Yowling cat still allows for a lot of emotion and dynamic.
2:08 PM · Jan 26, 2020·Twitter Web App