Classical music Joseph Haydn's
Symphony No. 47 in G is nicknamed "the Palindrome". In the third movement, a
minuet and
trio, the second half of the minuet is the same as the first but backwards, the second half of the ensuing trio similarly reflects the first half, and then the minuet is repeated. The interlude from
Alban Berg's opera
Lulu is a palindrome, as are sections and pieces, in
arch form, by many other composers, including
James Tenney, and most famously
Béla Bartók.
George Crumb also used musical palindrome to text paint the
Federico García Lorca poem "¿Por qué nací?", the first movement of three in his fourth book of
Madrigals.
Igor Stravinsky's final composition,
The Owl and the Pussy Cat, is a palindrome. The first movement from
Constant Lambert's
ballet Horoscope (1938) is entitled "Palindromic Prelude". Lambert claimed that the theme was dictated to him by the ghost of
Bernard van Dieren, who had died in 1936. British composer
Robert Simpson also composed music in the palindrome or based on palindromic themes; the slow movement of his
Symphony No. 2 is a palindrome, as is the slow movement of his
String Quartet No. 1. His hour-long
String Quartet No. 9 consists of thirty-two variations and a fugue on a palindromic theme of Haydn (from the minuet of his Symphony No. 47). All of Simpson's thirty-two variations are themselves palindromic.
Hin und Zurück ("There and Back": 1927) is an operatic 'sketch' (Op. 45a) in one scene by Paul Hindemith, with a German libretto by Marcellus Schiffer. It is essentially a dramatic palindrome. Through the first half, a tragedy unfolds between two lovers, involving jealousy, murder and suicide. Then, in the reversing second half, this is replayed with the lines sung in reverse order to produce a happy ending. The music of
Anton Webern is often palindromic. Webern, who had studied the music of the Renaissance composer
Heinrich Isaac, was extremely interested in symmetries in music, be they horizontal or vertical. An example of horizontal or linear symmetry in Webern's music is the first phrase in the second movement of the
symphony, Op. 21. A striking example of vertical symmetry is the second movement of the
Piano Variations, Op. 27, in which Webern arranges every pitch of this
dodecaphonic work around the central pitch axis of A4. From this, each downward reaching interval is replicated exactly in the opposite direction. For example, a G3—13 half-steps down from A4 is replicated as a B5—13 half-steps above. Just as the letters of a verbal palindrome are not reversed, so are the elements of a musical palindrome usually presented in the same form in both halves. Although these elements are usually single notes, palindromes may be made using more complex elements. For example,
Karlheinz Stockhausen's composition
Mixtur, originally written in 1964, consists of twenty sections, called "moments", which may be
permuted in several different ways, including retrograde presentation, and two versions may be made in a single program. When the composer revised the work in 2003, he prescribed such a palindromic performance, with the twenty moments first played in a "forwards" version, and then "backwards". Each moment is a complex musical unit and is played in the same direction in each half of the program. By contrast,
Karel Goeyvaerts's 1953 electronic composition,
Nummer 5 (met zuivere tonen) is an
exact palindrome: not only does each event in the second half of the piece occur according to an axis of symmetry at the centre of the work, but each event itself is reversed, so that the note attacks in the first half become note decays in the second, and vice versa. It is a perfect example of Goeyvaerts's aesthetics, the perfect example of the imperfection of perfection. In
classical music, a
crab canon is a
canon in which one line of the melody is reversed in time and pitch from the other. A large-scale musical palindrome covering more than one movement is called "chiastic", referring to the cross-shaped Greek letter "
χ" (pronounced /ˈkaɪ/.) This is usually a form of reference to the crucifixion; for example, the '''' movement of Bach's
Mass in B minor. The purpose of such palindromic balancing is to focus the listener on the central movement, much as one would focus on the centre of the cross in the crucifixion. Other examples are found in Bach's cantata BWV 4,
Christ lag in Todes Banden, Handel's
Messiah and Fauré's
Requiem. A
table canon is a rectangular piece of sheet music intended to be played by two musicians facing each other across a table with the music between them, with one musician viewing the music upside down compared to the other. The result is somewhat like two speakers simultaneously reading the
Sator Square from opposite sides, except that it is typically in two-part polyphony rather than in unison.
Biological structures A: Palindrome, B: Loop, C: Stem Palindromic motifs are found in most
genomes or sets of
genetic instructions. The meaning of palindrome in the context of genetics is slightly different, from the definition used for words and sentences. Since the
DNA is formed by two paired strands of
nucleotides, and the nucleotides always pair in the same way (
Adenine (A) with
Thymine (T),
Cytosine (C) with
Guanine (G)), a (single-stranded) sequence of DNA is said to be a palindrome if it is equal to its complementary sequence read backward. For example, the sequence is palindromic because its complement is , which is equal to the original sequence in reverse complement. A palindromic
DNA sequence may form a
hairpin. Palindromic motifs are made by the order of the
nucleotides that specify the complex chemicals (
proteins) that, as a result of those
genetic instructions, the
cell is to produce. They have been specially researched in
bacterial chromosomes and in the so-called Bacterial Interspersed Mosaic Elements (BIMEs) scattered over them. In 2003, a research genome sequencing project discovered that many of the bases on the
Y-chromosome are arranged as palindromes. A palindrome structure allows the Y-chromosome to repair itself by bending over at the middle if one side is damaged. It is believed that palindromes are also found in proteins, but their role in the protein function is not clearly known. It has been suggested in 2008 that the prevalent existence of palindromes in peptides might be related to the prevalence of low-complexity regions in proteins, as palindromes frequently are associated with low-complexity sequences. Their prevalence might also be related to an
alpha helical formation propensity of these sequences,
Computation theory In
automata theory, a
set of all palindromes in a given
alphabet is a typical example of a
language that is
context-free, but not
regular. This means that it is impossible for a
finite automaton to reliably test for palindromes. In addition, the set of palindromes may not be reliably tested by a
deterministic pushdown automaton which also means that they are not
LR(k)-parsable or
LL(k)-parsable. When reading a palindrome from left to right, it is, in essence, impossible to locate the "middle" until the entire word has been read completely. It is possible to find the
longest palindromic substring of a given input string in
linear time. The
palindromic density of an infinite word
w over an alphabet
A is defined to be zero if only finitely many prefixes are palindromes; otherwise, letting the palindromic prefixes be of lengths
nk for
k=1,2,... we define the density to be : d_P(w) = \left( { \limsup_{k \rightarrow \infty} \frac{n_{k+1}}{n_k} } \right)^{-1} \ . Among aperiodic words, the largest possible palindromic density is achieved by the
Fibonacci word, which has density 1/φ, where φ is the
Golden ratio. A
palstar is a
concatenation of palindromic strings, excluding the trivial one-letter palindromes – otherwise all strings would be palstars. == Notable palindromists ==