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A musical cryptogram is a cryptogrammatic sequence of musical notes, a sequence which can be taken to refer to an extra-musical text by some 'logical' relationship, usually between note names and letters. The most common and best known examples result from composers using ciphered versions of their own or their friends' names as themes or motifs in their compositions. Much rarer is the use of music notation to encode messages for reasons of espionage or personal security called steganography.
Because of the multitudinous ways in which notes and letters can be related, detecting hidden ciphers and proving accurate decipherment is difficult.
From the initial assignment by Western music theorists of letter names to notes in the 9th century it became possible to reverse the procedure and assign notes to the letters of names. However, this does not seem to have become a recognized technique until the Baroque period. From the mid-19th century it has become quite common. Sporadic earlier encipherments used solmization syllables.
It is believed that this method was first used by Josquin des Prez in his Missa Hercules Dux Ferrarie. It was named Soggetto cavato by the later theorist Zarlino. Under this scheme the vowel sounds in the text are matched to the vowel sounds of the solmization syllables of Guido of Arezzo (where 'ut' is the root, which we now call 'do'). Thus the Latin name of the dedicatee 'Hercules Dux Ferrarie' (Ercole d'Este, Duke of Ferrara) becomes re-ut-re-ut-re-fa-mi-re, which translates as D-C-D-C-D-F-E-D in modern notation with C as 'ut'. This is used as the cantus firmus of the mass setting. Josquin's method was imitated by several of his contemporaries and successors, including Adrian Willaert and Costanzo Festa.
Since the note names only cover letters A to G (reflecting the octave repetition of these names), the problem arises as to how to cipher the rest of the alphabet. Historically there have been two main solutions, which may be labelled for convenience the 'German' and the 'French' methods.
Because the development of note names took place within the framework of modes, in the German-speaking world B-flat was named 'B' and B-natural was named 'H'. The most common musical cryptogram is the B-A-C-H motif, which was used by Johann Sebastian Bach himself, by his contemporaries and by many later composers. Other note names were derived by sound, for example E-flat, 'Es' in German, could represent 'S' and A-flat the digraph 'As'.
Composers less fortunate than Bach usually seem to have chosen to ignore non-musical letters in generating their motifs. For example, Robert Schumann, an inveterate user of cryptograms, has just S-C-H-A (E-flat, C, B-natural, A) to represent himself in Carnaval. Sometimes phonetic substitution could be used, Schumann representing Bezeth by B-E-S-E-D-H. Johannes Brahms used B-A-H-S (B-flat, A, B-natural, E-flat) for his surname in the A-flat minor organ fugue, and the mixed language Gis-E-La (G-sharp, E, A) for Gisela von Arnim, among many examples.
The 'French' method of generating cryptograms arose late in the 19th century and was more akin to normal encipherment. The most popular version involved writing out the letters H-N, O-U and V-Z in lines under the original diatonic notes A-G, as follows:
so that A, H, O, and V are enciphered by note 'A', B, I, P and W by 'B' (flat or natural) and so on. This scheme was used by Jules Écorcheville, editor of the journal S.I.M., to solicit centenary commemorations of Joseph Haydn in 1909, except that he diverted the 'H' to B-natural, presumably to avoid too many repeated notes. Writing to Gabriel Fauré about the invitation, Camille Saint-Saëns said he was writing to Écorcheville asking him to prove that Y and N could signify D and G as "it would be annoying to get mixed up in a farcical business which would make us a laughing stock in the German musical world."  The many-to-one mapping of this method makes it more difficult to extract possible motifs from the musical score than the one-to-one correspondence (apart from 'As') of the German system.
A French tradition of celebratory uses developed from the Haydn centenary, with tributes to Gabriel Fauré by Maurice Ravel, Florent Schmitt, Charles Koechlin and others in 1922 (added to later by Arnold Bax, 1949) and to Albert Roussel by Francis Poulenc, Arthur Honegger, Darius Milhaud and others (using various ciphering schemes) in 1929. Honegger's system involved placing the letters after 'H' under sharpened and flattened notes, an example of how chromatic cryptograms could be more easily accommodated in 20th-century music.
Dmitri Shostakovich used the German scheme for his personal motto D-Es-C-H (D, E-flat, C, B-natural), representing D.SCH, which appears in many of his most characteristic works. Elliott Carter featured both a cryptogram for the last name "Boulez" in his piece Réflexions (2004) and a sonic symbol of the first name "Pierre".
Cryptograms were less common in England, but Edward Elgar, who was also interested in general cryptography and puzzles, wrote an early Allegretto for his pupils the Gedge sisters using G-E-D-G-E  and part of the 'enigma' in the Enigma Variations involves cryptograms.
A more sophisticated contemporary use of musical ciphering can be found in the popular online puzzle Cicada 3301. In the puzzle, the creators use dyads (two notes) to represent letters and numbers with word spacing based on prime numbers of beats. There are two notes, both in Ionian mode. The range of the top note is limited to a major 9th and there are 4 bottom notes, thus yielding 36 combinations, enough for 26 letters and 10 numbers. This makes for a naturally musical and melodic result while still permitting entire sentences to be encrypted.
In 1947 Friedrich Smend also been suggested that Bach enciphered significant numbers through methods including repetitions of a motif, word, or phrase; the notes played on the continuo; the use of sequence; and the notes played by the accompaniment. However, Ruth Tatlow has presented evidence questioning the plausibility of Smend's claims.
During the first quarter of the 20th century, American author and occultist Paul Foster Case established an esoteric musical cryptogram for the purposes of ceremonial magick. The system was a derivative of a cipher used by an affiliated magical order called the Hermetic Order of the Golden Dawn. Each note of the 12 tone system was assigned a set of correspondences including colors, planets, zodiacal signs, and Hebrew letters. The holy names of biblical characters were translated letter by letter into a linear sequence of musical notes, so that each letter could be sung by the congregation in unison.
Ezra Sandzer-Bell has written and published two books on this subject, describing how Paul Foster Case's system of musical cryptography could be applied to songwriting. Any word can be translated phonetically into Hebrew and converted using Case's cryptogram to generate a series of notes. Sandzer-Bell's project involves the conversion of the common and Latin names of plants, trees, and mushrooms into melodies. Each song was composed by consuming the plant in tea or tincture form, then using the physical effects of the plant to determine what kind of rhythm, harmony, instruments, and dynamics to use. A lengthy demonstration and proof concept is publicly available on the author's website.
The following list includes only motifs which are known to have been used in published works.