There are many varieties of ten-string guitar, including:
Harp guitars are guitars to which extra strings have been added which are never fretted but may be plucked or strummed. These strings are therefore played in a manner somewhat similar to those of the harp, while those of the principal neck are played as a guitar, hence the name.
Often but not always, a second neck, parallel to the fretboard, carries these extra strings. There have been many designs of harp guitar, but in the nineteenth century ten-string versions were particularly popular.
Information on nineteenth-century harp guitars comes from three main primary sources:
In the early 19th century Ferdinando Carulli and René Lacôte developed a harp guitar they called the Décacorde (French for ten-string).
Carulli played this type of guitar and wrote a method for it titled Méthode Complète pour le Décacorde. In it he describes the tuning as C-D-E-F-G-A-d-g-b-e' (strings 10 to 1), with the upper five strings A-d-g-b-e' fretted and the lower basses C-D-E-F-G not fretted.
Carulli also wrote divertissements for this instrument.
Two Décacordes by Lacôte are housed in the Music Museum of the Cité de la Musique in Paris:
There is also a Décacorde (attributed to Lacôte), that was in the workshop of Françoise Sinier de Ridder, which has 7 strings on the neck (fretted) and 3 sub-basses (unfretted strings).
Sinier and de Ridder have pointed out that the décacorde was made in three different string configurations. Those instruments that adhere to the Carulli patent have 5 strings on the fingerboard and 5 floating basses [...]. Other specimens that do not bear the patent stamp are known with 6 strings on the fingerboard and 4 floating, and 7 strings on the fingerboard and 3 floating. I now speculate that these latter may have been configured, not as true Carulli Patent Décacordes, but as similar-appearing Lacôte ten-strings tuned more traditionally, and perhaps, played "professionally."
Period harp guitars built by Johann Gottfried Scherzer survive. A copy of one of these, based on an original circa 1862, has six fretted and four unfretted strings.
Johann Kaspar Mertz is known to have played ten-string harp guitars. Based on surviving instruments and urtexts of music written for it, the tuning was AI-BI-C-D-E-A-d-g-b-e'.