Dear all Listers, there is a sort of paradoxical legend about Fourier : when his methods has so many applications in this field, he is said to have left no work on music theory. You can have a look at Champollion-Figeac, Louis Charbonneau, Gaston Darboux, Dhombres & Robert, Ivar Grattan-Guinness, J. W. Herivel, H. Yoshida and S. Takata, at all who wrote about Fourier's works and life, they all agree on this : Fourier did not work on music.
I am very happy to tell you they are wrong, and the legend is far-fetched. Indeed, I found in  p. 17-23 an abstract of Fourier's "Mémoire sur les vibrations des surfaces élastiques", which clearly deals with musical theory. As it is under Delambre's authority, it could easily have been unseen by the best commentators of Fourier. But where I do not find any excuse or explanation, is that the very same abstract is reproduced in  p. 258-264 by Fourier himself, as soon as he was elected as Secrétaire perpétuel after Delambre's death.
Under a similar title, an article by Fourier was well known. It is listed as . Some commentators have noticed it, but did not see its relation to a near homonym memoir, and, as it contains no reflection on music, their comments are of no help for us here.
The "Mémoire sur les vibrations des surfaces élastiques" was the first submitted by Fourier since he was elected at the Paris Academy. This is one more reason to be astonished it was unseen by all commentators, and it is a very strong argument to call for its rediscovering. Who could help so? Someone familiar with Fourier's manuscripts ? I wrote to M. Charbonneau, with no answer, but may be the adress I have is obsolete. Could someone tell him about ? I consider of the highest importance to find this memoir and to publish it. Just consider : Fourier's "lost" memoir on music ! ----------------------  DELAMBRE, [FOURIER] ; Analyse des travaux de l'Académie royale des Sciences, pendant l'année 1818. Partie mathématique. Par M. le Chevalier Delambre, Secrétaire perpétuel.
 FOURIER ; Analyse des travaux de l?Académie royale des Sciences, pendant l?année 1822.
 Note relative aux vibrations des surfaces élastiques et au mouvement des ondes : p. 129-136, livraison de septembre 1818 of the Bulletin des sciences par la Société Philomatique. ----------------------