Generalized keyboards are musical keyboards, a type of isomorphic keyboard, with regular, tile-like arrangements usually with rectangular or hexagonal keys, and were developed for performing music in different tunings. They were introduced by Robert Bosanquet in the 1870s, and since the 1960s Erv Wilson has developed new methods of using and expanding them, proposing keyboard layouts (and some notations) including any scale made of a single generator within an "octave" (or more generally, period) of any size. The generalized keyboard is one kind of symmetrical arrangement that represents pitches according to their relationship to each other - rather than their positions in specific scales such as in the familiar piano and organ keyboard - as well as in sequence of pitch, unlike arrangements such as duet systems for concertinas and the array system keyboard. Bosanquet used chains of fifths to generate the mapping of the keys, with the pitches transposed into the space of an octave (or more generally, period) so that pitch determined the lateral position of keys, and the number of fifths, or any other interval used as generators, from the starting pitch its vertical position in the keyboard. The coordinates of a key can be calculated using the size of the generator (in cents), where the vertical position is the number of generators away from the starting pitch, (positive or negative depending on whether they are in the sharp or flat direction), which Bosanquet rounded to the nearest semitone, giving 12 vertically aligned rows of keys in each octave. Wilson introduced alternate arrangements with combinations of seven or five (and in some cases, i.e., a division of the octave in 13 tones, eight) vertical lines of keys, and so Bosanquet keyboard sometimes is used to distinguish the original arrangement. The angle between keys originally depended on the shape of the key and relative size of the generating fifth, but Bosanquet discovered fingering could sometimes be improved by reversing this angle; the angle can also be straightened out, or the spacing changed to better represent the keys' pitches, which alters vertical alignment. Generalized keyboards usually duplicate pitches mapped close to the front and rear edges of the arrangement in order to organize pitches for consistent fingering. This can result in very large arrangements depending on the shape of the keys and any modifications made to the basic design. The possibility of the duplication of pitches occurs because the