The use of simple geometry and the local unit of measurement in the design of Italian stringed keyboard instruments:

an aid to attribution and to organological analysis


Grant O'Brien


Appendix 1 - Definitions of the Geometrical Functions


The geometry of the simple right-angled triangle is basic to the understanding of the design of Italian keyboard instruments. The definitions of the sine (sin), cosine (cos) and tangent (tan) of the angles of a right-angled triangle are based on the ratios of the lengths of the sides x, y and r in the diagram below:



  The angle α is measured in degrees and this angle, for baseboard corners of Italian stringed keyboard instruments, usually has a value between about 30 to 60. Here by definition:

sin α = cos α = tan α =


arcsin = α arccos = α arctan = α


Of these the most important factors involved in the determination of the unit of measurement used in the design of Italian stringed keyboard instruments are tan α and arctan = α. The actual values of the tan and arctan can be calculated using a normal scientific calculator, they can be found in tables of the geometrical functions, or they can be read from any good scientific slide rule.

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