The unequal division of a line such that the ratio of the smaller part to the larger is the same as that of the larger to the original whole. This ratio is approximately 1:1·618, and in mathematics is described as the ‘division in extreme and mean ratio’. The term ‘golden number’ is used today in the context of a natural phenomenon or a man-made object; it has often been held to produce harmonious proportions in, for example, architecture, fine art and sculpture, and there have been attempts to detect it in musical forms. Some 20th-century composers have used it consciously (see Numbers and music).
Roger Herz-Fischler has shown that the terms ‘golden section’ and ‘golden cut’ are relatively modern. ‘Goldene Schnitt’ (golden section/cut) was first used in 1835 by the mathematician Martin Ohm in the second edition of a text book, the first edition having described the ration with the usual term ‘stetige Proportion’ (continuous proportion). ‘Golden number’ has been a well-known and commonly used term since it was coined in 432bce by Meton the Athenian. The 19-year Lunar Cycle discovered by Meton (later known as the Metonic cycle) was originally written in golden numbers, hence the name. The Golden Numbers have been in constant use for centuries as the means by which to calculate the Ecclesiastical Paschal full moon, and thence Easter Sunday. It was only after the 1850s and the work of Zeising that the term ‘golden number’ became synonymous with ‘golden section’. The non-existence of the terms ‘golden section’ and ‘golden number’ (in the new sense) before 1830 should sound a note of warning to musicians and artists.
The ‘division in extreme and mean ratio’, on the other hand, is an ancient geometric ration, first described by Euclid. Herz-Fischler shows that very little attention was given to the ratio by the Greeks and argues that it is false to assume they advocated its use in architecture. He claims that the spread of golden numberism was aided by an error made in 1799 by Montucla and Lalande in the second edition of their Histoire des mathématiques in which they state that Pacioli (Divina proportione, 1509) advocated the use of the ratio in determining the proportions of works of art and architecture. This was false: Pacioli in fact recommended the use of simple ratios. The work of A. Zeising (1854) and F. Röber (1855) in Germany established the practice of ‘golden numberism’, and although it seems to have been limited to Germany, it gained international popularity after about 1910. Although some 20th-century composers have made deliberate use of the ratio, the term should be used cautiously in the context of music from before 1835. Historical evidence shows indisputably that composers would not have the term. The discovery of a 1:1·618 ratio in a work created before then may suggest that the composer consciously used the golden number, but it seems more likely to be, at best, an affirmation of Zeising’s theory, and at worst a fanciful imposition.
For bibliography see Numbers and music.
RUTH TATLOW