Key Transposition Calculator

Transposition is addition on the chromatic circle. Numbering the twelve pitch classes 0 through 11 from C, shifting by n semitones maps every note to (pitch plus n) modulo 12, and because the same offset applies to all of them the intervals between notes are unchanged. That invariance is the point: the major scale pattern of 2, 2, 1, 2, 2, 2, 1 semitones and the natural minor pattern of 2, 1, 2, 2, 1, 2, 2 survive the shift, so a I - V - vi - IV progression keeps its harmonic function and only its letter names change. In twelve-tone equal temperament each semitone multiplies frequency by the twelfth root of two, about 1.059463, so the ratio for n semitones is 2 raised to n over 12. A guitar capo is the physical version of the same operation: each fret raises every open string by one semitone. The main limitation of modulo-12 arithmetic is that it discards enharmonic spelling. Pitch class 6 can be F sharp or G flat, and which is correct depends on the key signature and harmonic context, not on the arithmetic. Notation software resolves this by following the circle of fifths and preferring signatures of seven accidentals or fewer, which is why G sharp major, needing eight sharps including F double sharp, is written as A flat major instead. Transposition also has consequences the numbers do not show: it moves a part relative to an instrument's tessitura and a singer's passaggio, and on guitar it changes which open strings ring.

Equal temperament, the tuning system that makes transposition arithmetically clean, was worked out independently in the late sixteenth century. The Chinese scholar Zhu Zaiyu published a calculation of the twelfth root of two in 1584, and Simon Stevin arrived at an equal division of the octave in the Netherlands around the same period in a manuscript not printed in his lifetime. Marin Mersenne discussed the numbers in Harmonie universelle in 1636. Adoption was slow because meantone and well temperaments made some keys sound sweeter than others, so transposing genuinely changed a piece's character. Bach's Well-Tempered Clavier of 1722 demonstrated writing in all twenty-four keys, and equal temperament became standard for European piano tuning in the nineteenth century. Transposing instruments have a separate origin. Natural horns and trumpets could sound only one harmonic series, so players inserted crooks of different tubing length to change the fundamental. Parts were written as though in C and the crook determined the sounding key, sparing players from relearning fingerings. After valves appeared in the early nineteenth century the convention was kept and extended to woodwinds, so a clarinettist can switch between B flat and A instruments using identical fingerings. This is why a written C on a B flat trumpet sounds a concert B flat, a major second lower, and why arrangers transpose parts in the opposite direction. Concert pitch was standardised at A equals 440 Hz in 1939 and later codified as ISO 16.