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Harmonograph

43 42 equal TemPeramenT changing keys made easy Although early tunings enabled many pure harmonic ratios to be played, it was hard to move into other keys, one could only really change mode see page 52. Musicians often had to retune their instruments, or use extra notes reserved for specific scales classical Indian tuning uses 22 notes. In the sixteenth century a new tuning was developed which revolutionised western music and which predominates today. The octave is divided into twelve fixed equal intervals, each chromatic semitone being 1.05946 times its neighbour 2 112 , roughly 1817. The twelve equally spaced notes are arranged in a circle below. Six flat wholetones now make an octave, as do four very flat minor thirds, or three sharp major thirds. The Pythagorean comma vanishes, as do all perfect intervals except the octave its a clever fudge which allows us to change key easily. It is slightly out of tune and we hear it every day. Triads are chords of three notes. Opposite top we see major and minor triads involving the note c, in the key of C. Use the mastergrid opposite below to navigate the equaltempered sea, and perceive any 345 triad a chord of three notes in three distinct keys after Malcolm Stewart.
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