Mathematics and Society - OS X Lion Server

THETWtLVt- TONEM[T~OO(1.All twelve tones of the chromaticscale are used in the twelve-tonemethod of composition. Each of thetwelve tones must be used beforeany of the tones can be repeated.The sequence of twelve tones is then varied in ways somewhat likemotions in geometry. Suppose the following ordered row of twelvetones is chosen. The same ordering of tones is indicated on pianokeys and in written music.2. G 5 3 108 7 1"'" 9 "l- 12.' 11NOie:NOTEtMIDDLE:C(II.The next twelve tones can be determined by reflecting the originaltwelve as shown below. This gives the original twelve tones readbackwards. (In music this is called the retrograde series.)ORIGINAL2. G 5 3 103088 7 l s, 9 4- 12T~ 111\\0",: '-lo,.t:MIDDLECtTHE TWELFTH NOTE BECOMES THESECOND MEASUREFIRST NOTE OFTHE FIRST NOTE BECOMES THE TWELFTH NOTE OF THE SECONDMEASURE(