Mathematics and Society - OS X Lion Server

III.THETW~LVt- TONtMET~DO(PAC,E. 2)The original twelve tones can be transposed (translated) up or downany number of half steps or piano keys.keys between each pair of tones remains the same.The number of half steps orhas been raised a whole step (two consecutive piano keys).The example belowopo o ~e ~o I2 1 5 3 11IV. An "inversion" of the original twelve tones can also be used. In aninversion the same number of half-steps are maintained between eachpair of notes, but the direction of pitch change is different. Theinversion of the first two notes of the original sequence is givenbelow.ORIG\IIlAL!I\lVl:RS10l-lAninterval of 3 half-steps.rfirst note isthe same.The second note is 2 half-stepshigher in pitchHere is the whole inversion of the original twelve tones.(Inversionsshould not be confused with inverted intervals in which one tone changesby an octave.)v.10 4 9 '1 8T T11T12 3T 5T1 GT~1 1MIDDLE:CThere are 39,916,800 different ways of ordering the original twelvetones (12! = 12 • 11 . 10 • 9 . 8 • 7 • 6 • 5 . 4 . 3 • 2 . 1 =479,001,600, but if transpositions are disregarded at this point,there are only II! or 39,916,800 ways.)any combination of the above variations can be used.IAfter an ordering is chosen,309