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    Tone rows and tropes

    Harald FripertingerKarl-Franzens-Universitat Graz

    International Congress on Music and MathematicsPuerto Vallarta, Mexico, November 2629, 2014

    Joint work with Peter Lackner,University of Music and Performing Arts Graz.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    Tone rows and tropes

    Harald FripertingerKarl-Franzens-Universitat Graz

    International Congress on Music and MathematicsPuerto Vallarta, Mexico, November 2629, 2014

    Joint work with Peter Lackner,University of Music and Performing Arts Graz.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    What is a tone-row?

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    What is a tone-row?

    O. Messiaen: Le Merle Noir: A piece for flute and piano composed in1951. The tone row appears in the coda of the piano part.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    What is a tone-row?

    O. Messiaen: Le Merle Noir: A piece for flute and piano composed in1951. The tone row appears in the coda of the piano part.

    The rhythm is not important for the analysis.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    What is a tone-row?

    O. Messiaen: Le Merle Noir: A piece for flute and piano composed in1951. The tone row appears in the coda of the piano part.

    The rhythm is not important for the analysis.

    Replacing the tones by representatives of octave equivalence.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    The chromatic scale

    Here is a short part of the chromatic scale together with the labelling ofthe tones in Z and pitch classes in Z12.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    Tone rows

    In Music: a sequence of 12 tones so that different tones are not octaveequivalent.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    Tone rows

    In Music: a sequence of 12 tones so that different tones are not octaveequivalent.

    In Mathematics:

    f :{1, . . . ,12} Z, f (i) 6= f ( j), i 6= j.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    Tone rows

    In Music: a sequence of 12 tones so that different tones are not octaveequivalent.

    In Mathematics:

    f :{1, . . . ,12} Z, f (i) 6= f ( j), i 6= j.

    Simple model:f :{1, . . . ,12} Z12, bijective.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    Tone rows

    In Music: a sequence of 12 tones so that different tones are not octaveequivalent.

    In Mathematics:

    f :{1, . . . ,12} Z, f (i) 6= f ( j), i 6= j.

    Simple model:f :{1, . . . ,12} Z12, bijective.

    Each pitch class occurs at most once as f ( j), j {1, . . . ,12}.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    JJ II

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    Tone rows

    In Music: a sequence of 12 tones so that different tones are not octaveequivalent.

    In Mathematics:

    f :{1, . . . ,12} Z, f (i) 6= f ( j), i 6= j.

    Simple model:f :{1, . . . ,12} Z12, bijective.

    Each pitch class occurs at most once as f ( j), j {1, . . . ,12}.

    Each pitch class occurs at least once as f ( j), j {1, . . . ,12}.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    Tone rows

    In Music: a sequence of 12 tones so that different tones are not octaveequivalent.

    In Mathematics:

    f :{1, . . . ,12} Z, f (i) 6= f ( j), i 6= j.

    Simple model:f :{1, . . . ,12} Z12, bijective.

    Each pitch class occurs at most once as f ( j), j {1, . . . ,12}.

    Each pitch class occurs at least once as f ( j), j {1, . . . ,12}.

    Each pitch class occurs exactly once as f ( j), j {1, . . . ,12}.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    How many tone rows exist?

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    How many tone rows exist?

    1. step: Choose a pitch class from 12 possible classes.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    How many tone rows exist?

    1. step: Choose a pitch class from 12 possible classes.2. step: Choose a pitch class from 11 possible classes.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    How many tone rows exist?

    1. step: Choose a pitch class from 12 possible classes.2. step: Choose a pitch class from 11 possible classes.3. step: Choose a pitch class from 10 possible classes.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    How many tone rows exist?

    1. step: Choose a pitch class from 12 possible classes.2. step: Choose a pitch class from 11 possible classes.3. step: Choose a pitch class from 10 possible classes....11. step: Choose a pitch class from 2 possible classes.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    How many tone rows exist?

    1. step: Choose a pitch class from 12 possible classes.2. step: Choose a pitch class from 11 possible classes.3. step: Choose a pitch class from 10 possible classes....11. step: Choose a pitch class from 2 possible classes.12. step: Choose a pitch class from 1 possible class.

    (Actually the last pitch class is already determined.)

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    How many tone rows exist?

    1. step: Choose a pitch class from 12 possible classes.2. step: Choose a pitch class from 11 possible classes.3. step: Choose a pitch class from 10 possible classes....11. step: Choose a pitch class from 2 possible classes.12. step: Choose a pitch class from 1 possible class.

    (Actually the last pitch class is already determined.)

    This leads to a total of

    12 11 2 1 = 479001600 = 12!

    tone rows.

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    O. Messiaen: Le Merle Noir

    Reduction to pitch classes (9,2,8,3,10,6,4,0,1,11,5,7).

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    Circular representation of a tone row

    f := ( f (1), . . . , f (12)) = (9,2,8,3,10,6,4,0,1,11,5,7)

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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    Circular representation of a tone row

    f := ( f (1), . . . , f (12)) = (9,2,8,3,10,6,4,0,1,11,5,7)we draw the 12 pitch classes as a regular 12-gon,

    http://www.uni-graz.athttp://www.uni-graz.at/~fripert/

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