Magic square, from twelve-tone row¶
The first function below accumulates the twelve transpositions of a twelve-tone row, ordered by the pitch-classes of the row’s inversion. The second function (and LilyPond preamble) notate the result. Three rows used in compositions of the 1950s conclude the example:
>>> def make_transpositions(row):
... transpositions = []
... for pitch_class in row.invert():
... n = pitch_class.number - row[0].number
... transposition = row.transpose(n)
... transpositions.append(transposition)
... return transpositions
...
>>> def make_score(row):
... score = abjad.Score(name="Score")
... transpositions = make_transpositions(row)
... for i, transposition in enumerate(transpositions):
... notes = [abjad.Note(_, (1, 4)) for _ in transposition]
... voice = abjad.Voice(notes, name=f"Voice_{i}")
... staff = abjad.Staff([voice], name=f"Staff_{i}")
... score.append(staff)
... number = notes[0].written_pitch.number
... string = r"\markup \larger \with-color #blue"
... string = string + r" { T \hspace #-0.75 \sub" + str(number) + "}"
... markup = abjad.Markup(string)
... instrument_name = abjad.InstrumentName(markup)
... abjad.attach(instrument_name, notes[0])
... for note in score["Voice_0"]:
... number = note.written_pitch.number
... string = r"\markup \larger { IT \hspace #-0.75 \sub "
... string += str(number)
... string += " }"
... markup = abjad.Markup(string)
... abjad.attach(markup, note, direction=abjad.UP)
... note = abjad.select.note(score, 0)
... time_signature = abjad.TimeSignature((12, 4))
... abjad.attach(time_signature, note)
... return score
...
>>> preamble = r"""#(set-global-staff-size 14)
...
... \layout {
... \context {
... \Staff
... \override VerticalAxisGroup.staff-staff-spacing.minimum-distance = 11
... }
... \context {
... \Score
... \override BarLine.stencil = ##f
... \override Clef.stencil = ##f
... \override SpacingSpanner.strict-spacing = ##t
... \override SystemStartBar.stencil = ##f
... \override Stem.stencil = ##f
... \override TextScript.color = #blue
... \override TextScript.staff-padding = 5
... \override TimeSignature.transparent = ##t
... proportionalNotationDuration = #(ly:make-moment 1 16)
... }
... }
... """
Example 1. Here are the 48 row forms of [7, 3, 8, 5, 4, 6, 0, 10, 11, 2, 9, 1]:
>>> row = abjad.TwelveToneRow([7, 3, 8, 5, 4, 6, 0, 10, 11, 2, 9, 1])
>>> score = make_score(row)
>>> lilypond_file = abjad.LilyPondFile([preamble, score])
>>> abjad.show(lilypond_file)
From the opening measure of Karlheinz Stockhausen’s Grüppen (1955-57) for three orchestras.
Example 2. Here are the 48 row forms of [3, 5, 2, 1, 10, 11, 9, 0, 8, 4, 7, 6]:
>>> row = abjad.TwelveToneRow([3, 5, 2, 1, 10, 11, 9, 0, 8, 4, 7, 6])
>>> score = make_score(row)
>>> lilypond_file = abjad.LilyPondFile([preamble, score])
>>> abjad.show(lilypond_file)
From Koblyakov’s analysis of Pierre Boulez’s Marteau sans maître (1955).
Example 3. Here are the 48 row forms of [9, 2, 11, 4, 1, 6, 8, 3, 5, 10, 7, 0]:
>>> row = abjad.TwelveToneRow([9, 2, 11, 4, 1, 6, 8, 3, 5, 10, 7, 0])
>>> score = make_score(row)
>>> lilypond_file = abjad.LilyPondFile([preamble, score])
>>> abjad.show(lilypond_file)
From Benjamin Britten’s The Turn of the Screw (1954).