Pitch Recipes

Babbitt Example

Derived tone rows from inversionally related interval cycles in Milton Babbitt’s Partitions for Piano:

Define helper function for creating interval cycles:

>>> import abjad
>>> def perle_cyclic_set(starting_pitch, interval):
...     returned_list = [starting_pitch]
...     for _ in range(10):
...         val = returned_list[-1] + interval
...         val %= 12
...         returned_list.append(val)
...     return returned_list
...

Define helper function to intersect cycles:

>>> def intersect_sequences(seq_1, seq_2, pattern_1, pattern_2, length):
...     returned_list = []
...     for index in range(length):
...         match_1 = pattern_1.matches_index(index, length)
...         match_2 = pattern_2.matches_index(index, length)
...         if match_1:
...             value = seq_1.pop(0)
...             returned_list.append(value)
...         elif match_2:
...             value = seq_2.pop(0)
...             returned_list.append(value)
...         else:
...             returned_list.append(None)
...     return returned_list
...

Create first intersection:

>>> ic1_p5 = perle_cyclic_set(starting_pitch=5, interval=1)
>>> ic1_p2 = perle_cyclic_set(starting_pitch=2, interval=1)
>>> ic1_p5_pattern = abjad.Pattern(
...     indices=[0, 2, 4],
...     period=6,
... )
>>> ic1_p2_pattern = abjad.Pattern(
...     indices=[1, 3, 5],
...     period=6,
... )
>>> intersection_1 = intersect_sequences(
...     ic1_p5, ic1_p2, ic1_p5_pattern, ic1_p2_pattern, 6
... )

Create second intersection:

>>> ic11_p10 = perle_cyclic_set(starting_pitch=10, interval=11)
>>> ic11_p1 = perle_cyclic_set(starting_pitch=1, interval=11)
>>> ic11_p10_pattern = abjad.Pattern(
...     indices=[0, 2, 4],
...     period=6,
... )
>>> ic11_p1_pattern = abjad.Pattern(
...     indices=[1, 3, 5],
...     period=6,
... )
>>> intersection_2 = intersect_sequences(
...     ic11_p10, ic11_p1, ic11_p10_pattern, ic11_p1_pattern, 6
... )

Create and override staff:

>>> row = abjad.TwelveToneRow(intersection_1 + intersection_2)
>>> staff = abjad.Staff([abjad.Note(_, (1, 8)) for _ in row])
>>> abjad.attach(abjad.TimeSignature((6, 8)), staff[0])
>>> abjad.Label(staff).with_intervals(prototype=abjad.NumberedIntervalClass)
>>> abjad.override(staff).Beam.stencil = "##f"
>>> abjad.override(staff).Flag.stencil = "##f"
>>> abjad.override(staff).Stem.stencil = "##f"
>>> abjad.override(staff).text_script.staff_padding = 4
>>> abjad.override(staff).TimeSignature.stencil = "##f"
>>> score = abjad.Score([staff])
>>> abjad.setting(score).proportional_notation_duration = "#(ly:make-moment 1 20)"

Show score:

>>> abjad.show(score)

Carter Example

Elliott Carter’s parallel-inverted all-interval collections:

Define appropriately invertible hexachords:

>>> import abjad
>>> score = abjad.Score()
>>> group = abjad.StaffGroup()
>>> source_hex = abjad.PitchClassSegment([0, 4, 9, 10, 8, 5])
>>> source_hexachords = [
...     source_hex,
...     source_hex.invert(),
...     source_hex.retrograde().transpose((0 - source_hex[-1].number)),
...     source_hex.retrograde().transpose((0 - source_hex[-1].number)).invert(),
... ]

Iterate over hexachords, combining with hexachord inversion transposed six semitones above the final hexachord pitch, stacking each pitch as an interval above the previous pitch:

>>> for hexachord in source_hexachords:
...     s1 = hexachord
...     s2 = s1.invert().transpose(s1[-1].number + 6)
...     full_sequence = abjad.PitchSegment(s1 + s2)
...     transposed_sequence = full_sequence.transpose(-24)
...     vertical_sequence = [-24]
...     for pitch in transposed_sequence[1:]:
...         pitch_number = pitch.number
...         while pitch_number < vertical_sequence[-1]:
...             pitch_number += 12
...         vertical_sequence.append(pitch_number)
...     staff = abjad.Staff([abjad.Note(_, (1, 16)) for _ in vertical_sequence])
...     abjad.attach(abjad.Clef("bass"), staff[0])
...     abjad.attach(abjad.Clef("treble"), staff[4])
...     abjad.ottava(staff[8:])
...     abjad.Label(staff).with_intervals(prototype=abjad.NumberedIntervalClass)
...     abjad.override(staff).text_script.staff_padding = 7
...     group.append(staff)
...

Add staff group to score and override settings:

>>> score.append(group)
>>> abjad.override(score).Beam.stencil = "##f"
>>> abjad.override(score).Flag.stencil = "##f"
>>> abjad.override(score).Stem.stencil = "##f"
>>> abjad.override(score).TimeSignature.stencil = "##f"
>>> abjad.override(
...     score
... ).StaffGrouper.staff_staff_spacing = (
...     "#'((basic-distance . 20) (minimum-distance . 20) (padding . 2))"
... )
>>> abjad.setting(score).proportional_notation_duration = "#(ly:make-moment 1 45)"
>>> abjad.Label(group[0][:6]).color_leaves("#red")
>>> abjad.Label(group[0][6:]).color_leaves("#blue")
>>> abjad.Label(group[1][:6]).color_leaves("#blue")
>>> abjad.Label(group[1][6:]).color_leaves("#red")
>>> abjad.Label(group[2][:6]).color_leaves("#blue")
>>> abjad.Label(group[2][6:]).color_leaves("#red")
>>> abjad.Label(group[3][:6]).color_leaves("#red")
>>> abjad.Label(group[3][6:]).color_leaves("#blue")
>>> score_block = abjad.Block(name="score")
>>> score_block.items.append(score)
>>> file = abjad.LilyPondFile(
...     items=[score_block],
...     includes=["abjad.ily"],
... )

Show file:

>>> abjad.show(file)

Hoffman Example

Non-octave-iterating scale in Joel Hoffman’s Piano Concerto:

Define source scale and interval of replication:

>>> import abjad
>>> interval_down = abjad.NamedInterval("-M9")
>>> cell = abjad.PitchSegment(
...     [
...         "bf''''",
...         "af''''",
...         "g''''",
...         "fs''''",
...         "f''''",
...         "ef''''",
...         "d''''",
...         "cs''''",
...         "c''''",
...         "b'''",
...         "a'''",
...     ]
... )

Collect transpositions of scales:

>>> cells = [cell]
>>> for _ in range(5):
...     new_cell = cells[-1].transpose(interval_down)
...     cells.append(new_cell)
... 
>>> full_scale = []
>>> for cell in cells:
...     full_scale.extend(cell)
... 
>>> full_scale.sort()
>>> final_set = abjad.PitchSegment([_ for _ in full_scale])

Create notes from pitch segment:

>>> staff = abjad.Staff(
...     [abjad.Note(abjad.NumberedPitch(_), (1, 16)) for _ in final_set]
... )

Attach extra attachments and override score settings:

>>> abjad.attach(abjad.Clef("bass"), staff[0])
>>> for note in abjad.select(staff).leaves():
...     if note.written_pitch == "c'":
...         abjad.attach(abjad.Clef("treble"), note)
... 
>>> abjad.ottava(staff[:11], start_ottava=abjad.Ottava(n=-1))
>>> abjad.ottava(staff[44:])
>>> abjad.override(staff).BarLine.stencil = "##f"
>>> abjad.override(staff).Beam.stencil = "##f"
>>> abjad.override(staff).Flag.stencil = "##f"
>>> abjad.override(staff).Stem.stencil = "##f"
>>> abjad.override(staff).TimeSignature.stencil = "##f"
>>> abjad.setting(staff).proportional_notation_duration = "#(ly:make-moment 1 25)"
>>> colors = [
...     "#red",
...     "#blue",
...     "#red",
...     "#blue",
...     "#red",
...     "#blue",
... ]
>>> leaf_group = (
...     abjad.select(staff)
...     .leaves()
...     .partition_by_counts(
...         [11],
...         cyclic=True,
...         overhang=True,
...     )
... )
>>> for color, leaves in zip(colors, leaf_group):
...     abjad.Label(leaves).color_leaves(color)
... 
>>> score_block = abjad.Block(name="score")
>>> score_block.items.append(staff)
>>> paper_block = abjad.Block(name="paper")
>>> paper_block.items.append("indent = 0")
>>> file = abjad.LilyPondFile(
...     items=[paper_block, score_block], includes=["abjad.ily"]
... )

Show file:

>>> abjad.show(file)

Nono Example

Double-stop creation from hexachord pairs in Luigi Nono’s Fragmente – Stille, an Diotima:

Define tone row and divide into hexachords:

>>> scale = abjad.PitchSegment(
...     [
...         "cs''",
...         "d''",
...         "ef''",
...         "e''",
...         "f''",
...         "fs''",
...         "g''",
...         "gs''",
...         "a''",
...         "bf''",
...         "b''",
...         "c'''",
...     ]
... )
>>> hexachord_1 = [_ for _ in scale[:6]]
>>> hexachord_2 = [_ for _ in scale[6:]]

Isolate diads from paired hexachords:

>>> diads = [list(_) for _ in zip(hexachord_1, hexachord_2)]
>>> reversed_indices = [1, 2, 4, 5]
>>> for index in reversed_indices:
...     diads[index] = (diads[index][1], diads[index][0])
... 
>>> staff = abjad.Staff()
>>> for diad in diads:
...     lower = diad[0]
...     higher = diad[1]
...     while higher < lower:
...         higher = abjad.NamedInterval("+P8").transpose(higher)
...     chord = abjad.Chord([lower, higher], (1, 8))
...     staff.append(chord)
...

Change octaves:

>>> staff[2].written_pitches = abjad.NamedInterval("+P8").transpose(
...     staff[2].written_pitches
... )
>>> staff[3].written_pitches = abjad.NamedInterval("+P8").transpose(
...     staff[3].written_pitches
... )
>>> staff[4].written_pitches = abjad.NamedInterval("-P8").transpose(
...     staff[4].written_pitches
... )
>>> staff[5].written_pitches = abjad.NumberedInterval("-24").transpose(
...     staff[5].written_pitches
... )

Override staff settings:

>>> abjad.override(staff).Beam.stencil = "##f"
>>> abjad.override(staff).Flag.stencil = "##f"
>>> abjad.override(staff).Stem.stencil = "##f"
>>> abjad.override(staff).text_script.staff_padding = 4
>>> abjad.override(staff).TimeSignature.stencil = "##f"
>>> score = abjad.Score([staff])
>>> abjad.setting(score).proportional_notation_duration = "#(ly:make-moment 1 20)"

Show score:

>>> abjad.show(score)

Stravinsky Example

Tone row rotation in Igor Stravinsky’s Abraham and Isaac:

Define tone row and row permutations:

>>> import abjad
>>> file = abjad.LilyPondFile()
>>> source = abjad.TwelveToneRow([5, 6, 4, 2, 3, 11, 9, 7, 8, 10, 0, 1])
>>> perms = [
...     source,
...     source.invert(),
...     source.retrograde(),
...     abjad.TwelveToneRow(source.retrograde()).invert(),
... ]
>>> labels = [
...     abjad.Markup(
...         r"\markup P",
...         literal=True,
...         direction=abjad.Up,
...     ),
...     abjad.Markup(
...         r"\markup I",
...         literal=True,
...         direction=abjad.Up,
...     ),
...     abjad.Markup(
...         r"\markup R",
...         literal=True,
...         direction=abjad.Up,
...     ),
...     abjad.Markup(
...         r"\markup IR",
...         literal=True,
...         direction=abjad.Up,
...     ),
... ]

Define rotation distances and iterate through permutations, creating charts:

>>> rotations = [0, -1, -2, -3, -4, -5]
>>> for perm, label in zip(perms, labels):
...     source_staff = abjad.Staff([abjad.Note(_, (1, 16)) for _ in perm])
...     abjad.attach(label, source_staff[0])
...     score = abjad.Score([source_staff])
...     group = abjad.StaffGroup()
...     hexachords = [
...         [_.number for _ in perm[:6]],
...         [_.number for _ in perm[6:]],
...     ]
...     margin_markups = [
...         abjad.StartMarkup(markup=label),
...         abjad.StartMarkup(markup="I"),
...         abjad.StartMarkup(markup="II"),
...         abjad.StartMarkup(markup="III"),
...         abjad.StartMarkup(markup="IV"),
...         abjad.StartMarkup(markup="V"),
...     ]
...     for r, m in zip(rotations, margin_markups):
...         staff = abjad.Staff()
...         sets = [
...             abjad.PitchClassSegment(hexachords[0]).rotate(r),
...             abjad.PitchClassSegment(hexachords[1]).rotate(r),
...             abjad.PitchClassSegment(hexachords[0])
...             .rotate(r)  # transpose to zero since stravinsky keyword is removed
...             .transpose(hexachords[0][0]),
...             abjad.PitchClassSegment(hexachords[1])
...             .rotate(r)  # transpose to zero since stravinsky keyword is removed
...             .transpose(hexachords[1][0]),
...         ]
...         names = [
...             abjad.Markup(r"\markup \box α", literal=True, direction=abjad.Up),
...             abjad.Markup(r"\markup \box β", literal=True, direction=abjad.Up),
...             abjad.Markup(r"\markup \box γ", literal=True, direction=abjad.Up),
...             abjad.Markup(r"\markup \box δ", literal=True, direction=abjad.Up),
...         ]
...         for set, name in zip(sets, names):
...             voice = abjad.Voice([abjad.Note(_, (1, 16)) for _ in set])
...             for leaf in abjad.iterate(voice).leaves():
...                 mark = abjad.Markup(
...                     f"\markup {abjad.NumberedPitchClass(leaf.written_pitch)}",
...                     literal=True,
...                     direction=abjad.Up,
...                 )
...                 abjad.tweak(mark).staff_padding = "3"
...                 abjad.attach(mark, leaf)
...             abjad.tweak(name).staff_padding = "3"
...             abjad.attach(name, voice[0])
...             abjad.attach(abjad.TimeSignature((6, 16)), voice[0])
...             staff.append(voice)
...         abjad.attach(m, abjad.select(staff).leaf(0))
...         group.append(staff)
...     score.append(group)
...     abjad.override(score).Beam.stencil = "##f"
...     abjad.override(score).Flag.stencil = "##f"
...     abjad.override(score).Stem.stencil = "##f"
...     abjad.override(score).TimeSignature.stencil = "##f"
...     abjad.override(
...         score
...     ).StaffGrouper.staff_staff_spacing = (
...         "#'((basic-distance . 10) (minimum-distance . 10) (padding . 2))"
...     )
...     abjad.setting(
...         score
...     ).proportional_notation_duration = r"#(ly:make-moment 1 25)"
...     file.items.append(score)
...

Show file of chart scores:

>>> abjad.show(file)

Xenakis Example

Pitch sieve in Iannis Xenakis’s Jonchaies:

Initialize periodic patterns and create union:

>>> import abjad
>>> x17_0 = abjad.Pattern(indices=[0], period=17)
>>> x17_1 = abjad.Pattern(indices=[1], period=17)
>>> x17_4 = abjad.Pattern(indices=[4], period=17)
>>> x17_5 = abjad.Pattern(indices=[5], period=17)
>>> x17_7 = abjad.Pattern(indices=[7], period=17)
>>> x17_11 = abjad.Pattern(indices=[11], period=17)
>>> x17_12 = abjad.Pattern(indices=[12], period=17)
>>> x17_16 = abjad.Pattern(indices=[16], period=17)
>>> sieve = x17_0 | x17_1 | x17_4 | x17_5 | x17_7 | x17_11 | x17_12 | x17_16

Iterate through boolean vector to create pitch list:

>>> pitches = []
>>> length = 56
>>> indices = [_ for _ in range(length)]
>>> vector = sieve.get_boolean_vector(total_length=length)
>>> for index, boolean_value in zip(indices, vector):
...     if boolean_value:
...         pitches.append(abjad.NumberedPitch(index))
...

Initialize note objects from pitch list:

>>> staff = abjad.Staff([abjad.Note(_ - 15, (1, 16)) for _ in pitches])
>>> abjad.attach(abjad.Clef("bass"), staff[0])
>>> abjad.attach(abjad.Clef("treble"), staff[7])
>>> abjad.ottava(staff[21:])
>>> abjad.override(staff).BarLine.stencil = "##f"
>>> abjad.override(staff).Beam.stencil = "##f"
>>> abjad.override(staff).Flag.stencil = "##f"
>>> abjad.override(staff).Stem.stencil = "##f"
>>> abjad.override(staff).TimeSignature.stencil = "##f"
>>> abjad.setting(staff).proportional_notation_duration = r"#(ly:make-moment 1 25)"

Show score:

>>> abjad.show(staff)