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)


    Webern Example
    --------------

    Derived tone rows in Anton Webern's `Concerto for Nine Instruments, Op.24`:

    Define trichord source and tone-row-forming transformations:

    ::

        >>> import abjad
        ...
        >>> score = abjad.Score()
        >>> group = abjad.StaffGroup()
        >>> source_trichord = abjad.PitchClassSegment([0, 1, 4])
        >>> webern_source = source_trichord.invert().rotate(1).transpose(-8)
        >>> first_part = webern_source.transpose(7)
        >>> second_part = webern_source.invert().retrograde().transpose(6)
        >>> third_part = webern_source.retrograde().transpose(1)
        >>> fourth_part = webern_source.invert()
        >>> row = abjad.TwelveToneRow(first_part + second_part + third_part + fourth_part)
        >>> perms = [
        ...     (
        ...         row,
        ...         abjad.StartMarkup(abjad.Markup(r"\markup \box P", literal=True)),
        ...     ),
        ...     (
        ...         row.retrograde(),
        ...         abjad.StartMarkup(abjad.Markup(r"\markup \box R", literal=True)),
        ...     ),
        ...     (
        ...         row.invert(),
        ...         abjad.StartMarkup(abjad.Markup(r"\markup \box I", literal=True)),
        ...     ),
        ...     (
        ...         row.invert().retrograde(),
        ...         abjad.StartMarkup(abjad.Markup(r"\markup \box RI", literal=True)),
        ...     ),
        ... ]
        ...

    Iterate through permutations, creating staves and labeling trichords:

    ::

        >>> counter = 0
        >>> for perm in perms:
        ...     cyc_tuple = abjad.CyclicTuple(["red", "blue"])
        ...     staff = abjad.Staff([abjad.Note(_, (1, 16)) for _ in perm[0]])
        ...     abjad.attach(perm[1], staff[0])
        ...     for trichord in (
        ...         abjad.select(staff)
        ...         .leaves()
        ...         .partition_by_counts(
        ...             [3],
        ...             cyclic=True,
        ...             overhang=True,
        ...         )
        ...     ):
        ...         pc_set = abjad.PitchClassSet([_.written_pitch for _ in trichord])
        ...         set_class = abjad.SetClass.from_pitch_class_set(pc_set)
        ...         abjad.attach(abjad.Markup(f"\markup {set_class}", literal=True), trichord[0])
        ...         abjad.Label(trichord).color_leaves(cyc_tuple[counter])
        ...         counter += 1
        ...         abjad.override(staff).text_script.staff_padding = 4
        ...     group.append(staff)
        ...

    Attach extra labels and override score settings:

    ::

        >>> abjad.attach(
        ...     abjad.Markup(
        ...         "\markup \parenthesize \concat \markup {P} \markup \sub {7}", direction=abjad.Up, literal=True,
        ...     ),
        ...     abjad.select(group[0]).leaf(0),
        ... )
        ...
        >>> abjad.attach(
        ...     abjad.Markup(
        ...         "\markup \parenthesize \concat \markup {RI} \markup \sub {6}", direction=abjad.Up, literal=True,
        ...     ).parenthesize(),
        ...     abjad.select(group[0]).leaf(3),
        ... )
        ...
        >>> abjad.attach(
        ...     abjad.Markup(
        ...         "\markup \parenthesize \concat \markup {R} \markup \sub {1}", direction=abjad.Up, literal=True,
        ...     ).parenthesize(),
        ...     abjad.select(group[0]).leaf(6),
        ... )
        ...
        >>> abjad.attach(
        ...     abjad.Markup(
        ...         "\markup \parenthesize \concat \markup {I} \markup \sub {0}", direction=abjad.Up, literal=True,
        ...     ).parenthesize(),
        ...     abjad.select(group[0]).leaf(9),
        ... )
        ...
        >>> 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)"
        >>> 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)


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)