def get_sequence_period_of_rotation(sequence, n):
'''Change `sequence` to period of rotation:
::
>>> sequencetools.get_sequence_period_of_rotation([1, 2, 3, 1, 2, 3], 1)
3
::
>>> sequencetools.get_sequence_period_of_rotation([1, 2, 3, 1, 2, 3], 2)
3
::
>>> sequencetools.get_sequence_period_of_rotation([1, 2, 3, 1, 2, 3], 3)
1
Returns positive integer.
'''
degree = get_sequence_degree_of_rotational_symmetry(sequence)
period = len(sequence) / degree
divisors_of_n = set(mathtools.divisors(n))
divisors_of_period = set(mathtools.divisors(period))
max_shared_divisor = max(divisors_of_n & divisors_of_period)
return period / max_shared_divisor
def test_mathtools_divisors_01():
r'''Positive divisors of integer n.
'''
assert mathtools.divisors(1) == [1]
assert mathtools.divisors(2) == [1, 2]
assert mathtools.divisors(3) == [1, 3]
assert mathtools.divisors(-4) == [1, 2, 4]
assert mathtools.divisors(-5) == [1, 5]
assert mathtools.divisors(-6) == [1, 2, 3, 6]
assert mathtools.divisors(7) == [1, 7]
assert mathtools.divisors(8) == [1, 2, 4, 8]
assert mathtools.divisors(9) == [1, 3, 9]
assert mathtools.divisors(10) == [1, 2, 5, 10]
def test_mathtools_divisors_01():
r'''Positive divisors of integer n.
'''
assert mathtools.divisors(1) == [1]
assert mathtools.divisors(2) == [1, 2]
assert mathtools.divisors(3) == [1, 3]
assert mathtools.divisors(-4) == [1, 2, 4]
assert mathtools.divisors(-5) == [1, 5]
assert mathtools.divisors(-6) == [1, 2, 3, 6]
assert mathtools.divisors(7) == [1, 7]
assert mathtools.divisors(8) == [1, 2, 4, 8]
assert mathtools.divisors(9) == [1, 3, 9]
assert mathtools.divisors(10) == [1, 2, 5, 10]
def is_compound(self):
r'''Is true when meter is compound. Otherwise false.
.. container:: example
**Example 1.** Compound meters written over ``4``:
::
>>> for numerator in range(1, 13):
... meter = metertools.Meter((numerator, 4))
... string = True if meter.is_compound else ''
... print(str(meter), string)
...
1/4
2/4
3/4
4/4
5/4
6/4 True
7/4
8/4
9/4 True
10/4
11/4
12/4 True
.. container:: example
**Example 2.** Compound meters written over ``8``:
::
>>> for numerator in range(1, 13):
... meter = metertools.Meter((numerator, 8))
... string = True if meter.is_compound else ''
... print(str(meter), string)
...
1/8
2/8
3/8
4/8
5/8
6/8 True
7/8
8/8
9/8 True
10/8
11/8
12/8 True
Compound meters defined equal to those meters with a numerator
divisible by ``3`` (but not equal to ``3``).
Returns true or false.
'''
if 3 in mathtools.divisors(self.numerator):
if not self.numerator == 3:
return True
return False
def greatest_common_divisor(*integers):
'''Calculates greatest common divisor of `integers`.
::
>>> mathtools.greatest_common_divisor(84, -94, -144)
2
Allows nonpositive input.
Raises type error on noninteger input.
Raises not implemented error when ``0`` is included in input.
Returns positive integer.
'''
from abjad.tools import mathtools
common_divisors = None
for positive_integer in integers:
all_divisors = set(mathtools.divisors(positive_integer))
if common_divisors is None:
common_divisors = all_divisors
else:
common_divisors &= all_divisors
if common_divisors == set([1]):
return 1
return max(common_divisors)
def _is_valid_definition(self, definition):
if not isinstance(definition, dict):
return False
elif 'divisors' not in definition:
return False
elif not len(definition['divisors']):
return False
elif not all(isinstance(x, int) and \
1 < x for x in definition['divisors']):
return False
elif not all(mathtools.divisors(x) == [1, x]
for x in definition['divisors']):
return False
elif 'max_depth' not in definition:
return False
elif not isinstance(definition['max_depth'], int):
return False
elif not 0 < definition['max_depth']:
return False
elif 'max_divisions' not in definition:
return False
elif not isinstance(definition['max_divisions'], int):
return False
elif not 1 < definition['max_divisions']:
return False
return True
def is_compound(self):
r'''Is true when meter is compound. Otherwise false.
.. container:: example
**Example 1.** Compound meters written over ``4``:
::
>>> for numerator in range(1, 13):
... meter = metertools.Meter((numerator, 4))
... string = True if meter.is_compound else ''
... print(str(meter), string)
...
1/4
2/4
3/4
4/4
5/4
6/4 True
7/4
8/4
9/4 True
10/4
11/4
12/4 True
.. container:: example
**Example 2.** Compound meters written over ``8``:
::
>>> for numerator in range(1, 13):
... meter = metertools.Meter((numerator, 8))
... string = True if meter.is_compound else ''
... print(str(meter), string)
...
1/8
2/8
3/8
4/8
5/8
6/8 True
7/8
8/8
9/8 True
10/8
11/8
12/8 True
Compound meters defined equal to those meters with a numerator
divisible by ``3`` (but not equal to ``3``).
Returns true or false.
'''
if 3 in mathtools.divisors(self.numerator):
if not self.numerator == 3:
return True
return False
def _is_valid_definition(self, definition):
if not isinstance(definition, dict):
return False
elif 'divisors' not in definition:
return False
elif not len(definition['divisors']):
return False
elif not all(
isinstance(x, int) and 1 < x for x in definition['divisors']):
return False
elif not all(
mathtools.divisors(x) == [1, x]
for x in definition['divisors']):
return False
elif 'max_depth' not in definition:
return False
elif not isinstance(definition['max_depth'], int):
return False
elif not 0 < definition['max_depth']:
return False
elif 'max_divisions' not in definition:
return False
elif not isinstance(definition['max_divisions'], int):
return False
elif not 1 < definition['max_divisions']:
return False
return True
def greatest_common_divisor(*integers):
'''Calculates greatest common divisor of `integers`.
.. container:: example
>>> abjad.mathtools.greatest_common_divisor(84, -94, -144)
2
Allows nonpositive input.
Raises not implemented error when zero is included in input.
Returns positive integer.
'''
from abjad.tools import mathtools
common_divisors = None
for positive_integer in integers:
all_divisors = set(mathtools.divisors(positive_integer))
if common_divisors is None:
common_divisors = all_divisors
else:
common_divisors &= all_divisors
if common_divisors == set([1]):
return 1
return max(common_divisors)
def recurse(n):
results = []
for key in n:
if not isinstance(key, int) or \
not 0 < key or \
not mathtools.divisors(key) == [1, key]:
results.append(False)
elif not isinstance(n[key], (dict, type(None))):
results.append(False)
elif isinstance(n[key], dict) and not recurse(n[key]):
results.append(False)
else:
results.append(True)
return results
def recurse(n):
results = []
for key in n:
if not isinstance(key, int) or \
not 0 < key or \
not mathtools.divisors(key) == [1, key]:
results.append(False)
elif not isinstance(n[key], (dict, type(None))):
results.append(False)
elif isinstance(n[key], dict) and not recurse(n[key]):
results.append(False)
else:
results.append(True)
return results
def rewrite_integer_tempo(integer_tempo, maximum_numerator=None, maximum_denominator=None):
r'''Rewrite `integer_tempo`.
Allow no tempo less than half `integer_tempo` or greater than double `integer_tempo`:
::
>>> pairs = tempotools.rewrite_integer_tempo(
... 58, maximum_numerator=8, maximum_denominator=8)
::
>>> for pair in pairs:
... pair
...
(Multiplier(1, 2), 29)
(Multiplier(1, 1), 58)
(Multiplier(3, 2), 87)
(Multiplier(2, 1), 116)
Returns list.
'''
# find divisors
divisors = mathtools.divisors(integer_tempo)
if maximum_denominator is not None:
divisors = [x for x in divisors if x <= maximum_denominator]
# make pairs
pairs = []
for divisor in divisors:
start = int(math.ceil(divisor / 2.0))
stop = 2 * divisor
numerators = range(start, stop + 1)
if maximum_numerator is not None:
numerators = [x for x in numerators if x <= maximum_numerator]
for numerator in numerators:
multiplier = durationtools.Multiplier(numerator, divisor)
new_tempo = fractions.Fraction(multiplier * integer_tempo)
assert mathtools.is_integer_equivalent_number(new_tempo)
new_tempo = int(new_tempo)
pair = (multiplier, new_tempo)
if pair not in pairs:
pairs.append(pair)
# sort pairs
pairs.sort()
# return pairs
return pairs
def respell_division(self, division):
r'''Respells `division` according to partition table.
.. container:: example
**Example 1.** Respells divisions according to table:
::
>>> table = rhythmmakertools.PartitionTable([
... (2, [1, 1]),
... (3, [2, 1]),
... (5, [3, 2]),
... (7, [4, 3]),
... ])
::
>>> for numerator in range(1, 15):
... division = mathtools.NonreducedFraction(numerator, 4)
... result = table.respell_division(division)
... print((division, result))
(NonreducedFraction(1, 4), [NonreducedFraction(1, 4)])
(NonreducedFraction(2, 4), [NonreducedFraction(1, 4), NonreducedFraction(1, 4)])
(NonreducedFraction(3, 4), [NonreducedFraction(2, 4), NonreducedFraction(1, 4)])
(NonreducedFraction(4, 4), [NonreducedFraction(2, 4), NonreducedFraction(2, 4)])
(NonreducedFraction(5, 4), [NonreducedFraction(3, 4), NonreducedFraction(2, 4)])
(NonreducedFraction(6, 4), [NonreducedFraction(4, 4), NonreducedFraction(2, 4)])
(NonreducedFraction(7, 4), [NonreducedFraction(4, 4), NonreducedFraction(3, 4)])
(NonreducedFraction(8, 4), [NonreducedFraction(4, 4), NonreducedFraction(4, 4)])
(NonreducedFraction(9, 4), [NonreducedFraction(6, 4), NonreducedFraction(3, 4)])
(NonreducedFraction(10, 4), [NonreducedFraction(6, 4), NonreducedFraction(4, 4)])
(NonreducedFraction(11, 4), [NonreducedFraction(11, 4)])
(NonreducedFraction(12, 4), [NonreducedFraction(8, 4), NonreducedFraction(4, 4)])
(NonreducedFraction(13, 4), [NonreducedFraction(13, 4)])
(NonreducedFraction(14, 4), [NonreducedFraction(8, 4), NonreducedFraction(6, 4)])
.. container:: example
**Example 2.** Respells divisions according to another table:
::
>>> table = rhythmmakertools.PartitionTable([
... (5, [1, 3, 1]),
... (7, [1, 2, 4]),
... ])
::
>>> for numerator in range(1, 15):
... division = mathtools.NonreducedFraction(numerator, 4)
... result = table.respell_division(division)
... print((division, result))
(NonreducedFraction(1, 4), [NonreducedFraction(1, 4)])
(NonreducedFraction(2, 4), [NonreducedFraction(2, 4)])
(NonreducedFraction(3, 4), [NonreducedFraction(3, 4)])
(NonreducedFraction(4, 4), [NonreducedFraction(4, 4)])
(NonreducedFraction(5, 4), [NonreducedFraction(1, 4), NonreducedFraction(3, 4), NonreducedFraction(1, 4)])
(NonreducedFraction(6, 4), [NonreducedFraction(6, 4)])
(NonreducedFraction(7, 4), [NonreducedFraction(1, 4), NonreducedFraction(2, 4), NonreducedFraction(4, 4)])
(NonreducedFraction(8, 4), [NonreducedFraction(8, 4)])
(NonreducedFraction(9, 4), [NonreducedFraction(9, 4)])
(NonreducedFraction(10, 4), [NonreducedFraction(2, 4), NonreducedFraction(6, 4), NonreducedFraction(2, 4)])
(NonreducedFraction(11, 4), [NonreducedFraction(11, 4)])
(NonreducedFraction(12, 4), [NonreducedFraction(12, 4)])
(NonreducedFraction(13, 4), [NonreducedFraction(13, 4)])
(NonreducedFraction(14, 4), [NonreducedFraction(2, 4), NonreducedFraction(4, 4), NonreducedFraction(8, 4)])
Returns list of new divisions.
'''
division = mathtools.NonreducedFraction(division)
result = [division]
divisors = mathtools.divisors(division.numerator)
for numerator, ratio in reversed(list(self.items())):
if numerator in divisors:
result = [division * _ for _ in ratio.multipliers]
break
result = [mathtools.NonreducedFraction(_) for _ in result]
return result
def list_related_tempos(
self,
maximum_numerator=None,
maximum_denominator=None,
):
r'''Lists tempos related to this tempo.
Returns list of tempo / ratio pairs.
Each new tempo equals not less than half of this tempo
and not more than twice this tempo.
.. container:: example
Rewrites tempo ``58`` MM by ratios of the form ``n:d`` such that
``1 <= n <= 8`` and ``1 <= d <= 8``:
...
::
>>> tempo = Tempo(Duration(1, 4), 58)
>>> pairs = tempo.list_related_tempos(
... maximum_numerator=8,
... maximum_denominator=8,
... )
::
>>> for tempo, ratio in pairs:
... string = '{!s}\t{!s}'.format(tempo, ratio)
... print string
4=29 1:2
4=58 1:1
4=87 3:2
4=116 2:1
.. container:: example
Rewrites tempo ``58`` MM by ratios of the form ``n:d`` such that
``1 <= n <= 30`` and ``1 <= d <= 30``:
::
>>> tempo = Tempo(Duration(1, 4), 58)
>>> pairs = tempo.list_related_tempos(
... maximum_numerator=30,
... maximum_denominator=30,
... )
::
>>> for tempo, ratio in pairs:
... string = '{!s}\t{!s}'.format(tempo, ratio)
... print string
...
4=30 15:29
4=32 16:29
4=34 17:29
4=36 18:29
4=38 19:29
4=40 20:29
4=42 21:29
4=44 22:29
4=46 23:29
4=48 24:29
4=50 25:29
4=52 26:29
4=54 27:29
4=56 28:29
4=58 1:1
4=60 30:29
Returns list.
'''
# assert integer tempo
assert isinstance(self.units_per_minute, int), repr(self)
# find divisors
divisors = mathtools.divisors(self.units_per_minute)
if maximum_denominator is not None:
divisors = [x for x in divisors if x <= maximum_denominator]
# make pairs
pairs = []
for divisor in divisors:
start = int(math.ceil(divisor / 2.0))
stop = 2 * divisor
numerators = range(start, stop + 1)
if maximum_numerator is not None:
numerators = [
x for x in numerators
if x <= maximum_numerator
]
for numerator in numerators:
ratio = mathtools.Ratio(numerator, divisor)
multiplier = durationtools.Multiplier(*ratio)
new_units_per_minute = multiplier * self.units_per_minute
assert mathtools.is_integer_equivalent_expr(
new_units_per_minute)
new_units_per_minute = int(new_units_per_minute)
new_tempo = type(self)(self.duration, new_units_per_minute)
pair = (new_tempo, ratio)
if pair not in pairs:
pairs.append(pair)
# sort pairs
pairs.sort()
# return pairs
return pairs
def list_related_tempos(
self,
maximum_numerator=None,
maximum_denominator=None,
):
r'''Lists tempos related to this tempo.
Returns list of tempo / ratio pairs.
Each new tempo equals not less than half of this tempo
and not more than twice this tempo.
.. container:: example
Rewrites tempo ``58`` MM by ratios of the form ``n:d`` such that
``1 <= n <= 8`` and ``1 <= d <= 8``:
...
::
>>> tempo = Tempo(Duration(1, 4), 58)
>>> pairs = tempo.list_related_tempos(
... maximum_numerator=8,
... maximum_denominator=8,
... )
::
>>> for tempo, ratio in pairs:
... string = '{!s}\t{!s}'.format(tempo, ratio)
... print(string)
4=29 1:2
4=58 1:1
4=87 3:2
4=116 2:1
.. container:: example
Rewrites tempo ``58`` MM by ratios of the form ``n:d`` such that
``1 <= n <= 30`` and ``1 <= d <= 30``:
::
>>> tempo = Tempo(Duration(1, 4), 58)
>>> pairs = tempo.list_related_tempos(
... maximum_numerator=30,
... maximum_denominator=30,
... )
::
>>> for tempo, ratio in pairs:
... string = '{!s}\t{!s}'.format(tempo, ratio)
... print(string)
...
4=30 15:29
4=32 16:29
4=34 17:29
4=36 18:29
4=38 19:29
4=40 20:29
4=42 21:29
4=44 22:29
4=46 23:29
4=48 24:29
4=50 25:29
4=52 26:29
4=54 27:29
4=56 28:29
4=58 1:1
4=60 30:29
Returns list.
'''
# assert integer tempo
assert isinstance(self.units_per_minute, int), repr(self)
# find divisors
divisors = mathtools.divisors(self.units_per_minute)
if maximum_denominator is not None:
divisors = [x for x in divisors if x <= maximum_denominator]
# make pairs
pairs = []
for divisor in divisors:
start = int(math.ceil(divisor / 2.0))
stop = 2 * divisor
numerators = range(start, stop + 1)
if maximum_numerator is not None:
numerators = [
x for x in numerators
if x <= maximum_numerator
]
for numerator in numerators:
ratio = mathtools.Ratio(numerator, divisor)
multiplier = durationtools.Multiplier(*ratio)
new_units_per_minute = multiplier * self.units_per_minute
assert mathtools.is_integer_equivalent_expr(
new_units_per_minute)
new_units_per_minute = int(new_units_per_minute)
new_tempo = type(self)(
duration=self.duration,
units_per_minute=new_units_per_minute,
)
pair = (new_tempo, ratio)
if pair not in pairs:
pairs.append(pair)
# sort pairs
pairs.sort()
# return pairs
return pairs