music21.common.numberTools¶
Functions¶
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music21.common.numberTools.
fromRoman
(num)¶ Convert a Roman numeral (upper or lower) to an int
http://code.activestate.com/recipes/81611-roman-numerals/
>>> common.fromRoman('ii') 2 >>> common.fromRoman('vii') 7
Works with both IIII and IV forms:
>>> common.fromRoman('MCCCCLXXXIX') 1489 >>> common.fromRoman('MCDLXXXIX') 1489
Some people consider this an error, but you see it in medieval documents:
>>> common.fromRoman('ic') 99
But things like this are never seen, and thus cause an error:
>>> common.fromRoman('vx') Traceback (most recent call last): Music21CommonException: input contains an invalid subtraction element: vx
Return type: int
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music21.common.numberTools.
toRoman
(num)¶ Convert a number from 1 to 3999 to a roman numeral
>>> common.toRoman(2) 'II' >>> common.toRoman(7) 'VII' >>> common.toRoman(1999) 'MCMXCIX'
>>> common.toRoman("hi") Traceback (most recent call last): TypeError: expected integer, got <... 'str'>
Return type: str
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music21.common.numberTools.
addFloatPrecision
(x, grain=0.01)¶ Given a value that suggests a floating point fraction, like .33, return a Fraction or float that provides greater specification, such as .333333333
>>> import fractions >>> common.addFloatPrecision(.333) Fraction(1, 3) >>> common.addFloatPrecision(.33) Fraction(1, 3) >>> common.addFloatPrecision(.35) == fractions.Fraction(1, 3) False >>> common.addFloatPrecision(.2) == 0.2 True >>> common.addFloatPrecision(.125) 0.125 >>> common.addFloatPrecision(1./7) == 1./7 True
Return type: float
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music21.common.numberTools.
almostEquals
(x, y=0.0, grain=1e-07)¶ almostEquals(x, y) – returns True if x and y are within grain (default 0.0000001) of each other
Allows comparisons between floats that are normally inconsistent.
>>> common.almostEquals(1.000000001, 1) True >>> common.almostEquals(1.001, 1) False >>> common.almostEquals(1.001, 1, grain=0.1) True
For very small grains, just compare Fractions without converting...
Return type: bool
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music21.common.numberTools.
approximateGCD
(values, grain=0.0001)¶ Given a list of values, find the lowest common divisor of floating point values.
>>> common.approximateGCD([2.5,10, .25]) 0.25 >>> common.approximateGCD([2.5,10]) 2.5 >>> common.approximateGCD([2,10]) 2.0 >>> common.approximateGCD([1.5, 5, 2, 7]) 0.5 >>> common.approximateGCD([2,5,10]) 1.0 >>> common.approximateGCD([2,5,10,.25]) 0.25 >>> common.strTrimFloat(common.approximateGCD([1/3.,2/3.])) '0.3333' >>> common.strTrimFloat(common.approximateGCD([5/3.,2/3.,4])) '0.3333' >>> common.strTrimFloat(common.approximateGCD([5/3.,2/3.,5])) '0.3333' >>> common.strTrimFloat(common.approximateGCD([5/3.,2/3.,5/6.,3/6.])) '0.1667'
Return type: float
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music21.common.numberTools.
cleanupFloat
(floatNum, maxDenominator=65535)¶ Cleans up a floating point number by converting it to a fractions.Fraction object limited to a denominator of maxDenominator
>>> common.cleanupFloat(0.33333327824) 0.333333333333...
>>> common.cleanupFloat(0.142857) 0.1428571428571...
>>> common.cleanupFloat(1.5) 1.5
Fractions are passed through silently...
>>> import fractions >>> common.cleanupFloat(fractions.Fraction(4, 3)) Fraction(4, 3)
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music21.common.numberTools.
contiguousList
(inputListOrTuple)¶ returns bool True or False if a list containing ints contains only contiguous (increasing) values
requires the list to be sorted first
>>> l = [3, 4, 5, 6] >>> common.contiguousList(l) True >>> l.append(8) >>> common.contiguousList(l) False
Sorting matters
>>> l.append(7) >>> common.contiguousList(l) False >>> common.contiguousList(sorted(l)) True
Return type: bool
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music21.common.numberTools.
decimalToTuplet
(decNum)¶ For simple decimals (usually > 1), a quick way to figure out the fraction in lowest terms that gives a valid tuplet.
No it does not work really fast. No it does not return tuplets with denominators over 100. Too bad, math geeks. This is real life. :-)
returns (numerator, denominator)
>>> common.decimalToTuplet(1.5) (3, 2) >>> common.decimalToTuplet(1.25) (5, 4)
If decNum is < 1, the denominator will be greater than the numerator:
>>> common.decimalToTuplet(.8) (4, 5)
If decNum is <= 0, returns a ZeroDivisionError:
>>> common.decimalToTuplet(-.02) Traceback (most recent call last): ZeroDivisionError: number must be greater than zero
TODO: replace with fractions...
Return type: tuple(int)
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music21.common.numberTools.
dotMultiplier
(dots)¶ dotMultiplier(dots) returns how long to multiply the note length of a note in order to get the note length with n dots
>>> common.dotMultiplier(1) Fraction(3, 2) >>> common.dotMultiplier(2) Fraction(7, 4) >>> common.dotMultiplier(3) Fraction(15, 8)
Return type: Fraction
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music21.common.numberTools.
euclidGCD
(a, b)¶ use Euclid’s algorithm to find the GCD of a and b
>>> common.euclidGCD(2,4) 2 >>> common.euclidGCD(20,8) 4 >>> common.euclidGCD(20,16) 4
Return type: int
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music21.common.numberTools.
groupContiguousIntegers
(src)¶ Given a list of integers, group contiguous values into sub lists
>>> common.groupContiguousIntegers([3, 5, 6]) [[3], [5, 6]] >>> common.groupContiguousIntegers([3, 4, 6]) [[3, 4], [6]] >>> common.groupContiguousIntegers([3, 4, 6, 7]) [[3, 4], [6, 7]] >>> common.groupContiguousIntegers([3, 4, 6, 7, 20]) [[3, 4], [6, 7], [20]] >>> common.groupContiguousIntegers([3, 4, 5, 6, 7]) [[3, 4, 5, 6, 7]] >>> common.groupContiguousIntegers([3]) [[3]] >>> common.groupContiguousIntegers([3, 200]) [[3], [200]]
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music21.common.numberTools.
lcm
(filterList)¶ Find the least common multiple of a list of values
>>> common.lcm([3,4,5]) 60 >>> common.lcm([3,4]) 12 >>> common.lcm([1,2]) 2 >>> common.lcm([3,6]) 6
Return type: int
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music21.common.numberTools.
mixedNumeral
(expr, limitDenominator=65535)¶ Returns a string representing a mixedNumeral form of a number
>>> common.mixedNumeral(1.333333) '1 1/3' >>> common.mixedNumeral(0.333333) '1/3' >>> common.mixedNumeral(-1.333333) '-1 1/3' >>> common.mixedNumeral(-0.333333) '-1/3'
>>> common.mixedNumeral(0) '0' >>> common.mixedNumeral(-0) '0'
Works with Fraction objects too
>>> from fractions import Fraction >>> common.mixedNumeral( Fraction(31,7) ) '4 3/7' >>> common.mixedNumeral( Fraction(1,5) ) '1/5' >>> common.mixedNumeral( Fraction(-1,5) ) '-1/5' >>> common.mixedNumeral( Fraction(-31,7) ) '-4 3/7'
Denominator is limited by default but can be changed.
>>> common.mixedNumeral(2.0000001) '2' >>> common.mixedNumeral(2.0000001, limitDenominator=10000000) '2 1/10000000'
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music21.common.numberTools.
nearestMultiple
(n, unit)¶ Given a positive value n, return the nearest multiple of the supplied unit as well as the absolute difference (error) to seven significant digits and the signed difference.
>>> print(common.nearestMultiple(.25, .25)) (0.25, 0.0, 0.0) >>> print(common.nearestMultiple(.35, .25)) (0.25, 0.1..., 0.1...) >>> print(common.nearestMultiple(.20, .25)) (0.25, 0.05..., -0.05...)
Note that this one also has an error of .1 but it’s a positive error off of 0.5 >>> print(common.nearestMultiple(.4, .25)) (0.5, 0.1..., -0.1...)
>>> common.nearestMultiple(.4, .25)[0] 0.5 >>> common.nearestMultiple(23404.001, .125)[0] 23404.0 >>> common.nearestMultiple(23404.134, .125)[0] 23404.125
Error is always positive, but signed difference can be negative.
>>> common.nearestMultiple(23404 - 0.0625, .125) (23404.0, 0.0625, -0.0625)
>>> common.nearestMultiple(.001, .125)[0] 0.0
>>> common.almostEquals(common.nearestMultiple(.25, (1/3.))[0], .33333333) True >>> common.almostEquals(common.nearestMultiple(.55, (1/3.))[0], .66666666) True >>> common.almostEquals(common.nearestMultiple(234.69, (1/3.))[0], 234.6666666) True >>> common.almostEquals(common.nearestMultiple(18.123, (1/6.))[0], 18.16666666) True
>>> common.nearestMultiple(-0.5, 0.125) Traceback (most recent call last): ValueError: n (-0.5) is less than zero. Thus cannot find nearest multiple for a value less than the unit, 0.125
Return type: tuple(float)
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music21.common.numberTools.
numToIntOrFloat
(value)¶ Given a number, return an integer if it is very close to an integer, otherwise, return a float.
>>> common.numToIntOrFloat(1.0) 1 >>> common.numToIntOrFloat(1.00003) 1.00003 >>> common.numToIntOrFloat(1.5) 1.5 >>> common.numToIntOrFloat(1.0000000005) 1
Return type: float
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music21.common.numberTools.
opFrac
(num)¶ opFrac -> optionally convert a number to a fraction or back.
Important music21 2.x function for working with offsets and quarterLengths
Takes in a number (or None) and converts it to a Fraction with denominator less than limitDenominator if it is not binary expressible; otherwise return a float. Or if the Fraction can be converted back to a binary expressable float then do so.
This function should be called often to ensure that values being passed around are floats and ints wherever possible and fractions where needed.
The naming of this method violates music21’s general rule of no abbreviations, but it is important to make it short enough so that no one will be afraid of calling it often. It also doesn’t have a setting for maxDenominator so that it will expand in Code Completion easily. That is to say, this function has been set up to be used, so please use it.
This is a performance critical operation. Do not alter it in any way without running many timing tests.
>>> from fractions import Fraction >>> defaults.limitOffsetDenominator 65535 >>> common.opFrac(3) 3.0 >>> common.opFrac(1.0/3) Fraction(1, 3) >>> common.opFrac(1.0/4) 0.25 >>> f = Fraction(1,3) >>> common.opFrac(f + f + f) 1.0 >>> common.opFrac(0.123456789) Fraction(10, 81) >>> common.opFrac(None) is None True
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music21.common.numberTools.
ordinalAbbreviation
(value, plural=False)¶ Return the ordinal abbreviations for integers
>>> common.ordinalAbbreviation(3) 'rd' >>> common.ordinalAbbreviation(255) 'th' >>> common.ordinalAbbreviation(255, plural=True) 'ths'
Return type: str
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music21.common.numberTools.
roundToHalfInteger
(num)¶ Given a floating-point number, round to the nearest half-integer. Returns int or float
>>> common.roundToHalfInteger(1.2) 1 >>> common.roundToHalfInteger(1.35) 1.5 >>> common.roundToHalfInteger(1.8) 2 >>> common.roundToHalfInteger(1.6234) 1.5
.25 rounds up:
>>> common.roundToHalfInteger(0.25) 0.5
as does .75
>>> common.roundToHalfInteger(0.75) 1
unlike python round function, does the same for 1.25 and 1.75
>>> common.roundToHalfInteger(1.25) 1.5 >>> common.roundToHalfInteger(1.75) 2
negative numbers however, round up on the boundaries
>>> common.roundToHalfInteger(-0.26) -0.5 >>> common.roundToHalfInteger(-0.25) 0
Return type: float
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music21.common.numberTools.
standardDeviation
(coll, bassel=False)¶ Given a collection of values, return the standard deviation.
>>> common.standardDeviation([2,4,4,4,5,5,7,9]) 2.0 >>> common.standardDeviation([600, 470, 170, 430, 300]) 147.3227... >>> common.standardDeviation([4, 2, 5, 8, 6], bassel=True) 2.23606...
Return type: float
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music21.common.numberTools.
strTrimFloat
(floatNum, maxNum=4)¶ returns a string from a float that is at most maxNum of decimial digits long, but never less than 1.
>>> common.strTrimFloat(42.3333333333) '42.3333' >>> common.strTrimFloat(42.3333333333, 2) '42.33' >>> common.strTrimFloat(6.66666666666666, 2) '6.67' >>> common.strTrimFloat(2.0) '2.0' >>> common.strTrimFloat(-5) '-5.0'
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music21.common.numberTools.
unitBoundaryProportion
(series)¶ Take a series of parts with an implied sum, and create unit-interval boundaries proportional to the series components.
>>> common.unitBoundaryProportion([1,1,2]) [(0, 0.25), (0.25, 0.5), (0.5, 1.0)] >>> common.unitBoundaryProportion([8,1,1]) [(0, 0.8...), (0.8..., 0.9...), (0.9..., 1.0)]
Return type: list(tuple(float))
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music21.common.numberTools.
unitNormalizeProportion
(values)¶ Normalize values within the unit interval, where max is determined by the sum of the series.
>>> common.unitNormalizeProportion([0,3,4]) [0.0, 0.42857142857142855, 0.5714285714285714] >>> common.unitNormalizeProportion([1,1,1]) [0.3333333..., 0.333333..., 0.333333...]
On 32-bit computers this number is inexact. On 64-bit it works fine.
#>>> common.unitNormalizeProportion([.2, .6, .2]) #[0.20000000000000001, 0.59999999999999998, 0.20000000000000001]
Return type: list(float)
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music21.common.numberTools.
weightedSelection
(values, weights, randomGenerator=None)¶ Given a list of values and an equal-sized list of weights, return a randomly selected value using the weight.
Example: sum -1 and 1 for 100 values; should be around 0 or at least between -30 and 30
>>> -30 < sum([common.weightedSelection([-1, 1], [1,1]) for x in range(100)]) < 30 True
Return type: int