# music21.common.numberTools¶

## Functions¶

music21.common.numberTools.fromRoman(num: str, *, strictModern=False) int

Convert a Roman numeral (upper or lower) to an int

https://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 and ancient roman documents:

```>>> common.fromRoman('ic')
99
```

unless strictModern is True

```>>> common.fromRoman('ic', strictModern=True)
Traceback (most recent call last):
ValueError: input contains an invalid subtraction element (modern interpretation): ic
```

But things like this are never seen, and thus cause an error:

```>>> common.fromRoman('vx')
Traceback (most recent call last):
ValueError: input contains an invalid subtraction element: vx
```
music21.common.numberTools.toRoman(num: int) str

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'>
```
```>>> common.toRoman(0)
Traceback (most recent call last):
ValueError: Argument must be between 1 and 3999
```

Given a value that suggests a floating point fraction, like 0.33, return a Fraction or float that provides greater specification, such as Fraction(1, 3)

```>>> import fractions
Fraction(1, 3)
Fraction(1, 3)
False
True
0.125
True
```
music21.common.numberTools.approximateGCD(values: Collection[int | float | Fraction], grain: float = 0.0001) float

Given a list of values, find the lowest common divisor of floating point values.

```>>> common.approximateGCD([2.5, 10, 0.25])
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, 0.25])
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'
```
music21.common.numberTools.contiguousList(inputListOrTuple) bool

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
```
music21.common.numberTools.decimalToTuplet(decNum: float) tuple[int, int]

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(0.8)
(4, 5)
```

If decNum is <= 0, returns a ZeroDivisionError:

```>>> common.decimalToTuplet(-.02)
Traceback (most recent call last):
ZeroDivisionError: number must be greater than zero
```
music21.common.numberTools.dotMultiplier(dots: int) float

dotMultiplier(dots) returns how long to multiply the note length of a note in order to get the note length with n dots

Since dotMultiplier always returns a power of two in the denominator, the float will be exact.

```>>> from fractions import Fraction
>>> Fraction(common.dotMultiplier(1))
Fraction(3, 2)
>>> Fraction(common.dotMultiplier(2))
Fraction(7, 4)
>>> Fraction(common.dotMultiplier(3))
Fraction(15, 8)
```
```>>> common.dotMultiplier(0)
1.0
```
music21.common.numberTools.groupContiguousIntegers(src: list[int]) list[list[int]]

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]]
```
music21.common.numberTools.lcm(filterList: Iterable[int]) int

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

Works with any iterable, like this set

common.lcm({3, 5, 6}) 30

Deprecated in v9 since Python 3.10 is the minimum version and math.lcm works in C and is faster

music21.common.numberTools.mixedNumeral(expr: Real, 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(-4, 5))
'-4/5'
>>> 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'
```
music21.common.numberTools.nearestMultiple(n: float, unit: float) tuple[float, float, float]

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(0.25, 0.25))
(0.25, 0.0, 0.0)
>>> print(common.nearestMultiple(0.35, 0.25))
(0.25, 0.1..., 0.1...)
>>> print(common.nearestMultiple(0.20, 0.25))
(0.25, 0.05..., -0.05...)
```

Note that this one also has an error of 0.1, but it’s a positive error off of 0.5

```>>> print(common.nearestMultiple(0.4, 0.25))
(0.5, 0.1..., -0.1...)
```
```>>> common.nearestMultiple(0.4, 0.25)[0]
0.5
>>> common.nearestMultiple(23404.001, 0.125)[0]
23404.0
>>> common.nearestMultiple(23404.134, 0.125)[0]
23404.125
```

Error is always positive, but signed difference can be negative.

```>>> common.nearestMultiple(23404 - 0.0625, 0.125)
(23403.875, 0.0625, 0.0625)
```
```>>> common.nearestMultiple(0.001, 0.125)[0]
0.0
```
```>>> from math import isclose
>>> isclose(common.nearestMultiple(0.25, 1 / 3)[0], 0.33333333, abs_tol=1e-7)
True
>>> isclose(common.nearestMultiple(0.55, 1 / 3)[0], 0.66666666, abs_tol=1e-7)
True
>>> isclose(common.nearestMultiple(234.69, 1 / 3)[0], 234.6666666, abs_tol=1e-7)
True
>>> isclose(common.nearestMultiple(18.123, 1 / 6)[0], 18.16666666, abs_tol=1e-7)
True
```
```>>> common.nearestMultiple(-0.5, 0.125)
Traceback (most recent call last):
ValueError: n (-0.5) is less than zero. Thus, cannot find the nearest
multiple for a value less than the unit, 0.125
```
music21.common.numberTools.numToIntOrFloat(value: int | float | Fraction) int | float

Given a number, return an integer if it is very close to an integer, otherwise, return a float.

This routine is very important for conversion of `Accidental` objects’ .alter attribute in musicXML must be 1 (not 1.0) for sharp and -1 (not -1.0) for flat, but allows for 0.5 for half-sharp.

```>>> common.numToIntOrFloat(1.0)
1
>>> common.numToIntOrFloat(1.00003)
1.00003
>>> common.numToIntOrFloat(1.5)
1.5
>>> common.numToIntOrFloat(2)
2
>>> common.numToIntOrFloat(-5)
-5
>>> common.numToIntOrFloat(1.0000000005)
1
>>> common.numToIntOrFloat(0.999999999)
1
```
```>>> sharp = pitch.Accidental('sharp')
>>> common.numToIntOrFloat(sharp.alter)
1
>>> halfFlat = pitch.Accidental('half-flat')
>>> common.numToIntOrFloat(halfFlat.alter)
-0.5
```

Can also take in a string representing an int or float

```>>> common.numToIntOrFloat('1.0')
1
>>> common.numToIntOrFloat('1')
1
>>> common.numToIntOrFloat('1.25')
1.25
```

Others raise a ValueError

```>>> common.numToIntOrFloat('one')
Traceback (most recent call last):
ValueError: could not convert string to float: 'one'
```

Fractions also become ints or floats

```>>> from fractions import Fraction
>>> common.numToIntOrFloat(Fraction(1, 2))
0.5
>>> common.numToIntOrFloat(Fraction(4, 3))
1.333333333...
```

Note: Decimal objects are not supported.

music21.common.numberTools.opFrac(num: None) None
music21.common.numberTools.opFrac(num: int) float
music21.common.numberTools.opFrac(num: float | Fraction) float | Fraction

opFrac -> optionally convert a number to a fraction or back.

Important music21 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 expressible 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(1/3)
Fraction(1, 3)
>>> common.opFrac(1/4)
0.25
>>> f = Fraction(1, 3)
>>> common.opFrac(f + f + f)
1.0
>>> common.opFrac(0.99999999842)
1.0
>>> common.opFrac(0.123456789)
Fraction(10, 81)
>>> common.opFrac(0.000001)
0.0
>>> common.opFrac(None) is None
True
```
music21.common.numberTools.ordinalAbbreviation(value: int, plural=False) str

Return the ordinal abbreviations for integers

```>>> common.ordinalAbbreviation(3)
'rd'
>>> common.ordinalAbbreviation(255)
'th'
>>> common.ordinalAbbreviation(255, plural=True)
'ths'
```
music21.common.numberTools.roundToHalfInteger(num: float | int) float | int

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
```

0.25 rounds up:

```>>> common.roundToHalfInteger(0.25)
0.5
```

as does 0.75

```>>> 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
```
music21.common.numberTools.strTrimFloat(floatNum: float, maxNum: int = 4) str

returns a string from a float that is at most maxNum of decimal 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'
```
music21.common.numberTools.unitBoundaryProportion(series: Sequence[int | float]) list[tuple[int | float, float]]

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, 0.25), (0.25, 0.5), (0.5, 1.0)]
>>> common.unitBoundaryProportion([9, 1, 1])
[(0.0, 0.8...), (0.8..., 0.9...), (0.9..., 1.0)]
```
music21.common.numberTools.unitNormalizeProportion(values: Sequence[int | float]) list[float]

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...]
```

Works fine with a mix of ints and floats:

```>>> common.unitNormalizeProportion([1.0, 1, 1.0])
[0.3333333..., 0.333333..., 0.333333...]
```

On 32-bit computers this number may be inexact even for small floats. On 64-bit it works fine. This is the 32-bit output for this result.

common.unitNormalizeProportion([0.2, 0.6, 0.2]) [0.20000000000000001, 0.59999999999999998, 0.20000000000000001]

Negative values should be shifted to positive region first:

```>>> common.unitNormalizeProportion([0, -2, -8])
Traceback (most recent call last):
ValueError: value members must be positive
```
music21.common.numberTools.weightedSelection(values: list[int], weights: list[int | float], randomGenerator=None) int

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 -50 and 50 (99.99999% of the time)

```>>> -50 < sum([common.weightedSelection([-1, 1], [1, 1]) for x in range(100)]) < 50
True
```