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librosa.plimit_intervals

librosa.plimit_intervals(*, primes, bins_per_octave=12, sort=True, return_factors=False)

Construct p-limit intervals for a given set of prime factors.

This function is based on the “harmonic crystal growth” algorithm of [3] [4].

[3]

Tenney, James. “On ‘Crystal Growth’ in harmonic space (1993–1998).” Contemporary Music Review 27.1 (2008): 47-56.

[4]

Sabat, Marc, and James Tenney. “Three crystal growth algorithms in 23-limit constrained harmonic space.” Contemporary Music Review 27, no. 1 (2008): 57-78.

Parameters:

primesarray of odd primes

Which prime factors are to be used

bins_per_octaveint

The number of intervals to construct

sortbool

If True then intervals are returned in ascending order. If False, then intervals are returned in crystal growth order.

return_factorsbool

If True then return a list of dictionaries encoding the prime factorization of each interval as {2: p2, 3: p3, …} (meaning 3**p3 * 2**p2). If False (default), return intervals as an array of floating point numbers.

Returns:

intervalsnp.ndarray or list of dictionaries

The constructed interval set. All intervals are mapped to the range [1, 2).

See also

pythagorean_intervals

Examples

Compare 3-limit tuning to Pythagorean tuning and 12-TET

>>> librosa.plimit_intervals(primes=[3], bins_per_octave=12)
array([1.        , 1.05349794, 1.125     , 1.18518519, 1.265625  ,
       1.33333333, 1.40466392, 1.5       , 1.58024691, 1.6875    ,
       1.77777778, 1.8984375 ])
>>> # Pythagorean intervals:
>>> librosa.pythagorean_intervals(bins_per_octave=12)
array([1.        , 1.06787109, 1.125     , 1.20135498, 1.265625  ,
       1.35152435, 1.42382812, 1.5       , 1.60180664, 1.6875    ,
       1.80203247, 1.8984375 ])
>>> # 12-TET intervals:
>>> 2**(np.arange(12)/12)
array([1.        , 1.05946309, 1.12246205, 1.18920712, 1.25992105,
       1.33483985, 1.41421356, 1.49830708, 1.58740105, 1.68179283,
       1.78179744, 1.88774863])

Create a 7-bin, 5-limit interval set

>>> librosa.plimit_intervals(primes=[3, 5], bins_per_octave=7)
array([1.        , 1.125     , 1.25      , 1.33333333, 1.5       ,
       1.66666667, 1.875     ])

The same example, but now in factored form

>>> librosa.plimit_intervals(primes=[3, 5], bins_per_octave=7,
...                          return_factors=True)
[
    {},
    {2: -3, 3: 2},
    {2: -2, 5: 1},
    {2: 2, 3: -1},
    {2: -1, 3: 1},
    {3: -1, 5: 1},
    {2: -3, 3: 1, 5: 1}
]