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librosa.pythagorean_intervals

librosa.pythagorean_intervals(*, bins_per_octave=12, sort=True, return_factors=False)[source]

Pythagorean intervals

Intervals are constructed by stacking ratios of 3/2 (i.e., just perfect fifths) and folding down to a single octave:

1, 3/2, 9/8, 27/16, 81/64, ...

Note that this differs from 3-limit just intonation intervals in that Pythagorean intervals only use positive powers of 3 (ascending fifths) while 3-limit intervals use both positive and negative powers (descending fifths).

Parameters:

bins_per_octaveint: The number of intervals to generate
sortbool: If True then intervals are returned in ascending order. If False, then intervals are returned in circle-of-fifths 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

plimit_intervals

Examples

Generate the first 12 intervals

>>> librosa.pythagorean_intervals(bins_per_octave=12)
array([1.      , 1.067871, 1.125   , 1.201355, 1.265625, 1.351524,
       1.423828, 1.5     , 1.601807, 1.6875  , 1.802032, 1.898437])
>>> # Compare to the 12-tone equal temperament intervals:
>>> 2**(np.arange(12)/12)
array([1.      , 1.059463, 1.122462, 1.189207, 1.259921, 1.33484 ,
       1.414214, 1.498307, 1.587401, 1.681793, 1.781797, 1.887749])

Or the first 7, in circle-of-fifths order

>>> librosa.pythagorean_intervals(bins_per_octave=7, sort=False)
array([1.      , 1.5     , 1.125   , 1.6875  , 1.265625, 1.898437,
       1.423828])

Generate the first 7, in circle-of-fifths other and factored form

>>> librosa.pythagorean_intervals(bins_per_octave=7, sort=False, return_factors=True)
[
    {2: 0, 3: 0},
    {2: -1, 3: 1},
    {2: -3, 3: 2},
    {2: -4, 3: 3},
    {2: -6, 3: 4},
    {2: -7, 3: 5},
    {2: -9, 3: 6}
]