Harmonic series segments

A harmonic series segment is a range of consecutive numbers divided by the smallest one. For example, the harmonic series segment between 6 and 12 is

6/6, 7/6, 8/6, 9/6, 10/6, 11/6, 12/6

or simplifying the fractions

1/1, 7/6, 4/3, 3/2, 5/3, 11/6, 2/1

If the largest number is double the smallest, the scale will span an octave.

Subharmonic series segment

If we take the reciprocals of a range of consecutive numbers and multiply by the largest one, we get a subharmonic series segment. For example, the subharmonic series segment from the range 6 to 12 is

12/6, 12/7, 12/8, 12/9, 12/10, 12/11, 12/12

or simplifying fractions and sorting

1/1, 12/11, 6/5, 4/3, 3/2, 12/7, 2/1

Further reading

• Harry Partch, Genesis of a Music, 2nd ed. Da Capo Press (1979)
• Augusto Novaro, Sistema Natural de la Música, pp. 36–37, (1951)

from fractions import Fraction


def harmonic_series_segment(m, n):
    """
    >>> harmonic_series_segment(6, 12)
    [Fraction(1, 1), Fraction(7, 6), Fraction(4, 3), Fraction(3, 2), Fraction(5, 3), Fraction(11, 6), Fraction(2, 1)]
    """
    return [Fraction(i, m) for i in range(m, n + 1)]


def subharmonic_series_segment(m, n):
    """
    >>> subharmonic_series_segment(6, 12)
    [Fraction(1, 1), Fraction(12, 11), Fraction(6, 5), Fraction(4, 3), Fraction(3, 2), Fraction(12, 7), Fraction(2, 1)]
    """
    return sorted(Fraction(n, i) for i in range(m, n + 1))