Equal Division Tunings

Equal division tunings divide an equivalency interval into a finite number of steps, with each pitch having the same interval ratio to its successor. The most well-known equal division tuning is 12-EDO, the standard contemporary Western system that divides an octave into 12 equal steps.

In general, equal division tunings can be created by defining the number of divisions and the equivalency interval in the constructor of the EDTuning class:

from xenharmlib import FrequencyRatio
from xenharmlib import EDTuning

# create a tuning that divides the frequency ratio of 3
# (a tritave) into 13 equal steps (this tuning is also
# known as the "Bohlen-Pierce Tuning")
ed13_3 = EDTuning(13, FrequencyRatio(3))

Equal division tunings that slice up the octave (having frequency ratio 2) are also called “Equal Division of the Octave Tunings” (or EDO for short):

from xenharmlib import FrequencyRatio
from xenharmlib import EDTuning

# create the modern Western tuning
ed12_2 = EDTuning(12, FrequencyRatio(2))

As a short form, EDOs can also be created by the EDOTuning class, which sets the frequency ratio of the equivalency interval implicitly:

from xenharmlib import EDOTuning

# this tuning is equivalent to the above ed12_2
edo12 = EDOTuning(12)

By default, xenharmlib sets the frequency of the 0-pitch of equal division tunings to the default frequency for C0 in an equally tempered Western system (about 16.35 Hz). You can change the frequency of the 0-pitch with the ref_frequency parameter:

from xenharmlib import Frequency
from xenharmlib import FrequencyRatio
from xenharmlib import EDTuning
from xenharmlib import EDOTuning

new_ref = Frequency(20)

ed13_3 = EDTuning(13, FrequencyRatio(3), ref_frequency=new_ref)
print(ed13_3.pitch(0).frequency.to_float())

edo24 = EDOTuning(24, ref_frequency=new_ref)
print(edo24.pitch(0).frequency.to_float())

20.0
20.0