Just Intonation / Prime Limit Tunings

Just Intonation refers to tunings that are based on “pure intervals”. A pure interval is characterized by a fraction of two integers, e.g. \(\frac{3}{2}\). Historically, most tuning systems were based on Just Intonation. It was not until the 18th and 19th centuries that equal temperament systems began to gain widespread acceptance in practice. Xenharmlib now introduces support for Just Intonation with the Prime Limit Tuning construct:

from xenharmlib import PrimeLimitTuning

limit11 = PrimeLimitTuning(11)

A Pythagorean 12-tone chromatic scale can, for example, now be generated by stacking 5 ascending pure fifths and 6 descending pure fifths from the root pitch:

from xenharmlib import PrimeLimitTuning
from xenharmlib import periodic

pythagorean = PrimeLimitTuning(3)

C0 = pythagorean.rs_pitch('1/1')
P5 = pythagorean.rs_interval('3/2')

iseq_a = pythagorean.interval_seq([P5] * 5)
iseq_b = pythagorean.interval_seq([-P5] * 6)
chromatic_scale = (
    C0.scale(iseq_a) | C0.scale(iseq_b)
).pcs_normalized()
print(chromatic_scale)

PrimeLimitPitchScale([1, 256/243, 9/8, 32/27, 81/64, 4/3, 1024/729, 3/2, 128/81, 27/16, 16/9, 243/128], 3-Limit)

Most software programs implement functions for Just Intonation not as a complete arithmetic system, but as a system of scalar transposition. In the case of a transposition by a fifth, for example, a definition of the chromatic scale (such as the one mentioned above) is typically used as a basis, and the transposition is understood as a shift of 7 scale degrees:

for i in range(0, 12):
    pitch = chromatic_scale[i]
    result = periodic.scalar_transpose(chromatic_scale, pitch, 7)
    interval = pitch.interval(result)
    print(f'{pitch.short_repr} --> {result.short_repr} ({interval.short_repr})')

1 --> 3/2 (3/2)
256/243 --> 128/81 (3/2)
9/8 --> 27/16 (3/2)
32/27 --> 16/9 (3/2)
81/64 --> 243/128 (3/2)
4/3 --> 2 (3/2)
1024/729 --> 512/243 (3/2)
3/2 --> 9/4 (3/2)
128/81 --> 64/27 (3/2)
27/16 --> 81/32 (3/2)
16/9 --> 8/3 (3/2)
243/128 --> 2048/729 (262144/177147)

As you can see with the last pitch of the output (\(\frac{243}{128}\)), transposition by scale degree in a Pythagorean chromatic scale does not always result in an actual transposition of the interval in question. This is a common problem when tuning a keyboard instrument (such as a piano or harpsichord) that has only a limited number of notes per octave. Mathematically speaking, this definition of transposition by a fifth is irregular: the size of the fifth changes depending on which note is chosen as the starting point.

Unlike a keyboard instrument, a stringed instrument without frets or the human voice does not have the problem of limited note selection. Xenharmlib therefore implements its default transposition not as scalar transposition, but as regular transposition, with the effect that the result of transposing the last note of the above scale has a pitch class no longer represented in the scale itself.

for i in range(0, 12):

    pitch = chromatic_scale[i]
    result = pitch.transpose(P5)
    interval = pitch.interval(result)

    try:
        scale_degree = periodic.index(chromatic_scale, result)
    except ValueError:
        scale_degree = 'not in scale'

    print(f'{pitch.short_repr} --> {result.short_repr} ', end='')
    print(f'({interval.short_repr}) - degree: {scale_degree}')

1 --> 3/2 (3/2) - degree: 7
256/243 --> 128/81 (3/2) - degree: 8
9/8 --> 27/16 (3/2) - degree: 9
32/27 --> 16/9 (3/2) - degree: 10
81/64 --> 243/128 (3/2) - degree: 11
4/3 --> 2 (3/2) - degree: 12
1024/729 --> 512/243 (3/2) - degree: 13
3/2 --> 9/4 (3/2) - degree: 14
128/81 --> 64/27 (3/2) - degree: 15
27/16 --> 81/32 (3/2) - degree: 16
16/9 --> 8/3 (3/2) - degree: 17
243/128 --> 729/256 (3/2) - degree: not in scale

This definition is not only mathematically more sound (since it has a single, consistent definition of a fifth), but it also allows us to deal with compositions that use the available tuning space to its full extent, not just a chromatic selection of pitches.

Custom Regular Temperaments

Custom regular temperaments can now be designed using the newly introduced Multi-Generator Tunings. The Quarter-Comma-Meantone Temperament that flattens the tritave by a quarter of the syntonic comma can, for example, be constructed like this:

from fractions import Fraction
from xenharmlib import MultiGenTuning
from xenharmlib import FrequencyRatio

g3 = FrequencyRatio(3) * FrequencyRatio(80, 81) ** Fraction(1, 4)
qcm = MultiGenTuning(
    (FrequencyRatio(2), g3),
    eq_diff_vec=(1, 0)
)

Full Interval Arithmetic

A feature necessary for many theoretical applications — one that was long missing from xenharmlib — is now being introduced in version 0.4.0. We’re talking about interval arithmetic:

from xenharmlib import WesternNotation

western = WesternNotation()

m3 = western.shorthand_interval('m', 3)
M3 = western.shorthand_interval('M', 3)

print(m3 + M3)
print(M3 - m3)
print(3 * M3)

WesternNoteInterval(P, 5)
WesternNoteInterval(A, 1)
WesternNoteInterval(A, 7)

To learn more, please consult the dedicated page on intervals.

Additional Harmonic Primitives

Version 0.4.0 comes with two new harmonic primitives for its analytical toolbox:

Sequences allow you to conceptualize a “succession of notes”, for example in segment analysis. Sequences come with typical transformation methods like retrograde and inversion.

from xenharmlib import WesternNotation
western = WesternNotation()

# melodic flow of the beginning of "marry had a little lamb"
seq = western.seq(western.note(pcs, 4) for pcs in 'EDCDEEE')

print(seq)
print(seq.retrograde())
print(seq.inversion())

WesternNoteSeq([E4, D4, C4, D4, E4, E4, E4])
WesternNoteSeq([E4, E4, E4, D4, C4, D4, E4])
WesternNoteSeq([E4, F#4, G#4, F#4, E4, E4, E4])

With Interval Fans, a new type of abstract scale object is introduced that describes the relative distances of the elements of a sequence or scale to a tonal center:

from xenharmlib import WesternNotation
western = WesternNotation()

# melodic flow of the beginning of "marry had a little lamb"
seq = western.seq(western.note(pcs, 4) for pcs in 'EDCDEEE')

# calculate relative distances to the tonic of C4
tonic = western.note('C', 4)
ifan = seq.to_interval_fan(tonic)
print(ifan)

WesternNoteIntervalFan([M3, M2, P1, M2, M3, M3, M3])

Interval Class Vectors

As an essential tool for scale analysis, especially in post-tonal analysis, xenharmlib now supports the calculation of interval class vectors:

from xenharmlib import EDOTuning
from xenharmlib.setc import ic_vector

edo31 = EDOTuning(31)

scale = edo31.index_scale([0, 5, 11, 15, 17, 21])
print(ic_vector(scale))

(0, 1, 0, 2, 1, 3, 0, 0, 0, 3, 1, 1, 0, 1, 2)

Closest Frequency Approximation

All harmonic primitives of one-dimensional tunings and notations now support closest frequency approximation, e.g. scales:

from xenharmlib import WesternNotation
from xenharmlib import Frequency

western = WesternNotation()

# overtone frequencies measured e.g. by
# a spectrogram plugin or librosa
frequencies = [
    Frequency(f) for f in [230, 410, 460, 500, 690]
]

scale = western.closest_scale(frequencies)
print(scale)

WesternNoteScale([A#3, G#4, A#4, B4, F5])