pythagorean-tuning.pypythagorean-tuning.py
#?##################################################################################################
#?#
#?# Musica - A Music Theory Package for Python
#?# Copyright (C) 2017 Fabrice Salvaire
#?#
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#?# GNU General Public License for more details.
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#!# =======================================
#!#  Pythagorean Tuning compared to 12-TET
#!# =======================================

#!# This section compares the Pythagorean tuning and the twele-tone equal temperament (12-TET).

####################################################################################################

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator

####################################################################################################

from Musica.Math.MusicTheory import *

####################################################################################################

prev_fifth = None
for i, pitch in enumerate(PythagoreanTuning):
    delta_numerator = delta_denominator = ''
    if isinstance(pitch, PythagoreanFifth):
        if prev_fifth is not None:
            # 3**7/2**11 apotome |2**8/3**5 limma
            delta_numerator, delta_denominator = pitch / prev_fifth
        prev_fifth = pitch
    print('{:3} {:3} | {:3} {:3} | {:6} / {:6} | {:4.2f} | {:6.0f}'.format(
        pitch.numerator_power, pitch.denominator_power,
        delta_numerator, delta_denominator,
        pitch.numerator, pitch.denominator,
        float(pitch), float(pitch.cent)))
#o#

figure1 = plt.figure(1, (20, 10))
axe = plt.subplot(111)
axe.set_title('Pythagorean Tuning / 12-tone Equal Temperament')
axe.set_xlabel('Cents')
axe.set_ylabel('Frequency Ratio')
axe.xaxis.set_major_locator(MultipleLocator(100))
axe.grid()
axe.plot([float(pitch.cent) for pitch in PythagoreanTuning],
         [float(pitch) for pitch in PythagoreanTuning],
         'o-')
axe.plot([float(pitch.cent) for pitch in ET12Tuning],
         [float(pitch) for pitch in ET12Tuning],
         'o', color='orange')
for frequency_ratio in (
        FrequencyRatio.unisson,
        FrequencyRatio.fourth,
        FrequencyRatio.fifth,
        FrequencyRatio.octave,
        ):
    axe.axhline(y=frequency_ratio, color='orange')
axe.axhline(y=float(PythagoreanTuning.wolf_interval), color='red')

plt.show()

#fig# save_figure(figure1, 'temperament.png')

8.3.1. Pythagorean Tuning compared to 12-TETΒΆ

This section compares the Pythagorean tuning and the twele-tone equal temperament (12-TET).

import numpy as np
import matplotlib.pyplot as plt
from matplotlib.ticker import MultipleLocator

from Musica.Math.MusicTheory import *

prev_fifth = None
for i, pitch in enumerate(PythagoreanTuning):
    delta_numerator = delta_denominator = ''
    if isinstance(pitch, PythagoreanFifth):
        if prev_fifth is not None:
            # 3**7/2**11 apotome |2**8/3**5 limma
            delta_numerator, delta_denominator = pitch / prev_fifth
        prev_fifth = pitch
    print('{:3} {:3} | {:3} {:3} | {:6} / {:6} | {:4.2f} | {:6.0f}'.format(
        pitch.numerator_power, pitch.denominator_power,
        delta_numerator, delta_denominator,
        pitch.numerator, pitch.denominator,
        float(pitch), float(pitch.cent)))

  0   0 |         |      1 /      1 | 1.00 |      0
  8   5 |         |    256 /    243 | 1.05 |     90
  7  11 |  11   7 |   2187 /   2048 | 1.07 |    114
  2   3 |   8   5 |      9 /      8 | 1.12 |    204
  5   3 |         |     32 /     27 | 1.19 |    294
  9  14 |  11   7 |  19683 /  16384 | 1.20 |    318
  4   6 |   8   5 |     81 /     64 | 1.27 |    408
  2   1 |         |      4 /      3 | 1.33 |    498
 11  17 |  11   7 | 177147 / 131072 | 1.35 |    522
 10   6 |         |   1024 /    729 | 1.40 |    588
  6   9 |   8   5 |    729 /    512 | 1.42 |    612
  1   1 |   8   5 |      3 /      2 | 1.50 |    702
  7   4 |         |    128 /     81 | 1.58 |    792
  8  12 |  11   7 |   6561 /   4096 | 1.60 |    816
  3   4 |   8   5 |     27 /     16 | 1.69 |    906
  4   2 |         |     16 /      9 | 1.78 |    996
 10  15 |  11   7 |  59049 /  32768 | 1.80 |   1020
  5   7 |   8   5 |    243 /    128 | 1.90 |   1110
 12  18 |  11   7 | 531441 / 262144 | 2.03 |   1223

figure1 = plt.figure(1, (20, 10))
axe = plt.subplot(111)
axe.set_title('Pythagorean Tuning / 12-tone Equal Temperament')
axe.set_xlabel('Cents')
axe.set_ylabel('Frequency Ratio')
axe.xaxis.set_major_locator(MultipleLocator(100))
axe.grid()
axe.plot([float(pitch.cent) for pitch in PythagoreanTuning],
         [float(pitch) for pitch in PythagoreanTuning],
         'o-')
axe.plot([float(pitch.cent) for pitch in ET12Tuning],
         [float(pitch) for pitch in ET12Tuning],
         'o', color='orange')
for frequency_ratio in (
        FrequencyRatio.unisson,
        FrequencyRatio.fourth,
        FrequencyRatio.fifth,
        FrequencyRatio.octave,
        ):
    axe.axhline(y=frequency_ratio, color='orange')
axe.axhline(y=float(PythagoreanTuning.wolf_interval), color='red')

plt.show()
../../_images/temperament.png