In [1]:
%matplotlib inline
import seaborn
import scipy, scipy.signal, IPython.display as ipd, matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (13, 5)

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Tuning Systems¶

  1. Introduction
  2. Unison
  3. Octaves
  4. Fifths
  5. Fourths
  6. Major Sixths
  7. Minor Thirds
  8. Major Thirds
  9. Minor Sixths
  10. Major Seconds
  11. Minor Sevenths
  12. Major Sevenths
  13. Minor Seconds
  14. Tritone, Augmented Fourths, Diminished Fifths

Introduction¶

Tuning Systems¶

In twelve-tone equal temperament (Wikipedia), all twelve semitones within the octave have the same width. With this tuning system, expressed as a frequency ratio, the interval of one semitone is $2^{1/12}$. Expressed in cents, this same interval is defined to be 100 cents. Therefore, the octave has 1200 cents.

In just intonation (Wikipedia), the frequency ratio is expressed as a fraction between two small integers, e.g. 3:2, 4:3. As a result, the higher harmonic partials between two notes will overlap, resulting in a consonant interval that is pleasing to the ear. In 5-limit just tuning, these fractions are expressed with prime factors no larger than 5, i.e. {2, 3, 5}. In 7-limit just tuning, these fractions are expressed with prime factors no larger than 7, i.e. {2, 3, 5, 7}. For example, 7:4 is a 7-limit interval, but it is not a 5-limit interval.

In Pythagorean tuning (Wikipedia), every frequency ratio is based upon the ratio 3:2. To find that ratio, from one note in the interval, step around the Circle of Fifths until you reach the other note in the interval, multiplying (if stepping forward) or dividing (if stepping backward) by 3/2 with each step. Finally, multiply or divide by 2 enough times to return to the octave of interest. Pythagorean tuning can also be considered 3-limit just tuning since every ratio only uses prime factors no greater than 3.

How To Use This Notebook¶

In the examples below, listen to each interval, and compare intervals both visually and aurally across tuning systems. In some tuning systems, the upper harmonics do not align. In such cases, try to listen for dissonance and beat frequencies. Then compare them to the tuning systems where the harmonics align perfectly.

Notebook Setup¶

Global parameters for this notebook:

In [2]:
T = 5     # duration in seconds
sr = 22050  # sampling rate in Hertz
fcutoff = 4000 # frequency cutoff for filter

Functions used to create the sounds and figures:

In [3]:
def simulate_tone(f0, harmonics=None):
    """Returns a tone with a specified fundamental frequency and harmonic amplitudes."""
    if harmonics is None:
        harmonics = [1]
    t = scipy.linspace(0, T, T*sr, endpoint=False)
    x = sum(v*scipy.sin(2*scipy.pi*i*f0*t) for i, v in enumerate(harmonics, 1))
    return x
In [4]:
def filter_tone(x, fcutoff=None):
    """Return a low-pass filtered signal."""
    if fcutoff is None:
        fcutoff = sr/2.0
    h = scipy.signal.firwin(55, 2*float(fcutoff)/sr)
    y = scipy.convolve(x, h)
    return y
In [5]:
def make_double_stop(ratio, f0=440, harmonics=None):
    """Listen to two tones played simultaneously, and plot the tones' spectra."""
    if harmonics is None:
        harmonics = [1]
    f1 = ratio*f0
    
    # Generate both tones.
    x0 = filter_tone(simulate_tone(f0, harmonics=harmonics), fcutoff=fcutoff)
    x1 = filter_tone(simulate_tone(f1, harmonics=harmonics), fcutoff=fcutoff)
    
    # Add both tones, and normalize.
    y = x0 + x1
    y = 0.5*y/y.max()
    
    # Generate both spectra.
    X0 = scipy.fft(x0)
    X1 = scipy.fft(x1)

    # Create frequency variable.
    N = len(X0)
    f = scipy.linspace(0, sr, N, endpoint=False)
    
    # Plot spectrum of both notes.
    plt.semilogy(f, abs(X0), marker='o', linewidth=1)
    plt.semilogy(f, abs(X1), color='r', linewidth=1)
    
    plt.xlim(xmin=0, xmax=(len(harmonics)+1)*(f1 if ratio > 1 else f0))
    plt.ylim(ymin=0.1)
    
    plt.xlabel('Frequency (Hz)')
    plt.legend(('f0 = %.2f Hz' % f0, 'f0 = %.2f Hz' % f1))
    
    # Output audio widget.
    print 'ratio between notes:', ratio
    print 'difference in cents:', scipy.log2(ratio)*1200
    return ipd.Audio(y, rate=sr)

Unison¶

Within each section below, the intervals are provided in order of interval width from lowest to highest.

In [6]:
make_double_stop(1)

ratio between notes: 1
difference in cents: 0.0
Out[6]:

Octaves¶

In [7]:
harmonics = [1.0, 0.5]

Just intonation or equal temperament, 12 semitones, 1200 cents:

In [8]:
make_double_stop(2, harmonics=harmonics)

ratio between notes: 2
difference in cents: 1200.0
Out[8]:

Pythagorean tuning, twelve steps forward on the Circle of Fifths. In Pythagorean tuning, multiply the fundamental frequency by 3/2 twelve times, and then divide by two enough times to return to the octave of interest:

$$ \left( \frac{3}{2} \right)^{12} \left( \frac{1}{2} \right)^6 = \frac{531441}{262144} \approx 2.0273 $$
In [9]:
make_double_stop(531441.0/262144, harmonics=harmonics)

ratio between notes: 2.02728652954
difference in cents: 1223.46001038
Out[9]:

The Pythagorean comma, the degree of inconsistency when trying to define a twelve-tone scale using only perfect fifths, is about 1.0136 when expressed as a frequency ratio:

$$ \left( \frac{3}{2} \right)^{12} \left( \frac{1}{2} \right)^7 = \frac{531441}{524288} \approx 1.0136 $$

Fifths¶

In [10]:
harmonics = [1.0, 0.1, 0.1]

Equal temperament, seven semitones, 700 cents:

In [11]:
make_double_stop(2**(7.0/12), harmonics=harmonics)

ratio between notes: 1.49830707688
difference in cents: 700.0
Out[11]:

Just intonation or Pythagorean tuning, one step forward on the Circle of Fifths:

In [12]:
make_double_stop(3.0/2, harmonics=harmonics)

ratio between notes: 1.5
difference in cents: 701.955000865
Out[12]:

Fourths¶

In [13]:
harmonics = [1.0, 0.01, 0.1, 0.1]

Just intonation or Pythagorean tuning, one step backward on the Circle of Fifths:

$$ \left( \frac{2}{3} \right) \left( \frac{2}{1} \right) = \frac{4}{3} = 1.\overline{3} $$
In [14]:
make_double_stop(4.0/3, harmonics=harmonics)

ratio between notes: 1.33333333333
difference in cents: 498.044999135
Out[14]:

Equal temperament, five semitones:

In [15]:
make_double_stop(2**(5.0/12), harmonics=harmonics)

ratio between notes: 1.33483985417
difference in cents: 500.0
Out[15]:

Major Sixths¶

In [16]:
harmonics = [1.0, 0.01, 0.1, 0.01, 0.1]

Just intonation:

In [17]:
make_double_stop(5.0/3, harmonics=harmonics)

ratio between notes: 1.66666666667
difference in cents: 884.358712999
Out[17]:

Equal temperament, i.e. nine semitones:

In [18]:
make_double_stop(2**(9.0/12), harmonics=harmonics)

ratio between notes: 1.68179283051
difference in cents: 900.0
Out[18]:

Pythagorean tuning, three steps forward on the Circle of Fifths:

$$ \left( \frac{3}{2} \right)^3 \left( \frac{1}{2} \right) = \frac{27}{16} = 1.6875 $$
In [19]:
make_double_stop(27.0/16, harmonics=harmonics)

ratio between notes: 1.6875
difference in cents: 905.865002596
Out[19]:

Minor Thirds¶

In [20]:
harmonics = [1.0, 0.01, 0.01, 0.01, 0.1, 0.1]

Pythagorean tuning, three steps backward on the Circle of Fifths:

$$ \left( \frac{2}{3} \right)^3 \left( \frac{2}{1} \right)^2 = \frac{32}{27} = 1.\overline{185} $$
In [21]:
make_double_stop(32.0/27, harmonics=harmonics)

ratio between notes: 1.18518518519
difference in cents: 294.134997404
Out[21]:

Equal temperament, three semitones:

In [22]:
make_double_stop(2**(3.0/12), harmonics=harmonics)

ratio between notes: 1.189207115
difference in cents: 300.0
Out[22]:

Just intonation:

In [23]:
make_double_stop(6.0/5, harmonics=harmonics)

ratio between notes: 1.2
difference in cents: 315.641287001
Out[23]:

Major Thirds¶

In [24]:
harmonics = [1.0, 0.01, 0.01, 0.1, 0.1]

Just intonation:

In [25]:
make_double_stop(5.0/4, harmonics=harmonics)

ratio between notes: 1.25
difference in cents: 386.313713865
Out[25]:

Equal temperament, four semitones:

In [26]:
make_double_stop(2**(4.0/12), harmonics=harmonics)

ratio between notes: 1.25992104989
difference in cents: 400.0
Out[26]:

Pythagorean tuning, four steps forward on the Circle of Fifths:

$$ \left( \frac{3}{2} \right)^4 \left( \frac{1}{2} \right)^2 = \frac{81}{64} = 1.265625 $$
In [27]:
make_double_stop(81.0/64, harmonics=harmonics)

ratio between notes: 1.265625
difference in cents: 407.820003462
Out[27]:

Minor Sixths¶

In [28]:
harmonics = [1.0, 0.001, 0.001, 0.001, 0.01, 0.001, 0.001, 0.01]

Pythagorean tuning, four steps backward on the Circle of Fifths:

$$ \left( \frac{2}{3} \right)^4 \left( \frac{2}{1} \right)^3 = \frac{128}{81} \approx 1.58 $$
In [29]:
make_double_stop(128.0/81, harmonics=harmonics)

ratio between notes: 1.58024691358
difference in cents: 792.179996538
Out[29]:

Equal temperament, eight semitones:

In [30]:
make_double_stop(2**(8.0/12), harmonics=harmonics)

ratio between notes: 1.58740105197
difference in cents: 800.0
Out[30]:

Just intonation:

In [31]:
make_double_stop(8.0/5, harmonics=harmonics)

ratio between notes: 1.6
difference in cents: 813.686286135
Out[31]:

Major Seconds¶

In [32]:
harmonics = [1.0, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.1, 0.1, 0.1]

In just intonation, there are two versions of the major second: the major tone a.k.a. greater tone (9:8), and the minor tone a.k.a. lesser tone (10:9). The difference between the two, expressed as a ratio of their widths, is called a syntonic comma and is equal to 81:80.

$$ \frac{10}{9} n = \frac{9}{8} \Rightarrow n = \frac{81}{80} $$

Just intonation (minor tone a.k.a. lesser tone). Observe how the tenth harmonic of the lower note aligns with the ninth harmonic of the upper note.

In [33]:
make_double_stop(10.0/9, harmonics=harmonics)

ratio between notes: 1.11111111111
difference in cents: 182.403712134
Out[33]:

Equal temperament, two semitones:

In [34]:
make_double_stop(2**(2.0/12), harmonics=harmonics)

ratio between notes: 1.12246204831
difference in cents: 200.0
Out[34]:

Just intonation (major tone a.k.a. greater tone) or Pythagorean tuning, two steps forward on the Circle of Fifths:

$$ \left( \frac{3}{2} \right)^2 \left( \frac{1}{2} \right) = \frac{9}{8} = 1.125 $$

Observe how the ninth harmonic of the lower note aligns with the eighth harmonic of the upper note.

In [35]:
make_double_stop(9.0/8, harmonics=harmonics)

ratio between notes: 1.125
difference in cents: 203.910001731
Out[35]:

Minor Sevenths¶

In [36]:
harmonics = [1.0, 0.01, 0.01, 0.1, 0.1, 0.01, 0.1, 0.01, 0.1]

7-limit just intonation, a.k.a. septimal minor seventh, harmonic seventh, or subminor seventh:

In [37]:
make_double_stop(7.0/4, harmonics=harmonics)

ratio between notes: 1.75
difference in cents: 968.825906469
Out[37]:

Pythagorean tuning, two steps backward on the Circle of Fifths, or small just minor seventh:

$$ \left( \frac{2}{3} \right)^2 \left( \frac{2}{1} \right)^2 = \frac{16}{9} = 1.\overline{7} $$
In [38]:
make_double_stop(16.0/9, harmonics=harmonics)

ratio between notes: 1.77777777778
difference in cents: 996.089998269
Out[38]:

Equal temperament, ten semitones:

In [39]:
make_double_stop(2**(10.0/12), harmonics=harmonics)

ratio between notes: 1.78179743628
difference in cents: 1000.0
Out[39]:

5-limit just intonation, a.k.a. large just minor seventh:

In [40]:
make_double_stop(9.0/5, harmonics=harmonics)

ratio between notes: 1.8
difference in cents: 1017.59628787
Out[40]:

Major Sevenths¶

In [41]:
harmonics = [1.0,] * 15

Just intonation:

In [42]:
make_double_stop(15.0/8, harmonics=harmonics)

ratio between notes: 1.875
difference in cents: 1088.26871473
Out[42]:

Equal temperament, eleven semitones:

In [43]:
make_double_stop(2**(11.0/12), harmonics=harmonics)

ratio between notes: 1.88774862536
difference in cents: 1100.0
Out[43]:

Pythagorean tuning, five steps forward on the Circle of Fifths:

$$ \left( \frac{3}{2} \right)^5 \left( \frac{1}{2} \right)^2 = \frac{243}{128} \approx 1.8984 $$
In [44]:
make_double_stop(243.0/128, harmonics=harmonics)

ratio between notes: 1.8984375
difference in cents: 1109.77500433
Out[44]:

Minor Seconds¶

In [45]:
harmonics = [0.01,] * 16
harmonics[0] = 1.0
harmonics[14] = 0.1
harmonics[15] = 0.1

Pythagorean tuning, five steps backward on the Circle of Fifths:

$$ \left( \frac{2}{3} \right)^5 \left( \frac{2}{1} \right)^3 = \frac{256}{243} \approx 1.0535 $$
In [46]:
make_double_stop(256.0/243, harmonics=harmonics)

ratio between notes: 1.05349794239
difference in cents: 90.2249956731
Out[46]:

Equal temperament, one semitone:

In [47]:
make_double_stop(2**(1.0/12), harmonics=harmonics)

ratio between notes: 1.05946309436
difference in cents: 100.0
Out[47]:

Just intonation:

In [48]:
make_double_stop(16.0/15, harmonics=harmonics)

ratio between notes: 1.06666666667
difference in cents: 111.73128527
Out[48]:

Tritone, Augmented Fourths, Diminished Fifths¶

Diminished fifth, Pythagorean tuning, six steps backward on the Circle of Fifths:

$$ \left( \frac{2}{3} \right)^6 \left( \frac{2}{1} \right)^4 = \frac{1024}{729} \approx 1.40466 $$
In [49]:
make_double_stop(1024.0/729)

ratio between notes: 1.40466392318
difference in cents: 588.269994808
Out[49]:

Augmented fourth:

In [50]:
make_double_stop(45.0/32)

ratio between notes: 1.40625
difference in cents: 590.223715596
Out[50]:

Equal temperament, six semitones, 600 cents:

In [51]:
make_double_stop(2**(6.0/12))

ratio between notes: 1.41421356237
difference in cents: 600.0
Out[51]:

Diminished fifth:

In [52]:
make_double_stop(64.0/45)

ratio between notes: 1.42222222222
difference in cents: 609.776284404
Out[52]:

Augmented fourth, Pythagorean tuning, six steps forward on the Circle of Fifths:

$$ \left( \frac{3}{2} \right)^6 \left( \frac{1}{2} \right)^3 = \frac{729}{512} \approx 1.4238 $$
In [53]:
make_double_stop(729.0/512)

ratio between notes: 1.423828125
difference in cents: 611.730005192
Out[53]:

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