"Our ears interpret a changing pattern in air pressure as sound. If the pressure pattern repeats periodically, we experience a sensation of **pitch**.\n",
"\n",
"Pitch is determined by the pattern's **frequency** (i.e. the number of times per second that the pattern repeats). The actual *shape* of the pattern only affects what we call the _timbre_ of the sound (note: _affects_ not _determine_ ; many more things go into our sense of timbre). "
"The actual mechanics of sound production will enter the story a bit later. For now, let's use the simplest form of periodic patterns, namely, **sinusoids**. "
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"### 1.2. Continuum and categorical perception\n",
"Ancient monophonic instrument used in ancient Greece (VI century BC). The motion of the vibrating string can be decomposed into the sum of harmonic standing waves with frequencies\n",
"Naturally occurring pitched sounds possess a _harmonic_ spectrum, with frequency content pooled around multiples of the fundamental frequency $f_1$. "
"It was known since antiquity that two vibrating strings sound well together if the length of the second string is **two thirds** of the first. This correspond of a frequency scaling factor equal to $3/2 = 1.5$\n",
"\n",
"The $3/2$ frequency ratio is called a **fifth** (for anachronistic reasons that will be clear later...)"
"Just intonation bypasses the geometric construction of Pythagorean tuning and defines the degrees of the scale in terms of \"what sounds good\". This intonation is also known as the Ptolemaic tuning.\n",
"\n",
"It turns out that most good intervals are described by simple fractions!"
"Consonance and dissonance for real-world musical sound has a lot to do with the harmonic nature of the associated spectra. Let's unpack this a bit at a time."
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"#### 4.2.1. Frequency beatings\n",
"\n",
"Two sinusoidal tones close in frequency produce so-called beatings: the sound is perceived as a single tone modulated in amplitude at a rate equal to the difference between frequencies:\n",
"If $\\omega_a \\approx \\omega_b$, then $(\\omega_a + \\omega_b)/2 \\approx \\omega_a $ and $(\\omega_a - \\omega_b)/2$ will have a very low frequency."
"The basilar membrane in the cochlea is composed of frequency-selective regions called _critical bands_. The ear can only resolve frequencies falling in distinct critical bands, whose spacing is logarithmic. Critical bands are also involved in the mechanism of _masking_.\n",
"\n",
"Beatings occur when a \"blurry\" spectral perception induces an unwanted temporal percpetion. Beatings are generally unwanted and unpleasant."
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"#### 4.2.3. Perceptual dissonance curves\n",
"\n",
"A classic psychoacoustic experiment by Plompt and Levelt (1965) produced a model for the relation between the perceptual concept of consonance/dissonance and the physical nature of critical bands. Listeners are asked to rate the level of dissonance between two sinudoidal tones like so:"
"The Plompt-Levelt curve applies to pairs of puire sinusoids. To determine the consonance of two harmonic spectra, we can accumulate the dissonance scores for all pairs of spectral lines in the two tones:"
"plt.xticks([n/d for n, d in intervals], ['{}/{}'.format(n, d) for n, d in intervals]);\n",
"plt.yticks([]);"
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"### 4.3. The dirty secret of Just Intonation: the comma pump\n",
"\n",
"Zarlino (XVI century) considered JI the only intonation suitable for singing. But a cappella groups singing in JT are prone to the so-called _comma pump_."