7. DESARROLLO
7.2 CONDICIONES Y VOLUMEN DE LOS SERVICIOS DE LAS ENTIDADES
7.2.3 Volumen de los servicios de las entidades financieras en Cali
All pitched instruments will produce harmonics in the same pattern of ratios relative to the fundamental; only the intensities of the harmonics will vary, giving each their characteristic timbre. So far we have only considered these harmonics in frequency values. When viewed on a musical scale (as in Figure 2.7) the harmonics reveal something interesting: their role as basis for musical scales and common chord progressions. This harmonic series is built off a low C2, and
visible in the first four harmonics is the bedrock chordal progression of western tonal music: I– V–I (C–G–C). The equal tempered scale—the scale with which we are most familiar—is largely based on the natural harmonic series, but with twelve equal divisions of an octave rather than the naturally occurring integer multiples of the fundamental.
Complex Waveforms
All sounds in nature and from acoustic instruments have waveforms that are considered complex. By this it is meant that they are a compound of harmonics caused by vibrating and resonating components. In Figure 2.8 we see several common instruments and choral sounds dissected into their constituent harmonics. Way back at the turn of the nineteenth century, Joseph Fourier first recognized that complex sounds are really just an amalgam of partials, each of which is itself a sine wave. Therefore the sine wave (shown earlier in Figure 2.3) is effectively the basic building block of all sound, and when we view the dominant harmonics in Figure 2.9, we are looking at individual sine waves working together to create the timbre.Phase
Phase describes the combination of two identical (or very similar) sounds separated by a time interval. Delay may be the unintended product of analog or digital audio system latency, sound picked up by separated microphones, or introduced deliberately with a delay-based-effects processor. Figure 2.9 shows how the degrees of rotation in a circle can be conveniently used to define the stages of development in a single wave cycle.
Phase can be described in more useful detail as one wave being combined with another at a certain degree point. Saying two waves are “out of phase” is less useful than saying they are 90º or 180º out of phase, both of which have very different consequences (see Figure 2.10). Figure 2.6 String vibration patterns (left) and spectral analysis of a plucked note on a nylon string acoustic guitar (right). Figure 2.7 Harmonic series displayed on a musical staff.
Remembering Fourier’s lesson that a complex wave is just a whole lot of sine waves on top of each other at different frequencies, when two of the same complex waves are combined at a short delay, some of those sine waves will meet closer to 0º and result in constructive interference, meaning the sum of the two increases the amplitude at that frequency, whereas others will combine closer to 180º and result in destructive interference, a decrease in amplitude at that frequency. When viewed on a spectrum analyzer, spikes in amplitude occur at the points of constructive interference and dips occur at points of destructive interference. With some imagination the spectrum has the appearance of a hair comb with the tines pointed upward, so the effect has become known as comb filtering (see Figure 2.11).
In synthesis, comb filtering can be used creatively to color a sound, particularly when the amount of delay applied is changed over time in either a repeating pattern (perhaps defined by a LFO or an envelope shape). Listen to the demonstration in Audio Example 2.1.
In the preceding example, comb filtering is created simply by misaligning two complex waves when they are summed. A similar and more common effect in synthesis is slightly detuning an oscillator relative to another with both set to the same waveform. The change in wavelength that is due to the pitch shift creates a comb filter. Listen to the demonstration in Audio Example 2.2.
The spectrographs in Figure 2.12 reveal distinct differences in the amplitude levels of the harmonic components as a result of the detuning. In particular, note the (a) presence of a subharmonic below the 87-Hz fundamental that wasn’t part of the original sound, the (b) attenuation of the fundamental, and the (c) increase in amplitude of the second harmonic. The before and after versions of this along with a video of the real time measurements are available at Audio Example 2.3.
Pitch
Pitch is a function of the frequency of a tone that is perceived when a sound is at least 2 ms in duration (this varies a bit with frequency, but is good enough for this conversation!). Shorter durations begin to sound more like noise and lose a strong pitch center.
Harmonics also play a role: We can hear a sine wave as having pitch, but it is much easier for us to identify when that sine wave is a fundamental reinforced by harmonics. Our ears are very good at discerning pitch: With experience, some musicians are able to hear the difference between a pitch that is “in tune” from one that is only a few cents sharp or flat. The term cents is used to describe the difference between two pitches separated by less than a semitone. For example, a note that is a halfway between a C and a C# is said to be 50 cents sharp of C (or 50 cents flat of C#). Figure 2.8 Spectral analyses of familiar musical sources. Figure 2.9 Degrees of a wave.
Figure 2.10 Summing sine waves at 0º, 180º, and 90º. When combined at 0º there is a doubling of amplitude, and thus a doubling of intensity. When combined at 180º there is a complete cancellation, thus no output. It is more common, however, that sine waves will meet slightly out of phase, somewhere other than 0º or 180º. At 90º there is attenuation but not complete cancellation. Figure 2.11 Comb filtering from two pink-noise tracks summed slightly out of phase as viewed on a spectrum analyzer.
Figure 2.12
Original sawtooth wave (top left) combined with another sawtooth slightly detuned. Notice in the summed version the change in amplitude of the harmonics.
Figure 2.13