3.4.3 Harmonic analysis
In order to produce useful timbres, an additive synthesizer user really needs to know about the harmonic content of real instruments, rather than mathematically derived waveforms. The main method of determining this information is Fourier analysis, which reverses the concept of making any waveform out of sine waves and uses the idea that any waveform can be split into a series of sine waves.
The basic concept behind Fourier analysis is quite simple, although the practical implementation is usually very complicated. If an audio signal is passed through a very narrow band-pass filter that sweeps through the audio range, then the output of the filter will indicate the level of each band of frequencies which are present in the signal (Figure 3.4.4). The width of this bandpass filter determines how accurate the analysis of the frequency content will be: if it is 100 Hz wide, then the output can only be used to a resolution of 100 Hz, whereas if the band-pass filter has a 1-Hz bandwidth, then it will be able to indicate individual frequencies to a resolution of 1 Hz.
FIGURE 3.4.4 Sweeping the center frequency of a narrow band-pass filter can convert an audio signal into a spectrum: from the time domain to the frequency domain.
For simple musical sounds that contain mostly harmonics of the fundamental frequency, the resolution required for Fourier analysis is not very high. The more complex the sound, the higher the required resolution. For sounds that have a simple structure consisting of a fundamental and harmonics, a rough 'rule of thumb' is to make the bandwidth of the filter less than the fundamental frequency, since the harmonics will be spaced at frequency intervals of the fundamental frequency.
Having 1-Hz resolution in order to discover that there are five harmonics spaced at 1-kHz intervals is extravagant. Smaller bandwidths require more complicated filters, and this can increase the cost, size and processing time, depending on how the filters are implemented. Fourier analysis can be achieved using analogue filters, but it is frequently carried out by using digital technology (see Section 5.8).
Numbers of harmonics
How many separate sine waves are needed in an additive synthesizer? Supposing that the lowest fundamental frequency which will be required to be produced is a low A at 55 Hz, then the harmonics will be at 110, 165, 220, 275, 330, 385, 440 Hz,... The 32nd harmonic will be at 1760 Hz and the 64th harmonic at 3520 Hz.
An A at 440 Hz has a 45th harmonic of 19,800 Hz. Most additive synthesizers seem to use between 32 and 64 harmonics (Table 3.4.2).
Table 3.4.2 Additive Frequencies and Harmonics
Harmonic and inharmonic content
Real-world sounds are not usually deterministic: they do not contain just simple harmonics of the fundamental frequency. Instead, they also have additional frequencies that are not simple integer multiples of the fundamental frequency.
The following are several types of these unpredictable 'inharmonic' frequencies:
- beat frequencies
Noise has, by definition, no harmonic structure, although it may be present only in specific parts of the spectrum: colored noise. So any noise which is present in a sound will appear as random additional frequencies within those bands, and whose level and phase are also random.
Beat frequencies arise when the harmonics in a sound are not perfectly in tune with each other. 'Perfect' waveshapes are always assumed to have harmonics at exact multiples of the fundamental, whereas this is not always the case in real-world sounds. If a harmonic is slightly detuned from its mathematically 'correct' position, then additional harmonics may be produced at the beat frequency, so if a harmonic is 1 Hz too high in pitch relative to the fundamental, then a frequency of 1 Hz will be present in the spectrum.
Sidebands occur when the frequency stability of a harmonic is imperfect, or when the sound itself is frequency modulated. Both cases result in pairs of frequencies which mirror around the 'ideal' frequency. So a 1-kHz sine wave which is frequency modulated with a few hertz will have a spectrum that contains frequencies on either side of 1 kHz, and the exact content will depend on the depth of modulation and its frequency. See Section 3.5.1 for more details.
Inharmonics are additional frequencies that are structured in some way, and so are not noise, but which do not have the simple integer multiple relationship with the fundamental frequency. Timbres that contain inharmonics typically sound like a 'bell' or 'gong'.
Many additive synthesizers only attempt to produce the harmonic frequencies, with perhaps a simple noise generator, as well. This deterministic approach limits the range of sounds which are possible, since it ignores many stochastic, probabilistic or random elements which make up real-world sounds.
The control of the level of each harmonic over time uses EGs and VCAs. Ideally, one EG and one VCA should be provided for each harmonic. This would mean that the overall envelope of the final sound was the result of adding together the individual envelopes for each of the harmonics, and so there would be no overall control over the envelope of the complete sound. Adding an overall EG and VCA to the sum of the individual harmonics allows quick modifications to be made to the final output (Figure 3.4.5).
FIGURE 3.4.5 Individual envelopes are used to control the harmonics, but an overall envelope allows easy control over the whole sound which is produced.
In order to minimize the number of controls and the complexity, the EGs need to be as simple as possible without compromising the flexibility. Delayed ADR (DADR) envelopes are amongst the easiest of EGs to implement in discrete analogue circuitry, since the gate signal can be used to control a simple capacitor charge and discharge circuit to produce the ADR envelope voltage. DADR envelopes also require only four controls (delay time, attack time, decay time and release time), whereas a DADSR would require five controls and more complex circuitry. If integrated circuit (IC) EGs are used, then the ADSR envelope would probably be used, since most custom synthesizer chips provide ADSR functionality.
Control grouping and ganging With large numbers of harmonics, having separate envelopes for each harmonic can become very unwieldy and awkward to control. The ability to assign a smaller number of envelopes to harmonics can reduce the complexity of an additive synthesizer considerably. This is only effective if the envelopes of groups of harmonics are similar enough to allow a 'common' envelope to be determined. Similarly, ganging together controls for the level of groups of harmonics can make it easy to make rapid changes to timbres – altering individual harmonics can be very time consuming. Simple groupings such as 'all of the odd' or 'all of the even' harmonics, can be useful starting points for this technique.
A more advanced use for grouping involves using keyboard voltages to give pitch-dependent envelope controls. This can be used to create the effect of fixed resonances or 'formants' at specific frequencies.