Tuesday, June 23, 2009

Triangle wave as a control function

An object oscillating in simple harmonic motion, such as a pendulum swinging or a tuning fork vibrating, has a "restoring force" that pulls the object toward its central position -- gravity in the case of a pendulum, and tension in the case of a tuning fork. As the object approaches maximum displacement from the center it loses momentum due to the restoring force pulling against it. It loses velocity until its velocity is 0 and it then begins to be pulled in the opposite direction back toward the central position; it reaches maximum velocity in the center, overshoots the center due to its accrued momentum, and then increases its displacement in the opposite direction. This deceleration as it reaches maximum displacement, and acceleration as it is pulled away from maximum, is reflected in the smooth sinusoidal curve when we graph its displacement.

If there were no change in its velocity -- if it were to somehow instantaneously change direction with no deceleration and acceleration -- the graph of its displacement would look triangular.

This sort of triangular function can be useful for representing constant periodic linear change -- back and forth, up and down, etc. -- between two extremes. In terms of sound, it's important to remember than the impression of linear change in the subjective phenomena of pitch and loudness actually correspond to exponential or logarithmic change in the empirical measures of frequency and amplitude.

It's important to understand that a triangle function in pitch or decibels is not the same as a triangle function in frequency or linear amplitude. For example, a triangular displacement up and down from a central pitch by a tritone (+ or - 6 semitones) is exponential in frequency, and is a greater change upward in Hertz than it is downward. That is, a shift of 1/2 octave is a greater number of Hertz at higher frequencies than at lower frequencies, and a linear pitch change corresponds to an exponential curve in frequency, as illustrated in the following two visualizations. The first image depicts a linear change in pitch, as occurs in the program.
The second image depicts the change in frequency that corresponds to that triangle function in pitch.
Note that the change in frequency is exponential rather than linear, and that a greater change of frequency is needed to go up a given pitch interval than is needed to go down the same pitch interval. In this case, to go from middle C (261.6 Hz) up to F# (370.0 Hz) is a difference of 108.4 Hz, while going down to F# (185.0 Hz) is a difference of 76.6 Hz. In Max the mtof or mtof~ object takes care of this translation from pitch to frequency for you. The formula in use is ƒ = 440.(2^(69.-p)), where p is the pitch in MIDI terminology (MIDI 69 = A 440 Hz).

A similar translation from linear to exponential is needed to go from the level in decibels, which is a logarithmic descriptor, to amplitude, which is on a linear scale. The formula used by the dbtoa and dbtoa~ objects to perform this translation is a = 10.^(d/20.), where d is the level in decibels relative to the maximum possible value, 1.

This program demonstrates the use of a triangular function to make periodic changes of pitch and loudness.
Instead of the cycle~ objects in this program reading from the default wavetable of 512 values in the form of a cosine function, we have to make a triangular function in a wavetable and read from that. The lower-right portion of the program creates and stores that function. When the program first starts up, it uses a mathematical expression to place 512 values in the form of a triangle wave into memory in a buffer~ object. (The size of the buffer -- 11.61 -- is expressed in milliseconds; at a sampling rate of 44,100 Hz, 11.61 milliseconds is 512 samples.)

The carrier oscillator is a tri~ object. Similarly to the saw~ and rect~ objects, tri~ generates a signal with a spectrum like that of a triangle wave, but one that limits its upper partials to avoid aliasing. This carrier oscillator's frequency and amplitude are controlled by three low-frequency oscillators -- the cycle~ objects reading from the triangle function in the buffer~. The triangle functions of the control oscillators are expressing values in terms of equal-tempered semitones and decibels, which correspond to our perception of pitch and volume, and those values are then converted to frequency and amplitude by the mtof~ and dbtoa~ objects.

Lets look at each of the uses of the triangle function here. At the top we have a triangle control oscillator with a frequency of 0.5 Hz, which means that it completes a cycle once every 2 seconds. Its amplitude is scaled by 6 and offset by 60, so it oscillates around the central value of 60, up as high as 66 and down as low as 54. This function is used to define the pitch, which is then translated into frequency by the mtof~ object and used to control the frequency of the carrier oscillator, the tri~ object. As in the other examples of classic control functions such as pulse wave, sawtooth wave, and sine wave, we use the triangle wave here as the carrier oscillator so that you can hear its timbral character. A triangle wave contains energy only at odd harmonics of the fundamental, with the amplitude of each partial proportional to the square of the inverse of the harmonic number. Thus, its timbre is richer than a sine tone, but mellower than a sawtooth wave.

The second triangle control oscillator has a frequency of 6 Hz, and serves to create an amplitude tremolo for the carrier oscillator. The control oscillates around -5 dB, going as high as 0 dB and as low as -10 dB. This level control is translated into amplitude by the dbtoa~ object and is used to control the amplitude of the carrier wave, creating a 10 dB fluctuation of amplitude 6 times per second.

The third triangle control oscillator has a frequency of 0.0625 Hz (1/16 Hz), so it completes a cycle only once every 16 seconds. This exerts a more formal function, creating a swell of the overall amplitude every 16 seconds, like a master volume knob, ranging as high as 0 dB and as low as -40 dB.

So in this one program we see the triangle function used 1) as a carrier waveform, for its timbral effect, with a frequency ranging from 185 to 370 Hz, 2) as a control function to create expressive amplitude tremolo at a rate of 6 Hz, 3) as a control function to create pitch glissando at a rate of 1/2 Hz, and 4) as a control function to create crescendo/decrescendo at a rate of 1/16 Hz.

Admittedly, the sound of an octave-wide pitch glissando is not terribly attractive; it's more like a siren than a musical gesture. But this program demonstrates several uses of this particular control function (timbre, glissando, tremolo, and dynamics), and shows the aesthetic effect of its characteristic shape.

Friday, June 19, 2009

Pulse wave, a binary control function

A pulse wave, also known as a rectangle wave, is a function that alternates periodically between two values, such as 1 and 0. This is a classic waveform of electronic music, and can be used as a control function to obtain an alternation between two states.

Max doesn't really provide a pulse wave generating object, but it's easy enough to create an ideal pulse wave by combining a phasor~ object, which ramps periodically from 0 to 1, and a <~ object, which will output only 1 or 0 depending on whether its input is less than a given value. For example, if you use <~ to test whether the value of the phasor~ is less than 0.5, the output will be 1 for the first half of the phasor~'s ramp, and 0 for the second half, resulting in a square wave.
We can then scale and offset the output of >~ to obtain a periodic alternation between any two desired values. In terms of musical pitch, a periodic alternation between two values is known as a trill or, in the case of a wider interval, a tremolo, and is usually at a fast but still sub-audio rate from about 8 to 18 notes per second. When the alternation takes place at an audio rate, the wave is heard as a pitched tone containing only odd harmonics, with the amplitude of each harmonic inversely proportional to the harmonic number. Thus, if the waveform has amplitude A, the fundamental (first harmonic) has amplitude A, the third harmonic has amplitude A/3, the fifth harmonic has amplitude A/5, etc. This means that upper harmonics may exceed the Nyquist frequency, possibly causing unwanted audible aliased frequencies. So for times when a rectangle waveform is desired for a carrier oscillator, there is an object called rect~ that generates a wave with a spectrum very similar to a rectangle wave, but that only produces audible harmonics up to the Nyquist frequency.

When a rectangle wave spends the same amount of time on 1 as it spends on 0, as in the above example, the wave is called a square wave. The amount of time the wave spends on 1, expressed as a fraction of one entire cycle, is called the duty cycle; in the case of a square wave the duty cycle is 0.5 because the wave spends 1/2 of its time at the value 1. However, a rectangle wave can have a duty cycle ranging anywhere from 0 to 1, which will allow a variety of alternation effects when the wave is used as a control function, and which will have a timbral effect when the wave is used as a carrier tone. For example, if we use 0.75 as the comparison value in the <~ object, the object's output will be 1 for 3/4 of each cycle of the phasor~, giving a rectangle wave with a duty cycle of 0.75.

This program demonstrates the use of a pulse wave as a control function and as a carrier waveform, using modulator pulse wave LFOs to control both the frequency and the amplitude of the carrier wave.

In this example a square wave at a rate of 6 Hz is used to modulate the frequency of the carrier oscillator back and forth between 220 Hz and 247 Hz, which gives the impression of a musical trill between A 220 and the B above that, at the rate of 12 notes per second. In this case, rather than use a <~, we use a >=~ object so that the square wave will start with a 0 value, thus starting the trill on A 220. Note that for the carrier oscillator, the one we actually listen to, we have used the rect~ object, which gives a band-limited pulse tone that resists the aliasing effects of an ideal rectangle wave.

To control the amplitude in this example we use a rectangle wave with a rate of 1 Hz and a duty cycle of 0.75. The effect is that we hear the sound for 3/4 of a second, followed by 1/4 of a second of silence; we hear 9 notes of trill followed by three notes' worth of silence.

Because an instantaneous switch from an amplitude of 0 to an amplitude of 1 or vice versa would cause a click, we use the rampsmooth~ object to interpolate linearly to the new amplitude value over 441 samples (10 milliseconds).

Monday, June 15, 2009

Sine wave as control function

Sine and cosine are trigonometric functions that come from graphing the y or x value, respectively, of a point as it traverses the circumference of a unit circle in a constantly changing radial angle from 0 to 2π radians. The cosine is actually exactly the same as the sine with a phase offset of π/2 radians, which is to say starting 1/4 of a cycle into the sine function. To talk about any such function, regardless of phase offset, we can use the noun sinusoid and the adjective sinusoidal.

It happens that the sinusoid is also the graph of simple harmonic motion, such as the natural oscillation of a pendulum or the simple back-and-forth vibration of the tine of a tuning fork or an alternating electrical current. Simple harmonic motion is oscillation at a single frequency, so the sinusoidal wave is the most basic "building block" or elemental unit of all sound.

The cycle~ object in Max acts as a wavetable oscillator for generating periodic signals, and by default it uses the cosine function.

(Internally it is actually reading from a 512-point lookup table, and interpolating between those points as necessary to generate a smooth signal at any frequency.) Its peak amplitude is 1; it oscillates in the range from 1 to -1. Every time you turn on audio signal processing in Max, all cycle~ objects begin in cosine phase--i.e., starting at 1. However, you can supply a phase offset in the right inlet, so to make a cycle~ start with sine phase, you must supply a phase offset of 0.75 to start 3/4 of a cycle into the cosine function.

The sinusoidal oscillator can be used at an audio frequency as a carrier oscillator, or as a control function at a low frequency (or indeed any frequency) to modulate other signals. Since the sinusoidal oscillator was a basic generator in almost all early electronic synthesizers, we have become very familiar with the sound of using one oscillator for sinusoidal modulation of another oscillator's frequency and amplitude to create vibrato and tremolo effects. A singer or flutist will generally use a combination of vibrato and tremolo -- modulation of both frequency and amplitude of the tone -- for expressive effect at a rate somewhere between 5 to 8 Hz. Electronically and digitally, of course, we can modulate a tone at any rate, from extremely slow (such as 1/20 Hz) to audio rates (such as 2000 Hz). So we can use these different rates for sinusoidal control functions at the phrase level, the note level, or the microsonic timbral level.

This program demonstrates the use of the sine function for low-frequency modulation of a tone. One sine wave is the carrier oscillator that we actually hear, and the other three sine oscillators are the modulators.

When MSP is turned on, that fact is reported by the adstatus object, and that report is used to set the phase offset of the cycle~ objects to sine phase. This ensures that they all start with the correct, identical phase offset. The frequency is modulated + and - 25 Hz around a center frequency of 440 Hz. That's + and - about a semitone, so it fluctuates between approximately Bb and Ab around A 440. The amplitude is modulated + and - 0.4 around a center amplitude of 0.5. That's a total range from 0.9 to 0.1, which is a factor of 9, which is about 19 dB. Both of those ranges are larger than most singers or instrumentalists would ordinarily use for expressive vibrato, but they're made deliberately large here so they'll be obvious. The tremolo (amplitude modulation) rate is 6 Hz, which is about normal for idiomatic instrumental vibrato. However, the vibrato (frequency modulation) rate is only 1 Hz, which means it glides fairly slowly -- slowly enough that we can track its pitch -- and we tend to hear its extremes, Bb and Ab, as the main pitches. Interestingly, if you adjust the vibrato rate to be 6 Hz like the tremolo, the vibrato will be so fast that -- at this still fairly narrow pitch interval -- we tend to hear its center frequency, A 440, as the main pitch.

A third, very slow modulating sinusoid is used to shape the overall amplitude over a longer period of time. Its frequency is 1/8 Hz, which means that it completes one full cycle every 8 seconds. So, every 4 seconds it goes from 0 to peak amplitude (+ or -) and back to 0. Therefore, we hear a complete crescendo-diminuendo every 4 seconds. Even though the waveform is inverted when this modulator is negative, we don't perceive that fact when we listen to the tone in isolation, so amplification by a negative factor (the second half of the cycle of the modulating sinusoid) sounds the same to us as amplification by a positive factor (the first half of the cycle).

You can experiment with some different rates of modulation, including very slow and very fast (even audio rate) speeds.

Sunday, June 14, 2009

Classic waveforms as control functions

When using a computer to compose and play music, the distinction between music composition and sound synthesis becomes blurred. Since 1950 or so, composers' ways of conceiving musical sound and musical structure have been influenced by technological developments in electronic and computer sound synthesis and processing. This chapter and the next few chapters will provide some simple illustrations of how the electronic music technique of "modulation" -- using a waveform as a control function for another audio signal -- bears resemblance to some more traditional compositional techniques and structures. For simplicity and clarity, we'll start by using only classic electronic music waveforms: sine wave, rectangle wave, triangle wave, and sawtooth wave. We'll first show some sonic/musical effects of using those waveforms as modulators, then in subsequent chapters we'll increase the complexity by modulating the modulators, and then finally we'll make explicit the relationship between those classic sounds of electronic music and algorithmic composition with notes of the twelve-tone equal-tempered scale.

As shown in other chapters, a straight line function has many uses. When normalized as a ramp from 0 to 1, it's easy to enlarge it or contract it (with multiplication) and move it to a desired range (with addition) and even change its direction (by multiplying it by -1 or subtracting it from 1). The phasor~ object in Max provides a signal that ramps repeatedly from 0 to 1 at whatever frequency you specify.

This repeating ramp effectively creates a "sawtooth" function. By scaling it and offsetting it -- for example, by multiplying it by 220 and adding 220 to it -- we get a repeating ramp from 220 to 440 that can act as a control function for the frequency input of a carrier oscillator.

This example program uses a sawtooth function to make repeating frequency glissandi from 220 Hz to 440 Hz 2 times per second, using a phasor~.
It uses the same phasor~ object to generate a simple amplitude envelope that goes immediately to 1 and ramps down to 0 at the same time as each glissando occurs.

The carrier oscillator (the oscillator that generates the signal that we actually listen to) is also a sawtooth waveform. A sawtooth waveform produces a harmonic spectrum, with the amplitude of each harmonic inversely proportional to the harmonic number. Thus, if the waveform has amplitude A, the fundamental (first harmonic) has amplitude A, the second harmonic has amplitude A/ 2, the third harmonic has amplitude A/3, etc. This means that when the fundamental frequency is 440 Hz -- and when the audio sampling rate is 44,100 Hz -- harmonics above the 50th will be above the Nyquist frequency, yet will have an amplitude that is, in many cases, still audible. (The 51st harmonic will have an amplitude about 34 dB less than the fundamental.) Those upper partials will cause aliasing, so when a sawtooth waveform is desired for a carrier oscillator, an ideal sawtooth wave such as that generated by phasor~ is not always the best choice. In Max, there is an object called saw~ that generates a wave with a spectrum very similar to a sawtooth, but that only produces audible harmonics up to the Nyquist frequency. So for the carrier oscillator we use the band-limited sawtooth sound of saw~ rather than the ideal sawtooth wave of phasor~.

So in this simple example, we use a sawtooth waveform as the carrier signal, we modulate its frequency with a low-frequency sawtooth wave, and we use an inverted sawtooth wave at the same rate to generate amplitude envelopes that start loud and fade to 0. The result is repeating notes that glide up from 220 Hz to 440 Hz as the amplitude ramps from 1 to 0.

N.B. Because the glissandi are linear in frequency, they will actually be perceived as a logarithmic curve in terms of pitch.