电子音乐原理和技术电子书

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"电子音乐理论和技术" 本电子书《The Theory and Technique of Electronic Music》由 Miller Puckette 撰写，是一本全面介绍电子音乐原理的电子书。下面是根据书中的内容生成的相关知识点：一、正弦波、幅值和频率 * Measures of Amplitude：幅值的度量单位有 dB、Watt、 Volt 等。 * Units of Amplitude：幅值的单位有 dB、Watt、 Volt 等。 * Controlling Amplitude：可以通过调整音量、增益等方式来控制幅值。 * Frequency：频率是指每秒钟内振荡的次数，单位是 Hz。 * Synthesizing a Sinusoid：可以通过加法合成或频率变换来合成正弦波。 * Superposing Signals：可以将多个信号叠加以生成新的信号。 * Periodic Signals：周期信号是一种重复出现的信号，例如正弦波。二、波表和采样器 * Wavetable Oscillator：波表振荡器是一种使用波表来生成音频信号的方法。 * Sampling：采样是指将连续信号转换为离散信号的过程。 * Enveloping Samplers：包络采样是指对采样信号进行包络处理以生成新的音频信号。 * Timbre Stretching：音色拉伸是一种将音色变化的技术。 * Interpolation：插值是一种用于生成新的音频信号的技术。三、Pd 软件介绍 * Quick Introduction to Pd：Pd 是一种基于图形化编程的音频处理软件。 * How to find and run the examples：如何在 Pd 中找到和运行示例程序。四、练习和更多内容 * Exercises：本书提供了一些练习题，以帮助读者更好地理解电子音乐理论和技术。 * More Additive Synthesis：加性合成是一种生成音频信号的技术。 * Conversion between Frequency and Pitch：频率和音高之间的转换。本电子书《The Theory and Technique of Electronic Music》为读者提供了电子音乐理论和技术的系统介绍，涵盖了正弦波、幅值和频率、波表和采样器、Pd 软件等多方面的知识点，是电子音乐爱好者和专业人士不可错过的良好资源。

1.1. MEASURES OF AMPLITUDE 3

the sample number n by Rt = n, or t = n/R. A sinusoidal signal with angular

frequency ω has a real-time frequency equal to

f =

ωR

2π

in Hertz (i.e., cycles per second), because a cycle is 2π radians and a second is

R samples.

A real-world audio signal’s amplitude might be expressed as a time-varying

voltage or air pressure, but the samples of a digital audio signal are unitless

numbers. We’ll casually assume here that there is ample numerical accuracy so

that we can ignore round-oﬀ errors, and that the numerical format is unlimited

in range, so that samples may take any value we wish. However, most digital

audio hardware works only over a ﬁxed range of input and output values, most

often between -1 and 1. Modern digital audio processing software usually uses

a ﬂoating-point representation for signals. This allows us to use whatever units

are most convenient for any given task, as long as the ﬁnal audio output is

within the hardware’s range [Mat69, pp. 4-10].

1.1 Measures of Amplitude

The most fundamental property of a digital audio signal is its amplitude. Unfor-

tunately, a signal’s amplitude has no one canonical deﬁnition. Strictly speaking,

all the samples in a digital audio signal are themselves amplitudes, and we also

spoke of the amplitude a of the sinusoid as a whole. It is useful to have measures

of amplitude for digital audio signals in general. Amplitude is best thought of

as applying to a window, a ﬁxed range of samples of the signal. For instance,

the window starting at sample M of length N of an audio signal x[n] consists

of the samples,

x[M], x[M + 1], . . . , x[M + N − 1]

The two most frequently used measures of amplitude are the peak amplitude,

which is simply the greatest sample (in absolute value) over the window:

peak

{x[n]} = max |x[n]|, n = M, . . . , M + N − 1

and the root mean square (RMS) amplitude:

RMS

{x[n]} =

P {x[n]}

where P {x[n]} is the mean power, deﬁned as:

P {x[n]} =

|x[M]|

+ ··· + |x[M + N − 1]|

(In this last formula, the absolute value signs aren’t necessary at the moment

since we’re working on real-valued signals, but they will become important later

1.4. FREQUENCY 7

(which we assume doesn’t change much over the N samples in the window).

In the more general case where both x[n] and y[n] are allowed to take negative

and positive values and/or to change quickly, the eﬀect of multiplying them can’t

be described as simply changing the amplitude of one of them; this is considered

later in Chapter 5.

1.4 Frequency

Frequencies, like amplitudes, are often measured on a logarithmic scale, in order

to emphasize proportions between them, which usually provide a better descrip-

tion of the relationship between frequencies than do diﬀerences between them.

The frequency ratio between two musical tones determines the musical interval

between them.

The Western musical scale divides the octave (the musical interval associated

with a ratio of 2:1) into twelve equal sub-intervals, each of which therefore

corresponds to a ratio of 2

1/12

. For historical reasons this sub-interval is called

a half-step. A convenient logarithmic scale for pitch is simply to count the

number of half-steps from a reference pitch—allowing fractions to permit us

to specify pitches which don’t fall on a note of the Western scale. The most

commonly used logarithmic pitch scale is “MIDI pitch”, in which the pitch 69

is assigned to a frequency of 440 cycles per second—the A above middle C. To

convert between a MIDI pitch m and a frequency in cycles per second f, apply

the Pitch/Frequency Conversion formulas:

m = 69 + 12 · log

(f/440)

f = 440 · 2

(m−69)/12

Middle C, corresponding to MIDI pitch m = 60, comes to f = 261.626 cycles

per second.

MIDI itself is an old hardware protocol which has unfortunately insinuated

itself into a great deal of software design. In hardware, MIDI allows only integer

pitches between 0 and 127. However, the underlying scale is well deﬁned for

any “MIDI” number, even negative ones; for example a “MIDI pitch” of -4 is

a decent rate of vibrato. The pitch scale cannot, however, describe frequencies

less than or equal to zero cycles per second. (For a clear description of MIDI,

its capabilities and limitations, see [Bal03, ch.6-8]).

A half-step comes to a ratio of about 1.059 to 1, or about a six percent

increase in frequency. Half-steps are further divided into cents, each cent being

one hundredth of a half-step. As a rule of thumb, it might take about three

cents to make a discernible change in the pitch of a musical tone. At middle C

this comes to a diﬀerence of about 1/2 cycle per second. A graph of frequency

as a function of MIDI pitch, over a two-octave range, is shown in Figure 1.4.

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