Do all Harmonics of a periodic sound start from a 0 base

I've been thinking about additive synthesis and I'm wondering if all sounds begin at (theoretically) phase 0?

• What do you mean by begin? Jul 31 '15 at 23:41
• What do you mean by "sounds"? Do you mean individual audio tones that have different frequencies? Aug 1 '15 at 2:20
• by 'begin' I mean at the start of the sound, for example, assume a bird could chirp a square wave, when the sound starts would the harmonics start from a relative base of zero? Aug 1 '15 at 10:03
• by sounds, I mean any periodic and harmonic sound. Aug 1 '15 at 10:03
• @RichardLyons enjoying my copy of understanding digital signal processing. Aug 1 '15 at 20:52

It's common to not start all harmonically related sinusoids with a phase of 0 at some initial time point. Varying the phase relationships helps reduce the peak-to-average power ratio, which allows a higher average volume without clipping or otherwise exceeding some peak limit.

• but doesn't this alter the wave shape? or do you mean maintaining a the fourier series ratios but starting at a different overall phase? Aug 1 '15 at 10:00

I'm not knowledgeable enough to comment on "all sounds" found in nature. But here's an interesting experiment. Use software to generate an audio square wave containing a fundamental freq and the next higher five odd harmonics (each with appropriate peak amplitudes). Play that signal through a speaker. Then have each of the five odd harmonics start at random initial phases. The resultant signal will have a drastically different wave shape, BUT the sound of the random-phase signal will sound VERY similar to your original square wave! It's an experiment to show that the human ear/brain combination is very tolerant of phase distortion.

i would say that is is not common that all harmonics start with a phase of zero w.r.t. any single, specific time (like MIDI NoteOn). BTW, to add to Rick Lyons answer, here is a MATLAB program that sorta demonstrates what you can and cannot hear regarding phase alignment of harmonics. give it a try. see what you can hear (or not).

%
%   square_phase.m
%
%   a test to see if we can really hear phase changes
%   in the harmonics of a Nyquist limited square wave.
%
%   (c) 2004 rbj@audioimagination.com
%

if ~exist('Fs')
Fs = 44100                      % sample rate, Hz
end

if ~exist('f0')
f0 = 110.25                     % fundamental freq, Hz
end

if ~exist('tone_duration')
tone_duration = 2.0             % seconds
end

if ~exist('change_rate')
change_rate = 1.0               % Hz
end

if ~exist('max_harmonic')
max_harmonic = floor((Fs/2)/f0) - 1
end

if ~exist('amplitude_factor')
amplitude_factor = 0.25         % this just keeps things from clipping
end

if ~exist('outFile')
outFile = 'square_phase.wav'
end

% make sure we don't uber-Nyquist anything
max_harmonic = min(max_harmonic, floor((Fs/2)/f0)-1);

t = linspace((-1/4)/f0, tone_duration-(1/4)/f0, Fs*tone_duration+1);

detune = change_rate;

n = 3;                               % continue with 3rd harmonic
while (n <= max_harmonic)
if ((n-1) == 4*floor((n-1)/4))   % lessee if it's an "even" or "odd" term
x = x + (1/n)*cos(2*pi*n*f0*t);
else
x = x - (1/n)*cos(2*pi*(n*f0+detune)*t);
detune = -detune;       % comment this line in an see some
end                          % funky intermediate waveforms
n = n + 2;                   % continue with next odd harmonic
end

x = amplitude_factor*x;

% x = sin((pi/2)*x);               % toss in a little soft clipping

plot(t, x);                      % see
sound(x, Fs);                    % hear
wavwrite(x, Fs, outFile);        % remember
• Thanks @Robert Bristow-Johnson. While I don't have Matlab. I've performed a similar experiment using C++. I can't hear the difference. I guess this also means that if I choose to start a periodic sound from a zero phase base in order to make it look like an expected waveform (square or saw) then it wouldn't really matter either. Thanks for the responses. Aug 3 '15 at 21:52