How to make pop-free, crisp two-beep ringtone?

Question

I hope this ultra-noob question is not entirely out of place here.

I'm trying to make a super simple ringtone: 0.5s silence + 1s "pure tone" + 0.5s silence + 1s "pure tone" + 0.5s silence. By "pure tone" I mean something like a sinusoidal or square wave with a fixed frequency (let's say 350Hz for concreteness). This profile results in a very standard beep-beep ringtone.

With Audacity, it's relatively simple to generate this profile (with Generate > Silence... and Generate > Tone...), and one even has a choice of wave shapes (sinusoidal, square, etc.).

If I pick the sinusoidal option when constructing the profile above, and export it as a WAV file, when I play the ringtone in my phone, I can hear a slight "pop" at the beginning of each beep.

I've tried to get rid of those pops, with little success. I figure that the pops are an artifact of the discontinuity of the wave at the beginning of each beep (though, curiously, I don't hear a pop at the end of each beep). So I tried fading the sound in and out, which does get rid of the pop, but, as one would expect, makes the beeps sound less crisp.

I've heard many ringtones of this sort that are crisp, pleasant-sounding, and pop-free, so there must be a way to get rid of these pops. Any suggestions would be appreciated!

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P.S. If I pick the square wave + no aliasing option (which, to my ear sounds a little "cleaner" than the option with aliasing, for some reason), the resulting ringtone does not have the "pop" at the beginning of each beep, but it sounds very harsh and loud. I tried to adjust the loudness of the beep by lowering the amplitude of the wave generated by Audacity, and in fact, when Audacity plays this reduced-amplitude wave it indeed sounds quieter, but when it is saved as a ringtone in my phone, the phone plays both versions of the square-wave ringtone equally loudly. I figure that the phone probably normalizes the ringtone's loudness somehow. What's surprising to me is that the ringtone made with the sinusoidal wave (the one with the slight "pops" at the beginning of each beep) sounds considerably less loud than the ringtones made with the square wave. In any case, even if I could find a way to reduce the volume of the square-wave ringtone that the phone did not "normalize away", I still would prefer a pop-free version of the sinusoidal-wave ringtone.

Answer 1

• The "click" is caused by the discontinuity in the waveform and its derivatives - even if the waveform stops at a zero-crossing you might still hear a pop if there's a discontinuity in the higher order derivatives! Its loudness depends on the amplitude at the discontinuity - and it is thus influenced by the length/frequency/phase of the sinusoidal tone. Note also that if the tone is contained within a clip in an audio editor, a short fade-in/fade-out might be added automatically by it, attenuating the problem. I know a few audio editors that dynamically apply a short fade (say 32 samples) at the boundary of each audio clip during mixing during playback.

• You hear the click clearly with the sine wave because the tone has very few harmonics (ideally: none). You might not hear it with the other waveforms (square) because they already have high frequency content that will cover part of it (frequency masking phenomenon). For example, you won't hear a click when a burst of white noise stops.

• The solution is to apply a short fade-in / fade-out. 1ms is a good start. Above 5ms it'll start feeling "slow". There is a trade-off to find between "clicky and snappy" and "slow" - this is a problem even well-established synth manufacturers have to educate the public about. With a sine-wave, an envelope curve like a raised cosine function ensures that the resulting signal has continuous derivatives throughout the fade.

• Aliasing is the bad thing to get rid off, so it is normal that the "no-aliasing" option produces a better quality output.

• A sine wave with an amplitude of 1V has twice as less energy as a square wave with the same amplitude. Assuming that the phone normalizes amplitude, it is normal that with a given dynamic range/amplitude budget, the sine wave will sound quieter. Energy/RMS is a rather bad proxy for the perceptual concept of loudness, but it's much more accurate at this task than peak-to-peak amplitude!