Ok, as I answer the question, "What is the loudest part of a sin wave?", let me begin by dispelling the myth of RMS as being the way to measure audio loudness. RMS stands for Root Mean Squared. It is computed by the following math equation.
RMS = sqrt[(s1 x s1 + s2 x s2 + s3 x s3, ..., + sN x sN) / N]
In other words, add up the square of each sample, average them by dividing by the number of samples, and then take the square root of the result. Note that RMS can be transformed to db by taking the logarithm of RMS (base 10) and then multiplying that by 10.
In this math equation, sqrt is the square root, s1 is the first sample in a set of N samples, s2 is the second one, s3 is the third one, ..., and sN is the last sample in the set. RMS is always computed over a collection of N samples where N could be directly specified or determined by a time window (i.e., N = sample rate x time window, so a 48,000 sample rate x 0.01 second time window would mean computing RMS with 480 samples over 10 milliseconds).
Now imagine we have a wav file where every sample is maxed out (not the sin wav example). It would be like a long, positive, square wave that never comes down. Remember what actually happens at the speaker when we play the wav file. Positive samples push the speaker out and negative samples pull the speaker in. The speaker's movement essentially mimics what's in the wav file. In our maxed-out sample case, RMS would be maximized to its largest value possible, the speaker would be pushed out (ultimately controlled by the volume at the amplifier), but the speaker is not moving, which means no sound would be produced. This should alarm anyone who has thought of RMS as being a good measure of loudness. Clearly, this example describes a case where RMS is maximized, yet no sound would be produced...
RMS cannot possibly be the correct measure of audio loudness!
So why does everybody in the industry use RMS as the standard measure of loudness? It took me a while to figure this out. I had to go back to basic physics involving electromagnetic circuits. It turns out that RMS describes average power in electrical circuits involving voltage, currents, and resistors. Think of it this way. Your stereo system would be doing a lot of work to push and hold a speaker out at its maximum position even though no sound is being produced. It would be like holding your arm out motionless and parallel to the ground. Your body would get very tired (just like your stereo would be doing a lot of work holding the speaker out) even though your arm is motionless (i.e., doing no kinetic work).
As it turns out, RMS measures electrical power (think powering a radio station), which can be a useful measure. But, it does not measure audio loudness. Does anybody care how much electricity their compositions use when mixing (answer: no), or is audio loudness the real focus (answer: yes)...
So, with this concept in mind (i.e., that the speaker's out/in position, amplified by your amplifier, mimics the wav file), go back to my original question, "What is the loudest part of a sin wave?" If you haven't thought of audio in the way I just laid out, the answer might completely stun you...
Again, I hope to hear your responses. I'll provide the actual loudness equation in my next post. And guess what... It explains why compression and limiting work so well...
RMS = sqrt[(s1 x s1 + s2 x s2 + s3 x s3, ..., + sN x sN) / N]
In other words, add up the square of each sample, average them by dividing by the number of samples, and then take the square root of the result. Note that RMS can be transformed to db by taking the logarithm of RMS (base 10) and then multiplying that by 10.
In this math equation, sqrt is the square root, s1 is the first sample in a set of N samples, s2 is the second one, s3 is the third one, ..., and sN is the last sample in the set. RMS is always computed over a collection of N samples where N could be directly specified or determined by a time window (i.e., N = sample rate x time window, so a 48,000 sample rate x 0.01 second time window would mean computing RMS with 480 samples over 10 milliseconds).
Now imagine we have a wav file where every sample is maxed out (not the sin wav example). It would be like a long, positive, square wave that never comes down. Remember what actually happens at the speaker when we play the wav file. Positive samples push the speaker out and negative samples pull the speaker in. The speaker's movement essentially mimics what's in the wav file. In our maxed-out sample case, RMS would be maximized to its largest value possible, the speaker would be pushed out (ultimately controlled by the volume at the amplifier), but the speaker is not moving, which means no sound would be produced. This should alarm anyone who has thought of RMS as being a good measure of loudness. Clearly, this example describes a case where RMS is maximized, yet no sound would be produced...
RMS cannot possibly be the correct measure of audio loudness!
So why does everybody in the industry use RMS as the standard measure of loudness? It took me a while to figure this out. I had to go back to basic physics involving electromagnetic circuits. It turns out that RMS describes average power in electrical circuits involving voltage, currents, and resistors. Think of it this way. Your stereo system would be doing a lot of work to push and hold a speaker out at its maximum position even though no sound is being produced. It would be like holding your arm out motionless and parallel to the ground. Your body would get very tired (just like your stereo would be doing a lot of work holding the speaker out) even though your arm is motionless (i.e., doing no kinetic work).
As it turns out, RMS measures electrical power (think powering a radio station), which can be a useful measure. But, it does not measure audio loudness. Does anybody care how much electricity their compositions use when mixing (answer: no), or is audio loudness the real focus (answer: yes)...
So, with this concept in mind (i.e., that the speaker's out/in position, amplified by your amplifier, mimics the wav file), go back to my original question, "What is the loudest part of a sin wave?" If you haven't thought of audio in the way I just laid out, the answer might completely stun you...
Again, I hope to hear your responses. I'll provide the actual loudness equation in my next post. And guess what... It explains why compression and limiting work so well...