Math – Building Your Band

Sound – Feedback

November 4, 2014 By Pete Bulanow

Microphones (and pickups) are sources of feedback

Seemingly one of the great mysteries of running sound is the source and cause of feedback. Perhaps the greatest sin one can commit behind the sound board is allowing feedback. Running sound truly is a thankless job. If everything is going right, no one takes notice. So thank your soundman today!

Since we paid attention at math in school, can we use math to understand feedback? The answer, to all of our relief, is a resounding “yes”. Incredibly, the language of mixing and sound is entirely one of engineering (as is perhaps all of reality), which makes me happy.

Feedback implies the idea of a loop. All the math we need to understand feedback is multiplication and the concept of “unity“, or 1, meaning if you multiply this number by itself, you get this number back again.

But, if you multiply this number by a number smaller than itself, you get a smaller number, and if you keep multiplying, the numbers keep getting smaller. Similarly, if you multiply this number by a number larger than itself you get a larger number, and if you keep multiplying, the numbers keep getting larger. This is the essence of a feedback loop and why it can seem to hang on a knife’s edge – because it does.

To be clear, the loop we are talking about is sound that goes into a microphone, then into a mixing board where it might get EQ’d, then over to an amplifier, and then out via main and/or monitor speaker.

The loop occurs when some of that sound leaks back into the microphone. If the amount of sound that leaks in is greater than 1x what it was originally, by even a tiny little bit like 1.001 x bigger, that sound will start feeding back on itself and continue getting louder. If it’s smaller, like .9999, that sound may ring momentarily, but it will die out.

Knowing what we know then about the nature of sound, that the atoms of sound are sine waves, this feedback could occur at any frequency that our sound system is capable of making, which is another reason we cut and try not to boost gain at a specific frequency using EQ.

Furthermore, the acoustics of the room will come into play as every room will have a bunch of resonant frequencies (just like a coke bottle or flute) that will be more prone to build gain. And even the angle of the microphone with respect to the speakers will have a role, as some mics reject on the side purposefully for this reason.

Positive feedback like we discussed above is ultimately unstable and applied socially can be unhealthy. Positive feedback can make a diva or a spoiled child. Negative feedback is required for stability.

Sound: Quiz

October 28, 2014 By Pete Bulanow

Unclipped sine wave compared to a sine wave 1dB higher that is clipped

What happens when a signal clips (runs out of headroom, or hits a digital ceiling, or an amplifier runs out of power)?

Well, when a sound (such as a sine wave) clips, we start to see a corner that looks like a square wave forming. So what is happening to that sound? We know that the sharp corners on a square wave are high frequencies consisting of odd harmonics – which is exactly what happens.

Spectrogram of 40 Hz sine wave 1 dB into hard clipping

So on the one hand, odd harmonics are not atonal, so as a signal starts to clip, the sound still could be pleasing / musical as it’s still related by integer harmonics – at the very least it’s not inharmonic!

But on the other hand, pushing that much power normally found in the low frequencies up into the higher frequencies which need/use less power is a formula for disaster.

THIS is how speakers get blown: when an amplifier runs out of power. As shown above, when (for example) a 40 Hz / low frequency signal meant for the big woofer clips because an underpowered amplifier runs out of power, basically a square wave is formed, converting much of that signal into typical square-wave odd harmonics. These odd harmonics are higher frequency, which get directed at the little tweeter speaker, which then fries.

Contrary to conventional wisdom, too little amplifier blows speakers. You can never have too much amplifier.

And now the term “total harmonic distortion” makes a lot more sense!

Sound 202 – Inharmonic Timbre

October 24, 2014 By Pete Bulanow

Welcome to your second semester of Timbre! I hope you have everything from first semester under your belt! 🙂

Integer Harmonics

Previously, we looked at the harmonic structure of some nice pretty harmonic sounding sounds. That is, sounds that seemed to have a very clear note (tonal center if you will) and a nice even pleasing timbre to them. We did this by looking at platonically ideal waveforms like square waves and sawtooth waves – which are actually common starting points in many synthesizers.

Acoustic instruments are generally pretty harmonic but a little richer sounding. They mostly follow these same integers for their arrangement of harmonics. Although often when I hear instruments from the far east I hear less harmonic, or inharmonic, sounds that sound “clangy” to my ears. I am not at all an expert on these instruments however so I’ll stop there.

White Noise

But I am somewhat of an expert at the piano, which employes stretch tuning, meaning that harmonics are progressively sharper as you go up the piano. This is done to align the fundamentals of higher notes to the slightly sharp harmonics of lower notes. This is also why you will see season stringed musicians tune their instrument to their harmonics.

So inharmonic sound starts on a continuum starting with strech-tuned pianos, extending to clangy sounds, and ending up with atonal sounds and finally random noise. To get that type of sound we start with harmonics that are increasingly not related by whole numbers to the fundamental, extending to atonal sounds such as a snare drum with rattles, through completely random pink or white noise.

Sound 201 – Timbre

October 17, 2014 By Pete Bulanow

Sine, square, triangle, and sawtooth waveforms

Before we dig deeper, let’s remind ourselves of some basics:

Typically we humans hear down to 20 Hertz (Hz) or vibrations per second) and up to 20,000 Hz (also said 20 KHz).

If we were to hear a note at 440 Hz, that note would be the A above middle C, also known as “Concert A” which is the note an orchetra tunes to.

The question we pose is, “How would we be able to tell if a 440 Hz ‘Concert A’ sound came from a violin or a clarinet?” The answer is, we can tell by the harmonics, or the mathematically related sine waves above 440 hz that give each instrument their characteristic sound or timbre.

Let’s understand this better by looking at a mathematically ideal square wave and sawtooth wave. For reference, a square wave sounds somewhat string-like -any early string emulation was built on these square waves. However, a sawtooth wave sounds somewhat reedy, like a clarinet.

Animation of the additive synthesis of a square wave with an increasing number of harmonics

So mathematically, a square wave contains the odd harmonics (1, 3, 5, 7, 9, etc), each one half as quiet as the previous while a sawtooth wave contains all harmonics (1, 2, 3, 4, 5, etc).

What we see as we add harmonics, is that the waveform gets less wobbly, more mathematically precise, and eventually (with the harmonics going out to a theoretical infinity requiring an infinite frequency response) we have a perfectly sharp corner.

Thinking about sound as sine waves lets us make sense of a lot of things which we will talk about soon.

Sound 101 – Sine Waves

October 16, 2014 By Pete Bulanow

Transforming Sound from the Time Domain to the Frequency Domain

Probably the most foundational thing any musician or sound engineer could take the time to understand is sound. And probably the most important way one could do that is to understand Fouriers theorem.

If you ever said to your math teacher “how am I ever going to use this in the real world,” you are about to eat those words. I hope they’re delicious.

Fouriers theorem says that any waveform (i.e. timbre) can be made by adding sine waves at various multiples (i.e. harmonics) of the fundamental (i.e. note).

More mathily – Fouriers Theorem transforms sound from the time domain (the way we see and experience it) and rotates it 90 degrees to look at it sideways in the frequency domain (which actually provides insight).

The six arrows represent the first six terms of the Fourier series of a square wave (they are sine waves!). The two circles at the bottom represent the exact square wave (blue) and its Fourier-series approximation (purple).

Put another way, Fouriers Theorem shows us that sine waves are the atoms of sound.

Isn’t that cool? It doesn’t get much more awesome than that people. All of a sudden, sound is much less mysterious.

And the more you think about it, the more situations it helps you make sense of, the more situations you see people who don’t understand this get things wrong, the less mysterious sound becomes.

(So that’s why they test our hearing with “pure” sign waves, because they’re checking our hearing at a given frequency and don’t want our ability to hear overtones to affect the results.)