This post is going to be a pretty short one, so here it is 😛

The Beating Effect

The beating effect occurs when you play two similar frequencies of sound at the same time.

Because playing two sounds at once means adding them together, and due to the fact that sound waves are made up of positive and negative values (aka positive and negative pressures), the sounds playing at different frequencies will sometimes add peaks and troughs together to get louder, and other times the peak of one wave will add to the valley of another wave and the result will be quieter. This is know as constructive and destructive interference respectively.

The end result is that the sound will have a pulsing quality to it, like a tremolo effect. If one sound is played at frequency F1 and the other sound is played at frequency F2, the pulsing sound will occur at 2*(F2-F1) times a second.

Here’s a demo of this in action where the first sound is 200hz, the second sound is 205hz, and the result of them played together has a 10hz tremolo effect!

monobeat.wav

Binaural Beat

The beating effect happens when sound waves physically mix together.

Believe it or not though, there is a part of your brain where it mixes (adds) the sounds from each ear together as well.

That means that if you play similar frequencies to different ears, they will mix INSIDE YOUR BRAIN, and you will perceive a different kind of beating effect.

Below is a demo of that. You might need a pair of headphones to get the full effect.

stereobeat.wav

If you think this is pretty neat, you might want to google “psycho-acoustics” for more stuff like this (:

Source Code

Here’s the C++ code that generated these sound files.

#include <stdio.h>
#include <memory.h>
#include <inttypes.h>
#include <vector>

// constants
const float c_pi = 3.14159265359f;
const float c_twoPi = 2.0f * c_pi;

// typedefs
typedef uint16_t    uint16;
typedef uint32_t    uint32;
typedef int16_t     int16;
typedef int32_t     int32;

//this struct is the minimal required header data for a wav file
struct SMinimalWaveFileHeader
{
    //the main chunk
    unsigned char m_chunkID[4];
    uint32		  m_chunkSize;
    unsigned char m_format[4];

    //sub chunk 1 "fmt "
    unsigned char m_subChunk1ID[4];
    uint32		  m_subChunk1Size;
    uint16		  m_audioFormat;
    uint16		  m_numChannels;
    uint32		  m_sampleRate;
    uint32		  m_byteRate;
    uint16		  m_blockAlign;
    uint16		  m_bitsPerSample;

    //sub chunk 2 "data"
    unsigned char m_subChunk2ID[4];
    uint32		  m_subChunk2Size;

    //then comes the data!
};

//this writes
template <typename T>
bool WriteWaveFile(const char *fileName, std::vector<T> data, int16 numChannels, int32 sampleRate)
{
    int32 dataSize = data.size() * sizeof(T);
    int32 bitsPerSample = sizeof(T) * 8;

    //open the file if we can
    FILE *File = nullptr;
    fopen_s(&File, fileName, "w+b");
    if (!File)
        return false;

    SMinimalWaveFileHeader waveHeader;

    //fill out the main chunk
    memcpy(waveHeader.m_chunkID, "RIFF", 4);
    waveHeader.m_chunkSize = dataSize + 36;
    memcpy(waveHeader.m_format, "WAVE", 4);

    //fill out sub chunk 1 "fmt "
    memcpy(waveHeader.m_subChunk1ID, "fmt ", 4);
    waveHeader.m_subChunk1Size = 16;
    waveHeader.m_audioFormat = 1;
    waveHeader.m_numChannels = numChannels;
    waveHeader.m_sampleRate = sampleRate;
    waveHeader.m_byteRate = sampleRate * numChannels * bitsPerSample / 8;
    waveHeader.m_blockAlign = numChannels * bitsPerSample / 8;
    waveHeader.m_bitsPerSample = bitsPerSample;

    //fill out sub chunk 2 "data"
    memcpy(waveHeader.m_subChunk2ID, "data", 4);
    waveHeader.m_subChunk2Size = dataSize;

    //write the header
    fwrite(&waveHeader, sizeof(SMinimalWaveFileHeader), 1, File);

    //write the wave data itself
    fwrite(&data[0], dataSize, 1, File);

    //close the file and return success
    fclose(File);
    return true;
}

template <typename T>
void ConvertFloatSamples (const std::vector<float>& in, std::vector<T>& out)
{
    // make our out samples the right size
    out.resize(in.size());

    // convert in format to out format !
    for (size_t i = 0, c = in.size(); i < c; ++i)
    {
        float v = in[i];
        if (v < 0.0f)
            v *= -float(std::numeric_limits<T>::lowest());
        else
            v *= float(std::numeric_limits<T>::max());
        out[i] = T(v);
    }
}

void GenerateMonoBeatingSamples (std::vector<float>& samples, int sampleRate)
{
    int sectionLength = samples.size() / 3;
    int envelopeLen = sampleRate / 20;

    for (int index = 0, numSamples = samples.size(); index < numSamples; ++index)
    {
        samples[index] = 0.0f;
        int section = index / sectionLength;
        int sectionOffset = index % sectionLength;

        // apply an envelope at front and back  of each section keep it from popping between sounds
        float envelope = 1.0f;
        if (sectionOffset < envelopeLen)
            envelope = float(sectionOffset) / float(envelopeLen);
        else if (sectionOffset > sectionLength - envelopeLen)
            envelope = (float(sectionLength) - float(sectionOffset)) / float(envelopeLen);

        // first sound
        if (section == 0 || section == 2)
            samples[index] += sin(float(index) * c_twoPi * 200.0f / float(sampleRate)) * envelope;

        // second sound
        if (section == 1 || section == 2)
            samples[index] += sin(float(index) * c_twoPi * 205.0f / float(sampleRate)) * envelope;

        // scale it to prevent clipping
        if (section == 2)
            samples[index] *= 0.5f;
        samples[index] *= 0.95f;
    }
}

void GenerateStereoBeatingSamples (std::vector<float>& samples, int sampleRate)
{
    int sectionLength = (samples.size() / 2) / 3;
    int envelopeLen = sampleRate / 20;

    for (int index = 0, numSamples = samples.size() / 2; index < numSamples; ++index)
    {
        samples[index * 2] = 0.0f;
        samples[index * 2 + 1] = 0.0f;

        int section = index / sectionLength;
        int sectionOffset = index % sectionLength;

        // apply an envelope at front and back  of each section keep it from popping between sounds
        float envelope = 1.0f;
        if (sectionOffset < envelopeLen)
            envelope = float(sectionOffset) / float(envelopeLen);
        else if (sectionOffset > sectionLength - envelopeLen)
            envelope = (float(sectionLength) - float(sectionOffset)) / float(envelopeLen);
        envelope *= 0.95f;

        // first sound
        if (section == 0 || section == 2)
            samples[index * 2] += sin(float(index) * c_twoPi * 200.0f / float(sampleRate)) * envelope;

        // second sound
        if (section == 1 || section == 2)
            samples[index * 2 + 1] += sin(float(index) * c_twoPi * 205.0f / float(sampleRate)) * envelope;
    }
}

//the entry point of our application
int main(int argc, char **argv)
{
    // generate the mono beating effect
    {
        // sound format parameters
        const int c_sampleRate = 44100;
        const int c_numSeconds = 4;
        const int c_numChannels = 1;
        const int c_numSamples = c_sampleRate * c_numChannels * c_numSeconds;

        // make space for our samples
        std::vector<float> samples;
        samples.resize(c_numSamples);

        // generate samples
        GenerateMonoBeatingSamples(samples, c_sampleRate);

        // convert from float to the final format
        std::vector<int32> samplesInt;
        ConvertFloatSamples(samples, samplesInt);

        // write our samples to a wave file
        WriteWaveFile("monobeat.wav", samplesInt, c_numChannels, c_sampleRate);
    }

    // generate the stereo beating effect (binaural beat)
    {
        // sound format parameters
        const int c_sampleRate = 44100;
        const int c_numSeconds = 4;
        const int c_numChannels = 2;
        const int c_numSamples = c_sampleRate * c_numChannels * c_numSeconds;

        // make space for our samples
        std::vector<float> samples;
        samples.resize(c_numSamples);

        // generate samples
        GenerateStereoBeatingSamples(samples, c_sampleRate);

        // convert from float to the final format
        std::vector<int32> samplesInt;
        ConvertFloatSamples(samples, samplesInt);

        // write our samples to a wave file
        WriteWaveFile("stereobeat.wav", samplesInt, c_numChannels, c_sampleRate);
    }
}

Links

For a more mathematical explanation of these, check out wikipedia (:
Wikipedia: Beat (acoustics)

Wikipedia: Binaural beats

If you are looking to synthesize the sound of a plucked string, there is an amazingly simple algorithm for doing so called the Karplus-Strong Algorithm.

Give it a listen: KarplusStrong.wav
Here it is with flange and reverb effects applied: KPFlangeReverb.wav

It works like this:

1. Fill a circular buffer with static (random numbers)
2. Play the contents of the circular buffer over and over
3. Each time you play a sample, replace that sample with the average of itself and the next sample in the buffer. Also multiplying that average by a feedback value (like say, 0.996)

Amazingly, that is all there is to it!

Why Does That Work?!

The reason this works is that it is actually very similar to how a real guitar string pluck works.

When you pluck a guitar string, if you had a perfect pluck at the perfect location with the perfect transfer of energy, you’d get a note that was “perfect”. It wouldn’t be a pure sine wave since strings have harmonics (integer multiple frequencies) beyond their basic tuning, but it would be a pure note.

In reality, that isn’t what happens, so immediately after plucking the string, there is a lot of vibrations in there that “don’t belong” due to the imperfect pluck. Since the string is tuned, it wants to be vibrating a specific way, so over time the vibrations evolve from the imperfect pluck vibrations to the tuning of the guitar string. As you average the samples together, you are removing the higher frequency noise/imperfections. Averaging is a crude low pass filter. This makes it converge to the right frequencies over time.

It’s also important to note that with a real stringed instrument, when you play a note, the high frequencies disappear before the low frequencies. This averaging / low pass filter makes that happen as well and is part of what helps it sound so realistic.

Also while all that is going on, the energy in the string is being diminished as it becomes heat and sound waves and such, so the noise gets quieter over time. When you multiply the values by a feedback value which is less than 1, you are simulating this loss of energy by making the values get smaller over time.

Tuning The Note

This wasn’t intuitive for me at first, but the frequency that the note plays at is determined ENTIRELY by the size of the circular buffer.

If your audio has a sample rate of 44100hz (44100 samples played a second), and you use this algorithm with a buffer size of 200 samples, that means that the note synthesized will be 220.5hz. This is because 44100/200 = 220.5.

Thinking about the math from another direction, we can figure out what our buffer size needs to be for a specific frequency. If our sample rate is 44100hz and we want to play a note at 440hz, that means we need a buffer size of 100.23 samples. This is because 44100/440 = 100.23. Since we can’t have a fractional number of samples, we can just round to 100.

You can actually deal with the fractional buffer size by stepping through the ring buffer in non integer steps and using the fraction to interpolate audio samples, but I’ll leave that as an exercise for you if you want that perfectly tuned note. IMO leaving it slightly off could actually be a good thing. What guitar is ever perfectly in tune, right?! With it being slightly out of tune, it’s more likely to make more realistic sounds and sound interactions when paired with other instruments.

You are probably wondering like I was, why the buffer size affects the frequency of the note. The reason for this is actually pretty simple and intuitive after all.

The reason is because the definition of frequency is just how many times a wave form repeats per second. The wave form could be a sine wave, a square wave, a triangle wave, or it could be something more complex, but frequency is always the number of repetitions per second. If you think about our ring buffer as being a wave form, you can now see that if we have a buffer size of 200 samples, and a sample rate of 44100hz, when we play that buffer continually, it’s going to play back 220.5 times every second, which means it will play with a frequency of 220.5!

Sure, we modify the buffer (and waveform) as we play it, but the modifications are small, so the waveform is similar from play to play.

Some More Details

I’ve found that this algorithm doesn’t work as well with low frequency notes as it does with high frequency notes.

They say you can prime the buffer with a saw tooth wave (or other wave forms) instead of static (noise). While it still “kind of works”, in my experimentation, it didn’t work out that well.

You could try using other low pass filters to see if that affects the quality of the note generated. The simple averaging method works so well, I didn’t explore alternative options very much.

Kmm on hacker news commented that averaging the current sample with the last and next, instead of just the next had the benefit that the wave form didn’t move forward half a step each play through and that there is an audible difference between the techniques. I gave it a try and sure enough, there is an audible difference, the sound is less harsh on the ears. I believe this is so because averaging 3 samples instead of 2 is a stronger low pass filter, so gets rid of higher frequencies faster.

Example Code

Here is the C++ code that generated the sample at the top of the post. Now that you can generate plucked string sounds, you can add some distortion, flange, reverb, etc and make some sweet (synthesized) metal without having to learn to play guitar and build up finger calluses 😛

#include <stdio.h>
#include <memory.h>
#include <inttypes.h>
#include <vector>

// constants
const float c_pi = 3.14159265359f;
const float c_twoPi = 2.0f * c_pi;

// typedefs
typedef uint16_t    uint16;
typedef uint32_t    uint32;
typedef int16_t     int16;
typedef int32_t     int32;

//this struct is the minimal required header data for a wav file
struct SMinimalWaveFileHeader
{
    //the main chunk
    unsigned char m_chunkID[4];
    uint32		  m_chunkSize;
    unsigned char m_format[4];

    //sub chunk 1 "fmt "
    unsigned char m_subChunk1ID[4];
    uint32		  m_subChunk1Size;
    uint16		  m_audioFormat;
    uint16		  m_numChannels;
    uint32		  m_sampleRate;
    uint32		  m_byteRate;
    uint16		  m_blockAlign;
    uint16		  m_bitsPerSample;

    //sub chunk 2 "data"
    unsigned char m_subChunk2ID[4];
    uint32		  m_subChunk2Size;

    //then comes the data!
};

//this writes
template <typename T>
bool WriteWaveFile(const char *fileName, std::vector<T> data, int16 numChannels, int32 sampleRate)
{
    int32 dataSize = data.size() * sizeof(T);
    int32 bitsPerSample = sizeof(T) * 8;

    //open the file if we can
    FILE *File = nullptr;
    fopen_s(&File, fileName, "w+b");
    if (!File)
        return false;

    SMinimalWaveFileHeader waveHeader;

    //fill out the main chunk
    memcpy(waveHeader.m_chunkID, "RIFF", 4);
    waveHeader.m_chunkSize = dataSize + 36;
    memcpy(waveHeader.m_format, "WAVE", 4);

    //fill out sub chunk 1 "fmt "
    memcpy(waveHeader.m_subChunk1ID, "fmt ", 4);
    waveHeader.m_subChunk1Size = 16;
    waveHeader.m_audioFormat = 1;
    waveHeader.m_numChannels = numChannels;
    waveHeader.m_sampleRate = sampleRate;
    waveHeader.m_byteRate = sampleRate * numChannels * bitsPerSample / 8;
    waveHeader.m_blockAlign = numChannels * bitsPerSample / 8;
    waveHeader.m_bitsPerSample = bitsPerSample;

    //fill out sub chunk 2 "data"
    memcpy(waveHeader.m_subChunk2ID, "data", 4);
    waveHeader.m_subChunk2Size = dataSize;

    //write the header
    fwrite(&waveHeader, sizeof(SMinimalWaveFileHeader), 1, File);

    //write the wave data itself
    fwrite(&data[0], dataSize, 1, File);

    //close the file and return success
    fclose(File);
    return true;
}

template <typename T>
void ConvertFloatSamples (const std::vector<float>& in, std::vector<T>& out)
{
    // make our out samples the right size
    out.resize(in.size());

    // convert in format to out format !
    for (size_t i = 0, c = in.size(); i < c; ++i)
    {
        float v = in[i];
        if (v < 0.0f)
            v *= -float(std::numeric_limits<T>::lowest());
        else
            v *= float(std::numeric_limits<T>::max());
        out[i] = T(v);
    }
}
//calculate the frequency of the specified note.
//fractional notes allowed!
float CalcFrequency(float octave, float note)
/*
	Calculate the frequency of any note!
	frequency = 440×(2^(n/12))

	N=0 is A4
	N=1 is A#4
	etc...

	notes go like so...
	0  = A
	1  = A#
	2  = B
	3  = C
	4  = C#
	5  = D
	6  = D#
	7  = E
	8  = F
	9  = F#
	10 = G
	11 = G#
*/
{
    return (float)(440 * pow(2.0, ((double)((octave - 4) * 12 + note)) / 12.0));
}

class CKarplusStrongStringPluck
{
public:
    CKarplusStrongStringPluck (float frequency, float sampleRate, float feedback)
    {
        m_buffer.resize(uint32(float(sampleRate) / frequency));
        for (size_t i = 0, c = m_buffer.size(); i < c; ++i) {
            m_buffer[i] = ((float)rand()) / ((float)RAND_MAX) * 2.0f - 1.0f;  // noise
            //m_buffer[i] = float(i) / float(c); // saw wave
        }
        m_index = 0;
        m_feedback = feedback;
    }

    float GenerateSample ()
    {
        // get our sample to return
        float ret = m_buffer[m_index];

        // low pass filter (average) some samples
        float value = (m_buffer[m_index] + m_buffer[(m_index + 1) % m_buffer.size()]) * 0.5f * m_feedback;
        m_buffer[m_index] = value;

        // move to the next sample
        m_index = (m_index + 1) % m_buffer.size();

        // return the sample from the buffer
        return ret;
    }

private:
    std::vector<float>  m_buffer;
    size_t              m_index;
    float               m_feedback;
};

void GenerateSamples (std::vector<float>& samples, int sampleRate)
{
    std::vector<CKarplusStrongStringPluck> notes;

    enum ESongMode {
        e_twinkleTwinkle,
        e_strum
    };

    int timeBegin = 0;
    ESongMode mode = e_twinkleTwinkle;
    for (int index = 0, numSamples = samples.size(); index < numSamples; ++index)
    {
        switch (mode) {
            case e_twinkleTwinkle: {
                const int c_noteTime = sampleRate / 2;
                int time = index - timeBegin;
                // if we should start a new note
                if (time % c_noteTime == 0) {
                    int note = time / c_noteTime;
                    switch (note) {
                        case 0:
                        case 1: {
                            notes.push_back(CKarplusStrongStringPluck(CalcFrequency(3, 0), float(sampleRate), 0.996f));
                            break;
                        }
                        case 2:
                        case 3: {
                            notes.push_back(CKarplusStrongStringPluck(CalcFrequency(3, 7), float(sampleRate), 0.996f));
                            break;
                        }
                        case 4:
                        case 5: {
                            notes.push_back(CKarplusStrongStringPluck(CalcFrequency(3, 9), float(sampleRate), 0.996f));
                            break;
                        }
                        case 6: {
                            notes.push_back(CKarplusStrongStringPluck(CalcFrequency(3, 7), float(sampleRate), 0.996f));
                            break;
                        }
                        case 7: {
                            mode = e_strum;
                            timeBegin = index+1;
                            break;
                        }
                    }
                }
                break;
            }
            case e_strum: {
                const int c_noteTime = sampleRate / 32;
                int time = index - timeBegin - sampleRate;
                // if we should start a new note
                if (time % c_noteTime == 0) {
                    int note = time / c_noteTime;
                    switch (note) {
                        case 0: notes.push_back(CKarplusStrongStringPluck(55.0f, float(sampleRate), 0.996f)); break;
                        case 1: notes.push_back(CKarplusStrongStringPluck(55.0f + 110.0f, float(sampleRate), 0.996f)); break;
                        case 2: notes.push_back(CKarplusStrongStringPluck(55.0f + 220.0f, float(sampleRate), 0.996f)); break;
                        case 3: notes.push_back(CKarplusStrongStringPluck(55.0f + 330.0f, float(sampleRate), 0.996f)); break;
                        case 4: mode = e_strum; timeBegin = index + 1; break;
                    }
                }
                break;
            }
        }

        // generate and mix our samples from our notes
        samples[index] = 0;
        for (CKarplusStrongStringPluck& note : notes)
            samples[index] += note.GenerateSample();

        // to keep from clipping
        samples[index] *= 0.5f;
    }
}

//the entry point of our application
int main(int argc, char **argv)
{
    // sound format parameters
    const int c_sampleRate = 44100;
    const int c_numSeconds = 9;
    const int c_numChannels = 1;
    const int c_numSamples = c_sampleRate * c_numChannels * c_numSeconds;

    // make space for our samples
    std::vector<float> samples;
    samples.resize(c_numSamples);

    // generate samples
    GenerateSamples(samples, c_sampleRate);

    // convert from float to the final format
    std::vector<int32> samplesInt;
    ConvertFloatSamples(samples, samplesInt);

    // write our samples to a wave file
    WriteWaveFile("out.wav", samplesInt, c_numChannels, c_sampleRate);
}

Links

Hacker News Discussion (This got up to topic #7, woo!)

Wikipedia: Karplus-Strong String Synthesis

Princeton COS 126: Plucking a Guitar String

Shadertoy: Karplus-Strong Variation (Audio) – I tried to make a bufferless Karplus-Strong implementation on shadertoy. It didn’t quite work out but is still a bit interesting.