June 14, 2014

So you’ve learned the basics of how digital audio works and it’s gone to your head. Your friends can’t stand you anymore. Your parents “forget” to tell you they’ve moved. The next step toward total alienation is to flex your knowledge further by writing a program to create some simple sounds.

Good thing I’m here to help! This article describes a simple command-line Ruby program called NanoSynth. It can create 5 different simple sounds, played at different notes and volumes. Although it’s very simple, it shows the basics of how to write programs that create sound.

Go ahead and check out the code on GitHub. The rest of this article goes into detail about how it works.

NanoSynth can create 5 different types of sound wave: sine, square, sawtooth, triangle, and white noise. These wave forms look and sound like this:

#### Sine

#### Square

#### Sawtooth

#### Triangle

#### White Noise

If you’ve spent some time with an 8-bit Nintendo, they probably sound familiar. They are often used as building blocks for more complex sounds. (In case it’s not clear, white noise is just random sample data).

When you run NanoSynth, you specify the wave type, frequency, and maximum amplitude (i.e. volume). NanoSynth then generates 1 second worth of sample data. This data is then written to a Wave file on disk using the WaveFile gem. To hear the sound, you can play the Wave file in whatever media player you like.

NanoSynth can be thought of as two main parts: the part that reads the command-line arguments, writes the generated sample data to a Wave file (using the WaveFile gem) and generally drives things, and the part that actually generates the audio data. Let’s get the first part out of the way, so we have a shell to fill out. Here it is:

```
gem 'wavefile', '=0.6.0'
require 'wavefile'
OUTPUT_FILENAME = "mysound.wav"
SAMPLE_RATE = 44100
TWO_PI = 2 * Math::PI
RANDOM_GENERATOR = Random.new
def main
# Read the command-line arguments.
wave_type = ARGV[0].to_sym # Should be "sine", "square", "saw", "triangle", or "noise"
frequency = ARGV[1].to_f # 440.0 is the same as middle-A on a piano.
max_amplitude = ARGV[2].to_f # Should be between 0.0 (silence) and 1.0 (full volume).
# Amplitudes above 1.0 will result in distortion
# (or other weirdness).
# Generate 1 second of sample data at the given frequency and amplitude.
# Since we are using a sample rate of 44,100Hz,
# 44,100 samples are required for one second of sound.
samples = generate_sample_data(wave_type, 44100, frequency, max_amplitude)
# Wrap the array of samples in a Buffer, so that it can be written to a Wave file
# by the WaveFile gem. Since we generated samples between -1.0 and 1.0, the sample
# type should be :float
buffer = WaveFile::Buffer.new(samples, WaveFile::Format.new(:mono, :float, 44100))
# Write the Buffer containing our samples to a 16-bit, monophonic Wave file
# with a sample rate of 44,100Hz, using the WaveFile gem.
WaveFile::Writer.new(OUTPUT_FILENAME, WaveFile::Format.new(:mono, :pcm_16, 44100)) do |writer|
writer.write(buffer)
end
end
# The dark heart of NanoSynth, the part that actually generates the audio data
def generate_sample_data(wave_type, num_samples, frequency, max_amplitude)
# We're gonna fill this in, but for now just return an empty list of samples.
return []
end
main
```

This is pretty bare bones. For example, the command-line arguments aren’t validated - if you give bad arguments when running NanoSynth, it will crash or not work right.

Save this to a file called `nanosynth.rb`

.

Next, install the WaveFile gem. If you’re using RVM or rbenv, you can install it like this:

```
gem install wavefile
```

You should ideally use RVM or rbenv, but if you’re using the built-in Ruby that comes with a Mac, you might need to use `sudo`

when installing the gem:

```
sudo gem install wavefile
```

Finally, from the command line navigate to the folder containing `nanosynth.rb`

. Run it like this:

```
ruby nanosynth.rb sine 440.0 0.2
```

This should create a file called `mysound.wav`

in the current folder, containing a sine wave at 440Hz and 20% full volume. If you try to play it though, you won’t hear anything. That’s because we haven’t filled in the `generate_sample_data`

method yet. When we’re done though, it will!

As the next step we need to implement the `generated_sample_data()`

method. If you remember from the digital audio primer, we need a method that will generate an array of samples between -1.0 and 1.0. Since NanoSynth uses a sample rate of 44,100Hz and generates 1 second of sound, the array should contain 44,100 values.

Our method will be able to generate sine, square, sawtooth and triangle waves, as well as white noise. It will also allow passing in a frequency to determine the pitch of the generated tone, as well as the amplitude to determine the loudness.

Let’s start with sine waves. They hold a hallowed position as the fundamental building block of sound. In fact, any possible sound (including the other waveforms in this article) can be re-created by combining the correct sine waves together. (Thanks to @fourier for figuring this out).

A sine wave looks like this:

You can generate values in a sine wave by using Ruby’s built-in `sin()`

function. It takes an x-value in radians, and returns the height of the sine wave at that x-value.

```
Math::sin(0.0) # Returns 0.0
Math::sin(Math::PI * 0.5) # Returns 1.0
Math::sin(Math::PI) # Returns 0.0
Math::sin(Math::PI * 1.5) # Returns -1.0
Math::sin(Math::PI * 2) # Returns 0.0
```

An important fact about the sine wave is that it has a period of 2π radians (about 6.28). So if you call `sin()`

with values from 0.0 to ~6.28, you’ll see the y-value returned go from 0.0 up to 1.0, down to -1.0, then back up to 0.0 when you get to ~6.28. If you then keep incrementing the x-value you’ll see the values repeat again, with another full cycle ending at ~12.57 (4π).

So let’s say we want to generate a sine wave with a frequency of 1Hz. Assuming we’re using a standard sample rate of 44,100Hz (i.e. 44,100 samples every second), we could do it like this:

```
position_in_period = 0.0 # Range of 0.0 to 1.0
position_in_period_delta = 1.0 / 44100.0 # Frequency divided by sample rate
# Initialize an array of samples set to 0.0. Each sample will be replaced with
# an actual value below.
samples = [].fill(0.0, 0, num_samples)
num_samples.times do |i|
samples[i] = Math::sin(position_in_period * TWO_PI) * max_amplitude
position_in_period += position_in_period_delta
# Constrain the position in period between 0.0 and 1.0
# I.e., re-loop over the same cycle again and again
if position_in_period >= 1.0
position_in_period -= 1.0
end
end
```

Think of it like this. We start at 0, and the finish line for one cycle is ~6.28 (2π). Each time through the loop, we increment `position_in_period`

, getting closer to the finish line of 2π. Once we go past the finish line, we’ll loop back around to the start of the cycle and keep going.

So the code above will give us a nice 1Hz sine wave, but uh, humans can’t hear anything below about 20Hz, so if you save it to a wave file and play it back, enjoy the silence. No problem though, when calculating `position_in_period_delta`

, we can just plug in a higher frequency to get something we actually *can* hear. Let’s try 440Hz, which is the same as middle A on a piano.

```
position_in_period_delta = 1.0 / 44100.0 # i.e. 0.000023
position_in_period_delta = 440.0 / 44100.0 # i.e. 0.009977
```

Notice how 440Hz makes `position_in_period_delta`

larger. This means each time through the loop we’ll get to 2π faster, and therefore complete one oscillation faster. So to change the frequency of a sine wave, all you need to is plug in a different frequency when calculating `position_in_period_delta`

.

The sine function returns values between -1.0 and 1.0. By convention these also happen to be the equivalent to the minimum and maximum amplitudes our speakers can play, so this means a generated sine wave will play at full volume. If we want to make it quieter, we just multiply each sample by a constant amount. For example to have it play at half volume, multiply everything by 0.5. If you multiply each sample by a number greater than 1.0, you’ll cause *clipping* to occur, which is a type of distortion. Unless you’re purposely trying to cause distortion or weird sound effects you don’t want to do that. (This applies to any waveform, not just the sine wave).

Let’s add this final code to our template:

```
def generate_sample_data(wave_type, num_samples, frequency, max_amplitude)
position_in_period = 0.0
position_in_period_delta = frequency / SAMPLE_RATE
# Initialize an array of samples set to 0.0. Each sample will be replaced with
# an actual value below.
samples = [].fill(0.0, 0, num_samples)
num_samples.times do |i|
# Add next sample to sample list. The sample value is determined by
# plugging the period offset into the appropriate wave function.
if wave_type == :sine
samples[i] = Math::sin(position_in_period * TWO_PI) * max_amplitude
end
position_in_period += position_in_period_delta
# Constrain the period between 0.0 and 1.0
if(position_in_period >= 1.0)
position_in_period -= 1.0
end
end
samples
end
```

Next, run NanoSynth, using a frequency of 440Hz, and an amplitude of 50% full volume:

```
ruby nanosynth.rb sine 440.0 0.2
```

This should write `mysound.wav`

to the current directory, and it should sound like this:

Next up, let’s create a square wave. In a certain sense square waves are simpler than sine waves, because they consist of switching back and forth between just two different amplitudes. For example a certain number of samples at amplitude -0.75, followed by an equal number of samples at amplitude 0.75, and then repeat. Square waves have an especially 8-bit Nintendo sound to them. They are also important in subtractive synthesis, since they have odd harmonics, while sine waves have none. However, that’s beyond the scope of this article.

The amplitude for a square wave determines the two amplitudes it switches between. For example, for an amplitude of 0.75, a square wave will toggle between 0.75 and -0.75. The rate at which the two amplitudes are toggled determines the frequency. The faster the toggle, the higher the frequency (and thus the higher the pitch).

A square wave looks like this:

Here are some more concrete examples:

- If we assume a sample rate of 44,100Hz, and a maximum amplitude of 0.75, then a square wave with a frequency of 1Hz would be represented by 22,050 samples with an amplitude of 0.75, followed by 22,050 samples with an amplitude of -0.75. This would then repeat for as long as the sound is played.
- Alternately, if the desired frequency is 441Hz, a square wave would be a repeating set of 50 samples of amplitude 0.75, followed by 50 samples of amplitude -0.75. See how that works? If the frequency is 441Hz and the sample rate is 44,100 samples per second, then each cycle is 100 samples long (44,100 / 441).

So to generate a square wave, we can take the `position_in_period`

variable we already have, which ranges from `0.0`

to `1.0`

. If it’s less that `0.5`

use the positive amplitude, and if it’s greater-or-equal to 0.5, use the negative amplitude.

With that in mind, here’s the updated `generate_sample_data`

method so that it generates a square wave for us. This code is the same as before, except for the changes highlighted in yellow.

```
def generate_sample_data(wave_type, num_samples, frequency, max_amplitude)
position_in_period = 0.0
position_in_period_delta = frequency / SAMPLE_RATE
# Initialize an array of samples set to 0.0. Each sample will be replaced with
# an actual value below.
samples = [].fill(0.0, 0, num_samples)
num_samples.times do |i|
# Add next sample to sample list. The sample value is determined by
# plugging the period offset into the appropriate wave function.
if wave_type == :sine
samples[i] = Math::sin(position_in_period * TWO_PI) * max_amplitude
```

```
elsif wave_type == :square
samples[i] = (position_in_period >= 0.5) ? max_amplitude : -max_amplitude
```

```
end
position_in_period += position_in_period_delta
# Constrain the period between 0.0 and 1.0
if(position_in_period >= 1.0)
position_in_period -= 1.0
end
end
samples
end
```

After adding this code go to the command line, and run:

```
ruby nanosynth.rb square 440.0 0.2
```

Open `mysound.wav`

in your favorite media player. It should sound like this:

Sawtooth waves are so named for their appearance, because their waveform looks like the edge of a saw. They also have a buzzy sound, which is appropriate. A naked saw wave makes a nice synth bass sound if used at lower frequencies (say 110Hz). Along with square and triangle waves, they are prized in subtractive synthesis since they contain all harmonics*.

The waveform for each period consists of a straight line ramp going from a sample with maximum amplitude to the a sample with minimum amplitude (or vice versa). When the next period begins, the waveform jumps back up to the starting point and repeats.

So to generate this, we need to calculate continuously increasing (or decreasing) numbers. We already have `position_in_period`

which ranges from 0.0 to 1.0, so we can just scale it to range between -1.0 and 1.0 instead.

That is, we just need to:

- Multiply
`position_in_period`

by 2 so it goes from 0.0 to 2.0 instead of 0.0 to 1.0 - Subtract 1, to make it go from -1.0 to 1.0
- Multiply by the amplitude to scale it accordingly. For example, if the amplitude is 0.25, multiplying by 0.25 will make it scale from -0.25 to 0.25.

The rest of our existing code that increments `position_in_period`

will do the rest.

```
if wave_type == :sine
samples[i] = Math::sin(position_in_period * TWO_PI) * max_amplitude
elsif wave_type == :square
samples[i] = (position_in_period >= 0.5) ? max_amplitude : -max_amplitude
```

```
elsif wave_type == :saw
samples[i] = ((position_in_period * 2.0) - 1.0) * max_amplitude
```

` end`

To test this out, run:

```
ruby nanosynth.rb saw 440.0 0.2
```

It should sound like this:

Triangle waves sound like a cross between a sine wave and a square wave. It resembles a sine wave made out of straight lines rather than an S curve.

One way to generate a triangle wave is to start with a saw wave. Notice how half of the samples in a saw wave will be below 0, and half will be above. If you take the absolute value of each sample, you’ll get a triangle pattern.

This is close to what we need, but it’s not scaled properly. For example, if our amplitude is 0.5, we want the samples in our triangle wave to span from -0.5 to 0.5. However, just taking the absolute value of a saw wave will give us samples that range from 0.0 to 0.5. No problem - we just need to multiple the samples in our proto-triangle wave by 2, and then subtract them from our amplitude.

```
if wave_type == :sine
samples[i] = Math::sin(position_in_period * TWO_PI) * max_amplitude
elsif wave_type == :square
samples[i] = (position_in_period >= 0.5) ? max_amplitude : -max_amplitude
elsif wave_type == :saw
samples[i] = ((position_in_period * 2.0) - 1.0) * max_amplitude
```

```
elsif wave_type == :triangle
samples[i] = max_amplitude - (((position_in_period * 2.0) - 1.0) * max_amplitude * 2.0).abs
```

` end`

To test this out, run:

```
ruby nanosynth.rb triangle 440.0 0.2
```

It should sound like this:

White noise is just random samples. It sounds like radio static. It can be useful for making snare drum sounds if you for example apply an envelope to it (beyond the scope of this article).

Generating white noise is simple, we just need to generate random numbers between the maximum and minimum amplitudes. This is really easy, just use Ruby’s built-in `Random`

class - the `rand`

method let’s you pass in a range that the random number should be within.

Note that the frequency argument doesn’t affect the samples generated, since they are random.

```
if wave_type == :sine
samples[i] = Math::sin(position_in_period * TWO_PI) * max_amplitude
elsif wave_type == :square
samples[i] = (position_in_period >= 0.5) ? max_amplitude : -max_amplitude
elsif wave_type == :saw
samples[i] = ((position_in_period * 2.0) - 1.0) * max_amplitude
elsif wave_type == :triangle
samples[i] = max_amplitude - (((position_in_period * 2.0) - 1.0) * max_amplitude * 2.0).abs
```

```
elsif wave_type == :noise
samples[i] = RANDOM_GENERATOR.rand(-max_amplitude..max_amplitude)
```

` end`

To test this out, run:

```
ruby nanosynth.rb noise 440.0 0.2
```

Note that the frequency doesn’t matter - since these are just random samples, the frequency won’t affect the output. It should sound something like this: