Introduction to Fourier
Sampling and the Discrete Fourier Transform
Short Time Fourier Transform and Windowing
Digital Tuner for iOS in Swift

I wrote this series after attending a course on Digital Signal Processing at my university. I will try not to go into too much detail. Thus there are many essential aspects that I deliberately don’t cover (such as complex numbers). You should be able to follow along even if you don’t have a technical background. Afterwards, if you find it interesting, I would recommend you to attend a DSP course at your university.

Fig. 1Screenshot of the Guitar Tuner app

In this last part of a series of articles on Fourier transform, we will bring theory into practice by building our own digital guitar tuner for iOS in Swift (fig. 1).

The code listings that are provided in this article are limited. Only code that is specific to this application is provided. A download link to the entire project, including everything that is necessary to run the application, in a zip archive is provided at the end of this article.

1. Getting Started

We will use AudioKit[1] to do the heavy lifting of gathering output from the built-in microphone and analyzing the data. I can’t stress this enough: always use frameworks and libraries for the computational heavy lifting. Do not ever implement those algorithms on your own, except for educational purposes of course.

In order to add AudioKit to your iOS app, you will want to use CocoaPods[2]. You can install it from your command line by running sudo gem install cocoapods.

After you created a new Xcode iOS project, run pod init in the directory of your project. Then in your Podfile, uncomment use_frameworks! and add pod 'AudioKit', '~> 2.2' between the first target '...' do and end. Finally, run pod install. See fig. 2 for a recording.

Fig. 2Recording of terminal when setting up Cocoapods and installing AudioKit.

2. Frequency Tracker

We start by capturing input from the microphone. Fortunately, AudioKit makes it really easy to implement this part. Add @import AudioKit to the top of your Swift file.

// Start application-wide microphone recording.

// Add the built-in microphone.
let microphone = AKMicrophone()

// Add an analyzer and store it in an instance variable.
self.analyzer = AKAudioAnalyzer(input: microphone.output)

// Start the microphone and analyzer.

Fig. 3Listing of code that demonstrates how to setup AudioKit in Tuner.swift.

Note that the AKAudioAnalyzer implements the algorithm that we examined last week. Fortunately, using AudioKit makes it really easy because fig. 3 is all the code we need for this.

3. Polling for Updates

The AKAudioAnalyzer has a property that contains the measured frequency. We will poll for this property and update the UI accordingly. The number of updates per seconds is a matter of choice. I choose to update each 100ms, which is 10 times per second. We initialize and schedule a NSTimer using its one-shot constructor in fig. 4.

timer = NSTimer.scheduledTimerWithTimeInterval(0.1, target: self,
                                               selector: "tick", userInfo: nil,
                                               repeats: true)

Fig. 4Listing of code that demonstrates how to initialize and schedule a new run loop timer in Tuner.swift.

What we want to do now is to match the frequency we track using the microphone, with the nearest tone and octave.

4. Tones and Octaves

In music, there are 12 notes in a chromatic scale. Each note starts with a letter (A-G) and optionally an accidental (sharp or flat). Although there are multiple naming conventions, this is the (most commonly used) English and Dutch convention. Each octave consists of those 12 notes. In fig. 5 I provide a reference table for the frequencies of each note in each octave.

Number Name Octave 2 Octave 3 Octave 4 Octave 5 Octave 6
1 C 65.406 Hz 130.81 Hz 261.63 Hz 523.25 Hz 1046.5 Hz
2 C♯ / D♭ 69.296 Hz 138.59 Hz 277.18 Hz 554.37 Hz 1108.7 Hz
3 D 73.416 Hz 146.83 Hz 293.66 Hz 587.33 Hz 1174.7 Hz
4 E♭ / D♯ 77.782 Hz 155.56 Hz 311.13 Hz 622.25 Hz 1244.5 Hz
5 E 82.407 Hz 164.81 Hz 329.63 Hz 659.26 Hz 1318.5 Hz
6 F 87.307 Hz 174.61 Hz 349.23 Hz 698.46 Hz 1396.9 Hz
7 F♯ / G♭ 92.499 Hz 185.00 Hz 369.99 Hz 739.99 Hz 1480.0 Hz
8 G 97.999 Hz 196.00 Hz 392.00 Hz 783.99 Hz 1568.0 Hz
9 A♭ / G♯ 103.83 Hz 207.65 Hz 415.30 Hz 830.16 Hz 1661.2 Hz
10 A 110.00 Hz 220.00 Hz 440.00 Hz 880.00 Hz 1760.0 Hz
11 B♭ / A♯ 116.54 Hz 233.08 Hz 466.16 Hz 932.33 Hz 1864.7 Hz
12 B 123.47 Hz 246.94 Hz 493.88 Hz 987.77 Hz 1975.5 Hz

Fig. 5Table with note names, octaves and their frequencies.

Obviously, you don’t want to hardcode these frequencies into your application. For one, you may want to adapt your app later to support custom standard pitches (e.g. = 435 Hz). For this reason, we need a formula instead that, given a note (C to B) and an octave (2 to 6 seem reasonable for guitars) can compute the corresponding frequency. Fortunately, there is.

Let’s look at octaves first. , , , etc. are the same note, but have a different frequency. Precisely, each octave starts at twice the previous frequency. Does that make sense? Absolutely, let’s look at an example. In fig. 6 we can see that each higher tone fits exactly twice in the lower tone.

Fig. 6Visualization of the same tone (A) in three octaves.

Fig. 7Visualization of distribution of notes within one octave.

So if we can get a different octave by multiplying with where is the number of octaves, we can reason what we need to do to get a different note in the same octave. Remember that there are 12 notes, so moving one octave up is a multiplication with . The notes within one octave are evenly distributed (as shown in fig. 7). So if we want to get from A to A♯ all we have to do is multiply with (or ).

In Swift, I implemented the notes using enumerations (fig. 8). The pitch is implemented using a class that has instance variables for the note, octave and computed frequency based on the standard pitch (A440), multiplied with where , and are the relative note and octave respectively (so this may result in a negative number). In fig. 9 I show some code for implementing the Pitch class.

enum Accidental: String {
    case Sharp = "♯"
    case Flat  = "♭"

enum Note: CustomStringConvertible {
    case C(_: Accidental?)
    case D(_: Accidental?)
    case E(_: Accidental?)
    case F(_: Accidental?)
    case G(_: Accidental?)
    case A(_: Accidental?)
    case B(_: Accidental?)

Fig. 8Listing of code that I use to implement musical notes in Note.swift.

class Pitch: CustomStringConvertible {
    static let all = Array(
        (2 ... 6).map {
            octave -> [Pitch] in
                note -> Pitch in
                Pitch(note:   note,
                      octave: octave)

Fig. 9Listing of code that I use to implement pitches of all notes in the 2nd to 6th octave.

Finally, to find the nearest pitch, I simply map each pitch to a tuple that contains the difference between the pitch frequency and the queried frequency. Then, all I have to do is return the first item in the resulting array. I show how to do this in fig. 10.

class func nearest(frequency: Double) -> Pitch {
    var results = { pitch -> (pitch: Pitch, distance: Double) in
        (pitch: Pitch, distance: abs(pitch.frequency - frequency))

    results.sortInPlace { $0.distance < $1.distance }

    return results.first!.pitch

Fig. 10Listing of code that I use to find the nearest pitch to a given frequency.

5. Rotating Knob View

Although the user interface of the guitar tuner app is really minimalistic, some thought went into creating the rotating knob view. You will notice that it consists of several separate layers, some of which rotate.

Fig. 11Individual layers of knob view from KnobView.swift. From left to right: the thin dash layer, the thick dash layer, the arrow layer, the stable layer (always facing north) and the text layers for each pitch.

Both dash layers, the arrow layer and the text layers are grouped in another layer that is rotated. That layer I call the turn layer. Only the rectangular “stable” layer is fixed and does not rotate so it is not placed in the turn layer.

5.1. Dash Layers

The first two layers, the thin and thick dash layers, we can get with a simple CAShapeLayer.

let frame         = CGRectInset(self.bounds, 8.0, 8.0)
let path          = UIBezierPath(ovalInRect: frame).CGPath

let layer         = CAShapeLayer()
layer.frame       = frame
layer.path        = path
layer.lineWidth   = 16.0
layer.fillColor   = UIColor.clearColor().CGColor
layer.strokeColor = UIColor.whiteColor().CGColor

Fig. 12Listing of code adapted from KnobView.swift that shows how to setup the two dash layers.

In iOS, for some reason the total circumference is expressed in 720 units. For the thin dash layer we want 120 dashes and each one should be 0.5 unit. Therefore, the dash pattern is . In other words: 0.5 painted segment and 5.5 unpainted segment. The dash phase is half the painted segment so that it is centered.

layer.lineDashPattern = [ 0.5, 5.5 ]
layer.lineDashPhase   = 0.25

Fig. 13Listing of code adapted from KnobView.swift that shows how to setup the dash pattern and dash phase of the thin dash layer.

layer.lineDashPattern = [ 1.5, 58.5 ]
layer.lineDashPhase   = 0.75

Fig. 14Listing of code adapted from KnobView.swift that shows how to setup the dash pattern and dash phase of the thick dash layer.

5.2. Arrow Layer

The arrow layer is also a simple CAShapeLayer with its path set to a triangle.

let path = UIBezierPath()
path.moveToPoint(CGPointMake(8.0, 0.0))      // Top
path.addLineToPoint(CGPointMake(16.0, 13.0)) // Right
path.addLineToPoint(CGPointMake( 0.0, 13.0)) // Left
path.addLineToPoint(CGPointMake( 8.0,  0.0)) // Back to top

let layer       = CAShapeLayer()
layer.path      = path.CGPath
layer.fillColor = UIColor.redColor().CGColor
layer.frame     = CGRect(
    origin: CGPoint(
        x: round(self.bounds.width / 2.0 - 8.0),
        y: -22.0
    size:   CGSize(
        width:  16.0,
        height: 13.0

Fig. 15Listing of code adapted from KnobView.swift that shows how to setup the red arrow layer.

In fig. 15 I have adapted some code from the repository that initializes the arrow layer.

5.3. Stable Layer

The rectangular stable layer is actually the easiest of all. In fig. 16 I show how to initialize it.

let layer             = CALayer()
layer.frame           = CGRect(
    origin: CGPoint(
        x: round(self.bounds.width / 2.0 - 2.5),
        y: -56.0
    size:   CGSize(
        width:   5.0,
        height: 72.0
layer.backgroundColor = UIColor.whiteColor().CGColor

Fig. 16Listing of code adapted from KnobView.swift that shows how to setup the rectangular stable layer.

5.4. Pitch Text Layers

Lastly, we still have to add text layers around the knob for each pitch that is near the tracked frequency.

let labels = (0 ... 4).map { _ in CATextLayer() }

let weight = UIFontWeightUltraLight

for (i, label) in enumerate(labels) {
    let offset: CGFloat   = 30.0 + (i == 2 ? 50.0 : 0.0)
    let angle: CGFloat    = CGFloat(i - 5) / 6 * 2 * 3.14

    let width             = self.bounds.width  / 2.0 + offset
    let height            = self.bounds.height / 2.0 + offset)
    let center            = CGPointMake(width  - width  * sin(angle) - offset,
                                        height - height * cos(angle) - offset)

    label.frame           = CGRectMake(center.x - 40, center.y - 20, 80.0, 40.0)
    label.alignmentMode   = kCAAlignmentCenter
    label.contentsScale   = UIScreen.mainScreen().scale
    label.foregroundColor = UIColor.whiteColor().CGColor
    label.font            = UIFont.systemFontOfSize(17, weight: weight)
    label.fontSize        = 17.0
    if i == 2 {
        label.font        = UIFont.systemFontOfSize(36, weight: weight)
        label.fontSize   *= 2.0


Fig. 17Listing of code adapted from KnobView.swift that shows how to setup all 5 text layers.

Then, every time the tracked frequency changes, we update the pitch labels to present the nearest pitches. In Pitch.swift I overloaded the + and - operators so if we want to get the next or previous pitch we can simply do pitch + 1 (see fig. 18). Note that the offset ranges from -2 to 2 and 0 is the big label at the top.

for (i, label) in labels.enumerate() {
    let offset   = i - 2
    label.string = (pitch + offset).description

Fig. 18Listing of code adapted from KnobView.swift that shows how to update the pitch labels.

5.5. Rotating the Knob

Every time the tracked pitch updates, we compute the distance between the nearest pitch and the tracked frequency in Tuner.swift. We then divide that distance by the total difference between the nearest pitch and the second nearest pitch in order to express it as a percentage. Additionally, we multiply the thing by two in order to get the knob angle between -0.5 and 0.5. In fig. 19 I show how to do these computations.

let previous   = pitch - 1
let next       = pitch + 1
let difference = distance < 0 ?
                 (pitch.frequency - previous.frequency) :
                 (next.frequency  - pitch.frequency)

knobView.distance = distance / difference / 2.0

Fig. 19Listing of code adapted from ViewController.swift that shows how to express the distance between the tracked frequency and the nearest pitch in a percentage.

In fig. 20 I show how to use an affine transform to rotate the group layer and the text layers.

let angle = CGFloat(distance * M_PI)


for label in labels {

Fig. 20Listing of code adapted from KnobView.swift that shows how to use the distance that we computed in fig. 19 to rotate the knob view clockwise and the text layers counter clockwise.

6. Wave Display View

The wave display view consists of a plot view with 5 graphs and a gradient mask that fades out horizontally at both ends.

Each plot has a different multiplier from 1.0 to -1.0 and a different opacity that ranges from 1.0 to 0.2. This creates a great looking effect. In fig. 21 I show how to do that.

let plots = (0 ... 4).map { _ in AudioPlot() }

for i in 0 ... 4 {
    plots[i].autoresizingMask = [ .FlexibleWidth, .FlexibleHeight ]
    plots[i].backgroundColor  = .clearColor()
    plots[i].lineColor        = .whiteColor()
    plots[i].lineWidth        = 1.0
    plots[i].frequency        = 0.0
    plots[i].amplifier        = abs(1.0 - Double(i) * 0.4) *
                                (i % 2 == 0 ? 1.0 : -1.0)
    plots[i].alpha            = 1.0 - CGFloat(i) * 0.2

Fig. 22Listing of code adapted from PlotView.swift that shows how to combine multiple plots into one plot.

6.1. Plot

We subclass AKAudioPlot to implement a custom plot. In bufferWithCsound we generate a new samplebuffer based on a sine of the tracked frequency and amplitude.

First we setup the float buffer with a C call from Swift (fig. 23). Note that 4 is the sizeof a float in C (at least it is safe to assume that it is on the architectures we are targeting).

let length = Int(AKSettings.shared().numberOfChannels) *
             Int(AKSettings.shared().samplesPerControlPeriod) * 4
let num    = length / 4
let floats = UnsafeMutablePointer<Float>(malloc(length))

Fig. 23Listing of code adapted from PlotView.swift that shows how to setup a float buffer.

Then we are going to fill it with num floats by using a for-loop and sampling a sine at each iteration (fig. 24).

// These are arbitrary float parameters to get a nice effect.
let phase     = (self.amplifier + 0.8) / 1.8

let amplitude = self.amplifier * amplitude 

var t         = (time + Double(i) / Double(num) * self.frequency + phase)
floats[i]     = Float(sin(t * 2 * 3.14))

// Again the 2 is a bit arbitrary but it determines the speed of the sine.
time         += self.frequency / 44100 / 2

Fig. 24Listing of code adapted from PlotView.swift that shows how to compute the result of a sine at each iteration.

Now we multiply the results with a sine of 0.5 Hz to fade out the horizontal ends and we multiply it with a power to make the plot look more dramatic at the right side and less symmetric (fig. 25).

t          = Double(i) / Double(num)
floats[i] *= Float(sin(t * 3.14) * amplitude
floats[i] *= 1 - pow(1 - Float(i) / Float(num), 2.0)

Fig. 25Listing of code adapted from PlotView.swift that shows how to multiply the intermediate results with a smoothing function.

Finally, at the end of each iteration we fade smoothly to the next frequency and amplitude that we track using the microphone (fig. 26).

self.frequency += (nextFrequency - self.frequency) / 44100.0 / 4.0
self.amplitude += (nextAmplitude - self.amplitude) / 44100.0 / 2.0

Fig. 26Listing of code adapted from PlotView.swift that shows how to fade the frequency and amplitude of the plot to the frequency and amplitude that are currently tracked.

6.2. Gradient Mask

Finally, to make both horizontal ends fade out smoothly, we use a gradient mask. In fig. 27 I show how to setup a new horizontal gradient mask.

let layer = CAGradientLayer()
layer.colors     = [ UIColor.clearColor().CGColor,
                     UIColor.clearColor().CGColor ]
layer.startPoint = CGPointZero
layer.endPoint   = CGPointMake(1.0, 0.0)
self.layer.mask  = layer

Fig. 27Listing of code adapted from DisplayView.swift that shows how to use a gradient mask to fade out both horizontal ends of a UIView.

This is it! I have upload the entire repository to GitHub. You can git clone the repository or download the latest release in a zip archive or tarball. If you stumble upon a bug or if you can improve the code (e.g. make it more Swift-y) please file a Pull Request (at Github) and send me an email, I really appreciate that!

8. References

  1. Prochazka, Aurelius   “ AudioKit ”  
  2.   “ CocoaPods ”