Just Intonation

Just Intonation

Just Intonation is a multiplatform application that allows the user to hear and play music in just intonation (pure frequency ratios) in any scale.

Just Intonation

Karolin Stange, Christoph Wick, and Haye Hinrichsen

Institute for Physics and Astronomy
Department of Theoretical Physics III
University of Würzburg
D-97074 Würzburg, Germany

Resources: Link:
Project Website just-intonation.org
Source Code sourcecode.just-intonation.org
Documentation doxygen.just-intonation.org
E-Mail info at just-intonation.org

JustIntonation is an open-source application for various platforms that allows the user to hear and play music in just intonation and other temperaments, independent of the musical key.

The problem

The Western musical scale is based on the Equal Temperament (ET), which is defined by constant semitone ratios of 21/12. As a geometrically organized temperament the ET has the great advantage to be scale-invariant, i.e., the interval frequency ratios are the same in all scales, no matter whether you play music e.g. in C-major or D-major.

The problem is that all intervals except for the ocatave are characterized by irrational frequency ratios. However, because of the ubiquitious harmonic series of higher partials in the spectra of musical instruments, which are characterized by integer multiples of the ground frequencies, intervals are perceived as well-tuned if they exhibit a rational frequency ratios. This means that in the ET all intervals except for the octave are more or less out of tune. Over the last two centuries, however, humanity got used to these discrepancies.

Static unequal temperaments

With the increasing awareness of harmony and the renewed focus on historical performance practices there is currently a great interest in so-called unequal temperaments. Historically there has been a fascinating development of musical temperaments, ranging from Pytharorean tuning over Meantone temperaments to the ET, to name only a few. All these temperaments represent static tuning schemes, reconciling the incompatibility of perfect harmony and scale invaraince to a different extent. The two extremes in this spectrum are on the one hand the Western ET (perfect scale invariance, lowest harmony) and the so-called just intonation, which has the highest degree of harmony but only in a single reference scale.

Dynamical tuning

To overcome the problem of the broken scale symmetry in static just intonation, a possible way out would be to tune dynamically, that is, each tone is tune instantly when the key of the instrument is hit. This is of course impossible with traditional mechanical instruments but it is very easy with electronic oscillators. The idea is to recognize the scale just in time and to tune the current chord in just intonation.

This project

In the past there has been a large variety of attempts to implement dynamical tuning schemes, most of them on a proprietary basis and adapted to the limited computer power in old times. Here we propose a new method as an open-source project that could be a basis for further development. This application allows one to test the method and to experience how it would be to hear Music in just intonation.

What is new

Whenever a key is pressed (or a Midi signal is received) the application sets up a linear system of equations to work out the optimal tuning. This system is instantly solved using the "eigen" library. The application comes with an inbuilt microtunal sampler that allows you to hear real instruments in just intonation. Our method has the great advantage that it does not only tune simple chords perfectly but it also finds reasonable compromises for self-contradicting chords such as the diminished Seventh that cannot be tuned consistently in just intonation.


JustIntonation is a multi-platform application based on Qt. It depends on the QMidi library developed by Christoph Wick. It includes the linear algebra package "Eigen" in the thirdparty folder.

General preparations

Before starting one needs to download and install Qt:

  • Download and install the open source version of Qt and Qt Installer Framework from qt.io -- For intallation select the components Desktop-GCC, Android-ARM, QtCreator and Installer Framework
  • ### Linux To compile the code on a Linux machine proceed as follows:
  • Download the open-source version of Qt and Qt Installer Framework from https://www.qt.io/
  • Make Qt ready for Android
    • Install openjdk-devel
    • Get JDK installation path with sudo update-alternatives --config javac
    • Add JDK installation path (e.g. /usr/lib64/jvm/java-1.8.0-openjdk) to Qt-Settings-Devices
  • Get the qtmidi package from https://gitlab.com/tp3/qtmidi
  • Open qtmidi project, add "make install" in the build settings
  • Compile qtmidi, this will install the qtmidi libraries in Qt


  • Compile source
    • Get QtMidi: clone https://gitlab.com/tp3/qtmidi
    • Get Just Intonation: clone https://gitlab.com/tp3/justintonation
    • Configure QtMidi in QtCreator, add "make install" (see QtMidi instructions)
    • Build QtMidi in Release mode, then check that QtMidi.framework in generated in the Qt/5.x/lib folder
    • Build and run JustIntonation. If there are no errors the application should start out of the box.
  • If the compilation fails because of the wrong XCode version please proceed as follows:
    • Add the following line to Qt/5.x/clang_64/mkspecs/qdevice.pri (replace version number)
    • !host_build:QMAKE_MAC_SDK = macosx10.12
  • Repeat the same steps for iOS

Structure of the software


The application folder contains the following subfolders:

  • instrument: Parts of the code which define a musical instrument (Piano, Organ, Harpsichord etc.), holding the sampled data.
  • sampler: Parts of the code responsable for real-time sampling and sound generation.
  • platforms: Platform-dependent additions (Android etc)
  • resources: Wave file, QML files, images and icons
  • system: Various system-related parts of the software
  • modules: Various parts of the software which could also be compiled separately, having a well-defined meaning even without the reset of the code.

To see all modules follow this link

Tuner module

The essential module is te 'Tuner'. This module is connected to the application exclusively via Qt signals and slots. This allows you to use the tuner in different projects very easily. It also makes it easier to understand how it works.