Here are some examples of tuning and mistuning using just intonation. They are designed to demonstrate the difference between just intonation and equal temperament and the importance of correct disambiguation of intervals when justly intoning music.

The audio is generated by my Harmonical, which can output midi files with tuning events for a set of tonal space points. The midi is synthesized using Timidity.

These examples come directly from articles by Christopher Longuet-Higgins:

• Two Letters to a Musical Friend, The Music Review, 1962

• The Perception of Melodies, Nature, 1976

Cliche

• Audio file: ogg, mp3

This is an example used, among other places, in The Perception of Melodies. The short melody is shown to the right.

The point of the example is that the Ab should be so written, not as a G#.

When this example is played in just intonation, the choice between Ab and G# affects the pitch of the notes. The Ab is a major 3rd below the tonic (C), whilst the G# is two major 3rds above it.

The example contains the phrase played:

1. in equal temperament
2. in just intonation with an Ab
3. in just intonation with a G# (incorrect)

Land of Hope and Glory

• Audio file: ogg, mp3

Longuet-Higgins uses this as a rare example of an imperfect fourth.

The equal temperament version, of course, doesn't sound too odd, since it's just playing an equally tempered fourth. When played in just intonation, the strangeness of the interval can be heard. Note that the first fourth (between the 5th and 6th notes) is an imperfect fourth, whilst that between the last two notes is a perfect fourth.

The third performance in the audio is also justly intoned, but makes the wrong choices for these intervals, swapping them round. The first fourth is played as a perfect fourth and the final one as an imperfect fourth.

After listening to the second and third performances, listen again to the first. The equal temperament fourth is very close to a perfect fourth, but a long way from an imperfect fourth.

The example contains the phrase played:

1. in equal temperament
2. in just intonation
3. in just intonation with the wrong fourths

Sevenths: Minor Seventh

• Audio file: ogg, mp3

Longuet-Higgins gives an example of a dominant seventh and a minor seventh (both in minor keys) in Two Letters.

This is his minor seventh example. The minor seventh occurs between the D and C in the penultimate bar.

For comparison, the audio contains the phrase played as it should be, with a minor seventh, and again with a dominant seventh instead.

The audio contains the phrase played:

1. in equal temperament
2. in just intonation (correctly)
3. in just intonation, with a dominant 7th instead of a minor 7th

Sevenths: Dominant Seventh

• Audio file: ogg, mp3

I've done the same with Longuet-Higgins' dominant seventh example. You can hear it played correctly in just intonation, and also with a minor seventh substituted where each dominant seventh should be.

The first and last intervals (each B to A) are the dominant sevenths.

The audio contains the phrase played:

1. in equal temperament
2. in just intonation (correctly)
3. in just intonation, with a minor 7th instead of a dominant 7th