Just Intonation is the tuning of pitches related by whole number ratios. The following will serve as a brief overview and introduction to the system’s theory and practice. The harmonic series is a good place to start when discussing Just Intonation (henceforth JI).

Partial numbers inside the staff, frequencies below

From a fundamental frequency (here the low C at ~65.4 Hz), the harmonic series ascends in multiples. If we refer to the partials of the tone, rather than the overtones, we can more easily see the math behind the present frequencies. Numbering the fundamental frequency as the 1st partial, the 2nd multiplies the frequency by 2, the 3rd partial by 3, ad infinitum. What does all this have to do with JI? JI deals in tuning intervals by simple whole number frequency ratios, and since we have demonstrated that the number of a partial is a multiplier of the fundamental frequency, we can use the harmonic series to find the intervals of these ratios. 2/1 is an octave, 3/2 the perfect fifth, 4/3 the perfect 4th, 5/4 the major third, 6/5 the minor third. What is significant about these intervals is that they deviate from the tempered intervals one finds on the modern piano. The simpler ratios sound beatless, and ‘pure’. If you are a choral singer or brass player, you probably are already used to finding this beatless sound by tuning wider fifths and lower thirds and sevenths in chords.

JI systems are sometimes referred to as “x-limit” systems, where x is some prime number. For example, a 5-limit system includes no prime numbers higher than 5 in any ratios, allowing for pure major thirds, but not true septimal or 7-limit consonances. 3-limit tuning is often referred to as Pythagorean tuning, and is composed entirely of just fifths. 5-limit tuning can approximate the major-minor system of Western music very well.

### A few notable composers using JI

Americans: Ben Johnston, Harry Partch, LaMonte Young, James Tenney. Europeans: Gérard Grisey, Georg Friedrich Haas, György Ligeti. This is of course, no exhaustive list of composers, but a guide for the novice to what might be more familiar music, and an easier entry into the system. For a further discussion of the motivation to use JI, see Colin Holter’s fantastic paper “The Spiritual Construction of Tuning in American Experimental Music” at the Search Journal.

### Notation of JI

The notation of JI varies depending on the composer and the circumstances. Ligeti often notated JI intervals simply by the fundamental on which the horn was to play (as in his last work, Hamburg Concerto). James Tenney sometimes notated JI intervals by writing cents deviation from ET. Perhaps the most exhaustive and most widely embraced system yet devised is Ben Johnston’s. Beginning from the assumption that the C Major scale is to be built from interlocking C F and G major triads, all tuned 6:5:4, Johnson introduces novel accidentals to shift these notes to different relationships. For example, in his C major scale, the supertonic D is ~4 cents higher than ET, while the A is ~16 cents low. If we are to use a properly tuned triad based on D, this requires an accidental to raise the A approximately 21 cents, Johnston’s +. Alternately, the D could be lowered, using -. The interval expressed by + or – is called the syntonic comma (81:80), and is the difference between a Pythagorean (or 3-limit) major third (81:64) and a 5:4 (5-limit) major third. Johnston’s system of accidentals continues similarly, with each successive accidental expressing a kind of fundamental shift related to a higher prime ratio.

### Learning JI

Just Intonation tunings appear in traditional drone-based musics, like North-Indian Classical music, and can be easily practiced over a drone. If you play any string instrument, you can accompany yourself with a drone for practice, use a recording of a tambura, or even an electronic tone (preferably one rich in harmonics). Singing a just interval correctly feels ‘anchored’ in the fundamental tone and its harmonic spectrum. Aside from going by feel, if you are a string player of any kind you can also use the open harmonics of a string to learn simple just intervals. Being a computer enthusiast, I prefer to practice my microtones with the aid of software. Rote learning is the basis of the oral traditions that function within a JI framework, and is an indispensable tool. I use a program called Scordatura for the playback of JI microtones, and it is extremely flexible. Using CSE, a companion program to Scordatura, I can designate tunings wholly via ratio from any given fundamental at any tuning. It has made the task as simple as entering ratios in scalar order, and assigning them to keys on my midi keyboard. From this point on, I work by transcribing the pitch notation of the score to the re-mapped microtones of the midi keyboard (one octave of pitch extended over more than 4 octaves of keyboard (!) in my most recent JI undertaking, Johnston’s arrangement of Partch’s Barstow). I have a similar setup prepared for learning the tuning of Randy Gibson‘s upcoming work for ekmeles.

Of course, this is but a simplified overview – there are manifold internet resources for learning more about JI theory and practice. I’m happy to address questions or requests for elaboration on any of these points (and to accept corrections from those more deeply involved in JI than I). But if you’re interested, I would recommend as a first step getting your hands on a CD – or better yet, attending a concert – featuring JI music and hearing the difference a few cents here or there can make! Ben Johnston’s fourth String Quartet is a melodic and beautifully lucid introduction to JI.