# Just intonation

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Just intonation (JI) is an approach to musical tuning where pitches are chosen in a way such that every interval can be expressed as a whole-number ratio of the frequencies of pitches. Just intervals naturally occur in the harmonic series as intervals between any two harmonics of a fundamental tone produced with a harmonic timbre. For instance, an interval with a frequency ratio of 3/2 appears between the 2nd and 3rd harmonics. Just intonation is particularly efficient when used with harmonic instruments, because it allows the tuning and the timbre to reinforce each other.

In theory, there are infinitely many just intervals, because each possible fraction corresponds to a just interval. In practice, however, additional constraints are used to reduce the number of intervals to a reasonable amount, but also in many cases to prioritize consonant intervals. Usual constraints include subgroups of generators (including prime limits), common denominators or numerators (as used in primodality), and complexity limits (usually height limits). Multiple constraints can be applied at the same time as well, such as the intersection of a prime limit and an odd limit.

In the context of Western music theory prior to the 20th century, the term just intonation used alone usually refers to 5-limit tuning. Extended just intonation, a term coined by Ben Johnston, usually refers to higher prime limits,[1] such as the 7-limit, the 11-limit and the 13-limit. The practice of just intonation without any particular constraint is sometimes referred to as rational intonation (RI) or as free style JI.

The structure of just intonation has several implications on music composition. Wolf intervals and commas, two kinds of dissonant intervals, may appear between distantly-related pitches. In addition, certain chord progressions are comma pumps, which may cause the tonal center of a piece to drift up or down in pitch over time. These effects can be treated either as features or as problems to be solved. Examples of approaches that try to solve these problems include adaptive just intonation and temperament.

## Just intonation explained

If you are used to speaking only in note names (e.g. the first 7 letters of the alphabet), you may need to study the relation between frequency and pitch. Kyle Gann's Just Intonation Explained is one good reference. A transparent illustration and one of just intonation's acoustic bases is the harmonic series.

In languages other than English, the original conceptions of "just intonation" are more obviously retained in the terms used in those languages: German Reine Stimmung (pure, that is, beatless, tuning), Ukrainian Натуральний стрій and French gamme naturelle (both referring to the "natural scale", that is, intervals derived from the harmonic series), Italian intonazione naturale (natural intonation, once again intervals derived from harmonic series), and so on.

In the English language, the term "just" once referred to "true, correct", and is still used today in this sense, in the crafts. In printing, to "justify" a line of type is to fit it precisely to a certain measure, for example. The original sense, then, was similar to that sense which is clearly retained in other languages as "natural".

Of course, a historical description of something as "natural" does not prove that something is "natural." Similarly labeling something "natural" without any ground, especially in the arts, is always very problematic. Nevertheless, the historical meanings of the terms for what we call "just intonation" do claim a "natural" status, and just intonation is indeed derived from genuine acoustic phenomena. How important, universal, etc., these phenomena are has been a matter of debate for thousands of years.

Specifying ratios of frequencies is another way of expressing the "natural scale", for it describes ratios between partials in the harmonic series (in their ideal form). So, contemporary usage of the term is in keeping with historical and international usages. However, just as harmonic vocabulary has expanded over the centuries, so has that which falls under "just intonation" expanded.

But, first things first. Let us take a look at why the idea of a "natural" or "just" tuning came about, and is still with us.

If we have a tone with a harmonic timbre and a fundamental frequency at 100 Hz (Hertz, or cycles per second), we will find the second harmonic component at 200 Hz, the third at 300 Hz, the fourth at 400 Hz...Yes, the harmonics are found at the fundamental frequency times 1, times 2, times 3...

The simplicity of it all can be difficult to believe at first. You can easily imagine people discovering this and getting carried away with ideas of "music of the spheres" and other mystical ideas. Yes, it IS amazing. Please keep in mind that not all sounds have a harmonic spectrum[2].

Of course we are describing an ideal tone - in real life, tones waver, certain harmonics are missing, etc. Nevertheless this is the harmonic series, and measuring the spectra of violins (or any other stringed instruments), human voices, and woodwinds, for example, will reveal that this is indeed the pattern, and even in our "fuzzy" and "flawed" reality, spectra adhere to this pattern with impressive consistency.

In a tuning "according to the natural scale", we have for example a "perfect fifth" as simply the ratio between the third partial and the second partial: "3:2". In our example tone, that would be the ratio of 300 Hz to 200 Hz. Were we to want a just intonation perfect fifth above our original tone, its fundamental frequency would be found at 3/2 times the fundamental frequency of our original tone. So, 3/2 times 100 gives us 150. Our example perfect fifth has a fundamental frequency at 150 Hz.

Now, let us play our two example tones together, and we shall see why the German term is Reine, "pure", and why you'll hear "pure" used in English and many other languages as well. Let's call our first tone "Do" and our second tone, a perfect fifth higher, "Sol".

```Tone   Frequencies of partials (Hz)
Do     100  200  300  400  500  600   700   800   900   ...
So     150  300  450  600  750  900  1050  1200  1350   ...```

You see that the tones share the frequencies of some of the partials. These partials will "meld" when our Do and Sol are played together. This goes by the wonderful name of Tonverschmelzung in German. It is a very distinctive "blending" sound. If our Sol was tuned to, for example, 148 Hz, its second harmonic component would be at 296 Hz, and the two tones played together would not "meld together" at 300 Hz, but would "beat". That is, we would hear a throbbing sound, the "beat rate" of which is found by reckoning the distance in Hertz between the two near-coincident partials. In this case, 300 - 296 = 4 Hz, so we'd hear a beating of four times a second (this is like a rhythm of eighth notes at a metronome marking of 120 beats per minute).

One does not need to know of the harmonic series, nor even know how to read, or even count, to sing this.

There is more to it than this, of course, but the basic principles of just intonation are very simple. Hundreds of years ago, when the intonation of a few well-known intervals was the concern, understanding and defining "just" was not difficult. These days, though, and going on from these basics, it can get a bit more complicated...

## Just intonation in use

To start off your exploration of just intonation scales, the Gallery of 12-tone Just Intonation Scales is a good place to start.

Look at notation systems for just intonation.

The use of just intonation could be divided into these two flavors:

### Free style just

Lou Harrison used this term; it means that you choose just-intonation pitches from the set of all possible just intervals (not from a mode or scale) as you use them in music. Dedicated page: FreeStyleJI

### Constrained just

(In need of a better name maybe) Here are six ways that musicians and theorists have constrained the field of potential just ratios (from Jacques Dudon, "Differential Coherence", 1/1 vol. 11, no. 2: p.1):

1. The principle of "harmonic limits", which sets a threshold in order to place a limit on the largest prime number in any ratio (cf: Tanner's "psycharithmes" and his ordering by complexity; Gioseffe Zarlino's five-limit "senario," and the like; Helmholtz's theory of consonance with its "blending of partials," which, like the others, results in giving priority to the lowest prime numbers). See 3-limit, 5-limit, 7-limit, 11-limit, 13-limit.
2. Restrictions on the combinations of numbers that make up the numerator and denominator of the ratios under consideration, such as the "monophonic" system of Harry Partch's tonality diamond. This, incidentally, is an eleven-limit system that only makes use of ratios of the form n:d, where n and d are drawn only from harmonics 1, 3, 5, 7, 9, 11, or their octaves.
3. Other theorists who, in contrast to the above, advocate the use of products sets of given arrays of prime numbers, such as Ervin Wilson, Robert Dussaut, and others.
4. Restrictions on the variety of prime numbers used within a system, for example, 3 used with only one [sic, also included is 2] other prime 7, 11, or 13.... This is quite common practice with Ptolemy, Ibn-Sina, Al-Farabi, and Saf-al-Din, and with numerous contemporary composers working in just intonation.
5. Restricting the denominator to one or very few values (the harmonic series, isoharmonic chords, AFDOs/overtone scales).
6. Restricting the numerator to one or a very few values (the subharmonic series, IFDOs/undertone scales).

1. The use of harmonic and arithmetic mediants as was common with the Ancient Greeks. This can also involve further divisions besides two parts as seen with Ptolemy sometimes using 3 parts. The Chinese have historically used as many as 10 parts.
2. While related to the above, the use of recurrent sequences is by some included under JI as it involves whole numbers. Wilson's Meru scales are a good example.
3. Choosing some set of relatively high overtones (disregarding prime limit or subgroups), and using each overtone as a root for extended harmony within the set (primodality, nejis).
4. The use of a tritriadic scale as described by John Chalmers in 1986.

## Instruments

• The array mbira was designed by Bill Wesley as a versatile just intonation instrument, covering a 5 octave range.
• Most of Harry Partch's instruments were designed to be for just intonation.