# Just intonation

## Contents

# Just Intonation explained

Just Intonation (JI) describes intervals between pitches by specifying ratios (of rational numbers) between the frequencies of pitches.^{[1]}

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. Where 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).*

*6. Restricting the numerator to one or a very few values (the subharmonic series or aliquot scales).*

to this can be added

*7. 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.*

*8. 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.*

# Variations on 'Just'

Regular Temperaments are just intonation systems of various harmonic limits with certain commas 'tempered out'

# Links

Gallery of 12-tone Just Intonation Scales

List of root-3rd-P5 triads in JI

58 note 11 limit JI scale - hyper-Partchian!

# Articles

- Wikipedia article on just intonation
- Just Intonation by Mark Nowitzky Permalink
- Just Intonation Explained by Kyle Gann Permalink
- Anatomy of an Octave by Kyle Gann Permalink
- What is Just Intonation? by David B. Doty Permalink
- What is "just intonation"? by Carl Lumma Permalink
- A Response to Julia Werntz by David B. Doty Permalink
- Comma Sequences by Gene Ward Smith Permalink

## References

- ↑ Just Intonation is sometimes distinguished from
*rational intonation,*by requiring that the ratios be lower than some arbitrary complexity (as for example measured by Tenney height, Benedetti height, etc.) but there is no clear dividing line. The matter is partially a question of intent. The rank two tuning system in which all intervals are given as combinations of the just perfect fourth, 4/3, and the just minor third, 6/5, would seem to be a nonoctave 5-limit just intonation system by definition. In practice however, it casually suggests a rank two 7-limit microtempering system because of very accurate approximations to the octave and to seven limit intervals: (6/5)^2/(4/3) = 27/25, the semitone maximus or just minor second; and (27/25)^9 is less than a cent short of an octave, while (27/25)^2 is almost precisely 7/6, the septimal minor third. - ↑ All manner of bells, gongs, percussion instruments, synthesizer sounds, have spectra that follow their own rules, usually very complex. Inharmonic tones can be found in otherwise harmonic spectra, and instruments with harmonic spectra may have inharmonic spectra during the attack portion of the sound. Loudly played brass instruments, for example, have a moment of extremely complex sound not unlike that of striking a piece of metal, followed by a moment in which the partials are "stretched" according to a more complex rule than simply multiplying by, 1, 2, 3, etc., before settling down into a harmonic series accompanied by various amounts of characteristic "noise". A breathily played flute has a large addition of inharmonic material, a "jinashi" shakuhachi flute is an excellent example of an instrument of varying harmonicity and inharmonicity.