The Just Intonation Issue

We were on a long drive to Maine when I said this to my wife. The trumpet player on the radio was wincingly sharp and I felt the need to defend him, somehow, by dusting off this pedagogical chestnut as a mea culpa. She raised one eyebrow—her “Let’s pretend you didn’t say that” look. I’m used to her incredulity—most of it properly attributed—but this particular raised eyebrow was more than the typical noting of my tomfoolery; this eyebrow carried with it an epiphany about, among other things, our specific relationships to the musical concept of tuning.

To say something is sharp or flat is to work only with the surface tension of intonation. Within the Western musical tradition, there is, conceptually, a fairly wide spectrum of tuning systems: ways of articulating a series of frequencies into harmonies and melodies that are pleasing to the ear. These sweep may include anything from an extremely flexible concept of intonation as a way to color sound to a more exacting system that attempts to make pitches fit in a mathematically exact architecture.

In this issue, we go deeply into work that is related in some way to one of these tuning systems: just intonation or JI. Just intonation is, many believe, a more “pure” way of tuning and offers greater timbral and sonic possibilities than equal-temperament—the de facto form of intonation in Western music today. Its foundation is ancient, but it is reemerging as an important conceptual tool in some of the more radical contemporary composition of the 21st century. 

This issue does not intend to offer just intonation as a perfect system. SA20 is simply a chance to understand music in a different way. No matter how you choose to engage with just intonation as it’s discussed in this issue, or any alternative to equal temperament, being exposed to different ways of imagining intonation and sound will only open your ears to new possibilities. Keep an open mind and your ears can’t help but follow.

For those unfamiliar with the concepts of tuning, intonation systems, just intonation, etc., an explanation of some basic concepts and terms may be helpful before diving into the issue. The information found below is just enough to give you a head start.

The earliest music theorists were mathematicians and natural scientists trying to understand the way the vibration of a sounding body creates an audible pitch. Their primary tool was the monochord, a one-stringed instrument that, when its string is plucked, produces a specific pitch depending on the length and tension of the string. That set pitch can then be raised by shortening its length with a fret or pressure from the player’s finger—similar to the way a guitar or violin works. 

These theorists used mathematical ratios to explain where the finger should be placed on the string to create a specific pitch. The use of ratios is very important for any discussion of just intonation, so a simple example may be helpful. 

Say a monochord is tuned so that it plays an A (vibrating at 440 Hz) when you pluck its open string. This A would be considered the fundamental and expressed as 1:1. If you wanted to then play the A one octave higher (vibrating at 880 Hz), you would need to make the string twice as short by somehow stopping the string precisely in the middle. This would be a ratio of 2:1, and therefore, the theorist would write that the ratio 2:1 represents a pitch one octave higher than 1:1, or the fundamental. 

To get a more in-depth explanation of the theory, practice, and art of JI, I suggest you follow this article with David Doty’s writing in this issue.

Tuning is the term used to describe the basic systems of intonation that will be discussed later in this introduction. A basic feature of tuning is that it consists of simple superparticular ratios—meaning both the numerator and the denominator of the ratio are whole numbers. For example, an octave is expressed as the ratio 2:1, a perfect fifth as 3:2, major third as 5:4—all simple whole number ratios. 

If a theorist feels that notes should be more evenly spaced, for example, or adjusted to accommodate the physics of a new instrument or compositional need, they may propose a modification to the ratios to readjust the basic tuning system. This modified tuning system is called a temperament, and these adjustments mean that we must abandon our lovely simple ratios for numbers a bit more gnarly such as 216:125, the diminished seventh in meantone temperament. 

As a way of guiding ourselves through the development of 216:125 from 3:2, it is helpful to understand why musicians and theorists have been hard at work with these systems. And, that is a story best told through the history of three major ways of conceptualizing intonation: just intonation, Meantone temperament, and, what we commonly use today in the West, equal temperament.

If you’ve studied basic geometry, you know the name Pythagoras from his biggest hit: a2 + b2 = c2. However, the Greek mathematician, alive roughly between 570 and 495 BC, also played a major role in early music theory. As in his eponymous triangle equation, his method of tuning (note: tuning versus temperament) bears his name. Pythagorean tuning is one of the most basic systems, built only on the ratios that produce octaves (2:1) and fifths (3:2). 

This system has an obvious limitation. If only octaves and fifths are derived, wouldn’t you just create music that is ultimately made of, well, octaves and fifths? The answer is yes, but there is a simple way to create melodic variety from Pythagoras’s limited set of ratios. A diatonic or chromatic scale can be created from an accumulations of fifths. 

Here’s another example using the monochord from above: with the A as our fundamental (1/1), one can come up with a perfect fifth E using the ratio 3:2. Another fifth can be built off of the E by applying that same 3:2 ratio and getting something like a B. Then, applying the 3:2 ratio to the B, we get an F#, and so on and so forth: A-E-B-F#-C#-G#-D#-A#-F-C-G-D-A follows the whole process through to a version of a chromatic scale.

Composers of this period could make melodies consisting of a small group of pitches derived from this process. Keep in mind that this music was to be performed on some derivation of the monochord. So simplicity was key. 

But there must be a problem with this system, or we’d still be using it, right? If you follow Pythagoras’s theory, building a chromatic scale from fifths, and then convert the notes derived from the simple ratios into their sounding frequencies (their vibrations per second, or cents), then you will see that the system doesn’t mathematically cycle back to its beginning, and pitches from points far apart on that cycle of fifths can be very harsh to our ears. In fact, by building a scale using the fifths in our example above, the major third (between the fundamental A and the C# built from the fifths) is about 1/9th of a tone sharp. This amount of derivation is called a diatonic comma. 

Another famous Greek, Ptolemy (c. 100–170) is, arguably, responsible for the next big innovation in intonation. Ptolemy built his system by providing superparticular ratios for the first five intervals of the overtone system: the octave (2:1), perfect fifth (3:2), perfect fourth (4:3), major third (5:4), and minor third (6:5). This system is called just intonation and allowed for a much greater set of melodic and harmonic possibilities that are still simple enough to be playable on basic string instruments. 

Since just intonation is featured in this issue, I’ll leave more elucidation of its history, theory, and uses to the experts. 

Composers working with mixed ensembles and larger forces need a standardized way to tune and, with equal temperament, all the keys were “in tune” in exactly the same way. The ability to modulate freely from one key to a different far-away key became viable. For these reasons, equal temperament remains the primary system used in Western music today.

The first exact calculation of equal semitones was published by Zhu Zaiyu in late 16th-century China, but the concept of equally tempered pitches reaches back through history to the early writings of Aristoxenus in the 4th century BC. While the math of equal temperament is much more complex, its conceptual explanation may be one of the simplest. Equal temperament simply begins with the 1:1 octave of Pythagorean tuning and divides it into twelve, equally spaced, semitones.  

However, equal temperament comes at a price. First, just because the intervals are all equal doesn’t mean they are all perfectly in tune. Remember that through this entire history, there have been adjustments made for the diatonic comma and a plethora of other discrepancies. When equal temperament forces each semitone into its position, equidistant from every other semitone around it, it erases the ability to adjust for intonation discrepancies. It also limits the specific character, or feeling, that came from the acoustic effects of that specific set of discrepancies. The sound of A major is now, more or less objectively, the same as the sound of G minor.

This issue is not a bible of all things JI. The theory and the ways in which it has been used, especially in the 20th century, is so vast that there is no single place that could ever amass anything close to a complete overview. We strive, instead, to give you enough information to spark an interest that grows into your own research project. There is a phenomenally colorful and complex body of work based on just intonation in the world, and not all of it will be of interest to you, but if this gives you an appreciation for the expressive capabilities of a non-equal tuning system or sends you down a deep and lasting rabbit hole of a specific composer that lights your fire, then we’ll consider it a success.

In the ensuing articles of SA20: The Just Intonation Issue, you’ll have a chance to explore one specific concept of building a tuning system in a multitude of ways—the technical, theoretical, historical, and pragmatic. There may be moments when the light comes on and you can appreciate a new way of conceiving intonation, and there will most definitely be moments when you are overwhelmed with the complexity required to explain the building of a composition using just intonation as a basis. I suggest that you take in what you can and concentrate on the actual music being talked about and made by our contributors for this issue. Find what makes you thrum and follow that to your own understanding.