In modern musical performance, the word intonation has many meanings. For a string player, it might be the daily chore of twisting the pegs on their instruments to just the right amount. For a woodwind or brass player, it might be the constant struggle to shape their mouths in precisely the right way so as to achieve the correct pitch. And for a pianist, it’s the one thing they don’t have to worry about. But here’s one definition you won’t find as often:
Intonation is a design decision.
We’re so used to thinking about notes and pitches as a fixed foundation, thinking that a note is attributed to a particular frequency, or that the distance between any two notes is always fixed. But in truth, there’s far more to it than that. Not only are there different tuning systems out there, but it turns out that not a single one of them has managed to capture the musical ideal: just intonation.
In this article, we’ll take a look at what exactly that ideal is, why it’s been such an elusive goal for so long, and some steps that have been taken to try and catch it.
The Purest of Intervals
Several millennium ago, Greek thinker Pythagoras came up with perhaps the west’s first ever tuning system. At its core lay the idea of the pure interval.
Pure interval: an interval whose notes’ frequencies can be related by a ratio of small whole numbers.
Specifically, he defined the ratio of frequencies between a note and its equivalent note an octave down to be 2:1, and the ratio for a perfect fifth to be 3:2.
For example, if the frequency of middle C was set to 262 hertz (in reality, it's roughly 261.626), then by the 2:1 ratio the C an octave above would have a frequency of 524 hertz. The G a perfect fifth above middle C would be 393 hertz.A couple centuries later another thinker, Claudius Ptolemy, defined ratios for the other five notes in the major scale to give us Ptolemy’s Intense Diatonic Scale. Here they are, in order:
These are the ratios you subconsciously attempt to match when you play an interval on the violin, or when you spontaneously try to harmonize with the guy singing next to you in the shower.
What’s more, the “purity” of an interval isn’t a binary label. We won’t get into the mechanics on why this is how it is, but the smaller the numbers used in the whole number ratio of an interval, the “purer” we perceive that interval to be. This is why we traditionally consider the perfect fifth (3:2) to be stronger or more harmonious than the minor second (9:8) or seventh (15:8).
Don't take my word for it! Open up two tabs of this online tone generator, and create a couple of intervals using the ratios above. Do you feel that some ratios create "purer" intervals than others?As you’ve probably guessed by now, just intonation is a tuning system whereby the intervals between notes are all pure intervals.
Interval Arithmetic and Pythagorean Tuning
With just our current description, we don’t have a tuning system yet. In order to do so, we need to define some method of coming up with the other notes in our scale.
Lets say we wanted to stack two intervals on top of each other and determine the ratio between the very top and very bottom notes. To do so, we multiply the two ratios together (instead of adding, which would have been the intuitive process). Likewise, if we wanted to subtract one interval from another, we would divide the second interval by the first.
We'll stack together a perfect fifth (ratio 3:2) and a major third (ratio 5:4) to get the frequency ratio for a major seventh interval. 3/2 * 5/4 = 15/8(Note that this is the same ratio used for the major seventh in Ptolemy's Intense Diatonic Scale!)
By using this interval stacking technique, we can start from a set note frequency (for instance, the pitch A is often set to 440 hertz), then add and subtract intervals to determine the frequencies of all other notes in the scale. This methodology was how Pythagoras, after only defining the octave and perfect fifth, was able to invent Pythagorean Tuning.
Here’s the process. Starting from a base note, we build up intervals of perfect fifths until we’ve encountered all twelve notes in a chromatic scale. For example, here’s a sequence that uses C as the base note:
C → G → D → A → E → B → F# → C# → G# → D# → A# → E#
Once we know where all the unique notes were, we then multiply/divide each of them by powers of two to obtain their frequencies in every octave. With that, we get pitch values for every single note of the scale.
The Problem
So what’s wrong with this tuning system? Sure, it attempts to utilize the strongest two pure intervals, the octave and the perfect fifth, and consequently fits in well with the ideals of just intonation. But in doing so, we run into a big problem:
There are multiple ways to stack intervals such that you arrive at the same note, but at a different frequency.
As an example of this, consider again the sequence of perfect fifths, this time with another fifth added to end to complete the cycle:
C → G → D → A → E → B → F# → C# → G# → D# → A# → E# → B# (C)
Lets try calculating the ratio between the low C and high B#/C. Since this sequence crosses exactly seven octaves, we can determine the frequency ratio by simply stacking seven octaves of top of each other:
(2/1)⁷= 128:1
However, we also know that the sequence is constructed out of perfect fifths. Hence, we can also determine the frequency ratio by stacking twelve (the number of intervals) perfect fifths on top of each other:
(3/2)¹² = 129.7463…:1
This first method generated the frequency of a C via octaves, the second generated the frequency of a B# via fifths, yet they resulted in different ratios. This discrepancy between the C and B# is known as the Pythagorean comma.
The result of this discrepancy is that intervals in the more extreme ends of the spectrum become distorted, falling farther and farther away from their ideal ratios. This effect is seen particularly strongly with the perfect fifth at the end of the sequence (between the E# and B#). The effect was so disoriented it gained the nickname of “wolf fifth”.
The Pythagorean comma is concrete evidence of the impracticality of just intonation. Notes approached from different intervals or constructed using different bases fail to have the same frequencies. Not all intervals can be satisfied with only twelve notes to an octave.
Compromise and Temperaments
So what can be done about it? Clearly, creating instruments that have multiple buttons for every key variant isn’t very practical. (That’s not to say it hasn’t been tried, however. Harry Partch defined his just intonation scale with 43 pitches in an octave. He then created specially designed instruments to play in that tonality, like the Chromelodeon below.)
The solution for western classical music was to compromise. If they couldn’t satisfy possible interval, they would manipulate values such that even if none of the intervals were truly pure, they would at least come close. This was the core idea behind temperament.
Temperament: the adjusting of intervals to improve overall quality in different keys and situations.
Many different temperament systems were tried with and abandoned throughout history, but the one that is almost exclusively used today in western culture is known as twelve-tone equal temperament (12-TET). This system is often attributed to Vincenzo Galilei from the 1500s, but in actuality had been developed in various forms in China centuries beforehand.
In 12-TET, some base note is again set, and all of its octaves are determined via the 2:1 ratio. Then for each octave, the twelve intervals are proportioned out such that each part is equal on a logarithmic scale — in other words, the ratios between two adjacent notes are equal for each partition. Here are the ratios, with the just intonation versions for comparison:
Notice that while a lot of them come very close, none of the ratios except for the octaves match their pure interval counterparts. The major third ratio is particularly off — you can see a great audio and visual representation of this, along with other comparisons between 12-TET and just intonation in the following video.
In our musical culture today, we’ve become so ingrained in the distorted major third interval used in 12-TET that we’ve more or less associated it as normal.
Closing Thoughts
We’ve seen the theory behind just intonation, why its fundamentally so difficult to achieve, and the predominant compromise used to deal with it today. Now, we look what the future could have in store:
The impracticality of just intonation has always lied in its physical constraints, the fact that it’s unfeasible to create an instrument that can produce the nigh infinite different pitches producible by stacking different combinations of intervals. But the relatively recent advent of computer technology has changed the game. With the power of flexible software, we can generate any tone or pitch imaginable, and people have already utilized the technology to create snippets of music in just intonation. For instance, consider this small piece in 12-TET:
And its computer-generated just intonation version:
Clearly, we are capable of replicating just intonation to some extent. The next step would be to create keyboards that could mold their pitches subtly while being played so as to meet the just intonation standard in live performance as well.
But perhaps a more fundamental question to ask is whether or not just intonation deserves its title as the “musical ideal”. Clearly there is artistic value in the equal temperament system that is commonly used today, as it’s contributed towards the creation of a vast amount of beautiful works that would not have been possible if composers had been tied down to only using intervals that sounded decent.
Maybe in the future, even if we develop the ability to easily implement just intonation, we’ll continue to use the system we have today, or perhaps switch to an entirely different one.
Works Cited
Chalmers, John H. “Pythagoras, Ptolemy, and the Arithmetic Tradition.” Divisions of the Tetrachord: A Prolegomenon to the Construction of Musical Scales. Hannover: Frog Peak Music, 1993. N. pag. Print.
Gann, Kyle. “An Introduction to Historical Tunings.” An Introduction to Historical Tunings. N.p., n.d. Web. 08 May 2016.
Gann, Kyle. “Just Intonation Explained.” Just Intonation Explained. N.p., n.d. Web. 08 May 2016.
Garrett, Gary. “Just Intonation | Untempered Music.” Untempered Music. N.p., n.d. Web. 08 May 2016.
Johnston, Ben, and Bob Gilmore. “A Notation System for Extended Just Intonation.” “Maximum Clarity” and Other Writings on Music. Urbana: U of Illinois, 2006. N. pag. Print.
“Just Intonation vs Equal Temperament.” YouTube. N.p., n.d. Web. 08 May 2016.
Moriarty, John. “Tuning Theory 1: Just Intonation (“Microtonal” Theory).” YouTube. YouTube, 10 Aug. 2014. Web. 08 May 2016.
“Ptolemy’s Intense Diatonic Scale.” Ptolemy’s Intense Diatonic Scale. N.p., n.d. Web. 08 May 2016.
Schulter, Margo. “Pythagorean Tuning and Medieval Polyphony.” — Table of Contents. N.p., n.d. Web. 08 May 2016.
Tisue, Seth. Chromelodeon. Digital image. Flickr. N.p., 13 Apr. 2008. Web. 08 May 2016.
“Wolf Fifth.” YouTube. YouTube, 20 Sept. 2014. Web. 08 May 2016.
