Equal Temperaments and Just Intonations as they apply to Fretted Instruments
In western music, there is one musical scale that is used almost (but not quite) universally. This is the familiar chromatic scale, technically known as twelve-tone equal temperament. Pianos are almost always tuned to this scale, electronic keyboards have their software set up for it, and the fret spacing of almost all guitars is made for it. But of course, many other scales are possible. Looking at the wider world of possible scales, there are two leading types in use by musicians and music theorists today: equal temperaments and just intonations. For definitions of these two terms, click here.
Of these two, just intonations more closely follow the ear’s way of interpreting musical intervals. For that reason, they are often considered to be the ideal. But in practice, they can be inconvenient or impractical, especially for fretted instruments (we’ll get to the reasons why in a moment). Equal temperaments don’t have the same inconveniences.
By chance, one very practical equal temperament — namely, 12-tone equal temperament — produces a scale that approximates fairly closely the most important intervals of a basic just intonation. For this reason (and over the objections of some purists), 12-tone equal temperament has become the standard scale in western music.
Equal temperaments, by definition, have all of their tones equally spaced. The 12-tone equal temperament scale, for instance, divides the octave into twelve equal steps. Just intonation scales, by contrast, have intervals of unequal sizes. This has implications for fretted instruments. With equal temperaments, the required fret locations will be the same regardless of which note of the scale the open string is to be tuned to. But with just intonations, the fret locations for a particular desired scale depend on the pitch of the open string. So if you have different strings on the instrument tuned to different open pitches, the fret locations under each string will have to be different; the frets cannot cross the neck in straight lines. Many makers and players have experimented with fretted instruments in just intonation using many short “fretlets” placed here and there under the strings, or using wavy frets, or even using movable frets and fretlets.
To set up ExMI’s fret-calculation program to handle just intonations would require more input from the user, more code from the programmer, and a more complicated interface and output for all concerned. That’s why we’ve decided to hold off on just-intonation fretting and limit this fret calculator to equal temperaments.
I mentioned a moment ago that 12-equal is widely used because it happens to do a decent job of approximating the most important just intervals. What about other equal temperaments — what would 11 equally spaced tones per octave sound like, or 13, or 8, or 17, or 36? Most non-12 equal temperaments stray rather far from the comfortable just intervals. This tends to give them a strange sound. The strangeness is compounded for stringed instruments by the fact that musical strings produce harmonic overtones (or close approximations to them), and the idiosyncratic intervals of non-12 equal temperaments are likely to clash with those overtones. Yet experimenters have produced some fascinating music with these non-12 ETs. Each one, as the late Ivor Darreg noted, seems to have its own characteristic musical mood. Here are some notes on different equal temperaments for builders who might want to make non-12 fretted instruments.
- Temperaments beyond 12 which approximate just intervals reasonably well include 19, 24, 31 and 53.
- 19 is highly recommended. Many very successful fretted instruments in 19 have been made. The music is attractive and natural-sounding; the number of frets is manageable; and the musical logic of 19 turns out to be not far removed from that of 12.
- 24 is the quarter-tone scale — two pitches for every one in 12-ET.
- 31 also does very well in approximating just intervals, and some successful 31-tone guitars have been made. 31 frets per octave is pretty tight though, as the fret spacings get uncomfortably close. You’ll do better in a case like this with instruments of relatively long string lengths, for which the frets don’t have to be so close.
- The same is true, but more so, for 53-equal.
- 6-tone equal temperament is the whole-tone scale — it’s already available on your standard guitar or mandolin or piano. Similarly: 4-ET also is a subset of 12-ET, comprising a diminished chord; and 3-ET is an augmented chord.
- Some people find 5-tone equal temperament to be appealing. The mood is different and exotic, yet peaceful.
- Several composers have experimented with odd and alien tempermanets such as 13 and 17. About these and other improbables, all I can say is, try them!
These scales types can be described mathematically. To do this, think of each note of any given scale in terms of its frequency — that is, the fundamental vibratory frequency for that musical note. The musical intervals that define the scale can then be described in terms of the mathematical relationships between the frequencies of the scale tones. In musical strings, other factors being equal, frequency is inversely proportional to string length, so there’s a simple relationship between the mathematics of scale frequencies and the geometry of fret placement. With these things in mind, here are the definitions:
Just Intonation: “Just Intonation” refers to any tuning system in which the frequencies of the scale degrees form simple ratios. The interval of the octave, for instance, corresponds to a frequency ratio of 2:1 between two frequencies; a perfect 5th is 3:2; a major third 5:4 and so forth. You can define a just scale by giving the frequency ratio for each degree of the scale relative to the tonic note. For example, here’s the most common form of major scale in just intonation over one octave: 1:1 (the unison, or first degree to which all the following ratios relate), 9:8, 5:4, 4:3, 3:2, 5:3, 15:8, 2:1. If we take A-440 as the starting point for this scale, we get the following frequencies for the notes of the scale over one octave: 1) 440Hz; 2) 9/8 x 440 = 495Hz; 3) 5/4 x 440 = 550Hz … and so forth up to the octave at 2/1 x 440 = 880.
By selecting different ratios, an endless variety of just scales can be devised.
The intervals created this way tend to be unequal — for instance, the “whole step” between the first degree in the major scale above (1:1) and the second degree (9:8) is not the same size as the “whole step” between the second degree (9:8) and the third (5:4). As described above, this unequalness makes the business of fret placement for fretted string instruments quite a bit more complicated.
Equal Temperament: Equal temperaments are scales in which the octave is divided into some number of scale steps of equal size. For instance, the standard western scale of 12-tone equal temperament contains twelve equally spaced scale degrees per octave.
While the idea of ratios still applies in theory, the simple ratios you find in just intonation are no longer relevent here, as the situation calls for a different sort of mathematical logic. The math normally used to describe an equal-tempered scale involves finding a scale factor, which is the constant by which the frequency of each step must be multiplied to arrive at the frequency of the next step. These factors typically turn out to be irrational numbers which can, for practical purposes, be rounded off to some convenient number of digits. For instance, the scale factor for 12-tone equal temperament, rounded off to 4 digits past the decimal, is 1.0595. Thus, if you establish the first degree of the scale as A at 440Hz, the frequency of the next degree would be 440 x 1.0595 = 466.18Hz; the next would be 466.18 x 1.0595 = 493.92Hz, and so on up through the A an octave above, after 12 applications of the factor, at 880Hz.
When it comes to fret placements for fretted stringed instruments, the equal spacings of equal temperaments make matters simpler. And there are other, more purely musical conveniences associated with uniformly equal intervals as well. But these are gained at the cost of the “naturalness” and the musical clarity of the simple ratios of just intonation.