A Beginner’s Guide to Tuning
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How to start experimenting with microtones today!

PART 1: Intervals as Ratios and the Ptolemaic Sequence

We all know about the twelve notes of the chromatic scale, but did you ever wonder why we use those particular notes and not others? Have you ever wanted to experiment with different notes but didn’t know how to do it? Or maybe you’ve played around a bit with altered tunings (some keyboard synthesizers, for instance, allow you to do this, though the reprogramming can be a bit laborious) but either lacked a theoretical basis for using it systematically or found it was hard to make music with it, perhaps because you couldn’t change scales and/or modulate to a different tonal center quickly and easily.

Tuning is a big topic, and the math can sometimes seem daunting, but the basics of tuning theory really aren’t that difficult. Moreover, tools now exist that let you experiment with tuning (and even use it in performance) much more easily than has been possible in the past. What’s more… and this should be particularly exciting for the cash-strapped musician or the frugal tinkerer… you can get these tools for free! Excited to try it out for yourself? I hope so, because a lot of very interesting musical effects are possible once you move beyond the twelve tones of equal temperament.

Recommended: a MIDI-equipped keyboard synthesizer or some other way of generating MIDI signals (an electronic wind instrument, for instance, or even Finale/Sibelius MIDI playback)

Frequency and Intervals

Musicians nowadays are used to thinking of intervals in terms of half steps and whole steps, of major and minor 2nds, 3rds, and the like, perfect 4ths, 5ths, and so on. This fits the music theory we’ve inherited. But it turns out there’s a simpler (and more precise) way of naming intervals–as a ratio of two numbers. In fact, in the early days of music theory (think ancient Greece), this was the only way to describe an interval.

Let’s back up and talk just a little about acoustics, because this is very helpful for understanding tuning. A musical pitch, as you may know, is a vibration at a particular frequency. What’s doing the vibrating doesn’t really matter–it could be a violin string, a drum head, the air in the room between you and the performer, the bones in your middle ear, etc. The faster the vibration, the higher the pitch. Musicians are used to talking about pitch in terms of note names like C, F-sharp, or B-flat, but a really precise measurement of pitch is in cycles per second, also known as Hertz (Hz). So when you hear the principal oboist play an A to tune the orchestra before a concert, the air inside the oboe is vibrating 440 times per second (hypothetically; in practice, orchestras are often tuning higher than that nowadays… because orchestras are weird).

So now we have a scientific measure for pitch; what about intervals? Let’s say we have two pitches sounding simultaneously, one at 100 Hz and one at 200 Hz. To find the interval between them, simply divide the bigger number by the smaller number:

200 Hz / 100 Hz = 2/1,

so the interval is 2/1. In other words, in the time it takes for the higher frequency to complete two cycles, the lower frequency will complete one cycle. The interval 2/1 is what musicians call an “octave,” and it’s the most basic interval in all traditional tuning systems (including, of course, ours).

How do we find the octave above 200 Hz? Simply multiply 200 Hz by the fraction 2/1:

200 Hz * 2/1 = 400 Hz

Notice that whereas the pitch increased by 100 Hz between 100 and 200 Hz, here the pitch increased by 200 Hz, even though we ascended by the same interval (an octave). This is because the relationship between frequency (as measured in Hz) and pitch (as our ears and brain perceive it) is exponential rather than linear.

As an example, successive octaves above 100 Hz are as follows: 100 Hz, 200 Hz, 400 Hz, 800 Hz, 1600 Hz, etc. As you can see, the higher up the frequency spectrum we go, the greater the distance (in Hz) between successive octaves (or any other interval, for that matter). It would of course be more convenient computationally if an interval were a fixed number of Hertz (if two pitches separated by an octave were always 100 Hz apart, for instance), but this does not match our perception. We perceive, then, a fixed interval not as a fixed number of Hertz but as a fixed ratio of frequencies.

The Harmonic Series

Let’s talk for a moment about the harmonic series, because it will allow us to derive all the intervals we need. If you’ve ever played a brass instrument or taken an orchestration class, the harmonic series should be a familiar concept. Perhaps the easiest way to conceptualize it is in terms of a vibrating string–a cello C-string, let’s say.

The string vibrates as a whole, of course (top of Fig. 1), generating the lowest or fundamental frequency (C2 in the case of the cello C-string). But one of the properties of all real-world vibrating objects is that they vibrate in several modes at once. In the case of our cello string, for example, while the string is vibrating as a whole, it’s also vibrating in half (second example in Fig. 1). This second mode or second partial produces a frequency double that of the fundamental. In other words, the interval between the fundamental and the second partial is 2/1. And what happens when we double the frequency? We go up an octave, of course, so the second partial is an octave above the fundamental (C3 in our cello example). The string is also vibrating in thirds (third example in Fig. 1), producing a frequency three times that of the fundamental (an interval of 3/1 above the fundamental, or an octave and a fifth). These fractional vibrations continue to higher and higher frequencies, and in theory there’s no upper limit. The smaller the fraction, though, the less energy there tends to be in the vibration, so that very high partials will tend to be inaudible.

But if all vibrating objects produce these partial vibrations, each of which generates a distinct frequency, why do we hear only one pitch? It’s because our brains perform a sort of spectral analysis on sounds without our even realizing it! Though you’re hearing many frequencies, your brain fools you into thinking you’re hearing a single pitch with a particular timbre or tone color. The tone color of a given acoustic sound, then, is simply its overtone structure, which is to say the relative strengths of the upper partials and how they change over time. Actually, it is possible to train your ears to distinguish individual overtones, but for most people this demands a certain degree of concentration.

Here’s the harmonic series of our cello C-string, up to the 16th partial. The number beneath each note indicates the number of the partial (i.e.: how many parts in which the string is vibrating to produce that frequency).

As it turns out, it’s very easy to figure out the ratio of a given interval simply by comparing partial numbers in the overtone series. As already noted, the interval between the second partial (2) and the fundamental (1) is 2/1 (by convention, we generally place the larger number in the numerator or top of the fraction). Similarly, the interval between the third partial (3) and the second partial (2), a perfect fifth, is 3/2. Easy, right? What about the interval between the fourth and second partials? Well, looking at it on the staff, we can already tell it’s going to be an octave (2/1). Comparing the partial numbers, we find that the interval is 4/2, which of course reduces to 2/1. We can now formulate a general rule:

To find the interval between any two partials of the harmonic series, use the higher number as the numerator of the fraction and the lower number as the denominator.

Just Intonation and the Ptolemaic Sequence

Let’s now examine how our familiar diatonic major scale was built. For historical reasons, this scale is sometimes known as the Ptolemaic Sequence, since it was first described by Claudius Ptolemy of Alexandria (the same man who gave us, among other things, the geocentric model of the solar system) in the second century CE. Note that Ptolemy himself called this the “intense diatonic” scale.

Here’s the scale, written as C major:

Note that the ratios given here describe the intervals formed between each note and what we’ll call (for lack of a better term) the “tonic” pitch, 1/1 (here, middle C) and not the intervals formed between successive notes of the scale. So, for instance, 5/4 is the ratio between C and E, 4/3 is the ratio between C and F, etc.

We’ll now build the scale one note at a time, using the harmonic series. It’s worth noting at this point, though, that the choice of this particular scale and indeed the choice of intervals within it is more or less arbitrary. There is nothing about this scale that makes it more basic or natural in a scientific sense than any other scale. It is not derived as a mathematical formula is derived. Instead, we simply decide we want a certain succession of intervals (major 2nd, major 3rd, perfect 4th, etc.) and then use ratios of the smallest integers we can manage to get those intervals.

We will also arbitrarily decide that, since the seventh partial sounds funny to our ears (i.e.: since it’s very much “out of tune” with respect to our familiar tuning system), we won’t use 7 (or its multiples, like 14) in any of our intervals. Since we’re not using 7, let’s also exclude prime numbers greater than 7, like 11 and 13 (again, these partials sound strange to our ears). Non-prime numbers, of course, are simply multiples of smaller numbers that we are using, so we’ll include numbers like 15 (multiple of both 3 and 5) and 16 (multiple of 2, 4, and 8). Because 5 is the largest prime number we’re using, we call this sort of tuning system 5-limit just intonation.

Examining the harmonic series in Fig. 2 above, and proceeding from bottom to top (smaller integers to larger integers), we find that the first major second we come to is between the 8th and 9th partials (remember 8/7 is not allowed in 5-limit JI), so let’s use 9/8 for our first interval in the scale, between C and D. Next we want a major 3rd between C and E. We use 5/4, as it’s the lowest major 3rd in the harmonic series. Now we need a perfect 4th. Well, in the harmonic series, there’s a perfect 4th between the 3rd and 4th partials, so let’s use 4/3. The perfect fifth 3/2 is an easy choice; after all, it’s the first interval after the octave in the harmonic series. To get a major 6th, we’ll have to use non-adjacent partials. Fortunately, there’s a major 6th quite low in the series, between the 3rd and 5th partials, so we use 5/3. Finally, we use 15/8 for our major 7th, since the major 7ths 11/6 and 13/7 aren’t permissible in a 5-limit system. Voilà! We have our major scale built from small-integer ratios! This scale is nice because it allows us to form very consonant major triads (1/1, 5/4, 3/2) on scale degrees 1, 4, and 5.

At this point we know all the intervals in our scale with respect to the tonic. It would, however, be nice to have a way of figuring out intervals with respect to other notes in the scale too. This would allow us to know, for instance, what the intervals are between successive steps of the scale. It would also let us verify that the triads built on scale degrees 4 and 5 are indeed major triads of the form 1/1, 5/4, 3/2 (rather than simply taking my word for it). Fortunately, it’s very easy to do this:

To add two intervals, multiply the ratios.

To subtract one interval from another, divide the larger ratio by the smaller ratio.

So to find the interval between scale degrees 2 and 3, for instance, divide 5/4 by 9/8. If you’re doing these computations manually (recommended for practice), an easy way to divide one fraction by another is to flip the numerator and denominator in the second fraction and then multiply the two fractions (so multiply 5/4 by 8/9). We get 40/36, which reduces to 10/9 (divide both numerator and denominator by 4), so the interval between D and E in our example is 10/9. If you look back at the harmonic series on C (Fig. 2), you’ll notice that 10/9 is the interval between the 10th partial (an octave above the 5th partial, which we used to build scale degree 3) and the 9th partial (which we used to build scale degree 2). Notice also that 10/9 is a slightly smaller major 2nd than 9/8, so in contrast to equal temperament, not all the whole steps in a just-intonation major scale are the same size.

As another exercise, let’s verify that the triad built on scale degree 4 is indeed a major triad of the form 1/1, 5/4, 3/2, just like the triad on scale degree 1. Let’s do it this time by adding intervals. We start with scale degree 4 (4/3) and add a major 3rd (5/4):

4/3 * 5/4 = 20/12 = 5/3

5/3 matches our scale degree 6, so from 4/3 to 5/3 is a 5/4 major 3rd. Now let’s try the perfect fifth (3/2) above 4/3:

4/3 * 3/2 = 12/6 = 2/1

2/1 is of course the upper octave of 1/1, so we can now say for certain that the triad built on 4/3 in our Ptolemaic Sequence is a 1/1, 5/4, 3/2 major triad. For practice, try doing the same thing with the triad built on 3/2. You might also try finding the intervals between all adjacent notes of the scale.

Next:

In the next part in this series we’ll look at some other interesting tuning systems, including Pythagorean tuning (the basis for Medieval music) and some bizarre and fascinating ancient Greek scales. We’ll also talk about equal temperament and why it has become so ubiquitous, we’ll introduce the cent as a unit of measure, and we’ll discover how to translate ratios into cents. Stay tuned!