The Harmonic Series

The founding concept behind almost all tuning systems is the Harmonic Series, the set of pitches that ring out when you play any note on an instrument. These other pitches are multiples of the base pitch, so if you play an A that’s 220 hz, you'll hear pitches of 440 hz, 660 hz, 880 hz, 1100 hz, and so on, with the pitches getting quieter as they get higher. These come out on a guitar, piano, saxophone, etc, and the differences in the weights of individual harmonics are a defining part of what makes different instruments sound different.

When we hear multiple notes together, our brain notices the ratios between them, so consonant chords should be made up of relatively simple ratios. The harmonic ratios are 1/1, 2/1, 3/1, 4/1, etc, and are the building blocks of all other ratios such as 3/2 or 5/4, so it's important to get them right.

So, why don't we just build a tuning system out of these simple ratios?

Well, that's where we come to our first example of a historical tuning system, Just Intonation, used notably in Ancient Greece.

Just Intonation

The idea behind Just Intonation, or JI for short, is that by using exact ratios to make music, the resulting intervals and chords will sound pure and simple to our brain. To see how this works, let's build a simple scale, the major scale. Specifically, this JI tuning is called Ptolemy's Intense Diatonic Scale.

We'll start with our root, or tonic, a 1/1 ratio. The ratio itself is called the unison, and it's what we're going to build other ratios from.

The next simplest ratio is a 2/1, or octave. Notes an octave apart sound so consonant together that our brain tends to hear them as different versions of the same note, a concept known as Octave Equivalency. Given this, we have a point of repetition for our scale, known as a period, so for any note in our scale, we'll include all notes that are octaves above or below it.

Next, we'll include the 3/1. This is above our octave period, so the note within the octave will be the one that's 3/2 above the unison, or the perfect fifth. This interval is a key part of most tuning systems, as it's extremely consonant, though less than the octave, while being small enough to allow chords using it to not feel too spread apart.

The perfect fifth is so quintessential that we'd like notes a fifth below our root and above our fifth, giving us ratios of 2/3 and 9/4 respectively. Bringing them into the octave, we get 4/3 and 9/8, our perfect fourth and major second. From here, we could continue expanding out in perfect fifths, which the next system we'll discuss will, but here, we'll try a different method.

The next prime harmonic above 3/1 is 5/1. Prime harmonics are interesting, because they're the only ones that can't be built from smaller harmonics, so with each one that you introduce, you get access to a new set of ratios. For this scale, 5 will be the highest prime we use, but later we'll go much higher. 5/1 can be brought down two octaves to 5/4, the major third. 1/1-5/4-3/2 is known as the most consonant standard chord, often written as 4:5:6 to show the ratios between individual notes, due to being composed of very simple ratios, with the distinct primes giving it character that octaves and perfect fifths alone don't have.

To round out our scale, we can include the notes a perfect fifth above and below our major third, giving us 5/6, or 5/3, and 15/8, the major sixth and major seventh. Our final scale is 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, characterized by abundant perfect fifths and major thirds, with simple chords such as the major triad, 4:5:6 and the minor triad, 6/5/4 (written with slashes to indicate that the numerator is shared, not the denominator) on multiple notes in the scale. This is clearly a good scale, with pure sounding intervals and chords, so what is the trade-off?

Well, not every fifth in the scale is perfect. From the major second, we have a 40/27 fifth between 9/8 and 5/3, significantly more complex than 3/2. This more dissonant interval makes the major second a very difficult note to build chords on, so when this scale is used, II chords (chords on the major second) tend to be avoided. This problem gets worse when you expand the scale:

Factor	⅓	1	3	9
⅕	16/15	8/5	6/5	9/5
1	4/3	1/1	3/2	9/8
5	5/3	5/4	15/8	45/32

The scale shown is the standard 5-limit chromatic scale in JI, 5-limit meaning no primes greater the 5. The notes are roughly equidistant, though not exactly, with 3 out of 12 fifths being 40/27s as opposed to 3/2s, and 4 out of 12 major thirds being 32/25s as opposed to 5/4s.

The issue of fifths has a potential solution in another tuning system, Pythagorean Tuning.

Pythagorean Tuning

Pythagorean tuning is a type of JI that came before standard 5-limit JI. The idea of pythagorean tuning is to maximize the number of perfect fifths in the scale, and we do this by using the perfect fifth as a generator. What this means is that we start with our root, and keep adding perfect fifths on either end until we have a 12-note scale. It looks like this:

With this type of scale construction, we get 11 out of 12 of our perfect fifths. The one note without a perfect fifth is our augmented fourth, as there is no augmented unison above it. However, there is a minor second, giving us a ratio of 262144/177147 between this 729/512 and 256/243. This interval is called a "wolf fifth", sounding significantly more dissonant than a 3/2 perfect fifth. However, in keys nearby to the one we construct the system out of, these notes rarely are used, so this issue doesn't often come up.

You may notice, however, that the intervals are significantly more complicated. The prime 5 is nowhere to be seen, as every interval comes from stacks of 3/2, leading to numbers growing far faster, having only one prime to work with. Additionally, no 5 means that an otherwise simple chord like the major triad is now a 64:81:96 chord instead of the ideal 4:5:6. It sounds similar, and its intervals are close to what we use today so you may not notice much of a difference, but the difference between the 81/64 and an ideal 5/4 is large enough to be a problem, at 81/80.

A third system came into use in the Renaissance with the aim of fixing this problem, known as Meantone tuning.

Meantone Tuning

The 81/80 ratio between 5/4 and 81/64 is noticeably large, and made thirds in Pythagorean tuning dissonant. To fix this, a Meantone fifth is flattened slightly. Any individual fifth hardly sounds different, but when you stack four of them, you get a fifth harmonic that's either exact or very close. The most common fifth tuning is 5^¼, close to 3/2's 1.5.

When this fifth is used to generate a scale, we get major triads significantly purer than in Pythagorean Tuning, while being equally abundant. And while JI has more in tune fifths, Meantone has two more in its 12 note scale, which is a compromise that was found to be worth it up until the early to mid 19th century.

The one major issue with meantone tuning is one that's found in Pythagorean tuning as well, but at a larger scale. One note has no perfect fifth, so there are two notes without major seconds, three without major sixths, and so on. This means that different keys have dramatically different tuning, and some, such as the key with no fifth, are much harder to use.

As music progressed through the 18th and 19th centuries, we began to experiment more and more with key changes and chromaticism, so Meantone became rather limiting. Keys sounding different was taken to be a major problem, leading to the system that we use today.

Equal Temperament

In the mid 1800s, we introduced Well Temperaments, a family of tuning systems with the goal of making every key usable, with no one interval too out of tune, but still distinct, with some keys having better thirds, others having better fifths, and so on. However, we eventually settled on Equal Temperament. The three systems discussed all produce chromatic scales whose twelve notes are roughly equidistant, and this fact gives us the ability to smooth over the issues that different keys produce by simply splitting the octave evenly into twelve steps of size 2^¹⁄₁₂. We lose any pure JI ratios besides the octave itself, but every key is now usable, with the same intervals as any other. The only issue is that every one of them is out of tune.

But how do we know how out of tune any interval is?

To do this, we've created a unit called cents. One cent is equal to one one-hundredth of a 12edo semitone, so there are 1200 cents in an octave, and 100 cents in a semitone. Using this logarithmic scale, we have a granular way to tell how close any given ratio comes to a 12edo interval.

Humans can, on average, tell a difference between intervals that different by at least five cents. The 3/2 perfect fifth is 701.96 cents, so 12edo's 700 is extremely close, generally sounding pure even to trained ears. However, the 5/4 major third is 386.31 cents, and 12edo's 400 is distant enough to be noticeably off. Most listeners are used to this difference, but it's the reason that you can hear "beating" or a wavering when listening to 12edo chords. Instruments that are fully in tune with 12edo are out of tune with the harmonic series.

So, what do we do with this information? 12edo is a great compromise, with every key usable, and 5-limit ratios represented fairly well. However, we don't need to stop at 12 notes. With each system mentioned, it's possible to continue adding more notes. By dividing the octave into finer gradations, we get the ability to approximate the harmonic series and other JI ratios even better than in 12edo. Many options exist, such as 17edo, 19edo, 22edo, or 41edo, but I'll be talking about 31edo. With a meantone perfect fifth, an essentially pure 5th harmonic and even 7th harmonic, and the ability to go even higher, 31edo is a spectacular tuning, with a wealth of chords and scales like nothing you've heard before.