### 1.2. Scales

Most modern musicians know a chromatic scale for their instrument. That scale, which sounds like an upward crawl, contains every note that the musician is likely to play in a given piece of Western music. But why is that chromatic scale, with twelve notes, the scale we learn and play today? Why not a scale with ten notes, or fifteen, or twenty? This article will explore the history of the twelve-tone equal tempered chromatic scale, and how certain ratios of sound waves sound pleasant to the ear.

#### 1.2.1. Pythagoras, the Pythagorean Scale, and the Circle of Fifths

Pythagoras, the namesake of the Pythagorean Theorem, was an ancient Greek philosopher, cult leader, and mathematician. He and his followers were obsessed with numbers, and they firmly believed in the power of whole-number ratios (like 1:2 or 5:3) to describe the world around them. Pythagoras discovered that certain ratios of string length made pleasant harmonies when plucked together. He found that, based on some arbitrary string, strings $$\frac{3}{4}$$ as long, $$\frac{2}{3}$$ as long, and $$\frac{1}{2}$$ as long (so long as linear mass density and tension are conserved, explored in a later article) would each produce pure sounding harmonies when plucked with the original string. He had discovered, respectively, the open fourth, the open fifth, and the octave.

Each of these intervals has a very open, almost hollow sound. The octave in particular has such an open sound that notes separated by some whole number of octaves are considered to be the same note, just higher or lower in pitch. It is this reason that Pythagoras is thought to have preferred the open fifth - it is the simplest interval that is not a unison (the exact same note, pitch and all) or an octave. (The ratio $$\frac{1}{3}$$ is constructed by combining an open fifth and an octave ($$\frac{2}{3}\times\frac{1}{2} = \frac{1}{3}$$), because intervals are multiplicative rather than additive.)

Any music constructed out of only three possible string lengths isn't likely to be particularly interesting, so Pythagoras or some of his followers decided to combine multiple fifths in order to devise the first formally defined scale. The thought was that, if fifths were stacked on top of each other and adjusted by octave, eventually one would get back to the starting pitch (approximately), with a full 12-tone chromatic scale created in the process. Let's build this scale like the Pythagoreans would have:

It might be helpful to sit at a piano to visualize this process. First we start with a reference pitch - I will use a C, but it doesn't matter. This C is a unison, so it is represented with the ratio $$\frac{1}{1}$$. The next note we'll derive is a G, a fifth from the C. If we're thinking in Pythagoras's strings, G is $$\frac{2}{3}$$ as long as the C string, and has a pitch with a frequency $$\frac{3}{2}$$ that of C (the reciprocal). If the C had a frequency of 100 Hz, that means the G will have a frequency of 150 Hz.

The next note is a fifth above the G, a D. This note's frequency is related to the starting pitch by a ratio of $$\frac{3}{2}\times\frac{3}{2} = \frac{9}{4}$$ (intervals multiply). Since we want our scale to be contained in one octave, we will take this ratio down an octave by multiplying it by $$\frac{1}{2}$$. This gives us the ratio $$\frac{9}{8}$$, a major second.

We will continue the process of multiplying by 3/2 to add fifths, and dividing by 2 to adjust octaves. In doing so, we will continue down what is called the circle of fifths. You can clearly see this process on the piano by stacking fifths. The notes we will derive, in order, are C G D A E B F# C# G# D# A# and E# (E# is enharmonic with F, so we shall call it F for convenience's sake). Some musicians will recognize this as the order of sharps.

Here is a list of all the ratios we get using this technique. Note that the interval column contains the fraction that must be multiplied with the previous note to get the current note - we perceive this as the space between the notes, more commonly called the interval. The cents column will be explored in section 1.2.3., for now you can simply ignore it. The scale has been ordered chromatically.

 Note Calculation Simplified Ratio Interval Cents C na $$\frac{1}{1}$$ na 0 C# $$\frac{3^7}{2^{11}}$$ $$\frac{2187}{2048}$$ $$\frac{2187}{2048}$$ 113.69 D $$\frac{3^2}{2^{3}}$$ $$\frac{9}{8}$$ $$\frac{256}{243}$$ 203.91 D# $$\frac{3^9}{2^{14}}$$ $$\frac{19683}{16384}$$ $$\frac{2187}{2048}$$ 317.60 E $$\frac{3^4}{2^{6}}$$ $$\frac{81}{64}$$ $$\frac{256}{243}$$ 407.82 F (or E#) $$\frac{3^{11}}{2^{17}}$$ $$\frac{177147}{131072}$$ $$\frac{2187}{2048}$$ 521.51 F# $$\frac{3^6}{2^{9}}$$ $$\frac{729}{512}$$ $$\frac{256}{243}$$ 611.73 G $$\frac{3^1}{2^{1}}$$ $$\frac{3}{2}$$ $$\frac{256}{243}$$ 701.96 G# $$\frac{3^8}{2^{12}}$$ $$\frac{6561}{4096}$$ $$\frac{2187}{2048}$$ 815.64 A $$\frac{3^3}{2^{4}}$$ $$\frac{27}{16}$$ $$\frac{256}{243}$$ 905.87 A# $$\frac{3^{10}}{2^{15}}$$ $$\frac{59049}{32768}$$ $$\frac{2187}{2048}$$ 1019.55 B $$\frac{3^5}{2^{7}}$$ $$\frac{243}{128}$$ $$\frac{256}{243}$$ 1109.78

I would like to point out several things. Notice how the order we derive the notes is not the order they are in this table. The order the notes are derived is simply the exponent that that 3/2 is raised to. For example, G# is derived 8th because it is calculated with 3 to the 8th power. F is derived last, with a power of 11. If you are familiar with cents, you probably have noticed that every single pitch on this scale is sharp when compared with equal temperament. This is because the perfect fifth is slightly sharp with respect to equal temperament.

A quick note on interval-preserving operations: if you've noticed, so far in this article I've been taking reciprocals and multiplying by 2 and $$\frac{1}{2}$$ a lot. What exactly do these operations do? Multiplying by 2 and $$\frac{1}{2}$$ takes a note up or down an octave - it preserves the note, but loses the interval. For example, an F is a fifth below a C, but if we transform it up an octave, it is a fourth above the C. Octaves preserve notes, but not intervals. Reciprocals do the opposite - they preserve intervals, not notes. If we take the reciprocal of the ratio defining a fifth above a reference pitch ($$\frac{3}{2}$$), then we get $$\frac{2}{3}$$, which is the ratio of a fifth below a reference pitch. For example, a C and G would turn into an F and C - the fifth is preserved, the note is not. By using these operations, we can transform and simplify some of these ratios. This will not be explored in this article.

By the F our ratio is looking pretty nasty ($$\frac{177147}{131072}$$, or about 1.352), not quite the $$\frac{4}{3}$$ that we would like from a perfect fourth. The next fifth up from an F is a C, so we would like the ratio $$\frac{177147}{131072}\times\frac{3}{2}$$ to equal exactly $$\frac{2}{1}$$. However, because of that pesky fundamental law of arithmetic, the only number that can be multiplied by $$\frac{3}{2}$$ to equal $$\frac{2}{1}$$ is $$\frac{4}{3}$$ - a perfect fourth. This forces us to make our last fifth not a perfect $$\frac{3}{2}$$, but rather a nasty and truncated $$\frac{262144}{177147}$$ (twice the reciprocal). This last fifth is called a wolf interval, presumably because it sounds mangy and disgusting and should be avoided at all costs - it is 23.46 cents flat of a perfect fifth.

Fortunately, some clever people figured out a way to hide the wolf interval in the seldom-used parts of the C chromatic scale (between the F# and the C#, for instance). But this meant that parts of the scale could never be used, hardly something that is convenient to a musician. Changing key is also made virtually impossible, because the wolf interval is lurking for scales that dare touch the forbidden notes. Additional problems rise out of this scale, now called the Pythagorean Scale - thirds don't sound particularly consonant. Thirds, which define our modern major and minor chords, are sacrificed by the Pythagoreans to keep the fifths pure and good in sound. This is why most music written before the 16th century seldom uses thirds, hence its open and "medieval" sound.

The lesson to be learned from the Pythagoreans is that simple ratios are far more pleasant to the ear. Unfortunately, because of the nature of prime factors and exponential growth, stacking fifths leads to unwieldy ratios and, therefore, unpleasant harmonies. A good scale, then, must have the simplest possible ratios. This is the idea behind the next scale we will discuss.

#### 1.2.2. The Just Scale

The most egregious fault of the Pythagorean scale, in my opinion, is its poor treatment of thirds. Luckily, a Roman man called Didymus adjusted the Pythagorean major third ratio of $$\frac{81}{64}$$ to the simple and beautiful $$\frac{5}{4}$$. This beautiful harmony is pure and happy in tone, characteristic of the major chords we are accustomed to today. He did this using what is called the syntonic comma, which is just a fancy way of saying the ratio $$\frac{81}{80}$$ (a musical comma is simply an adjustment to a scale). By multiplying $$\frac{81}{64}$$ by $$\frac{80}{81}$$ (spoiler, it equals $$\frac{5}{4}$$), Didymus established a core part of what is now called the Just Scale - the proper major third.

The Just Scale is made simply by adjusting the Pythagorean scale to nearby, and much simpler, ratios. There are a variety of ways to actually go about coming up with the precise ratios of the Just Scale, because there isn't just one standard way to go from each Pythagorean note to each Just note (the Just Scale can by derived from a number of other scales as well). The scale I use, and the one I include with Pocket Orchestra, is shown below:

 Note Ratio Interval Cents C $$\frac{1}{1}$$ na 0 C# $$\frac{25}{24}$$ $$\frac{25}{24}$$ 70.66 D $$\frac{9}{8}$$ $$\frac{27}{25}$$ 203.91 D# $$\frac{6}{5}$$ $$\frac{16}{15}$$ 315.64 E $$\frac{5}{4}$$ $$\frac{25}{24}$$ 386.31 F (or E#) $$\frac{4}{3}$$ $$\frac{16}{15}$$ 498.05 F# $$\frac{45}{32}$$ $$\frac{135}{128}$$ 590.22 G $$\frac{3}{2}$$ $$\frac{16}{15}$$ 701.96 G# $$\frac{8}{5}$$ $$\frac{16}{15}$$ 813.69 A $$\frac{5}{3}$$ $$\frac{25}{24}$$ 884.35 A# $$\frac{9}{5}$$ $$\frac{27}{25}$$ 1017.60 B $$\frac{15}{8}$$ $$\frac{25}{24}$$ 1088.27

It should be noted, however, that this scale is pretty much standard nowadays. There are some deviants, but this is the model to stick by.

Alas, the Just Scale has a fatal flaw as well - it has intervals that aren't standard. In the Pythagorean scale, all steps have one of two standardized sizes, as a result of its construction from $$\frac{3}{2}$$. The Just Scale is made by adJUSTments, so it loses this advantage. Ironically, one of the fatal flaws we were hoping to escape from the Pythagorean scale (the ability to change keys), is also present in the Just Scale, but manifest in a different form. Can we ever make a perfect scale?

NOPE. It's impossible to have a scale that both has Just-like ratios AND the ability to change keys. In modern tuning, we chose the ability to change keys over Just ratios. We can blame this decision on pianists, because pianos are not designed to have multiple Ab's on the keyboard. Piano builders "tempered" the Just scale so that intervals are equally spaced (like the Pythagorean scale), and so that pitches are ALMOST (but not quite) in Just ratios. This means that only one of each note is on the keyboard, thus allowing current piano design to exist, rather than the old harpsichord design with a 16 note scale (shown here):

 Note Ratio Interval Cents C $$\frac{1}{1}$$ na 0 C# $$\frac{25}{24}$$ $$\frac{25}{24}$$ 70.66 D- $$\frac{10}{9}$$ $$\frac{16}{15}$$ 182.40 D $$\frac{9}{18}$$ $$\frac{81}{80}$$ 203.91 D#- $$\frac{32}{27}$$ $$\frac{256}{243}$$ 294.13 D# $$\frac{6}{5}$$ $$\frac{81}{80}$$ 315.64 E $$\frac{5}{4}$$ $$\frac{25}{24}$$ 386.31 F (or E#) $$\frac{4}{3}$$ $$\frac{16}{15}$$ 498.05 F#- $$\frac{25}{18}$$ $$\frac{25}{24}$$ 568.72 F# $$\frac{45}{32}$$ $$\frac{81}{80}$$ 590.22 G $$\frac{3}{2}$$ $$\frac{16}{15}$$ 701.96 G# $$\frac{25}{16}$$ $$\frac{25}{24}$$ 772.63 A $$\frac{5}{3}$$ $$\frac{16}{15}$$ 884.35 A#- $$\frac{16}{9}$$ $$\frac{16}{15}$$ 996.09 A# $$\frac{9}{5}$$ $$\frac{81}{80}$$ 1017.60 B $$\frac{15}{8}$$ $$\frac{25}{24}$$ 1088.27

Notice how harpsichords are tuned to a combination of the Just and Pythagorean scales.

This new piano temperament, called Equal Temperament, has the additional effect of making every chord equally in tune (or out of tune, depending on your perspective).

#### 1.2.3. Equal Temperament

So how did those piano builders figure out a way to change keys? Because intervals combine by multiplication, the only way to divide an octave into twelve perfectly spaced half steps (without a wolf) is using the interval $$2^{\frac{1}{12}}$$. Equal temperament is constructed as shown in this table:

 Note Exact ratio Decimal Ratio Cents C $$\frac{1}{1}$$ 1.0 0 C# $$2^{\frac{1}{12}}$$ 1.05950 100 D $$2^{\frac{2}{12}}$$ 1.12250 200 D# $$2^{\frac{3}{12}}$$ 1.18918 300 E $$2^{\frac{4}{12}}$$ 1.25991 400 F (or E#) $$2^{\frac{5}{12}}$$ 1.33484 500 F# $$2^{\frac{6}{12}}$$ 1.41418 600 G $$2^{\frac{7}{12}}$$ 1.49832 700 G# $$2^{\frac{8}{12}}$$ 1.58741 800 A $$2^{\frac{9}{12}}$$ 1.68182 900 A# $$2^{\frac{10}{12}}$$ 1.78182 1000 B $$2^{\frac{11}{12}}$$ 1.88775 1100

Notice how in this table the cents are all precisely multiples of 100. This is because a cent is defined as one hundredth of an equal-tempered half step, or $$2^{\frac{1}{1200}}$$. To find how many cents wide an interval is, take the log base 2 of the interval ratio (like $$\frac{3}{2}$$) and multiply it by 1200. A perfect fifth is about 702 cents wide, while an equally tempered fifth is precisely 700 cents.

Unfortunately, equal temperament just destroys thirds, much like those Pythagoreans did. Major thirds are quite sharp, and minor thirds are flat, which leads to some out-of-tune chords. This is why ensembles adjust thirds so that chords are lovely and consonant and in tune. Ensemble tuning will be discussed more in the next section.