Music Theory from Basic Principles (Part 1)

2025-07-20

Disclaimer: I am not a music theory expert. This text summarizes my understanding, and is probably full of mistakes. Think of it as an exploration.

Note: I intentionally avoid standard music theory terminology. I do this to improve my communication with the people educated in music who are not trained in math. In my experience, when you use a standard term (e.g. "Pythagorean tuning"), people automatically bring hidden assumptions into the debate without realizing it. When we are building anonymous "Scale 0", we are forced to just use only the properties in the text without using external knowledge.

As a child, I played guitar for a few years. I was also supposed to visit music theory. But the theory teacher was unable to explain why I needed it. So I skipped the theory classes completely. Last year I started to play cello and thought it may be a good idea to give music theory a second chance. Moreover, compared to my childhood, I now have a PhD in computer science, so hopefully I am more ready for understanding theories.

I tried to read some books, papers, and watch videos, but it was quite hard for me. They usually avoid formal definitions or they use the same term in different contexts without proper explanation. My favorite is the argument by piano, "some property holds because on piano …" like a piano were a universal principle in the universe. I always imagine the LHC producing pianos among other elementary particles.

ChatGPT's visualization of a piano produced by LHC

Another confusion for me was how things that are culturally dependent, biological facts, physical properties, and mathematical consequences are often explained together without distinguishing their origins.

I am aware that music is about emotions and cannot be fully captured formally. I actually want music to remain mostly an emotional experience for me. But this does not discourage me from trying to explain music theory in a slightly unusual way: build it from basic principles.

In other words, I aim to give a few elementary principles, explain where they come from, and then derive music theory from these principles without introducing new, unexplained things along the way.

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Definitions

Sound is a vibration that travels as a wave through a medium (like air, water, or solids). It starts with something vibrating (e.g., vocal cords, guitar strings, speaker diaphragm). This creates alternating regions of compression and rarefaction, forming pressure waves that travel through the medium. Eventually, these waves reach a receiver (like your eardrum), which converts the vibrations into neural signals that your brain interprets as sound.

Principles

Principle 1: We perceive fundamental frequency as pitch

An object doesn't just vibrate at a single frequency. It vibrates at a primary frequency called the fundamental frequency (or first harmonic), which determines the pitch we perceive. Simultaneously, it vibrates at higher frequencies called overtones, often forming a harmonic series: $f, 2f, 3f, 4f, \dots$

The spectrogram when playing a single open string on my cello. y-axis is time; x-axis are frequencies.

In the following text, we will ignore overtones and represent a tone by a single number: its fundamental frequency.

Principle 2: Human hearing works on a logarithmic scale

Human perception often follows the Weber–Fechner law, which states that perception works on a logarithmic scale (or “log scale” for short). Hearing is no exception, both for loudness and pitch perception. We perceive pitch intervals not as linear differences but as ratios.

What does this mean? On a linear scale, the distance between 1 and 2 is the same as the distance between 10 and 11. On a logarithmic scale, the distance between points is about their ratio. So, the perceived "distance" between 100 Hz and 200 Hz (a 2:1 ratio) is the same as the perceived distance between 1000 Hz and 2000 Hz (also a 2:1 ratio).

Let's take an example of integers between 1 and 32. On a linear scale, they are evenly distributed points:

On a log scale, the same points look like this:

What is the difference? We have much bigger spaces between lower points, and they become more condensed for higher numbers. This provides us with “better” resolution for smaller numbers and lower resolution for bigger numbers. It is useful for perceiving a large range of stimuli.

How quickly does the spacing decrease? Proportionally to the value's magnitude. So instead of equally distributed points, let's plot points [1, 2, 4, 8, 16, 32] (where each number is generated by multiplying the previous number by 2). The blue points are on the linear scale; the yellow are on the log scale.

We see that when we multiply by 2, the points have the same distance on a log scale. This is a crucial property.

If we move a point by the same distance on a linear scale, we are adding the same number (+2 in the following example; all the red arrows have the same length):

When we move by the same distance on a log scale, we have to multiply by the same number (*2 in the following example; all the green arrows have the same length).

We will work almost exclusively on a log scale from now on. Thus, our "basic" operation for moving between pitches will be multiplication, not addition.

Principle 3: Octave Circularity

Two tones are an octave apart if their frequencies have a ratio of $2:1$ . We perceive these tones as being, in some sense, the "same" note, just higher or lower.

If we write our base pitch as $f$ , then its octave equivalents are $\dots, \frac{f}{4}, \frac{f}{2}, f, 2f, 4f, \dots$ .

Note that they lie evenly spaced on a log scale. Let's assume a pitch of 440Hz. We can generate “octave equivalent” tones as follows (the yellow point is 440, green arrows are multiplication by 2, blue arrows are division by 2):

Out of curiosity, let us look one last time at the linear scale and see the same points:

Octave circularity seems to be something that we share with some animals.

Mathematically speaking, this principle establishes a cyclic multiplicative group. "Cyclic" means that it behaves like a wall clock: when the hand reaches 12, it starts over. In our case, our range is not 0-12 but 1-2. "Multiplicative" in the name just means that we are moving by multiplication rather than addition (as in the case of the clock).

Principle 4: "Small ratios" sound good together

Two tones $f$ and $\frac{a}{b}f$ , where $a$ and $b$ are small integers, sound good when played together. Their waveforms align periodically, creating repeating patterns that our brains perceive as pleasant.

For example, take $a = 3$ and $b = 2$ .

A tone of $\frac{3}{2}f$ sounds harmonious with a tone of $f$ because their sine waves align every few cycles. Let's visualize this. The following image shows a signal with frequency $f$ . To emphasize repetition, the start of each period of the sine wave is marked by a blue dot.

Now, consider a signal with frequency $\frac{3}{2}f$ . The start of each new period is marked by a star.

If we overlay the two images, we get:

Here we can see that the blue dots and stars overlap at points 2, 4, 6, etc. (assuming the first point is at time 0), so both sine waves start a new period simultaneously at these points. It should not be surprising that if we add these two waves together, we get something that repeats from these points. The green line is the sum of the blue and the orange waves. Blue dots and orange stars still have their original meanings.

What is the general rule? If we have tones with frequencies $f$ and $\frac{a}{b}f$ , where $\frac{a}{b}$ is a fraction in simplified form, then the combined signal will have a fundamental frequency of $\frac{f}{b}$ .

So if $a$ and $b$ are small integers, then $f$ and $\frac{a}{b}f$ have frequencies that are close, and their combined signal also has a relatively close frequency (i.e., the resulting wave repeats almost as often as the original ones).

As $a$ and $b$ grow large (e.g., $\frac{211}{111}$ ), the repetition interval becomes too long, and the resulting sound is perceived as dissonant.

Harmonics (multiples of a fundamental frequency) fit this model well. Ratios like $\frac{2}{1}$ , $\frac{3}{1}$ , and $\frac{4}{1}$ sound consonant because their waveforms align neatly with the base tone.

For example, let us take $\frac{2}{1}f$ . The combined signal looks as follows:

Principle 5: Western music adds cultural constraints

For most of this principle, I have not been able to find exact reasons; it seems to be based mostly on historical and cultural factors.

a) Ratios use only prime factors 2, 3, and 5.

So $\frac{15}{16}$ is acceptable because $\frac{15}{16} = \frac{3\times5}{2\times2\times2\times2}$ . On the other hand $\frac{14}{15}$ is not, because $14 = 2 \times 7$ .

b) Two triples are musically significant: $1, \frac{5}{4}, \frac{3}{2}$ and $1, \frac{6}{5},\frac{3}{2}$ .

The first triplet has the nice property that its frequency ratios are 4:5:6. It is also connected to the first few harmonics $2f, 3f, 4f, 5f$ : 2 and 4 are whole octaves, 3 shifted by an octave is $\frac{3}{2}$ , and 5 shifted down two octaves is $\frac{5}{4}$ . So $\frac{3}{2}$ and $\frac{5}{4}$ are connected to the first two harmonics that are not simple octaves.

For the second triplet, the ratios are 10:12:15. The $\frac{3}{2}$ is the same as in the first one. And $\frac{6}{5}$ can be seen as a move by $\frac{5}{4}$ in the opposite direction from $\frac{3}{2}$ .

The blue points are the first triplet, the orange points are the second triplet, and the red arrow is multiplication/division by $\frac{5}{4}$ .

The second triplet also has a unique property. If we want a sequence of ratios 1, X, Y, 2 such that we minimize the largest denominator that occurs in any pairwise ratio, then X=6/5 and Y=3/2 is the optimal solution.

	$1$	$\frac{3}{2}$	$\frac{6}{5}$	$2$
$1$	$1$	$\frac{2}{3}$	$\frac{5}{6}$	$\frac{1}{2}$
$\frac{3}{2}$	$\frac{3}{2}$	$1$	$\frac{5}{4}$	$\frac{3}{4}$
$\frac{6}{5}$	$\frac{6}{5}$	$\frac{4}{5}$	$1$	$\frac{3}{5}$
$2$	$2$	$\frac{4}{3}$	$\frac{5}{3}$	$1$

c) Western music favors 7- and 12-tone scales.

Principle 6: Human hearing is not perfect

We don’t need exact ratios (like $\frac{3}{2}$ ) for notes to sound consonant. We only need to get "close enough."

Constructing Scales

If we want to play music, it is handy to select a set of tones (and give them names). The goal of this section is to build a set of notes — a scale. We'll represent notes as numbers relative to a starting frequency of 1.

We want a finite set of tones to play music. From Principle 4, ratios with small integers are good. But if we only include octaves, all tones will be perceived as the same.

So we start with the next best ratio: $\frac{3}{2}$ . Let our first attempt at a scale be the set ${1, \frac{3}{2}}$ . Let's visualize our first scale, naming the tones $t_0$ and $t_1$ .

From Principle 3, it is enough to select tones within the range of one octave—in our notation, tones with ratios in the range [1, 2). We can generate other tones from our minimalistic scale by applying octave circularity, multiplying our initial tones by $2^n$ where $n \in \mathbf{Z}$ .

Having just two tones makes for a poor scale, so let's explore how to add more tones.

Approach 1: What if $\frac{3}{2}$ is good enough?

Let us assume that $\frac{3}{2}$ is such a good ratio that we only need to work with it. How can we extend our minimalistic 2-tone scale? We can multiply $\frac{3}{2}$ by $\frac{3}{2}$ again to get $\frac{9}{4}$ , which will be our next tone, $t_2$ .

Since $\frac{9}{4}$ falls outside the interval [1, 2), we use octave circularity and fix it by dividing by 2, getting a value of $\frac{9}{8}$ that is inside the interval. In the following figure, the blue arrow represents division by 2; the orange arrows represent multiplication by $\frac{3}{2}$ .

This gives us a procedure for creating additional tones for our scale. We will take a power of $\frac{3}{2}$ and divide it by 2 enough times to get it into the interval [1, 2). The following picture shows how we get $t_3$ , i.e., $\frac{(\frac{3}{2})^3}{2}$ .

Note that we may need to divide by $2$ several times for larger numbers, e.g., $t_4 = \frac{(\frac{3}{2})^4}{2^2}$ . Therefore, you can see two blue arrows in the following figure.

Using this approach, we generate more and more tones. Let's do it up to $t_{11}$ and we get our scale:

You may wonder why we stopped at $t_{11}$ . Let's look at what happens for $t_{12}$ :

$t_{12}$ is very close to $t_{0}$ . The reason is that $(\frac{3}{2})^{12} = \frac{531441}{4096} \approx 129.75$ . This number is very close to $128 = 2^7$ . So when we are dividing by 2 seven times, we eventually get a value relatively close to 1.

Because of this, generating new tones this way would basically be starting over from scratch and generating tones similar to the previous steps ($t_{i}$ and $t_{i+12}$ are close together).

So this is it. We have created our first non-trivial scale: $t_{0}, \dots, t_{11}$ . We will name it Scale 0. Let's summarize its properties:

• It is quite evenly distributed across the interval [1, 2).
• Tones $t_{i}$ and $t_{i+1}$ (in order of generation) always have a ratio of $\frac{3}{2}$ or $\frac{3}{4}$ (i.e., $\frac{3}{2}$ moved by an octave).

Specifically: $\frac{t_1}{t_0} = \frac{3}{2}$ , $\frac{t_2}{t_1} = \frac{3}{4}$ , $\frac{t_3}{t_2} = \frac{3}{2}$ , $\frac{t_4}{t_3} = \frac{3}{4}$ , $\frac{t_5}{t_4} = \frac{3}{2}$ , $\frac{t_6}{t_5} = \frac{3}{4}$ , $\frac{t_7}{t_6} = \frac{3}{4}$ , $\frac{t_8}{t_7} = \frac{3}{2}$ , $\frac{t_9}{t_8} = \frac{3}{4}$ , $\frac{t_{10}}{t_{9}} = \frac{3}{2}$ , $\frac{t_{11}}{t_{10}} = \frac{3}{4}$ .

We can visualize this as follows. Yellow arrows are multiplication by $\frac{3}{2}$ . Blue arrows are multiplication by $\frac{3}{4}$ .

On the other hand, Scale 0 has some problematic properties. If we look at the ratios with respect to the initial tone $t_0=1$ :

$t_1 = \frac{3}{2}$ , $t_2 = \frac{9}{8}$ , $t_3 = \frac{27}{16}$ , $t_4 = \frac{81}{64}$ , $t_5 = \frac{243}{128}$ , $t_6 = \frac{729}{512}$ , $t_7 = \frac{2187}{2048}$ , $t_8 = \frac{6561}{4096}$ , $t_9 = \frac{19683}{16384}$ , $t_{10} = \frac{59049}{32768}$ , $t_{11} = \frac{177147}{131072}$ ,

The ratios between consecutive notes in the sorted scale are $\frac{256}{243}$ and $\frac{2187}{2048}$ . The former is the yellow arrow and the latter is the blue arrow in the following image:

The large numbers in these ratios are a problem according to Principle 4. In the next section, we can try to fix this.

Approach 2: Define goals, then search

In Approach 1, we defined a procedure that generated a scale and then observed its properties. We can turn this around: define the desired properties of the scale and then try to find ratios that best match these conditions.

We define the following conditions that the resulting scale must hold:

• We are looking for a scale with 7 tones (Principle 5c).
• Ratios can be factored only by 2, 3, and 5 (Principle 5a).
• Each tone is part of a triplet whose members are also part of the scale and have ratios of 4:5:6 (Principle 5b).
• Ratios in the scale must have a denominator of at most 100. This is generally motivated by Principle 4, but the constant 100 is an arbitrary choice to get some bounds on the search space of ratios.

Among all solutions that hold the conditions above, we will pick the one that has:

• (primary criterion) maximal evenness across the interval [1, 2)
• (secondary criterion) the minimal maximal denominator that occurs in any pairwise ratio of two tones in the scale.

Before we continue, let us clarify the primary optimization criterion. For optimizing spread, we need to be able to measure distance. It would be a bad idea to compute the distance between two ratios a and b as $a-b$ . Since we are on a log scale, we use $\log_2(a) - \log_2(b) = \log_2(\frac{a}{b})$ .

We define the evenness of a scale as the sum of squared distances between consecutive tones. For this computation, we also add $\frac{2}{1}$ into the set, so we are also measuring the distance from the highest tone in the scale to the next octave.

Now, if we create a simple program that runs through all combinations of fractions and finds the optimal one, we get:

$1, \frac{9}{8},\frac{5}{4}, \frac{4}{3}, \frac{3}{2}, \frac{5}{3}, \frac{15}{8}$

Let's call it Scale 1. This scale is commonly used in Western music. If we look at pairwise fractions, the largest denominator is 45, which is much better than the 131,072 we saw in the previous approach.

	$1$	$\frac{9}{8}$	$\frac{5}{4}$	$\frac{4}{3}$	$\frac{3}{2}$	$\frac{5}{3}$	$\frac{15}{8}$	$2$
$1$	$1$	$\frac{8}{9}$	$\frac{4}{5}$	$\frac{3}{4}$	$\frac{2}{3}$	$\frac{3}{5}$	$\frac{8}{15}$	$\frac{1}{2}$
$\frac{9}{8}$	$\frac{9}{8}$	$1$	$\frac{9}{10}$	$\frac{27}{32}$	$\frac{3}{4}$	$\frac{27}{40}$	$\frac{3}{5}$	$\frac{9}{16}$
$\frac{5}{4}$	$\frac{5}{4}$	$\frac{10}{9}$	$1$	$\frac{15}{16}$	$\frac{5}{6}$	$\frac{3}{4}$	$\frac{2}{3}$	$\frac{5}{8}$
$\frac{4}{3}$	$\frac{4}{3}$	$\frac{32}{27}$	$\frac{16}{15}$	$1$	$\frac{8}{9}$	$\frac{4}{5}$	$\frac{32}{45}$	$\frac{2}{3}$
$\frac{3}{2}$	$\frac{3}{2}$	$\frac{4}{3}$	$\frac{6}{5}$	$\frac{9}{8}$	$1$	$\frac{9}{10}$	$\frac{4}{5}$	$\frac{3}{4}$
$\frac{5}{3}$	$\frac{5}{3}$	$\frac{40}{27}$	$\frac{4}{3}$	$\frac{5}{4}$	$\frac{10}{9}$	$1$	$\frac{8}{9}$	$\frac{5}{6}$
$\frac{15}{8}$	$\frac{15}{8}$	$\frac{5}{3}$	$\frac{3}{2}$	$\frac{45}{32}$	$\frac{5}{4}$	$\frac{9}{8}$	$1$	$\frac{15}{16}$
$2$	$2$	$\frac{16}{9}$	$\frac{8}{5}$	$\frac{3}{2}$	$\frac{4}{3}$	$\frac{6}{5}$	$\frac{16}{15}$	$1$

In the construction of Scale 1, we only used the first triplet from Principle 5b (the third condition for our scale). What if we also allowed the second triplet (ratios 10:12:15)? The result would be the same: we would still obtain Scale 1. But what if we used only the second triplet? Then the situation becomes slightly more complex.

We get two optimal results:

A) $1, \frac{16}{15}, \frac{6}{5}, \frac{4}{3}, \frac{3}{2}, \frac{8}{5}, \frac{16}{9}$

B) $1, \frac{9}{8}, \frac{6}{5}, \frac{27}{20}, \frac{3}{2}, \frac{27}{16}, \frac{9}{5}$

Both of them also have 45 as the worst pairwise denominator. The (A) is again a well recognized scale. For (B), I was not able to find any information about practical usage. My guess is that $\frac{27}{20}$ and $\frac{27}{16}$ is not a good ratio for basic ratio in scale.

But following a cultural tradition of Western music, we will use a crossover between (A) and (B).

$1, \frac{9}{8}, \frac{6}{5}, \frac{4}{3}, \frac{3}{2}, \frac{8}{5}, \frac{9}{5}$

Let us call it Scale 2. It is actually the second-best solution from our optimization process, right behind (A) and (B). It has the same evenness as (A) and (B) but a slightly worse maximal pairwise denominator: 64.

	$1$	$\frac{9}{8}$	$\frac{6}{5}$	$\frac{4}{3}$	$\frac{3}{2}$	$\frac{8}{5}$	$\frac{9}{5}$	$2$
$1$	$1$	$\frac{8}{9}$	$\frac{5}{6}$	$\frac{3}{4}$	$\frac{2}{3}$	$\frac{5}{8}$	$\frac{5}{9}$	$\frac{1}{2}$
$\frac{9}{8}$	$\frac{9}{8}$	$1$	$\frac{15}{16}$	$\frac{27}{32}$	$\frac{3}{4}$	$\frac{45}{64}$	$\frac{5}{8}$	$\frac{9}{16}$
$\frac{6}{5}$	$\frac{6}{5}$	$\frac{16}{15}$	$1$	$\frac{9}{10}$	$\frac{4}{5}$	$\frac{3}{4}$	$\frac{2}{3}$	$\frac{3}{5}$
$\frac{4}{3}$	$\frac{4}{3}$	$\frac{32}{27}$	$\frac{10}{9}$	$1$	$\frac{8}{9}$	$\frac{5}{6}$	$\frac{20}{27}$	$\frac{2}{3}$
$\frac{3}{2}$	$\frac{3}{2}$	$\frac{4}{3}$	$\frac{5}{4}$	$\frac{9}{8}$	$1$	$\frac{15}{16}$	$\frac{5}{6}$	$\frac{3}{4}$
$\frac{8}{5}$	$\frac{8}{5}$	$\frac{64}{45}$	$\frac{4}{3}$	$\frac{6}{5}$	$\frac{16}{15}$	$1$	$\frac{8}{9}$	$\frac{4}{5}$
$\frac{9}{5}$	$\frac{9}{5}$	$\frac{8}{5}$	$\frac{3}{2}$	$\frac{27}{20}$	$\frac{6}{5}$	$\frac{9}{8}$	$1$	$\frac{9}{10}$
$2$	$2$	$\frac{16}{9}$	$\frac{5}{3}$	$\frac{3}{2}$	$\frac{4}{3}$	$\frac{5}{4}$	$\frac{10}{9}$	$1$

Now that we have established Scale 1 and Scale 2, we can explore them more.

If we look at the ratios between consecutive tones, we will see three repeating step sizes:

• $\frac{16}{15}$ (green)
• $\frac{10}{9}$ (blue)
• $\frac{9}{8}$ (orange)

Scale 1:

Scale 2:

For completeness, let us also look at Scale (A):

We can see that it has the same pattern as Scale 2, but shifted. If we start from $\frac{3}{2}$ in Scale 2 and cyclically write down the pattern, we get the pattern for (A).

Visually, you can observe that we have two kinds of steps: long ones ($\frac{10}{9}$ and $\frac{9}{8}$ ) and a short one ($\frac{16}{15}$ ), where the short one is about half the size of the long ones. We can also check this numerically in log2 distance: $\log_2(\frac{9}{8}) \approx 0.170, \log_2(\frac{10}{9}) \approx 0.152, \log_2(\frac{16}{15}) \approx 0.093$ .

These different step sizes and shifts will be important in another part of this blog post series. For now, we will continue to generate one more scale.

Approach 3: Equal Steps

In the previous two approaches, we have seen that the size of steps between consecutive tones varies. Let's try to fix this. Our goal is to create a scale with n tones such that there is an equal distance between consecutive steps; i.e., when we want to get from $s_i$ to $s_{i+1}$ , we always multiply by the same constant, c. Here is an example with a scale of 4 tones:

How do we compute the size of the step? If we have 4 tones, we want to multiply by c four times to get to 2 (a whole octave). This means:

$1 * c * c * c * c = 2$

that is:

$c^4 = 2$

so we get:

$c = \sqrt[4]{2}$

If we abstract from 4 to n tones, we get:

$c = \sqrt[n]{2}$

This result brings us a problem: for all n > 1, $\sqrt[n]{2}$ is an irrational number; i.e., it cannot be expressed as a fraction $\frac{a}{b}$ . Therefore, all tones $s_i, i \ge 1$ in such a generated scale will also be irrational. Here, Principle 6 saves us. We do not need exact ratios; we just need to get close enough.

The question now is what n we should choose. For starters, let's say that we want to approximate $\frac{3}{2}$ very closely. We can look at all scales where n ranges from 2 to 30 and see how close the nearest tone is to $\frac{3}{2}$ . (The range up to 30 is arbitrary, but a scale with more than 30 tones is probably impractical).

Let us plot the result:

The X-axis is the number of tones; the Y-axis is the squared log distance. Note that the Y-axis is shown on a log scale, so we are "zooming in" on the smaller numbers.

From the figure, we see that good candidates for "n" are 12, 24, and 29. But we want to optimize not only for $\frac{3}{2}$ but also for other "good ratios". As our set of good ratios, we take the union of ratios in Scale 1 and Scale 2. If we take the mean squared log distance to all of these ratios, we get the following figure:

We can see that good candidates for "n” appear to be 12, 19, 22, 24, 27, and 29.

We choose n = 12 for compatibility with Western music (this aligns with Principle 5c). Notes on other “n” values: historically, people have experimented with 19- and 22-tone music scales. A 24-tone scale is used in Middle Eastern music. Scales with 27 and 29 tones seem to be obscure and are not practically used.

So our Scale 3 is defined as follows:

$s_i = (\sqrt[12]{2})^i$ for $i \in \{0, 1, \dots, 11\}$

For comparison, we plot all scales together: Scale 3 = orange circles, Scale 1 = blue triangles, Scale 2 = red crosses, and Scale 0 = green boxes.

Conclusion

We have derived four different types of scales using different approaches, resulting in different properties. In the next part, we will explore the properties of the large and small steps that occur in Scale 1 and Scale 2.