Building the Pythagorean Scale with Pure Fifths

Introduction

In our last lesson, we established the octave and its 2:1 frequency ratio as the fundamental organizing principle in music. You learned that due to octave equivalence, we can focus on defining the notes within a single octave, knowing that the pattern can be repeated across the entire pitch spectrum. Now, we will begin to fill that octave with notes. To do this, we will turn to the next most fundamental interval in Western music: the perfect fifth.

This lesson addresses the learning outcome: Derive the Pythagorean tuning system from successive applications of the pure fifth (3:2 ratio). We will explore the physical and mathematical basis for the perfect fifth's importance and then use it as a generative tool. By repeatedly applying this single interval and folding the results back into our octave framework, we will construct one of history's most important tuning systems from first principles.

The Perfect Fifth: A Physical and Perceptual Anchor

After the octave, the interval of a perfect fifth is perceived as the most stable and consonant. Its frequency ratio is 3:2. Like the octave, this isn't an arbitrary cultural choice; it arises directly from the harmonic series. The third harmonic of a note has a frequency of $3 f_{0}$ , and the second harmonic has a frequency of $2 f_{0}$ . The interval between them is $3 f_{0} : 2 f_{0}$ , or simply 3:2.

Because of your background in physics and experience with vibrating strings on a guitar, you can appreciate the physical origin of this ratio. The following video demonstrates this beautifully.

🎵 Pythagorean tuning

The video "🎵 Pythagorean tuning" from the channel discovermaths provides an excellent physical intuition for the importance of the fifth.

First, watch the segment on consonant intervals to hear the distinction between stable (consonant) and unstable (dissonant) intervals. The fifth stands out as uniquely stable. Next, watch the explanation using a guitar string from physics of the fifth. The core insight here is that stopping a string at simple fractional lengths—like 1/2 for the octave and 2/3 for the fifth—produces these most consonant sounds. This observation, that simple integer ratios in physics lead to perceptual stability in music, was a foundational discovery for the Pythagoreans.

The Generative Principle: Stacking Fifths

The Pythagoreans reasoned that if the octave (2:1) and the fifth (3:2) are the most "perfect" intervals, a complete musical scale could be built using only these two ratios. The method is as follows:

Start with a base frequency, our tonic. Let's normalize it to a ratio of 1. The octave above it is at 2. All our notes must lie in the range [1, 2].
Generate a new note by going up a perfect fifth, i.e., multiplying the frequency by 3/2.
If this new note's frequency ratio is greater than 2, it has left our target octave. We bring it back down by dividing by 2 (applying octave equivalence).
Repeat this process, always starting from the newly generated note, to create a chain of pitches. This process is often called "stacking fifths."

Let's walk through the first few steps to build a 7-note scale. We will label our starting note C.

Note 1: C. Our starting point. Ratio: 1.
Note 2: G. Go up a fifth from C: $1 \times \frac{3}{2} = \frac{3}{2}$ . This is between 1 and 2, so we keep it. The interval C to G is a perfect fifth.
Note 3: D. Go up a fifth from G: $\frac{3}{2} \times \frac{3}{2} = \frac{9}{4}$ . This ratio is greater than 2, so we bring it down an octave: $\frac{9}{4} \div 2 = \frac{9}{8}$ . This is our D.
Note 4: A. Go up a fifth from D: $\frac{9}{8} \times \frac{3}{2} = \frac{27}{16}$ . This is between 1 and 2. This is our A.
Note 5: E. Go up a fifth from A: $\frac{27}{16} \times \frac{3}{2} = \frac{81}{32}$ . This is greater than 2, so we bring it down an octave: $\frac{81}{32} \div 2 = \frac{81}{64}$ . This is our E.
Note 6: B. Go up a fifth from E: $\frac{81}{64} \times \frac{3}{2} = \frac{243}{128}$ . This is between 1 and 2. This is our B.

So far, by moving upwards in a "chain of fifths" (C → G → D → A → E → B), we have generated 6 of the 7 notes of a major scale. What about the last note, F? We can generate it by going down a fifth from our starting C. Going down a fifth is equivalent to dividing by 3/2, or multiplying by 2/3.

Note 7: F. Go down a fifth from C: $1 \div \frac{3}{2} = \frac{2}{3}$ . This ratio is less than 1, so we bring it up an octave to place it within our [1, 2] range: $\frac{2}{3} \times 2 = \frac{4}{3}$ . This is our F.

This entire process is laid out systematically in the following table. Each numbered row represents one step of adding a new note to the scale by stacking fifths.

This table visualizes the step-by-step derivation of the Pythagorean scale. Row 2 generates G (3/2). Row 4 generates D (9/8 after octave reduction). The final ratios for the C major scale are shown in row 11.

The Complete Pythagorean Scale

By arranging the ratios we derived in ascending order, we get the Pythagorean tuning for a major scale:

Note	C	D	E	F	G	A	B	C
Ratio	1	$\frac{9}{8}$	$\frac{81}{64}$	$\frac{4}{3}$	$\frac{3}{2}$	$\frac{27}{16}$	$\frac{243}{128}$	2

Notice a key mathematical property: every ratio is of the form $2^{n} 3^{m}$ for integers $n$ and $m$ . This elegant system, built entirely from powers of 2 and 3, is the essence of Pythagorean tuning.

The following resources will help you consolidate your understanding of this derivation.

🎵 Pythagorean tuning

This segment of the "discovermaths" video provides a narrated walk-through of the derivation we just performed, starting with an actual frequency for Middle C.

Please watch the section demonstrating the scale construction. This will reinforce the process of multiplying by 3/2 and dividing by 2 to keep the notes within the octave.

[PDF] The physics of musical scales: Theory and experiment

This paper from the American Journal of Physics provides a concise, formal explanation of the Pythagorean scale's construction, perfectly suited to your background.

Read the section titled "III. THE PYTHAGOREAN SCALE," starting from the description. It summarizes the process of moving up and down in fifths and using octave adjustments to generate the scale.

The Unclosed Circle: A Preview of the Problem

A natural question arises from this process: if we keep stacking fifths, will we ever land exactly back on a C, but several octaves up? In other words, does some number of fifths equal some number of octaves? Mathematically, is there an integer solution for $n$ and $m$ to the equation $(\frac{3}{2})^{n} = 2^{m}$ ?

This is equivalent to asking if $3^{n} = 2^{n + m}$ . The left side of the equation is always an odd number, while the right side is always an even number. Since an odd number can never equal an even number, there is no integer solution.

This means the chain of fifths never perfectly closes. This "glitch" is a fundamental mathematical impossibility that has profound consequences for music.

The Mathematical Problem with Music, and How to Solve It

The video "The Mathematical Problem with Music" explains this core issue and presents the proof by contradiction we just discussed.

Watch the segment from the infinite piano. Focus on the process of generating notes and, crucially, the mathematical proof presented at the proof. This demonstrates why a tuning system based on pure fifths and octaves cannot be a closed, finite system.

Conclusion

In this lesson, you have seen how a complete musical scale can be rigorously derived from just two fundamental frequency ratios that are grounded in physics: the octave (2:1) and the perfect fifth (3:2).

Here are the key takeaways:

The Pythagorean tuning system is constructed by repeatedly "stacking" perfect fifths (3:2 ratio) and using octave reduction (2:1 ratio) to keep all notes within a single octave.
The frequency ratio of every note in the Pythagorean scale can be expressed in the form $2^{n} 3^{m}$ .
The process of stacking fifths is mathematically "open"—it never perfectly returns to the starting pitch class. An integer number of perfect fifths can never equal an integer number of octaves ( $(3/2)^{n} \neq = 2^{m}$ ).

We have successfully built a scale, but we've also uncovered a fundamental mathematical inconsistency at its heart. In our next lesson, we will explore the musical consequences of this "unclosed circle." We will define the Pythagorean comma—the small interval that quantifies this error—and see how it creates a "wolf interval" that plagued musicians for centuries, ultimately motivating the search for alternative tuning systems.