Number Theory and Ratio: The Pythagorean Bedrock and Its Limitations

At the heart of the Pythagorean tradition lies a simple yet profound assertion: musical intervals correspond to numerical ratios. The octave (2:1), the perfect fifth (3:2), and the perfect fourth (4:3) were not merely empirical observations but metaphysical truths, forming the bedrock of what was believed to be a cosmic harmony. However, a rigorous re-examination through the lens of modern number theory reveals inherent limitations within this closed system of small integer ratios. This chapter will demonstrate that the very elegance of the Pythagorean ratios—their expressibility as products of the first few primes—contains the seeds of their own undoing in a closed tuning system. We will dissect the prime factorization of interval ratios, introduce the concept of limit and approximation in the context of tuning, and show, through the case studies of the syntonic comma and the diesis, how numerical discrepancies inevitably arise. These discrepancies are not mere theoretical curiosities; they are the mathematical manifestation of the incompatibility between pure intervals and the practical demands of a modulating instrument. The analysis will establish why a more powerful algebraic framework—one that embraces vector spaces and group theory—is not just desirable but necessary for a consistent theory of harmony, paving the way for the subsequent development of harmonic calculus.

1.1 The Pythagorean Foundation: Ratios as Sacred Numbers

The Pythagoreans, operating in the 6th century BCE, discovered that harmonious sounds corresponded to simple integer ratios when produced by vibrating strings or air columns. This discovery was not an experimental accident but a philosophical cornerstone: the universe was fundamentally numerical, and music was the audible expression of this numerical order. The primary intervals were derived from the monochord experiment:

• **Octave**: String length ratio 2:1 (e.g., C to C′).
• **Perfect Fifth**: String length ratio 3:2 (e.g., C to G).
• **Perfect Fourth**: String length ratio 4:3 (e.g., C to F).

These ratios share a critical property: they are *superparticular* (of the form $n+1:n$) and, crucially, their numerators and denominators are products of the smallest primes: 2, 3, and 5. The octave and fifth involve only 2 and 3; the fourth is the inverse of the fifth (4:3 = (2:1)/(3:2)). The Pythagorean school extended this system by stacking perfect fifths to generate new pitches, a process known as the *circle of fifths*. Starting from a reference pitch (e.g., C), ascending by successive fifths yields:

$$ \mathrm{C} \xrightarrow{\times 3/2} \mathrm{G} \xrightarrow{\times 3/2} \mathrm{D} \xrightarrow{\times 3/2} \mathrm{A} \xrightarrow{\times 3/2} \mathrm{E} \xrightarrow{\times 3/2} \mathrm{B} \xrightarrow{\times 3/2} \mathrm{F}^\sharp \xrightarrow{\times 3/2} \mathrm{C}^\sharp \xrightarrow{\times 3/2} \mathrm{G}^\sharp \xrightarrow{\times 3/2} \mathrm{D}^\sharp \xrightarrow{\times 3/2} \mathrm{A}^\sharp \xrightarrow{\times 3/2} \mathrm{E}^\sharp \xrightarrow{\times 3/2} \mathrm{B}^\sharp $$

After twelve such steps, one expects to return to a pitch an octave above the starting C, i.e., $ \mathrm{C} \times (3/2)^{12} = \mathrm{C}' \times 2^k $ for some integer $k$. The required power of 2 is found by reducing the ratio to within one octave:

$$ (3/2)^{12} = \frac{3^{12}}{2^{12}} = \frac{531441}{4096} \approx 129.746 $$

The nearest power of 2 is $2^7 = 128$. Thus, twelve perfect fifths exceed seven octaves by the factor:

$$ \frac{531441}{4096} \div 128 = \frac{531441}{524288} \approx 1.01364 $$

This small excess is the Pythagorean comma, a fundamental discrepancy in the Pythagorean tuning system. Its prime factorization is:

$$ \frac{531441}{524288} = \frac{3^{12}}{2^{19}} $$

The Pythagorean comma is approximately 23.46 cents (where 1 cent = 1/1200 of an octave). In a closed system that insists on using only the primes 2 and 3 (i.e., only octaves and fifths), this comma cannot be eliminated; it is an intrinsic feature of the group generated by 2 and 3 under multiplication. The Pythagoreans were aware of this “comma” but treated it as a negligible imperfection or, in some mystical interpretations, as a divine signature. From a modern number-theoretic perspective, the comma emerges because the multiplicative group generated by 2 and 3 is *dense* in the positive reals but not *discrete* in the logarithmic scale; the logarithms of 2 and 3 are irrational multiples of each other (since $\log_2 3$ is irrational), so their integer combinations never exactly equal an integer. The equation $2^a 3^b = 2^c$ implies $3^b = 2^{c-a}$, which has no integer solutions except $b=0$, by the fundamental theorem of arithmetic. Thus, any product of fifths (3/2) and octaves (2) that should theoretically close to a unison will always miss by a factor that is a power of 3 divided by a power of 2, never exactly 1.

1.2 Prime Factorization as the Algebraic Signature of an Interval

Modern number theory allows us to formalize the Pythagorean insight. Any just (pure) interval ratio can be written in lowest terms as $p/q$ with $p,q \in \mathbb{Z}^+$. By the fundamental theorem of arithmetic, $p$ and $q$ have unique prime factorizations:

$$ p = 2^{a_2} 3^{a_3} 5^{a_5} 7^{a_7} \dots, \quad q = 2^{b_2} 3^{b_3} 5^{b_5} 7^{b_7} \dots $$

The interval ratio is then:

$$ \frac{p}{q} = 2^{a_2-b_2} 3^{a_3-b_3} 5^{a_5-b_5} 7^{a_7-b_7} \dots $$

We can represent the interval by an *exponent vector* $\mathbf{v} = (e_2, e_3, e_5, e_7, \dots)$, where $e_i = a_i - b_i$ for each prime $i$. Since octave equivalence is usually assumed (pitches differing by powers of 2 are considered the same note class), we can reduce the vector modulo the octave by ignoring the component $e_2$ or, equivalently, by requiring that the total exponent sum for 2 be within a fixed range. More commonly, we work in the *prime limit* sense: an interval is said to be *$n$-limit* if its prime factors are all $\leq n$. The Pythagorean intervals are 3-limit (only primes 2 and 3). The introduction of the prime 5 yields 5-limit intervals, such as the just major third $5:4$ and just minor third $6:5$.

The exponent vector representation is powerful because it translates musical problems into linear algebra over the integers. Consider the following just intervals and their vectors (ignoring the octave component $e_2$ for now, or treating it separately):

| Interval | Ratio | Prime Factorization | Vector $(e_3, e_5)$ | |----------|-------|---------------------|---------------------| | Perfect Fifth | $3:2$ | $3^1 \cdot 2^{-1}$ | $(1, 0)$ | | Perfect Fourth | $4:3$ | $2^2 \cdot 3^{-1}$ | $(-1, 0)$ | | Just Major Third | $5:4$ | $5^1 \cdot 2^{-2}$ | $(0, 1)$ | | Just Minor Third | $6:5$ | $2^1 \cdot 3^1 \cdot 5^{-1}$ | $(1, -1)$ | | Just Major Sixth | $5:3$ | $5^1 \cdot 3^{-1}$ | $(-1, 1)$ |

Notice that the minor third vector is the difference between the fifth and major third vectors: $(1,0) - (0,1) = (1,-1)$. This reflects the identity:

$$ \frac{3}{2} \div \frac{5}{4} = \frac{6}{5} $$

Thus, the set of all 5-limit intervals forms a lattice in $\mathbb{Z}^2$ generated by the basis vectors $(1,0)$ (fifth) and $(0,1)$ (major third). Any 5-limit interval is an integer linear combination of these two vectors. This lattice structure is the algebraic home of just intonation.

1.3 The Incompatibility of Pure Intervals in a Closed System

A closed tuning system for a fixed number of notes (e.g., the 12 notes of the chromatic scale) requires that we assign a frequency (or string length) to each note such that all consonant intervals between them are pure (just) ratios. Moreover, the system must be *self-consistent*: if we form a cycle of intervals that should theoretically return to the starting pitch (up to octaves), the product of the ratios must equal a power of 2. For example, the circle of fifths: twelve fifths should equal seven octaves plus a unison. In just intonation, we can