EP11

EP11: Comma Drift and the Impossibility of Perfect Tuning

Irrationality of log₂3, Syntonic Comma 81:80, Pythagorean Comma

▶ 8:05 Number Theory

后续拓展

EP12 The Circle of Fifths as a Folded Spiral

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Overview

Jacob Collier calls the piano’s major third a “cheat.” He is right, in a precise mathematical sense: the equal-tempered major third is 14 cents wider than the just major third. And the reason this discrepancy can never be eliminated — not by any tuning system, however ingeniously constructed — is a theorem in number theory.

中文: “Jacob说，钢琴的大三度比纯律高了14音分——他把这叫做骗局。他选择用耳朵去调律，追求纯律的融合。但这个选择有代价：调性会漂移。这集，我们从数学角度解释：为什么纯净必然导致漂移？因为三个素数，永不相交。”

The core result is that $\log_2 3$ is irrational — no power of a perfect fifth (3:2) equals any power of an octave (2:1). This means the “spiral of fifths” never closes into a circle. Every tuning system is a choice about which aspect of acoustic purity to sacrifice. 12-tone equal temperament chooses to sacrifice a little from every interval equally. Jacob Collier chooses to sacrifice transposability in order to keep pure intervals — and accepts that his pitch center drifts by 14 cents with each just major third progression.

This is the second episode in the Jacob Collier arc (of four).

Prerequisites

Basic number theory: prime factorization, the Fundamental Theorem of Arithmetic
Logarithm properties: $\log_b(xy) = \log_b x + \log_b y$ , $\log_b(x^n) = n\log_b x$
Cents: $1 \text{ cent} = 2^{1/1200}$ , so an interval of ratio $r$ equals $1200\log_2 r$ cents
Combination Tones (EP09) — why just intervals produce locked harmonic series

Definitions

Definition 11.1 (Just Intervals)

A just interval is a frequency ratio expressible as $p/q$ where $p, q$ are positive integers with small prime factors (i.e., drawn from the 5-limit prime set $\{2, 3, 5\}$ ):

Interval	Ratio	Cents
Perfect octave	2/1	1200.00
Perfect fifth	3/2	701.96
Perfect fourth	4/3	498.04
Just major third	5/4	386.31
Just minor third	6/5	315.64

The “just” in “just intonation” refers to the small integer ratios that produce perfect harmonic alignment (cf. EP09, Theorem 9.3).

Definition 11.2 (Syntonic Comma)

The syntonic comma is the ratio

\sigma = \frac{81}{80} = \frac{(3/2)^4 / 2^2}{5/4} = \frac{81/64}{80/64} = \frac{81}{80}

In cents: $1200\log_2(81/80) \approx 21.51$ cents.

It measures the discrepancy between two natural routes to the major third:

Via the harmonic series: $5/4$ (the 5th partial divided by the 4th partial) $\approx 386.31$ cents
Via four perfect fifths: $(3/2)^4 / 4 = 81/64 \approx 407.82$ cents

The difference is $407.82 - 386.31 = 21.51$ cents. This means the Pythagorean major third (four fifths up, two octaves down) is sharper than the just major third by exactly one syntonic comma.

Definition 11.3 (Pythagorean Comma)

The Pythagorean comma is the ratio

\kappa = \frac{(3/2)^{12}}{2^7} = \frac{3^{12}}{2^{19}} = \frac{531441}{524288}

In cents: $1200\log_2(531441/524288) \approx 23.46$ cents.

It measures how far 12 consecutive perfect fifths overshoot 7 perfect octaves:

(3/2)^{12} = \frac{531441}{4096} \approx 129.746 \quad \text{vs} \quad 2^7 = 128

The spiral of fifths misses closure by this amount after 12 steps.

Definition 11.4 (Equal Temperament)

12-tone equal temperament (12-TET) sets every semitone to the ratio $2^{1/12}$ , so that 12 semitones span exactly one octave. The equal-tempered fifth is $2^{7/12}$ and the equal-tempered major third is $2^{4/12} = 2^{1/3}$ .

Comparison with just intervals:

Interval	Just ratio	ET ratio	Difference
Fifth (P5)	3/2 = 1.50000	$2^{7/12} \approx 1.49831$	$-1.955$ cents
Major third (M3)	5/4 = 1.25000	$2^{4/12} \approx 1.25992$	$+13.686$ cents

The fifth is nearly pure (2 cents flat); the major third is noticeably wide (14 cents sharp). This is the asymmetry Collier objects to.

Definition 11.5 (Cents)

The cent is a logarithmic unit of musical interval: 100 cents = 1 equal-tempered semitone, 1200 cents = 1 octave. For a frequency ratio $r$ ,

\text{interval in cents} = 1200 \cdot \log_2 r

The cent scale linearizes the logarithmic frequency axis, making interval arithmetic additive: stacking two intervals means adding their cent values.

Main Theorems

Theorem 11.1 (Irrationality of log₂3)

\log_2 3

is irrational. Equivalently, there exist no positive integers

m, n

such that

2^m = 3^n

Proof.

Suppose for contradiction that $\log_2 3 = m/n$ for positive integers $m, n$ . Then

2^{m/n} = 3 \implies 2^m = 3^n

The left side $2^m$ has prime factorization containing only the prime 2. The right side $3^n$ has prime factorization containing only the prime 3. By the Fundamental Theorem of Arithmetic, two integers with distinct prime factorizations cannot be equal. Therefore $2^m \neq 3^n$ for all positive integers $m, n$ .

This is a contradiction. Hence $\log_2 3 \notin \mathbb{Q}$ . $\square$

中文: “假设 log 以 2 为底 3 等于某个分数 m 比 n。那么 2 的 m 次方等于 3 的 n 次方。但左边是 2 的 m 次方——素因子分解后，只有 2。右边是 3 的 n 次方——只有 3。一个数不可能既等于’全是 2 的乘积'，又等于’全是 3 的乘积'。矛盾。”

Theorem 11.2 (Irrationality of log₂5)

\log_2 5

is irrational. More generally,

\log_2 p

is irrational for every odd prime

p

Proof.

The same argument applies. Suppose $\log_2 5 = m/n$ , giving $2^m = 5^n$ . The left side factors as $2^m$ , the right side as $5^n$ . Since 2 and 5 are distinct primes, the Fundamental Theorem of Arithmetic gives a contradiction. $\square$

Musical consequence: Since $\log_2 2 = 1$ , $\log_2 3$ , and $\log_2 5$ are linearly independent over $\mathbb{Q}$ , the three interval generators — octave (2:1), fifth (3:2), major third (5:4) — cannot simultaneously be expressed as integer multiples of any single interval. No EDO (equal division of the octave) can make all three perfectly just.

中文: “八度即2的幂、纯五度即3的幂、纯大三度即5的幂，这三条指数曲线永不相交。不存在一种调律系统能同时保持它们。你必须选择牺牲哪一个。”

Theorem 11.3 (The Spiral of Fifths Never Closes)

The sequence of pitch classes generated by iterating the just perfect fifth

3/2

never returns to its starting frequency. Formally, there is no positive integer

n

such that

(3/2)^n

is a power of 2.

Proof.

This is an immediate restatement of Theorem 11.1. $(3/2)^n = 2^k$ iff $3^n = 2^{n+k}$ , which is impossible for $n > 0$ by the Fundamental Theorem of Arithmetic (the left side has 3 as a prime factor, the right side does not). $\square$

Corollary (Pythagorean Comma as a Measure of Non-closure): After 12 steps of the just fifth, we reach $(3/2)^{12} = 3^{12}/2^{12}$ . The closest power of 2 is $2^7 = 128$ . The ratio $(3/2)^{12}/2^7 = 3^{12}/2^{19} = 531441/524288 \approx 1.01364$ is the Pythagorean comma $\kappa \approx 23.46$ cents.

Theorem 11.4 (Continued Fraction Approximation and Good EDOs)

The continued fraction expansion of $\log_2(3/2)$ is

\log_2(3/2) = [0;\, 1, 1, 2, 2, 3, 1, 5, 2, 23, \ldots]

The convergents are $0/1,\, 1/1,\, 1/2,\, 3/5,\, 7/12,\, 17/29,\, 24/41,\, 31/53, \ldots$

The convergent $p/q$ means that $q$ equal divisions of the octave (q-EDO) approximate the fifth to within $1/(q^2 \cdot a_{k+1})$ octaves, where $a_{k+1}$ is the next continued fraction coefficient. In particular:

7/12: 12-TET has the fifth at 7 semitones $= 2^{7/12}$ , error $\approx 2$ cents — small enough to be nearly imperceptible in melodic context.
31/53: 53-TET has the fifth at 31 steps, error $< 0.1$ cents — essentially just.

The large coefficient $a_{10} = 23$ after the 53-EDO convergent means 53-TET is exceptionally good — the next meaningful improvement requires 306-TET.

Proof.

The continued fraction algorithm applied to $\log_2(3/2) \approx 0.58496$ :

$0.58496 = 0 + 0.58496$ ; $a_0 = 0$
$1/0.58496 \approx 1.7095$ ; $a_1 = 1$ , remainder $0.7095$
$1/0.7095 \approx 1.4094$ ; $a_2 = 1$ , remainder $0.4094$
$1/0.4094 \approx 2.4426$ ; $a_3 = 2$ , remainder $0.4426$
$1/0.4426 \approx 2.2593$ ; $a_4 = 2$ , remainder $0.2593$
$1/0.2593 \approx 3.857$ ; $a_5 = 3$ , remainder $0.857$
… and so on.

The convergents $p_k/q_k$ are computed by the standard recurrence $p_k = a_k p_{k-1} + p_{k-2}$ , $q_k = a_k q_{k-1} + q_{k-2}$ :

\frac{0}{1},\, \frac{1}{1},\, \frac{1}{2},\, \frac{3}{5},\, \frac{7}{12},\, \frac{17}{29},\, \frac{24}{41},\, \frac{31}{53}

The best-approximation property of continued fraction convergents guarantees that $q_k$ -EDO provides a better approximation to the just fifth than any $q$ -EDO with $q < q_{k+1}$ . $\square$

Prop 11.1 (Syntonic Comma Drift in a Comma Pump)

A comma pump is a chord progression that uses only just intervals and returns to the same nominal pitch class but has accumulated a net pitch shift of one syntonic comma (21.5 cents). The canonical example is:

C \to \text{Am} \to \text{Dm} \to G \to C

If each fifth and major third is just, the final C is $81/80 \approx 1.0125$ times the starting C frequency — 21.5 cents higher. After 4 repetitions of this progression, the pitch has drifted by $4 \times 21.5 = 86$ cents — nearly a semitone.

Numerical Example: Jacob Collier’s Drift

Collier’s “In the Bleak Midwinter” (a cappella) uses just intervals throughout. Consider four consecutive just major third chords in ascending transposition:

Each just major third raises the pitch center by $5/4$ . Four such steps:

(5/4)^4 = 625/256 \approx 2.4414

An equal-tempered four-major-thirds progression would be $2^{16/12} = 2^{4/3} \approx 2.5198$ , which is exactly a diminished seventh above the start.

The just version lands at $625/256 \approx 2.44$ , not $2.52$. In cents:

1200\log_2(625/256) \approx 1200 \times 1.2854 \approx 1542.5 \text{ cents}

vs. the equal-tempered $1600$ cents. The drift is $1600 - 1542.5 = 57.5$ cents, or approximately $4 \times 13.7 = 54.8$ cents (four times the major-third error in 12-TET).

中文: “四个连续的纯律和弦，每步令调性中心偏移14音分。四步累积56音分——约半个半音。从E大调出发，就到了G半升大调，一个钢琴键盘上不存在的音。”

Collier treats this drift as a compositional tool — the gradual brightening of pitch level is perceived as an emotional ascent, not a tuning error.

Musical Connection

音乐联系

The three incommensurable generators

Octave (2:1), fifth (3:2), and major third (5:4) are the three generators of 5-limit just intonation. Theorems 11.1 and 11.2 show that $\log_2 2 = 1$ , $\log_2 3$ , and $\log_2 5$ are linearly independent over $\mathbb{Q}$ . This means the “pitch lattice” $\{2^a 3^b 5^c : a, b, c \in \mathbb{Z}\}$ is a 3-dimensional free abelian group — no element can be reached from another by a finite sequence of transpositions unless the generators commute exactly, which they cannot (by the irrationality theorems).

Bach’s pragmatic solution: 12-TET

The Well-Tempered Clavier (1722) demonstrated that 12-TET (or a close approximation) makes all 24 keys usable. The price: every major third is 14 cents sharp. Bach presumably accepted this bargain as enabling the compositional freedom of free modulation.

中文: “具体的赢：任意转调。巴赫1722年的《平均律键盘曲集》，24个前奏曲与赋格，遍历所有大小调——这在纯律里不可能。具体的输：纯五度缩短2音分，几乎无感；但大三度拉宽14音分，明显粗糙。这不是最好的调律——是最公平的调律。”

Collier’s solution: deliberate drift

Where Bach sacrifices purity for transposability, Collier sacrifices transposability for purity. His voice-leading uses just intervals at each local step; the global pitch center drifts. The drift is reframed as a narrative arc — ascending pitch equals ascending emotional intensity. The mathematics of tuning becomes a compositional parameter.

Connection to EP33 (Physical Modeling)

The syntonic comma directly affects inharmonicity in stringed instruments (EP33). A real string’s overtone series is slightly stretched (inharmonic), so the “pure” fifth a string naturally produces is not exactly 3:2 but slightly wider. The comma thus interacts with the physical properties of strings, making just intonation on a bowed string instrument a dynamic target, not a fixed table.

Limits and Open Questions

Higher-limit just intonation. The 5-limit system (primes 2, 3, 5) covered here is the classical foundation. 7-limit (adding 7:4 as the harmonic seventh), 11-limit (adding 11:8), and higher limits introduce additional incommensurable generators. Each new prime adds a new dimension to the pitch lattice and new categories of commas. The “optimal” temperament depends on which prime limit the composer wishes to approximate.
Adaptive tuning in electronic music. Software systems (e.g., Hermode Tuning) attempt to dynamically adjust pitch to minimize beating in real time, effectively implementing just intonation on the fly. The challenge is that adjustments must be perceptually smooth (avoiding “jumps”) while tracking the drift — an optimization problem with conflicting objectives.
Perceptual threshold of comma drift. The syntonic comma is 21.5 cents. The just-noticeable difference (JND) for pitch in a musical context is roughly 5–10 cents. This means a single comma step is detectable, but gradual drift over many measures may be below the threshold of conscious perception. The exact threshold for “perceived drift” vs. “unnoticed drift” in a musical context is not well-established.
Transcendental extensions. Theorems 11.1–11.2 show $\log_2 3$ and $\log_2 5$ are irrational. Are they transcendental? By the Lindemann–Weierstrass theorem, $e^{\alpha}$ is transcendental for any nonzero algebraic $\alpha$ , but the transcendence of $\log_2 3 = \ln 3 / \ln 2$ follows from the Gelfond–Schneider theorem: $2^{\log_2 3} = 3$ is rational, so if $\log_2 3$ were algebraic and irrational, $2^{\log_2 3}$ would be transcendental — a contradiction. Hence $\log_2 3$ is transcendental. The tuning problem thus involves transcendental numbers at its core.

Academic References

Barbour, J. M. (1951). Tuning and Temperament: A Historical Survey. Michigan State College Press. (Reprinted Dover, 2004.)
Duffin, R. W. (2007). How Equal Temperament Ruined Harmony (and Why You Should Care). Norton.
Milne, A., Sethares, W. A., & Plamondon, J. (2007). Isomorphic controllers and dynamic tuning: Invariant fingering over a tuning continuum. Computer Music Journal, 31(4), 15–32.
Sethares, W. A. (2005). Tuning, Timbre, Spectrum, Scale (2nd ed.). Springer.
Euler, L. (1739). Tentamen novae theoriae musicae. St. Petersburg Academy. (The first systematic mathematical theory of consonance using integer ratios.)
Noll, T. (2008). Rolling the dice for musical scales. Journal of Mathematics and Music, 2(2), 105–127.
Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. Ch. 4 (Irrational numbers and the continued fraction algorithm).