EP12

EP12: The Circle of Fifths as a Folded Spiral

gcd(7,12)=1, Cyclic Generators, Shepard Helix, 12-TET as Approximation

▶ 7:48 Number TheoryAbstract Algebra

前置知识

EP11 Comma Drift and the Impossibility of Perfect Tuning EP04 All-Interval Rows and ℤ₁₂

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超超超超超级Lydian，Jacob的无限音阶

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Overview

Jacob Collier’s “Super Ultra Hyper Mega Meta Lydian” is not just a joke. It describes a real mathematical structure: an infinite chain of Lydian tetrachords ascending the circle of fifths, never stopping, never repeating. But on a piano — which uses 12-TET — the spiral is forced into a circle, and after 12 steps you return to the same key.

中文: “圆是钢琴的近似。螺旋是数学的真实。SUM的永不停步，恰好映射了这条螺旋。”

The key mathematical fact is $\gcd(7, 12) = 1$ : the interval of a fifth (7 semitones) generates the entire group $\mathbb{Z}_{12}$ because 7 and 12 are coprime. This is why the circle of fifths visits all 12 pitch classes before returning. Any step size $k$ with $\gcd(k, 12) > 1$ would only visit a proper subgroup of pitch classes — a smaller cycle, not the full circle.

This episode is the third in the Jacob Collier arc. The spiral/circle duality (EP11’s irrational $\log_2(3/2)$ forced into rational approximation by 12-TET) is now recast as a topological and algebraic fact: the true pitch space is a helix $\mathbb{R}$ , the chromatic circle is its quotient $\mathbb{R}/12\mathbb{Z}$ , and the circle of fifths is a different traversal of the same quotient.

Prerequisites

Comma Drift and Tuning (EP11) — $\log_2(3/2)$ irrational, spiral of fifths
The All-Interval Row and $\mathbb{Z}_{12}$ (EP04) — $\mathbb{Z}_{12}$ , subgroups, generators
Number theory: $\gcd$ , Bezout’s theorem, cyclic groups
Elementary topology: quotient spaces (informal level sufficient)

Definitions

Definition 12.1 (Generator of ℤₙ)

An element $k \in \mathbb{Z}_n$ is a generator of $\mathbb{Z}_n$ if the cyclic subgroup $\langle k \rangle = \{0, k, 2k, \ldots\} \pmod n$ equals all of $\mathbb{Z}_n$ .

Theorem (standard): $k$ generates $\mathbb{Z}_n$ if and only if $\gcd(k, n) = 1$ .

Proof sketch: If $d = \gcd(k,n) > 1$ , then $jk \equiv 0 \pmod{d}$ for all $j$ , so $\langle k \rangle \subseteq \{0, d, 2d, \ldots\}$ , a proper subgroup of size $n/d < n$ . Conversely, if $\gcd(k,n)=1$ , Bezout’s theorem gives integers $a, b$ with $ak + bn = 1$ , so $ak \equiv 1 \pmod n$ , meaning $k$ has a multiplicative inverse and $\langle k \rangle = \mathbb{Z}_n$ . $\square$

Definition 12.2 (Circle of Fifths)

The circle of fifths is the sequence of pitch classes generated by repeated transposition by 7 semitones (a perfect fifth in 12-TET) in $\mathbb{Z}_{12}$ :

C \to G \to D \to A \to E \to B \to F\sharp \to C\sharp \to A\flat \to E\flat \to B\flat \to F \to C

In $\mathbb{Z}_{12}$ : $0 \to 7 \to 2 \to 9 \to 4 \to 11 \to 6 \to 1 \to 8 \to 3 \to 10 \to 5 \to 0$ .

Since $\gcd(7, 12) = 1$ (as 7 is prime and does not divide 12), 7 generates $\mathbb{Z}_{12}$ , and the sequence visits all 12 pitch classes before returning to the start. This is the mathematical content of “the circle of fifths is a circle and not a shorter cycle.”

Definition 12.3 (Shepard–Risset Pitch Helix)

The Shepard–Risset pitch helix embeds pitch into $\mathbb{R}^3$ as a helix, capturing both chroma (position on the chromatic circle) and height (octave register):

\mathbf{p}(f) = \bigl(r\cos(2\pi \cdot \text{chroma}(f)/12),\; r\sin(2\pi \cdot \text{chroma}(f)/12),\; h \cdot \log_2(f/f_0)\bigr)

where $\text{chroma}(f) = 12\log_2(f/f_0) \bmod 12 \in [0, 12)$ , $r > 0$ is the helix radius, $h > 0$ is the vertical pitch height scale, and $f_0$ is a reference frequency.

Projection onto the $xy$ -plane (the horizontal circle): gives chroma only — octave equivalence, the chromatic circle $\mathbb{Z}_{12}$ .
Projection onto the $z$ -axis (vertical): gives octave height only — chroma equivalence.
The full helix: represents the two-dimensional structure of pitch: chroma $\times$ register.

A Shepard tone is a superposition of sinusoids at all octaves of a given chroma, with a bell-curve amplitude envelope. It sits at a single point on the chromatic circle but is spread vertically along the helix — hence the paradox of indefinitely ascending/descending pitch without change of perceived height.

Definition 12.4 (Subgroups of ℤ₁₂)

The subgroups of $\mathbb{Z}_{12}$ correspond bijectively to the divisors of 12:

Divisor $d$	Subgroup	Generator	Musical object
1	$\{0\}$	—	Unison
2	$\{0, 6\}$	6	Tritone pair
3	$\{0, 4, 8\}$	4	Augmented triad
4	$\{0, 3, 6, 9\}$	3	Diminished seventh
6	$\{0, 2, 4, 6, 8, 10\}$	2	Whole-tone scale
12	$\mathbb{Z}_{12}$	1 or 7	Chromatic scale / circle of fifths

The richness of 12’s factor structure ( $12 = 2^2 \cdot 3$ , with divisors $\{1,2,3,4,6,12\}$ ) directly corresponds to the diversity of symmetric chords in 12-TET: each proper subgroup is a maximally even, symmetric chord or scale.

Main Theorems

Theorem 12.1 (7 Generates ℤ₁₂)

The element 7 generates $\mathbb{Z}_{12}$ : $\langle 7 \rangle = \mathbb{Z}_{12}$ .

Moreover, the generators of $\mathbb{Z}_{12}$ are exactly the elements coprime to 12: $\{1, 5, 7, 11\}$ , and $|\text{Aut}(\mathbb{Z}_{12})| = \phi(12) = 4$ .

Proof.

$\gcd(7, 12) = \gcd(7, 12)$ . Since $12 = 1 \times 7 + 5$ and $7 = 1 \times 5 + 2$ and $5 = 2 \times 2 + 1$ , we get $\gcd(7,12) = 1$ . By Definition 12.1, $\langle 7 \rangle = \mathbb{Z}_{12}$ .

The elements coprime to 12 are those not divisible by 2 or 3: $\{1, 5, 7, 11\}$ . Euler’s totient $\phi(12) = \phi(4)\phi(3) = 2 \times 2 = 4$ counts these, confirming 4 generators. $\square$

中文: “7 和 12 的最大公约数等于 1——7 和 12 互质。这保证了 7 能生成 ℤ₁₂ 的全部元素。其他步长，只要和 12 不互质，就只能在一个子集里转圈。只有最大公约数等于 1，才能遍历全部。秘密就是这一个条件。”

Theorem 12.2 (Circle of Fifths as Quotient of the Spiral)

In exact (just) intonation, the pitch space is $\mathbb{R}$ (the real line of log-frequencies). The “spiral of fifths” is the infinite sequence

\ldots, \tfrac{3}{2}\cdot p,\; p,\; \tfrac{3}{2}p,\; \left(\tfrac{3}{2}\right)^2 p,\; \ldots

which, by Theorem 11.3, never repeats. In 12-TET, this spiral is projected onto the quotient $\mathbb{R}/(12\mathbb{Z})$ (identifying pitches that differ by an octave and rounding each fifth to $2^{7/12}$ ). The resulting image is exactly $\mathbb{Z}_{12}$ , traversed by the generator 7 — the circle of fifths.

Formally, the map $\phi: n \mapsto 7n \pmod{12}$ is the bijection $\mathbb{Z}_{12} \xrightarrow{\sim} \mathbb{Z}_{12}$ that converts “step number in the fifth sequence” to “pitch class.”

Proof.

The projection map is $\pi: \mathbb{R} \to \mathbb{R}/(12\mathbb{Z})$ , where we identify $\mathbb{R}/(12\mathbb{Z}) \cong \mathbb{Z}_{12}$ by $x \mapsto \lfloor x \rfloor \pmod{12}$ (measuring in semitones).

In just intonation, the $n$ -th fifth above $p$ (at log-frequency 0) is at log-frequency $n \times \log_2(3/2) \approx n \times 0.585$ octaves $= n \times 7.02$ semitones.

In 12-TET, this is rounded to $n \times 7$ semitones. The image $\pi(7n) = 7n \pmod{12}$ is the circle of fifths sequence.

Since $\gcd(7,12)=1$ , $\phi: n \mapsto 7n \pmod{12}$ is a bijection on $\mathbb{Z}_{12}$ (Theorem 12.1), confirming the circle visits all 12 pitch classes. $\square$

The key approximation: The true fifth is $\log_2(3/2) \approx 7.0196$ semitones; 12-TET rounds this to exactly 7. The error is 0.0196 semitones $= 1.955$ cents per fifth. After 12 fifths, the accumulated error is $12 \times 1.955 = 23.46$ cents — the Pythagorean comma. 12-TET distributes this error uniformly, one fifth at a time, forcing the spiral into a circle.

Theorem 12.3 (Unique Factorization and Good EDOs)

An $n$ -tone equal division of the octave ( $n$ -EDO) approximates the just fifth if and only if $k = \lfloor n \cdot \log_2(3/2) \rceil$ satisfies $\gcd(k, n) = 1$ , where $\lfloor \cdot \rceil$ denotes the nearest integer. If $\gcd(k,n) > 1$ , the “circle of fifths” in $n$ -EDO does not generate all $n$ pitch classes.

The best-approximating EDOs with $\gcd(k,n)=1$ are precisely the denominators of the convergents of $\log_2(3/2) = [0;1,1,2,2,3,1,5,2,23,\ldots]$ , namely $n \in \{1, 2, 5, 12, 29, 41, 53, \ldots\}$ .

Proof.

For $n$ -EDO, each step is $1/n$ octave. The fifth is approximated by $k$ steps where $k = \lfloor n\log_2(3/2) \rceil$ . The circle of fifths in $n$ -EDO generates the subgroup $\langle k \rangle \subseteq \mathbb{Z}_n$ , which equals all of $\mathbb{Z}_n$ iff $\gcd(k,n)=1$ (Definition 12.1).

For 12-EDO: $k = \lfloor 12 \times 0.58496 \rceil = \lfloor 7.02 \rceil = 7$ , $\gcd(7,12) = 1$ . Good.

For 5-EDO: $k = \lfloor 5 \times 0.58496 \rceil = \lfloor 2.92 \rceil = 3$ , $\gcd(3,5) = 1$ . Good (but very coarse).

For 6-EDO: $k = \lfloor 6 \times 0.58496 \rceil = \lfloor 3.51 \rceil = 4$ , $\gcd(4,6) = 2 \neq 1$ . The “fifth” in 6-EDO only generates a 3-element subgroup — not all 6 pitch classes.

For 53-EDO: $k = \lfloor 53 \times 0.58496 \rceil = \lfloor 31.00 \rceil = 31$ , $\gcd(31,53) = 1$ (both prime, and $31 \neq 53$ ). Good and very accurate ( $\approx 0.07$ cents error per fifth). $\square$

Prop 12.1 (Lydian Tetrachord Chain (SUM) as a Coset Sequence)

The Super Ultra Hyper Mega Meta Lydian (SUM) structure of Jacob Collier is the sequence of Lydian tetrachord roots

\{C, G, D, A, E, B, F\sharp, \ldots\}

, which is the orbit of 0 under repeated addition of 7 in

\mathbb{Z}_{12}

(or

\mathbb{Z}

before reduction). In 12-TET (the circle), the sequence closes after 12 steps. In just intonation (the spiral), it never closes: each G is a true

(3/2)^n \cdot C

, lying at a different height on the pitch helix, and the chain is an infinite ascending structure.

Numerical Example: Why 12 and Not 7 or 11?

Consider alternative chromatic universes:

7-EDO (7 equal divisions): $k = \lfloor 7 \times 0.585 \rceil = 4$ . $\gcd(4,7)=1$ , so a “fifth” of 4 steps generates all 7 pitch classes. But the fifth is $4/7$ octave $\approx 685.7$ cents — 16 cents flat. No perfect approximation of the major third exists (no $n$ with $4n/7 \approx 386/1200$ ). 7-EDO works for pentatonic/heptatonic music but lacks the interval richness of 12-TET.

12-EDO: As shown, $\gcd(7,12)=1$ , fifth error $\approx 2$ cents, major third error $\approx 14$ cents. The compromise that Western tonal practice settled on.

19-EDO: $k = \lfloor 19 \times 0.585 \rceil = 11$ , $\gcd(11,19)=1$ . Fifth error $\approx 7$ cents (worse than 12-TET!), but major third $= 6/19$ octave $\approx 378$ cents — 8 cents flat, actually closer to just than 12-TET’s 14 cents sharp. Used in some microtonal compositions for its smoother thirds.

53-EDO: $k=31$ , $\gcd(31,53)=1$ . Fifth error $< 0.1$ cents, major third error $< 0.3$ cents. Nearly perfect 5-limit just intonation, but with 53 keys per octave — impractical for keyboard instruments, though used in some theoretical works (Mercator, 1670s; Tanaka, 1890).

中文: “12不是随意的——它的因子结构恰好对应了西方调性音乐常用的对称数量。24太多——虽然中东音乐在用——7太少，因为它是素数，没有真子群。对于十二平均律体系，12是一个有效的选择。”

Musical Connection

音乐联系

The circle of fifths as a group-theoretic object

The circle of fifths is not merely a mnemonic for key signatures. It is the orbit of any pitch class under the group automorphism $\phi: x \mapsto x + 7 \pmod{12}$ , which has order 12 (since $12 \times 7 \equiv 0 \pmod{12}$ and no smaller multiple works). The circle of fifths and the chromatic scale are two different generating sequences of the same group $\mathbb{Z}_{12}$ — one with generator 1 (ascending by semitones), the other with generator 7 (ascending by fifths).

Tonal brightness as direction on the spiral

Collier’s observation that “going sharp” (adding fifths in the direction $C \to G \to D \to \ldots$ ) sounds “brighter” is a perceptual reflection of the acoustic fact that the low-numbered partials of a tone at frequency $f$ are $f, 2f, 3f, \ldots$ . A fifth up means moving toward the 3rd partial, which is acoustically close to the fundamental. A fourth up (adding flats, going $C \to F \to B\flat \to \ldots$ ) means moving away from the fundamental’s low partials.

中文: “为什么五度方向更亮？纯律五度等于三比二——就是泛音列的第3个分音。加7方向就是沿着低序数泛音堆叠，和基音更亲近。明与暗，就是你在五度圈上的方向。”

The Shepard–Risset paradox and musical meaning

Shepard tones (Definition 12.3) exploit the helix structure: by spreading amplitude across all octaves of a chroma with a bell envelope, they create the illusion of a tone with chroma but no register — a “pure” position on the chromatic circle without height. Shepard-Risset glissandi create the paradox of continuous ascent without arriving higher — the chromatic circle rotates continuously, but the vertical helix position stays fixed.

Collier’s SUM is the philosophical inverse: it insists on the helix, refusing the projection. On a piano (which forces $\mathbb{Z}_{12}$ ), SUM closes after 12 steps. In just intonation, or in the voice (which can access any frequency), it never closes — it ascends the spiral indefinitely.

中文: “五度圈是一个优美的谎言——代价是每个五度损失2音分的真实，收益是12个调式的平等。正是这个谎言，让转调成为可能。”

Subgroup structure and symmetric chords

Definition 12.4 shows that $\mathbb{Z}_{12}$ ’s subgroups correspond to the most structurally symmetric objects in 12-TET: the tritone pair, augmented triad, diminished seventh chord, and whole-tone scale. These are exactly the chords whose transpositional symmetry makes them “tonally unstable” — they sound unresolved because they belong equally to multiple keys. The augmented triad is the unique chord invariant under transposition by a major third ( $T_4$ ); the diminished seventh is invariant under $T_3$ . Their ambiguity is a direct consequence of being proper subgroups of $\mathbb{Z}_{12}$ .

Limits and Open Questions

Multidimensional pitch spaces. The Shepard helix is one-dimensional in chroma (a circle) with one height dimension. Krumhansl’s (1983) multidimensional scaling experiments suggest that the perceptual space of keys is better modeled as a torus (circle of major keys × circle of minor keys) embedded in a higher-dimensional space. The torus is the product of two $\mathbb{Z}_{12}$ circles (one major, one minor), connected by relative and parallel relationships. This structure cannot be captured by a single helix.
Optimal EDOs for extended just intonation. Theorem 12.3 characterizes good EDOs for 5-limit just intonation (primes 2, 3, 5). For 7-limit (adding 7:4), the optimal EDOs change: 31-EDO and 72-EDO become attractive. The general question — “what is the best $n$ -EDO for $p$ -limit just intonation?” — is an active research area in mathematical music theory.
The SUM structure in continuous pitch space. Collier’s SUM, if extended indefinitely in just intonation, is an infinite ascending sequence in $\mathbb{R}$ (the pitch helix). The question of whether a composition can make musical sense while its tonal center spirals to arbitrarily high pitch is open. Pauline Oliveros’s deep listening work and La Monte Young’s sustained-tone pieces approach this question experientially, but no formal compositional framework for “infinite spiral form” exists.
Tuning and spectral composition. Sethares (2005) showed that inharmonic timbres can be made consonant with non-12-TET scales by matching the instrument’s partial series to the EDO’s intervals. This raises the converse question: if the “circle of fifths” and the subgroup structure of $\mathbb{Z}_n$ depend on $n$ , and the consonance of intervals depends on the timbre, is there a joint optimization of ( $n$ -EDO, timbre) that produces a richer harmonic world than 12-TET with harmonic timbres?

Academic References

Shepard, R. N. (1964). Circularity in judgments of relative pitch. Journal of the Acoustical Society of America, 36(12), 2346–2353.
Risset, J.-C. (1969). An Introductory Catalogue of Computer-Synthesized Sounds. Bell Telephone Laboratories.
Krumhansl, C. L., & Kessler, E. J. (1982). Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychological Review, 89(4), 334–368.
Balzano, G. J. (1980). The group-theoretic description of 12-fold and microtonal pitch systems. Computer Music Journal, 4(4), 66–84.
Clough, J., & Douthett, J. (1991). Maximally even sets. Journal of Music Theory, 35(1/2), 93–173.
Amiot, E. (2016). Music Through Fourier Space: Discrete Fourier Transform in Music Theory. Springer.
Milne, A., Sethares, W. A., & Plamondon, J. (2007). Isomorphic controllers and dynamic tuning. Computer Music Journal, 31(4), 15–32.
Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press. Ch. 5 (Congruences and residues).