Tuning by Common Multiples and Common Divisors
Two experiments in treating six guitar strings as one resonant system.
Ege Batuhan Akgül MLKFS
Ask a violinist whether A♭ and G♯ are the same note, and the honest answer is: it depends on how you got there. On a piano they are the same key, by construction. But in tuning systems built from pure ratios, a G♯ reached as a major third above E and an A♭ reached as a chain of descending fifths land on different frequencies — and which one ends up higher depends on the system. In Pythagorean tuning G♯ sits about 23 cents above A♭; in quarter-comma meantone it sits about 41 cents below. The note is not a point. It is a decision.
Twelve-tone equal temperament is the decision that makes every such pair identical: split the octave into twelve equal steps, and A♭ and G♯ collapse into one key. The price is that only the octave remains a pure whole-number ratio; every other interval is displaced by a few cents from the simple ratios a vibrating string naturally produces. The equal-tempered major third, for instance, is 13.7 cents sharp of the pure 5:4 third.
Guitarists have all met this number, whether they know it or not — usually on the B string. Tune the open B so it makes a pure major third with the open G and it lands roughly 14 cents flat of equal temperament; the open G chord glows, and half the other chords on the neck sour. There is even a famous accidental recording of it: the intro of Red Hot Chili Peppers’ “Scar Tissue”. Guitarist Paul Davids, trying to reproduce its sound, measured John Frusciante’s B string at about 13–14 cents flat — almost exactly the just-intonation correction, which is why the major third between the D and the F♯ fretted on that flat B string rings the way the record does. Frusciante later confirmed the theory while waving away the mystique: “I just was out of tune… it sounded good so nobody ever said anything.” We will come back to that sentence at the end, because it deserves a closer reading than he gave it.
For now it is the right frame for everything below: a few cents of displacement, in the right direction, is not noise. It is a different tuning system briefly showing through.
Standard guitar tuning gives us E–A–D–G–B–E, but the frequencies underneath those names are compromises of exactly this kind.
What happens if we keep EADGBE as the identity and playable geometry of the instrument, but stop treating equal temperament as the only possible target for the six open strings?
Two opposite constructions emerge. One searches upward, for harmonics the strings can share. The other searches downward, for a frequency the strings could all be harmonics of. Neither requires moving any string more than about four cents — less than most players ever tune to in the first place.
Before going further, one honest disclaimer: none of the ideas below are individually new, and the names we use for them here — “common-multiple tuning”, “common-divisor tuning” — are our labels, not established terminology. What might be mildly interesting is the framing: treating the six open strings as one global optimization problem, with an explicit error budget in cents and an explicit preference for physically meaningful harmonics. We will come back to what already exists at the end.
What is established
A (nearly) ideal string vibrating at fundamental frequency f also vibrates at 2f, 3f, 4f, … — the harmonic series. When two strings with fundamentals f₁ and f₂ satisfy
n · f₁ = m · f₂for small integersn, m
some of their partials land on exactly the same frequency. This coincidence-of-partials view of consonance goes back to Helmholtz, and it is the reason simple ratios sound “locked in”: the third harmonic of a low E and the second harmonic of the B a fifth above it are the same note.
Guitarists already exploit this every time they tune with the fifth- and seventh-fret harmonics. That method silently produces whole-number ratios rather than equal-tempered intervals — which is exactly why it drifts audibly out of tune with a keyboard. Tune the B string to the seventh-fret harmonic of the low E and it lands about two cents sharp of equal temperament, because 3:1 is a pure twelfth and the equal-tempered twelfth is not pure.
Two more established facts constrain everything below:
- Real strings are slightly inharmonic. Stiffness pushes the actual partials sharp of the ideal
n·f, and the shift grows roughly withn²(the stiff-string wave equation). On a guitar this smearing reaches whole cents somewhere around the 8th–15th partial. Any scheme that depends on the 29th harmonic of a string landing somewhere exact is fiction: by that order, the partial isn’t where the ideal math says it is, and it carries almost no energy anyway. - Sympathetic resonance between guitar strings is real but weak. Strings couple through the bridge and body; an undamped string audibly rings when another string feeds its frequencies. Whether that coupling — or the feedback loop of an amplified guitar — behaves measurably differently under the tunings below is precisely the open question, not something we get to assume.
Experiment 1: common multiples (the feedback tuning)
On an electric guitar under gain, feedback settles at frequencies where the loop — string, pickup, amplifier, speaker, air, body, string — has enough gain to sustain itself. Open strings that share a partial offer the same frequency several paths back into the system. The hypothesis, and it is only a hypothesis: a tuning with deliberately coincident low-order partials gives feedback more places to lock, and changes how the instrument settles when everything rings.
The construction starts from the two E strings, kept exactly two octaves apart, and derives everything else as a whole-number ratio of the low E:
| String | Ratio to E2 | Frequency | Offset from 12-TET | Lowest shared partial |
|---|---|---|---|---|
| E2 | 1 | 82.41 Hz | 0.0 ¢ | anchor |
| A2 | 4/3 | 109.88 Hz | −2.0 ¢ | 3×A = open E4 (329.6 Hz) |
| D3 | 16/9 | 146.50 Hz | −3.9 ¢ | 3×D = 4×A (439.5 Hz) |
| G3 | 19/8 | 195.72 Hz | −2.5 ¢ | 8×G = 19×E2 (1565.7 Hz) |
| B3 | 3 | 247.22 Hz | +2.0 ¢ | B is the 3rd harmonic of E2 |
| E4 | 4 | 329.63 Hz | 0.0 ¢ | anchor |
No string moves more than 3.9 cents. The offsets are pure interval arithmetic, so they apply unchanged to a half-step-down E♭ tuning.
Some of what falls out is pleasingly low-order. The open B string sits exactly on the third harmonic of the low E, and 4×B = 3×E4 at 988.9 Hz, so the two E strings and the B share partials all the way up. The third harmonic of the A string is the open high E itself. The D and A strings meet at 439.5 Hz.
Here is the honest part: five of these six strings are just the old harmonics tuning. Ratios of 4/3, 16/9, 3 and 4 are exactly what the fifth- and seventh-fret harmonic method produces. The only deliberate choice in the table is G. The harmonics method would give G = 64/27 of E2, which lands 5.9 cents flat — the famous weak spot of tuning by harmonics. Choosing G = 19/8 · E2 instead keeps G within 2.5 cents and pins its 8th harmonic to the 19th harmonic of the low E at about 1.6 kHz — high, but still within the range where a guitar, its pickups and an amplifier carry meaningful energy.
An earlier draft of this construction related G to B through 29·G = 23·B, which is numerically closer to equal temperament (+0.65 ¢). We dropped it. The coincidence it buys lives at 5.7 kHz on the 29th partial of the G string — an order at which string stiffness has already shifted the real partial by more than the tuning precision being optimized, and at which almost no energy remains. A rational relation that only exists in the 29th harmonic exists on paper, not on a guitar. That trade — accept two extra cents of offset in exchange for coincidences the instrument can physically express — is the whole design principle here.
Whether any of this changes feedback behavior in practice is an experiment, not a result. We are not claiming it works. We are claiming it is cheap to try.
Experiment 2: common divisors (the subharmonic construction)
Now reverse the direction. Instead of asking where the harmonics meet above the strings, ask whether all six fundamentals can be written as integer multiples of a single frequency below them:
fᵢ = nᵢ · F₀for integersnᵢ
This is where a “greatest common divisor” analogy suggests itself — and it is worth being precise about the word. Equal-tempered frequencies have irrational ratios, so no common divisor exists at all. The moment you nudge the strings onto whole-number ratios, a common divisor exists automatically: the tuning in the table above already has one, at F₀ = E2/72 ≈ 1.14 Hz, with the strings sitting at multiples 72, 96, 128, 171, 216 and 288.
That reveals something the two-experiments framing slightly hides: the two constructions are mathematically the same thing viewed from opposite ends. Six strings share a common subharmonic exactly when every pair of strings shares coincident partials. The common-multiple view and the common-divisor view describe one family of tunings; the only physical question is whether the coincidences occur at orders low enough to matter.
Treated as a question in Diophantine approximation — how coarse can F₀ be while every string stays within ±5 cents of EADGBE? — the answer has a clean shape. Writing F₀ = E2/q and choosing the nearest integer multiple for each string:
| q | F₀ | Worst string error |
|---|---|---|
| 18 | 4.58 Hz | 7.6 ¢ |
| 24 | 3.43 Hz | 9.6 ¢ (D string fails) |
| 42 | 1.96 Hz | 3.8 ¢ |
| 69 | 1.19 Hz | 2.0 ¢ |
| 72 | 1.14 Hz | 3.9 ¢ (= the tuning above) |
| 206 | 0.40 Hz | 0.85 ¢ |
The elbow sits around q ≈ 42–72: below that, some string (usually D or G) cannot stay within five cents; beyond q = 69 the improvement stalls near two cents for a long stretch, because the pure fourth and pure twelfth that pin the A and B strings are themselves 1.955 cents from equal temperament. Finer subdivision buys almost nothing until the integers grow past 150 — deep into physically meaningless territory.
And here is where conservatism is required. A shared period of 1–3 Hz is infrasonic. It is not heard as a pitch: virtual pitch — the “missing fundamental” the ear reconstructs from upper partials — operates when the implied fundamental is in the audible range, not at 1.14 Hz. Nor does the shared period do anything mechanical on its own: the strings are not phase-locked, and real inharmonicity breaks the exact periodicity anyway. Any genuine acoustic consequence of a common subharmonic is carried entirely by the coincident audible partials — which is Experiment 1 again.
So the common-divisor construction should be read as what it is: a coordinate system, not a claim. It is an elegant way to parameterize the family of near-EADGBE rational tunings and to see the trade-off between integer size and tuning error. It earns a place in the mathematics. It does not earn a claim about sound.
What already exists
Almost every ingredient here has a literature. Coincident partials as the basis of consonance is Helmholtz; tuning toward aligned partials, including for non-standard spectra, is developed at length in William Sethares’ Tuning, Timbre, Spectrum, Scale and his work on adaptive tunings. Just-intonation guitars, microtonal fretwork, and “sweetened” tunings that offset open strings by a few cents are established practice, and commercial systems like Buzz Feiten and True Temperament already ship cent-level corrections — though those aim at making fretted notes more equal-tempered, roughly the opposite goal. Piano tuners have always tuned to the instrument’s actual partials rather than ideal ones; entropy-based tuning formalizes exactly that.
What we have not seen framed this way is the small, specific thing this post does: take the six open strings of a standard-tuned guitar as one system, impose a hard cents budget, and select the integer network — asking on one side which coincidences an amplified instrument could plausibly use, and on the other how the same tunings look as a single subharmonic lattice. Modest, but it seems to be an unclaimed corner.
Trying it
Any tuner that displays cents can set the table above: E strings at zero, A at −2, D at −4, G at −2.5, B at +2. (If you tune A, D and B with the old fifth- and seventh-fret harmonics method, you are already there; only G needs the tuner.)
What would count as evidence? Not “it sounds mystical.” Something more like: with identical gain staging, does the guitar enter feedback faster, or favor different pitches, than the same guitar tuned to strict equal temperament? Do open chords ring measurably longer? A spectrogram of both takes would settle more than any amount of description.
Equal temperament asks where each note should be. This experiment asks a slightly different question — where the energy of one particular instrument, with six strings ringing at once, might prefer to meet. The mathematics above guarantees only that the meeting points exist. Whether the guitar cares is what the experiment is for.
Coda: what the ear knew first
Go back to Frusciante’s sentence. “I just was out of tune… it sounded good so nobody ever said anything.”
Read literally, it is a disclaimer. Read carefully, it describes a measurement process. A string drifted. An ear — his, then Rick Rubin’s, then everyone’s who heard the record — evaluated the result and kept it. Nothing about that keeping was random. Out of all the directions and distances a string can drift, the one that stayed on the record was flat by almost exactly 13.7 cents: the precise gap between the piano’s compromise third and the third a vibrating string actually produces. The tuner said out of tune. The ear said finally in tune. They were both right, against different references — and the ear’s reference turned out to be the one with the physics behind it.
This is worth saying plainly, because “it was an accident” and “it reflects a deep regularity” are not competing explanations. The accident supplied the variation; the intuition did the selecting. Frusciante’s modesty is about intent, not about knowledge. A musician who has spent tens of thousands of hours listening carries a model of consonance far more precise than any number he could quote — precise enough to recognize a 13.7-cent correction as good in the half-second it takes to decide not to re-tune. The mathematics in this essay adds nothing to that judgment. It only transcribes it.
That pattern is older than this song. Singers, violinists and barbershop quartets have always bent thirds toward 5:4 without knowing the ratio. Piano tuners stretched octaves to fit each instrument’s inharmonicity generations before the stiff-string equation was written down. The formalism, in every one of these cases, arrived late — not to correct practiced intuition but to explain why it had been right all along.
Which is, finally, the honest way to read everything above. The tables in this essay are not instructions the ear must obey; they are a transcription of what a well-listened ear tends to choose, extended a little further than habit usually goes. If a tuning in this essay sounds wrong on your guitar, the tuning is wrong — the integers have no vote against the ear. But when something out of tune sounds this good, it is worth taking seriously the possibility that it isn’t out of tune at all. It is in tune with something you haven’t written down yet.