Eleven gentle fifths. One unruly interval. Listen around the circle, then make your accusation.
Wolf hunt / ¼-comma meantone
Tap a fifth to hear it. Select “Accuse this fifth” when you find the roughest one. Interval sizes stay hidden until the case is solved.
CASE OPEN / Listen to a fifth first.
THE PRICE OF BEAUTY
A perfect third has a price.
Quarter-comma meantone narrows each ordinary fifth by one quarter of the syntonic comma. Four such fifths produce a pure 5:4 major third after octave reduction. But twelve of these fifths fail to close the circle. On this twelve-note keyboard, the remaining gap becomes a wide wolf fifth.
In each new hunt we rotate the entire tuning map. The wolf moves, but its size does not. This is an educational transposition, not a claim about a particular historical keyboard.
737.64 ¢
The wolf in this model · ordinary fifth 696.58¢
EXHIBIT 03 / LISTEN · DISCOVER · UNDERSTAND
Twelve keys. Different shadows.
Transpose one chord through a fixed tuning. Numbers describe intervals; emotions belong to you.
C major
The major-third reading compares your interval to 5:4. Fifths are compared to 3:2. The beat estimate uses |5f(root) − 4f(third)|. It is one pair of partials, not a universal measure of consonance.
Just intonation here is a fixed C-based twelve-note 5-limit lattice, not adaptive harmony. Remote keys may sound rough. Equal temperament preserves interval sizes, but higher transpositions increase beat rates in Hz.
THE TWELVE-KEY THIRD MAP
See where the compromise lives.
Each tile shows the major third above that root in the selected tuning. A gold border marks your current key. This measures intervals, not emotion.
Near 5:4 Farther from 5:4386.31¢ reference
EXHIBIT 04 / LISTEN · DISCOVER · UNDERSTAND
One phrase. Six histories.
Choose a tuning, then play the same original eight-bar study. Timbre stays constant for a fair comparison.
Digital teaching models, not recordings of historical instruments. Dates and practices varied by place and musician. This independent project is not affiliated with temperamentlab.com.
EXHIBIT 05 / LISTEN · DISCOVER · UNDERSTAND
A chord leaves a fingerprint.
Follow the fifth partial of the root and the fourth partial of its major third as they approach each other.
HARMONIC FINGERPRINT / THEORETICAL MODEL
An idealized two-partial beat envelope, not a microphone recording or a live FFT. The drawing is time-scaled for readability.
When two nearby partials differ by 10 Hz, their interference envelope repeats about ten times per second. At an exact 5:4 third, the selected partials coincide. Other partials remain; a complex chord is not reduced to one beat rate.
Use the rich tone to hear upper partials. A pure sine tone contains only its fundamental.
EXHIBIT 06 / LISTEN · DISCOVER · UNDERSTAND
Spend your musical debt.
Twelve pure fifths overshoot seven octaves. Distribute the comma, then hear the consequences.
23.46001 ¢
Pythagorean comma = 1200 log₂(3¹² / 2¹⁹). Sliders set relative shares; the total correction is always normalized to exactly one comma.
Invent your imperfect perfection.
This workshop only distributes the Pythagorean comma with nonnegative shares. It does not reproduce quarter-comma meantone, which uses the distinct 21.51¢ syntonic comma and needs a widened closing fifth. Your result is labeled experimental.
Local saving is optional. No account or server upload.
EXHIBIT 07 / LISTEN · DISCOVER · UNDERSTAND
Trust your ears, not the labels.
An ABX trial: X repeats either A or B. Listen freely, choose once, then reveal the answer.
12-TET vs 5:4 / C–E
Which reference matches X? All three use the same root, level, timbre and duration.
Listen to A, B and X before choosing.
A LISTENING DIARY, NOT A DIAGNOSIS
Small differences take time.
Difficulty is normal. Headphones, playback level, hearing, attention and synthesis all affect the experience. This informal test does not measure hearing health or musical talent. Guessing in a two-choice trial succeeds half the time on average.
0 / 0
Correct / answered · skipped trials excluded
EXHIBIT 08 / LISTEN · DISCOVER · UNDERSTAND
Take the harmony with you.
Keep your tuning name and listening results as a small souvenir from the museum.
TEMPERAMENT LAB / TUNING PASSPORT
Midnight Cathedral
Your discoveries will appear here.
Share links include the experimental tuning, not your listening score.
EXHIBIT FIELD NOTES / LISTEN · DISCOVER · UNDERSTAND
The mathematics behind the mystery.
A compact guide to the choices made by every fixed twelve-note tuning.
What does a cent measure?
A cent is a logarithmic interval unit. An octave is 1200 cents. For two frequencies, cents = 1200 log₂(f₂/f₁). Equal temperament divides the octave into twelve 100-cent steps. A 5:4 third is 386.3137 cents, so the equal-tempered third is about 13.6863 cents wider.
Two commas, different questions.
The Pythagorean comma (531441:524288, 23.4600 cents) is the closure error of twelve pure fifths. The syntonic comma (81:80, 21.5063 cents) separates a Pythagorean major third from a 5:4 third. Confusing them leads to incorrect models of meantone.
A third is not always the same third.
TUNING
C–E MAJOR THIRD
12-TET
400.00 ¢
Just / 5:4
386.31 ¢
Pythagorean / 81:64
407.82 ¢
¼-comma meantone
386.31 ¢
Werckmeister III
390.22 ¢
Vallotti
392.18 ¢
A4 calibration is separate from tuning. Here “C-root locked” keeps C4 = A4 × 2^(−9/12) for controlled comparisons. “A4 fixed” shifts each entire tuning so that its A4 equals the selected reference. No stretch tuning or inharmonicity is modeled.
No single ranking fits every musical purpose. Exact low-integer ratios can align selected harmonics, while equal temperament offers consistent transposition. Timbre, register, musical context and preference matter.
Which historical recipes are used?
All maps are normalized to C. Pythagorean and meantone use a fifth chain F–C–G–D–A–E–B–F♯–C♯–G♯–D♯–A♯. Werckmeister III tempers C–G, G–D, D–A and B–F♯ by ¼ of the Pythagorean comma. Vallotti tempers F–C through E–B by ⅙ of that comma. Just ratios are 1, 16/15, 9/8, 6/5, 5/4, 4/3, 45/32, 3/2, 8/5, 5/3, 9/5, 15/8. Enharmonic spellings share a key in these twelve-note models.
Why do I hear nothing?
Press a play button to unlock browser audio, check the volume and silent-mode settings, and try a current browser. Stop any other playback if comparing carefully. The stop control ends every note scheduled by this page.
What happens to my data?
Synthesis and calculations run in your browser. No microphone is requested. Saving a tuning writes to localStorage on this device; share links put your tuning name and shares in the URL fragment. Blogger and Google advertising may use cookies under their own policies. See Google’s advertising privacy information.
