Suppose you tune one note, then tune another a perfectly pure fifth above it. Keep going. After twelve fifths you expect to return to a familiar note name, seven octaves higher. The arithmetic disagrees. The journey overshoots, leaving a small remainder that cannot be wished away by careful workmanship.
This is a design problem disguised as a mistake. A fixed keyboard asks a finite set of keys to serve many harmonic relationships. Some desirable frequency ratios refuse to fit together exactly. A tuning system decides where that disagreement will be heard.
Do the journey with ratios
In John Baez's mathematical account, a pure fifth has frequency ratio 3:2 and an octave 2:1. Twelve fifths multiply frequency by (3/2)12; seven octaves multiply it by 27. Their quotient is 531441/524288, approximately 1.01364. This discrepancy is the Pythagorean comma.
The two routes end about 1.36 percent apart in frequency. That percentage is calculated from the ratio, not measured in a performance here. Musical intervals are more conveniently compared logarithmically in cents: 1,200 cents make an octave. The same remainder is about 23.46 cents.
You can verify the issue with a calculator. Multiply 1.5 by itself twelve times and compare the result with 128. Now ask a different question: how much must each of twelve identical steps be adjusted if their combined journey is to close? The answer concerns a repeating ratio, so simply subtracting a fixed frequency from every note is the wrong operation.
A wolf is tied to a particular compromise
The Lehman College physics notes compare open, closed unequal, and equal temperaments. In a twelve-note Pythagorean arrangement, preserving eleven pure fifths leaves a substantially narrowed closing interval. It is often called a wolf fifth. Other temperaments can produce a different wolf with a different size and direction of error.
Keep that last distinction. The famous wide wolf in quarter-comma meantone is not numerically the same interval as the narrowed Pythagorean closure demonstrated below. “Wolf” is a family resemblance in the problem a listener encounters, not one universal frequency ratio.
Move the closing interval
A silent ratio calculator compares a Pythagorean closing link with equal-tempered fifths. No measured instrument data.
The model uses twelve abstract links in a closed chain. It compares eleven pure fifths plus one closing remainder with twelve equal-tempered fifths. It deliberately avoids a historical keyboard's spelling and octave-placement conventions. Moving the marked closure changes its location in the chain, not the amount of arithmetic left over.
Give the remainder an address
Imagine programming a small musical instrument for only one piece. If the piece never crosses a particular harmonic relationship, putting the largest mismatch there might be a practical choice. Now lend the instrument to someone playing in distant keys. The hidden compromise may become the first thing they notice.
Move the closing link in the model. Choose three neighboring links as your imaginary piece's territory. Can you place the special interval outside that territory? Expand the territory to all twelve. You can no longer keep one exceptional link out of every possible journey. The exercise turns an abstract comma into a question about intended use.
Equal does not mean all ratios are pure
Equal temperament makes each semitone the same ratio, 21/12. Its seven-semitone fifth is 700 cents, slightly narrower than the pure fifth's approximately 701.96 cents. The mismatch is spread instead of collected in a single closure. This removes the exceptional fifth within this ideal twelve-note model, while giving up pure 3:2 fifths.
The Tune Smithy creator's listening notes distinguish the Pythagorean example from the original meantone use of the term. Its examples can extend the arithmetic into listening, but compare the tuning names before comparing impressions. Different demonstrations may be talking about different compromises.
What the model leaves out
Real piano strings are not ideal harmonic oscillators, and practical tuning may stretch octaves. This calculator does not model that, individual instruments, historical tuning procedures, or judgments of musical beauty. It isolates the incompatibility between closing a twelve-link chain and keeping every fifth at 3:2.
There is another way to check the calculator: multiply all twelve displayed link ratios. In either arrangement their product should be 128, the seven-octave span. That check tests the closure independently of the displayed cents. The two systems agree on the journey's endpoint while disagreeing about the size of its individual steps.
The wolf is interesting because it exposes a system's priorities. A tuner can favor certain relationships, distribute discrepancies, or make additional pitches available. The question is not whether arithmetic can be defeated. It is which musical possibilities a particular compromise makes comfortable—and which relationship is asked to carry the remainder.
Sources and further exploration: John Baez: mathematics of tuning systems; Lehman College: temperament and pitch; Tune Smithy: different wolf fifths.
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