What makes two notes agree, and what must a tuning give up to keep them agreeing?
Level 1
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5 ideas. Read the hook, pick an answer, see the picture.
The Seven Liberal Arts, Hands-On
The Seven Liberal Arts, Hands-On: Logic, Proof and the Quadrivium
Euler's polynomial, n squared plus n plus 41, gives a prime for every n from 0 to 39. At n = 40 it gives 1681, which is 41 squared. Testing refutes a rule quickly; it cannot prove it for every n.
Pick one
You want a new square with exactly twice the area of an old one. What should its side be?
Doubling the side quadruples the area; one and a half times the side gives 2.25 times. Cut the old square along its diagonal: the square on that diagonal holds four such halves, the old square two. Both wrong options are the slave boy's guesses in Plato's Meno (82e, 83e); Socrates leads him to the diagonal (85a-b).
Your maths progress bar can read high while a whole domain is weak, because it rests on small samples. This wing's test is blunter: solve fresh problems unaided, explain why, and repeat it a week or more later.
Pick one
This wing proposes a spine of four stages for learning mathematics. Which stage comes first?
Logic and proof come first, for knowing when a claim is actually established. Calculus is not among the four stages at all. The page points to it through the Learn forward door.
Three Entry Doors
Repair gaps
Maths did not stick before
Test, repair only what fails
Learn forward
Foundations are solid
Use the map to choose next area
Learn to reason
Thinking clearly over computing
Begin with proof and discrete maths
Pick the door that matches where you are, not where you wish you were.
Square both sides of sqrt(x + 2) = x and you get two answers: 2 and -1. Only 2 survives the check, since sqrt(1) is 1, not -1. Squaring smuggled in an impostor.
Pick one
Cancelling turns (x^2 - 1)/(x - 1) into x + 1. For which values of x do the two actually give the same number?
At x = 1 the fraction divides by zero and has no value, but x + 1 gives 2. Simplifying does not erase that restriction.
Algebra's dependency groups
7Quadratics, growthFactoring, completing the square, the quadratic formula, exponential models; leans on groups 4, 5 and 6.
6FactoringCommon factors, difference of squares, trinomials; leans on groups 1 and 5.
5Exponents, radicalsLaws of exponents, rational exponents, simplifying roots; leans on group 1 only.
4FunctionsNotation f(x), domain, range, graph reading, composition; leans on groups 2 and 3.
3Lines, systemsSlope, intercepts, graphing, elimination and substitution; leans on groups 1 and 2.
2EquationsSolving, rearranging formulas, flipping an inequality when multiplying by a negative; leans on group 1.
1ExpressionsVariables, like terms, distribution, evaluating; rests on arithmetic.
↑ higher level
Seven groups, numbered as in the table on the full page. Each leans on earlier ones, so repair the group where the error began.
Shrink a right triangle to napkin size or stretch it across a football pitch. Keep one angle at 30 degrees, and the side opposite it stays exactly half the hypotenuse. That ratio is the sine of 30 degrees.
Pick one
With no interval stated, how many solutions does sin t = 1/2 have?
One turn holds two, 30 and 150 degrees (pi/6 and 5 pi/6). Adding or removing whole turns gives infinitely many more. In general, t = pi/6 + 2 pi k or 5 pi/6 + 2 pi k, for any integer k.
From geometry to precalculus
1SimilarityEqual angles give proportional sides; this defines sine.
2Pythagorasa^2 + b^2 = c^2, plus its converse and coordinate distance.
3Unit circlePoint at angle t is (cos t, sin t).
4Laws, identitiesLaw of sines, law of cosines, identities and equations.
5PrecalculusInverse functions, conics, vectors, series; then calculus.
Trigonometry grows from similarity, so learn proportion before treating sine as a button.
Proof and Precise Reasoning: From Arguments to Theorems
Euclid's proof that the primes never run out is often presented as a proof by contradiction. His own version is a recipe. Give it any finite list of primes, and it produces a prime not on the list.
Pick one
Take 'If it rained, the road is wet'. Its truth table has four rows: rain or not, wet or dry. In how many rows is it false?
Only rain with a dry road makes it false. Without rain it promised nothing, so it counts as true, wet road or dry.
What Counts as Proof
Example
Shows a claim holds once
Can fail at the next case
Computation
Checks particular cases
Refutes; proves only if every case is checked
Diagram or simulation
Suggests a truth
Can hide assumptions
Evidence about a model only
Proof
Each step follows from definitions
Covers every case in the domain
Only a valid proof establishes a universal claim (checking every case of a finite domain is one); the rest suggest or refute.
A string sounds 220 Hz when open. You stop it so that exactly half of its length vibrates, with tension unchanged. What is the new frequency?
Frequency is inversely proportional to vibrating length, so half the length doubles the frequency to 440 Hz. This is the 2:1 octave.
Question 2 of 8
A pure fifth (3:2, 701.955 cents) is stacked on a pure fourth (4:3, 498.045 cents), and their ratios multiply to 2:1. What happens to their cent values?
Cents add where ratios multiply: 701.955 + 498.045 = 1200, the octave. Subtracting cents matches dividing ratios, and 701.955 - 498.045 = 203.91 cents is the 9:8 tone.
Question 3 of 8
Why can twelve pure fifths never exactly equal seven octaves?
Equality would need (3/2)^12 = 2^7, which means 3^12 = 2^19, and an odd number never equals an even one. The pure fifth 3:2 is rational, intervals combine by multiplying ratios rather than by counting steps, and tuning accuracy has nothing to do with it.
Question 4 of 8
On a monochord, the stopped part of the string is two thirds of the open length. Compared with the open string, the stopped part sounds:
Frequency varies inversely with length, so two thirds of the length gives three halves of the frequency: a pure fifth, 3:2. A stopped length of three quarters would give the fourth, 4:3.
Question 5 of 8
The equal-tempered fifth is 700 cents and the pure 3:2 fifth is 701.955 cents. Which statement is correct?
701.955 - 700 = 1.955, so the tempered fifth is about 1.96 cents narrow. The tempered fifth is 2^(7/12), about 1.4983, not exactly 3/2.
Question 6 of 8
Which of these is the Pythagorean comma written as a frequency ratio?
A comma is a ratio, found by dividing the stack of twelve fifths by seven octaves: (3^12/2^12) / 2^7 = 3^12/2^19. Subtracting gives 1.746, which is not an interval, and 23.46 is the size in cents, not a ratio.
Question 7 of 8
Which statement about strings and hammers is physically correct?
Frequency varies inversely with length, so a 2:1 length ratio gives an octave, as the monochord shows. The hammer story is legend, tension acts through its square root, and the longer string is the lower one.
Question 8 of 8
The equal-tempered major third is 400 cents and the pure 5:4 third is 386.31 cents. Which statement is correct?
400 - 386.31 = 13.69, so the tempered third is wide of pure by 13.69 cents. A cent is a measure of interval, not a number of hertz.