The octave is a logarithm
Equal musical distances come from multiplication in frequency—and become addition after a logarithm.
Watch the reference Short · Original Atlas teaching model with linked sources.
LIVE PITCH FIELD
12 SEMITONES440.0 Hz
THE RELATIONn = 12 · log₂(f₂ / f₁)
MUSICAL VIEW
FROM220 Hz
TO440.0 Hz
DISTANCE12 semitones
MOVEMove the frequency ratio.
HEARPlay both pitches.
NOTICEEvery doubling adds exactly 12 semitones.
Listen beyond twelve notes
Three original experiments inspired by Levi McClain’s music and science channel. Move one control, listen, then explain what changed. These are our teaching models, not copies of his videos.
A third has more than two sizes
348.39 cents · ratio 1.22291. Both tones start together: 220 Hz and 269.04 Hz. The ratio is 2^(steps/31). A 12-EDO minor third is 300 cents; a major third is 400.
Practise: alternate 8, 9 and 10 steps. Sing the upper note before replaying. Teach: ask the class to describe each interval without requiring a right answer. Reference: subminor and neutral harmony.