The Tonnetz (tone network) maps musical intervals as geometry. Moving right multiplies frequency by 3/2 (perfect fifth); moving diagonally multiplies by 5/4 (major third). Click nodes to hear tones; click multiple to build chords.
Click nodes to hear intervals and chords
Just intonation uses pure integer frequency ratios: 5th = 3/2, major 3rd = 5/4, minor 3rd = 6/5. The 2D Tonnetz lattice (Euler 1739, Riemann 1880) encodes harmonic proximity as geometric distance. The Pythagorean comma (3/2)^12 ≠ 2^7 is why equal temperament was invented — but just intervals beat zero and sound purer.