Just Intonation vs Equal Temperament

This looks neat. I wish I could try it on my iPad. There is a lot of interesting mathematics behind music. I particularly enjoy reading about different temperaments, which are our attempts to compromise between just intonation (the harmonic-based scales the article mentions, which ostensibly sound the most "natural" and "correct" in a given key), and physical instruments, which we usually want to be able to play in multiple keys without significant adjustment. Twelve-tone equal temperament is the most common temperament in Western music, where the frequency of each semitone is the frequency of the previous semitone multiplied by the twelfth root of two. With equal temperament instruments, the error between just intonation intervals is the same regardless of the key. There is a rich history of temperaments. A early attempt, attributed to Pythagoras, illustrates the impossibility of constructing a temperament perfectly from the simple ratios generates by harmonics: [LINK] The Wikipedia article on the tuning of Bach's aptly-titled Well-Tempered Clavier is also fascinating: [LINK] [LINK] Edit: this video is a great demonstration of the errors between just intonation intervals and our equal temperament compromise: [VIDEO] .

With equal temperament you actually don't modulate precisely, so there is always weird disharmonic sound every single time. As I said, for some electronic music subgenres it's a welcome feature; it would shock classical composers if they ever heard such a version though. Have you ever thought why do you need to state scale at the beginning of a classical piece if you can just easily transpose song to whatever scale you like? Well, those scales sounded differently and each composer was exploring what sounds best in a given scale. Watch this: [VIDEO] Ever played on a top end concert grand and thinking it sounds weird? Then tried a different tuning and being shocked how much better it sounded?

Equal tempered tuning dissonant on oscilloscope: [VIDEO] Exploring the lattice, JI and ET: [LINK] [LINK] > the Tonnetz (German: tone-network) is a conceptual lattice diagram representing tonal space first described by Leonhard Euler in 1739. [LINK]