Comment by codeulike | Hacker News Reader

Comment by codeulike | Hacker News Reader

We know that notes with simple ratios to their frequencies sound 'good' together (e.g. an octave is a 2:1 frequency ratio, a 'fifth' is more or less a 3:2 frequency ratio), a perfect fourth is 4:3 ratio - https://en.wikipedia.org/wiki/Interval_ratio

Equal temperament messes with that a little but its still approximately there.

So why should simple ratios sound good? From what I've read, its probably overtones, and the way overtones excite the cilia in our inner ear. If you plot the overtones from the two notes of an octave or a fifth, they co-incide a lot.

And then why the pattern of notes on the piano keyboard? (More precisely 'why does the major scale use those 7 semitones?') ... I think with that scale, and the other scales, its something to do with squeezing the most amount of 'relationships' possible from a subset of notes without muddying the waters by having too many in play. Music is all about patterns and a mix of repetition and progression, and so having a finite number of notes that have intersting relationships with each other gives a good canvas to work with.

(Any why 12 semitones? Again, to do with getting the most interesting relationships without muddying things ... I think its possible to divide the octave into 19 or 43 parts and get some interesting ratios/relationships, but then it gets quite fiddly)

12 tones is the smallest approximately rational scale. There's a technological reason for preferring this. Up until fairly recent times, a musician had to be able to tune and maintain their own instrument. Also, we can't grow more fingers.

There may also be a benefit to more widely spaced notes, notably (!) that it's easier to distinguish if you're playing a "real" note or not, and if it's the same note as the one you're hearing someone else play. This would make it easier to learn musical ideas and pass them on to others, giving 12 tone music (fewer tones if using an agreed upon scale) a built in error correction code, and a sort of evolutionary advantage over time if you will.

With the exception of a couple instruments like the piano and electronic keyboards, don't musicians still tune their own instruments?

That's true.

When my dad got a harpsichord, part of the process was getting trained by the maker on how to tune it. There's an old joke: Q: What's a harpsichordist doing when they're not tuning their harpsichord? A: Playing out of tune.

I'll just put in that I think things like 12 tone scales and 7 note diatonic scales were local minima in a vast landscape that we happen to hit upon and settle into. The more I look at what music is from a neurological perspective, the more it feels/seems like the deepening and establishing of patterns is the main mode.

The major scale is just the most straightforward way to be able to construct the most usable 4:5:6 ratios and 10:12:15 ratios (i.e. major and minor chords) from the fewest possible notes. In equal temperament, those ratios are approximated rather than exact, but those are the mathematical relationships implied by the chords.

The desirability of "simple ratios" is based on the idea that if we play two pure sine waves at the same time, they sound good if they're exactly the same frequency and if they're some distance apart, but they sound bad if the two notes are close but don't line up. (This creates a beat frequency, which make the music sound unstable and noisy.)

Notes played on real instruments have harmonics, and so if you play two notes at once all those harmonics either need to be a long way from each other or they need to line up. Notes with frequencies that correspond to simple ratios are the ones where the harmonics also line up in the cleanest way.

Simple ratios like 2:1 or 3:2 are very stable and consonant. Larger ratios like 5:4 make modern music a bit more interesting. Still larger ratios like 7:4 and 11:8 can start to sound pretty alien and sort of dissonant and more complex.

Basically, the most consonant music, which is easy to play (imprecisely) in 12-tone equal temperament, is pretty well explored territory. There is only one major chord, and we won't find anything that sounds any more "major-chordish" than it. But there's a huge unexplored territory when it comes to larger ratios that can't be played accurately enough to be intelligible on 12-tone equal tempered instruments.

If you scroll down a bit on this page, I made a visualization of how 12-tone equal temperament lines up with a straightforward just-intonation scale based on the ratios implied by the 12-TET chromatic scale. It's really amazing both how well 12-tone equal temperament lines up, and at the same time how much better things could sound if you play the exact just intervals rather than this system that by some weird mathematical coincidence happens to be good enough for most simple musical purposes.

http://jsnow.bootlegether.net/cbg/justintonation.html

>> why 12 semitones?

As you noted, other numbers like 19, 43, and obviously also 24, gives interesting ratios.

My current understanding is that while human ear is easily able to distinguish finer frequency ratios, singers aren't able to match vocals to much higher precisions. 19 may perhaps still work, but 43 I think would be out of question.

The 7 note major scale is more-or-less an historical accident: https://en.wikipedia.org/wiki/Musical_system_of_ancient_Gree....

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