4 ms·
The centuries-old struggle to play in tune
- tkiley 16y agoQuick mathematical summary of the problem: An octave is a 2:1 ratio between frequencies, so 880 hz is one octave above 440 hz. A perfect fifth is a 3:2 ratio between frequencies, so 660 hz is a perfect fifth above 440. In the modern western system of music, twelve perfect fifths is harmonically equal to seven octaves. In other words, (2/1)7 == (3/2)12 Unfortunately, we know this is mathematically untrue. Furthermore, three major thirds is harmonically equal to one octave: (2/1) == (5/4)3 This also is mathematically untrue. Hilarity ensues.
- baddox 16y agoI'll also summarize the advantage of equal temperament: Regardless of what key the song is in, a certain interval is always the same exact ratio. A major third in the key of F is the same as a major third in the key of Bb. This is good for instruments like the guitar and piano, which aren't made or tuned for a single immovable key. Contrast that with harmonicas, for examples, each of which is made for only a certain key.
- btilly 16y agoYou are using multiplication where you should be using exponentiation. If one fifth is (3/2) then 2 fifths is (3/2) times (3/2) which is (9/4). Your math looked semi-plausible when comparing fifths and octaves, but comparing 3 thirds and an octave it was way off. Therefore the first comparison should be 7 octaves which is (2/1)^7 = 128, versus 12 perfect fifths which is (3/2)^12 = 531441/4096 = 129+3057/4096 = 129.746337890625. Similarly for thirds, you're comparing one octave (2/1) = 1 with 3 thirds (5/4)^3 = 125/128 = 1.953125. As you can see, the ratios are close, but not quite right. Hence the problem.
- mechanical_fish 16y agoI think we just lost some carats in the mix. The original comment meant to type exponentials but they didn't come out right on the page. Someday HTML will support TeX and we'll never have this problem again. ;)
- nandemo 16y agoA better explanation of the problem: http://www.yuvalnov.org/temperament/ http://www.yuvalnov.org/temperament/ Also, if you listened to samples in the Slate article and couldn't hear any difference, try this: http://www.youtube.com/watch?v=BhZpvGSPx6w&feature=related http://www.youtube.com/watch?v=BhZpvGSPx6w&feature=relat...
- danbmil99 16y agoterrible example, because he's using pure sine tones. What makes intervals sound in or out of tune are the harmonics, which clash if it's not just tuning. No one can hear the difference between just and tempered thirds on a tone with no harmonics -- our aural circuitry is just not that precise.
- baddox 16y agoReally? The aural difference was painfully obvious, especially with the thirds and fifths. With the actual songs it was less noticeable because any given chord didn't play for long before changing.
- nandemo 16y ago>What makes intervals sound in or out of tune are the harmonics, which clash if it's not just tuning. How come people use tuning forks and pipes for tuning?
- sp332 16y agoBecause it is convenient :-)
- barrkel 16y agoThere are two things at work here. There's harmonics, where you're working with a relationship between two or more frequencies; and then there's the absolute frequency, which is needed to get different instruments to harmonize. Tuning forks give you an absolute frequency.
- btilly 16y agoDupe: http://news.ycombinator.com/item?id=1283523 http://news.ycombinator.com/item?id=1283523
- tokenadult 16y agoPrevious submission, submitted with the canonical URL: http://news.ycombinator.com/item?id=1283523 http://news.ycombinator.com/item?id=1283523