Random note. One of my favorite instruments is the wire-strung Celtic harp. It was killed by several things. First of all it didn't include the sharps and the flats. Secondly it has a 10-30 second sustain. Because of those two factors it sounds better in the older tunings. Because of those two factors it sounds horrible playing more complex modern music.
That was a great article. It used multimedia wonderfully, was intriguing, had a subtly-humorous tone, told an interesting narrative, and most importantly it taught me something I knew absolutely nothing about in the meantime.
This is what Hacker News is all about, in my opinion.
For bonus difficulty: Single strings of the low and high end of pianos aren't even in tune with themselves. All those little terms that in Physics 101 you said "…and those approach zero so cross them out" come back and make the higher harmonics and lower harmonics not match. Pity the poor bass player attempting to choose his pitch for his note. [wikipedia:stretched tuning]
I generally tell pianists to keep their left hand out of my octave and no one will get hurt.
You know, I've tried to jam with bands as a non-fretted player. It's tough - especially since I haven't played with fretted/keyboard instruments in about 12 years (switched from duet recitals to chamber/symphonic).
It's really frustrating to try and figure out the precise notes - but I can improvise fairly well on my own or with another string. I definitely need to train my ear better, but knowing that there's this interval problem now - it'll probably help, because I know that there's something I'm going to have to adjust for.
Do we really need these for every multi-page article? I mean, on this one in particular you read down to the bottom of the first page and see:
1 | 2 | 3 | 4 | SINGLE PAGE
Clicking SINGLE PAGE even puts us at the correct position in the single-paged article! When I saw the link though, I couldn't help laughing to myself and thinking, "I should be one of those karma dudes who posts 'Single Page:' as a comment." Then I checked out the comment thread and here it is.
Who votes these up? Don't we all know by now that clicking the "Print this article" or "Single page" link on a multi-page article will transport us to that magical single page?
(In case you're wondering, I didn't vote the parent down.)
I'd prefer that the submitters submit the full page stories or printer friendly versions as the submission. Posting the single page link in the comments is friendly. I also check the comments on HN before clicking the submission link, as there is often a full page link, which always makes me happy. (The comments also tell me whether if I should take the time to actually check the submission out)
I kind of had the feeling that someone is jumping to the karma conclusion. Woe. Woe, I dare to say.
Oh, and if I said I didn't appreciate the karma, I'd be lying. The +3 boost this is giving me puts me at 660, and I'm annoyed when my karma isn't divisible by 5. (I'm serious)
Otherwise I don't really care. In fact, I often post riskier comments if my overall karma isn't divisible by 5. If I was at 901, and someone somewhere downvoted me to 900, it would make me happy.
So please don't screw up my karma, you'll make me sad.
I also remembered that I hate touching my mouse, and I prefer to keep my ring finger on "page down" when reading on a computer. Having to physically lift my hand, move it to the mouse, find the cursor on the screen, aim it at "next page" or whatever then click, and if I misfire... start over. This really breaks my flow and is quite disruptive.
I know I'm not the only one who cares about stuff like this.
Fair enough, thanks for answering my question. FWIW, I wouldn't have made my comment if I couldn't have made it (I hope!) a little humorous. I'm in an especially comment-y mood tonight, and I've often wondered why these "Single page" posts are so popular and if they stuck out to anyone else. I find the automatic-karma-race nature of such posts (analogous to the classic xkcd new-comic-insta-submissions on reddit) somewhat intriguing. That is, such posters are obviously being helpful, but in the back of your mind you can't help but question their motivation.
It includes music featuring some really intriguing concepts. For example, "Three Ears" isn't played in a fixed scale at all - instead the tuning is tweaked on the fly "for maximum consonance". The result is weirdly fascinating.
Virtually no untrained ear can hear the difference. It's perfectly normal. Now, if you really have no clue, I can give you a mathematical intuition, so your brain understand what your ear can't hear.
(1) Axiom: We hear at a logarithmic scale. What we perceive to be a difference (or interval) between 2 notes is actually a ratio of frequencies. (I think biology may explain this axiom.)
(2) Axiom: Say you hear 2 notes, of frequencies f1 and f2 respectively. When the f1/f2 ratio is a simple rational number (like 2) or (3/2), it sounds good. When the ratio is more complicated (like 19/17), it sounds worse. (Physics can explain that axiom.)
(3) Definition: when the f1/f2 ratio is 2, we call that an octave. The 3/2 ratio is a fifth. The 4/3 ratio is a fourth. As a side note these ratio were basically the only ones that were used. They didn't really used thirds or sixths, probably because of their more complicated ratios, which may have sounded bad to their ears. 
(4) Theorem: There is no way in hell you can make an octave out of fifths (they won't perfectly tune together). Informal proof: this is because you can't find any (i,n) ∈ ℤ², such that (3/2)ⁱ = 2ⁿ. As a side note, you can come relatively close: (3/2)¹² = 129,75 which is close to 2⁷.
So, a mathematical impossibility prevents you to perfectly tune the two most basic intervals ever. Ouch. We have to compromise, then. We can sacrifice a few chords (which if played will cause severe ear damage); or cheating a little bit on every ratio, so no chord sounds outright wrong, nor exactly right; or we can try to find a middle ground between these two extremes.
The "sacrifice" strategies was originally favoured. They sounded better, but restrained what you could play. Now, we favour the "cheating a little" strategy (also called "equal temperament"). They give you more liberty, but sounds rather dull on old music meant to be played with an old fashioned tuning (to trained ears, at least :-).
The actual note doesn't matter at all. Just the ratio. We do have a reference note, but this is only for convenience. (Indeed, the reference note in Europe progressively changed, from 415Hz to 440Hz).
I'm not sure the ratio of two notes played one after the other really counts by itself. However in many instruments, two consecutive notes will tend to overlap (piano, for instance). Also, many instruments (especially those with strings, like claviers and violins), resonate better when the note you play has a "good" ratio with the natural notes of the instruments, even if you don't play them! Some instruments were even designed around this principle. So you have to maintain a good ratio with respect to these "base" note at all times.
In practice, the ratio of two notes played one after another is indeed important.
I'm not sure I understand your last question. Actually, you can change the tune of a piano. So, the optimization you speak of is possible even on a modern piano. Just re-tune it.
A final note about why "simple ratios" sound better: When you make a string resonate, it doesn't do only one note. It does its base frequency (say f), and many others (every multiple of f). So, in a piano, when you play a G at 100Hz, it also plays at 200Hz (a G), 300Hz (a D), 400Hz (a G) etc. Note that there exist actual keys whose main frequencies are 200Hz, 300Hz, 400Hz and so on. They will resonate, which will make the sound richer, louder.
The problem with equal temperament is that the only ratio which is really respected is the octave (2 to 1). In such a case, the D I mentioned above won't be quite at 300Hz, and won't resonate well. So equal temperament became practical only when instruments became loud enough to make up for the loss in resonance.
you can see how the resulting values approximate certain ratios. Ratios matter because two tones that are related by a simple ratio will tend to have greater harmonic relationship and be more 'consonant'.
Other tunings emphasize different ratios and give up the equal spacing of notes -- equal spacing when plotted by frequency on a logarithmic scale.
For information about the temperament used in Watchorn's recording of the Well-Tempered Clavier, based on an examination of (I kid you not) the spirally ornaments on the title page of Bach's manuscript, see http://www.larips.com/ . (Why "larips"? Well, you see, the analysis begins by looking at that title-page upside down, and "larips" is "spiral" backwards. Again, I kid you not.)
It sounds pretty good, actually, but so do plenty of other temperaments.
I have done some work on this. I think the best way to tune a system for playing diatonic music is to ensure that the major and minor chords on tonic, dominant and sub-dominant are perfectly tuned in harmony. This sets exact frequencies for every note of the 12 except for the diminished dominant, which is not used much except in modulation or more atonal music. This tuning gives very good sounding harmonies. If the tune modulates, the best thing to do is to change the tuning of all the notes to match the new key. You might also shift the pitch of the new key note to its equal-tempered value, or to the value in the original "base" key, to which your piece will probably be returning. This will give excellent sounding music. Even the chromatic scales sound really brilliant with proper harmonic tuning like this, if you listen to a random segment of a chromatic scale, you can get a feel for where the tonic note is! equal-tempered scales just don't give that. I have written a simple computer-keyboard app that implements this tuning and modulation, it not complete, a work in progress.
this system does keep every key sounding the same, for me that is a desirable quality. Some variety of harmonies might be used to achieve different feels. The key of the system is that it is designed for an electronic synthesizer - a different tuning can be used for any piece or key within a piece, and the tuning can be varied at modulation.