Music theory for nerds

Music theory can't explain why a piece is designed in some way; it explains what patterns can be found within an existing piece. Designing any aesthetic is primarily about how patterns are prioritized, associated to other parts of culture and turned into conventional tropes or motives. As we get new genres of music the pattern languages tend to change. (The idea of music as "universal language" is only true in a basic sense of what things our ears and brains can comprehend and how we would perceive them in an ungrounded state. In the details, cultural differences will definitely matter.)

So, music theory "catches up" to the pattern language by associating it to human natural language, but it doesn't say why. I concur with the "music appreciation" recommendation for learning the whys. When you get deep into analysis of a work, all sorts of angles can be found to correlate "the thing in the work" with "the reason and context of its creation". For one song, maybe it's the lyrical content that is important. For another, it's about rhythm, or dynamics. The artist's life at that moment, sociopolitical context, and newly available technology are often considered as factors. In a complete work, these elements blend such that it can't be reduced to a singular "this word or phrase is definitely all this thing is" - analysis highlights parts of an experience that can't be fully conveyed in a different form, rather than trying to "spoil" or "solve" its mysteries.

That works, and would be an improvement, so long as your rhythms are based on constant integer subdivisions.

It's possible (and fun) to play music where the rhythmic structure changes smoothly and continuously. But it's bloody impossible to notate and very difficult to orally communicate, so music cultures that depend on notation or oral communication have left this territory largely unexplored.

The handful of classical composers that have attempted this (Steve Reich, Brian Current, etc) have either abandoned western notation (Reich) or hacked on their own bespoke glyphs with their own situation-specific explanations (Current).

Electronic musicians can easily explore this space by writing their own software (Autechre, your humble author, etc). When your musicians are mechanical, you can explore all sorts of otherwise impractical permutations of theory.

Most delightful, though, are the non-western cultures that communicate musical ideas entirely without written notation or spoken language, and instead communicate musical ideas through play (Indonesian Gamelan, Australian Aborigines, etc). You get the ineffable human qualities that make music most beautiful, and the freedom to explore structural spaces that are difficult to capture with discrete/unitized/quantized notation and language.

It seems like his questions were serious (not rhetorical), so I'll answer them for real. The answers pretend that the 20th century hasn't happened yet—there's no point explaining the interesting things people like Schoenberg did if you don't understand the mainstream tradition in place before them.

1) Why does notation allow for seven pitches, not 12? Because music is at most built out of 7-note scales, not 12. If something's in C major, you can expect to play a C, D, E, F, G, A, and B. If something's in B minor, you can expect to play a B, C#, D, E, F#, G, and A. The notation makes writing this fairly compact…and if you do need a pitch outside the scale, it's easy enough to write in the accidental #, b, or natural sign. If each semitone had a unique place in the staff (base-12 notation instead of base-7), sheet music would take up 70% more space for no good reason.

2) What about C# and F# is supposed to tell you 'D'? Um…the fact that the key of D has an F# and C# in it. You literally just memorize it. It can be constructed from the circle of fifths semi-elegantly, but at the end of the day any semi-competent musician should be able to tell you without thinking that the first two sharps are F and C, and the major key with two sharps is D.

3) Why do some have sharps and some have flats? Think of sharps as protons, flats as electrons, and the key as the overall charge. D has a charge of +2 (two sharps). F has a charge of -1 (one flat). Going up the circle of fifths adds charge (a fifth above D is A, which is +3…a fifth above that is E, which is +4). Going the other way around the circle of fifths removes charge (a fifth below D is G, +1. Fifth below that is C, 0. Fifth below that is F, -1). The most elegant way of talking about D is to say it's +2 and that that corresponds to two sharps. It's mathematically equivalent to write it with 3 flats and 5 sharps (so you'd have, for example, B-flat, sharpened) but that's not a useful way to model it. B-flat, sharpened, is the same as B natural, and B natural is much more fun to work with.

4) Confusion about major and minor It's worth introducing the word tonic. The tonic is the "home" note—the note you can play that makes the music sound like it could be finished. The tonic is also the key that you're in. So if you're in G major, the tonic is G, and the tune will either finish on a G or sound very incomplete (some composers exploit this effect, ending on not-the-tonic to catch the listeners by surprise). E minor has the same sharps and flats as G major, but it resolves to an E instead of a G. Für Elise is written in A minor, which does indeed have the same sharps and flats as C major. But it's "in" A—it resolves to A. If you rejiggered it to resolve to a C, some of the notes would sound out of tune. If you bent the notes until they sounded in tune, you'd realize that you were playing a Bb, Eb, and Ab instead of all naturals…and that means you're playing in C minor and all you did was transpose the thing up a third. Major and minor have very different feels (this is easily noticeable in the Für Elise video), and most people can listen to a fragment of a melody and instantly decide whether it felt major or minor. Major and minor aren't the only scale systems, by the way. Having a tonic of C and no sharps is major. C with one sharp is lydian. C with one flat is mixolydian. C with two flats is dorian. And so on and so forth.

5) Futzing about with hertz and intervals It's not quite fair to say that half steps "should" go by the 12th root of two or whatever. That results in "equal temperament", which is a relatively modern phenomenon. The ratios that it's close to are what the ear actually wants to hear—the most pleasant-sounding fifth will have the ratio 3:2, not 1.498:1. This is actually because of the interference of the waves—if you play 3hz against 2hz the waves will both be at 0 every 6 seconds and you get a very pure tone (actually 3hz is too low to hear, but the math is convenient). But if you play 1.498hz against 1hz they'll both be at zero again who knows when, and a good ear can hear the "beating" as the waves almost-but-not-quite line up. The same applies to all the other intervals. You would think that we could tool our way up by fifths to get the "best" tuning for everything, but the math doesn't quite work out. If you tune C to 100hz (and thus 200hz, 400hz, 800hz, etc), then the G a fifth above will be 150hz, D above that will be 225hz, A 337.5hz, E 506.25, B 759.375, F# 1139.0625, C# 1708.59375, G# 2562.890625, D# 3844.3359375, A# 5766.50390625, E# 8649.755859375, and B# (which is supposed to be the same as C) will be 12974.6337890625. But if we continue stacking octaves on top of the base C, we go 100-200-400-800-1600-3200-6400-12800-25600. Crap! 12800hz is almost-but-not-quite the 12974.6337890625hz we got from stacking fifths. The difference between the two is called "the comma", and figuring out what to do with it has plagued musicians for 500 years. Each tuning that deals with the comma is called a "temperament". The most common one NOW is "equal temperament", which is what was discussed above. In terms of the comma, it just distributes the comma equally across all 12 intervals, so that everything is equally out of tune. But that's not the only answer. "Quarter-comma meantone", for example, flattens the fifth (and messes with a few other intervals), but has a perfect major third—and sounds very different! And now, hold that thought…

6) Why A-B-C#-D instead of A-B-Db-D? Apart from the "every seven-note scale should have one instance of each note" maxim, this becomes a very practical question when dealing with temperament. C# is not Db…it's just that the modern piano likes to equate them. Let's say we're in quarter-comma meantone in A. So A (440hz) is in tune because we're in A, and C# (550hz) is in tune because the point of quarter comma is to make the major third in tune. But E (657.932hz) is a little bit flat (should be 660hz). Okay. And it also turns out that G# is 822.448hz. So you've tuned your keyboard thusly, and things are sounding pretty good in A major. Now you turn the page and—surprise!—the next piece is in Db minor. With the tuning on your keyboard, a Db minor chord (Db + Fb + Ab) would be 550hz (C# ~= Db) + 657.932hz (E ~= Fb) + 822.448hz (G# ~= Ab). The "ideal" values, based on A-440, would be 550hz, 660hz, and 825hz. Take it on faith, this doesn't sound good. And with a different temperament, it could've been worse—in this one at least you got the Db right! So why does it work like this? Inherent in unequal temperament is the notion of "good intervals" (A-C# as a major third is good here) and "bad intervals" (Db-Fb as a minor third is a little more icky). Violin (and similar) players work around this by fudging notes on the fly to be properly in tune (since it's an analogy instrument you can play whatever hertz you want). Through the 18th century, a C# would be played a little flatter than a Db, because that makes the harmonies as a whole turn out better (keep in mind that harmony is built on the desired interval: C#-Fb is a fucked-up fourth and isn't really supposed to sound good, Db-Fb is a minor third and should sound fine). Later on people started playing C sharps higher than D flats because it makes the melody line sound more compelling, harmony be damned (so-called "expressive intonation"). But at no point in time before the flowering of equal temperament was it ever acceptable to consider C# and Db as the same thing. Some early keyboards (much loved by Haydn and others) had much more than 12 keys per octave—they'd have the seven naturals, seven sharps, seven flats, and then specialist stuff like "C# when part of an A major chord", "C# when part of an F# chord", etcetera.

Hopefully this was a good balance of depth and brevity…let me know if anything's unclear or there are more questions.

Music theory describes music.

Your questions are very interesting, but theories answering them would describe humans.

As far as the math and wave theory components of music theory go... OP covered everything except a discussion of harmonics.

For example, third harmonic (3f in OP's terms) of a note is the perfect fifth of that same note. You can find similar relationships with higher harmonics.

This is why chords sound nice, playing the perfect fifth reinforces the 3rd harmonic, the major third reinforces the 5th harmonic, etc.