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Music theory for nerds

• lohankin 10y ago
  The explanation of the origin of major scale in the article is pure voodoo. Minor third is not a simple fraction - is that the reason to exclude it from the scale? How do you explain minor scale then? Maybe it should be excluded, too? Here're my thoughts on the subject. For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament, This system emerged as a result of long evolution of Western music, and experimentally proven to be very rich in possibilities. Scales used in Western music (of which jazz is a part of) are built on two simple principles: 1) interval between adjacent notes of the scale is either tone or semitone 2) there's no two semitones in a row. It's easy to check that all scales that satisfy these 2 rules are: major scale and its modes (7-note scales; 7 modes) melodic minor scale and its modes (7-note scales; 7 modes) diminished scale and its modes (8-note scales; 2 modes) whole-tone scale (6-note scale, single mode). (Whole tone scale is not used very often, except by T.Monk) But even after we "explain" scales, we need to figure out how to use them, what their role is, what the properties of each mode are. There's no hard science behind this, the properties just "emerge", and you have to experience them - theorizing is not of much help, math formulas don't explain anything, just lead to confusion. In short: you have to play AND think; thinking alone won't help. It's an experimental subject. Edit: forgot to say: scale is a very useful notion, but in some contexts, it's more convenient to think in terms of triads and interpolation. I know this all sounds hand-wavy, and it is! Unfortunately, without piano, it's impossible to to illustrate what it all means. The subject doesn't easily lend itself to verbalization.
  • rspeer 10y ago
    > Minor third is not a simple fraction - is that the reason to exclude it from the scale? The minor third is a 6:5 ratio. Does that change your mind? Where do your tones and semitones come from? You've just rejected the explanation for why we have a 12-tone scale (it is a local optimum of tuning, closed under the operation of transposition, that satisfies lots of nice ratios), so what do you propose instead? Saying "no one can explain" is a cop-out. To follow up: Major and minor scales are 7-tone subsets of the 12-tone scale that were discovered first. They allow for many of the same possibilities, but they're not closed under transposition, which is why we now have the 12-tone scale that includes all of them.
    • lohankin 10y ago
      Please read wikipedia article on temperament. 7-note scales were "discovered" long before equal temperament was introduced. Equal temperament is a relatively recent invention, initially was very controversial, and remained so for a long time. In other cultures, there's still a variety of scales and temperaments that have nothing to do with 7-note scales at all. (Jazz uses variety of scales, too - some even with 2 semitones in a row). There's only one thing people seem to agree on: the role of fifth (3/2).
      • rspeer 10y ago
        This is exactly what I'm telling you. Why are you being condescending to me about it? You're the one claiming that the semitones of the 12-tone scale are some sort of fundamental axiom. Why didn't you know where minor thirds come from? Why do you seem to be denying that simple harmonic ratios are where harmony comes from? (EDIT: he's not, it's fine) Why are you promoting a theory of music with absurd axioms, which manage to explain the octatonic scale (which almost nobody uses) better than the pentatonic scale (which almost everybody uses)? That's not a sign of a good theory.
        • lohankin 10y ago
          Please read the part of the article starting with "The twelfth root of two may be irrational, but it turns out to almost create several nice ratios". The author excludes minor third from the set of "nice ratios". I referred to this while calling it a voodoo.
          • sverige 10y ago
            Dude, if you can't hear it, it doesn't matter. You'll never listen to Coletrane's "Night Train" and have your head explode when you realize he's jumping from Dorian to Mixolodian to whatever the fuck was that!?
          • rspeer 10y ago
            A simplified explanation is not voodoo. Yes, he chose to list only the intervals in the major scale. Showing the intervals in the major and minor scale at the same time would have been very confusing. I understand now that you weren't trying to say that minor thirds were a hole in the theory he presented, but that they were a hole in the way he presented it. I took issue when I thought you were criticizing the theory itself, because what he presented is a simplified view of generally accepted foundations of harmony. I apologize for mis-characterizing your position there. There is more that could have been said -- and it sounds like we agree on what could have been said -- but I think it's not necessary to go into all the details of the construction of scales, including temperament, just to write a blog post. Describing temperament accurately, without handwaving, requires a book.
  • haberman 10y ago
    I agree very much with your post's thesis (you have to play AND think; thinking alone won't help. It's an experimental subject.) Just noticed one thing: > For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament It doesn't seems surprising to me. If you start from a pitch and go upwards in both octaves and perfect fifths (2:1 and 3:2, the two most fundamental intervals), the perfect fifth sequence will land on 11 distinct tones before (nearly) meeting the octave sequence. Mathematically, (3/2)^12 ≈ 2^7. So 12 semitones works out nicely because you can follow perfect fifths out in any direction as far as you want and never go outside the set of semitones. And most of the small-ratio'd intervals can be represented with pairs of notes inside this set.
    • lohankin 10y ago
      Interesting idea indeed. I need to think about it. Edit after thinking: still, it doesn't explain the number 12 IMO. It could be 17 or something else. Probably, it's a long chain of coincidences at play: Western music settled on 7-note scales long time ago (long before equal temperament was invented), and we should start looking for explanations from here. Another edit: one of the important coincidences is that number 12 makes possible the existence of diminished scale, which serves as a "universal glue" due to 2 tritones. (There's not enough space here to elaborate, but you probably know what I mean). And maybe tritone itself is one of factors leading to number 12.
      • fenomas 10y ago
        Quora (of all places) has some really good articles about this. Check for example the second answer in this page: https://www.quora.com/Why-are-there-only-12-pitch-notes-C-C-B-in-the-world https://www.quora.com/Why-are-there-only-12-pitch-notes-C-C-...
      • jacobolus 10y ago
        The number 12 is just a coincidence, that 3^12 = 531441 ≈ 524288 = 2^19 This means that 3/2 ≈ 2^(7/12) [accurate to about 0.1%] And also 4/3 ≈ 2^(5/12) [also accurate to about 0.1%] And also 9/8 = (3/2) / (4/3) ≈ 2^(2/12) [accurate to about 0.2%; putting two factors of 3 in makes the approximation only half as good] You also get another nice coincidence, that 5^3 = 125 ≈ 128 = 2^7 This means that 5/4 ≈ 2^(4/12) [accurate to about 1%] There are some people who have written music in a 41-note equal tempered scale, because then you get an even better approximation to the 3/2 ratio: 3^41 = 36472996377170786403 ≈ 36893488147419103232 = 2^65 (3/2) ≈ 2^(24/41) [accurate to about 0.03%]
  • midgetjones 10y ago
    I think you're being unfair on the whole tone there, it's the intro to 'Take the A Train', and people will often substitute it in a dominant chord when improvising.
• theseoafs 10y ago
  > It completely obscures the relationship between the pitches, though. It doesn't actually obscure the relationship between the notes -- it makes them clearer. For example, I see the notes C, E, and G on some sheet music, maybe with some accidentals on some of those notes. I know that I'm therefore supposed to play a C triad. Now, there are multiple kinds of triads, but once I know I'm supposed to play a triad, it's easy to use context to pick out which one I need (major, minor, diminished, augmented -- usually one of the first two). If I were supposed to play a C# major triad, though, and the written notes were (C#-F-G#) as opposed to what they should be (C#-E#-G#) then that's confusing because it looks like I should be playing an arpeggiated sus4 of some kind. So the written nature of scales on the staff engenders an understanding of the relationship between the notes. Basically we write things the way we do so that the people reading the music can more efficiently pattern-match. > C major is identical to A minor, and I don’t understand why we need both. They're not identical. C major and A minor have the same notes in their respective scales. But we say that a piece is in the key of C major when it resolves to the a C major chord at the end, and we say a piece is in the key of A minor when it resolves to an A minor chord at the end -- an important concept for reasoning about how a piece is supposed to be performed. > C minor: C D D# F G G# A# C Eb, Ab, and Bb, not D#, G#, and A#. > This has got to be some of the worst jargon and notation for anything, ever. It's really not. Keep practicing. It makes sense, I promise. I hear a lot of people -- usually people who have not been studying music for very long -- insist that the system would be more logical if there were no accidentals and there were 12 notes with distinct names and the staff had a bunch more lines on it. I've never bought it. The notation of music isn't arbitrary, it's informed by experience and it works.
  • theseoafs 10y ago
    Reading this over again, it seems like the author's not yet fully wrapped their mind around "big picture" stuff like keys, chords, basic compositional structure, etc. It's an... interesting choice to write a piece proclaiming that musical notation is absurd when you're only a beginner.
    • ksml 10y ago
• markbnj 10y ago
  I love pieces like this that explore the underpinnings of why we experience music the way we do. I'm a self-taught guitar player and its only now, in my 50's with the kids more or less grown and a lot more time, that I have started to dig in and understand what I've been playing for two decades. Good stuff. Thanks for the post.
• princeb 10y ago
  if you were hoping for a nerdy treatise on musical rhythm, harmony, form, cadences, sequences and motifs, and evolution from baroque to classical to romantic to modern... well this is not it. edit: just a comment about the temperament- before the octave was equally tempered intervals were rational fractions of the root. after the equal temperament was developed instruments could go across different keys without sounding too dissonant. the well tempered clavier was a series of preludes and fugues that JSB wrote for the equally tempered scale that goes across all the keys on the same instrument (the prelude in c major which is the very first piece in the collection is the famous one in Forrest Gump). but sometimes you want to return to the pure temperament - like if a passage focuses on a particular key for a while - and in the brass and wind instruments it is easy to squeeze the embouchure a little bit to slightly tweak the tuning of a particular note.
  • nazgul17 10y ago
    I would love something like that. Any pointer?
    • Ericson2314 10y ago
      A large part of it hasn't been researched very mathematically, to my knowledge. If you google music+grammar you can get some stuff with a linguistic approach, which may be less empirically grounded but certainly a richer theory than the alternatives. And don't get me wrong, science is good but as musician mathematician a richer theory that overreaches reality is much more fun, and besides most of this stuff is cultural not immutable so a wrong theory may become correct if is influential. A few stray ideas that may or not be correct: - Grammar of common-practice harmony is probably largely left-branching (see https://en.wikipedia.org/wiki/Branching_(linguistics) https://en.wikipedia.org/wiki/Branching_(linguistics)). This is why so such music makes a surprising transition, then resolves it, and it's orthodox overall. Music is about where it's going. - For a first approximation rhythm models the rational numbers, because all durations exist in a ratio to other duration. - Now the rational numbers are infinitely dense and clearly humans can't do that. So there is a minimum resolution which has been wonderfully termed the tatum. Whoever did this however may or may not have realized that there may be a few minima whose join (as in lattices, I'm sure it's a lattice but I never leaned which one) isn't realized. E.g. you might have 8th notes and triplet 8ths but never triplet 16ths (the join). - Form has all these stupid names when clearly it's a tree and similar structure repeated on every level. - People sometimes try do do harmony wit a markov model. This sucks because how the chords transition depends on the form, which I believe must largely (needed to be orthodox/fluent) have a fair regular rhythm (avoid those 5-bar phrases). I think something like a ruler function (https://en.wikipedia.org/wiki/Thomae%27s_function https://en.wikipedia.org/wiki/Thomae%27s_function) based on time + transitions would be a much better and relatively cheap model. If all the fancy grammar stuff is legit, then this too would be woefully inadequate. - That said, the "meaning" is definitely in the transitions not the states (production rules in grammar case). A chord with no context has very little meaning (in equal temperament everything can be transposed with 0 informational change except for timbre). Given a key, a chord still has relatively little meaning (also 1 and only 1 key is a reduction, look up "tonicization", there is probably a stack of broad-specific key-like things forming a context). now given a trace of chords, the meaning comes out---tension and release as they say. - For form and rhythm, 2 branching factor is the default. Everything else is extra work. Probably even in West Africa and other places where the polyrhythms are mad good. - Emotional relationship definitely changes over time within a culture. Up to the 18th century major=happy minor=happy was far from universal. This is probably why even in popular culture old music is stereotypically all dark and serious. - The big 5 components of (at least Western) music are (in no particular order) Melody, Harmony, Rhythm, Form, Timbre. Timbre has become more important over time, even before recorded music. Call me a cranky old cultural conservative, but you could say that Western Society is loosing it's collective understanding-of/ear-for Western music, and Timbre is the easiest to appreciate due to its lack of rich structure (as far as I can tell). And of course recorded and synthesized music means we can timbregasm all day.
• analog31 10y ago
  This has got to be some of the worst jargon and notation for anything, ever. Indeed. I'm a musician, and something non-musicians often ask (especially techies, it seems) is why we use such an archaic notation system. The reason is simply that a certain number of musicians have developed the skill of sight reading which is the ability to perform a composition directly from a written sheet, with little or no rehearsal. Those players, myself included, can't quite explain how we do it, and aren't going to learn a new notation system.
  • shabbyrobe 10y ago
    Sounds like vim!
    • mettamage 10y ago
      Sounds like... And is it? ;) In terminal run vimtutor. The sound will change. Does anyone know of a similar tutor for sheet music?
  • andrepd 10y ago
    >why we use such an archaic notation system. The answer is, because it works, and nobody has managed to propose a better one.
    • GirlsCanCode 10y ago
      It works well. I can sight read piano well. I do thus because the notation really does make sense; it's very dense, and I can read ahead many measures at a time. This guy hasn't even figured out what's different between major send minor, or why the equal tempered system is used on keyboard instruments.
    • hardmath123 10y ago
      One attempt was Hummingbird. http://www.hummingbirdnotation.com http://www.hummingbirdnotation.com
      • 10y ago
• verroq 10y ago
  >I suppose it’s possible to change the sound of an entire piece of music just by changing the key signature, but does anyone actually do that? >How would that work for music that also uses notes outside the scale? These seem more like questions of composition, which I definitely don’t know anything about. From wikipedia: Although transpositions are usually written out, musicians are occasionally asked to transpose music "at sight", that is, to read the music in one key while playing in another. Musicians who play transposing instruments sometimes have to do this (for example when encountering an unusual transposition, such as clarinet in C), as well as singers' accompanists, since singers sometimes request a different key than the one printed in the music to better fit their vocal range (although many, but not all, songs are printed in editions for high, medium, and low voice). There are three basic techniques for teaching sight transposition: interval, clef, and numbers ... [*] https://en.wikipedia.org/wiki/Transposition_(music)#Sight_transposition https://en.wikipedia.org/wiki/Transposition_(music)#Sight_tr...
  • ramshorns 10y ago
    I don't think the author was referring to transposition there. I think that was about leaving the notes on the staff the same but just changing the key signature. This is done sometimes as a novelty, and the best example is changing something from a major key to a minor key or vice versa. The article has a link to a recording of Für Elise in a major key [1], and there are many similar renditions of other pieces around. You could in principle do this with any of the seven modes, not just major and minor. As for notes outside the scale, it seems like these key-signature-changing compositions typically keep them the same (like the D# in Für Elise). [1] https://www.youtube.com/watch?v=Y-rZD2AsHbI https://www.youtube.com/watch?v=Y-rZD2AsHbI
• deleted 10y ago
  [deleted]
• andrepd 10y ago
  TLDR: a person who knows nothing about music or music theory gripes about music and music theory. Admittedly knows nothing, right off the bat and several times throughout the text. How can you declare something to be "terrible", "the worst notation for anything ever", and so forth if you admittedly know nothing of the subject? I know nothing about databases, would reading the Wikipedia page on NoSQL entitle me to declare it "absolute garbage"? o.O
  • phyzome 10y ago
    It can be clear to a newcomer that a system has problems without needing to understand everything, or even very much. Sometimes the newcomer is wrong, but a quick scan of the comments here indicate that a number of musicians agree.
    • rvense 10y ago
      Funny, I skimmed the comments and actually came to the conclusion that I did not need to write another one stating that music notation, not unlike many other complicated tools, is perhaps at first sight arbitrary and odd, but after working with it for a while you find that all the little quirks, like having multiple ways to write the same note, are actually very helpful.
    • andrepd 10y ago
      https://xkcd.com/793/ https://xkcd.com/793/
  • sebastos 10y ago
    If reading the Wikipedia page on NoSQL offered you a completely opaque and unhelpful window into how NoSQL worked, and if every tutorial and typical description of the subject seemed to assume you already understood NoSQL, then yes. It would.
  • tux1968 10y ago
    Interesting objection. To my eye it read as relatively humble and playfully thoughtful.
  • 10y ago
• jacobolus 10y ago
  This source is a bit wordy. Let’s summarize: The core idea of music made with harmonic sounds is that “notes” with frequencies at small-integer ratios will “harmonize”. Harmonic sounds means something like a vibrating string where the vibrations are integer multiples of some fundamental frequency, because other non-integer-multiple vibrations are damped out by the fixture of the string at two points. Different (non-harmonic) types of sounds often sound better with a different sort of scale, for details see this book http://sethares.engr.wisc.edu/ttss.html http://sethares.engr.wisc.edu/ttss.html * * * The “octave”, 2:1, is the simplest whole-number ratio, and makes many of the vibrations in two notes in such frequency ratio align with each-other, to the point that two harmonic sounds exactly an octave apart almost sound like the same sound. Other simple ratios like 3:1, 4:3, 5:4, etc. also “harmonize”, with (not quite as) many aligned overtones. The core idea of the 12-note musical scale (pretty much regardless of specific tuning) is the approximation: 3^12 = 531441 ≈ 524288 = 2^19 3/2 ≈ 2^(7/12) [this is accurate to about 0.1%] Or another way to say this: 7/12 of “doubling” on a log scale is very nearly “three-to-two”. Musicians call this ratio a “perfect fifth”. In the case of equal temperament, an octave is split into 12 precisely equal steps (on a log scale), each one the 12th root of 2. There’s one other nice approximation to take advantage of: 5^3 = 125 ≈ 128 = 2^7 5/4 ≈ 2^(4/12) [this approximation is only accurate to about 1%] Musicians call this ratio a “major third”. * * * Even outside music, these approximations can be useful for doing approximate computations. If only our society switched from decimal to “duodecimal” numerals, it would be very natural to use logarithms base two, notated with “duodecimal” fractions. If you have a number expressed in log base two, and you use duodecimal notation, approximately multiplying or dividing by 2, 3, 4, 5, 6, 8, 9, 10, 12, ... is very easy using addition/subtraction of easy-to-remember multiples of 2^(1/12). Unfortunately our society instead has slide rules and measurement scales (decibels, etc.) which are all built around logarithms base ten, and decimal notation.
• watermoose 10y ago
  This is great, but I would recommend Robert Greenberg's "How to Listen to and Understand Great Music". He goes through the history of western music in a way that makes it clear why Amin != Cmaj, and other questions that the OP has. Yes, sheet music is crap, and he explains how it evolved to be the way it is, after which you'll be much more forgiving. He's a great speaker who obviously knows the material inside and out.
• robbrown451 10y ago
  One thing that is particularly arbitrary is that C is named "C" rather than "A", which would make far more sense (given that it is the root note of the only major scale that has no sharps or flats).
  • dietrichepp 10y ago
    Well, it has to be somewhat arbitrary. But it is nice that the relative minor of C, which is two steps down, is A. Otherwise you would have to go five steps up (from A to F, hypothetically, if A B C D E F G A were a major scale).
    • robbrown451 10y ago
      If C was called A, that wouldn't be particularly arbitrary. It would certainly make more sense for people learning music. The C major scale is universally agreed to be the simplest, most basic scale, so wouldn't it make more sense for it to be ABCDEFG, rather than CDEFGAB?
      • gnaritas 10y ago
        No, because that's the A minor scale, which is the same notes as the C major scale but starting at A. A scale should logically always start with its root note.
        • dietrichepp 10y ago
          That's not really what the comment is saying. If we relabeled the notes CDEFGAB to ABCDEFG, then ABCDEFG would be the "A major" scale under our new system. The question is, "Wouldn't that make more sense?" My answer is, "It would make a tiny bit more sense to the most absolute beginner students, who would then have to learn other scales anyway."
          • djaychela 10y ago
            Yes, exactly. I teach Music Technology, and that's one of the questions I get nearly every year, and one I've not found a concrete answer for, so I always preempt it by saying that the C is arbitrary as far as my research has gone.
• dietrichepp 10y ago
  > Also, this notation has a slight problem. That problem is that sheet music is terrible. > This has got to be some of the worst jargon and notation for anything, ever. I hear it bandied around in these articles that music notation is awful. Can someone explain to me why? I'm not much of a musician, but I can work my way through sheet music if necessary, or write it when I want to. I don't remember what it was like to learn to read and write music so I'm not going to understand why it's so bad just by thinking about it. First of all, the lines and spaces make a ton of sense to me. When you're playing, reading, or writing a piece, you spend most of your time in a key with a diatonic scale. Maybe you're thinking about the chords: I, vi, ii, V7. The lines and spaces are great for giving you that kind of information. I guess this "completely obscures the relationship between the pitches". I guess that's the case if you forget what key you're playing in. The alternative would be to make the staff fully chromatic, but that sounds like it has a bunch of disadvantages for most music. The chromatic staff (http://musicnotation.org/ http://musicnotation.org/) looks terrible to me, since it spreads out the notes farther. I've already memorized all of the diatonic scales, and now the diatonic scales are jumping around, and the staff covers a shorter interval so you get more of those 8va or 15ma. Hummingbird (http://www.hummingbirdnotation.com/ http://www.hummingbirdnotation.com/) seems like classical notation, just with more redundancy and the symbols changed around a little bit. Maybe it's better but it's a minor improvement at best. Clairnote (http://clairnote.org/ http://clairnote.org/) looks the worst of them all, with half-steps being half-way between lines and spaces. It looks like a recipe for making mistakes and misreading music. I'm not about to say that it's easy to read music notation, just that the notation seems to have a reasonable amount of logic to it. The idea that somehow it's the notation that's terrible and music itself would be easier to understand or play if we just had better notation--I don't buy it. Maybe someone could convince me. (Addendum: If you think the chromatic scale is the "right" scale for notating most music, as in "Chromatic staff" or "Clairnote", as an exercise... why do you think that the chromatic scale is correct? Is it because it represents both sets of keys on the piano or the frets on a guitar? Is it because of the mathematical relationship between the notes? My counterargument, against the chromatic scale, is presented by Bobby McFerrin: https://www.youtube.com/watch?v=ne6tB2KiZuk https://www.youtube.com/watch?v=ne6tB2KiZuk)
  • hexane360 10y ago
• zeta0134 10y ago
  On the science side of things, Vi Hart put together just an excellent video that goes over harmonics, the overtone series, and why 440 Hz and 880 Hz sound so "indescribably similar" to this blog author. https://www.youtube.com/watch?v=i_0DXxNeaQ0 https://www.youtube.com/watch?v=i_0DXxNeaQ0
• leafo 10y ago
  If anyone is interested in learning to read music, I've been slowly building a tool to practice: http://sightreading.training/ http://sightreading.training/ source is here (built in react, es6): https://github.com/leafo/mursicjs https://github.com/leafo/mursicjs It's still lacking a lot (like rhythm), but the different generators definitely give my brain a workout. It works best if you hook up a midi keyboard.
  • jcurbo 10y ago
    As someone self-teaching themselves piano, this is really fantastic. I already play drums but have no notion of what notes are what on a staff and piano keys and have been slowly teaching myself. The MIDI keyboard support really makes this stand out.
    • estefan 10y ago
      I've been teaching myself piano after someone gave me one. Here's a great set of online lessons [1] (not free, costs about $20 per month if you put in your email address). The guy is really talented, and teaches non-classical stuff like pop, boogie woogie, etc. I'm super inspired by it! I also found this [2] which does have free lessons and also looks good. [1] https://pianowithjonny.com/ https://pianowithjonny.com/ [2] https://www.hoffmanacademy.com/ https://www.hoffmanacademy.com/
• ektimo 10y ago
  I've skimmed a lot of articles on music "theory" but none of them provide anything like what I'm looking for. A music theory should explain: 1. Why do we like pieces when played forward but not backward or inverted? 2. Why do certain sounds evoke certain emotions? 3. How could you write a program to pick out music that people find especially good (versus music that has surface similarities)? In other words, why does a particular sequence of sounds A, B, C lead to a mental state M that has particular internal qualities?
  • pdkl95 10y ago
    > 1. Why do we like pieces when played forward but not backward or inverted? Why do we like text when read forward, but not backward or inverted? There are, of course, works that are palindromic or otherwise written to be read/heard backwards, but most of the time that kind of global transformation tends to ruin the "spelling"/"narrative". > 2. Why do certain sounds evoke certain emotions? Just like text, evoking emotions needs some sort of narrative. A story isn't a single fact or statement (or a single sound); it's about how those facts (or sounds) flow or change. In music you might hear a brief bit of new melody that foreshadows something big later in the song. A clear rhythm or melody might be repeated to get the listener to follow along only to have it cut short at a key moment to deny the obvious resolution (similar to a melodrama that suddenly reveals a new twist in the plot as a cliffhanger). It's the story you tell that matters, and it takes a skilled composer to put sounds together to make a song emotionally evocative. The song that is mostly a 16 bar loop probably sounds boring (but not always!), while the song that introduces the same 16 bars and then plays with variations of it to create an initial conflict, rising action, and a climax is probably a lot more interesting. An obvious example might be Mozart playing Salieri's march in Amadeus[1]. It's not just that he embellished the simple march; Mozart adds a lot of variations that culminate at a comic ending. [1] https://www.youtube.com/watch?v=P5n0pkNpDWY https://www.youtube.com/watch?v=P5n0pkNpDWY
    • sverige 10y ago
      Actually, great composers such as Beethoven and Bach and Chopin had very definite ideas about what emotions are evoked by certain keys. They even argued about it with their peers. Music is not something that is reducible to mere quanta and waves and frequency. You all are missing the human part. Sorry, but it's true.
• gtani 10y ago
  Interested to see where this goes. The 5 blogs/books cited at the end look especially interesting. I never thought about music notation until i learned guitar. Previous to that i was "gifted" years of lessons in piano, woodwinds and percussion, and reading treble clef was easy. I never tried to read sheet music in guitar books because i never had to, i just read tabs and played. Besides some classical books and jazz books by Berklee profs and others (Leavitt, Martino, Goodrick), there's very few guitar books that don't have tabs. So you could argue that's a notation. You could also say that FL piano rolls and renoise timelines (what OP is trialling) are notations, as are lead sheets and chords charts in Nashville format (the Roman numerals like ii-V-I you see all over the place). They're sufficient for people to play music, they happen to be dynamically typed and GC'd vs static typed and manually alloc'd
• Ericson2314 10y ago
  Octave is not arbitrary, inner ear hairs should resonate more to octaves above/below (forget which) their maximally resonate frequency right?
• quadrangle 10y ago
  The author and anyone else who understands correctly that traditional music notation is shitty in lots of ways and is trying to understand how music really works should go get the book "Music and Memory: An Introduction" by Bob Snyder. It uses no music notation and explains music in terms of psychological principles of perception of time. It's not a complete theory of everything, but it shows the way you should be understanding the nature of music. Beyond that, check out Sweet Anticipation by David Huron, and Tuning Timber Spectrum Scale by William Sethares (and his other Rhythms and Transforms, see http://sethares.engr.wisc.edu/ http://sethares.engr.wisc.edu/ for web versions of first chapter of each). These sorts of resources are where real understanding of music comes from. Not from the "theory" stuff us music professionals had to deal with that fails to explain anything well.
• ngneer 10y ago
  The interested reader is referred to "Mathematics and Music" by David Wright, published by the American Mathematical Society. The realization that the circle of fifths is really there because 5 and 7 (aka -5) are co-prime to 12 was worth the entire thing! Together with 1 and 11 (aka -1) they generate the group Z_{n} for n = 12...
  • ktRolster 10y ago
    That explains why the circle of fifths is a circle that goes through all keys, but the really interesting thing about the circle of fifths is that a fifth sounds relatively close to the key next to it, and it doesn't really explain that, I guess.
    • djaychela 10y ago
      Any key a fifth away only changes one note - if you go from C Major (CDEFGABC) to G Major (GABCDEF#G) then only the F# has changed, and as a result there are chords which are common between the two keys - any chord without an F in it in C will be common to G as well. Works the other way if you go 'down' a fifth (to F), but has a flat instead. (Bb)
      • ngneer 10y ago
        More specifically, going up a fifth augments the fourth by a semitone to become the seventh and going down a fifth diminishes the seventh by a semitone to become the fourth. In your example, F the fourth became F# the seventh and B the seventh became Bb the fourth. A fifth away _IS_ a semitone away. Defining the following eight functions: SEMITONE = +1 mod 12; ANTI_SEMITONE = SEVENTH; FOURTH = +5 mod 12; ANTI_FOURTH = FIFTH; FIFTH = +7 mod 12; ANTI_FIFTH = FOURTH; SEVENTH = +11 mod 12; ANTI_SEVENTH = SEMITONE; We ask ourselves why the following holds: FOURTH(X) = ANTI_SEMITONE(SEVENTH(FIFTH(X))) SEVENTH(X) = SEMITONE(FOURTH(ANTI_FIFTH(X))) Substituting, the answer is clear: X + 5 = X + 7 + 11 - 1 X + 11 = X - 7 + 5 + 1 Keys one fifth apart differ by one note because: 1 = 11 + 7 - 5 mod 12 Algebraically it seems to work out...
• MrManatee 10y ago
  The article says that the "human ear loves ratios", but doesn't dig deeper into why. Here's my two cents. First of all, let's focus on harmony (notes played at the same time) as opposed to melody (notes played one after another). What sounds good in a melody is quite culture-dependent, but there are reasons why harmony is more universal. Second, let's focus on sounds that are produced by something long and narrow. In a guitar, violin, or piano it's a string, and in a flute it's a column of air. The physics of vibrations goes so that in such a case the sound is composed of harmonics: sine waves of frequencies f, 2f, 3f, 4f, ... If the shape is different (say, a circular membrane of a drum), then this may not apply. Suppose we add a second sound, whose fundamental frequency is, say, 3/2 f. This means that its harmonics are 1.5f, 3f, 4.5f, 6f, 7.5f, 9f, ... Half of these (3f, 6f, ...) coincide with the harmonics of the first sound, so the sounds "reinforce" each other. More generally, if the ratio of the frequencies is p/q for some integers p and q, then there will be overlap in the harmonics. And the smaller p and q are, the more overlap there will be.
  • hajiurtr12 10y ago
    An entire class of music, Carnatic music, has existed for thousands of years with millions of listeners that is based on melody alone. For a drum, interestingly, the fundamental vibration modes are all Bessel functions.
  • sirtaj 10y ago
    Melody in the form of the pentatonic scale is much more universal across cultures, though. Use of non-trivial harmony is significantly less widespread, it's usually no more than a melody played over a single root tone or chord. South-Asian classical and folk music is one example in which Western-style harmony is not used at all.
    • MrManatee 10y ago
      You're right, of course. My wording is was bit poor. What I should have said that harmony is less ad hoc; it has less "degrees of freedom". With regards to melody, there are tons of tuning systems that are quite close to the usual twelve-tone equal temperament. It would be hard to give a convincing argument that one of these sounds better than all others. Contrast this to the system of harmony where the basic principle is that ratios of small integers sound good together. This is not the only possible system of harmony, but it does seem to represent some kind of local optimum. And this makes it more amenable to the kind of purely theoretical reasoning that the article is trying to do.
• rspeer 10y ago
  If you want a series of books that constitute "music theory for nerds" -- building up music theory from a solid foundation of acoustics, and math -- try "Musimathics" by Gareth Loy. It is a great read. It takes very little for granted. Now, sometimes you have to say "this is just the way it turned out" to explain Western music, but the best way to do so is to show some other ways it could have turned out, and show their role in non-Western music. Musimathics does that often.
• intrasight 10y ago
  I've restarted my music brain after 30+ years. Two resources I found valuable are: http://andrewduncan.net/cmt/ http://andrewduncan.net/cmt/ http://oyc.yale.edu/music/musi-112 http://oyc.yale.edu/music/musi-112
• tsm 10y ago
  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.
• saynsedit 10y ago
  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.
• matchagaucho 10y ago
  No two Musicians take the same path to learning music. I enjoyed this perspective. Particularly the focus on sine wave theory and frequencies (which also serves Audio Engineering). Quite a departure from the traditional rhythm, melody, harmony, and form framework. But it works :)
• acjohnson55 10y ago
  This is an interesting article, which touches on a lot of interesting musical topics. However, it's wrong on so many levels that I would never recommend it to someone who wants to understand music theory. I'll try to summarize. I spent a long time researching where Western music "comes from". One important thing to understand is that the key feature of Western music is harmony. This is not the case for most other musical traditions. Harmony comes from a layering of independent melodic lines (called voices), such that they evolve in ways that create and resolve tensions. This is called counterpoint. The ways in which these tensions resolve are called cadences, and classical music was all them. Musical form arises around how a piece of music is organized to have an arc (and arcs of arcs) that leads to a cadence. It's kind of like how there are only a few basic narrative structures in storytelling, yet infinite variations in types of stories. The tension between the sounds in concurrent melodic voices emerges from the structure of the sounds. Harmonic sounds, like ones produced from wind and string instruments, sound good (consonant) when their fundamental frequencies are simple ratios. This is largely because of how the human auditory system works. Our scales and chords come from picking collections of notes that approximate simple ratios. We deal with approximate ratios using nth-roots for two reasons. First, simple ratios don't compose well -- stack them (i.e. multiply them) and you get ratios that aren't so simple. nth-roots do stack nicely. Second, it lets us merge nearby flat and sharp notes. This wouldn't be possible if we used real ratios. That merger, known as enharmonicity, lets us do all sorts of cool compositional tricks that are generally considered more worthwhile than the slight improvement in sound quality we'd experience by using the real ratios. There are various ways to pick just which notes merge, but by far the most popular approach is to use 12 notes. So, this is where we got our notes and scale from. This happened around the year 1800, give or take a couple decades. The history of Western music since that point in time largely revolves around coming up with new ways to play with those 12 notes, in many ways erasing the musical features that caused the 12 notes to emerge in the first place. This is why the system of note names, with the flats and sharps, feels rather arbitrary today. Unfortunately, there are not a lot of great resource that tell this full story. The two books I found most enlightening in piecing together the "real story" are: - Tuning, Timbre, Spectrum, Scale: https://www.amazon.com/Tuning-Timbre-Spectrum-William-Sethares/dp/1852337974 https://www.amazon.com/Tuning-Timbre-Spectrum-William-Sethar... - A Geometry of Music: https://www.amazon.com/Geometry-Music-Counterpoint-Extended-Practice/dp/0195336674 https://www.amazon.com/Geometry-Music-Counterpoint-Extended-...
• bbtn 10y ago
  Well, the first figure: frequency is NOT the period, and amplitude is not that. I guess there are lots of errors in the text if the first figure is completely wrong.
• raverbashing 10y ago
  Wow, that's a not good article. More like a rambling than anything else. > C major is identical to A minor, and I don’t understand why we need both. That's like saying house numbers are not important in finding an address... I'll just leave this here: http://openmusictheory.com/ http://openmusictheory.com/
• hellofunk 10y ago
  > C major is identical to A minor, and I don’t understand why we need both. Well, an orange and a carrot are the same color, but does that mean we don't need both? These two keys have the same signature, and that's it. To say they are identical is false. > I don’t know anything about music. You got that right. > I suppose it’s possible to change the sound of an entire piece of music just by changing the key signature, but does anyone actually do that? Only about every major composer who has ever lived. Look through any Beethoven sonata and see all the key signature changes throughout. Your article didn't even hit one of the most important parts of music, modulation, which if you understood it would make many of your other confusions go away.
  • dfabulich 10y ago
    > > I don’t know anything about music. > You got that right. This is gratuitous negativity. It adds nothing to your post except to emphasize disdain for the author. It's especially uncivil in response to the author's own statement of humility. This post would be better (and much more constructive) if you explained (or linked to an explanation of) how C major and A minor are different, or how the concept of modulation dispels the author's confusions.
    • hellofunk 10y ago
      I disagree. The article is not of high quality, and I am compelled to point that out. It is one thing to admit not knowing anything about a subject, it is something else entirely to suggest that the way things are in that subject don't make any sense. That is arrogance hiding behind a pretence of ignorance. If the author truly wanted to explore this subject, rather than suggest the key of A minor is superfluous and redundant, he would make an attempt to answer, not ask, that question. Western music theory has not existed for centuries without reason, and some of the attitude in this post suggests the author has superficial appreciation for how music works, as if the author could do it better. There are many more examples of this in the article. I found the article questionable in its intent and content, and the general tone throughout is worthy of my response.
    • denim_chicken 10y ago
• brunorsini 10y ago
  In case anyone is searching for a really smart, modern method for learning music theory, this is it: https://www.hooktheory.com https://www.hooktheory.com The author devised his own system for visually representing notes, it makes it much easier to understand things like scale degrees and relative notation (and thus the theory around famous harmonies, melodies, etc). I think music tools are in desperate need for improvement... Starting with notation, which is still a bit akin to forcing programmers to go straight to Assembly. Little is gained from it as most people just completely give up and then go on to live the rest of their musical lives "in the dark", without knowing how to read and write at all. This can actually be good for some but I'm sure it hinders the creativity of a lot more. I think we also need way better digital instruments... That make it easier to stay on scale (or to modulate, etc -- whatever the mood is), for instance, allowing people to just play away which is what actually matters. I've spent countless hours of my life learning scales on several different instruments and think a lot of that was wasteful. More often than not I'm just trying to stay in a given key anyway, nothing fancy... Instruments really need better interfaces :)
  • teen 10y ago
    > I think we also need way better digital instruments... That make it easier to stay on scale (or to modulate, etc -- whatever the mood is), for instance, allowing people to just play away which is what actually matters. I'm not sure what new stuff has come out in the last few years, but Ableton Push is exactly what you're describing. It's a grid where you can select a key, scale, and tone, and then you can apply effects in series / parallel. The notes in the scale light up. There's a bunch of other stuff you can do as well. https://youtu.be/0CdMvkBOUgs?t=69 https://youtu.be/0CdMvkBOUgs?t=69
    • brunorsini 10y ago
      Thanks! I've played with Push, it's pretty good... but as someone commented above it's fairly limited in many ways too. This is an interesting piece of software I have used before: http://autotheory.net http://autotheory.net. It simply translates incoming midi data so you can use whatever controller/instrument you're already familiar with. The creator is nice and responsive, he often attends shows like AES and NAMM.
• arnarbi 10y ago
  Why is this at the top of HN? While the author is right that western music notation has shortcomings, almost none of the (admittedly poorly understood) reasons they list are a factor in that. > [Key signatures] completely obscure the relationship between the pitches, though. Nope, they exactly allow the notes to show scale degree relationships. Without key signatures, music in any major scale other than C would be very confusing as the regular scale notes would be marked with accidentals. > I don’t think I entirely understand this, because it still seems so convoluted to me. You have to mentally translate that C to a C, and then translate the C to however that particular note is actually played on your instrument.What does this accomplish? Same. Because at some point a player stops playing by translating the exact pitch to a fingering (for example), they start playing on a scale. They practice the scale by itself, and other pointless "music" in that key and it's the fingerings of the scale that they learn. I.o.w. this system allows them to think at one higher level of abstraction. > I’m a bit out in the weeds from here on. C major is identical to A minor, and I don’t understand why we need both. This whole section reads "I don't understand this so the world must be stupid to have come up with it." Rest assured that C-major and A-minor are not the same. Yes they have the same pitches, but a different starting point (root). A note's relationship to the root of they key of the piece is much more important than the exact pitch. Fur Elise in C-major would sound just like it sounds in A-major. Not A-minor. > I only looked into this because I want to compose some music, and I feel completely blocked when I just don’t understand a subject at all. Here's the bright point: You don't need to understand notation to compose music. Just understand scales and chords (and their progressions), and use whatever notation makes sense for you.
  • Chinjut 10y ago
    > Nope, they exactly allow the notes to show scale degree relationships. Without key signatures, music in any major scale other than C would be very confusing as the regular scale notes would be marked with accidentals. Presumably, they have in mind to straightforwardly eliminate BOTH key signatures and accidental markers; e.g., a staff in which adjacent positions are always one semitone apart, and all pitches and intervals between them are thus always directly indicated without need for reference to a key signature for decoding. There are pros and cons to this scheme (as it privileges indicating pitch relationships over, as you put it, "scale degree" relationships), but at any rate, it would not involve confusing accidentals, nor any significant difference between working in C major scale vs. any other (even less bias in favor of C major than the current system!).
• anirudt 10y ago
  I preferred the following. Throwback to another ShowHN page. https://news.ycombinator.com/item?id=4295714 https://news.ycombinator.com/item?id=4295714
• marmaduke 10y ago
  This is only one aspect of music: there's timbre, dynamics, rhythm, groove, etc.
• skrebbel 10y ago
  I'd like to elaborate a bit, in the same "for nerds" manner, on where Eevee seems to get lost a bit with scales and notation. He (she? not sure) calls the A minor and C major scales the same, because they contain the same notes. That's not an odd thought, but it's like calling sine and cosine the same because both functions contain the same set of values, in the same order. The difference is phase. Basically, scales aren't just an ordered set of notes, they also have a starting point. This note, the note the scale is named after, is often the last note of a tune played in that scale, and often the first too (especially for more poppy tunes). So if you play Für Elise in C major, like Eevee suggests, the entire melody will be pitched 3 semitones higher than playing Für Elise in A major. And it'll sound awkward because you're supposed to play it on a minor scale. Once you understand this, the whole notation thing makes a lot more sense as well.
  • deleted 10y ago
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  • perlgeek 10y ago
    Thanks, that's an explanation I haven't heard that way before, but that makes sense to me. Follow-up question: how are phase of the scale and the piece of music synchronized? When I think in terms of a wave, I could Fourier-Transform it into sines or in cosines, or any other phase-shifted variant (a * sin(nx + const.))? Is it always the first note of the piece of music that "anchors" the piece in its scale? Does that question even make sense? :-)
    • abannin 10y ago
      This gets pretty complicated and subjective. Strictly speaking, a "note" cannot define a key. To determine a key, you really need at least 2 intervals, wherein interval a somehow resolves to interval b. Keys are just a mapping of relationships; how does each of the 12 notes relate to each other? If you just play a C major chord, you aren't really playing in the key of CMaj, you're just playing a CMaj chord. For example: Play B+G, then C+G. This is (arguably) C major. The B note resolves into the C note. That's the 7th moving into the Tonic (#1). This is an example of a V-I resolution, the strongest possible. If you'd like more examples, look up "cadences". There are 'principles' for determining the key, but they should not be understood as proofs. I think you're also asking if/how to change keys. This is not only possible, but often highly encouraged. There are two extremely common key changes in pop music: 1) Up one step: This is incredibly common and can be heard in "I Will Always Love You" (album version, right after the 3 minute mark when she starts singing the chorus again). Once you start hearing this, you can't stop. It's all over the place. If we're in the key of C, we'll just play a D chord with a lot of confidence, typically after a G chord. 2) Minor to Major (or reverse, moving from sine to cosine in the analogy): This is much more subtle than the previous example. Typically you'll recognize this as a change in 'mood' or 'feeling'. You'll find a lot of Am-Em changes, then the chorus will be a lot of C-G changes. The tonic changes from Am to C, even though all the chords and notes being used are still the same (although the duration of said chords will probably be different). Most "classical" music moves around keys pretty frequently, in extreme cases multiple times within a measure. And then there are chords/sections in which the key is debatable if not indiscernible (I would argue that the intro to Smashing Pumpkins "1979" is changing keys every 2 measures, but I think there's also a strong argument for it being in a single key) Thinking about keys in terms of phases is a good basic explanation, but ultimately phase is much easier to measure. Perhaps I see it this way because my understanding of music is much better than my understanding of physics.
• karlb 10y ago
  Many musicians much better than me are surprised at how I can play a song just by hearing it on the radio. My breakthrough came from understanding music was realizing that the real “meaning” of a note lies in its position relative to the tonic note (e,g, I-II-II, etc, also written do-re-mi). Suddenly, almost all of the clutter was removed, and the problem became manageable. Let's consider the three-note tune “do, re, mi”. If that tune were played in the key of C, it would become C-D-E. If it were played in G, it would become G-A-B. But in either case, it's the same tune but with each frequency increased by the same percentage. Trying to understand music by understanding the letters is like trying to read in a world where every article has been enciphered into a different “key”: e.g., the word "cab" in “the key of A” (the alphabet we normally use) would be written as "dbc" if the article were written in “the key of B”. In the latter case, you could discern meaning only once you realised that the letter “d” represented the third letter of the alphabet. There's nothing meaningful about a “d” but there is something meaningful about a “4th letter of the alphabet”. Once you start to “decipher” all music into I, II, III, IV, V, etc., the complexity becomes manageable. You can start to learn to recognize the sound of a III note, or of a VI minor chord. After all, there are only eight notes in the major scale.
  • andrepd 10y ago
    The gist is correct, but it's worth noting that, for example, if you played do-re-mi in E it would do E-F#-G#, not E-F-G, to preserve tone/semitone order. That's where the key signature comes in.
    • ajuc 10y ago
      And that shows how WTF the musical notation is.
      • andrepd 10y ago
        o.O why so? And what on earth do you propose?
        • ajuc 10y ago
• Cozumel 10y ago
  What a ridiculously uneducated article! The very first sentence starts 'I don’t know anything about music.' and IT SHOWS!! Music notation has evolved over several hundred years, it's not arbitrary or random. The author should take some music classes before trying to write about it.
• andrepd 10y ago
  "C major is identical to A minor, and I don’t understand why we need both." Cringe. If I understand nothing of some subject, I would do well to just shut my mouth about it.
  • almostarockstar 10y ago
    In fairness, the author said " I don’t understand why we need both", and not "We don't need both".
• tehwalrus 10y ago
  > I suppose it’s possible to change the sound of an entire piece of music just by changing the key signature, but does anyone actually do that? Much more interestingly, this touches on the reason why different pieces were given keys at all by their composers (think "foo in B flat minor by Chopin"). Musical instruments have to be tuned to a particular key. This is because of the frequency ratios: the circle of fifths is a lie, (3/2)^N != 2^M for any N and M, you can't be an in-tune fourth in one key and third in another, the frequency values are slightly off. Thus, some instruments (e.g. Piano) are tuned to be in C, and the other keys sound different played on that instrument, because the ratios are slightly different. Other instruments are in B flat (e.g. Trumpet) and the combination of them with C-tuned instruments sounds interesting. Indeed, stringed instruments are tuned to N different keys where N = number of strings! The mathematics of music is so cool, I found the dismissive tone in the article to be quite unnecessary and irritating. EDIT: thanks to replies informing me that I'm wrong about instrument tuning. This means I've been tuning my guitars wrong all these years! REEDIT: but perhaps I'm right about old-school music with a key in the name? Since instruments would have been tuned with a tuning fork and then harmonics pre-20th century according to one reply.
  • tgb 10y ago
    Is that true? I thought most instruments (or at least most pianos) these days were tuned to be equal tempered, so each scale would sound the same (modulo a pitch scaling).
    • endymi0n 10y ago
      Pianos actually often even employ a form of stretch tuning ( https://en.wikipedia.org/wiki/Stretched_tuning https://en.wikipedia.org/wiki/Stretched_tuning ) - which gets even more complex, as the OP only refers to sinus tones, which don't have any melodic overtones. Things get even more fun with this. Otherwise, you're right. Historic tunings often tried to optimize for a certain base scale (often times C major), while leaving behind a really bad sounding tuning at the other side of the circle of fifths ( https://en.wikipedia.org/wiki/Wolf_interval https://en.wikipedia.org/wiki/Wolf_interval ). Modern pianos are usually equal tempered, which is the strategy of leaving every scale equally bad off - but on the other hand being able to play in all keys without sounding too bad (The famous "well-tempered piano" is referring to this historic change in tuning strategies with small pieces written in all different kinds of keys, all of which would have sounded weird before). The more I learned about tuning, the more complex, beautiful and sad this topic became for me: http://blogs.scientificamerican.com/roots-of-unity/the-saddest-thing-i-know-about-the-integers/ http://blogs.scientificamerican.com/roots-of-unity/the-sadde... It's fascinating and even a little philosophical that we can get so close to perfection with the 12 step tuning - but never, ever completely attain it. By the way - That's part of the reason why choirs and orchestras often sound so good: As most other non-fixed-string instruments are able to do a slight adjustment in pitch, they can actually play perfect intervals in any chord, dynamically adjusting to any scale.
• ajuc 10y ago
  My main problem with music notation is the difference between C-C# and E-F interval. That is - there's no difference, but notations pretends there is one because that's how we put the keys closer to each other or some other historical reason.
  • theOnliest 10y ago
    This is (sometimes) a legitimate gripe about the standard notation system, which is that it's optimized for music in a key. Sometimes there is a difference between E-F and C-C#. In the key of F major, E-F is the leading tone moving to tonic, which is a diatonic interval (a diatonic half-step). C-C#, on the other hand, is a chromatic half-step: in F major, it represents an alteration of scale degree 5 (sol). If you see C# in F major, there's a good chance it's going towards D, as a temporary leading tone. This is a useful distinction! The E-F half-step in F major (or C major, or A or D minor) is completely typical and not at all remarkable, while the C-C# half-step is much rarer. In musics where there isn't a key, you're right that it doesn't make any sense to draw a distinction between the two. This is one reason that the music of the 2nd Viennese School (Schoenberg, Webern, Berg) is so impossible to look at: the structure of the music is obfuscated by the structure of the notation. Schoenberg was trying to come up with a 12-tone notation system for a while, but ultimately abandoned it.
    • komponisto 10y ago
      >This is one reason that the music of the 2nd Viennese School (Schoenberg, Webern, Berg) is so impossible to look at: the structure of the music is obfuscated by the structure of the notation Wrong. First of all, it's not "impossible" to look at by any means; I think it's beautiful to look at (as great music usually is). There is a good reason why Schoenberg abandoned his (briefly-held) ideas about new forms of notation (and went on to produce another three decades' worth of music in traditional notation). He, and his disciples Berg and Webern, were steeped in the Western art music tradition, of which they believed their work to be a natural continuation. They didn't have a very good theoretical understanding of the new music they were creating -- because, apparently, music theory is hard. But they could sense its intimate relationship to its historical predecessors; indeed, they specifically, cultivated that relationship, baking it into the music. This, in my view, is why they were never going to break away from the visual representation of that relationship, of that continuity -- namely, traditional notation. The idea that their music is not in a key is widespread, but incorrect. Inferential distance (https://wiki.lesswrong.com/wiki/Inferential_distance https://wiki.lesswrong.com/wiki/Inferential_distance) precludes me from being able to explain this concisely in a non-misleading way, unfortunately.
• mmcconnell1618 10y ago
  Here's a great video explaining the origins of equal temperament: http://www.dailymotion.com/video/x2ismsw http://www.dailymotion.com/video/x2ismsw
• paracarro 10y ago
  > body {width:100%; max-width:1120px;... Is this good?
• torrent-of-ions 10y ago
  He says the intervals are arbitrary, but they are not, or at least not as arbitrary as that. An octave is 2x the pitch, a fifth is exactly 3/2x the pitch, a major third is exactly 5/4x the pitch etc. This is fine for instruments which can play an arbitrary pitch like a violin or trombone. But for fretted instruments like guitar, or ones which use a different oscillator per note like piano, xylophone or harmonica, one has to make a decision. Either you tune it to play perfect intervals and can therefore play in only one key, or you tune it to a compromise which can play in any key and sound OK, but no key "perfectly". Equal temperament is one such compromise and is described in the article. For instruments which play often play long chords like strings, the difference between a perfect interval (as an orchestra would play) and an equal temperament imperfect interval (as a general purpose synthesiser will play) is stunning. Some synthesisers like those from Access have modes to automatically correct the tuning of the third, fifth and maybe seventh harmonics to make them perfect.
• dbrgn 10y ago
  Here's an e-book that has a similar approach to explaining music: http://pedrokroger.net/mfgan/ http://pedrokroger.net/mfgan/ I read it some time ago and liked it. No affiliation with the author :)
• SubiculumCode 10y ago
  the article is interesting but its style doesn't mesh with title...i.e. Music Theory for Nerd, but then explains what a sin wave is, and apologizes in advance for the 'scariness' of log scales. Nerds know and are not afraid of sin waves and log scales...its as if the author doesn't know his audience.
• calebm 10y ago
  Looks like almost exactly the same experience I had: http://calebmadrigal.com/music-theory-notes/ http://calebmadrigal.com/music-theory-notes/
• chrisarensky 10y ago
  A clear map of tonality is used in Mapping Tonal Harmony Pro. If you want to view the entire harmonic landscape is the best concept out there