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Music theory for nerds (2016)

367 points by marianoguerra 5y ago

chrisweekly 5y ago
Love it! Learning in public FTW. (I've been making music for over 40 years, and some of this gave me a fresh appreciation for how convoluted and borderline impenetrable music theory can be.)
taubek 5y ago
Math and music are similar in so many aspects. Are the people good at one also good at other? I mean, do mathematicians understand musical theory better than those that have no background in mah?
- Rochus 5y ago
  > Are the people good at one also good at other? do mathematicians understand musical theory better than those that have no background in math Music and music theory are two different things. A person who is an expert in music theory is not necessarily a good musician and vice versa. There are approaches to music theory based on mathematics (see book list below); here at the latest, your assumption is likely to be correct, but hardly in general. Book list: Mathematics and Music - Composition, Perception, and Performance Musimathics Volume 1 and 2 Music and Mathematics - From Pythagoras to Fractals Mathematical and Computational Modeling of Tonality - Theory and Applications
  - rotten 5y ago
    There is also "The Geometry of Musical Rhythm"
  - taubek 5y ago
    Yeah, I meant music theory. Playing a music instrument is a skill that can be learned (if you have what it takes :) ).
    - watwut 5y ago
      Music theory can be learned too. I mean, there is not other way how to get it then by learning.
      - Rochus 5y ago
        > Music theory can be learned too Sure, but it's an add on. Music theory is developed after the fact by analyzing music created by people who (mostly) didn't know the theory. It's a (more or less useful) means to understand what the original creator did.
        nescioquid 5y ago
        For Western classical music, this is often claimed and is probably only a little true these days (e.g. Schenker is truly post facto, though illuminating). However, the theory really was deeply tied to the practice of composition and performance. One of the first exercises Mozart set his composition students was to write out various cadences. Likewise, Bach started his pupils with voice-leading. Theory was also needed for various forms of performance -- look at any manual on continuo playing (this is where the figures on roman numeral notation come from). Handel wrote a manual on continuo playing in which he touches on how to improvise a fugue!
- schwartzworld 5y ago
  I don't know about mathematicians, but there's definitely an overlap with programmers and musicians.
- scottious 5y ago
  I don't really think so. The stuff that's covered in the first half of this article is certainly very math-y, but you can be a great musician and not know this stuff. I really think all you need to know is basic arithmetic and the absolute basics of modular arithmetic (akin to learning how 12 o'clock + 1 hour = 1 o'clock, not 13 o'clock). Becoming a great musician is more about developing your ear, technique, repertoire, etc. It's more similar to learning a foreign language than learning math, in my opinion.
- karmakaze 5y ago
  I've read that Einstein often improvised on violin. Here's a search result[0]. > Einstein's violin playing took a gypsy turn when he improvised, often while thinking about mathematics. He was a very good musician. He could navigate the piano well enough. But his real love was for the violin. Even so, Einstein's violin playing was not of the first rank. Yet such was his love for the violin that he played and practiced anyway. Such is the spell that musical instruments cast on people's minds. > In addition to studying the classic violin pieces of Bach and Mozart that he adored, Einstein was also given to improvisations. Some have told of the haunting, gypsy-like quality of these rhapsodies. Einstein himself stated that many of his greatest mathematical ideas came when he was improvising on the violin. [0] https://pianobynumber.com/blogs/readingroom/einsteins-violin-improvisations-in-gypsy-style https://pianobynumber.com/blogs/readingroom/einsteins-violin...
  - Rochus 5y ago
    > Einstein often improvised on violin Improvisation has little to do with music theory; most Folk and Jazz music is improvised; even what we know today as "classical music" was often improvised in the past; many great composers were also great improvisers.
    - watwut 5y ago
      Wut? Jazz is heavy on theory from the start precisely to facilitate the improvisation. You need theory to improvise, basically. You do not need theory to perform existing pieces all that much. Which is why you can learn to play classical music while ignoring theory.
      - sornaensis 5y ago
        Jazz theory came decades after jazz.
        lewispollard 5y ago
        Yes, to describe and group together the specific music theory elements that jazz musicians tended to use: sevenths, avoiding triads, modulation, etc. Not because there was no theory intentionally used in jazz music. If you think those guys didn't know what they were doing...
- lelandfe 5y ago
  My family is full of music theorists, including a professor of it. I’ve asked this before, and they’re of the thinking that no, one’s aptitude at music theory is not related to a knowledge of mathematics. The professor in my family has in particular seen no relationship between the two in their decades of teaching. But - importantly - a background of math can be a tremendous asset later in one’s career as a music theorist. It can help provide a unique way of looking at and improving on theory.
- gjulianm 5y ago
  I have a degree in mathematics and have studied music for some years. My personal opinion is that they're quite different things. Yes, the underlying theory of music requires some mathematics, but that's it. Once you get into it there's much more "feeling" and "intuition" and "subjectiveness" than math could ever accept.
  - env123 5y ago
    ^ This. Maths ignore the stuff you mentioned above just to get by The same can be said about science ignoring philosophy, morality and spirituality just to get by
strong_as_oak 5y ago
whiddershins 5y ago
I love this style of clear, unambiguous writing starting with no knowledge assumptions. [but] The author is missing or skipping key pieces of information about why we use the notes we use, how they were derived, and why they sound the way they do to us. Edit. There was zero sarcasm in my comment. I really love this style of writing. The missing concept here is the harmonic series. A pitched note is not generally a sine wave, it is properly modeled as a series of sines waves combined. These sine waves go up in mathematical multiples. So say we have a note at 100hz. Simplistically it is actually 100, 200, 300, 400 etc all at once. So if 100 were A, 200 is A, 400 is A, 300 is … E, so 150 is E. Etc. None of this is arbitrary because the When the waves combine the result is more ordered or more chaotic, depending on the ratios of the waves, which our ears hear as consonant or dissonant. Best I can do from a phone.
- chrisweekly 5y ago
  It took a moment to realize your first sentence was derisive sarcasm. From TFA: > "Here is what I gathered, from the perspective of someone whose only music class was learning to play four notes on a recorder in second grade. I stress that I don’t know anything about music and this post is terrible. If you you so much as know how to whistle, please don’t read this you will laugh at me." ... which makes your comment even more uncharitable and unrealistically demanding. As if any mere blog post could answer those questions in meaningful depth. IOw, I don't understand the point of your complaint.
  - whiddershins 5y ago
    I was not being sarcastic. The part about how the octave gets divided up is just missing some key concepts.
    - chrisweekly 5y ago
      Ah, ok, thanks for clarifying. (If your 2nd sentence began with "But, ", this would have been clear to me.) PS I love OP's writing style, too. Fearless learning in public is to everyone's benefit -- given proper disclosure/transparency.
ben7799 5y ago
He left off right where it starts to get really interesting. Most of what he covered is stuff you learn the first day playing and never really need to worry about again. The interactions of how different notes & chords in a scale draw our ears, how 7ths and other intervals work/sound. Modes, chord voicings, inversions, modulations.. all that stuff gets incredibly fascinating.
- PaulDavisThe1st 5y ago
  > Modes, chord voicings, inversions, modulations In defense of the article, I'd note that these are all choices you make within the framework set up by 12TET, not the framework itself.
- iamevn 5y ago
  (fyi, it's she)
whycombs 5y ago
Sorry, a lot of this is wrong. A major 7th is not 17:9 but 15:8. Our ear tries to represent all these rich resonances in as simple primes as possible (Think, combos of 2's 3's and 5's - and occasionally 7's in blue notes). A minor 6th isn't just a note in between, but a simple 4:5, The minor sound that we associate with sadness is simply the reciprocal nature of the ratio - the more complicated prime is in the denominator (as opposed to the overtonal nature of the M7 above) Short answer: read Harmonic Experience by W.A. Mathieu
- mrob 5y ago
  The ear does not care about ratios. Dissonance can be modeled using Plomp and Levelt's curve, which predicts the perceived dissonance of a pair of sine waves. Dissonance is minimum at unison, smoothly increases to a maximum at a narrow interval (dependent on absolute frequency), and smoothly decreases again as the interval increases. Sethares extended this model to arbitrary sounds by decomposing them into their sine-wave partials with a Fourier transform and then taking an amplitude-weighted average of the predicted dissonance of each pair of partials. If you apply this procedure to intervals using harmonic timbres (timbres composed of partials with frequency an integer multiple of the fundamental frequency), the dissonance curve has minima that happen to be at small integer frequency ratios. Harmonic timbres are most common in music, but there is an important class of musical instruments that are naturally inharmonic: tuned percussion. Unless substantial effort is put into designing them to approximate harmonic timbres, these sound most consonant when tuned outside small integer ratios. Indonesian gamelan music is a well known example of this. And with synthesized timbres you can have whatever inharmonicity you like, so timbres can be designed to suit any arbitrary tuning system. Non-technical explanation of Sethares' model: https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html Full mathematical details: https://sethares.engr.wisc.edu/paperspdf/consonance.pdf https://sethares.engr.wisc.edu/paperspdf/consonance.pdf
  - notahacker 5y ago
    > If you apply this procedure to intervals using harmonic timbres (timbres composed of partials with frequency an integer multiple of the fundamental frequency), the dissonance curve has minima that happen to be at small integer frequency ratios. In practice that's pretty much "the ear cares about ratios" isn't it? Sure, it's caring about the ratios because they align with harmonic timbres in non-percussive instruments, but various human cultures do listen to a wide variety of instruments and voice with harmonic timbres, and perceive melody and chords based around [reasonably close approxiaations] of simple integer ratios sounding more consonant. The implications for creating chordal music with synthesised inharmonic timbres designed to be consonant are interesting, but not widely applicable to what we actually listen to as music; and much of what we hear that goes outside simple ratios with the express intent of sounding dissonant...
abetusk 5y ago
I really don't understand all the hate in the comments for this article. This seems like a really fine introduction to a lot of the concepts. I had to scour the web a long time before understanding that the scales were constructed because of the simple/reduced ratio of root note to the others, which is clearly spelled out in this article. Music theory is, in my experience, typically taught as a list of facts to remember. Deriving it from first principles, insofar as it can be, is not common. This article is attempting to start that process. Some commentators are saying that the article doesn't address why we have 12 notes but I wonder if they even know why (one answer is because it's a good compromise between number of notes and that have simple/reduce ratios to each other [0]). I'm also skeptical of music theorists that can't even attempt an answer to basic questions about note length frequency, why some chords are "sad" or "happy" and other basic questions. It's difficult because music theory is a hodge podge of theory that's attempting to describe what's effectively an evolved language (with different music evolution for different regions), but there are some basic tenets that probably apply. I don't claim to have deep knowledge but there are a few key facts about music and music theory that are really obscure unless you know where to look. Any attempt at coming to music theory from a more rigorous foundation should be encouraged. [0] "Measures of Consonances in a Goodness-of-fit Model for Equal-tempered Scales" by Aline Honigh (https://github.com/abetusk/papers/blob/release/Music/measures-consonances_honingh.pdf https://github.com/abetusk/papers/blob/release/Music/measure...) EDIT: corrected spelling (thanks for the comments pointing it out)
- Infernal 5y ago
  > basic tenants Just in case it’s helpful - “tenets”
- chrisweekly 5y ago
  +1 / agreed. Also (tangent / tiny nit): "tenants" (inhabitants) -> "tenets" (rules)
- criddell 5y ago
  You seems to know a lot about the art and science of music so maybe you can answer a question I’ve had lately: what’s the difference between pitch and frequency? When a person talks about middle C on a piano, is that ‘C’ more closely related to pitch or frequency?
wildmanx 5y ago
This has about as much to do with "music" as analyzing grammar when talking about poetry. Musical notation, and lots of "rules" that come with it, came after the fact. Just like with natural language. And just like staring at grammar rules is a terrible way to learn to speak a new language, analyzing music "theory" at this level does not teach you anything about actual music.
- Aidevah 5y ago
  Agreed. The author mentioned that the desire to compose music is what spurred this post. It would be much quicker to read through a music textbook. The basics of western music notation can be taught in 15 minutes, and one could get used to it after an afternoon. There is no need to interrogate the origins and reasons behind conventional notation if creative work is the goal, since there are much more important concepts like harmony and voice leading which are more worth your time.
  - trasz 5y ago
    If by "western music notation" you mean the clef and stuff, then lol, no. There's a reason most musicians don't use nor know it.
    - Aidevah 5y ago
      The reason most musicians don't use staff notation is because most musicians do not need it unless they want to engage with the literate tradition of music, and are not disadvantaged if they can't. Not because of any inherent difficulties in the notation itself.
      - zozbot234 5y ago
        > and are not disadvantaged if they can't Literacy is just as helpful in music as in any other artform. One can certainly play and create music without learning how to read and write staff notation, but being literate can only help!
  - 5y ago
throw7 5y ago
"This has got to be some of the worst jargon and notation for anything, ever." Ummm, do you math? ;) My point of view of music theory is: sound is objective. music is subjective. where the two meet is music theory.
- H8crilA 5y ago
  Yeah, no, modern math notation is a lot cleaner than musical notation, and completely free of historic conventions. Pretty much the only thing "forced" into it by human preference is the standard of using base-10 numbers. Prove me wrong please :)
  - KineticLensman 5y ago
    > and completely free of historic conventions > Prove me wrong please :) Some of the notation goes back centuries [0]. Our current representation of calculus reflects a convention that was hammered out following significant disputes in the early 18th Century. [0] https://en.wikipedia.org/wiki/History_of_mathematical_notation#Early_arithmetic_and_multiplication https://en.wikipedia.org/wiki/History_of_mathematical_notati...
PaulDavisThe1st 5y ago
On the question of "why 12 notes", I STRONGLY recommend this video by David Bennett: https://www.youtube.com/watch?v=lvmzgVtZtUQ https://www.youtube.com/watch?v=lvmzgVtZtUQ There's a really important step missing in the article which has been aluded to by a couple of comments already. It's the step between: 1. a series of notes defined by the harmonic series, that is, exact integer multiples (or ratios, if you prefer) of a base frequency, a physical phenomena found in nature and 2. the well-tempered scales used by most western music, which adjusts the ratios to allow certain compositional "tricks" to be used without sounding dissonant. The set of notes defined by (1) are typically referred to as "Just Intonation", and was the basis for most western music (and some non-western music) until somewhere between about 1400 and 1600 (lots of room for discussion/debate there). The "problem" with just intonation, if indeed it is a problem, is that if you define two series of notes (call them a scale, if you like) using these integer ratios but starting from two different notes, you will trivially find cases where "the next higher E" is a different frequency. So starting from (say) A and moving by integer ratios gets you to a different (say) F than if you start from (say) B. This means that if you are writing in a just intonation scale with (say) A as the root, you have a set of notes that are not actually the same frequencies as if your scale started on (say) B. By itself, this is not a problem at all - there is all kinds of lovely music written through history that works just fine with notes and scales defined this way. You just stay in the same scale throughout, and there are no issues. There are even some scale changes you can make that still work, you just have to know what they are (and they depend on the root note and the set of integer ratios you're using, so it gets complicated). But ... somewhere in the period mentioned above, a subset of western musical culture started to want to experiment with "modulation" - changing from one scale to another in the midst of piece. On continuous pitch instruments (e.g. violin, voice), this is entirely possible to do, since they can play any frequency in their range at all. However, it does require significant skill on the part of the performer, since the pitch of a (say) "F" will differ depending on the scale currently in use in the piece. The breaking point, such as it was, came with the development of fixed pitch instruments (keyboards). These can only play the notes they are currently tuned to, and so if the (say) F in two scales is a different frequency, you cannot play in both scales without retuning - an obvious impossibility in the middle of a piece. So, "well-tempered" tuning was developed - the ratios described in TFA. These are tweaked by relatively small amounts so that the notes are close to where Just Intonation (integer ratios) would have placed them, but not precisely the same. These small shifts mean that the (say) "F" is the same frequency whether your scale began on (say) A or B. You can now play a fixed-tuning instrument like a harpsichord or a piano, change from one scale to another, and everything remains "in tune". Of course, to ears used to Just Intonation, "well-tempered" tuning sounds out of tune. But in the west, most people (even within our musical academies) have grown up used to the sound of well-tempered tuning, and it is Just Intonation that sounds "off".
bannedbybros 5y ago
>Not counting the octave, there are seven fairly nice fractions here. Hmm. Seven. What a conspicuous number. There's seven nice fraction because of how they were rounded. But there's no explanation of why he rounds 1.682 to 5/3 but doesn't round 1.587 to the even closer 8/5. It's because scales are a social construction, not a result of vague math.
calebm 5y ago
My quick notes on music theory: https://calebmadrigal.com/music-theory-notes/ https://calebmadrigal.com/music-theory-notes/
srcreigh 5y ago
> I don’t know why twelve in particular has this effect, or if other roots do as well Here's a 31-equal temperament song. Microtonal music has a reputation for sounding bad, but it doesn't have to. https://www.youtube.com/watch?v=u-PEhbSOh74&list=PLC6ZSKWKnVz3lBogeQ66N8W3x_GwUhTFw&index=17 https://www.youtube.com/watch?v=u-PEhbSOh74&list=PLC6ZSKWKnV... > ... but it’s probably why Western music settled on twelve. The curious part is that Western music settled on twelve a long time before equal temperament. They used to have 12 notes but use for example quarter-comma meantone. Great mozart meantone piano piece (mozart never used equal temperament): https://youtu.be/lzsEdK48CDY?t=700 https://youtu.be/lzsEdK48CDY?t=700
- jacquesm 5y ago
  Interesting, that sounds like music from some kind of medieval castle that somehow got stuck in a timewarp. I can't listen to it for very long but it sounded a lot better than I thought it would.
- hammock 5y ago
  What am listening to? Long time musician here, just not familiar with untraditional scales. I assume 31edo but using Bach means you are ignoring 19 of 31 possible semitones- correct? Less? (I could imagine leading tones going in the opposite direction, or something) To my ear it sounds like there are way more major thirds than there normally would be in a Bach piece like this... anything else I ought to be hearing?
  - srcreigh 5y ago
    I would guess you're right but I'm not 100% sure! This channel is pretty responsive, you ask on the video. This one is fascinating too - it's just intonation vs 12edo comparison. You can really hear all the beating in the harmonics in the EDO one. https://www.youtube.com/watch?v=AgB4TrV57AU https://www.youtube.com/watch?v=AgB4TrV57AU
  - lewispollard 5y ago
    I haven't listened to the piece, but afaik typically transposing existing 12TET pieces into microtonal tunings are taking the intervals in the original piece and applying them to scales in the new tuning, with some creative license/taste to embellish where it doesn't sound so good. So it's not ignoring or limiting any pitches to my (limited) understanding, other than the natural way that pieces written in a particular key/mode will skip over some tones in favour of others for harmonic or melodic reasons.
fugalfervor 5y ago
Music was taught as part of the quadrivium (alongside arithmetic, geometry, and astronomy), so it has a long history of being thought about in mathematical terms. There's nothing wrong with looking at music that way. What is so interesting about music is its ability to turn mathematical beauty into aesthetic, artistic beauty. Sometimes the math is just there in the background: "This chord progression is based on certain physical facts related to the harmonic series. That makes it sound good, and I like it". Sometimes it is brought to the forefront: "All of this music appears to be generated from a short pattern of notes which is recognizable even as it is subjected to rotations, elongations, truncations, etc. That is beautiful." Certain types of music play up the lovely-pattern-ness of music, and some don't. Neither is better than the other (denigrate the art of Justin Bieber who dare), but you can probably guess which I prefer :) Consciousness, pattern-recognition, spatial rotation of abstract objects, natural mathematical beauty, poetry, drama, acting/performance, nonverbal collaboration, shared rhythm: music is a nexus for all of these things. Music is nothing short of awesome, in the truest sense of that word.
- john61 5y ago
  What is interesting is that master composers often broke the "mathematic" rules. Schubert, Beethoven or Chopin for example have sometimes very strange and unbelievable beautiful harmonics sprinkled in. Whereas mediocre composer do not do this, they are predictable.
  - fugalfervor 5y ago
    I don't think it's helpful to think of music in terms of mathematical "rules", but more in terms of cultural variation on what is acceptable. Now this is my perspective, after spending a long time steeped in this stuff, but feel free to disagree: The history of music has been oversimplified and done a disservice by its characterization as "great men" breaking all the rules. Taken to its extreme, music gets better when a greater man breaks more rules. That's obviously a little silly if you ask me. Dissonance (a "broken mathematical rule") has as much a right to exist in music as consonance does. There is a constant tension between consonance and dissonance, and it is that dynamic tension which is exploited for affect or structural definition. So what we have in the case of Schubert, Beethoven, and Chopin is not so much the breaking of mathematical rules. In my opinion, they are just examples of expert judgment applied to the tension and release within an artistic work.
ngcc_hk 5y ago
Seems not knowing the struggle with the note system. The struggle eg ## which is not really the note above because the # and b is not perfect. Bath well pieces probably not in his understanding. Not really a music theory.
coldtea 5y ago
>If we say 440 Hz produces a note called A, then 880 Hz, 220 Hz, 1760 Hz, 110 Hz, and so forth will also produce a note called A. An important consequence of this is that all distinct notes we could possibly come up with must exist somewhere between 440 Hz and 880 Hz. Any other pitch could be doubled or halved until it lies in that range, and thus would produce a note in that range. That's inaccurate and backwards. 338 Hz is not in that range, and it can't be doubled or halved to fit in that, but it's a perfectly fine not. The range is more about tuning (which is a choice) than about 440-880Hz being something special. >All of these things messily overlap and create multiple conflicting names for the same things, because they’re attempts to describe human intention rather than an objective waveform. Nope, they don't overlap any more messily than 1+2 == 2+1, or in how 0.99999... == 1.
- quasisphere 5y ago
  338 * 2 = 676, which is between 440 and 880
  - coldtea 5y ago
    Yeah, bad wording. 338*2 is in the range, but is nowhere near the 12 tones we use when A=440 (and wont be with integral doubling or halving). The property "within the range A4=440 A5=880" doesn't have any particular special musical meaning. The idea I wanted to convey is that 440-880 is just a convention, starting from 440 as concept pitch, but concept pitch can be totally different (and outside the range). 432 is a famous standard which has and is been used today, but also tons of others, even widely outside the range... (also: "but it's a perfectly fine not" --> "note").
andrewzah 5y ago
Ultimately a lot of things around music are cultural and/or arbitrary. It's like languages in that sense. Personally I don't find this sort of analytical study to be that helpful for actually playing or improvising music, but it is neat information. You don't have to clinically study grammar and language theory in order to be a great writer; it's the same with music. Something that's very important to note is that music theory and playing music are very different. Music theory allows us to understand what someone played, and lets us communicate to other musicians in a compact manner. Theory by no means imposes rules, unless you are specifically wanting to write music exactly like Bach or something. As a logical-minded person myself, it's tempting to really dive into theory and the root of things, but I've found that it's a trap most of the time. For getting better at songwriting or composing, I've found the best thing to do is transcribe the songs you like (or portions of them) and see what they do, then incorporate those ideas into your own playing. I also get the frustration around enharmonics (e.g. C# vs Db), but honestly it's not an issue in practice. I had no opinion until I started typesetting songs myself, and it just looks cleaner to sometimes have Cb or B# depending on the context. The circle of 5ths also neatly puts scales in the order of how many sharps or flats they have. Also, C# is not a commonly used key anyways. From a theoretical perspective C major and A minor share the same notes, but when I improvise I do find the distinction meaningful. Having multiple ways to think about things is helpful. There are other things too: C minor 7 is the same as Eb major 6. Or if you play a rootless voicing, the harmony could be ambiguous. Etc. Yes, our music notation system has some issues. But this article is overanalyzing it from an armchair. It's mostly fine in practice.
cousin_it 5y ago
The article says: 0 1.000 = 1:1 (unison) 1 1.059 (semitone; minor second) 2 1.122 ≈ 1.125 = 9:8 (whole tone; major second) 3 1.189 (minor third) 4 1.260 ≈ 1.250 = 5:4 (major third) 5 1.335 ≈ 1.333 = 4:3 (perfect fourth) 6 1.414 7 1.498 ≈ 1.500 = 3:2 (perfect fifth) 8 1.587 (minor sixth) 9 1.682 ≈ 1.667 = 5:3 (major sixth) 10 1.782 (minor seventh) 11 1.888 ≈ 1.889 = 17:9 (major seventh) 12 2 = 2:1 (octave) This table has many omissions and some errors. The way people usually play these intervals on just intonation instruments, minor second should be 16:15, major second can be 9:8 or 10:9 depending on context, minor third is 6:5, minor sixth is 8:5, minor seventh can be 9:5 or 16:9 (or 7:4 in barbershop), major seventh is 15:8.
- kmill 5y ago
  You might like the book "Equal Temperament Ruined Harmony." It makes the case that there is a long history of differentiating sharps and flats in western music (flats are just a bit sharper than sharps), while arguing that in practice the whole tone was subdivided into nine parts, with the major and minor half steps respectively 5 or 4 of those nine parts. I don't think the precise tuning system described by the book really happened, but it does collect evidence that "enharmonic" used to mean "finesse your intonation to make this sharp be that flat." In my own experimentation, the just intonation intervals are jarringly perfect, and I'd say that the 5:4 major third is too narrow. Has anyone checked that barbershop quartets are actually singing such narrow major thirds? My understand is that many studies have shown that string instrumentalists and singers, even when they claim to be using just intonation, play their major thirds somewhere between 5:4 and equal temperament. It's also extraordinarly difficult to really tune basic harmony justly. The ii V I progression is a fun one... if you're not careful you shift the tonic by a syntonic comma.
  - tshaddox 5y ago
    > Has anyone checked that barbershop quartets are actually singing such narrow major thirds? I don’t know if anyone has checked, but why wouldn’t they want to? Surely the low integer ratios are best going to get the intended sound of barbershop harmony, unless there are other acoustic reasons why slightly changing the intervals would work better (something akin to octave stretching in pianos, although I’m not sure that sort of thing would be as relevant to barbershop harmony).
dataspun 5y ago
As an amateur musician (and amateur nerd) I found this article to be insightful in many ways. The author’s opening disclaimer renders most of the HN criticism moot.
toolslive 5y ago
Hold on. a C# and a Db are not the same note. They are a comma apart. https://en.wikipedia.org/wiki/Comma_(music) https://en.wikipedia.org/wiki/Comma_(music)
- asdfman123 5y ago
  Everyone knows the C# connects to the DB
- commandlinefan 5y ago
  Hm - maybe technically, but they are the same on any instrument you can play them on.
  - rdlw 5y ago
    Except for voice [0], any fretless stringed instrument [1], any fretted string instrument with separate frets for sharps and flats [2], keyboard instruments that allow expression [3], keyboard instruments with split flat/sharp keys [4], and electronic instruments that allow unorthodox tunings without huge hassle [5]... I'm sure I'm missing some, but other than that you're totally right ;) [0] https://youtu.be/XnGqVIW51JA https://youtu.be/XnGqVIW51JA, https://youtu.be/mPZn4x3uOac https://youtu.be/mPZn4x3uOac [1] https://youtu.be/PqR36Drhn_k https://youtu.be/PqR36Drhn_k [2] https://youtu.be/iRsSjh5TTqI https://youtu.be/iRsSjh5TTqI [3] https://youtu.be/P14JcRyJCEI https://youtu.be/P14JcRyJCEI [4] https://youtu.be/7GhAuZH6phs https://youtu.be/7GhAuZH6phs [5] https://youtu.be/xVZy9GUeMqY https://youtu.be/xVZy9GUeMqY
    - toolslive 5y ago
      trombones and horns.
dang 5y ago
Discussed at the time: Music theory for nerds - https://news.ycombinator.com/item?id=12528144 https://news.ycombinator.com/item?id=12528144 - Sept 2016 (378 comments)
streamofdigits 5y ago
> That problem is that sheet music is terrible. Something that people with casual exposure to musical notation probably don't know is that before the standard 5-staff notation became ubiquitus there were quite a few alternatives, maybe the most interesting from a mathematical perspective being the Byzantine notation Broadly speaking, while the modern system focuses on pitch values and an elaborate (modulated) map from frequency space to physical space (paper), older systems used in the Byzantium used "deltas" or the first differences of pitch values. So you start with a base note (lets say C) and then you go +1, +1, -2 to indicate pitch changes (in semitones). This is quite well adapted to monophonic chant. This notation was never developed to cope with the complexity of modern music but its not immediately obvious that it can't be done There is no easily accessible exposition of this musical notatin style, this cheatsheet gives a flavor http://www.byzantinechant.org/notation/Table%20of%20Byzantine%20Notation%20Symbols.pdf http://www.byzantinechant.org/notation/Table%20of%20Byzantin...
honkycat 5y ago
I read eev.ee's tutorial on platformer physics when doing a little gamejam. Worked great! What a talented person!
navane 5y ago
Why did we settle on 12 notes? Rule 1 (also stated in the article): Doubling frequency gives us practically the same note. It is so pleasant together with the original note, that it is boring. Lets make a set of notes that sound nice together, and are more or less equally distributed between a note and its frequency double (octave). The next most pleasant note, is a note with 3 times the frequency, then 4, etc... (Why is this pleasant? Because its a close natural harmonic. When you vibrate a string with frequency 1, it will start vibrating with a frequency of 2 as well, but softer. This doubling in frequency is because the string will behave both as a string with length L, and as two separate strings with length L/2. Because the length is halve, the frequency is doubled.) Now, 3 times the frequency, is the same as 1.5 times the frequency. If we do the same from that base note, we get 1.5 * 1.5 = 2.25 times the frequency. You could decide that that 2.25 is close to 2, call it a day, and settle on a two note scale. Or, you go further. 2.25/2 = 1.125, so now we have the notes 1, 1.125, 1.5 and 2. Not really evenly spaced yet. 1.125*1.5=1.6875...: [1, 1.125, 1.265, 1.5, 1.687, 1.89]. Oh, 1.89 is almost 2. Let's call it a day and settle on a five note scale. (there are plenty of cultures with a five not scale folk music tradition). Note that the error gets smaller, it was 2.25/2 for our 2 note scale, it's 1.89/2 for the 5 note scale. Now it happens that the next smallest error is with a 12 note scale. But there's still an error, you don't arrive at exactly 2. The intervals between your notes are not even, and your last interval is all retarded. That's why people settled on "equal temperament", where all the intervals between the notes are made equal, and the number 12 looks seemingly arbitrary. I did the math, see table below. The first column are the frequency of the notes, with base frequency 1, where I multiplied by 1.5, and divided by 2 to keep the notes between frequency 1 and 2. The 4th column gives you the "error", how close is the last note of the scale, to the octave of the base note. The closer it is, the better, it means we have an equally divisible range of notes that all have nice (with ratio 1.5) intervals with each other. The arrows indicate where we could stop adding notes to our scale, because we arrived at a local optimum: a five not scale is small but has quite an error. The 12 note scale is quite elaborate, but reduces the error a lot. The next best scale would be a 41 note scale, with only a slightly diminished error compared to the 12 note scale. Freq. n/1 2/n min(n/1;2/n) 1.500 1.50 1.33 1.333 1.125 1.13 1.78 1.125 1.688 1.69 1.19 1.185 1.266 1.27 1.58 1.266 1.898 1.90 1.05 1.053 <-- (5 notes) 1.424 1.42 1.40 1.405 1.068 1.07 1.87 1.068 1.602 1.60 1.25 1.249 1.201 1.20 1.66 1.201 1.802 1.80 1.11 1.110 1.352 1.35 1.48 1.352 1.014 1.01 1.97 1.014 <-- (12 notes) 1.520 1.52 1.32 1.315 1.140 1.14 1.75 1.140 1.711 1.71 1.17 1.169 1.283 1.28 1.56 1.283 1.924 1.92 1.04 1.039 1.443 1.44 1.39 1.386 1.082 1.08 1.85 1.082 1.624 1.62 1.23 1.232 1.218 1.22 1.64 1.218 1.827 1.83 1.09 1.095 1.370 1.37 1.46 1.370 1.027 1.03 1.95 1.027 1.541 1.54 1.30 1.298 1.156 1.16 1.73 1.156 1.734 1.73 1.15 1.153 1.300 1.30 1.54 1.300 1.951 1.95 1.03 1.025 1.463 1.46 1.37 1.367 1.097 1.10 1.82 1.097 1.646 1.65 1.22 1.215 1.234 1.23 1.62 1.234 1.852 1.85 1.08 1.080 1.389 1.39 1.44 1.389 1.041 1.04 1.92 1.041 1.562 1.56 1.28 1.280 1.172 1.17 1.71 1.172 1.758 1.76 1.14 1.138 1.318 1.32 1.52 1.318 1.977 1.98 1.01 1.012 <-- (41 notes)
mtalantikite 5y ago
> More “fake” notes exist than E♯, too; I hear rumor of such nonsense as G𝄪, “G double sharp”, which I would rather call “A”. It starts making more sense when you're working with harmony and chords. The wikipedia example section on enharmonic equivalents has a nice example of a Shubert sonata that makes use of this [1]. [1]https://en.wikipedia.org/wiki/Enharmonic https://en.wikipedia.org/wiki/Enharmonic
- zozbot234 5y ago
  Yes, if you think enharmonic notes are pointless or "fake" you should look into the derivation of diatonic and chromatic scales via "stacking" fifths, and what that implies for modulation along the circle of fifths. Also how sharp and flat are often theoretically conflated with mi and fa, since both proceed by half-step to their neighbor note.
- deleted 5y ago
  [deleted]
adamnemecek 5y ago
I’m working on an IDE for music composition https://ngrid.io https://ngrid.io
dan_mctree 5y ago
When I listen to the three sinewaves, two of which are supposed to feel more similar because they a factor two apart, I don't actually get any such similarity effect. They all seem just about equally similar to each other, even when played together. Is everyone here picking this sameness up? I'm curious what kind of mechanic underlies those feelings of sameness and what it feels like.
- whiddershins 5y ago
  Its very hard to hear this with sine waves. Sine waves work to visualize the concepts but real instrument notes almost always are better modeled as a large number of sine waves with a mathematical relationship that sound simultaneously and in varying amplitudes that combine to create the timbre of the note. Then when those notes sound together we get combinations of all those sine waves together to create a very complex wave that sounds a certain way to our ear based on the interactions between all of these waves at once. So to visualize it or reason about it, my comment here makes sense: https://news.ycombinator.com/item?id=30363653 https://news.ycombinator.com/item?id=30363653 But to hear it you would be much better off with triangle or sawtooth waves or samples of actual musical instruments.
- OscarCunningham 5y ago
  Right. Octaves sound consonant because the harmonics of one note line up with the harmonics of the other. Without the harmonics you expect to hear no effect.
buzzwords 5y ago
I have always struggled to learn music theory. If anyone has a good source please let me know.
ben7799 5y ago
I guess I fell into the same trap as a lot of others reading this and commenting it and having the programmer side of my brain see one thing and the musician side see something else. Computer programs must follow the rules... the rules get made up first and the programs get built to conform to the rules. Music theory isn't like that. The music comes first. The theory gets made up to try and describe & understand what is going on with the music. It's not at all like a program where the rules & theory come first and you build something that fits into that. If you write a piece of music that doesn't conform to some kind of arbitrary rule in music theory that's fine. If your music is successful and important enough someone will modify music theory to explain why your music was successful and important. There is a cycle where you learn some music theory to help learn music but the theory just helps you understand what's going on, it doesn't force you to do anything a certain way. You can't come at music and think about it like you're building a computer program.
- quaintquant 5y ago
  > You can't come at music and think about it like you're building a computer program. I think the absolutionist take here is unproductive. I would argue a writing a fugue is analogous to "building a computer program"; to say little of 12 tone chromatic atonal music compositions that usually sacrifice 7 tone based harmonies for interesting structure, like palindromic reflection of ascending and descending musical ideas, and optimising for a high number of pitches in a melody. Additionally, 7 note diatonic music theory can be completely summed up with two statements: 1) tick-tick-tock-tick-tick-tick-tock 2) permuate all the things! Most common western instruments are literally built to only play 7 tone diatonic music theory. Look at a piano. It can literally ONLY play "semi-tones". It's impossible to explore music beyond the 12 note chromatic scale without first modifiying the physical construction of the piano.
  - powersnail 5y ago
    The problem with coming at music like building a computer program is that, while it theoretically can work when you look at it from thousand miles away, no musician actually does that. It reminds me of when Dijkstra wrote that all computer science students should always start with manipulating symbols before touching a machine, and Knuth replied that nobody learned programming that way. There's no empirical data of such thinking actual work. What makes good music good is ultimately human ears. Fugues, for example, might seem to be programmatical at first glance, but get your hands dirty and compose one yourself, based on nothing but the mechanical rules. Doesn't sound good? It doesn't. It's like writing English based on syntactical rules. You'll end up with a music equivalent of "Colorless green ideas sleep furiously".
zoltar 5y ago
I really like 12tone's breakdown of pop songs. Here's an example with The Immigrant Song: https://www.youtube.com/watch?v=gwsfR5sJPbM https://www.youtube.com/watch?v=gwsfR5sJPbM
TuLithu 5y ago
Once you've understood the basics, get the Gradus Ad Parnassum by Johann Joseph Fux. A word to the wise is sufficient.
deltaonefour 5y ago
I wish there was a theory for why a certain set of sounds is music and why it isn't music. I'm positive I can follow all the rules according to standard "music theory" and produce sounds that aren't classified as music.
838812052807016 5y ago
Much of the confusion and misunderstanding is because when this subject is taught and explained, the teacher usually jumps from sine waves to music, where a musical note is thought of as a sine wave. But it is better to think of a musical note as a set of sine waves. Each sine wave in the set has a frequency which is an integer multiple of the frequency of the longest sine wave. These frequencies are known as harmonics. If we plot these frequencies on a logarithmic scale, they'll be evenly spaced. https://en.wikipedia.org/wiki/File:Spectrogram_of_violin.png https://en.wikipedia.org/wiki/File:Spectrogram_of_violin.png https://en.wikipedia.org/wiki/File:Human_voice_spectrogram.jpg https://en.wikipedia.org/wiki/File:Human_voice_spectrogram.j... Imagine a note as a comb, where each tooth represents the frequency of one of the sine waves. Now imagine a second comb where the teeth are spaced farther apart. Now imagine that you put one comb on top of the other so that the teeth line up. You'll see a nice pattern. This is the harmony of the combs. C4 | | | | | | | | G4 | | | | | | | | C5 | | | | | | | | E4 | | | | | | | | But these combs aren't only in an abstract frequency space. The cochlea inside your ear maps each frequency to a specific physical location where your ear hairs are tickled, forming a comb-like pattern. https://en.wikipedia.org/wiki/File:Journey_of_Sound_to_the_Brain.ogg https://en.wikipedia.org/wiki/File:Journey_of_Sound_to_the_B... Consider this chord composed of the notes G4, C5, and E5, which some of you may know as C 2nd inversion or G⁶₄. chord | | | | | | | | | | | | | | | | | | | | This chord will tickle your ears in many of the same places as the note C3. C3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | One way we can tell human voices apart is by the relative intensities of the harmonics. And the way we tell different vowel sounds apart is by the relative intensities of the harmonics. So our brains have lots of wiring for interpreting relative intensities of harmonics. https://auditoryneuroscience.com/sites/default/files/Fig1-16color.jpg https://auditoryneuroscience.com/sites/default/files/Fig1-16... (source: https://auditoryneuroscience.com/vocalizations-speech/formants-harmonics https://auditoryneuroscience.com/vocalizations-speech/forman...) So when we hear this chord, our brains are likely using some of the same wiring they would use for a human voice singing the note C3.

367 points by marianoguerra 5y ago

Love it! Learning in public FTW. (I've been making music for over 40 years, and some of this gave me a fresh appreciation for how convoluted and borderline impenetrable music theory can be.)

taubek 5y ago

Math and music are similar in so many aspects. Are the people good at one also good at other? I mean, do mathematicians understand musical theory better than those that have no background in mah?

Rochus 5y ago
> Are the people good at one also good at other? do mathematicians understand musical theory better than those that have no background in math Music and music theory are two different things. A person who is an expert in music theory is not necessarily a good musician and vice versa. There are approaches to music theory based on mathematics (see book list below); here at the latest, your assumption is likely to be correct, but hardly in general. Book list: Mathematics and Music - Composition, Perception, and Performance Musimathics Volume 1 and 2 Music and Mathematics - From Pythagoras to Fractals Mathematical and Computational Modeling of Tonality - Theory and Applications
- rotten 5y ago
  There is also "The Geometry of Musical Rhythm"
- taubek 5y ago
  Yeah, I meant music theory. Playing a music instrument is a skill that can be learned (if you have what it takes :) ).
  - watwut 5y ago
    Music theory can be learned too. I mean, there is not other way how to get it then by learning.
    - Rochus 5y ago
      > Music theory can be learned too Sure, but it's an add on. Music theory is developed after the fact by analyzing music created by people who (mostly) didn't know the theory. It's a (more or less useful) means to understand what the original creator did.
      - nescioquid 5y ago
        For Western classical music, this is often claimed and is probably only a little true these days (e.g. Schenker is truly post facto, though illuminating). However, the theory really was deeply tied to the practice of composition and performance. One of the first exercises Mozart set his composition students was to write out various cadences. Likewise, Bach started his pupils with voice-leading. Theory was also needed for various forms of performance -- look at any manual on continuo playing (this is where the figures on roman numeral notation come from). Handel wrote a manual on continuo playing in which he touches on how to improvise a fugue!
schwartzworld 5y ago
I don't know about mathematicians, but there's definitely an overlap with programmers and musicians.
scottious 5y ago
I don't really think so. The stuff that's covered in the first half of this article is certainly very math-y, but you can be a great musician and not know this stuff. I really think all you need to know is basic arithmetic and the absolute basics of modular arithmetic (akin to learning how 12 o'clock + 1 hour = 1 o'clock, not 13 o'clock). Becoming a great musician is more about developing your ear, technique, repertoire, etc. It's more similar to learning a foreign language than learning math, in my opinion.
karmakaze 5y ago
I've read that Einstein often improvised on violin. Here's a search result[0]. > Einstein's violin playing took a gypsy turn when he improvised, often while thinking about mathematics. He was a very good musician. He could navigate the piano well enough. But his real love was for the violin. Even so, Einstein's violin playing was not of the first rank. Yet such was his love for the violin that he played and practiced anyway. Such is the spell that musical instruments cast on people's minds. > In addition to studying the classic violin pieces of Bach and Mozart that he adored, Einstein was also given to improvisations. Some have told of the haunting, gypsy-like quality of these rhapsodies. Einstein himself stated that many of his greatest mathematical ideas came when he was improvising on the violin. [0] https://pianobynumber.com/blogs/readingroom/einsteins-violin-improvisations-in-gypsy-style https://pianobynumber.com/blogs/readingroom/einsteins-violin...
- Rochus 5y ago
  > Einstein often improvised on violin Improvisation has little to do with music theory; most Folk and Jazz music is improvised; even what we know today as "classical music" was often improvised in the past; many great composers were also great improvisers.
  - watwut 5y ago
    Wut? Jazz is heavy on theory from the start precisely to facilitate the improvisation. You need theory to improvise, basically. You do not need theory to perform existing pieces all that much. Which is why you can learn to play classical music while ignoring theory.
    - sornaensis 5y ago
      Jazz theory came decades after jazz.
      - lewispollard 5y ago
        Yes, to describe and group together the specific music theory elements that jazz musicians tended to use: sevenths, avoiding triads, modulation, etc. Not because there was no theory intentionally used in jazz music. If you think those guys didn't know what they were doing...
lelandfe 5y ago
My family is full of music theorists, including a professor of it. I’ve asked this before, and they’re of the thinking that no, one’s aptitude at music theory is not related to a knowledge of mathematics. The professor in my family has in particular seen no relationship between the two in their decades of teaching. But - importantly - a background of math can be a tremendous asset later in one’s career as a music theorist. It can help provide a unique way of looking at and improving on theory.
gjulianm 5y ago
I have a degree in mathematics and have studied music for some years. My personal opinion is that they're quite different things. Yes, the underlying theory of music requires some mathematics, but that's it. Once you get into it there's much more "feeling" and "intuition" and "subjectiveness" than math could ever accept.
- env123 5y ago
  ^ This. Maths ignore the stuff you mentioned above just to get by The same can be said about science ignoring philosophy, morality and spirituality just to get by

strong_as_oak 5y ago

whiddershins 5y ago

I love this style of clear, unambiguous writing starting with no knowledge assumptions. [but] The author is missing or skipping key pieces of information about why we use the notes we use, how they were derived, and why they sound the way they do to us. Edit. There was zero sarcasm in my comment. I really love this style of writing. The missing concept here is the harmonic series. A pitched note is not generally a sine wave, it is properly modeled as a series of sines waves combined. These sine waves go up in mathematical multiples. So say we have a note at 100hz. Simplistically it is actually 100, 200, 300, 400 etc all at once. So if 100 were A, 200 is A, 400 is A, 300 is … E, so 150 is E. Etc. None of this is arbitrary because the When the waves combine the result is more ordered or more chaotic, depending on the ratios of the waves, which our ears hear as consonant or dissonant. Best I can do from a phone.

chrisweekly 5y ago
It took a moment to realize your first sentence was derisive sarcasm. From TFA: > "Here is what I gathered, from the perspective of someone whose only music class was learning to play four notes on a recorder in second grade. I stress that I don’t know anything about music and this post is terrible. If you you so much as know how to whistle, please don’t read this you will laugh at me." ... which makes your comment even more uncharitable and unrealistically demanding. As if any mere blog post could answer those questions in meaningful depth. IOw, I don't understand the point of your complaint.
- whiddershins 5y ago
  I was not being sarcastic. The part about how the octave gets divided up is just missing some key concepts.
  - chrisweekly 5y ago
    Ah, ok, thanks for clarifying. (If your 2nd sentence began with "But, ", this would have been clear to me.) PS I love OP's writing style, too. Fearless learning in public is to everyone's benefit -- given proper disclosure/transparency.

ben7799 5y ago

He left off right where it starts to get really interesting. Most of what he covered is stuff you learn the first day playing and never really need to worry about again. The interactions of how different notes & chords in a scale draw our ears, how 7ths and other intervals work/sound. Modes, chord voicings, inversions, modulations.. all that stuff gets incredibly fascinating.

PaulDavisThe1st 5y ago
> Modes, chord voicings, inversions, modulations In defense of the article, I'd note that these are all choices you make within the framework set up by 12TET, not the framework itself.
iamevn 5y ago
(fyi, it's she)

whycombs 5y ago

Sorry, a lot of this is wrong. A major 7th is not 17:9 but 15:8. Our ear tries to represent all these rich resonances in as simple primes as possible (Think, combos of 2's 3's and 5's - and occasionally 7's in blue notes). A minor 6th isn't just a note in between, but a simple 4:5, The minor sound that we associate with sadness is simply the reciprocal nature of the ratio - the more complicated prime is in the denominator (as opposed to the overtonal nature of the M7 above) Short answer: read Harmonic Experience by W.A. Mathieu

mrob 5y ago
The ear does not care about ratios. Dissonance can be modeled using Plomp and Levelt's curve, which predicts the perceived dissonance of a pair of sine waves. Dissonance is minimum at unison, smoothly increases to a maximum at a narrow interval (dependent on absolute frequency), and smoothly decreases again as the interval increases. Sethares extended this model to arbitrary sounds by decomposing them into their sine-wave partials with a Fourier transform and then taking an amplitude-weighted average of the predicted dissonance of each pair of partials. If you apply this procedure to intervals using harmonic timbres (timbres composed of partials with frequency an integer multiple of the fundamental frequency), the dissonance curve has minima that happen to be at small integer frequency ratios. Harmonic timbres are most common in music, but there is an important class of musical instruments that are naturally inharmonic: tuned percussion. Unless substantial effort is put into designing them to approximate harmonic timbres, these sound most consonant when tuned outside small integer ratios. Indonesian gamelan music is a well known example of this. And with synthesized timbres you can have whatever inharmonicity you like, so timbres can be designed to suit any arbitrary tuning system. Non-technical explanation of Sethares' model: https://sethares.engr.wisc.edu/consemi.html https://sethares.engr.wisc.edu/consemi.html Full mathematical details: https://sethares.engr.wisc.edu/paperspdf/consonance.pdf https://sethares.engr.wisc.edu/paperspdf/consonance.pdf
- notahacker 5y ago
  > If you apply this procedure to intervals using harmonic timbres (timbres composed of partials with frequency an integer multiple of the fundamental frequency), the dissonance curve has minima that happen to be at small integer frequency ratios. In practice that's pretty much "the ear cares about ratios" isn't it? Sure, it's caring about the ratios because they align with harmonic timbres in non-percussive instruments, but various human cultures do listen to a wide variety of instruments and voice with harmonic timbres, and perceive melody and chords based around [reasonably close approxiaations] of simple integer ratios sounding more consonant. The implications for creating chordal music with synthesised inharmonic timbres designed to be consonant are interesting, but not widely applicable to what we actually listen to as music; and much of what we hear that goes outside simple ratios with the express intent of sounding dissonant...

abetusk 5y ago

I really don't understand all the hate in the comments for this article. This seems like a really fine introduction to a lot of the concepts. I had to scour the web a long time before understanding that the scales were constructed because of the simple/reduced ratio of root note to the others, which is clearly spelled out in this article. Music theory is, in my experience, typically taught as a list of facts to remember. Deriving it from first principles, insofar as it can be, is not common. This article is attempting to start that process. Some commentators are saying that the article doesn't address why we have 12 notes but I wonder if they even know why (one answer is because it's a good compromise between number of notes and that have simple/reduce ratios to each other [0]). I'm also skeptical of music theorists that can't even attempt an answer to basic questions about note length frequency, why some chords are "sad" or "happy" and other basic questions. It's difficult because music theory is a hodge podge of theory that's attempting to describe what's effectively an evolved language (with different music evolution for different regions), but there are some basic tenets that probably apply. I don't claim to have deep knowledge but there are a few key facts about music and music theory that are really obscure unless you know where to look. Any attempt at coming to music theory from a more rigorous foundation should be encouraged. [0] "Measures of Consonances in a Goodness-of-fit Model for Equal-tempered Scales" by Aline Honigh (https://github.com/abetusk/papers/blob/release/Music/measures-consonances_honingh.pdf https://github.com/abetusk/papers/blob/release/Music/measure...) EDIT: corrected spelling (thanks for the comments pointing it out)

Infernal 5y ago
> basic tenants Just in case it’s helpful - “tenets”
chrisweekly 5y ago
+1 / agreed. Also (tangent / tiny nit): "tenants" (inhabitants) -> "tenets" (rules)
criddell 5y ago
You seems to know a lot about the art and science of music so maybe you can answer a question I’ve had lately: what’s the difference between pitch and frequency? When a person talks about middle C on a piano, is that ‘C’ more closely related to pitch or frequency?

wildmanx 5y ago

This has about as much to do with "music" as analyzing grammar when talking about poetry. Musical notation, and lots of "rules" that come with it, came after the fact. Just like with natural language. And just like staring at grammar rules is a terrible way to learn to speak a new language, analyzing music "theory" at this level does not teach you anything about actual music.

Aidevah 5y ago
Agreed. The author mentioned that the desire to compose music is what spurred this post. It would be much quicker to read through a music textbook. The basics of western music notation can be taught in 15 minutes, and one could get used to it after an afternoon. There is no need to interrogate the origins and reasons behind conventional notation if creative work is the goal, since there are much more important concepts like harmony and voice leading which are more worth your time.
- trasz 5y ago
  If by "western music notation" you mean the clef and stuff, then lol, no. There's a reason most musicians don't use nor know it.
  - Aidevah 5y ago
    The reason most musicians don't use staff notation is because most musicians do not need it unless they want to engage with the literate tradition of music, and are not disadvantaged if they can't. Not because of any inherent difficulties in the notation itself.
    - zozbot234 5y ago
      > and are not disadvantaged if they can't Literacy is just as helpful in music as in any other artform. One can certainly play and create music without learning how to read and write staff notation, but being literate can only help!
- 5y ago

throw7 5y ago

"This has got to be some of the worst jargon and notation for anything, ever." Ummm, do you math? ;) My point of view of music theory is: sound is objective. music is subjective. where the two meet is music theory.

H8crilA 5y ago
Yeah, no, modern math notation is a lot cleaner than musical notation, and completely free of historic conventions. Pretty much the only thing "forced" into it by human preference is the standard of using base-10 numbers. Prove me wrong please :)
- KineticLensman 5y ago
  > and completely free of historic conventions > Prove me wrong please :) Some of the notation goes back centuries [0]. Our current representation of calculus reflects a convention that was hammered out following significant disputes in the early 18th Century. [0] https://en.wikipedia.org/wiki/History_of_mathematical_notation#Early_arithmetic_and_multiplication https://en.wikipedia.org/wiki/History_of_mathematical_notati...

PaulDavisThe1st 5y ago

On the question of "why 12 notes", I STRONGLY recommend this video by David Bennett: https://www.youtube.com/watch?v=lvmzgVtZtUQ https://www.youtube.com/watch?v=lvmzgVtZtUQ There's a really important step missing in the article which has been aluded to by a couple of comments already. It's the step between: 1. a series of notes defined by the harmonic series, that is, exact integer multiples (or ratios, if you prefer) of a base frequency, a physical phenomena found in nature and 2. the well-tempered scales used by most western music, which adjusts the ratios to allow certain compositional "tricks" to be used without sounding dissonant. The set of notes defined by (1) are typically referred to as "Just Intonation", and was the basis for most western music (and some non-western music) until somewhere between about 1400 and 1600 (lots of room for discussion/debate there). The "problem" with just intonation, if indeed it is a problem, is that if you define two series of notes (call them a scale, if you like) using these integer ratios but starting from two different notes, you will trivially find cases where "the next higher E" is a different frequency. So starting from (say) A and moving by integer ratios gets you to a different (say) F than if you start from (say) B. This means that if you are writing in a just intonation scale with (say) A as the root, you have a set of notes that are not actually the same frequencies as if your scale started on (say) B. By itself, this is not a problem at all - there is all kinds of lovely music written through history that works just fine with notes and scales defined this way. You just stay in the same scale throughout, and there are no issues. There are even some scale changes you can make that still work, you just have to know what they are (and they depend on the root note and the set of integer ratios you're using, so it gets complicated). But ... somewhere in the period mentioned above, a subset of western musical culture started to want to experiment with "modulation" - changing from one scale to another in the midst of piece. On continuous pitch instruments (e.g. violin, voice), this is entirely possible to do, since they can play any frequency in their range at all. However, it does require significant skill on the part of the performer, since the pitch of a (say) "F" will differ depending on the scale currently in use in the piece. The breaking point, such as it was, came with the development of fixed pitch instruments (keyboards). These can only play the notes they are currently tuned to, and so if the (say) F in two scales is a different frequency, you cannot play in both scales without retuning - an obvious impossibility in the middle of a piece. So, "well-tempered" tuning was developed - the ratios described in TFA. These are tweaked by relatively small amounts so that the notes are close to where Just Intonation (integer ratios) would have placed them, but not precisely the same. These small shifts mean that the (say) "F" is the same frequency whether your scale began on (say) A or B. You can now play a fixed-tuning instrument like a harpsichord or a piano, change from one scale to another, and everything remains "in tune". Of course, to ears used to Just Intonation, "well-tempered" tuning sounds out of tune. But in the west, most people (even within our musical academies) have grown up used to the sound of well-tempered tuning, and it is Just Intonation that sounds "off".

bannedbybros 5y ago

>Not counting the octave, there are seven fairly nice fractions here. Hmm. Seven. What a conspicuous number. There's seven nice fraction because of how they were rounded. But there's no explanation of why he rounds 1.682 to 5/3 but doesn't round 1.587 to the even closer 8/5. It's because scales are a social construction, not a result of vague math.

calebm 5y ago

My quick notes on music theory: https://calebmadrigal.com/music-theory-notes/ https://calebmadrigal.com/music-theory-notes/

srcreigh 5y ago

> I don’t know why twelve in particular has this effect, or if other roots do as well Here's a 31-equal temperament song. Microtonal music has a reputation for sounding bad, but it doesn't have to. https://www.youtube.com/watch?v=u-PEhbSOh74&list=PLC6ZSKWKnVz3lBogeQ66N8W3x_GwUhTFw&index=17 https://www.youtube.com/watch?v=u-PEhbSOh74&list=PLC6ZSKWKnV... > ... but it’s probably why Western music settled on twelve. The curious part is that Western music settled on twelve a long time before equal temperament. They used to have 12 notes but use for example quarter-comma meantone. Great mozart meantone piano piece (mozart never used equal temperament): https://youtu.be/lzsEdK48CDY?t=700 https://youtu.be/lzsEdK48CDY?t=700

jacquesm 5y ago
Interesting, that sounds like music from some kind of medieval castle that somehow got stuck in a timewarp. I can't listen to it for very long but it sounded a lot better than I thought it would.
hammock 5y ago
What am listening to? Long time musician here, just not familiar with untraditional scales. I assume 31edo but using Bach means you are ignoring 19 of 31 possible semitones- correct? Less? (I could imagine leading tones going in the opposite direction, or something) To my ear it sounds like there are way more major thirds than there normally would be in a Bach piece like this... anything else I ought to be hearing?
- srcreigh 5y ago
  I would guess you're right but I'm not 100% sure! This channel is pretty responsive, you ask on the video. This one is fascinating too - it's just intonation vs 12edo comparison. You can really hear all the beating in the harmonics in the EDO one. https://www.youtube.com/watch?v=AgB4TrV57AU https://www.youtube.com/watch?v=AgB4TrV57AU
- lewispollard 5y ago
  I haven't listened to the piece, but afaik typically transposing existing 12TET pieces into microtonal tunings are taking the intervals in the original piece and applying them to scales in the new tuning, with some creative license/taste to embellish where it doesn't sound so good. So it's not ignoring or limiting any pitches to my (limited) understanding, other than the natural way that pieces written in a particular key/mode will skip over some tones in favour of others for harmonic or melodic reasons.

fugalfervor 5y ago

Music was taught as part of the quadrivium (alongside arithmetic, geometry, and astronomy), so it has a long history of being thought about in mathematical terms. There's nothing wrong with looking at music that way. What is so interesting about music is its ability to turn mathematical beauty into aesthetic, artistic beauty. Sometimes the math is just there in the background: "This chord progression is based on certain physical facts related to the harmonic series. That makes it sound good, and I like it". Sometimes it is brought to the forefront: "All of this music appears to be generated from a short pattern of notes which is recognizable even as it is subjected to rotations, elongations, truncations, etc. That is beautiful." Certain types of music play up the lovely-pattern-ness of music, and some don't. Neither is better than the other (denigrate the art of Justin Bieber who dare), but you can probably guess which I prefer :) Consciousness, pattern-recognition, spatial rotation of abstract objects, natural mathematical beauty, poetry, drama, acting/performance, nonverbal collaboration, shared rhythm: music is a nexus for all of these things. Music is nothing short of awesome, in the truest sense of that word.

john61 5y ago
What is interesting is that master composers often broke the "mathematic" rules. Schubert, Beethoven or Chopin for example have sometimes very strange and unbelievable beautiful harmonics sprinkled in. Whereas mediocre composer do not do this, they are predictable.
- fugalfervor 5y ago
  I don't think it's helpful to think of music in terms of mathematical "rules", but more in terms of cultural variation on what is acceptable. Now this is my perspective, after spending a long time steeped in this stuff, but feel free to disagree: The history of music has been oversimplified and done a disservice by its characterization as "great men" breaking all the rules. Taken to its extreme, music gets better when a greater man breaks more rules. That's obviously a little silly if you ask me. Dissonance (a "broken mathematical rule") has as much a right to exist in music as consonance does. There is a constant tension between consonance and dissonance, and it is that dynamic tension which is exploited for affect or structural definition. So what we have in the case of Schubert, Beethoven, and Chopin is not so much the breaking of mathematical rules. In my opinion, they are just examples of expert judgment applied to the tension and release within an artistic work.

ngcc_hk 5y ago

Seems not knowing the struggle with the note system. The struggle eg ## which is not really the note above because the # and b is not perfect. Bath well pieces probably not in his understanding. Not really a music theory.

coldtea 5y ago

>If we say 440 Hz produces a note called A, then 880 Hz, 220 Hz, 1760 Hz, 110 Hz, and so forth will also produce a note called A. An important consequence of this is that all distinct notes we could possibly come up with must exist somewhere between 440 Hz and 880 Hz. Any other pitch could be doubled or halved until it lies in that range, and thus would produce a note in that range. That's inaccurate and backwards. 338 Hz is not in that range, and it can't be doubled or halved to fit in that, but it's a perfectly fine not. The range is more about tuning (which is a choice) than about 440-880Hz being something special. >All of these things messily overlap and create multiple conflicting names for the same things, because they’re attempts to describe human intention rather than an objective waveform. Nope, they don't overlap any more messily than 1+2 == 2+1, or in how 0.99999... == 1.

quasisphere 5y ago
338 * 2 = 676, which is between 440 and 880
- coldtea 5y ago
  Yeah, bad wording. 338*2 is in the range, but is nowhere near the 12 tones we use when A=440 (and wont be with integral doubling or halving). The property "within the range A4=440 A5=880" doesn't have any particular special musical meaning. The idea I wanted to convey is that 440-880 is just a convention, starting from 440 as concept pitch, but concept pitch can be totally different (and outside the range). 432 is a famous standard which has and is been used today, but also tons of others, even widely outside the range... (also: "but it's a perfectly fine not" --> "note").

andrewzah 5y ago

Ultimately a lot of things around music are cultural and/or arbitrary. It's like languages in that sense. Personally I don't find this sort of analytical study to be that helpful for actually playing or improvising music, but it is neat information. You don't have to clinically study grammar and language theory in order to be a great writer; it's the same with music. Something that's very important to note is that music theory and playing music are very different. Music theory allows us to understand what someone played, and lets us communicate to other musicians in a compact manner. Theory by no means imposes rules, unless you are specifically wanting to write music exactly like Bach or something. As a logical-minded person myself, it's tempting to really dive into theory and the root of things, but I've found that it's a trap most of the time. For getting better at songwriting or composing, I've found the best thing to do is transcribe the songs you like (or portions of them) and see what they do, then incorporate those ideas into your own playing. I also get the frustration around enharmonics (e.g. C# vs Db), but honestly it's not an issue in practice. I had no opinion until I started typesetting songs myself, and it just looks cleaner to sometimes have Cb or B# depending on the context. The circle of 5ths also neatly puts scales in the order of how many sharps or flats they have. Also, C# is not a commonly used key anyways. From a theoretical perspective C major and A minor share the same notes, but when I improvise I do find the distinction meaningful. Having multiple ways to think about things is helpful. There are other things too: C minor 7 is the same as Eb major 6. Or if you play a rootless voicing, the harmony could be ambiguous. Etc. Yes, our music notation system has some issues. But this article is overanalyzing it from an armchair. It's mostly fine in practice.

cousin_it 5y ago

The article says: 0 1.000 = 1:1 (unison) 1 1.059 (semitone; minor second) 2 1.122 ≈ 1.125 = 9:8 (whole tone; major second) 3 1.189 (minor third) 4 1.260 ≈ 1.250 = 5:4 (major third) 5 1.335 ≈ 1.333 = 4:3 (perfect fourth) 6 1.414 7 1.498 ≈ 1.500 = 3:2 (perfect fifth) 8 1.587 (minor sixth) 9 1.682 ≈ 1.667 = 5:3 (major sixth) 10 1.782 (minor seventh) 11 1.888 ≈ 1.889 = 17:9 (major seventh) 12 2 = 2:1 (octave) This table has many omissions and some errors. The way people usually play these intervals on just intonation instruments, minor second should be 16:15, major second can be 9:8 or 10:9 depending on context, minor third is 6:5, minor sixth is 8:5, minor seventh can be 9:5 or 16:9 (or 7:4 in barbershop), major seventh is 15:8.

kmill 5y ago
You might like the book "Equal Temperament Ruined Harmony." It makes the case that there is a long history of differentiating sharps and flats in western music (flats are just a bit sharper than sharps), while arguing that in practice the whole tone was subdivided into nine parts, with the major and minor half steps respectively 5 or 4 of those nine parts. I don't think the precise tuning system described by the book really happened, but it does collect evidence that "enharmonic" used to mean "finesse your intonation to make this sharp be that flat." In my own experimentation, the just intonation intervals are jarringly perfect, and I'd say that the 5:4 major third is too narrow. Has anyone checked that barbershop quartets are actually singing such narrow major thirds? My understand is that many studies have shown that string instrumentalists and singers, even when they claim to be using just intonation, play their major thirds somewhere between 5:4 and equal temperament. It's also extraordinarly difficult to really tune basic harmony justly. The ii V I progression is a fun one... if you're not careful you shift the tonic by a syntonic comma.
- tshaddox 5y ago
  > Has anyone checked that barbershop quartets are actually singing such narrow major thirds? I don’t know if anyone has checked, but why wouldn’t they want to? Surely the low integer ratios are best going to get the intended sound of barbershop harmony, unless there are other acoustic reasons why slightly changing the intervals would work better (something akin to octave stretching in pianos, although I’m not sure that sort of thing would be as relevant to barbershop harmony).

dataspun 5y ago

As an amateur musician (and amateur nerd) I found this article to be insightful in many ways. The author’s opening disclaimer renders most of the HN criticism moot.

toolslive 5y ago

Hold on. a C# and a Db are not the same note. They are a comma apart. https://en.wikipedia.org/wiki/Comma_(music) https://en.wikipedia.org/wiki/Comma_(music)

asdfman123 5y ago
Everyone knows the C# connects to the DB
commandlinefan 5y ago
Hm - maybe technically, but they are the same on any instrument you can play them on.
- rdlw 5y ago
  Except for voice [0], any fretless stringed instrument [1], any fretted string instrument with separate frets for sharps and flats [2], keyboard instruments that allow expression [3], keyboard instruments with split flat/sharp keys [4], and electronic instruments that allow unorthodox tunings without huge hassle [5]... I'm sure I'm missing some, but other than that you're totally right ;) [0] https://youtu.be/XnGqVIW51JA https://youtu.be/XnGqVIW51JA, https://youtu.be/mPZn4x3uOac https://youtu.be/mPZn4x3uOac [1] https://youtu.be/PqR36Drhn_k https://youtu.be/PqR36Drhn_k [2] https://youtu.be/iRsSjh5TTqI https://youtu.be/iRsSjh5TTqI [3] https://youtu.be/P14JcRyJCEI https://youtu.be/P14JcRyJCEI [4] https://youtu.be/7GhAuZH6phs https://youtu.be/7GhAuZH6phs [5] https://youtu.be/xVZy9GUeMqY https://youtu.be/xVZy9GUeMqY
  - toolslive 5y ago
    trombones and horns.

dang 5y ago

Discussed at the time: Music theory for nerds - https://news.ycombinator.com/item?id=12528144 https://news.ycombinator.com/item?id=12528144 - Sept 2016 (378 comments)

streamofdigits 5y ago

> That problem is that sheet music is terrible. Something that people with casual exposure to musical notation probably don't know is that before the standard 5-staff notation became ubiquitus there were quite a few alternatives, maybe the most interesting from a mathematical perspective being the Byzantine notation Broadly speaking, while the modern system focuses on pitch values and an elaborate (modulated) map from frequency space to physical space (paper), older systems used in the Byzantium used "deltas" or the first differences of pitch values. So you start with a base note (lets say C) and then you go +1, +1, -2 to indicate pitch changes (in semitones). This is quite well adapted to monophonic chant. This notation was never developed to cope with the complexity of modern music but its not immediately obvious that it can't be done There is no easily accessible exposition of this musical notatin style, this cheatsheet gives a flavor http://www.byzantinechant.org/notation/Table%20of%20Byzantine%20Notation%20Symbols.pdf http://www.byzantinechant.org/notation/Table%20of%20Byzantin...

honkycat 5y ago

I read eev.ee's tutorial on platformer physics when doing a little gamejam. Worked great! What a talented person!

navane 5y ago

Why did we settle on 12 notes? Rule 1 (also stated in the article): Doubling frequency gives us practically the same note. It is so pleasant together with the original note, that it is boring. Lets make a set of notes that sound nice together, and are more or less equally distributed between a note and its frequency double (octave). The next most pleasant note, is a note with 3 times the frequency, then 4, etc... (Why is this pleasant? Because its a close natural harmonic. When you vibrate a string with frequency 1, it will start vibrating with a frequency of 2 as well, but softer. This doubling in frequency is because the string will behave both as a string with length L, and as two separate strings with length L/2. Because the length is halve, the frequency is doubled.) Now, 3 times the frequency, is the same as 1.5 times the frequency. If we do the same from that base note, we get 1.5 * 1.5 = 2.25 times the frequency. You could decide that that 2.25 is close to 2, call it a day, and settle on a two note scale. Or, you go further. 2.25/2 = 1.125, so now we have the notes 1, 1.125, 1.5 and 2. Not really evenly spaced yet. 1.125*1.5=1.6875...: [1, 1.125, 1.265, 1.5, 1.687, 1.89]. Oh, 1.89 is almost 2. Let's call it a day and settle on a five note scale. (there are plenty of cultures with a five not scale folk music tradition). Note that the error gets smaller, it was 2.25/2 for our 2 note scale, it's 1.89/2 for the 5 note scale. Now it happens that the next smallest error is with a 12 note scale. But there's still an error, you don't arrive at exactly 2. The intervals between your notes are not even, and your last interval is all retarded. That's why people settled on "equal temperament", where all the intervals between the notes are made equal, and the number 12 looks seemingly arbitrary. I did the math, see table below. The first column are the frequency of the notes, with base frequency 1, where I multiplied by 1.5, and divided by 2 to keep the notes between frequency 1 and 2. The 4th column gives you the "error", how close is the last note of the scale, to the octave of the base note. The closer it is, the better, it means we have an equally divisible range of notes that all have nice (with ratio 1.5) intervals with each other. The arrows indicate where we could stop adding notes to our scale, because we arrived at a local optimum: a five not scale is small but has quite an error. The 12 note scale is quite elaborate, but reduces the error a lot. The next best scale would be a 41 note scale, with only a slightly diminished error compared to the 12 note scale. Freq. n/1 2/n min(n/1;2/n) 1.500 1.50 1.33 1.333 1.125 1.13 1.78 1.125 1.688 1.69 1.19 1.185 1.266 1.27 1.58 1.266 1.898 1.90 1.05 1.053 <-- (5 notes) 1.424 1.42 1.40 1.405 1.068 1.07 1.87 1.068 1.602 1.60 1.25 1.249 1.201 1.20 1.66 1.201 1.802 1.80 1.11 1.110 1.352 1.35 1.48 1.352 1.014 1.01 1.97 1.014 <-- (12 notes) 1.520 1.52 1.32 1.315 1.140 1.14 1.75 1.140 1.711 1.71 1.17 1.169 1.283 1.28 1.56 1.283 1.924 1.92 1.04 1.039 1.443 1.44 1.39 1.386 1.082 1.08 1.85 1.082 1.624 1.62 1.23 1.232 1.218 1.22 1.64 1.218 1.827 1.83 1.09 1.095 1.370 1.37 1.46 1.370 1.027 1.03 1.95 1.027 1.541 1.54 1.30 1.298 1.156 1.16 1.73 1.156 1.734 1.73 1.15 1.153 1.300 1.30 1.54 1.300 1.951 1.95 1.03 1.025 1.463 1.46 1.37 1.367 1.097 1.10 1.82 1.097 1.646 1.65 1.22 1.215 1.234 1.23 1.62 1.234 1.852 1.85 1.08 1.080 1.389 1.39 1.44 1.389 1.041 1.04 1.92 1.041 1.562 1.56 1.28 1.280 1.172 1.17 1.71 1.172 1.758 1.76 1.14 1.138 1.318 1.32 1.52 1.318 1.977 1.98 1.01 1.012 <-- (41 notes)

mtalantikite 5y ago

> More “fake” notes exist than E♯, too; I hear rumor of such nonsense as G𝄪, “G double sharp”, which I would rather call “A”. It starts making more sense when you're working with harmony and chords. The wikipedia example section on enharmonic equivalents has a nice example of a Shubert sonata that makes use of this [1]. [1]https://en.wikipedia.org/wiki/Enharmonic https://en.wikipedia.org/wiki/Enharmonic

zozbot234 5y ago
Yes, if you think enharmonic notes are pointless or "fake" you should look into the derivation of diatonic and chromatic scales via "stacking" fifths, and what that implies for modulation along the circle of fifths. Also how sharp and flat are often theoretically conflated with mi and fa, since both proceed by half-step to their neighbor note.
deleted 5y ago
[deleted]

adamnemecek 5y ago

I’m working on an IDE for music composition https://ngrid.io https://ngrid.io

dan_mctree 5y ago

When I listen to the three sinewaves, two of which are supposed to feel more similar because they a factor two apart, I don't actually get any such similarity effect. They all seem just about equally similar to each other, even when played together. Is everyone here picking this sameness up? I'm curious what kind of mechanic underlies those feelings of sameness and what it feels like.

whiddershins 5y ago
Its very hard to hear this with sine waves. Sine waves work to visualize the concepts but real instrument notes almost always are better modeled as a large number of sine waves with a mathematical relationship that sound simultaneously and in varying amplitudes that combine to create the timbre of the note. Then when those notes sound together we get combinations of all those sine waves together to create a very complex wave that sounds a certain way to our ear based on the interactions between all of these waves at once. So to visualize it or reason about it, my comment here makes sense: https://news.ycombinator.com/item?id=30363653 https://news.ycombinator.com/item?id=30363653 But to hear it you would be much better off with triangle or sawtooth waves or samples of actual musical instruments.
OscarCunningham 5y ago
Right. Octaves sound consonant because the harmonics of one note line up with the harmonics of the other. Without the harmonics you expect to hear no effect.

buzzwords 5y ago

I have always struggled to learn music theory. If anyone has a good source please let me know.

ben7799 5y ago

I guess I fell into the same trap as a lot of others reading this and commenting it and having the programmer side of my brain see one thing and the musician side see something else. Computer programs must follow the rules... the rules get made up first and the programs get built to conform to the rules. Music theory isn't like that. The music comes first. The theory gets made up to try and describe & understand what is going on with the music. It's not at all like a program where the rules & theory come first and you build something that fits into that. If you write a piece of music that doesn't conform to some kind of arbitrary rule in music theory that's fine. If your music is successful and important enough someone will modify music theory to explain why your music was successful and important. There is a cycle where you learn some music theory to help learn music but the theory just helps you understand what's going on, it doesn't force you to do anything a certain way. You can't come at music and think about it like you're building a computer program.

quaintquant 5y ago
> You can't come at music and think about it like you're building a computer program. I think the absolutionist take here is unproductive. I would argue a writing a fugue is analogous to "building a computer program"; to say little of 12 tone chromatic atonal music compositions that usually sacrifice 7 tone based harmonies for interesting structure, like palindromic reflection of ascending and descending musical ideas, and optimising for a high number of pitches in a melody. Additionally, 7 note diatonic music theory can be completely summed up with two statements: 1) tick-tick-tock-tick-tick-tick-tock 2) permuate all the things! Most common western instruments are literally built to only play 7 tone diatonic music theory. Look at a piano. It can literally ONLY play "semi-tones". It's impossible to explore music beyond the 12 note chromatic scale without first modifiying the physical construction of the piano.
- powersnail 5y ago
  The problem with coming at music like building a computer program is that, while it theoretically can work when you look at it from thousand miles away, no musician actually does that. It reminds me of when Dijkstra wrote that all computer science students should always start with manipulating symbols before touching a machine, and Knuth replied that nobody learned programming that way. There's no empirical data of such thinking actual work. What makes good music good is ultimately human ears. Fugues, for example, might seem to be programmatical at first glance, but get your hands dirty and compose one yourself, based on nothing but the mechanical rules. Doesn't sound good? It doesn't. It's like writing English based on syntactical rules. You'll end up with a music equivalent of "Colorless green ideas sleep furiously".

zoltar 5y ago

I really like 12tone's breakdown of pop songs. Here's an example with The Immigrant Song: https://www.youtube.com/watch?v=gwsfR5sJPbM https://www.youtube.com/watch?v=gwsfR5sJPbM

TuLithu 5y ago

Once you've understood the basics, get the Gradus Ad Parnassum by Johann Joseph Fux. A word to the wise is sufficient.

deltaonefour 5y ago

I wish there was a theory for why a certain set of sounds is music and why it isn't music. I'm positive I can follow all the rules according to standard "music theory" and produce sounds that aren't classified as music.

838812052807016 5y ago

Much of the confusion and misunderstanding is because when this subject is taught and explained, the teacher usually jumps from sine waves to music, where a musical note is thought of as a sine wave. But it is better to think of a musical note as a set of sine waves. Each sine wave in the set has a frequency which is an integer multiple of the frequency of the longest sine wave. These frequencies are known as harmonics. If we plot these frequencies on a logarithmic scale, they'll be evenly spaced. https://en.wikipedia.org/wiki/File:Spectrogram_of_violin.png https://en.wikipedia.org/wiki/File:Spectrogram_of_violin.png https://en.wikipedia.org/wiki/File:Human_voice_spectrogram.jpg https://en.wikipedia.org/wiki/File:Human_voice_spectrogram.j... Imagine a note as a comb, where each tooth represents the frequency of one of the sine waves. Now imagine a second comb where the teeth are spaced farther apart. Now imagine that you put one comb on top of the other so that the teeth line up. You'll see a nice pattern. This is the harmony of the combs. C4 | | | | | | | | G4 | | | | | | | | C5 | | | | | | | | E4 | | | | | | | | But these combs aren't only in an abstract frequency space. The cochlea inside your ear maps each frequency to a specific physical location where your ear hairs are tickled, forming a comb-like pattern. https://en.wikipedia.org/wiki/File:Journey_of_Sound_to_the_Brain.ogg https://en.wikipedia.org/wiki/File:Journey_of_Sound_to_the_B... Consider this chord composed of the notes G4, C5, and E5, which some of you may know as C 2nd inversion or G⁶₄. chord | | | | | | | | | | | | | | | | | | | | This chord will tickle your ears in many of the same places as the note C3. C3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | One way we can tell human voices apart is by the relative intensities of the harmonics. And the way we tell different vowel sounds apart is by the relative intensities of the harmonics. So our brains have lots of wiring for interpreting relative intensities of harmonics. https://auditoryneuroscience.com/sites/default/files/Fig1-16color.jpg https://auditoryneuroscience.com/sites/default/files/Fig1-16... (source: https://auditoryneuroscience.com/vocalizations-speech/formants-harmonics https://auditoryneuroscience.com/vocalizations-speech/forman...) So when we hear this chord, our brains are likely using some of the same wiring they would use for a human voice singing the note C3.