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Music theory for nerds (eev.ee)
766 points by hardmath123 on Sept 19, 2016 | hide | past | web | favorite | 378 comments

I'd like to elaborate a bit, in the same "for nerds" manner, on where Eevee seems to get lost a bit with scales and notation. He (she? not sure) calls the A minor and C major scales the same, because they contain the same notes. That's not an odd thought, but it's like calling sine and cosine the same because both functions contain the same set of values, in the same order.

The difference is phase. Basically, scales aren't just an ordered set of notes, they also have a starting point. This note, the note the scale is named after, is often the last note of a tune played in that scale, and often the first too (especially for more poppy tunes). So if you play Für Elise in C major, like Eevee suggests, the entire melody will be pitched 3 semitones higher than playing Für Elise in A major. And it'll sound awkward because you're supposed to play it on a minor scale.

Once you understand this, the whole notation thing makes a lot more sense as well.

Even though the key signature for a major and its relative minor are the same, that is only part of the story.

When you see that A Minor has the same key signature as C Major, what you are seeing is the "natural minor" scale. This is simply the notes implied by the key signature, and so is identical (but down a third) from the relative major. If you play purely in the natural minor, you are really closer to the Aeolian mode than a minor key.

There is also the "harmonic minor" and "melodic minor" which, in common practice period, are much more commonly used than natural minor. Harmonic minor has a raised seventh. This makes the dominant (V) triad a major triad which increases its need to resolve to the tonic (I). It is used for harmonies (obviously based on the name) for this reason, but it makes the step from 6 to 7 an augmented second (i.e. minor third) so its isn't used melodically very often. Which is why there is also a Melodic Minor.

Melodic minor is tricky because it varies depending on whether the melody is ascending or descending. Descending is easy, it is identical to natural minor. Ascending is similar to the harmonic minor with its raised seventh (creating what is known as a leading tone, i.e. a half step below the tonic that wants to resolve up to the tonic.) But the ascending scale also has a raised sixth, which eliminates the augmented second between 6 and 7.

I've never quite understood why you learn a melodic minor as being different depending on if it's ascending or descending.

Are there examples of this in music? Or is it just something invented to test young pianists in exams?

That's only in classical music. Jazzers play "pure" melodic minor both up and down.

Thank you for this. I learned to write natural, harmonic, and melodic minor scales when I was a kid, but I never knew this explanation. It makes perfect sense!

> they also have a starting point

Indeed. For anyone that is not sure about that, take a piano (like this virtual one[1]) and play a scale of all the white notes starting with C.

The play a scale of all those same white notes, but starting with A.

You should immediately notice that the second version is in a minor key (sounds "sadder").

1: http://virtualpianos.info/

Awesome example! Folks, do this if this stuff is new to you, it's beautifully enlightening.

Thanks, that's an explanation I haven't heard that way before, but that makes sense to me.

Follow-up question: how are phase of the scale and the piece of music synchronized? When I think in terms of a wave, I could Fourier-Transform it into sines or in cosines, or any other phase-shifted variant (a * sin(nx + const.))?

Is it always the first note of the piece of music that "anchors" the piece in its scale?

Does that question even make sense? :-)

Nice question, I think I can answer (although you'll soon notice that my understanding of this subject is limited as well)

First of all, this is a mathematically undecidable problem, as in that there exist tunes that could be in multiple scales. The extreme example is a melody that consists of only a single note. Ridiculous, but as a fellow nerd I'm sure you can see how I could call a single note a melody. So we have to lose the phase metaphor here a little bit.

Most songs "anchor" with the last note. If you really can't tell whether a song is in major or minor (songs in major are a bit happier, jollier; songs in minor a bit sadder and more melancholic), going for the very last note (or the last note of the chorus, if the song a chorus) is a very good bet. Find the core melody of the song, the thing everything is hung up on, the last note of that core melody is the ground note of the scale. This is a really safe bet.

There are, of course, exceptions, and then I think you'll need to do some mathematical analysis to find out which key the notes in the melody match the best to. I'm not 100% sure how this is defined but I'm sure people have researched it.

All that said, musicians don't really have this problem. After all, hardly any of this stuff was designed. The only genius piece of engineering in all this was the discovery that if you choose to 12 notes in an octave (and not 8 or 15 or whatever), you have a very versatile instrument that can play almost any melody at any pitch in a way that sounds pretty good to human ears (pretty good, not perfect, because of the all the ≈'s in the "Intervals" section of Eevee's article).

But the rest was discovered, not designed, just by fooling around. The analysis came after the music.

Musicians just start with a scale and then make the music they compose fit. This comes pretty natural to you with a bit of practice; most people have a pretty decent innate ability to hear which melodies "match" with a chord and which chords "match" with a previous chord. In all honesty, once you've understood things this far, I'd recommend fooling around with the easiest instrument or tool you know, rather than diving even deeper in to the mathematics of things :-)

> The extreme example is a melody that consists of only a single note. Ridiculous, but as a fellow nerd I'm sure you can see how I could call a single note a melody.

I got your single note melody right here:


I was half expecting the One Note Samba (which is sort of the pathological case for the key-finding heuristic skrebbel describes).

This gets pretty complicated and subjective.

Strictly speaking, a "note" cannot define a key. To determine a key, you really need at least 2 intervals, wherein interval a somehow resolves to interval b. Keys are just a mapping of relationships; how does each of the 12 notes relate to each other? If you just play a C major chord, you aren't really playing in the key of CMaj, you're just playing a CMaj chord.

For example: Play B+G, then C+G. This is (arguably) C major. The B note resolves into the C note. That's the 7th moving into the Tonic (#1). This is an example of a V-I resolution, the strongest possible. If you'd like more examples, look up "cadences". There are 'principles' for determining the key, but they should not be understood as proofs.

I think you're also asking if/how to change keys. This is not only possible, but often highly encouraged. There are two extremely common key changes in pop music: 1) Up one step: This is incredibly common and can be heard in "I Will Always Love You" (album version, right after the 3 minute mark when she starts singing the chorus again). Once you start hearing this, you can't stop. It's all over the place. If we're in the key of C, we'll just play a D chord with a lot of confidence, typically after a G chord. 2) Minor to Major (or reverse, moving from sine to cosine in the analogy): This is much more subtle than the previous example. Typically you'll recognize this as a change in 'mood' or 'feeling'. You'll find a lot of Am-Em changes, then the chorus will be a lot of C-G changes. The tonic changes from Am to C, even though all the chords and notes being used are still the same (although the duration of said chords will probably be different).

Most "classical" music moves around keys pretty frequently, in extreme cases multiple times within a measure. And then there are chords/sections in which the key is debatable if not indiscernible (I would argue that the intro to Smashing Pumpkins "1979" is changing keys every 2 measures, but I think there's also a strong argument for it being in a single key)

Thinking about keys in terms of phases is a good basic explanation, but ultimately phase is much easier to measure. Perhaps I see it this way because my understanding of music is much better than my understanding of physics.

Yes, I was trying to explain some theory to my sister and she had it in her head that "black notes" on the piano - or anything we call a sharp or flat - were "minor". It took a lot of explaining to get her to realize that a single note on it's own is in no particular key and no particular mode (scale).

That's funny because B+G, C+G to me says G major: a major third, then a fourth. Is the ear guided to keys by inversion, some inversions more natural or root-y than others?

I've been composing pop music for a long time without knowing stuff like this.

I can see the G major argument. That would be a I/3 into IV. My professor would have said that answer is wrong due to voice leading, the 7 to 1 is very powerful. And the last chord is C, in root, meaning that's where you have resolved. Had the second interval been G+B, it would make the GMaj argument stronger. Ultimately, it's really context. You're really just asking about cadences, very roughly translated means "how chords resolve".

B+G is a minor sixth, not a major third. If you invert it, it's a major third.

B+G followed by C+G is most likely going to sound like a cadence in C Major. You could claim it is G major only if you considered it unresolved transition to the subdominant.

>Is it always the first note of the piece of music that "anchors" the piece in its scale?

No it isn't. If you put any piece of music that is harmonious through a device that analyses the frequencies you'll find three main notes and those give you the key. In C-major for example the main notes are C, E and G. The section will be in C even if the first note is E for example.

You can do this with software. There's a nice video of this here https://youtu.be/xVmkIznjUPE?t=24s

or you can upload a recording to get the chords https://chordify.net/

You can indeed fourier transform the music. I imagine the above software does so.

My teacher always said "music is about tension and the resolution of tension". So in A minor if you play tense chords and then resolve them to A minor, you're in A minor. If you resolve them to C, you're in C major.

The phase of a sound wave doesn't play a role in the context of compositions or scales. It does of course play a role in audio in general though - in recording, and analog and digital signal processing etc.

they're not talking about phase of the sound wave but "phase" of the scale notes as an alternative way of describing what music theory calls modes

It's not even just that. They are only "the same" with equal temperament. If you use proper intervals then they are quite different.

I'm not sure this applies here. Equal temperament explains how an interval is in tune and does not determine what key a song is in.

Torrent-of-ions is correct. Equal temperament is a compromise in tuning (assigning pitches to notes) which makes the most keys sound ok. The intervals aren't perfect, but close enough. If you tuned a piano for perfect intervals in one key, those pitches would be too much off for another key.

See e.g. http://www.phy.mtu.edu/~suits/scales.html

Yes, equal temperament is basically a compromise of tuning methods. While it can help ensure that a piece which modulates into different keys is in tune (as close as possible), equal temperament cannot tell the key that is being played.

> He (she? not sure)

FTR the pronoun where gender is ambiguous is 'they' (It's also sometimes explicitly requested by nonbinary people).

From eevee's Twitter:

> she/they/he

I have a similar relationship with pronouns. My own perception of my gender jumps around so much that I don't even bother trying to figure it out. It usually leans in one direction, but I don't have many strong feelings about it.

> It completely obscures the relationship between the pitches, though.

It doesn't actually obscure the relationship between the notes -- it makes them clearer. For example, I see the notes C, E, and G on some sheet music, maybe with some accidentals on some of those notes. I know that I'm therefore supposed to play a C triad. Now, there are multiple kinds of triads, but once I know I'm supposed to play a triad, it's easy to use context to pick out which one I need (major, minor, diminished, augmented -- usually one of the first two). If I were supposed to play a C# major triad, though, and the written notes were (C#-F-G#) as opposed to what they should be (C#-E#-G#) then that's confusing because it looks like I should be playing an arpeggiated sus4 of some kind. So the written nature of scales on the staff engenders an understanding of the relationship between the notes. Basically we write things the way we do so that the people reading the music can more efficiently pattern-match.

> C major is identical to A minor, and I don’t understand why we need both.

They're not identical. C major and A minor have the same notes in their respective scales. But we say that a piece is in the key of C major when it resolves to the a C major chord at the end, and we say a piece is in the key of A minor when it resolves to an A minor chord at the end -- an important concept for reasoning about how a piece is supposed to be performed.

> C minor: C D D# F G G# A# C

Eb, Ab, and Bb, not D#, G#, and A#.

> This has got to be some of the worst jargon and notation for anything, ever.

It's really not. Keep practicing. It makes sense, I promise.

I hear a lot of people -- usually people who have not been studying music for very long -- insist that the system would be more logical if there were no accidentals and there were 12 notes with distinct names and the staff had a bunch more lines on it. I've never bought it. The notation of music isn't arbitrary, it's informed by experience and it works.

Reading this over again, it seems like the author's not yet fully wrapped their mind around "big picture" stuff like keys, chords, basic compositional structure, etc. It's an... interesting choice to write a piece proclaiming that musical notation is absurd when you're only a beginner.

Is there an article/book/video anyone can recommend that explains the advantages of modern staff notation in terms that beginners can understand?

In particular, I think everyone would agree that it's it's much harder for beginners to hunt and peck sheet music in modern staff notation than it is for a novice typist to type words on a QWERTY keyboard. (At least when the sheet music is written in any key except C major, since the configuration of black/white piano keys corresponds to C major.)

This prompts beginners to ask, "why is this unnecessarily hard?"

The general advice I see (especially in the comment thread here) is to just spend years practicing and then you'll "get it;" you won't just learn how to play, but you'll understand why staff notation is awesome.

Can the beginner's question be answered, except by saying, "uh, trust me, it's great, just keep practicing"?

I don't think you can explain it to someone who can't play music, because it's a notation for music. You could teach someone to play an instrument the old-fashioned way, person-to-person without any sheets, and once they understood music (i.e. what patterns of notes sound good or don't) ask them to come up with a way to write it down, and they'd come up with something much like staff notation. But that's a very labour-intensive way to learn.

Hunt-and-peck is always going to be hard on an instrument that supports multiple modes, because each mode only uses some of the modes. At my school music classes were taught on the xylophone/glockenspiel, which you configure for your piece at the start (putting the correct bars on for the mode/key you're working in), which is probably easier for beginners, but it's not a popular instrument (and nor is the harp, which is the only other example I can think of of that approach).

I think a lot of the favoring of staff notation comes down to what the instrument does well. A majority of acoustic instruments are monophonic(voice, wind, brass, many strings) and have some form of linearity in pitch, and the staff accommodates them optimally - the beginner's sheet music always starts out by accompanying the staff with fingerings for each note, to help you get some confidence that you are actually playing right notes, and then after that it's pretty straightforward to extend your knowlege to read more notes.

Chord structure just isn't even touched in most of these instruments, at least in classical performance, because they aren't capable of polyphony! That only becomes a topic as one starts moving into, e.g., jazz improvisation, where knowing how to recognize and play around a root is critical. Piano is exceptional here, since finger independence and chording practically define that instrument.

On the other hand, guitar tabs are a Big Thing in part because guitar chords transpose very well, so you don't have to learn too many unique fingerings to start accessing the others - and then if the song is defined as a sung melody plus rhythmic chord backing, as a lot of popular songs are, you don't need more than a tab and hearing the tune once to have a shot at covering it competently.

Guitar's 2D layout favors isomorphism - this is what gives it this extra power to transpose. Guitar is conventionally not tuned isomorphically, but some forms of accordion like bayan, and alternative keyboard layouts such as Wicki-Hayden, Harmonic Table or Janko, are fully isomorphic. In that case, you don't have to mode set or learn any unique scales or chord fingering per key: Learn any set of intervals(chords or scales) and you can reuse them in every key by moving your fingers over. This is wonderful for learning theory and doing composition, but it doesn't make the instruments a 1-to-1 replacement for similar instruments as in performance the distances are usually much smaller, fingerings can get tangled, it can be harder to find your place or maintain tempo, etc.

> Can the beginner's question be answered, except by saying, "uh, trust me, it's great, just keep practicing"?

Basically: not really.

Musical notation... notates a bunch of stuff, not only pitches.

Unless you have a basic understanding of all the factors involved in reading and writing music, you won't be able to understand the choices fully, and you'll have to accept them.

TLDR: music notation needs to express the five qualities of sound: pitch, duration, loudness, intention and timbre. You can't just look at pitch.

I'm working on a blog post now which I hope will answer that question (at least, in very general terms).

The way to learn the logic behind the scales in my opinion is to learn the theory of the early Western music, up to the Baroque era, that was around while the staff was being developed. That is: learn the seven modes of Western music (https://en.wikipedia.org/wiki/Mode_(music)).

Each mode is a diatonic scale (https://en.wikipedia.org/wiki/Diatonic_scale). Each mode has a particular combination of whole steps and half steps. Each mode has a "natural" key which can be represented in modern staff notation by all white keys.

In Gregorian chant, there are rarely any accidentals. Even by the Baroque era, while there are more accidentals, I would say the music is still very diatonic in nature.

Early music was more oriented towards just intonation (the whole integer ratio harmony mentioned in the article, which is what I would consider the more "natural" way of harmony). With just intonation, however, you can't just shift to any random key on the fly. Some keys sound nice and related. Some keys sound awful and horrible.

The article author made a statement: "If your music mostly relies on the seven notes from a particular scale, then it’s more compact to only have room for seven notes in your sheet music, and adjust the meaning of those notes when necessary… right?" I think that's an absolutely a correct way of framing early Western classical music, yes.

What makes staff notation harder probably is the corresponding rise in chromatic music.

The rise of 12TET instruments (equal temperament) like the piano, and the corresponding development of chromatic instruments (compare: the natural horn used in the Baroque era vs. the modern valved horn) allows for this sort of modulation at the cost of some chords being slightly out of tune. Modulating key signature all over the place is now possible. Yes, the author should realize that some composers do change the key in the middle of a song to change the mood. (For a nice pop example, here's the Temptations' "My Girl", which changes keys midway through -- https://www.youtube.com/watch?v=6IUG-9jZD-g)

Incidentally, the diatonic nature of pre-19th century classical music is the answer to the author's sharp / flat question as well. (http://jtauber.com/blog/2006/11/17/why_a-sharp_is_not_b-flat...)

This shift to chromaticism really isn't reflected in staff notation. In fact, I'm aware that some 20th century composers abandoned the use of key notation, ledger lines, etc. altogether, perhaps in part for this sort of reason.

Yes, I almost mentioned this point about A# and Bb on his blog but didn't. He gives an example of an A# major scale and it becomes immediately obvious that something's wrong. In order to write such a scale you either have to skip named notes (A#, C, D.. what happened to B?) or have a key signature that contains double sharps. That would be the "theoretically correct" way to do it but there are no key signatures with double sharps. So the correct name for the scale is B flat, which follows the diatonic major scale pattern of "wwhwwwh" and does not skip named notes.

But from his point of view (as any beginner) this is very confusing. They sound the same and they use the same physical keys on the keyboard, so what's the difference? There is no good answer to that except it satisfies the theory (the musical "rules") of western music.

I don't think the author was necessarily proclaiming that it's absurd, only that it seems absurd from his/her beginner perspective. He/she was very up front about not having much experience, so I think this is totally reasonable (and expected)

If a beginner can't immediately grasp musical notation, isn't that evidence that it is, in fact, absurd?

A beginner can grasp musical notation; its rules are simple and consistent, and the conventions are well-documented and reasonably straightforward. This blog post should be evidence of that: the beginner who wrote it clearly has the basics of notation down, and there are no inaccuracies at least with regard to the notation.

A beginner can't grasp the motivations behind the design of musical notation, and I don't expect a beginner to grasp the motivations behind the design of any complex system. That does not make the system absurd.

Maybe the notation is optimized for efficiency for experts, at the cost of beginner-friendliness. Maybe it's representing something genuinely hard. Absurdity is possible but not the only possible reason.

Ah, the hipster programmer attitude to learning, based on the idea that because something is:

a) hard (it takes more than a day to understand the complexities)

b) incredibly old

the community should declare it obsolete and immediately begin work on a replacement (preferably based on JavaScript).

HA! This made my day.

If a beginner can't immediately grasp mathematical notation, or a programming language - or a native language for that matter - is that evidence that it is, in fact, absurd? The musical notation has evolved (toward some local optimum) to be useful for practicing musicians and composers, at the expense of having a learning curve.

I'm curious, did you immediately grasp the syntax of every programming language you've ever learned?

Programming languages are a lot easier to grasp. ;)

The oldest programming languages are only 50 years old, whereas the oldest music notation is at least 4000 years old.

Most programming lanuguages haven't crossed any spoken languages, but modern music notation & terminology has been heavily developed by countries all over Asia and Europe.

This is a big reason why musical notation seems so weird at first, especially to engineers, because it is a legacy that comes from a different time, a different context, in a different language. The people who developed musical notation had different math, different logic and different musical motivations than we have now.

Think about this for a while and it starts to feel like a miracle that musical notation works at all, not to mention how well it works.

Programming languages were developed by people nearer to us in every way, and made to be logical and simple, so it makes sense that they're easier to grasp quickly.

Can't speak for the parent, but as far as I can recall, I did. Well, maybe not literally immediately, but to me, the syntax was generally the easiest part of learning a programming language. (It probably helps that they are often very similar to each other.)

The only things that were hard were forgetting to write semicolons after statements in C (I have previously mostly written Pascal), and C declarators — but even the latter were easy after I learned that "declaration reflects use".

Yes, but did you learn your first programming language immediately? Because that's closer to the author's situation.

Well, in our programming world new languages seem to come out every week, trying to find better ways to write programs. By this standard a new way to encode music is long overdue.

In that metaphor, coming up with a new way to encode music is like releasing a new programming language and then asking everyone to write their own compiler for it. Not impossible, but it would have to be a hell of an encoding!

Maybe not as many as programming languages, but new notations do appear time to time. It's just that none has ever got as popular as the traditional notation, I guess.

So if you can't read Italian after a half-arsed effort to learn it, is this proof Italian is broken?

Not as badly as English is broken.

I mean, it's generally agreed that English as a language has accrued so much historical baggage that it's shed any elegance or coherence it may or may not have once had. It would honestly be surprising to me if the same thing didn't happen to musical language.

> it's generally agreed

Buy people who know nothing about how languages and linguistics work, I can only assume.

well a beginner can't really grasp shorthand or kanji either, right? A beginner not being able to get things means that the learning curve is higher.

Most musical notation is meant for non-beginners, right? And a lot of the beginner confusion is from features that are useful for advanced users. So there's a tradeoff.

There might, of course, be changes to be had. But a lot of beginner confusion is because the higher-level abstractions are needed.

A good example in mathematical notation is order of operations. Why not just do left to right? Or just going with Polish Notation?

Standard order of operations work well for people working with large formula, because they allow you to write many common things without many parentheses. But mandatory parentheses + left-to-right only would be easier for beginners.

If you are mathematically literate but a beginner in music most texts on music won't tell you the simple stuff in here. And most music students can't tell you what an octave means (i.e. frequency doubles) besides something wooly like "(a C an octave up from another C) is the same note but a different pitch".

This text is what it claims to be: music theory for nerds (just not necessarily music nerds!)

Yes, it's like some of my students who wonder why I won't let them use a GOTO in their programming assignments.

I tell them that, if they are using assembly, then it is fair game.

It can also be worth it, in very specific limited circumstances, when you're emulating constructs not present in the target language.

Emulating a multi-level break statement and error handlers come to mind as examples.

Not to mention that A minor is generally considered to contain two additional notes compared to the A natural minor scale (AKA the A Aeolian mode).

Well, two _or_ one. The melodic (two) and harmonic (one) minor scales.

Sort of. They are separate scales, but if you compose (classical-style) music in a minor key, you don't typically pick one of those scales. The chords that appear might include any of those notes, even within individual phrases.

In my totally non-expert observation, contemporary music often is more modal, in the sense that it probably would just pick one of those scales (particularly natural minor) and stick to it.

In jazz A minor generally implies A Dorian

Yeah, in my reading, it never seems that there was a compelling reason to chose Aeolian vs. Dorian as the definitive "minor". I think it's largely because common practice evolved to much more frequently approach the tonic from below instead of above. That would favor harmonic minor over Dorian. I suspect that this is because it allows for very similar cadences as major keys.

I think a lot of the time (in a jazz context) the chords are approached as almost individual elements or as part of a small "local" progression instead of in the context of the global key of the song, since the key could be changing as often as every couple bars. In the case of a common minor progression: B-E7alt-Am7 the major third of the dominant chord is the same note as the major 7 in the harmonic minor scale of the target minor chord, which would be one option for playing straight through that progression, but you'll often find that players will use either A harmonic minor, diminished, E altered, G# whole tone, etc. over the V7 and resolve to dorian or aeolian or something with a b7. It doesn't make much sense to analyze that progression in a classical context since those chords aren't entirely contained within any single key, and you end up something that looks like a key change, or has B as a secondary dominant implying a key change to the E7, where a jazz player would immediately recognize it as a minor 2-5-1

That's wild. Jazz blows my mind!

I've heard it said that it's fundamentally actually very tonal, in the sense that the underlying progressions are typically 2-5-1. Which has always been hard for me to imagine :), but I guess it simply dispenses with the slavish dedication to a particular scale of the tonic. "A Geometry of Music", which I've linked to elsewhere in this thread, digs into jazz practice a lot.

> we say that a piece is in the key of C major when it resolves to the a C major chord at the end

Can you ELI5 that for me, please? I'm very interested to have a better understanding of that concept.

Broadly speaking, major chords sound bright and happy. Minor chords sound dark and sad.

The C major scale consists of C-D-E-F-G-A-B-C. The A-minor scale consists of A-B-C-D-E-F-G-A. Those are the same notes, but if you play each of those patterns on a keyboard, the first one sounds happy, the second one sounds sad.

A C-major chord (technically, triad) is made up of C-E-G, the first, third, and fifth notes of the scale. Again, sounds happy. The A-minor triad is A-C-E ... sad.

But if you play a melody -- that is, one note at at a time -- it isn't always clear whether it's happy or sad. In most western music, though (including virtually all pre-1900 classical music and the vast majority of modern pop), the piece will end ("resolve") with a clearer "happy" or "sad" type of chord. That final chord is what determines the key.

(In a huge amount of classical and popular music, the final chord is the same as the opening chord, but not always. When they're different, the final chord tells you the key.)

> the final chord tells you the key

That makes it sound like a definitive rule, but it's just a common convention.

It is just a convention, not a rule. It is very common in classical music for a piece in a minor key to end on a major chord (called a Picardy 3rd) or in some cases to end on the dominate 5th - which is a major chord. The latter though, isn't usually the absolute end of a piece because it leaves you hanging (rather like ending "Happy Birthday" on the word "to" - and leaving off the "you").

Resolving to a chord just means that the music sounds "complete" when it hits that chord. In other words, it wouldn't sound weird if the song ended right there.

For example, if you hear "Happy Birthday" played in C, the final chord is C major, and the song sounds done. If you heard it with the final chord changed to something else, it would sound like it was leading somewhere, and you'd expect another verse or a bridge or something to follow.

Just as it can be said that music requires at least 3 consecutive notes or beats to have a tempo and a rhythm, it can also be said that it requires a "cadence" which at bare minimum is two consecutive pitched notes but usually at least three.

C-G-C is a simple 'harmonic' cadence. It is, for example, the basis of oom-pah music (think military marches which repeat C-G in the bass with a melody on top, and end on C when the melody is done).

There is a form of musical analysis (Schenkerian) that can be used to show that nearly all music has the basic form C-G-C, which is usually expressed as I-V-I (chord I is C major and chord V is the 5th chord of C major, which is G major). It is of course nowhere near as simple as I've expressed here!

So resolution is basically the cadence of the piece's harmony, much like a story has beginning, middle and end, so does harmony.

The reason chord V and chord I are so powerfully related is that the 3rd note of V is the leading note of chord I and the 7th note of V is the 4th of chord I. This is actually significant to understand because it is how one can see the purest relationship between harmony and melody. In early music (eg medieval plainsong) the fundamentals of more complex harmony were first developed from melody.

In the case of C major with two voices, the notes would provide resolution from notes F and B to notes E and C. The FB is a tritone, which is dissonant and wants to be resolved in the human ear. The EC is the major triad of C major, which is harmonious and consonant and resolves the dissonance of FB.

Many musicians much better than me are surprised at how I can play a song just by hearing it on the radio. My breakthrough came from understanding music was realizing that the real “meaning” of a note lies in its position relative to the tonic note (e,g, I-II-II, etc, also written do-re-mi). Suddenly, almost all of the clutter was removed, and the problem became manageable.

Let's consider the three-note tune “do, re, mi”. If that tune were played in the key of C, it would become C-D-E. If it were played in G, it would become G-A-B. But in either case, it's the same tune but with each frequency increased by the same percentage.

Trying to understand music by understanding the letters is like trying to read in a world where every article has been enciphered into a different “key”: e.g., the word "cab" in “the key of A” (the alphabet we normally use) would be written as "dbc" if the article were written in “the key of B”. In the latter case, you could discern meaning only once you realised that the letter “d” represented the third letter of the alphabet. There's nothing meaningful about a “d” but there is something meaningful about a “4th letter of the alphabet”.

Once you start to “decipher” all music into I, II, III, IV, V, etc., the complexity becomes manageable. You can start to learn to recognize the sound of a III note, or of a VI minor chord. After all, there are only eight notes in the major scale.

I'm not a good musician by any means, never had music education apart from the primary school which was abysmal. I can't recognise pure tones (just the intervals). I still can play any song I hear on guitar or keyboard "good enough" so that people have fun singing to it.

The huge reveal to me was the same - notes doesn't matter - the intervals make the song recognizable. People change notes all the time when singing (jump octaves, start again lower to adjust to others, etc).

So on amateur level it's really just starting on random place on keyboard and guessing which note will sound "right" after that. Everybody hear if the next note is higher or lover, so it's just "was that +1, +2, or +3?" Usually you can guess, if not - start again. Very easy and makes playing instruments so fun.

I never understood why they bother kids with these complicated drawings and hashes and be-mols, if they could've just wrote all songs as "start at this note, and jump by +2, +3, -5, ...".

Some notation systems do exactly that. For example, the notation used for byzantine chanting: http://www.byzantinechant.org/notation/Table%20of%20Byzantin...

Many (but not all, or even the majority) music classrooms around the world teach students fluency in solfege (do-re-mi), and everyone I've spoken to about it agrees it helps out a lot in the way you describe.

That's interesting from a personal perspective. I was taught music in the UK where that is specifically not used (or at least wasn't when I learnt).

I took a year out of my Music degree to attend the Sorbonne. French music education places a heavy emphasis on solfege. When I started going to their undergraduate classes it was immediately apparent that the level was several years behind that of the UK (in classes for composition and orchestration most noticeably). To attend classes dealing with similar material to what I was used to as a UK undergraduate, I was attending Post-grad courses. Having just completed my first year on a UK BMus course it was quite an eye opener to see 19yr olds learning material I had been taught at 16.

I wonder if the UK system is better at sieving natural talent whereas the French system is better at teaching? Personally I hated music (in the UK) because we never seemed to get taught anything, we we largely expected to just know things or magically learn through awkward repetition.

Knowing what I know now, I think there are a lot of ways we could have practised music early on that would have helped those of us not born with perfect absolute pitch. Most people have perfect relative pitch (afaik I fall into this group). Perfect absolute pitch and tone-deafness are both quite rare.

It sounds like the French system is optimised for the majority, forcing everyone to practise interval differentiation, including those who don't need it (and the small minority who will never be able to do it).

Without trying to cite what currently happens in the UK, I think it is fair to say that kids are handed instruments like the recorder at a fairly early age, and those that do well are encouraged to progress. That's a sieving process for sure.

When they get to secondary school, music as a subject is most often just a minor inconvenience in the curriculum to most pupils, and those who have ability are pushed into learning flute/violin etc (at additional cost to themselves and outside of the timetable). In the course of going through the grades of music (performance exams), kids are taught aural skills and theory (grade 5 theory is required to take higher performance grades). This results in a select group of instrumentalists that have learned intervals, harmony and scales practically. Whether any of those skills are useful to a non musician is debatable, so one could say that it is the most efficient way of getting a rounded skill set into the brain of a musical 15-16 yr old.

The French system would, I agree, produce a broader spectrum of musically able people, but in practice it results in a lower level of specific and important knowledge. The UK system produces more complete performers whereas I would say the French system has large gaps which then get filled in at degree level.

Perhaps things are greatly different these days. I know for example that studio production is an option for A level music, and there is absolutely no musical theory knowledge required to produce a studio track. I wonder if A level students are even taught basic 4 part harmony any more.

Music, to an individual, has always been a matter of ability, discipline and perseverance to practice. In education, the solution to nourishing those qualities is never going to be perfect. I do recall the French students I was with were quite annoyed that my education was years ahead of theirs, but with perspective, I'm sure it didn't really matter then, and it surely doesn't now.

It's also interesting to compare and contrast the apparent results. The UK has a fairly long history of producing a far above average number of world-class musicians. France on the other hand seems to have a broader musical culture, or at least, so it seems in Paris during the Fête de la Musique.

I'm hesitant to afford any particular value to educational practices in this case. Let's take the specific musical skill of composition, which I believe is at the root of musical traditions in any country...

The French have a much richer and deeper musical tradition. For example the 'blood lines' of Renaissance Troubadours and Trouveres, or the French Operatic Style, to name a couple. English 'classical' music tradition had a golden age which had Tallis, Byrd and Dowland but kinda stopped with Purcell and didn't really flourish again until Elgar. Elgar himself was writing in an identifiably English way, but his language was very much based on the Germanic tradition (which been imprinted on the English style by the likes of Handel,Haydn,Schumann and Mendelssohn).

All that time, let's say broadly 1700-1900 the French were much closer to their own 'cutting edge' of musical development, although the Germanic style was still very much dominant throughout Europe. What the French had, was a progressive heritage that had somehow been preserved - Ravel and Debussy (the impressionist style) could only have come from France, which I think would relate to your reference to a broader musical culture.

Of course by the mid 20th Century, in Classical music at least, the French started leaping ahead of the UK again, with Stravinsky, Les Six, Satie and numerous others building a significant new tradition that still exists today through the legacies of Messiaen and Boulez. England had Britten, Vaughn Williams and few others of note.

I think the reason you see a broader base in French 'national' music (for example at la fete) is as much down to the fact that England produced and still does produce exemplary pop/rock with a worldwide market. The French pop culture is insular and that's a good thing IMHO because it maintains integrity and does not to attempt to compete in a global market that is pretty much a cultural vacuum these days.

As for the standard of professional performers, I think there isn't too much difference in numbers produced or quality. Being a pro musician is very hard and the attrition rates are not down to which country you make your career in.

I'm a very good musician and have never thought about it exactly that way before but I have to say, that's a very good explanation and way of thinking about it. I think it's a bit abstract for someone trying to learn to read music, but it's absolutely correct.

The gist is correct, but it's worth noting that, for example, if you played do-re-mi in E it would do E-F#-G#, not E-F-G, to preserve tone/semitone order. That's where the key signature comes in.

And that shows how WTF the musical notation is.

o.O why so? And what on earth do you propose?

C is 0. Each half-tone up is +1.

If you really want to keep it concise you can write it as base-12 numbers.

    C1 C1# D1 D1# E1 F1 F1# G1 G1# A1 A1# B1 C2 C2# D2 D2# E2 F2 F2# G2 G2# A2 A2# B2 ...
    00 01  02 03  04 05 06  07 08  09 0a  0b 10 11  12 13  14 15 16  17 18  19 1a  1b ...
First number is octave. Second number is half-tone in that octave. Translating is just mechanical addition.

EDIT: on second thought making it base-12 just to save some space makes no sense, people are good with base-10, just keep the numbers.

It looks better on paper for an engineer, but not for someone who actually plays an instrument and reads the notation. With most western music, not all 12 pitches in an octave are used most of the time, but only a subset determined by the key and scale. Although the currently used notation may look weird for a newbie, it takes just a quick look at the key signature and you know which pitches will be used in a piece of music. When you know the scale (and practicing scales is just a standard part of learning), then "decoding" a note by counting tones is much easier than counting individual semitones (12 seems just too many). After a little practice you get it intuitively and you really don't count; you just know where each tone (or chord) is in a given scale and what function it has. And then when you suddenly see an additional flat or sharp symbol before a note, you know that this is an out-of-scale note, so it is also easier to play it. Disclaimer: I'm an engineer.

That completely misses the point. Without wanting to sound brash, you know jack about music and you should act accordingly, that is, don't spout ill informed suggestions when you clearly don't play an instrument or studied this mater well enough to give an informed opinion.

Scales have 7 notes, not 12. A musician plays music in a scale, they aren't a computer outputting pitches, they are a person playing notes. Music notation has a reason to be this way: notes in the scale don't have flats or sharps next to them, accidentals do. Reducing everything to a number describing absolute pitches is the right thing to do for a computer to play (see midi). It's not the way to go for a person that actually has to understand the logic and patterns in the music.

What would D Flat be, 0.5? what about D Double flat?

Db would be 01. Dbb would be 00. (or 11 and 10 for the octave)

The point ajuc is making is that the flat-sharp accidentals aren't used or needed at all if you just assign numbers to each tone. There's no concept of flat or sharp, unless you want to deal with microtones.

The letters and flats/sharps give you key/value over frequencies, which is better than just a numeric index over a chromatic scale. Working with the keys allows for the same abstractions to be used with all 12 keys at the same time, on the same staff.

I'm not really arguing one way or the other, just pointing out that they're functionally equivalent. It's two different maps keying to the same set of frequencies. The BASE12 system described above would allow for keys as well, just with a different notation. Instead of flats/sharps marked next to the clef in traditional sheet music, the BASE12 system could start each line with a list of 'prohibited' notes. For example, when indicating that a piece is in the key of Gm, the staff in BASE12 could start with : [01,04,06,08,0b]. This indicates that the majority of the song will be made of the notes 7,9,b,0,2,3, and 5, and serves the same purpose as having two flats next to the treble clef, one on the middle line and one in the top space. Which of these systems would be easier for humans to grok is up for debate. I personally don't think either one is better.

Thinking about it, the traditional notation is just mapping to an octal system, with the key accidentals acting as modifiers to the map and the base-8 values being displayed graphically as vertical position on the staff.

You need to stop thinking about things in engineer terms. The scale going 7,9,b,0,2,3 instead of G,A,B,C,D,E,F with B and E flat may be functionally identical, but one is sure a better representation of the actual patterns in the music than simply referring to a number proportional to the log(frequency).

I started music with classical piano and only recently started transposing instrument (bamboo flute), and am trying to get used to movable do solfege. I'm still easily confused when key change occurs or borrowed chord appears. (For the latter there are limited patterns so I should remember them, I guess.)

@karb: mind if I shoot some questions about the "decipher” all music into I, II, III, IV, V" part in email? Thanks.

Indeed, the scales are isomorphic.

This has got to be some of the worst jargon and notation for anything, ever.

Indeed. I'm a musician, and something non-musicians often ask (especially techies, it seems) is why we use such an archaic notation system.

The reason is simply that a certain number of musicians have developed the skill of sight reading which is the ability to perform a composition directly from a written sheet, with little or no rehearsal. Those players, myself included, can't quite explain how we do it, and aren't going to learn a new notation system.

It's not merely sight reading that relies on the existing notation, jargon, and theory.

Every element of Western music builds on the same building blocks. Western harmony, as taught, relies on understanding chromatic and diatonic harmony; the author did a nice job figuring out and explaining the compromises inherent in equal temperament (without knowing about how ratios and tuning worked before equal temperament; people didn't always fudge the ratios to make them work out the same across all keys), which is cool. But, he still doesn't actually know much about western musical harmony, as evidenced by the assertion that C Major and A minor are "the same", because they share a key signature and the same notes. Had he known that equal temperament is a moden-ish invention, he might have also figured that C Major and A minor actually have (slightly) different notes if your instrument is tuned specifically for that key rather than with equal temperament.

In short, he's just not done learning yet. Most musicians and composers never really are, as it is a vast subject. And, what he's calling "music theory" is really more "equal temperament tuning math" with very little theory.

Not to say it's not interesting and well-presented. It may be useful for non-musical nerds to see it presented in this way.

Very important historical details that are missing completely from all music theory textbooks I have had the misfortune to read. They only give the rules, without explaining the reasons. Also, imagine using notation to write down, say, electonic music. Drums? Maybe, but which notation? Filters, effects? ugh

As a percussionist, yes, musical notation is used for drums. However, it ends up looking rather like a step sequencer with some standard notational effects (staccato/legato, rests or varying lengths instead of the absence of markings, etc.) This helped me out quite a bit later when I started getting into digital music.

I did learn to sight-sing in high-school with a lot of effort (but I sucked at sight-reading on my main instrument, the piano) and I agree with the article writer.

When you read these sentences, you can very easily speak them out loud, you don't have to mouth each and every vowel to hear it in your head. This is not the case for music notation, even for very experienced musicians. The clue is that they associate notes not with sounds or intervals, but with absolute positions (fingerings, etc.) on their instrument. Maybe they can sight-read a melody with a little effort, but for most of them it will be much simpler to just play it - and this goes triple if it's not tonal and idiomatic for a style you know. Music notation is really a somewhat instrument-independent tabulature.

Solfege or numeric notation systems, which exist in many versions, are much better for reading in your head once you learn them. They're terrific for singers. But for instrumentalists, usually your instrument is already biased to certain keys, and going from notation->sound in your head->fingers would therefore be much harder than going straight from notation->fingers.

This is true for me. I can go from notation to my fingers, faster than I can go from notation to singing a tune. It's a hyper specialized skill. I can sight-read treble clef on flute, and bass clef on the double bass, but it's hard for me to sight-read treble clef music on the double bass, even though the transposition of a couple octaves is no problem at all.

You can almost guess what instrument the composer played because of that bias. If you see a violin piece with 6 flats, it's a pretty good guess that the composer didn't play violin.

> The reason is simply that a certain number of musicians have developed the skill of sight reading ...

It's moer than just that though. The inertia to overcome also includes pretty much all sheet music ever printed, all the schools that have adopted the current system, all the educational materials... etc. etc. the list goes on.

In addition to all of that, the new system needs to convey most of (if not all) the information that the current system has. I have seen a few attempts at improvement, but they all fell short of the current system, which let's face it has been in development for several hundred years.

Once we get off printed paper, computerized notes should solve much of the inertia. problem

Displaying a song in whatever notation the musician/reader prefers should be as simple as setting a preference on your hyperPad.

That still leaves the difficult task of figuring out better notations, but it can be done incrementally. You don't need to convert the whole world or the entire notation at once.

> Once we get off printed paper, computerized notes should solve much of the inertia. problem

I don't know...ebooks still look a lot like treebooks.

The one example I've seen of what you're describing is I've seen jazz musicians play off ipads. This give them access to very large catalogs without lugging around giant binders, and also they can transpose their sheets into any key.

(That's more important than it might sound at first, because different instruments "play" in different keys -- if a pianist thinks the piece is in C major, the clarinetist thinks the piece is in Bb major. So now you don't have to have separate books for the different instruments in the band.)

But the music as written down in these electronic fakebooks is a lot less complex than your average classical piece, so it's a much more tractable problem.

I have those electronic materials. They come in two varieties. There's something called iRealB, which contains thousands of tunes, but is just chord changes due to a quirk of copyright law -- the melody and lyrics are copyrighted but not the harmony. That's the one where you can transpose tunes. But you don't get the melody.

Then there are large PDFs where somebody ran the old paper fake books through a scanner, but they are just images and are not in an actual computer readable format, so they can't be transposed.

So the computer has not solved the notation problem, yet.

Myself, I've memorized most of the standard jazz repertoire.

In fact I suspect that most of the written music repertoire that exists today will never be translated into computer readable form, because it's just too much work. And not enough new music is being composed to form a critical mass around some new notation system or computer format. When somebody composes a tune (I play in one band that does original jazz compositions), they send out a PDF.

Maybe software will eventually automate the process reliably enough to be useful.

Notation would tend to be one of the easiest things for which to develop automatic recognition / parsing?

It should be, but there's the problem of demand. Somebody has to be motivated to do it, meaning that they are probably equally passionate about music and image analysis.

Also, a lot of the written stuff is handwritten, not typeset. So it's a subset of the handwriting recognition problem.

It's pretty bad for the bassist, since the bass part is usually the second to last part to be copied, meaning that the copyist was probably drunk. ;-)

Just a nitpick, but the transposition is the other way around. Concert B flat is C for B flat instruments.

Argh. In my defense, I play a C instrument. :)

>why we use such an archaic notation system.

The answer is, because it works, and nobody has managed to propose a better one.

> nobody has managed to propose a better one

A new notation system will need to be a lot better to justify the change, because there is also a lot of value in compatibility with everything that already exists.

I'm not sure a sufficiently better system exists, because as you say, the traditional notation works. It has its quirks and rough corners, but music is complicated enough that any system would probably have similar imperfections.

One attempt was Hummingbird. http://www.hummingbirdnotation.com

Which in durations are represented spacially, which in my opinion has two negative effects.

The first is that spacial recognition takes more effort than symbol recognition, because it's comparative.

The second, and more important being that complex sequences of notes will be very dense on the page, and simple sequences of notes will take up a lot of space on the page, so suddenly there is a tradeoff between having sheet music that doesn't take 10 pages, and having enough space to represent hemi-demi-semiquaver sequences when they inevitably appear somewhere.

Yeah wtf. Like I don't think current notation is perfect by any means (and I am a PLer, I love new representations) but this and every other replacement I've seen blatantly sucks. ....As with so many things, know what your disrupting!

[In this case, I'd like to say go apprentice engraving if you are 100% serious.]

With Hummingbird's inclusion of a trailing line as part of the indicator of note duration you also have to read ahead in the score and then jump back if you want to use the trailing line to identify the duration, which is a lot harder than just identifying by local information in the form of a set of note tails and whether or not a note is filled in. While there are additional parts to the glyphs for half and whole notes, these aren't the dominant part of the symbol.

Another question is whether or not the length of a trailing line is absolute or relative to the bar itself.

On the subject of filled notes, Hummingbird is also conveying a lot of information that for performance of a score is useless. Note letters (A-G) aren't actually important for performance, only the action or position that they map to for each instrument. No musician parses a score and translates each note to a letter and then each letter to an action, instead going directly from note to action.

Essentially, telling you the note letter with a glyph shape on top of the position on the stave is adding noise to the signal.

I'll admit I'm coming at it from a position where I'm perfectly comfortable with traditional notation, so part of the reason that it appears difficult is simply because it's unfamiliar, however the terseness of traditional notation and ability to read in one "parse" without forward- and back-skipping seems to give it the advantage.

That and over 300 years of existing music, too ;)

Your second point seems pretty valid, and to address your first point:

"There are multiple cues to the same information. Everything has both a symbol and spatial element, for all kinds of thinkers."

Indeed, in addition to the spatial length of the notes, there is also a symbol next to the notes denoting their life. You can see this on the linked page, in the second section. (Next to "Intuitive.")

Try writing hummingbird notation by hand though!

Their pages claims: "It’s quick and easy enough to write with an unsharpened pencil. You can scrawl it on a napkin in a pinch."

I don't know if it's true or not, as I view Hummingbird to be fixing things that aren't broken (is the difference between a whole note and a half that hard to suss?) and doesn't fix the things that are.

I think the more difficult part writing by hand would be the redundancies -- not only do I have to know which line to put the note on, I have to know which symbol to draw. Which could be tough, as it forces the composer to be consciously aware that e.g. "this note is D#" rather than "this note should be two steps above the previous note in the current key".

The thing I most want to change is how time signatures are notated: I'd like to specify branching factor all the way down the rhythm tree.

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

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

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

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

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

> But it's bloody impossible to notate and very difficult to orally communicate

It's really easy to orally communicate. You can just sing it!

(I know what you mean, is it's difficult to describe using a computer keyboard. Using a pen and paper it is really easy, given that you can just draw notes and time signature changes in the margin.)

I'd like to specify branching factor all the way down the rhythm tree.

I'm not really sure what this means, but it sounds interesting.

By default the next note is half as long as the previous. But sometimes that's inconvenient. For example 9/8 is usually 3-3-2-2-2-2-2-2-2... A bar is devided into 3 dotted quarters and a dotted quarter into 3 8ths, and anything further down is split in half as is normal.

Oh that sounds great, but very difficult to read/play

Sometimes when there's a bunch of triplets the editor will just write "simile" and drop the triplet notation.

This could work like that and only the time signature would need a brand new notation, so no I don't think it would be super hard to read.

Triplets are tough to play haha

And then most musicians will have to learn both notation systems.

Have you played with trackers?

2d grid, time is down, instruments(channels) are right, each cell can have a note or "stop this channel". There also can be modifiers in each cell (louder, start tremolo, slide this note into the next, etc).

It's most applicable to keyboard music, I think. Sight-reading this without rehaersal would be trivial, right?


For better or worse, the only way to guess if this notation can be sight-read, is for someone to train themselves to do it. And to make matters worse, learning to sight-read for adults is so hard that it's virtually prohibitive. Every musician I know who can sight-read fluently, learned it as a kid.

So it's virtually impossible to try out a new notation system.

But my impression is that this would be phenomenally hard to read, especially in a live performance situation where your attention is divided between the sheet music and other stuff. If I were staring at a solid grid of text, and were to glance away for a split second, I'd be lost. Part of sight-reading for me is being able to read ahead by a few notes or even a few bars.

It may also be that conventional notation displays a lot more density on a single page, because a 16th note takes up no more space than a whole note.

The main problem I can see with that sort of system is that it probably would get quite unwieldy for decently complex chord patterns (try scoring a chord pattern up the keyboard, say, like what's at the beginning of Tchaikovsky's Piano Concerto 1 for a start -- http://imslp.org/wiki/File:PMLP02744-Tchaikovsky-Op23v1FSmuz...).

For other styles I think it would be pretty good; although the score won't be as compact, it might be easier to understand a "fake book" scored piece (ala what's used a lot in jazz) written this way, let alone pop (which often can be represented with a melody line and Roman numeral chords).

It's a lot like why the QWERTY keyboard remains dominant.

Yeah, it would be really hard to change all the millions of documents written with QWERTY.

It'd still probably be hard to convert thousands of existing computer keyboards to anything else. Every keyboard at every school, office, home, etc.

And it'd be harder still to convert billions of existing people to another layout. Inertia's hard to overcome, even if another keyboard layout or music notation system had ever been convincingly proven superior than the respective dominant ones.

No, it's not the same. Current music notation is a result of long evolution, unlike QWERTY. Try to invent usable alternative - then you will appreciate the convenience of current notation.

What are you talking there? Qwerty is indeed not that old as the musical notation but it is ~150 years old, and it was invented for the typing machines in order to avoid jams, so it does have some history and evolution [0]

[0] https://en.wikipedia.org/wiki/QWERTY

"In order to avoid jams" is a popular story but it's credibility has been challenged [1]. It may be worth noted that Qwerty was made before people used all fingers and touch-typed---they were likely to use just two fingers and typing speed might be quite slow in today's standard.

[1] http://gizmodo.com/qwertys-origin-story-is-a-big-fat-lie-493... (This is just a first one I found in English, but Prof. Yasuoka referred in the article publishes quite a few articles about early typewriters in Japanese, in which he lists several evidences that "jamming" wasn't the reason.)

This has got to be some of the worst jargon and notation for anything, ever.

I'd like to argue that as inefficient as it seems, it's really a reflection of how broad and complex music actually is. This notation and organization system is all a perspective or reference to take when actually trying to comprehend/play music. It's not a fixed set of rules and there are exceptions everywhere.

It's one of those systems where you learn the rules only to know when you're breaking them. So by all means learn to do this, but don't get hung up on real-life deviations.

It actually works well for people performing together, especially when led by a conductor.

I think you sum it up pretty well. As a techie who has never understood music, and who is fascinated by watching my own kids learn and understand music theory (something I was never exposed to as a child), I found this an interesting article.

Clearly history, culture, notation and reproduction technology have all conspired to produce a certain flawed, but accepted, jargon that you just have to bite the bullet and learn. If we could start again using colours, numbers, augmented reality, etc., then it seems obvious we could come up with a better system, but that would be tantamount to proscribing a new alphabet, or telling everyone to start using base-16.

I can sight read for winds and voice, but not for Piano, despite being able to play Piano relatively well. Thus, this explanation will probably apply mostly to single-voice musicians.

1. Find Do. This is the first note of the Major scale in the key which you are playing. Find it on the page, and burn it into your mind; everything you're about to do centers on the location of Do. In the key of C (white keys on a piano) Do is right in the middle of the grand staves, keeping things simple. If you know where Do is, you don't have to care about the key signature for like 97% of the notes you're going to read, and you can make educated guesses about what the sharps and flats do by using your musical intuition. Once you've found Do on the page, figure out where Do is musically, and keep it in your head. Make sure that when you're singing or playing any other note, you know where you are relative to Do, and can jump back there without losing your place.

2. Learn your intervals, and learn Solfege. If you're a musician, you already know these things in principal, even if you don't know the words or the terminology. In particular, you should be able to "hear" the distance between Do-Mi-Sol without too much difficulty, because you hear those distances in music a lot.

3. When you're sight reading the first note in a passage, find the note, then step it off visually to the nearest Do. If you're good, you can also step it off to the nearest Sol. I usually root myself on Do and Sol both in a given passage, as it gives me less visual separation between any note I need to find in a hurry.

4. For every note after that first note, pay mind to the spacing on the page:

4a. Moving up and down the scale should be easy; you're a musician, you already know what a scale sounds like, just sing along.

4b. Moving up or down from one line to the next line is a third. Most often you'll hear these relative to Do-Mi-So-Ti in major keys, and La-Do-Mi in minor keys. You'll hear these a lot, so get used to reading them quickly.

4c. Get a good feel for common jumps while you're at it. As a Bass singer, I got really good at reading jumps between Do-Fa and Do-So, because those chord progressions exist in everything, and their sole purpose in life is to make bass singers bored out of their mind. You're a human, in addition to a musician, so you'll pick up on the patterns in any given composition and have the common tricks down before you even realize what your brain is doing. Don't question that, let your brain be awesome all on its own.

That sounds like a lot of steps, until you realize that you're performing most of them already to make sense of any piece of music that you play. Don't worry about the names or the terminology, you can use numbers (1-3-5) if that's easier, or just "hear" it if you don't feel like you need an aide. Remember, the goal is to drop the aide and be able to just perform these steps on your own eventually.

I'm analytical in nature, so I'm always thinking about these things. I suspect the reason most musicians can't describe this process very well is because the process of analyzing our craft detracts from the quality of its output. The more you can commit to "muscle memory" as a musician, the more you can pull your analytical mind out of the process of a performance, the easier it becomes. Your body's built in reflexes are faster than your analytical brain will ever be, and good musicians tap into that biological strength through loads of practice.

"2. Learn your intervals, and learn Solfege. If you're a musician, you already know these things in principal, even if you don't know the words or the terminology. In particular, you should be able to "hear" the distance between Do-Mi-Sol without too much difficulty, because you hear those distances in music a lot."

I believe the Solfege named intervals (Do-Re-Mi-Fa-So-La-Ti-Do) are only taught as a historical oddity these days (or maybe used more in classical training?). All of my musical instruction, at the high school and college level, used numeric interval names (1-2-3-4-5-6-7-8). Most of the serious musicians I've played with also used the numeric scale rather than Do-Re-Mi. We learned how to sing Do-Re-Mi, "Just in case", but we never used it.

Were you taught in the US, or somewhere else? Maybe it is a regional thing.

I was taught in the US, and Solfege was used prominently in my voice classes, but rarely in my theory and piano classes. I think it's a system that works well when you don't have to think about the actual note name that you're singing, as it describes intervals very well, but relates somewhat poorly to pitch.

I grew up attending a Church of Christ, which used full congregational singing with four part harmony. Our songbooks used a variation on music notation that used shape notes, and the shapes corresponded to Solfege. If you knew your musical intervals, you could completley ignore the key signature, because a Do was always drawn as a triangle, a Sol was a circle, a La was a square, etc etc. This was especially handy because the song leaders were always men, and could not always sing as high as the written music required. They picked whatever key they could sing comfortably, and the congregation adjusted to them. Drove the music majors in the audience nuts. :D


Growing up with that system meant that Solfege was simply the easiest system I had to understand music. To this day, I struggle with pieces in unusual modes, and with passages that modulate their key and make use of unusual progressions, because it breaks down my innate understanding of music and requires me to think in a different way.

Shape note seems really clever! But, I can also see where it would break down in many cases. I learned music predominantly in a Jazz context. That'd be very tricky to use for jazz...especially the various modal types of jazz.

Solfege/solfège is still used prominently in France, or at least it was ~25 years ago. I took music lessons when I lived in France as a teenager. My piano instructor was very confused by the fact that I spoke good French, was at a solid intermediate level at the piano, and yet I could not at all follow his solfège commands (like you say, I had only used numeric intervals back home, although teachers usually called the note by its alphabetic name, like most here I think).

I tried reviewing in my head each week before class what I remembered from the Sound of Music, but made the mistake of thinking that Do-Re-Mi etc. was a static C-D-E instead of realizing that my music instructor was simply describing intervals depending on the key we were in.

In the present day, I would just look it up online, but back in 1991, in a small city in northern France, I didn't have that privilege. It took me several months of twice-weekly instruction before I finally figured out that he was using solfège for intervals, I'm embarrassed to say (to my instructor's frustration and confusion). I'm wincing even now when I think of it.

I asked around at the time and was told it was pretty universal to use solfège there.

I do think using solfège to indicate notes is a much better system for students, since it emphasizes the importance of intervals and keys. It's probably harder at the beginning that just learning static A-B-etc., but worth it.

Solfege is definitely still in strong use, especially in public school choirs. They are convinced that it helps in sight-reading competitions (yes, sight reading is one of several areas in which a choir can compete).

Having said that, I absolutely hate solfege. But my bachelor's was in piano performance, not vocal.

Interesting note: quite a few countries use a "fixed Do" system rather than "A-B-C". It is quite confusing (and humorous to observe) when a solfege disciple tries to sing with a fixed Do native.

Fixed Do sounds awful! I just spent some bit of time reading the wiki on Solfege, as I realized I have a limited view of it. Interestingly, there are additional syllables beyond the 7 I learned! There are also syllables for flattened and raised notes, which is really nice. I have always been bugged by saying "flat three" or "minor three" when singing intervals, and it's hard to make the voice actually make it minor (for me) because of the muscle memory for three being so firmly set.

So, I may have to somewhat rethink my dismissal of the solfege, at least for singing intervals. Unless there's a secret system for singing with intervals by number that accommodates accidentals.

As a computery person, what bothers me the most is the 1-based indexing of intervals. I get that music theory predates zero, but it makes it incredibly frustrating to work with (e.g. add a third to a fourth and you get a... sixth).

When do you need to do math like that with intervals?

As a computer nerd myself, I can understand the argument for 0-based indexing in this case, but I don't recall ever being stumped by it being 1-based. When would you need to add a 3rd and a 4th to get a 6th? Harmonic theory doesn't use addition like that. e.g., playing a 6th is not the same as playing a third and a fourth. So, why do that kind of math with intervals?

Whenever you play three notes in sequence, no? I'll read a passage and think "tonic, up a third, up a fifth so that puts me up to the tonic again... nope."

Huh. That's interesting. In a "people's brains think surprisingly differently sometimes and we rarely think about those differences when the resulting behaviors look the same" kinda way.

I mean, I guess that's not so foreign...But, I tend to think of it as pulling out the notes I need from the scale, and not actually counting up to them. e.g. in my brain I'm grabbing the third and the octave (well 7th, if you've got a third and then the fifth of that third, which I guess is why you're preferring 0-based) that are already there...not climbing up them to find there's the tonic there. I mean, I can see that it's a fifth interval if I go from E to B (in C), but unless I'm building a chord on E, I don't care..it's either the 7 in C, and I'm not so much thinking of its relation to E as I play it, and it's a phrase in C, with maybe an Em chord (either implicit or explicit) underneath; or I'm playing jazz, or some other very chord-based music, and I want my phrase to be relative to the chord we're currently playing (so we're inside that Em, and the key is less relevant).

Sight reading is different, as well, in my brain, but, I think it even bypasses the intervals to some degree and is just distances and shapes and an awareness of the key I'm in. I don't read much these days, but I recall it working best (or at least fastest and most accurately) when most of the theory was turned off in my brain and I just let the shape of the notes (their distance from each other) guide me. But, I feel like it's only in improvising and composition where one would be doing any sort of interval math. But, maybe I'm wrong.

When are you doing this kind of math? When reading, improvising, playing memorized pieces, or composing?

The main context I was thinking of was learning a new piece, particularly the initial read-through of something I haven't heard - I sometimes try to think what it "should" sound like ahead of playing it.

> (e.g. add a third to a fourth and you get a... sixth)


In equal temperament, log base two: add 4/12 to 5/12 and you get 9/12.

In ratios (depending on tuning, these could be loose approximations): multiply 5/4 by 4/3 and you get 5/3.

Yes, I know. But having the log base 2^(1/7)-ish built in is useful. If we could just subtract 1 from all the numbers (i.e. what we call a fifth should be 4) then we would have a measure that actually made sense.

Eh. Calling it a “fifth” is making it clear that the label is an ordinal number: first second third fourth fifth ...

You need to think of it as “if the bottom note was the first note of a scale, where in the scale would the top note be?”

As with many questions of indexing, off by one errors are tricky.

It’s a system that confuses names for notes in a scale with names for intervals between notes. You’d rather they called them by cardinal numbers representing some kind of “distance”, instead of a count starting at one.

But that would be applying a later mathematical understanding on the earlier system. If that’s what you want, you should just use a log scale and count twelfth roots of two.

Ideally we’d switch all our indexing to start at zero, and use half-open intervals everywhere. Start at 0 AD, call the ground floor of a building “0”, start spreadsheets with row 0, switch Matlab to index from 0, et cetera. This is pretty unlikely to happen though.

> If that’s what you want, you should just use a log scale and count twelfth roots of two.

I don't want to count in twelfths, I want to count up the scale.

> call the ground floor of a building “0”

I'm a Brit, we do that here already.

The intervals in a 7-note scale inherently don’t add up like that, because they’re not based on even divisions. So regardless we need to have 12 different named intervals for various numbers of semitones:

For instance, “minor second”, “major second”, “minor third”, “major third”, “perfect fourth”, “augmented fourth”, “perfect fifth”, “minor sixth”, “major sixth”, “minor seventh”, “major seventh”, “octave”.

Reducing all those ordinal numbers by one really doesn’t help all that much. You still have to remember how the “minor” and “major” labels interact for every interval in the scale, and remember that sometimes the interval between the same two notes is given multiple names depending on the key, etc., which is all horribly confusing mess.

> Reducing all those ordinal numbers by one really doesn’t help all that much. You still have to remember how the “minor” and “major” labels interact for every interval in the scale, which is a horribly confusing mess.

Those interactions are pretty intuitive. Where defined, major + minor = perfect (considering an octave as perfect), perfect + major/minor = major/minor. As long as you remember which notes exist, you can't get it wrong, so you'll never get confused by a piece of arithmetic in an actual piece.

They’re not remotely “intuitive”. They only make sense to someone with years of training.

If instead you used digits from –5 to 6, using arithmetic mod 12, it becomes obvious that e.g.:

  -2 + -3 = -5
  4 + 3 = -5
  5 + 5 = -2
  -5 + -1 = 6
  4 + -3 = 1
The “perfect” intervals are just ±5 (ratios very close to 3:2 and 4:3). The “major” intervals are –3, –1, 2, 4 (approx. ratios of 5:3, 15:8, 9:8, 5:4). The “minor” intervals are –4, –2, 1, 3 (approx. ratios of 8:5, 16:9, 16:15, 6:5).

Then it’s easy to see that your “major + minor = perfect” formula only works for some intervals, Etc. Overall the simple heuristics are more obfuscatory than helpful IMO.

> Then it’s easy to see that your “major + minor = perfect” formula only works for some intervals, Etc.

Where does it go wrong? Do those cases come up in practice?

Counting up and down the scale is a core use case for a notation for intervals. It absolutely needs to be well-supported. A 12-semitone approach is never going to match the usability of even the existing system.

>A 12-semitone approach is never going to match the usability of even the existing system

I am obliged to point out that a 12-semitone approach is in fact part of the "existing system" (see: pitch-class set theory).

(Mind you, I of course think its usefulness is overrated, because I think the "atonal" repertory is tonal.)

Nearly all aural skills classes (learning to hear/sing music) for music majors use solfege or something like it. Some people use scale degree numbers instead (so a IV chord in a major key is "4-6-1" rather than "fa-la-do"), but the concept is still very useful.

What I'm questioning is the popularity of Solfege vs numeric interval names.

My music classes of ~20 years ago treated Solfege as being of historic interest, but not particularly common. While numeric intervals were used daily. It came up somewhat more in sight singing and vocal training than in any of the instrument or theory oriented classes. But, I think it was mostly students who were used to it using it rather than instructors teaching it.

But, maybe I just so strongly preferred numbers that I immediately discarded any instruction involving Solfege as being silly and a waste of my time.

Still, I can only recall seeing numbers (and Roman numerals) in writings on theory and such.

Fair enough. It's still pretty common...it's hard to pin down any numbers exactly, but I'd guess it's probably half and half for solfege vs. other systems. We teach solfege at my school (although I prefer numbers myself). And you're right that it's used mostly in sight-singing/aural-skills classes; I mention solfege much less often in my written theory classes.

Exact opposite for me. I learned do-re-me as a kid and then ever saw it again, until I started reading about music theory recently. Then I saw it a lot - e.g. voice leading rules saying that a voice must start from Do and end on a ti-do step, etc.

Doesn't everyone learn Do-Re-Mi from "The Sound Of Music"?

Well, sure, but what's that got to do with actual music instruction and how musicians talk about music? Movies aren't always entirely accurate representations of the world, particularly on highly technical topics.

Knowing solfege because you heard it in a movie and using it on a daily basis in the process of teaching or making music are independent concepts.

Anyway, conversation here has brought it to my attention that it is still pretty common, there are some areas where it is useful (maybe even better than numbers, as in the singing and vocal training area; I personally have recognized the limitation of numbers when singing minor notes, for example), and that my own experience was only partly representative of music pedagogy in the US and elsewhere. That said, when I'm teaching people about music, I still plan to only use numbers...the areas where solfege would be useful are pretty advanced, and require more than watching Sound of Music to understand.

> 2. Learn your intervals

> 4c. Get a good feel for common jumps

i think that's why the current system works so well. most musicians have intervals burned into their muscles. to use an excel reference, reading R1C1 from the staves is much faster than reading A1, because you can read R1C1 from any line in the staves.

Sounds like vim!

Sounds like... And is it? ;)

In terminal run vimtutor. The sound will change.

Does anyone know of a similar tutor for sheet music?

In case anyone is searching for a really smart, modern method for learning music theory, this is it: https://www.hooktheory.com

The author devised his own system for visually representing notes, it makes it much easier to understand things like scale degrees and relative notation (and thus the theory around famous harmonies, melodies, etc).

I think music tools are in desperate need for improvement... Starting with notation, which is still a bit akin to forcing programmers to go straight to Assembly. Little is gained from it as most people just completely give up and then go on to live the rest of their musical lives "in the dark", without knowing how to read and write at all. This can actually be good for some but I'm sure it hinders the creativity of a lot more.

I think we also need way better digital instruments... That make it easier to stay on scale (or to modulate, etc -- whatever the mood is), for instance, allowing people to just play away which is what actually matters.

I've spent countless hours of my life learning scales on several different instruments and think a lot of that was wasteful. More often than not I'm just trying to stay in a given key anyway, nothing fancy...

Instruments really need better interfaces :)

No, instruments really don't. The point of conventional instruments is that once you learn them - which takes years - you can instantly express almost any musical idea using all the possible degrees of freedom available on that instrument.

With something like Ableton Push, you're one step removed from the sound generation, because you're triggering automata with a very limited expressive repertoire. (With Push, it's often just a triggered sample, which has almost no expressive potential at all.)

You can change keys instantly on a piano. You can play any chord you can get your fingers around, in any inversion, using any voicing, with fine control of the relative level of each note in the chord.

With button controllers the best you'll get is one chord per button with no fine shading of levels, no control over inversions or voicings, and so on.

It's absolutely fine to make music like this, but it's not fine to demand that all music be made like this.

Controllers like Push are good for performing effects - filter sweeps, and such - which aren't possible on a keyboard. But that's a different skill to learning scales, and much more expressively limited.

Electronic art forms generally are more rigid and less expressive than non-mechanised media. In theory you should be able to do more, but in practice no one has cracked the problem of building high-bandwidth expressive automata that are as physically responsive and open as traditional instruments/media.

Aesthetically, that can be a problem. A lot of machine-assisted art is either chaotic and formless, or formulaic and repetitive. The best classical music and classical performance lives in an expressive and creative sweet spot between those extremes, and it's incredibly hard to hit that spot with machine assistance.

You seem to be pegged on what current controllers can do... And that's exactly what I am saying: they often suck!

But they can improve and I am confident they will. When I am learning a brand new instrument I can literally feel my brain knowing exactly what I want to do way before my fingers/mouth/feet are able to perform the task at hand. How is this not an interface problem?

With button controllers the best you'll get is one chord per button with no fine shading of levels, no control over inversions or voicings, and so on.

No way. If you don't have to be memorizing stupid things such as "where is the minor 7th again on this one particular instrument?" maybe you could use your free mental cycles (and fingers, feet, mouth) to control that instead... And who knows, maybe you could now do 4-5 inversions in the same amount of time it would take you to do a single one on a piano. Or maybe you can do inversions way more effortlessly on another instrument and focus on really nailing the vibrato.

It's absolutely fine to make music like this, but it's not fine to demand that all music be made like this.

I never said this, I'm just saying that a lot more can be done with a lot less effort if instrument/controller interfaces improve.

Once you actually practice a physical instrument for a reasonable amount of time things like the concept of "memorizing where the minor 7th is" quickly become non issues - the only memorization involved is that of your muscles i.e. the cognitive load is essentially nonexistent. Involving more parts of your body than your cognition is one of the joys of playing a physical instrument, versus pressing a button and thinking a lot.

I play several! But of course I'm not proficient in all of them, which is the whole point. There isn't an "universal controller" that is expressive enough across a variety of timber types... Yet if that existed one could master one interface and do a lot more musically with that acquired skill.

"The point of conventional instruments is that once you learn them - which takes years - you can instantly express almost any musical idea using all the possible degrees of freedom available on that instrument."

But the degrees of freedom of conventional instruments are severely limited compared to what is possible.

Let us also recall that every "conventional instrument" was at one time not only unconventional, but even radically new. The piano, is itself only a few hundred years old. I'm sure when it was invented there were some people who argued against its use and that one should instead stay with "conventional instruments", which then did not then include the piano.

I strongly recommend a talk[1] by Jordan Rudess, who is widely considered to be one of the greatest living keyboard players.

In this talk, Rudess discusses and vividly demonstrates the greatly expanded possibilities that innovative keyboards bring to the table.

Novel instruments that somewhat resemble conventional instruments like the keyboard are only the tip of the iceberg of music interface possibility, however. There are plenty of novel music expression technologies that don't have even the remotest resemblance to conventional instruments, and allow ways of expression that were hardly imaginable a hundred years ago. Things like whole body position tracking, which allows you to make music through dance.

Of course, mature musicians like Rudess who've spent their entire lives learning and practicing on traditional instruments will be unlikely to switch to something radically different, as they'll be starting from ground zero on those instruments. But others with less to lose will be more open to learning something completely new.

It's impossible to tell which novel instrument will become the conventional instrument of tomorrow, but it's very likely some will, because that's how we got all of the conventional instruments of today.

[1] - https://www.youtube.com/watch?v=8h-TsqoSWgo

Let us also recall that every "conventional instrument" was at one time not only unconventional, but even radically new.


Thanks for sharing the talk! I'm a proud owner of one of those keyboards he is playing, a ROLI Seaboard. It is indeed an amazingly expressive, fantastic product

This is utter BS, you can play the Push as if it were a piano and the sound design options are endless.

Saying electronic music is either too chaotic or too repetitive is not only entirely subjective but completely impossible for you to say. Artists like Kiasmos or, famously, Aphex Twin, just to name a couple amongst hundreds, make music that can be neither repetitive or chaotic, for example.

> I think we also need way better digital instruments... That make it easier to stay on scale (or to modulate, etc -- whatever the mood is), for instance, allowing people to just play away which is what actually matters.

I'm not sure what new stuff has come out in the last few years, but Ableton Push is exactly what you're describing. It's a grid where you can select a key, scale, and tone, and then you can apply effects in series / parallel. The notes in the scale light up. There's a bunch of other stuff you can do as well.


Thanks! I've played with Push, it's pretty good... but as someone commented above it's fairly limited in many ways too.

This is an interesting piece of software I have used before: http://autotheory.net. It simply translates incoming midi data so you can use whatever controller/instrument you're already familiar with. The creator is nice and responsive, he often attends shows like AES and NAMM.

Thanks for the recommendation, I just bought the Hook Theory book and it looks quite good. I've been searching for something like that for a while!

I've not read the book but the tool is amazing. I believe the authors have been working on a HTML rewrite for some years now (currently it's Flash). The whole site design got redone a month or two ago, so there may be more changes on the way.

> In case anyone is searching for a really smart, modern method for learning music theory, this is it: https://www.hooktheory.com

I really want to buy the books but they're all DRM'd :-(.

Here's hoping someone from there reads this page and releases it as an epub...

I've skimmed a lot of articles on music "theory" but none of them provide anything like what I'm looking for. A music theory should explain:

1. Why do we like pieces when played forward but not backward or inverted?

2. Why do certain sounds evoke certain emotions?

3. How could you write a program to pick out music that people find especially good (versus music that has surface similarities)?

In other words, why does a particular sequence of sounds A, B, C lead to a mental state M that has particular internal qualities?

> 1. Why do we like pieces when played forward but not backward or inverted?

Why do we like text when read forward, but not backward or inverted?

There are, of course, works that are palindromic or otherwise written to be read/heard backwards, but most of the time that kind of global transformation tends to ruin the "spelling"/"narrative".

> 2. Why do certain sounds evoke certain emotions?

Just like text, evoking emotions needs some sort of narrative. A story isn't a single fact or statement (or a single sound); it's about how those facts (or sounds) flow or change.

In music you might hear a brief bit of new melody that foreshadows something big later in the song. A clear rhythm or melody might be repeated to get the listener to follow along only to have it cut short at a key moment to deny the obvious resolution (similar to a melodrama that suddenly reveals a new twist in the plot as a cliffhanger).

It's the story you tell that matters, and it takes a skilled composer to put sounds together to make a song emotionally evocative. The song that is mostly a 16 bar loop probably sounds boring (but not always!), while the song that introduces the same 16 bars and then plays with variations of it to create an initial conflict, rising action, and a climax is probably a lot more interesting. An obvious example might be Mozart playing Salieri's march in Amadeus[1]. It's not just that he embellished the simple march; Mozart adds a lot of variations that culminate at a comic ending.

[1] https://www.youtube.com/watch?v=P5n0pkNpDWY

Actually, great composers such as Beethoven and Bach and Chopin had very definite ideas about what emotions are evoked by certain keys. They even argued about it with their peers. Music is not something that is reducible to mere quanta and waves and frequency. You all are missing the human part. Sorry, but it's true.

> what emotions are evoked by certain keys

Yes, choice of key is one of the tropes that is useful when composing a song's "plot".

> Music is not something that is reducible to mere quanta and waves and frequency.

That's my point; interesting aspects of a song are not derived from specific sounds (and their frequency/etc). Those are the atoms that can be used to create the larger plot.

While it is possible to reduce music to the frequency and timing of its atomic structure, it's similar to analyzing the phonetics of speech or the glyphs of text in isolation. A low level perspective may be useful, but misses the larger structure we call a "song" or "essay".

That, and they didn't all necessarily use a pure Equal Temperament, either. Different temperaments can give more distinct feelings to certain keys more so than the modern equal temperament. (Note: I used to tune pianos.)

Music is waves and frequency. Music appreciation is what you are describing. And appreciation is very dependant on culture. That is why Bach is not (as) appreciated in certain cultures.

Just like photography. Why is one photograph more meaningful than another? it has nothing to do with photography, per se, it has everything to do with the culture of the person doing the appreciation.

There is a link between the two, between creation and appreciation, and those who understand it generally fare better. But it is not required to be a musician, or a photographer or a poet or anything really.

> Music is waves and frequency.

Sound is waves and frequency. Music is a collection of sounds arranged in a specific sequence.

> Music appreciation is what you are describing. And appreciation is very dependant on culture.

Music relies on various "tropes" to construct a narrative. This includes the choice of key/scale (or none at all), ideas about timing and harmony, etc. These "standard parts" of music are usually from the local culture, just like how a play or movie will use standard character archetypes ("tropes") that are culturally derived.

"music is waves and frequency" in the same way that "spoken language is waves and frequency"---not very usefully. I think bringing in" appreciation" muddies the waters.

I was just responding to the parent who claimed that music was not "waves and frequency".

My point is music is (mostly) independent of its appreciation. Machines can, and do, make music based entirely on the theory of music.

Music theory is an accepted and used term of art for the category of things that this article talks about. There are courses, books, university departments, and degree programmes that use the term. Nobody is going to stop using it because you skimmed it but it doesn't describe some other thing that you think it should.

Some examples:

* http://www.music.msu.edu/areas-studios/music-theory

* https://en.wikipedia.org/wiki/Music_theory#Fundamentals_of_m...

Grand parent is saying he wants an article that covers the "why does music sound good" part of music theory, something I want too. I think most people have a basic grasp of notes, scales and that a middle C is air oscillating at 440 cycles per second. How can you make something sound good is the interesting part of music theory for me and I still haven't found a good intro to it!

> How can you make something sound good is the interesting part of music theory for me and I still haven't found a good intro to it!

As someone who majored in music composition, I have a very simple answer. I'd have some sort of idea in my head of what I wanted the music to sound like (or the emotion to evoke). Then I'd fiddly around for quite some time, discarding the things that didn't meet my criteria.

That's sort of a glib answer, but the fact is that no one really knows exactly why certain things evoke certain emotions, even though most composers understand various building block ideas like "odd meters like 5/8 and 7/8 generally evoke intensity and tension" or "brass chorale in a major key sounds triumphant" or "gong crescendo roll is scary". And of course, even then, we could find counter-examples for every one of those things.

Also, all music theory will tell you is why something in some piece of historical music sounded the way people expected it to sound at the time. It will definitely not tell you how to write good original music (though it may be a good guide on how to imitate past composers if that's useful for what you're trying to do).

> like "odd meters like 5/8 and 7/8 generally evoke intensity and tension" or "brass chorale in a major key sounds triumphant" or "gong crescendo roll is scary"

Can you recommend any books that teach these sorts of general rules, or the emotive feeling generally associated with different keys and modes?

AFAIK there are no such books. As for whether certain keys evoke certain emotions, that's highly debated other than "major happy, minor sad (other modes weird)".

The way I learned about how composition worked was mostly two things. One, listening to lots of music, ideally with the score in front of me so I could zoom in on some particular bit I really liked. Two, writing music and seeing how it turned out in practice.

How can you make something sound good is the interesting part of music theory

You need something like this: https://www.amazon.com/Alfreds-Essentials-Music-Theory-Self-...

It's not something you're going to learn in an afternoon or a weekend, it's hard work and Beethoven was still working on it at the end of his life.

Last month I spent an evening analyzing and discussing a passage of Rachmaninoff, trying to understand how he knew how to write a certain sequence.

That may be a very good book, but I'll be honest, it doesn't look like a very gentle introduction.

It's not :(. Although the average person could probably get a lot of good high-level info by browsing it.

I haven't yet come across anything that is a good gentle intro. Most resources that approach music and math make the mistake of treating music theory like the law, without any rationale for it provided. Music history textbooks typically give a lot more context of how our music theories emerged, but they don't talk about why that might be, based on acoustics, psychoacoustic, and math.

Maybe one day, I'll write the comprehensive intro I wish I'd had.

Does that book provide anything helpful to someone who already understand music theory?

Oh most definitely! I'd describe it as an extension of what's taught in the standard music theory curriculum. It makes the very ambitious claim of developing a framework that can be used to analyze all tonal music, from the renaissance to the present.

middle C isn't 440hz. that's generally an A.

Good point, not sure where I got that from then. Record updated.

When we have discordant sound (e.g. a collection of plucked strings where the fundamental frequency ratios are not in nice simple ratios, so that none of the overtones align), there’s a great deal of complexity to the sound, and you get interference patterns between them, similar to moiré patterns in images.

This causes “musical tension”.

Some types of discord are mild, and cause a bit of mild annoyance or “sadness” in the sound. Other types are aggressive and cause serious anxiety.

When you return most of the sounds to be in harmony, that tension is relieved. This causes a more positive emotional response. The greater the former tension, the more satisfying the release.

Imagine you’re in a crowd of applauding people, each clapping at a different rate, so that the sound is like a cacophony. Your brain can’t make out any pattern except a wave of sound. Now imagine the people start clapping in rhythmic unison, with some kind of structure. Suddenly your brain can make sense of the pattern.

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

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

An aesthetic just seems like a unconscious favorable reaction to stimuli based on genetics and culture. Why our genes and culture have favored certain forms, I suspect there isn't a way to reduce it to something satisfactory.

I used to wonder why I felt good when looking at sunsets, landscapes and clouds. But I figured that our aesthetics probably evolved in response to what was around us. Happier people probably survive better.

The way we hear music seems like such an aesthetic, as it might've occurred within ancient cultures as a form of play and release. I suspect that our random genetic hunger towards different aesthetics might have created an incredible developmental feedback loop. I'm not sure where I'm going with this incredibly complicated topic, but I have a lot of very unrefined thoughts about them, that are probably overly-reductive and wrong.

You are looking for a "theory of music", as opposed to "music theory", ie an explanatory or scientific theory.

I offer my own efforts in this direction at http://whatismusic.info/.

Thanks! This sounds like good terminology and your writing attempts to answer the question in ways I haven't seen elsewhere.

Also I think the fact that songs can get stuck in your head suggests some kind of mental reinforcement exercise for patterns over time.

Those qualities aren't universal in people and as such what you are describing is more of a study of culture than musical theory. You would probably enjoy a music appreciation course, and possibly one taught by a philosophical professor.

Music theory describes music.

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

Not an article, but I think this talk hits a lot of interesting points:


Answer to question 1 is simple: good resolution is a semitone up, but scale step down (normally, whole tone). No symmetry here. If you play backwards, you won't get resolutions - it will sound as nonsense.

Harmony Explained: Progress Towards A Scientific Theory of Music


3. How could you write a program to pick out music that people find especially good (versus music that has surface similarities)?

Be an artist.

I don't think really we know the answers to those questions.

> 1. Why do we like pieces when played forward but not backward or inverted?

Would you like a movie played backward?

> 2. Why do certain sounds evoke certain emotions?

Large part of this boils down to if the waves representing the sounds meet at zeroes or not.

> 3. How could you write a program to pick out music that people find especially good (versus music that has surface similarities)?

I think that this is currently impossible. The music composition search space is actually extremely large, larger than say the search space of Go. You can restrict the search space quite a bit but it's still large.

> In other words, why does a particular sequence of sounds A, B, C lead to a mental state M that has particular internal qualities?

Think of it as design. It's the same sort of problem.

If anyone is interested in learning to read music, I've been slowly building a tool to practice: http://sightreading.training/

source is here (built in react, es6): https://github.com/leafo/mursicjs

It's still lacking a lot (like rhythm), but the different generators definitely give my brain a workout. It works best if you hook up a midi keyboard.

As someone self-teaching themselves piano, this is really fantastic. I already play drums but have no notion of what notes are what on a staff and piano keys and have been slowly teaching myself. The MIDI keyboard support really makes this stand out.

I've been teaching myself piano after someone gave me one.

Here's a great set of online lessons [1] (not free, costs about $20 per month if you put in your email address). The guy is really talented, and teaches non-classical stuff like pop, boogie woogie, etc. I'm super inspired by it!

I also found this [2] which does have free lessons and also looks good.

[1] https://pianowithjonny.com/

[2] https://www.hoffmanacademy.com/

If you want a series of books that constitute "music theory for nerds" -- building up music theory from a solid foundation of acoustics, and math -- try "Musimathics" by Gareth Loy. It is a great read.

It takes very little for granted. Now, sometimes you have to say "this is just the way it turned out" to explain Western music, but the best way to do so is to show some other ways it could have turned out, and show their role in non-Western music. Musimathics does that often.

On the science side of things, Vi Hart put together just an excellent video that goes over harmonics, the overtone series, and why 440 Hz and 880 Hz sound so "indescribably similar" to this blog author.


This is great, but I would recommend Robert Greenberg's "How to Listen to and Understand Great Music". He goes through the history of western music in a way that makes it clear why Amin != Cmaj, and other questions that the OP has. Yes, sheet music is crap, and he explains how it evolved to be the way it is, after which you'll be much more forgiving. He's a great speaker who obviously knows the material inside and out.

The explanation of the origin of major scale in the article is pure voodoo. Minor third is not a simple fraction - is that the reason to exclude it from the scale? How do you explain minor scale then? Maybe it should be excluded, too?

Here're my thoughts on the subject.

For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament, This system emerged as a result of long evolution of Western music, and experimentally proven to be very rich in possibilities.

Scales used in Western music (of which jazz is a part of) are built on two simple principles: 1) interval between adjacent notes of the scale is either tone or semitone 2) there's no two semitones in a row.

It's easy to check that all scales that satisfy these 2 rules are:

major scale and its modes (7-note scales; 7 modes)

melodic minor scale and its modes (7-note scales; 7 modes)

diminished scale and its modes (8-note scales; 2 modes)

whole-tone scale (6-note scale, single mode).

(Whole tone scale is not used very often, except by T.Monk)

But even after we "explain" scales, we need to figure out how to use them, what their role is, what the properties of each mode are. There's no hard science behind this, the properties just "emerge", and you have to experience them - theorizing is not of much help, math formulas don't explain anything, just lead to confusion.

In short: you have to play AND think; thinking alone won't help. It's an experimental subject.

Edit: forgot to say: scale is a very useful notion, but in some contexts, it's more convenient to think in terms of triads and interpolation. I know this all sounds hand-wavy, and it is! Unfortunately, without piano, it's impossible to to illustrate what it all means. The subject doesn't easily lend itself to verbalization.

I agree very much with your post's thesis (you have to play AND think; thinking alone won't help. It's an experimental subject.) Just noticed one thing:

> For some reason no one can explain, Western music settled on a system of 12 tones with equal temperament

It doesn't seems surprising to me. If you start from a pitch and go upwards in both octaves and perfect fifths (2:1 and 3:2, the two most fundamental intervals), the perfect fifth sequence will land on 11 distinct tones before (nearly) meeting the octave sequence. Mathematically, (3/2)^12 ≈ 2^7.

So 12 semitones works out nicely because you can follow perfect fifths out in any direction as far as you want and never go outside the set of semitones. And most of the small-ratio'd intervals can be represented with pairs of notes inside this set.

Interesting idea indeed. I need to think about it.

Edit after thinking: still, it doesn't explain the number 12 IMO. It could be 17 or something else. Probably, it's a long chain of coincidences at play: Western music settled on 7-note scales long time ago (long before equal temperament was invented), and we should start looking for explanations from here.

Another edit: one of the important coincidences is that number 12 makes possible the existence of diminished scale, which serves as a "universal glue" due to 2 tritones. (There's not enough space here to elaborate, but you probably know what I mean). And maybe tritone itself is one of factors leading to number 12.

The number 12 is just a coincidence, that 3^12 = 531441 ≈ 524288 = 2^19

This means that 3/2 ≈ 2^(7/12) [accurate to about 0.1%]

And also 4/3 ≈ 2^(5/12) [also accurate to about 0.1%]

And also 9/8 = (3/2) / (4/3) ≈ 2^(2/12) [accurate to about 0.2%; putting two factors of 3 in makes the approximation only half as good]

You also get another nice coincidence, that 5^3 = 125 ≈ 128 = 2^7

This means that 5/4 ≈ 2^(4/12) [accurate to about 1%]

There are some people who have written music in a 41-note equal tempered scale, because then you get an even better approximation to the 3/2 ratio:

3^41 = 36472996377170786403 ≈ 36893488147419103232 = 2^65

(3/2) ≈ 2^(24/41) [accurate to about 0.03%]

If you start off from assuming that the Do-Sol (fifth, 3:2) harmony is a "pleasing" one, and also the Do-Mi (third, 5:4) one, you can create new "mostly pleasing" harmonies by for example taking the fifth of a fifth (9:4, which can be transposed an octave to get 9:8, which is Re or a second), and doing similar things (you can also do things like finding the note whose fifth is Do, which is 2:3, or 4:3, a fourth or Fa).

Repeat this process and you start getting a bunch of notes which fall on the 7-note scale. In the blog post the major seventh is listed as 17:9, but by this method you get a 16:9. Basically the same thing.

At this stage, you may notice that the notes are roughly equidistant, except for Mi-Fa and Ti-Do, which are at ~half the distance. This is the first hint of the 12-note scale. We could have stopped earlier in the notemaking process and had a 6-note or a 5-note scale or whatever, but it wouldn't be so equidistant.

Now pick each note, and build an octave from it. The new notes created will invariably be very close to existing notes, or very close to the midpoint between existing notes. This gets us the 12 note scale (5 midpoints + 7 notes, the aforementioned half-step notes don't have midpoints), if you choose a canonical note for each part. The number 12 just happens to be the number where simple harmonic ratios can get you a mostly-equidistant scale.

At this stage, different music systems do different things.

One kind of Chinese scale uses a 2:3 ratio and generates ratios involving these numbers that form a 12-note (roughly equidistant) division.

Indian music does something similar, though it instead generates a 22-note scale, where many of the 12-note scale notes have two forms. It is rare that a given piece of music will use both forms of the same note.

Western music goes ahead and invents the piano, realizes that the piano is hard to tune/transpose, and settles on the twelfth-root-of-two stuff so that transposing becomes dead easy.

A bit better modification of the argument: continue cycle of 5th. After 12 steps, you get (3/2)^12=129.7, which is really close to 128=2^7 (whole number of octaves). That's where 12 steps come from!

And from here, the natural idea follows: what if we take not exactly 3/2 for fifth, but value x such that x^12 is exactly equal to 128? This leads to equal temperament.

Yeah, that might be it! (Not sure that it's true historically though).

Quora (of all places) has some really good articles about this. Check for example the second answer in this page:


I think that there are several thing that brought us to 12. First it is very easy to divide, this is why we use 12 hours clocks and between 2 octave the ear is able to distinguish 1/12 of an octave as 2 different sounds, it might be possible to do better but it would be unpractical because it would be more difficult to found chords that sound good. Arabic music use quater tones (24 quater tones per octave).

> Minor third is not a simple fraction - is that the reason to exclude it from the scale?

The minor third is a 6:5 ratio. Does that change your mind?

Where do your tones and semitones come from? You've just rejected the explanation for why we have a 12-tone scale (it is a local optimum of tuning, closed under the operation of transposition, that satisfies lots of nice ratios), so what do you propose instead? Saying "no one can explain" is a cop-out.

To follow up: Major and minor scales are 7-tone subsets of the 12-tone scale that were discovered first. They allow for many of the same possibilities, but they're not closed under transposition, which is why we now have the 12-tone scale that includes all of them.

Please read wikipedia article on temperament. 7-note scales were "discovered" long before equal temperament was introduced. Equal temperament is a relatively recent invention, initially was very controversial, and remained so for a long time. In other cultures, there's still a variety of scales and temperaments that have nothing to do with 7-note scales at all. (Jazz uses variety of scales, too - some even with 2 semitones in a row). There's only one thing people seem to agree on: the role of fifth (3/2).

This is exactly what I'm telling you. Why are you being condescending to me about it? You're the one claiming that the semitones of the 12-tone scale are some sort of fundamental axiom.

Why didn't you know where minor thirds come from?

Why do you seem to be denying that simple harmonic ratios are where harmony comes from? (EDIT: he's not, it's fine)

Why are you promoting a theory of music with absurd axioms, which manage to explain the octatonic scale (which almost nobody uses) better than the pentatonic scale (which almost everybody uses)? That's not a sign of a good theory.

Please read the part of the article starting with "The twelfth root of two may be irrational, but it turns out to almost create several nice ratios". The author excludes minor third from the set of "nice ratios". I referred to this while calling it a voodoo.

A simplified explanation is not voodoo.

Yes, he chose to list only the intervals in the major scale. Showing the intervals in the major and minor scale at the same time would have been very confusing.

I understand now that you weren't trying to say that minor thirds were a hole in the theory he presented, but that they were a hole in the way he presented it. I took issue when I thought you were criticizing the theory itself, because what he presented is a simplified view of generally accepted foundations of harmony. I apologize for mis-characterizing your position there.

There is more that could have been said -- and it sounds like we agree on what could have been said -- but I think it's not necessary to go into all the details of the construction of scales, including temperament, just to write a blog post. Describing temperament accurately, without handwaving, requires a book.

Dude, if you can't hear it, it doesn't matter. You'll never listen to Coletrane's "Night Train" and have your head explode when you realize he's jumping from Dorian to Mixolodian to whatever the fuck was that!?

BTW, please read something like "Musimathics", it'll be better than trying to learn music theory from Wikipedia.

I'm not learning it from wikipedia. I'm playing for 50+ years. Best resource is Jazz Piano Book by M.Levine. (I read lots of others, but... everything I learned, I learned from this one).

Okay, yeah. I bristled at your suggestion that I should go learn temperament from Wikipedia, when I was making an effort to not bring up temperament to keep my reply focused.

I thought you didn't understand harmony, you thought I didn't understand temperament, I was talking from a historical point of view, you were talking from a jazz point of view.

We might disagree on two kind of minor points:

* How this blog post should have been written

* I'm uncomfortable with describing fundamentals of music from a jazz point of view, because that seems to me to be putting the effect before the cause.

I think you're being unfair on the whole tone there, it's the intro to 'Take the A Train', and people will often substitute it in a dominant chord when improvising.

"C major is identical to A minor, and I don’t understand why we need both."

Cringe. If I understand nothing of some subject, I would do well to just shut my mouth about it.

In fairness, the author said " I don’t understand why we need both", and not "We don't need both".

>I suppose it’s possible to change the sound of an entire piece of music just by changing the key signature, but does anyone actually do that?

>How would that work for music that also uses notes outside the scale? These seem more like questions of composition, which I definitely don’t know anything about.

From wikipedia:

Although transpositions are usually written out, musicians are occasionally asked to transpose music "at sight", that is, to read the music in one key while playing in another. Musicians who play transposing instruments sometimes have to do this (for example when encountering an unusual transposition, such as clarinet in C), as well as singers' accompanists, since singers sometimes request a different key than the one printed in the music to better fit their vocal range (although many, but not all, songs are printed in editions for high, medium, and low voice).

There are three basic techniques for teaching sight transposition: interval, clef, and numbers ...

[*] https://en.wikipedia.org/wiki/Transposition_(music)#Sight_tr...

I don't think the author was referring to transposition there. I think that was about leaving the notes on the staff the same but just changing the key signature. This is done sometimes as a novelty, and the best example is changing something from a major key to a minor key or vice versa.

The article has a link to a recording of Für Elise in a major key [1], and there are many similar renditions of other pieces around. You could in principle do this with any of the seven modes, not just major and minor.

As for notes outside the scale, it seems like these key-signature-changing compositions typically keep them the same (like the D# in Für Elise).

[1] https://www.youtube.com/watch?v=Y-rZD2AsHbI

The author and anyone else who understands correctly that traditional music notation is shitty in lots of ways and is trying to understand how music really works should go get the book "Music and Memory: An Introduction" by Bob Snyder. It uses no music notation and explains music in terms of psychological principles of perception of time. It's not a complete theory of everything, but it shows the way you should be understanding the nature of music.

Beyond that, check out Sweet Anticipation by David Huron, and Tuning Timber Spectrum Scale by William Sethares (and his other Rhythms and Transforms, see http://sethares.engr.wisc.edu/ for web versions of first chapter of each). These sorts of resources are where real understanding of music comes from. Not from the "theory" stuff us music professionals had to deal with that fails to explain anything well.

Well, the first figure: frequency is NOT the period, and amplitude is not that. I guess there are lots of errors in the text if the first figure is completely wrong.

This source is a bit wordy. Let’s summarize:

The core idea of music made with harmonic sounds is that “notes” with frequencies at small-integer ratios will “harmonize”. Harmonic sounds means something like a vibrating string where the vibrations are integer multiples of some fundamental frequency, because other non-integer-multiple vibrations are damped out by the fixture of the string at two points. Different (non-harmonic) types of sounds often sound better with a different sort of scale, for details see this book http://sethares.engr.wisc.edu/ttss.html

* * *

The “octave”, 2:1, is the simplest whole-number ratio, and makes many of the vibrations in two notes in such frequency ratio align with each-other, to the point that two harmonic sounds exactly an octave apart almost sound like the same sound.

Other simple ratios like 3:1, 4:3, 5:4, etc. also “harmonize”, with (not quite as) many aligned overtones.

The core idea of the 12-note musical scale (pretty much regardless of specific tuning) is the approximation:

3^12 = 531441 ≈ 524288 = 2^19

3/2 ≈ 2^(7/12) [this is accurate to about 0.1%]

Or another way to say this: 7/12 of “doubling” on a log scale is very nearly “three-to-two”. Musicians call this ratio a “perfect fifth”.

In the case of equal temperament, an octave is split into 12 precisely equal steps (on a log scale), each one the 12th root of 2.

There’s one other nice approximation to take advantage of:

5^3 = 125 ≈ 128 = 2^7

5/4 ≈ 2^(4/12) [this approximation is only accurate to about 1%]

Musicians call this ratio a “major third”.

* * *

Even outside music, these approximations can be useful for doing approximate computations.

If only our society switched from decimal to “duodecimal” numerals, it would be very natural to use logarithms base two, notated with “duodecimal” fractions.

If you have a number expressed in log base two, and you use duodecimal notation, approximately multiplying or dividing by 2, 3, 4, 5, 6, 8, 9, 10, 12, ... is very easy using addition/subtraction of easy-to-remember multiples of 2^(1/12).

Unfortunately our society instead has slide rules and measurement scales (decibels, etc.) which are all built around logarithms base ten, and decimal notation.

The article says that the "human ear loves ratios", but doesn't dig deeper into why. Here's my two cents.

First of all, let's focus on harmony (notes played at the same time) as opposed to melody (notes played one after another). What sounds good in a melody is quite culture-dependent, but there are reasons why harmony is more universal.

Second, let's focus on sounds that are produced by something long and narrow. In a guitar, violin, or piano it's a string, and in a flute it's a column of air. The physics of vibrations goes so that in such a case the sound is composed of harmonics: sine waves of frequencies f, 2f, 3f, 4f, ... If the shape is different (say, a circular membrane of a drum), then this may not apply.

Suppose we add a second sound, whose fundamental frequency is, say, 3/2 f. This means that its harmonics are 1.5f, 3f, 4.5f, 6f, 7.5f, 9f, ... Half of these (3f, 6f, ...) coincide with the harmonics of the first sound, so the sounds "reinforce" each other. More generally, if the ratio of the frequencies is p/q for some integers p and q, then there will be overlap in the harmonics. And the smaller p and q are, the more overlap there will be.

An entire class of music, Carnatic music, has existed for thousands of years with millions of listeners that is based on melody alone.

For a drum, interestingly, the fundamental vibration modes are all Bessel functions.

The really wierd thing is that intervals such as a fifth, major and minor thirds, and sevenths occur regularly in bird song, whale song, and a bunch of other non-human sounds. I assume that evolution favours constructive interference as the communication will generally travel further, but I also feel that human music is in some way influenced by our pre-lingual history.

In some way our brains are hard wired to appreciate and recognise these intervals, and to infer certain emotions from them.

Melody in the form of the pentatonic scale is much more universal across cultures, though. Use of non-trivial harmony is significantly less widespread, it's usually no more than a melody played over a single root tone or chord. South-Asian classical and folk music is one example in which Western-style harmony is not used at all.

You're right, of course. My wording is was bit poor.

What I should have said that harmony is less ad hoc; it has less "degrees of freedom".

With regards to melody, there are tons of tuning systems that are quite close to the usual twelve-tone equal temperament. It would be hard to give a convincing argument that one of these sounds better than all others.

Contrast this to the system of harmony where the basic principle is that ratios of small integers sound good together. This is not the only possible system of harmony, but it does seem to represent some kind of local optimum. And this makes it more amenable to the kind of purely theoretical reasoning that the article is trying to do.

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