
Music theory for nerds - hardmath123
https://eev.ee/blog/2016/09/15/music-theory-for-nerds/
======
skrebbel
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.

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

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

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

Does that question even make sense? :-)

~~~
abannin
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.

~~~
msg
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.

~~~
abannin
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".

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

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

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

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

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

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

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

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

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

~~~
theseoafs
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.

~~~
blake8086
If a beginner can't immediately grasp musical notation, isn't that evidence
that it is, in fact, absurd?

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

~~~
UweSchmidt
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.

~~~
fenomas
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!

------
karlb
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.

~~~
ajuc
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, ...".

~~~
entropy_
Some notation systems do exactly that. For example, the notation used for
byzantine chanting:
[http://www.byzantinechant.org/notation/Table%20of%20Byzantin...](http://www.byzantinechant.org/notation/Table%20of%20Byzantine%20Notation%20Symbols.pdf)

------
analog31
_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.

~~~
andrepd
>why we use such an archaic notation system.

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

~~~
hardmath123
One attempt was Hummingbird.
[http://www.hummingbirdnotation.com](http://www.hummingbirdnotation.com)

~~~
ludston
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.

~~~
Ericson2314
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.]

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

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

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

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

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

Instruments really need better interfaces :)

~~~
TheOtherHobbes
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.

~~~
brunorsini
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.

~~~
palimpsests
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.

~~~
brunorsini
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.

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

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

2\. Why do certain sounds evoke certain emotions?

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

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

~~~
pdkl95
> 1\. Why do we like pieces when played forward but not backward or inverted?

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

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

> 2\. Why do certain sounds evoke certain emotions?

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

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

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

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

~~~
sverige
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.

~~~
redial
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.

~~~
pdkl95
> 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.

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

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

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

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

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

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

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

[1] [https://pianowithjonny.com/](https://pianowithjonny.com/)

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

------
rspeer
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.

------
zeta0134
On the science side of things, Vi Hart put together just an _excellent_ video
that goes over harmonics, the overtone series, and why 440 Hz and 880 Hz sound
so "indescribably similar" to this blog author.

[https://www.youtube.com/watch?v=i_0DXxNeaQ0](https://www.youtube.com/watch?v=i_0DXxNeaQ0)

------
watermoose
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.

------
lohankin
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.

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

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

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

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

~~~
lohankin
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.

~~~
Manishearth
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.

~~~
lohankin
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).

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

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

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

------
verroq
>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...](https://en.wikipedia.org/wiki/Transposition_\(music\)#Sight_transposition)

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

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

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

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

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

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

------
bbtn
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.

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

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

* * *

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

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

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

3^12 = 531441 ≈ 524288 = 2^19

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

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

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

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

5^3 = 125 ≈ 128 = 2^7

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

Musicians call this ratio a “major third”.

* * *

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

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

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

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

------
MrManatee
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.

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

~~~
MrManatee
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.

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

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

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

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

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

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

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

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

~~~
theOnliest
> Why does notation allow for seven pitches, not 12? Because music is at most
> built out of 7-note scales, not 12.

This answer is good, but I wanted to pick one tiny nit, which is that not all
music is built out of 7-note scales. A lot of music is, but music that isn't
doesn't often lie well on the staff. That's true, incidentally, whether there
are more than 7 notes in the scale (12-tone music, lots of octatonic music
from people like Stravinsky, Scriabin, etc.) or fewer (whole-tone music comes
to mind, as does the slightly more esoteric hexatonic scale). Pentatonic music
fits well on the staff because the pentatonic scale is a subset of the
ordinary diatonic.

~~~
soundwave106
Probably the correct way to phrase this is Western sheet notation was designed
around the diatonic scale. Western art music rarely deviated a lot from
diatonic until the late 19th / early 20th century.

Another way would be to say, "most music historically has been built out of
7-note scales", because from my perspective, that would be correct... the vast
majority of music systems in the world _are_ either heptatonic or pentatonic.
There are obviously exceptions (Gamelan scales for a start --
[https://en.wikipedia.org/wiki/Slendro](https://en.wikipedia.org/wiki/Slendro)),
but to be honest I can't think of any historical octatonic or higher scales at
the moment. I'm sure they exist, but they seem pretty rare (until the late
19th / early 20th century classical era that is).

------
gtani
Interested to see where this goes. The 5 blogs/books cited at the end look
especially interesting.

I never thought about music notation until i learned guitar. Previous to that
i was "gifted" years of lessons in piano, woodwinds and percussion, and
reading treble clef was easy. I never tried to read sheet music in guitar
books because i never had to, i just read tabs and played. Besides some
classical books and jazz books by Berklee profs and others (Leavitt, Martino,
Goodrick), there's very few guitar books that don't have tabs.

So you could argue that's a notation. You could also say that FL piano rolls
and renoise timelines (what OP is trialling) are notations, as are lead sheets
and chords charts in Nashville format (the Roman numerals like ii-V-I you see
all over the place). They're sufficient for people to play music, they happen
to be dynamically typed and GC'd vs static typed and manually alloc'd

------
markbnj
I love pieces like this that explore the underpinnings of why we experience
music the way we do. I'm a self-taught guitar player and its only now, in my
50's with the kids more or less grown and a lot more time, that I have started
to dig in and understand what I've been playing for two decades. Good stuff.
Thanks for the post.

------
raverbashing
Wow, that's a not good article. More like a rambling than anything else.

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

That's like saying house numbers are not important in finding an address...

I'll just leave this here:
[http://openmusictheory.com/](http://openmusictheory.com/)

------
intrasight
I've restarted my music brain after 30+ years. Two resources I found valuable
are:

[http://andrewduncan.net/cmt/](http://andrewduncan.net/cmt/)

[http://oyc.yale.edu/music/musi-112](http://oyc.yale.edu/music/musi-112)

------
dbrgn
Here's an e-book that has a similar approach to explaining music:
[http://pedrokroger.net/mfgan/](http://pedrokroger.net/mfgan/) I read it some
time ago and liked it. No affiliation with the author :)

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

Well, an orange and a carrot are the same color, but does that mean we don't
need both? These two keys have the same signature, and that's it. To say they
are identical is false.

> I don’t know anything about music.

You got that right.

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

Only about every major composer who has ever lived. Look through any Beethoven
sonata and see all the key signature changes throughout.

Your article didn't even hit one of the most important parts of music,
modulation, which if you understood it would make many of your other
confusions go away.

~~~
dfabulich
> > I don’t know anything about music.

> You got that right.

This is gratuitous negativity. It adds nothing to your post except to
emphasize disdain for the author. It's especially uncivil in response to the
author's own statement of humility.

This post would be better (and much more constructive) if you explained (or
linked to an explanation of) how C major and A minor are different, or how the
concept of modulation dispels the author's confusions.

~~~
hellofunk
I disagree. The article is not of high quality, and I am compelled to point
that out. It is one thing to admit not knowing anything about a subject, it is
something else entirely to suggest that the way things are in that subject
don't make any sense. That is arrogance hiding behind a pretence of ignorance.

If the author truly wanted to explore this subject, rather than suggest the
key of A minor is superfluous and redundant, he would make an attempt to
answer, not ask, that question. Western music theory has not existed for
centuries without reason, and some of the attitude in this post suggests the
author has superficial appreciation for how music works, as if the author
could do it better. There are many more examples of this in the article.

I found the article questionable in its intent and content, and the general
tone throughout is worthy of my response.

------
robbrown451
One thing that is particularly arbitrary is that C is named "C" rather than
"A", which would make far more sense (given that it is the root note of the
only major scale that has no sharps or flats).

~~~
dietrichepp
Well, it has to be somewhat arbitrary. But it is nice that the relative minor
of C, which is two steps down, is A. Otherwise you would have to go five steps
up (from A to F, hypothetically, if A B C D E F G A were a major scale).

~~~
robbrown451
If C was called A, that wouldn't be particularly arbitrary.

It would certainly make more sense for people learning music. The C major
scale is universally agreed to be the simplest, most basic scale, so wouldn't
it make more sense for it to be ABCDEFG, rather than CDEFGAB?

~~~
gnaritas
No, because that's the A minor scale, which is the same notes as the C major
scale but starting at A. A scale should logically always start with its root
note.

~~~
dietrichepp
That's not really what the comment is saying. If we relabeled the notes
CDEFGAB to ABCDEFG, then ABCDEFG would be the "A major" scale under our new
system. The question is, "Wouldn't that make more sense?" My answer is, "It
would make a tiny bit more sense to the most absolute beginner students, who
would then have to learn other scales anyway."

~~~
gnaritas
How does that make any sense at all? I don't see it making any more sense at
all, not even a little bit. There's nothing arbitrary about C major/A minor
being taught first as it's just the simplest scale with no sharps and flats.
And calling a clearly minor scale major under a new system doesn't clear
anything up for anyone.

~~~
dietrichepp
The original question is frequently misinterpreted, so I can understand the
confusion. I've seen the same question asked on forums a few times and someone
always comes out of the woodwork saying that, "Well, ABCDEFG is a minor scale,
not a major scale, what are you even going on about?" Let's ignore the part
about why the scales we call C major / A minor being taught first, because we
all agree that it makes sense to teach those scales first.

The question is, "Why are the notes labeled the way they are, instead of some
_different_ labeling (which would change their relationships between each
other)?" This has nothing at all to do with "calling a clearly minor scale
major". Let's suppose we relabel the notes, so that C is now named "A", D is
now named "B", E is now named "C", et cetera. In this alternate universe, Ab
is enharmonically equivalent to G, and the scale "ABCDEFGA" is a major scale,
and "FGABCDEF" is the corresponding minor scale.

Part of the question is, "Why did we name the notes the way we do, instead of
that other way?" That's actually an interesting question, once you get down to
it.

Another part of the question is, "Isn't our system kind of arbitrary, and
doesn't this alternative universe make more sense?" The answer is "No, the
alternative universe isn't inherently better or less arbitrary, for the
various reasons we talked about in this thread."

The main two reasons why we wouldn't prefer one universe over the other are
because (1) both the Ionian and the Aeolian modes are important in western
music, and it's difficult to claim that one is _more_ important than the
other, and (2) you have to learn a bunch of other scales anyway, and if you
have a hard time with CDEFGABC then you're never going to make it through
other basic scales like G major or F major.

(Bug... the "flat" symbol seems to be getting stripped out of my post... so
forgive me for using "b" instead U+266D)

~~~
gnaritas
Ah, OK that's much clearer, I understand now. I'd say this, picking A for what
is now C would be just as arbitrary as there is no "first" scale, they're all
equally important and trying to match up a first scale to the first letter of
the alphabet would be just as arbitrary. For those who feel the need to start
with A, then start with minor scales, A minor first, artificial newbie
weirdness solved.

------
arnarbi
Why is this at the top of HN? While the author is right that western music
notation has shortcomings, almost _none_ of the (admittedly poorly understood)
reasons they list are a factor in that.

> [Key signatures] completely obscure the relationship between the pitches,
> though.

Nope, they exactly allow the notes to show _scale degree_ relationships.
Without key signatures, music in any major scale other than C would be very
confusing as the regular scale notes would be marked with accidentals.

> I don’t think I entirely understand this, because it still seems so
> convoluted to me. You have to mentally translate that C to a C, and then
> translate the C to however that particular note is actually played on your
> instrument.What does this accomplish?

Same. Because at some point a player stops playing by translating the exact
pitch to a fingering (for example), they start playing on a scale. They
practice the scale by itself, and other pointless "music" in that key and it's
the fingerings of the scale that they learn. I.o.w. this system allows them to
think at one higher level of abstraction.

> I’m a bit out in the weeds from here on. C major is identical to A minor,
> and I don’t understand why we need both.

This whole section reads "I don't understand this so the world must be stupid
to have come up with it." Rest assured that C-major and A-minor are _not_ the
same. Yes they have the same pitches, but a different starting point (root). A
note's relationship to the root of they key of the piece is much more
important than the exact pitch. Fur Elise in C-major would sound just like it
sounds in A-major. Not A-minor.

> I only looked into this because I want to compose some music, and I feel
> completely blocked when I just don’t understand a subject at all.

Here's the bright point: You don't need to understand notation to compose
music. Just understand scales and chords (and their progressions), and use
whatever notation makes sense for you.

~~~
skrebbel
> _Why is this at the top of HN?_

What the ..? Because it's a cool blog post of course! It carefully and clearly
explains things the author understands about music (in a much better way than
I've seen before), and then clearly and carefully goes on to describe and
comment on some things the authors _doesn 't_ understand as well.

Basically, it's a Julia Evans blog post but about music instead of system
programming. I've never seen a comment like yours on top of an HN thread about
any of her posts. What's going on, somehow because this is about music and not
programming we need to get all pedantic?

~~~
Angostura
I don't think the commentor is suggesting that music is an unworthy topic to
be on the top, rather that the post is insufficiently insightful to be on the
top.

~~~
arnarbi
Yes, this.

------
dietrichepp
> Also, this notation has a _slight_ problem. That problem is that sheet music
> is terrible.

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

I hear it bandied around in these articles that music notation is awful. Can
someone explain to me why? I'm not much of a musician, but I can work my way
through sheet music if necessary, or write it when I want to. I don't remember
what it was like to learn to read and write music so I'm not going to
understand why it's so bad just by thinking about it.

First of all, the lines and spaces make a ton of sense to me. When you're
playing, reading, or writing a piece, you spend most of your time in a key
with a diatonic scale. Maybe you're thinking about the chords: I, vi, ii, V7.
The lines and spaces are great for giving you that kind of information. I
guess this "completely obscures the relationship between the pitches". I guess
that's the case if you forget what key you're playing in. The alternative
would be to make the staff fully chromatic, but that sounds like it has a
bunch of disadvantages for most music.

The chromatic staff ([http://musicnotation.org/](http://musicnotation.org/))
looks terrible to me, since it spreads out the notes farther. I've already
memorized all of the diatonic scales, and now the diatonic scales are jumping
around, and the staff covers a shorter interval so you get more of those 8va
or 15ma.

Hummingbird
([http://www.hummingbirdnotation.com/](http://www.hummingbirdnotation.com/))
seems like classical notation, just with more redundancy and the symbols
changed around a little bit. Maybe it's better but it's a minor improvement at
best.

Clairnote ([http://clairnote.org/](http://clairnote.org/)) looks the worst of
them all, with half-steps being half-way between lines and spaces. It looks
like a recipe for making mistakes and misreading music.

I'm not about to say that it's easy to read music notation, just that the
notation seems to have a reasonable amount of logic to it. The idea that
somehow it's the _notation_ that's terrible and music itself would be easier
to understand or play if we just had better notation--I don't buy it. Maybe
someone could convince me.

(Addendum: If you think the chromatic scale is the "right" scale for notating
most music, as in "Chromatic staff" or "Clairnote", as an exercise... why do
you think that the chromatic scale is correct? Is it because it represents
both sets of keys on the piano or the frets on a guitar? Is it because of the
mathematical relationship between the notes? My counterargument, against the
chromatic scale, is presented by Bobby McFerrin:
[https://www.youtube.com/watch?v=ne6tB2KiZuk](https://www.youtube.com/watch?v=ne6tB2KiZuk))

~~~
cturner

        > I hear it bandied around in these articles that music
        > notation is awful. Can someone explain to me why?
    

I've played a lot with alternatives, and disagree. But I can explain why
people might object to it, but then defend the settlement.

There's so much harmonic music in our lives that most people take harmony for
granted. Separately, someone who was interested in only a subset of western
music might decide that western notation was unnecessarily complex. e.g. piano
can be represented with a role, single-voice instruments can be represented
with solfa, guitar with chord sheets.

The Bobby McFerrin video shows us that human music starts with rhythm and a
single-voice pentatonic melody.

Western music is many layers more complex than this.

* Uses a standard template of notes (diatonic scale, aka set of black and white notes on the keyboard). Hence, instead of five tones, we have 12.

* Polyphony/harmony. Multiple voices singing against one another.

* Due to the way the tuning of this scale has been standardised, you can play a piece in the key in C, or in F#, and it is recognisably similar. This is a non-trivial achievement, and is post-Bach.

* The "circle of fifths" and the standardised system of tuning that has been in place since Bach allows us to modulate (change key) during the course of a piece. This is a theoretical accomplishment that builds on the standardised tuning.

The optimisations here are so competently selected and executed that "people"
take all of this for granted. But Harmony is a product of engineering.

Western sheet music has evolved to give us mastery over harmony. Tracking
harmony requires us to track at least two dimensions: (1) time; (2) chords
that sit within it at any point. Then, the system needs to scale to multiple
voices, and support complex metadata such as vocal words, note ornaments and
musical dynamics.

Western notatation is a strong settlement for capturing harmonic music, and
it's optimised for the player's readability. (Consider the rule that says you
are not allowed to have a crotchet rest across the middle of a 4/4 bar; you'd
have to have two semicrotchets and a tie)

Where I think music could be improved is if it were easier to compose. An
analogy: consider if you wrote papers using PostScript. Can it be done? Yes.
Is the output effective? Yes. But what you really want is a composition-
oriented markup that then renders to the form.

I'm currently playing with an approach to composition that feels like literate
programming. You build up melody, voices and chords using ascii
representations like solfa and figured bass, and a DSL to glue the elements
together.

Having done this, I'll want to render it. Render to Clojure/Overtone so it can
be played by a computer; render to sheet music so an orchestra could perform
it.

My opinion is that there is only glaring flaw in the western musical system:
we count notes from one rather than zero. The thing we call "a third" should
have been called "a second". But it's a superficial flaw, an awkwardness that
makes thinking about intervals more difficult than they should be but is
otherwise harmless.

    
    
        > why do you think that the chromatic scale is correct? 
    

It's not that it's correct so much as that it allows you to do certain things
that aren't otherwise possible. The key thing is modulation. As you step away
from the current chromatic-scale settlement, you lose the ability to modulate
between keys.

Consider this alternative-world settlement. All composers draft all music by
pulling levers. The levers drive a pipe organ. It has a strict tonal scale
(the seven white notes on the piano, no black notes). The tones are the same
as they are currently: B to C is a half tone, C to D is a whole tone. As they
play into this pipe organ, the notes are punched into a piano roll, and it can
be played back, or a choir can read it.

In this world, you could represent the early catholic church music, although
you'd have to play some of it at a different key than it was originally
played.
([https://en.wikipedia.org/wiki/Neume](https://en.wikipedia.org/wiki/Neume))

You could get nice harmony going. You could perform the Tallis canon both as a
piece with chords ([https://www.youtube.com/watch?v=_PqV-
iL44wk](https://www.youtube.com/watch?v=_PqV-iL44wk)) or as a true canon
([https://www.youtube.com/watch?v=pMNMKxOyGq8](https://www.youtube.com/watch?v=pMNMKxOyGq8)).

In this sparse world, you'd write stuff in alternative musical modes to get
variety. As they did. You could arrange the Third Mode Melody.
([https://www.youtube.com/watch?v=T8oKEx1-J1w](https://www.youtube.com/watch?v=T8oKEx1-J1w))
But you wouldn't get the consoling resolution on the final note.

Now let's consider an evolution. Say you allowed the levers to swing left and
right a bit so that you could get blues notes by leaning on levers to move
them to side or the other. The world we're in here is kind of equivalent to
the renaissance. You could perform a fair bit of Palestrina and Victoria on
this instrument.

But the Monteverdi Vespers and everything after it is impossible. Bach,
Beethoven, Brahms, Jazz, Beatles, Coldplay and everything in between rely on
extensive use of chromatic scale, particularly for modulation.

Bach lived after the main transition, but is interesting because the tuning
settlement hadn't been worked out yet. He experimented with the frequency
differences between the notes. Some say that (some of) his pieces play better
if you tune your keyboard for "mean temperament" rather than the now-standard
"even temperament".

The rise of electronic music gives us room to go back and play with these
fundamentals a lot. The chromatic scale can be challenged: you could design a
microtonal scale that followed the same rules but had more subtlety, without
needing a craftsman capable of producing a piano keyboard with extra black
keys, or someone to play it. You can represent complex rhythms without relying
on a human that can perform them. And you can change the temperament of a
piece while it's playing.

~~~
dietrichepp
I think this comment may have been written for someone else to read, it's less
of a "head start" and more of a "let's review the fundamentals".

My point about the chromatic scale not being "correct" is that it's
inconvenient to make the chromatic scale the primary scale we use in notation.
We certainly make use of the chromatic scale, but we notate its use with
accidentals and key changes.

That doesn't work well for some of Schönberg's works and it doesn't work at
all for some of Wendy Carlos, but traditional music notation seems to work
pretty well most of the time, and anything based on a chromatic scale would be
inferior most of the time.

~~~
cturner

        > I think this comment may have been written for someone
        > else to read, it's less of a "head start" and more of a
        > "let's review the fundamentals".
    

Early hours of morning here, no doubt flawed.

    
    
        > My point about the chromatic scale not being "correct"
        > is that it's inconvenient to make the chromatic scale
        > anything based on a chromatic scale would be inferior
        > most of the time.
    

Yeah, I strongly agree.

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

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

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

------
ngneer
The interested reader is referred to "Mathematics and Music" by David Wright,
published by the American Mathematical Society. The realization that the
circle of fifths is really there because 5 and 7 (aka -5) are co-prime to 12
was worth the entire thing! Together with 1 and 11 (aka -1) they generate the
group Z_{n} for n = 12...

~~~
ktRolster
That explains why the circle of fifths is a circle that goes through all keys,
but the really interesting thing about the circle of fifths is that a fifth
sounds relatively close to the key next to it, and it doesn't really explain
that, I guess.

~~~
djaychela
Any key a fifth away only changes one note - if you go from C Major (CDEFGABC)
to G Major (GABCDEF#G) then only the F# has changed, and as a result there are
chords which are common between the two keys - any chord without an F in it in
C will be common to G as well. Works the other way if you go 'down' a fifth
(to F), but has a flat instead. (Bb)

~~~
ngneer
More specifically, going up a fifth augments the fourth by a semitone to
become the seventh and going down a fifth diminishes the seventh by a semitone
to become the fourth. In your example, F the fourth became F# the seventh and
B the seventh became Bb the fourth. A fifth away _IS_ a semitone away.

Defining the following eight functions:

    
    
      SEMITONE =  +1 mod 12;  ANTI_SEMITONE = SEVENTH;
      FOURTH   =  +5 mod 12;  ANTI_FOURTH   = FIFTH;
      FIFTH    =  +7 mod 12;  ANTI_FIFTH    = FOURTH;
      SEVENTH  = +11 mod 12;  ANTI_SEVENTH  = SEMITONE;
    

We ask ourselves why the following holds:

    
    
       FOURTH(X) = ANTI_SEMITONE(SEVENTH(FIFTH(X)))
      SEVENTH(X) = SEMITONE(FOURTH(ANTI_FIFTH(X)))
    

Substituting, the answer is clear:

    
    
      X +  5 = X + 7 + 11 - 1
      X + 11 = X - 7 +  5 + 1
    

Keys one fifth apart differ by one note because:

    
    
      1 = 11 + 7 - 5 mod 12
    

Algebraically it seems to work out...

------
matchagaucho
No two Musicians take the same path to learning music.

I enjoyed this perspective. Particularly the focus on sine wave theory and
frequencies (which also serves Audio Engineering).

Quite a departure from the traditional rhythm, melody, harmony, and form
framework. But it works :)

------
ajuc
My main problem with music notation is the difference between C-C# and E-F
interval.

That is - there's no difference, but notations pretends there is one because
that's how we put the keys closer to each other or some other historical
reason.

~~~
theOnliest
This is (sometimes) a legitimate gripe about the standard notation system,
which is that it's optimized for music in a key.

Sometimes there _is_ a difference between E-F and C-C#. In the key of F major,
E-F is the leading tone moving to tonic, which is a diatonic interval (a
diatonic half-step). C-C#, on the other hand, is a chromatic half-step: in F
major, it represents an alteration of scale degree 5 (sol). If you see C# in F
major, there's a good chance it's going towards D, as a temporary leading
tone. This is a useful distinction! The E-F half-step in F major (or C major,
or A or D minor) is completely typical and not at all remarkable, while the
C-C# half-step is much rarer.

In musics where there _isn 't_ a key, you're right that it doesn't make any
sense to draw a distinction between the two. This is one reason that the music
of the 2nd Viennese School (Schoenberg, Webern, Berg) is so impossible to look
at: the structure of the music is obfuscated by the structure of the notation.
Schoenberg was trying to come up with a 12-tone notation system for a while,
but ultimately abandoned it.

~~~
komponisto
>This is one reason that the music of the 2nd Viennese School (Schoenberg,
Webern, Berg) is so impossible to look at: the structure of the music is
obfuscated by the structure of the notation

Wrong. First of all, it's not "impossible" to look at by any means; I think
it's beautiful to look at (as great music usually is).

There is a good reason why Schoenberg abandoned his (briefly-held) ideas about
new forms of notation (and went on to produce another three decades' worth of
music in traditional notation). He, and his disciples Berg and Webern, were
steeped in the Western art music tradition, of which they believed their work
to be a natural continuation. They didn't have a very good theoretical
understanding of the new music they were creating -- because, apparently,
music theory is hard. But they could sense its intimate relationship to its
historical predecessors; indeed, they specifically, cultivated that
relationship, baking it into the music. This, in my view, is why they were
never going to break away from the visual representation of that relationship,
of that continuity -- namely, traditional notation.

The idea that their music is not in a key is widespread, but incorrect.
Inferential distance
([https://wiki.lesswrong.com/wiki/Inferential_distance](https://wiki.lesswrong.com/wiki/Inferential_distance))
precludes me from being able to explain this concisely in a non-misleading
way, unfortunately.

~~~
theOnliest
I was being a bit hyperbolic; certainly it's not impossible to look at, and I
quite like a lot of it. My experiences showing it to students is that people
often find it foreboding on first glance, and part of that has to do with all
of the accidentals used. It doesn't _look_ like music they're familiar with,
even though, as you point out, it comes from the same tradition.

> The idea that their music is not in a key is widespread, but incorrect.
> Inferential distance
> ([https://wiki.lesswrong.com/wiki/Inferential_distance](https://wiki.lesswrong.com/wiki/Inferential_distance))
> precludes me from being able to explain this concisely in a non-misleading
> way, unfortunately.

I'd be interested to hear your thoughts on this. (I have a PhD in music
theory, so the distance may not be as great as you'd imagined.)

~~~
komponisto
>My experiences showing it to students is that people often find it foreboding
on first glance, and part of that has to do with all of the accidentals used

But, of course, other late-Romantic and early-modern music has almost as many
accidentals. (Try the music of Max Reger, to take my favorite example du
jour.)

I personally think accidentals ought to be retrospectively regarded as
implicitly or explicitly attached to every note, with a convention of omitting
them for brevity in passages that stay in a single diatonic area for long
stretches. This 'retconning' of notational convention makes them seem much
less forbidding to me.

>I have a PhD in music theory, so the distance may not be as great as you'd
imagined

(Indeed, I wasn't expecting that!) That does cut down on it significantly,
though it still needs more exposition than can be given in a comment.

Very briefly, the idea is that if (following Schenker and Westergaard, and for
that matter the implications of staff notation itself) you take a line-based
view rather than a chord-based view, tonality doesn't depend on "classified
chords". And if, furthermore, you discard the peculiar non-Bayesian notion of
tonality characteristic of German theory at turn of the twentieth century
(where a key must be "established" or "confirmed" by a cadential ritual in
order to be said to exist), you find that you can always read local keys if
you zoom in enough; and out of these local keys grow the global ones.

There's a pernicious confusion that persists in music theory between _tonal
function_ and the chord-based view (to the point where the former is most
commonly referred to as "harmonic function", as if the two were conceptually
inseparable). But a tone has a scale-degree value independently of its
participation in vertical "chords". This should have been clear ever since
Schenker ; yet it is so poorly understood that, for example, Daniel Harrison
could write a whole book advocating this view, all the while under the
impression that he is doing something new and non-Schenkerian, when in fact
this is part of the core of Schenkerian theory (as the origin of the
circumflex notation testifies).

------
marmaduke
This is only one aspect of music: there's timbre, dynamics, rhythm, groove,
etc.

------
mmcconnell1618
Here's a great video explaining the origins of equal temperament:
[http://www.dailymotion.com/video/x2ismsw](http://www.dailymotion.com/video/x2ismsw)

------
calebm
Looks like almost exactly the same experience I had:
[http://calebmadrigal.com/music-theory-notes/](http://calebmadrigal.com/music-
theory-notes/)

------
anirudt
I preferred the following. Throwback to another ShowHN page.

[https://news.ycombinator.com/item?id=4295714](https://news.ycombinator.com/item?id=4295714)

------
Ericson2314
Octave is not arbitrary, inner ear hairs should resonate more to octaves
above/below (forget which) their maximally resonate frequency right?

------
chrisarensky
A clear map of tonality is used in Mapping Tonal Harmony Pro. If you want to
view the entire harmonic landscape is the best concept out there

------
SubiculumCode
the article is interesting but its style doesn't mesh with title...i.e. Music
Theory for Nerd, but then explains what a sin wave is, and apologizes in
advance for the 'scariness' of log scales. Nerds know and are not afraid of
sin waves and log scales...its as if the author doesn't know his audience.

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

Much more interestingly, this touches on the reason why different pieces were
given keys at all by their composers (think "foo in B flat minor by Chopin").

Musical instruments have to be tuned to a particular key. This is because of
the frequency ratios: the circle of fifths is a lie, (3/2)^N != 2^M _for any N
and M_ , you _can 't_ be an in-tune fourth in one key and third in another,
the frequency values are slightly off.

Thus, some instruments (e.g. Piano) are tuned to be in C, and the other keys
_sound different_ played on that instrument, because the ratios are slightly
different. Other instruments are in B flat (e.g. Trumpet) and the _combination
of them with C-tuned instruments_ sounds interesting. Indeed, stringed
instruments are tuned to N different keys where N = number of strings!

The mathematics of music is _so cool_ , I found the dismissive tone in the
article to be quite unnecessary and irritating.

EDIT: thanks to replies informing me that I'm wrong about instrument tuning.
This means I've been tuning my guitars wrong all these years!

REEDIT: but perhaps I'm right about old-school music with a key in the name?
Since instruments would have been tuned with a tuning fork and then harmonics
pre-20th century according to one reply.

~~~
tgb
Is that true? I thought most instruments (or at least most pianos) these days
were tuned to be equal tempered, so each scale would sound the same (modulo a
pitch scaling).

~~~
endymi0n
Pianos actually often even employ a form of stretch tuning (
[https://en.wikipedia.org/wiki/Stretched_tuning](https://en.wikipedia.org/wiki/Stretched_tuning)
) - which gets even more complex, as the OP only refers to sinus tones, which
don't have any melodic overtones. Things get even more fun with this.

Otherwise, you're right. Historic tunings often tried to optimize for a
certain base scale (often times C major), while leaving behind a really bad
sounding tuning at the other side of the circle of fifths (
[https://en.wikipedia.org/wiki/Wolf_interval](https://en.wikipedia.org/wiki/Wolf_interval)
).

Modern pianos are usually equal tempered, which is the strategy of leaving
every scale equally bad off - but on the other hand being able to play in all
keys without sounding too bad (The famous "well-tempered piano" is referring
to this historic change in tuning strategies with small pieces written in all
different kinds of keys, all of which would have sounded weird before).

The more I learned about tuning, the more complex, beautiful and sad this
topic became for me: [http://blogs.scientificamerican.com/roots-of-unity/the-
sadde...](http://blogs.scientificamerican.com/roots-of-unity/the-saddest-
thing-i-know-about-the-integers/)

It's fascinating and even a little philosophical that we can get so close to
perfection with the 12 step tuning - but never, ever completely attain it.

By the way - That's part of the reason why choirs and orchestras often sound
so good: As most other non-fixed-string instruments are able to do a slight
adjustment in pitch, they can actually play _perfect_ intervals in any chord,
dynamically adjusting to any scale.

------
acjohnson55
This is an interesting article, which touches on a lot of interesting musical
topics. However, it's wrong on so many levels that I would never recommend it
to someone who wants to understand music theory. I'll try to summarize.

I spent a long time researching where Western music "comes from". One
important thing to understand is that the key feature of Western music is
harmony. This is not the case for most other musical traditions. Harmony comes
from a layering of independent melodic lines (called voices), such that they
evolve in ways that create and resolve tensions. This is called counterpoint.

The ways in which these tensions resolve are called cadences, and classical
music was all them. Musical form arises around how a piece of music is
organized to have an arc (and arcs of arcs) that leads to a cadence. It's kind
of like how there are only a few basic narrative structures in storytelling,
yet infinite variations in types of stories.

The tension between the sounds in concurrent melodic voices emerges from the
structure of the sounds. Harmonic sounds, like ones produced from wind and
string instruments, sound good (consonant) when their fundamental frequencies
are simple ratios. This is largely because of how the human auditory system
works.

Our scales and chords come from picking collections of notes that approximate
simple ratios. We deal with approximate ratios using nth-roots for two
reasons. First, simple ratios don't compose well -- stack them (i.e. multiply
them) and you get ratios that aren't so simple. nth-roots do stack nicely.
Second, it lets us merge nearby flat and sharp notes. This wouldn't be
possible if we used real ratios. That merger, known as enharmonicity, lets us
do all sorts of cool compositional tricks that are generally considered more
worthwhile than the slight improvement in sound quality we'd experience by
using the real ratios. There are various ways to pick just which notes merge,
but by far the most popular approach is to use 12 notes.

So, this is where we got our notes and scale from. This happened around the
year 1800, give or take a couple decades.

The history of Western music since that point in time largely revolves around
coming up with new ways to play with those 12 notes, in many ways erasing the
musical features that caused the 12 notes to emerge in the first place. This
is why the system of note names, with the flats and sharps, feels rather
arbitrary today.

Unfortunately, there are not a lot of great resource that tell this full
story. The two books I found most enlightening in piecing together the "real
story" are:

\- Tuning, Timbre, Spectrum, Scale: [https://www.amazon.com/Tuning-Timbre-
Spectrum-William-Sethar...](https://www.amazon.com/Tuning-Timbre-Spectrum-
William-Sethares/dp/1852337974) \- A Geometry of Music:
[https://www.amazon.com/Geometry-Music-Counterpoint-
Extended-...](https://www.amazon.com/Geometry-Music-Counterpoint-Extended-
Practice/dp/0195336674)

~~~
Manishearth
> This is not the case for most other musical traditions.

Do you have examples of musical traditions that don't do this? Both the Indian
and Chinese musical systems do something similar.

\----

I'm not really sure if this post is "wrong". It gets the chronology mixed up a
bit -- talking of twelfth roots of two before talking about ratios, but I
thought that was just done to help explain things by reverse-engineering them.
I don't think it claims that the notes "came from" roots of two.

~~~
acjohnson55
I'm not an expert, but Indian classical music isn't known for having harmony
in the same sense [1]. Modern Indian music is influenced by Western music. I
know even less about Chinese music, which isn't mentioned in that article. But
in all the sources I've read, the concept of deeply developed chordal harmony
is pretty unique to the Western music tradition.

[1]
[https://en.wikipedia.org/wiki/Harmony](https://en.wikipedia.org/wiki/Harmony)

~~~
Manishearth
Right, it doesn't involve playing notes together the same way it's done in
Western music.

I guess I misinterpreted what you were saying -- the Indian notes are still
derived by looking at notes which sound harmonious when played together,
roughly the same way Western notes are derived. They are not usually played
together on the same instrument like you would with a piano or a guitar.
Harmony between vocal parts is not uncommon as is harmony with background
instruments like the tanpura. But yes, Indian music doesn't have harmony in
the same sense as Western music. But the origins of the notes are very
similar.

(I'm talking about classical Indian music, btw)

~~~
acjohnson55
Yep! Many traditions got their notes the same way Western music did. The
_Tuning, Timbre, Spectrum, Scale_ book does a great job illuminating this.

~~~
Manishearth
Ooh, that sounds like a fun book to read!

I spent a lot of time studying this myself because I was fascinated with the
fact that these 12 notes in particular seemed ubiquitous in many music
systems, and that there wasn't any immediately obvious reason as to why these
notes sounded pleasing. Ultimately learned some music theory in the process
(I've studied Indian and some Western music, so a lot of it was stuff I
already knew, but not in concrete form), but never in a very structured
fashion -- more along the lines of asking people and reading snippets.

Put this book on my list, sounds like a structured way to learn something I've
been super interested in for years.

------
princeb
if you were hoping for a nerdy treatise on musical rhythm, harmony, form,
cadences, sequences and motifs, and evolution from baroque to classical to
romantic to modern... well this is not it.

edit: just a comment about the temperament- before the octave was equally
tempered intervals were rational fractions of the root. after the equal
temperament was developed instruments could go across different keys without
sounding too dissonant. the well tempered clavier was a series of preludes and
fugues that JSB wrote for the equally tempered scale that goes across all the
keys on the same instrument (the prelude in c major which is the very first
piece in the collection is the famous one in Forrest Gump). but sometimes you
want to return to the pure temperament - like if a passage focuses on a
particular key for a while - and in the brass and wind instruments it is easy
to squeeze the embouchure a little bit to slightly tweak the tuning of a
particular note.

~~~
nazgul17
I would love something like that. Any pointer?

~~~
Ericson2314
A large part of it hasn't been researched very mathematically, to my
knowledge. If you google music+grammar you can get some stuff with a
linguistic approach, which may be less empirically grounded but certainly a
richer theory than the alternatives. And don't get me wrong, science is good
but as musician mathematician a richer theory that overreaches reality is much
more fun, and besides most of this stuff is cultural not immutable so a wrong
theory may _become correct_ if is influential.

A few stray ideas that may or not be correct:

\- Grammar of common-practice harmony is probably largely left-branching (see
[https://en.wikipedia.org/wiki/Branching_(linguistics)](https://en.wikipedia.org/wiki/Branching_\(linguistics\))).
This is why so such music makes a surprising transition, then resolves it, and
it's orthodox overall. Music is about where it's going.

\- For a first approximation rhythm models the rational numbers, because all
durations exist in a ratio to other duration.

\- Now the rational numbers are infinitely dense and clearly humans can't do
that. So there is a minimum resolution which has been wonderfully termed the
tatum. Whoever did this however may or may not have realized that there may be
a few minima whose join (as in lattices, I'm sure it's a lattice but I never
leaned which one) isn't realized. E.g. you might have 8th notes and triplet
8ths but never triplet 16ths (the join).

\- Form has all these stupid names when clearly it's a tree and similar
structure repeated on every level.

\- People sometimes try do do harmony wit a markov model. This sucks because
how the chords transition depends on the form, which I believe must largely
(needed to be orthodox/fluent) have a fair regular rhythm (avoid those 5-bar
phrases). I think something like a ruler function
([https://en.wikipedia.org/wiki/Thomae%27s_function](https://en.wikipedia.org/wiki/Thomae%27s_function))
based on time + transitions would be a much better and relatively cheap model.
If all the fancy grammar stuff is legit, then this too would be woefully
inadequate.

\- That said, the "meaning" is definitely in the transitions not the states
(production rules in grammar case). A chord with no context has very little
meaning (in equal temperament everything can be transposed with 0
informational change except for timbre). Given a key, a chord still has
relatively little meaning (also 1 and only 1 key is a reduction, look up
"tonicization", there is probably a stack of broad-specific key-like things
forming a context). now given a trace of chords, the meaning comes out---
tension and release as they say.

\- For form and rhythm, 2 branching factor is the default. Everything else is
extra work. Probably even in West Africa and other places where the
polyrhythms are mad good.

\- Emotional relationship definitely changes over time within a culture. Up to
the 18th century major=happy minor=happy was far from universal. This is
probably why even in popular culture old music is stereotypically all dark and
serious.

\- The big 5 components of (at least Western) music are (in no particular
order) Melody, Harmony, Rhythm, Form, Timbre. Timbre has become more important
over time, even before recorded music. Call me a cranky old cultural
conservative, but you could say that Western Society is loosing it's
collective understanding-of/ear-for Western music, and Timbre is the easiest
to appreciate due to its lack of rich structure (as far as I can tell). And of
course recorded and synthesized music means we can timbregasm all day.

~~~
zhemao
> Melody, Harmony, Rhythm, Form, Timbre

I think dynamics (or at least dynamic contrast) is also an important element.

> Timbre is the easiest to appreciate due to its lack of rich structure

It does have a rich structure, we just don't really understand it yet. Timbre
is essentially the linear combination of harmonics of a single pitch. You
could express it as a Fourier series. But we don't have a good model of how to
play with specific timbres, since acoustic instruments don't really allow
fine-grained control of timbre.

~~~
Ericson2314
Fourier analysis indicates there is a huge space of timbre, but not
necessarily a rich one. In my experience while timbres can be annoying, they
aren't _wrong_ like a an arbitrary harmonic progression is. Harmony being such
a minefield is what makes its space so rich.

~~~
zhemao
Ah, I see what you mean. I guess you can think of timbre as a mixture of
dynamics and a very simplified version of harmony in which you can only have
intervals of an octave.

~~~
anentropic
you can have fifths and thirds too I think... you can pluck those harmonics on
a single guitar string.

I don't know if any other intervals are possible as harmonics... perhaps the
ratios for other intervals are too big to make it practical, they are too weak
to sound

------
andrepd
TLDR: a person who knows nothing about music or music theory gripes about
music and music theory. Admittedly knows nothing, right off the bat and
several times throughout the text. How can you declare something to be
"terrible", "the worst notation for anything ever", and so forth if you
admittedly know nothing of the subject? I know nothing about databases, would
reading the Wikipedia page on NoSQL entitle me to declare it "absolute
garbage"? o.O

~~~
phyzome
It can be clear to a newcomer that a system has problems without needing to
understand everything, or even very much. Sometimes the newcomer is wrong, but
a quick scan of the comments here indicate that a number of musicians agree.

~~~
rvense
Funny, I skimmed the comments and actually came to the conclusion that I did
not need to write another one stating that music notation, not unlike many
other complicated tools, is perhaps at first sight arbitrary and odd, but
after working with it for a while you find that all the little quirks, like
having multiple ways to write the same note, are actually very helpful.

------
torrent-of-ions
He says the intervals are arbitrary, but they are not, or at least not as
arbitrary as that. An octave is 2x the pitch, a fifth is _exactly_ 3/2x the
pitch, a major third is _exactly_ 5/4x the pitch etc.

This is fine for instruments which can play an arbitrary pitch like a violin
or trombone. But for fretted instruments like guitar, or ones which use a
different oscillator per note like piano, xylophone or harmonica, one has to
make a decision. Either you tune it to play perfect intervals and can
therefore play in only one key, or you tune it to a compromise which can play
in any key and sound OK, but no key "perfectly". Equal temperament is one such
compromise and is described in the article.

For instruments which play often play long chords like strings, the difference
between a perfect interval (as an orchestra would play) and an equal
temperament imperfect interval (as a general purpose synthesiser will play) is
stunning. Some synthesisers like those from Access have modes to automatically
correct the tuning of the third, fifth and maybe seventh harmonics to make
them perfect.

------
Cozumel
What a ridiculously uneducated article! The very first sentence starts 'I
don’t know anything about music.' and IT SHOWS!!

Music notation has evolved over several hundred years, it's not arbitrary or
random. The author should take some music classes before trying to write about
it.

------
paracarro
> body {width:100%; max-width:1120px;...

Is this good?

