These are all good exercises that help you build a solid foundation, but they can sometimes cause motivation to dip being somewhat clinical in nature.
So what I usually do is compile a list of melodic hooks from popular songs my students enjoy. Every so often, we’ll play them and let the student try to pick them out on the piano or their instrument of choice. I find that the satisfaction they get from being able to recreate a familiar pop‑culture melody really helps spark their interest in getting better at playing by ear, which in turn motivates them to stick with the exercises.
Shameless plug but I built a unique game specifically to help some of my more classically trained friends get better at playing piano by ear.
It's a free piano game in the style of the old "Simon" toy which presents players with increasingly longer sequences of musical notes and challenges them to reproduce the sequence using either an on-screen piano or connected MIDI keyboard. It also works with acoustic instruments through the mic.
Just testing out practice mode, I found what I really wanted was to be able to stay at a certain level until I felt I was getting good at sequences of that length, not immediately get pushed to the next level every time even when it took me 8 tries to get the 4-note sequence right. Give me a chance to feel like I'm improving! Don't just keep giving me harder things when I keep struggling with the existing ones.
It already has that feature! :) It’s just not very obvious. If you click the small lock icon near the top, it will snap and to that difficulty so you can practice only sequences with that specific number of notes.
I remember in jazz class the instructor had a stack of jingles that showcased particular "weird" intervals. The only one I really remember now is the "Alway Coca Cola" jingle and a M6, but this was a _long_ time ago.
I'm completely new to ear training. Could you give some advice on what a newbie should think while doing this? For example, should I try to sing the thing in solfeges in my head, or it's considered bad practice? And if I do, should I sing the first note as Do?
So the only problem is that until you've internalized the intervals a bit better, you might get frustrated trying to sing out solfege since you might say "re" when the note was "mi" in the context of the key signature and that might reinforce a bad intervalic relationship. However you could still hum/whistle the pitches as an assistive tool.
If you’d like to make things a bit easier, you can go into the options and restrict the key signature. That way you can keep it simple and just practice in a more common keys like one major scale like C major and its relative minor, A minor.
Where I really recommend "singing" out each note is when I'm teaching my students to improvise on the piano since it creates a sort of intentionality about what you’re about to hear and sing.
For example, if they had a chord progression or melodic idea in mind but accidentally played a wrong note, they’ll notice right away because what they’re singing won’t match what they’re playing.
Whereas if you don’t sing or whistle the notes as you play your instrument, you might not notice that you’re drifting off from what you actually intended to play because within the confines of the key signature it might still sound melodically acceptable (if that makes sense).
This is really simple and great!! Thanks for not stuffing it with ads.
Is there a way to make it work a bit better for phones? On mobile Safari, just tapping to enable sound doesn’t seem to work until I reload and tap again.
I think this is smart. I have never done ear training apps because I just don't like to learn music that way - it doesn't "stick" for me.
I like to learn in the context of a song. Here's what a melody sounds like when you start it over the 1 of a chord. Here's a melody when you start it on the 3 over a chord. But, again, in the context of a known song.
I just don't think "non-musical" exercises have ever moved me forward as a musician, if that makes sense.
I’m with you. I rarely use mechanical drills, even though I recognize that they can sometimes have value (cough Hanon cough), especially when you’re focusing on ergonomics. I tend instead to focus on things my student already enjoys, because it gives them a grounding.
Another trick I like to do is take a popular song, rip out the melody, and keep the chord progression. Then I’ll usually scaffold a nice accompaniment using Band-in-a-Box so the student has something looping in the background while they try to piece out the original melody themselves on their respective instrument.
That can sometimes give them more guidance, since it locks them into a specific key signature and helps them feel the flow and explore the space.
Tip: you don't have to recognize sounds in order to compose music.
I have aphantasia and can't memorize sounds or recall them. For decades I thought I'm deaf (Ockham say hi). But I picked up piano, play for 3 years, can't discern C from G if my life depended on it but my friends tell me I'm pretty decent composer.
Writing this so people don't get disappointed about themselves just because they can't pick ear skills.
I've been doing these exercises regularly for years and haven't made much progress ;). I still perform music regularly and get positive feedback. I do hope that some day I'll get to where these exercises seem easy - I haven't made much progress, but I do make a tiny bit at a time.
can you tell if a piano is out of tune? I'm curious since I don't know anyone with aphantasia. Can you tell if someone is playing your composition correctly?
I was expecting this was a tool to help people who couldn't distinguish sounds in a noisy environment to train their ears. But its for musical training. (which may or may not help?)
Hey this seems like a nice tool. I would request if you can also add examples/demo for a listener before they begin the test, like intervals(what is P5, m6,etc how they sound), chords(major/minor chord in different octave), etc. That way listener would know about each facet of music, and then they can take the test.
This site insisted on having write access to my connected MIDI devices, which is a bit concerning as it's not required for what the site actually does.
They're asking for midi device control & reprogram, my guess is that for some of the exercises you can use your own midi keyboard to input the note (dictation etc.)
I figured the same, but I did not encounter such exercises, and regardless having the ability to send sysex to my MIDI devices is not required for such a thing.
I think we would be a lot more musically verbal as a society if our musical notation had a more objective foundation in math and reason. For example, A to B is a different distance than B to C. We have a 12 note system with only 7 names for them; 12 names would make sense, and even 6 names would make sense, but 7?
We could be teaching notes to children objectively like how we teach colors, but we're not.
Mapping twelve letters onto a piano keyboard would then look something like this:
B D G I K
A C EF H J LA
Which means an A major scale in this notation would be ACEFHJLA, which is actually less intuitive than understanding the circle of fifths etc. and arriving at ABC#DEF#G#A (to use this universe's notation)
To +1 the "no objective foundation": browse music theory research a bit! There's a ton of caveats and poor-sounding fits and whatnot for literally every approach, and there's endless discussion of it. E.g. have a MinutePhysics take on how the common 12 note western scale falls apart: https://www.youtube.com/watch?v=1Hqm0dYKUx4
(maybe to clarify: there are objective aspects, in that sound is measurable. but there is nothing like a "grand unified theory" that covers all music, nor are roughly any of the popular ones internally consistent - it's far, far too varied for that, and physics often doesn't allow the desired consistency, causing more variety)
In your traditional system, if you want to play something a step up, you have to actually think about it; which notes will now become sharps, which won't, etc.
In my system (A though L, or more simply, 1 through 12), you simply add 2 to each note. It's easier to work about and isn't as rigidly defined by the culture it came from.
You just described https://en.wikipedia.org/wiki/Whole-tone_scale. Nothing stopping anyone from writing and playing music in such a scale, but it won't help you with nearly all the music you are likely to be familiar with.
Sure, if path dependency was were not a thing, this might make more sense. But it takes an extraordinary amount of time to really get good at music and you don't want to be the only person who speaks a completely different language to the people around you. So it makes sense to stick with what everybody speaks.
I skimmed the article but the parallel did not seem obvious to me. I play 'string' instruments (bass guitar, guitar, open tuned resonators) but I guess my approach is fairly standard - I read music but don't site read real-time very well. But When I "think" about music it's the same way most western musicians think about music (I assume) and that's diatonically in relation to the major scale as a reference point.
If you want an excellent explanation of why proposals for new notation systems like you’re suggesting have been developed (and failed) Tantacrul has made an excellent video describing the history and tradeoffs that lead us to our current system:
He’s product lead at MuseGroup developing notation software and his expertise lies at the intersection of music composition, UX design, and programming.
>We could be teaching notes to children objectively like how we teach colors, but we're not.
Do you mean trying to teach all children perfect pitch even though society has no expectation of that? Unlike knowing at least your primary colors which is expected of everyone. I suspect that could be unnecessarily stressful for many.
Or do you mean as some kind of metaphor or analogy? If the latter, I think it would be quite confusing as there are aspects of vision and hearing that are quite different. Pitch classes have no analog in vision that I can think of. Color vision is roughly 3 dimensional but sound is not. The aspects of timbre don't map to color.
I think that understanding music theory does require work. It emerges from physics and physiology and a very long history including a bunch of culturally specific things. Did your ancestors make music with long skinny strings or pipes with nice integer-ish overtones? Here are some tuning systems for you (among them the set of C-D-E-F-G-A-B you mentioned). Did they use bells or gongs with decidedly non-integer ratio overtones? Here's a set of different systems for you!
Anyway, if you have a mapping/analog/metaphor you think is useful between music or sound and color I would be interested to hear it / see it!
None of this is objective. The construction of our scale is subjective, other countries use different scales. Even with this same scale, there are multiple different tuning systems. There is micro-tonal music. Musical keys are arbitrary too. We teach the way we do because it is a culmination of musical history, our particular western musical history, and it's own arbitrary decisions that western musicians have made over the past hundreds, even thousands of years. If you like, there is also solfege, which replaces letter names with just sounds, "do re mi" etc,
Also what makes you think colours are objective? Different languages have words for a single colour that map onto multiple other words in other countries, does not seem very objective to me, unless you are talking about notating colours as hex or RGB number values, or wavelengths (which we don't do in natural language)
You think music— essentially expressing ideas and emotion through sound— isn’t approachable enough to children because the notation lacks a solid foundation in math? I very much disagree. In fact, I think teaching children to experiment more with musicality well before they’re introduced to things like notation and formal theory would do far more to pique their interest. I happily played guitar for a decade before I learned to read music.
Indeed, likely the most widespread teaching method today is the Suzuki method, and it doesn't introduce reading until after at least the first couple of years. There are books, but they're more for the parents to follow along than for the kids.
American teachers were horrified by this idea when I was a kid. But the Suzuki method has been successful, and I think it has raised the level of playing overall. Many famous musicians self-identify a "Suzuki kids." On the other hand, many of them admit to not being the strongest readers, but reading takes practice. You can also pick up repertoire by following the sheet music while listening to a recording. Like many skills, it fades if it isn't used. I'm fortunate to be a fluent sight-reader, but not a virtuoso.
In my view the notation is what it is. Changing it would be hard. "Standard" notation creates a kind of symbiosis between composers and players. If a composer uses a nonstandard notation, nobody will play their stuff. And the standardization is why musicians can learn the skill of reading.
Do you have much experience reading musical notation?
I've found that engineer types tend to immediately bristle at the weird parts of how notes are named because the system seems really kludgy until you realize that there's actually a utility in the weirdness - namely, that scale patterns look roughly similar in any given key and so sight reading is counterintuitively easier with the current system than it would be in a system which assigned a different position on the staff (or a different name) to each note.
Furthermore - we have seven note names because there are seven notes in the major scale, so changing this count would definitely not make sense.
To be clear there are definitely warts in the current system, lots of confusing stuff around enharmonics. But there's definitely babies in the bathwater and any alternate system would not want to toss them out.
Yeah, I work in programming languages and always liked the idea of notation do-overs. But after getting more into music these days, I've returned to learning and appreciating musical notation. For better or for worse, it is the standard way of writing music. If you want to get serious at music, you need to know it.
There's a lot to hate about musical score, but the A-G notes and sharps and flats aren't all that bad once you realize that everything is based on the 7 note diatonic scale. In C major, it's just the names of the letters with no sharps or flats. On the piano, C major is just the white keys, which will get you pretty far--tons of songs are in C major. You have to remember B-C and E-F are the short intervals, and memorize the 2-2-1-2-2-2-1 semitone pattern, but after that, a lot follows. Then minor is just starting a different note in this pattern, as are all the other modes. There are other scales too, but this one main pattern is going to cover 98% of all music you run across.
There's a huge amount of stuff that gets unlocked when you just give up fighting the standard and instead learn to go with it. Music is a language, and the way we write it down is maybe a little suboptimal, but then again, the "optimal" way to write it down has a maximum on how much better it could possibly be.
I do have a beef with the notation for rhythm, because as it is, the standard musical notation is just a shorthand form for fitting more music horizontally. For computer-based music, I find it a lot easier to follow a display where horizontal length is proportional to time. We've got infinite screen space, so no need to compress anymore.
There's also a huge amount of math behind music that is fascinating.
The first-approximation engineer realization about music (which I suspect the GP is going off of) is "okay, there are 12 notes in the chromatic scale, each octave doubles frequency, therefore the frequency ratio between two adjacent notes is the 12th root of 2 and we should just have 12 names for the notes". This is what's called an "equal-tempered scale"; the gap between each note is the same ratio, and you have a simple geometric progression upwards.
Except we don't actually have an equal-tempered scale. If you try to play on an equal-tempered scale, it'll sound subtly "off", and certain chords will result in "beats" (pulsing) where the frequency ratios are off just enough to cause an unpleasant modulation in loudness.
The modern diatonic scale is based on the circle-of-5ths [1], where the fundamental ratio is the 5th at 3/2 the frequency. It works like this because now chords are an even multiple of frequencies, while you would get an irrational number with the equal-tempered scale. Going up from the root (C), the next 5th up is G at a ratio of 3/2. Then you go up to D (9/4); when you reduce this to lowest terms because you've ascended a full octave, it gives a ratio of 9/8, which is one whole tone above. Next 5th up is A (27/16), which is the ratio in frequencies of a 6th. And then you get E (81 / 32 = 81/64), a major 3rd. And so on. The frequency ratios of the diatonic scale come from repeatedly reducing powers of 3/2 to lowest terms after dividing out the octave.
We do have an equal temperament scale! That’s what most music uses. What we don’t have is just intonation which uses simple ratios/intervals for the notes. It’s just intonation without the beating, equal temperament, which was listen to all day has the beating but we’re adjusted to it.
Do we? My impression was that strings, woodwinds, choirs instinctively tend to use just intonation, and Wikipedia entries for both just intonation [1] and equal temperament [2] seem to back that up. That's why symphony orchestra players will often have a different flute, clarinet, or oboe for different tunings. It's just fretted instruments like guitars that are by nature equal-tempered.
On a side note, both Wikipedia entries reinforce my original point that the mathematics of this is fascinating.
Not just fretted instruments, pianos as well. Anything which lacks ability to adjust tuning on its own, while being played (things that wind instrumenst and non-fretted strings can do), for example a steel pan drum, would be equally tempered (assuming you want to be able to play in all possible keys).
>That's why symphony orchestra players will often have a different flute, clarinet, or oboe for different tunings.
Not sure what you are referring to here? Clarinets don't come in different "tunings" unless you mean different keys - like Eb or Bb clarinet, but those aren't there for intonation, they play in different ranges of pitch than one another.
Edit: in addition, you don't HAVE to equally temper a guitar. You can choose just intonation. The problem is that you can only have just intonation on a single KEY for that instrument. So if you tune justly to C major, a key like B major, will sound horrendous!
I agree that there are some quirks to music theory but ultimately I think it's a very good system that was been refined over hundreds of years.
As for your point about A->B being a larger interval than B->C. There are two half-steps in the white keys (B->C and E->F) because there are two half steps in the major scale! This way, you can play C to C with all white keys and get a major scale.
A major scale is probably one of the most fundamental building blocks in western music theory and it's encoded right onto the keyboard layout itself.
The oddities of music theory are no more strange than all of the strange things in the English language that we just shrug about and move on once we learn it.
We actually have multiple names for all the notes - which have a 'reason' to exist. A B-double-flat and an A-natural, and G double-sharp exist, distinct for notational purposes... yes, it sounds dumb. Music IS arbitrary in a lot of ways.
For example: 12 tone equal-temperament was chosen/invented (nearly) (by Bach) over just intonation because of 'musical gags' like https://en.wikipedia.org/wiki/The_Musical_Offering (also written by Bach).
Music making neat, orderly, mathematical sense is the struggle, and reality doesn't play nice with harmonics like we would like... (much like with irrational numbers throwing a wrench with Pythagoras' ideals) so stop being a Pythagoras :p
Music IS weird: no matter how you try to quantify it.
It exists in some California schools, but this is one of those things that exists because in some districts the parents setup a 501(c)3 whose only job is to fill the gap left by Prop 13 and Prop 98. My kid's had it every year, but in kinder it was a parent volunteer teaching it and in 1st/2nd the music teacher was entirely funded out of parent donations.
Literally everyone in my country did, and nobody knows theory except a subset of musicians, not even all of them. Thus highlighting my point that the theory is inadequate, subjective, and immature in nature.
Whatever else music theory is, "immature" isn't one of them.
Western music theory has evolved over literally thousands of years. You can put a very rough start of it to Pythagoras, around 550BCe ish, which gives us 2,500 years of evolution and refinement.
But even if you want to start with the popularity and adoption of the major scale, that was around 1500CE ish, which gives us a solid 500 years. It handles many, many corner cases quite gracefully.
It undoubtedly has its quirks, but any notational system for this will also have its quirks (cf, the difference between systems of intonation). There is just no way around it.
I agree, but it seems this is something that will never change, because of tradition.
I tried many times to "understand" music rationally, because I kept people use the term "music theory". I reached a conclusion that there is no "theory" whatsoever: music notation is a hodgepodge of various traditions stacked one on top of the other (we started with 8 notes but then realized that 12 would be better, for example, hence all the mess with flats and sharps). I actually feel better now knowing that you just have to accept it for what it is and go with the flow :-)
So what I usually do is compile a list of melodic hooks from popular songs my students enjoy. Every so often, we’ll play them and let the student try to pick them out on the piano or their instrument of choice. I find that the satisfaction they get from being able to recreate a familiar pop‑culture melody really helps spark their interest in getting better at playing by ear, which in turn motivates them to stick with the exercises.
Shameless plug but I built a unique game specifically to help some of my more classically trained friends get better at playing piano by ear.
It's a free piano game in the style of the old "Simon" toy which presents players with increasingly longer sequences of musical notes and challenges them to reproduce the sequence using either an on-screen piano or connected MIDI keyboard. It also works with acoustic instruments through the mic.
https://lend-me-your-ears.specr.net
Side note but I'd love to see a nicely printed stack of physical cards with popular melodic hooks/jingles, the demonstrated intervals, notation, etc.
I'm completely new to ear training. Could you give some advice on what a newbie should think while doing this? For example, should I try to sing the thing in solfeges in my head, or it's considered bad practice? And if I do, should I sing the first note as Do?
If you’d like to make things a bit easier, you can go into the options and restrict the key signature. That way you can keep it simple and just practice in a more common keys like one major scale like C major and its relative minor, A minor.
Where I really recommend "singing" out each note is when I'm teaching my students to improvise on the piano since it creates a sort of intentionality about what you’re about to hear and sing.
For example, if they had a chord progression or melodic idea in mind but accidentally played a wrong note, they’ll notice right away because what they’re singing won’t match what they’re playing.
Whereas if you don’t sing or whistle the notes as you play your instrument, you might not notice that you’re drifting off from what you actually intended to play because within the confines of the key signature it might still sound melodically acceptable (if that makes sense).
Is there a way to make it work a bit better for phones? On mobile Safari, just tapping to enable sound doesn’t seem to work until I reload and tap again.
I like to learn in the context of a song. Here's what a melody sounds like when you start it over the 1 of a chord. Here's a melody when you start it on the 3 over a chord. But, again, in the context of a known song.
I just don't think "non-musical" exercises have ever moved me forward as a musician, if that makes sense.
Another trick I like to do is take a popular song, rip out the melody, and keep the chord progression. Then I’ll usually scaffold a nice accompaniment using Band-in-a-Box so the student has something looping in the background while they try to piece out the original melody themselves on their respective instrument.
That can sometimes give them more guidance, since it locks them into a specific key signature and helps them feel the flow and explore the space.
I have aphantasia and can't memorize sounds or recall them. For decades I thought I'm deaf (Ockham say hi). But I picked up piano, play for 3 years, can't discern C from G if my life depended on it but my friends tell me I'm pretty decent composer.
Writing this so people don't get disappointed about themselves just because they can't pick ear skills.
I've been working on a voice trainer that uses DSP to analyze a signal and do real-time pitch tracking: https://htmlpreview.github.io/?https://github.com/titzer/Rif...
The latter I've been using to suck just a tiny bit less at singing.
https://play.google.com/store/apps/details?id=com.evilduck.m...
It has interactive exercises and singing practice
We could be teaching notes to children objectively like how we teach colors, but we're not.
Mapping twelve letters onto a piano keyboard would then look something like this:
Which means an A major scale in this notation would be ACEFHJLA, which is actually less intuitive than understanding the circle of fifths etc. and arriving at ABC#DEF#G#A (to use this universe's notation)And that's before even getting into completely alternate approaches, or how strongly harmonics affect perception: https://www.youtube.com/watch?v=VAgXpCK_4gA or https://www.youtube.com/watch?v=AZ8qZCGg4Bk
(maybe to clarify: there are objective aspects, in that sound is measurable. but there is nothing like a "grand unified theory" that covers all music, nor are roughly any of the popular ones internally consistent - it's far, far too varied for that, and physics often doesn't allow the desired consistency, causing more variety)
Well, there is a number of "objective" factors which play a significant role. For example, see: https://www.youtube.com/watch?v=tCsl6ZcY9ag
In my system (A though L, or more simply, 1 through 12), you simply add 2 to each note. It's easier to work about and isn't as rigidly defined by the culture it came from.
Sure, if path dependency was were not a thing, this might make more sense. But it takes an extraordinary amount of time to really get good at music and you don't want to be the only person who speaks a completely different language to the people around you. So it makes sense to stick with what everybody speaks.
https://youtu.be/Eq3bUFgEcb4
He’s product lead at MuseGroup developing notation software and his expertise lies at the intersection of music composition, UX design, and programming.
Do you mean trying to teach all children perfect pitch even though society has no expectation of that? Unlike knowing at least your primary colors which is expected of everyone. I suspect that could be unnecessarily stressful for many.
Or do you mean as some kind of metaphor or analogy? If the latter, I think it would be quite confusing as there are aspects of vision and hearing that are quite different. Pitch classes have no analog in vision that I can think of. Color vision is roughly 3 dimensional but sound is not. The aspects of timbre don't map to color.
I think that understanding music theory does require work. It emerges from physics and physiology and a very long history including a bunch of culturally specific things. Did your ancestors make music with long skinny strings or pipes with nice integer-ish overtones? Here are some tuning systems for you (among them the set of C-D-E-F-G-A-B you mentioned). Did they use bells or gongs with decidedly non-integer ratio overtones? Here's a set of different systems for you!
Anyway, if you have a mapping/analog/metaphor you think is useful between music or sound and color I would be interested to hear it / see it!
American teachers were horrified by this idea when I was a kid. But the Suzuki method has been successful, and I think it has raised the level of playing overall. Many famous musicians self-identify a "Suzuki kids." On the other hand, many of them admit to not being the strongest readers, but reading takes practice. You can also pick up repertoire by following the sheet music while listening to a recording. Like many skills, it fades if it isn't used. I'm fortunate to be a fluent sight-reader, but not a virtuoso.
In my view the notation is what it is. Changing it would be hard. "Standard" notation creates a kind of symbiosis between composers and players. If a composer uses a nonstandard notation, nobody will play their stuff. And the standardization is why musicians can learn the skill of reading.
I've found that engineer types tend to immediately bristle at the weird parts of how notes are named because the system seems really kludgy until you realize that there's actually a utility in the weirdness - namely, that scale patterns look roughly similar in any given key and so sight reading is counterintuitively easier with the current system than it would be in a system which assigned a different position on the staff (or a different name) to each note.
Furthermore - we have seven note names because there are seven notes in the major scale, so changing this count would definitely not make sense.
To be clear there are definitely warts in the current system, lots of confusing stuff around enharmonics. But there's definitely babies in the bathwater and any alternate system would not want to toss them out.
There's a lot to hate about musical score, but the A-G notes and sharps and flats aren't all that bad once you realize that everything is based on the 7 note diatonic scale. In C major, it's just the names of the letters with no sharps or flats. On the piano, C major is just the white keys, which will get you pretty far--tons of songs are in C major. You have to remember B-C and E-F are the short intervals, and memorize the 2-2-1-2-2-2-1 semitone pattern, but after that, a lot follows. Then minor is just starting a different note in this pattern, as are all the other modes. There are other scales too, but this one main pattern is going to cover 98% of all music you run across.
There's a huge amount of stuff that gets unlocked when you just give up fighting the standard and instead learn to go with it. Music is a language, and the way we write it down is maybe a little suboptimal, but then again, the "optimal" way to write it down has a maximum on how much better it could possibly be.
I do have a beef with the notation for rhythm, because as it is, the standard musical notation is just a shorthand form for fitting more music horizontally. For computer-based music, I find it a lot easier to follow a display where horizontal length is proportional to time. We've got infinite screen space, so no need to compress anymore.
The first-approximation engineer realization about music (which I suspect the GP is going off of) is "okay, there are 12 notes in the chromatic scale, each octave doubles frequency, therefore the frequency ratio between two adjacent notes is the 12th root of 2 and we should just have 12 names for the notes". This is what's called an "equal-tempered scale"; the gap between each note is the same ratio, and you have a simple geometric progression upwards.
Except we don't actually have an equal-tempered scale. If you try to play on an equal-tempered scale, it'll sound subtly "off", and certain chords will result in "beats" (pulsing) where the frequency ratios are off just enough to cause an unpleasant modulation in loudness.
The modern diatonic scale is based on the circle-of-5ths [1], where the fundamental ratio is the 5th at 3/2 the frequency. It works like this because now chords are an even multiple of frequencies, while you would get an irrational number with the equal-tempered scale. Going up from the root (C), the next 5th up is G at a ratio of 3/2. Then you go up to D (9/4); when you reduce this to lowest terms because you've ascended a full octave, it gives a ratio of 9/8, which is one whole tone above. Next 5th up is A (27/16), which is the ratio in frequencies of a 6th. And then you get E (81 / 32 = 81/64), a major 3rd. And so on. The frequency ratios of the diatonic scale come from repeatedly reducing powers of 3/2 to lowest terms after dividing out the octave.
[1] https://en.wikipedia.org/wiki/Circle_of_fifths
On a side note, both Wikipedia entries reinforce my original point that the mathematics of this is fascinating.
[1] https://en.wikipedia.org/wiki/Just_intonation
[2] https://en.wikipedia.org/wiki/Equal_temperament
>That's why symphony orchestra players will often have a different flute, clarinet, or oboe for different tunings. Not sure what you are referring to here? Clarinets don't come in different "tunings" unless you mean different keys - like Eb or Bb clarinet, but those aren't there for intonation, they play in different ranges of pitch than one another.
Edit: in addition, you don't HAVE to equally temper a guitar. You can choose just intonation. The problem is that you can only have just intonation on a single KEY for that instrument. So if you tune justly to C major, a key like B major, will sound horrendous!
https://en.wikipedia.org/wiki/Solf%C3%A8ge might be easier for some people.
Haven't heard about CDEFGAE up until I was in my mid 20s trying to learn guitar (after 7 years of music school and musical calsses in regular school)
As for your point about A->B being a larger interval than B->C. There are two half-steps in the white keys (B->C and E->F) because there are two half steps in the major scale! This way, you can play C to C with all white keys and get a major scale.
A major scale is probably one of the most fundamental building blocks in western music theory and it's encoded right onto the keyboard layout itself.
The oddities of music theory are no more strange than all of the strange things in the English language that we just shrug about and move on once we learn it.
For example: 12 tone equal-temperament was chosen/invented (nearly) (by Bach) over just intonation because of 'musical gags' like https://en.wikipedia.org/wiki/The_Musical_Offering (also written by Bach).
Music making neat, orderly, mathematical sense is the struggle, and reality doesn't play nice with harmonics like we would like... (much like with irrational numbers throwing a wrench with Pythagoras' ideals) so stop being a Pythagoras :p
Music IS weird: no matter how you try to quantify it.
Western music theory has evolved over literally thousands of years. You can put a very rough start of it to Pythagoras, around 550BCe ish, which gives us 2,500 years of evolution and refinement.
But even if you want to start with the popularity and adoption of the major scale, that was around 1500CE ish, which gives us a solid 500 years. It handles many, many corner cases quite gracefully.
It undoubtedly has its quirks, but any notational system for this will also have its quirks (cf, the difference between systems of intonation). There is just no way around it.
Also, in your own color analogy: we have a small number of main color names, then a bunch of in betweens.
I tried many times to "understand" music rationally, because I kept people use the term "music theory". I reached a conclusion that there is no "theory" whatsoever: music notation is a hodgepodge of various traditions stacked one on top of the other (we started with 8 notes but then realized that 12 would be better, for example, hence all the mess with flats and sharps). I actually feel better now knowing that you just have to accept it for what it is and go with the flow :-)