Beginner music theory for people OK with some maths?

[motivation]

Hi all and thanks for all your efforts - I've already learned some useful things from this forum.

Quick version: any entry-level music theory books / websites / apps / etc. that don't try to hide mathematical concepts where they come up, and assume only a bare minimum of musical literacy?

Not looking for "the maths of music" - just basic music theory but presented by somebody who "gets" how to explain things to people who know nothing yet about music but enough about maths.



Long version:

My background: I'm in my mid forties, didn't learn any instrument as a child, and I'm now learning from Faber "Adult Piano Adventures" without a teacher, wanting to play classical music.  Slow progress at the moment due to only 2 practice sessons of 40 mins per week - I need to increase that.  I started 15 months ago, slipped into just learning / playing only Messiaen's "Regards du Pere" instead of actual practice (I know, that's weird and I've ruined my learning process, you're probably right - but it was and is very motivating!).  5 months ago started on the Fabers' book, I'm only 1/3 through still.

I think I would benefit from starting to learn about music theory, but right now I'm doing nothing on that.

I'm a hard science/maths sort of person and I think music theory connects with that well, but my very strong impression from my limited past experience is that many (most?) authors/teachers/learning materials don't do well communicating music theory to people like me: no music background but not scared of maths.  From my point of view, musicians write in a very quirky foreign language, which obscures, for musical illiterates like me, what are sometimes relatively simple concepts behind a wall of "the curse of knowledge" https://en.wikipedia.org/wiki/Curse_of_knowledge.

To be clear, I'm happy to eventually learn that foreign language over time, I don't expect music theory to be 100% maths about music, and nor do I want to learn about the maths of music for its own sake.  But I do think at my level the "language issue" and perhaps a presumption that "obviously nobody wants to see ANY maths" are potentially major hindrances for me in making some rapid initial and motivating progress, given that I've seen from occasional glimpses that some of the musical content is far easier for me to start to grasp with the help of somebody who has some mathematical background and knows how to talk to other people with that sort of background.  In fact I have a strong memory of being put off music as a child by exactly this kind of problem (not the only problem, but certainly one).

Any advice for somebody like me?  Books, general advice/tips, suggestions how to stay motivated, online tools, ... whatever you think is relevant.

One note: I'm keen to hear about online things, but to avoid annoyance let me be open up front about one of my peculiarities: I'm also a free (as in freedom) software geek who's keen on decentralisation, and often steer away from many online services (music or otherwise).  So I'm actually very keen to hear about such things, but I'm interested much more in what they do, and not so much in simple links to websites or apps that are rather opaque until you sign up - I *might* not sign up, but I think I might well still learn something very useful from hearing about what it is they're doing!

Thanks again

[mad_max2024]

to be honest I find watching youtube videos on theory concepts to be both fun and educational (Though you always need to be critical about them)
Some channels I follow are Musictheoryforguitar, Jazzpianoschool, Rick Beato, Charles Cornell, Adam Neely, Noah Kellman, Jen Larsen. There are probably many others.

When it comes to books I found Mark Levine's books on Jazz Harmony to be helpful (but maybe not very accessible for beginners)

Some things to keep in mind for music theory:
The point of theory (imo) is to explain the sounds you hear and to build a system that allows you to understand, replicate and change those sounds to express what you want. Different people will often have their own systems and their own ways of thinking using their own labels which can make things confusing if you don't understand the concepts and focus on the labels they use.

Also don't think of theory as a set of rules that you can't break. It is more like a map of what people before you found that works and that you can use to guide yourself to get the sound you want. If you ever find something that sounds good to you but doesn't "fit" the rules you learned just go with it. You can try to find a way to explain it or analize it later (There usually will be one). The sound is always more important than the theory. Theory is not there to restrict you.

Also, music is subjective and relative and you should always keep in mind that there is usually not one way to interpret something. Very often you can ask the same question to two different people and get two different answers that can both be right. It is a bit unlike maths in that regard. Chords are not the notes you play or the physics of the sounds you make, they are the sounds you hear in your brain. And your brain will fill in gaps and interpret things as it wants, and different brains might fill in the gaps in different ways.
So... while it's good to analyze and build a system try to keep in mind that things are often not black and white and there might be none or more than one correct answer. ie You may be taught that C, E, G, B make a Cmaj7 chord (which is true) but it can also be a rootless Am9 depending on context. Bb, Db, E(Fb), Ab is a Bbm7b5 chord but it's also a voicing I use all the time for C7alt or C7b9b13. I remember transcribing a Bill Evans solo where he phrased an Fm with just an Ab and a G.
Sometimes even in the same context I can disagree with my teacher on what a chord is and we can both just understand each other's point of view and agree that we are just hearing it differently or thinking about it in a different way and that we are both "right".
So try to understand the concepts and make your own judgements. Trust your ears (and brain) and stay away from internet arguments.
Musicians love to argue.
Good Luck

[motivation]

to be honest I find watching youtube videos on theory concepts to be both fun and educational (Though you always need to be critical about them)
Some channels I follow are Musictheoryforguitar, Jazzpianoschool, Rick Beato, Charles Cornell, Adam Neely, Noah Kellman. There are probably many others.

Thanks!

Different people will often have their own systems and their own ways of thinking using their own labels which can make things confusing if you don't understand the concepts and focus on the labels they use.

So... while it's good to analyze and build a system try to keep in mind that things are often not black and white and there might be none or more than one correct answer.

This seems good advice, but though I sneaked in a very broad request for random good advice and am grateful for it, it's true the main focus of my question is on those parts of music theory whose explanations are fundamentally mathematical.  It's not that I want to focus on those parts of music theory, rather that I don't want to suffer through needlessly non-mathematical attempts to explain them.

So try to understand the concepts and make your own judgements and stay away from internet arguments.
Musicians love to argue.

Wait surely people on web forums never get into arguments??

[mad_max2024]

yes, I know I ranted a bit. That happens.

[motivation]

yes, I know I ranted a bit. That happens.

It didn't come across as one.

In fact perhaps I need to be clearer just how basic where I'm starting from is: maybe the phase "music theory" is too grand?  I don't even have the knowledge to explain what I don't know, because I don't know any of it.  Basically whatever your expectations are of my knowledge level, lower them and you won't be far off

Here's an example of what I had in mind when I said I'd had glimpses of understanding as an adult of some things I suspect I was frustrated about as a child because of technical musical language obscuring simple concepts (and you can see "mathematical" is also perhaps too grand a phrase in this case) - scroll down to "Diatonic Chords" "Chords that fit in a particular Key.":

https://www.lightnote.co/

But I looked at that website before I got my piano and have since forgotten even what little I learned from this!

[bwl_13]

I agree with the statements about theory and pretty much everything mad_max2024 said.

I might be missing something but mathematical background isn't really something that is needed for music theory, especially introductory stuff. Perhaps tuning theory or the physics of sound but that stuff isn't really the sort of stuff that musicians use on a day to day basis. If mathematics is required to teach a concept, most resources and teachers will explain it as such. However, most music theory does not require it and if it were to use it, then it would be needlessly mathematical. If you insist that these concepts are fundamentally mathematical, I would appreciate you giving me the name of an introductory concept like that.

If you're frustrated about technical language in theory, it's kind of like being frustrated with doctors using medical terminology or any other system that professionals use. This stuff is really necessary when you're trying to be specific and communicate your ideas to another musician, and again, most of this language is best expressed in musical terms and not mathematical. If you prefer to call perfect fifths "3:2s" then you're be missing the point of the terminology. Music is just not as universal as maths, so explaining that "A simple ratio between the notes means they sound nice together" is a statement assumed to be true in Western tonal music, it's not the case across the globe. The tuning systems and intervals across the world do tend to follow some universals (harmonic series for instance) and I'm also pretty sure fifths are quite universal as intervals, but regardless this stuff isn't really introductory and won't help expand your knowledge of the piano.

Feel free to let me know if I missed something in your post or misunderstood it. I'm still confused as to why this thread focuses on mathematical explanations of theory.

[lelle]

Hi all and thanks for all your efforts - I've already learned some useful things from this forum.

Quick version: any entry-level music theory books / websites / apps / etc. that don't try to hide mathematical concepts where they come up, and assume only a bare minimum of musical literacy?

Not looking for "the maths of music" - just basic music theory but presented by somebody who "gets" how to explain things to people who know nothing yet about music but enough about maths.



Long version:

My background: I'm in my mid forties, didn't learn any instrument as a child, and I'm now learning from Faber "Adult Piano Adventures" without a teacher, wanting to play classical music.  Slow progress at the moment due to only 2 practice sessons of 40 mins per week - I need to increase that.  I started 15 months ago, slipped into just learning / playing only Messiaen's "Regards du Pere" instead of actual practice (I know, that's weird and I've ruined my learning process, you're probably right - but it was and is very motivating!).  5 months ago started on the Fabers' book, I'm only 1/3 through still.

I think I would benefit from starting to learn about music theory, but right now I'm doing nothing on that.

I'm a hard science/maths sort of person and I think music theory connects with that well, but my very strong impression from my limited past experience is that many (most?) authors/teachers/learning materials don't do well communicating music theory to people like me: no music background but not scared of maths.  From my point of view, musicians write in a very quirky foreign language, which obscures, for musical illiterates like me, what are sometimes relatively simple concepts behind a wall of "the curse of knowledge" https://en.wikipedia.org/wiki/Curse_of_knowledge.

To be clear, I'm happy to eventually learn that foreign language over time, I don't expect music theory to be 100% maths about music, and nor do I want to learn about the maths of music for its own sake.  But I do think at my level the "language issue" and perhaps a presumption that "obviously nobody wants to see ANY maths" are potentially major hindrances for me in making some rapid initial and motivating progress, given that I've seen from occasional glimpses that some of the musical content is far easier for me to start to grasp with the help of somebody who has some mathematical background and knows how to talk to other people with that sort of background.  In fact I have a strong memory of being put off music as a child by exactly this kind of problem (not the only problem, but certainly one).

Any advice for somebody like me?  Books, general advice/tips, suggestions how to stay motivated, online tools, ... whatever you think is relevant.

One note: I'm keen to hear about online things, but to avoid annoyance let me be open up front about one of my peculiarities: I'm also a free (as in freedom) software geek who's keen on decentralisation, and often steer away from many online services (music or otherwise).  So I'm actually very keen to hear about such things, but I'm interested much more in what they do, and not so much in simple links to websites or apps that are rather opaque until you sign up - I *might* not sign up, but I think I might well still learn something very useful from hearing about what it is they're doing!

Thanks again

I'm quite the music theory nerd, but I wouldn't necessarily say that there is a lot of math involved. It's just kind of a logical system where things fit neatly together in a very satisfying way once you get it - which can take a while.

I'm curious if you could write an example or two of this:

I'm a hard science/maths sort of person and I think music theory connects with that well, but my very strong impression from my limited past experience is that many (most?) authors/teachers/learning materials don't do well communicating music theory to people like me: no music background but not scared of maths. From my point of view, musicians write in a very quirky foreign language, which obscures, for musical illiterates like me, what are sometimes relatively simple concepts behind a wall of "the curse of knowledge"

I could try to offer some explanations of the things that were unclear to get you going!

[mad_max2024]

Yes, if there are any specific things you don't understand you can ask in this forum and you're likely to get a few replies.

[mad_max2024]

I don't think I can help you with a mathematical language or application for music theory.  Like I said music is highly subjective and I don't really see how you can apply mathematics to explain it (and I also like maths).

That being said, if the physics behind the sounds interest you you can try to do some research on the harmonic series. Or acoustics in general. That is a very interesting read.

[frodo1]

I'm sorry, but I just can't resist asking:

Why "maths"?  We say "math" in English?

Actually I just googled "maths vs math" and it took google 6 seconds to come up with this.  It usually takes google a millisecond.  Must be a tough question 

Math is preferred in the U.S. and Canada, and maths is preferred in the U.K., Australia, and most other English-speaking areas of the world.

[ranjit]

There is actually not that much mathematics involved in music, people who say there is usually have a pretty poor and superficial understanding of math. It's unfortunate, because I had hoped there would be more math involved as it would allow me a shortcut to learn the subject (just as math students often breeze through physics courses for example).

It's a logical system for the most part, that's all there is to it. Perhaps you need to use some fractions to line up pitches and rhythms, but that's hardly enough to claim that math is present in music, unless you're in elementary school yourself! (As a side note, it might reflect on our education system that it's a common sentiment that music involves quite a bit of math...)

I would say it actually involves more ear training. You need to be able to hear the chords and process voice leading and such in your head. It requires a bit of visualization to be able to see the keyboard with shifting chords in your mind's eye.

The closest I have seen to math is by our very own Ted, using the Polya theorem to enumerate all possible distinct chords, if I remember correctly. But that in of itself probably won't be useful to most in order to create music.

I do think there is some overlap among people who are good at math and those who are good at music/music theory. However in my experience this is mostly due to a preference for logical thinking which speeds up the learning of certain systems and general smarts. I do think mathematics students are better than average at learning and processing music in most cases, but I don't see a direct link from math to music. It's probably somewhat better pattern recognition ability when it comes to musical structures.

[motivation]

If you insist that these concepts are fundamentally mathematical, I would appreciate you giving me the name of an introductory concept like that.

Hi, thanks for replying!

Not sure how to make this not sound prickly, but: did you look at the post of mine immediately before yours?  I'm talking about the part that starts with "Here's an example"

If you prefer to call perfect fifths "3:2s" then you're be missing the point of the terminology. Music is just not as universal as maths, so explaining that "A simple ratio between the notes means they sound nice together" is a statement assumed to be true in Western tonal music, it's not the case across the globe. The tuning systems and intervals across the world do tend to follow some universals (harmonic series for instance) and I'm also pretty sure fifths are quite universal as intervals, but regardless this stuff isn't really introductory and won't help expand your knowledge of the piano.

I think you might be hearing more in my request than I actually intended, or likely I was unclear 

I don't want somebody to write a book in which everything is renamed - really I'm looking for material from people who understand where somebody like me is coming from, so that I don't struggle needlessly with concepts described in terms of a lot of musical background knowledge involving quirky conventions and language.  I'm sure often concepts don't really depend on those things, and may be actually obscured by it.  I don't insist that then a standard English musical language term must not be introduced, having explained the concept - more the opposite.  I think there's nothing special to music here, it's common to communication of all knowledge.  My guess is that somebody with the same mindset will do better at that job.  That is going on my actual, if limited, experience - even my limited experience about music theory! - as I've explained.

And as I say, I know basically nothing about music theory, so I'm sure you wouldn't expect me to dump a long series of music theory examples to back up my hunch here.  That's not how this works 

Feel free to let me know if I missed something in your post or misunderstood it. I'm still confused as to why this thread focuses on mathematical explanations of theory.

Just one other example: the one you gave above about perfect fifths being related to ratios (being vague here since I don't know what a perfect fifth is!).

One more: I very vaguely recall doing a tiny bit of scribbling to see how the 12-note thing works out sort of neatly mathematically.  I honestly don't remember enough right now to tell you exactly what I did.

[frodo1]

Pythagoras was a brilliant mathematician but unfortunately did not believe irrational numbers existed.  I think he drowned a person that believed otherwise.  And so he was not able to come up with the formula:

2^(1/12) is the ratio of 2 frequencies that are a semitone apart.

[motivation]

I'm quite the music theory nerd, but I wouldn't necessarily say that there is a lot of math involved. It's just kind of a logical system where things fit neatly together in a very satisfying way once you get it - which can take a while.

Ha, I think some people are latching on to the word maths as if I'm talking about knot theory or something.  But I hoped I'd already corrected that misconception with the first example I posted (search for "and you can see "mathematical" is also perhaps too grand a phrase in this case") and again, I'm talking much more about people's backgrounds making it easier or harder for them to explain concepts to people of other backgrounds than I am about anything fancy.

kind of a logical system where things fit neatly together in a very satisfying way once you get it

Yep, sounds like maths to me 

I could try to offer some explanations of the things that were unclear to get you going!

Thanks, I might take you up on that later!  As I say, I've not studied anything like this for years, so naturally I'm not coming at this with a list of specific questions.  However, my experience is that often musical people suffer badly from the curse of knowledge when writing notionally introductory explanations - as everybody does - and that on the occasion when I've come across people with some other background in addition to music (or just having thought about or tested their explanations carefully perhaps) explaining similar concepts, I've understood them and thought: that's night and day from other experiences I've had where more musical people tried to explain similar concepts to me.  I'm sure you'll agree that those experiences over 40 years are not really something that can easily be dumped out onto a web page.

I'm curious if you could write an example or two of this:

As I say, I've done really nothing on this for years, so my position here is based on past experience of being perplexed, then enlightened on a few little things over past years (some of this from childhood!).

But some things from this thread:

https://www.pianostreet.com/smf/index.php?topic=68835.msg718024#msg718024
https://www.pianostreet.com/smf/index.php?topic=68835.msg718037#msg718037

[frodo1]

There is actually not that much mathematics involved in music, people who say there is usually have a pretty poor and superficial understanding of math. It's unfortunate, because I had hoped there would be more math involved as it would allow me a shortcut to learn the subject (just as math students often breeze through physics courses for example).

It's a logical system for the most part, that's all there is to it. Perhaps you need to use some fractions to line up pitches and rhythms, but that's hardly enough to claim that math is present in music, unless you're in elementary school yourself! (As a side note, it might reflect on our education system that it's a common sentiment that music involves quite a bit of math...)

I would say it actually involves more ear training. You need to be able to hear the chords and process voice leading and such in your head. It requires a bit of visualization to be able to see the keyboard with shifting chords in your mind's eye.

The closest I have seen to math is by our very own Ted, using the Polya theorem to enumerate all possible distinct chords, if I remember correctly. But that in of itself probably won't be useful to most in order to create music.

I do think there is some overlap among people who are good at math and those who are good at music/music theory. However in my experience this is mostly due to a preference for logical thinking which speeds up the learning of certain systems and general smarts. I do think mathematics students are better than average at learning and processing music in most cases, but I don't see a direct link from math to music. It's probably somewhat better pattern recognition ability when it comes to musical structures.

I agree with all here.  Polya theorem - that's impressive!  Is Ted a mathematician?

Edit:  Polya theorem?  Can you explain how combinatorics can be used in a meaningful way for chord construction?? 

[motivation]

There is actually not that much mathematics involved in music, people who say there is usually have a pretty poor and superficial understanding of math. It's unfortunate, because I had hoped there would be more math involved as it would allow me a shortcut to learn the subject (just as math students often breeze through physics courses for example).

I'm happy that you know some fancy maths.  Well done. 

It's a logical system for the most part, that's all there is to it. Perhaps you need to use some fractions to line up pitches and rhythms, but that's hardly enough to claim that math is present in music, unless you're in elementary school yourself!

Think "curse of knowledge" and "mindset", not "abstract algebra".

It does seem a little bit ironic that I've showed up proclaiming my total ignorance of music theory, only to find the response is that I'm... wrong about music theory!

I guess I over-used the word maths in my first post - put that down to my not having studied any music theory.  But I hope I've clarified the thought and experiences behind it?  I think those are unchanged.  And heck, I even stick by the word "mathematical", ha.

I would say it actually involves more ear training. You need to be able to hear the chords and process voice leading and such in your head. It requires a bit of visualization to be able to see the keyboard with shifting chords in your mind's eye.

Can you explain what "process voice leading" means?

The closest I have seen to math is by our very own Ted, using the Polya theorem to enumerate all possible distinct chords, if I remember correctly. But that in of itself probably won't be useful to most in order to create music.

Thanks, but definitely not what I have in mind!  Again, think "mindset" not "Polya theorem"!

I do think there is some overlap among people who are good at math and those who are good at music/music theory.

I think this question is quite distinct from the one I raise in this thread.

[frodo1]

OP - If you are able to read and understand any standard pre-calculus book - you will have no problem reading and understanding ANY standard beginner level music theory book. 

[lelle]

Here's an example of what I had in mind when I said I'd had glimpses of understanding as an adult of some things I suspect I was frustrated about as a child because of technical musical language obscuring simple concepts (and you can see "mathematical" is also perhaps too grand a phrase in this case) - scroll down to "Diatonic Chords" "Chords that fit in a particular Key.":

https://www.lightnote.co/

But I looked at that website before I got my piano and have since forgotten even what little I learned from this!

Honestly I'm not a fan of how that website lays everything out, starting from frequencies and frequency ratios and all.

When you start out learning music theory, I think you'll profit more from digging into some basic terminology until it gets familiar, which will make it easier to get into more advanced concepts. A lot of these basic terms simply consists of "we decided to give this particular thing this particular name because... we just did". (There is often some logic to it too, but it's not always immediately obvious and still it's all ultimately based on subjective things we just sort of decided to use as basic building blocks to construct music with). I would start out with just learning what intervals, scales, and chords are and how they are named.

This applies to the "perfect fifth" mentioned earlier. On the piano, there are a bunch of keys. The distance from any key to any other key has a general name - an interval. The interval created by pressing one key, for example a C, and the next key of the same name either above or under it (so another C) is called an octave. Each interval that can be created by pressing two keys within that range from for example C to the next C has a specific name - for example, if you press a C and the G directly above it you get an interval called a "perfect fifth". There are reasons why each interval has the name it has, but in the beginning you are better off just memorizing the names, and getting into the why:s later. Beginner resources/courses on music theory should have a part covering the names of all the intervals. If you are reading something that's just namedropping the names of intervals, or the word interval without explaining what it is, it's not suitable for beginners.

Thanks, I might take you up on that later!   As I say, I've not studied anything like this for years, so naturally I'm not coming at this with a list of specific questions.  However, my experience is that often musical people suffer badly from the curse of knowledge when writing notionally introductory explanations - as everybody does - and that on the occasion when I've come across people from other, I've understood them and thought: that's night and day from other experiences I've had where more musical people tried to explain similar concepts to me.  I'm sure you'll agree that those experiences over 40 years are not really something that can easily be dumped out onto a web page. 

Anytime! 

[motivation]

I don't think I can help you with a mathematical language or application for music theory.  Like I said music is highly subjective and I don't really see how you can apply mathematics to explain it (and I also like maths).

I think (admittedly like every other responder so it must be my fault!) you're focussing entirely on the word "maths".  It's genuinely not the focus of the question, as I've tried to explain in my replies.  I think once that word was out of the bag that's all anybody could see, my mistake I guess.

That being said, if the physics behind the sounds interest you you can try to do some research on the harmonic series. Or acoustics in general. That is a very interesting read.

Thanks again, but, as I tried to express in my first post, I'm just feeling I should learn some basic music theory, rather than about the maths or physics of music.

[motivation]

OP - If you are able to read and understand any standard pre-calculus book - you will have no problem reading and understanding ANY standard beginner level music theory book.

LOL.  At this point I think I'm perhaps justified to wonder: It's almost as if you haven't really read my replies carefully.  I think it's time (for me) to step back from this thread for a bit!

[frodo1]

LOL.  At this point I think I'm perhaps justified to wonder: It's almost as if you haven't really read my replies carefully.  I think it's time to step back from this thread for a bit!

I can vouch for Lelle.  He seems willing to help you.  Lelle knows theory well and is good at explaining.  I wish you the best of luck! 

[motivation]

I can vouch for Lelle.  He seems willing to help you.  Lelle knows theory well and is good at explaining.  I wish you the best of luck! 

Yes contrary to what I just said in fact I can't resist replying to their interesting post in a minute... I was only responding just now to your last post frodo1, not everybody, though much but not all of the rest of the thread has a similar problem - as I say I'm sure it's my fault with my opening post.

[motivation]

Honestly I'm not a fan of how that website lays everything out, starting from frequencies and frequency ratios and all.

When you start out learning music theory, I think you'll profit more from digging into some basic terminology until it gets familiar, which will make it easier to get into more advanced concepts. A lot of these basic terms simply consists of "we decided to give this particular thing this particular name because... we just did".

That's interesting, I think you get to the core of what prompted my original question here.

I agree completely about terminology: it's important to learn the same language as everybody else.

Where we seem to differ is on learning the concepts to which those terms are attached.  Though people vary, I think it's not merely a matter of taste whether it's better to start by "digging into some basic terminology" or learning missing concepts to which terminology can be attached: the latter is better.  Once you have a concept you can immediately attach a word to it without much effort.  That isn't a claim that learning is ever a simple process of pouring knowledge into people's heads, it isn't and always involves guesswork and backtracking.  But there is such a thing as an elementary explanation, and they're very useful for people starting out in an area of knowledge new to them!

(There is often some logic to it too, but it's not always immediately obvious and still it's all ultimately based on subjective things we just sort of decided to use as basic building blocks to construct music with).

All subjective knowledge is objective knowledge, and in music more clearly than anything I can call to mind in the moment.

I would start out with just learning what intervals, scales, and chords are and how they are named.

This applies to the "perfect fifth" mentioned earlier. On the piano, there are a bunch of keys. The distance from any key to any other key has a general name - an interval. The interval created by pressing one key, for example a C, and the next key of the same name either above or under it (so another C) is called an octave. Each interval that can be created by pressing two keys within that range from for example C to the next C has a specific name - for example, if you press a C and the G directly above it you get an interval called a "perfect fifth". There are reasons why each interval has the name it has, but in the beginning you are better off just memorizing the names, and getting into the why:s later.

I'm quite sure this response of childhood me is a matter of personal variations - and I don't mean this as a personal attack (!) - but if I'd have come across somebody without the approach you just advocated ("in the beginning... better off just memorizing..."), I think it's possible I might have learned about music and acquired musical skills largely under my own steam decades ago.  Motivation is important.

But also: this very basic concept of intervals is an example of something that's already confused me and perhaps is an instance of just the kind of thing I was loosely referring to in my first post.  I recall I've cleared this up once already with the help of a family member of a similar technical mindset to me who knows about music, but I've already forgottten, so I'll ask again here!

Here we go: I've read terms like "fifth" (not sure whether I mean "perfect fifth" or just fifth - I won't try and google any of this to make my train of thought apparent) being defined as "the interval between N white notes" (probably it was seconds and thirds rather than fifths where I read that).  But of course the interval in semitones between N white notes depends on where you start on the keyboard: 7 semitones * 2 ** (1/12) is about a ratio of 1.5 in your example of C to G, but B to F is only six semitones, so 2 ** (1/2), about 1.41.  Are intervals (fifths, thirds, seconds etc.) constant intervals in log(frequency), or do they refer to different physical intervals depending on where you start on the keyboard?

Beginner resources/courses on music theory should have a part covering the names of all the intervals. If you are reading something that's just namedropping the names of intervals, or the word interval without explaining what it is, it's not suitable for beginners.

Anytime! 

I think these "curse of knowledge" problems are more pervasive and much harder to spot than we give them credit for, and it's hard to take them as seriously as they should be.  As soon as we learn the right concepts it's hard to keep a grasp on the misconceptions we had before.

[frodo1]

Here we go: I've read terms like "fifth" (not sure whether I mean "perfect fifth" or just fifth - I won't try and google any of this to make my train of thought apparent) being defined as "the interval between N white notes" (probably it was seconds and thirds rather than fifths where I read that).  But of course the interval in semitones between N white notes depends on where you start on the keyboard: 7 semitones * 2 ** (1/12) is about a ratio of 1.5 in your example of C to G, but B to F is only six semitones, so 2 ** (1/2), about 1.41.  Are intervals (fifths, thirds, seconds etc.) constant intervals in log(frequency), or do they refer to different physical intervals depending on where you start on the keyboard?

This math is not important to your understanding of basic music theory - but I need to correct your math.

2^(7/12) = 2 to the 7/12 power is approx 1.498 is the correct ratio of frequencies for an interval of a 5th under equal temperament tuning and is a little less than Pythagoras 1.5 = 3/2 ratio of 2 integers.

7 semitones * 2 ** (1/12) - What is this?  7 times 2 to the 1/12 power??  The math is 2 to the 7/12 power - not 7 times 2 to the 1/12 power.

Your question "Are intervals (fifths, thirds, seconds etc.) constant intervals in log(frequency), or do they refer to different physical intervals depending on where you start on the keyboard?"

Intervals are determined by the ratio of 2 frequencies.and the tuning method.  Equal temperament is the standard tuning method.  Under this method, a semitone has 2^(1/12) as the ratio of frequencies.

Again this is not important to basic music theory.

[frodo1]

Lelle - please help motivation.  Sorry - I do not intend to interfere.  I am only going to comment on math concepts that I see are incorrect.  But basic music theory for beginner musicians does not get into defining intervals as a ratio of frequencies and the math behind this.

[motivation]

This math is not important to your understanding of basic music theory - but I need to correct your math.

2^(7/12) = 2 to the 7/12 power is approx 1.498 is the correct ratio of frequencies for an interval of a 5th under equal temperament tuning and is a little less than Pythagoras 1.5 = 3/2 ratio of 2 integers.

7 semitones * 2 ** (1/12) - What is this?  7 times 2 to the 1/12 power??  The math is 2 to the 7/12 power - not 7 times 2 to the 1/12 power.

Whoops very sloppy/wrong use of "*" there, sorry.  Happy to have my maths corrected too, but I don't think I made a mistake in understanding.  I will just paste what I typed at my python prompt in the hope that's clear - at least I can't easily mess up my symbols this way:

Code: [Select]

>>> r = 2 ** (1/12.)
>>> r ** 12
2.000000000000001
>>> r ** 7
1.498307076876682
>>> r ** 6
1.4142135623730954


Your question "Are intervals (fifths, thirds, seconds etc.) constant intervals in log(frequency), or do they refer to different physical intervals depending on where you start on the keyboard?"

Intervals are determined by the ratio of 2 frequencies.and the tuning method.  Equal temperament is the standard tuning method.  Under this method, a semitone has 2^(1/12) as the ratio of frequencies.

Again this is not important to basic music theory.

I think perhaps you have in mind that the differences between tuning methods aren't important to basic music theory?  But what musicians mean by "interval" is important to basic music theory, right?  Talking about semitones makes it very concrete and much harder to misunderstand.  And to explain what a semitone is just is to describe and understand them in terms of exponents and logarithms and their physical and psycho-acoustic counterparts I think (I'm using "explain" here to mean to describe one thing in terms of something simpler).  I find it demotivating to needlessly slog through that sort of confusion about a simple concept that could be explained better in that sense.

I'm honestly still confused what an interval is!  If you have a "third", is that sometimes three semitones and sometimes four?  Or is it always the same numbers of semitones?  I think you're saying that it's alway
s the same number of semitones?

[frodo1]

I think perhaps you have in mind that the differences between tuning methods aren't important to basic music theory?  But what musicians mean by "interval" is important to basic music theory, right?  Talking about semitones makes it very concrete and much harder to misunderstand.  And to explain what a semitone is just is to describe and understand them in terms of exponents and logarithms and their physical and psycho-acoustic counterparts I think (I'm using "explain" here to mean to describe one thing in terms of something simpler).  I find it demotivating to needlessly slog through that sort of confusion about a simple concept that could be explained better in that sense.

I'm honestly still confused what an interval is!  If you have a "third", is that sometimes three semitones and sometimes four?  Or is it always the same numbers of semitones?  I think you're saying that it's always the same number of semitones?

Here are the intervals and the number of semitones:
Interval = # semotones
Minor 2nd = 1 semitone
Major 2nd = 2 semotones
Minor 3rd = 3 semitones
Major 3rd = 4 semotones
perfect 4th = 5 semitones
tritone = 6 semitones
perfect 5th = 7 semitones
minor 6th = 8 semitones
major 6th = 9 semitones
minor 7th = 10 semitones
major 7th = 11 semitones
octave = 12 semotones

[motivation]

Here are the intervals and the number of semitones:
Interval = # semotones
Minor 2nd = 1 semitone
Major 2nd = 2 semotones
Minor 3rd = 3 semitones
Major 3rd = 4 semotones
perfect 4th = 5 semitones
tritone = 6 semitones
perfect 5th = 7 semitones
minor 6th = 8 semitones
major 6th = 9 semitones
minor 7th = 10 semitones
major 7th = 11 semitones
octave = 12 semotones

Aha!  Now I understand, thanks.

I think you also just demonstrated with your explanation exactly why you're wrong about maths being unimportant in music theory 

[motivation]

Aha!  Now I understand, thanks.

I think you also just demonstrated with your explanation exactly why you're wrong about maths being unimportant in music theory 

... and now that many of you have explained at length how I'm quite mistaken about my own experiences and ways of understanding the world (as I well could be - though I'm pretty sure I'm not), you do still have the option of taking my original question seriously

[frodo1]

Aha!  Now I understand, thanks.

I think you also just demonstrated with your explanation exactly why you're wrong about maths being unimportant in music theory 

The Pianostreet spell checker shows that you misspelled math!  It's math, not maths! 

[bwl_13]

Hi, thanks for replying!

Not sure how to make this not sound prickly, but: did you look at the post of mine immediately before yours?  I'm talking about the part that starts with "Here's an example"

Not a problem, I don't mean to seem rude in my responses, I just am trying to get some clarity. I think you're looking into it, because I did see your example and I don't really think there's a mathematical explanation for this. Again, there's math in music, but it's more in the physics of sound and it's mostly irrelevant to musicians using theory to strengthen their understanding of music.

I didn't realize this thread would blow up so it's a bit hard to dig through your other quotes, but it seems you've got some pretty good answers. However...

Aha!  Now I understand, thanks.

I think you also just demonstrated with your explanation exactly why you're wrong about maths being unimportant in music theory 

Just because there are numbers involved, doesn't mean it's necessarily mathematics (we can argue semantics all day). You're bound to come across numbers in just about any field, and there's different ways of understanding things. I can't give you the exact number of semitones in a given interval because it's just not useful. The interval is useful. Again, it appears in music theory, but it's not anything you need a background in to understand. There are children who understand many of these concepts before they learn simple addition.

lelle seems to know what they're talking about. With theory it's easy to view it as a hard science, but the terminology and rules are pretty much all arbitrary (though they've been chosen for good reason). The numbers just aren't as universal as in mathematics

[frodo1]

Again, there's math in music, but it's more in the physics of sound and it's mostly irrelevant to musicians using theory to strengthen their understanding of music.

I agree with this statement and your spelling of math.

However, I can understand an adult that is learning theory at a late age and has strong math background wanting to pursue the math behind it at the start.  However, it won't add much to their understanding of music as you say.  It may help quench their natural curiosity though.

[frodo1]

Here is a Pianostreet link to ear training for intervals.  In addition to understanding the math for intervals - why not train your ear for intervals?  Maybe motivation can spend time and do these exercises to train his ear?  Just a suggestion.

https://www.pianostreet.com/smf/index.php?topic=68711.0

[lelle]

That's interesting, I think you get to the core of what prompted my original question here.

I agree completely about terminology: it's important to learn the same language as everybody else.

Where we seem to differ is on learning the concepts to which those terms are attached.  Though people vary, I think it's not merely a matter of taste whether it's better to start by "digging into some basic terminology" or learning missing concepts to which terminology can be attached: the latter is better.  Once you have a concept you can immediately attach a word to it without much effort.  That isn't a claim that learning is ever a simple process of pouring knowledge into people's heads, it isn't and always involves guesswork and backtracking.  But there is such a thing as an elementary explanation, and they're very useful for people starting out in an area of knowledge new to them!

That's true. I'm guess I'm just trying to avoid overloading you with information.  You can get VERY deep into chords, scales and intervals but it is a mouthful when you're just starting out. Perhaps it would be better to say, start out by learning intervals, scales and chords with some basic explanations of what they are and why they work the way they do. But if you feel more motivated by getting deeper explanations, of course there should be sources out there that can provide

Here we go: I've read terms like "fifth" (not sure whether I mean "perfect fifth" or just fifth - I won't try and google any of this to make my train of thought apparent) being defined as "the interval between N white notes" (probably it was seconds and thirds rather than fifths where I read that).  But of course the interval in semitones between N white notes depends on where you start on the keyboard: 7 semitones * 2 ** (1/12) is about a ratio of 1.5 in your example of C to G, but B to F is only six semitones, so 2 ** (1/2), about 1.41.  Are intervals (fifths, thirds, seconds etc.) constant intervals in log(frequency), or do they refer to different physical intervals depending on where you start on the keyboard?

frodo1 gave you an overview of the intervals already. I think it illustrated the point I was trying to make in that you kinda just look at the list which says "this is what it is" and you quickly get the basic gist. But I'll give you a deeper explanation about the naming convention. It's correct that a perfect fifth is always 7 semitones, so B to F is not a perfect fifth, however B to F# is. (B to F is called a diminished fifth, assuming it's notated as a B and F; if it's notated as a B and E# it's an augmented fourth, but all of this is another whole can of worms).

Are you familiar with scales? A scale in general is basically defined as a sequence of notes in ascending order. In western music, there is a set of specific scales that are used as a basic building block from which to construct music, divided into two types called major and minor, both which feature seven individual pitches before you come back to the first pitch again an octave up, perhaps you have heard of this? The simplest scale to remember is perhaps C major so I'll use that one to explain interval naming conventions - it starts on C and uses only the white keys on the piano:

C D E F G A B C

If you look at the piano, you'll see that the scale is made from a series of 2nds, either major or minor, ie either 2 semitones (a whole tone) or one semitone, starting from C and then going to D, E and so on:

whole whole semitone whole whole whole semitone

This is the pattern we call a "major" scale. You can start from any note on the piano, and as long as you play following the pattern above, it'll be a major scale. For example, B major is B C# D# E F# G# A# B

The minor scale exists in three variants, but we'll use the simplest one for now, the "natural minor". (The other two variants, "harmonic" and "melodic" minor, feature minor changes to the scale to better suit the ways chords and melodies are typically used/written in traditional western music). The simplest minor scale to remember is natural A minor, which also uses just the white keys, starting from A:

A B C D E F G A

The natural minor scale instead follows the pattern

whole semitone whole whole semitone whole whole

If you look at frodo1's summary of the intervals you'll see that they all either are, or are some type of second, third, fourth, fifth, sixth, seventh or octave. This "generic" name is taken from how the intervals fit into the scales western music uses, for example in the C major scale above

C to D is called a second, because D is the 2nd step (often called degree) of the scale
C to E is called a third, because E is the third step of the scale
C to F is called a fourth, because F is the fourth step of the scale
C to G is called a fifth, because G is the fifth step of the scale
etc.

Regardless if you start from the C in the C major scale or A in the A minor scale and play each interval in turn, they're all seconds, thirds, fourths, fifths, sixths, sevenths and octaves. The fourth, fifth and octave consist of the same amount of semitones in both the C major and A minor scale, we call them "perfect". Take any major or minor scale (using the patterns outline above) and go up to the fifth step, and you'll have a perfect fifth between the first step and the fifth step. For example, in F major:

F G A Bb C D E F ----- F and C are a perfect fifth.

Compare, however, the third, sixth and seventh in the C major vs the A minor scale - they're major thirds, sixths and sevenths in the major scale, and minor thirds, sixths and sevenths in the natural minor scale. Take any major scale, and the interval between the first and third step will be a major third, and take any natural minor scale and the interval between the first and third step will be a minor third, etc. The only exception to this "rule" is the first second, which is a major second both in the major and minor scale.

In fact, the difference between a major chord and a minor chord, ie C major, which consists of CEG played simultaneously, and A minor, which consists of ACE, is if the third above the starting note is a major third or a minor third. So C major is kinda short for "the scale or chord which starts on C and has a major third", and A minor is kinda short for "the scale or chord which starts on A and has a minor third". It's this difference mainly between the major/minor thirds and sixths, around the framework of the perfect fourths, fifths and octaves, which gives the major vs minor keys their distinct flavor/color.

Why intervals called perfect, diminished/augmented, or major/minor is answered better than I can here by Rita Karpati: https://www.quora.com/Music-Theory-Why-do-we-call-a-fifth-perfect-but-other-intervals-%E2%80%9Cmajor%E2%80%9D

I hope this was helpful and not confusing, because it became a bit longer than I thought it would 

[bwl_13]

I agree with this statement and your spelling of math.

However, I can understand an adult that is learning theory at a late age and has strong math background wanting to pursue the math behind it at the start.  However, it won't add much to their understanding of music as you say.  It may help quench their natural curiosity though.

I fully agree with that, I understand it as well. I also think it could be quite the overload of information, and music theory is notorious for overloading information with need for background etc.

On a side note, maths makes more sense to me as I'm guessing the word comes directly as a shorter way to say mathematics, thus math = mathematic... I've grown up using math but will use the other spelling in a discussion for consistency's sake.

[keypeg]

This thread is already big so I skimmed through some of it.  Understanding concept, then name, is an order I prefer too.   
Intervals - my take.
In Western music intervals are the distance between two notes in the range of an octave, and the smallest unit is a semitone.

One aspect of intervals is that they have different sound qualities which we feel.  Play on your piano: C D# -- EF -- BC - F# G ..... These are all a semitone apart.   Listen to what they have in common, a kind of grating vibrationy not pleasant feel.   Play C E, F A, D F# .... also C Eb, F Ab, D F ... the first of these are "major thirds", the 2nd are "minor thirds" .... both will have a smoother and more pleasant feel to them; the first group will all have a same quality you might hear, and the 2nd group has another kind of same quality - one perhaps feeling "happy", the other "sad". You may or may not hear it that way, or even hear the sameness.

Another important aspect of intervals is the "names" aspect, versus the "what it is" aspect.  Names depending on spelling: which notes are written into the music so which letters; "what it is" is independent of that.  You should be aware of this fact, so that it doesn't confuse you later.  For example, play C Eb on the piano; listen to the sound, and look at the piano keys you pressed.  Now play C D# and do the same thing.  You have the same piano keys, and the same sound.  But they'll have a different name in standard formal theory.   If you wanted an unchanging measuring stick, with the smallest unit, you'd get semitones, or adjacent piano keys; just like a ruler has centimeters in some countries, and inches in some countries, as the commonly used smallest unit of measure.

NAMING:  The conventions are idiotically simple once you get the idea, but it's also sort of antiquated.  The traditional names of intervals are based on the major scale, and hearken from an earlier time.  You always start with the lowest note of the major scale, and count up from there.  This has already been given:
C D ..... from first to 2nd note - it's a major 2nd
CE ....  from first note to 3rd note - it's a major 3rd
etc.

In the key of D major that is:
D E = M2
D F# = M3, because F# is the 3rd note in a D major scale

If you take the notes of a major scale, you will always have a major interval.  The first task is: "what number do I have?"  (unison, 2nd, 3rd, 4th, 5th, 6th, 7th, octave).  D F is "some kind of 3rd" ... D# F is "some kind of 3rd", D F# is "some kind of 3rd".  ...... The only one that is a major 3rd is the when both top note belongs to the major scale of the bottom note. So D F# is a major 3rd.

With this ' major scale bias' - If the top note belongs to the major scale, the 2nd, 3rd, 6th, 7th will be called major.  unison, 4th, 5th, octave will be called Perfect.  It's that way due to history.  It just is.

----------------------
For 2nd, 3rd, 6th, 7th, if the top note is lower by a semitone, you'll have a minor.
D F#  = M3 ....... D F = m3
C E = M3 ....... C Eb = m3
C# A# = M6 ......... C# A = m6

For 2nd, 3rd, 6th, 7th, if the top note is lower by two semitones, you'll have a diminished.
D F#  = M3 ....... D F = m3 ....... D Fb = dim3 (which on the piano looks and sounds like DE - dim3 = M2)
C E = M3 ....... C Eb = m3 ....... C Ebb = dim3 (same as CD which would be called M2)
C# A# = M6 ......... C# A = m6 .....  C# Ab = dim6 (same as C# G# = P5 .... do we get dim6's in music?)

For 2nd, 3rd, 6th, 7th, if the top note is raised by a semitone, you'll have augmented
D F#  = M3 ....... D F## = aug3 (which is the same as D G = P4)
C E = M3 ....... C E# = aug3 (like CF, = P4)
C# A# = M6 ......... C# A## = aug6 (like C# B = m7)

In other words, your baseline is the major scale, and then looking at the difference.

Perfects don't have minors.  If you lower a Perfect by a semitone, you get diminished.
E A = P4,  E Ab = dim4 (same as E G# which is M3)
E B = P5, E Bb = dim5

I won't touch unison or octave

If you raise a perfect by a semitone, you get augmented.
EA = P4, E A# = aug4
EB = P5, E B# = aug 5 which is like E C which is m6

Your named intervals first have a number attached to them, and then a quality.  2nd 3rd, 4th etc. is the number; major, minor, diminished, augmented, perfect, are qualities.

Quite a few people have ranted about this system, which we're stuck with along with the way notes are spelled the note names, and the rest.

A word about the aug4 and dim5 - they are both given the name "tritone" and have unique properties.  CF# and CGb are aug4 and aug5, and make exactly the same sound.  CF# and F#C (inverted) are both a tritone.

One unique thing about perfects is that when inverted, they form another perfect.  CF = P4; FC = P5.  But not so with majors and minors: CE = major 3, EC = minor 6.

[keypeg]

The Pianostreet spell checker shows that you misspelled math!  It's math, not maths! 

It probably uses American (US) English.  It's like you guys write "labor", we write "labour" and we pay with cheques and not "checks".  The spell checker has also flagged my "labour". 

[bwl_13]

It probably uses American (US) English.  It's like you guys write "labor", we write "labour" and we pay with cheques and not "checks".  The spell checker has also flagged my "labour".

It's weird because in Canada we use British/UK spelling for almost everything, but we don't use maths, only math (at least in Ontario)

[keypeg]

It's weird because in Canada we use British/UK spelling for almost everything, but we don't use maths, only math (at least in Ontario)

I'm in Canada too.  You are right on both counts.

I'm also a linguist and work as a trained translator (into English).  I always have to check in what country it will be read, and so, which English to use.  Music is also a mix.   The on-line sites all seem to use US English, and so they refer to "parallel" keys, whereas in the RCM it's Tonic vs. Relative keys (C minor and C major are Tonic keys (parallel), A minor and C major are relative keys.  Personally I don't like "parallel" because the notes are not "parallel", are they?  They dip at different points of semitones to the degrees.  There are a few things like that - and not just "hemidemisemiquaver" (not Canada).

Again in Canada I was once in a choir where the French Canadian person beside me was lost as the conductor talked about the part in "A minor" - so I quickly scribbled "La mineur" because in French they use fixed Do syllables, because that's what's done in French. 

I think that the RCM's book, sold as "Celebration" series, has Americanized some of its terms, maybe giving both names? (parallel and Tonic)

[keypeg]

Here are the intervals and the number of semitones:
Interval = # semotones
Minor 2nd = 1 semitone
Major 2nd = 2 semotones
Minor 3rd = 3 semitones
Major 3rd = 4 semotones
perfect 4th = 5 semitones
tritone = 6 semitones
perfect 5th = 7 semitones
minor 6th = 8 semitones
major 6th = 9 semitones
minor 7th = 10 semitones
major 7th = 11 semitones
octave = 12 semotones

That's part of it.

Take:
Minor 3rd = 3 semitones

C Eb = 3 semitones and is a minor 3rd
C D# = 3 semitones, and is called an augmented 2nd

C Db = 2 semitones and is called a minor 2nd
C C# = 2 semitones and is called an augmented unison

The semitones are an absolute measurement, like inches or centimeters.  The names are based on notation, and the letter names.  Our C Eb involves a total of three letters in a row: C, D, E so it is "some kind of 3rd" since E is the 3rd note over from C.  Then its quality is minor = minor 3rd.  Our C D# is a total of two letters, since D is the 2nd note over from C, so it is "some kind of 2nd" = augmented 2nd.

[motivation]

You can get VERY deep into chords, scales and intervals but it is a mouthful when you're just starting out. Perhaps it would be better to say, start out by learning intervals, scales and chords with some basic explanations of what they are and why they work the way they do. But if you feel more motivated by getting deeper explanations, of course there should be sources out there that can provide

Basically I don't really, at the moment.

But I think there's not much point in further repeating my earlier replies! ¯\_(ツ)_/¯

For what it's worth, to be closer to the truth but more confusing: I'm interested in anything like that along with a million other things I'll probably never spend any real time on, but if I started getting into that I don't think I'd see it as the best bang for buck for learning to play the piano.

frodo1 gave you an overview of the intervals already. I think it illustrated the point I was trying to make in that you kinda just look at the list which says "this is what it is" and you quickly get the basic gist

This reads to me like you may be getting that I find a problem with that list, maybe as "not being mathematical enough"?  It was great!  It's exactly this kind of thing that I've found lacking in my little previous contacts with books that touch on basic music theory, in fact.

But I'll give you a deeper explanation about the naming convention. It's correct that a perfect fifth is always 7 semitones, so B to F is not a perfect fifth, however B to F# is. (B to F is called a diminished fifth, assuming it's notated as a B and F; if it's notated as a B and E# it's an augmented fourth, but all of this is another whole can of worms).

Are you familiar with scales?

long snip...

The only exception to this "rule" is the first second, which is a major second both in the major and minor scale.

I think I basically knew the content here, but not the language, and not the explanation of how the generic "third" (etc.) terminology connects with the scales - thanks, very clear!

In fact, the difference between a major chord and a minor chord, ie C major, which consists of CEG played simultaneously, and A minor, which consists of ACE, is if the third above the starting note is a major third or a minor third. So C major is kinda short for "the scale or chord which starts on C and has a major third", and A minor is kinda short for "the scale or chord which starts on A and has a minor third".

Just to make sure I get it: "third" in those last two sentences in the "scales" sense you explained earlier, right?  They both have a major third in the sense of semitone intervals (C to E is in both chords)?

Of course when I see this sort of thing, I do immediately wonder what explanations people have come up with for the affect of major vs. minor chords   But I guess that's a google away

Why intervals called perfect, diminished/augmented, or major/minor is answered better than I can here by Rita Karpati: https://www.quora.com/Music-Theory-Why-do-we-call-a-fifth-perfect-but-other-intervals-%E2%80%9Cmajor%E2%80%9D

I hope this was helpful and not confusing, because it became a bit longer than I thought it would 

Yes it was, and thanks for the link, I'll check it out later.

[motivation]

With this ' major scale bias' - If the top note belongs to the major scale, the 2nd, 3rd, 6th, 7th will be called major.  unison, 4th, 5th, octave will be called Perfect.  It's that way due to history.  It just is.

I understood up to here, then got lost.  What do "2nd" "3rd" etc. refer to here, and why is the top note in particular relevant?

Sorry I'm sure I'm missing an obvious interpretation, if I read it tomorrow I'll probably understand.

[keypeg]

I understood up to here, then got lost.  What do "2nd" "3rd" etc. refer to here, and why is the top note in particular relevant?

Sorry I'm sure I'm missing an obvious interpretation, if I read it tomorrow I'll probably understand.

I probably explained the first premise poorly.

There are two major concepts about intervals.  One is what I call "what it is", in absolute terms.   Play C  Eb on the piano - look at the keys you are playing, listen to the sound.  Now play C D# on the piano and do the same.  You have the same sound and played the same piano keys.  You'll see that they are the same number of piano keys away from each other, and each piano key from black to white, and white to black (EF and BC are the only two adjacent white keys), are each a semitone apart.  You have the same number of semitones (like measuring in inches, except semitones, as a smallest unit).

The second concept is "what it's called" - the naming of intervals.  It's a convention based on how notes are written and thus their "spelling" and goes by letter names.  The 2nd, 3rd, 4th has to do with the naming.  If you think of the absurdity of English spelling (through, though, tough, with ough making different sounds due to history; gait and gate; to too two - same sound represented by different spelling) it's a bit like that.

In naming we have two elements: (i) 2nd, 3rd, 4th etc. up to octave - (ii) quality (major, minor, [perfect], diminished, augmented.  So you can have a major 6th, minor 6th, diminished 6th, or augmented 6th.

For formal naming of intervals we go by letter names: how the notes appear on the page.  The first thing we determine is the numbers part (2nd, 3rd) and here we go from lower to higher.

CD .... D is the 2nd note up from C, so it's  "2nd"
CDb .... same thing; it's another kind of "2nd"
CD# .... another kind of 2nd
C# D# ... another kind of 2nd

because if C and D stand beside each other, C is the first note, D is the 2nd note.

CE .... E is the 3rd note over (C,D,E) .... it's a 3rd
CF ..... F is the 4th note over (C,D,E,F) .... it's a 4th
until you get to CB .... B is the 7th note over (C,D,E,F,G,A,B) ... it's a 7th

The first thing I learned was just to be able to call an interval by its number (2nd, etc.)

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Back to our CD# and CEb that you played in the beginning.  They were the same piano keys and gave you the same sound.  For "what it is" in reality - they're the same interval.  For "what they're called" for traditional naming:

CD# --- some kind of 2nd
CEb .... some kind of 3rd

CD# is an "augmented 2nd":  CEb is a "minor 3rd"
This comes later.  But I found it good to know that there are these two aspects - just like I know that "gait" and "gate" sound the same, but they are spelled differently with a different meaning.

[ranjit]

Check this out.

[keypeg]

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