Emeton In an elementary presentation of intervals peculiarities such as augmented intervals can be mentioned. In this ex. note how the top voice in the ii 4/3 chord moves chromatically upward. ......
Emeton, we cross-posted. Would you like to start a new thread on the topic of progressions and movement on chords? For the reason in my post where we cross-posted.
The topic of intervals for learners which goes all the way to dims and augs is already a vast and complex one.
The purpose is to help to understand the aug and dim intervals. I hope that someone will play through my example; it’s a pretty basic progression.
Keystring's knowledge of theory is impressive and exhaustively explained in this post, and may be helpful to those who want to dig into the intricacies of this aspect of music theory, and I applaud his efforts and his willingness to share.
I'm a practical guy and I have a rather practical approach to playing music. If someone is going to primarily play piano (as an example, since this is mostly a piano oriented forum) by reading notes on a page, little to no knowledge of music theory is required; you just play what's written.
If a player wants to improvise, a knowledge of chords, scales and/or modes, is helpful, but certainly not necessary. Knowing how scales and chords fit the key of the song offers a framework of what's available in terms of tonal choices, and whether the player wants to play "inside" or "outside" of the changes. But again, players with good ears may be able to improvise without knowing much theory.
If a player wants to venture into composing, theory is certainly helpful in being able to construct a piece that makes sense, technically, and tonally, speaking. But let's face it, there are plenty of song writers who have composed, or constructed, songs that have been accepted by the public; some who couldn't even read music, let alone understand the theory.
For those scholarly folks who want to want to discuss the intricate workings of music theory, things like intervals, diminished 3rds, and how they're called, and when to call them, can probably fill countless hours of discourse, much like what famous writers did at their local pub with regard to storytelling, syntax, grammar, or plot. But for the majority of people who want to play the piano, or any other instrument, none of it matters.
GaryD The traditional system goes back centuries. I think your idea of talking about trees is a good explanation of how it works.
I was grasping, but I have used trees, or houses, or any row of "things" multiple times in the past to sort out what this naming system is actually about. In discussions among adults and teens, this has worked. It also solves a problem created by how interval names are created, when we're told that these names describe a "distance" between intervals. That's a glitch, because when we describe actual distance, we measure using standard unchanging units like inches, centimeters, ounces, and the starting point is always zero. People see on the piano that CE involves two whole tones, or four semitones, and "3" doesn't add up (literally). The word "distance" is misleading. (That's what SP had been trying to get at, and why he invented the word "span" for this).
Ok, so we have "distance" and this peculiar way of naming it. A student trying to understand rather than just memorizing stuff can be confused by what they actually perceive. So I wondered whether in real life we have something similar, and yes, we do:
My parents farmed, and when you're out in the field you make do with what's around you, and you're also repeatedly working with familiar landmarks. You may gauge the "distance" for this many trees without ever measuring anything. I may indeed need to stretch a rope or wire and for whatever reason I'm counting out trees, or rocks, or whatever. There are plenty of times in real life that we "measure out" things without every using a unit of measure.
The old naming system we inherited is sort of like the old farmers and their fields and familiar landmarks. Instead of unevenly spaced trees and "1st tree, 2nd tree, 3rd tree" we have unevenly spaced notes, always in the same sequence of spacing regardless of where you start. It is not really "distance" in how we "measure" distance.
An important step may be simply realizing that the interval names are not measured distances in how we understand measurement, but this other thing. And then linking that to actual measurement, whether semitones. Some people do this instinctively. There are two things. This "1st 2nd 3rd" naming thing; the actual distance of intervals.
GaryD ..... The more modern system and the one I always use could be called the sound interval system. And this system only cares about what we hear.
Yes. This touches on "actual distance" (as in standard measurement) and in this, C D# and C Eb are in actuality the same distance apart (because they are) and "on the piano" have the same sound quality. I have seen a few ways of expressing intervals in alternate ways - the one I saw first was by semitones. All of them touch on the reality of our "1st tree, 2nd tree" system with unevenly spaced trees is quite awkward, esp. as music gets less diatonic.
I'm only going as far as the idea of two views: one being how the interval naming system is set up .... so as to be able to use it, and know what it's not - and the "realness" where C D# and C Eb are the same piano keys emitting the same sound quality played on the piano.
And then ... in this thread ....... how to learn to use that system. (My first three posts)
Emeton The purpose of my contribution is to help the various participants who seem to be trying to understand the aug and dim intervals and writing a lot about it. I know this is your thread and I will stop contributing if you wish. My last post however was a response to Dire Tonic who asked me a direct question. I do hope that someone will play through my example; it’s a pretty basic progression. The aug 6th resolves in the opposite direction from a mi 7th in a dom 7th chord.
I would not want to stop you. 
(shuddering at the thought of becoming a gatekeeper) I was thinking ahead that this topic can spawn a number of related topics which are also important, and the idea that maybe other threads might spring from that.
The aug6 is indeed a good example.
fwiw, I don't "own" the thread I started. Nobody can own a thread.
PianoMonk I'm a practical guy and I have a rather practical approach to playing music. If someone is going to primarily play piano (as an example, since this is mostly a piano oriented forum) by reading notes on a page, little to no knowledge of music theory is required; you just play what's written. .......
I agree with the essence of what you're saying, and to a point. 
The reality is that many learners embarking on the journey, and taking lessons or self-teaching using method books - these learners will be given theory to learn. These often seem to be taught hastily and shoddily. This topic came about precisely because a fellow student asked about resources for learning theory, and then the intervals tangent happened.
I tried to fill a gap here in regard to intervals because of what I saw in the other thread. My first three posts here, I hoped might be a rough guideline of essential points. This was for those students especially.
For your broader point, I don't totally agree. In my experience as a learner, some degree of theory also helps in reading music, making it faster, or in working on a new piece. Just a few examples: recognizing a V7-I cadence often marking the end of a phrase; knowing what key I'm in; common chord progressions making the music go exactly where I expect it to go (when it does). If your piece has a simple A B A' form, where the middle part is in the Dominant key and the end is almost totally the same as the beginning - if you recognize that, then your job of learning the piece is cut down by a third. All this is "theory".
For the teacher: how easy is it to point out things without a name or reference? Supposing the student plays a C chord when the music shows Cm. Yes, he can point to the "Eb", or demonstrate what it's supposed to sound and look like by playing it for the student to imitate. He may say to the student, "Can you hear the difference between what you played and what I played?" without defining that difference. But what if the teacher teaches the student to distinguish between a major and minor triad? He has to give it a name, and that again is theory.
The problem I've seen with theory is that it often seems to be taught as a separate abstract topic - like algebra. A thing to be learned filling out exercise books and passing exams on things separated from music. It never connects to music. It's this scary isolated thing. You learn to play music. And you do your theory work. Never the twain shall meet.
PianoMonk Keystring's knowledge of theory is impressive and exhaustively explained in this post, and may be helpful to those who want to dig into the intricacies of this aspect of music theory,
I had another thought, and it's a thing that matters a great deal to me.
I actually tried to get at the foundations - the "basics" - rather than complex intricacies. This is going to get a tad philosophical and maybe personal.
My first music studies was when I was a middle aged adult. The 'boring' things were rushed through, or "simplified" for "fast progress". Most of that involved physical playing with an instrument which was technically difficult - but theory was there too. When I crashed a few years in due to this wobbly foundation, the solution turned out to be getting at basic things. .... unsurprising actually. When I left teaching for another field but still tutored on and off, the grade 7 student struggling with algebra often had a weak grasp of what multiplication or addition actually was about: they had memorized times tables and such.
We see basics taught in a "get-it-over-itis" manner. You get little tricks for getting the right answer, and you never really grasp what the thing really is. The basics of what a thing really is, is actually the hardest thing to learn. In public school grades 1 & 2 and maybe kindergarten are the "hardest" grades because you're not just reading, writing, doing arithmetic - you're picking up the concept of these things. It's crucial.
What I really tried to get at in my first three posts was the basics of written intervals, and what an interval is; how the naming system works. It is extremely condensed and is not meant to "study and learn" and especially not in a few days or a week or two. In my own quest, grasping the foundation of things, has seemed to pull me out of almost every hole. Basics and foundations are not "baby-stuff" - and getting at them is potent stuff.
I'm not going after intricacies. However, when you get the essence of a thing, it might lead to intricacies.
That said, the teaching and learning of such things are not done in written summaries. It's exploration, maybe guided exploration. This thread is no substitute or at best a poor one.
Emeton In an elementary presentation of intervals peculiarities such as augmented intervals can be mentioned. In this ex. note how the top voice in the ii 4/3 chord moves chromatically upward. The bVI+6 chord to an elementary student studying theory, as in this thread, might spend way too much effort convincing themselves that it just like a dominant 7th chord. It’s a waste of time doing that and if you don’t have an actual theory instructor that can tell you, ‘wait until 3rd semester and it will be explained’. The bVI+6 chord doesn’t function as a dominant 7 chord whatsoever but more as a convenient artifice brought about from the chromatic voice movement between all voices moving from the ii 6/4 through the bVI+6 to the I 6/4.
It is true that to understand why a dominant 7th chord is respelled as a German 6th chord you have to know what the next chord is. This is where you get the augmented second at the top of the chord. Every chord like that will have an augmented 6th and an augmented 2nd.
But every diminished 7 chord with an octave will contain an augmented 2nd. You don't have to know anything about theory to grasp this because it's simply a fact which is built into a notation system going back many centuries.
In my view you don't have to know any of these complicated names to write such music. You simply have to know that whatever chord you write is going to be shaped in the way that is is written by what the next chord is. For me the so called augmented 6th chord is simply a dominant 7th chord that goes to a different place than expected and is usually respelled at the top to make more elegant notation.
I'm a practical guy and I have a rather practical approach to playing music. If someone is going to primarily play piano (as an example, since this is mostly a piano oriented forum) by reading notes on a page, little to no knowledge of music theory is required; you just play what's written.
I am just as practical. For me anything theoretical that does not help me play better, improvise better, compose better or memorize better is useless.
That said, you have to ask yourself what theory is. If you simply learn to play major scales in 12 keys, that's theory.
If you learn how to play major and minor chords in root position in all 12 keys, that's theory.
If you learn how to invert chords, that's theory.
Depending upon your background you may or may not have gotten everything that you need. I most certainly did not. What I lacked gave me a serious problem in memorization because I didn't understand what I was playing. I also could not instantly identify why something was wrong what sounded wrong. In my middle 20s I went into a dance band situation armed with so called classical music theory. The other musicians had all sorts of pop and jazz theory that I was missing, and they had to help me fill in the gaps.
In my mind the word theory covers a lot of ground. For the record I'm still learning.
keystring
The problem I've seen with theory is that it often seems to be taught as a separate abstract topic - like algebra. A thing to be learned filling out exercise books and passing exams on things separated from music. It never connects to music. It's this scary isolated thing. You learn to play music. And you do your theory work. Never the twain shall meet.
When I am explaining music to people with less experience, I often make connections to our native languages. There are people who shudder at the mere mention of the word grammar.
But in my opinion all sorts of basic things that we absolutely need for writing text requires basic knowledge that most people will not think of as grammar, but we might call these things language theory.
For instance, what words need to be capitalized? What is basic punctuation? What is a complete sentence, and how do you recognize when you haven't written one? When is it time to start a new paragraph?
When are things like "gonna" and "wanna" appropriate in both informal speech and writing? When can we use phrases like "I don't give no never mind?"
This is the way I look at music theory. For me it is all about what is practical and what helps me become a better musician. For me it has nothing to do with getting a grade or passing a test.
Emeton The purpose is to help to understand the aug and dim intervals. I hope that someone will play through my example; it’s a pretty basic progression.
I didn't see the "I hope..." part when I responded initially. This is actually a good thing for anyone to do. A good idea.
PianoMonk If someone is going to primarily play piano (as an example, since this is mostly a piano oriented forum) by reading notes on a page, little to no knowledge of music theory is required; you just play what's written.
PianoMonk, I know I responded before. Something came into my feed which brought me back to this conversation. I hope it won't be too off topic.
I watched a violinist who critiqued his own playing when he was 13 years old, and showed himself playing those same passages as a concert violinist. What struck me is that as he was critiquing his old interpretation, he kept referring to theory - what he had done with a suspension and should have done, for example. Food for thought: Can theory at some level help us with interpretation and expressiveness? I can think of instances as pianists where it can.
Timestamp should be about 2:18
