Intervals - explanation (let's see how it goes)

keystring

The other thread is theory in general. I'm going to try to present intervals separately and see what happens.

What we have is a naming system for intervals occurring in written notation. It evolved in earlier times and is thus kind of antiquated, but we're stuck with it if we're going to communicate in a "common language" or understand theory books and such referring to them.

PART I

These names have two parts: a number (2nd, 3rd, 4th etc.) and a "quality" (major, minor etc.)

Number

This comes first. You can think of a row of notes, 1st note, 2nd note, 3rd note, 4th note, 5th note etc. So if we have CE, our row is C, D, E, and E is the 3rd one over. (count them) If we have C, A, our row is C, D, E, F, G, A - and A is the 6th one over. So for CE, the interval is "some kind of 3rd". For CA, the interval is "some kind of 6th". The "some kind" part is our quality (major, minor etc.)

This part is easy, we're just counting. It doesn't matter whether we have A, Ab, A#, Abb, Ax - It's still the letter A. If the bottom note is C of some kind, we have some kind of 6th because it's the 6th one over.

Quality

That's the tricky part where students, teachers, and books, tie themselves into knots.

We start with the idea of "major" and "perfect". These are our reference intervals.

You start with the bottom note. If the top note belongs to the major scale of that bottom note, then your interval is major for 2nds, 3rds, 6ths, 7ths. Example:

CA = M6 because A is the 6th one over from C, and A belongs to the C major scale.
CD = M2 because D is the 2nd one over from C, and D belongs to the scale of C major scale.
CB = M7 because B is the 7th over from C, and B belongs to the scale of C major scale.

Perfects are unisons, 4ths, 5ths, octaves. If the top note belongs to the scale of the bottom note, then we have a perfect.

C unison - d'uh 😉
C F = P4 because F is the 4th over from C, and F belongs to the C major scale.
CG = P5 because G is the 5th over from C, and G belongs to the C major scale.
C C' (octave) = P8 (perfect octave) - another d'uh.

SouthPark

keystring CA = M6 because A is the 6th one over from C, and A belongs to the C major scale.

'A' is the 5th one over from C actually (in the C major scale). This is the exact reason why I prefer 'span' of notes. Because C to A spans 6 notes (including themselves).

But if you say sixth one 'over from' C, then first one over from C is D. And eventually ... get to the sixth one 'over from' C ..... which is B.

With spans ... C and E ... spans 3 notes in C major scale. Major 3rd. C and A ... spans 6 notes in C major scale ... major 6th.

E and G ...... E major scale ... we have E, F#, G#, which doesn't have G. But E minor scale has E, F#, G ..... minor third. Or alternatively, E and G# is major third in E major scale, so E and G is one semitone less ... minor third.

keystring

PART II

(For this part, when I studied theory, minors came later after having the majors/perfects solid. The augmented & diminished came at still another level later on)

Qualities that are not major or perfect

Again, your reference intervals are major and perfect. You derive the other qualities from those intervals, and how they're different.

from majors

For 2nds, 3rds, 6ths, 7ths

i) If your interval is a semitone lower than a major, then it is a minor

CD = M2
C Db - the notes are closer together by a semitone. So that's m2
(You could also say that C# D is closer by a semitone, and is also m2 - I don't know if that will mix people up)

CE = M3
CEb - the notes are closer together by a semitone, so that's m3

CA = M6, so CAb = m6
CB = M7, so C Bb = m7

If the 2nd note is two semitones lower (= a whole tone lower) than the major interval, then it is "diminished".

CA = M6
CAb = m6
C Abb = dim6

(We still always start with "numbers" - 1st, 2nd, 3rd etc.)

If the 2nd note is a semitone higher than than major, then it is "augmented".

CA = M6
CA# = aug6

From Perfects

If the 2nd note is a semitone lower than a perfect, it's called diminished (there are no minor 4ths, 5ths, unisons, or octaves)

C Cb' (an octave above but a tad lower) = dim octave

C Gb = dim5

C Fb = dim4

If the 2nd note is a semitone higher than perfect, it's called augmented

C G# = aug5

C F# = aug4

That's a summary. The basis is i) counting 1st, 2nd, 3rd ii) majors and perfects as starting point. It's also a cumbersome system and more than one person will question it.

keystring

PART III (misc. thoughts)

The main reference I learned when I started was, "Does the top note belong to the major scale of the bottom note?" as I wrote. Unless you know your scales super-well, that gets tedious (it did for me back then).

You might also see how many semitones are involved, and I've seen that presented.
Your ear may be able to recognize things like M3, P4, octave, m3. (If it's spelled C D# and you hear a minor 3rd, you may know it's "aug 2" if you know how they relate).

Rather than just referring to the major scale, since pianists play chords, that might help recognize some of them. Here are intervals that I can recognize from chords:

major & minor triads in root position - P5
major triads - M3 (CEG gives us CE)
minor triads - m3
sus4 chord - P4
M2 and m2 is something we'll just recognize since that's a whole tone and a semitone

So there we have at our fingertips: unison, m2, M2, m3, M3, P4, P5.

The piano itself is our reference, if we use the "white major scale" (C major) Any interval from C which has a white key will give us majors and perfects.

The "7" is a bit of a troublemaker, because we are so used to the 7 in the dom7 (G7 going to C for example) that it sort of feels like a default interval, but GF is m7, because in the G major scale, F# is the 7th note.

I don't know how much jazz chords, extensions, and such, help or hinder. (have been pondering that)

I'd love to see additions to this by others, questions, and what not. Also don't know if this was a good or bad idea. 🤓

ranjit

While the explanation is good, I feel like there should be a way to structure it more concisely. Images would also probably make it much more efficient.

I know I just picture a piano keyboard in my mind's eye and think of the C as "1". Everything else falls into place. What exactly do people find difficult about intervals?

keystring

ranjit While the explanation is good, I feel like there should be a way to structure it more concisely. Images would also probably make it much more efficient.

My time is limited and so are the resources.

keystring

Ranjit, I wanted to thank you for your response - I'll be able to do it here because the topic is still related to intervals. (The thread where I'd normally respond is locked).

Ranjit wrote
Right, I meant to address the reason why you sometimes use "augmented second" instead of "minor third". Typically it has to do with scale degree relationships.

(You had given the example of a C#m harmonic scale.) We've been exploring how to recognize and name intervals. Another aspect is where do we see them, esp. the ones that will be less familiar to those starting out, like the augmented and diminished. Yours is a good example. Now I understand why you presented that scale.

We also find that in the dim7 chord in "inversions". Whereas the chord itself is absolutely symmetrical with the same "minor 3rd" going up the piano forever, if it's inverted in the same spelling, or you go up one more note, you'll get an aug2nd.

Visually it looks like we have "seconds" in there but the sound is continuous m3's (what we hear), and ofc those adjacent notes are aug2's. (If this makes sense) All of these are the same dim7 chord, shown as inversions for the example (that in itself is a kettle of fish).

SouthPark

keystring (The thread where I'd normally respond is locked).

They should not have locked it.

In any case - regarding intervals - particularly the 'diminished third' ..... one way of looking at it is - if a person has a 'visual' only on the keyboard - or if a person has 'sound' (audio) only, and regardless of what 'symbols' are written ....... if I get you to play C# and E-flat - regardless of their symbolism, and if others are not 'told' or shown the symbols --- as they only have a visual or audio, and when everybody then uses their 'theory' (involving scale degree - either major or minor) without any other relative assumptions, then 100% of people will arrive at the same result. Major second interval.

And yes -- if somebody wants to define an interval called a 'diminished' third - in which there is certainly a definition for it, then 'diminished third' interval simply means an existing minor third cut down by 1 semitone, or a major third cut down by 1 tone (aka two semitones).

What that guy on Quora is trying to get people to 'know' is the 'on-paper' method.

This is not his fault - or our fault. It's once again - those 'geniuses' that don't have the 'important' information about their musical theory 'rules/convention' (even if it is along the lines of pseudo-science) stated clearly on the front page or at the start of their chapter ..... explained clearly, with adequate examples covering all the relevant cases. The aim is to teach everybody the convention(s) ..... 'properly'. In fact ... I don't think they even teach it in their books etc at all.

From 'their' perspective, they are trying to say that - 'on paper' (not pure visual or pure audio -- but on score sheet paper) -- for example C# and E-flat are written on a score sheet -- their 'rule/convention' is to temporarily forget or ignore any sharp or flat symbols associated with the pair of notes. In this case, C and E are brought forward for referencing purposes. And then we keep that 'reference' condition in mind - a major third. It's a reference. Then the next step is to observe the symbols next to the letters - which operate on each of those two reference notes. And depending which operations (flatting or sharping) are carried out on each 'letter' (eg. on the 'C' and also on the 'E'), they then arrive at the particular defined interval name in their 'theory system'.

So in their system - once again - it's not because we can't understand it -- but because those geniuses don't write it at the 'beginning' of their theory book (or anywhere for that matter) -- it's actually on them (and the teachers). And it is no wonder nobody was able to refer anybody to anything official on this particular aspect of theory - which is ----- on 'paper' (the score sheet) -- regardless of the sharp or flat symbols next to the pair of notes being compared (ie. flats or sharps etc), simply focus on the note symbol (the round things) to begin with. That information - without the flats or sharps - provides 'their' starting point or starting interval. For example, if on paper they write 'C# and E-flat', then the 'starting' point will be C and E (relating to a major third).

And if they had written C# and D# (on the score sheet) - then they will use C and D as the starting reference (relating to a major second).

It is only after that - the next stage is then to focus on the particular operations done on those base letters (eg. C and E; or the C and D), which then determines whether the BASE 'third' or BASE 'second' is going to be reduced (diminished) or expanded (augmented) etc.

One example which the theory books probably hasn't got a definition for is what I was wrote in the 'other' thread.

Eg. let's take middle C (C4), and then we take G4, and remember this pair. And then we put four sharp symbols on the C4 (even if it doesn't fit well), and then we put three 'flat' symbols on the G4. According to 'their' convention, the base interval (ignoring the symbols) will be a fifth --- ok a 'perfect fifth. So by 'their' music theory convention - note 'convention' -- they define THEIR interval as 'referenced' to a fifth. And it is going to be cut down by SEVEN semitones. And their result - which is what they call 'enharmonically' equivalent to a 'unison' interval ----- well, this particular result is probably undefined (as in there is probably no name for it such as 'super-duper-diminished fifth' or 'normalised fifth'. But it is - regardless - 'enharmonically' equivalent to a 'unison'.

They can use that convention if they want. The important thing is for those 'geniuses' to explain it properly - such as in a way that is coherent, and understandable. That is - they don't even explain any of this at all in their courses or books.

But - let's put it this way. When somebody physically plays a C4 with four sharps and a G4 with three flats, and the people (receivers) are only getting a visual or audio (not paper) - and when they (the receivers) apply the theory 'rules' associated with intervals, and they (the receivers) are not allowed to interpret those two notes in other ways (such as sharped or flatted versions of other notes) - then the result is going to be 100% the same for every person (that correctly applies their theory) ...... a unison.

Same with C# and E-flat. If getting a visual and/or audio only ----- every single person applying their interval theory correctly (and not allowed to interpret the two pressed notes on the keyboard as sharped or flatted versions of other notes) - will 100% always arrive at major second.

I'm also going to turn the tables on the guy in Quora that said - the teacher will give 'red ink' (ie. a cross etc). It's more like I will fail the theory books and all teachers that do not explain the system to students in a coherent way. This is the life story of a lot of books and teachings. It's the people that write the books and the documents that drop the ball. Not explaining clearly, and not providing various examples and cases to get the picture across very clearly to everybody.

In my opinion ... even with symbols of sharps etc, the interval associated with C# and E-flat is NOT a type or kind of 'third'. It is simply 'referenced' to a major third interval. And the way they convey it ..... is they had to give a name that contains details about the operation as well as the 'reference' ...... that is ..... 'diminished third'. That is ... the reference interval is set to be a major third, and the operation is a reduction. A reduction of 1 tone (aka 2 semitones).

Similarly ... the interval associated with C# and D-flat is not a type of second, but can be referenced to a major second. And that reference is cut down ... and a name is given (with details about the operation together with the reference - that's if a name is defined ... as not all cases have a name as such) ... eg. diminished second, having absolute equivalence to a 'unison'.

Also - interestingly - a 'diminished third' is equivalent (in absolute terms) to a major third reduced by TWO semitones, while a diminished perfect fourth (ie. diminished fourth) is equivalent to a perfect fourth reduced by ONE semitone. And a diminished perfect fifth (ie. diminished fifth) is equivalent to a perfect fifth reduced one ONE semitone.

On the other hand, some people could consider a diminished third being equivalent to a MINOR third reduced by ONE semitone. So everyone can see that some details need to be rote learned, which is no problem. As long as the teachers tell the students about these various details, then everybody can have a better idea about how 'their' system works - when explained properly.

ranjit

Yes, that is a good example too.

keystring

ranjit Yes, that is a good example too.

The fully diminished chord was an important one for me, because at a quick glance we may think we have an M2 or m2 and totally miss the type of chord we have - it fools the eye. I already "knew" how everything worked and still was almost fooled one day with a new piece until I played that particular chord on the page. Anticipating what the music should be because of how it goes and should logically progress helps too, of course.

keystring

To Ranjit and anyone - I think this thread may continue working if one decides to ignore and refrain from responding to OT-type things.

This thread was intended to help teach learners how the traditional naming of intervals, which is universally known and used.

SouthPark

keystring OT-type things.

It's far from O.T. This makes the significant difference between a proper education and going blindly without understanding particular important aspects - due to it being undocumented mainly, or never spelled out clearly or at all.

A proper foundation, which is the reason for making notes about the gaps or weaknesses in the written/documented theory and courses. There are holes.

It's not because the students aren't able to understand it. It's because very few people have actually got around to properly 'teach' certain aspects ... which actually doesn't take long to do. But they either didn't bother, or they just didn't know it at all.

It is then up to us to address and fix that.

I reckon a lot of people will indeed find what I wrote to be very useful in terms of understanding the terminology/definitions, and they can have a good think about the intervals system convention, and then decide for themselves what they think or feel about particular aspects of the system/convention.

WieWaldi

SouthPark I would prefer a "go 5 notes up" notation instead of having a span of 6 notes (including first and last).
And I would name the frequency doubling step not octave like 8, but septive like 7. There are only 7 different notes (qualifies excluded). But this discussion has been discussed to death over in PW. And it was fun to see how both camps argued to their side.

WieWaldi

SouthPark keystring CA = M6 because A is the 6th one over from C, and A belongs to the C major scale.

'A' is the 5th one over from C actually.

Southpark is right. And your post is the proof that the existing system is confusing. Even for experts and teachers.

SouthPark

WieWaldi C and A is a major sixth interval. So if we use the span number in C major scale, then the value '6' aligns excellently.

Similarly ... C and G spans 5 notes in C major ... perfect 5th ... aka 5th.

keystring

WieWaldi I would prefer a "go 5 notes up" notation instead of having a span of 6 notes (including first and last).
And I would name the frequency doubling step not octave like 8, but septive like 7. There are only 7 different notes (qualifies excluded). But this discussion has been discussed to death over in PW. And it was fun to see how both camps argued to their side.

I tried to give a 3-stage lesson for anyone who might need it for what it is and how it is done, for the existing system. Could we please not discuss alternative systems that might exist exist? Another new thread could be started for that. I'd like this thread not to become confusing and convoluted like the other one did, for the sake of learners.

In my 3-part presentation, the first part defined how the "numbers" part works. It represents "1st note, 2nd note, 3rd note, 4th note" as though you were looking at racers coming in, or a row of houses saying "I live in the 3rd house from here." CA --- you're standing at the C house, and we're going to the 6th house over.

Please look at the very first post. It was organized that way for a reason.

keystring

I presented my terms and premises carefully before using them. The first thing I defined was "numbers" meaning what we call 2nd, 3rd, 4th etc.

me wrote
This comes first. You can think of a row of notes, 1st note, 2nd note, 3rd note, 4th note, 5th note etc. So if we have CE, our row is C, D, E, and E is the 3rd one over. (count them) If we have C, A, our row is C, D, E, F, G, A - and A is the 6th one over. So for CE, the interval is "some kind of 3rd". For CA, the interval is "some kind of 6th". The "some kind" part is our quality (major, minor etc.)

What is meant is defined there. Therefore in the next part when I give examples and write "CA - A is the 6th one over...." this goes with the definition already given. If someone glosses over the intro and jumps into the middle and then gets to the wrong conclusion of what is meant by "6th one over" - that's a problem, and THE problem here. If you want to give it a specific name yourself, such as "span" - go for it. but please start with what I defined, rather than assuming.

I usually teach this in stages, making sure that each concept is understood and can be worked with, before going on. It is not confusing when done that way. When it is all written out, there is the propensity to:

a) try to get it all in one quick read
or
b) extrapolate things you heard before because of trigger words in there
or
c) jump to bits that catch your attention and assume you know what was meant

Nobody is disagreeing that it's an awkward system. I called it "antiquated". However, there are people who want or need to learn this system. Given how complicated the topic got in the other thread, I tried to present something organized. I had no idea whether it would work, however.

i.e. this:

wiewaldi wrote
And your post is the proof that the existing system is confusing.

It is definitely an awkward system. I said as much in my opening post. I don't find it confusing, but I also don't like it that much.

SouthPark

K.S. I know what you mean. It was a long time ago - I thought the issue was me - but the issue is actually with the system. The way that people are teaching it - which is messed up.

For example, for intervals - people are often teaching things like ---- 'From C to E'. And students are always thinking the obvious -- and intuitively -- sure -- from C --- in C major scale, the 'E' is two notes 'away' from 'C', right?

Well ---- of course --- yes it is. E is TWO notes away from C. In other words, that is TWO houses from C. Not three houses from C.

Although, if we think of it as --- houses on a street, with the first house (labeled number 1), second house (labeled number 2), third house etc. Then we can say things like the 'third house'. We don't need to introduce uncertainty by saying 'from'. So when we say sixth house - we know EXACTLY which one it is .... it's the sixth one. And this is extremely clear. This is the 'span' method.

For the span method -- we say eg. C and E. The C being the lower note. The E being the higher note. The lower note sets the scale, and is also assigned temporarily to be the tonic (root note of that scale). In this case - C major scale, which goes C, D, E etc.

E happens to exist in the C major scale. C to E spans THREE notes in the C major scale. Major third.

Had E been an E-flat instead. Then we can just go with C to E-flat is one semitone less than the major third. So -- that would be a minor third interval. But to get formal about it -- C minor scale goes C, D, E-flat. So E-flat exists in C minor scale. C to E-flat is a minor third interval, because C to E-flat spans three notes in the C minor scale. It spans three notes in the minor scale --- minor third.

keystring

SouthPark K.S. I know what you mean. It was a long time ago - I thought the issue was me - but the issue is actually with the system. The way that people are teaching it - which is messed up.

For example, for intervals - people are often teaching things like ---- 'From C to E'. And students are always thinking the obvious -- and intuitively -- sure -- from C --- in C major scale, the 'E' is two notes 'away' from 'C', right?

There is a much bigger picture that I'd like to go into for teaching as a whole. I find that when things are taught, esp. theory, "basic things" are glossed over, or gotten at in a utilitarian superficial way. By "utilitarian", I mean let's give a quick name to this so that we can get back to playing music - or let's memorize these rules so the exam can be passed. The teacher may have learned in the same manner and nobody has ever really gone to the heart of anything. We can pass exams nicely without ever grasping the essence of a thing.

In 2006 I studied the whole RCM rudiments, did the middle and upper level exams passing the middle with 99.95% and felt quite dissatisfied. I restudied it all, this time at the piano or tapping rhythms to make it more real: then I taught the whole thing over 18 months, this time the way that I would have wanted to do it. I was still near the start of my own journey, but it went fairly well.

When I worked with that student, who had been playing for some years, one of the elementary concepts that had gotten mixed up involved time. There is note value, the relative note value of say half note vs. quarter note, beats, and meter. For some of this there were "what is this really?" moments to sort out. (Just as an example - I don't want to start a new topic. 😉 )

The elementary concepts are "so easy", so "babyish" - why spend any time on them. And yet we build everything on top of them and if the bricks are misshapen, we'll flounder here and there without ever knowing the cause as we advance. OR we actually succeed with high grades in exams, be able to pontificate to the next crop of students if we become teachers, without ever really connecting any of this to music.

In my own training as a teacher at the "formative" level (primary grades), I already had the concept of going from concrete to abstract. The "centimeter cubes" example is that. So I went in that direction. My own first experience in lessons, which happened as an adult, collapsed largely because the "baby things" were rushed through. While a lot of this was physical instruction, the same principle applied. When I went back to basics and the "real" version, problems in advanced music also often resolved themselves.

This has gone way past intervals, but the issue with intervals that you described I think is part of that picture.

keystring

On the numbers part. Maybe we can even get to a common point here.

When we measure something, the first number on a ruler is "0". We start at point 0. When something is 4 centimeters long, we started at 0. When I used to teach primary school the kids got "centimeter cubes" which were each 1 cm. long. They stuck together their centimeter cubes and visually got to see how many centimeters long something was. How many centimeters fit. This is the best image I could find:

This is NOT what our interval names reflect. They don't tell us "how long the length of the interval is". For that, our equivalent to centimeter cubes might be semitones.

Instead, we've got something quite primitive. You're at a row of trees. How far are you supposed to stretch that rope? "Up to the 3rd tree". You're counting your trees: 1st tree, 2nd three, 3rd tree. You start at 1, not 0. That's what the interval names do.

Supposing your trees are 1 yard apart (they're small trees). The distance from the 1st tree to the 3rd tree would be 2 yards. We've got a distance of two yards, but we've gone to the third tree.

The "length" of an interval - or "size" of an interval - is done in a primitive way. "It's over yonder, 4th gopher hole over." and if you stretch your rope first the 1st to the 4th, you get the right length of your rope - this is also a way of "measuring" but not how we think of measuring.

keystring

South Park, we cross posted. I did the centimeter cube one while you were writing. 🙂 We might actually be getting somewhere. I'm going to respond to what you wrote - this may be going somewhere.