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.

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.

K.S. --- the two 'proper' ways they generally use to define intervals is a 'distance' approach -- measured as an integer multiple number of semitones. The other proper way -- is the SPAN method.

Ok ...... I do know there is the way that a lot of people try to teach or push onto the students, which involves people telling students things like 'G is FIVE notes from C'. And then they will START their count AT C, Beginning the count AT C is not an example of FROM C.

In C major scale, beginning the count (1) at 'D' is the correct example of 'FROM' C. But the issue there is -- G is four notes from (ie. away from) C in the C major scale.

It's ok to use the 'FROM' method if we're using SPAN of notes though. That works excellently. Eg. in the C major scale, from C to D, this pair of notes spans (including themselves) 2 notes. Major 2nd.

K.S. - also --- very importantly, I totally understand what the teachers have been teaching. I notice there is a fundamental issue with that approach. But on the other hand, I definitely know that if the students are prepared to accept the 'G is 5 notes away from C' thing (and begin the counting from the tonic) --- then it obviously isn't going to ruin them. And the thousands of people that learned it that way certainly have not been ruined at all - and in fact are very successful. The main thing is that there is an understanding that teachers saying things like 'G is five notes from C' (and starting the count from C) is ----- I think we all know that it's 'technically' wrong. But I don't mind at all. I'm just trying to point that out only.

    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.

      SouthPark Ok ...... I do know there is the way that a lot of people try to teach or push onto the students, .....

      I'd like to get past what "people do" when they teach. I'm trying to get past that here. I presented it as how I teach it rather than what "they" teach. Either it helps others or it doesn't. You and I seem to be on the same page, where you've decided to use the word "span" to describe what I call "1st house, 2nd house, 3rd house..." which is less abstract than the word "span" which frankly I'd have to look up and might still get wrong.

      I'd be interested to know whether my presentation is useful to anyone. What others do, wrongly or otherwise, not so much. That's not the purpose.

          SouthPark The main thing is that there is an understanding that teachers saying things like 'G is five notes from C' (and starting the count from C) is ----- I think we all know that it's 'technically' wrong. But I don't mind at all. I'm just trying to point that out only.

          That puts us on the same page. Good enough.

              keystring That puts us on the same page. Good enough.

              We're on the same page now K.S. The centimetre (distance) units is equivalent to the integer multiple of semitone units.

              When we use the definitions of major and minor intervals in the form of unison, major second, minor second, major third, minor third etc --- there is a direct but non-linear mapping. Eg. unison maps to zero semitone. Minor second maps to 1 semitone. Major second maps to 2 semitones. Minor third maps to three semitones. Major third maps to four semitones. Perfect fourth maps to FIVE semitones. Perfect fifth maps to SEVEN semitones etc.

              Some people do take advantage of the above mapping.

              I don't have a problem with you or teachers teaching what you want to teach.

              And earlier, I was only pointing out that when you wrote 'A' is sixth one over from 'C', it should be 'A' is the sixth one in the set of notes 'C to A' in C major scale, which is the application of the SPAN-of-notes method.

              Another example of that is E4-sharp and C5. Since E4-sharp is the lower note, we go for E4-sharp major scale, which has this sequence E♯, F, G, A♯, B♯, C, D -- and then finally back to E# of course. And the E4-sharp and C5 spans six notes. Major sixth interval. Here, we can say that from E4-sharp to C5, we have a span of notes equal to six. We have a major sixth interval.

              I think it was great that we had this conversation, and also the discussion in this thread and the other thread about the diminished third interval. It looks like everyone has learned something - which benefits all of us.

              keystring You and I seem to be on the same page, where you've decided to use the word "span" to describe what I call "1st house, 2nd house, 3rd house..." which is less abstract than the word "span" which frankly I'd have to look up and might still get wrong.

              Once you have been taught what span-of-notes means --- which probably takes 10 seconds or less to convey (ie. span of notes being the number of notes covered/contained in a set/sequence of notes) --- then you will never forget it for the rest of your life.

                    We have some common ground but I'm afraid you slipped up on the "E# major scale" 😏

                    SouthPark E♯, F, G, A♯, B♯, C, D -- and then finally back to E#

                    We don't usually have a scale written out as "E# major scale", which doesn't make it impossible. Did you try playing what you wrote out at the piano, before posting? (I always double check) Try it - it doesn't sound like a major scale. A quick indication is that E#, F .... are the same piano keys, and B#, C are also the same piano keys.

                    I've written out an E# major scale, and juxtaposed F major. If you play these on the piano you should hear the same thing.

                    E# to C is dim6 (what you were after)
                    F to C is P5
                    That's a quick check, because you'll be playing the same piano keys. We know that dim6 = P5, and we also see and hear it.
                    However, in the E# major scale there is no C. There is a B# ---- and E# to B# is a P5. It also fits neatly.

                    This was kinda fun. 🙂

                    The problem happened in trying to write out this interesting scale. It happens. That's why I try to play what I write out before sending, to make double sure. 🙂

                        keystring We don't usually have a scale written out as "E# major scale"

                        K.S. There indeed is an E sharp major scale. As long as we follow that pattern ... tone, tone, semitone etc .... the major scale will sound like .... a major scale. It's all relative.

                            SouthPark K.S. There indeed is an E sharp major scale. As long as we follow that pattern ... tone, tone, semitone etc .... the major scale will sound like .... a major scale. It's all relative.

                            Are you taking the time to carefully and fully read what others write? I take care in my posts, double checking before sending. I literally wrote out an E# major scale. Did you not see it?

                            The notes you wrote out do not give us a major scale because you made some mistakes: If you play what you wrote out at the piano you'll hear it. I also did a sound file letting you hear that.

                            Please do go back and take your time reading everything. Otherwise we're not communicating. thx.

                              keystring We have some common ground but I'm afraid you slipped up on the "E# major scale"

                              Earlier .... I was meaning to write e-flat major scale.
                              E♭, F, G, A♭, B♭, C, and D.

                              The two notes I was meaning to set the example for is ... E-flat and C.

                              So now we see the value of myself laying out the foundations ... and explaining my procedure comes in handy.

                              The interval E5-flat and C5 in the E-flat major scale is a major sixth. Spanning 6 notes.

                                  SouthPark Earlier .... I was meaning to write e-flat major scale.

                                  It is important when writing in a forum to double check before posting. Otherwise time is wasted, including the reader's, and you can be taken less seriously when you end up making gross mistakes. Before I posted my notation, I played it to make double sure.

                                  The rest is the same as what I presented which is unsurprising since it's basic. The answer of M6 is correct.

                                      keystring It's fine K.S. In my post .... I did mention the pattern .... tone, tone, semitone etc. So you will know that the sequence I had earlier doesn't follow that pattern. And you know full well my example intention is to demonstrate a major 6 interval. And as I told you .... it should have been E-flat and C as the example pair. I was thinking of E-flat ...... not E-sharp, since E sharp is an F in absolute terms.

                                      Hence the importance of me writing the process ... so it's easy to see that cross checking indicates that my example really was to meant to be what I wrote. E4-flat and C5.

                                      Likewise .... when you make mistakes like writing A is 6 notes over from C ..... same deal.

                                        I think all has been written that can be written on this tangent. I'm done.

                                        Almost done. E# major scale will be F major scale. F, G, A, B♭, C, D, E

                                        If considering F4 and C5, then the procedure is to take the lower note. It is F. So use F major scale. If the lower note had been F# instead, then would use F# major scale. But we're dealing with the lower note being F. So F major scale.

                                        In this case .... F4 and C5 interval spans 5 notes in F major scale. Perfect fifth interval.

                                        Better cover F4-sharp and C5 as well. Lower note is F#, so choose F# major scale.

                                        F♯, G♯, A♯, B, C♯, D♯, E♯

                                        F# and C# spans five notes ... perfect fifth, while we want F# and C, which is 1 semitone less than the perfect fifth interval. So F# and C is associated with a diminished fifth interval.

                                          SouthPark Almost done. E# major scale will be F major scale. F, G, A, B♭, C, D, E

                                          In other words, this, which I posted a day ago.

                                          keystring If you play these on the piano you should hear the same thing.

                                          (acknowledgment would have been nice 🙂 )

                                          Your conclusions are all correct, and the way you approached it was valid.

                                              Hi

                                              This is a very simple way of thinking about the diminished 3rd that Keystring and I discussed privately whilst this thread was locked.

                                              If you take the C major scale that we all know and love...
                                              C D E F G A B C

                                              And if we zoom in on the first 3 notes of the scale C D E
                                              These are the root, 2nd and 3rd of the scale.

                                              Any combination of notes that involve C and D will always be called a 2nd.
                                              So C and D# is an augmented 2nd, even though it sounds the same as a minor 3rd; but it cannot be called a minor 3rd because it does not contain any form of E - the 3rd note of the scale.

                                              Any combination of notes that involve C and E will always be called a 3rd.
                                              So C# and Eb is a diminished 3rd, even though it sounds the same as a major 2nd; but it cannot be called a major 2nd because it does not contain any form of D - the 2nd note of the scale.

                                              That is how I think about the theory behind these intervals when looking at written music. You have to consider the key signature, the accidentals and how they relate to the root, 2nd and 3rd. That is the music theory we have! It isn't perfect, but it's very unlikely to change.

                                              Of course if you only hear the intervals and you have good musical ears you would name the intervals the other way around. You will hear a major 2nd when it's actually written as a diminished 3rd on paper. Though maybe if someone had perfect pitch and they heard the interval in context, I suppose it's just possible they might work out that it's not a major 2nd??

                                              All that said I'm not an expert 🙂, and it's 16 years since I last studied theory seriously.

                                              Cheers

                                              Simon
                                              All round average Jazz, Blues & Rock player.
                                              Currently working towards ABRSM grade 8.

                                                A diminished third in chorale music, string quartet or other ensembles that can employ ‘just’ intonation is not enharmonic to a major second.

                                                  Emeton A diminished third in chorale music, string quartet or other ensembles that can employ ‘just’ intonation is not enharmonic to a major second.

                                                  This is a good point.

                                                    Emeton A diminished third in chorale music, string quartet or other ensembles that can employ ‘just’ intonation is not enharmonic to a major second.

                                                    You are correct. That is why in discussing this I try to say "when played on the piano" - i.e. an instrument where pitches are fixed and can't be created by the player. It's probably a good idea to keep it at that simple level.