Teaching Intervals

David Newman and Greg Ristow chat about four ways to teach intervals in music theory, as well as how to overcome some of the challenges of teaching intervals.

 

Transcript

[music]

0:00:20.8 Greg Ristow: Welcome to Notes from the Staff, a podcast from the Creators of uTheory, where we dive into conversations about music theory, ear training, and music technology with members of the uTheory staff and thought leaders from the world of music education.

0:00:33.4 David Newman: Hi, I'm David Newman, and I teach voice and music theory at James Madison University and I create content and code for uTheory.

0:00:42.0 Greg Ristow: Hi, I'm Greg Ristow, founder of uTheory and Associate Professor of conducting at the Oberlin Conservatory.

0:00:47.8 DN: Thank you listeners for your comments and episode suggestions. We love to read them, send them our way by email at notes@utheory.com and remember to like us and subscribe wherever you get your podcasts.

0:01:00.1 GR: Today we'll be talking about teaching intervals, teaching reading and writing intervals more so than teaching the oral side of intervals, which we'll save for another episode and this David, I find to be one of the most challenging things to teach in fundamentals of theory.

0:01:17.0 DN: There's certainly a lot of approaches to use and picking one is hard.

0:01:22.5 GR: Yeah.

0:01:23.1 DN: And doing them all is also hard.

[chuckle]

0:01:24.8 GR: Yeah, for sure. For sure. And there's just a lot involved too, right? I mean, it's like you've got the generic interval size. You've got the specific size or quality, you've got the inversions, you have the compound intervals, consonance and dissonance. It just adds up and it's one of those concepts that it seems like, I often forget to budget enough time for because there's almost always a next step.

0:01:56.0 DN: And it's one of those foundational concepts that if you are struggling with it, you're gonna struggle with everything else.

0:02:02.7 GR: Yeah, for sure. For sure. And I find it's also one that can be hard to motivate students to learn, because if you're not careful, it feels really like a terrible set of math problems.

0:02:13.2 DN: Yeah. Also, it's really easy to do if you have plenty of time but it's, you need to know it better than that. You need to know it so well that it doesn't take you time.

0:02:26.2 GR: Yeah and I think for a lot of us who are teaching it there's also that challenge that both Leigh VanHandel and Melissa Hoag talked about of we already know it so well, it's really hard to conceive of what it's like not to know it.

0:02:40.2 DN: Yeah. And there's all these extra concepts involved, the letter names and...

0:02:46.2 GR: So I guess one of the things that I think about a lot is this challenge of how do we keep the teaching of intervals musical, right? 

[laughter]

0:02:55.6 GR: I think I'm gonna be really honest here and say, I think the first 15 times I taught intervals it was the dullest thing. I remember saying to classes, I'm sorry, this is gonna seem really boring, but it's really critical. As like a motivator, and I guess it's an okay motivator but there may be better ways the more I've taught it recently I've focused a lot on ways to keep it musical and make it musically relevant.

0:03:25.3 DN: Yeah. Yeah. And of course we've also had... We've talked to people this just recently about other ways of making it fun, but making it fun is not necessarily the same thing as making it musical. And making it musical is certainly more compelling.

0:03:40.4 GR: Yeah. Yeah. And if you can do both, all the better, right? Yeah.

0:03:44.5 DN: Right.

0:03:45.2 GR: Yeah. So I guess I don't know. I'd be curious your take on this. For me, I spend a lot more time teaching when I teach intervals now than I used to talking about consonance and dissonance really early on and talking about intervals as a way to get into how notes work together or work against each other.

0:04:06.3 DN: Ooh. I think that's fabulous. Yeah. And I think we we're... We said, we're just gonna talk about the written theory of things today, but the intervals that are most fundamental to our physical world are those early low notes in the Harmonic series. So the octave and the fifth and the fourth. Well arguably not well. Yeah. But the fourth.

[laughter]

0:04:36.4 GR: Yeah, certainly before the third on the Overton series.

0:04:38.9 DN: Right. And those intervals are those Pythagorean, those things that are closest to the Pythagorean ideal.

0:04:49.6 GR: I feel like we should unpack some of these things, right? Like, so...

0:04:53.1 DN: Yeah. Yeah, yeah, yeah, yeah, yeah. Yeah.

0:04:54.0 GR: So when we say the Pythagorean ideal, we're talking about the, that these intervals come from ratios of frequencies or ratios of length of strings.

0:05:02.3 DN: And so when you have the, especially those notes that are close, those simplest ratios, so one to two.

0:05:08.9 GR: Which is the octave.

0:05:09.7 DN: And two to three.

0:05:11.2 GR: Which is the fifth.

0:05:12.3 DN: And three to four.

0:05:13.8 GR: Which is the fourth.

0:05:14.8 DN: Those are the ones that we're going to perceive most readily when they're perfectly in tune. And those are sounds that the, that we gravitate to. And if we can build everything else from that. And of course, when you tune a harps accord, you tune it by fifths because that's the interval that you can hear.

[chuckle]

0:05:39.7 GR: I mean, to be fair, you have to adjust some of those fifths along the way.

0:05:44.3 DN: Yes you have to adjust.

0:05:45.4 GR: Exactly.

0:05:45.7 DN: Whether you put all of that mess in one of the fifths or spread it out among four or more. Yeah. Yeah. We should do an episode on tuning systems. That would definitely be worth doing. We should add that to our list. In any case you were saying, although we said we're gonna talk about this from a written concept, we can't fully separate it out from the oral concept. And I'm getting this right in that those early intervals, which are effectively the perfect intervals on the Harmonic series have a particular sound and when we connect to that, it starts to make a lot more sense.

0:06:25.8 DN: And to some level, not in all cases. But to some level, when we're talking about consonance and dissonance, we're talking about resolution back to one of those simpler ratios.

0:06:40.1 GR: Yeah. It's interesting, right? Because of course, the third is... And especially when we talk about if we're resolving, say to a major chord versus a minor chord, and of course that minor chord, we don't get on the overtone series.

0:06:57.0 DN: Right...

0:06:57.3 GR: And it's still...

0:07:00.0 DN: Not very low down. [laughter]

0:07:00.8 GR: Yeah, 'cause it's not early on, yeah. Not as a collection of the first in notes on the series. Yeah.

0:07:07.1 DN: Although I'm discovering through some other methods that Riemann was on to something when he was making up SAP harmonics.

0:07:19.0 GR: The undertones? 

0:07:21.1 DN: The undertones. [laughter]

0:07:21.7 GR: Yeah. And that, we should also save for another episode.

[laughter]

0:07:28.1 DN: This feels like a teaser for a whole other season.

0:07:30.9 GR: Right. Doesn't it? Yeah, it's always good to have ideas for our coming episodes. Maybe we just dive into the ways that we teach this because, I know for myself, when I was first starting to teach fundamentals of theory, I taught this like I had learned it, and like I'd always taught it, and I remember observing another teacher teaching intervals completely differently, and thinking, oh, that's totally wrong, the right way to do it is this, right? And I think...

[laughter]

0:08:03.9 GR: Like everything else, the more I've done it, I'm like, oh, there are, maybe right elements to all the different ways that we can teach this, so...

0:08:12.5 DN: Right, and there are some philosophical things to consider, 'cause you need to know about intervals early...

0:08:21.4 GR: For sure.

0:08:21.5 DN: But you also don't want to learn to build your entire schema around intervals.

0:08:28.7 GR: Yeah, this is a great question. At what point do we start teaching intervals? 

0:08:35.1 DN: And of course, in most curriculum, we start teaching them very early.

0:08:40.1 GR: Usually immediately after scales, typically...

0:08:42.4 DN: Yeah.

0:08:43.3 GR: Or key signatures. The typical arrangement is, you learn your letter names, you learn things on the piano, you learn half steps and whole steps, scales, maybe major key signatures or perhaps interval since it's a classic fundamental structure.

0:09:01.7 DN: And there's a lot of chicken and the egg here. How do you start, because you need some knowledge to talk about the other things? 

0:09:09.7 GR: Right. And we don't get to introduce scales without first talking about half and whole steps. Right? So we've already touched intervals.

0:09:19.3 DN: So I don't... If I still have not solved the problem for myself, except that I think the idea of spiraling your curriculum so that you keep revisiting subjects in more depth is the only solution I've come up with.

[laughter]

0:09:33.6 GR: Yeah. Which is hard to do, especially if you're in a one semester kind of fundamental sequence.

0:09:41.7 DN: Right, and yeah, when you only get control over one little tiny part of the education.

[laughter]

0:09:51.2 GR: Yeah. I've experimented. I've moved it around actually. I have experimented with teaching intervals after triads. The good thing about doing it that way is triads are pretty inherently musical things, and there are so many ways you can connect the teaching of triads to real music and to music that students are listening to, from your very first day teaching triads, and if you teach intervals after triads, then you can talk about the intervals as growing out of triads. Oh look, we already know our perfect fifths. We already know our major and minor thirds. I'm not sure I'd do it that way again, but I will say I found it very musical to do that way.

0:10:35.2 DN: Yeah. And the tricky part is teaching all the things without intervals up to that point. [laughter]

0:10:43.0 GR: Yeah, exactly right, exactly right. Yeah.

0:10:45.4 DN: But again, we can spiral, we can talk about them, and then we can talk about them in more depth.

0:10:49.9 GR: Yeah.

0:10:50.0 DN: And we can talk about them in more depth.

0:10:52.9 GR: Yeah.

0:10:52.9 DN: If we've learned scales, then we have that first step of being able to just identify a generic interval, and we can learn that we can count one, two, that's a second, one, two, three, that's a third. Even that is one of those tricky things that is probably hard for a beginner student, is to remember to include the note that you started on when coming up with the number for the interval.

0:11:24.7 GR: Yeah, for sure.

0:11:26.7 DN: Yeah, that's definitely a common fundamentals mistake. [laughter]

0:11:31.3 GR: It absolutely is. And even with someone experienced in fundamentals, if I say, "What's a third plus a third?"

0:11:39.3 DN: Oh, that is... The math then gets ridiculous because you're saying... Yeah, it's like saying two plus two equals five and...

0:11:45.1 GR: All right, yeah.

0:11:46.8 DN: Except in this case two plus two equals three, yeah. [laughter]

0:11:51.5 GR: Yeah.

0:11:52.2 DN: Here, everything you've learned at elementary school is wrong.

0:11:56.6 GR: Yeah, so let's talk about the pedagogy of generic intervals, so when we say generic intervals, we're talking about intervals just including the letter name, so we're not concerned about whether something's major, or a minor, or perfect, etcetera, just the distance on the staff.

0:12:13.3 DN: Right.

0:12:13.9 GR: And as you said, the first challenge there is the tendency to say, okay, yeah, E minus C should be two 'cause it's two below E.

[laughter]

0:12:26.0 GR: But of course, we use one-based counting for our interval system instead of zero based counting, and so it's three.

0:12:35.0 DN: And we just take it for granted.

0:12:36.5 GR: And we just take it for granted, yeah.

[laughter]

0:12:40.8 GR: Once you get past that basic challenge of, Oh yeah, we have to count the outer notes, both of them, then doing generic intervals is pretty easy, but it's not necessarily fast.

0:12:57.2 DN: Right. Especially if you're doing letter names, because then you have that... It's almost... It's easier to look at it on a keyboard or a staff, and see, because you have a very clear visual, but letter names means you have to have a schema for that. You have to have that pattern in your head.

0:13:16.9 GR: Yeah.

0:13:17.7 DN: Which I guess is one of the reasons why we hear a lot of advocating for use of keyboard.

0:13:26.1 GR: Although I actually think figuring out the generic interval, looking at a keyboard, is harder, because...

0:13:31.9 DN: Truth. Because on a staff...

0:13:33.2 GR: Because the keyboard, of course, by itself doesn't...

0:13:34.8 DN: And the reason why it's so easy on a staff, is because the staff is, and I love to remind my students of this, a tonal notation system.

0:13:42.7 GR: Oh, yeah. Very true.

0:13:43.2 DN: The tonality is built into that notation system.

0:13:46.5 GR: Yeah.

0:13:47.2 DN: And so, yeah, if you see it... If you see it on a staff, you know exactly what it is... There's no question and if you see a tritone on the staff. I tend to use in my classes, and I don't know if this is legit, but I tend to use tritone generically for that sound... Because out of context, you don't know whether it's an augmented fourth or a diminished fifth. But on a staff, you see right away, whether it's an augmented fourth or a diminished fifth.

0:14:19.2 GR: Yeah, yeah. On a staff, we're in the context of a key. Yeah, for sure.

0:14:22.8 DN: Yeah.

0:14:22.9 GR: So, ways we can get students to fluency with generic intervals... I think you've hit up upon something really important, which is that looking at it on the staff, versus working in letter names, can be two different things. And we need students to attain fluency in both of those. So, I certainly have ways that I like to try and help students get there. We're speeding up interval writing, generic interval writing, and identification on the staff. One of the classic tricks, of course, is to help students to notice that if both notes are on a line, or both notes are on a space, it's going to be an odd-numbered interval, no matter how big or small the interval is. And if they're not, then it's an even-numbered interval.

0:15:13.9 DN: And we've talked before about pattern matching... And I definitely know that I look for patterns on the staff.

0:15:25.2 GR: Yeah.

0:15:26.3 DN: It's more advanced concept to think about what a triad looks like in various inversions.

0:15:34.9 GR: Yeah.

0:15:35.0 DN: But I guess if we learn quickly to identify thirds and fourths... Or fifths and sixths as intervals, then we're more likely to be able to also see how those fit into patterns that we're likely to see in triadic... In intuition harmony.

0:15:50.1 GR: For sure. Yeah. Yeah. I like to do a silly game of... I put up my hand sideways with my fingers spread, so it makes a staff, and then I just take two other fingers and I flop in an interval, and after a beat I nod my head and the class calls out, "Fifth" or "Third" or "Second".

[laughter]

0:16:18.8 GR: But you know, it just... And then to have students drill each other that way on their little hand staff, just to pair up and, yeah.

0:16:27.8 DN: And not only is that very handy, because you always have it with you. Yeah, that's great. And I have also used... I've done similar things with the cinder blocks on the walls of my school.

[laughter]

0:16:45.8 GR: Where the blocks themselves are the spaces and the gravel is the other lines? 

0:16:50.4 DN: Yeah, yeah. You can find staff analogies all over the place. [laughter]

0:16:56.0 GR: Yeah, yeah. And just being able to see that, right? It's so much easier to see those intervals on the staff than it is to think about them in abstract letter names... An exercise that I use, that I learned from a student of a student of Nadia Boulanger...

[laughter]

0:17:19.8 GR: It's theoretically attributed back to Nadia Boulanger, is that she would talk about the idea of an infinite piano... It just spread infinitely to the left, low, infinitely to the right, high. And she would say, just take a finger and put a finger out, and now let the piano slide left or right under you and you can come to any note. And so, at first just imagine that... Just see your finger over C, and then see the keyboard sliding under it until your finger is at A, for instance. And so now we're going to set the keyboard sliding at kind of a slow rate, and we're going to say the name of every note that passes under our finger. So, we'll slide it and we'll say, C, D, E, F, as we're seeing the keyboard pass under. Great, easy enough. We get back to C, we come back down. So now, let's slide it a little faster. And let's say the name every of every note, we'll wind up skipping one of the notes that passes under our finger each time.

0:18:36.0 DN: Oh.

0:18:36.6 GR: And so now we're saying, C, E, G, E, D, F, A, C. And we do that coming down, C, A, F, D, B, G, E, C. And now we're saying our thirds right. And then... But the important thing in this is that you are actually, you're seeing the keyboard sliding under your fingers, that you're not just memorizing some abstract set of letters. And so her idea was to make late. So when you're thinking about your thirds or your fourths or your fifths, you're not just thinking about some abstract, oh, I know G is a fifth above C, but in knowing that you are also seeing everything that came in between it.

0:19:24.6 DN: Right. And you're connecting those two worlds that we were talking about, the alphabet world and the visual world.

0:19:32.2 GR: Yeah. Yeah.

0:19:33.0 DN: So that they're not separate concepts.

0:19:36.2 GR: Yes. Yes. And the great thing about all of the intervals, which maybe not everyone realizes, but there are many great things about all the intervals, [laughter] But one handy thing for this particular exercise is that for any interval, if you repeat that interval up or down enough times, you will get back to where you started.

0:19:56.8 DN: Right. Yeah. And we're still talking generic intervals.

0:20:00.7 GR: And we're still talking generic intervals. That's right. Because that is actually also the case with specific intervals. You will eventually get back where you started. It may have a different name. You'll get back where you started.

0:20:12.8 DN: Yeah. Yeah. Sometimes faster than you expected.

[laughter]

0:20:17.1 GR: That's right. Exactly. Yeah. Yeah. Tritone C, F-sharp, C done [laughter]

0:20:25.6 DN: A hypothetically harder concept, which is actually, in fact, easier.

0:20:29.6 GR: Right. Yeah.

0:20:30.6 DN: Major thirds or, yeah. Yeah. Well, not major thirds 'cause then they would get wildly complicated. You, wouldn't get back to where you're starting 'cause you...

0:20:38.9 GR: Oh no, you do very quickly. You do, yeah. C, E, G-sharp, C.

0:20:46.3 DN: But, no, because you're C, E, G-sharp, B-sharp.

0:20:46.7 GR: Oh right. To do this version, you have to allow yourself to re-spell, [laughter] something. That's right. Yeah. Yeah. The thing about that exercise is you can't do it for more than 45 seconds or you lose your class [laughter] Right.

[laughter]

0:21:02.6 DN: It's true. And I studied with several people, as I'm sure you did, you studied with Nadia Boulanger and, some of them, the ones who more distinctly tried to emulate her definitely posed challenges that left the class in the dust [laughter] at times.

0:21:26.8 GR: Yeah. Yeah. That might be a fun episode too. The, yeah.

0:21:32.2 DN: The Nadia Boulanger teaching Legacy. Gosh, she was so influential.

0:21:37.4 GR: Right. So many composers in particular studied with her. Yeah. So, okay. So I mean, generic intervals. I think maybe the other thing worth saying as you were talking about spiraling, with any of these concepts, students will not attain fluency and speed within the first day, 2, 3, 4, that they experience. It takes repeatedly coming back to something to build those connections, to build that fluency.

0:22:11.5 DN: And so how are they gonna build that fluency? And I think like the same way we learn language, you just have to use it constantly and probably engage them in ways where they need to come up with it quickly. I mean, the way that they hate is timed quizzes although that's valuable. Of course we love the fact that computers are a way of giving that kind of feedback at a rate that a student can handle and progress with the student. 'Cause computers are much more patient than we are.

0:22:56.0 GR: For sure.

0:22:57.0 DN: But also through games, and that competitive spirit.

0:23:00.8 GR: Along with that, another thing I love to do is take something that's fairly easy to sing and to have students sing it on generic interval names. Like happy birthday, for instance. You go, haha, unison, second, second, fourth, second, third, unison. Second, second, fifth, second. Fourth unison, octave, third, third, second, second. Sixth unison, second, third, second, second. And it's wonderfully hard. It's even hard for me. But it's certainly easier when you're looking at it on the page, but it can be, and I always will pick a song that most everyone knows like to do this, this is not how we want students to sight sing by thinking about the interval from the note they're on to the note they're not on. That can build up, that can cripple fluency in sight singing for life. If you're thinking about sight singing that way.

0:24:10.1 DN: And it's super common.

0:24:11.7 GR: And it is super common.

0:24:13.1 DN: I so often am doing sight singing with class and they're already in their third or fourth semester of, and they will get a big leap and they'll all choke.

0:24:30.2 GR: And they're trying to summon the interval to mind.

0:24:32.0 DN: Yeah. Because they're trying to think of the interval when I'm like, it's T, you know where T is just sing T.

[laughter]

0:24:38.6 DN: Yeah. Yeah, yeah. Yeah. I feel like this is going to be the hill I die on [laughter] is don't sight sing by intervals. I mean, there are times when the intervals are helpful, but they are rare and or where, we wind up needing to really think primarily in an interval context. But those situations are really rare in actual music. Yeah. Yeah. Anyone who wants to hear us opine more about this should go back to our very early episode on Sailfish systems.

0:25:16.9 GR: As if we're opinionated or something.

0:25:19.0 DN: No, I know, right imagine, imagine. Okay. So we've talked about generic intervals and that's the, "easy part" of teaching intervals where people, and I think most people kind of teach generic intervals the same way. It's a simple concept. They're not that many different ways of doing it. But where pedagogy really differs is in how we teach specific intervals. So I've seen four ways of doing this. And I'm... Just to enumerate them, teaching intervals by half steps, which is to say learning that a perfect fifth has seven half steps and four letter names, or five letter names, depending on whether you're coming outside. Teaching intervals by scales, which is to say that from scale, in a major scale from one up to any node in the scale is a major or perfect interval, and from one down to any node in the scale, is a minor or perfect interval. Teaching intervals by natural intervals, in other words, intervals from, if both of the notes were natural as in C natural, E natural, what would that interval be? And then, teaching intervals by scale degrees. So knowing that anytime I'm going from scale 3, 4 to scale 3, 6, that's a major third, and that can transfer across various keys. So those were the four I came up with. David, do you have any others that I missed in that list? 

0:26:48.0 GR: Not that I'm aware of, but those... And the thing is, except for the first one, I think you just need to know all of them.

[laughter]

0:26:56.7 DN: Right. So, okay. Yeah. And the first one being intervals by half steps, knowing that your major thirds have four half steps.

0:27:03.8 GR: Right. Oh God. I don't even know what that is. I would have to look and go. Right. Okay. I would've to count. I don't have that information in my brain.

[laughter]

0:27:14.5 DN: Yeah. It's funny. I don't either except from set theory. From teaching oral skills 4 so many times and, you know, just, yeah. Yeah. But, okay. So this first approach, teaching intervals by half steps, effectively the idea here is that you learn that any given interval has a certain number of half steps in it, right? And a certain number of letter names. So we learn our perfect fourth has five half steps and four letter names. Four letter names total. That's including the outer notes. And you learn this, most people who teach this way teach all the major and perfect intervals, and then teach how to alter from the major and perfect intervals to get any of the other intervals. And...

0:28:10.0 GR: That makes sense.

0:28:10.9 DN: Yeah. Yeah. I have to say, one of the things that I love about this approach, is you can teach intervals in a way that someone who has failed every previous quiz in your fundamentals class will still understand.

0:28:28.3 GR: Gotcha.

0:28:29.6 DN: Right? Because it's just, it's you find the note on the piano and you count your half steps.

0:28:36.4 GR: Right.

0:28:36.5 DN: And then you find the version of that note that matches the letter name, that's however many letter names apart.

0:28:43.5 GR: You know, I think the thing that's so tricky about this though, is that it requires so many transformations. It won't work if... It won't work just looking at a staff, unless you have an already inherent knowledge of how that staff translates to whole steps and half steps. In which case you're already advanced enough that maybe you didn't need this process. And if you're looking at the keyboard, it doesn't... Yeah. You're gonna have to do a lot of translation. You're gonna have to count 1, 2, 3, 4, 5, and then go, okay. But that's the third. [laughter] And it's... That's a lot of...

0:29:31.3 DN: Yeah, it is a slow...

0:29:33.3 GR: That's a of steps.

0:29:34.0 DN: It is a slow process because of all those transformations, right? 

0:29:38.1 GR: Yeah.

0:29:39.0 DN: I will say, and this was the method that when I saw that teacher, this was back in 2005, I saw a teacher teaching it this way, and I thought, this is the worst. I will never do it this way. But I also think if we skip teaching this concept, sometimes students, what this concept makes really clear is that when we're using intervals, what we're really doing is counting musical distance two ways. We're counting it by letter names and we're counting it by distance in half steps. And those are different. Even though we kind of merge them into one naming system. So I guess for me, this approach is one that I introduce in about 15 minutes, just to be sure students have this idea that like, and I introduce it usually by saying, okay, find me a white note fifth on the piano. And how many half steps are in there? And can you find me a white note fifth on the piano that has a different number of half steps. And of course there's only one white note fifth on the piano that has six half steps in it instead of seven half steps.

0:30:57.3 GR: Well, that is a good experiential learning process for them to find that one natural, as we call it interval.

0:31:06.1 DN: Yeah. Diminished fifth interval. Yeah.

0:31:07.7 GR: Yeah. That is different, rather than just saying, Hey, there is this one natural interval that's different.

[laughter]

0:31:15.2 DN: Right. Right. But for me, the important thing is that they understand that not all fifths are created equal, not all whatevers are created equal, right? This concept is a way for me of getting to why we need these specific interval names.

0:31:34.2 GR: So I love that. I love that as a tool to get yourself going. And then you've, and then they're aware of the concept. I'm not sure I would wanna personally dive too deeply in there, at least in a fundamental class.

0:31:50.5 DN: Yeah. And you know there are people who this is the way they teach intervals. And so if they really do that, then students memorize that major second, two half steps, major third, four half steps. Perfect fourth, five half steps. Perfect fifth, seven half steps, major six, nine half steps, major seventh, 11 half steps, right? Just...

0:32:12.1 DN: But I am with you that, to be fluent, that approach involves way too many transformations that just slow us down. So I think these other three approaches that we're talking about are effectively abstractions above that basic concept. That get you more quickly to a place of being able to identify your name intervals. So second one, intervals by scales, which is this wonderful trick that if you start from one in a major scale and you go up to any node in the scale, you're going to have a perfect or major interval. And if you do the same thing coming down, you're going to have a minor or perfect interval. And then everything is just a transformation by half step away from that.

0:32:58.9 GR: Which is also one of the ways that's closest to the way we may hear things. Wonderful.

0:33:04.2 DN: For sure. Yeah. And it's also, it's a wonderfully easy one to connect to oral training as well. 'Cause you can do the classic 1, 2, 2 is a major second. 1, 2, 3 is a major third and so on and so forth. Which of course is in building up students for, functional hearing. Or do do re, whatever language you wanna use of recognizing. Okay. Oh, where is my fa? Oh yeah. Do do fa there's my fa 'Cause I can just call do do mind and have, I've sung my Do do fa enough times that. Yeah.

0:33:42.8 GR: So yes, if we have a schema of what a major scale sounds like if we have a schema of, we keep the, I love this word now.

0:33:49.8 DN: Yeah. Thank you Lee.

0:33:50.9 GR: Yeah, exactly. If we have a schema of what a major scale looks like, we also have our answers.

0:33:57.6 DN: Yeah. And I think there you're getting to another of the reasons that I think all of these approaches, other than the first one we talked about of counting by half steps are maybe frankly stronger is because as we learned in our episode with Leigh VanHandel, a lot of learning is about making connections to things that are already known. And the greater number of connections, the easier a concept is to call to mind. And so yeah. As you're saying schema right. The technical term for that and this connects so directly to that schema of scales orally and in written theory.

0:34:39.8 GR: Yeah. And then we learn other fundamental concepts and fundamentals that will make this easier of when we learn key signatures, we quickly learn what notes are going to be in a major scale we can, and that is, that becomes an easy way to calculate things.

0:35:00.6 DN: Yeah. Now, one of the downsides of this approach is that there are notes for which there is no major scale or minor scale. For instance, David, what's a minor third above G-sharp? 

[laughter]

0:35:16.3 GR: Well, I can look at it on the piano.

[laughter]

0:35:18.0 DN: Yeah. But also you probably know right away, right? A minor third above G-sharp is B that's a pretty common interval. But if you've learned this approach, then suddenly you're like, oh gosh, do I know G-sharp major? Nope I don't.

0:35:30.5 GR: Yeah. I don't have a G-sharp major scale in my head.

0:35:34.2 DN: Or for instance, minor third above F-double sharp. Something that exists in the key of G-sharp minor.

0:35:39.6 GR: There are many situations in which this is gonna be cumbersome.

0:35:42.9 DN: Yeah. And in those cases, the classic approach is either you say, okay, I've gotta do a minor third above F-double sharp. So to do that, I'm going to cancel, hold off my double accidental, I'm gonna pretend it's F-natural. Oh yeah. Minor third above F-natural, A-flat easy. Okay. Bring back my double sharp. So I have to go up to accidental. So my, A flat becomes natural then sharp. And so my minor third above F-double sharp is A-sharp. One other challenge of intervals by scales, which is that if I'm working in a key, say I'm working in E-flat major, and I'm looking at, D to F in that key.

0:36:26.8 DN: And someone asks me what interval it is, if I only know my intervals by scales, I have to erase the E-flat major key and scale from my mind. And call to mind a different key and scale and...

0:36:41.1 GR: Yeah. That would become... That's a one too many things to do.

0:36:45.2 DN: Yeah.

[laughter]

0:36:48.7 DN: And problematic because it causes us, because we want to be thinking functionally always with whatever key we're in. And this causes us to have to let go of that to calculate an interval.

0:37:03.0 GR: Right. So that's why we have these other two methods that work much better in other situations.

[laughter]

0:37:08.5 DN: That's right. That's right. Yeah. Absolutely. So David, can you tell us about natural intervals? 

0:37:14.9 GR: I don't think that I prefer, I either of these last... Any of these last three methods, I think that you need to have all of them at your disposal.

0:37:25.8 DN: And it really... It comes back to what you were saying earlier about spiraling. That depending on our context as fluent musicians, we're going to use a variety of different techniques to recognize and create intervals. So natural intervals, usually what we're talking about there is learning, for instance, where all the major thirds are on the white notes or in letter names without accidentals, and then altering from there.

0:37:58.4 GR: Which teaches you some valuable tools anyway, that you should know what all those intervals are because they'll help you understand.

0:38:08.5 DN: And there's some great, really beautifully obvious visual things when you do it this way. So if we take our thirds, and if you see C, D, E on the piano, you'll notice right away there are two black notes between that. And our major thirds are white note, our natural interval major thirds all have two black notes between them. But if you take... But find a third on the piano that only has one black note between it, and that's going to be a minor third. And so if you're teaching interval qualities, if your primary way of teaching interval qualities is this natural intervals approach, then or when you're dealing with notes, other than the white notes, the classic way of doing it is to imagine the white note interval and to determine how it's altered based on that. So let's say I'm asked to write a minor third above F-sharp. I'm going to say, okay, I know my white note third F to A is major.

0:39:08.1 DN: And the given note is F-sharp that shrinks the interval by a half step, oh, good F sharp to A is minor. So you learn your intervals as alterations from the white notes.

0:39:19.0 GR: Yeah, and then that also ends up working well for core qualities and...

0:39:25.3 DN: Yeah, absolutely, and it's pretty darn quick, and it doesn't require a knowledge of scales or keys, which in this case can be a positive, because when we're working in the context of a key to figure out an interval, we don't have to erase that key context.

0:39:45.5 GR: Right.

0:39:46.2 DN: But we may have to erase an accidental which...

0:39:50.2 GR: Yeah and I think if you're working in a key with a lot of... If you're trying to do it visually on a piece of paper and you're working in a key with a lot of accidentals, you also then have to do a lot of cross-checking to make sure that you've accounted for all the accidentals...

0:40:08.0 DN: All the accidentals that you cancelled to figure it out. Yeah. This is... Yeah, for sure. Yeah. And David I think one of the things that you said earlier gives us a clue on how we can get past this challenge of canceling and adding accidentals, which as you were saying that you want students to know that, for instance, in a major scale, the third on one is major, the third on two is minor, the third on three is minor, et cetera, et cetera.

0:40:36.7 GR: Right.

0:40:37.4 DN: And which takes us to the fourth approach of intervals by scale degrees, which is to say knowing with all of the intervals whether a fifth on two, what kind of fifth is that? Well, in major it's perfect and in minor, it's diminished. And actually, in each of these, the complexity of how many things you know has gone up. There's a lot of information to remember, if we're doing intervals by scale degrees. You have to remember seven qualities or seven intervals, and so in one sense, it's harder, but the reality is that, I think most musicians who are fluent at these things are using this approach.

0:41:38.0 GR: And if I'm playing piano and I'm in a major key, I just know that a two chord is minor. I just know that. I don't...

0:41:48.2 DN: And with that knowledge comes the knowledge of the third and the fifth.

0:41:56.5 GR: And yeah... And I just know what those quality... And I don't have to think about them. They're just embedded in my schema, and I guess I do want to embed that schema in my students as well.

0:42:07.5 GR: Yeah. Yeah and I do things like I think many of us do right of... Especially with the smaller intervals of saying one to two is a major second, two to three is a major second, three to four is a half step, or one to three is a major third, two to four is a minor third of like actively singing various warm up patterns to make those things explicit and start to build up those schema. So this method of knowing intervals by where they are within... By scale degrees within a key, it's really wonderful and it's really fast, but it does require real fluency with scale degrees in keys. Knowing if I'm in any key and I have this note, what's the scale degree for that note or vice versa? What's scale degree two in the key of A major.

0:43:09.0 DN: Yeah And there are... I think there are challenges and how do you... How you develop that fluency. I think there are not challenges if everybody has a keyboard, because if everybody has a keyboard, you can have them play through it, and then they can see, "Oh, that's what I have to play."

0:43:31.4 GR: I'd also argue that for me, that particular skill, the ability to go between a scale degree and the name of a specific note in a key, or the name of a specific note in a key and a scale degree, that skill to me is the most important skill that I want students to learn in anything I'm teaching, in any written theory, in any oral skills, I want them to have that fluency of moving between functional identification of scale degree and letter name because it opens up sight reading and dictation, because as long as you can hear the scale degree, call to mind what the scale degree sounds like, or solfege, these are different names for the same concept or location within a scale, then you gain fluency in the oral side of things, and if you know your theory concepts by relation to solfege or scale degree. Then those become fluent and connected to hearing as well.

0:44:45.8 DN: So it's a broader schema that we're building, and the thing is that... So a lot of these are patterns and my goodness, we've had a great season of episodes, but so much of what we do is pattern recognition. So if you have this pattern of in your... If you have that pattern and you know that pattern and you can connect those patterns to notes, is what you're saying. Then you have the world at your fingertips.

0:45:32.6 GR: And, you do, literally at your fingertips. That's it. Whatever you can hear. If you can hear all the scale degrees you can write down or play.

0:45:41.3 DN: And I do. I remember having a teacher speaking of Nadia Boulanger, one of my teachers who studied with Nadia Boulanger. He could do things that seemed absolutely magical. And I just thought, I'll never be able to do that. But then some of this skill, some of my students look at it as if it's some magical skill. And I say, literally, I wasn't good at this before and I'm just trying to get you to be... There's no magic in this. This is just something that you can learn to do.

[laughter]

0:46:16.5 GR: Yeah. Yeah. I keep coming back to what you said at the beginning about spiraling because I think if on day one of teaching intervals we try and teach them by scale degrees, you lose the class. It's too advanced at that point.

0:46:34.4 DN: We can't teach them everything at once.

0:46:37.4 GR: At once. That's right. Yeah. And we have to, especially with intervals, gradually build it up, so that each step is a manageable step and not overwhelming. But I think, in that process as we've talked about, gradually connecting to schema that they've already mastered or begun to master. They connect to scales, to connect to their knowledge of the keyboard, to connect eventually to scale degrees and functional hearing. And when I think about teaching intervals this way, I hate it a lot less than when I think I'm going to teach them one way of doing it. But it's a... Pacing is a challenge too. You don't want to give them a new way until the prior way has become pretty automatic.

0:47:45.2 DN: I think it's a... Yeah, that's a good argument for really careful curricular design. And if you're in control of the whole curriculum, woo. Great. And if you're not in control of the whole curriculum, whoof. You've just gotta figure out how to fit it in.

0:48:04.5 GR: Especially if you're lucky enough to have your students over a more extended period of time, let's say you're working with them for two years or even, like middle school and high school teachers, three, sometimes four years there's a real opportunity to just sequentially layer these things up.

0:48:22.6 DN: And a little bit, there's some forgiveness in this too, that if you inherit students from another person who used a different method, this is not necessarily bad. This just means that they have that additional perspective [laughter] and now you can give them more additional perspective.

[laughter]

0:48:42.9 GR: Yeah. Yeah. And you know, when a student quickly knows what I hope they know, I don't particularly care [laughter] which process they used to get there.

0:49:00.5 DN: And it's just... It's more important when they're not quite getting there or not getting there quickly enough. And if they... To use a voice analogy, it's interesting because in order to build a consistent instrument vocally, sometimes you have to break down the strategies that they've already developed that aren't good long-term strategies. [laughter], there's a... Maybe this is not the place to talk about it, but young singers tend to raise their larynx as they get higher. And that's an intuitive strategy. It works kind of a little bit, but it is not a good long-term strategy. And convincing a body to stop doing that can be tricky. And it can be frustrating for a student who has learned to rely on that not so good strategy or on that less, that strategy that will take them less far.

0:50:11.4 GR: Yeah. Yeah.

0:50:15.3 DN: Maybe that was not a useful analogy.

0:50:18.3 GR: No I think it's a... But I think we all know cases of this in our own practice of where we may have had to rework something. I think in the case of these strategies for identifying and writing intervals, one of the good things is we don't have to convince students to give something up. We're just sharing an additional approach that can be faster and progressively more musical.

0:50:54.1 DN: And these things expand into other things. And as soon as I started talking about harmony, they're going to have to know that pattern with the exception, that of course the chords are gonna be major, minor, minor, major, major minor diminished.

0:51:15.8 GR: You're right. Yeah. Yeah.

0:51:16.9 DN: Plus theirs. [laughter] Yeah.

0:51:18.4 GR: And it's funny that you think I've never actually taught that as a linear pattern except in various warm up songs. Like one three, major third, two four, minor third. Whereas I've just always taught it as the major thirds are on one, four and five. Everything else is minor. And similarly. And then the expansion to the triads, major triads are one, four and five. Everything else is minor except seven.

0:51:46.3 DN: Right. Right. And although with intervals and perfect fifths or and fifths, it's fifths and fourths. If the two notes are not B and F...

0:52:06.2 GR: Yeah, this is the brilliant...

0:52:07.2 DN: That is perfect.

0:52:07.7 GR: We should just... We should just take a second and explain that shortcut because it's a wonderful shortcut, that for any perfect interval, both notes will have the same accidental.

0:52:16.9 DN: Right.

0:52:17.6 GR: Unless the two notes involved are F and B, in which case F will always have one accidental higher than B.

0:52:24.4 GR: Right. Or depending on... Yes.

0:52:28.6 DN: No. That's always true.

0:52:29.7 GR: Not depending on. [laughter]

0:52:30.9 DN: Yeah, yeah. For instance, if you have B flat to make a perfect interval, one accidental higher than F is natural. Yes, B flat to F perfect fifth, or F to B flat perfect fourth, yeah.

0:52:41.7 GR: Right.

0:52:42.4 DN: So yeah. A good shortcut. And just to wrap up David, what's your favorite interval? 

0:52:49.7 GR: Oh, what is my favorite interval? I can I...

[laughter]

0:52:55.8 GR: I really love a well-tuned fifth. Yeah, you know I love... This is so cheesy in the best way. A well-tuned major seventh.

0:53:10.2 DN: Oh, oh, all right.

0:53:12.6 GR: Yeah.

0:53:13.4 DN: Cool. That was not what I was expecting you. I was expecting a sacred...

0:53:17.0 GR: I wasn't expecting the perfect fifth...

0:53:17.7 DN: Major fifth, major sixth, or something.

0:53:18.7 GR: Oh, yeah, yeah, yeah. Yeah.

0:53:22.0 DN: Sixths are great though. Oh my gosh.

0:53:24.1 GR: Sixths are really great. Yeah, they are. They are. But there's nothing for me like the rub of singing a major seventh.

[laughter]

0:53:32.2 GR: It's just... It's just delicious. Whichever note I'm singing. I'm just... I'm just, yeah.

0:53:41.9 DN: Yeah. Well...

0:53:42.2 GR: A well-tuned perfect fifth I also love. Well, this has been fun. And we came out with a lot of ideas for future episodes.

0:53:55.7 DN: I know. Well, we know that. We already know in our own relationships that you are very good at staying on track and I am very good at taking us off.

[laughter]

0:54:05.7 DN: David, your creativity adds so much that...

[laughter]

0:54:13.6 GR: So if we rambled it's my fault.

0:54:15.9 DN: Otherwise I'll blindly follow my checklists.

[laughter]

0:54:20.9 DN: Well, this is why we're good together. All right.

0:54:23.1 GR: Yeah. Yeah. Awesome. And so listeners if you have a favorite way of teaching intervals that we didn't talk about, please let us know. And we'll be circling back to this concept when we talk about teaching intervals in ear training as well. So thanks so much for tuning in.

[music]

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