Interval Ear Training

In this episode, Greg Ristow and David Newman talk about the value and role of intervallic ear training, why it's time to move beyond Here comes the bride, and ways of teaching intervallic hearing that build fundamental skills for sight singing and dictation.

Links:

Karpinski, Gary. "A Cognitive Basis for Choosing a Solmization System," Music Theory Online, Vol. 27, No. 2. June 2021. https://mtosmt.org/issues/mto.21.27.2/mto.21.27.2.karpinski.html

 

Transcript

[music]

0:00:21.2 Gregory 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.

0:00:35.5 David Newman: Hi, I'm David Newman, and I teach voice and music theory at James Madison University. And I write code and create content for uTheory.

0:00:43.4 GR: Hi, I'm Greg Ristow. I conduct the choirs at the Oberlin Conservatory of Music, and I'm the founder of uTheory.

0:00:49.9 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.

0:01:01.7 GR: So today we'll be talking about interval ear training. And interval ear training is central to many teachers' and textbooks' approaches to sight singing and dictation. But the title of this episode is maybe a little bit misleading because research in music cognition suggests that for most common aural skills, ear training tasks we process notes by their relationship to a tonic or by their position in a scale rather than by actually hearing adjacent note to note intervallic relationships. So in our conversation today, we'll look at this research on how we hear and the role that intervals play in that hearing. We'll talk about why classic techniques we're teaching intervals can actually undermine students' reading skills. And we'll look at ways of teaching intervals that instead compliment and strengthen students' aural skills. It's a lot to get through in the course of an hour. [chuckle]

0:02:03.2 DN: It is.

0:02:04.5 GR: But David and I have agreed to play particular roles on this. So I'm going to, I'm gonna be sort of the the playback, keep us on track role and David's gonna be the the color commentary, [chuckle] role.

0:02:14.0 DN: Playing to our strengths.

0:02:15.4 GR: Playing to our strengths for sure, for sure. It is hard to talk about or even to think about how we hear, so much of how we hear music is really innate, that we don't, especially for someone with a well-developed ear, "how do I know how I know what I'm hearing?" is a hard question to answer.

0:02:40.1 DN: Yeah.

0:02:40.8 GR: And fortunately we have scientists and researchers who've been looking at exactly this question for a little, I don't know little over 40 years now. And what they have pretty consistently found is that when someone who is experienced in a particular musical culture, and so let's say broadly Western music, music that exists within the notes on a Western piano.

0:03:17.7 DN: An equal tempered scale.

0:03:19.2 GR: Yeah. A tempered major-y minor-y or rotation of its scale as opposed to for instance, some of the Turkish collections that have more notes in the scale than we have and notes that don't exist on our piano. So when someone is encultured in a musical system, when first they start hearing notes, the primary thing that their brain does is seek to determine a central pitch, what we would call a tonic and that's known in music cognition as the primacy hypothesis. The idea being that David, if I throw a few notes at you, before you're going to do anything with those notes, your mind is going to say, "what could potentially be tonic given these notes?" And we're gonna hold onto them.

0:04:20.8 DN: We contextualize it.

0:04:22.5 GR: Exactly. We seek to find the context in which that's occurring and will tend to hold onto our belief of that central note as long as we reasonably can even through the first few notes that contradict it.

0:04:39.1 DN: Yeah. I even think this is central to so much of why we enjoy music. And so if you enjoy music, you probably do this.

0:04:47.8 GR: Yeah, absolutely, absolutely. Yeah, yeah. And it should be said we're saying this and let's just imagine that if I'm someone with really strong absolute pitch. And even in those cases although yes, someone with absolute pitch will know immediately, yes I'm hearing these particular letter names. They are also still working to contextualize them within some sort of tonal framework. If that's something that you're interested in reading about, one of my favorite articles on this is by Gary Karpinski and it's his, it just came out a couple of years ago in Music Theory Online. We'll put the link in the show notes, but this is freely accessible online and it's "A Cognitive Basis for Choosing a Solmization System." And in the first, I don't know, 15 or so paragraphs of it Karpinski goes through and just summarizes all of the research that has occurred over the past 40 years to this.

0:05:53.0 GR: And the big conclusion that he lists there is, and I'm gonna quote here, "These studies and observations lead to the conclusion that while attending to the pitches of tonal music, the first and most fundamental process listeners carry out is tonic inference. And from that, we can conclude that the single most immediately knowable tonal characteristic is the tonic." Now, what does all this have to do with intervallic ear training? What it really comes down to is this question of how do we actually hear music? And we really hear music based on how the notes relate to a sense of tonic. And we don't actually hear music based on the pitch relationship of immediately adjacent notes or even of vertical notes sounding together.

0:06:45.6 DN: And in fact, I know even for myself that if I'm singing a tonal melody I probably could very easily tell you what generic intervals I'm singing at any given time, but I would have to stop and think about what specific intervals. And when I say generic intervals, I just mean, I could tell you that I'm singing a 5th. I could tell you that I'm singing a 6th, but 6th especially, I would've to stop and think for a second to tell you what quality of 6th that was that I was singing, because I'm just going between notes in the scale.

0:07:24.5 GR: That's right.

0:07:24.9 DN: That's the simple thing. And to add intervals to that would be an additional step for me.

0:07:31.3 GR: A really great example of this is to even, to ask someone who believes they're sight singing by intervals, to take a song they know and to sing that song on the specific intervals of the song. Sing the Star Spangled Banner on specific intervals. And you get. First note, obviously, Unison. 'Cause you have nothing before. So unison, minor 3rd, major 3rd, major 3rd, minor 3rd, perfect 4th.

0:08:02.7 DN: Major 3rd, major 2nd, major 2nd. Oh gosh. Yeah, that would be a minor 6th. [chuckle] Oh, whole step, half step.

0:08:21.4 GR: Yeah. It is not how we hear. On the other hand, as you said, I think totally, yeah, very often we're we're of, oh yeah, I'm singing a 3rd, I'm singing another 3rd. Those two thirds were different, right? But, to us...

0:08:40.2 DN: They were thirds in the key.

0:08:42.3 GR: They were thirds in the key. Yeah. And as we've talked about on a number of previous episodes, our musical notation system reflects this. Our staff system with its use of key signatures is designed to show us very quickly, generic intervals, interval distance within a key and not specific interval distance or chord quality. It's really... It makes primary, this idea of our seven note key collections.

0:09:11.8 DN: Yeah. It was designed for tonal music.

0:09:15.6 GR: Because that's what it reflects.

0:09:16.6 DN: That's what music was. [laughter]

0:09:18.8 GR: Yeah. And largely still is.

0:09:21.5 DN: Yeah.

0:09:22.5 GR: And what we're getting to here is two approaches to learning and thinking about intervals. Intervals in the context of a key, which throughout this episode we'll refer to as contextual interval hearing and intervals as pure relationships between any two notes or what we'll call acontextual interval training. And if we look at the classic way intervals are taught, which I've started to call the naive approach to teaching intervals. It blends these two, it blends contextual and acontextual interval hearing without being explicit about which is which, which can lead to some real problems.

0:10:08.2 DN: Yeah.

0:10:09.9 GR: So what I mean by the classic or naive approach is the approach of saying, okay, a perfect 4th is "Here Comes The Bride." And of course, "Here Comes the Bride" comes along with context. Because it's 5, 1, 1, 1. And there are six perfect 4ths within our diatonic collection, and they don't all feel like 5, 1 and they can feel very different than that.

0:10:33.5 DN: I have a song about that. [laughter]

0:10:34.9 GR: Yeah. [laughter]

0:10:35.4 DN: We should... Yeah. That should go in the show notes too. There's an interval song as specifically about 4ths actually.

0:10:44.7 GR: That's...

0:10:45.0 DN: Some say that ascending 4th sound like, "Here Comes the Bride," but change the context and that perfect 4th may not sound the same and your song won't help as planned [laughter]

0:10:56.9 GR: Shall we just take a moment and pause and listen to it? 

0:11:00.0 DN: Oh sure.

[music]

[laughter]

0:12:18.0 GR: Yeah. And that's exactly it. That they... That these 4ths are very different. And so let's now carry that out to its dangers with dictation. So if I'm trying to do dictation and I come across a leap and I'm like, "Okay, does that sound like here comes the bride?" If it was from five to one it sure does, but if it was one of those other 4ths, it does not. And frankly, the more sensitive musical listener you are, the less another 4th will sound like the 4th of five to one. And so if we are using this approach it will disadvantage our most sensitive listeners. Okay. And so now let's flip and let's say what if we're using this for sight singing. And this is... I call this the how do you measure a mile problem? So, David, we can see each other on video, our listeners can't. But, if you wanted to measure the size of the room that you're in there, what tool would you want? 

0:13:25.4 DN: I would want a really long tape measure.

0:13:28.4 GR: Yeah, absolutely.

0:13:31.1 GR: I'm sorry, David, you only have a ruler. What's going to go... A foot long ruler, what could go wrong? 

0:13:37.2 DN: Well, it sounds like when I try and measure a room by pacing it or by measuring my foot lengths, assuming that they're about a foot, but since my feet are probably not exactly a foot, even if I had an exact ruler, I'm gonna make a little mistake every time.

0:13:57.2 GR: Mm-hmm.

0:13:58.4 DN: And if I'm lucky, the mistakes will kind of cancel each other out by the time I get to the end. But if I'm not lucky, I'm gonna end up with an incorrect number.

0:14:11.0 GR: Yep. And you will have... And if you're lucky you'll buy more carpet than you need. And if you're unlucky, you'll wind up with a couple feet of un-carpeted room. Right? Yeah. So there's this element, if we're sight singing by interval, that if I ever-so-slightly misjudge an interval, the inaccuracies can add up. But it's more than that because in fact, as we know, and especially as you know, singing early music, there are many different ways that are correct of tuning intervals.

0:14:47.7 DN: Yeah. Every single one.

0:14:49.8 GR: Every single one, absolutely.

0:14:51.4 DN: But although in early music, yeah, I know every single one. Yep.

0:14:55.7 GR: Yeah.

0:14:56.1 DN: And some more than others. Some notes are more equal than others. Like Animal Farm. Some intervals are... Yeah.

0:15:04.2 GR: And we're gonna do an episode on tuning systems coming up soon. But to give a really strong example of that, if we tune a major 3rd like it appears on the overtone series.

0:15:19.6 DN: Right.

0:15:21.4 GR: First off, we should say, the distance between a half step on a piano or any half step on a piano can be called 100 cents. And the difference between a major third as it appears on the overtone series, and the major third as it appears on the piano is 14 cents.

0:15:40.0 DN: Mm-hmm.

0:15:42.0 GR: So if, for instance, you ask someone to arpeggiate an augmented triad until they come back to the starting point with overtone major 3rds. Right? So each of those, each of those intervals is a major third.

0:15:58.4 DN: Right.

0:16:00.3 GR: You're going to add 14 cents of difference from the piano each time, which comes up to 1, 2, 3... 42 cents difference, in other words, all just pretty darn close to a quarter tone away by the time you get to the top.

0:16:16.5 DN: Right. And it's gonna be 42 cents flat. Right? 

0:16:18.9 GR: That's right. Yep.

0:16:20.1 DN: Yeah.

0:16:20.7 GR: So yeah. So we have that issue, right? With sight singing by intervals it... And then we have the issue that's more related to what we were talking about before, which is to say that, if I come across a 4th when I'm sight singing, if that's five to one, "Here Comes the Bride" is a really useful tool. But if it's not five to one, and I summon to mind, "Here Comes the Bride," I have brought with it the context of whatever note I'm starting from being five, even though it's not actually five in the key I'm in, and whatever note I'm going to being one, even though it's not actually one in the key I'm in and I've caused myself, I've had to briefly erase my tonal context to sing that perfect 4th, which is problematic.

0:17:00.6 DN: That's the most important thing about this whole conversation. If there was a clip to take out, that's the one to share and say, "Here's the main point."

0:17:12.5 GR: For me... Absolutely. Right? And this... I was joking last episode. I was like, this is the hill I will die on. Is that we cannot primarily do dictation of sight singing by intervals. It works too strongly against tonal context, which is where so much of the meaning of our music comes from.

0:17:31.5 DN: So tell me if I'm distracting from the stream of thought here. But the issue then, the problem is that when we're teaching early skills in music theory, even in aural skills, we'd still need to refer to intervals. So we kind of need the people to know what intervals are. And I guess the trick then is, how do we introduce intervals without creating these problems? 

0:18:05.3 GR: Yeah. Yeah. Absolutely. For sure. And a related, but different thing is, I think that it seems, it is such a simple idea that if I know all of my intervals, I should be able to sight sing anything or do any dictation.

0:18:21.4 DN: Right. That's a lovely thing in imagination.

0:18:25.1 GR: In... But that ends up failing in practice. That's right. That's right.

0:18:29.1 DN: Yeah.

0:18:29.2 GR: Yeah. Yeah.

0:18:32.6 DN: So instead we teach frameworks.

0:18:35.5 GR: Right.

0:18:35.6 DN: For tonal music, we teach for frameworks.

0:18:38.5 GR: Yeah. And you might be rightly at this point saying, should we even bother to teach intervals in an ear training context? 

0:18:48.6 DN: And I think we have to, but the question is how do we do it? 

0:18:53.2 GR: Yeah. No, I will tell you, in all of my undergraduate ear training, we had zero intervallic training.

0:19:01.1 DN: Uh-huh.

0:19:02.9 GR: That's not true. In the last semester when we got to atonal stuff.

0:19:07.6 DN: But did you do that? 

0:19:07.8 GR: There was some intervallic training.

0:19:09.9 DN: But did you have it in written theory at least? 

0:19:12.5 GR: Oh, for sure. For sure. Yeah.

0:19:14.7 DN: Yeah. And then I think, again, like it's not so hard in that exercise that we did earlier to name the generic intervals.

0:19:25.3 GR: Right.

0:19:25.3 DN: And I think that's something that's pretty straightforward. And of course, if I know the generic intervals and I see a third going up and I see that the second note is raised from what it was going to be in the key, then I know, "Oh, that's gonna be different."

0:19:42.5 GR: Yeah.

0:19:42.5 GR: Yeah, yeah.

0:19:44.3 DN: It's probably gonna be a major third.

0:19:47.3 GR: Right, but you know for me, even when I sing a song I know on specific interval names, if I'm being honest, I'm not really hearing the specific intervals, I am hearing...

0:20:00.4 DN: No.

0:20:01.1 GR: The generic intervals, intervals within a key and using my knowledge of music theory to label those specific intervals.

0:20:07.0 DN: Yeah, yeah, that's a thing that I don't even think about.

0:20:10.3 GR: Yeah, yeah, so let's... So again coming back to this idea of the classical or naive approach to interval training, the learn a name of a tune for each interval approach.

0:20:20.8 DN: Oh, I just try and get people not to do that.

0:20:25.0 GR: I do too, I do too, because I think it conflates contextual intervallic hearing, intervals within a key with acontextual intervallic hearing, pure relationships between two notes. But I do think there's value, and as you were just saying, in learning intervals within the context of the framework of a key or scale. And also, I think there is value in learning intervals out of that context as pure sonorities of relationships between notes, especially if we're concerned about tuning in context. For instance our... If we're playing in an orchestra, if we're, basically anything other than a pianist.

0:21:12.2 DN: Right, barbershop.

0:21:14.6 GR: Then we have to make decisions about how we tune our intervals. So I thought it would be just, it would be great to be really practical and go through a bunch of exercises of contextual intervallic practice and of acontextual intervallic practice. Sound good? 

0:21:36.5 DN: Sure. Yeah.

0:21:37.8 GR: Okay, great. So these exercises, although we're gonna demonstrate these pretty much throughout in major keys, in practice, I also like to do these in harmonic minor as well, because that'll introduce fun intervals like the augmented second and diminished seventh that don't exist in the major scale. And although we'll generally demonstrate these using scale degrees, movable solfège works the same way, pick your system of choice. And although this is a little bit evil, when I do these exercises with my students, I like to switch between a functional system like scale degrees or movable solfège, and a fixed system like letter names or fixed Do, because I want my students at all times to be thinking...

0:22:32.2 DN: To suffer...

0:22:33.9 GR: To suffer, yes, yes.

[laughter]

0:22:35.8 GR: Although I want them at all times to be thinking about both the, sort of, the abstraction that function gives us, and the specificity that note names or playing it on an instrument give us.

0:22:55.8 DN: And I have just started atonal music, aural skills with my class in the last week, and I told them if their brains are hurting, it just means they're growing neurons, that their brain should hurt a little bit, it's probably good.

0:23:10.8 GR: Yeah. Yeah.

0:23:12.4 DN: Growing neural pathways, they already have neurons.

0:23:15.6 GR: Yeah. Yes. Yeah. So let's just dive in and let's... Shall we do some exercises together, David? 

0:23:23.3 DN: I am always game for exercises.

0:23:25.5 GR: Okay, great, I'm gonna give you a pattern, and would you just take it and continue it as you know it, yeah? 

0:23:30.6 DN: Okay.

0:23:31.3 GR: Great. [Singing] 1 to 2 is a Major 2nd, 2 to 3 is a Major 2nd

0:23:42.6 DN: [Singing] 3 to 4 is a half step. Or I should say... minor 2nd. 4 to 5 is a Major 2nd. 5 to 6 is getting high... Major 2nd. 

[vocalization]

0:23:55.5 GR: Why did I pick this key is...

0:23:57.5 DN: [Singing, drops octave] 6 to 7 is a Major 2nd, 7 to 1 is a minor 2nd.

0:24:02.3 GR: Great, excellent. We're in the key of D, would you do the same game on letter names, please? 

0:24:06.2 DN: Okay. Is that D? 

0:24:08.8 GR: Yes.

0:24:09.4 DN: Okay. D... Oh gosh.

0:24:14.3 DN: It's morning.

0:24:14.4 GR: It is morning.

0:24:15.1 DN: D to E is a Major 2nd, E to F# is a Major 2nd, F# to G is a minor 2nd.

0:24:25.1 GR: Et cetera, yeah, yeah. Good. And of course, and for me, by doing that first with numbers or solfège, right? Do-to-RE, is a major second, RE-to-MI is a major second, the words are always the same no matter what major key you're in if you're using a movable system, but they change when you're using letter names or fixed Do. And to me, there's huge value in thinking about that. I also love to do this game with instrumentalists and have them play on their instrument D, E and then sing major second, E, F sharp and then sing major second.

0:25:05.9 DN: Great.

0:25:06.2 GR: So the analytical part they sing and the note naming part they play.

0:25:10.3 DN: Well, I noticed two things about doing that is, having just gone through it with just saying the interval names, which of course, I know very well. When I did it again with letter names, I've noticed two things happen is one, I already knew what that pattern was 'cause I had just done it. And in thinking about what the next note was, I already knew what that pattern was and therefore I had another contextual clue, just in recognizing my own brain process, I had another contextual clue about what that next note would be. And secondly, because I play the piano, I also didn't have to think about what I just, I know what that next interval is going to feel like. And I think I was envisioning a piano keyboard, and I know that we have the keyboard advantage...

0:26:15.6 GR: Sure.

0:26:16.2 DN: In music theory, but also a keyboard bias sometimes but I just was recognizing that both of those skills were...

0:26:26.1 GR: Coming into play.

0:26:27.0 DN: Contributing to my ability to do the exercise.

0:26:30.1 GR: Yeah.

0:26:30.3 DN: Just in terms of thinking of how, how we think and that's likely to be the skills that are also being developed in the students that are working with it.

0:26:38.1 GR: That's right. And the other thing is in giving it the specific letter names, whether we realize it or not, we are practicing interval building in a written theory context, aurally.

0:26:51.3 DN: Right.

0:26:51.9 GR: And we're also practicing interval recognition in a written theory context aurally.

0:26:57.2 DN: Yeah.

0:26:58.4 GR: That we are... Not only are we acknowledging that we have all these different seconds in a scale, we're looking at them in various different scales and naming them within the context of those keys.

0:27:11.7 DN: Can I also just explicitly add what this is doing is that you were talking about measuring a room. And so we're sitting here, we're naming a bunch of intervals in a row, and each of them is a small interval, but we're always, and it's not like measuring with a ruler, it's like taking the tape measure and extending it another foot.

0:27:32.8 GR: Yeah. Yeah. And sort of looking at each number on the tape measure as you go through the room.

0:27:38.2 DN: But we're still measuring from the same starting point.

0:27:41.8 GR: Absolutely. Yeah.

0:27:43.3 DN: Yeah.

0:27:44.2 GR: Yeah. Yeah. And you know, and with this, and with all these exercises, if you find you have a student or a class that loses tonic, that tonic shifts after a while or that has intonation problems, these are all great exercises to just, to have a drone going. Right.

0:28:01.5 DN: Right.

0:28:02.9 GR: And to just have, have a, like in this case it was D Major, have a D drone going in the background to play it against that and to just, and which can also make really beautifully explicit the feeling of each note against tonic. Of course, we just did this with seconds and we could do this with any other interval. We could do it with thirds 1 to 3 is a major third. 2 to 4 is a minor third, 3 to 5 is a... And so on and so forth.

0:28:33.6 DN: And as we said in the last episode those... That builds patterns that they need to know for other things. They? We.

0:28:43.0 GR: Absolutely.

0:28:44.1 DN: We all need to know those patterns for other things.

0:28:47.9 GR: Right. Especially the thirds. Right. Which get us towards triads, which are such a fundamental building block...

0:28:53.8 DN: Right.

0:28:54.2 GR: ...of our harmonic system. So those are... I think of those as being I call them Intervallic walks. We're gonna walk in interval up or down the scale. Right. You know, the sequencing of that seems pretty darned straightforward. Start with the second, go to the third, et cetera, gradually extend by the time you get to the seventh, things are really fun and there are only certain keys that work well. Right. Like, you gotta start on a pretty low key, it works pretty well with like a key of G A-flat, A, beyond that things get a bit ridiculous.

0:29:27.9 DN: Right.

0:29:28.7 GR: But it's still pretty darn fun. A complement to that is taking a note of the scale and finding all the intervals from that note of the scale. So let's be in this key since it's morning.

0:29:40.3 DN: Yeah.

0:29:41.7 GR: And David, here's the start of this pattern. Do to Re is a Major 2nd. Do to Mi is a Major 3rd. Would you continue?

0:29:51.3 DN: Do to Fa is a Perfect 4th. Do to Sol is a P5. Do to La is a M6. Do to Ti is a M7. Do to Do is a P8.

0:30:07.0 GR: Great. And then shoot the upside down version starting... "Do to Ti..."

0:30:08.3 DN: Do to Ti is a m2, Do to La is a m3, Do to Sol is a P4.

0:30:18.8 GR: Great. And so on and so forth. Yeah, absolutely. Absolutely. We're in the key of B flat. Would you do the version coming down on letter names? 

0:30:23.2 DN: Oh, okay. Bb to A is a m2. Bb to G is a m3. Bb to... 

0:30:33.7 DN: Wait, what I'm I doing? 

0:30:35.7 GR: Yep. You're right. You're right.

0:30:38.0 DN: Bb to F is a P4. Bb to Eb is a P5. Bb to D is a m6. Bb to C is a m7. Bb to Bb is a P8.

0:30:57.0 GR: Yeah, yeah, yeah. It engages your brain in a very different way, doesn't it? 

0:31:00.7 DN: It does.

0:31:02.0 GR: And I love that. That's why I will pretty much always do this both ways.

0:31:08.6 DN: And again, that pattern is really obvious. But then having established that pattern there's, you have another tool to do the... Note naming pattern that you've just done if you do it in that order.

0:31:25.9 GR: Right. And you know, the other thing about this, going to note names like that is a preparatory exercise for sight singing. Where we are looking at notes on the staff that have names. And yes, we could do it just by spotting the intervallic distance. This is some sort of 4th, right. But or saying, oh, yeah, I recognize I'm going to, I've spotted which line Sol is on, or which space in this case so is on if we're thinking in treble clef... But here, by doing specific names, we're reinforcing the note names of the staff we're hopefully causing all of these things to come together to be thought about in all those different ways at once. And did you stop thinking about your solfege as you were singing letter names?

0:32:22.3 DN: I wasn't explicitly thinking about solfege.

0:32:25.3 GR: Were you aware of your solfege? 

0:32:28.0 DN: If you had asked me at any moment where I was then I would... Yes, I would know.

0:32:32.0 GR: Yeah. This to me, is, I think the big truth about this right? Is that exercises like this, we have to know where we are in the scale to do them.

0:32:40.0 DN: Oh, right. Yeah.

0:32:40.5 GR: Even if we're not explicitly naming out loud where we are in this game.

0:32:46.0 DN: Yeah. Yeah, that's good. And then that solves the dilemma that we were talking about earlier, of how do I teach intervals without confusing people about... Yeah. Without introducing a problem into the situation.

0:33:06.0 GR: Yeah, yeah. Now, that exercise, it's a little less obvious of how you continue from there. Like what's the next step with that exercise? I love to do, to continue this one in two different ways. To take Do and we still do intervals to Do or to tonic. But let's start the pattern. Let's work in the octave of five to five instead of in the octave of one to one.

0:33:35.8 DN: All right. That's a very Dalcroze thing.

0:33:38.0 GR: It is a very Dalcroze thing. So let's imagine that we were in this key. [Piano] So there's a one. We're gonna work in the octave of one to five. So we're gonna go 1 to 5 is a perfect 4th, 1 to 6 is a minor third, would you continue? 

0:33:54.5 DN: Oh. 1 to 7 is a minor second. 1 to 1 is a unison. Ooh, that rhymes. That should be a song. 1 to 2 is a major second. 1 to 3 is a major third. Then the rest feels like just what we were doing the...

0:34:13.8 GR: The same. Exactly. Exactly. Yeah, yeah, yeah, exactly. And so, and as you're discovering the farther up we move 1, the harder that becomes, right? So that if we say, now, let's say, this is 1.

0:34:27.8 DN: 1.

0:34:31.8 GR: And okay. And we're going to work in the octave of 3 to 3. So starting from 1 down to 3. Go ahead, David.

0:34:38.2 DN: 1 to 3 is a minor sixth. 1 to 4 is a perfect fifth. 1 to 5 is a perfect 4th. 1 to 6...

0:34:48.1 GR: Etcetera, etcetera. Now, do that in the key of A flat. Can you do that in letter names? 

0:34:52.3 DN: Oh gosh. A to C is a minor sixth. Oh sorry, A flat to C, A flat to D is a...

0:35:02.9 GR: D? What kind of D? 

0:35:03.4 DN: D flat. Thank you. A flat to... Yeah. Woof. You know, gotta visualize my keyboard.

[laughter]

0:35:08.8 GR: Yeah. Yeah, yeah. And it's, you know, and this is also a good one because, it works pretty well to do this game, as you introduce the various keys, right? So like you can... Like, you know, once you get two sharps and two flats, this game is really easy to do from like B flat as a low one just going up, or D, right? And yeah. Okay. So there's this version of it, and then the related version of it is, instead of moving 1 up or down, take any note of the scale as the note we're bouncing to and from. So for instance, let's come to this key. [Piano]

[laughter]

0:35:52.9 GR: And now we're going to go to 5 always. Like this. 1 to 5 is a perfect fifth, 2 to 5 is a perfect 4th, would you continue? 

0:36:03.8 DN: 3 to 5 is a minor third. 4 to 5 is a major second. 5 to 5 is a unison, which doesn't rhyme. 5 to 6 is a major second. 5 to 7 is a major third. 5 to 1 is a perfect 4th.

0:36:25.4 GR: Yeah, exactly. Exactly.

0:36:26.2 DN: Did I do that right? 

0:36:27.1 GR: You did that. Yes. You absolutely did. Yeah. Yeah. And then the reverse is just 1 to 5 is a perfect 4th. 7 to 5 is a major third. 6 to 5 is a major second, etcetera, right? And similarly, we can switch to note names here. I'm in C major, so that...

0:36:41.4 DN: I should practice these.

0:36:43.3 GR: Pretty easy. Yeah, yeah, yeah. Right. And there're...

0:36:46.1 DN: I could use some refresher course.

[laughter]

0:36:50.6 GR: And so right of course you have all the various varieties of those and yeah, and I quite love those. And if you wanna get really crazy, you can combine the two of them. And were we working in the compass of 1 to 1, right? The range of 1 to 1. But we could work instead in the compass of like say, 5 to 5, going to 2 every time.

0:37:13.1 DN: Right.

0:37:13.4 GR: Right. So in other words, let's say we're in this key, [piano] 1 and we're gonna do, 5 to 2 is a perfect fifth. 6 to 2 is a...

0:37:25.7 DN: Oh my gosh. I don't even think of that. All right. Wait. 5 to 2, 5 to 2 is a perfect fifth. This requires so many levels of thinking.

0:37:40.4 GR: Yes, it does. Yes, it does.

[laughter]

0:37:42.1 DN: So 6 to 2 is a perfect 4th. 7 to 2 is a minor third. 1 to 2 is a major second, 2 to 2 is a unison. 2 to 3 is a major second, 2 to 4 is a minor third, 2 to 5 is a perfect 4th.

0:38:06.7 GR: And sing 1.

0:38:08.3 DN: 1.

0:38:09.3 GR: Yes. Yes. Always sing 1 at the end of that exercise to give a sense of closing it off. Yeah.

0:38:18.2 DN: And, yeah, yeah. And then just to make sure that you've kept your tonic in mind, that would be a good diagnostic...

0:38:25.3 GR: Yes. Yes.

0:38:25.8 DN: Tool to see if you've lost track of your tonic.

0:38:29.2 GR: Yeah. Yeah. And if we were in F sharp major, right? So...

0:38:38.6 DN: Oh God! 

0:38:39.2 GR:  But you see, right. And you just see how you can layer this up and up and up and effectively.

0:38:45.6 DN: Yeah.

0:38:46.8 GR: I think about these as just being exercises to prepare us for whatever might come at us in the context of a key.

0:38:55.2 DN: It's great. And listen, this is absolutely tangential, except that I'm totally obsessed with Duolingo. And I'm also going to Poland soon. And so I've been doing Polish and Polish is hard. And, unfortunately I started learning Polish on Duolingo a couple years ago, and then I kind of stopped doing it for a while and now I started again. And now it thinks I know more than I know.

0:39:18.6 GR: Than you actually know. Yeah.

0:39:20.6 DN: And throws really hard things at me. But I was just thinking this feels like that, you sit and you do something and you go, well, I don't remember. And you... But the process of doing it, the process of suffering through it a little bit, is the process that makes us better. It's just great. I'm... That's probably just stating the obvious but, when I think of something like this that where when it feels challenging, is the part where a part of us tends to want to go, ooh that's scary. I'm gonna avoid it. Those are the times when we should dive in and try it, and by doing it, we're gonna get better and it will get easier.

0:40:07.0 GR: Yeah. Just lean in a little bit to that. Yeah totally. There is, the mind is very good at self-protection, right? It wants to, it will happily divert from things that make us recognize our limitations. And it does take a little bit of self managing, to lean into those. Or a teacher. Right? I mean, I think this is the real value of a teacher who picks something for us to do, rather than letting us just keep playing the music that we already play. Well, because it's so fun to do.

0:40:46.3 DN: Right.

0:40:50.0 GR: Okay. So, these can be extended. The other thing of course, you can go from these to then doing this with triads, right? Like, let's say we're in this key and doing, [piano] "one, three, five, major I, two, four, six minor ii, three, five, seven, minor iii,' and so on and so forth.

0:41:12.2 DN: Yeah.

0:41:15.7 GR: Which of course is preparing students to, both... Is helping remind students of the qualities of each triad within the scale. But also of their spellings in scale degrees or, solfege. Do Mi Sol, Major I; Re Fa La, minor ii; Mi Sol Si, minor iii. Fa La Re...

0:41:39.2 GR: Right? And especially if you're a theory teacher who is, who loves scale degrees, but whose aural skills classes are taught in solfege, this is a great way to help your solfege fluent students with their roman numeral spelling and identifications.

0:41:55.4 DN: So it just occurred to me that I can make another version of my court spelling song, which I wrote to teach court spelling, but I could, change it from.

Do Mi Sol, Fa La Do, So Ti Re Ti Sol, Do Do Do.

0:42:15.6 DN: You could then do major I, major IV, major V,' and then 1,1,1.

0:42:20.3 GR: Yeah, totally yeah.

0:42:25.0 DN: Interesting.

0:42:25.9 GR: And so, that gets pretty easy, pretty quickly. Though. Same rule applies, right? [piano chord] 'B, D sharp, F sharp, major I; C sharp, E sharp, G sharp, minor ii; D sharp, F sharp, A sharp Right? 'Cause we're, because I put us in B major.

0:42:44.6 DN: But I can hear that you're already, but you're already recruiting other things that you know.

0:42:49.4 GR: Right? 

0:42:50.4 DN: To do that.

0:42:51.2 GR: Yeah. And then, having done that, I love to then do that in inversion. So let's move first inversion triads up the scale, right? I called these triadic walks, in the same way that we had interval walks, right? So, one, three, six, minor vi; two, four, seven, diminished vii; three, five, one major I, four, six, two, minor ii. And then B D sharp, G sharp, minor vi, C sharp, E, A sharp diminish vii, D sharp F sharp, B major I, E, G sharp, C sharp, minor ii, right? Or second version, one, four, six, major IV, two, five, seven major V, et cetera, right? 

0:43:58.2 DN: Yeah.

0:44:00.6 GR: And then, spelling them as well. And then of course, dealing with the seventh chords, and the seventh chords in inversions are just delightful, right? Especially like your 4/2 position chords.

0:44:12.4 DN: Right? 

0:44:13.1 GR: Right. Yeah. So anyway.

0:44:16.7 DN: There's certainly unlimited potential for making your students suffer in a good learning fashion.

0:44:25.5 GR: Yeah. And I always like to say to my students, well, you've always gotta walk places. You've always gotta drive places. Or, you're making dinner or you're doing dishes, or you're vacuuming or whatever, folding laundry. Here's something you can just do like, just pick one of these and go and gradually build up your fluency.

0:44:49.6 DN: Now there's another pattern. I don't know if it fits with these, but it's an alternative. I did... John Peterson showed me a pattern where you, look at all the half steps and you sing the half steps first. So you sing, you take toe and you sing Mi to Fa is a half step and Ti to Do is a half step. And then you do, Do to Re is a major second Re to Me is a major second. You do Fa to So is a major second. So to La is a major. You do all the half steps, you do all the minor seconds, then the major seconds, then the minor thirds.

0:45:26.2 GR: Yeah.

0:45:26.8 DN: Then the major thirds. But you're doing them out of order.

0:45:30.7 GR: This is beautiful within this. I love that. I love that. Oh. That is great.

0:45:38.6 DN: But then you really have to think about, it's particularly useful just right at the beginning that you're cementing where those half steps are.

0:45:47.4 GR: Yeah. Well, but then also to, with each of the intervals. I'm gonna add this to my... David, thank you for that. I'm gonna add this to my list. And it is worth just being really explicit here about the fact that these are exercises, these are not especially musical things and I trained as a pianist and so to me these are like the Hanon exercises in piano [plays piano]  they're just these different little finger wiggling patterns, that effectively force us to get familiar with all the possible ways of wiggling our fingers. And this is that for ear training.

0:46:43.9 DN: And on top of that, sorry, that just connected to a thing that I was just talking to. I have a former student who's a wonderful, middle school choir director and college choir director in Virginia, Tim Drummond. And we were talking about intervals 'cause I told him we were doing this podcast and we were talking about intervals and solfege and we were talking about the fact that you really, the goal, if you compare it to reading, just doing one interval at a time is reading one phoneme at a time or one letter at a time. And, ideally we want to broaden to seeing words. And those Hanon exercises allows you to see this finger wiggling pattern as a word. And similarly we want to develop some of that with singing, maybe that's out of the context of intervals.

0:47:40.2 GR: Okay. So David, one other fun thing I like to do with this, is what I call coded melodies. And so, I've written here, do you see on our shared Google doc, do you see my outline for a coded melody? 

0:47:53.5 DN: I do. Yeah.

0:47:53.6 GR: So a coded melody, you're given the starting solfege or scale degree. And then you're given a series of generic intervals from that and part of the challenge is like you'll see plus one and your instinct is to go up a second, but plus one means just a unison. So just beware.

0:48:12.6 DN: Okay.

0:48:13.8 GR: Right. Which is our weird system. So I'll just read off this one to our listeners.

0:48:19.9 DN: This could get confusing then.

0:48:21.8 GR: It does, but so are intervals. Right? So, here's the coded melody for you. Do plus one plus two minus three plus two plus two plus two plus one plus two minus two minus two minus two plus two minus two minus two plus two. And so now...

0:48:40.3 DN: Okay, well as soon as I got to the third interval, I recognized it.

0:48:42.8 GR: Of course. Of course. But would you sing it for us on those interval? And actually would you turn it into specific interval names and let's put it in this key.

0:48:52.9 DN: So Do unison, major second, minor third, minor second, major second, major second. Or no, sorry. Major second, unison, minor second, minor second. Major second, major second. I have to put my finger on them. So I keep track of myself. Major second, major second, minor second, minor second.

0:49:26.0 GR: Yes. And part of the challenge of that is David's looking at like plus two, minus two minus two, minus two plus two. And each of those twos are different. There are different kinds of twos. And so you really have to keep track as you're doing that of, and there was that moment where you sang major second unison, Maj--minor second. Right? 

[laughter]

0:49:53.8 GR: As you went from three to four, from me to fa there.

0:49:57.4 DN: And what happened in my brain? I didn't think, ooh, this sounds like a minor second. I thought, I'm going from mi to fa, I have to sing a minor second.

[laughter]

0:50:07.1 GR: Yeah. But it does then start to draw on both of those things of, "Oh right, that's mi to fa. That does sound a minor second."

0:50:14.3 DN: Which, and the moment we start to sing more complicated music, we need to have sort of both of those strategies at hand because the music that we sing is going to modulate. [laughter]

0:50:26.6 GR: Yeah. And here, I want to come back to why do I torture my poor students and have them sing these things both on letter names and on scale degrees. It's because by going through this kind of series, when a student is doing something like, [piano chord] D to F sharp is a major third, E to G is a minor third, they're also thinking two to four is a minor third. But there's also this awareness that that could be three to five is a minor third, at which point we could have modulated down a whole step, granted it's a weird modulation. But there becomes this explicitness of, "Oh yeah, I recognize that. I've seen D two F sharp as one to three. I've seen it as four to six. I've seen it as five to seven. I've seen it as six to one in F sharp minor. I've seen it as all these different things." And what that opens up then as we talk about more complex music, is this potential to take any combination of two notes and to recontextualize them in a different collection, a different scale, a different key which opens up then that world of modulation and the world of mixing between major and minor and all of that wonderful stuff.

0:51:51.7 DN: Yeah. And re-contextualization is the source of all goodness in music.

0:51:56.8 GR: It is. That is heaven for me. Honestly.

0:52:00.1 DN: Hyperbolic, but... [laughter]

0:52:01.6 GR: I don't know. Hyperbolic, yeah. But like... it's when you stand still and the world changes around you, that is an experience that we don't get mostly in life, but we can have in music, and it's magical.

0:52:19.1 DN: And Taylor Swift exploits it all the time. [laughter] Although she's just exploiting... Here's what this little melodic fragment sounds like in this harmony. And now I'm gonna change the harmony and now I'm gonna repeat this melodic fragment one more time with a completely different harmony. And we obviously love that, she sells a lot of albums.

0:52:43.1 GR: Yeah. No, I love that and let's be honest, that trick is, we find that in Beethoven as well like, it's... That's great.

0:52:53.4 DN: Yeah.

0:52:56.1 GR: So, okay, all of these have been contextual intervallic training exercises of intervallic training within keys. We can also do acontextual intervallic training. And I think there's real value to this, especially for learning intonation. And so I thought maybe just to share a couple of quick exercises for this, brass players are especially familiar with this game. There's a series of exercises called the Remington exercises. And the way the Remington exercises work is you start on a note and you work chromatically down going between that note and a half step below back to the note, whole step below. Why am I describing this when I could just be singing it, David? But as far as the Remington exercises usually on instruments you go, doo-doo-doo, which is a minor second, doo-doo-doo major second, minor third, major third, etcetera. And I love to do... So I learned these from brass players, by the way, Emory Remington was a professor of trombone at the Eastman School of Music for ages and ages and ages, and has a room named after him, the Remington room on the ninth floor of Eastman's Annex, which is totally random, which was at one point the basketball court. And prior to that was the dance studio when Martha Graham, when Eastman had a dance program, and when Martha Graham was on the dance faculty of Eastman.

0:54:33.3 DN: I met her.

0:54:35.1 GR: You met Martha Graham? 

0:54:36.2 DN: I did.

0:54:36.8 GR: Oh my gosh.

0:54:37.5 DN: In Spoleto in 1989, in Charleston.

0:54:41.4 GR: Yeah. Near the end of her life.

0:54:43.3 DN: Spoleto Festival in 1989, I think.

0:54:47.3 GR: Amazing. Amazing. Anyway, so these Remington exercises, I do these all the time with my choirs. And the way I do them is we start from a mid-range note, usually F or F sharp, let's take F sharp. And I have them sing half steps working their way out. So we go. Half step, half step, whole step, whole step, minor third, minor third, major third, major third etcetera, up to the seventh. And as they go get good at that, I have them do it in canon. So in other words, canon at one note. So I'm gonna do that with a piano and myself. Minor, let me put my... Here. Minor second, minor second, major second, major second. Or go in opposite direction. So some go up, while some go down, etcetera, etcetera. And, always as I'm doing this, when I'm doing with a choir, I conduct it. And anything that's ever so slightly out of tune, I just freeze on that until we find the exact tuning of it and then we come back. Yeah.

0:56:24.7 DN: Of course, the danger is so within, what is the exact tuning? [laughter]

0:56:29.2 GR: That's right. That's right. And I'm so glad you asked this. And so when I'm doing this exercise, I will take the just intervals for each, so the minor third higher than the piano, the major third lower than the piano, etcetera. And yeah, we're gonna do an episode on tuning systems and so we can dive into that more there, but, Yeah. For me, that is really about helping a choir to hear... and this is why brass players do it so much because they are really trained to hear those overtone intervals.

0:57:09.3 DN: And they have to.

0:57:10.8 GR: And they have to, because of how their instruments work, the fundamental is so present that we hear any lack of tuning much more strongly. And because of the shape of the bell, the dampening of every other partial, we hear intonation so much more clearly with those. So yeah, anyway, Remington exercises, and you can see how those can play out in a variety of ways. So I learned this from a student of a student of Nadia Boulanger which as I said before is like everyone in the world is a student of a student of Nadia Boulanger. So, she had this exercise I am told, where she would play an interval at the piano and you had to name it, using the distance in half steps of that interval. So a minor second would be called a one, a major second, a two, because they're two half steps in and a minor third a three, a major third a four, a perfect 4th a five, an augmented 4th a six, and so on and so forth. And she started with just half and whole steps. So one versus two. And the way this works is I'm going to play and you are to listen, not for the separate notes, this is very important in this exercise. Do not concern yourself, but intentionally try to pay no attention to hearing the two notes, the two distinct notes you hear. Instead, would you listen to how they rub with each other? 

0:58:46.4 DN: Right.

0:58:47.4 GR: Would you listen to the sound of their relationship rather than to the sound of each note? 

0:58:53.0 DN: So should I be explicitly listening for the beating between them? 

0:58:58.5 GR: Effectively, yes. Imagine you were kind of a tuner and you're listening to... Yeah. So shall we try this? 

0:59:04.5 DN: Sure.

0:59:05.2 GR: Just ones and twos. So just the half and whole steps. Here we go.

0:59:07.8 DN: Should I answer? 

0:59:12.9 GR: Yes. You should say it loud. Either one or two, yes.

0:59:14.9 DN: Right. One. Oh yeah, two. One. Oh, sorry, Two. One. Two. Two. Two. Two. One.

0:59:45.8 GR: And so on and so forth.

0:59:46.7 DN: Suddenly that becomes more... That's interesting. That is not an exercise I've done before, especially on the last two, the beating became very apparent.

1:00:01.9 GR: Yeah. Yeah. And the more you do this, to me, each interval has its own color. And in this kind of exercise, you really can't... You can't be thinking about key so much. 'Cause the key is constantly changing around you.

1:00:20.5 DN: Right.

1:00:21.9 GR: And you actually start to focus in on what the interval itself sounds like.

1:00:29.0 DN: It is, it's a focusing exercise. That's great.

1:00:34.0 GR: Yeah. I love this exercise. I adore it. And so she would do ones, twos, then she'd do threes, fours. So major minor thirds. Then she'd do one, two, three, four.

1:00:44.8 DN: Right.

1:00:45.4 GR: Then she would do fives and sevens. So perfect 4th versus perfect fifth.

1:00:53.6 DN: Right. Ooh, that's... Those are astoundingly tricky.

1:00:58.7 GR: Those are astoundingly tricky. Absolutely so. Absolutely so. And to be clear, I believe those are astoundingly tricky for one primary reason, if I play a perfect 4th, this note in it has the overtone, an octave higher. So you're hearing this perfect 4th and this perfect fifth. They're both present, but the perfect 4th is under. If I play now a perfect fifth, this note has this octave in it. So you're hearing a perfect fifth and a perfect 4th. So it's impossible to play on an overtone rich instrument. It's impossible to play a perfect 4th and not hear a perfect fifth. It's impossible to play a perfect fifth and not hear a perfect 4th.

1:01:44.8 DN: Right.

1:01:45.4 GR: So you really have to focus down to the fundamentals of each note. Okay. So then... Yeah. So five and seven, perfect 4th, perfect fifth, then after that, six, eight, and nine altogether, the tritone, the minor six and the major six and then 10 and 11 together. So at which point... And then finally... Yeah, you've got everything. Right? And then having done that, you go back, and you do the whole cycle again with three notes and you're to name the bottom interval. For instance, if I play [piano cluster], that's two, two, but if I play; that's two, one, or if I play, that's one, two. And you gradually build it up that way so that you're...

1:02:45.7 DN: And you're still hearing the beating of those twos separate? 

1:02:48.1 GR: That's right. You listen for the beating. Yeah.

1:02:51.7 DN: Alright.

1:02:55.5 GR: And this training as painful as it is, that training has made me more effective than I care to admit at hearing wrong notes and out of tune notes, in choirs and orchestras. I owe a huge amount of the work I do on the podium to that kind of training of saying, oh, I should have been hearing four, three, four. Right? Or four, three, three. And I'm hearing it, right? Or I'm expecting to hear two, three, two, and instead, I'm hearing something different. And especially in non tonal contexts, this is... Yeah, in my mind I hear a chord and those Boulanger numbers popped out at me. There is a two in that chord. There is a five in that chord. There is a seven in that chord.

1:04:05.6 GR: And yeah. Anyway, it's a different way of hearing and thinking about intervals, but the combination of that with tonal contextual stuff, I find very powerful.

1:04:17.7 DN: Well, I think, as long as a strategy is useful, then having multiple strategies is just gonna make you a better musician. [laughter]

1:04:28.5 GR: For sure. For sure.

1:04:31.9 DN: Yeah.

1:04:32.8 GR: Well, we got through everything in about an hour. I feel like this was a lot. My mind actually feels a little bit tired of... I need to stop thinking about intervals for a while now. I don't know about yours. [laughter]

1:04:43.0 DN: Mine is... Yeah, mine is also running at a million miles a minute, but I'm also thinking like, [chuckle] I think I know what I'm doing in class tomorrow morning.

[laughter]

1:04:55.2 GR: Right. So, yeah. We'll soon be talking about tuning systems, and temperament, and as we found, you can't really talk about intervals without talking about that. So I think that'll be a lot of fun. But... Yeah. David, it's been fun to do some interval stuff with you.

1:05:16.4 DN: I feel once again, enlightened.

[laughter]

1:05:19.6 GR: Excellent. And yeah, I am going to take that exercise from John Peterson, and I am going to go and do it through the full interval cycle, 'cause that, what a delight that is.

1:05:33.3 DN: Yeah.

1:05:34.1 GR: Yeah. Excellent. Awesome. Well, see you next time.

1:05:37.1 DN: All Right. See you soon.

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