Tuning Systems

Notes from the Staff

In this episode we explore the difference between just intonation and equal temperament, and take a field trip to learn about an organ with 15 keys per octave (instead of the usual 12), that makes it possible to play purely tuned chords in almost all of the keys.

Links:

Tuning systems synthesizer: https://utheory.com/experiments/tuning-systems

Organ demo field trip video: https://youtu.be/A9Enpt8hREg

Chrome Music Lab: https://musiclab.chromeexperiments.com/

Intonation/Tuning System "Puzzle Pieces": https://casfaculty.case.edu/ross-duffin/just-intonation-in-renaissance-theory-practice/benedettis-puzzles/

Show Notes:

0:00:20.9 - Introduction

0:01:03.5 - Topic Introduction: Tuning Systems
Greg Ristow explains the complexity of tuning systems and opens the discussion on various aspects related to them.

0:01:58.4 - Just Intervals and Singing in Choirs
Discussion on how choirs tend to move towards just intervals in their performances.

0:02:43.2 - Historical Context of Tuning Systems
We delve into the historical variations of the A note’s frequency and introduces the concept of just intonation tuning.

0:03:22.2 - The Law of Superposition and Sound Waves
Exploration of how sound waves interact and affect what we hear, using sine waves as an example.

0:07:00.8 - Beating in Sound Waves and Tuning
Demonstration of how close frequencies can create a beating effect in sound, relevant to tuning and intonation.

0:11:25.7 - Constructing Scales with Tuning Systems
Examining how scales can be formed using different intervals, like perfect fifths, and the issues that arise in tuning systems like Pythagorean tuning.

0:21:16.4 - Assigning frequencies to notes
A practical approach to assigning frequencies to piano notes, highlighting the challenges in tuning.

0:24:00.8 - The Pythagorean Tuning System and the Wolf Fifth
Understanding the Pythagorean tuning system, its implications, and the infamous 'wolf fifth.'

0:27:24.5 - Equal Temperament Tuning System
Discussion on equal temperament tuning, its compromises, and comparison with just intonation.

0:28:27.3 - Comparison of Major and Minor Thirds in Different Tuning Systems
Exploring how major and minor thirds differ in various tuning systems, including just intonation and equal temperament.

0:31:04.3 - Just Intonation and Chord Tuning
Demonstration of how chords are tuned in just intonation, contrasting with equal temperament.

0:35:56.5 - Field Trip to Oberlin's Brombaugh Organ
Greg Ristow and David Kazimir discuss the unique features of the Brombaugh Organ at Oberlin, demonstrating quarter-comma meantone tuning.

0:41:12.1 - Split Black Keys and Specialized Tuning on the Brombaugh Organ
Exploration of the organ's split black keys and how they impact tuning and playability.

0:47:08.4 - Musical Composition and Tuning Systems
Discussion on how composers historically wrote music with specific tuning systems in mind, affecting the musical expression.

0:49:00.3 - Wrapping Up
We preview the next episode's topic, focusing on teaching intonation, and Leah Sheldon's experience in this area.

Transcript:

[music]

0:00:20.9 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.

0:00:35.4 David Newman: Hi, I'm David Newman. I teach voice and music theory at James Madison University, and I write code and create content for uTheory.

0:00:42.4 Leah Sheldon: I'm Leah Sheldon. I'm head of teacher engagement for uTheory.

0:00:45.7 GR: And I'm Greg Ristow. I'm the founder of uTheory, and I direct the choirs at the Oberlin Conservatory of Music.

0:00:51.9 DN: Thank you listeners for your comments and episode suggestions. We love to read them. Send them our way by email at [e-mail]. And remember to like us and subscribe wherever you get your podcasts.

0:01:03.5 GR: So, today our topic is tuning systems, and this one has taken us a while to put together, partly because we wanted to build some tools to show them off. It's a concept that I think a lot of us have heard little bits about, but maybe one that not all of us know mathematically and musically in a really deep way. So, I'm excited to talk about this one. Maybe to kick it off, what kinds of things come to mind when you think of tuning systems, Leah and David?

0:01:32.2 DN: Oh, man. I let the harpsichordists decide what tuning system we're using. [laughter]

0:01:37.7 GR: That's right, 'cause you worked so much in the early music world, yeah.

0:01:40.7 DN: I trust that whatever I'm hearing is the one that we're supposed to be using. But I think definitely when we're singing in choirs, and I think this is probably true no matter what choir you're singing in, that you find yourself moving towards just intervals for reasons that may become apparent soon.

0:01:58.4 GR: Which of course, brings up this concept of just intervals. And that's something we'll have to unpack for sure. Yeah. How about you, Leah?

0:02:05.5 LS: I know that there are different tuning systems, but what are the differences and when do you use which?

0:02:11.4 GR: And so, maybe we should just talk about what we mean when we say tuning systems first off. Actually, we're deciding what frequencies to assign to our 12 notes on the piano. We can call A440, but we don't have to, right. We know that A has moved throughout history and locations. In northern Germany in the Baroque era it was probably somewhere pretty close to 415. A little bit earlier than that, down in Italy, it was probably somewhere closer to 465.

0:02:43.2 GR: So, what we think of as very much being a fixed thing today hasn't always been a fixed thing at all. And I think a lot of us, as David mentioned, right, have heard when singing in ensembles or playing in ensembles that we should strive for using more of a just intonation tuning. And that's a whole thing to unpack as well. So, shall we dive into some of this? I want to apologize in advance, listeners. This is going to be a little bit of a math-heavy episode, but hopefully in a fun, nerdy kind of way, so.

0:03:22.2 DN: But too, I think even us, definitely as singers, because we don't have a fingering system for our voices. A lot of what we do is, is literally by ear and not mathematical or necessarily carefully considered except for singing it until it sounds right. And I mean you have some plans to show us why that is. [laughter]

0:03:46.7 GR: Yeah, yeah. So to start off, I thought for all of this to make sense, there's one central principle of physics that we have to understand. And this is called the law of superposition. Have you guys heard of this?

0:04:03.4 DN: I am not familiar with this.

0:04:05.3 LS: Me either.

0:04:06.2 GR: Okay.

0:04:06.7 DN: I feel like I should be.

0:04:08.1 GR: So the law of superposition has to do with what happens when you add two waveforms to each other. So, for instance, imagine that we were graphing a sine wave, right. And the sine wave starts from zero and goes up, then comes back down to zero, goes under, and then comes back up to zero, right. That's sort of the cyclical nature of a sine wave. Now, imagine if you took that same sine wave and shifted it over half of its cycle, which is to say that now the shifted version starts at zero but goes down and then comes up. So these two graphs are now mirrored.

0:04:47.9 DN: Right.

0:04:48.0 GR: What was up is now down and vice versa.

0:04:50.9 DN: Yes.

0:04:51.3 GR: Can you sort of picture that?

0:04:52.5 DN: We're moving at exactly half a cycle?

0:04:54.6 GR: Yeah. And so, picture a sine wave and its mirror image upside down.

0:05:02.8 DN: And this is how we used to remove lead vocals from recorded audio tracks.

0:05:08.7 GR: Good, David. Hold that thought. [laughter] Yes, yes, yes, yes, yes. Exactly, exactly right. So now if at any point in that graph we added the two graphs together, where we would have a positive number in the first graph, we'd have the same negative number in the second graph, and they'd add up to be zero.

0:05:25.5 DN: Yeah.

0:05:28.3 GR: Right. And so this is called the law of superposition, that to add waves together, you sum up the waves, and you get a composite wave. Now, how this works in sound is, sound is actually air pressure waves. It's air pressure traveling through the air, hitting our eardrum being translated into cycles. And if we were listening to a sine wave, and we mirrored it, and added exactly that same sine wave, as you said, David. This would cancel out the sine wave and subtract it from the sound. And this is, as you said, how we used to subtract lead vocals from recordings to make karaoke tracks. That's more usually done with machine learning these days.

0:06:19.6 DN: Right.

0:06:20.3 GR: It's also as David points to his earphones, go ahead.

0:06:22.9 DN: It's also how, how noise canceling earphones work.

0:06:26.9 GR: Exactly. By creating the negative image of the sound wave coming to your ear and adding that to... And then playing that negative image to cancel out what's coming to your ear. So yeah, so That's the law of superposition. It's how waves interact with each other. Imagine, for instance, I've got a little simple synthesizer I made up where I can play different frequencies. I'm gonna play just a 440 for us. Here it is. Easy enough. Right?

0:07:00.8 DN: Mm-Hmm.

0:07:00.9 GR: Now imagine if we had a slightly out of tune version of that with it. Instead of 440 cycles per second, we had 441 cycles per second. Here's a 441, just listen to it by itself. And here's a 440. Almost imperceptibly different. Right?

0:07:21.0 DN: Yeah.

0:07:21.8 GR: Now if we play them together, they're gonna sort of line up. They're gonna be, they're gonna start, they're gonna be going up from zero and coming down from zero together. But basically, each second or so, they are going to get more and more out of whack with each other until there will be a point where as the one is all the way at the top, the other is all the way at the bottom. And we'll hear a moment of silence and we'll hear that recurring as the waves line up with each other in opposite positions roughly once a second. You ready to hear this?

0:07:58.0 DN: Mm-Hmm.

0:07:58.9 GR: Okay. So here is a 440 and here is our second slightly out of tune sine wave joining it at a 441. And you can hear how as the waves support each other. Right. As they're both going up or both going down, we hear noise and as they cancel each other, we hear silence and hear that wah wah wah sound, which we call beating.

0:08:29.4 DN: And so if they're off by, this sounds mathy, but it's really a not mathy. If it's off by one, will it be one cycle per second that we get beating?

0:08:41.8 GR: That's exactly right.

0:08:42.6 DN: That is a, if we count the number of beats per second, we'll know how many cycles that is off.

0:08:48.3 GR: Correct yeah. And to prove that, here again is off by 1 Hz and now let's go to off by 2 Hz, which is gonna give us twice as many of those per second.

0:09:05.1 DN: Right.

0:09:07.5 GR: And similarly, you wanna do a couple more to hear them. Here's off by 3 Hz. [laughter] You see where this is going? Here's off by 4 Hz, right?

0:09:24.0 DN: Wow.

0:09:25.4 GR: So, however many Hz those are off from each other, you'll hear that number of beats per second.

0:09:30.1 DN: And this isn't random, because Hz literally means cycles per second.

0:09:35.2 GR: That's exactly right. That's exactly right. Yeah.

0:09:36.9 DN: Yeah.

0:09:40.2 GR: Yeah. And so that's the law of superposition at work. That's how waves combine to reach our ears. They add up and subtract from each other. So yeah. Does that make sense.

0:09:52.1 DN: As far as tuning a note with itself? Yes. [laughter]

0:09:56.3 GR: With itself it sure does yeah yeah. Now it's worth pointing out that, we're listening to right now to pure sign waves, but most waves have, most sounds are more complex than that. They have overtones to them. And you won't hear these, these beats as obviously. Just to... Just so that we, hear what that's like, I'm going to turn the, harmonicity of this up a little bit. I'm gonna add a couple of overtones to the sound just so we hear it. Here's now a much fuller overtony sound.

0:10:30.0 GR: Alright? It's a richer sound, more buzziness to it. And then here is one Hz off with it. We still hear the beat, but because there're more overtones in the sound, it's not fully canceling it out. There's other stuff happening.

0:10:51.8 DN: Right.

0:10:54.7 GR: Yeah. And that's just when I've added some rich overtones to the sound, which I will take away for most of these demos because...

[laughter]

0:11:01.5 GR: It's much easier to hear when we're dealing with the pure sine wave.

0:11:05.5 DN: Cool.

0:11:08.4 GR: Okay. Making sense so far?

0:11:09.2 DN: Sure is.

0:11:10.1 GR: Great. Okay. So We discovered when we offset by one Hz when we had 440 and 441, we got those awful beats. Let's do a little math experiment. Let, could we come up with a couple of numbers that would generally sound good with each other?

0:11:25.7 DN: 880 [laughter]

0:11:28.0 GR: 880? So We're gonna double it. And effectively what that means is right, with our one sine wave, the other one's gonna be going twice as fast. Will there ever be a moment when they're exactly in opposite states?

0:11:43.4 DN: Nope.

0:11:45.3 GR: No, there won't. And so we won't hear any beating between 440 and 880. Here's 440, and here's 880. And they're in perfect tune with each other. And by the way, what interval was that? Here's 440, here's 880.

0:12:06.8 LS: An octave.

0:12:07.6 DN: It's an octave.

0:12:09.8 GR: Yeah. It's an octave. Right? [laughter] Yeah yeah Good.

0:12:13.6 DN: We're all waiting for the kids to respond. [laughter]

0:12:18.5 GR: [laughter] Exactly. Always good to leave time in the classroom for, everyone to answer. Good. 880 works with it as well. Really about anything that's, a nice, multiplication or divisor of it will work pretty well also, like 550 sounds great with 440. Here's 440, here's 550.

0:12:41.3 GR: And the interval is.

0:12:45.1 LS: Major 3rd.

0:12:46.9 GR: Major 3rd. Yeah. And a really beautiful one at that, right?

0:12:52.4 GR: Yeah. So, and we can do that with 660 as well. Let's see, what do we get when we do 440 and 660?

[laughter]

0:13:01.1 GR: We get... Yeah. David holds up his hand...

0:13:06.2 LS: Perfect fit.

0:13:06.8 GR: With five fingers...

[vocalization]

0:13:08.5 GR: Because we get a perfect fit. Yeah, yeah, exactly. Exactly. And so it turns out that these natural multipliers make, the really basic intervals that we're familiar with. And that's largely why when we talk about consonance and dissonance, that's largely where that comes from, is the presence or absence of beats in a sound. So, yeah. Now this brings us to overtones and [laughter] that's kind of exciting and always weird and magical. But effectively think of a string, and if you pluck it in the middle it, and let it vibrate like a guitar string, for instance, right? The middle of that string is gonna go back and forth and back and forth, some number of times per second. That's determined by how long the string is and how tense it is.

0:14:04.5 GR: Easy enough. Right? But that's not gonna be the only vibration on that string. It turns out that there's also a vibration that happens from sort of a still point in the middle of the string where either half vibrates twice as fast as the length of the full string. And both of those vibrations are going on within the same string, the full length and the half length, which is twice as fast. And in fact, any ratio that you can divide the string evenly by, that's true. If you divide the string in three you get two still points in the middle and the 3rds around those are vibrating at three times the speed of the full length of the string. And that then that's actually the interval of an octave and a 5th at that point.

0:15:04.4 DN: Yeah.

0:15:05.2 GR: And if we go another one, right? We have the string divided in four. We've got two octaves from the original, the string divided in five. We have two octaves and a major 3rd from the original. So shall we build those up? Shall we imagine that we have a low guitar string at one 10. The low guitar string that happens to sound like a sine wave. It also vibrates in the middle at two 20. And then if we divide it into 3rds, and then if we divide it into fourths, and then finally if we divide it into 5ths, we could keep going. We could divide it into six and so on and so forth. But you hear we're basically building up a giant, major triad, which are the first notes on the overtone series.

0:16:08.5 DN: Yeah. And you were reducing the volume of the upper harmonics as we went. Right?

0:16:12.8 GR: I wasn't actually, those were all, those are all set equally on this synthesizer.

0:16:18.1 DN: Interesting. They started sounding loud and then they seemed to dip afterwards. But of course on a guitar string each, every time you divide the string in half, yes it's... Or into the more sections you divide the string into, also the less amplitude you're gonna get from each of those...

0:16:39.9 GR: That's right.

0:16:40.6 DN: Harmonics. So that they will be less and less loud. But still contributing to the tamber of the sound that we hear.

0:16:48.1 GR: Yeah. Absolutely. And so we stopped, I stopped here, I went all the way up to two octaves in a perfect 5th. If I had gone one more, we get the first note that is not on a major triad. And let me just, actually hold on a second. So here is up to the octave in the 5th, and then I'm gonna go one more, right. And lemme play just that note. So that is the, right, so here's our root, and here's that note.

0:17:34.1 GR: Yeah. So that's, we finally get a minor seventh up there on top of all of that. So, and that's the overtone series. And when we talk about, when we talk about hearing overtones in a sound, that's what we're talking about is hearing the combination of those different ones. And of course, that's significant for brass players on brass instruments. The pitch is determined by the length of the instrument, and then how many times that's divided. Right? How many of those standing nodes you create within it, in the same way that the guitar string is vibrating at multiple waves at the same time, on a brass instrument you buzz your lips and you can create a tone at the frequency, the length of the instrument, or twice the length, or three times or four times, et cetera. And that is the overtone series on there. Now there is a slight difference because some brass instruments are conical. And with those instruments, you actually lose the very lowest overtone because instead of having, low pressure, high pressure, low pressure, a full wave cycle, you only have low pressure, high pressure for the length of it. So yeah.

0:18:48.4 DN: So I've gone to, musiclab.chromeexperiments.com, or you can just Google Chrome Music Lab and it'll take you to this. And this has a whole bunch of different sort of ways of exploring music. And one of them has an explanation, not an explanation, just a experiment. Place a playground for harmonics. And if you go to the playground for harmonics, it shows you some things that look like they could be waving strings. And it shows you that. And if you click on the one string, you get the sound of one string. And if you do that and it's octave, you can hear them as you touch them, and you can sweep through them and get something that sounds a little more like a complex tone. There is also a, spectrogram.

0:19:56.4 GR: A spectrogram shows in, on a graph by the intensity of the different frequencies within a sound. The different overtones within a sound, generally red, representing more intense and blue, or black representing less intense.

0:20:17.3 DN: And so it gives you a number of different presets as well as a microphone, which if you allow it access to your microphone will show you the spectrum of the sounds that you're making.

0:20:29.1 LS: And there are also strings and sound waves and arpeggios and lots of lovely experiments.

0:20:36.2 DN: Yeah. There are lots of games to play, there.

0:20:39.3 GR: Yeah. Excellent. So we'll put that in the show notes. But you can just google, Chrome Music Lab experiments and that will come up for sure. Great. Okay. So with all that, I think we're in a pretty good place to actually turn and talk about, well, how do we get to a scale with all of this? So let's say we wanted to create all of the notes on the piano. Can we think of an interval that's on the overtone series where if we followed it through over and over, we'd eventually get all 12 notes on the piano?

0:21:16.4 DN: I mean, obviously you could just do half steps, right?

0:21:20.2 GR: You could absolutely do half steps.

[chuckle]

0:21:22.4 GR: You have to do half steps. Now, finding, it's gonna be a long time on the overtone series before we find a half step.

[laughter]

0:21:31.7 GR: So it's, that's a good thought. But maybe we can find one a little lower down little earlier on in the Harmonic series.

0:21:37.9 DN: And I think the first one that makes, the first one that's gonna make a difference is that 5th, that's the first one that we hear as a different note. Or if you're gonna use fancy speak, the a different pitch class.

0:21:51.5 GR: Yeah, absolutely. And we know also that if we go around by perfect 5ths, we're gonna get all 12, because circle of 5ths is exactly that, right? It's a collection of all the 5ths.

0:22:04.5 DN: And also because the harmonics get generally softer as you go up, in the series on any instruments, on most instruments. Generally the 5th is gonna be the one that's gonna come out the strongest. Likely to come out the strongest.

0:22:21.5 GR: Yeah.

0:22:21.6 DN: Right?

0:22:21.7 GR: Yeah. Yes, that's right. Yeah. Okay. So now our goal is to assign to the notes on the piano frequencies based on going around the 5ths. And so our 5th, when we found on the harmonic series, when we had, we multiplied by, we divided our string in three, right. And that, and it vibrated three times as fast as the original. That's actually an octave in the 5th. So we're gonna divide that frequency in half, to make it just a 5th. So we're multiplying by three dividing in half, and that gives us our 5th each time. So yeah. So let's sort of work our way through this. Here's A, and I'm actually gonna take us up a 5th and down an octave. So down a 4th to E and down a 4th to B, up a 5th to F sharp, down a 4th to C sharp, up a 5th to G sharp, down a 5th, down a 4th to D sharp which let's call E-flat just to make our life easy. Up a 5th to B flat, down a 4th to F, down a 4th to C, up a 5th to G, down a 4th to D. And finally up a 5th back to A. And now we've gone through our whole series and we have all 12 notes of the piano. But here's the bad news. This is the A that we ended on, and this is the A that we started on. Hear that difference?

0:24:00.8 DN: [laughter]

0:24:01.8 GR: Here's the A we ended on, and here's the A we started on, and if we play them together, we get that right. [laughter] And so this way of tuning is called the Pythagorean Tuning System. We tune by 5ths throughout, and it leaves a problem in there somewhere, right? Because, when we get back to A, it's no longer A, and the end result is that the 5th from, that the second to last 5th is pretty rough with itself. It's not well in tune with itself.

0:24:55.8 GR: You're sort of going wah-wah-wah-wah-wah in there.

0:25:00.7 DN: Right.

0:25:01.8 GR: And just to compare that, the 5th that we should have had instead would've been, and here is what's called the wolf 5th, hear that difference that out of tune ness of it?

0:25:23.4 DN: It also through the headphones. I don't know if it's gonna come through in the recording, but it through the headphones, it also gets a whole nother set of tones that sound along with it.

0:25:36.1 GR: Yeah. From the, yeah. The distortion.

0:25:38.6 DN: Yeah.

0:25:40.8 GR: The sound could by the distortion. Yeah. And that's called the wolf 5th, because it sounds a bit like a wolf howling. And actually it sounds even more like a wolf howling if we turn up the overtones of the sound itself. So remember I had no overtones in each of those. Those are pure sine waves. But so now here is our good 5th with some overtones in it.

0:26:06.1 GR: And here's our wolf 5th.

[laughter]

0:26:13.8 GR: Yeah, right? Good 5th.

0:26:20.4 GR: Wolf 5th.

0:26:23.7 GR: Isn't that awful?

[laughter]

0:26:28.5 GR: And because when we go by perfect 5ths all the way around, we come out significantly higher than where we started. In this case, we start from a 440. We wind up coming out exactly six Hz higher. We have to make a compromise. And this is true of every tuning system that we have to... That there's always a compromise to make some intervals sound better and some intervals sound worse. And so the compromise that we make in equal temperament is we say, let's divide the problems of that wolf 5th, equally around all of the intervals. And that does that... That works out pretty well. Unfortunately, it means that in equal temperament, there's no interval anywhere in equal temperament that is a pure overtone interval.

0:27:24.5 DN: Yeah. And that it's funny because I have a colleague who has done a lot of work on Bach's tuning system and the imagination, I think that is in the public sphere is that they didn't know about equal temperament at that time, but they did know about equal temperament and they didn't want it because they so valued the sound of pure chords.

0:27:55.1 GR: Yes. Yeah. That's absolutely right. And so just we compare these a little bit, right? So here's in equal temperament here is actually, let's start with if we did a just major 3rd, here's a just major 3rd. So that's the overtone major 3rd, and here's what we actually get on the piano.

0:28:27.3 DN: With sine waves. It sounds shockingly bad.

[laughter]

0:28:29.7 GR: It's exactly, it does, it does. It very much does with sine waves because they're so pure, it becomes so obvious when there's basically no overtones to balance that out. The more overtone rich something is, the harder it is to hear these detailed tuning things. And that, I mean, for instance, like on organ, depending on which pipe it is, some of them sound very... Have really simple overtone structures. And so you can really hear these things a lot. Yeah. So, and that's what we call equal temperament, where we compress every 5th a tiny bit so that it comes out with all of them being equal. And there are lots of... So the three intonations as we've talked about, I guess we've only talked about two really.

0:29:22.9 GR: We've talked about, Pythagorean, where we go around by perfect 5ths. And we have one terrible, perfect 5th wolf 5th. And we have equal temperament where we have 12, not quite perfect 5ths. But there are lots of other tuning systems that try and adjust for this. Quarter-comma meantone is a Pythagorean system where we compress, we spread out the wolf 5th across four different 5ths. And that sort balances some of the 3rds, so they sound pretty decent. And then, when we talk about just intonation itself, right? If we don't have an instrument that plays with fixed pitches, we can actually adjust in real time to tune different chords in a just way. So, and I've created, and we'll share this link, I've created a little synthesizer where you can build a chord. So for instance, here is, I don't know, B flat major in just intonation and it automatically adjusts and makes it just. For comparison, here's that same cord in equal temperament. You hear how wobbly it is compared to here's just, yeah. And actually if we... Let's hear that same cord in a Pythagorean temperament. And you'll hear the 5th is really beautiful in the Pythagorean temperament. The 3rd is quite high.

0:31:04.3 GR: Let me take the 3rd out of this. So it's a true perfect 5th. And now here's that Pythagorean 3rd in there.

[laughter]

0:31:16.2 GR: As we all cringe, right?

0:31:18.1 DN: This is the reason why society is so tense today.

0:31:21.5 GR: Yeah. [laughter]

0:31:22.1 GR: And so, okay, here's...

[laughter]

0:31:24.3 DN: It's all the fault of our tuning system.

0:31:27.7 GR: It's all the fault of these tuning systems. So just intonation has that same 5th, exactly the same as Pythagorean, but listen to its 3rd, right? And that's significantly lower than our equal temperament, major 3rd. Now I want to share another one with you, which always feels crazy to me. So here's that 5th again, and let's have a minor 3rd and that's a just tuned minor triad. Compare that with our equal temperament minor triad. And I'm gonna take the 5th out so we can hear it more. Obviously. Here's our equal temperament minor 3rd, and here's our just intonation minor 3rd.

0:32:19.9 GR: Which 3rd is bigger? Here's equal temperament. And here's just intonation.

0:32:28.7 LS: Just intonation?

0:32:32.0 DN: Yeah.

0:32:32.8 GR: The just intonation, that's right, yeah. So it turns out our minor 3rd in just intonation is about 14 cents higher. There are 100 cents in every half step. So it's about like, what's that? Like a seventh of a step higher than, a seventh of a half step higher than on the equal temperament version. And similarly our major 3rd is 14 cents lower...

0:33:02.6 DN: Lower, yeah.

0:33:04.0 GR: In just intonation compared to equal temperament. Yeah. Isn't that cool and wacky and wild?

[laughter]

0:33:11.6 DN: So the thing is, and if you're tuning by ear, you're just going to nudge it to the thing that sounds good.

0:33:20.8 GR: Yeah, yeah.

0:33:22.3 DN: If you're listening carefully, you're going to nudge it to the thing that sounds good. Which is also one reason sometimes why choirs go out of tune, is 'cause pieces have been written so that the adjustments that they keep making keep going in one direction.

[laughter]

0:33:36.6 GR: Yeah, exactly, exactly. That's true. There are these wonderful puzzle pieces from the Renaissance era that cause you to modulate up bit by bit. And I'll share a link. Ross Duffin, who's a retired Musicology Professor from Case Western University, has a wonderful article on that. He also has a wonderful book called How Equal Temperament Ruined Harmony and Why You Should Care, which is a delightful read. All of this is great and feels maybe a little bit abstract, so I thought it'd be fun to take a field trip. And so I got together with Oberlin's Curator of Pipe Organs. Oberlin has one of the largest pipe organ collections in the world, David Kazimir to take us to the Brombaugh Organ in Fairchild Chapel, which actually has, instead of five black keys per octave, it has eight black keys per octave, because three of those black notes are split with a sharp version and a flat version, which lets it be in a quarter-comma meantone that works around different places. Quarter-comma meantone, remember, is where we split up the wolf 5th into four of the 5ths in the Pythagorean tuning. So shall we go visit with David?

0:35:07.5 DN: Let's go visit!

[music]

0:35:56.5 GR: That was great. What was that?

0:35:58.1 David Kazimir: That was the opening verse of the Magnificat on the 9th tone by German composer Samuel Scheidt.

0:36:05.6 GR: So, that's excellent.

0:36:06.4 GR: I'm here with David Kazimir, who is the curator of pipe organs at Oberlin. It's amazing that Oberlin has a curator of pipe organs. How many pipe organs do we have and what do you do?

0:36:17.4 DK: There are 34 pipe organs on the campus. My job is like a family practice physician. I have a little bit of instruments who are newborns and some that are a little bit older, and no one's in palliative care, so I'm doing my job. But a lot of it is just tuning, regulation, general maintenance, and working with the facilities to keep everything and everyone moving and playing smoothly.

0:36:43.2 GR: This organ, that you're at, is a pretty special instrument. Now, most of our audience is listening, but we'll also post a video of this online. Can you tell us a little bit about this organ, and why it's so special?

0:36:55.8 DK: This instrument was built in 1983 by John Brombaugh, one of the 20th century's great American organ builders. It utilizes the tuning system called quarter-comma meantone tuning, and is unique amongst organs in that it has split sharps to accommodate those unique interval tuning systems that we find in quarter-comma meantone. Well, on a normal piano keyboard, there are 12 notes in an octave. Not here. There are 15. I'm going to start at C and go up, so middle C, C sharp, D, D sharp, E flat, E, F, F sharp, G, A flat, G sharp, A, A sharp, B flat, B, and C an octave higher.

0:38:12.9 GR: Wow, that's crazy. So if I heard right, that means that what we think of as being the enharmonically equivalent notes in equal temperament, like D sharp and E flat, always the flat version was higher than the sharp version.

0:38:27.0 DK: Correct. And as they are oriented on this keyboard, the more commonly used of those enharmonic notes, E flat, G sharp, and B flat, they are at the front of the key. And behind it, and a little bit higher, is the less commonly used one. So you have D sharp behind E flat, you have A flat behind G sharp, and you have A sharp behind B flat.

0:38:56.9 GR: So I wonder, maybe, could you just play some chords with the right and wrong black notes, so we can hear the difference?

0:39:03.8 DK: Sure. E major is E, G sharp, and B. This is the right version with the G sharp.

0:39:15.9 GR: And that, as we listen to that, that G sharp is significantly lower, that E to G sharp interval is significantly smaller compared to our modern piano.

0:39:24.3 DK: Correct. The 3rds on this organ, and in quarter-comma meantone tuning are left pure. So they don't beat as much. If you listen to that, there's no oscillation. The thing that gets interesting in quarter-comma meantone to make those 3rds pure, you end up with 5ths that are narrower.

0:39:51.3 GR: Yeah. So we hear a little bit of the beats in that 5th. Can you hold that 5th out a bit longer? That's that wah-wah-wah-wah-wah... [vocalization] effect we hear in it. Yeah. Yes

0:40:01.9 DK: You don't hear that in equal temperament, which is twelfth-comma meantone. Pure 3rd. Now I'm going to play it with the wrong note. Hold on to something.

0:40:19.2 GR: Oh, that's painful. So that was E, A flat, and B.

0:40:23.0 DK: Yes.

0:40:23.9 GR: Wow. Could you just play the E and A flat together? And now the E and G sharp. That's so much better. [chuckle] That is amazing. But then that lets you get, I imagine, other places that would use A flat. Can you play an A flat major on this organ?

0:40:46.7 DK: You can play an A flat major, so.

0:40:48.0 GR: And it's really, it's quite nice, actually.

0:40:55.3 DK: Yes, it's very sweet. Now I can use the wrong key, in that case, D sharp.

0:41:06.5 GR: Which is a little bit diminished. What if we use both G sharp and D sharp there? So what if...

0:41:12.1 DK: G sharp and D sharp.

0:41:16.3 GR: Wild.

0:41:18.1 DK: It gets really crazy.

0:41:21.8 GR: So normally in like a quarter-comma meantone, the farther you get away from the normal keys, the weirder it sounds.

0:41:32.6 DK: Right.

0:41:33.4 GR: Is that still true with this organ? Or because of the split keys, are you able to keep things sounding normal-ish?

0:41:41.3 DK: Well, with the split keys, you have more options to keep intervals that use a pure 3rd running. So with the option of F to F sharp to A sharp, that can be pure and comfortable because the difference between F sharp and B flat.

0:42:04.3 GR: Yeah.

0:42:05.9 DK: The wolf tone that we have in this...

0:42:08.1 GR: So the wolf tone is usually where you put all the leftover 5th change from adjusting the 5ths to be smaller. You still have one 5th that has to be way out.

0:42:21.4 DK: Right.

0:42:21.8 GR: And normally that's between, G sharp and E flat.

0:42:27.4 DK: Yes. So if we listen to that G sharp and E flat.

0:42:33.0 GR: Yeah. And it's called the wolf because it kind of howls.

0:42:35.5 DK: It howls. There's no way of getting around that. The theory practice of this time is through the use of these pure 3rds. It strengthens in some keys the function of the dominant going into the tonic. 'Cause you have pure 3rds in the dominant and you also have it really pure in the tonic when you resolve.

0:42:58.0 GR: Could you maybe just play some tonic dominant tonics for us and tell us what keys you're playing in there?

0:43:01.9 DK: Alright, so tonic, dominant tonic, this is C major. The weight of G major in that case really has, strength, but it also sets you up for... You really enjoy your resolution when you have those crunching, dissonances, like. The arrival at the cadence just goes... And that tension you hear in the music of this time period as well.

0:43:43.1 GR: David, could I give it a try?

0:43:44.5 DK: Of course you can.

0:43:45.0 GR: I'm just really curious to like, to explore this and... Okay. You may have to remind me which ones are the sharps and which ones are the flat. So I'm gonna try playing something in E major.

0:43:54.1 DK: Okay.

0:43:54.5 GR: And I think you said the part of the G sharp...

0:43:57.3 DK: Front?

0:43:57.7 GR: A flat closest to me is the G sharp.

0:44:00.5 DK: Correct.

0:44:00.9 GR: Okay. So in other words, I can have a beautiful E major triad...

0:44:04.3 DK: Triad.

0:44:05.2 GR: Like this and then the front of the E flat D sharp, the wait don't [0:44:12.4] ____ the front of it is the E-flat, right?

0:44:14.4 DK: Correct.

0:44:14.8 GR: And the back is the D sharp.

0:44:16.1 DK: Correct.

0:44:16.7 GR: So if I'm playing in E major and I wanna play 151. In my one chord, I have to use the front part...

0:44:22.8 DK: Right.

0:44:24.3 GR: Of the G sharp, A flat.

0:44:26.4 DK: Yes.

0:44:26.7 GR: But in my five chord, I have to use the back part...

0:44:29.6 DK: Correct.

0:44:30.3 GR: Of my... And if I get that wrong, it's going to be really spectacular. Like there.

0:44:41.5 DK: Yeah.

0:44:43.0 GR: And there's a much smaller target area.

0:44:46.1 DK: Yes.

0:44:47.0 GR: For these, is this an E down here on the bottom?

0:44:50.1 DK: Yes. Oh no. Short octave.

0:44:53.4 GR: Thank you.

0:44:53.9 DK: Yes.

0:44:54.5 GR: Got it.

0:44:54.8 DK: We should probably point out the short octave to them.

0:44:56.8 GR: Yeah. So... Oh, that's great. You know the question that we get all the time when we're teaching beginning students, is why do we need a D sharp and an E flat? And this really answers that.

0:45:11.1 DK: Yes.

0:45:11.6 GR: Because we can hear, right? Like, so if I play C minor using C, E flat, by G. But if I play C, D sharp, G that's just, it's just, that's just out.

0:45:27.2 DK: And the body reacts accordingly.

0:45:30.0 GR: Yeah. And that difference between the D sharp and the E flat, I mean that's like what? Quarter? Less than a quarter total.

0:45:40.0 DK: It's less than a quarter.

0:45:41.1 GR: Yeah.

0:45:42.0 DK: It really gets into micro tonal adjustments.

0:45:45.3 GR: Yeah.

0:45:46.6 DK: And we use this specialized tuning machine to make sure that we're getting right to the core of where that pitch needs to be. 'Cause there is a huge difference.

0:45:54.4 GR: David, this is amazing. Do you wanna play something for us to take us out?

0:46:00.4 DK: Okay.

[music]

0:46:57.2 GR: It's just really cool to hear the organ playing in pretty constant just intonation because of, yeah, because of those different black notes, so.

0:47:08.4 DN: And the funny thing is though that people were writing with these intonations in mind and therefore there are pieces that are deliberately flexing on the out-of-tuneness of a thing that then resolves into more pleasant tuning.

0:47:27.9 GR: That's right, that's right. Yeah, and think of like the farther you modulate away from a normal key, a key that's central to the tuning, the more and more out-of-tune it becomes. And if you think of like even classical sonata form, right, where in the exposition we go up to the 5th, which would be a little more out-of-tune, but in the development we go to some really remote places, which would be very out-of-tune. And equal temperament as we think of it today really didn't fully come about until the 20th century, right? The tunings were decidedly more flexible and preferenced more normal keys prior to that. Anyway, having said that, kind of cool to hear it on a real live organ, right?

0:48:15.5 DN: Yeah.

0:48:16.2 GR: And if you wanna try this experience out for yourself, you can try out our Tuning System Synthesizer, where if you plug in a MIDI keyboard and set it to automatic just intonation and play any chord, or click the notes on your screen, it will tune that chord in just intonation for you. And some of them I find really shocking. Like for me especially, first inversion triads are just weird beasts. So here is a first inversion F major triad, which sounds really beautiful, compare it to equal temperament.

0:49:00.3 DN: Whoa!

0:49:01.6 GR: Right? Shocking, isn't it? Yeah. And if we spread that out a little bit more on the keyboard, so I'm gonna give us a really beautifully spaced now, first inversion major triad. Here we go. That's just intonation, here's equal temperament. Right? Isn't it something that these things we don't think of as out of tune in equal temperament, when we compare them to just intonation, we can really hear, especially with a simple tone like this, how many beats there are in it, which is kind of cool and amazing. Now this has been a really almost esoteric episode for us, we haven't really talked at all about what do we do with any of this, but fear not, because our next episode is actually gonna be on teaching intonation. And Leah, I'm so excited that you're gonna help with that, because that's something you do quite constantly in your day-to-day life, yeah?

0:50:10.9 LS: I live and breathe it, yes.

0:50:16.5 DN: Wow. Excellent. Excellent.

0:50:17.8 GR: Well, thank you all for joining, and a special thanks again to David Kazimir for showing off the Brombaugh organ at Oberlin's Fairchild Chapel, and we'll see you all next time.

[music]

0:50:34.6 LS: Notes From the Staff is produced by uTheory.com.

0:50:34.7 GR: UTheory is the most advanced online learning platform for music theory.

0:50:39.1 LS: With video lessons, individualized practice, and proficiency testing, uTheory has helped more than 100,000 students around the world master the fundamentals of music theory, rhythm, and ear training.

0:50:49.9 GR: Create your own free teacher account at uTheory.com/teach.