Like most Suzuki youngsters, I went through a phase where I had little stripes of tape placed on my fingerboard to guide where my fingers ought to go. Which I’m guessing was my first lesson on intonation, and the idea that there’s a precise place on the fingerboard, where if you put your finger right there, all is good in the world because the resulting sound that comes out of your instrument rings, and just sounds right.

Of course, the rebellious part of me resented those lines, and I remember getting into an argument with my mom one day because she said I wasn’t putting my fingers square on the lines, and I insisted that the lines must not be in the right place because it didn’t sound right (I know – maybe sort of cute in hindsight, but probably not so much at the time).

Of course, years later, as my repertoire grew more difficult, and intonation became more of a challenge, there were many a day when I wished there had been frets on my fingerboard to help guide my fingers (even though, as you’ll learn in today’s episode, that actually would have made things harder, not easier).

In any case, if you’ve ever struggled with intonation, or got into an argument about intonation with your quartet while tuning chords, or diligently practiced with a piano or electronic tuner or drone and were still told that your intonation needed work, today’s episode is for you. =)

Meet Minna Chung

Cellist Minna Chung is a fellow Oberlin grad, who serves on the faculty of the University of Manitoba, and is co-author of the book CelloMind – a book which gets into all the details and intricacies of intonation, from the different types of intonation (and why that matters) to practical exercises for improving one’s intonation.

In this episode we’ll explore:

  • Why using a piano to find the right pitch can be problematic (4:12)
  • The three basic string intonation systems that we ought to know – and when to use each (6:09)
  • A few things to know about harmonic intonation (9:34)
  • Can Noa handle a basic music theory question? Spoiler alert: yes – but it’s not pretty (13:58)
  • Why some chords need to be played flat, and others sharp, to ensure that they sound in tune (14:34)
  • I try to sum up the CliffsNotes version of what to do with each interval (19:05)
  • What to do when you have a series of chords with a melodic line going through them (23:53)
  • What should you do when playing with piano? (28:25)
  • Is it true that if in doubt, we should err on the side of playing sharp? (33:13)
  • Which type of intonation do your ears prefer? Listen to these three samples and see which one you like the best. (33:47)
  • When playing in a quartet, who should be deferring to whom, in terms of intonation? (40:40)
  • How do we practice getting better at intonation? (spoiler alert: maybe this is why ear training is part of the music curriculum?) (44:13)
  • A potential problem with drones, and an alternative – the “Tartini tone” (45:01)
  • Why do cellists tune from their top string (A) to their lowest string (C), while violinists tune from the second string (A) down to the lowest string (G) and then add the top string (E) last? (55:07)
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Noa
So when I was growing up, I assumed that intonation was pretty straightforward like you either play in tune or you don’t play in tune. And it was just as simple as that. But I think it was probably high school, my teacher introduced me to the idea that intonation was an absolute, but relative to what other notes you’re playing, like, and this may not make sense verbally, but I’ll maybe try to put a little audio clip in, but so you can tune an E on the D string and first position to the open A, that’s adjacent to it right. But that fourth, while in tune when you play it that way, if you try to tune that same E with the G on the other side, the E is going to be sharp relative to the G. For instance, listen to this clip that Mina provided, pay attention to how the E sounds relative to the open A.

Noa
Totally fine, right? Now listen to how that E sounds relative to the open G. It’s going to be sharp. So you’ll hear the cellist adjust the E down a little bit,

Noa
it’s even more obvious in the other direction. Listen to how low the E sounds by itself, but then pay attention to how right it sounds when played together with the open G, and how incredibly flat it sounds relative to the open A.

Noa
And so that kind of blew my mind. Not in a good way that kind of made me go a little insane because it was a level of complexity about intonation that I didn’t really want to deal with. I just wanted to have things be in tune or not in tune, end ofstory. So then, you know, fast forward a few years in grad school, I was working on the Bach Chaconne and I got sucked into this whole other intonation black hole because I was trying to get all the chords perfectly in tune with themselves. But then I realized or I didn’t realize my teacher pointed out well, you know, the melodic line there isn’t necessarily in tune from one note to the next. So he started throwing out these words like, “just intonation” and “pythagorean”. And, this was all new to me and it just made me a little bit more annoyed. And I spent probably a month just on that first page, trying to sort everything out until my teacher found said, okay, you know, I’m getting a little too OCD, we should probably move on to the other pages in the Chaconne. All this to say, I came to understand over time that intonation isn’t as straightforward as we might like it to be. So I don’t know if this is the best place to start or not. But I wonder if you might be able to walk us through the sort of “intonation for dummies version” of what we need to know

Minna
My background is very similar to yours. And I was blissfully unaware of my accuracy all through high school, but there definitely are times when the intonation system that you are using may not be best suited to whatever passage you’re working on. And we all know growing up that there are millions of pianos out there that are completely out of tune, especially if you practice at a church. So the idea of equal tempered intonation, also known as like keyboard intonation is not even within itself an accurate representation of a pitch you’re looking for. There are many times I think both you and I studied at Oberlin, and at Oberlin, fortunate for us, we all had a piano in each of those practicing rooms. But searching for a pitch, this was way before the smartphone. We didn’t have cell phones back then. You couldn’t even rely on the piano in your room to help you find a note. So a lot of times, young musicians are going off of their inner hearing. Maybe many of us were obsessed with our tape recorder and that Sony Walkman and listening to the concertos over and over again. And you just developed that way, a very intuitive sense of like your melodic passage, I was unaware that you could play leading tones higher, sharps higher, flats lower. The whole idea of just intonation, I’m impressed that your teacher mentioned it to you. No one had ever really mentioned it to me, except for Hans Jensen at Northwestern and this was, I was in my mid 20s by then. But there are three basic string intonation systems that we present in Cellomind now there are many, many more, but these are the most simplified and useful ones in order to cultivate an inner hearing to increase your sensitivity to frequencies and vibrations. And we use the equal temperament as the control system. Now I want to mention that we do have a measurements for us acoustically, just like woodworkers have like the ruler. Even time, 60 seconds is a minute, we have 100 cents to our half step. And that unit of measuring is dedicated and works best for equal temperament. In fact, it was built to explain equal temperament. So by having that keyboard intonation of 100 cents per half step as a measurement tool that helps us clarify the differences the very subtle differences between melodic and just. So there are three: equal temperament is the standardized intonation system that keyboards are often tuned to but then I have since come to understand that piano tuning is very, very complex. It’s not like your piano tuner comes to your home and in this watching a needle for every single note, and it’s 100 cents is far more complex than that. So in theory, we just use 100 cents per half step, as an easy way to decipher the differences that one would need if they are tuning, double stops or chords, which we call harmonic intonation. And the fancy word is just intonation. And then the melodic one is Pythagorean and that’s probably the most common way most of us listen to because we’re so melodically inclined, the human ear likes to hear things that move horizontally. So it’s valuable to know certain things though like, sharps are played higher, flats are play lower, it can instantly improve your expression in my, in my opinion.

Noa
So it sounds like it just to make sure that I get the terms right, maybe it doesn’t matter. But so Pythagorean intonation is sometimes called melodic intonation. And then just intonation is called harmonic intonation.

Minna
Yes.

Noa
The Pythagorean and just I don’t… i’m sure those have interesting origins for it, melodic and harmonic kind of make more sense to me, just intuitively,

Minna
we can totally stick to that one.

Noa
Okay. Although I guess it sounds like we’re smarter if we use Pythagorean and just, but, so it sounds to me if I’m understanding that when we’re playing single lines, perhaps melodic intonation is kind of how we go and then if we’re playing chords, then we need to like switch to harmonic intonation or how does that work?

Minna
That’s exactly how it works.

Minna
So, most students are blissfully unaware of harmonic intonation, like yourself, the term has been brought up to them. And you know that certain notes need adjusting in order for the double stop or the chord to sound better in tune, more harmonious, less hitting in the vibrations, but very rarely has it ever been taught in a way that the student or the musician can figure out how to make those adjustments. Now there are plenty musicians who intuitively have wonderful ears and they just do it naturally. I don’t believe I was one of those because I would play my chords out of tune consistently. And I had to be told by either, a teacher here or there or a masterclass clinician, Hey, bring that E lower when you play it against the open G, no one bothered to tell me why that was and I didn’t care to find out. I just moved it lower. And I would just hope that my finger landed in the right place. The second time I had to play it, but there’s just a few little tweaks we have to make. And if you can understand the natural harmonic series it’s pretty much right in front of you. And that harmonic series, we introduced in Cellomind and it we go to the eighth partial and in the presentation, we show examples of how certain natural harmonics depending on what interval they represent, are out of tune, compared to equal temperment. So that is supposed to be a lightbulb moment. And we’re hoping that many people who read the book or are interested in intonation realizes at that precise time. Oh, that’s what my teacher meant 50 years ago.

Noa
Is there a way to because I imagine that knowing how to tune a chord is a common problem. And it’s easiest, obviously, to go to the piano, but you explain why that’s a problem. So if we all have our own instrument to deal with, and even going to the phone, I imagine could be a problem too for a tuner. Like is there like a set of principles or rules as far as like how to know when that E should have been lower? In that chord that you’re referring to a moment ago, like are there like, like, what do we use to to figure out what to tune ourselves to?

Minna
I think the best way start answering that question is to realize that most of the time when you are playing a chord, you have the tonic, major or minor third, and then the fifth, right. So let’s keep it simple for both our sakes. Okay, so it’s so important to realize that when you play a major third interval that is a major third. And you’re playing against another note, it could be a double stop or within the chord, that major third is always going to be flat, like you have to move your finger lower and it’s completely the opposite of what you would think. Because we think melodically and a major third, in melodic intonation, we always want to make big major intervals or big minor intervals or small. Harmonic intonation, in the most simplest terms is opposite of that, and you just have to buy into it. So a major interval no matter what it is, a major third, major sixth, a major seven is all going to be smaller compared to the measurement that we assigned an equal temperament. So I’m going to just throw it out there, like Noa, do you know how many half steps, don’t get nervous, I’m going to make it a small interval. Do you know how many has this are in a major third?

Noa
Oh I’ll have to count. This may take longer than I want it to, but let’s see here.

Noa
I lost count already. Hold on. (a lot of silence…)

Minna
A major third… How many half steps

Noa
I counted too high… (more silence) Four ?

Minna
Oh my gosh, you’re right. Yes. Very good. So, in equal temperament, if you have four half steps, and each half step is 100 cents, then an equal temperament, a major third is 400 cents. And let’s say you, you know how to an acoustic ruler or like you’re on your app. That’s how the needle tells you if you’re bang on or a little bit higher, a little bit low. But in harmonic intonation, a major third is 386 cents. And we know this because there’s a major third natural harmonic that exists on our strings. I was like 40 years old when I figured that out. I didn’t figure it out, I was told. So it’s 14 cents smaller, than equal temperament. So that gives us an idea. 14 cents, how small that increment is. And that’s where like a little bit of math can kind of help guide your ears to go higher or lower. So these are things that you just have to know. And in Cellomind, we begin the whole idea of harmonic intonation by finding those natural harmonics and doing a little bit of math to show you how much smaller or how much bigger they are compared to equal temperament. So that’s a huge one a major third anytime you have a major third in a chord you’re going to play it 14 cents flat. And then the other big one, which is really fun to show is the seventh partial, which is two octaves and a minor seventh above your fundamental note. Aside from that, let’s just look at that minor seven. I’m not going to ask you how many half steps… I’m going to do you a solid, and a minor seventh is nine half steps. So minor seventh, in in equal temperament is 1000 cents. But in harmonic intonation it’s minus 31 cents. So you subtract, you know, 1000 minus 31, whatever that number is. That’s how much smaller your minor sevenths are. So that means whenever you have a minor seventh in a chord, you’re going to have to play that super, super flat. And that’s where it really will sound much more harmonious. If you play a minor seventh, wider, like let’s say you play it in equal temperament. You will hear stopping vibrations against a pair of notes, which will cause you some distress like, Wow, my instrument is telling me I’m not in tune. So those who are very comfortable with their inner hearing will automatically make that note flat. For those of us who need to have more theory explained to us or who are just, perhaps if you have absolute pitch, oh, this is really hard if you have absolute pitch because there’s something internally that’s not allowing you to make these adjustments. So it’s much more challenging, but even those with absolute pitch, this book, knowing how much lower to play a note can really loosen any anxiety or frustrations you have, because it’s theoretical. What we’re doing is finding common frequencies of multiple pitches. And and this is how it’s done. So, in order to make your way through this book, you really just need to know how to multiply, divide, add, and subtract. And we all have calculators handy, so it won’t take too much time.

Noa
Right? And it’s mostly just to kind of make sure we understand the theoretical reason why our teachers have told us where to place notes over the years anyway, so that they’re not just making it up like there’s a reason for why their ears tell them. This is where it goes.

Minna
Yes, and but so many. So many great musicians, our friends, you know, Grammy Award winners. They may not really understand why. But they just do it. And if you’re naturally gifted like that, is very hard to teach something like that. You know, you would have to comb through the music with the student, or the student might just figure out, oh, whenever I play an E against an open G, I’ve got to play that lower. But what if you are playing an E flat against the open C? Then what do you do? Well, you have to play that E flat, a little bit higher. So the book is very streamlined. But there’s an appendix in the back. And for those who really enjoyed math and science, they can really buy into it. And there are some musicians who love this kind of stuff. And it’s really rewarding to to meet those people because I fall into the more the category you do, whereas just… Just tell me the least amount of things I need to know before I can go on.

Noa
Because I tried to go through this talk that I watched you give and try to summarize for myself, what’s like the CliffsNotes version of what I need to know. So that I can hang on to it and actually remember it. So you’ll have to tell me if I did this right or not, -crosstalk-. But if I’m understanding Pythagorean or melodic intonation means we can basically play higher leading tones, higher major thirds, maybe higher major six, but lower minor thirds, lower minor six, and now you’re adding sevenths into it. And I don’t know what to do with that. But it’s… what was it with sevenths higher? or lower?

Minna
Well, minor seventh, in just, they’re super, super small. And major seventh in just is also small but not as small as the minor seventh. So luckily, we lay out like a charts about all the intervals in like a C major scale. What the just math is compared to equal temperament so you can see how small or how large, certain intervals are. We put that right in the beginning so you don’t have to search for whatever interval you’re looking for.

Minna
But that’s absolutely correct.

Minna
And the presentation that you saw, called “you aren’t really listening, revealing the mystery behind great intonation.” That’s as concise as I could make it. And while I was putting together that presentation, while it took me a couple months, I think over 80 hours of crying to the tech people, trying to figure out why a video wouldn’t load, double checking the math. That’s probably as streamlined as I could funnel all the information into and it still is a lot to digest, even for professional musicians. I have friends who’ve seen me presented five, six times, and catching on to new information here and there. And that’s wonderful. The good news is that if you know, if you’ve committed to memory like one of the harmonic intervals, you can find this melodic component via the syntonic comma, you just add 22 cents up or down. So that gives me a lot of relief that, like I said, most musicians, we are intuitively always hearing more melodically which does pose a problem when we hit that double stop.

Noa
which fortunately doesn’t happen super often.

Minna
Unless you’re playing Bach,

Noa
right. And that’s where…

Minna
And we need to study it right or go through it with your teacher. Yeah,

Noa
and that I know did actually solve it because it seemed like what I needed to do is start with a melodic line that I needed to project and then tune the chords around that.

Minna
Yes, that’s very true

Noa
But basically, I couldn’t handle that, like my brain just couldn’t or didn’t want to deal with it.

Minna
So, when you have a series of chords that have a voice that is melodic. So many decisions have to be made, which voice are you going to be adjusting? The golden rule is if you have any note that’s an open string in that four note, three note chord. That’s your main note. That’s the note you will not be adjusting and you’re adjusting everything outside of that. But it’s also very possible to let’s say you have a low A on chord number one. And then that is a fingered A, okay? So it’s not an open string. And then chord number four has an open A. It’s very possible that your ‘A’s in chord number one is different from the A in chord number four. But the glorious thing is, when played in tune, no one’s gonna notice, your teacher will not be like, hold up Noa, your ‘A’s aren’t matching. Because each note in the melodic line functions differently. So once students realize, yes, there’s more than one place for a certain note. Hopefully that gives them some excitement to do some exploration and experimenting. At least that’s the in the case when I teach. Maybe they’re faking it, I’m not sure. But there are certainly many, many different places where a note can exist.

Noa
I want to make sure that I think I heard you say something and I want to make sure I heard what I thought I heard, if I remember correctly, and I just might be making this up, but I swear, there were times where I play a chord. And I would actually adjust the open string like I put my finger back there and kind of make the open string a little sharper to make it match other notes. Which just, I don’t know if my teacher told me to do that, or I just got frustrated, but is that what you’re referring to, like? An open string can be in multiple locations at the same time.

Minna
I just see an open string is like you’re you’re on level 10. That’s very creative. This is really cool. Sometimes, for cellos, probably any string instrument, you are needing to adjust the open string if it’s like an enharmonic note. So for the cello let’s say you have a low B sharp. That’s supposed to be sitting underneath your open C string Well, we don’t have any notes lower than a C. But when we play a B sharp and you’re playing it, melodically a B sharp is actually supposed to be higher in pitch than your open C, but of course, on the piano, a B sharp and a C is the same white key, right? But for string playing, there’s a difference of 24 cents between each enharmonic note. So for us, we would then put our finger right against the nut to sharpen that open string to make it sound as a B sharp. And you can of course, do that on any string, especially Noa, if you realized, okay, I have to make this open string a little bit higher. I think that’s genius. Never had someone do that. But that’s great. I mean, that that’s the thing. You decide which notes you need to adjust. But typically, we don’t adjust the open string so much because we might run out of fingers to play.

Noa
I think it really was sort of moot anyway, because I play plenty of other notes out of tune. So it’s like nobody was doing it anyway. So actually, that I should have asked what this whole comma thing is, but I feel like that’s gonna make my brain explode. So I might just kind of skip over that and link to somewhere else. So it seems like there are considerations then, when it comes to playing with piano because obviously, the piano is not going to be able to adjust anything. So I’m thinking in terms of piano trios at the moment, maybe we will then go to quartets but so if you’re just playing with piano, just you and a pianist, what are the considerations that we need to make in that scenario?

Minna
That’s a really good question. And that’s the number one question people jump to within two minutes of my presentation. What about piano? What about playing with piano. First of all, I find playing anything with piano very, very difficult because as I mentioned first thing in this podcast. Rarely is a piano ever in tune. But here’s the gist. What do you need to take into consideration? You need to take into consideration the key of the piece. Now for most string players, most of our repertoire, sonatas, unaccompanied music are all in our open string keys or relative major/minors of those keys. So the composers are really helping us out here. If you’re playing unison a lot in some of these like romantic sonatas or classical sonatas. If you are playing in unison with the like your lowest open string for cellos that would be like the open C string. The Sonata that comes to mind is the Prokofiev Sonata in C major We play many, many times with a piano in a melodic situation where the piano is also playing in the same register that we are. I can’t tune my C anywhere, but where the piano C is, in my experience, there are five different piano Cs. So usually what I do is have the pianos play the chord, the biggest C major chord, and I actually find the medium of where I think that C is going to be, as to not offend any ears. But in general, you can tune your outer strings with the piano. So for us, the A and then the low C. And then you can tune your middle strings via the harmonics. Now if you’re tuning carefully with the harmonics, Your strings are going to be slightly flat if you’re watching your tuner app, because a tuner app is tuning to equal temperament most of the time, some apps, you know, get fancy, and they say Pythagorean or meantone or Pythagorean just, I have never ever been in tune with my tuner when I’m tuning with my ears with frequencies. It’s pretty close, but it’s never exact. So when you’re not playing in unison with the piano, oftentimes, the audience ears are kind of zoned into the violin sound or the string sound or the other instrument because we sustain longer than the piano does. So the piano has that attack, decay, symptom and from what I know they obsess about that as much as we do about intonation, so all is fair in the world. But because they have this decay of sound, and we can sustain that the ears of the audience are tracking our line. So you can continue to play very melodically. Even if the piano is outlining a chord, or maybe even playing a sub melody. It will still sound very expressive and gorgeous. One of my very close friends who’s a concert artist, mentioned to me that his worst fear was sounding flat. And I thought that was a weird thing to say. But then I then I realized it’s true. Because I played a concerto and I sounded flat, pretty much the whole way through because I was tuning to myself. I was wasn’t tuning with the orchestra. Granted, the orchestra’s is just pitch was kind of all over the place too. But it was a good learning experience. It is true. Like, we tend to hear things on the brighter side. So a little bit sharp. So let’s say your chord is a little bit out of tune, and you didn’t really lower that major third, it would probably wouldn’t bother us as much as if you played very flat. Like if you play that F sharp way too low, like the dog who start barking in the neighborhood. But kind of a little sharp, no one’s really going to mind. So the theory behind that is, most of the time you can get away with playing Pythagorean.

Noa
I feel like I’ve heard somewhere that essentially if in doubt, err on the side of being sharp. Does that fair? Is that like a thing?

Minna
I would totally put that on a plaque. Yeah, I mean, I never heard of that, but from my experience, absolutely. err on the side of being sharp

Noa
So that’s considerations on play with piano. So tuning your outside strings to the piano, and I assume that’s just to make sure that your open strings are close to what the piano is going to be doing. And then, make adjustments everywhere in between those extremes.

Minna
Yeah. So actually I should mention so when you are playing melodically when you play for the first time in rehearsal with the piano, be mindful that your sharps aren’t too high, you might have to temper it slightly on the lower side, so that your half steps or whole steps are more in line with the piano like in that realm of acceptable. So the same thing happens with flats instead of playing too low. Go a little bit higher on the flat so that you are trying to play more or less equal temperament but still coloring it with some tasteful nuances. No one’s really going to walk out, you know that’s rarely going to happen. In the presentation we have three different clips of one passage being played just harmonically, melodically, and equal tempered.

Noa
Here’s the audio of that clip that Minna was talking about. Pay attention to which of these three versions you prefer, as Mina will tell you which ones most people preferred when they did a poll.

Noa
Pause the audio for a second and take a moment to reflect: Which one did you like the most? Number one, number two or number three? Okay, just so you know, number one was just or harmonic. Number two was Pythagorean or melodic. And number three was equal temperament. Now let’s hear Minna describe what the results of the poll were.

Minna
And you know what most people preferred just as much as they did equal temperament, so they weren’t told which one was which. They just had to raise their hand when I call out which episode they liked. And then you have some who really liked the melodic, but it was interesting. I thought most people would pick melodic, but most people actually picked just and equal tempered. So it’s very subjective.

Noa
So that’s true that it’s not like that’s something that people say when they have difficulty playing in tune like, there is some degree of subjectivity in our preference for what what’s in tune or not.

Minna
Oh yes, I think so. And, you know, playing with piano is a minefield if the piano is not tuned well, I recently played a concert in Panama, which was such an amazing experience right before COVID. And all was well with the piano, we just hit some spots, everything was good. I happen to be playing the Prokofiev then the pianist goes down to like the one of the the lowest octave and was hitting an F natural. And there is this bizarre sound that came from the piano, I think the string was broken. So I look at her and we just start laughing, because there’s nothing you can do. And she kept having to hit this low note. And then finally, she like just played it an octave higher.

Minna
It was a nice surprise.

Noa
So now I want to add another instrument into this mix. So if you doing a piano trio. The example that came to mind this morning. And this might not be the best one was the opening of the E minor Shostakovich, piano trio with the artificial harmonics. And then, you know, the violin comes in and then the piano comes in. And again, I might be misremembering, because it’s like 25, 30 years ago, but I feel like I was trying to match the cellist, whatever the notes were, and maybe what I don’t remember how artificial harmonics work, maybe you can adjust the intonation on the artificial harmonics based on whatever finger stopping it. But in any case, I was trying to match that. Then the piano comes in after the violin comes in. And so then I’m like, stuck between, like, Who am I supposed to match? Like, what do I do because I don’t know. I mean, do you have any…

Minna
remind me the violin is doesn’t play any artificial. Just regular notes. Yeah, yeah. So the cellist should be tuning, in that specific case. Equal temperment to the piano, so both string players should be using as much equal temperment memory. Where those notes are. Now it’s profoundly difficult on the cello, just one of those reoccurring nightmares. But here’s the thing, some natural harmonics that exists within that melody, cellos tend to, if there’s a natural harmonic cellists would tend to opt for the natural harmonic rather than making it artificial. They rather play with one finger than two. But those natural harmonics are grossly out of tune compared to equal tempered. So now that I understand the harmonic series, I would tell every cellist out there, keep playing artificial at all times, even when there’s an opportunity to not. Because you’re either going to be ever so sharp to the piano or you will be grossly flat to the piano. So now as a cellist when a violinist is comes in, you bow to the violinist, you know, you, you go to the violin frequency, especially because you’re not playing harmonics, so your timbre is stronger. And usually when I play with unison passage with violin, I like to have my cello sound either supports the violin sound, or I go right into your sound. So I think it depends on the group itself. If you have a very strong violinist who has an incredible sense of inner hearing, always follow that person. That’s probably way way better than touting your theory knowledge and why this note has to be flat because let me tell you, I think have very close friends just told me to shut up. They want that F sharp higher. So then I was like, okay,

Noa
So if we’re talking with quartets, then it was like arguing, like, Who’s gonna tune to whom? And like, when? Yeah, so it sounds like that’s one of the principles of getting along and the quartet when it comes to intonation, or there’s some other things that are useful for us as far as managing disagreements and knowing…

Minna
Well, I would say to everyone out there give more respect to the violist. Because I find that violas are superior superior musicians just because of the instrument and the role the instrument plays. I would say yes, follow the cello’s pitch. If the cello is the lowest line or has an open string anyone in that string quartet whoever is holding an open string. Everyone else has to adjust around it. But I find that in, in my experience, if there’s often disagreements in terms of pitch between the cello and viola, because their their voices are so closely linked, or either always playing in thirds, unisons, or fifths with viola, that has to be worked out first. And then the higher register you go, like you said, the sharper the instrument can play. So, if violin one has the melody, everyone is kind of listening to violin one’s progression, you know the melodic line, but it’s always helpful to just keep in mind, even if the cellist is holding an open C string, and the violin has a melodic line, that violin’s C natural does not at all have to match mine, because I’m a drone, but everyone’s listening to them, the melodic line. So wherever the violinist, wants to play that C, as long as it’s fleeting, is going to be okay. But at the violin then sits on a C chord, of course, violin has to adjust to the cello. But in my experience, the violist holds the key to a lot of success. Both personally in the quartet as well as technically. I was also says the second violin also, you know, is, is the, what is it called, like the, the invisible hero of the string quartet.

Noa
So, this seems to speak to the importance of because I think we’ve been talking a lot about what adjustments to make and when, which speaks to it being a really valuable skill, this ability to adjust really quickly, and I have no idea if Heifetz actually ever said this or not, because nowadays with the internet and quote It’s impossible to know almost who said what, but I feel like I read somewhere. He said something like he doesn’t actually play more accurately than anybody. He just adjusts faster. And so maybe there’s something to it. But in any case, I wondered. I think there are exercises in the book as far as how to get better at playing in tune, technically speaking, so not just the understanding of it, but like in terms of the mechanics of it, I wonder if you could share some, some exercises or suggestions on ways to actually practice not so much tuning chords, but playing things more in tune by adjusting faster as needed? Or if there may be other technical things that are relevant to playing in tune?

Minna
Well, I think a lot of it comes down to the training of the inner ear, because that’s really where the mind helps the body adjust. So fine tuning your inner hearing is kind of a phenomenon is hard to like write a paper about. But for me, what’s helped me increase the accuracy for some of my students who have absolute pitch. It was strange because they played the least in tune. And they couldn’t really explain to me what was going on in their mind. Other than I can only imagine is like a needle, you know, or some kind of radar. But for me, the secret to like building the inner ear is being open to vibrations, and frequencies. And just taking your eyes off any machine like, like a tuning app. I was a strong advocator of you know, using drones. And I still like to accept it’s very taxing on the ear. And at some point you zone out like you don’t hear that drone anymore. It’s like a continuous metronome. At some point your body can’t take it. So, for me like what was really was a turning point was when I was told about differential tones, Tartini tones and trying to hear that third differential tone emanating from my instrument. When I played any kind of double stop. It was, it was a revelation. And then I would spend hours finding double stops and chords and like the Dvorak Concerto or the Elgar Concerto trying to hear a third differential tone to me that third tone sounds like a helicopter. Other people say that that tone sounds like a crystal… Like when you lick your finger and move it around a crystal wineglass. That kind of sound is very pure is very circular, is very ringing. But depending on what register you’re in, it can either be a very thunderous kind of vibration or a very clear sounding bell. Once you’re keen on identifying that vibration or that frequency, then you’re always searching for that. So Noa, you asked me in an email about strings, like certain strings might encourage overtones, but I found that great instruments, the older instruments. They produce so many more overtones than the newer ones. Obviously, they’ve been around longer. And that can be so helpful for anybody to experience. Oh my gosh, I could hear it now because some instruments just don’t produce many vibrant overtones. But when you can lock into it when you understand what that sound is, I find you’re always looking for it. And for me, I don’t have absolute pitch. But in my lessons, I know when a certain note is out of tune because it doesn’t ring. It doesn’t spin it doesn’t have that sound and most of my students, they are unaware of it until they really start listening and the first thing I do is have them play a unison. I can’t tell you how many people don’t do not know how to play a unison. People. It’s supposed to sound like one note.

Minna
So I heard this thing about Heifetz that his octaves were so in tune, no one knew, or no one believed he was playing octaves. So he purposely played them out of tune so you could tell that’s, that’s a rumor I heard and I just thought that was brilliant. Yes, so when you play a unison you should hear just one vibration, not 20. And a good thing to mention to students is, the closer you are to the bullseye, the worse it sounds. So it really encourages them to like really find that nasty dissonance and then then the rainbow comes, you know, so like really encourage people to play out of tune is really fascinating. I think too many times young people are so scared of playing a tune. They don’t want to take that tape off their fingerboard. But I think if you are comfortable playing out of tune, then you’re really sensitive to those vibrations. You know, I think it’s that fine line, you know, love and hate in tune, not in tune.

Noa
I wonder if you can give me an example of, one I want to hear more about what the tartini tones or whatever this is, but also maybe like an example of specific notes so that people could listen for like try to actually see search for what notes should they be listening for that’s kind of hidden underneath, the underneath the mix that they’re playing.

Minna
Okay, so tartini tones is one of my all time favorites. And I, did a little write up in Strad magazine about it. I don’t think many people saw it because I think I over shared the information. But tartini tones is that third differential that emanates, so it’s chapter 26. It’s one of my favorite chapters. Noa, because it literally is two pages. For me, tartini tones is a great way to play double stops in tune or to train your ear. So I’m just going to read you the introduction from cello mind. “The famous composer and music scientists, Giuseppe Tartini, discovered in 1714, that when two notes are played simultaneously, either by a single instrument or by two instruments, a third lower note can be heard. These notes are either called tartini tones or difference tones. Their frequency, a difference tone, is the difference between the fundamental frequencies or the two notes that are actually being played.” So you’re taking, if we go back to math, you know, and one tone is 884 cents and the other tone is 368 cents. You know, what is the common denominator, the lowest denominator? Whatever that is, is that third pitch that comes out. So you asked like, what are some of the tube sounds two pitches you can play. So there’s a chart that I have tried to memorize, but it leaves me overnight, every single time. But here’s the thing. So let’s say you’re playing a perfect fifth. So you’re playing your strings like you’re tuning your strings. So you have the E and the A for violinists, that is a perfect fifth. When your E and A are perfectly in tune together, you are going to hear one octave below the lowest note. So you’re going to hear an A one octave lower. So basically the cello A. So if you’re A is 440, you’re going to hear a 220 A emanating from your violin. So for a major third, you’re going to hear two octaves below the low notes. And sometimes it’s not that simple. Sometimes, you’re going to hear notes that are like one octave plus one fifth below the high note. This is why I can’t memorize it. So if you have a minor third interval, so let’s say we’re going to play give me a minor third, A and C natural. Okay, let’s say A is on the bottom and C natural is on top. So if you’re playing that minor third completely in tune, and you’re going to have to use the harmonic intonation, you’re going to hear one octave plus one fifth below the high note. So what’s one octave below C, C what’s a fifth below C, you’re going to hear F, and that’s as complicated as it goes. Another one is a minor sixth, you would hear in the tartini tone, a major six below the low note. And in Cellomind, we give you mock examples. So you play the chord with a finger and we tell you and then underneath on a separate staffline, we tell you the tartini pitch that’s supposed to emanate in exactly that right octave. So it is, I mean, I’m not a nerd in any sense, but this somehow kept me completely focused. Because I knew if I play my double stops in tune, it was going to be a good lesson. There’s nothing worse than your teacher just yelling at you that you’re out of tune, and you have no idea why.

Minna
But this is a great way to develop your vibrations and your frequencies in your inner ear. And it really covers all the double stops, we cover major thirds, through the perfect fourth and fifths, the major six, minor six, and the 10th. So that’s most of the double stops you would ever find in a chord structure. It can really elevate quite quickly, your desire to want to play more in tune harmonically.

Noa
So this is basically a way to hone your ears to listen for something that clues you into whether you’re in tune or not. When you’re playing double stops. It sounds like?

Minna
Yes. And also repetitive chords, you know, so you can break up any chord into like a perfect fifth. And from there, try to find that tartini tone, or maybe that major sixth and try to find that tartini tone. And then from there you build. But it’s, I find that it’s an excellent way to really build up your confidence as well.

Noa
These might seem like random questions, but I’m just out of curiosity. I don’t know if they’re practical, or not.

Minna
I didn’t study before the podcast, and I just, I was just praying that I hope it just becomes a natural thing.

Noa
But I’m curious about the thing that I see bass players and cellists do and then the thing that violinists do in terms of tuning their instrument and to be honest, I feel bad. I don’t remember how violists tune. I said, I haven’t…

Minna
They tune the same way as you. Well, they probably tune the same way as us.

Noa
Okay, that’s what I’m thinking of

Minna
They get their A and then they go down.

Noa
Right. So first off, I see cellists do the harmonics thing, like they tune their strings using harmonics and bassits do, I don’t know what violists do… But what’s like, what’s that about? Because I remember you saying something about how harmonics are always flat. So like, does that mean? I don’t know. Is that a thing that cellists are supposed to do, to like tune with harmonics, or?

Minna
That’s a really good question. And this is something I, when I talk about intonation to like high school students who are really keen. If you continue to use your app to tune you’re actually tuning equal tempered. And that is not going to give you all the overtones because you’re already out of tune. So our strings are in fifths. So when we tune our strings, we need to tune it a little bit differently than how your app or how the piano was tuned. We rely on the just intonation to tune our fifths. I’m just going to tell you again, there are seven half steps in a perfect fifth. Okay, don’t feel bad my colleague is a prodigy violinist and I asked him mistakenly in a room full of young musicians in Vancouver. Hey, how many half steps in a perfect fifth? And he was surprised I called on him because nobody else raised their hand. He might have been sleeping, but he woke up and he goes, I don’t know 10. And I chuckled because it was cute, but also No, not at all. And then he said, Well, you know, Mina, I just never thought of it before. You can’t just ask me that. There are seven half steps in a perfect fifth. So in equal temperament, that would be 700 cents, but in both harmonic and melodic tuning, both, it’s 702 cents, two cents bigger. So that means when you’re tuning with your app, it’s not adjusting for those two cents. If I wanted to tune harmonically or melodically my perfect fifths are going to be fat each time I go down the string. So my D string is going to be two cents bigger, bigger meaning flatter, wider. And then by the time I get to my C string, I am suddenly six cents flat, two cents each string, D, G, C, six cents flat is big enough to for professional musicians to definitely notice, Hey, you’re flat, but this is also the reason why your conductor or your teacher might have told you bring your C string up. Because if you’re tuning from harmonic to harmonic, which cellist that’s what we’re doing and we’re doing that only because we can easily access that big tuner behind our heads. So violas they tune like violinists. They use their big tuner. Double Bass, cellos we rely on our fine tuners more or less, because the adjustments are not quite as big. But let me tell you there are so many times that the string tension pulls this way, that way, it’s just easier to use the peg. So you can really, you guys lock in immediately, right? When you hear that harmony, you’re like, oh, there it is. You guys don’t even think you just wahwahwah and then it gets there for cellists like, it takes a lot of precision to make sure that you’re hearing the same exact frequency of that harmonic. So we’re playing the harmonic of the string, which will have no deviation of cents. And then when we play the A on the D string, D, E, F, G, A, that’s a perfect fifth. Our perfect fifth naturally is going to be two cents higher, I should say two cents wider. So we are actually tuning ever so flat compared to that tuning app each time we go down the string, and then at some point, we’re all trained to bring that C up a little bit. So, I would say throughout my entire career, excluding the last five years, I have not been tuning my strings at all in just intonation or melodic intonation. I’ve been a little bit out of tune. I’ve been tuning equal tempered, and when you tune equal tempered, you’re not getting those overtones as readily or as clearly or brightly as you would if you actually started tuning two cents wider. So that’s the whole kind of dilemma in a nutshell, like how do you tune your strings, but you know, I I have clear tune. I don’t know if we can say these things on your podcast, but it will show you that needle being slightly to the left of that dark line, and I tried it out and it actually did a really good job. But a lot of us just hear sharp so maybe that’s where it came from, you know, you’re, you’re just tuning to the piano so we we tend to hear sharp.

Noa
The other tuning question was, yeah, why do violinists tune from the A down the lower strings and then go to the E and add E and relative to the A instead of just starting with the top string and going all the way down

Minna
if you started with the top string, your A would be flat to our’s. So all the string instruments should should start with A that’s the highest string of the cello. But bass, yeah, the E is on the bottom. And a bass sometimes they are given a note separately, at least in some orchestras. They don’t really tune with the strings they they get their own pitch. But I think that’s because the cello has our top string is an A . So then when you tune your E your E is 702 cents higher so you’re, you’re a little bit brighter on that E

Noa
At this point in the interview Minna’s dog started barking and mine started whining as well. So we took it as a sign that we had probably talked long enough about intonation for one day. I hope you found this enlightening, helpful, and maybe even a tiny bit entertaining. For more information about the CelloMind book, and links to other things that came up in conversation, visit bulletproofmusician.com/blog.

Notes

  • We allude to the book CelloMind a couple times. You can learn more about it here: CelloMind (6:15)
  • Minna refers to the “syntonic comma” (28:03) which we don’t explore at all, but if you’re wondering what that’s all about, you can learn more about in this video: Can You Tell These Notes Apart? Or in this reddit thread: What, in plain English, is a syntonic comma?
  • And if you’re wondering what the Tartini tone (49:40) is or sounds like, check out Minna’s Strad article (includes video examples): Minna Rose Chung on double stopping and Tartini tones Or this video: How To Play Notes That Aren’t There
  • Here’s a quick TLDR summary/overview/tips on intonation by Kurt Sassmannshaus: Intonation @ViolinMasterClass

Learn more about CelloMind (or ViolinMind)

You can take a look at what’s inside CelloMind, and pick up a copy here: CelloMind

And if you’re a violinist, wishing there were such a thing specifically for you, there is! Check it out here: ViolinMind

And if you’re a violist…umm…I believe there is a special viola-centric version coming out for you too, but it hasn’t been released quite yet. =(

Connect with Minna

You can learn more about or contact Minna here: Minna Rose Chung @University of Manitoba

About Noa Kageyama, Ph.D.

Performance psychologist and Juilliard alumnus & faculty member Noa Kageyama teaches musicians how to beat performance anxiety and play their best under pressure through live classes, coachings, and an online home-study course. Based in NYC, he is married to a terrific pianist, has two hilarious kids, and is a wee bit obsessed with technology and all things Apple.

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