My blog stats have made it crystal clear that very few of you want to read about tuning systems. However, a vocal minority of you do love reading about them, and I definitely enjoy writing about them. So, let’s dig in and see how much Western harmony we can derive from the natural overtone series!
Imagine that you are building a guitar. You start by putting on a string tuned to a note called C3. When you pluck your C string, it vibrates at a frequency of 1 Hz, meaning that it vibrates back and forth once per second. (In real life, a guitar string tuned to C3 would vibrate back and forth at 130.8128 Hz, so multiply all the numbers in this post by 130.8128 if you want to think in terms of actual frequencies.)
The string doesn’t just vibrate along its entire length. Subsections of the string also vibrate independently. These subsection vibrations are called harmonics. Each harmonic produces its own distinct pitch. When you pluck a string, you are hearing a lot of pitches from the complex blend of all of the harmonics. This means that every note you play on the guitar is really a chord!
Here is an animation of the first four harmonics of a string.
The first harmonic of a guitar string is its vibration along its entire length (also known as the string’s fundamental.) The second harmonic is the string’s vibration in halves. Each half vibrates twice as fast as the whole string, so it produces a higher pitch than the first harmonic does. Specifically, the pitch of the second harmonic is one octave higher than the pitch of the first harmonic. If the string is tuned to C3, then its second harmonic will produce C4. But they are totally different pitches! Why on earth are they both named C?
To understand why we consider octaves to be equivalent, imagine adding a second string to the guitar that’s half as long as the first one. This new string will play C4. We’ll call it the high C string, and we’ll call our original string the low C string. If you strum both C strings, their harmonics will overlap extensively.
- The high C string’s first harmonic will be the same as the low C string’s second harmonic.
- The high C string’s second harmonic will be the same as the low C string’s fourth harmonic.
- The high C string’s third harmonic will be the same as the low C string’s sixth harmonic.
Your ear will have no trouble detecting the overlap between the two strings’ harmonic series. This spectacular agreement is the reason why Western music theory considers C3 and C4 to be “the same” note (technically, the same pitch class.) The important concept here is that because octaves are equivalent, you can double or halve the fundamental frequency of any note and it will remain the “same”. This will be important later.
Anyway, let’s go back to the harmonic series of the original C string.
- The string’s third harmonic is its vibration in thirds. Each third vibrates three times as fast as the whole string, producing a new note called G.
- The string’s fourth harmonic is the vibration in quarters. Each quarter vibrates four times as fast as the whole string, producing yet another C. (Even-numbered harmonics are boring because they always produce higher-octave copies of the pitches produced by odd-numbered harmonics.)
- The string’s fifth harmonic is its vibration in fifths. Each fifth vibrates five times as fast as the whole string, producing a new note called E.
Here’s a schematic diagram of the notes you get from the first five harmonics of the C string. The third harmonic, G, has a frequency of 3 Hz. The fifth harmonic, E, has a frequency of 5 Hz.
The next thing we’re going to do is to add two new strings to the guitar. One will be tuned to G, and the other will be tuned to E. It will sound better if we move the G down an octave to 3/2 Hz, and the E down two octaves to 5/4 Hz.
Strumming the C, G and E strings makes a lovely chord called C major. Why does it sound so good? For one thing, the G string’s second, fourth, sixth and eighth harmonics produce the same pitches as the C string’s third, sixth, ninth and twelfth harmonics.
For another thing, the E string’s fourth, eighth, twelfth and sixteenth harmonics produce the same pitches as the C string’s fifth, tenth, fifteenth and twentieth harmonics.
And for yet another thing, the G string’s fifth, tenth, fifteenth and twentieth harmonics produce the same pitches as the E string’s sixth, twelfth, eighteenth and twenty-fourth harmonics.
That is extraordinary alignment! Your ear hears all these harmonics dovetailing together and thinks, this can’t be a coincidence, it must mean something.
We can find more related notes by examining the first five harmonics of the G and E strings.
The G string’s third harmonic produces the note D, and its fifth harmonic produces the note B. The E string’s third harmonic produces that same B, and its fifth harmonic produces G-sharp. Let’s add three more strings to the guitar, tuned to D, B, and G-sharp.
The D string’s third harmonic produces A, and its fifth harmonic produces F-sharp. The B string’s third harmonic produces F-sharp, and its fifth harmonic produces D-sharp. The G-sharp string’s third harmonic produces D-sharp, and its fifth harmonic produces, uh, B-sharp. That last note is an oddball. Pianos don’t have B-sharp! This note is only 3/64 Hz flatter than C. Weird.
Anyway, now let’s use a new strategy to find more notes. We will move up the family tree rather than down it. In other words, rather than looking at the harmonics of C, we are going to identify notes whose own harmonic series include C. If we add an F string, its third harmonic will produce C. If we add an A-flat string, its fifth harmonic will produce C.
The A-flat and F strings also have some new notes in their harmonic series, E-flat and A. Notice that E-flat is not quite the same pitch as D-sharp. They are not interchangeable with each other!
E-flat is interesting for another reason too. When you combine it with C and G, you get a C minor chord. E-flat’s fifth harmonic is the same as C’s sixth harmonic and G’s fourth harmonic. (C and E-flat have the same relationship as E and G, pictured above.) That is not as dense an overlap as C major, but it’s still a lot of alignment. Again, Richard Dreyfuss says it best:
Let’s keep working backwards up the family tree and figure out the harmonic “parents” of A-flat and F. That gives us three new strings tuned to F-flat, D-flat, and B-flat.
F-flat is another weird and unfamiliar note. It’s not quite the same pitch as E. You can’t play F-flat on the piano.
Hear all the intervals in the diagrams above:
It’s possible to keep expanding this harmonic family tree infinitely in all directions. Here are C’s eighteen closest relatives.
Here’s a visually simplified version:
This diagram is called a tonnetz, and it is rich in musical structure. Every group of three contiguous notes is a chord! The triangles that point upwards are major triads, and the triangles that point downwards are minor triads. You can make great chord progressions just by moving around the tonnetz via adjoining triangles.
If you examine the family tree diagram closely, you will notice that it has some contradictions in it.
- There is a “grandchild” D at 9/8 Hz, but also an “uncle” D at 10/9 Hz.
- There is a “sibling” A at 5/3 Hz, but also a “great-grandchild” A at 27/16 Hz.
- There is a “nephew” B-flat at 9/5 Hz, but also a “grandfather” B-flat at 16/9 Hz.
What do we do? We could decide to run with the grandchild D, but then it will clash with the uncle’s closest relatives. If we use the uncle D, then it will clash with the grandchild’s closest relatives.
It gets worse. If we expand the family tree further, it produces more conflicts.
One solution to the problem of conflicting notes would be to use fewer harmonics. Throughout this post, I have been describing a tuning system called five-limit just intonation, because it uses harmonics one through five. Medieval Europeans used a simpler system called three-limit just intonation, which only used harmonics one through three. In three-limit, you start with some note, then generate a new note from its third harmonic, then generate a new note from that note’s third harmonic, and so on. Here’s a diagram of three-limit just intonation, arranged in a handy configuration called the circle of fifths. (Except that in just intonation, it isn’t a circle, it’s a spiral.)
Three-limit just intonation has no conflicts. There is only one D, there is only one A, there is only one B-flat, and so on. Every note has one and only one tuning. C-sharp and D-flat are still different notes, but that’s okay, you can work around that. So, great! Why bother with five-limit at all? The problem is that three-limit produces terrible-sounding thirds.
Remember that your ear wants notes to line up with each others’ harmonic series. Your ear likes hearing the E at 5/4 Hz together with the C at 1 Hz because E’s fourth harmonic is the same as C’s fifth harmonic. However, in three-limit, E is tuned to 81/64 Hz. That note’s harmonics won’t line up with the low harmonics of C at all; it’s way too sharp. E-flat is similarly problematic. Your ear likes the five-limit E-flat at 6/5 Hz because its fifth harmonic is the same as C’s sixth harmonic. In three-limit, however, E-flat is 32/27 Hz, and that is too flat to have any good harmonic overlap with C. The three-limit versions of A and A-flat are similarly bad-sounding. No wonder medieval Europeans considered thirds and sixths to be dissonant. In three limit, they are dissonant.
What if you go higher in the harmonic series instead? The sixth harmonic is boring, it just produces a higher-octave G. The seventh harmonic is interesting, however: it gives you a B-flat at 7 Hz. That is much flatter than the three-limit B-flat at 16/9 Hz or the five-limit B-flat at 9/5. I put a downward arrow next to it to indicate that it’s a different note.
Harmonic sevenths give you beautifully consonant seventh chords. A C7 chord with a harmonic seventh in it feels stable and resolved. This sound is a likely basis for blues tonality.
However, putting the seventh harmonic into the mix makes the family tree extremely complicated. In addition to the conflicting values of the three-limit and five-limit versions of notes, you also get even more conflicts from the seven-limit versions. The diagram below shows just a small part of the seven-limit harmonic family tree.
In the blues, this complexity is not such a problem, because you rarely venture far from the tonic anyway. If you are playing the blues in C, you only need to use the seven-limit harmonics of C and F, and maybe G. That is plenty of fine subdivisions of the octave already. But Western Europeans like to jump around among the keys, and that requires a smaller and more standardized set of pitches to work from.
There is another reason to want to have a smaller collection of pitches. I have been blithely talking about adding strings to the imaginary guitar, but real instruments have limits. You only want to have so many strings, so many frets, so many keys on the keyboard. Designing a keyboard or fretted instrument for just intonation is very difficult, because you need a lot of keys or frets to play notes that are very close to each other. Remember that in five-limit, C-sharp and D-flat are two slightly different pitches, as are D-sharp and E-flat, E and F-flat, F-sharp and G-flat, G-sharp and A-flat, A-sharp and B-flat, B and C-flat, and B-sharp and C. Some historical keyboard makers did try making instruments with split black keys and multiple manuals that could distinguish between, say, F-sharp and G-flat, but they were too complicated to catch on. Imagine tuning one of those things!
Western Europeans tried making all kinds of tweaks and adjustments to five-limit just intonation to resolve its many conflicts and awkwardnesses. The eventual consensus solution was twelve-tone equal temperament (12-TET), the system we all use today. In 12-TET, you don’t use harmonics to derive your notes. Instead, you build intervals out of uniformly sized semitones. To go up a semitone from any note, you multiply its frequency by the twelfth root of two. To go down a semitone from any note, you divide its frequency by the twelfth root of two. This gives an approximation of just intonation intervals, while also limiting the possible pitches to only twelve per octave. When you play piano or guitar, the adjacent keys and frets play 12-TET semitones.
All the conflicts from five-limit get resolved by brute force in 12-TET. There is only one D, at 2(1/6) Hz. There is only one A, at 2(3/4) Hz. The problem with endlessly ramifying enharmonics is solved too; they are all regularized out of existence. B-sharp and C are identical at 1 Hz. C-sharp and D-flat are identical at 2(1/12) Hz. D-sharp and E-flat are identical at 2(1/4) Hz. E-sharp is the same as F at 2(5/12) Hz. And so on. Unlike the near-circle of three-limit (really an infinite spiral), in 12-TET the circle of fifths is an actual closed circle.
The benefit of 12-TET is that it’s nice and simple. The bad news is that no intervals are really in tune except octaves. Some of them are close. The 12-TET G at 2(7/12) Hz is almost indistinguishable from G at 3/2 Hz. However, the 12-TET E at 2(1/3) Hz is noticeably sharper than E at 5/4 Hz. Like three-limit just intonation, 12-TET gives you great-sounding fifths, but not-so-great-sounding thirds. Maybe this is why modern musicians like power chords so much.
Western European music is reasonably well suited to 12-TET, but the blues is not. You need finer pitch gradations than the piano-key notes can give you. This is why blues and jazz musicians prefer instruments with fine pitch control, like trumpet, trombone, violin, and the voice. This is also why blues guitarists bend strings “out of tune” and use slide: they are looking for those in-tune notes between the frets.
As with any discussion of this material, the big question is always: who cares? Music theory is confusing enough, why even bother engaging with a historical tuning system that no one even uses anymore? My motivation for wanting to learn all of this is that I like things to make sense. You don’t need to understand all the nuances of historical tuning systems, but it is good to know they exist. Think of it like English grammar, which seems insane and nonsensical until you find out that it’s an awkward hybrid of Latinate and Germanic languages, the result of French invaders colonizing England. So it is with music theory. Why are music theory teachers so persnickety about writing B-sharp instead of C in the key of C-sharp minor, when it makes no difference at all in how things sound? Now that I know that B-sharp and C used to be totally different notes, I’m more at peace with the spelling convention. Just intonation may be arcane, but at least it makes all the 12-TET arcana more logical and explicable.
“Remember that your ear wants notes to line up with each others’ harmonic series. Your ear likes hearing the E at 5/4 Hz together with the C at 1 Hz because E’s fourth harmonic is the same as C’s fifth harmonic.”
Several points I want to make that I’m afraid will sound like I’m heavily nitpicking your choice of wording, but I’ll try to explain why I think the distinction matters. I’m a theory novice who loves all kinds of experimental stuff, so I’m interested in digging around the edges of what’s possible. I’ve listened to a lot of stuff like Harry Partch and The Mercury Tree, and I love a lot of it, but personally, I’ve come to think of microtonal music as being similar to modal music. It can be interesting, but it all kind of has one sound to it. If I open one of these tracks and just skip about randomly, there aren’t going to be many unexpected harmonic changes surprising me. It’s all going to sound to me like an elaboration of that one sound that I’ve already more or less gotten the gist of. This has led me to wonder if microtonal systems just sound less versatile than 12tet because there aren’t as many people working as hard on exploring their possibilities. And TL;DR, I’ve come around to the conclusion that, no, that isn’t it, microtonal systems do in fact have these inherent limitations. There’s really just never going to be, say, a Stevie Wonder or McCoy Tyner working harmony magic in a purely microtonal system.
So first of all, isn’t it really the case that your ear likes hearing the E and C together (or even more properly, thinks they sound consonant – personal preference for consonance vs. dissonance is a separate thing) because they have a simple mathematical relationship to each other? The gist of how we hear harmony is still preserved when we hit the pitches with pure synth tones that have no overtones at all, right?
I had the idea of mocking the process we used to develop Western tuning off the harmonic series by repeating the kind of process described in this post, but using instruments (real as in the hyperpiano, or virtual) with wonky overtones. Gamelan doesn’t really have “harmony,” but what would it sound like if it did? Could we build consonant and dissonant Gamelan chords and make progressions that sound “functional” in their own fresh way? The hyperpiano wasn’t tuned to accord with its wonky overtones, but what would the chords sound like if it was? The major and minor chords already sound dissonant because of the weird overtones in them… so by corollary, could we even make augmented or diminished chords or something else entirely sound consonant the way normal major/minor triads do on normal pianos? What would harmony sound like in a state where the augmented or diminished chords sound like the most natural “home?” What would it sound like to venture “out” to a major chord from an augmented home?
Basically, couldn’t we repeat this process of getting E and G from C with inharmonic strings or funky virtual instruments and get entirely different notes and chords? And if we did, would our ears pick up on a certain consonance the same way it does with diatonic progressions? Could we then easily make “functional harmonies” that move through new kinds of consonances and dissonances with new kinds of tensions and resolutions? Would this be a better way of “grounding” an approach into microtonality, for creating tuning systems that don’t JUST sound “wonky” but actually seem like viable alternatives to the systems we’re used to, with spicy options but new and different versions of vanilla too?
The empirical answer, as best as I can tell, is “no” to all of this, because these microtonal notes and chords are all going to sound “off” purely by virtue of the fact that their direct numerical relationships to one another are complex. And if that’s the case, then it would seem that THAT’S also the reason why the harmonics have the effect that they do, too: they’re just more notes present in the chord that will have either simple or complex relationships to the rest of the notes present. If they align perfectly as in these examples, then 1:1 is (obviously) as simple as the relationship can get.Isn’t it also a bit wrong to think we “hear” the overtones aligning, as your wording suggests here? On a real instrument that necessarily has overtones, it seems the effect we hear is less about adding something that our ear likes (or even “hears as consonant”) into the picture, and more about NOT adding in something our ear will hear as dissonant. If you virtually removed the G harmonic from a C note and then played a G along with it, I’d bet no one besides a trained listener that knows what they’re looking for would hear anything wrong. It will still sound every bit as consonant as it did before – simply because the C and G stand in simple relationship to one another. And the 1:1 ratio in the overtone wasn’t increasing the consonance, it was just “nulling” the overtone out so that it wasn’t contributing any dissonance.
If that’s right, then there’s really little hope of making a wildly different microtonal tuning system that’s not going to just sound “wonky” all the way around. Sure, you can throw in some quarter tones, or “modulate to G half-sharp major,” but the C:G ratio here is 2:3, the C:E ratio is 4:5, the E:G ratio is 5:6… a major chord is 4:5:6… these are the simplest numerical relationships possible.
So quite literally all the conceivable consonant-sounding ratios are taken because JI/12TET has them systematically covered. So maybe you can hit a major/minor triad and then substitute something microtonal in places where you’d use augmented type sounds, throw in a quarter-flat second a la Misirlou, you can do different things like this from a base of 12tet major/minor tonality to add some spice to it, but you’re never going to reinvent a new way of writing consonant home chords that move into tense dissonances which resolve back into consonant home chords from scratch. Even Harry Partch, who explicitly writes about dividing his 51-note system into notes of approach, notes of tension, etc. – Delusion of the Furies, which I absolutely love, still gives me a very “modal” impression where the tonality all just kind of “sounds the same” when I just skip around the album at random.
It is fine to nitpick my choice of wording, this is why I post these things publicly.
Stevie Wonder is already the Stevie Wonder of a microtonal system, namely, the blues. It’s true that it isn’t a “purely” microtonal system, because it uses 12-TET notes also, but Stevie sings plenty of blue notes that fall between the piano keys. He also bends notes on the harmonica to reach them.
By the time you hear a pure synth tone with no harmonics in it, you have already had a lifetime of training in hearing the natural overtone series, for example, in the human voice. You are detecting the harmonic series in your mother’s voice long before you are born. So by the time you hear your first synthesizer, your brain is well trained to imagine the missing harmonics.
I am careful to say in the post and elsewhere that while many human societies base their tuning preferences on the natural overtone series, they don’t all do it. Gamelan uses “inharmonics”, the complex pitch ratios produced by bells and gongs. Gamelan music uses harmony, in the sense that there are multiple notes simultaneously with specific relationships between the notes, but you are right that it sounds very different from Western European harmony.
It isn’t that the major and minor chords in 12-TET sound intrinsically more stable than augmented chords; we consider them stable because of cultural convention. If you had heard thousands of pieces of music since birth that started and ended on augmented chords and consistently placed augmented chords in metrically emphasized positions, you would hear them as perfectly stable. People who only listen to European classical music hear dominant seventh chords as unstable, but people who listen to blues and rock hear dominant seventh chords as stable. You could probably train someone to hear just about anything as stable, but they would need to be hearing this hypothetical system everywhere they go, repeated across contexts, and that’s not an easy experiment to carry out. Nothing in our experience of music is intrinsic to the physics of sound; it’s all enculturation.
It often happens that an instrument will blend into another instrument’s overtone series and virtually disappear. This kind of consideration happens all the time (usually unconsciously) when you are making decisions about orchestration, arrangement, mixing, sound design and so on.
Anyway, the point is that 12-TET is in no way the last word in a systematic treatment of harmony. Look into Hindustani classical tradition or various Arabic traditions and you will find vast and rich systems of harmony and melody that are thousands of years old. The ubiquity of 12-TET around the world is an artifact of European (and American) cultural imperialism, not because it is any better (or worse) than Hindustani or Arabic tuning systems.
Last thing: Harry Partch’s music all “sounds the same” to you (and to me) because we haven’t spent enough time with it. Even without formal study, you have spent thousands of hours immersed in 12-TET and European tonal harmony. Every time you sing along with a song on the radio, you have been practicing your aural skills and music theory. If you put that kind of time into Harry Partch’s tuning (as Partch himself did) I’m sure you would have no trouble hearing all the richness of structure that Partch heard.
First of all, equating just intonation and 12edo in the same breath is super weird. 12edo is already very much “off” in the sense you’re talking about. It’s full of things that are struggling to make sense in terms of 5-limit harmony, and while it’s not too terrible about it overall, the only reason it could be considered an acceptable approximation is because that’s such a small number of notes per octave.
Apart from its major thirds being really sharp and minor thirds being really flat which most people know about (though maybe they don’t really feel it if they haven’t played something where those are better tuned), 12tet’s semitones, particularly the ones that occur in the major scale, are 11 cents flat, which is a pretty terrible difference to have when the relationship that 16/15 is trying to convey is already so subtle. The impact of this is not as appreciated as I think it ought to be.
43edo has a semitone which is about 0.1 cents off from just, and its major third and perfect fifth are both about 4 cents out, which gives it a really stable and nice feeling. This makes major 7th chords shine beautifully. Even in 31edo, where the semitone is 4 cents sharp, the difference is really remarkable — it’s so much more consonant than buzzy 12edo! Of course, the thirds being better tuned helps a lot also, but I think maj7 chords really highlight the difference in a much bigger way.
But the real thing which makes me sad about 12 equal is how many beautiful chords it just can’t even discuss, because it treats any ratio with a 7 in it so poorly (let alone any higher primes).
The thing which makes me love 31edo a little more than 43edo, is that 31edo’s 7-limit is much more convincing (nevermind that its 5/4 is so nice). Not only that but all the (otonal) 7-limit stuff just a little bit sharpward on the circle of fifths, so when you change keys or borrow notes from adjacent keys, you get all this 7th harmonic magic happening where the intervals are magically more consonant than they would be in 12 equal or even a 5-limit just intonation. You get these relationships which, while they might seem a little alien at first, are really very nice, and do a good job of lining up relevant low-order partials on a harmonic instrument.
The tritone in 31edo is a very good approximation to 7/5, and so e.g. if you start at C, and go up a tritone from your major third E, you get a 7/4 harmonic seventh at A#, and then surrounding that are basically all the simple intervals with a 7 in the numerator, for example, D# is a 7/6 septimal minor third up from C. If you head flatward instead, you get approximations utonal septimal stuff (with 7 in the denominator).
So in 43edo, I might feel a little more trapped in one key, though this may have more to do with the fact that the white keys obtain the same magic on a 43edo piano that the black keys have on the ones everyone’s familiar with… but in 31edo, I’m free to move around quite a lot. I can steal from a very wide swath of the circle of fifths and have to be careful about a small handful of the notes. It’s like each key is virtually so much *wider* on the circle of fifths, or at least, there are luxuriously large shoulders on the road we’re driving on.
But it’s hard to describe all this with words. If you ever have the chance to sit down at something like a Lumatone to be able to play in these larger tuning systems, try it. I have a hard time believing that someone could actually try playing in something like 31edo for even half an hour and come away with the impression that sticking entirely to 12 equally spaced notes per octave is enough forever. There are so many beautiful chords that just don’t exist in 12, and the intonation of the chords that you already know has a vastly different character, and you might not want to go back.
Here, check this out https://www.youtube.com/watch?v=z40RWIN-0cc — Mike Battaglia doing a little cover of Jacob Collier’s “The Sun is in Your Eyes” in 31edo. The other stuff on his channel is also great, I’d also recommend his cover of Herbie Hancock’s “Speak Like a Child”. Maybe also check out some of Zhea Erose’s stuff — she’s mostly into specially-crafted just intonations where the denominators all have a large common factor, which lends a really different character to things depending on what that common factor is thanks to the difference tones. But here’s a nice little improv from her in 31edo that shows off a few more of the interesting things available there: https://www.youtube.com/watch?v=cvaymi7x1cM
We’re only just getting the tools to begin to properly explore this stuff, but there’s so much more to what music can exist than what is confined to any one system.
I agree with you that people should be trying out alternative tuning systems for themselves. And folks should be aware that this is not just for avant-garde weirdos! Every guitarist who has ever bent a string has stepped outside of 12-TET. So has every singer who has ever given some bluesy inflection to their notes. Black American musics like blues, jazz and rock are loaded with just intonation intervals that fall between the piano keys.
For people interested in more details on the microtonality of blues scales, I recommend this YouTube playlist: https://www.youtube.com/watch?v=NVWX10c0bNc&list=PLnafBBFyC1Qg2kGpp2YeiiKjmMPiYz2s5
Thank you, this playlist is excellent!
I’m good with the tuning systems discussions, Ethan!