F-sharp vs G-flat in just intonation

As I gear up for teaching music theory in the fall, I’m still refining my explanation of Western music’s arcane naming system for enharmonics. Why is the note between F and G sometimes called F-sharp and sometimes called G-flat? Why do we sometimes call the interval between that note and C an augmented fourth, and sometimes call it a diminished fifth? What difference does it make if they sound the same?

I had a major “aha” moment when I learned about the history of Western tuning systems, and found out that F-sharp and G-flat were originally two different and non-interchangeable notes. I have enjoyed seeing that same “aha” look on my students’ faces when I explain it to them. But tuning systems are hard to understand, and my explanation still requires a lot of refining. This post is one in a series of iterations.

The twelve-tone equal temperament (12-TET) standard in Western societies is a relatively recent historical development. Before 1900 or so, Western Europe had no single standard tuning system. From about 1500 to 1900, most European tuning was based on five-limit just intonation. When you tune using this system, F-sharp and G-flat are not the same note at all. I made a track using Ableton Live and MTS-ESP so you can compare them for yourself.

Here’s a chart, if you want to follow along. I begin the track by alternating the notes F-sharp and G-flat. Then I alternate some chords that include those two notes: F# and Gb, B and Cb, D#m and Ebm, G#m7 and Abm7. Finally, I play a series of V-I cadences from F#7 to Gb to demonstrate how, in just intonation, there is no such thing as the circle of fifths. I put all this over some breakbeats sampled from “Hey Pocky A-Way” by the Meters, because funk makes tuning theory so much more enjoyable.

Before I started experimenting with MTS-ESP, I knew that F-sharp and G-flat were originally tuned differently, but I was not prepared for how far apart they are. The interval between them is called a diesis, and it’s almost a quarter tone, about half of the distance between any two keys on the (equal-tempered) piano. That is not a subtle difference! In 12-TET, the only reason to specify F-sharp versus G-flat is for notational clarity. In just intonation, however, it matters very much which one you use. If you play a just intonation D chord using G-flat rather than F-sharp, it will sound horrendous.

This next section of the post goes into detail about how I tuned my track. If you are allergic to math, skip to the bottom for some thoughts about what this all means. First, let’s set the table. Just intonation uses the simple pitch ratios found in the natural overtone series. Imagine a string tuned to middle C at 1 Hz. (The actual frequency of middle C is 262.626 Hz; multiply everything in this post by that number for actual frequencies.) I derive the notes in my track from C’s harmonics, and from some notes whose harmonics include C. I multiply and divide frequencies by two as needed to keep everything in the same octave.

Let’s start by finding F-sharp.

• The third harmonic of C produces G at 3/2 Hz.
• The third harmonic of G produces D at 9/8 Hz.
• The fifth harmonic of D produces F-sharp at 45/32 Hz.

Now let’s find G-flat.

• C is produced by the fifth harmonic of A-flat at 8/5 Hz.
• The third harmonic of A-flat produces E-flat at 6/5 Hz. 
• The third harmonic of E-flat produces B-flat at 9/5 Hz.
• B-flat is produced by the fifth harmonic of G-flat at 36/25 Hz.

Here’s a harmonic family tree showing all these notes, plus some related ones.

I tuned the white-key notes in my track to Ptolemy’s intense diatonic scale.

• C – 1 Hz
• D – 9/8 Hz (third harmonic of G)
• E – 5/4 Hz (fifth harmonic of C)
• F – 4/3 Hz (third harmonic is C)
• G – 3/2 Hz (third harmonic of C)
• A – 5/3 Hz (fifth harmonic of F)
• B – 15/8 Hz (fifth harmonic of G)

Here’s a graph of these pitches compared to their 12-TET equivalents. The numbers inside the circle are cents; each equal-tempered semitone is 100 cents.

The graph shows that the just intonation D, F and G are tuned about the same as they are in 12-TET, while the just intonation E, A and B are a bit flatter than you’re used to.

Here are the black key notes on the sharp side.

• C-sharp – 25/24 Hz (third harmonic of A)
• D-sharp – 75/64 Hz (fifth harmonic of B)
• F-sharp – 45/32 Hz (fifth harmonic of D)
• G-sharp – 25/16 Hz (fifth harmonic of E)
• A-sharp – 225/128 Hz (fifth harmonic of F-sharp)

These are all flatter than their 12-TET equivalents.

Here are the black keys on the flat side, plus the bonus C-flat that I needed for a few chords.

• D-flat – 16/15 Hz (its fifth harmonic is F)
• E-flat – 6/5 Hz (its fifth harmonic is G)
• G-flat – 36/25 Hz (its fifth harmonic is B-flat)
• A-flat – 8/5 Hz (its fifth harmonic is C)
• B-flat – 9/5 Hz (its fifth harmonic is D)
• C-flat – 48/25 Hz (its fifth harmonic is E-flat)

These are all sharper than you are used to, especially when you compare C-flat to 12-TET B.

And here’s a family tree of all the pitches in the track, plus some related ones.

When I made my track, I had to enter everything by hand into the piano roll. There is no way to play a standard MIDI keyboard in just intonation. You would need to have separate keys for F-sharp and G-flat, for C-sharp and D-flat, for D-sharp and E-flat, for G-sharp and A-flat, and for A-sharp and B-flat. Depending on how advanced you were, you might also need separate keys for C-flat and F-flat and B-sharp and E-sharp, not to mention (shudder) the double sharps and double flats. This is impractical, to put it mildly.

Enharmonics are not the only challenge of just intonation. The reason that people started using just intonation in the first place is that every pitch is gorgeously consonant with the original C. However, the problem is that the pitches are not necessarily consonant with each other. This is where things get very hairy, so hang in here with me.

You might notice that the family tree includes two different tunings for D. The one I used in the track, tuned to 9/8 Hz, is 3/2 times the frequency of G at 3/2 Hz, so they form a perfect fifth. However, that D is 27/20 times the frequency of A at 5/3 Hz. That is a wolf fourth, a much narrower interval than the 4/3 fourth you would want, and the effect is gratingly dissonant. If you wanted to make good fourths with A, you could tune your D to 10/9 Hz. However, that D makes a nasty 40/27 wolf fifth with G. You get into similar problems when you try to make thirds with these D’s. So your just intonation MIDI keyboard is going to need two different keys for D, and you will need to somehow keep track of which one to use in which situation.

This is not just a problem for D, either. My family tree also shows two possible tunings each for F-sharp and B-flat, and they pose similar conflicts with each other. If you expand the family tree further out, the conflicting note tunings just keep multiplying infinitely.

Wouldn’t it be nice if you could average the conflicting notes out so that there would only be one of each of them? What if you only had one D that was reasonably close to being in tune with both G and A and everything else? This idea motivated the development of meantone temperaments, which were widely used in Europe during the canonical classical era. I won’t explain how these tunings work; you should read Kyle Gann’s web site, or Rudolf Rasch’s chapter on tuning in the The Cambridge History of Western Music Theory. The bottom line with meantone is this: if you want in-tune fifths, then your major thirds have to be too wide. Conversely, if you want in-tune major thirds, then your fifths have to be too narrow. Europeans usually opted for tunings that preserved in-tune thirds, at the expense of introducing a horrible wolf fifth (really a diminished sixth) between G-sharp and E-flat.

Meantone temperaments also didn’t solve the problem of the black keys. Remember that if you want to play in just intonation, you need separate black keys for F-sharp and G-flat, for C-sharp and D-flat, for G-sharp and A-flat, and so on. This is also true for meantone. Some instrument builders did use split black keys, where, for example, the upper half of the key played D-sharp and the lower half played E-flat.

This is a clever solution, but you can see why the idea didn’t catch on. To play all the notes in my track, you would need eighteen keys or frets per octave!

Usually, keyboard makers only gave you single black keys. If you tuned the key between F and G to play F-sharp, then you had to avoid playing anything that required G-flat.

But what if you could smooth out the black-key discrepancies too? What if you could average out F-sharp and G-flat to be the same pitch, so you could play either of them with the same piano key or guitar fret? Neither one would be perfectly in tune, but could you at least get them to sound adequate? This idea motivated the creation of the well temperaments. These tunings reduce the infinitely ramifying universe of enharmonics down to twelve pitch classes total. Bach used one of the well temperaments for The Well-Tempered Clavier (thus the title), though no one knows which specific temperament he preferred.

Twelve-tone equal temperament is a well temperament. It spreads the required out-of-tune-ness evenly across the intervals. However, Bach and his contemporaries did not use 12-TET. They tuned so that the out-of-tune-ness was distributed unevenly. This meant that the keys closest to C major sounded more pure and sweet than they do in 12-TET, while the more distant keys sounded darker and edgier. This is why Bach wrote preludes and fugues in every key: to show how each one had its own particular character. I like this tuning concept, but sadly, it’s not the one that caught on.

Aside from a few nerds and oddballs, everyone in Western cultures eventually adopted 12-TET, with its uniform keys built from uniform semitones. Nothing in 12-TET is exactly in tune except for octaves. The fourths and fifths are very close to just intonation, so that’s nice, but the major thirds are too wide, and the minor thirds are too narrow. The good news is that no key is any worse than any other, so F-sharp major/G-flat major sounds just as good (or bad) as C major, or F major, or G major. You can transpose a tune to any key without changing its musical character. When you use a guitar tuner, MIDI, notation software or Auto-Tune, you are using 12-TET. 

Convenient though 12-TET is from an instrument design standpoint, we are still stuck with the note and interval naming conventions from just intonation. There is no auditory need to distinguish F-sharp from G-flat, but our notation system requires that we maintain the separate names. If we designed a new notation system from scratch around 12-TET, we would probably come up with something different, maybe a mod12 integer notation. But once conventions get established, it’s hard to change them.

So, what does all of this mean, and why should you care? For one thing, it’s important to know that Western note naming is a path-dependent kludge, not a flawless logical system. If you find it confusing, you’re right to! I come from the jazz world, where people are casual about their enharmonics, and from the rock world, where they barely even know what note names are, much less how to apply them correctly. When I learned my Western tonal theory properly, I found it oppressive. I tended to always call the note you play using the second fret on the guitar’s E string “F-sharp” regardless of context. Why bother remembering when to call it G-flat? When I learned about just intonation and meantone, I understood why the names mattered.

Just intonation is not just for music history class. Not everyone plays keyboards and fretted instruments! Singers and players of continuous-pitch instruments can play just intonation as easily as they can play 12-TET, and they often instinctively play more in tune than they are “supposed” to. Guitarists sometimes tune their B string a little flat so it plays the just third in the key of G. When blues musicians bend their notes, they are probably aiming to make them sound more in tune, not out of tune.

Beyond the specific technical aspects of musical tuning, there’s a larger philosophical point here. Students need to know that 12-TET is not the only possible tuning system. Even within Western Europe, there have been many alternatives. My track doesn’t represent just intonation very well; it intentionally jams together conflicting notes and chords for demonstration purposes. But just intonation can sound exquisitely beautiful when used for actual music, and it’s worth experiencing it. Breaking loose from the constraints of 12-TET can expand your musical imagination significantly. I tried writing and producing a bunch of music in different just intonation systems, and when I play these tracks for my students, they are immediately intrigued.

Some people think that 12-TET causes us actual psychological harm. Kyle Gann explains:

I’ve had interesting experiences playing just-intonation music for non-music-major students. Sometimes they will identify an equal-tempered chord as “happy, upbeat,” and the same chord in just intonation as “sad, gloomy.” Of course, this is the first time they’ve ever heard anything but equal temperament, and they’re far more familiar with the first sound than the second. But I think they correctly hit on the point that equal temperament chords do have a kind of active buzz to them, a level of harmonic excitement and intensity. By contrast, just-intonation chords are much calmer, more passive; you literally have to slow down to listen to them. (As Terry Riley says, Western music is fast because it’s not in tune.) It makes sense that American teenagers would identify tranquil, purely consonant harmony as moody and depressing. Listening from the other side, I’ve learned to hear equal temperament music as a kind of aural caffeine, overly busy and nervous-making. If you’re used to getting that kind of buzz from music, you feel the lack of it as a deprivation when it’s not there. But do we need it? Most cultures use music for meditation, and ours may be the only culture that doesn’t. With our tuning, we can’t.

My teacher, Ben Johnston, was convinced that our tuning is responsible for much of our cultural psychology, the fact that we are so geared toward progress and action and violence and so little attuned to introspection, contentment, and acquiesence. Equal temperament could be described as the musical equivalent to eating a lot of red meat and processed sugars and watching violent action films. The music doesn’t turn your attention inward, it makes you want to go out and work off your nervous energy on something.

This aligns with my own listening experience. I also found just intonation to be murky and subdued at first, and thought that 12-TET sounded better. After listening to a lot of just intonation, my ears acclimated, and I started finding 12-TET to be anxiety-producing. I won’t go so far as to say that 12-TET is destroying civilization or anything, we have vastly larger problems, but Kyle Gann and Terry Riley did get me thinking about my own listening and music-making experiences in a new way.

Taking another step back: what does it even mean for something to be “in tune” or “out of tune”? When you start reading about tuning systems, you very quickly discover guys online (they are always guys) who use the mathematical basis of their preferred system to “prove” its objective correctness and profundity. These arguments inevitably lead to intellectual colonialism. Most cultures around the world have historically tuned by ear rather than through any kind of numerical system; sometimes they do arrive at just intonation that way, sometimes they don’t. Auditory experience is ultimately the only thing that matters, and the ear likes what it likes for reasons we can barely even articulate.

Even within Anglo-American pop, there is no consensus on what constitutes “good” tuning practice. Millions of Rolling Stones fans enjoy listening to wildly out-of-tune guitars and singing that is casually tuned at best. Fans of the Grateful Dead seem to prefer to hear out-of-tune singing paired with (more or less) in-tune guitars. Jazz fans have to make peace with a lot of out-of-tune pianos. Hip-hop and electronic dance music producers detune their synths and samples on purpose. Whether you are interested in the specifics of Western European tuning history or not, it’s good to know that 12-TET is just one possible system, and not necessarily the best one.