Bach's schematic, as it appears on the page, © Bradley Lehman, 2005-22, all rights reserved.
All musical/historical analysis here on the web site is the personal opinion of the author,
as a researcher of historical temperaments and a performer of Bach's music.

An informal talk about the basic ideas.

What is all this, and why does it matter?

To begin, play "Twinkle Twinkle Little Star" on a keyboard.
C C G G A A G, F F E E D D C.
Twinkle twinkle little star, how I wonder what you are. That song is at least 250 years old, maybe more, at least the notes of it. People sing it in different languages.

Now play that melody again on different notes:
F# F# C# C# D# D# C#, B B A# A# G# G# F#.
All those black keys!

Now play it once again on different notes:
Eb Eb Bb Bb C C Bb, Ab Ab G G F F Eb.

You will hear about some weird and wild stuff here today. A long treasure hunt of at least 50 years and maybe 250 years. Music notes called different things and sharing invisible space. History that got lost. Math that looks more like a picture than numbers. Secret ways to make music sound better. A guy who hated math and spelling doing something brilliant, that other people have had a really hard time figuring out. Cooking oatmeal. Drawing a pretty shape on paper that is really a music lesson.

Equal tuning

There are hundreds of different ways to tune keyboard instruments (pianos, organs, harpsichords, clavichords, synthesizers). The most common way today is "equal" tuning where all the notes are exactly the same distance apart in pitch. All the notes on the piano are the same distance apart. That seems pretty obvious if you have ever had music lessons on piano.

Tuned this way, music sounds basically the same all the time even if it is written in different keys, other than being a little higher or lower in pitch. All the keys such as C major or D major or G major or B-flat major are the same amount "in tune" or "out of tune" as one another. If music is changed from one key to another, this "sameness" simply moves around to be higher or lower.

That is, all those versions of "Twinkle Twinkle" sound exactly the same in character, on equal tuning. But, some of those notes in the examples above really had two different names, such as A# and D# in one version, against Bb and Eb of the other. They "really" are different notes, that just happen to be sharing the same space on a keyboard.

A# and Bb (A-sharp and B-flat) are different notes from each other (and the difference is called "enharmonic"), but the keyboard has only one key to play them. Therefore there has to be some sort of compromise or work-around so that both of them can sound decent enough.

We'll come back to all that, don't worry about it. They really are different notes, Bb and A#! The main reason why we don't notice that is that today's equal tuning has trained us not even to think that way, let alone do anything about it. They're the "same" note, right?

All of keyboard tuning boils down to that basic problem (along with some other smaller problems): how can it be set up so the notes with different names can be used, and sound decent enough, in the ways they need to be used? How can more keys and scales than C major be played, so that everything sounds decent and (we hope) beautiful and interesting?

The solution now is to set it up so all those scales sound exactly the same. "Twinkle Twinkle" is always "Do Do Sol Sol La La Sol, Fa Fa Mi Mi Re Re Do", if you know your Do-Re-Mi's. The tuning when the piano tuner comes is set up so those notes are the same distance apart, exactly, no matter what note we started on (C or F# or Eb or something else).

In those Do-Re-Mi's today, we expect "Sol" to "La" to sound exactly the same as "Do" to "Re", or "Re" to "Mi". Those are all "whole steps". The other kind of steps, "half steps", are in there as "Mi" to "Fa". That's the notes of "How I wonder" in the song, that closer sound. And we always expect them to sound the same as one another, in all the keys. All whole steps are the same as each other. All half steps are the same as each other. Period.

Question mark?

Some animals are more equal than others

So we are used to equal tuning now being normal. All whole steps are the same as each other. All half steps are the same as each other. All the notes on the piano sound exactly the same amount apart in pitch.

It was not always so. That way of tuning is only a recent thing to be standard, since about the 1880s to 1920s. That is one of the Industrial Revolution ideas, going for standard machines and interchangeable parts, and assembly-line methods of building things. Factories. Something breaks, you replace just that broken part that some machine made, snap it into place. Life is easy that way because things are the same every time, easy to know what to expect everywhere...but also not as interesting as it might be otherwise. That is different from the old ways where everything had to be made one at a time, by hand, and the things made came out a little different from each other.

Time for a little trip into your imagination here. Try to forget everything you know about the way equal tuning sounds on pianos!

In the unequal ways of tuning, the keyboard ends up looking exactly the same, but the sounds it makes are not quite evenly spaced. For example, in "Do Re Mi" the notes "Do" to "Re" might not be exactly the same distance in sound as "Re" to "Mi". Or, if we play "Do Re Mi" in some other key, there might be even more difference in the way those notes are spaced. The point is that not all the major and minor scales sound exactly the same as each other. Some scales and chords are "better in tune" than others. Variety is a spice of life.

In all the unequal tunings, the whole steps are not all the same size as one another. And the half steps are not all the same size as one another. There are at least two different names for the half steps: "Chromatic" and "Diatonic". It is tied to the way music is written, the way the scales work, which names are used in which situations. In the 17th and 18th centuries, singers and players were trained to know this difference, which was very important in the way they sang or played pitch.

I mention that here mostly to point out that it existed, don't worry about it. The reasons for it all, and any decisions about "better" or "worse" compared to our system today, get into some complicated physics and mathematics. Equal tuning today is an easy way out, a compromise so none of those problems matter very much.

The disadvantage of equal tuning is that it is tricky to set up, getting it exactly right, and it's boring to listen to, compared with some other ways of tuning. Some of the notes in equal tuning, especially going directly from "Do" to "Mi" in all the keys, are pretty far out of tune (according to "harmonics" in wind and string instruments, going to the math and physics of it). That is the price to pay for modern convenience, interchangeable parts, interchangeable scales.

Another disadvantage of equal tuning today is that it makes people forget that there ever were any differences in the whole steps and half steps, or that different musical keys ever sounded seriously different from each other in the chords and scales.

People 300 years ago knew about that equal tuning option, and some other instruments such as lutes (like an early guitar) used it, but for general use on keyboards it was not the only way. People knew about equal tuning but most did not use it, on purpose.

Cats and dogs are built differently! Some of the other tuning methods were (and still are) easier to set up, and they made the keyboard instrument sound better in tune, when played in the easy keys that do not have a lot of sharps or flats. And, the other reason to prefer some unequal tunings is that the music is more fun and more beautiful to listen to if there is some variation in it.

Equal tuning is like saying that all dogs are the same. And all cats are the same. And all cats are the same thing as dogs. Everybody knows that's silly. So, why should all musical keys act like they're exactly the same dog or cat? Isn't it a lot better to have a bunch of different dogs and cats, as long as we can keep track of them? Isn't the cool thing to watch how different cats act differently, with different personality? My four cats and my dog do not act at all the same, ever.

20th century detective work

Especially in the 1960s and 1970s it became popular to tune keyboards in the old ways again, to play older music. The "Do Re Mi" scales in the different keys sound different in these old tunings. Sometimes the difference is so small, so subtle, that it's more about feeling the difference than about hearing and noticing it.

That turn back to older ways of tuning was done by people who specialize in playing and singing older music, from before the piano was invented: organ and harpsichord music especially. Part of that was trying to set things up the way they were in history, to give the older music a fair chance to sound the same way it did when it was written. Part of that, also, was simply to have it sound better and more engaging.

Most of that earlier music is in the easy keys, and therefore it works well if the instruments are tuned to sound best in the easy keys. We do not care quite as much about the keys we rarely play in (such as A-flat major or C-sharp minor). If we are going to spend most of our time playing with only two or three sharps, or two or three flats, the strategy is to make those scales sound as good as possible. (This also works with autoharps, putting the best tuning into the chords that get used most often.) We want our "Do Re Mi" and our "Twinkle Twinkle" and everything else to sound best, most of the time, in the scales we actually use most of the time. Good strategy?

Meanwhile, people playing the music of Bach on pianos and most organs still heard it only in equal tuning, and didn't know to think about this other stuff with scales sounding different. The modern way is so normal now that we hardly think about anything else. The older tunings were used mostly by specialists, trying to figure out which of the many unequal tunings to use in which music.

Part of that detective work was looking into the time and place and customs around the music when it was written. It is very complicated, and most keyboard players get tired of looking for more of these possible answers after they have five or six ways that sound pretty good. Tuning a harpsichord or an organ or a piano takes a lot of patient work, and who wants to remember a lot of different ways to do it?

It was also popular in some history books, especially in the 20th century, to give Bach the credit for "inventing" equal temperament. He wrote his "Well-Tempered Clavier" book of music in all the possible keys, and some of his other music also sounds badly out of tune if it is played in any tuning that is very far away from equal tuning. This credit to Bach also fit in nicely with making him sound like a modern hero, ahead of his time by inventing the tuning that is in most common use today. All those preludes and fugues, one in every key, to help people learn how to play keyboard (and read music!) in all the different keys. This is basic in learning how to play piano and the other instruments.

Did Bach use equal tuning?

In the 1980s and 1990s, some of the researchers and old-instrument specialists came back around, deciding that maybe Bach's music did have to be played in modern equal tuning after all. All the other unequal tunings that were recorded in historical documents turned out not to work very well in his music. The music would sound pretty good most of the time, but then it would run into "dead ends" where it was suddenly rough or weird.

If Bach ever had a preference for some different tuning, nobody alive today knew about it. It was concluded that either:

  • (1) It couldn't be figured out at all anymore for sure because there aren't enough clues, or
  • (2) Bach really used equal tuning after all, at some time in his life.
Or maybe both.

Harpsichord and organ puzzles

Birthday music I played harpsichord and tuned it myself for more than 20 years, using many of these other ways of tuning. I enjoyed this adventure, both as math problems and in getting the music I was working on to sound good. That's a lot of different music in different keys, written 500 years ago or 300 years ago or 100 years ago, or modern songs too. (I like to write music myself...sometimes even in rare keys like the B major of the piece seen here!) My training is in harpsichord as a specialty. Organ too. I've also tuned them for other people sometimes.

But, none of those tunings I studied and tried on the harpsichord and organ quite worked right for all of Bach's music. They didn't feel quite right and didn't sound like they were organized very carefully. Sometimes really sweet, sometimes ugly, but no reasonable logic to it. And all the reference books I looked at said it couldn't be figured out (or else somebody would have done it already). I spent a lot of time just getting used to things that sound weird, and talking myself into sort of liking them.

I convinced myself, and so did a lot of other people (in books and recordings and real life), that we have to change the tuning to play different pieces. I had to pick pieces for concerts that agreed with the way I wanted to tune. Or, I had to pick tunings that fit the pieces I had already decided to play. It might be different on different days. Kind of fun and kind of annoying to keep track of. Why did Bach's music not sound right in these other tunings, the ones that the people around him used (like Vivaldi and Buxtehude and Telemann, and those other guys)?

Sometimes it sounded great, and other times just awful, seeming almost randomly bad. Some of Bach's music especially ran into problems, except in equal tuning: and then in equal tuning those pieces are boring enough that they don't get played very often! If they have lots of flats or lots of sharps, using the unequal tunings in the history books, the pieces of music end up sounding harsh. Or if we use equal tuning to play them, they just seem to have no character and it's dull, like all cats being the same. Why bother writing music in interesting wild keys if it's just going to sound the same boring way as all the other keys?

Clearly, something was not right about this situation...but it seemed impossible to figure out. The music shouldn't sound either boring or randomly out of tune, with nasty dead ends in it! It doesn't make much sense that Bach would write music that sounds bad, on purpose. It seemed especially bad to me when I was playing music that has at least two or three flats in it. Sharps seemed to work better, more or less. But flats could be pretty ugly, whenever they came up. Eww. I have nothing against flats personally, in principle, but they sound bad in those unequal tunings. I talked myself into getting used to it.

How did Bach tune his own keyboards?

According to my present theory (or hypothesis) that a preferred tuning of Bach's has now been discovered:

As it turns out, Bach didn't write music that sounds bad on purpose! His music sounds amazingly beautiful, smooth and colorful, if we tune the harpsichord or organ exactly the same way I believe he did. It gets a little spicy at times, but is never harsh. Just set it up his way, because he surely knew what he was doing. And the flats are especially beautiful colors in his music, not ugly ones! I am going to tell you exactly how to set this up, stick with it.

Yeah right! But how do we know what Bach's way to tune was?! The history books say there is no way to know.

The legend was that Bach could set up the tuning on his whole harpsichord in 15 minutes. I reasoned that it had to be something pretty simple, easier to do quickly than equal tuning is. And it's without little electric machines of course; Bach didn't have electricity or batteries or any of that stuff. Just something like a tuning fork, to get one note from and then figure out all the rest of the notes from there. A very interesting puzzle. This is what we normally do anyway in harpsichord tuning, all the notes derived from one point. But which method is most appropriate?

The legend was also that nobody could ever tune keyboards to Bach's personal satisfaction, but he always had to do it for himself. And, he reportedly made "all the thirds sharp". How much sharp? The sound he got reportedly impressed a lot of people, being smooth and exciting. He wanted a tuning that sounded right all the time, his own way, so he could write as much different music as he wanted to. That is a big contrast against normal 17th century tuning, where only some of the major and minor keys can be played at all.

He also wanted to play it on keyboards that could be set up once and then left that way. To tune a big organ it takes weeks of work and a lot of money, and you don't want to do it more than every couple of years! You don't want to do anything serious with organ pipes, except fix a couple of them individually if they get bumped or something. To do the whole organ from scratch is a big job for professionals who know exactly what they're doing.

To tune a clavichord to some other way you have to bend some parts of it around, hopefully only once(!) with a pliers, so it works out the next time its strings are tuned. And that takes care, not to break it or bend it too much. I have one here and did it, part of one afternoon, carefully taking out each black key and bending its "tangent" (a piece of metal) slightly sideways, to hit the new right place on the strings.

A harpsichord is easier to tune quickly every couple of days, and has to be done anyway if the weather changes too much. The strings change tension and pitch whenever the wood of the instrument gets too much humidity or temperature change. We harpsichord tuners get good at it by having to do it a lot, keeping the harpsichord in tune! But, organs and clavichords stay pretty much in tune for months or a year or more.

So, it is nice to have a way to tune them so they sound good in all the music you're going to play for a whole year. Harpsichords too, while we're at it.

That's the basic keyboard tuning problem. Picking or finding some way to tune it, so all the scales and chords we care about can be played and sound decent. Even if (remember from what I said earlier!) C-sharp and D-flat really are different notes, but looking like only one note on the keyboard.

Mathematical patterns to tune keyboard instruments are called "temperaments". They are recipes to start from a single note and get all the other 11 notes, in relationships of pitch so the whole thing sounds good in the music that is to be played. Temperaments. Recipes.

As I said above, there are hundreds of ways to do this. It is like playing a card game, sorting the cards you have and then figuring out what best order to play them in, so you win the most you can. It might matter very much what order you play them in. Tuning is like that, too. It is strategy and figuring out math puzzles, and trying to get the best sound you can get out of the notes they give you to work with. Do it in exactly the right order with careful strategy.

Play break

OK, take a break here. Go play "Twinkle Twinkle Little Star" a couple more times and try it in more keys, so you get the feeling how things move around. The goal here is to have some setup where we can play it in all twelve possible starting positions, and have it make musical sense.

And get a snack or something.

Bach's flourish from the title page of the Well-Tempered Clavier

Welcome back.

Now we get to the really good part. Don't worry if some of that first part was weird, we'll come back to the good stuff in it.

A hand-drawn puzzle

In April 2004 I was looking closely at the title page that Bach wrote out for his main copy of "The Well-Tempered Clavier". I noticed that his border decoration on the page looks like this, and it is not the same all the way across:

Bach's schematic, as it appears on the page

Why would it not be the same? Was he just sloppy at drawing decorations? Or, is there some hidden meaning in there? Why do some of those mini-loops in there have just a plain loop, while others have one or two little knots in them? And why is there that capital letter C under the arch, the second from the right side?

I realized that this is not "only" decoration at all. (I'm not the first person to suspect that there's some meaning about Bach's tuning here in this drawing, but I'm the first to read it in the specific way that I describe here.) I believe Bach wrote down the math of his tuning method here in a simple way, not by a chart of numbers but by drawing a picture of it! This picture tells us exactly how the tuning should be set up unequally, so the music in that book and his other books sounds best. He wrote this down to apply for a job teaching children and teenagers in another city.

Bach disliked "dry mathematical stuff" but preferred to teach music through practical examples. One of his sons said so, later, for a biography after Bach had died. Bach also wasn't very good at arguing with people in words, or strong in spelling either. But Bach was an excellent music teacher. Never mind the math and the words, Bach wrote and taught music that took advantage of the available sounds. He knew exactly what he wanted.

Look again at that spirally thing. That is the picture of Bach's math! Math isn't only numbers and equations, it's shapes and sizes and ways to describe ideas. Math--in shapes such as blueprints--tells us how things are designed and built. It is the science of describing things and getting them done the same way each time. It is easier to understand and remember a shape once you've seen it, than to remember a whole page of numbers. That's Bach's point.

When something works really well, you write it down in some way that makes enough sense so it can be done again later by somebody else. You write it down, so somebody who is considering your job application is impressed enough to give you the job.

That is also part of Bach's point, writing this down to try to get a new job. This book ("Well-Tempered Clavier") plus a book of inventions for harpsichord or clavichord, plus a book of organ preludes. All three of these books together, played in Bach's specific tuning, make the application material for his job. These are the music textbooks he wrote, for use with his future students in Leipzig.

If we turn the book around, like peeking at the answers in a book printed upside-down, the drawing looks like this: Bach's schematic, rotated for use

Try copying both of those with old-fashioned pen and dipping ink. That is a pain. You will see that it is much easier to draw the second one, getting the loops to go the right directions. It is exactly the same drawing, but with the paper turned around the other way. This second one here can be drawn in just a few seconds, getting the flair of it, from left to right. The first one, higher above, is harder.

Sometimes it is easier to draw things, or copy them, by turning the paper some different direction. Try it with a drawing in an old coloring book, for practice. Turning it upside down makes you look closer at it, in different ways. The shapes look different and you notice them better, making it easier to copy them exactly, carefully.

Bach's drawing turned upside down (to make a copy of it!) means this. It tells us how to set up all the notes in the scale in this order:

Bach's schematic, rotated and mapped

I wrote in all the rest of the notes in the scale there. Bach wrote in the "C" (turn your head upside down) and I wrote in the rest of them, to keep track of where I was. (And knowing what's pretty normal in harpsichord tuning, too, starting with F and C and G and D, and so on.)

We do all the keyboard's white notes first, and then all the black notes. It's a simple and logical way to go about it. F has to be our starting note, if we're going to set up the whole keyboard by tuning 5ths in turn. (Try playing 5ths starting on any other white note, and it doesn't work out: we get into the black notes too soon and have some unfinished white notes left over.)

The note names are supposed to stick down into the spaces between the loops. Draw more of the little lines or arrows into it if you have to, to see what I mean.

Those loopy spirals between the notes

The different types of shapes between some of these tell the harpsichord tuner or organ tuner to make those notes almost in tune together, but slightly out of tune by a little amount on purpose. There is an art to this but it is basically science, knowing how much to put it a little bit out of tune, on purpose. It is study and experience. It is like dashes of spices in cooking: one dash there, two little sprinkles over there, nothing in a third place. This drawing is Bach's recipe to set up all twelve of the notes exactly right!

  • If it has two of those little bumpy knotty things in the loop, it means make it "double" some little amount out of tune, on purpose. You see there are five of those, from F to C, C to G, G to D, and so on until we get over to E. This sets up all of our notes C-D-E-F-G-A, not in that same order of course but they end up in the right places doing it this way. Those six notes are the result of these tuning steps. Our plain white notes on the keyboard, the basic notes of C major. (That group of six notes going up is called a "hexachord" but don't worry about it. That's the way that people before the 1700s were trained to read and sing music, with these "hexachord" groups of six notes like this!)
  • Then there are three plain loops with no knotty things in them. That means make those exactly in tune. E to B exactly in tune, no cheating. B to F#, and F# to C# exactly in tune. (That is a listening skill we work on to tune harpsichords, getting it exactly in tune and knowing when it is.) Those are "zero" amount out of tune, if you think of it that way!
  • Then finally you see three more loops that have one knot in them each. Well, you can guess the rest. They are "single" amounts out of tune for those particular notes, tuned carefully from the previous ones.

There is a total of 13 of these little bumps out of tune, count them up. 2 + 2 + 2 + 2 + 2 + 0 + 0 + 0 + 1 + 1 + 1. After we have tuned all those notes in that order, that amount carefully out of tune on each one (except on the three in the middle where it says get them exactly in tune), we are done. We have all our notes and we can play our music.

All those little amounts of "out of tune on purpose" like spice make it come out right when we are finished. Cook your oatmeal with no salt at all, or with too much salt, and it comes out badly. It all has to be very careful balance, whether it is cooking oatmeal or tuning keyboard instruments.

That "out of tune on purpose" is "temper" for the word "temperament". Temper, the verb, means to adjust. It means to moderate something, to make a good compromise. The compromise is done to make all the notes sound decent in all the music we want to play. They are made a little higher or lower in pitch as the compromise, so it lets them be used lots of different ways all at once. Temperament.

Equal temperament has similar adjustments on all the notes, by the way. All the notes are some amount out of tune, on purpose, to make it all come out smoothly. But it is much more difficult to get these adjustments balanced exactly evenly, especially without the help of a machine.

The drawing turns into sound

As soon as I translated Bach's drawing back into the math numbers, and set it up with the careful listening skills to do these fine adjustments sitting at a keyboard, my harpsichord suddenly sounded amazing. (Well, yeah, I also had to know how to tune harpsichords in the first place, from years of practice doing it. A nice head start, always doing it "by ear" with careful listening and counting, instead of copying from electric machines. But this way of tuning was something I never heard of or even thought of before. It looks so weird that it seemed it could not possibly work, it's so different from all the others. It is so different from the other historical ways. But I tried it anyway.)

The way it sounded, so beautiful and unexpected, made me cry.

I pulled out music books that I had not played for 10, 20, even 30 years. Everything works.

When the keyboard is set up exactly this way, all of Bach's music can be played, without any of the problems I had known from more than 20 years of playing it! It was like hearing all of this music for the first time, finding new treasures in it, and it is absolutely not boring.

I tried it many more times, and Bach was right. With practice doing it this way, I can tune the whole harpsichord in 15 minutes like him and it sounds amazing, without being very hard work after all. Just follow the instructions in Bach's drawing, and it comes out exactly right and it is easy with a little bit of practice. A teenaged student could get this, taking tuning lessons sitting at the harpsichord with a good teacher, and using this as a textbook of the right way to tune. That was the point.

(My research paper in Early Music explains all that. The history lessons, and the tuning instructions that come out of the drawing, and why it matters so much in the music Bach wrote for people to sing and play. The paper is crossover from history to art to math to physics and back to music, all at once. I enjoy figuring out problems that really belong in a couple of different fields at the same time.)

This tuning method was the answer to a puzzle I had worked on for literally 20 years when tuning harpsichords. Sitting right there, this silly little picture of a spirally snaky thing that Bach wrote down. Sitting right there, top thing on a title page, for a book that says it's about tuning, and playing in all the keys!

As a conductor friend said to me when I showed it to him the first time: "DUH!"

But I and everybody else for 250 years had missed it, even when working seriously on the problem. Yet the answer is sitting right there in the most obvious place Bach could have put it, his book about tuning.

There's a really fancy word that means "hiding something in plain sight." It is called "Steganography". Just in case you wondered if there was a word for that, now you know. People have looked at that Bach drawing for some hundred years and thought it was just a bunch of border decoration. The official Bach research edition of music even said so, three times starting in 1963. They call it "ornamental flourish" (in German, the nifty word "Schnörkel"!).

Say that fast five times, "Schnörkel steganography!"

I was born after 1963. So, officially, for my whole lifetime this spirally drawing has been declared meaningless, just a bit of decoration. No wonder the history books have paid no attention to it! History is so focused on things that are described in words, not in pictures. Names and dates and written books in words. Pictures are just for children or just for fun. Right?

When the people that Bach taught (his own kids and other people's) eventually died out, this secret tuning method got forgotten. So did the understanding that Bach even wrote it down at all, or that this spirally drawing is it. People were more interested by the 1760s to 1790s in some of the other ways to tune, including more use of equal tuning, which was starting to get more popular than it was before.

Bach's really amazing-sounding way to tune got lost. Well, one other musical scientist I found in my research (a man named Sorge) also wrote down something very close to it, in 1758, and he knew Bach personally. They were buddies together in a society of musical scientists, and joined it the same month!

But eventually Sorge died out too and anyway, it all got lost. The organs got wiped out or retuned other ways, and it was forgotten that Bach ever wrote down his semi-secret method at all, for that job interview way back when he was only 37, in 1722.

By the way, Bach got that job.

Checking out the answer

I did more research into all the reports of Bach's concerts, and his pickiness in getting things exactly the way he wanted. I realized that nobody else could tune the way he liked because his way sounds different from everybody else's. This way.

I contacted several other harpsichord players, organ builders, and professors of music history and mathematics. I showed them my historical research and the mathematical instructions I had figured out from the drawing. They tried this method on their own and confirmed that it works.

Testing the Goshen College organ using this tuning One organ builder named George Taylor even set up a whole new organ this way, lots of work, to test out if it really works and (most importantly) because it makes his organ sound better in the music. He invited me to his workshop to play it, and it made me cry again because the sound of this tuning is even more beautiful than I imagined it would be on the organ.

Bach knew better than almost everybody around him how to build organs and make sure they are set up right. Well, that is in all the history books too. All these pieces of music history about Bach make sense together, now hearing the way this sounds, and seeing what those people meant back then in describing this particular sound!

This organ is finished as of January 2005, and installed in a new concert hall for college students to take organ lessons and have concerts and church. (That's me in the picture there, testing it in December during the installation process.)

I checked out Bach's other music, both earlier and later than 1722, playing through all of it. It became clear that a lot of it works only when tuned this way, or in equal tuning where nothing matters.

It all comes back around to the "Well-Tempered Clavier". The whole book is about tuning, and having all the scales and chords sound different, all those "Do Re Mi" and all the "Re Mi Fa". Bach said so himself, right there on the page, that he's composing music in all the "Do Re Mi" and all the "Re Mi Fa". And that drawing at the top of the page tells how to set it up exactly. All these pieces of history fit together, like this giant puzzle being solved.

There are some other organs that have been built or retuned since early 2005, and concerts and recordings on other instruments too. I am trying to keep track of these, with a list over here. My own recordings using this are also published, January 2006, giving people more chance to hear it.

Wait, why is this so smooth?

Why does Bach's way of tuning work so nicely, while all the more familiar ones do not?

All the other tuning instructions I had used for 20 years come out sounding pretty much the same as one another, as to deciding which scales and keys should be best. The history on all that is very clear. They all go back to normal tuning methods in the 1600s, and then they are little variations on one another, going into the 1700s. They all have different adjustments but the same few strategies. All those other tunings make the major keys of D-flat major, F-sharp major, B major, and A-flat major be the worst out of tune, so all the other keys can be better.

Bach's way was different because he picked different keys to be "worst". In his tuning, C major and F major are the best in tune (like the other tunings of his time also have). But his end of the line is the surprising E major! That is only four sharps. Then, B major (5 sharps) and F-sharp major (6 sharps) are set up to sound better in tune than E major is. When we get around into the flat keys they also sound smoother than E major and A major do.

The result is: the chords with sharps such as D major, A major, and E major sound especially exciting, because the note written with a sharp on it (like the G# in the E-G#-B chord) is especially high in pitch. The "out-of-tuneness" gives the music tension that has to relax as the music moves forward. All very careful and tasteful, not too much out of tune to make us go "Eww!" but just enough to be interesting. On the other side of things, music written in the flat keys sounds especially calm and mellow, warm and rich.

Personally, I am very fond of those sharp keys too, like E major and A major and C-sharp minor. They don't really seem out of tune at all, but just especially bright and exciting, sort of like looking at yellow and orange in a picture. The flat keys sound more like blues and purples and deep dark red, sort of. I don't see those colors but they make me feel the same way I feel when listening to music played in these keys, with Bach's tuning.

The special secret spice is the black notes

Bach's big difference here is the way he treated sharps and flats! All the "white" notes on the keyboard sound normal according to the other tunings of his time. That goes back to the way they taught singers and players how to do their whole steps and half steps, remember? The half steps that use only the white keys, like "Mi" to "Fa", "how I wonder" coming down in "Twinkle Twinkle", those are called the "diatonic" half steps. Those are all tuned the normal way from the late 1600s, called "1/6 comma meantone" or just "regular one-sixth".

Then Bach's way does some tasteful little things, adjusting the rest of the notes so it all works out. We already have all the notes C-D-E-F-G-A, our white notes, in their normal places from "regular" tuning. That is the starting point and we are halfway done, with those six notes out of the total 12.

He puts the note B in its regular place, but then just a slight bit higher in pitch. Hardly noticeable. But if we call the notes B to C "Mi" to "Fa" like in G major, it is very slightly different in sound than if we used E to F, "Mi" to "Fa" in C major. Very slightly, just enough to make the C major and G major scales a little different from each other. The B also makes a smooth transition up to the high sharps.

Then he does similar adjustments to the five black keys on the keyboard. F#, C#, and G# get higher and higher as compared with their normal spots, where most of the other tunings put them. E-flat comes out to be at a pretty normal spot. B-flat is put at a normal spot and then dinked down just a little bit, so we can also play it like an A# (remember, they're "really" different in music!).

The G# is so high out of its normal spot as a "real" G#, that we can now play it also as an A-flat. And likewise for C# being high enough that it sounds good when the music says D-flat. (As a matter of fact, the pitch C#/Db is exactly halfway in the hole between A to F, going first from A up to C#, and then the rest of the way from Db to F. That is the strange thing about C# in Bach's tuning, and it's not true of the other black notes here, only C#/Db.)

Don't worry too much about this point, I know it's already pretty weird to think of G# and Ab as being different notes at all. But that's the way children were trained to sing back in 1700, that those two notes are absolutely different and have different pitch. I checked a singing textbook from back then, to make sure. And Bach's tuning here solves the problem by putting these black notes somewhere in between where they belong with their two different names, like G# vs Ab.

Each of these black notes has its own personality, as to how much it's tastefully dinked out of place, compared with the place we expected to hear them. A little bit higher or lower, on each of them differently, exactly according to the recipe in Bach's spirally drawing. The drawing tells exactly how much to dink them out of tune, on purpose, so it all works out. This lets all these notes work very well when we play them as either sharps or flats.

And, it helps all of our major scales and minor scales to sound slightly different from each other. When the music is written in different keys, it sounds different! Hardly enough to notice in thinking about it, but it feels different to listen to. That is described by the German word "Affekt". The Affekt affects your emotions, your affections, your feelings. And it's built right into Bach's tuning!

Music is supposed to help you feel happy and sad and lovely and angry and calm and excited, at different times, sometimes a bunch of those things all in the same song. It is supposed to. If it doesn't, somebody isn't doing their job right. The people in the 1700s, especially, wrote reports that music makes people cry and laugh, and feel excited and relaxed and surprised. That is expression, doing music with feeling. One of Bach's sons wrote that you have to feel it inside yourself when you play music, before you can help other people feel it too. Play it with emotion and don't sound like a trained bird!

The cool thing is, I'm pretty sure that he tuned keyboards this same way that his dad, his teacher, did. This special tuning works for all of his music too, and makes it sound totally colorful and emotional.

Due to the very special and rare thing this tuning does with the black notes and the B, they work nicely whether the music says it is C# or D-flat, sharing the space on our keyboard. Each black note works smoothly both ways, no matter what else is around it.

And each one of those black notes brings some special emotional spice to mix into the music. The music sounds seriously different as it goes along, whenever it changes keys temporarily. The scales and emotions keep moving around, and they all sound different and they all work smoothly. That seems amazing just talking about it, but you really have to try it at the keyboard playing all the scales and chords, to get what I mean by this.

That specially uneven handling of the black notes is the brilliant compromise that lets this tuning sound so beautiful. It makes there be no "dead ends" where the music would sound too strange. We can play and learn music in all the keys, and it all works, all of the time.

And because of the way all the keys sound a little different from each other, all those "Do Re Mi" are different depending where we started: the music comes out sounding more beautiful and interesting, with variety in it. And it makes our feelings do different things that we don't experience in equal tuning. It is like seeing a picture in full color, where we are used to seeing only the black and white that equal tuning gives us. It is so much more emotional. It is built right into the music.

Another thing I've noticed is that this tuning makes improvisation at the keyboard much easier, and feel more natural. The sound is inspiring. In itself it suggests the ways the melodies can go next, and the ways the harmonies move most naturally. And sure enough, Bach (like Mozart and Beethoven) was especially famous as an improviser too.

The art of playing keyboard is playing and improvising and composing and tuning, all parts of the same art, the same bunch of skills all woven together. It is not merely reading through other people's music and working on that carefully. It is playing music as if it is new, and maybe it really is different every time.

Bach's tuning does all that?

This Bach tuning gives the advantage of being able to play everything, all the time, without changing the tuning...and to have it sound more colorful and interesting in all music, all the time. It is just as flexible as equal tuning is, to play everything. But it is a lot more intense and colorful and beautiful.

That is what belongs in the history books, about how great a musician Bach was and his demonstration of music in all the keys. This is all important because the music sounds amazingly rich, and because it helps us appreciate and understand Bach even better.

Even if Bach wasn't fond of math and spelling and words, he was one of the best musicians that has ever lived. His special way of tuning makes that even clearer.

I'm sorry this explanation was so long, but there is a lot to it. There is a lot more to it than this, yet. It is very exciting. The way it all ties into music with words (cantatas, masses, passions, solo songs) is amazing too...and all the chamber music and orchestra music, not only the keyboard and organ solos.

The published paper from 2005 is even longer (two parts of about 20 pages each!), and much more technical and condensed. It shows how the music and math and history puzzle was even more complicated than this, and shows more deeply why this tuning works in Bach's music (and other people's music).

I wrote another one in 2021-22, "The Notes Tell Us How to Tune", to present everything more clearly as direct musical practice (i.e., with as little math as possible). It's adjusting the sharps and flats as compromises so everything will work out. It's not secret codes, and it's not merely interpreting a drawing. It's musical work, hands on, tuning a harpsichord quickly by ear.

I want everybody to understand this, at whatever level they look at it. The point is that the music sounds so beautiful, and Bach was brilliant to know exactly how to set it up.

Here is that drawing one more time, below. It's flipped so it can be read straight out, with C there in the second position from the left. It's all so obvious now once you've seen it, right? Maybe, maybe not. But it's obvious once you've heard the way it sounds set up on keyboards. That drawing is pure math turned into musical sound.

Sample recordings to listen to

Here are a few short recordings so you can hear what this sounds like.

From a 2005 practice session, here's a performance by me of the Prelude in C major from Bach's "Well-Tempered Clavier" book 1. That's available in better sound on a CD that is now available for sale, since January 2006. A web sample of that is here, including the fugue and some other pieces. I made some recordings on organ tuned this way, too.

Lullaby, click to enlarge this... Here also are recordings of the little lullaby that is printed here, some easy three-part music. These were played into a cheap video camera and then copied down to MP3 files. First I played it in the original B major where I wrote it (five sharps), and then again in B-flat major (two flats). Click on the music to enlarge the picture of it.

Other musicians have been making CDs using this tuning, which is exciting!

When I listen to these, it seems almost like equal tuning but I notice some small differences. The basic difference is more in character, like the "color" or how cool or warm it feels, bright or mellow, spicy, sweet, or whatever. Everybody will think up different words to describe this. The important thing is that there really is difference in there that can be measured, next to the way it makes people feel listening to it.

As Bach said about the way he played music himself, just hit all the right notes at the right time, and the instrument does the rest of the work. That's true, as long as the instrument is set up the right way to begin with, in his tuning!


Additional samples are here, and some whole CDs (by myself and some other people) are listed here.

See also the paper "Bach's Art of Temperament" (spring/summer 2006) for another introduction to this topic. That one brings up some of the other styles of tuning keyboards. It says some more about the "Do Re Mi" setup, emphasizing melody and scales instead of math.


Bach's schematic, rotated for use