I guess I must be dumb or something, but I'm simply not seeing the problem.
Imagine the piano had only white keys, no problem right? Now just place the black keys at the back, between some of the white keys, right in the middle, such that each black key takes like a quarter of the width of the sandwiching white keys.
Now what's the problem with this again? Can someone explain in clearer terms?
If the issue is that we are trying to make the white key all have the same width at the back, well, why should that matter? Pianists don't press the white keys all the way at the back, do they?
Pianists don't press the white keys all the way at the back, do they?
The good ones do it all the time because moving the entire hand forward and back can be significantly more fluid than contorting to play another way…the keyboard is three dimensional.
Pianists sometimes do have to play white keys with their fingers between the black keys. For example, if you want to play E flat major (Eb-G-Bb), you might put your thumb on Eb, your little finger on Bb, and then your middle finger on G, right up the back between the black keys. As the keys are parallel, the width of the white keys at the back is the same as the width all the way down to the front of the black keys. Apart from the need to fit the hand to the keyboard, a piano key is a lever and it gives more resistance further back, which may be useful to the pianist in controlling the attack and dynamics.
If you place the black keys right in the middle, as you suggest, the space between them is too narrow for a finger, while there is wasted space for those white keys that only have one adjacent black key. So piano makers push the C# and D# keys and the F# and A# keys further apart.
The mathematical problem discussed in the article is that there is no way to distribute the space equally between the keys, so various compromises are considered. The "B/12 solution" is practical and widely used. The suggested "optimum arrangement" is amusing to consider but unlikely to be worth the trouble.
That design you describe is what is pictured at the top of the article.
Problem is that then the keys are not equally spaced chromatically (e.g. larger spacing between B and C than between C and C#).
You could probably get used to play like that, but it would be ineficient in terms of space for both the fingers and the mechanics of the piano (hammers, strings).
So what you do, in reality, is move some of the black keys down a bit (C#, F#) and some up (Eb, Bb) so that the spacing between the center of the keys is regular.
I don't think that's what's described in the article though?
Yes, you do need to press white keys further back sometimes. Imagine trying to play on black keys with your thumb and pinky finger while playing a white key with your middle finger. You won't be pressing all the way at the back, but your finger will have to press between the black keys.
On my (accoustic) piano, the black keys are just as wide as the back ends of the white keys. This is achieved by shifting the position of the black keys a bit, instead of centering them right between the white keys.
The point that the article is addressing (but you have to ignore the image and study the equations to see this!) is that this sort of shifting can't equalize everything. In the span of 3 white keys C to E at the front, you have 2 black keys at the back, so if you take r to be the ratio of back-width to white key front-width then you have 3 = 5r. But in the 4 keys F to B, you've got 3 black keys so 4 = 7r. No single ratio works! So the article investigates various compromises. The B/12 solution is what seems to me the most straightforward, divide white keys in each of the sections C to E and F to B equally at the back, and don't expect anyone to notice the difference.
I don't see the problem... Use one unit of width per semitone. Then raise the black keys up a bit. Then for the white keys, elongate them and append some extra stuff on the sides of their fronts so the white keys' fronts' all have one same width as well. They are two separate "problems", not interdependent.
I also have a MIDI keyboard (M-Audio Hammer 88) which follows the same model.
I'd like to see a photo of someone's piano that uses a different system, really I thought they were always this way. It's a good system because it lets the black keys be spaced a little further apart, while also reducing the jump between black key clusters.
As the article says, it is impossible to make the back ends of the white keys equal in width so piano makers have to compromise. The difference is about a millimetre so many pianists may never have noticed.
The naive approach of placing each black key at the midpoint of its adjacent white keys makes B, C, E and F quite wide at the base, but it is harder to fit a finger between the black keys to play D, G or A, which can be necessary when the hand has to stretch or play both black and white keys. On a real piano, therefore, the C# and D# keys and the F# and A# keys are offset a little from the midpoints of their adjacent white keys. G# is the only black key that is actually at the midpoint of its adjacent white keys.
In the photograph of the Yamaha piano, you can see that the cutouts on the D key are symmetrical but less than half the width of a black key because the C# and D# keys are offset. Looking at the G key, the right cutout, at half the width of a black key, is deeper than the cutouts on the D key, so to compensate the left cutout is less deep than the cutouts on the D key. As a consequence, the cutout on the F key is deeper than the cutout on the E key, so the E key is wider at the back than the F key. Similarly, the C key is wider at the back than the B key.
As described in the article, on some keyboards just the C and E keys are wider at the back, with D the same width at the back as F, G, A and B. More often, however, the C# and D# keys are placed a little further out to spread the extra width equally between the C, D and E keys.
Good point about the ~1mm difference. Interestingly on my MIDI keyboard the difference between white key widths is even larger (maybe ~3mm max), showing some variation in style.
By the way your two comments in this discussion had been voted down and hidden (marked [dead]) for some reason. I voted for them to return and I'm happy to see they've now been reinstated.
There is a good [dead] comment reply to my comment here by user sefn where they point out that in fact the spacing is not exactly equal, it varies by about 1mm (and they're correct).
If you've got "showdead" on, you can see it, click the timestamp on it, and click "vouch" if you want to vouch for it to come back. Seems inappropriately killed to me.
They have a second good [dead] comment in this discussion which I've also vouched to return.
This is fascinating! I’ve never played a YC seriously, but I have played several Yamahas and currently play on a Kawai and Baldwin. I’ve often wondered if the Baldwin or Yamaha is laid out slightly differently than the Kawai because I feel like playing broken 4 note chords the fingering can feel off on one piano vs another. It’s a slight stretch but the 4-5 on the second 4 note chord can be uncomfortable on the Kawai and comfortable on the Yamaha. I never play them in the same room, one is at my teachers and one at home.
Very interesting! Is there a spec for this? Or a layout description? Surely something as precise as piano would note this.
Generally speaking fumbling on a piano doesn’t bode well for performance… it’s a little bit like Olympic gymnastics, you only get one chance to stick the landing!
I never understood why the piano keyboard isn't regular. It forces players to remember different positions for the same chord transposed to start at different notes.
Like why do I have to remember the shape for C major and D major chords? It should be the same shape just starting at C vs D.
It's not even that hard to fix. There's 12 semitones in an octave. Just make it 6 white 6 black keys.
It forces players to remember different positions for the same chord transposed to start at different notes.
The piano was developed well before equal temperament came to dominate tuning. [1] So each musical key would have different harmonic relationships between the intervals within it. And musical keys were not thought of as equal.
Generally, the musical keys based on “black keys”/“sharps and flats” would be farther from an ideal tuning and there were better and worse sounding keys depending on which musical keys a piano was tuned for.
Historically in Western European music, there were preferred keys and intervals inherited from Plain Chant (roughly C,G, & F and octave, perfect 4th, perfect 5th, and the 6th).
Of course using an electronic instrument that can be electronically transposed up and down by half steps might be an easy way to avoid learning lots of fingerings.
With the irregular layout we have now, some keys are easy (e.g. C) and some are hard (e.g. B). If you make the layout regular, putting a black note between every white note, then every key becomes the same, but also quite hard, because every major scale is now played like this: https://i.imgur.com/6EmW8eU.gif
It's not a particularly good tradeoff. If you got rid of the black keys entirely instead, you'd have to remember which keys to skip. Harder for beginners than just playing in C.
That regular layout also makes it likely to feel better to the hand to play diminished or augmented chords, while comparatively punishing major/minor chords. It would be an odd choice considering the traditions of western music.
There's also something to be said about each key having a specific motor pattern/spatial layout. Sure, it makes it harder to move knowledge of one key into another, but during playing it also makes it easier to not accidentally completely change the key unless you mean to. It's all tradeoffs.
Can you elaborate on your thinking here? I really feel C is the easiest (in standard major tonality - obviously C minor, blues scale etc are a different story).
The white keys form a sequence of notes (frequencies) that is known as the diatonic scale. It's the foundation underlying all popular western music. It is not random or arbitrary, it has some nice dual mathematical and musical properties: intervals between the notes in the scale have special frequency ratios that sound pleasing to the ear (read Helmholtz's "On the sensations of tone" for a fascinating physically-based take on why it is like that -- he is known as "the father of acoustics", and that book contains the distillation of 8 years of deep, smart research way before we had the means or understanding we hav today). A ton, if not most, of popular music can be played using only the white keys.
There used to be keyboards with other different arrangements, which were actually extremely cumbersome and actually didn't allow very rich and interesting musical excursions like modulations (look up "microtonal keyboards"). Today's standard keyboard and tuning is a compromise between those fundamentally mathematical and perceptual acoustic relations (the tonic, the fourth, the fifth, the sixth, the major and minor third, the "sensible" or subtonic...) and the ability to perform those trans-tonality excursions. A fully regular keyboard like you propose would lend itself more easily to those excursions, at the cost of being less apt at the foundational diatonic model and most popular music.
Interestingly also, the notes used by modern keyboards and all modern instruments, and to which we are all so accustomed that we thing it "just is", is an imperfect compromise that needed a lot of selling back in the day, much of which was done by Bach (the compromise scale is called the "tempered scale", and Bach authored the arch-famous "Well-tempered clavier" pieces to show it off -- impossible to perform on keyboards with other tunings).
And of course, there is a tradition factor. English isn't written like this because it's optimizing for any easily describable or measurable optimization metric, more like it minimized a socio-perceptual function covering many centuries of UX.
Finally, if you want an instrument where all keys are equal, you can always move to a fretboard based one like the guitar. Funnily, it has a one-semitone-short jump between strings 3 and 2 that will throw off the desire of full regularity... again due to diatonic leanings. A bass guitar is fully regular, even when they add a 5th and 6th string, so that may fulfill your wish of a fully regular instrument... and it sounds awesome! Just can't do the same things as a piano or a guitar.
I agree, the white keys on a piano represent a diatonic scale, but because today’s pianos are rarely tuned to anything other than 12TET, there are few interesting mathematical relationships between notes in practice (and pianos are normally tuned with high notes sharp and low notes flat because that’s how piano strings tend to produce their partials anyway).
Also worth noting the black keys represent a major pentatonic scale and the major pentatonic scale is how many of the earliest bone flutes are tuned.
>Interestingly also, the notes used by modern keyboards and all modern instruments
Vast majority of fretted instruments since the death of the lute are untempered.
Edit: Which is not to suggest that lutes were tempered. Lutes and other tied fret instruments allow for unequal fret spacing so you can temper one string at the cost of more notes being more off from the temperament on other strings, or the frets being at an angle so you could find a bit of a compromise. But often they were EDO or in the ancient tradition of fretted instruments, close enough for rock and roll.
I never heard someone describe a tuning system as "untempered", but I guess it would mean something like just intonation -- which sounds really great for playing anything in a specific key but falls horribly apart if you try to change the key (which is why it has seen very little use since the renaissance).
Equally divided octave (EDO) with no tempering which is distinct from Equal temperament. Tempered scales are generally EDO with tempering. Other methods like just intonation don't really need to be tempered and generally are not in my experience, but it has been years since I was into just intonation and may just not remember. Historically speaking, the advantage of justly tuned scales is there is no need to temper it because it is already just and perfect, things may have changed in 20th century as far as just intonation is concerned and tempering, don't recall.
Edit: ET and EDO are essentially the same in the case of most fretted instruments, I am dredging long forgotten stuff from memory here and somewhat off above.
Edit2: Refreshing my memory some and seeing how much things have become muddled in my head over the years. Clearly I did not even consider what came out of my memory and just regurgitated it verbatim. ET scales are not tempered but do not mean EDO. Guitar and the like are both ET and EDO. ET and EDO are untempered in the sense that notes are not shifted slightly away from the EDO/ET as they are on the piano and many instruments.
I don't see how you can divide the octave equally and not end up with equal temperament: that's exactly what equal temperament is!
> Tempered scales are generally EDO with tempering.
That's not historically accurate. EDO wasn't used until very recently (about the middle of the 19th century I think), tempering was used way before that.
For example, the first widely used temperament (which became popular in the Renaissance) was the quarter-comma meantone, which shrinks each fifth (from the natural 3/2) so that the major thirds are perfectly 5/4. The name "quarter-comma" means that the amount of shrinkage is 1/4 of the "syntonic comma", which is the difference you get beteween going up 4 fifths (e.g. C->G->D->A->E) and a major third plus 2 octaves (C->E->E->E). Those final Es can only be the same if you shrink the fifth or stretch the third (or both). What this tempering does is shrink each fifth by 1/4 of the difference (so that going up 4 fifths closes it) and doesn't touch the major third. That means the major thirds are beautiful, and the fifths are a little off. For a chosen key, that is -- everything sounds horrible as soon as you try to change the key too far away from the chosen key.
In the Baroque period a lot of other temperaments were invented, the Werckmeister temperaments were very widely used in (what is today) Germany for example (a lot of people believe Bach had one of these in mind when writing the Well-Tempered Clavier). Those temperaments were also defined by how much each fifth is changed from the "normal" 3/2, but each fifth was to be changed by some different amount in some complicated way.
It was only much later that EDO (12-TET, or "equal temperament") started to be widely used. You can think of it (and people do!) as a "temperament" because it just means you shrink the fifth from the "normal" 3/2 = 1.5 to be instead 2^(7/12) =~ 1.4983, so that going up 12 fifths lands you exactly 7 octaves above (since 2^(7/12)^12 = 2^7). That also means that the octave is divided exactly equally, because going up 12 fifths goes through every one of the 12 notes before going back to the original note.
EDO on fretted instruments goes back to at least the 16th century and was essentially the standard well before the mid 19th. Equal temperament is an EDO scale whose divisions approximate justly tuned scales. The western 12TET scale is not actually 12TET or 12EDO, we temper the scale itself and tweak some notes to make it work better unless you play fretted instruments and then it is up to the guitarist to make small adjustments in their playing technique so their untempered 12TET is in tune with the pianos tempered 12TET.
> The western 12TET scale is not actually 12TET or 12EDO, we temper the scale itself and tweak some notes [...]
I think you and I must be using words differently. To me (and to Wikipedia, and everything else I've ever read, including[1] which I just consulted to make sure I'm not crazy), 12TET is a way to specify by how much you have to multiply the frequency of the first note of the scale to get the other notes' frequencies. Wikipedia[2] has a table with the numbers for 12TET (the column "Decimal value in 12-ET"), but it's very simple: you just multiply the value of the preceding note by 2^(1/12). If you take 12TET and adjust/change the notes a bit, then it's not 12TET anymore.
> EDO on fretted instruments goes back to at least the 16th century
I'd love to see a reference for that. I just consulted [1], it has a chapter called "Non-Keyboard Tuning" and it doesn't mention that (although admittedly it spends most of its time talking about violin, with a ton of references to stuff that Mozart said). The book does say that equal temperament was known for centuries before it was used, but the people who first discovered it simply didn't think it sounded good.
[1] "How Equal Temperament Ruined Harmony (and Why You Should Care)" by Ross W. Duffin
Try the wikipedia pages for Equal Temperament and Musical Temperament, they explain all of this.
Here is a 1688 Stradivari[1] guitar with fixed frets and a EDO octave, they were reasonably common by that point. Much of the information regarding this is looking at the fixed frets that many lutes and guitars had applied to their soundboard and comparing that to how composers used those fixed frets, either the tied frets adhere to the scale of the fixed frets or they are out of tune. The history of EDO/ET in fretted instruments goes back to at least Vincenzo Galilei[2] (father of Galileo) who developed the rule of 18 for fret spacing. If memory serves we have a few early steel string instrument (cittern, bandora, orpharion) from around ~1600 with equal spaced frets and this orpharion[3] looks it but it is difficult to tell from that photo. Going back earlier things get more difficult since we have so few intact and unaltered instruments but we do have a fair amount of ingravings and art plus writing on the topic such as Galilei's.
There is a paper going into great depth on all this that is just out of reach in my memory and I can't seem to trick the search engines to give it to me, I will post it if I remember/find it. No time to dig more right now.
If you remember and have the time, please do! (And thank you for the links you already posted).
I see now that everything I read about this was way too focused on keyboard and violin, since I had never heard any of this about fretted instruments. I'm glad I get to correct a bit of my understanding, so thank you. Now I'm left wondering about wind instruments.
Western theory is focused on the keyboard with violin as second fiddle, the rest of the instruments do their own thing unless they are forced to kowtow to a piano or violin. Each instrument is ultimately tuned to the physics which dictate how it makes sound and this is part of why our tempered 12TET works and why we see the rise of the big orchestras of unlike instruments with the move away from meantone, it provides a compromise which works quite well for all instruments with the exception of the brass (with the exception of the trombone) who are forever out of tune (sort of).
Part of the reason beginners sound bad is because most instruments have to bend notes to be "in tune," I can teach anyone to play a chord on the guitar and get them having each note sounding clearly in a couple of minutes but my DMaj will sound better than theirs simply because I have played that DMaj thousands of times and my fingers have learned to adjust the pressure on each string in just the right way to make it sound "right" just as the woodwinds learn to bend certain notes and the brass learns to live with being out of tune.
Also part of why the lute became such a dominant instrument is that it could retune in ways other instruments can not, which was a major advantage for the working musician back in the days when every city had its own idea about tuning; nudge a few frets, retune a few strings and accept that certain notes were now out of bounds and you could play with anyone like you were playing in your native tongue. As tuning became more standard the lute started to die.
Meantone Temperaments on Lutes and Viols might be of interest to you, it is aimed towards lutenists and violists but has some more general stuff as well and I think does a good job of showing the compromises the lute (and viol) had to make in moving away from equal temperament.
I don't really think brass is forever out of tune, I love the brass and used to play trumpet but the brass section is more under the influence of the physics of its instrument than anyone but the pianist but the pianist is "in tune" because western theory is built around the keyboard.
To offer something better than this mess and correct. Fretted instruments are tehnically unequally tempered do to the physics the string, the fret board is EDO but in fretting the string we stretch it and raise its pitch. Fretted instruments use various compensation tricks to lessen this effect but most notes are off from both 12EDO and 12TET, we get the open string and the 12th fret but the rest are off by varying amounts.
Thank you for offering something clearer than the mess I made. Been 25 years since I studied this stuff and finally learned to just accept (and love) the modern standard.
There are such things are Chromatic keyboards (for instance, the Chromatone synth had one), and there was the Janko piano keyboard and there are people that have made keyboards without the spaces in at the B/C and E/F at the back of the keyboard by alternating the white and the black keys such that those spaces just disappear.
The advantages are:
- you need less reach for the same chords
- transposition becomes trivial
The disadvantages are:
- your muscle memory will be invalidated
- the number of instruments set up like that is really small
- no accomplished pianist will want to switch
But if you really wanted to you could adapt an existing instrument to use a different keyboard and it isn't even all that complicated (medium complexity wood working project).
I'm picky about layouts - I type in Dvorak, learned Janko via Chromatone, currently playing harpejji.
Coming from a classical piano background, there was definitely a learning curve, but I feel like it was worth it. Every chord shape is identical across all keys (C major and D major would be played the same way), which makes it much easier to learn jazz voicings or modulate a song.
If anyone ever builds a quality grand piano with Janko layout, I'm buying! Hacks on hacks become unnecessary if you start with the right design.
Chromatic button accordions have each octave in three rows of four instead of two rows of five and seven like a piano. It's very regular, but doesn't match up with major or minor scales or with sheet music. A major scale is a zig-zag.
I play both piano and button accordion and they're just different. Neither one has a compelling advantage.
having the keyboard the way it is also allows you to more easily orient yourself, you can feel with the sides of your fingers if you are next to E/F or B/C and with the corner of your eye it's also straightforward to figure it out. I don't think it'd be possible (or anyways even more difficult than it is now) to play large jumps accurately if the whole keyboard looked the same
There are multiple alterative layout that some advocate for. they generally do sonething else for orintation. putting a bump on middle c and other places.
That's a bit like complaining that there are six different kinds of chess pieces and you have to memorize how each moves. The truth is, if you have trouble remembering how a knight moves, you can't be that good at Chess anyway.
Remembering twelve different ways of playing a scale is a vastly small part of learning how to play a piano.
> if you have trouble remembering how a knight moves, you can't be that good at Chess anyway.
Non sequitur.
It’s also still a valid question. I play the scales really well on the guitar. And because the frets are all laid out straight, shifting up by one fret means I’m just playing my chords and scales sharp. It makes transposing music incredibly easy.
I still don’t understand why the piano can’t be laid out like that.
Guitar still has the problem of last 2 strings being shifted 1 semitone down for ergonomics reasons, which makes them irregular. But yeah, it's better than piano keyboard.
Except it would render a large swath of the repertoire from the common practice period (“classical music“) more difficult to play because it is written with the presumption that the keyboard is just so and that some future generations won’t try to optimize it.
The organization of the keyboard does necessitate certain fingering choices that are particular; but knowing this, composers have (usually) written in a way that respects that geometry.
The geometric relationship between the note frequencies of a C major chord and a D major chord on the piano is not the same. The key of piece is responsible for some of its “feel”. So it’s not unreasonable that they have different “representations” on the piano although the differences may be subtle.
There aren't 12 shapes. My fingers say C maj, F maj and G maj are the same shape. D maj, E maj, A maj too. And A min is the same shape as C/F/G maj, but would be a different pattern on the proposed "regular" layout. That may not matter when all you want is transposition, but as I said before: transposing is not so important. And there are "shape" based keyboards, as mentioned frequently in the comments. You can even retrofit some on your own keyboard. They're just not popular.
C maj, F maj are the same, but C maj7 and Fmaj7 are not. Ultimately you have to remember which chords are the same and which aren't - which is about the same amount of information as just remembering all the combinations.
It's just an ugly irregular hack. I understand that you can get good at working around it, but eventually it's worth it to fix the underlying problem.
I always thought the canonical way to place the black keys was to divide the octave in the two parts that have the sequential black keys (C-D-E and F-G-A-B), and then simply place the black keys so they're the same distance away from each other and the edge of the parts.
That means that the white keys in each of the groups have "mirrors": for example, C is a mirror of E, D is a mirror of itself (it's the only key like that), F is a mirror of B, and G is a mirror of A.
I just looked at the keyboards I have around me (a slightly-above-low-end digital piano, a small midi controller, and a small 90s synth), and they all seem to fit that description.
ETA: note that the image in the article doesn't fit this description: for example the D is way too narrow (the black keys around it should be much further apart).
ETA2: I just noticed that this seems to be the "B/12 solution" described in the article.
I suspect there is no such fussing about "back of white key widths" in actual piano design.
What's actually going on, when you look at any piano keyboard, is that the groups of black keys are given a substantially wider spread: they are not exactly centered on the dividing lines between the white keys.
What most likely matters to playability is the width of the black keys and this spread amount.
Now if you cover the front of the keys (e.g. put your fallboard felt over it) you can visualize the back of all the keys as a kind of barcode: alternating black and white strips, sometimes with two adjacent white strips. It so happens that these semitone strips do look about equal width, thanks to the sizing of the black keys and their spread. It might not be exact though.
A good starting configuration might be to start with a keyboard in which all the semitones are strips of equal width. Then we identify the C major keys, and paint them white, making the others black. Next, we shorten the black keys, and adjust the frontage of the white keys to be of equal width. Say, by keeping the division between every B and C exactly where it is and interpolating the others divisions. You will find that the E-F division does not fall exactly halfway between the surrounding black keys, Eb and F#. (That's what's observed on real piano keyboards!)
I guess I must be dumb or something, but I'm simply not seeing the problem.
Imagine the piano had only white keys, no problem right? Now just place the black keys at the back, between some of the white keys, right in the middle, such that each black key takes like a quarter of the width of the sandwiching white keys.
Now what's the problem with this again? Can someone explain in clearer terms?
If the issue is that we are trying to make the white key all have the same width at the back, well, why should that matter? Pianists don't press the white keys all the way at the back, do they?
Pianists don't press the white keys all the way at the back, do they?
The good ones do it all the time because moving the entire hand forward and back can be significantly more fluid than contorting to play another way…the keyboard is three dimensional.
Pianists sometimes do have to play white keys with their fingers between the black keys. For example, if you want to play E flat major (Eb-G-Bb), you might put your thumb on Eb, your little finger on Bb, and then your middle finger on G, right up the back between the black keys. As the keys are parallel, the width of the white keys at the back is the same as the width all the way down to the front of the black keys. Apart from the need to fit the hand to the keyboard, a piano key is a lever and it gives more resistance further back, which may be useful to the pianist in controlling the attack and dynamics.
If you place the black keys right in the middle, as you suggest, the space between them is too narrow for a finger, while there is wasted space for those white keys that only have one adjacent black key. So piano makers push the C# and D# keys and the F# and A# keys further apart.
The mathematical problem discussed in the article is that there is no way to distribute the space equally between the keys, so various compromises are considered. The "B/12 solution" is practical and widely used. The suggested "optimum arrangement" is amusing to consider but unlikely to be worth the trouble.
That design you describe is what is pictured at the top of the article.
Problem is that then the keys are not equally spaced chromatically (e.g. larger spacing between B and C than between C and C#).
You could probably get used to play like that, but it would be ineficient in terms of space for both the fingers and the mechanics of the piano (hammers, strings).
So what you do, in reality, is move some of the black keys down a bit (C#, F#) and some up (Eb, Bb) so that the spacing between the center of the keys is regular.
I don't think that's what's described in the article though?
Yes, you do need to press white keys further back sometimes. Imagine trying to play on black keys with your thumb and pinky finger while playing a white key with your middle finger. You won't be pressing all the way at the back, but your finger will have to press between the black keys.
> Pianists don't press the white keys all the way at the back, do they?
I do, and I'm not even a good pianist. Many chords will need it, just pick any chord that required the thumb or the pinky (or both) in black keys.
On my (accoustic) piano, the black keys are just as wide as the back ends of the white keys. This is achieved by shifting the position of the black keys a bit, instead of centering them right between the white keys.
The point that the article is addressing (but you have to ignore the image and study the equations to see this!) is that this sort of shifting can't equalize everything. In the span of 3 white keys C to E at the front, you have 2 black keys at the back, so if you take r to be the ratio of back-width to white key front-width then you have 3 = 5r. But in the 4 keys F to B, you've got 3 black keys so 4 = 7r. No single ratio works! So the article investigates various compromises. The B/12 solution is what seems to me the most straightforward, divide white keys in each of the sections C to E and F to B equally at the back, and don't expect anyone to notice the difference.
I don't see the problem... Use one unit of width per semitone. Then raise the black keys up a bit. Then for the white keys, elongate them and append some extra stuff on the sides of their fronts so the white keys' fronts' all have one same width as well. They are two separate "problems", not interdependent.
> Then for the white keys, [...] append some extra stuff on the sides of their fronts so the white keys' fronts' all have one same width as well.
There's no way you can achieve that.
Same, but I had to look. I wonder how badly this affects muscle memory?
In my experience, you fumble for a minute and then you adapt. I had a Young Chang that followed this model, and a Yamaha at school that didn’t.
Interesting, my piano is a Yamaha (specifically a 1980 Yamaha YUA, basically a U3), and it does follow the model (same width everywhere).
Photo: https://i.imgur.com/ulsLUoG.jpeg
I also have a MIDI keyboard (M-Audio Hammer 88) which follows the same model.
I'd like to see a photo of someone's piano that uses a different system, really I thought they were always this way. It's a good system because it lets the black keys be spaced a little further apart, while also reducing the jump between black key clusters.
As the article says, it is impossible to make the back ends of the white keys equal in width so piano makers have to compromise. The difference is about a millimetre so many pianists may never have noticed.
The naive approach of placing each black key at the midpoint of its adjacent white keys makes B, C, E and F quite wide at the base, but it is harder to fit a finger between the black keys to play D, G or A, which can be necessary when the hand has to stretch or play both black and white keys. On a real piano, therefore, the C# and D# keys and the F# and A# keys are offset a little from the midpoints of their adjacent white keys. G# is the only black key that is actually at the midpoint of its adjacent white keys.
In the photograph of the Yamaha piano, you can see that the cutouts on the D key are symmetrical but less than half the width of a black key because the C# and D# keys are offset. Looking at the G key, the right cutout, at half the width of a black key, is deeper than the cutouts on the D key, so to compensate the left cutout is less deep than the cutouts on the D key. As a consequence, the cutout on the F key is deeper than the cutout on the E key, so the E key is wider at the back than the F key. Similarly, the C key is wider at the back than the B key.
As described in the article, on some keyboards just the C and E keys are wider at the back, with D the same width at the back as F, G, A and B. More often, however, the C# and D# keys are placed a little further out to spread the extra width equally between the C, D and E keys.
Good point about the ~1mm difference. Interestingly on my MIDI keyboard the difference between white key widths is even larger (maybe ~3mm max), showing some variation in style.
By the way your two comments in this discussion had been voted down and hidden (marked [dead]) for some reason. I voted for them to return and I'm happy to see they've now been reinstated.
There is a good [dead] comment reply to my comment here by user sefn where they point out that in fact the spacing is not exactly equal, it varies by about 1mm (and they're correct).
If you've got "showdead" on, you can see it, click the timestamp on it, and click "vouch" if you want to vouch for it to come back. Seems inappropriately killed to me.
They have a second good [dead] comment in this discussion which I've also vouched to return.
This is fascinating! I’ve never played a YC seriously, but I have played several Yamahas and currently play on a Kawai and Baldwin. I’ve often wondered if the Baldwin or Yamaha is laid out slightly differently than the Kawai because I feel like playing broken 4 note chords the fingering can feel off on one piano vs another. It’s a slight stretch but the 4-5 on the second 4 note chord can be uncomfortable on the Kawai and comfortable on the Yamaha. I never play them in the same room, one is at my teachers and one at home.
Very interesting! Is there a spec for this? Or a layout description? Surely something as precise as piano would note this.
Generally speaking fumbling on a piano doesn’t bode well for performance… it’s a little bit like Olympic gymnastics, you only get one chance to stick the landing!
I never understood why the piano keyboard isn't regular. It forces players to remember different positions for the same chord transposed to start at different notes.
Like why do I have to remember the shape for C major and D major chords? It should be the same shape just starting at C vs D.
It's not even that hard to fix. There's 12 semitones in an octave. Just make it 6 white 6 black keys.
It forces players to remember different positions for the same chord transposed to start at different notes.
The piano was developed well before equal temperament came to dominate tuning. [1] So each musical key would have different harmonic relationships between the intervals within it. And musical keys were not thought of as equal.
Generally, the musical keys based on “black keys”/“sharps and flats” would be farther from an ideal tuning and there were better and worse sounding keys depending on which musical keys a piano was tuned for.
Historically in Western European music, there were preferred keys and intervals inherited from Plain Chant (roughly C,G, & F and octave, perfect 4th, perfect 5th, and the 6th).
Of course using an electronic instrument that can be electronically transposed up and down by half steps might be an easy way to avoid learning lots of fingerings.
[1] https://www.math.uwaterloo.ca/~mrubinst/tuning/tuning.html
Somewhat related:
https://strandbergguitars.com/true-temperament/
With the irregular layout we have now, some keys are easy (e.g. C) and some are hard (e.g. B). If you make the layout regular, putting a black note between every white note, then every key becomes the same, but also quite hard, because every major scale is now played like this: https://i.imgur.com/6EmW8eU.gif
It's not a particularly good tradeoff. If you got rid of the black keys entirely instead, you'd have to remember which keys to skip. Harder for beginners than just playing in C.
There is the Janko keyboard though: http://en.wikipedia.org/wiki/Janko_keyboard
That regular layout also makes it likely to feel better to the hand to play diminished or augmented chords, while comparatively punishing major/minor chords. It would be an odd choice considering the traditions of western music.
There's also something to be said about each key having a specific motor pattern/spatial layout. Sure, it makes it harder to move knowledge of one key into another, but during playing it also makes it easier to not accidentally completely change the key unless you mean to. It's all tradeoffs.
C is one of the hardest keys to play in on the piano
Can you elaborate on your thinking here? I really feel C is the easiest (in standard major tonality - obviously C minor, blues scale etc are a different story).
It's the easiest to learn because it's just the white keys in sequence.
It's not the most physically comfortable to play because your hand is not a rectangle.
The white keys form a sequence of notes (frequencies) that is known as the diatonic scale. It's the foundation underlying all popular western music. It is not random or arbitrary, it has some nice dual mathematical and musical properties: intervals between the notes in the scale have special frequency ratios that sound pleasing to the ear (read Helmholtz's "On the sensations of tone" for a fascinating physically-based take on why it is like that -- he is known as "the father of acoustics", and that book contains the distillation of 8 years of deep, smart research way before we had the means or understanding we hav today). A ton, if not most, of popular music can be played using only the white keys.
There used to be keyboards with other different arrangements, which were actually extremely cumbersome and actually didn't allow very rich and interesting musical excursions like modulations (look up "microtonal keyboards"). Today's standard keyboard and tuning is a compromise between those fundamentally mathematical and perceptual acoustic relations (the tonic, the fourth, the fifth, the sixth, the major and minor third, the "sensible" or subtonic...) and the ability to perform those trans-tonality excursions. A fully regular keyboard like you propose would lend itself more easily to those excursions, at the cost of being less apt at the foundational diatonic model and most popular music.
Interestingly also, the notes used by modern keyboards and all modern instruments, and to which we are all so accustomed that we thing it "just is", is an imperfect compromise that needed a lot of selling back in the day, much of which was done by Bach (the compromise scale is called the "tempered scale", and Bach authored the arch-famous "Well-tempered clavier" pieces to show it off -- impossible to perform on keyboards with other tunings).
And of course, there is a tradition factor. English isn't written like this because it's optimizing for any easily describable or measurable optimization metric, more like it minimized a socio-perceptual function covering many centuries of UX.
Finally, if you want an instrument where all keys are equal, you can always move to a fretboard based one like the guitar. Funnily, it has a one-semitone-short jump between strings 3 and 2 that will throw off the desire of full regularity... again due to diatonic leanings. A bass guitar is fully regular, even when they add a 5th and 6th string, so that may fulfill your wish of a fully regular instrument... and it sounds awesome! Just can't do the same things as a piano or a guitar.
I agree, the white keys on a piano represent a diatonic scale, but because today’s pianos are rarely tuned to anything other than 12TET, there are few interesting mathematical relationships between notes in practice (and pianos are normally tuned with high notes sharp and low notes flat because that’s how piano strings tend to produce their partials anyway).
Also worth noting the black keys represent a major pentatonic scale and the major pentatonic scale is how many of the earliest bone flutes are tuned.
>Interestingly also, the notes used by modern keyboards and all modern instruments
Vast majority of fretted instruments since the death of the lute are untempered.
Edit: Which is not to suggest that lutes were tempered. Lutes and other tied fret instruments allow for unequal fret spacing so you can temper one string at the cost of more notes being more off from the temperament on other strings, or the frets being at an angle so you could find a bit of a compromise. But often they were EDO or in the ancient tradition of fretted instruments, close enough for rock and roll.
Do you mean equal-tempered?
I never heard someone describe a tuning system as "untempered", but I guess it would mean something like just intonation -- which sounds really great for playing anything in a specific key but falls horribly apart if you try to change the key (which is why it has seen very little use since the renaissance).
Equally divided octave (EDO) with no tempering which is distinct from Equal temperament. Tempered scales are generally EDO with tempering. Other methods like just intonation don't really need to be tempered and generally are not in my experience, but it has been years since I was into just intonation and may just not remember. Historically speaking, the advantage of justly tuned scales is there is no need to temper it because it is already just and perfect, things may have changed in 20th century as far as just intonation is concerned and tempering, don't recall.
Edit: ET and EDO are essentially the same in the case of most fretted instruments, I am dredging long forgotten stuff from memory here and somewhat off above.
Edit2: Refreshing my memory some and seeing how much things have become muddled in my head over the years. Clearly I did not even consider what came out of my memory and just regurgitated it verbatim. ET scales are not tempered but do not mean EDO. Guitar and the like are both ET and EDO. ET and EDO are untempered in the sense that notes are not shifted slightly away from the EDO/ET as they are on the piano and many instruments.
I don't see how you can divide the octave equally and not end up with equal temperament: that's exactly what equal temperament is!
> Tempered scales are generally EDO with tempering.
That's not historically accurate. EDO wasn't used until very recently (about the middle of the 19th century I think), tempering was used way before that.
For example, the first widely used temperament (which became popular in the Renaissance) was the quarter-comma meantone, which shrinks each fifth (from the natural 3/2) so that the major thirds are perfectly 5/4. The name "quarter-comma" means that the amount of shrinkage is 1/4 of the "syntonic comma", which is the difference you get beteween going up 4 fifths (e.g. C->G->D->A->E) and a major third plus 2 octaves (C->E->E->E). Those final Es can only be the same if you shrink the fifth or stretch the third (or both). What this tempering does is shrink each fifth by 1/4 of the difference (so that going up 4 fifths closes it) and doesn't touch the major third. That means the major thirds are beautiful, and the fifths are a little off. For a chosen key, that is -- everything sounds horrible as soon as you try to change the key too far away from the chosen key.
In the Baroque period a lot of other temperaments were invented, the Werckmeister temperaments were very widely used in (what is today) Germany for example (a lot of people believe Bach had one of these in mind when writing the Well-Tempered Clavier). Those temperaments were also defined by how much each fifth is changed from the "normal" 3/2, but each fifth was to be changed by some different amount in some complicated way.
It was only much later that EDO (12-TET, or "equal temperament") started to be widely used. You can think of it (and people do!) as a "temperament" because it just means you shrink the fifth from the "normal" 3/2 = 1.5 to be instead 2^(7/12) =~ 1.4983, so that going up 12 fifths lands you exactly 7 octaves above (since 2^(7/12)^12 = 2^7). That also means that the octave is divided exactly equally, because going up 12 fifths goes through every one of the 12 notes before going back to the original note.
EDO on fretted instruments goes back to at least the 16th century and was essentially the standard well before the mid 19th. Equal temperament is an EDO scale whose divisions approximate justly tuned scales. The western 12TET scale is not actually 12TET or 12EDO, we temper the scale itself and tweak some notes to make it work better unless you play fretted instruments and then it is up to the guitarist to make small adjustments in their playing technique so their untempered 12TET is in tune with the pianos tempered 12TET.
I admitted to making a mess in that post.
> The western 12TET scale is not actually 12TET or 12EDO, we temper the scale itself and tweak some notes [...]
I think you and I must be using words differently. To me (and to Wikipedia, and everything else I've ever read, including[1] which I just consulted to make sure I'm not crazy), 12TET is a way to specify by how much you have to multiply the frequency of the first note of the scale to get the other notes' frequencies. Wikipedia[2] has a table with the numbers for 12TET (the column "Decimal value in 12-ET"), but it's very simple: you just multiply the value of the preceding note by 2^(1/12). If you take 12TET and adjust/change the notes a bit, then it's not 12TET anymore.
> EDO on fretted instruments goes back to at least the 16th century
I'd love to see a reference for that. I just consulted [1], it has a chapter called "Non-Keyboard Tuning" and it doesn't mention that (although admittedly it spends most of its time talking about violin, with a ton of references to stuff that Mozart said). The book does say that equal temperament was known for centuries before it was used, but the people who first discovered it simply didn't think it sounded good.
[1] "How Equal Temperament Ruined Harmony (and Why You Should Care)" by Ross W. Duffin
[2] https://en.wikipedia.org/wiki/12_equal_temperament
Try the wikipedia pages for Equal Temperament and Musical Temperament, they explain all of this.
Here is a 1688 Stradivari[1] guitar with fixed frets and a EDO octave, they were reasonably common by that point. Much of the information regarding this is looking at the fixed frets that many lutes and guitars had applied to their soundboard and comparing that to how composers used those fixed frets, either the tied frets adhere to the scale of the fixed frets or they are out of tune. The history of EDO/ET in fretted instruments goes back to at least Vincenzo Galilei[2] (father of Galileo) who developed the rule of 18 for fret spacing. If memory serves we have a few early steel string instrument (cittern, bandora, orpharion) from around ~1600 with equal spaced frets and this orpharion[3] looks it but it is difficult to tell from that photo. Going back earlier things get more difficult since we have so few intact and unaltered instruments but we do have a fair amount of ingravings and art plus writing on the topic such as Galilei's.
There is a paper going into great depth on all this that is just out of reach in my memory and I can't seem to trick the search engines to give it to me, I will post it if I remember/find it. No time to dig more right now.
[1]https://lsaguitarshop.substack.com/p/gear-27-the-stradivariu...
[2]https://en.wikipedia.org/wiki/Vincenzo_Galilei#Acoustics_and...
[3]https://i0.wp.com/earlymusicmuse.com/wp-content/uploads/2017...
> I will post it if I remember/find it
If you remember and have the time, please do! (And thank you for the links you already posted).
I see now that everything I read about this was way too focused on keyboard and violin, since I had never heard any of this about fretted instruments. I'm glad I get to correct a bit of my understanding, so thank you. Now I'm left wondering about wind instruments.
Western theory is focused on the keyboard with violin as second fiddle, the rest of the instruments do their own thing unless they are forced to kowtow to a piano or violin. Each instrument is ultimately tuned to the physics which dictate how it makes sound and this is part of why our tempered 12TET works and why we see the rise of the big orchestras of unlike instruments with the move away from meantone, it provides a compromise which works quite well for all instruments with the exception of the brass (with the exception of the trombone) who are forever out of tune (sort of).
Part of the reason beginners sound bad is because most instruments have to bend notes to be "in tune," I can teach anyone to play a chord on the guitar and get them having each note sounding clearly in a couple of minutes but my DMaj will sound better than theirs simply because I have played that DMaj thousands of times and my fingers have learned to adjust the pressure on each string in just the right way to make it sound "right" just as the woodwinds learn to bend certain notes and the brass learns to live with being out of tune.
Also part of why the lute became such a dominant instrument is that it could retune in ways other instruments can not, which was a major advantage for the working musician back in the days when every city had its own idea about tuning; nudge a few frets, retune a few strings and accept that certain notes were now out of bounds and you could play with anyone like you were playing in your native tongue. As tuning became more standard the lute started to die.
Meantone Temperaments on Lutes and Viols might be of interest to you, it is aimed towards lutenists and violists but has some more general stuff as well and I think does a good job of showing the compromises the lute (and viol) had to make in moving away from equal temperament.
I don't really think brass is forever out of tune, I love the brass and used to play trumpet but the brass section is more under the influence of the physics of its instrument than anyone but the pianist but the pianist is "in tune" because western theory is built around the keyboard.
To offer something better than this mess and correct. Fretted instruments are tehnically unequally tempered do to the physics the string, the fret board is EDO but in fretting the string we stretch it and raise its pitch. Fretted instruments use various compensation tricks to lessen this effect but most notes are off from both 12EDO and 12TET, we get the open string and the 12th fret but the rest are off by varying amounts.
An untempered instrument would be one that is tuned to maintain the perfect intervals of a specific root tone.
Temperament is adjusting tuning for musical practicality. 12 TET is simply one set of compromises/benefits in a constellation of alternatives.
https://en.m.wikipedia.org/wiki/Musical_temperament
Thank you for offering something clearer than the mess I made. Been 25 years since I studied this stuff and finally learned to just accept (and love) the modern standard.
There are such things are Chromatic keyboards (for instance, the Chromatone synth had one), and there was the Janko piano keyboard and there are people that have made keyboards without the spaces in at the B/C and E/F at the back of the keyboard by alternating the white and the black keys such that those spaces just disappear.
The advantages are:
- you need less reach for the same chords
- transposition becomes trivial
The disadvantages are:
- your muscle memory will be invalidated
- the number of instruments set up like that is really small
- no accomplished pianist will want to switch
But if you really wanted to you could adapt an existing instrument to use a different keyboard and it isn't even all that complicated (medium complexity wood working project).
I'm picky about layouts - I type in Dvorak, learned Janko via Chromatone, currently playing harpejji.
Coming from a classical piano background, there was definitely a learning curve, but I feel like it was worth it. Every chord shape is identical across all keys (C major and D major would be played the same way), which makes it much easier to learn jazz voicings or modulate a song.
If anyone ever builds a quality grand piano with Janko layout, I'm buying! Hacks on hacks become unnecessary if you start with the right design.
Chromatic button accordions have each octave in three rows of four instead of two rows of five and seven like a piano. It's very regular, but doesn't match up with major or minor scales or with sheet music. A major scale is a zig-zag.
I play both piano and button accordion and they're just different. Neither one has a compelling advantage.
having the keyboard the way it is also allows you to more easily orient yourself, you can feel with the sides of your fingers if you are next to E/F or B/C and with the corner of your eye it's also straightforward to figure it out. I don't think it'd be possible (or anyways even more difficult than it is now) to play large jumps accurately if the whole keyboard looked the same
I think both of those concerns were addressed by the Dvorak of piano keyboards: https://en.wikipedia.org/wiki/Jank%C3%B3_keyboard
Has the symmetry of GP while large jumps are accomplished by shifting up a row or two.
I assume it didn’t take off for the same reason Dvorak didn’t.
There are multiple alterative layout that some advocate for. they generally do sonething else for orintation. putting a bump on middle c and other places.
Make a dimple on every C key and paint it red.
That's all fine and dandy, except early apple keyboards have the dimples one key over :p
Historically before twelve tone equal temperament playing in another key on a keyboard instrument would sound different.
That's a bit like complaining that there are six different kinds of chess pieces and you have to memorize how each moves. The truth is, if you have trouble remembering how a knight moves, you can't be that good at Chess anyway.
Remembering twelve different ways of playing a scale is a vastly small part of learning how to play a piano.
> if you have trouble remembering how a knight moves, you can't be that good at Chess anyway.
Non sequitur.
It’s also still a valid question. I play the scales really well on the guitar. And because the frets are all laid out straight, shifting up by one fret means I’m just playing my chords and scales sharp. It makes transposing music incredibly easy.
I still don’t understand why the piano can’t be laid out like that.
Guitar still has the problem of last 2 strings being shifted 1 semitone down for ergonomics reasons, which makes them irregular. But yeah, it's better than piano keyboard.
Criticizing chess for lacking elegance compared to for example go is very valid (and a good analogy here).
Same. I recently tried to find a MIDI keyboard like that for sale and got nothing. Apparently this is what it's called:
https://en.m.wikipedia.org/wiki/Dodeka_keyboard
> It's not even that hard to fix.
Except it would render a large swath of the repertoire from the common practice period (“classical music“) more difficult to play because it is written with the presumption that the keyboard is just so and that some future generations won’t try to optimize it.
The organization of the keyboard does necessitate certain fingering choices that are particular; but knowing this, composers have (usually) written in a way that respects that geometry.
https://en.wikipedia.org/wiki/Jank%C3%B3_keyboard available since 1882!
The geometric relationship between the note frequencies of a C major chord and a D major chord on the piano is not the same. The key of piece is responsible for some of its “feel”. So it’s not unreasonable that they have different “representations” on the piano although the differences may be subtle.
I believe this is not the case in today’s ubiquitous equal temperament?
You are correct - my mistake.
There still would be two "shapes": C vs C#. Transposing would only be easier by two semitones.
But for most music and musicians it isn't that interesting. Transposing is rather niche. If it's too hard, an electronic keyboard can do it for you.
There would be 2 shapes instead of 12.
And the C shape would be vertically flipped C# shape.
There aren't 12 shapes. My fingers say C maj, F maj and G maj are the same shape. D maj, E maj, A maj too. And A min is the same shape as C/F/G maj, but would be a different pattern on the proposed "regular" layout. That may not matter when all you want is transposition, but as I said before: transposing is not so important. And there are "shape" based keyboards, as mentioned frequently in the comments. You can even retrofit some on your own keyboard. They're just not popular.
C maj, F maj are the same, but C maj7 and Fmaj7 are not. Ultimately you have to remember which chords are the same and which aren't - which is about the same amount of information as just remembering all the combinations.
It's just an ugly irregular hack. I understand that you can get good at working around it, but eventually it's worth it to fix the underlying problem.
It's not actually that much to remember. There are 3 shapes that cover most of the major chords, and 3 special cases (F♯, B♭, and B).
And then you add 4th note and you break the accidental 3-note symmetries :)
I always thought the canonical way to place the black keys was to divide the octave in the two parts that have the sequential black keys (C-D-E and F-G-A-B), and then simply place the black keys so they're the same distance away from each other and the edge of the parts.
That means that the white keys in each of the groups have "mirrors": for example, C is a mirror of E, D is a mirror of itself (it's the only key like that), F is a mirror of B, and G is a mirror of A.
I just looked at the keyboards I have around me (a slightly-above-low-end digital piano, a small midi controller, and a small 90s synth), and they all seem to fit that description.
ETA: note that the image in the article doesn't fit this description: for example the D is way too narrow (the black keys around it should be much further apart).
ETA2: I just noticed that this seems to be the "B/12 solution" described in the article.
I didn’t understand this post? More pictures needed?
I suspect there is no such fussing about "back of white key widths" in actual piano design.
What's actually going on, when you look at any piano keyboard, is that the groups of black keys are given a substantially wider spread: they are not exactly centered on the dividing lines between the white keys.
What most likely matters to playability is the width of the black keys and this spread amount.
Now if you cover the front of the keys (e.g. put your fallboard felt over it) you can visualize the back of all the keys as a kind of barcode: alternating black and white strips, sometimes with two adjacent white strips. It so happens that these semitone strips do look about equal width, thanks to the sizing of the black keys and their spread. It might not be exact though.
A good starting configuration might be to start with a keyboard in which all the semitones are strips of equal width. Then we identify the C major keys, and paint them white, making the others black. Next, we shorten the black keys, and adjust the frontage of the white keys to be of equal width. Say, by keeping the division between every B and C exactly where it is and interpolating the others divisions. You will find that the E-F division does not fall exactly halfway between the surrounding black keys, Eb and F#. (That's what's observed on real piano keyboards!)
I wish the page had a picture with a-b-c-d-... written on the buttons
I wish the page had pictures for the proposed widths of the buttons
[dead]