Piano Keys

https://www.mathpages.com/home/kmath043.htm

by gametorch • 1 year ago

74 69 1 year ago

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.

o11c• 1 year ago

Same, but I had to look. I wonder how badly this affects muscle memory?

madaxe_again• 1 year ago

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.

mlochbaum• 1 year ago

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.

bruce3434342• 1 year ago

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.

ajuc7• 1 year ago

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.

tetraodonpuffer• 1 year ago

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

paulgerhardt• 1 year ago

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.

ajuc7• 1 year ago

Make a dimple on every C key and paint it red.

bluGill• 1 year ago

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.

krallja• 1 year ago

https://en.wikipedia.org/wiki/Jank%C3%B3_keyboard available since 1882!

jng• 1 year ago

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.

ofalkaed7• 1 year ago

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.

brudgers4• 1 year ago

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.

brudgers4• 1 year ago

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

rollcat• 1 year ago

Somewhat related:

https://strandbergguitars.com/true-temperament/

IsTom• 1 year ago

Historically before twelve tone equal temperament playing in another key on a keyboard instrument would sound different.

skybrian• 1 year ago

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.

yayitswei• 1 year ago

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.

ncake• 1 year ago

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

Nition5• 1 year ago

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

drabbiticus• 1 year ago

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.

sambapa• 1 year ago

C is one of the hardest keys to play in on the piano

Snarwin• 1 year ago

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).

ajuc7• 1 year ago

And then you add 4th note and you break the accidental 3-note symmetries :)

yongjik• 1 year ago

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.

amingilani• 1 year ago

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.

ajuc7• 1 year ago

Criticizing chess for lacking elegance compared to for example go is very valid (and a good analogy here).

tgv2• 1 year ago

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.

ajuc7• 1 year ago

There would be 2 shapes instead of 12.

And the C shape would be vertically flipped C# shape.

derriz2• 1 year ago

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.

codyd51• 1 year ago

I believe this is not the case in today’s ubiquitous equal temperament?

jacquesm• 1 year ago

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).

kashunstva• 1 year ago

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.

moefh5• 1 year ago

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.

ralfd• 1 year ago

I didn’t understand this post? More pictures needed?

prmph• 1 year ago

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?

scarecrw• 1 year ago

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.

sefn2• 1 year ago

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.

brudgers4• 1 year ago

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.

javcasas• 1 year ago

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.

monomers• 1 year ago

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?

NooneAtAll3• 1 year ago

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

kazinator• 1 year ago

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!)

Piano Keys

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