# Layout optimisation and similarity to QWERTZ

## Introduction

On the Internet, every now and then, the question arises what small modifications to QWERTZ or QWERTY would improve the layout as much as possible. Often, the question is motivated by the assumption that using a layout similar to QWERTZ would reduce the effort of relearning and still provide most of the advantages of more radical alternatives. In my opinion, the practical value of such layout alternatives for wimps is limited. However, from the technical perspective, the question which layout that differs from QWERTZ in $n$ positions is the best one is an appealing, and at the very least, a nice opportunity to demonstrate the optimiser.

Results obtained with the optimiser always have to be taken with a bit of scepticism, as they are obtained based on evaluation criteria that are not scientifically justified, but mere heuristics. This is even more true in the following, as the experience with the evaluation criteria I am going to use have been obtained with “normal” optimisations, for which one basically has to differentiate among good layouts, whereas in the present case, we want to evaluate and compare poor layouts as well. It is not clear that criteria that are made to compare good layout are suited for comparison of bad ones.

## Preparation

I assume that you have downloaded the optimiser and are roughly familiar with it. The optimiser and its documentation can be found on the overview page. For the present article, I have used version 1.239. Here you can find a collection of files that help to reproduce this article more easily:

This web page uses MathML to display a few mathematical symbols. MathML has been a web standard since 2001, and is supported by Firefox, for example.

As in the following, we want to obtain 31 optimised layouts that differ from from QWERTZ at 2 to 32 positions, a significant amount of computational effort is required. Therefore, we will perform the computations using digrams only. In this case, we can ignore the space character that is usually fixed on the space key. To this end, we remove the corresponding key and the corresponding symbol in the configuration file, see basis.cfg in the supporting material. Then, we compile the optimiser for one key less as by default:

```g++ -std=c++11 -Wall -Ofast -DENGLISH -DNDEBUG -DOHNE2SHIFT \
```

I will come to -DEXPERIMENTELL later. For the optimisations, we want to consider German and English with equal weight. Therefore, we sum the corpora shipping with the optimiser

```./opt englisch.txt deutsch.txt gemischt.txt
```

and will use gemischt.txt in the following.

## Optimisation

A naive mind could come up with the idea to start from QWERTZ, apply the “best” key swap to it, then apply the best key swap to the layout so obtained, and so on. However, the sequence of layouts obtained in this manner has no meaning whatsoever, starting with the second step. That is because the layout that is created in one step is severely limited by what has been produced in the preceding steps. There is no guarantee that it is the best layout that can be obtained from QWERTZ with the given number of key swaps, or even the best layout that differs from QWERTZ at the given number of positions. We have to work a bit harder.

### Few differences from QWERTZ

As long as we tolerate only few differences, we can use the option -V to try out all layout which differ in exactly $n$ positions from QWERTZ. First, we create a layout file qwertz.txt which lists only QWERTZ, and then start with layouts that differ in exactly two positions from it (that is, layouts that can be obtained from QWERTZ by a single key swap):

```./opt -2 gemischt.txt -K basis.cfg -r qwertz.txt -V 2 | sort -rn
```

Then, we use the smallest total effort found as a threshold to limit the output of layouts that differ at three positions from QWERTZ. And so on.

```./opt -2 gemischt.txt -K basis.cfg -r qwertz.txt -V 3 -m 471.4257|sort -rn
./opt -2 gemischt.txt -K basis.cfg -r qwertz.txt -V 4 -m 441.3500|sort -rn
```

However, the computational effort increases very quickly with increasing $n$, and for $n>6$, it becomes obvious that this approach does not lead much further.

### The general case

The optimiser permits to specify preferences for certain symbols to be mapped to certain keys. If you specify for each symbol a preference $\lambda$ that it is located at its QWERTZ position, by choosing a the value for $\lambda$, you can enforce a similarity to QWERTZ more or less strongly. To this end replace the XXX in qwertzig.cfg as found in the supporting material by $\lambda$ and include the configuration file created in that manner when running an optimisation:

```sed -e "s/XXX/0.02/" qwertzig.cfg > qwertz.cfg
./opt -2 gemischt.txt -K basis.cfg -K qwertz.cfg -k
```

If for a given $\lambda$, you obtain a layout as optimum that differs from QWERTZ at $n$ positions, this is indeed the best layout for this $n$, as the difference of the total efforts for two layouts that differ from QWERTZ at the same number of positions does not depend on $\lambda$.

Using this approach, not all possibilities will be tried out. This allows, on the one hand, to cover the range of larger $n$. On the other hand, one looses the guarantee to have found an actual optimum for a given $n$. Initially, I just played around using different values for the preference $\lambda$, and this way found optima for some $n$. For other $n$, I could not find an optimum; depending on the value of $\lambda$, I found layouts with more or less deviations from QWERTZ.

### Curvature of the graph and linearity of preferences

For explanation, the following figure shows the results of my investigation, the total effort $A$ (without preferences) of the best layout for a given number $n$ of differences with respect to QWERTZ, plotted over $n$.

The black little squares represent the layouts. The layouts that could be found with my simplistic approach are connected by the blue polyline. The layout of the non-discoverable layouts are all located above this polyline, which illustrates the reason why they could not be found: If you draw a line with a negative slope $-\lambda$ through one of those points above the blue polyline, always at least one of the neighbouring point on the blue polyline is located below that line, and the respective layout is evaluated as better for that value of preference $\lambda$. Or, in other words, as the contribution of preferences is linear in $n$, by variation of $\lambda$, only layouts can be found that are located where the $A\left(n\right)$ graph has a left curvature.

To obtain the missing points, one must give up on linearity. To this end, I have introduced a new effort parameter VorliebeKnick which only becomes available if the optimiser has been compiled using the option -DEXPERIMENTELL. The parameter specifies an upper bound for the sum of all preferences. If you intend to look for a layout with $n$ differences, and you have already selected a value for $\lambda$, set the VorliebeKnick parameter to $\left(N-n\right)\lambda$, where $N$ denotes the number of keys in the layout considered. This way, the preference line towards the left is replaced by a horizontal line. This would not be sufficient if $A\left(n\right)$ would be not strictly monotonous, as it is in our example.

And how to select $\lambda$? In principle, one just could make it very large, to make it impossible that layouts with an $n$ larger than desired are found. However, according to my observation, this increases the number of iterations required to find the optimum. In practise, it seems to be preferable first to look for preliminary solutions for $n and then to choose $\lambda$ slightly larger than $\mathrm{\left[A}\left(n\right)\mathrm{-A}\left(m\right)\right]/\left(n-m\right)$, that is, just large enough that the actual best solution for $n$ presumably will be judged slightly better than that for $m$.

## Results

### Discussion of the figure

One can clearly see in the figure that small changes from QWERTY initially bring big gains, and that further changes bring additional gains, however, that those additional gains, by tendency, get smaller and smaller the further on gets away from QWERTZ. I suspect that this observation is not specific to my evaluation scheme, but holds more generally. To a certain degree, this observations confirms the wimp’s point of view. In our case, however, on must change about half of the keys until the graph is levelling off; the corresponding layouts are not very QWERTZ-like anymore.

With my corpus and my evaluation criteria, the graph is strictly monotonous, that is, the more deviations one from QWERTZ one is willing to tolerate, the better is the layout one gets; this might not be the case for all corpora or evaluation schemes.

There are ranges of $n$ (the regime where the points are above the blue polyline), for which no interesting layouts exist. I expect that such a behaviour could be observed with other evaluation schemes as well.

### The layouts found

The optimal layouts are listed in belegungen.txt, sorted and named according to their number of differences from QWERTZ:

```./opt -2 gemischt.txt -K basis.cfg -r belegungen.txt -Z

QWERTZ           568.134 total effort   349.408 positional effort    left right
8.204 same finger rp   9.526 shift same finger top 28.7 17.3
qwert zuiopü    52.035 hand alternat.  49.617 shift hand alter. mid 20.9  9.7
asdfg hjklöä     1.039 inward/outward  37.311 inward or outward bot  7.9 15.4
8.9  7.8 20.6 20.2 --.- --.- 18.8  9.4  9.5  4.7 Sh  1.5  2.6

Compared to first: 2 keys reassigned, cycles:  le
Compared to previous: 2 keys reassigned, cycles:  le

2                471.426 total effort   308.230 positional effort    left right
5.240 same finger rp   9.045 shift same finger top 18.9 17.3
qwlrt zuiopü    56.757 hand alternat.  42.113 shift hand alter. mid 20.9 19.6
asdfg hjkeöä     1.148 inward/outward  35.553 inward or outward bot  8.0 15.4
9.0  7.8 10.7 20.2 --.- --.- 18.8  9.4 19.3  4.7 Sh  1.6  2.5

Compared to first: 3 keys reassigned, cycles:  ule
Compared to previous: 2 keys reassigned, cycles:  ul

3                441.350 total effort   307.822 positional effort    left right
3.824 same finger rp   9.653 shift same finger top 18.2 18.0
qwurt zliopü    55.149 hand alternat.  42.292 shift hand alter. mid 20.9 19.6
asdfg hjkeöä     1.140 inward/outward  38.576 inward or outward bot  8.0 15.3
9.0  7.8 10.0 20.2 --.- --.- 19.5  9.4 19.3  4.6 Sh  1.6  2.5

Compared to first: 4 keys reassigned, cycles:  nd  le
Compared to previous: 4 keys reassigned, cycles:  nd  ul

4                426.217 total effort   291.348 positional effort    left right
4.133 same finger rp   9.704 shift same finger top 18.9 17.3
qwlrt zuiopü    59.427 hand alternat.  41.126 shift hand alter. mid 24.9 19.6
asnfg hjkeöä     0.968 inward/outward  33.990 inward or outward bot  8.1 11.2
9.2  7.8 14.7 20.2 --.- --.- 14.9  9.4 19.3  4.5 Sh  1.7  2.3

Compared to first: 5 keys reassigned, cycles:  rkule
Compared to previous: 6 keys reassigned, cycles:  nd  ulrk

5                400.971 total effort   299.580 positional effort    left right
2.281 same finger rp   9.322 shift same finger top 16.2 18.0
qwrkt zliopü    55.908 hand alternat.  40.934 shift hand alter. mid 20.9 21.6
asdfg hjueöä     1.205 inward/outward  39.361 inward or outward bot  8.0 15.4
9.0  7.8 13.4 14.8 --.- --.- 19.5 11.4 19.3  4.7 Sh  1.6  2.5

Compared to first: 6 keys reassigned, cycles:  njrd  le
Compared to previous: 7 keys reassigned, cycles:  njrlukd

6                383.975 total effort   265.695 positional effort    left right
3.580 same finger rp   8.606 shift same finger top 16.6 17.3
qwldt zuiopü    56.177 hand alternat.  40.400 shift hand alter. mid 24.9 25.8
asnfg hrkeöä     1.281 inward/outward  37.792 inward or outward bot  8.0  7.4
9.0  7.8 14.7 17.9 --.- --.- 17.1  9.4 19.3  4.6 Sh  1.6  2.5

Compared to first: 7 keys reassigned, cycles:  rdnka  le
Compared to previous: 4 keys reassigned, cycles:  rjka

7                365.943 total effort   268.742 positional effort    left right
2.629 same finger rp   7.233 shift same finger top 16.6 17.3
qwldt zuiopü    67.060 hand alternat.  33.925 shift hand alter. mid 24.8 25.0
rsnfg hjaeöä     1.109 inward/outward  27.860 inward or outward bot  8.2  8.1
9.1  7.8 14.7 17.9 --.- --.- 11.7 14.9 19.3  4.5 Sh  1.8  2.3

Compared to first: 8 keys reassigned, cycles:  rkiule  nj
Compared to previous: 9 keys reassigned, cycles:  iulra  kjnd

8                349.460 total effort   251.085 positional effort    left right
2.281 same finger rp   9.322 shift same finger top 16.2 13.8
qwrkt zluopü    55.908 hand alternat.  40.934 shift hand alter. mid 20.9 33.7
asdfg hnieöä     1.205 inward/outward  39.361 inward or outward bot  8.0  7.4
9.0  7.8 13.4 14.8 --.- --.- 19.5 11.4 19.3  4.7 Sh  1.6  2.5

Compared to first: 9 keys reassigned, cycles:  rd  le  uki  nj
Compared to previous: 4 keys reassigned, cycles:  rlkd

9                332.702 total effort   243.015 positional effort    left right
1.954 same finger rp   9.045 shift same finger top 16.6 11.1
qwldt zkuopü    56.757 hand alternat.  42.113 shift hand alter. mid 23.2 33.7
asrfg hnieöä     1.204 inward/outward  38.839 inward or outward bot  8.0  7.4
9.0  7.8 13.0 17.9 --.- --.- 16.8 11.4 19.3  4.7 Sh  1.6  2.5

Compared to first: 10 keys reassigned, cycles:  rd  le  tjnkuf
Compared to previous: 6 keys reassigned, cycles:  tjkf  ui

10               319.653 total effort   228.256 positional effort    left right
1.890 same finger rp   8.172 shift same finger top  9.7 16.1
qwldj zfiopü    55.619 hand alternat.  42.919 shift hand alter. mid 28.5 29.5
asrtg hnueöä     1.215 inward/outward  40.040 inward or outward bot  8.1  8.2
9.1  7.8 13.0 16.3 --.- --.- 18.4 11.4 19.3  4.6 Sh  1.7  2.4

Compared to first: 11 keys reassigned, cycles:  rd  le  tjnkiuf
Compared to previous: 2 keys reassigned, cycles:  ui

11               300.812 total effort   212.850 positional effort    left right
1.890 same finger rp   8.172 shift same finger top  9.7 11.9
qwldj zfuopü    55.619 hand alternat.  42.919 shift hand alter. mid 28.5 33.7
asrtg hnieöä     1.215 inward/outward  40.040 inward or outward bot  8.1  8.2
9.1  7.8 13.0 16.3 --.- --.- 18.4 11.4 19.3  4.6 Sh  1.7  2.4

Compared to first: 12 keys reassigned, cycles:  rkiuleftöa  nj
Compared to previous: 8 keys reassigned, cycles:  rdkjöa  lf

12               294.001 total effort   201.120 positional effort    left right
1.920 same finger rp  13.408 shift same finger top  4.6 13.8
qwfkö zluopü    56.323 hand alternat.  43.097 shift hand alter. mid 26.2 40.1
rsdtg hnieaä     1.286 inward/outward  39.306 inward or outward bot  8.3  7.1
9.2  7.8  8.8 13.2 --.- --.- 19.5 11.4 19.3 10.7 Sh  1.9  2.2

Compared to first: 13 keys reassigned, cycles:  rjdnkela  töiuf
Compared to previous: 9 keys reassigned, cycles:  iea  nd  lf  kj

13               282.413 total effort   202.418 positional effort    left right
1.729 same finger rp  12.563 shift same finger top  5.6 11.9
qwljö zfuopü    63.080 hand alternat.  39.493 shift hand alter. mid 30.2 36.1
rsntg hdeaiä     0.859 inward/outward  32.741 inward or outward bot  8.5  7.7
9.5  7.8 14.7 12.3 --.- --.- 14.5 17.7 12.2 11.3 Sh  2.1  2.0

Compared to first: 14 keys reassigned, cycles:  rcjdnkela  töiuf
Compared to previous: 2 keys reassigned, cycles:  jc

14               273.150 total effort   196.844 positional effort    left right
1.634 same finger rp  12.563 shift same finger top  8.3 11.9
qwlcö zfuopü    63.080 hand alternat.  39.493 shift hand alter. mid 30.2 36.1
rsntg hdeaiä     0.858 inward/outward  32.835 inward or outward bot  5.8  7.7
9.5  7.8 12.0 15.0 --.- --.- 14.5 17.7 12.2 11.3 Sh  2.1  2.0

Compared to first: 15 keys reassigned, cycles:  rcjdnkiuftpöa  le
Compared to previous: 5 keys reassigned, cycles:  eia  öp

15               265.684 total effort   196.128 positional effort    left right
1.238 same finger rp  14.062 shift same finger top  9.6 10.6
qwlcp zfuoöü    63.953 hand alternat.  38.988 shift hand alter. mid 30.2 36.1
rsntg hdieaä     0.819 inward/outward  32.359 inward or outward bot  5.7  7.9
9.3  7.8 12.0 16.3 --.- --.- 14.5 11.4 19.3  9.4 Sh  2.0  2.1

Compared to first: 16 keys reassigned, cycles:  srcjdnkiuftpöa  le
Compared to previous: 2 keys reassigned, cycles:  sr

16               262.891 total effort   195.872 positional effort    left right
1.245 same finger rp  10.269 shift same finger top  9.6 10.6
qwlcp zfuoöü    63.953 hand alternat.  38.988 shift hand alter. mid 30.2 36.1
srntg hdieaä     0.772 inward/outward  32.352 inward or outward bot  5.7  7.9
9.0  8.1 12.0 16.3 --.- --.- 14.5 11.4 19.3  9.4 Sh  2.0  2.1

Compared to first: 17 keys reassigned, cycles:  srcjdnmkiuftpöa  le
Compared to previous: 2 keys reassigned, cycles:  mk

17               260.663 total effort   194.374 positional effort    left right
1.245 same finger rp  10.269 shift same finger top  9.6 10.6
qwlcp zfuoöü    63.953 hand alternat.  38.988 shift hand alter. mid 30.2 36.1
srntg hdieaä     0.772 inward/outward  32.352 inward or outward bot  5.7  7.9
9.0  8.1 12.0 16.3 --.- --.- 14.5 11.4 19.3  9.4 Sh  2.0  2.1

Compared to first: 18 keys reassigned, cycles:  srcjdnmkiuftp.öa  le
Compared to previous: 2 keys reassigned, cycles:  .ö

18               258.580 total effort   194.564 positional effort    left right
1.098 same finger rp  10.630 shift same finger top  9.6 11.5
qwlcp zfuo.ü    63.953 hand alternat.  38.988 shift hand alter. mid 30.2 36.1
srntg hdieaä     0.764 inward/outward  32.499 inward or outward bot  5.7  7.0
9.0  8.1 12.0 16.3 --.- --.- 14.5 11.4 18.4 10.3 Sh  2.0  2.1

Compared to first: 19 keys reassigned, cycles:  srcmkiuftp.öa  jdnb  le
Compared to previous: 3 keys reassigned, cycles:  jmb

19               257.501 total effort   194.876 positional effort    left right
1.045 same finger rp  11.697 shift same finger top  9.6 11.5
qwlcp zfuo.ü    65.387 hand alternat.  38.013 shift hand alter. mid 30.2 36.1
srntg hdieaä     0.756 inward/outward  31.118 inward or outward bot  6.5  6.2
9.0  8.1 14.3 14.8 --.- --.- 13.6 11.4 18.4 10.3 Sh  1.9  2.2

Compared to first: 20 keys reassigned, cycles:  snbjdrcwmkiuftp.öa  le
Compared to previous: 4 keys reassigned, cycles:  wm  rn

20               255.851 total effort   192.967 positional effort    left right
1.060 same finger rp  11.697 shift same finger top 10.5 11.5
qmlcp zfuo.ü    65.387 hand alternat.  38.013 shift hand alter. mid 30.2 36.1
snrtg hdieaä     0.718 inward/outward  31.103 inward or outward bot  5.6  6.2
9.0 10.7 11.7 14.8 --.- --.- 13.6 11.4 18.4 10.3 Sh  1.9  2.2

Compared to first: 21 keys reassigned, cycles:  snbjhdrcwmkiuftp.öa  le
Compared to previous: 2 keys reassigned, cycles:  hd

21               254.836 total effort   192.149 positional effort    left right
1.060 same finger rp  11.697 shift same finger top 10.5 11.5
qmlcp zfuo.ü    65.387 hand alternat.  38.013 shift hand alter. mid 30.2 36.1
snrtg dhieaä     0.718 inward/outward  31.103 inward or outward bot  5.6  6.2
9.0 10.7 11.7 14.8 --.- --.- 13.6 11.4 18.4 10.3 Sh  1.9  2.2

Compared to first: 22 keys reassigned, cycles:  srcjdnbqzp.öa  le  tvmkiuf
Compared to previous: 11 keys reassigned, cycles:  hd  qzpvmwj  rn

22               253.478 total effort   191.213 positional effort    left right
1.071 same finger rp  11.045 shift same finger top  9.6 12.3
zwlcv pfuo.ü    65.226 hand alternat.  37.279 shift hand alter. mid 30.2 36.1
srntg hdieaä     0.758 inward/outward  31.252 inward or outward bot  5.8  6.1
9.7  8.1 12.0 15.8 --.- --.- 14.4 11.4 18.4 10.2 Sh  2.0  2.1

Compared to first: 23 keys reassigned, cycles:  srcß.öa  jdnb  le  tvmkiuf  qzp
Compared to previous: 4 keys reassigned, cycles:  ß.qj

23               252.504 total effort   191.139 positional effort    left right
1.049 same finger rp  11.047 shift same finger top  9.6 11.3
zwlcv pfuoqü    65.265 hand alternat.  37.510 shift hand alter. mid 30.2 36.1
srntg hdieaä     0.762 inward/outward  31.235 inward or outward bot  5.8  7.0
9.7  8.1 11.9 15.9 --.- --.- 14.4 11.4 18.4 10.2 Sh  2.0  2.0

Compared to first: 24 keys reassigned, cycles:  r.bödejsiltägocüknv,wufa
Compared to previous: 28 keys reassigned, cycles:  dsrineta  vägoc.ßükm,wuljöb  zqp

24               249.575 total effort   189.595 positional effort    left right
1.418 same finger rp   7.526 shift same finger top  4.6 11.4
quj.ä zflcpk    65.202 hand alternat.  24.259 shift hand alter. mid 38.4 32.3
rieao hsntdg     0.840 inward/outward  30.930 inward or outward bot  5.5  7.8
10.3 10.4 14.0 13.8 --.- --.- 16.3 13.4 11.6 10.1 Sh  3.0  1.1

Compared to first: 25 keys reassigned, cycles:  r.bödeufa  üknv,wyjsiltägoc
Compared to previous: 3 keys reassigned, cycles:  uyj

25               246.175 total effort   188.172 positional effort    left right
1.454 same finger rp   6.929 shift same finger top  5.3 11.4
qyu.ä zflcpk    65.202 hand alternat.  24.259 shift hand alter. mid 38.4 32.3
rieao hsntdg     0.852 inward/outward  30.893 inward or outward bot  4.8  7.8
9.6  8.2 16.9 13.8 --.- --.- 16.3 13.4 11.6 10.1 Sh  3.0  1.1

Compared to first: 26 keys reassigned, cycles:  r.wyjsiltägocüknfa  ödeub  v,m
Compared to previous: 5 keys reassigned, cycles:  wmvfb

26               243.591 total effort   187.506 positional effort    left right
1.406 same finger rp   6.929 shift same finger top  5.3 11.3
qyu.ä zblcpk    65.202 hand alternat.  24.259 shift hand alter. mid 38.4 32.3
rieao hsntdg     0.839 inward/outward  30.942 inward or outward bot  4.8  7.9
9.6  8.2 16.9 13.8 --.- --.- 15.5 14.3 11.6 10.1 Sh  3.0  1.1

Compared to first: 27 keys reassigned, cycles:  sa  öhdeob  äfilnguc  tükr,wzj  v.p
Compared to previous: 25 keys reassigned, cycles:  ia  cb  hd  gf  yzj  wpvm  rstn  uo  üä  ,.

27               242.590 total effort   188.854 positional effort    left right
1.106 same finger rp   2.265 shift same finger top  6.8 10.3
qzo,ü jclbvk    68.604 hand alternat.  24.783 shift hand alter. mid 36.5 32.0
saeiu dtrnhf     1.089 inward/outward  27.839 inward or outward bot  5.1  9.2
9.8  7.2 18.5 12.9 --.- --.- 19.1 11.7 11.2  9.6 Sh  2.7  1.4

Compared to first: 28 keys reassigned, cycles:  hdeäfa  ösicükr.mpjtzv,wugolnb
Compared to previous: 19 keys reassigned, cycles:  ia  lcgb  sh  vj  pm  äüzuo  ,.

28               240.485 total effort   187.240 positional effort    left right
1.207 same finger rp   7.670 shift same finger top  5.0 11.0
quä.z vgcljk    71.358 hand alternat.  23.306 shift hand alter. mid 36.4 33.8
hieao dtrnsf     1.668 inward/outward  24.984 inward or outward bot  5.5  8.3
8.3 10.4 14.1 14.1 --.- --.- 17.6 10.8 14.3 10.4 Sh  2.9  1.2

Compared to first: 29 keys reassigned, cycles:  hdeüzv,wugolnbqkr.mpjtäfa  ösic
Compared to previous: 6 keys reassigned, cycles:  zäüöqk

29               238.169 total effort   187.075 positional effort    left right
1.065 same finger rp   7.543 shift same finger top  5.7 10.6
kuü.ä vgcljz    71.404 hand alternat.  24.118 shift hand alter. mid 36.4 33.8
hieao dtrnsf     1.794 inward/outward  25.080 inward or outward bot  5.2  8.4
9.2 10.4 14.0 13.7 --.- --.- 17.6 10.8 14.3 10.1 Sh  2.9  1.2

Compared to first: 30 keys reassigned, cycles:  hdeüzv,wugolnbqkr.mpßjtäfa  ösic
Compared to previous: 2 keys reassigned, cycles:  ßj

30               238.121 total effort   187.054 positional effort    left right
1.065 same finger rp   7.543 shift same finger top  5.7 10.5
kuü.ä vgclßz    71.404 hand alternat.  24.118 shift hand alter. mid 36.4 33.8
hieao dtrnsf     1.794 inward/outward  25.080 inward or outward bot  5.2  8.5
9.2 10.4 14.0 13.7 --.- --.- 17.6 10.8 14.3 10.1 Sh  2.9  1.2

Compared to first: 31 keys reassigned, cycles:  hdeüzv,wugolnbqkr.mpjtäfa  ösic  yx
Compared to previous: 4 keys reassigned, cycles:  ßj  yx

31               236.945 total effort   187.075 positional effort    left right
1.029 same finger rp   6.976 shift same finger top  5.7 10.6
kuü.ä vgcljz    71.404 hand alternat.  24.118 shift hand alter. mid 36.4 33.8
hieao dtrnsf     1.796 inward/outward  25.117 inward or outward bot  5.2  8.4
8.4 11.2 14.0 13.7 --.- --.- 17.6 10.8 14.3 10.1 Sh  2.9  1.2

Compared to first: 32 keys reassigned, cycles:  hdeüzv,wugolnbqkr.mpßjtäfa  ösic  yx
Compared to previous: 2 keys reassigned, cycles:  ßj

32               236.897 total effort   187.054 positional effort    left right
1.029 same finger rp   6.976 shift same finger top  5.7 10.5
kuü.ä vgclßz    71.404 hand alternat.  24.118 shift hand alter. mid 36.4 33.8
hieao dtrnsf     1.796 inward/outward  25.117 inward or outward bot  5.2  8.5
8.4 11.2 14.0 13.7 --.- --.- 17.6 10.8 14.3 10.1 Sh  2.9  1.2
```

Atop of each layout there is a short summary how it differs from QWERTZ and from the layout displayed before, how many keys have changed their symbols and how this change can be represented by cyclic permutations.

As expected, with increasing distance from QWERTZ, the optimal layouts do not evolve stepwise, pair swap by pair swap. For example, in layout 4, the “u” is located at its QWERTZ position, even though in layout 3, it already had been moved away from there. The step from 23 to 24 is particularly dramatic, as the vowels move from the right to the left hand in this step.

Version 25. Apr 2016Impressum