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    • Showing only topics with the tag "measurements". Back to normal view

    2. Piano key dimensions are a math puzzle

      ~design Ask
      Piano keys are familiar and easy enough to draw if you're not trying to be exact, but if you want label the dimensions with their exact measurements (like in a CAD drawing), it turns into a math...

      Piano keys are familiar and easy enough to draw if you're not trying to be exact, but if you want label the dimensions with their exact measurements (like in a CAD drawing), it turns into a math puzzle. The problem comes from the groups of two and three black keys.

      This article explains it like this:

      If you've ever looked closely at a piano keyboard you may have
      noticed that the widths of the white keys are not all the same
      at the back ends (where they pass between the black keys). Of
      course, if you think about it for a minute, it's clear they
      couldn't possibly all be the same width, assuming the black keys
      are all identical (with non-zero width) and the white keys all
      have equal widths at the front ends, because the only simultaneous
      solution of 3W=3w+2b and 4W=4w+3b is with b=0.

      To unpack that a bit: in that equation, 'W' is the width of each white key at the front (which should all be the same), 'w' is the width of a white key at the back, and 'b' is the width of a black key.) The first equation is for the group of two black keys (separating C, D, and E) and the second equation is for the three black keys separating F through B.

      Since it's mathematically impossible, a constraint needs to be relaxed. The article describes ways to make the white keys have slightly different widths at the back.

      If we set c=e=(W-5B/8) and a=b=d=f=g=(W-3B/4) we have a maximum
      discrepancy of only B/8, and quite a few actual pianos use this
      pattern as well. However, the absolute optimum arrangement is to
      set c=d=e=(W-2B/3) and f=g=a=b=(W-3B/4), which gives a maximum
      discrepancy of just B/12. This pattern is used on many keyboards,
      e.g. the Roland PC-100.

      When actually building a musical instrument (instead of just drawing the keyboard), there is a further constraint, described in this article:

      The black keys on a piano keyboard, instead of always being centered on the dividing line between the two white keys they lie between, are spaced so that the twelve keys which make up an octave are spaced equally as they enter the internal mechanism of the instrument.

      But this means that the "key caps" for the white keys should be slightly off-center compared to whatever rod or lever they're attached to. The author speculates about how to divide this up using various units.

      (They seem quite annoying to 3D print.)

      19 votes

    12. Programming Challenge: Convert between units

      ~comp Text 333 words
      Hi everyone! It's been a long time since last programming challenge list, and here's a nice one I've encountered. If you search for something like 7km to AU, you'll get your answer. But how is it...

      Hi everyone! It's been a long time since last programming challenge list, and here's a nice one I've encountered.

      If you search for something like 7km to AU, you'll get your answer. But how is it done? I don't think they hardcoded all 23 units of distance and every conversion factor between them.

      If you were programming a conversion system - how would you do it?

      First of all, you have input in format that you can specify, for example something like this:

      meter kilometer 1000
mile kilometer 1.609344
second minute 60
...

      Then you should be able answer queries. For example 7 mile meter should convert 7 miles to meters, which is 11265.41.

      Can you design an algorithm that will convert any unit into any other unit?

      Edit: Some conversion rates I extracted from wikipedia:

      ångström
0.1nm
astronomical unit
149597870700m
attometre
0.000000000000000001m
barleycorn
8.4m
bohr
0.00846
cable length (imperial)
185.3184m
cable length
185.2m
cable length (US)
219.456m
chain (Gunters)
20.11684m
cubit
0.5m
ell
1.143m
fathom
1.8288m
femtometre
0.00000000000001m
fermi
0.00000000000001m
finger
0.022225m
finger (cloth)
0.1143m
foot (Benoit)
0.304799735m
foot (Cape) (H)
0.314858m
foot (Clarke's) (H)
0.3047972654m
foot (Indian) (H)
0.304799514m
foot,metric
0.31622776602m
foot,metric (long)
0.3m
foot,metric (short)
0.30m
foot (International)
0.3048m
foot (Sear's) (H)
0.30479947m
foot (US Survey)
0.304800610
french
0.0003m
furlong
201.168m
hand
0.1016m
inch
0.0254m
league
4828m
light-day
25902068371200m
light-hour
107925284880m
light-minute
17987547480
light-second
299792458m
light-year
31557600light-second
line
0.002116m
link (Gunter's)
0.2011684m
link (Ramsden's; Engineer's)
0.3048m
metre
1m
m
1metre
km
1000m
mickey
0.000127
micrometre
0.000001
mil; thou
0.0000254
mil
10km
mile (geographical)
6082foot (International)
quarter
0.2286m
rod
5.0292m
rope
6.096m
shaku
0.303 0303m
span (H)
0.2286m
stick (H)
0.0508m
toise
1.949 0363m
twip
1.76310
yard
0.9144m
      17 votes