6 ms·

Two things I hate but can't fight: Tetrahedra almost tessellate but don't quite do it: https://www.pnas.org/content/pnas/103/28/10612/F1.large.jpg https://www.

by pgn674 5y ago

Two things I hate but can't fight: Tetrahedra almost tessellate but don't quite do it: https://www.pnas.org/content/pnas/103/28/10612/F1.large.jpg https://www.pnas.org/content/pnas/103/28/10612/F1.large.jpg In music, the frequency ratio of a semitone is ideally 2^(1/12), but without some tiny fudging (called tuning), you can't make harmonies as the frequencies almost but don't quite line up right. I forget exactly how this one works so I may have something off.

saghm 5y ago
> In music, the frequency ratio of a semitone is ideally 2^(1/12), but without some tiny fudging (called tuning), you can't make harmonies as the frequencies almost but don't quite line up right IIRC correctly, it's not just harmonies; the range of a piano is big enough that if you tune each octave exactly based on that ratio, you'll end up with the first and last octaves sounding off from each other.
- lmm 5y ago
  If you tune with exact 2^(1/12) semitones then all your octaves will be in tune for obvious reasons. (And pianos are normally tuned this way, equal temperament, so I don't know what the grandparent is talking about; for any tuning system some intervals are just and others are not. Equal temperament gives you just octaves, Pythagorean tuning gives you just fifths, meantone temperaments give you just thirds).
  - adrianmonk 5y ago
    > If you tune with exact 2^(1/12) semitones then all your octaves will be in tune for obvious reasons. Hypothetically, or on an electronic instrument, you could. But if you did all 2^(1/12) ratios, your octaves wouldn't be in tune. Strings on a piano do not behave like an ideal string. Their overtones are not 2X, 3X, 4X, 5X, etc. times the fundamental frequency. Instead, the actual overtones are higher than the ideal frequencies. This is called inharmonicity (https://en.wikipedia.org/wiki/Inharmonicity https://en.wikipedia.org/wiki/Inharmonicity). So when tuning a piano, you have to tailor the way you tune it to each different piano if you want that piano's lower strings to be in tune with its higher strings.
    - resters 5y ago
      > This is called inharmonicity I think I've been hearing this for a long time but didn't realize it was real so questioned my perceptual system. Thank you for the info!
YetAnotherNick 5y ago
Basically at the least you want to create exact frequency ratio of 3/2(major 5th) and 4/3(major 3rd). And also all the power and inverse power which is not achievable in one scale. Instead we substitute 3/2 = 1.5 to 2^(7/12) = 1.4983 and 4/3 = 1.3333 to be 2^(5/12) = 1.3348
mhh__ 5y ago
There's a pretty big community of musicians who do away with almost all constraints wrt to tuning now. Some of it, of course, sounds like cats screeching but some of it is genuinely astonishing.
- bruce343434 5y ago
  For anybody wanting to hear this, look for "microtonal" music, or the artist Sevish is one that I know of.
rcxdude 5y ago
It's more the opposite: equal temperment is a compromise that fudges but mostly preserves the important harmonies (thirds and fifths) in all keys.
Waterluvian 5y ago
“12 tone equal temperament” is a good string to Google if you want to learn about that. Most pianos are intentionally out of tune just enough that they’re good enough for every key. Other coincidences that drive me wild: speed of light is almost but not quite 3.0E8m/s And the fine structure constant being almost but not exactly 1/137.
- BoxOfRain 5y ago
  >speed of light is almost but not quite 3.0E8m/s On that subject, if you decided to make your base unit of length the distance light travels in a nanosecond it'd almost be a foot but not quite.
  - Y_Y 5y ago
    I know that the speed of light is c, or 1 or 3e8 m/s, but I really feel like it's a foot per nanosecond, because that's how I measured it the lab one day messing around with BNC cables amd an oscillscope. Similarly the earth-sun distance is 8 light-minutes.These feel right, like measuring mass in stone for people, kilos for sugar, carat for diamonds, electron-volts for particles etc.
- FreeFull 5y ago
  Pianos are more complicated, because they also use stretched octaves that aren't just a simple 2:1 ratio, due to inharmonicity of the thick bass strings. There are only a few exceptions, where instead of thicker strings the piano was made really long.
- GuB-42 5y ago
  12 tone equal temperament is all wrong, but never by much, and it is very convenient, so almost everyone uses it.

by pgn674 5y ago

saghm 5y ago

> In music, the frequency ratio of a semitone is ideally 2^(1/12), but without some tiny fudging (called tuning), you can't make harmonies as the frequencies almost but don't quite line up right IIRC correctly, it's not just harmonies; the range of a piano is big enough that if you tune each octave exactly based on that ratio, you'll end up with the first and last octaves sounding off from each other.

lmm 5y ago
If you tune with exact 2^(1/12) semitones then all your octaves will be in tune for obvious reasons. (And pianos are normally tuned this way, equal temperament, so I don't know what the grandparent is talking about; for any tuning system some intervals are just and others are not. Equal temperament gives you just octaves, Pythagorean tuning gives you just fifths, meantone temperaments give you just thirds).
- adrianmonk 5y ago
  > If you tune with exact 2^(1/12) semitones then all your octaves will be in tune for obvious reasons. Hypothetically, or on an electronic instrument, you could. But if you did all 2^(1/12) ratios, your octaves wouldn't be in tune. Strings on a piano do not behave like an ideal string. Their overtones are not 2X, 3X, 4X, 5X, etc. times the fundamental frequency. Instead, the actual overtones are higher than the ideal frequencies. This is called inharmonicity (https://en.wikipedia.org/wiki/Inharmonicity https://en.wikipedia.org/wiki/Inharmonicity). So when tuning a piano, you have to tailor the way you tune it to each different piano if you want that piano's lower strings to be in tune with its higher strings.
  - resters 5y ago
    > This is called inharmonicity I think I've been hearing this for a long time but didn't realize it was real so questioned my perceptual system. Thank you for the info!

YetAnotherNick 5y ago

Basically at the least you want to create exact frequency ratio of 3/2(major 5th) and 4/3(major 3rd). And also all the power and inverse power which is not achievable in one scale. Instead we substitute 3/2 = 1.5 to 2^(7/12) = 1.4983 and 4/3 = 1.3333 to be 2^(5/12) = 1.3348

mhh__ 5y ago

There's a pretty big community of musicians who do away with almost all constraints wrt to tuning now. Some of it, of course, sounds like cats screeching but some of it is genuinely astonishing.

bruce343434 5y ago
For anybody wanting to hear this, look for "microtonal" music, or the artist Sevish is one that I know of.

rcxdude 5y ago

It's more the opposite: equal temperment is a compromise that fudges but mostly preserves the important harmonies (thirds and fifths) in all keys.

Waterluvian 5y ago

“12 tone equal temperament” is a good string to Google if you want to learn about that. Most pianos are intentionally out of tune just enough that they’re good enough for every key. Other coincidences that drive me wild: speed of light is almost but not quite 3.0E8m/s And the fine structure constant being almost but not exactly 1/137.

BoxOfRain 5y ago
>speed of light is almost but not quite 3.0E8m/s On that subject, if you decided to make your base unit of length the distance light travels in a nanosecond it'd almost be a foot but not quite.
- Y_Y 5y ago
  I know that the speed of light is c, or 1 or 3e8 m/s, but I really feel like it's a foot per nanosecond, because that's how I measured it the lab one day messing around with BNC cables amd an oscillscope. Similarly the earth-sun distance is 8 light-minutes.These feel right, like measuring mass in stone for people, kilos for sugar, carat for diamonds, electron-volts for particles etc.
FreeFull 5y ago
Pianos are more complicated, because they also use stretched octaves that aren't just a simple 2:1 ratio, due to inharmonicity of the thick bass strings. There are only a few exceptions, where instead of thicker strings the piano was made really long.
GuB-42 5y ago
12 tone equal temperament is all wrong, but never by much, and it is very convenient, so almost everyone uses it.