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There was 12 notes before there was even division. It was Bach that pushed equal temprament (equal spacing). Before that, the ratios were actual ratios (perfe

There was 12 notes before there was even division. It was Bach that pushed equal temprament (equal spacing). Before that, the ratios were actual ratios (perfect 4ths and 5ths), though you couldn't just transpose music and expect to sound good.

sabotista 4y ago
Pythagoras is believed to have come up with the just intonation (exact rational) figures. At the time, irrational numbers were distrusted and despised so, as you noted, the perfect fifth really was exactly 3:2. But it’s likely that a 12-tone system won out because lg(3/2) is so close to 7/12, even if this was never a conscious decision. 19, 31, and 53 are also credible candidates per continued fraction expansion, but unwieldy for physical instruments (although some computer music does use 53-TET).
- not2b 4y ago
  Pythagoras and his followers at first thought that irrational numbers didn't even exist, though the story that they drowned a guy for proving by contradiction that sqrt(2) is irrational is probably not right. Rather, strings with length ratios made of small integers, like 2/3 or 3/4, sound good (harmonize) when played together. So the started with the ratios, because that's what made sense. Not to use ratios was considered, well, irrational. :-)
  - dhosek 4y ago
    It may not be true, but I still loved telling that story to my students.
thirteenfingers 4y ago
IIRC the system Bach was pushing wasn't actually equal temperament, but "well temperament" which was some sort of compromise between equal temperament and having pure fifths everywhere. The result was that all twelve keys sounded acceptable, but some keys had purer fifths or thirds than others. Some musicians/scholars say that Bach composed the different preludes and fugues specifically to use the resulting different characters of the different keys to the best possible advantage. I can't speak to this personally, I keep my piano at equal temperament ;) (Big fan of your videos btw)
- joegahona 4y ago
  This is my understanding too. I have a Kurzweil K2500 keyboard from around 1999 that has a bunch of the alternate tunings, including three from that era. The Bach-era tunings weren't what we know of as "equal temperament." Truly equal temperament didn't come around until well after Beethoven was dead. I've always interpreted the "Well-Tempered" in Bach's title to mean that he was brining out the strength of each key. Some of those keys sound really "out of tune" to modern ears -- I have a friend with perfect pitch who legit can't listen to them. Owen Jorgensen's "Tuning the Historical Temperaments by Ear" is good bedtime reading on this topic, if you have a couple hundred bucks burning a hole in your pocket. https://www.amazon.com/Tuning-historical-temperaments-ear-eighty-nine/dp/091861600X https://www.amazon.com/Tuning-historical-temperaments-ear-ei...
  - wizofaus 4y ago
    > Truly equal temperament didn't come around until well after Beethoven was dead Source for that? The concept and practice certainly existed well before Beethoven's time but it's less clear at which point it became the norm. Even the wikipedia article on 12 TET has "citation needed" for the claim that it happened in the early 19th century.
    - joegahona 4y ago
      I don't remember where I learned this, but what you said is more accurate than what I said -- the concept was definitely known well before Beethoven, but my understanding is it wasn't the standard tuning on keyboard instruments until much later, and came to its fruition with all the atonal music of the early 20th century. I think composers even went so far as to assign emotions/moods to various keys based on each key's sound. E.g. "E-flat major is austere, D-minor is sad," etc. Might be worth reading: - https://books.google.com/books/about/The_Effects_of_Unequal_Temperament_on_Ch.html?id=O6b2tgAACAAJ https://books.google.com/books/about/The_Effects_of_Unequal_... - https://www.proquest.com/openview/b22142f819768ae82464aa2679f4ef68/1?pq-origsite=gscholar&cbl=18750&diss=y https://www.proquest.com/openview/b22142f819768ae82464aa2679...
dahart 4y ago
> it was Bach that pushed equal temperament (equal spacing). Not really. Equal temperament was being advocated both in China and Europe long before Bach was born. In Europe, it was the lute players that pushed for it, because it matters more for fretted instruments, where any temperament other than equal causes conflicts and inconsistencies on the neck. https://en.m.wikipedia.org/wiki/12_equal_temperament https://en.m.wikipedia.org/wiki/12_equal_temperament

sabotista 4y ago

Pythagoras is believed to have come up with the just intonation (exact rational) figures. At the time, irrational numbers were distrusted and despised so, as you noted, the perfect fifth really was exactly 3:2. But it’s likely that a 12-tone system won out because lg(3/2) is so close to 7/12, even if this was never a conscious decision. 19, 31, and 53 are also credible candidates per continued fraction expansion, but unwieldy for physical instruments (although some computer music does use 53-TET).

not2b 4y ago
Pythagoras and his followers at first thought that irrational numbers didn't even exist, though the story that they drowned a guy for proving by contradiction that sqrt(2) is irrational is probably not right. Rather, strings with length ratios made of small integers, like 2/3 or 3/4, sound good (harmonize) when played together. So the started with the ratios, because that's what made sense. Not to use ratios was considered, well, irrational. :-)
- dhosek 4y ago
  It may not be true, but I still loved telling that story to my students.

thirteenfingers 4y ago

IIRC the system Bach was pushing wasn't actually equal temperament, but "well temperament" which was some sort of compromise between equal temperament and having pure fifths everywhere. The result was that all twelve keys sounded acceptable, but some keys had purer fifths or thirds than others. Some musicians/scholars say that Bach composed the different preludes and fugues specifically to use the resulting different characters of the different keys to the best possible advantage. I can't speak to this personally, I keep my piano at equal temperament ;) (Big fan of your videos btw)

joegahona 4y ago
This is my understanding too. I have a Kurzweil K2500 keyboard from around 1999 that has a bunch of the alternate tunings, including three from that era. The Bach-era tunings weren't what we know of as "equal temperament." Truly equal temperament didn't come around until well after Beethoven was dead. I've always interpreted the "Well-Tempered" in Bach's title to mean that he was brining out the strength of each key. Some of those keys sound really "out of tune" to modern ears -- I have a friend with perfect pitch who legit can't listen to them. Owen Jorgensen's "Tuning the Historical Temperaments by Ear" is good bedtime reading on this topic, if you have a couple hundred bucks burning a hole in your pocket. https://www.amazon.com/Tuning-historical-temperaments-ear-eighty-nine/dp/091861600X https://www.amazon.com/Tuning-historical-temperaments-ear-ei...
- wizofaus 4y ago
  > Truly equal temperament didn't come around until well after Beethoven was dead Source for that? The concept and practice certainly existed well before Beethoven's time but it's less clear at which point it became the norm. Even the wikipedia article on 12 TET has "citation needed" for the claim that it happened in the early 19th century.
  - joegahona 4y ago
    I don't remember where I learned this, but what you said is more accurate than what I said -- the concept was definitely known well before Beethoven, but my understanding is it wasn't the standard tuning on keyboard instruments until much later, and came to its fruition with all the atonal music of the early 20th century. I think composers even went so far as to assign emotions/moods to various keys based on each key's sound. E.g. "E-flat major is austere, D-minor is sad," etc. Might be worth reading: - https://books.google.com/books/about/The_Effects_of_Unequal_Temperament_on_Ch.html?id=O6b2tgAACAAJ https://books.google.com/books/about/The_Effects_of_Unequal_... - https://www.proquest.com/openview/b22142f819768ae82464aa2679f4ef68/1?pq-origsite=gscholar&cbl=18750&diss=y https://www.proquest.com/openview/b22142f819768ae82464aa2679...

dahart 4y ago

> it was Bach that pushed equal temperament (equal spacing). Not really. Equal temperament was being advocated both in China and Europe long before Bach was born. In Europe, it was the lute players that pushed for it, because it matters more for fretted instruments, where any temperament other than equal causes conflicts and inconsistencies on the neck. https://en.m.wikipedia.org/wiki/12_equal_temperament https://en.m.wikipedia.org/wiki/12_equal_temperament