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.
(Big fan of your videos btw)
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-ei...
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.
Might be worth reading:
- https://books.google.com/books/about/The_Effects_of_Unequal_...
- https://www.proquest.com/openview/b22142f819768ae82464aa2679...
I have this CD [1] in a box somewhere but can't find it on Youtube. It's a few Beethoven sonatas in the temperament he would've used. Just sounded out-of-tune to me in certain parts (especially during the Waldstein), but I don't have perfect pitch. The booklet that came with that CD is really helpful in understanding all this, and I think that might be where I learned about that Owen Jorgensen tome.
There's no shortage of similar experiments on Youtube. This one [2] has a wild one in just intonation, but I doubt that temperament was still used when Mozart was composing.
[1] https://www.amazon.com/Beethoven-Temperaments-Historical-Tun...
This is truly a fascinating world. People who argue that we should be using the original temperaments that composers used do have an argument. Imagine, for example, if Shakespeare were "tuned" to be in modern UK English rather than the English of its time.
The general idea is to achieve consistent beat rates for a given interval -- major thirds being the most useful for its relatively high beat rate compared to other equally tempered intervals -- as you play it chromatically. By that I mean play A and C#, then Bb and D, then B and D#, etc. and if the beat rate hardly changes (but does consistently climb) then you've achieved equal temperament.
Say you've got A tuned to a reference (tuning fork). Then set the A above that so there's no beating, since octaves are always perfectly 2:1 regardless of temperament (until the extremes, when you need to stretch a bit, but I digress). Then tune the C# and then tune the F. Basically it's an augmented triad, or a stack of three major thirds (including from F back up to A). Get all of them to beat by about the same amount, but the higher ones just slightly faster than the lower ones. The fact that this feat utilizes two A's is the key to pulling it off. You frame out the octave and then fill in the augmented triad.
But now you need to do the next set: Bb, D, F#, Bb. How to get here without another external reference? Well, well, well. We have our ways. The perfect fourth between A and D is an option, but be careful not to make it actually a perfect integer (no beating), as that wouldn't be equal tempered; it should beat maybe about half as fast as the nearby major thirds, IIRC.
Notably:
> In equal temperament, all perfect fifths are “contracted”, while all perfect fourths as “expanded”. Minor thirds are contracted, while major sixths are expanded. Major thirds are expanded, while minor sixths are contracted. The piano tech must have knowledge of the approximate beat rates the intervals of equal temperament in the temperament octave: The beat rate of perfect fourths within the temperament octave may be about 1 beat per second. The beat rate of perfect fifths within the temperament octave may be about 1/2 beat per second. The beat rate of F3-A3 major third is about 7 beats per second and that of higher thirds are faster.
In my previous comment, my memory was a bit off when I said "about half the beat rate"... that's the difference in rate between fourths and fifths, apparently.
> Example: to check the tuning of D4 within the temperament octave, play A3-D4 and G3-D4. The fourth should beat faster than the fifth. If the fifth is too fast and the fourth too pure perhaps the D4 is flat; if the fourth and fifth beat at the same rate, perhaps the D4 is flat; if the fourth beats too fast and the fifth is too pure, perhaps the D4 is sharp.
> you can use more and more checks as one progresses through the sequence and tune each new note as a “best compromise” with all the previous notes, that is, each new note will not depend only on the last note tuned, so there will be more of a chance that errors will not accumulate in the later notes tuned.
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.
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).