npub1knzsux7p6lzwzdedp3m8c3c92z0swzc0xyy5glvse58txj5e9ztqpac73p (npub1knz…c73p) Thanks for this write up, it's quite accurate both from a historical and musical perspective too!
Btw, Western music since about the times of Bach uses neither the Pythagorean nor the just intonation. The problem with both is that they are optimized for some specific intervals (the 5th and the major 3rd respectively) and they build all the remaining intervals in such a way that the small integer ratio constraint keep being satisfied for those intervals.
The problem with the Pythagorean approach is that any interval that isn't a perfect 5th (or a perfect 4th) sounds quite dissonant and unresolved. To a trained ear, a Pythagorean major 3rd sounds actually quite harsh and unresolved, because of that ugly 81/64 frequency ratio - and that's also why early Medieval scholars who were into Pythagorean music considered the major 3rd an unresolved interval full of tension, not something you'd resolve a music passage to.
Just intonation solves the large integer ratios problem of the Pythagorean system, but everything starts to fall apart when you move to different keys - that's the reason why, to this day, many instruments whose history can be traced back to the Middle Ages or Renaissance can only perform the notes in a particular scale.
The equal temperament system we use today fixes the transposition problem, but in order to do so it had to give up on the main feature of both the Pythagorean and just systems - the integer ratios between frequencies.
The only interval that can be represented through a rational ratio is the octave (2/1). The octave is then split into 12 equal parts, so instead of a linear equation you end up with a logarithmic one - given a base frequency f(0), the frequency f(n) of the note that is n semitones away from the reference is (2**(n/12))*f(0).
This means that we have also lost the 3/2 ratio of the perfect fifth (it becomes 2**(7/12) =~ 1.498307) and the 4/3 ratio of the perfect fourth (it becomes 2**(5/12)=~1.334839). The major third ratio becomes 2**(4/12)=~1.259921 - not as nice and consonant as the 1.25 of the just intonation, but definitely better than that sharp and dissonant 1.265625 ratio that you get in the Pythagorean system.
Fabio Manganiello /
npub1v7…ukv0u
2023-11-01 22:08:01
in reply to nevent1q…3mxf
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npub1v78mmuz20p6qd6nve30axhqu74avwn4f6z4grhug7755rat7wh3syukv0uPublished at
2023-11-01 22:08:01Event JSON
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"content": "nostr:npub1knzsux7p6lzwzdedp3m8c3c92z0swzc0xyy5glvse58txj5e9ztqpac73p Thanks for this write up, it's quite accurate both from a historical and musical perspective too!\n\nBtw, Western music since about the times of Bach uses neither the Pythagorean nor the just intonation. The problem with both is that they are optimized for some specific intervals (the 5th and the major 3rd respectively) and they build all the remaining intervals in such a way that the small integer ratio constraint keep being satisfied for those intervals.\n\nThe problem with the Pythagorean approach is that any interval that isn't a perfect 5th (or a perfect 4th) sounds quite dissonant and unresolved. To a trained ear, a Pythagorean major 3rd sounds actually quite harsh and unresolved, because of that ugly 81/64 frequency ratio - and that's also why early Medieval scholars who were into Pythagorean music considered the major 3rd an unresolved interval full of tension, not something you'd resolve a music passage to. \n\nJust intonation solves the large integer ratios problem of the Pythagorean system, but everything starts to fall apart when you move to different keys - that's the reason why, to this day, many instruments whose history can be traced back to the Middle Ages or Renaissance can only perform the notes in a particular scale.\n\nThe equal temperament system we use today fixes the transposition problem, but in order to do so it had to give up on the main feature of both the Pythagorean and just systems - the integer ratios between frequencies.\n\nThe only interval that can be represented through a rational ratio is the octave (2/1). The octave is then split into 12 equal parts, so instead of a linear equation you end up with a logarithmic one - given a base frequency f(0), the frequency f(n) of the note that is n semitones away from the reference is (2**(n/12))*f(0).\n\nThis means that we have also lost the 3/2 ratio of the perfect fifth (it becomes 2**(7/12) =~ 1.498307) and the 4/3 ratio of the perfect fourth (it becomes 2**(5/12)=~1.334839). The major third ratio becomes 2**(4/12)=~1.259921 - not as nice and consonant as the 1.25 of the just intonation, but definitely better than that sharp and dissonant 1.265625 ratio that you get in the Pythagorean system.",
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