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To fill in the bit about temperament: The problem is, once you construct a diatonic scale using these "Pythagorean" ratios, trying to play a diatonic scale tha

To fill in the bit about temperament: The problem is, once you construct a diatonic scale using these "Pythagorean" ratios, trying to play a diatonic scale that starts and ends on a different pitch, but using the same set of pitches, will have intervals that aren't in tune and sound terrible. Musicians and theorists struggled with this for years, coming up with various compromises that sounded good in some "keys" and awful in others, until math provided the ultimate compromise: base your logarithmic scale on twelfth roots of two and every key will be exactly the same more-or-less-in-tune.

lebuffon 7y ago
In my musical training I was taught that musicians figured it out in the west by trial error. Bach then wrote the Well Tempered Klavier to demonstrate pieces in all 12 major and 12 minor keys, in 1722. There is a reference here however to a chinese mathematician who worked it out in 1584. I don't know if his work made it across the continent and influenced anyone. It is not likely that many musicians would have understood the math in the 16th century. https://en.wikipedia.org/wiki/Well_temperament#targetText=Origins,(Duffin%202007%2C%2037) https://en.wikipedia.org/wiki/Well_temperament#targetText=Or....
- IggleSniggle 7y ago
  Notably, well-temperament is not equal-temperament. Also under-appreciated is that you can change the pitch of each strike of a key on a clavichord by striking harder/softer or pressing into the key. It uses a “hammer-on” like action, so if you press into the key after striking a note, you will increase the tension on the string and raise the pitch. For me, realizing this gave me a whole new appreciation for CPE Bach’s music. There’s probably a dissertation in that somewhere for somebody. A klavier is not necessarily a clavichord, but an organ has some interesting tuning tricks up its sleeves with its various stops.
  - jacquesm 7y ago
    Organs are not really tuned with the stops, the stops engage or disengage banks ('ranks' in organ terminology) of related pipes. The only way this would affect tuning is if you have a secondary rank with an alternate tuning, you'd have to disable one rank and enable another. Organs are tuned directly at the individual pipes (just like string instruments are tuned at the strings). Each pipe, depending on the kind of pipe used has a slide that goes in and out on one end to match the pipe length or, alternatively, a tiny tongue of metal that is rolled up or extended. Tuning (or 'voicing') an organ is super labor intensive and time consuming. For a large organ it can take weeks.
    - IggleSniggle 7y ago
      I am well aware. I have participated in the tuning of a (small) organ, as I am an organist. You can affect the perceived tuning of the instrument by engaging mutation stops / aliquots. Further, speaking from experience, in practice, stops have a tendency to go out of tune at different rates. Most organists know their instrument very well, have have an intuition for what keys will sound best with what stops at any given time on their instrument.
      - jacquesm 7y ago
        That makes a lot more sense than the way you wrote it first :) Which organ do you play?
      - phlakaton 7y ago
        Oh yes. The things you do to avoid that out-of-tune trumpet pipe. :-)
madhadron 7y ago
> Musicians and theorists struggled with this for years, coming up with various compromises that sounded good in some "keys" and awful in others, until math provided the ultimate compromise: base your logarithmic scale on twelfth roots of two and every key will be exactly the same more-or-less-in-tune. This is inaccurate in two ways. First, only keyboards and those playing with them, use a fixed temperament. Everyone else, strings, voices, brass, winds, adjusts the pitch of individual notes based on vertical and horizontal context. A c does not have a fixed number of Hz throughout a piece. Second, math did not provide the ultimate compromise. Equal temperament was known as far back as the fourth century BC, and people were advocating for and composing in equal temperament in the sixteenth century. Rejecting equal temperament in favor of meantone temperings was a conscious decision, not a compromise from ignorance.

lebuffon 7y ago

In my musical training I was taught that musicians figured it out in the west by trial error. Bach then wrote the Well Tempered Klavier to demonstrate pieces in all 12 major and 12 minor keys, in 1722. There is a reference here however to a chinese mathematician who worked it out in 1584. I don't know if his work made it across the continent and influenced anyone. It is not likely that many musicians would have understood the math in the 16th century. https://en.wikipedia.org/wiki/Well_temperament#targetText=Origins,(Duffin%202007%2C%2037) https://en.wikipedia.org/wiki/Well_temperament#targetText=Or....

IggleSniggle 7y ago
Notably, well-temperament is not equal-temperament. Also under-appreciated is that you can change the pitch of each strike of a key on a clavichord by striking harder/softer or pressing into the key. It uses a “hammer-on” like action, so if you press into the key after striking a note, you will increase the tension on the string and raise the pitch. For me, realizing this gave me a whole new appreciation for CPE Bach’s music. There’s probably a dissertation in that somewhere for somebody. A klavier is not necessarily a clavichord, but an organ has some interesting tuning tricks up its sleeves with its various stops.
- jacquesm 7y ago
  Organs are not really tuned with the stops, the stops engage or disengage banks ('ranks' in organ terminology) of related pipes. The only way this would affect tuning is if you have a secondary rank with an alternate tuning, you'd have to disable one rank and enable another. Organs are tuned directly at the individual pipes (just like string instruments are tuned at the strings). Each pipe, depending on the kind of pipe used has a slide that goes in and out on one end to match the pipe length or, alternatively, a tiny tongue of metal that is rolled up or extended. Tuning (or 'voicing') an organ is super labor intensive and time consuming. For a large organ it can take weeks.
  - IggleSniggle 7y ago
    I am well aware. I have participated in the tuning of a (small) organ, as I am an organist. You can affect the perceived tuning of the instrument by engaging mutation stops / aliquots. Further, speaking from experience, in practice, stops have a tendency to go out of tune at different rates. Most organists know their instrument very well, have have an intuition for what keys will sound best with what stops at any given time on their instrument.
    - jacquesm 7y ago
      That makes a lot more sense than the way you wrote it first :) Which organ do you play?
    - phlakaton 7y ago
      Oh yes. The things you do to avoid that out-of-tune trumpet pipe. :-)

madhadron 7y ago

> Musicians and theorists struggled with this for years, coming up with various compromises that sounded good in some "keys" and awful in others, until math provided the ultimate compromise: base your logarithmic scale on twelfth roots of two and every key will be exactly the same more-or-less-in-tune. This is inaccurate in two ways. First, only keyboards and those playing with them, use a fixed temperament. Everyone else, strings, voices, brass, winds, adjusts the pitch of individual notes based on vertical and horizontal context. A c does not have a fixed number of Hz throughout a piece. Second, math did not provide the ultimate compromise. Equal temperament was known as far back as the fourth century BC, and people were advocating for and composing in equal temperament in the sixteenth century. Rejecting equal temperament in favor of meantone temperings was a conscious decision, not a compromise from ignorance.