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They’re trying to solve various issues with harmony. You want to pick a set of notes that produce good harmonies with each other, but you don’t want to pick too

They’re trying to solve various issues with harmony. You want to pick a set of notes that produce good harmonies with each other, but you don’t want to pick too many notes. If you “fudge” some notes, which compromises the harmony, you can keep a small set of notes in the octave (like twelve!), and if you’re careful about it, the music will still sound consonant / harmonious when it is supposed to sound that way. Historically, the older systems were some derivation of Pythagorean or just intonation, which are generated from small integer ratios like 3:2, 4:3, 5:4. If you continue building a tuning system out of these integer ratios, you get weird ratios like 81:80 (syntonic comma) or 531441:524288 (Pythagorean comma); you can then choose to include these ratios in your tuning system (too complicated), stop before you reach them (too simple), or divide up the ratios and distribute them in other parts of your scale. This last approach is called “musical temperament”. https://en.wikipedia.org/wiki/Musical_temperament https://en.wikipedia.org/wiki/Musical_temperament Werkmeister devised a set of musical temperaments in the late 1600s. Alternative temperaments include mean-tone temperament (various flavors) and equal temperament. Mean-tone temperament divides the commas equally apart parts of the octave. Equal temperament divides the commas equally across the entire octave. Werkmeister temperaments divide the commas unequally. This results in some intervals which sound more consonant and some intervals which sound more dissonant, relative to equal temperament. Unlike equal temperament, different keys will have different character. Just to illustrate a more concrete problem: systems which improve the fifth often require sacrificing the third. Pythagorean has exactly perfect fifths but the thirds are not so good. Quarter-comma meantone goes the opposite direction and gives you exactly just thirds, but the fifths are not so good. Equal temperament is between the two, but much closer to Pythagorean. If you want to play with it immediately, and you have Logic Pro installed, you can go menu diving into the tuning settings for your project. Other DAWs too. Various synthesizers also have Werkmeister buried in the menus—some brands more than others (I know Yamaha has had them for ages).

brudgers 2y ago
The manual for my Yamaha EW20 Windjamm’r is where I stumbled upon it. I find non-12TET temperaments anthropologically interesting in the context of European classical music culture…e.g. what does the exalted status of “perfect pitch” imply?
- klodolph 2y ago
  “Perfect pitch” or “perfect interval”? Someone with perfect pitch can tell what note you are playing just by listening to that note in isolation. Most people can’t do that. “Absolute pitch” is probably a better name, but it’s usually called “perfect pitch”. Perfect intervals are the intervals which don’t have a smaller and larger version of an interval with the same name. It’s nothing more than a naming convention for intervals. The 4th and 5th are perfect. So are the octave (8th) and unison (1). The other intervals (2nd, 3rd, 6th, and 7th), rather than being “perfect”, come in minor and major versions. Basically, every interval either has minor/major, or perfect. It’s a naming convention for intervals based on the western diatonic scale. Any interval which is smaller than minor/perfect is called “diminished”, and an interval which is larger than major/perfect is “augmented”. For example, C to Gb is a diminished fifth, and Ab to F# is an augmented sixth. Which number you use to name an interval is strictly based on the letter names of the note, ignoring sharps and flats.
  - brudgers 2y ago
    I used "perfect pitch" in the ordinary way ordinary people ordinarily use the phrase. Temperaments are interesting because the pitches (and intervals) in 12TET are different from the pitches (and intervals) in Mean Tone are different from the pitches (and intervals) in Werckmeister -- and within Werckmeister because Werkmeister made several temperaments. Never mind 31TET etc.
    - klodolph 2y ago
      > I used "perfect pitch" in the ordinary way ordinary people ordinarily use the phrase. I’m a little confused by what you are saying here, because most people who say “perfect pitch” mean that you can tell which note is being played in isolation. That’s what “perfect pitch” means to ordinary people, as far as I can tell. It’s something you have or don’t have. Like, “I have perfect pitch” or “I don’t have perfect pitch”. If you mean something other than that definition, maybe you could find some alternative way to express it because I don’t understand what meaning you are getting at. I’m not trying to be pedantic here, but “perfect pitch” is right there in the dictionary and it’s most commonly used to mean that you can recognize pitch without a reference, i.e., absolute pitch recognition.

brudgers 2y ago

The manual for my Yamaha EW20 Windjamm’r is where I stumbled upon it. I find non-12TET temperaments anthropologically interesting in the context of European classical music culture…e.g. what does the exalted status of “perfect pitch” imply?

klodolph 2y ago
“Perfect pitch” or “perfect interval”? Someone with perfect pitch can tell what note you are playing just by listening to that note in isolation. Most people can’t do that. “Absolute pitch” is probably a better name, but it’s usually called “perfect pitch”. Perfect intervals are the intervals which don’t have a smaller and larger version of an interval with the same name. It’s nothing more than a naming convention for intervals. The 4th and 5th are perfect. So are the octave (8th) and unison (1). The other intervals (2nd, 3rd, 6th, and 7th), rather than being “perfect”, come in minor and major versions. Basically, every interval either has minor/major, or perfect. It’s a naming convention for intervals based on the western diatonic scale. Any interval which is smaller than minor/perfect is called “diminished”, and an interval which is larger than major/perfect is “augmented”. For example, C to Gb is a diminished fifth, and Ab to F# is an augmented sixth. Which number you use to name an interval is strictly based on the letter names of the note, ignoring sharps and flats.
- brudgers 2y ago
  I used "perfect pitch" in the ordinary way ordinary people ordinarily use the phrase. Temperaments are interesting because the pitches (and intervals) in 12TET are different from the pitches (and intervals) in Mean Tone are different from the pitches (and intervals) in Werckmeister -- and within Werckmeister because Werkmeister made several temperaments. Never mind 31TET etc.
  - klodolph 2y ago
    > I used "perfect pitch" in the ordinary way ordinary people ordinarily use the phrase. I’m a little confused by what you are saying here, because most people who say “perfect pitch” mean that you can tell which note is being played in isolation. That’s what “perfect pitch” means to ordinary people, as far as I can tell. It’s something you have or don’t have. Like, “I have perfect pitch” or “I don’t have perfect pitch”. If you mean something other than that definition, maybe you could find some alternative way to express it because I don’t understand what meaning you are getting at. I’m not trying to be pedantic here, but “perfect pitch” is right there in the dictionary and it’s most commonly used to mean that you can recognize pitch without a reference, i.e., absolute pitch recognition.