4 ms·

Also, if I remember correctly, every note in the major scale is a 3/2,2/3 ratio from a note in the lower or higher octave following the standard order of key si

Also, if I remember correctly, every note in the major scale is a 3/2,2/3 ratio from a note in the lower or higher octave following the standard order of key signatures as you jump across consecutive octaves, which, I assume, is how they arrived at that standard order.

klodolph 5y ago
That’s called “Pythagorean tuning” and it’s possible but it’s not especially common. The system we use is a compromise derived from that system. If you use pure 3:2 ratios you get the so-called “wolf tones”, but we don’t get those any more. The current most common system is equal temperament, which is a type of meantone tuning, where you take the 3:2 ratio and alter it. Equal temperament alters it to 2^(7/12) ≈ 1.4983, and it has the nice property that you get twelve notes and it’s all perfectly symmetrical, it doesn't matter which note you start on. Before equal temperament, there were various other systems. One popular one was quarter-comma meantone, which took four stacked 3:2 ratios (81:16) and squashed them to equal two octaves and a 5:4 ratio (80:16), which results in a value of 5^(1/4) ≈ 1.4953. You see it in some old organs—which may have, say, separate keys for G# and Ab. Then there are various non-meantone tunings, various “just” intonations, etc. In just intonation, a chord like C E G will have a 4:5:6 ratio. You can’t do this with all major triads, though, so you end up with some major triads that sound correct and others that sound out of tune. Our modern system is a compromise where all major triads are equally out of tune, rather than having some better and some worse.
- lucas_codes 5y ago
  Wolf tones are certain notes which don't sound very well which vary between instruments due to their construction. I think you mean the Pythagorean comma.
  - bonzini 5y ago
    You are talking of the same thing. The typical wolf tone in quarter comma meantone was G#-Eb, which sounded so bad that some instruments added a separate key for Ab.
  - klodolph 5y ago
    Actually, "wolf interval". Pythagorean comma is the difference between enharmonic notes in Pythagorean tuning, e.g., difference between Ab and G#. Not what I was talking about. With any 12-tone system, you might have eleven fifths which are 700-x cents wide, on average... and then a diminished sixth (rather than a fifth), which is necessarily 700+11x cents wide as a result. The "wolf interval" is the diminished sixth (commonly G# to Eb, or C# to Ab). In equal temperament, the diminished sixth is equal to a fifth. You might also call something other than a fifth a wolf interval. Basically, an interval which normally sounds good, except you chose an enharmonic version of it, which sounds bad. "The" wolf interval is the quarter-comma meantone diminished sixth, which is 2^7 / 5^(11/4) = 1.5312. The Pythagorean wolf interval is 2^7 / (3/2)^11 = 1.4798. It's equal to a perfect fifth, minus the Pythagorean comma.
    - lucas_codes 5y ago
      I see. I was confused because string players talk a lot about our wolf notes/wolf tones (especially violists and cellists), but I hadn't come across the term wolf interval.
      - klodolph 5y ago
        That's because these tuning systems have been out of favor since about 1700. You might encounter some organists who know about it, but even then, there are only a handful of organs tuned this way in North America. https://en.wikipedia.org/wiki/Meantone_organs_in_North_America https://en.wikipedia.org/wiki/Meantone_organs_in_North_Ameri... It was Werckmeister who popularized and advocated systems which don't have wolf intervals (so-called "well temperament", as in "Bach's well-tempered clavier". The composers who cared about wolf intervals generally predate Bach, to give you an idea. Nowadays (since the 1980s) I can press a button and put my synthesizer or electronic piano in whatever tuning I want, so I can play with quarter-comma meantone temperament without finding one of the organs on the list, and experience it myself.

klodolph 5y ago

That’s called “Pythagorean tuning” and it’s possible but it’s not especially common. The system we use is a compromise derived from that system. If you use pure 3:2 ratios you get the so-called “wolf tones”, but we don’t get those any more. The current most common system is equal temperament, which is a type of meantone tuning, where you take the 3:2 ratio and alter it. Equal temperament alters it to 2^(7/12) ≈ 1.4983, and it has the nice property that you get twelve notes and it’s all perfectly symmetrical, it doesn't matter which note you start on. Before equal temperament, there were various other systems. One popular one was quarter-comma meantone, which took four stacked 3:2 ratios (81:16) and squashed them to equal two octaves and a 5:4 ratio (80:16), which results in a value of 5^(1/4) ≈ 1.4953. You see it in some old organs—which may have, say, separate keys for G# and Ab. Then there are various non-meantone tunings, various “just” intonations, etc. In just intonation, a chord like C E G will have a 4:5:6 ratio. You can’t do this with all major triads, though, so you end up with some major triads that sound correct and others that sound out of tune. Our modern system is a compromise where all major triads are equally out of tune, rather than having some better and some worse.

lucas_codes 5y ago
Wolf tones are certain notes which don't sound very well which vary between instruments due to their construction. I think you mean the Pythagorean comma.
- bonzini 5y ago
  You are talking of the same thing. The typical wolf tone in quarter comma meantone was G#-Eb, which sounded so bad that some instruments added a separate key for Ab.
- klodolph 5y ago
  Actually, "wolf interval". Pythagorean comma is the difference between enharmonic notes in Pythagorean tuning, e.g., difference between Ab and G#. Not what I was talking about. With any 12-tone system, you might have eleven fifths which are 700-x cents wide, on average... and then a diminished sixth (rather than a fifth), which is necessarily 700+11x cents wide as a result. The "wolf interval" is the diminished sixth (commonly G# to Eb, or C# to Ab). In equal temperament, the diminished sixth is equal to a fifth. You might also call something other than a fifth a wolf interval. Basically, an interval which normally sounds good, except you chose an enharmonic version of it, which sounds bad. "The" wolf interval is the quarter-comma meantone diminished sixth, which is 2^7 / 5^(11/4) = 1.5312. The Pythagorean wolf interval is 2^7 / (3/2)^11 = 1.4798. It's equal to a perfect fifth, minus the Pythagorean comma.
  - lucas_codes 5y ago
    I see. I was confused because string players talk a lot about our wolf notes/wolf tones (especially violists and cellists), but I hadn't come across the term wolf interval.
    - klodolph 5y ago
      That's because these tuning systems have been out of favor since about 1700. You might encounter some organists who know about it, but even then, there are only a handful of organs tuned this way in North America. https://en.wikipedia.org/wiki/Meantone_organs_in_North_America https://en.wikipedia.org/wiki/Meantone_organs_in_North_Ameri... It was Werckmeister who popularized and advocated systems which don't have wolf intervals (so-called "well temperament", as in "Bach's well-tempered clavier". The composers who cared about wolf intervals generally predate Bach, to give you an idea. Nowadays (since the 1980s) I can press a button and put my synthesizer or electronic piano in whatever tuning I want, so I can play with quarter-comma meantone temperament without finding one of the organs on the list, and experience it myself.