7 ms·

That book looks really interesting - thanks for the recommendation. What still twists my melon about harmony is the Pythagorean Comma. Harmony equating to int

by mhax 11y ago

That book looks really interesting - thanks for the recommendation. What still twists my melon about harmony is the Pythagorean Comma. Harmony equating to integer ratios seems so right... But the fact that a perfect 5th on a piano isn't really a perfect 5th troubles me at some existential level.

luch 11y ago
The comma is not a big deal : it's a problem caused by tone equalizing (aka temperament). Think of instruments as ladders : they have all the same steps, but start at different heights. When combining several types of instruments, the resulting steps are not aligned. Equal temperament solve that problem by normalizing the steps' absolute height. It's a trade-off between the instrument's perfect harmony, and the orchestra's overall one.
- haberman 11y ago
  I don't mean to be rude, but I'm not sure you understand the comma. It has nothing to do with combining types of instruments. It's a problem that exists even with a single instrument. The problem is that, even on a single instrument (say a piano), if you start on some note (say C0), going up seven octaves should land you on exactly the same note as going up twelve perfect fifths (C7). The problem is that the precise tuning of that C7 is different depending on if you followed the perfect octaves or perfect fifths. If you followed perfect octaves, which have a ratio of 2/1 (or just 2), C7 should have a tuning of precisely (2^7)·C0 = 128·C0. If you followed perfect fifths, which have a ratio of 3/2, C7 should have a tuning of precisely ((3/2)^12)·C0 = ~129.74·C0. The difference between these two is the Pythagorean comma. Equal temperament solves this problem by keeping the octaves perfect (2:1), but compromising on everything else. It defines a half-step as the 12th power of 2, so a fifth ends up being a little narrower than a perfect 3:2 perfect fifth.
  - dcsommer 11y ago
    I'd recommend "How Equal Temperament Ruined Harmony" if you are interested in the nuances and trade offs of the imperfections of tuning.
  - graycat 11y ago
    Nice. I'd go with the 2^7 and not the (3/2)^(12). > It defines a half-step as the 12th power of 2 Uh, instead, "12th root of 2".
  - BurningFrog 11y ago
    I understand that pianos have to pick a frequency for each key. I wonder if there is any effort in electronic instruments to produce "better" frequencies on the fly? I'll explain: On a piano hitting C+G plays something close to a perfect fifth. An electronic instrument could move the G slightly to produce a perfect fifth when that key combination is played.
    - jpitz 11y ago
      You cant do that, without a large effort in music education or ensembles of all electronic instruments implementing this feature. Everyone else will be out of tune. Edit: you could do it, but you'd better implement it on a feedback system that can re-tune in a hundred milliseconds or so.
      - BurningFrog 11y ago
        I'm just talking about an individual keyboard, though if all instruments are digital, which is pretty common these days, you could have a whole orchestra dynamically adjusting pitch to make "perfect" harmonies. What I wonder is mostly if that would sound any better.
    - mhax 11y ago
      It's not just a matter of 'better' frequencies. It's trivial to have a softsynth or something similar set up to use just intonation (and many modern composers have experimented with this), but that has practical problems of it's own. Problems which were the very reason that equal temperement took over in the first place. This page has some wonderful examples http://www.nesssoftware.com/home/asn/homepage/teaching/exp-lectureNotes/110412-young/young.html http://www.nesssoftware.com/home/asn/homepage/teaching/exp-l...
analog31 11y ago
In my view all it means is that an approximation of perfection is good enough. And the notion of temperament only really applies to keyboard instruments. Strings are tuned to perfect intervals, and even 'good' intonation of fingered notes is a matter of interpretation. Wind instruments are a hodgepodge of compromises.
- meggar 11y ago
  I think there is a similar situation with guitar tuning where the G string can either be tuned to a 4th above the D string, or a Maj3 below B string, which are not exactly equal, so guitar chords are usually a tiny bit out of tune.
  - neonscribe 11y ago
    No, the Maj3 interval between the 2nd and 3rd strings on the guitar is unrelated to this issue. It's just the fret spacing on the guitar that causes it to have equal temperament and therefore makes chords slightly "out of tune".
    - elihu 11y ago
      Incidentally, a just (non-tempered) guitar fingerboard might look something like this: http://jsnow.bootlegether.net/cbg/fingerboard_at_a.svg http://jsnow.bootlegether.net/cbg/fingerboard_at_a.svg I swapped fingerboards on an old Harmony Archtone with one based on the above design. Admittedly, 20 notes per octave may have been a bit much; it's kind of awkward to play. The next one I do will be simpler. Sounds nice, though.
      - gnaritas 11y ago
        Just remove the frets, problem solved.
        elihu 11y ago
        Fretless is difficult to play accurately, especially for chords. Also, you get a louder sound from frets because your fingers aren't muting the strings quite so much.
bsder 11y ago
22 frets can make some of the extended harmonies much, much closer to ideal. https://www.youtube.com/watch?v=qHHv3mwJTlg https://www.youtube.com/watch?v=qHHv3mwJTlg
im3w1l 11y ago
Just intonation is just a theory. In practice slightly detuned intervals sound good. Imo they can even sound better than when they are perfectly in tune. Sometimes musicians deliberately use detuning to create a sound, for instance in the "chorus effect". http://forum.audacityteam.org/viewtopic.php?f=42&t=68007 http://forum.audacityteam.org/viewtopic.php?f=42&t=68007 http://www.dancemusicproduction.com/forum/index.php?showtopic=4408 http://www.dancemusicproduction.com/forum/index.php?showtopi... EDIT: Another interesting fact is that pianos aren't actually tuned to equal temperament. The intervals are tuned slightly bigger than it would dictate, to accommodate pianos' slightly inharmonic overtone series. http://en.wikipedia.org/wiki/Piano_tuning#Stretch http://en.wikipedia.org/wiki/Piano_tuning#Stretch
foobarrio 11y ago
I've always seen equal temperament similarly to digitizing or any other form quantization. It's "close enough". Any type of dissonance heard in what are supposed to be perfect intervals is like grain in film.
m-photonic 11y ago
The Pythagorean comma is nothing to worry about -- it's tiny and more or less negligible. It's the syntonic comma that's the killer. The fifths in equal temperament are completely fine, but the thirds are really quite a ways off from the harmonic 5/4 major third. I would recommend experimenting with the traditional meantone tuning system if you're interested in this -- it offers almost as much flexibility as equal temperament (with 6 adjacent keys on the circle of fifths being free of wolf intervals), and it lets you play with "real thirds".

by mhax 11y ago

luch 11y ago

The comma is not a big deal : it's a problem caused by tone equalizing (aka temperament). Think of instruments as ladders : they have all the same steps, but start at different heights. When combining several types of instruments, the resulting steps are not aligned. Equal temperament solve that problem by normalizing the steps' absolute height. It's a trade-off between the instrument's perfect harmony, and the orchestra's overall one.

haberman 11y ago
I don't mean to be rude, but I'm not sure you understand the comma. It has nothing to do with combining types of instruments. It's a problem that exists even with a single instrument. The problem is that, even on a single instrument (say a piano), if you start on some note (say C0), going up seven octaves should land you on exactly the same note as going up twelve perfect fifths (C7). The problem is that the precise tuning of that C7 is different depending on if you followed the perfect octaves or perfect fifths. If you followed perfect octaves, which have a ratio of 2/1 (or just 2), C7 should have a tuning of precisely (2^7)·C0 = 128·C0. If you followed perfect fifths, which have a ratio of 3/2, C7 should have a tuning of precisely ((3/2)^12)·C0 = ~129.74·C0. The difference between these two is the Pythagorean comma. Equal temperament solves this problem by keeping the octaves perfect (2:1), but compromising on everything else. It defines a half-step as the 12th power of 2, so a fifth ends up being a little narrower than a perfect 3:2 perfect fifth.
- dcsommer 11y ago
  I'd recommend "How Equal Temperament Ruined Harmony" if you are interested in the nuances and trade offs of the imperfections of tuning.
- graycat 11y ago
  Nice. I'd go with the 2^7 and not the (3/2)^(12). > It defines a half-step as the 12th power of 2 Uh, instead, "12th root of 2".
- BurningFrog 11y ago
  I understand that pianos have to pick a frequency for each key. I wonder if there is any effort in electronic instruments to produce "better" frequencies on the fly? I'll explain: On a piano hitting C+G plays something close to a perfect fifth. An electronic instrument could move the G slightly to produce a perfect fifth when that key combination is played.
  - jpitz 11y ago
    You cant do that, without a large effort in music education or ensembles of all electronic instruments implementing this feature. Everyone else will be out of tune. Edit: you could do it, but you'd better implement it on a feedback system that can re-tune in a hundred milliseconds or so.
    - BurningFrog 11y ago
      I'm just talking about an individual keyboard, though if all instruments are digital, which is pretty common these days, you could have a whole orchestra dynamically adjusting pitch to make "perfect" harmonies. What I wonder is mostly if that would sound any better.
  - mhax 11y ago
    It's not just a matter of 'better' frequencies. It's trivial to have a softsynth or something similar set up to use just intonation (and many modern composers have experimented with this), but that has practical problems of it's own. Problems which were the very reason that equal temperement took over in the first place. This page has some wonderful examples http://www.nesssoftware.com/home/asn/homepage/teaching/exp-lectureNotes/110412-young/young.html http://www.nesssoftware.com/home/asn/homepage/teaching/exp-l...

analog31 11y ago

In my view all it means is that an approximation of perfection is good enough. And the notion of temperament only really applies to keyboard instruments. Strings are tuned to perfect intervals, and even 'good' intonation of fingered notes is a matter of interpretation. Wind instruments are a hodgepodge of compromises.

meggar 11y ago
I think there is a similar situation with guitar tuning where the G string can either be tuned to a 4th above the D string, or a Maj3 below B string, which are not exactly equal, so guitar chords are usually a tiny bit out of tune.
- neonscribe 11y ago
  No, the Maj3 interval between the 2nd and 3rd strings on the guitar is unrelated to this issue. It's just the fret spacing on the guitar that causes it to have equal temperament and therefore makes chords slightly "out of tune".
  - elihu 11y ago
    Incidentally, a just (non-tempered) guitar fingerboard might look something like this: http://jsnow.bootlegether.net/cbg/fingerboard_at_a.svg http://jsnow.bootlegether.net/cbg/fingerboard_at_a.svg I swapped fingerboards on an old Harmony Archtone with one based on the above design. Admittedly, 20 notes per octave may have been a bit much; it's kind of awkward to play. The next one I do will be simpler. Sounds nice, though.
    - gnaritas 11y ago
      Just remove the frets, problem solved.
      - elihu 11y ago
        Fretless is difficult to play accurately, especially for chords. Also, you get a louder sound from frets because your fingers aren't muting the strings quite so much.

bsder 11y ago

22 frets can make some of the extended harmonies much, much closer to ideal. https://www.youtube.com/watch?v=qHHv3mwJTlg https://www.youtube.com/watch?v=qHHv3mwJTlg

im3w1l 11y ago

Just intonation is just a theory. In practice slightly detuned intervals sound good. Imo they can even sound better than when they are perfectly in tune. Sometimes musicians deliberately use detuning to create a sound, for instance in the "chorus effect". http://forum.audacityteam.org/viewtopic.php?f=42&t=68007 http://forum.audacityteam.org/viewtopic.php?f=42&t=68007 http://www.dancemusicproduction.com/forum/index.php?showtopic=4408 http://www.dancemusicproduction.com/forum/index.php?showtopi... EDIT: Another interesting fact is that pianos aren't actually tuned to equal temperament. The intervals are tuned slightly bigger than it would dictate, to accommodate pianos' slightly inharmonic overtone series. http://en.wikipedia.org/wiki/Piano_tuning#Stretch http://en.wikipedia.org/wiki/Piano_tuning#Stretch

foobarrio 11y ago

I've always seen equal temperament similarly to digitizing or any other form quantization. It's "close enough". Any type of dissonance heard in what are supposed to be perfect intervals is like grain in film.

m-photonic 11y ago

The Pythagorean comma is nothing to worry about -- it's tiny and more or less negligible. It's the syntonic comma that's the killer. The fifths in equal temperament are completely fine, but the thirds are really quite a ways off from the harmonic 5/4 major third. I would recommend experimenting with the traditional meantone tuning system if you're interested in this -- it offers almost as much flexibility as equal temperament (with 6 adjacent keys on the circle of fifths being free of wolf intervals), and it lets you play with "real thirds".