Equal Temperament / Just Intonation

[westknife]

I want to understand this. Here's what I think I understand:

- Scales/chords are more naturally consonant in just intonation, no "beats" when 2 notes are played together. But you can only play in one key, otherwise the intervals will be off.
- Equal temperament was invented because it's more practical; you can play in any key on any instrument.
- The compromise is that all of our intervals are just a little bit off; the most noticeable is the major third interval, which is "sharp."

Thing is, major thirds have never sounded sharp to me. And if just intonation sounds so much better, how come more music isn't written for it? I am curious to not only understand this theoretically, but more to actually hear what the difference sounds like.

[Scarpia]

Thing is, major thirds have never sounded sharp to me. And if just intonation sounds so much better, how come more music isn't written for it? I am curious to not only understand this theoretically, but more to actually hear what the difference sounds like.

You've grown accustomed to it.  Just intonation is of very little use.  More common were various schemes of well-tempered intonation in which some keys sounded purer than others but most keys sounded ok.

There's a funny theory that the series of squiggles that Bach apparently used as an ornament on his WTC manuscript was actually in recipe for tuning the harpsichord.


http://www-personal.umich.edu/~bpl/larips/index.html

[eyeresist]

no "beats" when 2 notes are played together.

Is this really true? Surely two different notes will "beat" regardless of their harmonic relation.

[petrarch]

Is this really true? Surely two different notes will "beat" regardless of their harmonic relation.

Yes, two notes will always beat; they will beat at a frequency corresponding to the difference between them.

[petrarch]

And if just intonation sounds so much better, how come more music isn't written for it? I am curious to not only understand this theoretically, but more to actually hear what the difference sounds like.

String players will tend to play in just intonation when on their own or when not playing with instruments that impose a specific tuning (like, e.g. a piano).

[eyeresist]

String players will tend to play in just intonation when on their own or when not playing with instruments that impose a specific tuning (like, e.g. a piano).

Your source?

[Luke]

Your source?

What's certainly often true is that string players will emphasize the sharpness of leading notes, or the flatness of a 7th of a V7, etc. etc. etc. etc as voice-leading suggests. But the rigid fixity of the open strings will always keep them rooted in the right place.

[chasmaniac]

Tuning a guitar by ear will produce a temperament that is effectively just. Back when I dabbled, if the song was in C, I'd start with a correct E or A and tune until the big C/G chord was perfect and so on. Of course, electronic tuners are the norm now.

[petrarch]

Your source?

Helmholtz, On the sensations of tone; but some more recent results (published in the Journal of the Acoustical Society of America) show that string players and choral groups, when playing solo or in an ensemble, tend toward the Pythagorean intervals rather than the just intervals. At the end of the day, they will play what sounds best, rather than what the tuning system mandates.

[jochanaan]

Yes, two notes will always beat; they will beat at a frequency corresponding to the difference between them.

But, in "just intonation," the intervals are tuned to an integral ratio: the relationship between the notes' frequencies can be expressed in integers, that is, whole (non-fractional, non-decimal) numbers.  A major third, for example, has a theoretical intervalic ratio of 5/4.  There is no audible beat if the two notes are tuned to frequencies with such a ratio, any more than if they were tuned to the same frequency.  Yet, as you say, you can only tune one scale at a time into just intonation; the moment you play in another key without retuning, you get beats.  (Some composers like to take advantage of these audible beats; one of the finest examples is Stravinsky's The Rite of Spring with its shimmering dissonances.)

String players will tend to play in just intonation when on their own or when not playing with instruments that impose a specific tuning (like, e.g. a piano).

Actually, as Luke says, many string players, rather than playing in "naturally" just intonation, will emphasize certain intervals by "stretching" them, even at the cost of some audible beats.  For instance, they will play the top note of a major third sharp to emphasize its "majorness," adding a desirable harmonic spice to what would otherwise be a bland sound.  (I know this by talking to string players and hearing conductors' instructions to them during orchestra rehearsals. )

[escher]

String players will tend to play in just intonation when on their own or when not playing with instruments that impose a specific tuning (like, e.g. a piano).

But what does it happens when they play in different keys?

[escher]

I am curious to not only understand this theoretically, but more to actually hear what the difference sounds like.

http://www.youtube.com/watch?v=BhZpvGSPx6w

[Superhorn]

   Why can't you teach fish to sing ?  That's because it's awfully hard to tuna fish .









                                                                                                                                                 

[Szykneij]

Your source?

String players.

[petrarch]

But, in "just intonation," the intervals are tuned to an integral ratio: the relationship between the notes' frequencies can be expressed in integers, that is, whole (non-fractional, non-decimal) numbers.  A major third, for example, has a theoretical intervalic ratio of 5/4.  There is no audible beat if the two notes are tuned to frequencies with such a ratio, any more than if they were tuned to the same frequency.

Beats work in absolute terms, not relative, so to hear beats with frequencies not very close together you have to transpose down in order to bring the beat frequency to less than about 16 Hz. Above that, the beat meshes with the sound and becomes part of the 'timbre', just like doing fast amplitude modulation.

[petrarch]

But what does it happens when they play in different keys?

They adjust the individual intervals as they play. Reminds me of the anecdotal episode of the string ensemble playing Ligeti's Ramifications: The piece divides the ensemble into two groups, one tuned a quarter-tone higher than the other, and there was a natural tendency to try to tune to each other as they played, therefore eliminating the quarter-tone difference.

[Szykneij]

String players will tend to play in just intonation when on their own or when not playing with instruments that impose a specific tuning (like, e.g. a piano).

Yes, that's a pretty basic concept. This morning I was emphasising to my high school orchestra string players (an average group of public school music kids) that, in the introduction to Pirates of the Caribbean, they needed to play their half-position D-string first fingers lower during the repeated D-natural/E-flat figure (with the note as a flatted second in D minor) than they would if the note was a D# leading tone in the key of E.

[jochanaan]

Beats work in absolute terms, not relative, so to hear beats with frequencies not very close together you have to transpose down in order to bring the beat frequency to less than about 16 Hz. Above that, the beat meshes with the sound and becomes part of the 'timbre', just like doing fast amplitude modulation.

Or you have to tune the intervals so that the "beats" are less frequent.  I am also a piano tuner, and I work with these "beats," which are simply discrepanices in the pitches relative to each other, whenever I tune to a tempered scale.  I don't use an electronic tuner; I take the overall pitch from a single tuning fork at A440, then set the tempered scale from that pitch.  The beats I hear "vibrate" at from about one per second to about ten, depending on which intervals I'm tuning.  And yes, I do count them out, though after more than thirty years as a tuner I have a "feel" for the beat speeds and often don't have to actually count them.

I should add, though, that if the notes of a piano were sine waves, I could not tune by ear.  Each audible piano note is actually a complex of pitches organized in the harmonic series.  If the notes were tuned justly, a certain harmonic in one note would be identical in pitch to a harmonic in the other note.  But in a tempered scale, those particular harmonics differ slightly in pitch, causing the audible "beat" I use to temper the scale.

[Cato]

I want to understand this. ..

Thing is, major thirds have never sounded sharp to me. And if just intonation sounds so much better, how come more music isn't written for it? I am curious to not only understand this theoretically, but more to actually hear what the difference sounds like.

Check out the Piano Concerto by Lou Harrison.  It uses a type of just intonation.

[eyeresist]

But, in "just intonation," the intervals are tuned to an integral ratio: the relationship between the notes' frequencies can be expressed in integers, that is, whole (non-fractional, non-decimal) numbers.  A major third, for example, has a theoretical intervalic ratio of 5/4.  There is no audible beat if the two notes are tuned to frequencies with such a ratio, any more than if they were tuned to the same frequency.

Not sure about this. I think of the issue in terms of graphical representation of the waveforms, without regard for ratio. Where the wave peaks synchronise, there is a beat. Isn't that right? Why would that not be audible?