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.
Quote from: westknife on June 06, 2011, 05:51:44 PMThing 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
Quote from: westknife on June 06, 2011, 05:51:44 PM
no "beats" when 2 notes are played together.
Is this really true? Surely two different notes will "beat" regardless of their harmonic relation.
Quote from: eyeresist on June 06, 2011, 06:08:55 PM
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.
Quote from: westknife on June 06, 2011, 05:51:44 PM
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).
Quote from: petrarch on June 07, 2011, 02:20:56 AM
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?
Quote from: eyeresist on June 07, 2011, 02:41:30 AM
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.
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.
Quote from: eyeresist on June 07, 2011, 02:41:30 AM
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.
Quote from: petrarch on June 07, 2011, 02:17:31 AM
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.)
Quote from: petrarch on June 07, 2011, 02:20:56 AM
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. 8))
Quote from: petrarch on June 07, 2011, 02:20:56 AM
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?
Quote from: westknife on June 06, 2011, 05:51:44 PM
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 (http://www.youtube.com/watch?v=BhZpvGSPx6w)
Why can't you teach fish to sing ? That's because it's awfully hard to tuna fish .
;D ;D ;D ;D
Quote from: jochanaan on June 07, 2011, 08:44:59 AM
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.
Quote from: escher on June 07, 2011, 10:09:42 AM
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.
Quote from: petrarch on June 07, 2011, 02:20:56 AM
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.
Quote from: petrarch on June 07, 2011, 04:37:29 PM
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. 8)
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.
Quote from: westknife on June 06, 2011, 05:51:44 PM
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.
Quote from: jochanaan on June 07, 2011, 08:44:59 AM
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?
Quote from: eyeresist on June 08, 2011, 06:23:10 PM
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?
The two representations are mathematically identical. If the two tones have an integer ratio of frequency n:m one will complete n cycles while the other completes m and they will arrive at the same point after a fixed time interval, (time required for n cycles of the first tone) = (time required for m cycles of the second tone). They will therefore be stationary on the oscilloscope. The most obvious audible beat is when you have two tones that almost match, their ratio is almost 1:1 and you can hear them going in and out of synchronization.
Two tones will have a subtle beat if they are close to one of the simple ratios because their overtones are not meshing. For instance, you play a C and a G which is not quite in a 3:2 ratio with the C. The third harmonic of C is G, and the second harmonic of G is G, and those two harmonics will beat with each other if the original C and G are not in the proper 3:2 ratio.
However, these "just" intervals are inconsistent. Suppose you start with C. Then you tune G so that it is in a perfect 3:2 ratio with C. Then you tune D as a 3:2 ratio to G, then you tune A as a 3:2 ratio with D, continuing with E, to B, to F#, to C#, to G#, to D#, to A#, to F, to C. That C you arrive at after going around the circle of fifths is not in tune with your original C. Compounding those ratios of 3:2 you get a final ratio of (3^12)/(2^12) = 531441/4096 = 129.75. The problem is that C seven octaves above your original C should be in ratio 2^7, or 128. Not 129.75. That C is way out of tune with the C you started with, and came be be called "the wolf" because it seemed to howl. The bottom line is you have to compromise. If you pick your intervals to make one chord sound just, chords based on other tones will sound off.
Quote from: Il Barone Scarpia on June 08, 2011, 06:56:41 PM
If the two tones have an integer ratio of frequency n:m one will complete n cycles while the other completes m and they will arrive at the same point after a fixed time interval
This should be true of any two tones, regardless of the ratio relationship.
Quote from: eyeresist on June 08, 2011, 07:36:17 PM
This should be true of any two tones, regardless of the ratio relationship.
Not true. Suppose one tone has frequency 100 cycles per second, it takes 1/100th of a second to complete, or 10 milliseconds. Another tone has frequency 200 cycles per second, it takes 1/200th of a second to complete or 5 milliseconds. After 10 milliseconds the first tone has completed one complete cycle, the second has completed two complete cycles, they are synchronized again.
Now suppose the second tone has frequency 199 cycles per second instead of 200. It takes 5.025 milliseconds to complete a cycle. After 10 milliseconds, tone one has completed a complete cycle, but tone two is just shy of completing its second cycle. They are out of synchronization. After every 10 millisecond cycle of the first tone, the second tone falls a little more behind. However after one full second, tone one has completed 100 cycles, tone two has completed 199 cycles. It has fallen a full cycle behind and the two tones are finally synchronized again. A frequency error of 1 cycle per second means it takes one second for them to come back into synchronization.
I guess the point is that notes out of ratios produce an audible beat. But it's still the same phenomenon occuring in either case.
Quote from: eyeresist on June 08, 2011, 07:55:01 PM
I guess the point is that notes out of ratios produce an audible beat. But it's still the same phenomenon occuring in either case.
You will not hear an audible beat unless the tones are very close. For instance, 100 Hz and 101 Hz will produce a very strong beat as the two tones go in and out of synchronization.
The combination of 100 Hz and 199 Hz will not directly produce a direct audible beat. However, there will be an
indirect beat. Any musical instrument will produce overtones at multiples of the fundamental pitch. So a 100 Hz note contains overtones at 200 Hz, 300 Hz, 400 Hz, 500 Hz, etc. The 199 Hz pitch will contains overtones at 398 Hz, 597 Hz, 796 Hz, etc. So if you play 100 Hz and 199 Hz together, you will hear a beat between the second overtone of the first note (200 Hz) and the fundamental of the second note (199 Hz). You may also hear beats between different combinations of overtones. These beats make use of by people who tune instruments (as was mention above in this thread).
Quote from: Il Barone Scarpia on June 08, 2011, 09:09:17 PM
You will not hear an audible beat unless the tones are very close. For instance, 100 Hz and 101 Hz will produce a very strong beat as the two tones go in and out of synchronization.
The combination of 100 Hz and 199 Hz will not directly produce a direct audible beat. However, there will be an indirect beat. Any musical instrument will produce overtones at multiples of the fundamental pitch. So a 100 Hz note contains overtones at 200 Hz, 300 Hz, 400 Hz, 500 Hz, etc. The 199 Hz pitch will contains overtones at 398 Hz, 597 Hz, 796 Hz, etc. So if you play 100 Hz and 199 Hz together, you will hear a beat between the second overtone of the first note (200 Hz) and the fundamental of the second note (199 Hz). You may also hear beats between different combinations of overtones. These beats make use of by people who tune instruments (as was mention above in this thread).
Nicely explained! I would only add that for tuning purposes, there is little effective difference between a "direct" beat and an "indirect" beat; they're both equally useful. The difference is that in the equal-tempered scale you have to put up with the indirect ones, except for the ones resulting from an octave.
When the "beat" reaches 16-20 Hz, it begins to sound like another note. How audible that other note is relates to which overtones are beating. For a "unison," if you have that many beats per second, it's very unpleasant! :o But for, say. a major 3rd, it's much less obvious since what's beating is the 5th harmonic overtone of the lower note and the 4th harmonic overtone of the upper note--overtones that usually aren't even heard consciously. In many cases, such fast beating sounds like an audible undertone. These are usually unwanted in the Old Masters' music, but I'm sure some recent composers have taken advantage of just such effects... 8)
Quote from: jochanaan on June 09, 2011, 09:33:41 AM
In many cases, such fast beating sounds like an audible undertone. These are usually unwanted in the Old Masters' music, but I'm sure some recent composers have taken advantage of just such effects... 8)
I think that is the effect that gives elementary-school and middle-school bands their characteristic sound. :)
Quote from: escher on June 07, 2011, 10:10:22 AM
http://www.youtube.com/watch?v=BhZpvGSPx6w (http://www.youtube.com/watch?v=BhZpvGSPx6w)
That link is really cool. As a kid, I usually could not stand the sound of a newly tuned piano. Later on when working with string players, I found it was quite a task for them to adjust their intonation to the piano. Maybe over time we make the adjustments with our ear and brain to hear more perfect harmony, just as we can reconstruct a telephone conversation while hearing less than 50% of it.
Equal temperament seems to have been bolluxed up with just temperament at least in the 20th century. Otherwise, why would Schoenberg say that composing in tonality is like writing everything in the key of C, assuming all intervals being equal? Now, though, it is generally agreed that "well tempered" is not equal tempered.
This article was quite a revelation. http://www.eunomios.org/contrib/francis2/francis2.pdf
Apparently decorative curls or glyphs from the title page by Bach of the WTC had a more esoteric meaning, that is, the beats of tuning various notes on the keyboard. The significance is that keys like C and closely related ones were nearly perfectly tuned but others like Eb minor or B minor had an unsettled quality built into the tonality.
It's nice that electric keyboards can be tuned electronically so we can actually hear what Bach had in mind.
ZB