Posted: 30 Apr 2005 11:08 pm
Let me go out on a limb here and re-argue Bobbe Seymour's original point about the preferability of not having everything absolutely, perfectly in tune, Paul Franklin has given the perfect opening in his last paragraph. Clearly, when two tones are identical in pitch, the combined effort (phase of the sound waves not being considered) sounds in unison and roughly double in amplitude. Now this isn't perfect, because there may be harmonics, phases aren't precisely coherent, and in fact amplitude and phase relationships may be even fluctuating. But pitch-wise, even if we're talking about a long sustained tone, the overall effect is that of one unison voice.
On the other hand, suppose things are slightly out of tune. Label these distinct frequencies f1 and f2, which are now mixed together. Then there is a well-known beat frequency generated at the frequency of the difference between the two frequencies divided by two, which can be viewed as modulating (multiplying) a signal which is the at the average of the two frequencies. The difference sine-wave is known as the envelope of the signal. This arises from the trigonometric sum formula (sorry, the math is necessary) </p>
A*cos(w1*t) + A*cos(w2*t) = 2A*cos{(w1-w2)*t/2}cos{(w1+w2)*t/2}, where wi are the angular frequencies, or 2*PI*fi, for i = 1, 2.</p>
When this beat interval is short enough to be clearly audible, the notes clash and sound dissonant. Another way to view this is if the note duration is long enough for the beat to become audible, it's probably a problem. Now let me also say that there is no such thing as two physical notes of exactly the same pitch. If one waits long enough, there will be a beat. </p>
But what if the beat interval is very long relative to the duration of the note? Then the envelope of the signal is roughly constant over the note interval, and that signal has a frequency (f1 + f2)/2, or the average frequency. So perfect pitch is less critical for faster passages, as I'm sure most of us know practically. Think of the two unison pairs on a 12-string guitar. If they're out of tune enough for the beats to be noticed over the note interval and sound dissonant, they're too far out. But if they're close, they sound fine. All this considers unison pitches, but similar arguments can be made for harmonics, which is the basis of JI.</p>
Now if the phases of the two signals are not coherent (they never are in practice) and amplitudes vary (as they will in practice), the beats will be even further 'schmeared' and the average frequency will be biased towards the louder instrument, which makes this even more tolerable. A side benefit is that an ensemble of players will be more fault tolerant to pitch imperfections, and there will be additional freedom to creatively vary pitches slightly, since the additional notes just get averaged in. On the other hand, if everyone else is exact and someone comes in a little bit off, it sticks out. This is the mathematics of what Paul observes above, and what Bobbe mentioned earlier. I believe they have hit that nail on the head. I also argue that this produces a richer sound, since what is actually happening is two different tones around the average pitch, sort of the difference between the 12-string guitar and a 6-string guitar. I don't know all the psychoacoustics behind this, but I think the brain can sense this additional richness. There is a lot of research going on these days on the effect on the brain of low-beat-frequencies and their ability to calm or excite the brain. </p>
As I said in the other tuning post by Bob C., blues musicians absolutely work with pitch creatively, that's an integral part of the style. IMO, that's a big part of the beauty of any slide-oriented guitar, like a simple slide guitar or a steel guitar. The ability to creatively mess around with pitch. It's why slide is so popular among blues guitarists, IMO. Listen to Earl Hooker, Robert Nighthawk, and others for complete mastery of this. If I didn't want to mess around with pitch, I wouldn't play a variable-pitch instrument. In fact, I'd have stuck with classical piano, ironically basically ET, which I started on at the age of 7. What drove me to switch to guitar was Mike Bloomfield and the microtonal blues world he opened up to a lot of people my age, and sent me scurrying in the late 60s to find and listen to old blues records and live performances by the masters.</p>
Everything I've said so far has strictly to do with the actual tones produced. Another question is: "How does the ear perceive two very closely-pitched notes?". That's a psychoacoustic problem, as alluded to above. The ear is nonlinear, which means that it does not faithfully reproduce pitches. Typically, pitches are added, and additional filtering takes place. The ear cannot respond instantly to acoustic stimuli. This complicates things immensely. In addition, it guarantees that no two people will hear even the exact same input the same.</p>
Ultimately, I generally prefer older music, roughly recorded in, relative to today's standards, crude studios, as compared to most of what I hear on the radio today. Think Chess studios, Sun studio, RCA Studio B, Bradley's Barn, etc. These folks worked with what they had to produce an artistic result. They had no electronic tuners or digital recording studios with unlimited tracking and editing capability. Some of the instruments, especially steels, were crude by today's standards, but they made it work. I like the almost human-vocal-quality of the sounds. This is no denigration to the fine musicians who make music today. I think technical prowess and artistry among studio musicians is better than ever. As Paul points out, the tuning bar has been raised considerably by precise production values. That said, that's one of my problems with contemporary production values. I'd rather see players have more room to work with pitch, I think the overall result would be more pleasing (to me). I won't get on my soapbox about this, it's just my opinion.</p><font size="1" color="#8e236b"><p align="center">[This message was edited by Dave Mudgett on 01 May 2005 at 12:13 AM.]</p></FONT>
On the other hand, suppose things are slightly out of tune. Label these distinct frequencies f1 and f2, which are now mixed together. Then there is a well-known beat frequency generated at the frequency of the difference between the two frequencies divided by two, which can be viewed as modulating (multiplying) a signal which is the at the average of the two frequencies. The difference sine-wave is known as the envelope of the signal. This arises from the trigonometric sum formula (sorry, the math is necessary) </p>
A*cos(w1*t) + A*cos(w2*t) = 2A*cos{(w1-w2)*t/2}cos{(w1+w2)*t/2}, where wi are the angular frequencies, or 2*PI*fi, for i = 1, 2.</p>
When this beat interval is short enough to be clearly audible, the notes clash and sound dissonant. Another way to view this is if the note duration is long enough for the beat to become audible, it's probably a problem. Now let me also say that there is no such thing as two physical notes of exactly the same pitch. If one waits long enough, there will be a beat. </p>
But what if the beat interval is very long relative to the duration of the note? Then the envelope of the signal is roughly constant over the note interval, and that signal has a frequency (f1 + f2)/2, or the average frequency. So perfect pitch is less critical for faster passages, as I'm sure most of us know practically. Think of the two unison pairs on a 12-string guitar. If they're out of tune enough for the beats to be noticed over the note interval and sound dissonant, they're too far out. But if they're close, they sound fine. All this considers unison pitches, but similar arguments can be made for harmonics, which is the basis of JI.</p>
Now if the phases of the two signals are not coherent (they never are in practice) and amplitudes vary (as they will in practice), the beats will be even further 'schmeared' and the average frequency will be biased towards the louder instrument, which makes this even more tolerable. A side benefit is that an ensemble of players will be more fault tolerant to pitch imperfections, and there will be additional freedom to creatively vary pitches slightly, since the additional notes just get averaged in. On the other hand, if everyone else is exact and someone comes in a little bit off, it sticks out. This is the mathematics of what Paul observes above, and what Bobbe mentioned earlier. I believe they have hit that nail on the head. I also argue that this produces a richer sound, since what is actually happening is two different tones around the average pitch, sort of the difference between the 12-string guitar and a 6-string guitar. I don't know all the psychoacoustics behind this, but I think the brain can sense this additional richness. There is a lot of research going on these days on the effect on the brain of low-beat-frequencies and their ability to calm or excite the brain. </p>
As I said in the other tuning post by Bob C., blues musicians absolutely work with pitch creatively, that's an integral part of the style. IMO, that's a big part of the beauty of any slide-oriented guitar, like a simple slide guitar or a steel guitar. The ability to creatively mess around with pitch. It's why slide is so popular among blues guitarists, IMO. Listen to Earl Hooker, Robert Nighthawk, and others for complete mastery of this. If I didn't want to mess around with pitch, I wouldn't play a variable-pitch instrument. In fact, I'd have stuck with classical piano, ironically basically ET, which I started on at the age of 7. What drove me to switch to guitar was Mike Bloomfield and the microtonal blues world he opened up to a lot of people my age, and sent me scurrying in the late 60s to find and listen to old blues records and live performances by the masters.</p>
Everything I've said so far has strictly to do with the actual tones produced. Another question is: "How does the ear perceive two very closely-pitched notes?". That's a psychoacoustic problem, as alluded to above. The ear is nonlinear, which means that it does not faithfully reproduce pitches. Typically, pitches are added, and additional filtering takes place. The ear cannot respond instantly to acoustic stimuli. This complicates things immensely. In addition, it guarantees that no two people will hear even the exact same input the same.</p>
Ultimately, I generally prefer older music, roughly recorded in, relative to today's standards, crude studios, as compared to most of what I hear on the radio today. Think Chess studios, Sun studio, RCA Studio B, Bradley's Barn, etc. These folks worked with what they had to produce an artistic result. They had no electronic tuners or digital recording studios with unlimited tracking and editing capability. Some of the instruments, especially steels, were crude by today's standards, but they made it work. I like the almost human-vocal-quality of the sounds. This is no denigration to the fine musicians who make music today. I think technical prowess and artistry among studio musicians is better than ever. As Paul points out, the tuning bar has been raised considerably by precise production values. That said, that's one of my problems with contemporary production values. I'd rather see players have more room to work with pitch, I think the overall result would be more pleasing (to me). I won't get on my soapbox about this, it's just my opinion.</p><font size="1" color="#8e236b"><p align="center">[This message was edited by Dave Mudgett on 01 May 2005 at 12:13 AM.]</p></FONT>