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Are there any prevailing theories for how Paleolithic man discovered and shared knowledge about harmonic intervals?
EDIT: I'm not referring to the mathematical characterization of the overtone series in music, which is due to Pythagoras at ~ 500 BC. I'm assuming the Paleolithic man (before 10,000 BC) must have realized that two "instruments" (strings, bone flutes, or whatever) resonate when they are tuned at harmonic intervals.
How Paleolithic man discovered and shared knowledge about harmonic intervals?
Their musical knowledge was discovered through observation and ingenuity, and communicated through their musical instruments and musical traditions, no differently than people and cultures all over the world, since time immemorial - people who knew nothing about the theoretical aspects of music to which you are referring. There is no reason to believe that Paleolithics were any different.
Some interesting background:
National Geographic News June 24, 2009 - Bone Flute Is Oldest Instrument, Study Says
A vulture-bone flute discovered in a European cave is likely the world's oldest recognizable musical instrument and pushes back humanity's musical roots, a new study says.
Found with fragments of mammoth-ivory flutes, the 40,000-year-old artifact also adds to evidence that music may have given the first European modern humans a strategic advantage over Neanderthals, researchers say.
The bone-flute pieces were found in 2008 at Hohle Fels, a Stone Age cave in southern Germany, according to the study, led by archaeologist Nicholas Conard of the University of Tübingen in Germany. With five finger holes and a V-shaped mouthpiece, the almost complete bird-bone flute—made from the naturally hollow wing bone of a griffon vulture—is just 0.3 inch (8 millimeters) wide and was originally about 13 inches (34 centimeters) long.
Music may have been one of the cultural accomplishments that gave the first European modern-human (Homo sapiens) settlers an advantage over their now extinct Neanderthal-human (Homo neanderthalis) cousins, according to the team.
The ancient flutes are evidence for an early musical tradition that likely helped modern humans communicate and form tighter social bonds, the researchers argue.
Based on that report, Paleolithic man was indeed musical. However, no information leading to an answer to your question seems to be found there. In fact, there is no hint at all that they had the knowledge that you assert they did.
Are there any prevailing theories for how Paleolithic man discovered and shared knowledge about harmonic intervals?... I'm assuming the Paleolithic man (before 10,000 BC) must have realized that two "instruments" (strings, bone flutes, or whatever) resonate when they are tuned at harmonic intervals.
First, prove that they had that knowledge in a theoretical form that needed to be communicated somehow other than through the musical instruments themselves and the knowledge of how to play them. I believe your assumption is unwarranted without clear empirical evidence proving it to be so: One does not need such knowledge to make or play musical instruments such as those described above, and arguably the same is true of any acoustic instrument. If I am wrong, I am open to correction, but I don't believe it's necessary (or ever was necessary) to know about the overtone series to build a guitar or a flute or a violin, although the knowledge might be helpful. Far more important are a good ear (or a good tuner...), learning from a master builder, and good craftsmanship.
Without clear evidence proving otherwise, it's perfectly reasonable to assume that Paleolithics learned that blowing through a hollow bone or stone could make an interesting sound, and that the pitch and quality of the sound produced varied, depending on the properties of the object being used, and the count and position of its holes. A bone with a wide cavity sounded deeper than one with a narrow cavity; a stone with three open holes sounded deeper than one with only two; if you covered some of the holes, the pitched changed. A clever and curious child might easily figure out such things in a few hours.
Then they could have created instruments through trial and error by drilling holes at different points in a bone until they were able to achieve sounds that were pleasing or attention grabbing. All it takes is a bit of persistence, a decent ear and some craftsmanship. The same process can also give rise to stringed instruments and certainly drums other percussion instruments.
They likely also have emulated the sounds of birds and animals when building and tuning their instruments, and they could have learned to play music by ear, just as millions of people have done since the dawn of history and long before that - back to Paleolithic times.
Once they had that knowledge, it could be transmitted from person to person orally and mechanically, without any theoretical knowledge at all.
The musical instruments and traditions of indigenous people the world over prove all this:
I believe we can be quite certain that the Australian Aborigines did not have to learn or understand anything about the overtone series as we know it in order to build and perform on the didgeridoo.
Supposedly this dates back to Pythagoras using a monochord. However, there are some early cuneiform writings that may be musical notation.
