r/math • u/RingularCirc • 4d ago
Precise etymology of "harmonic" (including h. mean and h. series)?
It seems to be very much a common knowledge now that harmonics of an (almost) periodic audio signal do evidently stem from Ancient Greek investigations into musical harmony, which was intertwined and likely gave us harmonic means and harmonic sequence/series, means likely being derived from the sequence like it can be done with cases of arithmetic and geometric sequences and means.
The details are hovewer not clean at all. Presumably, not even in the Hellenistic period had it been known about harmonics in sound. One question is then, if a noun parallel to harmonic (which should stem from Greek, obviously) was in use, what meanings did it have in mathematics and music theory of the time. I see at least two options as very probable:
- (a) it hadn't been used in a way parallel to how today's harmonic is used;
- (b) it denoted sounds or musical pitches that you can derive from the "original" sound of an open string by subdividing it into k parts.
Note how right now, a very much related technique of playing stringed instruments is indeed called "harmonics" in English (but "flageolets" in some other languages). The second question (that's less related to this sub) is precise etymology of this name: I would expect flageolets should be a technique older than both the discovery of Fourier series and connecting string and air channel lengths to their fundamental frequencies.
Also a question 1.5 is relating the answer to the first question to harmonic means and harmonic series. For example if (a) held, the answer is very much warranted and shouldn't be deduced without historical sources.
There are probably a bunch of additional minor questions that I can't refine out of the mess in my head right now, that would be obvious when one tries to set things straight on this entire topic and encounters important details that answer them. For example as far I know, frequencies of sound weren't known in antiquity (but that's really a question to answer definitively as well!), so we can't attribute "reciprocalness" of harmonic mean and so on to the inverse relation between length and frequency. Instead it should probably be entirely due to there being a sort of a natural order to harmonics (in the sense of (b)) coming one after another at 1/1, 1/2, 1/3, ... of the string's length —but is that the factual reason that had been used?
You see even a mathematicians' brains would otherwise be glad to fill in the "obvious" blanks and I feel that a great lot of people think they sorta know the story about this. I'd of course expect it of non-historical music theorists even more because in their case it's actually quite important to talk about time to time —whereas in mathematics, harmonic mean and harmonic series (and wasn't there something else?) is just a neat and somewhat old-fashioned (as is, again, common) term flavoring that evokes associations that no one has to unpack to work with the actual math. But that may still give birth to mythology when one gets interested just a little. So it's a good point to look around and try remembering while the ancient sources may be still laying around! And if there's already a piece of work on this very topic already written, the easier it is to find it again (I couldn't, but my work with references is abysmal).
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u/Anaxamander57 4d ago
Why would the ancient Greeks not know about harmonics? They had stringed instruments.
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u/RingularCirc 4d ago
They didn't have Fourier analysis and an idea of decomposing sounds into separate frequencies, to my knowledge.
Right now I'm reading through a translation of Ptolemy's Harmonica with comments from Porphyry (3th century CE) and I've yet to even see anything about even fundamental frequencies, though there's discussion of speeds relating to pitches and of repeatedly hitting the air. Fascinating stuff, but you have to be very careful in not ascribing contemporary ideas when reading ancient texts.
I've yet to see a direct confirmation a ratio of such 'speeds', in that way of writing it, was considered the same as an inverse ratio of lengths on a monochord, for example. It might be implicit. Even then, it's miles to get understanding that, for example, we can envision a particle of air travelling in a superposition of what we call now harmonic motions.
Mind, there was even a common reverse understanding how sight works (travelling into the things we're looking at), with comparisons how hearing is unlike that and instead doesn't go anywhere and waits until sound gets to the ear, despite this correctly explained how the lightning is seen before the distant thunder, coupled with correct observation that actually lightning and thunder are generated at the same time (I guess they had enough close-but-survivable strikes, only fair.)
Do also remember that "harmonics" (flageolets) picked on a string by hand are actually sounds that still consist of multiples of a given harmonic, not just a harmonic in isolation. It's impossible to excite only a single harmonic without doing something more complicated (like playing a sine wave out of a speaker near the instrument). Though practically very high harmonics will be excited with lower power and also decay very fast, so if we can play very high-pitched flageolets, in practice they may as well mostly be a pure harmonic of that string. For playing octaves and fifths though, absolutely won't be a pure sine.
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u/Anaxamander57 4d ago
Euclid didn't have the ability to draw perfect circles but I don't think there's any question of how he managed to write so much about geometry. The lack of perfect knowlege of the physics of vibrations is likewise irrelevant to the mathemtical model of harmony developed by the Greeks. Greek philosophy is in fact particularly famous for putting forward the idea that the world of experience is a lesser version of a true perfect world that is accessible by reason.
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u/Kered13 4d ago
Ancient Greeks may not have known about sound waves or Fourier analysis, but they know that a string of length L and a string of length 2L played the same note one octave apart. They knew that strings of length 2L and 3L played a pleasant sounding chord which they called a Fifth. They knew that a string of length 2L would resonate with a string of length L, and could be induced to produce the same note as a string of length L. Thus they could deduce that the harmonics of a string of length L were the strings of length L/2, L/3, L/4, etc.
So they knew everything that they needed to know to talk about harmonics, harmonic means, the harmonic series, etc.
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u/SubjectAddress5180 4d ago
The string was divided into integer segments.,1:1, 1;2, 1:3, ... Its the reciprocal of arithmetic divisions.
Major chord, 4:5:6, minor chord: 10:12:15 = 60/6, 60/5, 60/4.
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u/RingularCirc 4d ago
This is surface-level, it doesn't answer any question I posed.
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u/RingularCirc 4d ago edited 4d ago
I'm not sure what the downvoters are trying to express. Maybe the wording of my post wasn't clear enough? Happens; then comment on that explicitly for the greater good.
But I think I was verbose enough to show I set the bar high here: we're to uncover what exactly did people mean and when —to the extent that's possible.
Contrary to that, enumerating particular justly-intoned chords is completely orthogonal to that. There's depth to that topic as well (people has different opinions on chords through times and in different countries, mind you), not exhibited in just putting 4:5:6 and 1/6:1/5:1/4 here. Also I plain presume it's known the string was divided in n equal parts for different positive integer numbers n. I think I even mentioned it in the OP?.. Anyway that doesn't solve anything I posed either.
Go on, downvote further. You know everything already, I presume.
EDIT: Well, I might've been more polite, I can own that. Sorry. I don't have all the time in the world to compose ideal responses to
low-qualitylow-effort comments. It's not that I asked gibberish and reaped what I've sown, is it?9
u/solartech0 4d ago
Look, I think others have already covered this, but my first thought on reading this post was:
Presumably, not even in the Hellenistic period had it been known about harmonics in sound.
Why would you ever believe that? It's challenging to even come up with what you could have meant, if that were intended to be some form of reasonable statement. What we call Harmonics (at least, my understanding of that term) are all the notes that get played alongside a given note, when you play that note. They are all the frequencies that "fit" on the "string"; on certain kinds of instruments, you can intentionally play a higher harmonic if you know what you're doing. If a people had instruments, then they would be able to come up with some form of understanding -- whether intuitive or explicitly reasoned out -- about what was going on.
If you want to ask people for historical records or thoughts, it's probably better to do that a different way.
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u/AMobius1832 3d ago edited 3d ago
Pythagoras "discovered" the consonances of the octave (in music). As a result, he created the "Pythagorean tuning (scale?)" in music. (Today, we use the equal temperament tuning 12th root(2).) The sequence of partial harmonics (multiples of the fundamental wavelength) form an arithmetic progression, the "harmonic series."
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u/Miguzepinu Number Theory 4d ago
What's Harmonic about the Harmonic Series?
There's also harmonic progressions and harmonic numbers.