I worry from this brief description that you may be ignoring cultural aspects to our perception of consonance and dissonance.

Also, harmony itself is a distinctly western concept whose musical role expanded dramatically with the advent of polyphony. Many (most!) musical cultures around the world don't give harmony much, if any role, and place higher importance on melodic structure.

Also, music harmony and the other kinds of harmony you mention seem to me to be related only by the language used for this particular metaphor. There seems to me to be no likelihood of there being any interesting relationship between musical consonsance and "harmony with technology and harmony with nature".

I agree that the central scientific question is whether harmony is a metaphor or mechanism, e.g., for psychological constructs like Cognitive Dissonance. My guess is that harmony is so pervasive that it ceases to have original meaning -- like in the manner that every atom is a harmonic oscillator or how brainwaves are based on an "octave" structure (e.g., Beta is double Alpha, Gamma is double Beta, etc). However, this question of metaphor or mechanism should be resolvable with science, eh?

However, I disagree that harmony is primarily a western phenomena. It is a central feature to Confucianism and Daoism. It also plays a major role in Native American philosophy.

I was talking specifically about harmony in a musical context as being very western.

Well, the point still applies. The small integer ratios of harmony apply even to the rhythms of gamelan. And, of course, octaves are more-or-less universal across cultures.

This starts to get silly.

"Small integer ratios" apply to huge numbers of natural phenomena, because "small integers" and "ratios" are properties of the world we live in. Pointing out that "harmony is another example of small-integer ratios" seems fairly devoid of content to me, especially when the primary user of musical harmony has long abandoned those pure integer ratios to allow modulation and other desired compositional techniques.

Octaves are not universal across cultures. Byzantine music has no octave equivalence, for example. There was even a paper cited here on HN recently showing lack of octave awareness in different cultures (the paper may have had some flaws, but was interesting).

Silly indeed.

The fact that integer relations are mathematical relations that bear so heavily on physical phenomena is precisely the point. That's what the ancients got so damned excited about. You may not find meaning in it, but it certainly drove the development of science and philosophy from Plato to Galileo to Kepler to Descartes to Newton, etc.

Now, in recognizing that integer relationships are, in fact, an imperfect description of a pervasive phenomena -- that's why the ongoing investigation is so interesting and challenging. Don't we expect what is pervasive in physics to apply to psychology, culture and economics? I'd argue that the mathematical "imperfections" of modern music point to better models of what universal harmony really is.

Overall, the point is that we haven't yet solved harmony. Not even remotely. And i'd love a reference on why you think Byzantine music doesn't have octaves when the Greek music it grew out of most certainly did.

1) "Do not mistake your models for reality" (Lord Kelvin, sometime in the late 1800s) ... many of things that we describe with "small integers and ratios" are not in fact small integers and the ratios are rooted in our observations and thinking. The natural world is more fractal than integral, but the notion of "oh! that thing is a lot like two times that other thing" was a notion more accessible to natural philosophers and early philosopers.

2) No, I see absolutely no reason to "expect what is pervasive in physics to apply to psychology, culture and economics".

3) I don't have a citation for you on the Byzantine stuff, but have been discussing it a lot recently with a musician who grew up on it and continues to perform it, and he was explaining to me how they have no notion of octave equivalency and that because of their tuning and scale systems, when you go up or down the number of steps that "should" correspond to an octave, you end up somewhere other than 2*freq or freq/2. Remember, this is in part why equal tempered tuning was developed: if you stick to "pure" just intonation (precise integer ratios), you can't construct intervals that fit nicely into the octave (e.g. pick a note, go up some number of intervals. Pick the ending note, go down the same intervals, you don't end up back where you started). This stuff is all covered in basic music theory. My understanding from this Byzantine musician is that their musical tradition basically just said "we don't care", and went with a tuning/scale system where you don't just (as in our western system) go up or down N (12 in our case) and end up an octave from where you started.

* On Byzantine music (this discusses a paper discussing the octaves) https://spinstrangenesscharm.wordpress.com/2017/04/29/making...

* Your friend seems to be discussing the Pythagorean pure interval tuning system where, indeed, going up is different from going down --

* In my own empirical work, I've found that pure intervals are not preferred compared to slightly dissonant intervals (in 3 tone sawtooth chords). You can try it yourself -- perfect consonance sounds much worse! Understanding this conflict with mathematical intervals is part of trying to "solve" harmony

* I agree we should remain disinterested in our models, such that we are driven to improve them and not espouse them as reality.

* Yet, I think it is a mistake if we aren't inspired by simple models -- or at least take their hypotheses seriously enough to test empirically

* For instance, is cognitive dissonance involving actual dissonance of some kind? The brain is incredibly rhythmic and even has octaves in the coupling between brainwave bands. It would seem to be natural to test these theories that Plato laid out thousands of years ago -- for instance, that musical rhythm entrains neural rhythms through resonance effects. Maybe that sounds silly, but I'd argue that we are foolish to avoid gathering empirical evidence for these ideas!

* Similarly, harmony had a major effect on astronomy. It actually still does, at the level of the cosmic microwave background radiation, where the presence of perfect harmonic peaks in the signal were the conclusive proof that the universe is "flat"

* Using the concept of sympathetic resonance has been extremely generative in psychology (see Adam Smith's first book) as has harmony in economics. Expecting what is pervasive in physics to apply to these domains doesn't mean it should apply in exactly the same manner -- just that one should expect analogous effects -- at least to the extent that one is looking for explanatory theories to test! We are so far from understanding these domains that, if we don't at least consider these natural and ancient theories (because they seem silly), we are doing ourselves a disservice. Let us be inspired by the past and test, test, test!

From a story a couple of weeks ago, octaves, while common, seem to be learned: https://news.ycombinator.com/item?id=23624596

That may be, but they are still learned according to https://www.quantamagazine.org/perceptions-of-musical-octave...

> Harmony was also the objective function for the first deep learning neural network, Paul Smolensky's Harmonium.

Here is the 1986 work in Parallel Distributed Processing, along with G. Hinton and Rummelhart. Don Norman wrote a chapter, too!

http://93.174.95.29/main/3C9F2A7AEF5BE7461DECE4AE766F94B1

I got a little lost when you tried to compare musical harmony to societal harmony.

With regards to musical harmony, is it possible that it's more or less random? I know multiple cultures have different definitions of musical harmony. I suspect the evolution of hearing also contains random elements. Similar to language, it's not so much about an inherent universality, just a universality we can all learn and agree on.

Thoughts?

The human ear has an intimate relationship with the octave 2:1 and its ratios, so its very hard to believe the convergence of appreciation globally to be random. More dramatically, visualizing the harmonious ratios on objects such as Chladni plates (a field called Cymatics) reveals that there is something deeper to consonance and harmony than meets the number line.

There is some evidence that ratio/octave recognition and preferences for consonant sound are culturally contingent

http://mcdermottlab.mit.edu/papers/McDermott_etal_2016_conso...

http://mcdermottlab.mit.edu/papers/Jacoby_etal_2019_Bolivia_...

Preferences vary individually as well: modern music may feature or favor dissonance (e.g., some forms of metal, electronic)

"visualizing the harmonious ratios on objects such as Chladni plates (a field called Cymatics) reveals that there is something deeper to consonance and harmony than meets the number line"

Ok I'll bite. What does it reveal? There's nothing inherently meaningful here. We know that 'dissonant' sounds (those that create interference patterns) create wavelets that are smaller and with less contrast than the more 'coherent' patterns from ratios that are closer to whole numbers.

But in what way is this meaningful or useful?

It means we find consonance pleasurable and see a distinct "signal" in it. At least, that's the information-theoretical way of looking at it.

When dealing with cosmology one often seeks to make a big deal out of a simple concept like a duality, a cycle, or a ratio. These are concepts recurring through the world, and looking for them in more places sometimes reveals knowledge.

Check out your voice on a spectrogram app. See all the harmonics? Our voices evolved this way, probably due to sexual selection.

It's not so different between cultures, in fact.

This comparison is delved into in the sections in Ernest McClain's The Pythagorean Plato [1] which analyze Plato's Republic. McClain summarizes his somewhat remarkable claims thus:

> From a musician's perspective, Plato's Republic embodies a treatise on equal temperament. Temperament is a fundamental musical problem arising from the incommensurability of musical thirds, fifths, and octaves. The marriage allegory dramatizes the discrepancy between musical fifths and thirds as a genetic problem between children fathered by 3 and those fathered by 5. The tyrant's allegory dramatizes the discrepancy between fifths and octaves as that between powers of 3 and powers of 2. The myth of Er closes the Republic with the description of how the celestial harmony sung by the Sirens is actually tempered by the Fates, Lachesis, Clotho and Atropos, who must interfere with planetary orbits defined by integers in order to keep them perfectly coordinated. In Plato's ideal city, which the planets model, justice does not mean giving each man (men being symbolized by integers) “exactly what he is owed,” but rather moderating such demands in the interests of “what is best for the city” (412e). By the 16th century A.D., the new triadic style and the concomitant development of fretted and keyboard instruments transformed Plato's theoretical problems into pressing practical ones for musicians and instrument makers. With the adoption of equal temperament about the time of Bach we made into fact what for Plato had been merely theory. Musically the Republic was exactly two thousand years ahead of time.

It would be easier to dismiss McClain's thesis as Bible Code crackpottery if he didn't have such interesting things to say about the seemingly arbitrary numbers[2] appearing in Plato's writing.

For example, on 5040

> “Our songs have turned into laws!” Plato exclaims in one of his relentless puns in the dialogue Laws, this time on nomoi meaning both laws and traditional melodies for the recitation of the epics (799d). [...] The absolute population limit of 5,040 “landholders” will be analyzed as the tonal “index” of a tuning system “fathered” by four primes, 2, 3, 5, and 7; the number 5040 = 2^4 x 3^2 × 5 × 7 defines a tuning system like that of Plato's friend Archytas, who is the earliest theorist credited with using 7 as a tone generator. Since 5,040 is also factorial seven (7! = 1 × 2 × 3 × 4 × 5 × 6 × 7)—i.e., just 7 times larger than factorial six (6! = 720) which defines the calendar octave, or Poseidon and his ten sons (cf. fig. 6) — we have a clue as to the identity of Plato's 37 guardians, 18 from the “parent city” and 19 “new arrivals,” for new arrivals among Plato's products are those generated by 7. [...]

Or, again, when McClain points out the relation between the number 729 - in Plato: the "distance" between the happiness of a king and that of a tyrant - and the simplest expression of the ratio of the Pythagorean comma: 531441::524288 (531441 is the square of 729)

Looked at this way, Plato's conclusion in the republic, that a society is just when each member of it deviates from their own interests just sufficiently to optimize for the interests of the society as a whole, is intended to map onto a truism of music theory - that an instrument, say a modern piano, is optimally in tune when each of its individual notes deviates from its own true value by the small amount required to keep the instrument as a whole sounding good.

This idea is the basis of various tuning systems including the predominant modern system known as "equal temperament."

1: https://ernestmcclain.files.wordpress.com/2017/05/plato_opti...

2: see, e.g. https://mathworld.wolfram.com/PlatosNumbers.html

> why consonant chords that are mathematically slightly dissonant actually sound much better than chords with perfect mathematical consonance.

Probably because we're used to hearing music in equal temperament, so we associate it with "correct" harmony, whereas something like just intonation sounds a little weird and off (but I ultimately find myself preferring it? Wtf?)

Regards Plato, I'm curious if you've looked at Ernest McClain's book The Pythagorean Plato? (See my other comment[1] here for some details if not.)

1: https://news.ycombinator.com/item?id=23746057

You may be interested in de Waal, Peacemaking Among Primates.

An under-appreciated aspect to our educational system is that school desks are shared between two children, so people grow up learning how to interact relatively harmoniously with at least one other small child.

Absent a vaccine, Covid may alter that to the one-child-per-desk model, so I may discover in another 20-30 years how important shared desks were or weren't.