Neo-Riemannian Theory: An Introduction

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0:00 : Who was Hugo Riemann?
1:38 : What's This Theory Trying to Do?
3:50 : Basic Neo-Riemannian Transformations (P, R, and L)
6:52 : Some Secondary Transformations (N, S, and H)
8:27 : Uniform Triadic Transformations (UTTs)
12:15 : Why Isn't This Taught?
📚 Selected Further Reading:
"Uniform Triadic Transformations" by Julian Hook (Journal of Music Theory, 46:1/2, Spring-Autumn 2022)
"Alfred Schnittke's Triadic Practice" by Christopher Segall (Journal of Music Theory, 61:2, Oct. 2017)
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Classical Nerd is a video series covering music history, theoretical concepts, and techniques, hosted by composer, pianist, and music history aficionado Thomas Little.
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Music:
- Thomas Little: Dance! #2, performed by Rachel Fellows, Michael King, and Bruce Tippette
- Thomas Little: Invasion of the Minor Seconds for Pierrot Ensemble and Percussion
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All images and audio in this video are for educational purposes only and are not intended as copyright infringement. If you have a copyright concern, please contact me using the above information.

[@reubennb2859]

I'd love to see an analytical take on how the harmony of Allan Holdsworth's music functions on a voice-leading level. Pieces such as 'Distance vs Desire', 'Home' and 'Sand' exhibit very sophisticated harmony akin to early 20th C. composers. Holdsworth had a unique way of finding unconventional interval structures to harmonise modally across a scale, and it allowed him to fully express tonalities like harmonic major and various symmetrical scales. Perhaps not appropriate here, but he is totally overlooked in the classical world despite making a genuinely original contribution to use of harmony.

[@neilsaunders9309]

Holdsworth, with his "parsimonious" voice leading, would be almost the perfect subject for Neo-Riemannian analysis.

[@bigdog38au]

A friend of mine is just finishing a PhD on precisely that subject :)

[@reubennb2859]

@@bigdog38au this is so cool, I'd love to read it

[@bigdog38au]

@@LouizSlein I will ask.

[@flashback4588]

@@no-rq7fp thats harsh bro

[@royschwaben9646]

I love that I can jump on KZbin and find super niche stuff like this that would never make it on the intellectual cesspool of TV in a million years. I've toyed around with these concepts for my own writing without knowing the names. Many thanks. I'm super excited to dig into the ultra-nerd stuff.

[@Z-Diode]

True words! In any case, I‘d suggest to every intellectual to dispose of the domestic TV set (if still available) once and for all.

[@seanriedy]

I've been waiting for someone to make a video like this, thanks!

[@jcrouse7461]

I’ve never heard of this theory before, thank you for introducing me to it!

[@brendaboykin3281]

Thanx, Maestro 🌹🌹🌹More lectures like this, please.

[@webstergadbois8606]

Have literally never heard of this theory before and I went to music school for years. Thank you!

[@space_dog_beats9240]

Many thanks, this is a fantastic introduction to an interesting topic. Just wanted to show my appreciation, my essay should prove easier to write now. Great channel!👍

[@gammarays3683]

This channel is lowkey helping me get through twentieth century and music history. Thanks Classical Nerd 👍

[@CharlesAustin]

Why indeed, isn’t this more commonly taught..? Thank you for the outline. Awesomely done as always !!

[@helenamarie4337]

fantastic video as always. Best channel on youtube!

[@tristanpaxton51]

Great video! So happy to discover this

[@TheSutov]

In Slovenian, teachers use term 'parallel' for relative relation and for true parallel relation, they use term 'istoimenski' which can translate as 'having the same name'. Switching parallel major to minor or vice versa it called 'mutation' in Slovenian practise. There you have it...otherwise, it is useful to regard horizontal symmetries in ancient modes on D or g#/Ab (beside trichords and seventh chords) in set theory as follows: Ionian and frygian, locrian and lydian, aeolian and mixolydian, and-you guessed it-dorian onto itself!

[@jazzloopjazz]

I think this parallel and relative conundrum is a common one also for Romance languages. In french for instance, when speaking of relative keys (tonalités relatives) one will refer to their relationship as parallelism, and when speaking of parallel keys (in the English sense: C major/c minor) they are called "homonymes", a term obviously taken from linguistics to mean things that have the same name, but different "meanings". It took me a while to realize that English speakers mean something completely different when they talk of parallel keys... 🤔🤔

[@dionizos0665]

We have the same situation in Poland. The "parallel"key will be C major and a minor but for "relative" relations it will be C major/c minor and we call it "jednoimiennie" which also mean "having the same name".

[@chutton988]

Fascinating topic. I’ve been having questions about Debussy chords and other composers’ harmonies and it’s great that this addresses them. Love the video, thank you!

[@TAP7a]

Commenting for engagement because this is the good stuff My instinct (tentatively reported by the concentration of it in a basic KZbin search for neo-riemannian theory) is that this is strongly correlated with music that is interested in a wider gamut of character to make functional than tonal theory - more in the vein of film music or as you mentioned impressionism. Tonic-dominant is one character but this gives us a framework to investigate how to connect *all* available characters to the set of pairs of triads in a 12-tone system. I can't wait for more work to be done on the larger scale structures, like is there a meaningful analogue to a modulation or other such smooth changes in base, what happens if you limit your choices in a serial composition to only those informed by neo-riemannian theory, how can we use this to respond to or generate musical structure and more global relationships, etc etc etc. Now I wish I lived closer to a university to see if I could sneak in the back of lecture in the topic!

[@ClassicalNerd]

Someone could do a dissertation on John Williams' film scores through this lens, for sure!

[@lerippletoe6893]

Commenting for algorithm, giving support because I love this topic

[@BRNTmodularmusic]

There’s a very popular module around the eurorack system called ‘Ornament & Crime’ which has 2 apps that uses the neo riemannian transformations for generating triads (‘Harrington 1200’ and ‘Automatonnetz’), so this system is more popular that one could expect (if people would understand it better -me being one of them 😅). But yeah, if you know someone with an eurorack you could use a musical instrument that would be based aroung P,L,Rs and N,S,H transformations 🥰🥰🥰. Thanks so much!

[@wellurban]

The reason that I started getting interested in music theory again a few years ago, many decades after I studied it, was another Eurorack module called Tonnetz Sequent, which is entirely dedicated to neo Riemannian transformations. The L and R transformations sound very sweet to Western-trained ears, but introducing the P transformation could be jarring. Yet after a while, I found the triadic drifting became strangely compelling when you combined all three, even though I’d mostly stuck to simple diatonic writing in my own music. Reading up on the Tonnetz appealed to my mathematical brain, and once I found a 12 Tone video on the topic I started watching more and more music theory video channels, including this one, from which I’ve learned so much. I’ve learned why I needed to understand traditional functional harmony more than I did, but it also justified for me why I didn’t like using strongly functional harmony in my own music. I’ve found that frameworks such as Philip Tagg’s and Patricia Taxxon’s make more sense for understanding popular music, and that various modal and quartal/quintal approaches also help break me out of my old diatonic rut. But it all started with a weird module that did neo-Riemannian transformations!

[@megamankicksyourass]

nice, i was watching this for modular synth application!

[@Marre2795]

11:50 that progression is beautiful! I'm hearing a triumphant Bb major chord at the end as well. I might use that as a fancy way of modulating down a whole step (or maybe up a minor third if i use the Bb at the end as a dominant chord)

[@brucebaldy]

I really had to crank up the volume to hear you, but, it was interesting. Keep it up.

[@rayphenicie7344]

The music of Bruckner is a perfect study for neo-Riemannian theory.

[@de31.8]

Thanks for the vids! Would you please make more “Great Composer” videos about Verdi and Puccini? I LOVE those guys 😍

[@baloothedrummer]

Love your vídeo man. Greetings from México. I Also think You Made the right call. Lets just make some more music using this interesting concept.

[@parsa.mostaghim]

great insight, although I'm not a musician and don't understand these topics deeply I really enjoy the niche topics you cover

[@musicalintentions]

That’s right. Keep us theorists employed. 😊 Seriously though, I really appreciate this video. Thank you! 💕🎵💕

[@contrustmusic]

Very helpful thank you! 😊

[@enriquesanchez2001]

My brain has officially exploded after this. :(

[@vaneesorn]

Thank you for the great content!!!!

[@AtomizedSound]

Interesting. I love theory talk and I wish I learned some of this in college but they didn’t teach any of this. I get it’s not prevalent in the musical world as a whole but at least learning about it as a device would have been great.

[@pantheon777]

Love the little rant at the end, lol

[@SevagDerGhougassian]

There is one thing I would like to add: It is true that not all music containing triads are considered as tonal, but also some atonal music containing triads could have central functions. These central functions could be considered as "Home Key" where the composer can create a shape or shapes that lead to re-invented "home keys". In order to understand my compositions (sorry - I don't mean to advertise; just example) check my 24 preludes and fugues in all keys.

[@Markrspooner]

Great video! I found it fascinating and would happy to see more on Mauchline examples that use it

[@SilverlingVirtualStudio]

Good stuff! I’d be really interested to see this kind of video on Schenker’s Theories, but also a video on Schoenberg’s Harmonielehre (which I still think is one of the best theory books written).

[@ClassicalNerd]

I'm skeptical of how well Harmonielehre would transfer to a video; there's too much there to boil down well into a single introductory video. Otherwise, duly noted: lentovivace.com/classicalnerd.html

[@SilverlingVirtualStudio]

@@ClassicalNerd oh for sure. It’s a beast of a book-there’s so much stuff packed in there.

[@gon9684]

There's more to neo-Riemmannian theory than I thought!

[@aviv312]

please upload more videos about this topic and maybe do a deeper dive ? ? ?? ? ? ? ?

[@omniphoriusvcf907]

Great video! As a university trained composer who uses these relationships often in my music, I will add that another reason they might not be widely taught is that combining triads in non-diatonic combinations is not difficult to conceptualize if you've spent years using triads...it's easy for me to predict how it will sound when moving from any one triad to any other, and I can quickly see how smooth the voice leading will be. In other words, for those of us using these juxtapositions creatively in our music, there doesn't seem to be a huge payoff for learning the theory. At least I personally don't get inspired by the theory or the honeycomb. This is in contrast to the way composers I know use set-theory-they find it quite useful. It may be that this is just my personal experience, but that's my "theory." ;-)

[@NickBatinaComposer]

Hot video, def my favorite technique in the werld

[@timbeaton5045]

Somehow i get the impression that Galois would have found this UTT idea, interesting. Well if he had any musical leanings!

[@user-qv3xs1li1s]

11:38 omfg that sounds v v v similar to the progression in dirty boy by the cardiacs

[@jdanielcramer]

Very cool

[@roses2155]

Riemann’s concepts explain the micro-theory but struggle to define over-arching harmonic structures. The non-tonal progressions often carry relationships that do establish cadential characteristics in some form. While the harmonic relationships are not traditional, the standardization of human expectations in music make the various leading tone aspects vital in driving expectations regardless of expected resolutions. I view Riemann’s work as cellular in scope versus the anatomy of a given work.

[@davehenry6997]

Love the videos. Any chance for one on Milton Babbitt?

[@ClassicalNerd]

I'd love to! The exact timing will be influenced by his place in the request pool: lentovivace.com/classicalnerd.html

[@idrisbalavakos]

I think we could condense the curriculum of Music Theory from 4 levels to 3 and reserve the 4th for other analytical systems. Then again, I was a theory whiz and can't understand why anyone struggles learning it.

[@108grog]

[I also hadn't heard of this. Only bachelor level studies.] Could you post links to music that best exemplifies this analysis method ? Whether written intentionally or not. Perhaps , compose a little etude , .... and then show the analysis. ??

[@davidantonioripolllopez5335]

Any book to study deeper?

[@danielharris9403]

Can't believe I never came across this. This reminds me of a concept on related harmony/voicing that I discovered through jazz: V7/bVII7/bII7/III7 = ii°/iv°/bvi°/vii° IV7/bVI7/VII7/II7 = i°/biii°/bv°/vi° I7/bIII7/bV7/VI7 = bvii°/bii°/iii°/v° I∆= iii(7) = V* = vi(7) IV∆ = vi(7) = I* = ii(7) V∆ = vii(7) = II* = iii(7) *may be treated as a dom.7 (tonic in Ionian) or nat.7 (tonic in Lydian). Major 7s traditionally resolved to the 6. This, with octatonic scales, seems like Barry Harris's diminished approach. b9s can resolve chromatically to majors tonics too (parallel major/minors; both following the same system of shifting tonal centres based on cadences/'changes').

[@nolanmccafferty9932]

“Common practice tonality” (western imperial tonality)

[@slowbreathe]

Do you know Aphex Twin? Can you make a video about him? We can say that he is composer and there are photos of Aphex Twin & Stockhausen / Penderecki.

[@trygvelebakkencomposer]

No way!! My theory professor was Brian Hyer!

[@thomasdavis8117]

I'm more of a Bernard Riemann fan myself.

[@tesset8828]

Yeah this dude is like 2nd 3rd tier Riemann. Bernhard Riemann S tier.

[@pablom.5698]

Lmao yes.

[@boazmecham5101]

amazing explanation! You should do a Video about Kapustin

[@ClassicalNerd]

Duly noted: lentovivace.com/classicalnerd.html

[@ericrakestraw664]

Has anyone requested Rzewski yet? He was sort of the American version of Kapustin.

[@boazmecham5101]

good idea!

[@rudigerk]

Check out some Music by Max Reger, who was a Pupil of Riemann.

[@ClassicalNerd]

And a man who quite literally wrote the book on modulation!

[@augustinejazznerd]

Hello. I'm Jazznerd.

[@ericglickrieman939]

Is there a text book that you think is really good on this topic? I see a few, and I'm wondering if you have experience with any of these - I'm guessing you might. Thanks

[@ClassicalNerd]

This theory is too new to have really made it into the curriculum in that way; I think the reason it's largely relegated to grad school today is because the existing literature is mostly in the form of papers in theory journals.

[@ericglickrieman939]

@@ClassicalNerd Thanks, I'll see if I can find some of those.

[@dr.eville]

I get why a major key can't have a parallel major or a minor key cant have a parallel minor ....but why cant a major key have a relative major or a minor key have a relative minor (Cmaj to Amaj or Cmin to Amin) ? Thank you for this video. I am very interested in Neo-Riemannian theory. Before I knew Neo Riemannian theory existed I actually drew the tonnetz grid as a way to explain Beatles chord progressions. Then when I found out what Neo-Riemannian theory was I felt satisfied I wasn't crazy and have been hooked on it since. I think a lot of popular music today can be explained with Neo-Riemannian theory...whether the song writer was using it intentionally or just going by ear. A lot of stuff is classified as modal interchange but I think it makes more sense explained with Neo-Riemannian theory. Can you recommend any books to learn more?

[@ClassicalNerd]

Relative relationships share notes. C Major and A (natural) minor have all of the same notes, whereas C Major and A Major differ by three notes (the ones that A Major's key signature sharpens).

[@adriansuchomusic]

I wonder if Philip Glass makes conscious use of Neo-Riemannian theory. I’m starting to think it could be what makes his music so easily recognizable!

[@PirataEscondido]

Which makes sense because this wasn’t designed to analyze music that used tonal cadence - oh snap!!

[@elba9066]

Please can you recommend a book/books that discuss this. TIA

[@ClassicalNerd]

Most of my sources for this are analytical papers, but I have heard very good things about Richard Cohn's book _Audacious Euphony: Chromatic Harmony and the Triad's Second Nature._

[@elba9066]

@@ClassicalNerd Many thanks, I did not see it when I searched so appreciate your suggestion

[@radu6772]

Is there a particular reason you haven't made a great composers episode about liszt? I'd love to see that

[@ClassicalNerd]

Because the sheer number of requests I get means I can't do it all: lentovivace.com/classicalnerd.html

[@MertCalkan]

waiting a video about Ivan Wyschnegradsky !

[@ClassicalNerd]

Duly noted: lentovivace.com/classicalnerd.html

[@stephenjablonsky1941]

There must be something in the German/Austrian diet that makes great thinkers like Riemann, Freud, Marx, Schoenberg, Einstein, Heisenberg, and Kafka perform mental gymnastics that end up in a Gordian knot. It is actually easier to understand the relationship of triads if you assign them to the twelve positions on the Circle of Fifths with C major/A minor at 12 o'clock and F major/D minor at 1 o'clock. I posted such a clock on the front wall of my theory classrooms and it helped students to appreciate the quality of various harmonic progressions from Bach to Debussy.

[@ClassicalNerd]

Fifth-relationships are very common and useful for understanding all sorts of styles of chord progressions, but neo-Riemannian operations can lend a new understanding to how certain passages of certain composers work. I highly suggest David Lewin's article "Amfortas's Prayer to Titurel and the Role of D in _Parsifal"_ [from _19th-Century Music_ (1984) 7 (3): 336-349] as a really good example of how neo-Riemannian operations are the best "lens" through which to view progressions that avail themselves of symmetrical, equidistant relationships around the 12-tone space.

[@stephenjablonsky1941]

@@ClassicalNerd As an author of several books and articles about music theory I know for sure that nothing is more narcoleptic than reading about music theory. If you give the same piece of music to 10 theorists you will receive back 10 different theories about its construction. That tells you everything you need to know about music composition. You are a composer and know full well the games that composers play with themselves as well as the role of inspiration or the unconscious. Describing music is much like analyzing the formation of clouds. It is very personal.

[@emanuel_soundtrack]

it is simple: they want to give rule to the world, and have problems with their fears. I observe it for 10 years long in loco. You forgot Witttgenstein I-II

[@emanuel_soundtrack]

@@stephenjablonsky1941 total. I would like to see some of these people doing gymnastics to understand the most complicated moments of my music, that in my head were absolutely no big deal more than some concepts and some sonic intuition

[@bremlquan]

Gosh great info and format but the piano sounds crunchy

[@lordwilksy]

A great video idea would be a music analysis of one of your pieces 🎵🎶

[@ClassicalNerd]

Hmm ... an interesting idea, but I'm not sure there's much of an audience for it.

[@charlesboyle9223]

@@ClassicalNerd Yes! I would definitely be interested!

[@charlesboyle9223]

Great suggestion!

[@MaximilianMKGill]

You should do I video about Sir Edward Elgar.

[@ClassicalNerd]

Duly noted: lentovivace.com/classicalnerd.html

[@MaximilianMKGill]

@@ClassicalNerd Thanks.

[@johnnylch18]

I don't think Neo Riemannian theory is that elegant, though I'm very interested in it. I think it also lacks the bigger picture level analysis and gives a lesser intuition on the relationships of chords to the experience of the whole piece. Though again I'm very interested in it and what it can explain, I think as it is now it feels more of a niche theory

[@sashakindel3600]

The idea of using a theory that only deals in major and minor triads to talk about Wagner, Strauss and Debussy seems strange given that they use diminished and augmented triads as well as 7th, 9th, etc. chords all the time. I guess the idea that this theory is applicable to some passages in their works, while other theories are applicable to the rest? I suppose it's not surprising that composers who weren't *trying* to work all under one theoretical roof didn't happen to do so by accident.

[@DavideVerotta]

It is a bit surprising to be told that composers should use Riemann, or Neo-Riemannian theory, when the “theory” actually boils down to nothing more than a list of (trivial) operations over triads in closed form. The "theory" does not contain any compositional "instruction", nor indication on how actually go from a chord to another, or why one would do so. You can make nice graphs, out of an existing composition, although their significance is unclear. A good example would be the graph for the Tonnetz Pole discussed in the video. It is elected to be the wildest chord because it takes the most UTT steps to go to it ... but one has to wonder what is the actual significance of that. What does it matter how many UTT steps there are between two chords when you do not actually hear the steps, when clearly more dissonant chords exist, and even more worrisome for the "theory" when it is obvious that the UTT path is not the only one one could take?

[@donkeykong4983]

R.I.P. 777

[@xKarenWalkerx]

Lady Gaga’s latest album is called Chromatica and incorporates a lot of Riemann’s theoretical contributions mainly his theory of harmonic functions, dualism and negative harmonies.

[@ShaharHarshuv]

Why does it feel like a computer science class

[@damoon57]

At the end got very complicated 🥵🥵🥵

[@rooyaedo]

can you put subtitles in spanish please

[@ClassicalNerd]

Sadly, I don't speak Spanish, I don't have the time to translate subtitles into _any_ language, and I don't have the funds to pay anyone to translate subtitles either. :(

[@rooyaedo]

@@ClassicalNerd que tristeza :(((

[@larsjohansson7954]

So the only thing this theory explains is how a major or minor chord moves to another major or minor chord...? Let's call a major chord D (dur) and minor chord M (moll). Then for example MD+4 would stand for minor to major chord, one major third up (four half steps). And DD-3 would stand for major to major, one minor third down. Moving from C major to B flat minor would be analysed DM-2. Easy as nothing!

[@richardhines8622]

Tonality can dictate rhythm, but rhythm can dictate tonality.

[@JosephElsherbini]

Have you seen Patricia Taxxon's video on Four Chord Loops? (kzbin.info/www/bejne/gV67hIeJo9aLrtE) In it, Patricia talks about another hard-to-analyze classically music phenomenon, and extends Phillip Tagg's "Everyday Tonality" to try to break down successful and unsuccessful 4 chord loops. As a layperson when it comes to Music Theory, it feels like maybe Neo-Riemannian theory could also shed some light on 4 chord loops.

[@timbeaton5045]

That is a fascinating video, indeed. Highly recommended, even if "Our Axis" might be bridge too far for some classical fans!

[@dc8955]

Let's simplify this. Any sound or silence may be preceded or followed by any other sound or silence. Ex Tone cluster, M chord, silence, silence, parallel fifths. Nothing is right or wrong, the test should be does it work well, because if it does not then it fails as music. Rather we should ask why does that particular sound stir specific emotions? If you can master that you will write something exciting. The sound is the stimulus the emotional response is the music.

[@bobpetters3449]

If it's not major/minor or pantonal, it's ambiguous.

[@donkeykong4983]

777

[@annelouisemaclellan485]

Nirvana

[@abrahamherrera4573]

I feel stupid

[@marceloleal1957]

Ucrazy?! Don't flatten the C unless it's Gb, it's G#min not Abmin. I dunno, whatever...

[@quartertoneharmony7100]

What's the theory part, though? As you've presented it, this is just a notation system for arbitrary chord sequences. Riemannian theory makes predictions about strong chord progressions, doing modulations, etc. which work reasonably well in practice. (OK, technically they're mostly postdictions since practical usage mostly long predated him. But still, it's pretty impressive.) What does Neo-Riemannian theory say about preferred progressions, emotional effects of various choices, or other musical consequences?

[@ClassicalNerd]

I think you're working with a different definition of "theory." By yours, Roman numeral analysis and pitch-class set theory don't have anything to say about emotion, either (and arguably, Schenkerian analysis doesn't either). Theories are just the lenses through which we view music; their universal downside is that it doesn't tell you _how_ a chord progression makes you feel, just terminology to describe it.

[@alphago9397]

I find it funny that if you get high enough in any given subject; it eventually "resolves" into math.. music and computer science are perfect examples.

[@emanuel_soundtrack]

blitz too fast, there is a huge amount of important information to think about there

[@jordancarpenter676]

We’ve made theory a thing of such grandeur. Quit the delusions. You don’t need to know why you write what you do. I’ve never met a musician that said theory made them write something or even tell them where to go. They wrote and let the rule writers say why it works. Write what comes to you and quit worrying bout this stuff.

[@ClassicalNerd]

My goofy little rant at the end notwithstanding, I want to emphasize that _all_ theory-including this one!-is at its best when it's _description_ and not _prescriptive,_ but that doesn't undermine its value to musicians (composers included).

[@wonder6789]

Very interesting tho I'm not sure I want all of this unorthodox triadic magic to be explained, catalogued and theorized. I kind of like it to stay a well-kept irrational secret.

[@hoon_sol]

There's nothing magic or irrational about it. Music is a mathematical structure that can be fully explained if you study it at the deepest levels.

[@wonder6789]

@@hoon_sol The artistic value of a given music is determined by the pleasurable emotional responses it produces, an entirely subjective and non-rational phenomenon.

[@hoon_sol]

@@wonder6789: *WRONG.* Those pleasurable emotional responses are a result of the mechanism by which the ear perceives sound, which is universal to all humans; it's an entirely objective and perfectly rational phenomenon.

[@dang5874]

@@hoon_sol no, you create models for music that will have more or less use for you depending on your experiences and interests, such "explanations" are how you understand it, not the uncovering of an underlying objective truth that was absolute even before humans (and so music) existed.

[@hoon_sol]

@@dang5874: *WRONG.* The models are modelling the underlying objective facts of how sound is perceived by the ears; it's due to the harmonic series formed by successive detective hairs in the cochlea that music is based on specific intervals, and why only certain ways of creating and resolving tension sound good. In other words, you literally couldn't be more wrong.