Set Theory: An Introduction

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I learned this material from private composition lessons during my undergrad and then from a theory professor who was a student of Dr. Allen Forte at Yale. Additional information has come from private composition lessons during discussions of serial techniques.
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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 in E minor, performed by Rachel Fellows, Michael King, and Bruce Tippette
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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.

[@ClassicalNerd]

This video attempts to distill a good chunk of a 20th-century analysis class into about twenty minutes. If there are sufficient questions, I'll try to cover them in a segment of my next _Composition Masterclass_ video.

[@loganm2924]

Pure maths students, this isn’t what you were looking for

[@martinhazard5982]

why?

[@juicedelemon]

@@martinhazard5982 it's music's set theory, not math's set theory

[@thats_inaccurate]

It's still very cool! I mean pitch classes look exactly like Z/12Z, interval classes are exactly a discrete metric on the pitch classes (given a 0 IC) which makes then have some sort of topology. I'm a sense Z-relationships are exactly given by modding out by an equivalence relation (namely two subsets of the set of n tuples of pitch classes are equivalent if their interval vectors are the same. I'll leave this to the reader to verify that this is indeed an equivalence relation) . It's very algebraic and topological feeling; hence quite familiar. It's lovely

[@loganm2924]

@@thats_inaccurate I agree entirely!

[@thats_inaccurate]

@@loganm2924 In fact, Im a maths student in university at the moment, and in our math club one day, a friend said some offhanded comment about how muscicians have their own set theory, and that its not at all the math kind of set theory. Initially I just found it mildly amuzing, but on a whim decided to look into it today and was extremely pleasently surprised to find that due to my education, I was already familair with all of these concepts, just in different words with wildly different uses. I guess its a great example of how basic and fundamental some of this stuff is! Also, its a great example that sometimes ideas of pure maths can have effect outside of the basic examples like physics, engineering, or cybersecurity or something. I can certainly think of a few people that would start to see math in a new light if I show them this stuff. Sorry for rambling, I'm just excited to see another person who also had the same discovery!!!!

[@TheSutov]

I was honored to have studied with prof. Forte. It is necessary to add three facts to this fine introduction: 1) Set theory offers insight into super/sub set relations within ATONAL and MODAL structures not just 12-tone. (This is how to progress in analysis of larger pieces of music) Master Forte already in the 90ies combined Set theory with schenkerian approach in observing the counterpoint and marginal pitches within atonal or modal texture. (Set theory serves us to ANALYZE and LISTEN with good knowledge of the material) 3) Interval vector information offer an incredible tool for composers: I call it THE GOLDEN RULE of invariant pitch classes under certain transpositions or inversions (two rules, actually). And, the interval vector accounts also for the INVERSIONS of interval classes: m2/M7, M2/m7 etc.(also, the set-table offers a comprehensive overview of ALL possible complementary sets within 12 tones!). I wrote my doctoral thesis (DMA) on Messiaen's harmony (Harmonic Density in Messiaen, Cornell 99', mentor R. Sierra) precisely with set theory tools. In Messiaen's bibliography (I think by Vincent Benitez) my last name Savli is misspelled into Salvi. (dyslexia not allowed here!)

[@edgarmatias]

Really great summary. I learned this stuff 30 years ago, and did some work building on it at the time. Agree with you that the IC vector is the most useful part of it all - especially ic5 and ic6, since all commonly used scales can be derived from some combination of ic5 and ic6 intervals. Love your channel!

[@jareddavismusic]

Great video and presentation! You have compressed a lot of info into a short digestible and entertaining clip! Bravo 👏

[@CharlesAustin]

Awesome !! Will view many times !,

[@UtsyoChakraborty]

Fabulous analysis!

[@maxdible]

Oh this is just so good. Thank you.

[@doricdream498]

i just recently finished up my general music theory studies, and set theory was by far my favorite part. it also resonates with me greatly because my musical background comes from chiptune and demoscene, which are usually written in programs called trackers. trackers actually make use hex values, and as a result one actually ends up thinking about and applying intervals in terms of set theory. the music is vastly more tonal in these circles so its not serial by any means, but its a fun parallel that made learning set theory enjoyable for me.

[@ClassicalNerd]

One thing that I didn't talk about is how set theory can absolutely be applied to tonal music, it's just not always the best option. I did not know about trackers, though!

[@wids]

Very dope

[@briansanders8748]

Shostakovich is a composer that mixes Dodecaphonic/Set Theory characteristics with tonality/modality/atonality. Check out String quartet No.12 👏🏼👏🏼

[@musicalintentions]

Thank you so much for sharing this. It has helped a lot!

[@aakashchakrabarty4262]

Hey this is the most comprehensive tutorial on set theory. I would also be interested in a video where you show your book collection.😆

[@ClassicalNerd]

I did a bookshelf tour back in the day but I may end up doing another one once I get back to my regular set!

[@Phaseish]

KJING! @@ClassicalNerd

[@davidmenchell5876]

Thanks!

[@sanchopanza9907]

I'm studying fundamental mathematics and find this very interesting and natural. Related terms might include modular arithmetic, reduced binary quadratic forms and, of course, prime numbers.

[@ClassicalNerd]

If you liked _this,_ you'll love it when I cover the tuning theories and system of Harry Partch's 43-tone extended-just-intonation scale. (That'll be in a few videos, though.)

[@Mr.Nichan]

@@ClassicalNerd Trying to come up with a neutral way to think about xenharmonic music, as well as other ways to think about it for con-cultures and just whatever, is probably the main thing that keeps motivating me to sporadically go into weird mathematical rabbit holes about rational numbers, primes, (prime factorizations, prime powers, primorials), bases, logarithmic, linear, and inverse scales, binary and trinary sequences, etc.

[@theodoremann1461]

I was at Eastman when Howard Hanson wrote the book that supposedly initiated interest in this approach. Although I majored in music theory, I didn't understand it then, and I don't understand it now. However, I have no doubt at all that it's a legitimate and significant contribution to music analysis.

[@charlesmoomaw9388]

Hanson's classification method is similar to set theory's interval vectors. Hanson classified chords according to their PMNSDT, i.e., the number of Perfect fifths, Major thirds, miNor thirds, major Seconds, minor seconds (Dissonances), and Tritones (or their inversions, reduced to the same octave) contained in a chord. For example, major and minor triads are both PMN, the quartal harmony C-F-Bb is P2S (C to F and F to Bb are both P's, hence P2), and a major/minor seventh chord is PMN2ST. A set theory interval vector counts intervals in the order m2, M2, m3, M3, P, and T, and it uses zeroes when a particular interval is not present, so Hanson's P2S is the same as interval vector [010020], and PMN2ST is the same as interval vector [021111].

[@cameronmurray3223]

this is cool thanks

[@rafaellazo1725]

Love your videos bro, you inspire me! thank you for bringing classical music closer to people!

[@neocleouscomposer]

Great video! Would love to see a similar explanation of Schenkerian analysis!

[@ClassicalNerd]

Duly noted: lentovivace.com/classicalnerd.html

[@filipaviana8794]

Does anyone know any analysis method directed to performance or interpretation?

[@ClassicalNerd]

Most analyses can be beneficial to a performer's interpretation, but none (that I'm aware of, at least) are specifically geared towards that.

[@boyisun]

I'd love to see a Schenkerian analysis video too!

[@williamhively3295]

For goodness’s sake, why?

[@tysongholstondaviscomposer]

Now this, this is epic.

[@rko2016]

i kept hearing about how important set theory and counterpoint, etc. are to composing your own music, but it's stuff like this that clearly shows me that, to me, it wouldn't be. So thanks for making such a comprehensible video. i dodged a bullet not going to music school.

[@landrypierce9942]

When you go to music school, at first all the rules seem to only be holding you back. At a certain point, however, you understand better why the rules exist, and you can choose which techniques you wish to include in your writing, and come out with a composition that is simply better written. Composition isn’t just about expression. There is a craft to it as well that is rarely straightforward.

[@pawncube2050]

I loved it! The explanation and the concept. I remember I always wanted to analyse music using something like this 5:32 Programmers will be pleased. The only thing is using A and B instead of T and E 18:46 I remember 12tones did a video on this. He explained basically all of this without using any of the terminology or mentioning set theory

[@ClassicalNerd]

Ah yes, hexadecimals ... definitely a related concept (and one that would be applicable to set theory in 16-TET).

[@pawncube2050]

@@ClassicalNerd 16 tet sounds evil, I wonder if prime numbers would make more exotic music, say the 17 TET

[@ClassicalNerd]

16-TET is similar to 12-TET in that it can be conceptualized as a set of four non-overlapping diminished-seventh chords (i.e. the 0369 set) as opposed to the three found in 12-TET. So it basically splits the minor third into one more part, which ultimately doesn't gain xenharmonic composers a whole lot. 17 is very creepy to my ear; I did some experiments in that temperament for a choreographer some years ago and she was spooked at the result so much that we scrapped it (much to my chagrin).

[@RobertWildling]

Great to find a video about set theory here! In Austria, were I am from, and maybe even in Europe, set theory is not accepted very well. The most common statement is "Set theory looks at music as if it was Switzerland without mountains." - BTW, there is **only one set** that has a different number on each interval class position. Who knows which one that is?

[@redvine1105]

Fun fact about those all interval tetrachords 0146 and 0137 as they relate to tonality: If you think of 0146 as the scale degrees of the harmonic minor scale you get #7, 1, 3, and 4 (I.e. G#, A, C, and D). If you think of 0137 as the scale degrees of its relative major scale you get scale degrees 3, 4, 5, and 7 (I.e. E, F, G, and B). That’s all 8 notes of the major/relative harmonic minor scale (aka the “bebop scale”) appearing once and only once. Just goes to show how colorful the good old fashioned major scale really is.

[@mickeyrube6623]

I don't know why we don't use set classes to teach music. Kids can still learn to play Mary Had a Little Lamb on piano, singers can still use Do-Re-Mi etc. But for people who want to advance in their abilities, or learn more complex musical idea, it can simply be explained if you were brought up to learn music this way.

[@ToneCanyon]

Im still confused, but this is the best one ive seen on the subject! I gotta bust out some manuscript paper!,!

[@alexgrunde6682]

The irony in this is that this way of thinking about music is probably closer to the way a lot of musicians who “don’t know music theory” approach it than traditional Roman numeral oriented theory. Which intervals apart of notes or particular chords sound good, as opposed to triadic groups in particular keys. Like, describing music as 0-1-3-6 or something similar, that looks like guitar tab, the way of reading music for guitarists that don’t know the “proper” way of reading music. Of course, that’s only a 1 to 1 if the open string in question is the root note, but it has similarities in that the tab would look identical for a song in D in D standard tuning as it would for the same song in E in standard tuning; the relative intervals on each string remain the same, even if they’d be completely different notes and key in traditional sheet sheet music.

[@marcmosteirin7575]

I think there's an error in the graphic at 8:18 - IC2 should say m7, not m6?

[@redvine1105]

6:30 nice

[@jorgeluiz1698]

Wow that was really interesting, I've never heard of it before and surprinsingly I reached some similar conclusions a while ago when I tried to summerize in a equation how intervals could be inverted. I did that just for fun because of something that the teacher of my music theory class said got me thinking of it, but I stopped working in that project eventually.

[@amorpheousstone]

Hi, Great video! Would you recommend George Perle's book 'Serial Composition and Atonality' ? Many Thanks.

[@ClassicalNerd]

Thank you! Unfortunately, I've not actually read that book (although I have read excerpts from it for class before). I think it would definitely be a worthy read if you can get your hands on it.

[@danielgonzalezjr8350]

I too am scared of the number 13

[@wids]

Same

[@VaSavoir2007]

ANton Webern was killed by an American soldier, which soldier them fell into chronic alcoholism before the enormity of his crime and died young. Was this soldier also a cook?

[@ClassicalNerd]

Yes-and I have a whole video on Webern with more on the story.

[@nartoomeon9378]

I can say that an interval vector looks like a simplified notation of a multiset ( set with repetitions) of interval differences. Is like an uncommon operation called Minkovski sum or substraction, but repetition is considered. We calculate intervals by subtracting pitch classes. Then, we group by some standard value of intervals, and make a formal sum as products of those intervals by the number of repetitions for each. I couldn't figure out for days why the backward motion is taken into count. But then I realized that the distance usually is |a - b|. But in modular math (-a = 12-a) So if (3 - 5 = -2) then (-2) = 10 mod(12). *In terms of integers |-2| should be 2, not 10.* This is the reason for the mutually inverted series being undistinguishable in this system. I think I'm right... But I don't like it... Inversion doesn't save the chords, most of them. The direct connection of regular and negative harmony does not seem pleasant to the ear. Matching interval vectors for different series seems unlikely to be a good thing. Maybe I don't know the main tasks for which it was created and therefore I don't see any perspective. I'll do a post later with a more interesting problem with some related concepts.

[@ClassicalNerd]

I think you're coming at this from a more mathematical angle, while music theorists adapt concepts in a more fast-and-loose style. The modulo-12 thing usually screws with people not used to it, but it works and is self-consistent. IC vectors are useful, but in a pretty limited sense. Elliott Carter's music, for instance, would not sound the same without his favorite all-interval tetrachords, and its unique properties are best shown in the IC vector. It also shows Z-relations pretty well. Since this is an analytical system for atonality, the goal isn't to preserve audible tonal relationships (inversion working totally differently in atonality than it does in tonal theory) or to be "pleasing" (which is highly subjective).

[@nartoomeon9378]

@@ClassicalNerd Well, I probably agree with you. I haven't listened to atonal music myself, except for 2-3 works by Alfred Schnittke, because I've listened to more microtonal music. I'm more interested in the question of the possibility of adapting the tonal system to it. Or something in between, a kind of bridge between them..or em.. like that.

[@ClassicalNerd]

I'm fully convinced that xenharmonic music is the way of the future, for both weird takes on tonality and for atonality. It would be fantastic to have an analytical system that could be used across different equal and unequal octave divisions, but that might be too much to ask of any one system.

[@nartoomeon9378]

@@ClassicalNerd We could agree with that, because music is, after all, about a man. It is the human who finally decides which music is good and which music is bad. I have long thought that music was able to go the other way because of people. And harmony is mostly a habit. Maybe the tritone sounded very dissonant in the past indeed. But now it's everywhere. Still, I'm not sure that psychology answers all questions about music. I'm sure theories can be constructed in different ways, depending on the goal. You don't have to calculate everything, you can build a theory about a *way* of calculating, for example. ✨ So I came to questions like this: Can humans get used to non-classical scales? (Relate dissonant intervals and chords to some emotion.) Maybe one should look for some structural properties of the scales, softer than the carefully and precisely tuned ones, as in JI? I already have a suggestion for what that might look like, but the message lines are too narrow for it. ) ✨From lectures by a very high level algebraist, I heard a criticism of his direction(homological algebra, etc.): "in recent years, constructions have been replaced by constructors". This, too, seems like a different way for theory - to increase the level of abstraction. Maybe this is bad for mathematics...I can't judge that.

[@ToneCanyon]

Hey Classical Nerd ! I am watching this video on my school account. I turn my head for a second, I look back, and the dislike button is flashing non stop and the screen says “feedback shared with creator”. I am really sorry, the school computer has been glitchy all day: (. So if you just got a bunch of dislikes on this video, they are regretfully from me by accident. I actually LOVE this video, and I am going to try counterbalance with a bunch of likes!! Sorry, keep up the great work : )

[@ClassicalNerd]

No worries! Most dislikes, according to my analytics, are from people who don't watch the videos at all or they're a side effect of Thailand-based bot farms. They're incapable of substantive critique, which I'm open to! (By the way, KZbinrs have no way of knowing _who_ likes or dislikes a video, and most of us don't really care-either one boosts engagement, which is a plus!)

[@grayson9839]

Hi there, I was wondering if you could do a biography on the composer Thomas Little?

[@ClassicalNerd]

Y'know, I usually don't cover living composers, but you're in luck: lentovivace.com/bio.html

[@grayson9839]

@@ClassicalNerd Thank you.

[@alcyonecrucis]

What’s your grad school about? Will you write a thesis? Great videos!

[@ClassicalNerd]

At this time, I was getting my MFA in Composition and Theory at Brandeis University. Now, I'm getting my PhD in Composition at the University at Buffalo! My MFA thesis was a piece, and I will be on the hook for a dissertation several years from now. More info on my Web site: lentovivace.com

[@mitodrumisra8972]

Hey there Mr.Little, it's me again. You would remember me as the guy who told you that your and Henryk Wieniawski's birthday have a nearly 2-day difference. Well, I've got 3 more requests for you: 1. Classical Music during WW1, WW2 and Cold War 2. Henri Vieuxtemps 3. A composer from India - Rabindranath Tagore (I think you might have heard his name) Is it possible for you to make these in the near future, at your own convenient speed? P.S. - THANKS FOR THE COMPOSITION MASTERCLASSES!!

[@ClassicalNerd]

Your first request is really too long for a single video. In fact, I would say that even covering the musical developments that happened in line with (or because of) those conflicts would go far beyond the scope of what I've done in my longest and most substantive videos. You are the second to request Vieuxtemps and the first to request Tagore [ lentovivace.com/classicalnerd.html ]. Given the way the request queue system works, combined with the number of requests I get and the sheer impossibility of keeping up the pace with my production schedule, I would strongly advise you to not get your hopes up about seeing either of those composers covered any time in the near future.

[@mitodrumisra8972]

@@ClassicalNerd It's ok. I know that you have a hell lot of requests.... Besides, I am not like the common person who plague KZbinrs to do their thing😂😂🤣🤣. But anyways, I'll cjheck back in 2022 😉😉 P.S. - For the first, you can concentrate upon the works of the Russian composers....after all, they were the angry young men..!

[@JanCarlComposer]

wow, this is super super nerdy content, but I somehow like it and will stick around ... maybe it's useful someday ... ;)

[@arne__]

So far, I struggle to see how this is a theory of music, looks like a system to sort and name subsets of arbitrary cyclic sets - what makes it a theory of music rather than a theory of the numbers on the clock? Does it make any prediction about how we perceive music?

[@gonzoengineering4894]

It describes a fairly fundamental relationship between what we would elsewhere call chord qualities in a generalized way -that major and minor triads are members of the same set seems to suggest something fundamental about tertiary harmony. This video would do a lot better with audio examples, but we're at past the 20 minute mark on a very imposing subject as it is. It is historically limited by being anchored to serial music, but I think there's an untapped potential here for a much more powerful generalized framework. Pentachords and septachords aren't very interesting to the serial composer, but they are incredibly useful to one interested in scale structure and synthetic scales.

[@ClassicalNerd]

I would argue that few (if any) analytical frameworks make predictions about how we perceive music. Generally they form either a lens through which listeners can come to better understand and appreciate a fundamentally aural experience, or a toolbox for composers to construct music in a way that has a logical underlying structure. In this case, the listening experience is recontextualized as a mathematical one using set theory, while its biggest functional use is for composers seeking to write self-consistent atonal music without either throwing all caution to the wind or using a 12-tone system.

[@arne__]

@@ClassicalNerd I think I disagree: If we think of pitch classes not as elements of a cyclic group but as letters in the alphabet and then read some piece of music out loud, trying to recognize words or names, this will certainly give us listeners a new perspective on the piece, and it is also something that has inspired many composers - but that does not make the linguistic theory of how to form words from letters that refer to some componist or a biblical verse a theory of music same goes for any theory about the non-musical phenomena that inspired xenakis music - it may be a theory that inspired music and that helps us understand music inspired by that music, but it's not a theory of music I think music is first and foremost a perceptual phenomenon (ignoring cultural aspects) and hence a theory of music should explain musical perceptions like dissonance or a sense of direction etc and isn't this exactly what a concept like leading tones or the harmony of Helmholtz, which is rooted in his theory of our perception of complex tones, are trying to do? to be fair - this is basically all I know about music theory, and it's all geared towards tonal music. maybe a theory of non-tonal music could look somewhat like the ligeti-video david bruce uploaded today, describing and explaining phenomena like recognition of motives that play an important role in non-tonal music as well in analogy to visual perception or in terms of psychological concepts like gestalt I don't wan't to arrive at the conclusion that all music theory that is not basically neuroscience is invalid though, Ill have to think about all that some more... Anyway! All this is not to say that I don't think this set theory thing isn't interesting - I would not invest in to a comment like this if I thought so, and I'm obviously very thankfull for your clear and thought provoking video! cheers!

[@ClassicalNerd]

I'm not sure we're approaching some of these terms from the same point of view but it's a fascinating discussion to have nonetheless! From a compositional perspective (which is where I come from), analysis is a tool that peels back the perceptual layer to something more concrete. In particular, set theory is just an approach towards naming and labeling harmonic or melodic motifs, which is why it means nothing in and of itself; one must be judicious in segmenting the music into sets that make contextual sense and then look all over the music to find other incarnations of the same or similar set. If the perceptual phenomenon is something akin to-in an atonal/serial piece, for instance-hearing something that has no discernible form or structure and yet still feeling as though the piece is something of a unified whole, set theory might be something that could help explain the details as to why it might be perceived that way.

[@ninreck5121]

how do you spell the other two analyses you mention at the beginning other than roman numeral analysis?

[@ClassicalNerd]

Schenkerian and neo-Riemannian.

[@DavidA-ps1qr]

Another fantastic explanation of the theory of music that, leads me to ask the $64,000 question: Is music just simply a branch of mathematics?

[@ClassicalNerd]

Most would say that just because of all the math involved with tuning theory, if nothing else!

[@danielgonzalezjr8350]

When will the Khachaturian video come out.

[@ClassicalNerd]

It'll be the next video, as he's at the top of the pool right now.

[@steventung5138]

Please do a video about Sergei Bortkiewicz

[@ClassicalNerd]

Duly noted: lentovivace.com/classicalnerd.html

[@SunriseFireberry]

What's next, Group Theory?

[@ClassicalNerd]

I'm not familiar with the use of group theory as a musical term ...

[@SunriseFireberry]

@@ClassicalNerd Set Theory & Group Theory are both found in mathematics. Music has Set Theory too, so I queried as to whether it also had Group Theory. If u aren't familiar, GT probably doesn't exist, yet anyway.

[@ClassicalNerd]

If it does, I'll probably eventually make a video about it.

[@robbyr9286]

@@ClassicalNerd Group theory is invoked, but in a fairly rudimentary way, in alot of 12 tone & set theory, in the sense that the integers mod 12 form a group under addition, & the transposition & inverse & retrograde forms of a row form groups. Are you aware of the cycle of 5ths transform as multiplication by 5 or 7?

[@anetaglinkowska6149]

Is understanding the set theory like tuning your own piano? Either way, watched it to the end with pleasure. Also, you're getting closer to having your face in focus.

[@ClassicalNerd]

Tuning an instrument is its own can of worms to which I'm not sure set theory would be very helpful. Also, my poor camera is really not designed for any of what I've put it through, and to be honestly, sometimes I'm just happy it turns on. (You're more than welcome to buy me a new one if you like.)

[@anetaglinkowska6149]

@@ClassicalNerd If you can manually set the focus on your face and lock it, it should stay there, since you don't move. But then it also depends if your camera has a setting like that. I enjoy your chanel and I'm only a parent of a very young pianist.

[@adrianoseresi3525]

Set theory? I did not know you were a mathematician as well!

[@schniT_T]

Base twelve is a lot like base ten, really.... If you have two extra fingers.

[@ClassicalNerd]

Your feet can be extra fingers if you believe in yourself.

[@scottgilesmusic]

Tom Lehrer! Love the quote!

[@schniT_T]

@@scottgilesmusic so happy you recognized it!

[@authenticbaguette6673]

Some guy who does like computer stuff at a microsoft told me how to count to 7 : 98 , 99 , millenium , Xp , 7 ? Yea sounds about right , pardon the pun ..

[@samuelmincarelli5051]

It is a shame so many people believe music has to be simply an expression of beauty as mathematical music is a taboo. But music at its core is just wiggly air and can be whatever the composer fancies.

[@scottgilesmusic]

Isn’t it Allen For-tay and Alberto Ginastera Heen-ast-er-ah? Outside of your pronunciations, (a problem with most of your videos) your presentation was quite clear and well organized. The Forte book is rather frustrating. Forte himself made many mistakes and the book is littered with them.

[@ClassicalNerd]

The theory professor who taught me this actually studied with Forte. So I would say the pronunciation of that is as accurate as I can make it.

[@michaelhall7499]

6:25 You did that on purpose

[@user-mo8xr9sf7b]

Nah, you're thinking too much

[@ammyvl1]

wait a minute this isn't a math channel

[@RecycleBin0]

pitches still are not always thought of as an exact frequency for example in microtonal or polychromatic music, music thats slightly flat or sharp but is still diatonic, or different tuning systems of which equal temperament is the most common. notes are generalisations for approximate frequencies

[@jonaskoelker]

I'm a mediocre composer on too many instruments, including guitar, and I have a mathematical bent. To me, counting in numbered semitone intervals feels quite natural. Mapping intervals onto integers feels natural to me; integers have quality I'll describe as flat, homogeneous and/or uniform, without defining further. Note that unlike a piano, a guitar has no "black frets". For those who found it interesting that the 0146 chord had '1' in every interval, you may be interested in Golomb ruers, see en.wikipedia.org/wiki/Golomb_ruler. I don't know if they're explicitly mentioned in this TED talk, but if you like the interface of music and wonky math, you may enjoy kzbin.info/www/bejne/iHaxnGyGgJVpd7M. I have never fully understood the function of developing a conceptual apparatus in order solely to analyze a piece of music: it seems iike the result of the analysis always gets stored in write-only memory. At least when I did it in school. This video contains an interesting semi-implicit claim: that interval vectors can in general be used to predict when chords sound similar-suggested by the application of this general idea to the specific case of 0146 vs. 0137. Hm. I guess I'll have to mull that one over. Also, I guess a minor vs. major chord sound relatively similar, compared to a diminished or augmented chord or a 7-chord or [insert any amount of frills here], even though major/minor is simultaneously a rather (ahem) key distinction. [Pun not intended. At least not at first.]