Tritone Substitutions
johncarlosbaez.wordpress.com
johncarlosbaez.wordpress.com
It’s also worth noting that when comping, the most important notes to be sure to hit are the 3rd (which expresses whether we’re major or minor) and the 7th (which gives the flavor of the chord), since the root is likely covered by the bass player and the fifth is implied by the 3rd (if you’re at a piano, try you can see this by, e.g., playing C-E-G-B in the right hand and C in the left, and comparing that to E-B in the right hand and C in the left. A jazz pianist soloing will likely do their melody line in the right hand and hit 3-7 with the left), so the tritone substitution will be made/implied by whatever the bass player does against that.
The other fun thing is to just vamp on a II♭7-V7 sequence. Some notable places you’ll encounter this in music you’ve heard include the Simpsons theme and the Beatles’ “I Am the Walrus.”
If and only if the chords all stay in the same inversion and the bass line doesn't make leaps. That might be what you want and it is idiomatic in some genres/styles. But it might not be what you want or even what you get from the notation depending on how the musicians interpret the notation.
Depending on context, it can take a whole paragraph of music theory to describe a handful of chords that would otherwise be instantly recognizable to a trained ear.
https://www.youtube.com/watch?v=Nf3_NfuvK8Y
There are two tritones in 7-limit just intonation (7/5 and 10/7) and therefore two possible tritone substitutions. Here they're played successively
http://lumma.org/music/theory/demo/progs/TritoneProgressions...
More broadly, a lot of people like to point out wild 20th century chords in baroque music, but they really didn't think in terms of chords, and as such these pseudo-chords don't have the same function that modern versions of chords do. In particular, Scarlatti was a prolific user of the partimento method of composition, which is a slightly of abstracted version of counterpoint, and his sonatas are pretty good examples of pieces written with this in mind. The method revolves around intervals and movement between voices rather than chords. It's normal for someone thinking in counterpoint to produce some very "modern" "chords" because that's a common consequence of following nothing but voice leading to produce a piece of music.
https://en.wikipedia.org/wiki/Partimento
See also Rick Beato mis-analyzing Bach's counterpoint as containing a very "modern" maj7#5 chord: https://www.youtube.com/watch?v=d1f_tzBx6ko&t=1s
Pick an example everyone knows: Bach's Prelude in C. I wrote in chord symbols over every bar of that, except for that one very weird bar near the end. It helped me a lot in memorizing it.
By the way, a 19th century functional harmonist would also find that prelude sort of odd. Not in terms of the chords themselves, but they don't always obey their usual functions.
Also, many modern players and theorists reach toward Schenkerian analysis and its derivatives to sort of explain why music sounds the ways it does. Unlike counterpoint and harmony, Schenkerian analysis is almost entirely descriptive, not prescriptive. I have no idea what your teacher would have suggested.
I don't even understand this paragraph. I'm not arguing.
As someone with a significant background in historical performance, my books of Bach preludes are still full of Roman numerals because that is a really good way to "compress" information about what notes to play.
I'm actually interested in other ways of analyzing it.
* Schenkerian analysis is probably the modern method of looking at Bach. A sister comment also indicates that Schenker was a controversial figure (he was a German fascist), but I believe the follow-ups from other theorists on his methods are the modern ones to use for analysis.
* Analyzing otherwise in the context of counterpoint and intervals - see Gradus ad Parnassum and Kennan's book on counterpoint.
It’s also, interestingly, illustrative of your argument. Musical analysis is necessarily socially contextual and therefore revealing of the author’s values and priorities, and Schenkerian arguments often imply, or directly come with, some really quite right-wing positions. Schenker was very much into motivated reasoning to defend his deep-rooted racism; many Schenkerian analysts have, deliberately or otherwise, wound up following in those footsteps.
It's true though that Schenker's treatises included plenty of political asides of such an extreme chauvistic character that to call them "quite right wing" is a huge understatement. (It's also true that, as scholarly research has pointed out in recent years, he seems to have expressed similarly extreme views in his private correspondence and other private writings.) Part of this might perhaps be explained as Schenker's awkward overcompensation for what would've been his remarkably humble origins back then (he came from a small village in what was then Austrian Galicia, now in modern Ukraine). Regardless, I think we nowadays have so many sources proving the relevance of melodic reduction/elaboration approaches (some of them quite early indeed, from the 16th-17th centuries) that to tie these analytical approaches polemically to Schenker and his specificities is really quite pointless, perhaps even misleading.
I'm curious what you would suggest to perform a modern analysis of Bach (what the original comment chain on this was about) if not Schenkerian analysis. I didn't see any alternative analysis tools proposed in your post, even though it sounds like you are educated on the subject.
Andras Schiff playing BWV 903 (there are some REALLY bad performances of this piece out there): https://www.youtube.com/watch?v=SNWOhm5iXxs
BWV 1079 wasn't necessarily written with musical instruments proscribed, but here's a good French group's performance: https://www.youtube.com/watch?v=tfMQ-AYiuJw
My piano teacher pointed me to this recording when I studied the piece with him. It was scandalizing at first, but the free improvisation in the arpeggiated chords is almost miraculous. (I also love Paul Badura-Skoda's take on the piece: https://www.youtube.com/watch?v=Xx5gojkXdeI)
When theorists talk about Scarlatti laying the foundation of harmony, partimento is probably the foundation they are referring to. Functional harmony of the 19th century is just a more powerful abstraction.
Second, while they did think differently about harmony, you can describe it in modern terms. A description like Cmaj7 doesn't mean anything else than a set of pitches.
A tritone substitution is somewhat different, since it depends on what the analyzer thinks the chord could/should have been. There's a recent YouTube video (by David Bennett, IIRC) that shows tritone substitutions in songs, but he mentions that he doubts Paul McCartney knew he was writing one. But does that mean it isn't? No, it means you can see it as a tritone substitution.
Music theory is a subjective tool to describe and compare, nothing else.
I think this might be missing the point a little bit - the bassline in figured bass has more primacy than any of the sonorities above it. The most relevant feature of "chord"-based approaches to music (as found historically, e.g. in performance methods for the Baroque guitar, and to a far lesser extent in repertoire for the lute) is that the vertical sonority (such as "Cmaj7") is all that matters; individual "voices" or "parts" are of secondary importance at best. This maps well enough to how lute- or guitar-like instruments might be played in an accompaniment role (since these instruments obscure the sense of a "melodic line" or a "part" the most, for a variety of reasons) but music includes a whole lot more than that.
It's very clear that the man wrote chords. It's also very clear that harmony (ie vertical music theory) was not the tool used to write those chords. If you listen to BWV903, for example, there is a long section of arpeggiated chords that defy 18th and 19th century harmony.
The OP's idea of a tritone substitution from Scarlatti does not have the same harmonic function as a tritone substitution in modern music. I would agree with you that if the function matches, you can certainly analyze things in the context of a later theory, but the function does not match.
Finally, figured bass notates intervals, not chords. A crude realization can neglect voice leading in the upper voices, but more skilled realizers of figured bass will preserve voice leading between the intervals in the upper voices. The idea that figured bass is about chords sort of maps backwards the modern abuses of figured bass (which IMO should not be taught as a "basic music theory" subject - but that is for another day) onto the original method.
author: spice up your jazz by simply replacing one dominant 7th chord with another a tritone away!
Scarlatti: You thought I was aiming toward D-flat? Bitch, I'm surfing in G major! Your puny diatonic expectations cannot contain me.
Put simply-- you can use regular expressions to try out the author's substitutions, but you'll need to build your own parser to try out Scarlatti's.
Alternatively, one could just hold one note fixed and move the other three, so you end up with both chords being self-consistent. That seems more mathematically satisfying at least.
Ultimately, though, I suppose the only real standard of correctness in music is whatever you can get away with.
There can be only I.
I, bII, etc. are chords not pitches. II-bII-I is a chord change. Pitches may go up or down and voices may go away or arrive. They may be part of those chords or not. And there may not even be a root pitch.
Or if you want formal theory, chords are scales [1] and there ain't no such thing as a scale with a sharp I. If you sharp the one it’s a different scale starting in a different place.
The b in bII is not an accidental and bII is how to communicate the chord in passing. I/bII is also possible to communicate a modulation (as would be vii/II, etc. if the modulation is elsewhere).
There are “enharmonics” for chords that aren”t the root of course, e.g. #IV and bV depending on intent, convention, or playability. But #I will be marked wrong on the entrance exam and only raise you standing among an avant guard that isnt accepting new members.
[1] see Russell’s Lydian Chromatic Concept.
It depends on what's the function. If you want to use the tritone-substituted G7 to modulate to the sharp fourth, you'd write it C#7 and resolve it to F# (or F#-).
The Roman Numeral analysis would be V7/#IV if you are modulating in C. And something else if you aren't in C. Roman Numeral Analysis is independent of key or tonal center. The Roman Numeral Analysis would be bII if you are not modulating because that's how Roman Numeral Analysis communicates.
Thinking about Phrygian mode. The two is always a bII.
{0} As an aside, in The Common Practice the bII is often notated N6 (for Neapolitan 6th) to set up a perfect cadence, i.e. I-N6-V-I.
Working with a MIDI file and without much grounding in music theory, I had and have no real way of determining whether to represent notes flat or sharp. I consulted several people and generally got a response of "Oh, interesting stuff! It's not clear how you should transcribe that."
I'm still interested in opinions! If you want to check it out, the score is here: https://musescore.com/user/36584999/scores/7810499
- Match the type of accidentals used by the key signature.
- Use sharps when the melody is moving up, and flats when it's moving down.
- Use whatever accidentals give you a visually pleasing spacing of notes in "thirds" as represented on the staff. (As advocated here.)
- Use the accidentals that are physically compatible with other notes in the chord. (Only a concern for certain instruments, but e.g. I understand that the pedals on a harp affect many strings at once, so playing a particular combination of notes may require a clever approach to the pedals, indicated by what might otherwise look like an unnatural set of accidentals. This wouldn't apply to a piano, where G# is just a separate key from G, not a modification of the G key.) I think this is what you're going for?
But I guess you could get the flat 5 if you did the tritone substitution of the secondary dominant of the IV, which would be bV7.
And then I think you may be talking specifically about measure 11. I'm pretty rusty on my analysis, but it seems like it's the iv7 chord, but starting and ending with a chromatically raised root, which is slick because it acts as a passing tone from the i chord in measure 10 and the V in measure 13. I would probably do Gb -> F -> Gb, because the F is the chord tone, and the Gb is the nonchord ornament. But yeah, probably doesn't really matter!
I can read a piece of code and know how the bits affect one another. When I read basically anything on music theory it feels like a string of non-sequiturs basically. Maybe there's just some skill floor where that all starts to feel organized?
If your goal is to be able to improvise then the purpose of theory is really ear training: it allows you to develop an instinct about what to do with your body to produce a desired result. Improvisation is a bit like "composition in the moment", like speed chess.
There are contrived situations where people treat programming as a performance activity but it's not the way most programmers spend their time.
The composition vs performance framing is very useful I think. Some musicians might just want to do composition, and can spend a little more time between the notes figuring out what ought to come next. (Although even in this case, it's easy to get bogged down in analysis and forget about what feels right). But a lot of musicians want to be able to improvise - and the difference is like being able to write down a compelling argument vs being able to make a compelling "speech" off the cuff.
Yeah, in both cases, you're going to need a good understanding of grammar and vocabulary - but when you're trying to make that speech (or improvise a solo), you're just not going to have the time to think about "the sound I want to hear next is the sound of the 5th of the chord the band is implying." All our practice and theory homework is in service of the fingers just doing what's in our heads automatically - and sometimes, even surprising ourselves with what comes out.
"The purpose of all theory is ear training" is actually a Warne Marsh quote, although one that I can't verify. My dad (who is/was a jazz musician) read it in an interview in Coda Magazine in 1976 and while I can find references to the interview I can't find a copy of it online.
My dad used the quote "The purpose of all theory is ear training" as the inscription on his self-published basic and advanced keyboard harmony books. It's certainly a quote I've heard him repeat many times over the years.
Jazz is 99% theory 1% magic. The difference between the greats is that they’ve forgotten all the theory
I’d say “music theory” as a general concept is closer to understanding computer science principles like managing complexity or the memory hierarchy. Individual disciplines, such as orchestrating for a specific kind of ensemble (string quartet, big band…) or writing in specific forms (fugue, AABA popular song…) are closer to mastering practical programming tasks like working fluently in a given programming language.
Other people can probably figure out better analogies, but I’d say it’s something like the difference reading an article about generics or monads vs actually using them every day. To an outsider these seem like a ton to think about, but when you use them every day they’re just part of your toolbox.
You wouldn't use music theory to write music, same as you wouldn't use a grammar book to write a novel. You write music by intuiting the sequence of moves you want to make, guided by your ear and listening experience, and maybe even coming up with some new ones. Then if you're successful, theorists will make theories about it.
Some people will naively say "All music has pentatonic! Universal!" but ignore vital cultural context. One example in the video is ragas: they can look a lot like Western scales and modes, but they have a time and context.
Western music also has a time and context, but it doesn't stand out as much to us unless someone goes full ham breaking that context like with the Imperial March in a Major mode: https://www.youtube.com/watch?v=B9MShtCg4fk
It changes the mood from "this is the bad guys!" to something more positive and triumphant.
This adds additional confusion to comparing scales/modes between cultures.
We do have at least one such mode/scale in the western european classical tradition, but it is not widely used.
Rather it's that things that hit one tone either also tickle the hairs in your organ of Corti at 2x, 3x, 4x, 5x the frequency (strings) or at 3x, 5x, 7x, 9x the frequency (half-open acoustic resonators—guitar bodies, panflutes), and your brain feels comfortable when those tickle together as they are familiar.
I would also say that the obvious impact of the ear and/or the tools you’re using to play music make it very different than programming. Without the knowing how you would actually play a piece of music, it can be difficult to visualize or interpret notation, whereas code is pretty straightforward in how it’s made or what it actually does. The “feel” is really abstract and interacting with the piece is more akin to observing a running system than it is to touching code.
That said, making music is not merely a technical exercise - there could be multiple theoretical ways to describe a harmony. You still have to listen to it and ask, "what makes the most sense? Is this functioning as a resolution to some tension?" It's easy in a classroom to lose sight of the fact that the theory was invented after the fact.
Sure, music theory involves learning the "syntax" of music notation, the "semantics" of e.g. key and genre, and the "programming pearls" of common musical idioms like tritone substitutions. There are analogs.
But what distinguishes music theory is:
1. The language and notation is hundreds of years old. It's been with us in one form or another 200 times longer than COBOL has. Maybe longer.
2. You don't have the peculiar challenge of having to speak it in the very precise form that a computer requires of a programming language. The terminology and communication between musicians is much fuzzier than that, much more like a human language.
But this game has 1) very short levels (1-2 seconds), 2) repeated patterns in the levels and sequence of levels, 3) strong incentives to solve levels in novel and interesting ways, but 4) there are hidden mines everywhere which can damage you, but 5) you get points for hitting the mines in the right way so that you avoid damage. The mines are “wrong notes.”
Music theory gives you power ups to see the mines, quickly plot paths, and treat sequences of levels as part of a meta-level. But you have to be FAST.
(I have been known to stretch an analogy too far..)
Functional harmony is usually what people mean by music theory.
Functional is the key to my understanding. Functional Harmony is what has been done harmonically and melodically in styles of music derived from The Period of Common Practice. Functional Harmony describes what some people did. Using it makes music that sounds like what those people did.
From the street where I live, I see cultural institutions that ascribe moral value to the conventions of The Period of Common Practice. There’s an industry built up around The Period of Common Practice and the flow of money means there are jobs for those of jobbing age and careers for those of careering age.
It is a disjointed mess like any jobs program.
Music theory is descriptive, not prescriptive. Oodles of people make music without knowing a lick of it, and people who learn it in academic environments still typically make music starting from other concepts. "Listen to records and vibe with what they are doing" is typically a much more effective way of learning to improvise seriously than "learn the theory rules", for example.
"Music Theory" also typically is very tightly bound to harmonic analysis modes that are focused on about two centuries of western european music. This is necessarily going to limit its effectiveness when applied in other spaces (even here, when applied to jazz).
You can certainly code lots of ingredients. For example, you can use embeddings to find better rhymes faster when composing a lyrics, or even LLM help to check if your lyrics make sense (If ChatGPT can guess what your lyrics are about, a human may be able too). You can use a host of computer utilities to find harmonies, and even use voice synthesizers to sketch the singing. But at the end, what makes a piece of music interesting is how well it meshes with its listener...you can model that with statistics, if you are pedantic enough, but at that point the maws of hell open and suck your soul out.
On the left, they're similar because on the surface level, you're basically executing scripts and there's a lot of jargon to describe higher levels of abstraction.
In the middle, they're different because, as others have said, music theory only loosely maps to how people write music, whereas software development seems much more algorithmic.
But as I became more experienced, I realized software is way more of an art than a science, and that all the best practices are contextual.
But I still wish someone would write a history about "what kinds of harmonies do people think sound good/melodious/interesting, and which do they consider bad/ugly/weird/useless at a given moment." Or if that history already exists, I wish I knew how to find it.
https://youtube.com/playlist?list=PLuQqHfLobLUIzDiIGrtE41KvQ...
That being said, Tritone substitutions sound good in some circumstances, but seem over the top in others. Use with care, as always. They'll always seem more appropriate in jazz genres
Pythagoras and Descartes had more to do with criticism and theory than anything anyone was listening to or composing, which was instrumental in the development of western theory. I don't think we have any music following the ideas of the Pythagoreans until the mid/late 20th century, anyone know of earlier examples?
I would suspect that many of the ones from the early periods were far more "mathematical" than 20th and 21st century composers, but there is very little writing either way.
That is a very crude view of the relation between math and music in 20th and 21st centuries. The puzzles Bach was working out had more to do with the theory than the math and we can support this, the math aspect is more interesting conjecture than something we can demonstrate, at least from what I have read on the subject which tends to be filled with assumptions and generalizations.
Schoenberg and his ilk are more music influenced by math, set theory in music is not set theory in math. We don't really see math literal in music until the rise of computer music and they are often doing things more than just seeing what the math results in. The thing with much computer music is that it is deeply tied into how music works on a computer right down to the way sound is generated which requires us to throw away traditional ideas of orchestration and form to understand it as something other than just a musical representation of math. I lack the math to get much of this stuff but I find it very interesting and it certainly provides me with a good number of ideas.
I define math as math.
> The puzzles Bach was working out had more to do with the theory than the math and we can support this, the math aspect is more interesting conjecture than something we can demonstrate, at least from what I have read on the subject which tends to be filled with assumptions and generalizations.
This sentence structure is very much beyond me, but from what I gathered, you do not consider propositional logic as created by harmony/counterpoint to be "math" and you believe that those rules are arbitrary and not based on math at all. That is sort of a fringe view that is facially false. Around the world, rules of harmony have a lot to do with the harmonic series, scales, tunings, and other building blocks that are hugely based on math. A lot of that was actually up to the composers and producers of music in the 19th century and earlier (around the world).
> We don't really see math literal in music until the rise of computer music and they are often doing things more than just seeing what the math results in.
I see that you have defined "math as math" in terms of the totality of the math involved in the process of composition. Most 21st century composers and producers I know do absolutely no math (defined in any way) when writing music. I will also say that the producers of computer music absolutely do not throw away "traditional ideas of orchestration and form" unless they are lazy or bad. Many of them very clearly understand that stuff and riff on it in ways that can only be done when you combine a deep artistic instinct (0 math) with a computer (that does all the math for you).
We've got way more composers and variety today. But part of that variety is https://en.wikipedia.org/wiki/Math_rock
Written about by this guy (not peer reviewed, but has good presentation of the original Aristoxenus) https://siementerpstra.com/aristoxenus/
Music was part of the quadrivium as well, on par with astronomy and math.
I played in middle and high school concert band and took private lessons in addition to that for a few of those years.
I never learned a lick of theory from any of my instructors. Maybe because as a horn player you can't play chords. All I learned was to read notes and play my part. After a year of high school band I came to the conclusion that this was just work and gave me no enjoyment at all so I quit. Everyone seemed surprised.
Edit: Also it really doesn't take much effort or theory to understand most of what would be interesting. Like many subjects, the 80% of what's helpful can be written on just a few pages and learned in a few hours - but whole college curriculums can be made of the details and nuances. Go read up on 7th chords, cadences, and simple chord progressions if interested.
I don’t have an answer for why most music teachers don’t teach theory (and improv) other than it’s more challenging and they might not know it themselves, and it’s easier to teach rote memorization.
There are plenty of people who learn piano (which obviously doesn’t suffer from the single-note “limitation”) and also learn almost no theory.
Typically, piano is part of post-secondary music program requirements and keyboards make it practical to apply music theory to make sounds...arpeggiation will get your ear so far. Polyphony gives a much clearer picture of what music theory is describing.
If you're interested in learning music theory, a polyphonic keyboard that makes sounds without a computer (maybe one with speakers, but at least with a headphone jack) is my recommendation. Maybe it's better if it is small and portable. A MIDI controller with a phone or tablet might be ok, but phones and tablets have email and the internet and those can interrupt what you set out to do.
Anyway, theory usually comes later in the US system. There are exceptions of course.
Not just theory though. I've met people who continue to perform in orchestras decades after high school and college, who play with really good groups, and who have literally no interest whatsoever in the music they play, let alone its theory. It's always bizarre to me to talk excitedly to someone about the piece their orchestra is playing to only be met with confused indifference about it and the composer. For some people it's just an activity to keep busy and nothing more. This is not necessarily a bad thing, it's nice that there are many ways to engage with music and be able to choose how involved you want to be.
I know tritone subtitution from music theory, and trying to grok this explanation. What does the "roation by 180 degrees" represent in this case?
It seems that the article would be clearer if the 180 degree rotation were to put F# at the top of the drawing.
I’m sure he didn’t mean it literally, but to be pedantic a tritone is only moderately dissonant. A minor second is very dissonant.
How many people consider the blues dissonant?
It reminds me of my first counterpoint assignment. "Fourths aren't dissonant" I said, and it came back covered in red marks
To say blues doesnt sound dissonant sounds more like only adding the relative minor b5. Or British blues rock.
And other cultures like tritones just fine, featuring them even more prominently.
There's no inherent "tritones are dissonant" human universal, it's a common practice harmony prejudice.
But the chord function is only one third of the picture. Dominant chords can have greater or lesser tension depending on which extensions are used. A closed voicing will sound more dissonant than an open one.
I didn't mean to suggest that dissonance has any correlation with good/bad.
Avoiding dissonance entirely can also be unpleasant. e.g. barbershop harmony can sound saccharine because the human voice can hit those simple ratios much closer than 12TET can. 12TET gives us some dissonance for free, which people have learned to appreciate. Like spice tolerance, dissonance tolerance is learned not inate.
A minor second is very dissonant, you can hear pronounced beating between the tones: https://en.wikipedia.org/wiki/Beat_(acoustics). This makes it stick out more than a tritone, so it has to be balanced more carefully.
But in the context of the dominant 7th chord, the only note dissonant with the bass is the 7th. The tritone itself is not so prominent since it occurs between two non-bass notes.
“I see you shiver with antici——”
It’s not dissonance that makes you want to hear the completion of the sentence, rather, it’s grammar. If you play a tritone out of context, there is no need for it to resolve, but you can cast context around it by adding a resolution to a tonic just after. The tritone has a culturally-learned grammatical role in western music; and that role is not a universal law of aesthetics.
“——pation.”
Always love to see music theory content here. Nice find!
[0] thank you RWR (1981)
[1] (wrt AG) note that Alphane saints tend to come from the Heebs
not sure why you had to prefix.. :)
Thanks for the PKD tip!!
(EDIT: this is the only dried bark PKD in my possession, which suggests that while PKD admired the schizos he might have selfdiagnosed as OCD?)
On that note, among my favourite anekdoty, "nasha" Natasha Ilyinichna is at a dance. (the following should be shaggy-dog extended when told, but I'll spare you all) First she dances with Andrei Nikolayevich, but when she notices a fleck on his shirt cuff he apologises profusely and begs pardon to leave so he can change. Then she dances with Pierre Kirillovich, but when she notices a crumb on his cheek he apologises profusely and begs pardon to leave so he can wash up. Then she starts dancing with a Hussar, Rzhevsky, but when she notices a large splotch of mud on his boot...
— Oh baby, don't you worry your pretty little head. That's not mud, it's shit, so when it dries up it'll fall off all by itself.
Dropping the innuendos a bit, I'm truly sorry for the hamfisted way I go about doing it (what with the shit on my uniform from transporting manure)
All I can say for now is, I see why you'd think it's impt that I rewrite that bit of firmware (and that three-way with mrmann was a precious moment in time --- but it could still be a moment that can rewrite firmwares in the future..)
L:https://littlebritain.fandom.com/wiki/Mr_Mann
(How do 2 firmwares overcome a race condition (e.g.))