How Chordcat works – a chord naming algorithm
blog.s20n.dev
blog.s20n.dev
My hunch is that any algorithm that analyzes chords in isolation (and not in the context of other chords in the song) is going to miss out on overall readability. Good chord notation has a sort of elegance that can highlight the similarities/differences between sequential chords that can make things very intuitive for the musician
In “An Introduction to Musical Analysis”[2], Nicholas Cook says something that I have always considered very profound and applies widely, not just in music, which is
> All notation is analysis
So when you write Cma7 for a jazzer, that’s an instruction that you are generally in the major tonal area and they may well play a Cma9 or a Cma7#11 for example. This is why my functional harmony teacher used to get annoyed by people writing the “Crosstown Traffic”/“Purple Haze” Hendrix chord as E7#9 (which you see a lot). He would say it is E7b10 because that g natural is coming from the minor modes so it’s actually the fourth degree which has been flattened. If you call it a #9 you’re telling players the wrong scale for improvisation (pretty much everyone else in the whole world calls it E7#9 though).
Likewise going back to your point about slash chords, a lot of people learning this stuff get hung up on “what is this slash chord really?” (Eg if I’ve got B/C or whatever what actually is that) whereas when I’ve talked to really serious musicians in that world who play that kind of intense modal music they really are thinking about what the slash chords are in terms of where they come from and what they are leading to. Because that gives a sense for what the underlying tonality is. You can’t get it just from the notes of the chord vertically in that one instant.
If you look at baroque music and earlier, if you just do a vertical chord analysis (eg something like Gesualdo if you really want an extreme example) the chords make absolutely no sense in many cases but that’s because they work in terms of voice leading (ie context) rather than vertical relationships. Analysts used to call that feature “vertical false relations” because they found the vertical chord analysis troubling.
[1] ex-professional musician with a degree and postgrad in jazz, contemporary and popular music here. Wife is a professional musician who teaches at a couple of conservatoires and mostly plays baroque and early music with some 20th and 21st C music thrown in for good measure. Lots of pro musician friends. Not trying to argue purely from authority but I do talk music a lot with people who know a lot, and this topic comes up a lot.
[2] Which stands out as a great book in a field with lots of terrible books btw.
you mean the third degree I think?
at first I thought your teacher seems overly pedantic here, but it does kinda make sense... I'm surprised I've never heard anyone argue that before!
Also as a sibling to your comment points out it's a blend of major and minor, so also in jazz you'll often see something like E7b9#9b13 written, but that really makes no sense in terms of a scale. The 9th is either flattened or sharpened, can't be both. So the next degree of the scale is the b10, which also makes sense when you think about something like a "diminished" scale which people play over this chord often.
He also used not to like the name diminished for the scale btw, but would refer to it as one of Messiaen's modes of limited transposition. I can't remember Messiaen's nomenclature exactly but the scales which people call the wholetone/semitone diminished and semitone/wholetone diminished are one of his symmetrical system of scales. IIRC there are 3 distinct "diminished" scales and all the other ones are just modes of these three. So you can see that the st/wt diminished starting on D is just a mode of the wt/st diminished starting on C for example.[1]
[1] This is him https://williamkinghorn.org/Biography he was an awesome guy and I learned a lot from him
Ultimately, I think it would depend on whether Hendrix included F or F# in the scales that he used to solo over The Hendrix Chord. If he includes F, E7#9 would be the better notation, since it suggests an altered mode: I, IIdim, IIImi, IVdim(and probably Vdim). (E, F, G, Ab, Bb, C D) . If he includes F#, then you could make a case for E7b10, which I would read as implying a blues scale, although I've never ever seen that notation: I, II, IIImi, IIImaj, IV, V (E, F#, G, G# A B C D). My bet is that Hendrix plays F, not F#, given that he was a jazz musician before he was a rock musician.
The argument for Em7b11 would be that the major third is actually the FOURTH note of an altered scale. . Better still: Em7b9b11. Or I suppose you could be uncontroversial and just write Ealt.
Chords derived from altered scales, and chords based on quartal stacks are both severely broken cases in modern chord notation. Neither seems to have settled chord notations, and the chord notations that do exists vary greatly by dialect.
(Jazz guitarist who very much prefers modern functional chord notation, that implies both scale and chord, to the older Broadway jazz notation style).
But you know, obviously do whatever makes sense to you. As Schonberg says in Harmonielehre, "There are no rules of music, only rules of style".
I'm not at all into music theory and such, but your passage just reminded me of what AlphaPhoenix mentioned in his latest video[1], where he analyzed two frames from an experiment, both with a piece of string stretched horizontally across the screen.
They look identical, but they are not. One is a stationary string and the other in an oscillating string which just happens to be flat at that moment in time, but the pieces of the string still carry momentum.
At least that's how I interpreted him :-)
I mean, B3 D4 G4 in isolation probably does sound like G/B to western ears, and maybe the overtone explanation is acoustically relevant, but it seems like there are not too many other things that those notes could be, and the spelling clearly spells a Gmaj triad.
I'm guessing you mean Emin7b10.
It's in interesting take, but a bit weird considering you wouldn't stack the chord that way. You want to voice the G# below the G, E G# D G, making a major 7 interval which sounds good. If you instead voice it E G D G#, then the G to G# will form a minor 9th which is pretty clashy, generally avoided, and doesn't sound like the Hendrix chord anymore. Try it out, tab (0 7 6 7 8 x) vs (0 7 5 7 9 x). This is a standard jazz voicing rule of thumb, although I forget the name of it.
You can at least get much further by naming a key (what a key is, is not etched in stone either, but there are some useful enough definitions for most cases). That would be a bare minimum for a useful algorithm though. Especially if your input is just a set of notes.
Roughly, what if we consider x_i the sequence of latent chords, and y_i the observed notes. If we reframe the accidental-based sorting this uses to act like a likelihood function e.g. P(y_i | x_i) ~= 1/exp(# accidentals), then we "only" need a transition function P(x_i | x_i-1) which describes what chords typically follow which preceding chords, and then we have an HMM system for which we can do inference easily. You can also imagine extending the order (number of preceding chords regarded as context), or training some higher order latent state z which might carry information about mood or genre and what chord sequences are appropriate in context e.g. P(y) = P(z_0) \Product P(z_i | x_i-1, z_i-1) P(x_i|z_i) P(y_i|x_i)
Without other context I would say that you can probably label G/C as Cmaj9 or even Cadd9, even though the E was not in the original label, but to be sure you do need to check whether the E is indeed okay instead of or in addition to the B.
⸻
1. This chord and its relatives in other keys is my favorite chord.
- Chords are definitely not always built on thirds. Mainly the traditional western sounds are.
- Even more importantly, chords are never named in isolation, they are named with respect to their musical context. An Am7 could more correctly be called a C/A depending on the overall harmonic picture - the melody, the chords before and after and even the emphasis created by the performing musicians. Voicings are basically ignored but could be said to come under this point.
When I teach harmony to students I always point out the key thing to understand is how the harmony is changing, and to look at it dynamically not statically. It's more about the relative motion of notes comprising chords than names although names are obviously a useful construct.
Except for the ones built on seconds ("cluster chords"), fourths ("quartal chords"), fifths ("quintal chords"), the chords that make jazz work, and for obsessive/completists, polychords.
Although:
a) I have only given this a moment's thought and...
b) The term "fiction" here is relative as all music theory is a fiction to some degree.
c) "Ears" here is short hand for a weird mixture of maths, physics, brain structure and learned cultural expectations.
I'm kinda clutching at a valid idea here though - despite the above flakiness.
I don’t really see a basis for this one. I think if we really dig into it, the use of thirds is somewhat arbitrary and has more of a historical basis to it than any other foundations (rather than physical or physiological foundations).
Thirds really do fall apart when you start looking at quartal harmony, tone clusters, and all sorts of modern harmony. Thirds even get stretched to the limit in a lot of very traditional, “safe” harmony. I think on some level, we are just using thirds because of the mathematical coincidences in our systems of harmony.
Cultural explanations are the biggest ones.
Fundamental - octave - fifth - octave - third - fifth - flat 7
The minor 7 is not in the harmonic series. It’s just not there. This is not a nitpick or a quibble—the minor 7 is just a completely different interval, separate from the just intonation 7:4 interval. To my ears, it is like saying that red and yellow are the same color.
What about augmented or diminished chords? How do they fit into this theory? What about, say, the 9? Why do we put a 9 in a chord, and not a 2? Why do we like the b9? Why do we use a #11?
There’s a million little historical and cultural factors going on here. You can come up with a sort of apologia to explain why thirds make sense, but this doesn’t mean that thirds are in any sense fundamental in some physical or physiological sense.
You can discount the fundamentalness of thirds, but it’s there.
https://www.youtube.com/watch?v=CrmnaiyS5EE
This thread reminded me of the "Conclusions" slide.
As you get further along the series you're moving from simple to complex ratios. Thirds are next in line after 4ths/5ths and they have a function that solves some obvious harmonic problems.
A lot of this has echoes in the historical development of music and the commonalities between harmony in different parts of the world. There's a reason cultures independently discover similar constructs i.e. the pentatonic scale
It’s not a leap to go from the fifth (as an interval, two notes) to the major triad, three notes. However, if you consider major triads, you’re just dipping your toe in the water as far as chords are considered.
The question is not, “Why does the major third above the root appear in chords?” but rather “Why do we build chords out of stacked thirds, which can be minor, major, or even diminished or augmented?” At this point, the part about the major third appearing in the harmonic series lacks explicatory power.
Consider, for example, that a chord “built out of thirds” can contain a major or minor seventh, a minor second, or an augmented fourth. Just as examples.
I really don't think it does. Fifths (and their inverted form fourths) are the most consonant interval (ignoring octaves as the trivial case).
Most cultures develop musical processes that involve fifths very early on in their history.
Where do you go from fifths if you want to add another voice? If you have three notes and you want the most consonant result - you put a note that forms a third between the root and the fifth.
Given a few basic assumptions ("prefer simpler integer ratios") there's no better way to play 3 non-octave notes together.
Sure, if you look at major triads only, and ignore extended harmony, it works fine.
But it’s also not that useful if you look only at triads, because there just aren’t that many triads.
According to modern harmonic rules those are valid. For a long time, these were not harmonies you'd hear in music. Really only since counterpoint fell out of style, since it forbids augmented intervals.
I think you're spot on with learned cultural expectation though. As a musician, you have to learn how to hear quartal harmony and polychord harmony. The sounds and the logic of how things fit together are different from triadic harmonic theory. As an audience, though, I think you're probably so used to quartal harmony and (to a slightly lesser extent) polychord harmony that I don't think you'd find it at all unusual to hear a chord without a third in it.
That particular chunk of harmonic theory was added near the end of the 19th century by composers like Debussy, and subsequently became completely embedded in hollywood movie music, so you are completely culturally used to it. You probably wouldn't even notice that what you're listening to has no 3rd. 19th century audiences weren't so comfortable about it.
The only real problem is that classical music theory gets quite broken when it comes to quartal and polychordal music. No accident, because compositional techniques involving quartal and polychordal music were deliberately designed to get away from classical music theory.
Thirds appear to me to be pretty much the fundamental particle of western harmony.
Ok, gotta remove multiples of factors of twelve, so 2 (whole), 3 (dim), 4 (aug), 6 (tritone), 8 and 9 are out.
That leaves 1, 5, 7, 10, and 11, plus all the variants from adding n*12 (e.g. 13 or 25). So my bold assertion didn’t work out too well.
For example, there are only three distinct diminished “loops”. These each define tonal centers because each pair of tritones uniquely identifies a 5 chord. You can view a fully diminished chord as a “gateway” of sorts into 4 different keys, depending on what root note you slap on it.
> For example, the notes C, E and G (The C major chord) can have any of the following names, based on what is considered to be the root note.
> root note chord name
> C C Major
> E E Minor #5
> G G sus4 (13)
> (To the best of my knowledge, I'm no music theorist)
is disqualifying for any usefulness in this program-- it will lead to more misguidance than clarity. without theory, there is no reason.
i would analyze these inversion chords in the frame of the key of C major: CM, C6, and C64 based on Figured Base Notation https://en.wikipedia.org/wiki/Inversion_(music)#Figured_bass
Coming from a cover band/local small-time musician, C6 means C-E-G-A. In other words, C with an added 6 on top.
The charts I play from generally don't have any inversion information on them, and I probably play different inversions each time.
In jazz/pop notation inversions are indicated with slash chords.
The key problem with this implementation in addition to not simply taking the root note as input (or a key or scale), appears to be the chord database is pretty spartan, since it doesn't appear it would identify C-E-G as anything other than CMaj. If you want figured bass/slash notation then you'll need to specify that as well (bass note).
A number of chord namers take these as additional inputs.
Another example to illustrate how identifying the root is important: C E G A. It's either a C6 or an amin7.
I think the point that root and bass should not be conflated is also valid.
Inverted chords have, in my ears, never sounded anything like the chords they are supposedly an inversion of. It's like saying "10 is just really 3+7". It's true, and it's also useless. Let 10 be 10. Let 3 be 3. Etc.
Most composers don't even use this system as they compose, it's just for after the fact analysis. Composers hear a tune in their head and then transcribe it. Unless they're writing cookie cutter shit without inspiration, then they might synthesize some soulless crap that intentionally follows some well known chord progression as picked by a dart thrown at a poster of the circle of fifths.
Ok. Then what do you call it?
Known in western classical harmonization theory as "Em+" (E chord with minor third, augmented 5th). C first inv is actually Gsus4add6no5 or as I like to call it G+5+4. It has an incredibly warm and rich tone, nothing like C+4+3 (C major triad).
It works in C because all of its notes are also present in the C major scale, but that's just a coincidence. Calling it a C inverted chord is like making astrological or numerological conclusions.
And no, shifting up or down by octaves is not the same sound. It's close but not equal. You can probably get away with octave substitutions in a busy song but you can not say it sounds exactly the same. A 9th is not a 2nd. 11 not 4, 13 not 6. When you play La Campanella, is that your excuse for skipping those octave jumps? :)
As I said, unpopular opinion :)
If you are already ignoring the actual chord "voicing" as written in the "classical" system, then just keep doing that in my system??
If you are already able to intuit that a note should go elsewhere, then keep doing that for your jazz sets? What's the issue again?
"Gsus4x5" is just as unreadable and ignorable and malleable as "G+5+4".
I know this is considered off-topic, but I’ve refrained a thousand times and now I can’t help it…
Please stop abusing commas! This sentence deserves a semicolon or a dash or a hyphen or a couple of full stops/periods or… anything else! With a comma it’s actually harder to read and is semantically indistinguishable from the sentence ‘I’m no music theorist, to the best of my knowledge’, which is obviously not what is meant.
</unpopular rant>
Em-dashes are typically used for separating clauses. Hyphens and en-dashes are used for hyphenating terms and indicating ranges (among a couple of other uses).
Not that this really matters in practice, but the visual cue is useful for reading rapidly and accurately
My point was that anything other than a comma would be preferable.
Language is extremely fluid and is defined by usage and changing over time. It is not defined by style guides or dictionaries or what we think we learned in school. Personally I think it has become far more fun to be a language pedant once I’ve embraced that fact that there’s rarely ever such thing as wrong usage, and started to look for the interesting uses in the past and the in present to think about how things have changed before and how they might change in the future.
Other examples of incorrect rants include how to use the semicolon (lots of ways), whether to use two spaces or one between sentences (either, who cares), whether using literally figuratively is wrong (it’s not), whether using the word really as an emphatic rhetorical question is acceptable (it is), whether the word myriad can be followed by the word of (it can), etc., etc., etc..
Come join the side of enjoying how weird language is and how many different flavors it comes in, and how as long as someone gets their point across, nothing else matters, though some people make really beautiful language and also communicate well while breaking all the “rules”.
> as long as someone gets their point across, nothing else matters
This is just classic. It’s the standard boilerplate response to any criticism of bad grammar (which this is, I’m afraid). I’m not foolish enough to believe that there are God-given rules for language, but when the non-conformity adds nothing and simply makes the text more difficult to read it’s worth pointing out.
I mean, seriously, why is a comma better than any of the alternatives here? It isn’t.
> why is a comma better than any of the alternative here? It isn’t.
Your complaint boils down to the writing has other alternative choices that you would prefer. That’s fine. It’s pure opinion, but you are entitled to your opinion. I’m not arguing the sentence in question is the best way to write it, I’m pushing back on your incorrect claim that it’s wrong. It wasn’t wrong, it wasn’t non-conforming, the usage of comma in the article matches what sources say about comma. Maybe before criticizing or arguing further, take a moment to review what a few sources say about the comma and it’s legitimate uses. [1] For the comma to be wrong, it must break all of the acceptable uses, not just any of them. IMO it’s also worth reading about why language prescriptivism itself is considered bad by many people. [2] Reasons include prescritivism being overly dogmatic, classist, racist, regional preference, and doesn’t adapt with changing language.
It’s fair for you to say it could be better written. It’s not fair to try to justify your opinion by claiming valid language use is breaking language rules.
And my actual point was the part about leaning into descriptivism and away from language policing other people. I have done both, and I’m just saying you’ll probably be happier — I am — if you realize that following prescribed language rules aren’t what makes writing good or bad, and that legitimate usage is broader than you imagine. It’s also helpful when discussing language clarity to take the viewpoint that some of the burden of successful communication is on the receiver, and that when the reader or listener has an open mind and assumes there are multiple interpretations, it becomes easier to choose a good faith interpretation that makes the most sense in context.
[1] https://en.wikipedia.org/wiki/Comma
[2] https://en.wikipedia.org/wiki/Linguistic_prescription#Critic...
Exactly! So I’m not even sure why you commented.
> dogmatic, classist, racist
Oh dear…
Why did you? Since you asked, I shared my opinion, same as you shared yours. You invited the sharing of opinions. Don’t criticize others if you can’t handle being criticized.
If you agree your comment was pure opinion, that implies you knew all along that the article was not “abusing” the comma per se.
I don’t know what ‘oh dear’ is supposed to mean, but did you read either Wikipedia article? The first one supported the article’s use of comma, and the second one explains my brief summary you quoted. It’s not my opinion, but that of many language experts that policing other people’s use of language necessarily imposes class, race, and region preferences on others. You should probably be aware of that fact when you choose to share your opinion on other people’s writing, especially when your opinion doesn’t seem to be backed up by the encyclopedia article on commas.
it's perfectly readable to me, I just read any comma as a short pause. When saying that out loud, you would make a short pause. It's read the same whether there's an ellipsis, semicolon, hyphen, or whatever.
Also, no, it isn't semantically indistinguishable from that latter sentence, what? You literally know this, you said that that's "obviously not what is meant". The order of the sentences, and the pause there, is what conveys semantic information, not whether that pause is a comma or not.
I don’t know what you mean about a low bar. Is that a bad thing? What do you want to have happen, either here or with commas in general?
Merriam Webster like descriptivism: https://www.merriam-webster.com/grammar/descriptive-vs-presc...
The more interesting (to me) question is whether spoken English and written English share the same conventions. I think it’s abundantly clear that in most languages, they are divergent, to the point that some spoken languages have multiple written forms, each with their own ruleset for handling topics like formality or gender.
So to me, saying “gonna” all day long doesn’t automatically transfer to writing, where most people write “going to.” Same with commas—the written conventions are not, as far as I’m concerned, about mimicking speech pauses. If that rule works for you, great. It doesn’t for me, and I fall back to patterns that are drawn from reading the written word.
For low bar, I didn’t mean to sound like I was taking a dig. What I meant was that a written sentence which isn’t comprehensible isn’t functional. Once you get past functional, there’s a world of expression and nuance that come from different writing styles.
I don’t think anyone would disagree with that. But what bothers me (enough to eventually write a comment) is not that others’ conventions differ — it’s that they don't have conventions at all. Words are just strewn all over the page with abandon and anyone who questions it is told to piss off, usually via a long counter-rant laced with moral superiority and mentions of the standard buzzword ‘prescriptivism’. I suspect it’s because it’s deeply insulting and humiliating to be told you’re ignorant about something so seemingly basic, and almost everyone is ignorant when it comes to effective punctuation.
> If that rule works for you, great.
You give people too much credit. It’s not that they’ve tried several styles and eventually landed on this one as a matter of personal taste. They’ve never even noticed the difference.
Again: that there are ‘no formal rules’ for language use does not imply there’s zero value in caring how language is used and trying to keep it consistent and logical.
So why do you think we have all of those different punctuation marks instead of just one?
I understood what was meant by reading carefully and guessing. If you punctuate more ‘correctly’ (and I’m aware that’s a controversial and triggering suggestion for some), readers don’t have to be careful or guess what you mean — it just flows.
You may think this stuff is surprising because you’ve never noticed it before, but if you look you’ll find that it matters to editors everywhere. If this were to be published, it’d be corrected immediately.
What we're doing right now is essentially a hard-coded database of chord names. Every single new chords requires a new entry into this database, which currently makes adding chords with many accidentals impractical. Your approach of tracking the accidentals separately to the database of chords seems very useful and it may allow me to add 'jazz chords' into the app!
(If only I could find the code)
My chord naming algorithm works somewhat like the one described here but is based on an intentionally more limited set of chords which I've carefully curated here:
https://github.com/seancolsen/music-theory-data/blob/master/...
But it can help you understand how most chords are formed.
https://github.com/ericfortis/web-projects/blob/main/chord-m...
Chord naming is not particularly useful as simply a function mapping a set of undifferentiated notes to a name. The post interestingly gets that the root note is important, but then sort of misses the point in the end. You either have to specify the root, or if the procedure is generating all combinations then one needs to know the implied root in the output (and then this of course leaves the hard bit of identifying what is the correct root to elsewhere). As an example, an input of {A, C, E, G} will spit out C6 and Am7. Which one it is, is the important part. And these are both relatively common chords.
Looking at prior art you will see that chord namers either take something like a key to infer the root, they take the root explicitly, or they treat the notes as an ordered set, some then improperly assume the first element is the root (conflating root and bass). In a GUI that is super misleading.
https://muted.io/chord-identifier/
eg: If you put in E-G-C it identifies as Em#5. Which is a completely false sense of "oneness" and would usually be incorrect. In practice E-G-C played on the keyboard in pop music is more likely a first inversion Cmaj chord, or C/E.
There are better tools out there (https://www.scales-chords.com/), but the example above would be how not to do it. This is not to dunk on it (or the post), but to motivate thinking more about the problem.
In this case the posted algorithm won't identify Em#5, simply because the database doesn't have an those entries. If you scale the algorithm by augmenting the database with more spellings then the output set is going to become larger without being more useful. It will be technically correct, yes that C-E-G might be a Csus4♭4, a C, a Em#5, but what do I do with that.
Good chord symbols indicate not only function, but also the composer's intention.
Debussy probably needs different chord symbols from a 1930s Broadway tune, which needs different chord symbols from 1960s jazz leadsheet, which needs different chord symbols for a 21st century jazz composition inspired by Dave Leibman's See "A Chromatic Approach to Jazz Harmony", which is where chord symbols go do die. Seriously. Liebman's chord notation style for quartal chords is influential, if not authoritative.
But I think it should be noted from the context of that comment (and the original article) that the names are necessarily ambiguous. So the names E-G-C (or whatever set you choose, how about F#, A#, C#) are pitch positions and don't disambiguate between enharmonic equivalents, eg in isolation we have no reason to prefer F# over Gb. Which is the problem.
Gb-Bb-Db can be obtained in the key of Db: Bb, Eb, Ab, Db, Gb. It is that key's subdominant.
F#-A#-C# is in the key of B: F#, C#, G#, D#, A#, and is that key's dominant.
Five sharps or five flats. Same distance from C major.
but why, oh why, does 12 but not 12*1 nor 6+6, not even 5+7 but precisely as 2+2+1+2+2+2+1=12 is so important for music !?
how can we make a general object that describes such a "folded" structure?
this strtucture arises naturally from repeating 3^1 all the way to 3^12 and inserting the appropriate 2^n (where n is negative integer) to keep all between 1 and 2 (or 2 and 4)
is 2212221 a group?
if we stop at 3^7 instead of going all the way to 12 we have a diatonic major, if we stop after 5 iterations of multiplying 3*3 it's a pentatonic.
https://sethares.engr.wisc.edu/consemi.html
Note how figure 3 shows dissonance minima close to degrees 3, 4, 5, and 6 of the major scale when played against degree 1. Although there are many other possible combinations of notes, combinations with degree 1 are very important because degree 1 provides a kind of "home" that the music often returns to (the "tonic"). But the goal isn't just to maximize consonance, so these notes alone would make for a boring scale. For better musical value, we need some more dissonant intervals to build tension before it can be resolved with the consonant intervals. The obvious place to put them is in the big gaps. Putting degree 2 a single semitone above degree 1 is probably overdoing the dissonance a bit, but putting it two semitones higher gives a useful mild dissonance. Considering only combinations with degree 1, degree 7 could theoretically also go a single semitone above degree 6, which gives the mixolydian scale, but the major scale is more popular. I assume this is because of combinations that don't involve degree 1.
he answers why it works well, but I don't think he really addresses why the "folding" pattern (as I called it) arises
I'm also looking at Erv Wilson's work (in the book by Nakamura) and he gets closer to addressing what I'm focusing on.
Indian music theory identifies an octave 7 notes, which are modified further to make 12. Beyond these 12 some of them have finer divisions that map onto an underlying 22 tone system.
This diagram summarizes it:
https://22shruti.com/wp-content/uploads/2023/03/revised-evol...
Notice something peculiar. The P note which corresponds to perfect fifth in western theory, is the same through those layers. It does not split into multiple fine or grain pitches.
Those pitches that we would identify as flat fifths are related to M, not P. And in fact, jazz theorists take a similar view: that in many contexts flat fifths are actually augmented fourths, related to the Lydian.
To make a 7 note scale, you would need an M. In the 12 tone layer, there are two flavors of it: M and m. Something analogous to your perfect fourth and augmented fourth. Then each of these has one of two possible representations in the 22-tone layer. The P is always just P.