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What is the minimum amount of information necessary to describe all chords?
This might seem a strange question. It might not have an answer.
What is the minimum set of parameters that can describe a chord - all chords. I’m guessing perhaps a vector, or perhaps a curve? I’m also thinking it should avoid mention of discrete note numbers, instead focusing on a description of a way of getting to them, using the minimum set of info necessary.
Obviously a triad can be described as only three discrete note numbers, which isn’t a lot. But this doesn’t apply to all chords. Is there a single set or structure of information that can describe all chords (by varying the parameter values)?
Comments
For what purpose?
Might Forte numbers (set theory) help? They describe intervallic relations between all pitches in an aggregate or "pitch class set" (chord becomes irrelevant if you do not distinguish between dissonance and consonance). Two numbers describe very complex structures by using an array of possible combinations of pitches. https://en.wikipedia.org/wiki/Forte_number
I'll just say it. I'm totally confused. I gave it some thought and I can't seem to avoid mentioning numbers when describing intervals. The only other option I know is notes which vary depending on key etc. I'm probably totally missing the point though. Or is this just a brain bender type question for the sake of mental torture?
Yeah, I’d think the intervals between the notes.
Yes. Yes there is.
He’s right. There definitely is.
? I’m also thinking it should avoid mention of discrete note numbers, instead focusing on a description of a way of getting to them, using the minimum set of info necessary.
—-
This part I could use you expanding on. Maybe I’m being dense at the moment, but it seems “focusing on a description of a way to get to them” requires a set of discrete markers (even though each marker is always necessary to get where you wanna go). It’s like driving from avenue C to D#dim7 boulevard. You’re always starting from a “home location” as an orientation point. Then I need some landmarks or street names to help me find a way.
Not on one of my best explaining what I truly mean to say days, maybe I’ll say it better later after a good ol nap.
I'd also guess using math to describe ratios between notes might be most efficient.
The only unknown is the root note. Provide a root note value, and a sequence of ratios yield the frequency of the other notes for a given chord.
Assuming it is to replicate what a human can play on piano, you would need 10 structures for each chord (potentially 12 if you allow for each thumb to play 2 notes).
It could also get more complex when you give consideration to use of the sustain pedal.
For each structure you would need a Boolean value to show if it is played or not, and a value to show the pitch.
It’s probably easier to use discrete note numbers for each note that is played, as descriptive values for chords are very dependent on context (key, mode, modulation, etc.).
The Forte numbers thing is highly interesting, although that's probably a different direction.
I'm wondering if there's a way of describing any chord through, well, perhaps root note or perhaps not even that (treating it as simply another note that this description intersects). I still can't help thinking of it as a vector — either a straight line, or a gradient — or perhaps even a bezier, and this vector or curve intersects something else and produces a group of viable hit points, or a set of possible 'yes'es among all the other nearby 'no's. To be honest I'm not even properly picturing which direction things go, what the things are that go, and how it's arranged. But – intersections. Something to do with planar intersections, or curves cutting across lots of parallel lines. Or something. Don't ask for specifics, there aren't any yet.
And this curve is very compact in description, compared with the resultant chords it can describe declarative (rather than imperative)ly
What a great question… as we define terms we might converge on an answer.
To describe a 3 note chord typically we use the chord “quality”:
Major
Minor
Diminished
Then there are extensions:
+2nd
6th
7th
9th
11th/4th
13th
And + or - augmentations of the extensions.
In practice a subset of the possible combinations are used with jazz harmony using the largest set, IMHO.
People going beyond that are typically thinking in terms of clusters where intervalic possibilities are the way to get to an answer.
I can’t think of a useful way to map curves, which are mappings of a dependent variable to an output variable and are by definitition continuous allowing an infinity between 2 points.
The music of the oscillator guy does fit into the curve way of organizing sound structures since he bases his ideas on knobs = x and sound = y and then created pallets of curves.
I took a trip down the MIDI path of chord recognition and had some interesting discoveries. I used the modulo function in Mozaic and reduce all notes in a chord to a single octave (0 to 11). Then compared the notes present to a table for 3 notes (easy) and 4 notes.
Using that simplification I can get a chord to generate a description. But the other way opens the question to “voicing” and inversions.
Here’s a completely bullshit example, false in almost every way, but imagine if something a bit like this from a distance was viable, and where it sat across a grid, turning the intersections into true + falses, and produced usable chords with a very compact description
From your graphs, I know you are thinking on a level I cannot say I understand. For me, the simplest definition would using 2 or more notes - that's a chord. I chose "using" since you can arp a chord instead of playing it all at once.
But my main reason for responding is to share the time my mind's understanding of music theory was blown when my teacher said you didn't need to play the root note for the major chord, because it is implied by the context of the song. So the major chord didn't have to be 1 3 5; it could just be 3 5 (or maybe even 3). Classic example of the more you know, the more you realize that you don't know.
Pretty sure I'm not thinking on the same level as this thread, so carry on. I'll curiously watch.
I love the astonished chord. I just didn't see it coming.
Someone in a video helped me think of Major as "bright/outgoing" and Minor as "dark/introverted" but I'm wondering what a good astonished chord might be made of
note-wise.
You also helped me remember there's an "augmented" chord.
The 4 basic chords are all 4 combinations of Major and Minor 3rd stacks:
Major = M3 + m3
Minor = m3 + M3
Dim = m3 + m3
Aug = M3 + M 3
Looking at your images, I think you're putting pitch on the Y axis and maybe time
on the X axis? That makes me think you're thinking of describing "when" specific chords
are played... the "piano roll" is an example of this type of X-Y description of chord events.
I just don't know what to think of the curves, unless they are related to something like volume where X is volume and Y is time, again.
Of course, all information gets mapped onto my world view so I could be unable to "see"
what your hearing because my training gets in the way.
I'm not clear on what the question is, but in my mind this is a major chord:
in hz
Root freq x 1.259899097997248 = 3rd
3rd x 1.189227272727273 = 5th
If I was to graph that.. My intuition tells me that any drawn shape I use to intersect a number of points on incrementally marked Hz vector markers, wont scale linearly with a change in root pitch.
But if a non-linearly marked Hz vectors were used, then I think it might work.
But what's it for?
This is a fascinating subject. I don’t have anything to add, but all this talk about chord voicings and tonalities is just in time while I’m getting acquainted with this (new to me!) app I’ve just discovered.
‘Impro Master”
https://apps.apple.com/us/app/impro-master/id1552035106
Not to stray off topic, but I think this is going to be the perfect practice tool for my guitar improvising practice! I tried it once, and instantly purchased the full version! How did I not know about this amazing app?
There’s less regularity as you extend beyond 3 notes too, as you then get to the concept of half-diminished and fully -diminished.
Also the stacking goes awry with 7ths as instead of the expected 9 types of 7ths based on stacking all the combinations, you only get 8, as three Major 3rds have no 7th quality when stacked.
Then there are other alterations such as sharpening or flattening intervals, or not including notes when you play shell chords or rootless voicings.
There are also all the inversions, which might have the same notes, but have a different sound. Then there’s adding different root notes, which is not uncommon for guitarists.
We’re talking about a lot of chords when it comes down to it, and it’s not easy to figure out simple rules to build them.
And it's very simple to use arrays of note numbers to define chords, so I don't understand what the practical purpose of this inquiry is supposed to be.
Although, if you wanted a simple way to indicate all permutations of a single chord, you could start with a 3 note array, say, [60, 64, 67], and then say that any other 3 number arrays where the corresponding elements differ by a multiple of 12 are also C Major Triads. So by that rule, e.g., [72, 64, 67] is also C Major Triad. By that rule the array [12, 52, 79] is also a C Major Triad, so maybe you'd want to refine the rule; to me it sounds strange to call that set of notes a chord. (This is all using MIDI note numbers, of course.)
For Western, 12ET, just use an array of half-step intervals with the first entry being the number of intervals. I don’t know off-hand if I’d do intervals from the root or from the last note. So a major triad would look like 2, 4, 3 or 2, 4, 7. To generate a closed chord you just need a note number for the root and add the intervals for the rest of the notes. Inversions would be handled when building, not specifying the chord. First inversion would just add 12 to the first note. Second inversion add 12 to the first and second notes. A sus2 would be 2, 2, 5 and a 13th would be 6, 4, 3, 3, 4, 3, 4 (hopefully those intervals are right). Maybe use negative numbers to identify optional notes (like the 11th) or just allow alternative configurations. Some other voicing like Bb2 C3 D3 E3 F3 G3 A3 might be easy to request from the chord builder algorithm but beyond super dense you might only have the option of inversion and octave range and maybe a description of the distribution of the notes (group more at the bottom or top or equally dispersed) and let the algorithm build what it will.
I too am curious where this came from. Homework?
Showing these frequency relationships as ratios is also worth mentioning. The ratios
fallout of the harmonic series where an oscillating string or air column generates
overtones of some fundamental frequency.
1
2 (octave up)
3 (octave + 5th)
4 (2 octaves up)
5 (2 octaves and a major 3rd)
6 (2 octaves and a fifth)
The ratio 4:5 describes the Major 3rd. (1.25)
and 5:6 describes the Minor 3rd. (1.2)
The more detailed values provided by @horsetrainer are "tempered" values typically
used to "stretch a piano" to play reasonably well in all 12 keys. Chomatic music requires
using these approximations to make sense.
For anyone that followed this and wants to keep going down the "well tempered" rabbit hole, I'd watch this video... the explanation shows up at after the 2 minute mark.
I think you could get there with a 3 digit hex number. 12 semitones x maybe somewhere less than 100 variations of each. Let's say it's 100, there' 1200 individual chords. 3 hex digits is 4095 individual values. leaving a lot of room for things I haven't thought about.
In that system each number is a code for a chord... Catchy sounding concept but not very musically intuitive.
If you want precision though I guess you'd just have to take the full range of playable notes, let's say 88 for a piano range, and whether each note was on or off. Which would allow for some pretty crazy chords admittedly, like the chord that is every note on a piano. So my maths is rusty and I tried to look up what that might be and got lost in a discussion of permutations and combinations and couldn't work it out for the life of me.
I fear my contribution may not be the most productive.
Why not just the intervals, as mentioned? 1, 3, 5 is a major chord. That’s about as simple as it gets. If you start abstracting from that, you need to define those abstractions, which makes it much less human-readable.
An example. If you abstract “1, 3, 5” to “major chord,” then you’ll have to come up with a name for every possible chord.
Intervals seems simplest. You’ll need to know the root note, too.
Very interesting, but what’s the purpose of all this?
Actually, having not thought about it at all overnight, here’s a different diagram. Still bullshit though
I really think the distance between notes is the best to do this. Depending on what "chords" mean to you.
For traditional, tonal music, chords are considered a tertian harmony, built from intervals of third, usually a major or minor third, although there are some exceptions, like the augmented sixth which, if stacked in a tertian form, also contains a diminished third. The largest chord then will have 7 members.
If you think of chords as simply a collection of pitch-classes, as in a set theory, then, of course, this opens for more possibilities: An example of 012 is the collection G-G#-A; 013 = G-G#-Bb or G-A-Bb; 014 = G-Ab-B or G-A#-B, etc. And these are just trichords. You also have many larger pitch-class collections. Interestingly, abstract as this concept seems to be, it can describe complex music in a simple way, like, Stravinsky's neoclassic pieces favor 025, while Schoenberg's atonal works often emphasize 014.
So, I think it's the number of chord members and the distance between the members, from 1-6 semitones, that should be the best and most precise way to describe "chords". An interval vector, stating the number of occurrences of each interval in the chord, is useful but not as exclusive.
I'm bad at maths, though, so there's that...😅
That’s a good point. What they mean to me is sidebands and spread of harmonics.
In my more arrogant times I say that chords are for people who don’t know how to do additive synthesis. But, I do maintain that a lot more brain-time is spent on chords than there should be, that music theory makes it seem more complicated and gatekeepery than it should be, and that a chord can (and probably usually is, to the consuming public) perceived as a single one unit thing, and not a group of discrete notes.
If it is one thing, then what is it? It doesn’t have the single tonal frequency that a note does, but wait, what if my note is two VCOs tuned a 5th apart, is that a single note? It is if I hit one note and that’s the result.
What chords are, as far as I’m concerned, is a spread into the harmonic range. It is either a spread upwards in frequency from a lowest fundamental, or it might be a spread either way, above and also below, around a centre fundamental (which sounds like a conflict in terms). I think there’s a way of describing the flavour of a chord by describing what pattern of spacing those sidebands are away from some fundamental. Of course in current chord playing, you can’t really tell which is the fundamental and which are the harmonics because they’re all played about the same volume. But, I don’t think that’s too important.
This pitch-classes thing is increasingly becoming very interesting
Interesting take! I don't have anything to add that's useful, but there is an interesting anecdote that you might enjoy. Around the 8th/9th centuries, they sang Gregorian chants in monophonic style, sometimes doubling at an octave. Later, the Notre Dame composers wrote organum using parallel perfect 5th and 4th above and below the chants, while the third and sixth were considered more dissonant. The 17th-19th century composers used the third and sixth as consonances, the previous perfect intervals became old-fashioned, and the second, seventh, and tritone being dissonances. The 20th-century composers loved the second, seventh, and tritone, avoiding the triadic—third/sixth—sound because it's old-fashioned. So, musicians have been developing new music by reaching up to the higher overtones over millenniums :-) And we do have spectral music finally in the 70s!
It's indeed a rabbit hole. You've been warned.
Intervals may work, but you would also need to define the root in some way. For example, the intervals in your 1, 3, 5 above could also be the 3, 5, 7 in a rootless voicing.
Because this appears to be more of an exploration of music in terms of harmonic constructions.
My current personal view, is that human perception of music has more to do with how the ear and mind process what is heard, and less to do with our scientific ability to deconstruct music and explain it using maths.
After a lot of rethinking this past year about what makes good sounds sound good. I've concluded that evolution developed the human ear and mind to "Separate" all that the ears hear, to attempt to divide every sound up, and then mentally separate and identify each part from the "whole" of what is being heard.
This is exactly how a neolithic hunter-gatherer would need be able to process sound in order to survive. It would provide them the ability to walk through their natural environment and pick out sounds from predators and prey, and to separate the important information from the drone of all else that may be making noise at the same time.
I think when people hear music their minds search for ways to separate it into parts.
Chord changes provide the mind with information on how to separate the chord into its constituent parts (notes).
Music is what the mind hears after it has separated all the parts of the overall sound, and is then keeping track of each part as an individual sound within the whole.
Chords may share harmonic structure with notes that contain the same harmonics. But I think the mind will understand chords as separate notes by remembering having listened to music in the past, that used notes that changed to build different chords .
Once the mind remembers how to hear chords, it hears chords when ever anything sounds familiarly like a chord.
The mind may tend to hear unchanging sounds as a single unit, unless and until the sound exhibits some characteristic that allows the mind to divide it into separate sounds.
In the case of pads, I doubt the consuming public break down a warm thick slabby layer of a pad droning away as what it contains, but rather, what it encapsulates. In fact, not even that, probably what it means semantically, but that’s hard to tell.
Wait, that all sounds like waffle. I don’t even know what I meant.
A thick warm pad bed is a single thing, not a ‘chord’ made of notes at intervals (although we know it is, but only because we’ve been indoctrinated with how music is made). A washing machine has a complex sound when it spins up partway through the cycle and then down again later. Nobody analyses that, it’s one thing. You’d recognise it as your washing machine and not the one you used to have or someone else’s.
A pad (which is a chord) is not going to sound overly different from a one note pad on a more complex synth (eg additive) that has all the same partials as the chord. The difference is that it won’t be easy to hop from one ‘chord’ equivalent to another, as easily as you can jump from one single note to another on this complex chord-making oscillator. It’d be set to one chord until you change it somehow in the settings.
What if there was a parameter that allowed changing of that arrangement of partials? It’s unlikely to be a single scalar value, but it might turn out to be describable as a more complex vector or arc or spline or something, that, well, ‘rearranges’ the distances between the partials and produces a different ‘chord’ equivalent.