The "Science" of Chord Construction?

[zedrein]

Lately I have been pouring over various "chord theory" lessons fruitlessly searching for any secrets or theories that might be hidden in their pages regarding the principles behind "nice/stable/pleasing/good" sounding chords. Ultimately, I acknowledge music is an art and there is always going to be some conjecture about what sounds "nice", but also consider that this sort of logic could be applied to melody even though musicians tend to use the same scales that have been repeated throughout history that are functionally the foundations of said melodies. For clarification: when I inquire about the construction of a chord I am referring to where the voices are placed next to each other; how the various voices jive with one another in an octave and beyond. Sometimes when I am hunting and pecking around the neck of my Taylor I'll stumble upon a real gem of a harmony but I can't quite justify into any kind of logic why it sounds so pleasing (besides of course that typically these sorts of chords tend to -but not always- be spaced in 3rds). Surely there exists a sort of devoted study to such things...can you all direct a brother to such a resource?

[walternewton]

There are some mathematical frequency relationships between pitches and overtones/harmonics that probably factor into why intervals such as octaves, fifths, thirds etc. sound, well, "harmonious" together...overall though I think what sounds "nice/stable/pleasing/good" is a much more matter of culture rather than science.

[JumpingJack]

You might want to research The Harmonic Series.

[Malcolm]

Keep this as reference material. http://www.smithfowler.org/music/Chord_Formulas.htm In the following charts R = root or the 1 note of the scale/chord.

Major Scale Box Pattern

E|---7---|--R(8)-|-------|---2---| 1st string
B|-------|---5---|-------|---6---|
G|---2---|-------|---3---|---4---| 
D|---6---|-------|---7---|--R(8)-|Next octave up
A|---3---|---4---|-------|---5---|
E|-------|---R---|-------|---2---| 6th string

Basic Chords
• Major Triad = R-3-5
• Minor Triad = R-b3-5
• Diminished Chord = R-b3-b5

7th Chords
• Maj7 = R-3-5-7
• Minor 7 = R-b3-5-b7
• Dominant 7 = R-3-5-b7
• ½ diminished = R-b3-b5-b7
• Full diminished = R-b3-b5-bb7

Scales
• Major Scale = R-2-3-4-5-6-7 Home base
• Major Pentatonic = R-2-3-5-6 Leave out the 4 & 7
• Natural Minor Scale = R-2-b3-4-5-b6-b7 Major scale with the 3, 6 & 7 flatted.
• Minor Pentatonic = R-b3-4-5-b7 Leave out the 2 & 6.
• Blues = R-b3-4-b5-5-b7 Minor pentatonic with the blue note b5 added.
• Harmonic Minor Scale = R-2-b3-4-5-b6-7 Natural minor with a natural 7.
• Melodic Minor Scale = R-2-b3-4-5-6-7 Major scale with a b3.
Let the major scale be your home base then change a few notes and you have something different. No need to memorize a zillion patterns. Let the major scale pattern be your go to pattern - then adapt/adjust from there.



I think your first thing to work on should be what chords get in each scale/key. If you learn how to stack 3rds you will understand chords and chord progressions better. It may be Jell-O right now, if you need help with stacking 3rds just ask. Here is the A scale stacked. Why did I pick the A scale? Wanted some sharps in the mix so you could see how they effect the chords.

Purpose of this paper is to call attention to the skip-a-note method
 of harmonizing a scale.   Take the scale then by skipping a note you build
 the chords for that scale.  Skip a note is an easy way to identify the notes
 within the chords of that scale.  From that you can then using the chord’s 
spelling to name the chord, i.e. 3 = major, b3 = minor, 7 = maj7,
 b7 = dominant seventh or minor seventh, m7b5 = ½ diminished, b5 bb7 = 
a full Diminished, etc.  From there you can identify that specific chord’s 
function within the key, i.e. I-IV-V, etc.  This chart can be used as a study 
of how chords are formed. 
A Major Scale
Note	Chord		Spelling	     Chord Name		Function
A	A-C#-E-G#	R-3-5-7	     Amaj7		I	Tonic		
B	B-D-F#-A	         R-b3-5-b7     Bm7	                   ii
C#	C#-E-G#-B	R-b3-5-b7    C#m7			iii
D	D-F#-A-C#	R-3-5-7	     Dmaj7		IV     Sub dominant
E	E-G#-B-D	         R-3-5-b7     E7			V      Dominant
F#	F#-A-C#-E	R-b3-5-b7    F#m7			iv
G#	G#-B-D-F#	R-b3-b5-b7  G#m7b5		vii     ½ diminished

Question -- Why is the B chord minor?  Were did that b3 come from?  Well 
the B major scale has a D# and an A# in it's scale so when you ended up with 
B-D-F#-A for the chord tones you flatted the 3rd and the 7th - and a 
flatted 3rd and 7th gives a Bm7 chord -- Drum Roll!!  Get a clean sheet of paper and
stack 3rds for the C scale.  Use the above as your Rosetta stone. Once you've 
stacked one scale you can stack any scale.

Good luck. Ask specific questions. Lot's of help here.

[Malcolm]

Lately I have been pouring over various "chord theory" lessons fruitlessly searching for any secrets or theories that might be hidden in their pages regarding the principles behind "nice/stable/pleasing/good" sounding chords. Ultimately, I acknowledge music is an art and there is always going to be some conjecture about what sounds "nice", but also consider that this sort of logic could be applied to melody even though musicians tend to use the same scales that have been repeated throughout history that are functionally the foundations of said melodies. For clarification: when I inquire about the construction of a chord I am referring to where the voices are placed next to each other; how the various voices jive with one another in an octave and beyond. Sometimes when I am hunting and pecking around the neck of my Taylor I'll stumble upon a real gem of a harmony but I can't quite justify into any kind of logic why it sounds so pleasing (besides of course that typically these sorts of chords tend to -but not always- be spaced in 3rds). Surely there exists a sort of devoted study to such things...can you all direct a brother to such a resource?

Most of the following comes from http://www.musictheory.net/ I recommend you bookmark that site.

Couple of things.
Certain chords like to move to certain other chords. The following is a short study about chord progressions.

I is a major chord. The tonic chord. It is the tonal center of the key, the name of the key the song is played in. The chord that the song wants to end on. When you use the I chord you resolve all tension you have built (tension is good) so -- are you ready to return to rest? If so I's the chord to use.

ii is the minor super tonic. In a major key the ii is a sub-dominant chord that wants to move to a dominant chord. The ii-V7-I is the go to progression in jazz.

iii is also a minor chord it likes to act as a lead somewhere chord. It normally drags the vi chord with it on the move.

IV is a major chord. It is also a sub-dominant chord and wants to move to a dominant chord. It and the ii having the same task in life can substitute for each other.

V is a major chord. It is also the dominant chord. When you add a b7 note to it as an extension it increase the tension in the chord and it becomes the dominant seventh chord, i.e. the climax chord, i.e. it wants to move to the tonic I chord right now.

vi is a minor chord. It wants to move to a sub-dominant chord.

viidim is the diminished chord. It is also a dominant chord and wants to move to the I tonic chord, however, it is not in a hurry like the V7 chord. Both being dominant chords they can substitute for each other. So if you want to resolve and return to the I tonic quickly use the V7, however if you would want to start a turn-a-round use the viidim, i.e. viidim, iii, vi, ii, V7, I for example. The viidim is a lead to somewhere chord.

Notice I use upper case and lower case to identify my chords. ii, iii, vi for minor chords and I IV V for major chords. Some of us do that, other do not. and use things like bIII to indicate a minor chord. Your choice. One more thing -- Roman numbers (I-IV-V) for chords and Arabic numbers (1, 2, 3) for notes.

Chords and melody. The melody line and the chord line should share some like notes for harmonization to take place. So it's a balancing act between the movement of I rest to tension IV to climax V7 to resolution and return to rest I, that the verse needs to make - so the story being told has a start, middle, climax, and ending. Continuing with the balancing act -- with out the melody line and the chord line sharing like notes you will never have harmonization, and that is kinda important (sounds good when it happens).

That is why we change chords. The melody line has moved on to notes not found in the old chord so it's time to insert a chord that does have some harmonizing notes - or - insert a harmonizing note into the old chord, as an extension, sus, augment, etc. Inserting a harmonizing note instead of bringing in a new chord and disrupting the movement flow is desired. That IMO is why you see all the fancy chords...... and would start another discussion of when to use the 9 note or the 11 note, which you asked abut, but, I'm not going to get into that here.

The Pentatonic scale will have three chord tones and two safe passing notes. That is why people tell you to look to the chord's pentatonic scale for your melody notes.

Is there more? Sure, but, that is enough for now.

Have fun.

[JonR]

Lately I have been pouring over various "chord theory" lessons fruitlessly searching for any secrets or theories that might be hidden in their pages regarding the principles behind "nice/stable/pleasing/good" sounding chords. Ultimately, I acknowledge music is an art and there is always going to be some conjecture about what sounds "nice", but also consider that this sort of logic could be applied to melody even though musicians tend to use the same scales that have been repeated throughout history that are functionally the foundations of said melodies. For clarification: when I inquire about the construction of a chord I am referring to where the voices are placed next to each other; how the various voices jive with one another in an octave and beyond. Sometimes when I am hunting and pecking around the neck of my Taylor I'll stumble upon a real gem of a harmony but I can't quite justify into any kind of logic why it sounds so pleasing (besides of course that typically these sorts of chords tend to -but not always- be spaced in 3rds). Surely there exists a sort of devoted study to such things...can you all direct a brother to such a resource?

As Jumping Jack says, the harmonic series is the physical factor at the root of our perceptions of consonance and dissonance. (I won't explain that, as you can google it well enough )

Notes that make consonant intervals have fundamental frequences which are in simple ratios (or very close to simple ratios). Eg:
Octave = 2:1 ratio
Perfect 5th = 3:2
Perfect 4th = 4:3
Major 3rd = 5:4
Minor 3rd = 6:5
The reason this makes the notes sound like they "belong" together is that they share some overtones (which is where the harmonic series comes in). The fewer overtones two pitches share, the more dissonant they will sound.

The fly in the ointment (for a mathematician) is that our tuning system of "equal temperament" means that tuned notes are not in these exact ratios, but slightly "out of tune". (Only the octave is perfectly in tune.) However, it seems our ears have a certain tolerance, as long as the tuning is near enough.

BTW, you can check these ratios on a guitar. If you measure fret distances compared to string length (nut to bridge) you'll find the following:
12th fret (octave) = 1/2 string length
7th fret (perfect 5th) = 2/3 string length
5th fret (perfect 4th) = 3/4 string length
4th fret (major 3rd) = 4/5 string length

You can also check how "out of tune" the major 3rd is. If you make sure your string is in tune, then play the harmonic over the 4th fret, it will read as around 14 cents flat (even if your tuner doesn't register cents, the needle should slow flat). The 4th fret harmonic is an exact 1/5 of the string, equivalent to a "pure" major 3rd. But the tuner tells you it's wrong because it's set to equal temperament, which demands that a major 3rd be slightly sharp - in order for the octave to divide equally into 12.

Equal temperament is just one of the "cultural amendments" we've made to nature over the centuries in order to be able to make the kind of music we want to make.
As well as physics, your question also touches on culture, because sounds we find "pleasant" are down to their familiarity as much as any physical attributes. IOW, the music we like is dependent on the music we grew up listening to.
And it's worth pointing out that dissonance is not necessarily "bad" - we are used to certain functional dissonances, which give contrast and meaning to music.

But it's only in the west that we care about harmony at all, that we have a "theory" of it. Other cultures have no harmony, or very little, and care much more about melody, rhythm and timbre. Ie, they have diverged in different directions from the facts of the harmonic series, in order to produce the sounds they like. Some aspects of music are shared worldwide, but harnony is not one of them; it's pretty much unique to the west (ie classical Europe originally).
It wouldn't therefore be true (btw) to say western music is more "advanced" than that of other culures. It only leads in the area of harmony and is actually a lot more undeveloped in terms of melody, rhythm and timbre than other cultures are.

If you really want to study the basis of our chord system (why we use the chords we do), it's not physics and acoustics you should look at, but European music history: going back to the earliest system of putting notes together simultaneously - "organum" (generally 5ths and 4ths only) - through to counterpoint, and then SATB harmony, which developed around the Renaissance.

[ragasaraswati]

Chord construction is a blend of intervals. Or combination of two notes.

Apart from unisons and octaves that represent the same note, a thing that has been called "the miracle of music", there are 2 types of intervals. Those who have a root, or stronger note, and those who don't.

1) Those who have a root, are subsequently divided into two groups. "Complete" intervals, whose root is the lower note and the "incomplete" intervals, whose root is the higher note or other note (e.g. M6 can suggest the IV as the root).

a. Complete intervals: M2, m3, M3, P5, m7, M7

b. Incomplete intervals: P4, m6, M6

Inversions of complete intervals result in incomplete intervals and vice versa. And of course octave distance plays but a small role as the character remains the same.

2) Rootless intervals. They are the only remaining two: m2 and Tritone. No tone of these intervals commands priority over the other and so they are ambiguous and free to interpretation.

The consonance of the intervals follows the same order 1.a, 1.b and 2.

If we're thinking outside of chord progressions and voice leading the above can help in constructing any chord close to the sound we're looking for. Also the positioning of dissonant intervals octaves apart softens their sound.

[rbarata]

I think this questions belongs here...at least as a practical example...
Why is the interval of the tritone has a tendency to resolve to the root and 3rd of the tonic? How can it be explained via the harmonic series?

BTW, when comparing two notes sharing some overtones, usually the frequencies of the shared overtones are not exactly the same. What's the range that allows us to consider them the same overtone?

[JonR]

I think this questions belongs here...at least as a practical example...
Why is the interval of the tritone has a tendency to resolve to the root and 3rd of the tonic? How can it be explained via the harmonic series?

It can't, fully. The simple answer is that we hear the tritone as dissonant: and the fact that other scale steps are a half-step away from each note - and those other two scale steps make a more consonant interval - means it has a strong tendency to move in that direction.
The diminished 5th resolves inward to a major 3rd (B-F > C-E); and the augmented 4th goes outward to a minor 6th, which is perceived as an inverted M3 (F-B > E-C).

The tritone is actually not too far away from a reasonably simple ratio, but you have to invoke a factor of 7. Ie, frequencies in the rato of 5:7 are quite close to a tritone in sound.

In terms of the harmonic series a 5:7 interval represents the 5th and 7th harmonics of a virtual root note. Eg, an A of 110 Hz (guitar 5th string) has harmonics of 550 and 770 Hz. Those frequencies are close to an equal tempered C# (554, 9th fret top E) and G (784.0, 15th fret top E). So those two notes would represent an "in-tune" tritone.

However, while 550 is 14 cents flat of 554 (not too bad), 770 is 31 cents flat of 784. So while we would hear 550 and 770 as "something like" a C#-G tritone - and they would sound very consonant with the low A root - I don't think that's how we perceive a tempered C#-G interval.
Eg, in the context of an A7 chord, the tempered G is a lot closer to a pure 6:5 minor 3rd with E. The tempered E in that region is 659 Hz, and 6:5 with that is 791: 15 cents sharp of G. Still some way from exact, but closer than 32 cents, and within normal tolerances.

Still, the question is open (IMO) whether (or how much) the 7th harmonic affects how we hear a dominant 7th chord. Certainly a 4:5:6:7 chord would seem (theoretically) to be more consonant than the closest harmonics if we ignore the 7th partial: 20:25:30:36. (Larger figures generally imply a morec complex sounding - ie more dissonant - harmony.)
The lower 3 figures are the same ratio, 4:5:6; it's the last one that means they need to be multiplied up by 5. The equivalent to the last one would be 20:25:30:35 - so you can see that the "36" means that that minor 7th is somewhat sharp of the one represented by "35" - but closer to the ET equivalent.
In terms of the tritone, the one in the second chord is 25:36 - rather more complex than 5:7.

So the short answer is as I said to begin with: the math doesn't fully explain it! It suggests the tritone will be perceived as dissonant, and certainly more dissonant in its tempered form.
But that ignores the fact that purer tritones were still regarded as horribly dissonant in the middle ages.

IOW, culture (and increasing harmonic sophistication since medieval times) are more relevant than simple math!

BTW, when comparing two notes sharing some overtones, usually the frequencies of the shared overtones are not exactly the same. What's the range that allows us to consider them the same overtone?

That's a good question.
It seems - judging from the amount of tempering we find acceptable - that up to around 15 cents out seems to be OK. The discrepancy in the major 3rd - as explained - is 14 cents, and we "get away with" that as long as instrumental timbres are relatively clean. IOW, as long as the overtones in question are not too prominent.
As any rock guitarist knows, once you add distortion to electric guitar, the major 3rd of a tuned chord starts to muddy up the sound; because the compression of the signal means all the overtones have become a lot more prominent, and that out-of-tune 5th harmonic of the root is starting to rub against octave harmonics of the 3rd. So we leave the 3rd out of the chord and make a "power chord", which immediately sounds stronger, purer and sustains better; because the harmonics of the 5th of the chord are only a negligible 2 cents out from the equivalent harmonics of the root.

(BTW, it's easy to confuse harmonics with chord tones here. The 3rd harmonic of a root represents a perfect 5th interval - in fact a 12th, octave plus 5th - and vice versa: the 5th harmonic represents a major 3rd interval, in fact 2 octaves plus M3.)

But as well as variations in instrumental timbre, tolerance can also vary between people. Some seem to have more sensitive ears than others. Eg, some guitarists (even on acoustic) find it hard to get the G and B strings sounding in tune. That's probably because they perceive those faint upper overtones (5th harmonic of G, 4th harmonic of B) clashing. The tuner says they're OK, the ears say they're not. (In fact the tuner will tell you the 5th harmonic is "out of tune", if you can get it to sound.)
And yet if you tune them so your ears like the open strings, then other intervals will sound equally out of tune. Eg, if a B tuned 14 cents flat sounds OK with the G, it won't sound OK with the E. And if (instead) you tune the G 14 cents sharp, so that B, E and G work OK together, then fretted chords like C or E will sound out of tune.
IOW, we simply have to learn to tolerate the tempered 3rd. (And most of us have no trouble with that.)

[rbarata]

So the short answer is as I said to begin with: the math doesn't fully explain it!

Right! That's what I suspected. I calculated several ratios between fundamental and overtones of different intervals and the results pointed me to the opposite direction of what I was expecting for.

As any rock guitarist knows, once you add distortion to electric guitar, the major 3rd of a tuned chord starts to muddy up the sound; because the compression of the signal means all the overtones have become a lot more prominent, and that out-of-tune 5th harmonic of the root is starting to rub against octave harmonics of the 3rd. So we leave the 3rd out of the chord and make a "power chord", which immediately sounds stronger, purer and sustains better; because the harmonics of the 5th of the chord are only a negligible 2 cents out from the equivalent harmonics of the root.

That's a very good explanation about how the power chord came up. I never thought it that way (so scientifically, I mean).

But as well as variations in instrumental timbre, tolerance can also vary between people. Some seem to have more sensitive ears than others. Eg, some guitarists (even on acoustic) find it hard to get the G and B strings sounding in tune. That's probably because they perceive those faint upper overtones (5th harmonic of G, 4th harmonic of B) clashing. The tuner says they're OK, the ears say they're not. (In fact the tuner will tell you the 5th harmonic is "out of tune", if you can get it to sound.)
And yet if you tune them so your ears like the open strings, then other intervals will sound equally out of tune. Eg, if a B tuned 14 cents flat sounds OK with the G, it won't sound OK with the E. And if (instead) you tune the G 14 cents sharp, so that B, E and G work OK together, then fretted chords like C or E will sound out of tune.

I always find a bit problematic tunning guitar strings from G to high E. I always find a compromise between the tunner and my ears. But I'm so used to dao it that way that is easy and fast (but if you ask me what is my technique, I would have to say "I don't know, I never thought about it".)