Why are triads considered perfect chords and the basis which all extensions are built off of?

Question

Why are triads considered perfect chords and the basis which all extensions are built off of?

I'm sort of posting the question from an ignorant standpoint but I do already have some prior knowledge. I wish I could buy the book From Polychords To Polya. Which kind of touches on my question. I’ve already read a bunch of it online but a lot of it is above my current level of understanding so I have to take some time to digest the information. Also the book Audacious Euphony seems to touch on the subject with depth but I’m having a lot of trouble understanding most of the information in that book (I ordered a physical copy today).

Why are dyads not the basis or even considered a chord by a lot of music theorists?

Why are tetrads (pentads etc), only considered extensions of the triad?

People say extensions and add chords only provide more “color” or “dissonance”. By color I don’t fully understand what they mean, I do understand what added dissonance would mean though, based on the harmonic series. With 3 notes you can have all consonant pitches, (Root, P5, M3/M6) or even (Root, P5, P4 /I like sus chords/). But once you get to 4 notes you have to add the 2nd or 7th which are more dissonant. Is this main reason Triads are as far as you can go before beginning to add dissonance to a consonant combination of pitches? Basically, are Tetrads the beginning of where we start to add “dissonant” pitches?

Lastly, usually triads are played and accompanied by vocals (a 4th pitch) which is basically extended and added harmony. Meaning, the triads are definitely the harmonic foundation, leaving the voice to play around with consonance and dissonance on top of the triads. Do I have this correct also?

Answer 1

Why are dyads not the basis or even considered a chord by a lot of music theorists?

They certainly used to be. The modern theory of chords evolved gradually over several hundred years during which the primary theoretical framework was dyadic. Music in three or more parts that modern ears would recognize as chords had already existed for a few centuries before theorists started talking about chords as such.

That some theorists distinguish dyads from chords is a bit arbitrary, but it does reflect the harmonic ambiguity of dyads. If the interval is perfect, the harmony could be major or minor (for example, to C and G you could add either E or E♭), and if the interval is major or minor, you can add a perfect interval related to either element (for example, to C and E you can add G or A; to E and G you can add B or C). The dyad is therefore somewhat problematic in harmonic analysis, and the theorists who exclude it from the definition of "chord" are mostly theorizing about harmonic analysis.

Why are tetrads (pentads etc), only considered extensions of the triad?

The historical perspective is also helpful to consider here. Until maybe 100 years ago, it was highly irregular to end with anything more than three distinct pitch classes. Such chords were considered dissonant because they contained at least one second (or seventh). The dissonance was not only an acoustic property but a contrapuntal one: it had to be resolved, so it couldn't be present in the final chord. When Rameau developed his theory that equates the chord EGC with CEG, this led to a chordal theory based on the two possible triads that contain a perfect fifth, the major and minor triads.

Much later, people started regarding "extra" tones in chords as color. If you add a sixth or seventh to a major triad, you no longer need to hear it as tension requiring resolution; it could just be a different sonority that is still restful. In this context, however, the underlying harmonic relationships can still be described in terms of triads, which leads us to refer to the "extra" tones as "color."

Surely someone will think of an exception to the preceding paragraph, but to illustrate what I mean, consider the chord progression Em7-Am7-Dm7-G7-C7. Any song using this very tonal circle-of-fifths progression can also be played without any of the sevenths: Em-Am-Dm-G-C. The harmonic functions of the chords do not change, but the color is very different.

People say extension’s and add chords only provide more “color” or “dissonance”. By color I don’t fully understand what they mean, I do understand what added dissonance would mean though, based on the harmonic series. With 3 notes you can have all consonant pitches, (Root, P5, M3/M6) or even (Root, P5, P4 /I like sus chords/). But once you get to 4 notes you have to add the 2nd or 7th which are more dissonant. Is this main reason Triads are as far as you can go before beginning to add dissonance to a consonant combination of pitches?

This is more or less it. The thing about "color" is, as I've said above, a change in the role of the acoustic dissonance: where before it provided from contrapuntal tension (also called "dissonance," which is sometimes confusing), it now serves to give the chord a different sonority.

But...

Basically, are Tetrads the beginning of where we start to add “dissonant” pitches?

Remember that the perfect fourth is frequently said to be consonant or dissonant depending on context. Also consider the diminished fifth/augmented fourth, which is always dissonant whether there is a third tone (or more) added to it.

Lastly, usually triads are played and accompanied by vocals (a 4th pitch) which is basically extended and added harmony. Meaning, the triads are definitely the harmonic foundation, leaving the voice to play around with consonance and dissonance on top of the triads. Do I have this correct also?

In some styles, yes. But more generally, the solo melody (whether a voice or an instrument) will basically have one of the chord tones, too. In fact, you will often see songs where a chord symbol seems to have been chosen specifically to reflect the fact that the melody has a certain pitch. It's fairly rare that the melody's main pitch at any point is not reflected in the chords. For example, you're not likely to see a G7 with the melody on A; instead, the chord symbol would be some variety of G9.

The melody might have some figures that use non-chord tones ("play around with consonance and dissonance") but in an ensemble context any part can do that. For example, if the chord is C then the melody can have C-D-E without the chord having to change. But an inner part could also have E-F-G at the same time. If the passing tones (D and F) happen quickly enough then we would not analyze that as a change of chord.

Even with a lead sheet, depending on the degree of precision indicated by the given chords, a pianist or guitarist can play around with consonance and dissonance, for example seeing C and playing C-Csus4-C or adding a little scale or some other lick, such as the passing tones mentioned in the previous paragraph. Similarly, a good bassist can embellish the indicated bass notes with scales and neighbor tones. The chords just specify the harmonic framework. They aren't a precise indication of everything anyone should be playing. A composer or arranger who wants that degree of precision writes a score in staff notation, not a lead sheet.

Answer 2

There is a general rule for intervals, which could be phrased like:

If a tone intersects a given interval in an arithmetic way (at the arithmetic mean) it will support the base tone, if it intersects in an harmonic way (at the harmonic mean) it will weaken the base tone.

And, since i have probably not explained what I mean very clearly (English is not my first language), let me explain what I mean by that:

Take the most basic interval, the octave. The proportion of the frequencies of the two tones are 1:2. Now, add a third tone, which intersects this interval at the arithmentic mean (in other words: "halfways" in terms of frequency). This intersecting tone will be the fifth of the base tone, with a proportion of 2:3. This fifth will sound supporting to the base, e.g. when you play C-G-c-g-c'-... on a piano your impression will be that "c" is the base tone and that impression is reinforced by the g.

If you instead use a third tone which intersects the octave interval at the harmonic mean, which means the fourth (3:4, which is near the golden ratio), then the tone will weaken the impression of the base tone being the base and try to establish itself as the base. In the above example, playing C-F-c-f-c'-... instead of C-G-c-g-c'-... the "c" will not be heard as undisputed base tone any more, but the f will try to esablish itself as base tone with c as its fifth.

In light of this, what is a major triad? It is the base tone, the fifth (which, as we said above, is the arithmetic mean of the octave interval) and the major third - 4:5 - which itself is is the arithmetic mean of the base-fifth-interval. This is why the major triad sound so "in rest" - it only consists of intervals, which reinforce the base tone in its role.

This also explains why the minor triad is less reinforcing the base tone - C major is in rest, but C minor "directs" or "hints" towards Eb major.

To sum it up: (major) triads are considered "basic" (and also "in rest"), because they only consist of tones supporting the base tone in its role.

Everything added to that will establish some sort of tension - "tension" here means no longer reinforcing the base tone in its role as base but in one way or the other poining away from it. Hear a C-triad major and c is undisputedly the base. Add a Bb to it you get a C7 and it points away from C, probably to F-major. Dyads are not considered to be basic in this sense, because they do not support the base to the maximum possible (by leaving out the major third).

Answer 3

The explanation I know as the usual explanation is Klang or the Chord of Nature. The harmonics of a string, the overtones of a fundamental pitch, outline a root position major triad for the strongest harmonics. This acoustic phenomena is thought to be the origin or many aesthetic preferences regarding harmony.

Why are dyads not the basis or even considered a chord by a lot of music theorists?

To some degree this is just a matter of terminology. When a textbook defines a chord as three tones, then that's your working definition. A broader view is that any number of simultaneous pitches is harmonic material, and that harmonic material can display harmonic function regardless of the exact number of simultaneous pitches. Those harmonic entities are functionally chords.

So, some consider two simultaneous tones to be chord, or partial/incomplete chords, because they can have recognizable harmonic function. In other words, when doing harmonic analysis, if you are able to, for example, label only two pitches as having function V, you have a dominant chord.

But, to be less abstract, the important thing that happens when adding a third chord tone is you can unambiguous know the chord root. Ex. with only tones C E you could have either a partial, root position C major chord, or a partial first inversion Am chord. Depending on the harmonic context one or the other is more strongly implied, but adding a third tone, either G or A will make the two cases clear.

Why are tetrads (pentads etc), only considered extensions of the triad?

One way of explaining this is a historic, contrapuntal approach. Assuming tertian harmony, the tetrad is a seventh chord, the seventh in the historic, contrapuntal context is a dissonance, not a chord tone, and part of some melodic voice movement.

V7 is an example, where some theory explains the origin of that chord being the melodic movement, the passing motion, of scale degrees ^5 of the V chord, through dissonant, non-chord tone, technically non-triadic tone, ^4, to ^3 of the I chord.

In other words the third and fifth above a root, or to be very historic above the bass, are consonant, and therefore actual chord tones, while the higher extension tones of seventh, ninth, and eleventh are dissonances. That leaves the thirteenth (or added sixth) the odd one out with additional history and theory discussing it, but let skip that for now. Historically, dissonances are not chord tones, therefore the fundamental chord ends with a third and fifth above a root/bass. Everything else will be either explained by chord inversions or non-chord tone motion.

I'm glossing over a lot of detail, because when taking a historic view of these topics you must remember that it goes back to a history before the concept of a chord root existed. It's hard to be complete and concise at the same time.

Answer 4

A fundamental property of a chord is that it has a functional root through which the other notes are viewed, with the note octaves being mostly unimportant.

A dyad is ambiguous: the two notes in a dyad form an interval with both ends of the interval being equally important. The exception is a (straight) fourth or fifth that are harmonically so close that they define a root note and overtones. The lower note of a fifth or the higher note of a fourth is the root.

If you now add a third note, this note stands in relation to the root note and thus provides color, defining a chord.

The most important color to fill in is major/minor: that creates a full chord entity in some inversion. Other colors are seventh/major seventh and suspensions. When those are missing major/minor, the result is not really satisfactory as a chord: with suspensions, that usually is the idea.

A diminished chord does not actually have the root-defining fourth or fifth. For that reason, it tends to work in the manner of a suspended chord as well, namely posing a question that only the context can answer. The difference here is that it does not even nail down a root note.

Jazz chords are more complicated since they tend to add significant more coloring and since the stacking is detailed enough that octave relations start making a significant contribution to the fingerprint of the chord.

One also has to see that the perception of chords differs over musical periods: in figured bass, the lowest note is considered the root note even if you stack a 4 and 6 above it: as a pop chord, this would rather be seen as the lowest note not being the root and the chord being an inversion, or a chord with a replaced base note.

Answer 5

My theory: Those styles of music for which it is true that triads form the basis of harmony heavily depend on the major-minor-dualism. Which means that you definitely need the (major or minor) third in a chord, to make it a distinct major or minor chord respectively. As to the fifth, in the history of many music cultures this interval has always had the special status of being the purest interval (following the prime and the octave, which are already in the chord according to the octave equivalency of the root). So it is somewhat natural to include also the fifth in the chord. Just look how naturally the fifth is used in drones, in bagpipe music for instance but also in Indian or Persian classical music; whenever one wants to establish a note as the root of something (chord, scale, piece), you often see that it is sounded together with its fifth.

That is my explanation of why it's all about the triads, composed of the root, the third and the fifth; instead of choosing dyads or tetrads composed of other intervals as the basis. Now regarding your other questions:

People say extension’s and add chords only provide more “color” or “dissonance”. By color I don’t fully understand what they mean, I do understand what added dissonance would mean though, based on the harmonic series.

"Color" refers to the idea of musicians in the beginning of the 20th century who had the feeling that the traditional euroclassical music with its three main pillars rhythm, harmony and melody has become hackneyed. So the idea arose to include timbre as a new pillar equitable to the three other ones. That way, styles like jazz and avantgarde music became born. Especially in jazz you add notes to chords to create new timbres, quite similar to additive synthesis of tones to create new timbres.

With 3 notes you can have all consonant pitches, (Root, P5, M3/M6) or even (Root, P5, P4 /I like sus chords/). But once you get to 4 notes you have to add the 2nd or 7th which are more dissonant. Is this main reason Triads are as far as you can go before beginning to add dissonance to a consonant combination of pitches? Basically, are Tetrads the beginning of where we start to add “dissonant” pitches?

Thinking of chords as layered thirds yes. This thinking developed through the establishment of 3rd-5th-triads, which are layered thirds, so people added the 7th and 9th as the next layers, which are commonly perceived as dissonant.

Lastly, usually triads are played and accompanied by vocals (a 4th pitch) which is basically extended and added harmony. Meaning, the triads are definitely the harmonic foundation, leaving the voice to play around with consonance and dissonance on top of the triads. Do I have this correct also?

This is a good way to think about it. Usually, on the hard beats this 4th pitch is one already included in the triad, but on the weak beats it's often a new pitch not in the triad. In this case the chord has a tetrardic character. This not only applies to the vocal strands on top of the chords, but all instruments, including the bass line.