Many theoretical systems have been erected to explain Consonance, but the vast majority of these systems have been dogmatic, autistic, and hyper-logical.
What I proposed in my last blogpost is not so much a system as it is a simple analogy: Consonances are Tone-like. For a sound to be "consonant," there are neither strict requirements nor arbitrary mathematical principles for it to meet; it just has to roughly resemble a Tone.
"Roughly" is the key word here. Our ear is very used to the sound the Tone and is able to locate it out anywhere. Consonance is extremely flexible. Just as an endless number of images can be produced which, though they radically differ in style, all clearly portray the same person, so too can endless sounds be produced which, though they may seem entirely different, all resemble the Tone. The droning of an Indian Citar, A C-major chord played on guitar, the hum of an oboe, the earthy growl of a didgeridoo - these are wildly different sounds and yet they all conform to the basic shape laid out in the Tone; they are all consonances.
The number of possible elaborations upon the Tone, and hence possible consonances, is virtually endless. We cannot afford to constrain this wide expanse of possibilities into a narrow set of classifications and rules; rather, we should be observing these various tonal shapes and describing them as they occur.
First of all, every consonant chord is Tone-like. Although many separate tones go into the making of a chord, they combine in such a way that they impress the feeling of a single tone.
Say that a church-organist is having an off-day, and he accidentally starts the hymn by playing a pretty harsh dissonance:
In this sound we hear 6 separate tones, 6 pitches all fighting for the ear's attention. This is an acoustical clutter and, consequently, belongs with Noise.
So, quickly fixing his mistake, the organist shifts each voice slightly changing the chord to a powerful consonance - a towering C-major chord:
Just by slightly shifting each of these pitches, the organist has eliminated that unsettling feeling of 6 pitches fighting against each other. But why? Why do these 6 tones feel unified whereas the previous 6 tones do not? It is because, although the organist is technically still playing 6 tones, all of these tones conform to a single Harmonic Series.
The lowest tone, the C1, has a harmonic series which serves as the basis of the entire chord. Although there are 5 other tones which go into this chord - the C2, G2, C3, E3, and G3 - every frequency in every one of these upper tones falls along the harmonic series established by C1. As a consequence, these upper tones are absorbed into the C1's series, ENHANCING it rather than detracting from it. Rather than perceiving 6 distinct pitches, we only comprehend a single Tonal structure rooted in the pitch of C1!
All consonant chords are, like this one, just elaborations upon a single tone. Given the proper arrangement, many tones can combine to create the impression of a single Tone. This, and nothing else, is what causes tones to be consonant with each other.
But, does all this sound ridiculous to you? Is "many tones creating the impression of one tone" just a bit of sophistry I have conjured up to push my dogshit theory? I would ask the reader: on what basis do we call the chord above "6 tones?" Just because something is written as 6 notes, and the organist hits 6 keys on the organ, does that necessarily mean our ear will recognize the sound as such? If our goal is to understand the world of acoustical phenomena, then we need to appreciate how great the discrepancy often is between music as it written and music as it is actually heard. Written music is, after all, just a guide for the performer to play his instrument correctly. But the moment these inked notes are played, they become something entirely new; they become ACTUAL SOUND. We music theorists should take care to study this actual sound, and not the notes which represent it.
Once the hypnosis of sheet music is broken, it becomes obvious that the consonant organ-chord bears far more resemblance to a C1-tone than it does to the dissonant organ-chord which came before it. How many tones we INTEND to play is irrelevant; as far as our tone-whore of an ear is concerned, when many frequencies conform to a single harmonic series, they are ONE tonal sound.
This fact is made more obvious by the similiarity between a "single note" and an "octave":
Here is a C1-tone:
And here is a C1-tone and a C2-tone played simultaneously (an octave interval):
Sound-wise, the two are nearly identical. It almost feels like, instead of adding another tone, we just changed the timbre of the initial tone. Indeed, an octave can easily be confused for a single note, if you aren't already accustomed to what a "single note" is supposed to sound like on a particular instrument. THAT is consonance. THAT is what it means for a sound to be Tone-like. It means that, although a "single tone" and an "octave" are easily distinguishable on paper, in terms of ACTUAL SOUND, the distinction between the two is almost completely arbitrary.
Once we start combining tonal molecules into larger tonal compounds, harmony becomes multi-layered.
Consider the following chord:
On one hand, this chord is composed of 4 separate tones, each with their own series: C1, G2, C3, and E3. But on the other hand, the harmonic series of C1 is kind of a broader "meta-series" which all of these lesser "sub-series" are assimilated into. When interpreting tonal structures, then, we need to distinguish between these two layers of tonality: the Meta-tonal which is the over-arching tonality, and Sub-tonal which represents the separate tones making up this body. Sub-tonally, the chord above is 4 separate tones but, Meta-tonally, it is a single tonal body.
Meta-tonality and Sub-tonality relate to each other as Form and Embellishment. The sub-tonal layer embellishes upon the tonal structure established in the meta-tonal layer.
In the example above, the pitch of each sub-tone aligns with a particular meta-tonal harmonic; G2 aligns with C1's 3rd harmonic, C3 aligns with C1's 4th harmonic, and E3 aligns with C1's 5th harmonic. Each of these harmonics - the 3rd, the 4th, and the 5th - is accentuated by a sub-tone. It is as if the C1's series forms the base of the structure, and the sub-tonal series are all "blossoming" out of different points of this series.
There is a well-known phenomenon among acousticians called "phantom pitch." Basically, even if you take away a tone's tonic-frequency (its 1st harmonic), we will still hear the tone at the pitch of that tonic.
Here's a pure Tone:
Now here's that same tone with its tonic removed.
The pitch remains the same, despite no such frequency representing that pitch. This phenomenon only affirms what we have discussed: that a tone's pitch is an acoustical feeling created not just by its tonic, but collectively by its entire series of harmonics. The entirety of the Harmonic Series - not any particular frequency - bolsters the pitch-feeling. Naturally then, removing any single harmonic from a tone, even one as important as the tonic, will not cancel this feeling out, but will only slightly weaken it.
What is remarkable is just how many harmonics we can take away from a tone before its pitch-feeling begins to noticably disintegrate.
Here is what happens when we remove the 1st - 4th harmonics:
And the 1st - 8th harmonics:
But phantom-pitch has its limits. The more of the harmonic series we remove, the more vague the pitch-feeling becomes.
If we only leave the 6th, 7th, and 11th harmonics of the previous tone, for instance, we can still just barely hear that underlying pitch, but we have to really strain our ear to do so:
And finally, if we only leave the 9th and the 13th harmonics, it becomes impossible to hear the pitch. Each harmonic removed is a brick pulled from the over-arching tonal structure until, at some crucial point, everything collapses, and all we are left with is a jumbled heap of Noise.
But all of this is to say that, within certain limits, even FRAGMENTS of tonal structures will be heard as consonant. Just as, if many segments are removed from a visual image, we can still sort of imagine what that image is SUPPOSED to look like, so too can we, even given the absence of key harmonics, fill in the blanks to imagine what a complete TONAL IMAGE would sound like. The sound of the Tone is so intuitive to us, so deeply carved into our psyches from millions of years of hearing out tonal-voices in the wilderness, that even the smallest fragment of a Harmonic Series causes us to hallucinate the underlying pitch of that tone.
Naturally, since consonant chords are tonal, the concept of "phantom pitch" applies to them as well.
Here is a consonant chord:
Here is the same chord with the C1-tone (the meta-tone) taken away:
Despite no meta-tone actually being played, the consonance is preserved. Just as we imagine a phantom pitch for a tone, we will also imagine a meta-tonal series to justify any consonance. This is why we feel an unmistakable "root" or "key-center" beneath every chord and every harmony; these are nothing other than meta-tonal phantom-pitches.
The wide range of possible "chord voicings" are just different meta-tonal fragments:
For instance, the major triad and all of its inversions:
Every "consonant interval" is just a fragment of a meta-tone.
An interval between 300 Hz and 400 Hz, a perfect-fourth, is heard as the 3rd and 4th harmonics of a tone at 100 Hz:
We do not hear the sound above as a "4/3 ratio," as such, but as the 3rd and 4th sub-tones of a meta-tone at 100 Hz. This is an important nuance. It is never the intervals, themselves, which are consonant, but the overarching meta-tonal structures which they imply.
The most widely accepted psychoacoustical explanation for Consonance is Helmholtz's theory that two tones are consonant with each other because of their overlapping harmonic series. Dissonance, according to Helmholtz, is the direct result of harmonics from different series beating with each other. Overlapping harmonics mitigates this beating and, hence, mitigates dissonance. But this is only a half-explanation. It doesn't get to the core of what's really happening. Helmholtz misses the fact that whenever 2 separate harmonic series overlap with each other they necessarily create a larger, more fundamental (meta) series through their union:
It is by virtue of this fundamental series that the tones are bound together; it is THIS series which is the source of thier consonance. So it is not just that the two tones are "consonant with each other"; they literally become one tone!
This is why playing the meta-tone afterwards sounds like a resolution:
By actually playing a tone at 100 Hz, we fill in the empty spaces in the series, blatantly affirming the meta-tonic which was already implied. What our ear imagined becomes actual. The 100 Hz Tone reaches up and absorbs the two sub-tones into itself, eradicating that slight feeling that something is "missing" from the Harmony.
Therefore, it is not the intervals themselves which are consonant, but the over-arching tonal structures which they imply through their union. The ear doesn't recognize "intervals"; it only recognizes Tone-feeling and Noise-feeling. If an interval CAN be heard as part of a larger tonal body, then it will be heard as consonant; if it cannot, it will be heard as two conflicting pitches. To recognize any interval as consonant without acknowledging this implicit tonality is to completely misunderstand the interval!
Certain intervals feel more dissonant because they are more tonally ambiguous.
A perfect 12th is a very straightforward consonance, very clearly fitting into the 1st and 3rd harmonics of a series. A major third is slightly less so, but still easily comprehensible as the 4th and 5th harmonics.
An interval of 10/9 though??? Now THIS is where our inner-ear starts to short-circuit. The thing is, an interval of 10/9 is similiar enough to both 8/9 and 10/11 to be easily confused for either of the two. Our ear is built to detect tonal structure; not minute differences in intervallic lengths. So, the ear, caught between the vertigo of these 3 separate harmonic possibilities, is unable to fully rationalize the 10/9 interval. We just end up hearing it as two separate tones.
But the intense tonal ambiguity of this interval only makes its resolution more satisfying.