# Harmonic Centricity



## millionrainbows

What is "natural" is harmonic centricity, and this should not be exactly equated with Western tonality. Harmonic centricity can loosely and generally be called "tonality," but this tends to confuse the issue, so I'll call it what it is: _harmonic centricity, which_ _can be the perception of one note only, _or more.

Western CP tonality is based on harmonic centricity. The C major scale is the given, and triads are built on each of the scale steps. These triads are then ranked in order of their relation to the tonic triad (I), in terms of consonance/dissonance. G major is the next most important triad, being related to the most prominent harmonic, the fifth.

Each of these triads are harmonic centricities in themselves; each has a fifth, a third, and so on. Thus, _triads_ are the device which creates a centricity on the scale steps, each triad being a small "harmonic model" of the natural overtones which occur in any pitch. Theoretically, and practically, each of these triads can become its own centric/root station, but in Western tonality, all the other triads on the scale steps are subordinated to I (C major), which is the tonic station. 
In jazz, any scale step in any scale can become a main root station, thus "modal" jazz in dorian mode, mixolydian mode, and so on.

In Western CP tonality, this is not so; unless there is definite modulation to a new root station (scale step triad), all of these other triads are subservient to the home key of C. Thus, when a progression goes from C to, say, G major, G major is heard as a centricity or new station, but there is "expectation" that it will return us to the home key of C. Thus, "expectation" and "anticipation" of the G triad is _not_ based solely on the ear's perception of centricity, but on a _cognitive_ process which occurs over the span of the progression. This system of expectation and anticipation is learned by repetition, by recognizable procedures of resolution and tension, and is based on the style of the music which is produced using this system.

The fact that all of these subordinate steps "away" from the home key are based on triads, with fifths reinforcing their centric identity, is the way the ear is "convinced" that the subordinate triad _might_ be a new root station; but _tension_ and _resolution_ are the ways the ear's natural tendency to hear harmonic centricity is _"overcome;"_ tension and resolution are the _cognitive_ devices, which require ear/_brain_ perception of progressions and events over spans of time. 
Thus, the 'natural' tendency of the ear to hear harmonic centricity is "overcome" by cognitive, narrative sequences of events which we call "Western tonality."

So to say atonal music, like Anton Webern's, is "unnatural" because it does not refer every centricity which occurs to a root or key area, is mistaken. All music (using pitch) has tone centricity, even when it is melodic. Centricity becomes more obvious and compelling when there is more than one note occurring, since we tend to hear sequences of pitches as "melody" rather than harmony.


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## Woodduck

millionrainbows said:


> When a progression goes from C to, say, G major, G major is heard as a centricity or new station, but there is "expectation" that it will return us to the home key of C. Thus, "expectation" and "anticipation" of the G triad is _not_ based solely on the ear's perception of centricity, but on a _cognitive_ process which occurs over the span of the progression. This system of expectation and anticipation is learned by repetition, by recognizable procedures of resolution and tension, and is based on the style of the music which is produced using this system.
> 
> The fact that all of these subordinate steps "away" from the home key are based on triads, with fifths reinforcing their centric identity, is the way the ear is "convinced" that the subordinate triad _might_ be a new root station; but _tension_ and _resolution_ are the ways the ear's natural tendency to hear harmonic centricity is _"overcome;"_ *tension and resolution are the cognitive devices, which require ear/brain perception of progressions and events over spans of time.
> Thus, the 'natural' tendency of the ear to hear harmonic centricity is "overcome" by cognitive, narrative sequences of events* which we call "Western tonality."
> 
> *So to say atonal music, like Anton Webern's, is "unnatural" because it does not refer every centricity which occurs to a root or key area, is mistaken. All music (using pitch) has tone centricity, even when it is melodic.* Centricity becomes more obvious and compelling when there is more than one note occurring, since we tend to hear sequences of pitches as "melody" rather than harmony.


This is a bit of obfuscation, and putting "overcome" in quotes is deceptive.

The mere fact that the note G is heard as the root of the G major triad has nothing to do with whether or not the G chord as a whole is felt as subordinate to the tonic C. We don't have to "overcome" a perception of a thing seen in isolation to see its relationship to something else. Perceiving tonal progression doesn't require "overcoming" any other perception. We are not tempted by the occurrence of the dominant triad to stop the music and rest content.

I don't think anyone has called atonality "unnatural" for the reason you give. And the business about all music having "tone centricity" is just not relevant.


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## millionrainbows

Woodduck said:


> ...The mere fact that the note G is heard as the root of the G major triad has nothing to do with whether or not the G chord as a whole is felt as subordinate to the tonic C.


Then the reason the G is felt to be "subordinate" is if it's a G7, and the F (seventh) is perceived as a dissonance which is to resolve to E, third of the C major; but *this perception of a dissonance involves comparison to the C, and this is established in memory by comparison or repetition, not by purely harmonic sensation.*

There is nothing inherently dissonant about a G7 chord; it is used all the time as an unresolved root chord, as in blues. _Only by comparison in time do we feel a need to resolve.

_(one must understand by this that dissonance and consonance are not absolute states, but exist only comparatively)

We have to *"learn"* to hear the G as subordinate, and dissonant, by _comparison_ in the context of a tonal progression _in time._ The "harmonic centricity" and "harmonic independence" of the G triad is "overcome" in the context of a tonal progression by _"learning"_ that it is subordinate to C, by means of a progression _designed to convince us_ of this.

_So, the reason the G chord is felt to be "subordinate" to C is due solely to cognitive comparison over time, and not to any self-contained harmonic sensation of the chord itself._

The point of all this is to say: *Thus, Western tonality is seen to be a system which is entirely based on cognitive comparisons, not any sort of "natural" sensation of centricity. *

As you yourself said: "The...fact that the note G is heard as the root of the G major triad has _nothing_ to do with whether or not the G chord as a whole _is felt as subordinate_ to the tonic C."


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## Woodduck

_All_ tonal systems are learned! In purely melodic music, there is nothing in any single tone that leads us to another tone. So what? Why is this significant?

You begin this thread by saying, "What is 'natural' is harmonic centricity, and this should not be exactly equated with Western tonality." But who equates them? Who needs to be warned about this? And how do you get to this idea of "overcoming" our supposed sense of "harmonic centricity"? I'm pretty sure that most listeners to music have never struggled with any such problem.


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## millionrainbows

*Western tonality is seen to be a system which is not primarily based on cognitive comparisons spread out over time, but an ultimately "natural" sensation of centricity, which all "centric" music is based on.

*I will show that Western tonality, and its comparative functions, can be traced back to _vertical _factors, and the way we hear harmonics.

Thus, the "narrative" aspects of Western tonal progressions do not establish tonality; these 'narrative' excursions are based on the original _vertical _model of harmonics, only _assisted_ by narrative procedures.

Western tonality is about establishing a key area, and this means we have to "move away" from the tonic chord (C) in order to "return" to it, and establish the tonality, unless we want to drone on in one chord. Western tonality likes to move around, unlike Indian raga music. So how is this "movement" away from, and back to, the key tonic station accomplished?

The main way that Western tonality establishes key areas is by root movement. These roots are the scale steps, and the functions assigned to them within that scale. 
But why do we hear (in the key of C Major) a G chord as being the "dominant" (V) function, as "subordinate" to I (C)? What is this perception based on, and why does it convince our ear that "G" needs to resolve, and that "C" is our home key?

Woodduck might say that this perception is based on various factors, which it is, such as repetition, phrasing, etc., but I'm saying that the _main _way tonality is "played with" is by root movement.

This "root movement" is really a harmonic interval. The progression from C major to G major is the interval of a fifth, just spread out over time. So all root movement can be traced back to our harmonic intervals, which is the source of all pitched sound.

What evidence do we have of this? Simply our ears, and the way we hear harmonics.

Let's say that the root movement from C to G is a fifth up. Harmonically, we hear a fifth with the "root" on the bottom note, so C-G is heard as being rooted on "C." Since Western tonality is largely based on root movements of a fifth, and the 'circle of fifths' is further evidence of this, then this explains much of tonal root movement.

Conversely, we hear a fourth (the inversion of a fifth) as having its "root" on the top note; so G up to C establishes the root as C.

This is all based on vertical factors; the way we hear the natural harmonic series. Root movement in Western tonality simply "spreads this out" over spans of time.

This can also show how the diatonic C major scale, the chosen scale for most of our music, is also inherently unstable as far as being "totally tonal." It's built for movement, for unrest.

The interval C-F is a fourth; if we hear this as "root on top," then F Major is established, subordinating C, supposedly the "home" key. All this is due to the fact of the tritone F-B in the C major scale. 
In this light, we can see the truth of George Russell's assertion that the Lydian scale is "more tonal" if one wants to establish the scale root as the key. The F lydian scale cycles through all 7 in fifths before it circles back around to F, its key note: F-C-G-D-A-E-B (F).

This is also why piano tuners start on F and tune by fifths. If we try to "stack fifths" starting on C, we get C-G-D-A-E-B-(F#?). It doesn't work for a C major scale, as it has an "F." 
As the Pebber Brown video on Youtube shows, when he sustains all the notes C-G-D-A-E-B, the consonance of perfect fifths falls apart when the clunker "F" is added on top.

The C major scale is structured so that there is a "leading tone" E-F (establishing F), as well as B-C (establishing C).
The C lydian scale has a leading tone F#-G (establishing the more closely related V step of G) and B-C (establishing the scale key).

I'm not criticizing the C major scale; it's perfectly suited for what it is used for: to travel to other key areas.


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## millionrainbows

Woodduck said:


> The mere fact that the note G is heard as the root of the G major triad has nothing to do with whether or not the G chord as a whole is felt as subordinate to the tonic C.


If we see G is subordinate in a "fifth" relationship to C (C-G), then G's subordinate role is easily demonstrated. We hear fifths with the "dominating note" on bottom.


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## BabyGiraffe

Woodduck said:


> _All_ In purely melodic music, there is nothing in any single tone that leads us to another tone.


There exist universal cadence like melodic gestures involving semitones (in scales with leading-notes; also flamenco/arabic type scales resolve 1 semitone down). And this is not something exclusive to Western music scale and tempered system.

Millionrainbows: "Thus, Western tonality is seen to be a system which is entirely based on cognitive comparisons, not any sort of "natural" sensation of centricity."

Not really, I suggest reading some papers on modern scale theory and diatonic set theory. Current research in mathematical music theory offers surprisingly deep connections to algebraic topology, graph theory, word combinatorics, group theory, projective geometry.

https://en.wikipedia.org/wiki/Diatonic_set_theory

This guy did some really interesting research (imo).

https://www.researchgate.net/profile/Marek_Zabka


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## millionrainbows

The fact that C-G root movement in Western tonality establishes G as subordinate to C (in the key of C) is based on the vertical factor derived from harmonic hearing; the fact that the C-G is spread out over time is dependent on the "memory" of that C-G vertical relationship. So the fact that the C-G is spread out as a "narrative" device really in itself has nothing to do with establishing the relationships in tonality; only vertical factors, which we cognitively remember.


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## millionrainbows

Woodduck said:


> _All_ tonal systems are learned!


That's misleading. The _harmonic _nature of any fundamental pitch is heard, and requires no learning to intuitively understand.



Woodduck said:


> In purely melodic music, there is nothing in any single tone that leads us to another tone. So what? Why is this significant?


The harmonic series, which is the model we base our other models on, is a simultaneity, not a melodic interval. So the reason G is sensed as subordinate to C is because of this.



Woodduck said:


> You begin this thread by saying, "What is 'natural' is harmonic centricity, and this should not be exactly equated with Western tonality." But who equates them? Who needs to be warned about this?


Any "tonality" is based on a harmonic model. This is vertical in nature, and is 'modeled' by different scales.

This should not be confused with the Western tonal system, which elaborates and 'plays' with this verticality by spreading it out over "progressions" of chords (root centers). This 'elaboration" and narrative 'playing with' is what makes Western tonality unique, and very advanced. But other than this, Western tonality is just a tonality based on harmonic centricity, like all other 'tonalities.'

Who equates "tonality" with harmonic centricity? You do, all the time, in all your statements about it. You make statements which confuse basic, primal "tonality" (harmonic centricity), which is ubiquitous in most musics of the world, with our particular Western form of 'harmonic elaboration' which plays narratively with the notion of subordinate functions, moving to new key areas, and 'functions' of triads. All of these methods and practices are dependent on long narratives or progressions of chords, and comparatively long-term goals.



Woodduck said:


> And how do you get to this idea of "overcoming" our supposed sense of "harmonic centricity"? I'm pretty sure that most listeners to music have never struggled with any such problem.


We have to listen to music with our memory. A fifth is heard with the root on bottom, as a harmonic interval. When this interval gets translated into a horizontal movement of a fifth, we hear the first area as the "root" area, with the move a fifth away from it as a 'secondary' area or function, by using our memory of the interval of a fifth. This takes it out of the vertical 'now'into the horizontal world of time. This is like 'reading' a sequence of events. This has to be learned.


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## fluteman

millionrainbows said:


> The fact that C-G root movement in Western tonality establishes G as subordinate to C (in the key of C) is based on the vertical factor derived from harmonic hearing; the fact that the C-G is spread out over time is dependent on the "memory" of that C-G vertical relationship. So the fact that the C-G is spread out as a "narrative" device really in itself has nothing to do with establishing the relationships in tonality; only vertical factors, which we cognitively remember.


Not quite, I think. C and G are indeed closely related for the reasons you state. I would put it more simply: every note in the C major triad is in the G major scale and vice versa (and of course they have one note in common). That is always and only true of notes that are a perfect fifth apart, i.e., next to each other on the circle of fifths.

When you hear a C or G major triad in isolation, you have no way of knowing if the music is in the key of C major or in G major. The only way to figure that out is to put the triad in context and see which scale predominates, C major or G major. Of course, there are a lot of ways to establish a key or root. For example, if you hear a lot of F-sharp, it's likely G major, and if you hear a lot of F-natural, it's more likely C major. But not necessarily. Composers can be very subtle and sophisticated about establishing a key. But it requires horizontal (to use your term) motion.

Western harmony is all about horizontal progressions. That's why when you listen to the traditional Indian raga I linked to earlier, which is based on a drone and doesn't progress anywhere, it sounds so strange and foreign. In Western music many centuries ago, only unisons were considered consonant. Then the perfect fifth was accepted, and finally the third, creating the triad that has long been the basis of Western harmony.

None of that is natural or inevitable. The natural overtone series with frequency ratios 1 - 1/2 - 1/3 - 1/4 - 1/5 - 1/6 - 1/7 roughly corresponds to C1 - C2 - G2 - C3 - E3 - G3 - Bb3 in equal temperament. But only roughly, as the equal tempered third is 14 cents sharp, the fifth is 2 cents flat and the minor seventh is 31 cents sharp. And without equal temperament, much of Western music from the early to mid-19th century on would sound pretty strange. That's because it relied on increasingly elaborate and complex (one could say "chromatic") harmonic progressions.

Music like other art forms and languages at their most basic level make use of natural phenomena of which the natural overtone series is one, but far from the only one. The natural human affinity for rhythm and cadence is another. But it takes a lot of complex manipulation before a human art form or language emerges. I think it's much too simplistic to conclude that G is naturally and inevitably subordinate to C due to its place in the overtone series, or the triad. Its subordinate position, when it is subordinate, generally arises from how it is used in harmonic progressions in Western music, which makes use of, but doesn't strictly conform to, the overtone series. An hour-long drone on G with only occasional and momentary forays to C would establish the primacy of G for reasons that have very little to do with the overtone series, as would repeated fortissimo blasts of G and only a pianissimo hint of C. Etc.


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## millionrainbows

fluteman said:


> Not quite, I think. C and G are indeed closely related for the reasons you state. I would put it more simply: every note in the C major triad is in the G major scale and vice versa (and of course they have one note in common). That is always and only true of notes that are a perfect fifth apart, i.e., next to each other on the circle of fifths.
> 
> When you hear a C or G major triad in isolation, you have no way of knowing if the music is in the key of C major or in G major. The only way to figure that out is to put the triad in context and see which scale predominates, C major or G major.


"In context" means root movement in a progression. The root movement C to G (a fifth up or a fourth down) suggests C as the root, because we hear fifths as root on bottom and fourths as root on top; the root movement C to F (a fourth up or a fifth down) suggests F as root, because of the same reason. These intervals are originally vertical, not spread out over time. This is a harmonic phenomenon, not a matter of progression per se.



fluteman said:


> Of course, there are a lot of ways to establish a key or root. For example, if you hear a lot of F-sharp, it's likely G major, and if you hear a lot of F-natural, it's more likely C major. But not necessarily. Composers can be very subtle and sophisticated about establishing a key. But it requires horizontal (to use your term) motion.


Not primarily, for the reasons stated above.



fluteman said:


> Western harmony is all about horizontal progressions. That's why when you listen to the traditional Indian raga I linked to earlier, which is based on a drone and doesn't progress anywhere, it sounds so strange and foreign.


Western harmony is about horizontal _elaborations_ of phenomena which originate vertically and harmonically. I don't buy the comparison with Indian raga, which is a rhythmic/melodic music with no functional chords.



fluteman said:


> In Western music many centuries ago, only unisons were considered consonant. Then the perfect fifth was accepted, and finally the third, creating the triad that has long been the basis of Western harmony. None of that is natural or inevitable. The natural overtone series with frequency ratios 1 - 1/2 - 1/3 - 1/4 - 1/5 - 1/6 - 1/7 roughly corresponds to C1 - C2 - G2 - C3 - E3 - G3 - Bb3 in equal temperament. But only roughly, as the equal tempered third is 14 cents sharp, the fifth is 2 cents flat and the minor seventh is 31 cents sharp. And without equal temperament, much of Western music from the early to mid-19th century on would sound pretty strange. That's because it relied on increasingly elaborate and complex (one could say "chromatic") harmonic progressions.


The comparisons of equal temperament and 'just' intervals is irrelevant for now. The 12-division of the octave is based on the _interval_ of the fifth in a more general sense, because it _nearly_ coincides after 12 fifths and "closes" the octave. The fact that tempered and 'just' fifths are different is of little consequence, and only 2 cent's worth. The fifth is the favored interval, because it closes the octave and because it is the dominant overtone.



fluteman said:


> Music like other art forms and languages at their most basic level make use of natural phenomena of which the natural overtone series is one, but far from the only one. The natural human affinity for rhythm and cadence is another. But it takes a lot of complex manipulation before a human art form or language emerges.


You're talking now about the _Western_ 'language' of music, which is, indeed, a complex narrative of progressions and procedures. But _all _music is traced back to vertical phenomena. This includes Western music and Indian raga.

When you see that universal element, all music becomes similar, not different.



fluteman said:


> I think it's much too simplistic to conclude that G is naturally and inevitably subordinate to C due to its place in the overtone series, or the triad.


Simple, but not simplistic. Schoenberg, I think, would agree with me, since this idea came from his book "Structural Functions of Harmony."



fluteman said:


> Its subordinate position, when it is subordinate, generally arises from how it is used in harmonic progressions in Western music, which makes use of, but doesn't strictly conform to, the overtone series. An hour-long drone on G with only occasional and momentary forays to C would establish the primacy of G for reasons that have very little to do with the overtone series, as would repeated fortissimo blasts of G and only a pianissimo hint of C. Etc.


I'm not impressed by this example, as there can be built-in ambiguity to any progression, especially simple as the ones you gave. An hour-long-drone on G, with only occasional forays to C, would probably be heard as being in G mixolydian. Likewise, repeated fortissimo blasts of G and only a pianissimo hint of C would most likely be heard as being in G, with an occasional "C" as a suspended note.
This does not "disprove" the tendency of the ear to hear fourths as "root on top" and fifths with "root on bottom." Go to any piano and try it yourself.

Progressions such as these are really "arguments of persuasion" which must be interpreted cognitively. In pop music, the ambiguity of repeating progressions is often used. For example, the progression C-C-C-C/F-F-F-F (repeat ad infinitum), is C the root (I), and F the IV, or is C the V, and F the I? 
With progressions, this is ambiguous, and open to interpretation.

But with root progressions "abstracted," _apart_ from progressions, and analyzed for their primary tendencies, as Schoenberg did in "Structural Functions of Harmony," we see that the primary tendency of a root movement up a fourth is to establish the top note as the root.


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## fluteman

millionrainbows said:


> "In context" means root movement in a progression. [....]
> But with root progressions "abstracted," _apart_ from progressions, and analyzed for their primary tendencies, as Schoenberg did in "Structural Functions of Harmony," we see that the primary tendency of a root movement up a fourth is to establish the top note as the root.


Yes, but Schoenberg is analyzing Western harmony, no? My main point is that there is more to music than Western harmony, or even harmony generally, and concepts such as root movement. It's hard for us to understand that since harmony reached such a dominant role in Western music by the end of the 19th century. One thing many modernist composers did in the 20th century was to explore non-harmonic aspects of music more deeply, but in most cases without abandoning traditional Western harmony altogether. Schoenberg certainly didn't.

You also seem to miss my point regarding temperament. Yes, the equal-tempered fifth is only 2 cents flat. But stack twelve of them together, as you do to construct a 12-tone scale, and you end up 23.56 cents flat, which is significant. As Western harmony became more elaborate, systems of temperament were devised to avoid this discrepancy and keep intervals as "pure" as possible, such as just and well temperament. But these are all only clever compromises to favor the more commonly used intervals. An especially clever one that scholars call Werckmeister III is supposedly the one Bach had in mind when he wrote The Well-Tempered Clavier. But in the end, harmony became so elaborate, thanks in part to the increasingly dominant role of keyboard instruments resulting from the perfection of the modern piano, all those attempts at compromise got thrown out the window with equal temperament. There, the octave is simply divided into 12 equal parts and every note in every key is more or less off, but there is the crucial advantage that every scale and every chord sounds exactly the same in every key, with exactly the same errors. Our ears have fully adjusted to those errors and accept them as right, while a scale based strictly on the natural harmonic series would sound weird and dissonant.

I don't know if Schoenberg deals with psycho-acoustics or how the ear can become so accustomed to bad or dissonant intervals it hears them as beautifully pure. But the equal tempered scale is a prime example of that. And it shows how the "natural laws" of music don't fully dictate the finished result.


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## millionrainbows

fluteman said:


> Yes, but Schoenberg is analyzing Western harmony, no? My main point is that there is more to music than Western harmony, or even harmony generally, and concepts such as root movement. It's hard for us to understand that since harmony reached such a dominant role in Western music by the end of the 19th century. One thing many modernist composers did in the 20th century was to explore non-harmonic aspects of music more deeply, but in most cases without abandoning traditional Western harmony altogether. Schoenberg certainly didn't.


I'm simply standing behind the reasons that root movement (as intervals) establishes direction to or from a root. The fifth up establishes root as being on bottom, the fourth up establishes root on top.



fluteman said:


> You also seem to miss my point regarding temperament. Yes, the equal-tempered fifth is only 2 cents flat. But stack twelve of them together, as you do to construct a 12-tone scale, and you end up 23.56 cents flat, which is significant.


Yes, but that discrepancy is not heard all at once; only in two cent increments. But as I already said, the difference is irrelevant to this discussion. What is important is to see that the fifth is the favored interval in the 12-note division of the octave, and that root movement by fifth is the most common, and that the 'top note' of a fifth movement (C-G) is a move 'away' from key center, thus establishing G as subordinate to C. I say this again, in response to your statement:
_
"When you hear a C or G major triad in isolation, you have no way of knowing if the music is in the key of C major or in G major. The only way to figure that out is to put the triad in context and see which scale predominates, C major or G major. Of course, there are a lot of ways to establish a key or root. "
_
By this you seem to be obscuring the real, primary acoustic reason that key areas are established, which is by intervallic/root movement, and going off into 'narrative' areas of convincing _cognitively_ the key area, which is secondary.

All music, whether it is static (Indian raga) or full of root movement (Western) establishes its primary tonal center harmonically. That's where the term "harmonic centricity" comes from._

_


fluteman said:


> As Western harmony became more elaborate, systems of temperament were devised to avoid this discrepancy and keep intervals as "pure" as possible, such as just and well temperament.


After the octave was divided into 12 pitches (using the fifth), many of the tempered tunings were designed to get better major thirds, but this gave only a limited range of keys which sounded passable, such as Eb-Bb-F-C-G-D-A. The fifths were always "bearable." It was only when all 12 keys were desired that the desire for an equal temperament became the prime consideration, as in Bach's Well-Tempered Clavier.



fluteman said:


> But these are all only clever compromises to favor the more commonly used intervals. An especially clever one that scholars call Werckmeister III is supposedly the one Bach had in mind when he wrote The Well-Tempered Clavier.


It has since been discovered that Bach used his own tuning. See Larips.com



fluteman said:


> But in the end, harmony became so elaborate, thanks in part to the increasingly dominant role of keyboard instruments resulting from the perfection of the modern piano, all those attempts at compromise got thrown out the window with equal temperament. There, the octave is simply divided into 12 equal parts and every note in every key is more or less off, but there is the crucial advantage that every scale and every chord sounds exactly the same in every key, with exactly the same errors. Our ears have fully adjusted to those errors and accept them as right, while a scale based strictly on the natural harmonic series would sound weird and dissonant.


Again, you should see that the fifth is the favored interval.



fluteman said:


> I don't know if Schoenberg deals with psycho-acoustics or how the ear can become so accustomed to bad or dissonant intervals it hears them as beautifully pure. But the equal tempered scale is a prime example of that. And it shows how the "natural laws" of music don't fully dictate the finished result.


I'm not convinced by the comparison of perfect 'just' intervals to tempered intervals as a way of 'disproving' the tendency of the ear to hear a fifth up as 'root on bottom.' It works in any tuning or temperament. That is irrelevant.

Tonality is established primarily _vertically i_n all music, not by horizontal sequences of events (which only _elaborate_ what is already established harmonically and are used to "prove an argument" of key area.


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## fluteman

millionrainbows said:


> After the octave was divided into 12 pitches (using the fifth), many of the tempered tunings were designed to get better major thirds, but this gave only a limited range of keys which sounded passable, such as Eb-Bb-F-C-G-D-A. The fifths were always "bearable." It was only when all 12 keys were desired that the desire for an equal temperament became the prime consideration, as in Bach's Well-Tempered Clavier.
> 
> Again, you should see that the fifth is the favored interval.
> 
> I'm not convinced by the comparison of perfect 'just' intervals to tempered intervals as a way of 'disproving' the tendency of the ear to hear a fifth up as 'root on bottom.' It works in any tuning or temperament. That is irrelevant.
> 
> Tonality is established primarily _vertically i_n all music, not by horizontal sequences of events (which only _elaborate_ what is already established harmonically and are used to "prove an argument" of key area.


Yes, the fifth is a "favored interval", though as the third harmonic perhaps less so than the unison or octave. And yes, that is true regardless of temperament or tuning, at least any tuning I know of. But no, tonality, or at least key, is not established vertically, as you demonstrated with your own example. A C major triad could be in either C or G major. The same is true for a G major triad. Neither triad by itself establishes by itself the key the music is in. You could argue, and I suppose you are arguing, that if you play a C major triad and nothing else, like the drone of an Indian raga, that means the music is in the key of C major, or at least, C is the dominant tone, as C is the fundamental of the harmonic series leading to the third harmonic G.

That's fine, but that's not how today's Western music generally works, at least not how it worked before some modernists, post-modernists, avant-gardists and experimentalists of the 20th century. And that's not how most Western ears have been trained to hear music.

Finally, your comment that "It was only when all 12 keys were desired that the desire for an equal temperament became the prime consideration" is exactly right, and exactly what I meant by my comment above. The thing is, all twelve keys are desired these days, and all intervals have to sound the same in every key. That is every bit as true with popular music over the past century as with classical music, and again, that is what our ears are trained to hear. Anything else would sound foreign and strange to most.

Respectfully, it seems to me that arguing here over and over about what is or isn't inevitable, essential and natural to music doesn't get us very far. We can only identify what our listening habits are as Western listeners, and decide whether we are interested in expanding beyond that. I see a current trend of "world music" incorporating African, Asian and Latin elements into Western music, including in the popular music realm. So perhaps that expansion is gradually happening whether we realize it or not.


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## millionrainbows

fluteman said:


> ...But no, tonality, or at least key, is not established vertically, as you demonstrated with your own example.


You give no context to this statement. I accept it as a given (as did Schoenberg) that the ear tends to hear a fifth as "root on bottom," thus establishing a movement away from the root to a subordinate root, as in C-G. The converse is true for a fourth.



fluteman said:


> A C major triad could be in either C or G major.


Not if viewed as a root movement. In a root movement of C-G (up a fifth), C (bottom note) will be established as the root. This is equivalent to a fourth down, C down to G, where the C (top note) will again be heard as the root.

If the root movement is down a fifth, as in G-C, C (bottom note) will be heard as root. Equivalent to a fourth up, as in G-C, where C (top note) is heard as root.



fluteman said:


> The same is true for a G major triad. Neither triad by itself establishes by itself the key the music is in.


I never said that "a triad by itself establishes by itself the key the music is in." I said that root movement in fifths and fourths did this.

The interval of the fifth (inverted as the fourth) is reflected in the harmonic series, and is a vertical interval which establishes the root by its tendency to be heard as a harmonic relation to the fundamental (root) tone.



fluteman said:


> You could argue, and I suppose you are arguing, that if you play a C major triad and nothing else, like the drone of an Indian raga, that means the music is in the key of C major, or at least, C is the dominant tone, as C is the fundamental of the harmonic series leading to the third harmonic G.


Yes, by your example the music would be harmonically centered on C, but that's not what I'm arguing. The relations of the harmonic "G" to a fundamental "C" is the model which establishes harmonic centricity.



fluteman said:


> That's fine, but that's not how today's Western music generally works, at least not how it worked before some modernists, post-modernists, avant-gardists and experimentalists of the 20th century. And that's not how most Western ears have been trained to hear music.


You need to clarify this conclusion.



fluteman said:


> Finally, your comment that "It was only when all 12 keys were desired that the desire for an equal temperament became the prime consideration" is exactly right, and exactly what I meant by my comment above. The thing is, all twelve keys are desired these days, and all intervals have to sound the same in every key.


I don't think that exactly tuned perfect fifths are necessary, and other intervals could deviate as well. The "general" interval of a fifth, whether it be two cents off or not, is what establishes the root in all triads. The equal tempered scale has twelve of these, and they are close enough to establish all twelve keys with no problem.

Even in the Bach/Lehman tuning (see Larips.com), which is tuned "by ear" and counting "beats," the fifths and thirds are not all exactly precise, but it was close enough, and sounded good enough for Bach to write his preludes and fugues in all 12 keys.

True and precise equal temperament was not achieved until about 1917, when electrical frequency counters were developed. Up until then, stopwatches were used to "time" the beats. It was close, but not perfected.



fluteman said:


> ...The thing is, all twelve keys are desired these days, and all intervals have to sound the same in every key.That is every bit as true with popular music over the past century as with classical music, and again, that is what our ears are trained to hear. Anything else would sound foreign and strange to most.


Probably true, but what does this have to do with whether or not intervals can establish tonality?



fluteman said:


> Respectfully, it seems to me that arguing here over and over about what is or isn't inevitable, essential and natural to music doesn't get us very far. We can only identify what our listening habits are as Western listeners, and decide whether we are interested in expanding beyond that. I see a current trend of "world music" incorporating African, Asian and Latin elements into Western music, including in the popular music realm. So perhaps that expansion is gradually happening whether we realize it or not.


I can concur with that, but I still stand behind the statement that basic harmonic centricity (basic tonality) is established by using models and tendencies which are derived from the harmonic series, and this is true of all music.


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## fluteman

millionrainbows said:


> I can concur with that, but I still stand behind the statement that basic harmonic centricity (basic tonality) is established by using models and tendencies which are derived from the harmonic series, and this is true of all music.


OK, but the first part of that is just a tautology. Tonalities are based on harmonic centricity by definition and are thus derived from the harmonic or natural overtone series -- in significant part, but as often is the case, including in Western music, with significant modifications as well. 
You say this principle applies to "all music", including music not based on any tonal hierarchy, a/k/a (by some) atonal music, which is fine, though some musicologists say the opposite, such as this guy I just found on the 'net: 
http://www.austinpattytheory.com/music-theory-2/centricity-and-atonality/
No need to debate that one endlessly. Suffice it to say at the very least that harmonic centricity is not the only game in town with music such as the Webern example he cites, even if hints of it remain.
More importantly, as Woodduck has already said, all systems of tonal music (and atonal too) are learned. They are not based solely on the natural harmonic or overtone series, among other reasons because harmonic centricity is not the only thing that comprises even tonal music, or on any other single innate natural principal. There is an intellectual process that comes into play with human sound expression that involves the manipulation of natural sounds to specific human purposes. And you can shrug off psycho-acoustic phenomena all you want, but the divergence of various Western systems of temperament from the natural harmonic series is a wonderfully simple and clear example of how humans hear what they want and expect to hear, really what they have learned to hear (to use Woodduck's term), rather than what you might expect them to hear from the principles you cite.


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## millionrainbows

fluteman said:


> ...Tonalities are based on harmonic centricity by definition and are thus derived from the harmonic or natural overtone series -- in significant part, but as often is the case, including in Western music, with significant modifications as well.


OK, we agree on this point, with the caveat that the "actual" harmonic series can be the "model" that a scale is based on, and need not be slavishly adhered to. In this sense, all scales and divisions of the octave are "harmonic models."



fluteman said:


> You say this principle applies to "all music", including music not based on any tonal hierarchy, a/k/a (by some) atonal music, which is fine, though some musicologists say the opposite, such as this guy I just found on the 'net:
> http://www.austinpattytheory.com/music-theory-2/centricity-and-atonality/
> No need to debate that one endlessly. Suffice it to say at the very least that harmonic centricity is not the only game in town with music such as the Webern example he cites, even if hints of it remain.


No, the overall long-form in serial music is not based on a tonal hierarchy, but any pitched sound is tone centric within itself. Therefore, at each moment and sound event, serial music presents a tone centric event, but only at that moment. If this is too literal fore you, so be it.



fluteman said:


> ...More importantly, as Woodduck has already said, all systems of tonal music (and atonal too) are learned.


The _system_ is learned, not the fact of tonality/tone centricity. If a simple folk music is based only on simple harmonic concepts, such as the way its scale divides the octave, then there is no "system" to be learned; the ear understands it intuitively.
Western music, on the other hand, has one of the most complicated and cognitively demanding "systems" of any music, with its drawn-out narrative progressions and procedures.



fluteman said:


> They are not based solely on the natural harmonic or overtone series, among other reasons because harmonic centricity is not the only thing that comprises even tonal music, or on any other single innate natural principal.


I never said that harmonic centricity was the only thing that comprises various systems of music; only that all music stems from this harmonic phenomenon. Where it goes from there is anybody's guess.



fluteman said:


> ...There is an intellectual process that comes into play with human sound expression that involves the manipulation of natural sounds to specific human purposes. And you can shrug off psycho-acoustic phenomena all you want, but the divergence of various Western systems of temperament from the natural harmonic series is a wonderfully simple and clear example of how humans hear what they want and expect to hear, really what they have learned to hear (to use Woodduck's term), rather than what you might expect them to hear from the principles you cite.


I don't think that the divergence of the tempered fifth by two cents from the "actual" harmonic series makes any difference; fifths and fourths still do what they do.

Please remember that the "actual" harmonic series is used as a model, not to be taken literally. Any division of the octave _(such as an arbitrary length of bamboo chosen to make a flute, and pierced with an arbitrary number of holes, making an arbitrary scale) _creates a "model" of the harmonic series, because it has a fundamental (root) note and subsidiary notes which are in relation to that.


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## fluteman

millionrainbows said:


> The _system_ is learned, not the fact of tonality/tone centricity. If a simple folk music is based only on simple harmonic concepts, such as the way its scale divides the octave, then there is no "system" to be learned; the ear understands it intuitively.
> Western music, on the other hand, has one of the most complicated and cognitively demanding "systems" of any music, with its drawn-out narrative progressions and procedures.


There is the important point on which we disagree, and where I agree with Woodduck. All music, whether "simple folk music" or sophisticated "Western music" (a rigid, artificial, and rather condescending distinction, I think) is learned. Exactly as with spoken language, no music is intuitively understood, at least not fully. But the human mind has an amazing capacity to learn, especially when it is young.

Edit: As for your comment that "any pitched sound is tone centric within itself", I cited a musicologist I found after 20 seconds of searching on google who goes to great lengths to say the opposite. You can argue that with him, it doesn't mean much to me. And as for "simple folk music", why not take a break from Schoenberg and consider Bartok's folk music studies, for example Mikrokosmos and 44 Duos for two violins. It isn't quite as simple as you suggest.


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## millionrainbows

fluteman said:


> There is the important point on which we disagree, and where I agree with Woodduck. All music, whether "simple folk music" or sophisticated "Western music" (a rigid, artificial, and rather condescending distinction, I think) is learned. Exactly as with spoken language, no music is intuitively understood, at least not fully. But the human mind has an amazing capacity to learn, especially when it is young.


"If a simple folk music is based only on simple harmonic concepts, such as the way its scale divides the octave, then there is no "system" to be learned; the ear understands it intuitively."

Intuitive: That's not quite the same as "learned," yet it does require an intuitive grasp of sound.



fluteman said:


> As for your comment that "any pitched sound is tone centric within itself", I cited a musicologist I found after 20 seconds of searching on google who goes to great lengths to say the opposite. You can argue that with him, it doesn't mean much to me.


I did not interpret it that way. The musicologist called prominent tones in atonal music "focal pitch points," which amounts to the same thing. A single note taken in isolation belongs to no hierarchy of other pitches, and has no relation to other pitches. It is "centric within itself," as I said.



fluteman said:


> And as for "simple folk music", why not take a break from Schoenberg and consider Bartok's folk music studies, for example Mikrokosmos and 44 Duos for two violins. It isn't quite as simple as you suggest.


That's quite different than the example I cited, which would represent Man's simple intuitive grasp of pitched sound. Conversely, it's not as complicated as you suggest.


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## fluteman

millionrainbows said:


> " Conversely, it's not as complicated as you suggest.


But complicated enough that it must be learned, and can't simply be intuited. That is the whole point behind Bartok's Mikrokosmos and For Children, both of which are educational works that make use (very intelligent and sophisticated use) of folk music.


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## millionrainbows

fluteman said:


> But complicated enough that it must be learned, and can't simply be intuited. That is the whole point behind Bartok's Mikrokosmos and For Children, both of which are educational works that make use (very intelligent and sophisticated use) of folk music.


Be that as it may, I still maintain that simple harmonically centered music and its creation are a basic part of Man's birthright, just like spiritual awareness is, and that musical sense is natural and intuitive; not a product of being "civilized," or learned by enculturation. It transcends all culture, and is universal.


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## millionrainbows

Also, it occurs to me that some people like to think that "Man created Man" and that Man is an isolated self-contained crature who is not "tuned in" to any sort of universal knowledge or awareness. This is certainly not the view of most composers, like Beethoven, Messiaen, Cage or any number of the "greats." They all had some awareness of 'the sacred' and 'spiritual' dimension of music.
What, then is the meaning of music which does not include the human elements of "resonance" and empathy? It seems that it would be on the verge of being emblematic or simply an agreed-upon term, with no real connection to human psychology or the fact that "God gave us brains."

In my view, "psychological phenomena" can be linked to music and to spirituality, or a sacred sense of being, before it has to be connected to any religion, which in my view comes after the fact.

In fact, it sounds like you are saying that music works by itself apart from any connection to our psychology as humans. How could that be?


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## fluteman

I was trying to say exactly the opposite of that.


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