Chord-Scale System

[JonR]

Jeff, I realise the thread has moved on somewhat since the post I'm responding to here. I felt I wanted to come back, but feel free to ignore this one while I catch up...

Originally Posted by Jeff Brent

Radial symmetry is based on perfect consonances.

Thanks, that's clear enough - I may have misunderstood earlier.

Originally Posted by Jeff Brent

Choosing our very first primal note to be D, there are two (and only two) perfect consonances with D.

Those primary perfect consonances are G and A. (This represents the first level of RS consonance.)

Understood. My query is based on the concept of an acoustic root. That works downward in 5ths, not outward (I mean the root is always the bottom of a 5th, or top of a 4th).
So the acoustic root of D, G and A is G. This is the case whatever order of pitch they are arranged in - although of course if G is lowest its root quality will be more evident.

This seems to be where disagreement about significance arises.

If D is the nominal root, then G is a dissonance, because it's not in the harmonic series of D. (D-G sounds consonant because we perceive G as root.)

Same applies for further notes in that direction - ie 5ths below G (C, F). These are increasingly more disruptive to the root identity of D.

Originally Posted by Jeff Brent

I don't see how that can be construed to be an artificial constraint, when all we are doing is combining D's natural primary and secondary consonances. (This represents the second level of RS consonance.)

Those are natural constraints.

Understood. It's not a constraint, as I see it, more an artificial system based on symmetry from a central note. The question is: why is radial symmetry considered important to begin with? Or rather, how does it deal with the discrepancy with the nature of the harmonic series?

As I understand it, a P5 above a given pitch is the strongest consonance (after the octave), because it aligns with the 3rd harmonic of the given pitch - or rather is an octave below that harmonic (which is actually a 12th above); ie, a 3:1 frequency ratio reduced to 3:2.
A P5 below a given pitch has no such relationship. Or rather, the relationship is inverted, so the lower pitch becomes the root.

I can see that radial symmetry (of 5ths at least) produces a set of notes with relatively simple ratios to one another - good consonances - but not that it results in the reference note being a natural root of any kind.

Originally Posted by Jeff Brent

What is your definition of tonal scales?

Well, I was maybe using "tonal" a bit vaguely. I meant a scale with a note that can be easily recognised as a natural root note.
I realise that perception is largely down to familiarity, cultural acclimatisation.

Ie, in our culture, we'd perceive the major scale as the primary tonal scale, and (IMO) the major pentatonic as a strong set of consonances derived (apparently) from it.
Ie., the major pent removes the half-steps (the tendency tones) that establish the sense of tonality in the major scale. But it retains the notes with the strongest internal consonances.
From the other perspective, the major pent seems like a more primitive basic scale, to which we add tendency tones (P4, M7) to strengthen the tonal (key) effect.
(I know this is not a historically accurate perspective )

When it comes to the minor scale, the minor pent obviously has all the same internal consonances as the major pent. But it's more of a problem to make it a tonal (minor "key") scale. Adding the b6 is OK (resolves down to 5), but the 2nd doesn't do much (goes up to b3?). The 7th of the scale is the issue, and needs raising to provide the leading tone to the octave of the root.
(I realise I'm talking about the classical western key system, itself somewhat artificial.)

Phrygian mode is an interesting variant, because the added notes are both tendency tones: a half-step down to root and 5th respectively.

And phrygian mode is also interesting as an "opposite" scale to the major scale, being radially symmetrical with it. (Measure the major scale's intervals downwards, you end up with phrygian mode. C major becomes C phrygian.)
But of course, phrygian mode still relates downward to its own root note, not in reverse to what would be the major root.
IOW, the radial symmetry is interesting, but doesn't seem to have a bearing on the sound or usage of the scale.

Originally Posted by Jeff Brent

Adam Neely says that the distinction between tonal and modal is artificial, because in modern musics when one lands on / comes to rest on a final tonic (tonal tonic or modal final), the effect is identical to that of a functional tonal cadence.

That makes sense. The difference - IMO - is simply one of strength - ie quantitative rather than qualitative, as it were. Functional cadences feel a lot stronger than modal ones, because of the half-step moves, particularly the leading tone up to the tonic.
IOW, that began as an artificial alteration to modal scales, precisely to underline the "finality" of the final. The practice became embedded in the later key system.

Originally Posted by Jeff Brent

We all know that the wholetone and diminished scales are considered to be atonal, and that the bebop major is simply the major scale plus a #5/b6. In addition, the jazz minor is not used as a stand-alone tonality in its own right, rather solely as part of a composite minor composition.

Sure.

Originally Posted by Jeff Brent

That leaves only the diatonic heptatonic tonal/modal keys to be labelled as "tonal scales" (which are all inversions of the doubly symmetrical ionian set), and the artificial man-made composite minor.

OK. (Still not sure about the "doubly symmetrical", but I probably just need to re-read your earlier posts...)

Originally Posted by Jeff Brent

The reason the major tonality is so popular is due to the great pull of the tritone in that heptatonic set to resolve so forcefully to the two defining notes of the major triad

Agreed.
Except I might disagree on the word "popular". The tritone resolution certainly feels like the reason the major scale is so tonally strong. Its popularity is another issue...

Originally Posted by Jeff Brent

(which, btw, is a radially symmetrical motion centered on D - see Modalogy pg 10 in your Amazon preview).

Again, interesting, but I'm not sure I see the relevance.
Eg, the tritone (as aug 4th F-B) also resolves outwards to a minor 6th (E-C). That's radially symmetrical about G# (tritone away from D). Is that relevant or significant? If so, how?

Originally Posted by Jeff Brent

This also has an intimate effect on the minors, because the tritone pulls toward the m3 and P5 of the minor triad.

Yes, that's also interesting, in that the keynote is not involved. The tonal minor scale requires the addition of the raised 7th, to pull to the tonic; otherwise the tritone seems to pull more convincingly to the relative major all the time.

Originally Posted by Jeff Brent

RS is only based on the harmonic series inasmuch as it uses P5 consonants surrounding a central note. The note D begins life at the center of the RS universe and remains there. In this case, the relationship of the notes is not vertical, rather radiating outwardly.

Right. But why?

Originally Posted by Jeff Brent

Although I could have mentioned this earlier in the discussion, I didn't want to get into nit-picky minutiae. I originally believed as you do, but Dogbite was kind enough to point out to me that the ratio between C and E (for example) is identical whether those notes are derived by stacking 5ths outwards OR upwards.

Yes, but I don't think that addresses the issue. C is still the acoustic root of that interval, whether it's below or above E.
A major 3d below C is Ab, which is the root of the C-Ab interval (in either inversion).

Originally Posted by Jeff Brent

This contradicts an earlier quote of yours in this post, and so now I guess you agree with *me* ...

Well I agree in the sense that - yes - we can construct a good sounding set of notes by working with 5ths in radial symmetry (better than working solely upwards). I don't deny that.

The issue is that that reference note is not the acoustic root of the scale.

Of course, with modes, we can nominate (almost) any note as the "final", and establish its governing role by how we compose the melody or harmony. The radially symmetrical Dorian mode was mode I in the middle ages - they clearly felt it was primary, although that may simply be because its final was the lowest of the four authentic modes.

Lydian is the only mode that (seemingly) obeys acoustic principles fully, in that it's the mode you get by working in 5ths upwards. But we all know what problems that causes... (not least the increasingly complex ratios with the root as those 3:2s are multiplied).

Interestingly (IMO), Lydian and Dorian are the two modes that jazz theorists regard as having no "avoid notes". But that's due (prosaically) to the absence of half-steps above triad chord tones.

Originally Posted by Jeff Brent

I am unaware of what point in the history of western music that musicians finally decided that the best place for the root is at the bottom of the scale.

The four medieval authentic modes established that principle.
(The Greek modes were measured downwards from an upper root, and medieval theorists may have misunderstood that.)
Whether it was logical (or scientific) or not, it was certainly established by the key system a few centuries ago.

Originally Posted by Jeff Brent

But since that convention is now *law*, we must abide by it.

It's "common practice". That's the only reason we abide by it (when we do).

Acoustically it makes sense, because acoustic roots are always low notes, although sometimes so low as to be virtual. The final tonic of the average classical piece is always in a really low octave in the bass, even if it's elsewhere too. Psychologically that just seems to really "nail it down".

Originally Posted by Jeff Brent

So, *by definition* the A at the bottom of the arrangement of notes A C D E G is the root of that scale.

Well yes, if we define it as that, then that's how we define it.
I thought you were invoking a definition from elsewhere. (Apologies for misunderstanding.)

IOW, an acoustic root is a physical entity, the result of frequency ratios and the harmonic series. A nominal root can be anything. We can define A as the "root" of ACDEG if we like. Doesn't make it better or stronger (tonally) than any other.
IOW, with some root nominations we need to support them acoustically - some more than others.

Eg, it doesn't do much good to say "B is the root of B locrian" - because it's impossible to support it acoustically.
So, modes like Ionian, Aeolian, Dorian and Mixolydian have a fairly strong root identity - it doesn't take much to establish their roots.
Ionian is strongest; Aeolian is weaker, but has a strength due to familiarity; Dorian and Mixolydian have become increasingly strong thanks to familiar practices in jazz and pop/rock over the last 50 years (and some residual familiarity from British folk music).
Lydian and Phrygian are weaker - probably due simply to lack of use. (Modern music in either of these modes needs to spend a lot longer on the root chord to avoid the scale's tendency to pull elsewhere.)
But all six have a perfect 5th, which is the main thing that confirms the root identity. (Of course, all except Lydian have a P4 too.) It's Locrian's lack of a P5 which means its "root" can't be convincingly established.

Originally Posted by Jeff Brent

Note that my complete qualifier reads: "its most compact form surrounding central D", and not the most compact arrangement possible.
Another 'more compact arrangment' is also G A C D E. Whaddaya call that one?

Fair point .

Originally Posted by Jeff Brent

We don't have to go there today or any time soon, but we will have to go there at some point before our Sun supernovas and destroys this planet ...

Not "we". I doubt you or I will be around when that happens .

[jazz oud]

To clarify my earlier statement, I should have said that maqamat are not typically named after numbers, but some notes are named after positions and so one could say that the number is included. The numbers used, though, are Persian and not Arabic. Sikah ("third position") and Jiharkah ("Fourth position") come from Persian numbers.

[Ramo]

Well, it seems, if you will allow me, I can continue.

“Intelligent improvising depends on a working understanding of the relationship between chords and melodic lines” (Joe Pass)

Chord-scale system in general is perhaps the most important system in the course of studying harmony. It reflected the structure of relationships: a) sounds in a chord, b) sounds in a scale, c) scales and chords to each other. (The letter should be understood as: scale and scale, scale and chords, chord and chord).

As you know, a chord in music is any harmonic set of two or more notes that is heard as if sounding simultaneously. If we talk about chords tonal music, it should be added that these harmonic set is formed from the diatonic modality.

Diatonic modality in Treatise DMC is absolutely exclusive concept that does not exist in Russia or in Western theories of harmony. Diatonic modality allows a complex structure represented as a sequence of simple structures. “Genius is taking the complex and making it simple". J.Brent. I think so too.

But I will not theorize now, and try to show the practical benefits of this approach. Suppose. we need to harmonize a melody. Take, for example, this: Alban Berg. Wozzeck (Marie's Lullaby)

This was done so. Whole melody is represented as a sequence of its parts, each of which is defined by its own modality. That is exactly one that best fits a natural diatonic. Here are four of these parts: in C, in Eb, in Db and in Bb.

Numbers under each note - this is the modal degrees. In the own modality melody should consist mainly of white (no sharps or flats) degrees. Degrees H, M and U are considered as own alterations. They will not change the modality. Degrees +1 (#4) and -1 (b7) change it, moving it the one step in the direction of sharps or flats. In the own modality modality they are nonharmonic tone.

Introducing the song in their own modal steps we can easily sing it on the notes (solmization). Thus, the modal degrees are functions of sounds in the horizontal communication. No not tonic, but the own modality defines horizontal communication sounds.

Knowing own modality of a segment melody, we know the range of possible standard chords, as well as of standard harmonization techniques. Here we are left with little choice for the invention of the "new wheels". And that's just fine. Modal degrees of a chords easily recognizable by ear too, they are also functions of a chords.

In West tonal music, has historically turned out that music began and ended with a one of the two modal degrees: I (major) or VI (minor). This defined the trend, which is now called "gravity", and these degrees - tonics.

Can we talk about this melody in such harmonization as tonal? I do not know. But in each of these four segments it is possible to assume a key (on I - major or on VI - minor). Here all in major: C, Eb, Db and Bb.

Now about scales.Any own modal degree of the sound determine the scale (its number), which can be played from it as from the starting point. This scale can be applied to all the chords segment, changing with considering alterations in their exponents.

For example, in the first segment, from G (5 degree) we can play G mixo N, changing it to the G mixo H (for Fm) and to the G mixo U (for F7).

In the second segment is similar: Bb mixo N and G mixo H. In the third segment, if you think from Gb (4 degree), we will play Gb lyd U (for Gb7) and Gb lyd N (for Gbmaj7 and Ab7). In the fourth segment, everything is clear: Bb ion N through the entire segment.

Of course, here we can consider also the vertical dimension of chords. But more about that after the pause, during which I suggest you look at the "Ipanema" with the position that I have shown in the previous example.

Continue, if we stay in the topic

[Jeff Smith]

Originally Posted by Ramo

Continue, if we stay in the topic

Please continue to enlighten us.

[Ramo]

Originally Posted by Jeff Smith

Please continue to enlighten us.

Consider that the bomb thrown by a terrorist (whose post has been removed) worked.

To all of you good luck!

[jazzman1945]

Originally Posted by Ramo

But I will not theorize now, and try to show the practical benefits of this approach. Suppose. we need to harmonize a melody. Take, for example, this: Alban Berg. Wozzeck (Marie's Lullaby)

Yes, I have also harmonized the same song - with courtesy of Ramo , which on the one hand recall lullaby of Mozart, on the other hand - "Frankenstein" of Grachan Monkur 3 -th.

http://www.mediafire.com/view/?hfw99hy4sdska3e

http://www.freejazzinstitute.org/sho...04_jazzman1945

In harmonizing my approach differs from Ramo. First I remembered that Berg had a favorite chord, which he enjoyed very much: triton plus perfect fourth, that for jazz ear sounds like rootless 7 sharp 9.
So I killed two birds with one stone.

Then I analyzed the melody to find a stable notes, they can serve as tonics, or be part of the tonic, besides always remember the phrase Ornette Coleman: "Tonic where I am at this moment".
The first half of the melody demonstrates this very clearly: G in first 2 bars , Bb in second two . Berg's favorite chord I used as unsustainable - in fact as sub5 or approach .
In bars 5 - 6 tune offers clear movement C - F - Bb, but for that very reason I abstained, holding Bb for the last chord.
My chords in the fifth bar - the most problematic, and I feel that they fall out of the selected style. Moreover the first chord of this bar, rather Db +/#9 than the declared inversion of F+/maj 7 .

In constructing the voicings themselves guided me, anyway tried to use the method of Juzef Kon, which has already said in the past:

http://forums.allaboutjazz.com/showt...t=Herb+Pomeroy

Under this method ratio densities in a number of chords is is a leading, and not a voicings function .

[jazzman1945]

Another version of the harmonization measures 5 - 6:

http://www.freejazzinstitute.org/sho...04_jazzman1945

[Jeff Brent]

JR: As I understand it, a P5 above a given pitch is the strongest consonance (after the octave), because it aligns with the 3rd harmonic of the given pitch - or rather is an octave below that harmonic (which is actually a 12th above); ie, a 3:1 frequency ratio reduced to 3:2.

A P5 below a given pitch has no such relationship. Or rather, the relationship is inverted, so the lower pitch becomes the root.

So the acoustic root of D, G and A is G. This is the case whatever order of pitch they are arranged in - although of course if G is lowest its root quality will be more evident.

My query is based on the concept of an acoustic root. That works downward in 5ths, not outward (I mean the root is always the bottom of a 5th, or top of a 4th).

I can see that radial symmetry (of 5ths at least) produces a set of notes with relatively simple ratios to one another - good consonances - but not that it results in the reference note being a natural root of any kind.


1. In relation to the three notes G D A, an ‘acoustic root’ of G explicitly implies the II-V-I progression via extension of the V-I perfect cadence: D > G.

A > D > G
II > V > I


Taking the ‘acoustic root hypothesis’ out to its logical conclusions:

A > D > G > C
VI > II > V > I

Now C is the acoustic root.


A > D > G > C > F
III > VI > II > V > I

Now F is the acoustic root.


A > D > G > C > F > Bb
VII > III > VI > II > V > I

Now Bb is the acoustic root.


A > D > G > C > F > Bb > Eb
IV# > VII > III > VI > II > V > I

Eb Lydian: Eb F G A Bb C D Eb

Now Eb is the acoustic root.

The ‘acoustic root’ chain bolded above is the veritable backbone of George Russell’s derivation of the Lydian scale, his main justification for that scale’s ‘primacy’, and a founding principle of the entire Lydian Chromatic Concept.

from George Russell’s LCC 1980’s edition (pg 50-iii)

According to Hindemith, the tonic of an interval of a fourth is the upper note, while the tonic of an interval of a fifth is the lower note. Consequently, in viewing the major scale as a chord, both intervals indicate that the tone on the fourth degree is the tonic of the chord. The C major scale, therefore, is actually an F major chord. […]

The presence of the interval of a fourth in the lower structure of the major scale subdivides the tonality of that scale into two tonalities; the one on its fourth degree, and the one on its root.

The absence of a fourth in the structure of a Lydian Scale enables the whole scale to emphasize the root tonality.


from GR’s LCCOTO volume 1, 4th edition 2001 (Page 3)

Tonal Gravity, or “tonal magnetism,”within a stack of intervals of fifths flow in a downward direction […]; the tone F# yields to B as its tonic – F# and B surrender “tonical” authority to E, and so on down the ladder of fifths – the entire stack conferring ultimate tonical authority on its lowermost tone, C. In this way, an order of six fifths represents a self-organized GRAVITY FIELD.

But, you knew all that already, right?


The fatal flaw here is that there is no inherent limiting factor in the acoustic root cadential chain (GR: order).

There is no reason to stop at three notes, four notes, five notes, six notes, seven notes, eight notes, nine notes, … or a hundred notes, etc. It could keep going on and on forever.

A > D > G > C > F > Bb > Eb > Ab
#I > IV# > VII > III > VI > II > V > I

A > D > G > C > F > Bb > Eb > Ab > Db
#V > #I > IV# > VII > III > VI > II > V > I

A > D > G > C > F > Bb > Eb > Ab > Db > Gb
#II > #V > #I > IV# > VII > III > VI > II > V > I

A > D > G > C > F > Bb > Eb > Ab > Db > Gb > Cb
#VI > #II > #V > #I > IV# > VII > III > VI > II > V > I

A > D > G > C > F > Bb > Eb > Ab > Db > Gb > Cb > Fb
#III > #VI > #II > #V > #I > IV# > VII > III > VI > II > V > I

The acoustic root hypothesis (if I truly understood your meaning of “works downwards in fifths”) quickly descends down a slippery slope into a bottomless abyss.

There is no ‘logical conclusion’, because the sequence never actually concludes.


2. The ‘acoustic root hypothesis’ also ignores other cadential forces at work.

Even in just these 3 primeval consonants (G D A), already exist very fundamental dynamic relationships.


Consider the Tonic, Dominant and Subdominant of the tonal realm.

The Tonic lies a perfect fifth equidistant between the Subdominant and Dominant.

S T D
G D A
IV I V

or

D T S
A D G
V I IV

Is D not clearly the tonic root to anyone’s ears in this context here?

As is well-known, the most important factor in any chord progression is root motion.

Playing either of the root sequences below clearly identifies D as the tonic among these three notes:

D G D A D
T S T D T

D A D G D
T D T S T

and validates the IV-I plagal (beta) cadence.

A huge majority of songs use the TDS musical principles to generate their chord sequences (at least in the occident).


3. There are certainly ways to make the G sound like the tonic root, if limited to playing combinations of solely the 3 notes G, D, and A.

In theory/composition class, I was taught that the lowest note defines all those above.

Playing A D G or A G D as a block chord, I hear A as the root.

Playing block G D A or G A D, I hear G as the root.

Playing D G A or D A G I hear D as the root.


Which of the above four 3-note voicings exhibit tendencies and what kind, I wonder?

A D G and A G D

The A7sus4(omit5) feels pretty happy right where it is, and if I play A minor pentatonic (for example) over this quartal (especially with an A pedal point in the bass), the melody unquestionably wants to resolve to A.

Proof that the natural acoustic root of these three notes is A?


If this chord were to want to go anywhere though, the most typical motion would be up the 4th to D. In fact, moving to the D major triad makes a very satisfying (but not absolutely necessary) resolution after a lengthy AQ drone.

Proof that the natural acoustic root of these three notes is D?


[tangential observation: In between vamping on this 3-note A quartal chord, playing an occasional G in the bass elicits the gamma cadence pull to resolve back to the established A tonic (bVII-I).]


G D A and G A D

The quintal G D A and the G2 voicings both feel at rest. Definitely not much tendency there.

Proof that the natural acoustic root of these three notes is G?

If this chord were to want to go anywhere though, the most typical occurrence would be to resolve the sus2 up to the major third of the G major triad. In fact, moving to the G major triad makes a very satisfying (but not absolutely necessary) resolution after a lengthy G quintal drone.

More proof that the natural acoustic root of these three notes is G?


D G A and D A G

This one has the strongest tendency of the three. Like most sus4 chords, the perfect fourth of the D4 wants to resolve to the major third of the D major triad.

More proof that the natural acoustic root of these three notes is really D?


My point here is that BOTH circular motion (eg. II > V > I, etc) and theTonic/Dominant/Subdominant movements are already inherent in the primordial stacking of only two fifths!

This complementary ambiguity is one of the important factors in giving us so many possibilities for making music.

BUT both of these functional musical properties have a tendency to break down when extended out to infinity. They seem to work best when the number of fifths involved in the stacking is fairly limited.


You make the point that stacking more and more fifths results in increasing disruption to the original roots’ identities.

I agree, and I see that as a GOOD THING in the same way that I see the complementary ambiguity in the first level of two stacked fifths (circular motion & TDS movements) mentioned above to be very advantageous to composers by giving us more options to fool around with.

At the second level of RS consonance, the root focus shifts from the D (or G, if you prefer) of the first level’s set of {G D A} to either of the key root notes A or C of the {A C D E G} set.

At the third level of RS consonance, the root focus shifts much more strongly to the C of the {A B C D E F G} set (due to the introduction of the “new” tritone and its tensions’ tendencies), yet does not eliminate the A root option (which can also form cadences using the “new” tritone), and additionally delivers a good number of other new keys that are completely unavailable in the 1st and 2nd levels.

These options are some of the basic properties that make music so malleable, versatile, and such a wonderful way to express ourselves.


JR: If D is the nominal root, then G is a dissonance, because it's not in the harmonic series of D. (D-G sounds consonant because we perceive G as root.)

The perfect fourth is too in the harmonic series! It’s found between the 3rd and 4th harmonics.

D is the 3rd harmonic of G (aka perfect fifth/twelfth).

D is consonant with G and reciprocally G is consonant with D (the third harmonic is consonant with the 1st, 2nd, 4th, 8th octave harmonics, etc).

All perfect intervals are consonant. Both perfect fourths and perfect fifths are consonant.


There are many definitions by as many theorists on the subject of what constitutes dissonance as well as distinctions under that umbrella. There is imperfect consonance, mild dissonance, harsh dissonance; and then there are tensions.

Earlier this year, I had a lengthy discussion with a Dutch theorist (whom Levine introduced me to), regarding the centuries-old debate of whether of not the perfect 4th is consonant or dissonant.

If you would like to read about the historical controversy regarding “dissonant perfect 4ths”, I suppose I could redact that conversation down into an article for you (and for other interested readers as well), although I’ve previously covered some of it here: http://forums.allaboutjazz.com/showt...096#post592096


JR: When it comes to the minor scale, the minor pent obviously has all the same internal consonances as the major pent. But it's more of a problem to make it a tonal (minor "key") scale. Adding the b6 is OK (resolves down to 5), but the 2nd doesn't do much (goes up to b3?).

In both the common tonal V7b9-Im cadence, and the modal bVII-Im cadence, the 2nd of the aeolian resolves to the b3 of the tonic minor.

JR: The 7th of the scale is the issue, and needs raising to provide the leading tone to the octave of the root.

JB: The tritone pulls toward the m3 and P5 of the minor triad.

JR: Yes, that's also interesting, in that the keynote is not involved. The tonal minor scale requires the addition of the raised 7th, to pull to the tonic; otherwise the tritone seems to pull more convincingly to the relative major all the time.

Either 7th works well to resolve as a strong minor cadence (raised 7th in the case of a tonal V-Im motion, or the natural b7 in the case of a modal bVII-Im or bVII7-Im motion). This usage is up to the composer’s discretion.


JB: I don't see how that can be construed to be an artificial constraint, when all we are doing is combining D's natural primary and secondary consonances. (This represents the second level of RS consonance.) Those are natural constraints.

JR: Understood. It's not a constraint, as I see it, more an artificial system based on symmetry from a central note.

The initial question is: Which notes are perfectly consonant with D?

D is perfectly consonant with A, and reciprocally A is perfectly consonant with D.

G is perfectly consonant with D, just as D is reciprocally perfectly consonant with G.

Excluding the octave and unison, there are two and only two notes that are perfectly consonant with D.

There is not only one other note perfectly consonant with D, nor are there three notes consonant with D either. There are EXACTLY two notes that are perfectly consonant with D, no more and no less.

This is this primary level’s inherent limiting factor.

If you see this as an ”artificial system”, please explain to me what is artificial about the fact that there are two and only two notes perfectly consonant with D?



JR: Still not sure about the "doubly symmetrical"

JR: the tritone (as aug 4th F-B) also resolves outwards to a minor 6th (E-C). That's radially symmetrical about G# (tritone away from D).

All RS pitch sets exhibit dual symmetry, example :

D = primary axis

|
A B C D E F G
|


G#/Ab = secondary axis

|
D E F G|A B C D
|

JB: RS is only based on the harmonic series inasmuch as it uses P5 consonants surrounding a central note. The note D begins life at the center of the RS universe and remains there. In this case, the relationship of the notes is not vertical, rather radiating outwardly.

JR: Right. But why?

Up until the discovery of this radial symmetry in music, theorists considered that there were only two ways to stack fifths, ie. either upwards or downwards.

Above in this same post here, the problems of stacking fifths downwards has already been dealt with (Acoustic Root Hypothesis - Hindemith/Russell).

The problems associated with stacking fifths upwards is covered in this '08 article of mine: Outwards vs Upwards.

Out of the three ways to stack fifths, RS is the only one that gives the answers to questions like Why are there exactly five notes in the primordial pentatonic? Why are there exactly seven notes in the ubiquitous ionian set?, etc).


JR: why is radial symmetry considered important to begin with?

Its importance is not only in the many items that I’ve already listed in several earlier posts (the TDS Triumvirate with its evolution into the 5-note pentatonic scale, the 7-note heptatonic scale and the jazz minor; the chromatic cube, etc), but also in its consequences. IOW, the effects its ripples produce. The beautiful colorful winged creatures that emerge from the chrysalis.

If the tree of scales’ trunk, branches, leaves, flowers and fruits are important, then it follows that the seed must be important as well.


One example of this symmetry’s outgrowths are the guitar fretboard fingerings in the Fingering Mastery series of books.

The Chromatic Cube scale fingerings for guitar are the easiest and most practical way ever discovered to visualize, manipulate, and mutate scalar pitch sets on the fingerboard.

Viewing the world of stringed instruments through the lens of music’s interlocking symmetrical structures brings into focus the nature of the close relationships between the occidental scales and modes, and possessing this knowledge translates into cool and useful information for making music in the real world.

Look, if you will, at the Fingering Mastery guitar handbook’s preview.

All those perfectly logical and comfortable fingerings are the direct result of looking at scales from the viewpoint of their inherent symmetry.

Who could deem the resulting guitar fingerings below as unimportant?

We also wrote books on bass, mandolin, and violin (please see their previews as well).

If these practical applications of the principles of symmetry are only important to musicians who are interested in such matters, then that’s perfectly fine by us.

This knowledge is now public, each will make use of it what they will.


[Sidebar: Who else has written a book that even remotely resembles the Fingering Mastery series with its symmetry-based fingerings?

No one – according to our research, and we’ve done plenty. As part of *my* extensive research phase on the subject, I bought every single guitar scale book in the entire Hal Leonard catalogue, plus I teach at a music store with an excellent selection of instruction books too, so I’ve pretty much seen ‘em all, you know?

Every one of those other scale books just regurgitates the same exact stuff over and over.

Not ours. Now that’s important (at least to me and dogbite).]


JR: how does it deal with the discrepancy with the nature of the harmonic series?

There is no shortage of discrepancies in the harmonic series. It is common knowledge that the series breaks down completely at the 11th harmonic. In addition, the b7 of the 7th harmonic is considered by most theorists to not be the “ideal b7”, and there are even those willing to debate whether the M3 present between the 4th harmonic and 5th harmonic suffers from a similar quirkiness (“flaws”) as those evidenced in the 7th and 11th harmonics.

“… a just third may at first seem […] too narrow for equal-tempered ears.“
~ W.A. Mathieu “Harmonic Experience” pgs 48-49

“Equal-tempered ears” = 99.999% of the western world …


JR: phrygian mode is also interesting as an "opposite" scale to the major scale, being radially symmetrical with it. (Measure the major scale's intervals downwards, you end up with phrygian mode. C major becomes C phrygian.)

The phrygian is the mirror of the ionian, just as the aeolian and mixolydian are mirrors of each other, the lydian and locrian are mirrors of each other, and the dorian is the mirror of itself.


JR: Lydian and Dorian are the two modes that jazz theorists regard as having no "avoid notes".

The entire issue of “avoid notes” has been debunked. The concept, when reduced down to its basic premise, simply revolves around which notes in a chord-scale suck when played over said scale’s tonic chord.

There is only ONE so-called “avoid note” in the entire ionian set of modes and that is the 4th degree of the ionian.

AND that note is only an “avoid note” in two modes: The ionian (F against a C major chord), and aeolian (F against an A minor chord).

There is an article here on AAJ regarding this that you might be interested in revisiting http://forums.allaboutjazz.com/showt...023#post538023

AND even then, the notes don’t actually have to be avoided. How many times have you seen Charlie Parker play a big fat perfect 4th heavy on the first beat of a chord containing a major triad as its bottom three notes, for example? Lots, huh?

There are no “avoid notes”. You just have to know what the hell you’re doing.


JR: … Phrygian [is] weaker

You’ve surely spent time in Spain, haven’t you?

During the three years that I lived in that country, I gained a seriously life-changing appreciation for the incredible depth of power and passion that those gypsy guitarists put into the playing of the phrygian (along with its variations). Those guys play their guitars like the fate of the world depended on it – a matter of life or death! Try telling one of those macho cats that what they’re doing is “weak”.

As regards phrygian cadences and progressions, the bII-I cadence is one of the strongest cadences in all music!


JR: it doesn't do much good to say "B is the root of B locrian" - because it's impossible to support it acoustically.

JR: It's Locrian's lack of a P5 which means its "root" can't be convincingly established.

Playing any one individual note out-of-the-blue, immediately establishes in the listener’s ear that THAT is now our “home note”.

While this perception of “home” can certainly be (and often is) altered to play games with the listener’s initial assumptions, it does not change the fact that all that is needed to establish “home” is one single solitary note.

This single note does not require a P5 to be present in order to establish the root (“home note”) of a musical piece.

(This is certainly related to the axiom that “the most important factor in any chord progression is root motion”.)

In fact, playing an open fifth out-of-the-blue does nothing at all to enhance the “home note” feel generated. At most, all that open fifth can tell you is that the music which follows will not be in the locrian mode.


I *have* encountered locrian being used as a tonal center. A student of mine brought in a Black Metal tune that he wanted to transcribe. I was quite surprised to discover during course of the analysis that it was, in fact, written completely in Locrian!

Granted, the number had a very Halloween feel to it, concentrating mostly on the interplay between the I and the bII and the bV and IV (even though over the course of the tune every note in the locrian got used). Whenever the phrases finished, they landed very very convincingly on the Locrian Tonic.

If you would like to musically channel Satan and his minions, Locrian is your #1 mode of choice.


Regarding ‘establishing root identity in the ionian set of modes’, please see Modalogy pg 17 (paragraph 2).


I want to thank you, Jon, for your interesting discourses over the many years we’ve known each other, and also for giving me the opportunity to clarify issues that might be a bit cloudy for other readers too.

[JonR]

Jeff, that's a hell of a response, and it may take me some time to digest it. But thanks for the careful consideration, and I may respond in a PM, rather than bloat this thread even more!

But - OK - just a couple of things while I'm here:

The ‘acoustic root’ chain bolded above is the veritable backbone of George Russell’s derivation of the Lydian scale, his main justification for that scale’s ‘primacy’, and a founding principle of the entire Lydian Chromatic Concept.

- and IMO is where his whole edifice falls down (or at least wobbles dangerously...). That's because, if we base the stacked 5ths process on pure 3:2 ratios, we end up with a horrendously out of tune 4th of the scale. (The so-called "#4" is in fact exactly in between a P4 and an augmented 4th.)
So lydian is not "natural" in that sense. Or rather, the "natural" lydian scale is not compatible with western instruments, nor with western scale and chord theory. (What would we call a pitch midway between 4 and #4? how would we use it?)

IOW, his system depends on equal temperament - which is fair enough. The 5ths in ET are all tempered to 2 cents flat, which is how and why they work together (2 cents is a negligible discrepancy). But you can't then appeal to any "naturalness" in support of lydian mode. The P4 of Ionian is just as close to that 11th harmonic.

The acoustic root hypothesis (if I truly understood your meaning of “works downwards in fifths”) quickly descends down a slippery slope into a bottomless abyss.

There is no ‘logical conclusion’, because the sequence never actually concludes.

True. We have to stop somewhere, and traditionally we stop at 7 notes within the octave, to make things manageable. That doesn't make the acoustic principle redundant.

The tradition - as I'm sure you know - is based on the tetrachord, which is actually an inverted 5th. It's what led to the I-IV-V-I basic octave divisions - the "perfect" intervals - that we still largely use today. The positions of 2nds, 3rds, 6ths and 7ths were always somewhat flexible, with no obvious preferred consonances. Consequently, we ended up with "minor" and "major" versions of each, according to practical divisions roughly aligning with that whole step between IV and V, and approximate half-steps filling the gaps.
The exact positions of those intervening intervals could be determined by various mathematical principles. You could have a Pythagorean M3, an 81:64 ratio based on 5ths above the root (C-G-D-A-E, 3:2 x 3:2 five times). Or you could have a 5:4 ratio based on the 5th harmonic of C (5:1 divided by 4). The latter (most agree) sounds better, because the acoustic relationship implied by 81:64 is so impossibly remote; we can't hear the harmonic it's based on, where we can (just about) hear the one 5:4 is based on.
IOW, 81:64 is the result of an artificial reductive philosophy, that said the greatest simplicity is best as a guiding principle; even if it results in idiotic complexity in the end. (AFAIK, the Pythagorean system might explain why major 3rds were considered dissonant in primitive medieval harmony. But I've probably got my history all screwed up here...)

Consider the Tonic, Dominant and Subdominant of the tonal realm.

The Tonic lies a perfect fifth equidistant between the Subdominant and Dominant.

... [G - D - A]

Is D not clearly the tonic root to anyone’s ears in this context here?

Good point.

But it depends on other factors. If we simply play those 3 notes together, as a chord of stacked 5ths, then G will sound like the root.

The reason D becomes "tonic", and the others become IV and V, is to do with how we manipulate the scale, how we use chords.

In the major key, the 4th is the main dissonance, the one "rogue" note. The "avoid note" on the tonic chord. In a primitive Christian analogy, it's the "devil", that seeks to rule the tonality as the true acoustic root of the whole pitch set. So it has to be fought. The dissonant tritone it makes with the 7th is resolved by movement to the more consonant interval between root and 3rd of the tonic chord.

It may be worth mentioning that the major key was a relatively late development in western music. The modal system (lacking both Ionian and Aeolian) ruled for around 1000 years, despite increasingly giving way to key-type alterations towards the end.
By any account, the major key is an artificial construct. The modes before that were too, of course, but it was the peculiar qualities of Ionian that lent itself well to the developing craft of harmony, and the narrative purposes that were beginning to be demanded of music.
What the major key does, classically, is represent the struggle between dissonance and consonance, showing how consonance always wins out (as long as we keep control). Major-minor key tonality was practically designed for the imperialist, explorative culture of Renaissance Europe, a parallel to the impulse to conquer, subdue and exploit nature. It represented order being imposed or restored on potential chaos, through conflict with natural forces.
The P4 is thus a necessary part of the scale, the inbuilt awkward note that sets up movement away from rest, presenting us with the dissonance that we then force back towards rest: "drawing the sting" of the tritone by moving the half-steps.
We make sure, at the end, that we put our tonic note right at the bottom of the harmony, preventing the 4th from going any lower than that. We force the tonic to rule, IOW. (Think of all those classical symphonies that end with hammered-out V-I chords, often repeated: dammit, THIS is the tonic! It's like a hero stamping on the dragon's head after spearing it to death. Would they have really needed to underline the point so vehemently, if the major scale really was a natural phenomenon...?)

Playing A D G or A G D as a block chord, I hear A as the root. 

Playing block G D A or G A D, I hear G as the root. 

Playing D G A or D A G I hear D as the root.

I disagree, actually. I don't think it's quite as clear as that.
The only one I fully agree with is G D A.
In the others, yes you can argue the bottom note sounds like a root (sort of), but the chord is clearly unstable - contains a suspension.
The suspension is the hint that that bass note is not a true root. We are attempting to mimic the effect of a true root by simply selecting a low note. (And the mimicry works up to a point.)

In a sense, it's the same idea as when we enforce the tonic of the major scale as its true root. We've done that for so long (centuries) that by now it does - indeed - feel "natural". Few would deny (at least among those acculturated to western music) that random noodling on the notes ABCDEFG will lead to a sense that C is the natural home note of them all, not F. We've been governed by the "do re mi" for so long...

To stretch the argument, this familiarity with an "artificial" root (C not F) has enabled us to countenance other modal roots more easily.
Eg, the implied (probable) mixolydian root of an A7sus4 (A-G-D) chord. That's if we don't hear it as a V chord in D major...

Of course I realise I'm somewhat conflating the concepts of "root" and "tonic" . But that's kind of the point. We can nominate various notes as either "root" (of a chord) or "tonic" or "keynote" (of a scale or mode).
But in some cases, we need to do more work to make those notes sound secure in that role; and that's because of acoustic factors.

(I agree with many of your other points on the comparisons between AGD, DGA, and GDA. I think we do indeed hear them as you say. But how much of that is cultural familiarity - with major key cadences - and how much is down to acoustic properties?)

You make the point that stacking more and more fifths results in increasing disruption to the original roots’ identities.

Just to be clear, I meant 5ths downwards. It's stacking 4ths upwards that makes the original bottom reference note less and less stable as a root. (IMO that's behind the preference for quartal harmonies in modal jazz - the point of that music was to subvert the clear tendencies of triadic chords, by using ambiguous, open-ended, "rootless" harmonies.)
5ths upwards tends to support the bottom root acoustically, by echoing its lowest odd harmonic (3rd).

The perfect fourth is too in the harmonic series! It’s found between the 3rd and 4th harmonics.

Huh? But there is no harmonic between the 3rd and 4th! (or was that a joke?)
The closest harmonic to the perfect 4th is the 11th harmonic, which is around 50 cents sharp of P4 (and 50 cents flat of #4, of course).

Taking A (110) as a reference note, the 11th harmonic is 1210. In that octave, an equal tempered D is 1174.7 and D#/Eb is 1244.51. 1210 is not exactly half way between, but almost.
Going up another octave, the 21st harmonic comes out as 2310, which is closer to D, but still 29 cents flat of the ET D at 2349.3.

What about the "pure" D, at 4:3 to A? That's 146.67, just 2 cents flat of the ET D. The 11th harmonic of A is actually a little further away from the nearest octave of that (1173.3, 53 cents flat of 1210). Going up another octave, the pure D is now 27 cents flat of the 21st harmonic.

Quite simply there is no harmonic of A that is close enough to a D, whether it's an ET one, or a 4:3 relative to A.

Of course, the fact that it does have a 4:3 ratio with A (or near as dammit in ET) means it sounds consonant! But D is the acoustic root.

I need to stop there... work to do!

[jazz oud]

Originally Posted by JonR

- and IMO is where his whole edifice falls down (or at least wobbles dangerously...). That's because, if we base the stacked 5ths process on pure 3:2 ratios, we end up with a horrendously out of tune 4th of the scale. (The so-called "#4" is in fact exactly in between a P4 and an augmented 4th.)
So lydian is not "natural" in that sense. Or rather, the "natural" lydian scale is not compatible with western instruments, nor with western scale and chord theory. (What would we call a pitch midway between 4 and #4? how would we use it?)

Just to clarify--the 11th harmonic is not the same interval as what you get from stacking 3:2 fifths (which is obvious when you think about it for a second, since 11 is a prime number). The #4 arrived at by stacking 5ths is higher than the one normally used in Lydian, not lower.


As for Jeff's post, I will leave that to you . . .

Except for his (joking?) contention that the overtone series contains a P4: of course suggesting the P4 between 3:1 and 4:1 only reinforces the fact that the upper note is the acoustical root, not the lower one.

[JonR]

Originally Posted by jazz oud

Just to clarify--the 11th harmonic is not the same interval as what you get from stacking 3:2 fifths (which is obvious when you think about it for a second, since 11 is a prime number). The #4 arrived at by stacking 5ths is higher than the one normally used in Lydian, not lower.

thanks, j.o.

Should be obvious really. Each 3:2 5th is 2 cents sharp of ET, so by the time you've gone through 6 of them, you'll be 12 cents sharp - which is in fact how it turns out - I just checked.
So not that far out, tuning-wise, although a horrendously complicated ratio with the root.

[Jeff Brent]

JR: If D is the nominal root, then G is a dissonance, because it's not in the harmonic series of D.

JB: The perfect fourth is too in the harmonic series! It’s found between the 3rd and 4th harmonics.

JO: As for Jeff's post, I will leave that to you . . .

Except for his (joking?) contention that the overtone series contains a P4

Not Joking.

It is universally accepted that "the ratio 4:3 is the ideal perfect fourth".

This is explicit recognition that the distance between the 4th harmonic and the 3rd harmonic is a perfect fourth, or rather the perfect fourth.

Example:
Choosing A110 as our fundamental (first harmonic), the 3rd harmonic is E330 and the 4th harmonic is A440 (the 4:3 ratio, n'est-ce pas?)


What about this: You have to go out to the 25th or 26th harmonic to get a "natural harmonic series" note that even approximates a b6 interval, yet 8:5 is still the ideal ratio, right? That's because the 8th harmonic and the 5th harmonic are "in the harmonic series" - and therefore the minor6 interval is in the harmonic series, wouldn't everyone agree?

The 8:5 ratio is precisely the description of the distance between the 8th harmonic's root and the 5th harmonic's M3 in the harmonic series.

Would anyone like to stand up and say that a m6 interval is dissonant because its first possible surrogate doesn't occur until at least the 25th harmonic?


If you want to get really picky, the 21st harmonic (which, ahem, comes before the 25th) is pert near a perfect fourth:

21st harmonic of A110 (2A)  = 2310Hz

tempered 7D                 = 2349.32Hz (39.32Hz from 2310Hz)

pythagorean 7D below A3520  = 2346.66Hz (36.66Hz from 2310Hz)

If approximation is allowed in the case of the b6 example above, then approximation must be allowed for the fourth as well.

If approximation is not allowed, then attempting to base a definition of dissonance on whether or not the "ideal interval" is directly generated by the fundamental is pinning the tail on the wrong donkey.

That's not what the consonance or dissonance of the perfect fourth is based on. It's based on the ratios in the harmonic series.

Why is that not obvious?

[motherlode]

*****

[jazz oud]

Jeff, JonR said that G was not in the harmonic series of D, not that no perfect 4th interval occurs anywhere between any two notes of the harmonic series.

What you are saying is that D is the harmonic series of G, which is quite different than saying that G is in the series of D.

And you conveniently left out my statement, which exactly addresses the issue, and which you do not acknowledge in your response:

Originally Posted by jazzoud

of course suggesting the P4 between 3:1 and 4:1 only reinforces the fact that the upper note is the acoustical root, not the lower one.

Your contention is that acoustical roots are irrelevant (with respect to radial symmetry, in that you can designate any tone the radial center), mine and JonR's contention is that they are not musically irrelevant and that if radial symmetry treats them like they are irrelevant then it is flawed with regard to the acoustical function of tones (while remaining potentially valuable as an organizing and pedagogical tool).

When JonR says that "G is a dissonance", I believe that he means that in the traditional sense (needs resolution), not in the non-technical sense of "sounds bad" or "is out of tune". It's very clear in the rest of his post that he is talking about the technical, functional meaning of "dissonance" and the role that the 4th has in functional tonal harmony. I believe you've misinterpreted him and your entire following post is a result of that misinterpretation. I don't believe anyone has suggested that notes must be "in the harmonic series" to be usable.

To summarize: if you think that I or JonR were suggesting that there is something wrong with the P4 or that it has no acoustical basis (either the JI or ET version, which differ only by about 2¢), then you have misunderstood. The difference of opinion is simply this:

Me, JonR: The fact that a group of tones may be arranged in a radially symmetric way is of no acoustic or harmonic significance if the center of the symmetry is not also the center of the key. This does not mean that it is not of technical, theoretical or pedagogical interest or utility.

Jeff Brent: Radial symmetry is of more than technical/theoretical/pedagogical interest/utility; it is an acoustically significant phenomenon in music even if the center of the key is not at the center of the symmetry.


Everything else in your post is a distraction, arguing against things that no one has proposed.


To clarify, since you seem to be arguing against something no one has said:

"Dissonance", as used by JonR and myself, is the technical music theory term for a note that requires resolution. While it is often used by non-musicians and musicians in a non-technical sense to mean "unpleasant", "out-of-tune", "chaotic", "noisy", "atonal" etc., this is not what it means in a discussion of music theory. A P4 above is regarded as dissonant interval relative to an acoustical root and has been for the history of tonal harmony.

In the case where an interval has the acoustical root as the upper note, it may be a perfectly pleasant sound, but the fact that the harmonic series has been inverted is a form of dissonance (this is very clear in Messiaen's writings, among others). So if, in the key of C we have C/E, we indeed have a m6 interval, one that occurs in the harmonic series of the key, but is nonetheless unstable to a degree because the bass note is not the acoustical root.
It is notable that a second inversion I chord, based on the use of the P4 in the harmonic series is one of the most unstable (i.e., dissonant) chords in tonal harmony, functioning as a suspended V chord.

So we are not saying that "usable tones" must not be dissonant. The fourth, while obviously a usable tone, is nonetheless dissonant if it occurs above the acoustical root.

And we are not saying that dissonant tones can't be of interest in a radially symmetrical arrangement, either. Just that one cannot ignore the phenomenon of key when discussing symmetry.

Yes, F C G D A E B is radially symmetrical. But if the center of the symmetry is D and the key is C, what is going on acoustically?
It is obviously asymmetrical with respect to the key. The keynote is the most acoustically significant note, how can you consider it irrelevant when considering symmetry?

(Note that acoustically, as I've mentioned previously, I do not believe that P5ths are the correct measure of symmetry and that any symmetry should consider at least a two-dimensional axis of both thirds and 5ths, and that on such a 2-dimensional axis, the major scale is more nearly symmetrical with the key)

Why is that not obvious?

[jazz oud]

Jeff, I will address some of your comments, as they are interesting in their own right, although not germane to the central discussion and I don't believe we have any meaningful disagreement here.

b6: the reasoning is a bit sideways, from my perspective.

While it is true that a m6 occurs between the 5 and 8 (in other words, between the major 3rd and the octave above, i.e., inverting the pure major third), it is (arguably, anyway) not "because" it occurs in the series that it is the preferred JI interval.

JI typically allows for ratios derived from the first 5 tones of the series and their inversions: octave (2:1, 1:2), fifth (3:1, 1:3) and 3rd (5:1, 1:5), ignoring 4:1 since it is just two octaves and is therefore redundant.
To simplify, since 3:1 contains an octave plus a fifth and 5:1 contains two octaves plus a thirds, we use the octave-reduced ratios (which we are allowed, since octave is one of the ratios we are allowed to use) and arrive at:
2:1, 1:2, 3:2, 2:3, 5:4, 4:5
Going down a major 3rd (4:5) and up an octave (2:1) gives us (4x2:5x1=8:5)

With such a simple interval (merely an inversion of one of the 3 basic intervals), it doesn't make much difference if you approach it that way or from the inversion that occurs in the harmonic series. But when you get into more complex intervals, the latter approach does not work very well, so

I would regard the interval's presence in the harmonic series as coincidental (not a random coincidence--of course its presence is due to obvious physical properties, but coincidental in the sense that there are many JI intervals that can't be found in the series and many intervals in the series that are not ordinarily considered typical ratios).

Regarding the 21st harmonic:

To compare intervals in terms of the way we hear them, it is more clear to compare the number of cents (100¢ being a semitone)

21st harmonic of A2 (110Hz) = 2310Hz
octave-reduced ratio: 21/16 or 1.3125, 470.781¢

ET D6 above A6 = 2349Hz
no rational ratio, approx 1.335 or 267/200 500¢

Pythagorean D above A6 = 2346.667Hz
ratio: 4/3, 498.045¢

So we see that the 21st harmonic is actually ~27.3¢ away from the pure fourth (~29.2¢ away from ET), or well over an 8th of a tone. This is quite a large difference.

If we compare ratios (using a common denominator), it does indeed seem like they are very close:

Pythagorean 64:48
21st Harmonic 63:48

But an 1/8 of a tone difference is not a minor discrepancy.

FWIW, I don't (and I suspect neither do you) think that the 21st harmonic has anything whatsoever to do with the P4 as used in Western tonal music.

Also, regarding this:

there are even those willing to debate whether the M3 present between the 4th harmonic and 5th harmonic suffers from a similar quirkiness (“flaws”) as those evidenced in the 7th and 11th harmonics.

“… a just third may at first seem […] too narrow for equal-tempered ears.“
~ W.A. Mathieu “Harmonic Experience” pgs 48-49

A bit disingenuous, considering that, in context, Mathieu's conclusions are exactly the opposite of your claim, and therefore not supporting your statements.