i have been drawn to this book a couple of times but have not as yet gone as far as buying it - it is expensive.

i like the sound of the lydian mode and i am interested in the fact that it is closer to the naturally occurring overtone series than the ionian, being based as it is on 7 intervals of a perfect fifth.  

i think i also am drawn to the fact that george russell has spent a lifetime working on and revising this theory. he has stuck to his guns.

i am currently working with mark levine's books and getting a huge amount from those, but i wondered if anyone has experience of george russell's book and can offer any insights into it's usefulness over and above levine's books?
There are 30 comments, leave a comment.

the origins of the discovery of the minor pentatonic scale
and its transformation into the major scale.
https://www.jeff-brent.com/lessons/evolutionofthemajorscale.html
that is interesting 7, thank you.

i have been doing bit of digging around re. the lydian sound and the overtone series and here are a couple of things:

first, b7 appears in the overtone series before the m7:
https://cnx.org/content/m11118/latest


the first naturally occurring 7 note scale would be what levine calls the lydian dominant scale - the 4th mode of the melodic minor scale.
digging around for information on this i came across this article:
https://www.lydiandominant.com/theory/lydian-dominant_theory.html
which is very interesting and immediately applicable, also echoing levine's material.  

it seems that george russell uses the lydian, not the lydian dominant as the main point of reference for some reason, in spite of that b7 implied by the overtone series:
https://www.lydianchromaticconcept.com/lccoto_2.html
(although i should point out that there is a disclaimer on mr russell's website which says that the pieces of sample text shown are taken out of context).

in any case equal temperament tuning (and therefore i suppose the chromatic scale) as we have with the piano is entirely artificial, some keys being more out of tune than others - a compromise necessary as i understand it for this system of tuning to work acceptably well:
https://www.comp-unltd.com/~jimloy/physics/scale.htm

i suspect the uncompromising goal of a deep understanding of musical truth and the reality of the piano - an instrument with a tuning system that makes many compromises - don't really go hand in hand!
perhaps there are many musical 'truths', dependent on context - middle eastern music probably has it's own 'truths', which contradict western music 'truths' which contradict blues 'truths' etc etc.

perhaps i will leave mr russell alone for now.

i'm not interested in theory per se., more in expressing myself/ improving how i sound, but i do like to keep an open mind you see.
one benefit of the lydian system is it deals with the problem of what michael garrick calls "auntie dolly" - finishing your improvised phrase over the major chord by plonking onto the fourth (e.g f on top of cmaj7, a truly honking dissonance that alone has been responsible for turning whole generations of beginners off jazz for life).  it's called auntie dolly because she comes over every christmas and gets the family to sing round the piano.  her speciality is "jingle bells", performed over a c major chord throughout.  try it - when you reach "oh what fun", that's when you hear auntie dolly in its full majesty.  anyway, the point is - iv in the major scale is in a fight to the finish with i over the tonality of the piece.  the lydian solution is either to kill iv by sharpening it, or to let it win, in which case i major transforms into iv lydian.

there's an awful lot more to the george russell system than this, but for practical purposes, particularly when teaching or learning, it's very handy.

sid
sid - so instead of  
iim7, v7, im7  

we either have:  
iim7, v7, im7#4 or  
iim7, v7, ivm7#4

is this what you are saying?

7 - those patterns are interesting to note, although the last one is not without some degree of dissonance...
im7#4 - yes.  but i don't mean ivm7#4 is somehow a substitution for im, it's just that sharpening the fourth decisively stakes the claim of i as the tonal centre.  for what it's worth, i've tried to put some of this elementary lydian stuff in some kind of context here:
https://www.sidthomas.net/spoonbill_music/harmony/index.htm

sid
sid, i've bookmarked your lessons page and had a quick look through. it looks like a really good resource - thank you.  

i know re-adjusting my ear to lydian harmony will take a while. i think this has a lot to do with the fact that i have grown up with auntie dolly and her ionian fantasies. she was there all through my childhood and she haunts me still.
7 i just sent you a email that you will find relevant to all of the above.  in my opinion comparing levine and russell is to compare apples and oranges or more correctly apples to steak, george russell is in a world of genius way beyond levine.  'but in fairness to levine they are covering two different subjects.  russell is much more pure theory than levine.  levine spends more time on preactical application.  russell really only spends time on practical application as far as chord scales go.  but what russell teaches about chord scales is pure revelation and genius. its funny when i teach my students
about why all music is based on the major scale and i teach them all the modes...  i laugh and i tell them i hope they will be studying long enough to learn why it is basically wrong.  but if i taught them george russell right from the start the average musician would never be able to communicate with them verbally.
another opinion on gr.  i think george russell is as advanced as it gets when it comes to music theory.  his book is for people who feel  
like they have learned everything there is to learn and find themselves
frequently wishing there was someone with a fresh outlook on things.
his theories are not simple.  it requires work to begin to understand  
what he is getting at and then a whole lot more work to begin to understand how mile davis and bill evans applied his theories and a whole lot more work to figure out how you can make his stuff work for you.   the book never never says "do this and this and this stuff will work for you too"  or anything remotely like that.  you gotta figure  
that out for yourself.  it is very adult like that.  i would say it is
x rated like that.  the book is not for kids.
if i say so myself i think this post might be worth reading:

i have a hunch:

i'm no scientist but i wonder if the reason the 'snake' does not quite 'eat it's tail', in other words that 12 intervals of a perfect fifth do not quite result in a complete circle... has to do with time. i wonder if it is related to einstein's theory of relativity.

i know that if two people synchronise watches and then one stays put while the other zooms round the world a certain number of times in a jet plane, then when the two people compare watches there is a discrepancy. so we know that time is relative.

now here's something:

perhaps the lydian scale is the mother scale:

i looked at 7's article about the evolution of the major scale (link above in 7's post) and although i have no doubt it is a true account of how this scale came to be, it seems to me to contain something misleading, right at the very start, a red herring which could explain where traditional western music went 'wrong':

using 7s example, it is stated that: "if we begin at d, then due to laws of physics, the two most harmonically consonant notes with d (as with any note) are either a perfect fourth above or a perfect fourth below". ie.:
a  d  g

however, if we look at these two intervals separately then the problem becomes apparent - to my ear at least.

hold down the a and d: my feeling is that this interval wants to resolve towards the d (the strongest harmonic tendency is down a 5th and a is a perfect 5th above d, although in this case the interval is inverted.)

do you agree?

next, hold down the d and the g.
where does this want to resolve? to my ear the same thing applies, the tendency towards resolving down a fifth. so here is the problem, this interval wants to resolve not to the d, which we are using as the centre of our hypothetical tonal world in this example, but to the g, because d is a perfect 5th above g, although the interval is inverted in this case.

so to my ear what we have is a conflict of tonal centre. the interval  a, d pulls, as is required, towards the d, but the interval d, g, i think pulls towards the g.

do you know what, i think i might be beginning to understand something here...

now mike, your posts make george russell's theory's sound very complex, but perhaps the root of his theory is in fact very very simple, as simple as the wheel. in fact is it related to the wheel?

i tend to agree with the assumption that stacked perfect fifths always gravitate to the root, but this should also apply even if they are inverted, rearranged etc. after all, chord function does not change when chords are inverted.

and perhaps, just perhaps, the overtone series is misleading because of the fact that time is relative? if time stood still, perhaps the overtone series would perfectly reflect the lydian feel and the circle of fifths would be a perfect circle, a perfect wheel.

anyway i tend to agree that g, d, a always gravitate back towards g, no matter in what inversion they are played, which of course includes the order a, d, g (as used in 7's example as related to the hypothetical and i think ultimately inaccurate tonal centre of d.)

i think those 3 notes are related to a root of g, not d.
so it follows that the very first sentence of 7's explanation of how the major scale came to be, highlights the essential problem of major scale and therefore western harmony!!

"the two most harmonically consonant notes with d (as with any note)
are either a perfect fourth above or a perfect fourth below"

i say this is false and the truth is:

"the two most harmonically consonant notes with d (as with any note)
are a perfect fifth above and a perfect fifth above that" (ie. a and e)

7, i think the 'laws of physics' mentioned in your first sentence point towards the e, rather than that g.

hmmmm this is already proving to be something of a revelation to me. did you know they used to think the world was flat?

by the way, i should say 7 i am very grateful to you for all of the help you give on this forum, and for the online materials you make freely available - you are a great chap - thanks.
i wonder if these could be true:

1/ einstein's theory of relativity would explain the "pythagorean error factor", which prevents the circle of fifths from closing. i have a strong feeling that the only reason that the circle of fifths does not close in on itself is because of the influence of time, which is not constant in nature.

2/ the tuning compromises needed for equal temparament (which are only necessary because the circle of fifths does not quite make a complete circle) are therefore red herrings.

3/ the slightly 'out', confusing nature of the overtone series - the way it almost but not quite implies the lydian feel, is also distorted by the influence of time. the discrepancies are red herrings. the true nature of the overtone series points toward the lydian scale.

4/ intervals of a perfect fourth/fifth always gravitate to the note which is the lower by a fifth after any necessary inversion has been carried out.

5/ stacked fifths (eg. c,g,d,a,e) always gravitate towards the bottom of the stack - in this case the c, even if the intervals are rearranged, inverted etc. this is true right up to the 11th superimposed fifth:
c,g,d,a,e,b,f#,c#,g#,d#,a#

if these notes are all inverted or rearranged, the ultimate resolution is still back to the c.

however:

6/ if twelve intervals of a fifth are stacked:
c,g,d,a,e,b,f#,c#,g#,d#,a#,f
the complete circle is implied and the stack becomes weightless.

---

could i be onto something here or do you think i'm a crackpot?

as i say, i'm no scientist, and i wouldn't know how to go about trying to prove that the pythagorean error factor is explicable by relativity but i wonder, i just wonder...
it would seem that the term "pythagorean error factor", which i borrowed from norm vincent: https://www.lydiandominant.com/theory/lydian-dominant_theory.html , is not a commonly used term. this margin of error apparently goes by various names.
yeah, that theory of relativity theory isn't why the circle of fifths doesn't close; it's a lot simpler. mathematically, this is why.  

in pythagorean tuning, the interval of a fifth is the mathematical relationship of 3/2, figured for each consecutive note.
  
so... a fifth higher than a 440 would be 660, a fifth higher than that would be 990, etc. once you go through this operation 12 times, you reach the end of the circle of fifths and return to the original note; in our understanding of tuning in western music (equal temperament), the twelve perfect fifths and seven octaves should be the same interval.

however, the pythagorean tuning results in an error; if the fifths are tuned mathematically, you drift sharper, and the resulting error when you get to the end is called the "pythagorean comma." even though each interval is justly in tune with the note that precedes it, we continue to get higher and higher in pitch.  

when equal temperament was designed, they made every fifth lower by the pythagorean comma divided by twelve -- this compensates for that error.

it's not time, or objectivity; it's just mathematical relationships.

i don't know if these things are red herrings - equal temperament is pretty crucial for us to have things like chromatic music, or for instruments to play with each other without resetting the tuning every time a key is changed.

if you played a piano in pythagorean tuning, which you can in synthesizer software, you'd hear that although your c major chord would be in tune (if you based the tuning in c), all your other chords would sound really out of tune to our ears, getting more and more out of tune the farther you diverted from diatonic chords in c major.


also, the overtone series doesn't distinctly point to the lydian scale, i don't think...it points by the time you have consecutive steps, the notes are so out of tune you can barely make analogies to equal tempered notes. i'd say it's closer to bb c d e f# g a than a lydian scale, but up there we're basically just grasping at really out of tune notes.

hm
hepcatmonk,

i am not saying that equal temperament is a red herring, quite the opposite, i am suggesting that ambiguities in the overtone series/circle of fifths are red herrings.
i would like to briefly clarify my position on a few things here, before anybody gets any wrong ideas (if anyone is still reading that is).

firstly, today i think is going to be something of a milestone in my musical understanding because i now grasp something about the lydian mode which i previously had not. a large piece of the puzzle has definitely clicked into place.

i would like to make clear that i don't know anything about einstein or pythagoras - my point in all that is basically to assert that it seems quite possible to me that the minor discrepancies in the circle of fifths and the slight skew of the harmonic series away from the lydian tendency could be down to a 'glitch', a 'quirk', call it what you will, that will probably eventually be explained by science, and that the underlying truth is for 12 intervals of a perfect 5th to form a complete circle.

as i stated above, i think a, d, g implies g, not d, no matter in what order/inversion those notes are played.
this is because the basic harmonic tendency (other than the octave) is down a fifth.
the a is pulled to the d, the d to the g.

the complete circle of fifths is:
c,g,d,a,e,b,f#,c#,g#,d#,a#,f,c

if you play:
c,g,d,a,e,b
in any inversion, mixed up any old how, i believe the underlying pull is back to the c.

more to come later... it's dinner time
i just try to make nice melodies and not waste too much time alternative methods.
to reiterate:
if you play:
c,g,d,a,e,b
in any inversion, mixed up in any order, i think the underlying pull is back to the c, because all of the intervals of a fifth collapse into each other leading down to the c.

the pull remains if we remove any or all of the notes (apart from the c of course, which needs to be in there to provide it's gravity).

i first tried this by just playing intervals, eg. c and d or c and b, but i think an easier way to hear what i'm talking about is to strike various combinations of several of the notes:

eg. b,c,d,a
or: d,a,c,e,b
or any other combination.
hold the 'chord' down and then try resolving it to each of the individual component notes by taking your fingers off all the keys but one.
try all the possibilities for resolution for the 'chord' you are using. eg. in the first example try resolving the chord to each of the single notes b,c,d, and a separately) then try another cluster of notes and do the same thing.

what i hear when i do this is that although the clusters can resolve, in some cases comfortably, to other notes, the strongest tendency is for them to resolve to the c. this seems to be true however many notes the initial 'chord' comprises, although it is more obvious when using  several notes rather than individual intervals. it also works regardless of inversion and spacing.

so my theory here is that striking a c and then any one or more of the following 5 notes : g,d,a,e,b ,in any inversion, space in any manner, tends to want to resolve back to the c. now you might say this depends on what mood you're in - if you're feeling minor perhaps you'd say it wants to go towards the a, but i think more often than not we end up at the c. i think the strongest pull is towards the c.

now comes the straw that breaks the camel's back:
add the 7th note in the circle of fifths sequence: the f#, the lydian note. we have now reached the opposite side of the circle from the c which we are using as our root. we are on the other side of the world.  the c and the f# form a tritone. the f# does not want to resolve to the c, in fact it is imposing an equal and opposite force on the c. the c is just as likely to be tempted towards the f# as the f# is to the c.

so the lydian mode represents exactly one half of the circle of fifths - a half moon. the lydian mode is as far as you can go without bursting out onto the dark side of the moon. go beyond this by adding the c# or the g# and the pull of the c goes beyond it's state of stasis/synergy with the f# and becomes itself attracted towards the c#
(because the c is the sixth in a series of perfect fifths up from the c#:
(c#,g#,d#,a#,f,c)
so if the f# didn't quite manage to overpower the c with it's own gravity then this c# certainly does).

to think of the times i have looked at that circle of fifths and i've never seen this before.
jazz+ : i know i'm getting a bit serious here, but i have always wanted to try and crack this, and this does feel like a revelation to me. the sort of thing that inspires a shift in thinking/approach.

of course i'm sure george russell goes a lot lot lot lot further but i'd wager that his basic premise is the same as i have outlined above (excepting the spacetime stuff).
here's a story:
i used to live in exeter (in devon, england). i was a student there and my friends and i would go out at the weekend - or in fact any night we had a few pounds in our pockets. sometimes a man was in town playing an unusual amplified instrument. it was a kind of wooden box with metal keys, which must have had pickups. it sounded like a sort of electric harp. he only played simple music but there was something heavenly, something ethereal about it. it floated around the city centre streets, echoing off the buildings. town seemed to be calmer when he was there and the overall effect of the sound was very uplifting - very positive. it was beautiful, simple music and yet there was something about it which was different than all the music i was used to hearing.

this man was a very calming character. he set his 'box harp' (for want of a better name) on a little table, sometimes he would have candles in jars and would also give out sheets of poetry on coloured paper.

it was only years later, after hearing a piece of music that i realised he had been playing the lydian scale.
i should alter that last sentence and say that he had been playing music based on the lydian scale, not just the scale itself.


the laws of physics don't support either of those statements.

paul, with yours, try playing the d with the e an octave above (without playing the a in the middle). the interval is definitely  not consonant. the relationship is much more complex than the simple 4th or 5th, or octave.

also, the laws of physics i think you might be referring to are the natural occurring tones in the overtone series? in that case, there are neither two consecutive fifths nor consecutive fourths. there is however, an octave, a fifth, then a fourth, through which we could say that besides an octave, the most consonant notes notes  would be a fourth above it, and a fourth below it.

something like for g, the most harmonious tones are the c a fourth above it and c a fifth below it. the resulting intervals would be a fourth (g c) and a fifth (c g).

so we can take from this that fourths and fifths are harmonious, as fourths and fifths invert to each other. in jeff's example on his website that you quoted, he is correct in saying that a to d is a perfect interval and d to g is a perfect interval, and that both notes are harmonious in relationship to the d.

however, the contention that d to a is a consonance and d to e is a consonance is not; d to a is the only consonance.

it is misleading, as a to e is a perfect interval, so when playing all three notes simultaneously the dissonance can be somewhat hidden by the structure of it. playing each interval isolated, however, reveals the nature of it.

in any case, it's been really interesting to talk and think about all this stuff.

hm
hm,

i really appreciate your involvement in this. i will continue looking at this later but for now i have sleep in mind and then a busy day tomorrow, so may be friday before i pick up the thread.

thanks again,

paul
now that was cool!  thanks hepcatmonk
for me the real impact of george russell comes after all the talk of the overtone series and all that physics.  when i read his book i kept saying "ok what if i just take your word for it... what if i just accept  
that everythng should be based on the lydian scale instead of the major  
scale.... then what.. where are you taking me?  "   where does he take you...  it is amazing .. a complete reorganization of tonal organization
that opens up a whole new world for creativity in improvisation and composition.... thats all.
hepcatmonk,

you just saved me a whole lot of typing by establishing that there are certain indisputable facts which are quite simply immutable laws of physics.

thanks!

==========================================

however on the subject of the pythagorean comma, it is interesting to note that piano tuners virtually always use stretch tuning (the notes get a bit sharper in relation to pure tempered tuning as the notes get higher on the piano, and the notes get a little flatter as you go lower on the piano keys in relation to tempered tuning).

this manner of tuning retains the ability of notes close together to be able to play in twelve keys, yet preserves the ear's expectation (due to the pythagorean comma) that notes do get a little sharper as they get higher and higher.

in essence, tempered tuning fixes the natural tuning's inability to play equally well in twelve different keys, and then stretch tuning "fixes the fix" (if you get my drift).
i think this will probably be my last post on this subject, but i do have a bit to say, so here goes, i hope you have 5 minutes for a read:

thoughts on previous comments:

hepcatmonk: you said:
"so we can take from this that fourths and fifths are harmonious, as fourths and fifths invert to each other. in jeff's example on his website that you  quoted, he is correct in saying that a to d is a perfect interval and d to g is a perfect interval, and that both notes are harmonious in relationship to the d.

however, the contention that d to a is a consonance and d to e is a consonance is not; d to a is the only consonance."

i think this is debatable:

consider this: jazz players use the following chords all the time, regularly resolving to any of them:
dm7 add 9
d7 add 9
dm7 add 9
dm6 add 9
lydian chords also support the 9th

i would therefore argue that e makes a good case for itself to be considered as somewhat consonant with d.


now consider the g, which in jeff's example is stated to be consonant with d.

jazz players don't very often play a g over a d without wanting to resolve the g down to the f#.

the g is also the "avoid note" in the key of d major. this isn't a good start for it when considering it's case for being regarded as consonant with d.

the interval g > d on it's own is consonant of course, but i think the truth is the d is being attracted down the circle of fifths to the g, not the other way round.

d is consonant in relation to root g, but g is actually dissonant in relation to root d and wants to resolve.



---

now to the pythagorean comma:

i don't know much about maths but i don't doubt for a moment the calculations you supply are correct.  

i also do not doubt that the pythagorean comma exists.

i have a strong feeling, although i can't prove it of course, that the downwards tug of the circle of fifths is somehow related to the pythagorean comma.

the most common chord progressions in jazz follow the circle of fifths, down one fifth at a time. relatively few jazz chord progressions go the other way - down a fourth at a time or up a fifth. i'm not saying these progressions don't happen, but look at: 2,5,1. look at 3,6,2,5,1, look at the bridge in the rhythm changes.

the history and language of jazz is telling us to follow the circle of fifths downwards.

a footnote to this: the general tendency towards downwardly resolving fifths might or might not be exaggerated to a certain degree by equal temperament tuning, i don't know.

in other systems of tuning, where instruments are designed to be played in just one key, perhaps 4,1 cadences work more strongly than they do with equal temperament. perhaps not. i don't know. certainly old religious music and early classical used a lot of 4,1 cadences.



---


now to the overtone/harmonic series:

hepcatmonk: contrary to what you mentioned in your last post, there is no 'natural' 4 in at least the first 20 places of the overtone/harmonic series (apologies for using numbers instead of roman numerals, it's quicker).


these web pages all agree that the 11th harmonic is a #4:

https://cnx.org/content/m11118/latest
https://www.lydiandominant.com/theory/lydian-dominant_theory.html
https://www.smu.edu/totw/overtone.htm
https://en.wikipedia.org/wiki/harmonic_series_(music)

they show the harmonic series is as follows:

fundamental (1)
1
5
1
3
5
b7  
1
2
3
#4
5
6 (a very sharp b6 according to wikipedia, which i think may be inaccurate)
b7
7
1


if you reorder the above series using a piano, disregarding changes of register, the series becomes this:

fundamental(1),1,1,1,1,2,3,3,#4,5,5,5,6,b7,b7,7


one thing to note is the b7 which crops up twice before the natural 7 does.
no wonder jazz is so full of b7s. the harmonic series is pointing to it.

the other thing to note is that there is a 2 in there. this might more appropriately be thought of as a 9 as there are 1,3,5 and b7 tones sounding below it.

to relate this back to my earlier point about e having a case for being considered consonant with root d, the harmonic series agrees with me.

again to support my earlier point, note the lack of the 4, which doesn't appear until later in the series.


now look at the presence of all of the essential tones of a 7#11 chord. no wonder jazz is so full of this chord, it is right there in the harmonic series.


it is also interesting to note that the first chords actually formed by the overtone series (here shown from root c) are as follows:

c
c7
c7 add 9
c7 add 9 #11
c7 add 9 add #11 add 13

all chords common to the jazz idiom.




-------


i think it's fair to say the harmonic series doesn't give many clues as to minor tonality though.


-------


one final thought on all this...

having spent too much time over the past few days vainly and fruitlessly trying to draw parallels between the pythagorean comma and the kind of skewed nature of the overtone series, i have of course drawn a complete blank and made a complete fool out of myself. i still can't help but feel though that the two things are somehow linked, however remote and unquantifiable that link may be. perhaps in centuries to come someone will shed some light on this.

in the meantime, harmony must remain wholly subjective, being influenced as it is by culture, context and personal taste.

thanks for your patience one and all.
good post, paul

i think you do a good job of making harmony seem at least partly subjective rather than wholly subjective. the overtone series occurs naturally and its influence on the evolution of western harmony is obvious dating from the middle ages, the renaissance, and up through modernism.

i see a clue to minor in the overtone series, notice a g minor maj7 chord in the overtone series for c.

now how to put all of this to practical use? i remember asking the  great pianist art lande, a master of free improvisation,  how he thinks about his harmonic shapes when he does free pieces. his gave me a clue when he answered "i know the overtone series very well."
typo correction, meant to write:

i think you do a good job of making harmony seem at least partly subjective rather than wholly objective.
double typo correction

i meant to write:

i think you do a good job of making harmony seem at least partly objective rather than wholly subjective.
say that the interval of a perfect fourth appears between the third and fourth partials; perhaps you were misinterpreting what i wrote.

i think you may have been thinking i was referring to scale degrees when i was talking about that there is an octave, a fifth, and a fourth (consecutively).

to address your other point. d to e is a dissonance, not a consonance. i know that jazz players resolve to chords containing the natural ninth all the time, and i'm not debating the validity of this. i love to put ninths in my major and minor chords.

yet it is not a "consonance" even though it sounds very pleasing to the ears. consonance isn't a subjective term in music theory; it refers to major and minor thirds and sixths, et al., with perfect consonances being fifths, octaves, unisons, and in some cases fourths. dissonances are major and minor seconds, fourths in tonal cases, and major and minor sevenths, major and minor ninths, et al.

should these theoretical terms be applied to jazz? i don't know, but that's merely why i was using them; i was using the standard music theory definitions of those terms to avoid confusion.

what you say about fifths resolving downwards is true. v-i is extremely prevalent in all of western music. i wasn't intending to argue that fifths resolve downwards; merely that mathematics didn't support the claims about the comma.

i think the discrepancies that are coming up are that you are applying the function of harmony to a discussion that seemed (at first) to only have to do with intervals.  

yes, dm9 sounds great, and as the natural 9th e functions great in the chord. however, it's still a dissonance.  

about fourths: modern music theorists consider the fourth dissonant in some places, consonant in others.  

particularly, as you say, talking about the "avoid note" of g in d major, yes, a fourth is dissonant. but this is in the context of tonality, where we're in d major, and we're talking about the fourth above the tonic; i was just speaking about the intervals in general, without regard to tonality or any type of contrapuntal function.

i agree that the most exciting music takes place without being concerned with theory, and just thinking about sound, not rules. i didn't intend with my comments to turn this into a conversation about what "works" musically, or what people do or should enjoy when it comes to music.

merely by labeling a dm9 chord as "dissonant", i'm not saying it's bad or undesirable, or out, or anything...merely that by music theory this chord is "dissonant". most of the chords we play in jazz are, and it keeps it exciting.

hell, some of my favorite music is messiaen, stravinsky, webern and takemitsu; very dissonant. in all of that music, the dissonance "works". it sounds great.

i wasn't meaning to say anything about music, or what harmony sounds good, which i feel is wholly objective, merely to talk about the overtone series and the classic definitions of consonance and dissonance.
thanks hepcatmonk and jazz+

all points taken. it's been very interesting to discuss and investigate all this - i have learned new things.
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