Author Topic: Dogbite's Music Theory  (Read 6282 times)

dogbite

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Dogbite's Music Theory
« on: January 02, 2013, 11:08:32 PM »
first things first:

happy new year!

second things second:

i've been avoiding music forums for a while due to a number of circumstances, one being a series of most unpleasant interactions at a particular jazz guitar forum (which will forever remain unnamed) where civility had for some reason become little more than non-existent and i discovered that several genres of musicians are held in low regard by those who consider themselves "true jazz players."

in particular, players held in lesser regard seem to include teachers, rock players and even, get this, fusion players! funny thing is that although i don't consider myself a fusion player, i am definitely a rock player (with a heavy blues influence) and am most definitely a teacher (been doing that for close to thirty years) and i play jazz when jazz is called for, but (yes, i babble) what what the heck is wrong with rock, blues and fusion players that earned such disrespect in the "jazz community" anyway???

trivia: steve lukather is coming out with a new album, yes?

so, what's this thread about anyway? well, i got a lot of music theory i'd like to discuss and i have no wish to quarrel with some jazz purist about the relevance of things i apply every time i pick up a guitar. therefore, with all due respect to the jazz idiom (i really do love joe pass, george russell, and miles davis) i would like to continue a dialog begun with m7b5 (halfdim7) some time ago regarding altered scales and melody which steps outside the diatonic scale traditionally associated with a particular harmonic environment. (too wordy?) in any event, i tend to write very long posts so forgive me if i go on and on about things which i truly hope will shed light on some potentially confusing topics.

btw, my favorite players include ian bairnson, martin barre, steve howe, and buck dharma. look 'em up if you're unfamiliar with them, that is, if you're inclined to do so as i perfectly well understand that the current musical climate lends itself to what i call "musical overload," where we are constantly surrounded by music, both good and bad, to the end of aural numbness :) :) :)

i promise my next posts will have a thread of commonality, even a purpose, with an emphasis on compositional and improvisational music theory so stay tuned and enjoy!

s/aka/db
s/aka/db

dogbite

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Re: Dogbite's Music Theory
« Reply #1 on: January 03, 2013, 12:10:54 AM »
funny thing also, that i found this, from a dialog i had with million rainbows about exactly what i wanted to start with, the interval most commonly known as the perfect fifth:

Quote
Dawg, or anybody who is listening: I'm finally figuring out how Hanson, Elliott Carter, and other "pseudo-serialists" are using the sets that Forte indexed. Elliott Carter came to these sets independently.
The key lies in which sets they are interested in. For Carter, these were the "all-interval tetrachords" and the "all-triad hexachords." This allows them to break-down the sets into triads, and "stack" them to control (somewhat) the vertical or harmonic aspects of a row, to create music easier on the ear, while remaining true to serial sets.
The idea started with Babbitt, but he was not interested in reconciling tonality with serialism.
George Perle's book "Twelve-Tone Tonality" deals with this subject. His music still sounds quite modern, but more ear-friendly than, say, Boulez.

And Dawg, your response to "why are the Forte sets reduced to normal order" left me wondering. You said it was for convenience, which I can see, but this "normalizing" of the interval-relations is biased, I think, towards atonal or serial considerations of ordered rows and interval relations, not as "tonal entities" or scales with pitch-based starting points.

What do you think?

Here's what dawg said:

 "...biased, I think, towards atonal or serial considerations of ordered rows and interval relations, not as "tonal entities" or scales with pitch-based starting points."
 
i agree. all of the "normalized" lists (and/or "descriptions" if you will) that i've seen are ordered according to only numerical (as in, say, a list of zip codes) criteria - ordered by numerical order rather than actual location; therefore, the complete lack of a tonal context is manifest...
 
one way to fix this is by (try this, really) applying the set theory description (such as 01234) to the circle of fifths rather than the circle of half-steps, thus 01234 = C G D A E rather than C C# D D# E. perhaps not perfectly unbiased in the tonal realm but most certainly "better."
 
do it though: apply 01234 and 01235, etc... to the circle of fifths and see if this does not fit better in terms of tonal consonance. it is very "russellian," as LCC is based initially on fifths.
 
 
dawg :o)

so, what is it with this fifth anyway and why is it so prevalent in music theory discussions as well as the music itself?

this seemingly simple question may actually take up a bit of space, so bear with me as i share my beliefs about this. for one, let us accept the equivalence of the octave as a cornerstone of many aspects of music, that which allows little girls and portly baritones to share the same stage as it would be physically impossible for them to sing the same pitches; however, they can sing the same pitch classes:

common name: the octave
meaning: the eighth diatonic scale step
frequency ratio, in both equal as well as "just" temperament: 2/1

so far so good? the overtone series itself provides this in the form of, for example, a guitarist's open string compared with its twelfth fret harmonic, since the length of [the vibrating portion of] a string is inversely proportional to the frequency of [the vibrating portion]

the next "pure" interval produced by the series is found by comparing the twelfth fret harmonic with the one closest to the 7th fret (as we shall see, it's not exactly located at the 7th fret) and is known as the perfect fifth:

common name: perfect fifth
meaning: the fifth diatonic scale step
"ideal" frequency ratio, in just temperament: 3/2
"actual" frequency ratio, in twelve-tone equal temperament: 2 raised to the power of (7/12) or approximately 1.498307*

*this works out to approximately 7.01955 half steps by the formula F=12logR/log2 [F=frets or half steps, R=harmonic ratio]

so...

1) there will be some players who simply don't care about the .01955 (just under two cents) half step discrepancy between the equal tempered and the pure fifth and i'm okay with that, since much wonderful music has been (and continues to be) composed by players with no such knowledge and/or interest. conversely,

2) there are other players who may be keenly interested in the observable phenomena arising from this discrepancy, so i hope y'all will be patient with me as i share with you my two cents about all this.

for one, i believe that equal temperament works so well because the human ear easily reconciles this small error on a continuing basis and that [for the most part anyway] the effects are not cumulative - this belief of mine has started many an argument online; however, this is why i am stating it here as a belief and not a fact so i'd appreciate it if we'd all be mindful of the difference.

this 0.01955 half step comma is exactly one twelfth of the pythagorean comma of approximately 0.2346 half steps (about 23½ cents)

so, what is the purpose of this (yeah, i'll freely admit overly lengthy) post???

to embrace the fifth as a founding tenet of the music theories of classical and modern western european tradition. i know some of you will already know some, perhaps all, of this but i wanted to start from this particular foundation, before i really do propose some wacky ideas!!!

bear with me. until next post, play a lot of octaves and fifths and fifths of fifths to really get the sound into your muse.
s/aka/db

millions

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Re: Dogbite's Music Theory
« Reply #2 on: January 03, 2013, 07:53:37 PM »
Dogbite said:"...for one, i believe that equal temperament works so well because the human ear easily reconciles this small error on a continuing basis and that [for the most part anyway] the effects are not cumulative - this belief of mine has started many an argument online; however, this is why i am stating it here as a belief and not a fact so i'd appreciate it if we'd all be mindful of the difference."

What is meant by the assertion that equal temperament (hereafter referred to as ET) "works so well"? ET allows all 12 key areas to sound "good," and identical as far as inter-octave relations; but the Bach-Lehman tuning, not an "equal" but a "well" tempered tuning, also sounds good in all 12 keys, though not identical. An advantage? Affeckt?

True, in ET the fifths are slightly flattened to absorb the "overspill," at 700 cents instead of the "perfect" 3/2 of 702 cents. The interval that really suffers is the major third, the "just" being 386 cents opposed to our ET 400 cents. That's a full 14 cents!

Mean-tone tunings were developed in order to create better sounding major thirds. This limited the key ranges, but music did not modulate to distant keys, or get chromatic until later.

So, yes, in certain respects ET works well. In other areas, such as sonority and consonance, it has weaknesses. In performing period music such as Bach, ET is not the ultimate solution.


Dogbite said: "...so, what is the purpose of this...post?...to embrace the fifth as a founding tenet of the music theories of classical and modern western european tradition."


Fifths establish key stations, and give harmonic stability to chords. I think the more compelling reason the fifth is the basis of Western tonality is that (root movement by) fifths and fourths encourage root movement. The diatonic 7-note major scale divides the octave unequally, by 4ths and 5ths.

Look at all the intervals: all of them have complementary intervals which add up to an octave (min. 3rd/maj. 6th, etc.), and the smaller of these two complements generates a cycle which divides the octave symmetrically: one cycle of m2, two cycles of M2, three of m3, four cycles of M3, and six cycles of tritones; except the p4 and p5: this complementary interval does not generate a cycle which divides the octave symmetrically, but must extend through many octaves in order to reach its initial starting point again. Thus, there is only one cycle of perfect fourths, or perfect fifths.

Perhaps this is why the 4th & 5th are different; instead of dividing the octave fractionally, they are expansive by nature; they go 'outward' past one, past the octave, into other 'root' stations. Hence, the use of 4ths & 5ths to create root movement.

So Western classical music, as it developed this "harmonic function in time" aspect, and developed "away" from Gregorian chant, became less centered, less "droney," and more varied and moving. The Baroque, and the Age of Enlightenment are perhaps the reason why. As science developed, and we learned that our Earth was not the center of the universe, and thinking developed, a new emphasis on the "nobility of Man" emerged, leading to the gradual loss of power by The Church, then Kings and nobility, then finally, Democracy, and the Rise of the Common Man.

Man was more conscious now, more "outward directed," more cerebral. He did not need to submit to the drone's "inner-directed" power like he used to; he wanted to actually go "outward" do "God's work" (God was "out there", not "in us") and dominate and conquer his world, in the name of God. Besides that, the "drone" had also always been associated with inner-directed "primitive" Eastern religion and foreign musics. The new, enlightened and rational Western Man was an active, outward-moving, conscious man, and his developed harmonically restless and moving music reflected this. There was no need to sit in front of a candle, sing droney chants, focus "inward" and "lose one's ego in submission to God." We had bigger fish to fry, and our new harmonic juggernaut would aid us in spreading the glory.

The drone was now seen as what it always was: an inner-directed, dark, sombre vision bordering on nothingness: the cessation of conscious will, stillness, quiet, meditative, "inner," lacking movement. Perhaps a little too close to the heretical, inner-directed, forbidden "nothingness" of Eastern religions and rogue, uncontrolled "spirituality." Too close to the Devil!

Now, Western music had become elaborate, full of detail, magnificent in form. Quite a bit of conscious cerebral effort was needed to follow these long developments; not a task for the zoned-out monks who chanted their way to ecstacy.

So here we are in the 21st century. What has happened since Gregorian chant appeared? A lot of harmonic development, that's what, finally culminating in the late-Romantic chromatic wanderings of Schoenberg, Strauss, and Mahler, and jazz.

So, as in my other blog about the "universes" of music, we see that Man's attitude toward his world, himself, and his God, have shaped his expressions of it, through his art.
« Last Edit: January 04, 2013, 08:02:56 AM by millions »
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"The trouble with New Age music is that there's no evil in it."-Brian Eno

dogbite

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Re: Dogbite's Music Theory
« Reply #3 on: January 04, 2013, 01:01:01 PM »
Quote
Dogbite said:"...for one, i believe that equal temperament works so well because the human ear easily reconciles this small error on a continuing basis and that [for the most part anyway] the effects are not cumulative - this belief of mine has started many an argument online; however, this is why i am stating it here as a belief and not a fact so i'd appreciate it if we'd all be mindful of the difference."

What is meant by the assertion that equal temperament (hereafter referred to as ET) "works so well"? ET allows all 12 key areas to sound "good," and identical as far as inter-octave relations; but the Bach-Lehman tuning, not an "equal" but a "well" tempered tuning, also sounds good in all 12 keys, though not identical. An advantage? Affeckt?

True, in ET the fifths are slightly flattened to absorb the "overspill," at 700 cents instead of the "perfect" 3/2 of 702 cents. The interval that really suffers is the major third, the "just" being 386 cents opposed to our ET 400 cents. That's a full 14 cents!

Mean-tone tunings were developed in order to create better sounding major thirds. This limited the key ranges, but music did not modulate to distant keys, or get chromatic until later.

So, yes, in certain respects ET works well. In other areas, such as sonority and consonance, it has weaknesses. In performing period music such as Bach, ET is not the ultimate solution.

yes, major thirds (as well as their inversions, the minor sixths) are "out" by close to 14 cents (i was gonna get to that :)) and minor thirds (and their inverted major sixths) are out by that same 14 plus the two cents error produced by the fifths, or 16 cents. further, the septimal intervals such as 7/4 introduce an error close to an entire third of a half step, so yes; 12TET has its drawbacks. so, to the extent that the music requires and/or otherwise allows microtonal adjustments to blend harmonically with such situations, yes - i champion that. for my (as the thread is called, dogbite's) theory though, i require free and unfettered access to all twelve keys and intervals in the chromatic scale. the one thing i do really embrace in 12TET is that yes, certain intervals are not as accurately presented, but they are all equally represented and/or misrepresented. so, whether i do that thing in Db major or F major, i don't have to worry about the wolf fifth in between the second (re) and sixth (la) steps in either case because all intervals are equally "wolfy" :)

check this out:

if the fifth is the first non-octave interval produced by "harmonic theory," than let's see what we can generate from a center nucleus, "C" by going up (or down) a fifth (or fourth) and get F C G (or G C F), now what can we do with this? for one, the root progression of the authentic and plagal cadences are described: I IV (and IV I) as well as I V (and V I). secondly, it describes the ambiguous "consonance"* of the unresolved sus4 chord and it's inverted sus2 and 7sus4(no5) cousins but let's go but one step further:

Bb F C G D (or D G C F Bb if you prefer) and look at the possibilities:

D
Bb
F
C
G

the quartally-derived "so what" chord as well as the linear G Bb C D F or G minor pentatonic and Bb C D F G or Bb major pentatonic scales. side note, the Bb D interval if defined by the series of pure (3:2) fifths would differ from the pure (5:4) major third by an interval of 81:80, the so-called "syntonic" comma of approximately 21½¢.

but commas on top of commas, all derived for differences between pure octaves and pure fifths, thirds and septimal sevenths, etc, i'm sure glad that in my own playing, i don't have to worry about it (because of 12TET) other than to know that the effects (wolf intervals, etc) are there and that i can manipulate certain tones microtonally (if i have the opportunity) to improve the overall harmonic situation.

i guess what i'm really saying is that my theory is based upon the pure fifth being rounded by that mere two cents on a continuing basis within the circle of fifths and that although i know that especially the thirds (and sixths) take the most pounding in the form of as much as a sixth of a half step, the availability of twelve pitch classes and twelve tonalities is worth the "wolf in the room."

next up, i'd like to take those fifths one step further into the seven-tone world. the purpose of this post? to embrace root progression in fourths and fifths and the introduction of the pentatonic scale as a linear manifestation of a sequence of fifths. i'll cover tertiary harmony (thirds) in a different dialog though, as i want to wrap up a few things here first.

look sharp!
s/aka/db

dogbite

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Re: Dogbite's Music Theory
« Reply #4 on: January 05, 2013, 12:36:34 AM »
i could have sworn i saw a post by millions after my last one; did you remove it, millions? in any case i wanted to clarify a couple of things that may help alleviate some of your concerns regarding what i was getting at. let me tell it this way:

one thing i tell students who have lots of questions is, "don't worry; it'll only take you about seventy-five years of serious study to master all of this stuff," and then i lean them into this one slowly: "i really meant instead of "master," how 'bout "barely scratch the surface of but one of its myriad concepts, this music thing..." or some shit like that :)

and i most certainly didn't mean for the teaching metaphor to give millions to get the impression i was trying to "teach" him anything as much as to share some ideas that we may all apply in as simple manner as possible...

i would also like to share this: i have discovered that in such a forum as this, it is difficult to balance the two extremes of either 1) "looking at an elephant through a microscope," 2)"trying to illuminate the entire elephant," and everything in between. i am not trying to do #2, which would involve a whole lot of discussion about history and evolution of musical theory and practice, which for the purposes of this particular discussion are not terribly useful. my purpose is to present useful snippets in a "modular" fashion, particularly for the purposes of improvisation, but which may also be applies to composition and arranging.

here's one: by extending the nucleus outward into a seven tone scale, Eb Bb F C G D A, where C was used as the center of "fifths radiating outward," one thing that may be derived from this is three pentatonic scales themselves stacked in fifths:

Eb major [C minor]
Bb major [G minor]
F major [D minor]

and one of things i remember from million's missing post was a reference to "directionality" or "gravity" where the bottom note in the fifths dyads (or the top note of an interval of a fourth) is perceived as the tonic of the two tones:

for example, C up to G (or G up to C) where "C" is perceived as the tonic. (see russell's "lydian chromatic concept" vol. one (2002, concept publishing) or hindemith's the "craft of musical composition" (schott, 1942) which leads to a bias in direction. here's how this works in the pentatonics:

Cm7 groove; you're playing happily along with C minor pentatonic and by going up (to G minor pentatonic) or down (to F minor pentatonic) a fifth, you get the two most closely related pentatonics to the original, but the G clearly sounds better because the F minor pentatonic "weakens" (million's word as i recall) the original tonality by making the F want to be the tonic because F is the tonic of the dyad C F and C is the tonic of the dyad C G.

so here's the snippet: go up a fifth (or even a fifth of a fifth) with those pentatonics for a real cool m7(9) sound:

Cm7 chord

C minor pentatonic C (root) Eb (b3) F (4 or 11) G (5) Bb (b7)

G minor pentatonic G (5th) Bb (b7) C (1) D (9th) F (4 or 11...)

practice regimen for this snippet: work around the circle with those pentatonics; iow, every time you play a pentatonic, play also the one next to it on both sides of the circle of fifths: C G D A E B Gb Db Ab Eb Bb F C...

one more example:

Ebmaj7 chord

Eb major pentatonic Eb (root) F (9th) G (3rd) Bb (5) C (6)

Bb major pentatonic Bb (5th) C (6) D (7th) F (9th) G (3)

i know that these devices can be obtained independently of what i've shown but follow me for a spell; there are more snippets to be extracted from a simple expansion of fifths, so let's go have some fun with fifths!

cya next time :)
« Last Edit: January 05, 2013, 12:39:23 PM by dogbite »
s/aka/db

millions

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Re: Dogbite's Music Theory
« Reply #5 on: January 05, 2013, 08:13:22 AM »
Sorry about that! After composing a rather lengthy post-reply, I decided that my observations about "strong" and "weak" progressions might have been distracting or perceived as too argumentative. I'm surprised that Dawg was able to use that idea as an improvisational strategy!

Part of my hesitancy to post was Dog's use of the bi-directional "radiating" build-up of fifths (C-G-F) vs. a directional clockwise projection of fifths (C-G-D etc). Why exactly are you building your circle this way? I know that piano tuners often start on F in order to get all the white-note keys taken care of. I know that part of this might be that F-B is the only white-note tritone, and this physical design of the keyboard is part of that.
« Last Edit: January 05, 2013, 08:29:47 AM by millions »
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dogbite

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Re: Dogbite's Music Theory
« Reply #6 on: January 06, 2013, 10:31:38 PM »
form (structure) vs function (theory)

is the answer (or non-answer) as regards the "why build outwards" [F<C<G<D>A>E>B] as opposed to "upwards" [F>C>G>D>A>E>B] and i dare say there is merit in both approaches. russell takes the upwards approach to a logical limit; however, i didn't start the thread to discuss LCC (inside sarcasm: maybe somebody should start an LCC thread :))

the purpose of the outwards approach is to take advantage of structural symmetry in the form of subsets and alterations. this is in no way meant to deny the harmonic importance of embedded triads, such as:

F C G D A, the F major / D minor pentatonic where the consonant triads F and Dm provide a relatively clear sense of tonality within the stacked fifths; C G D A E with C and Am; and G D A E B, with G and Em - but what of it?

what i'm saying is that although the F (not only the single tone but also the triad) may be the primal source of tonality while using the F, C and G major pentatonics over Fmaj7/6/9/#11, the structural symmetry of the C pentatonic being the center of the group F, C and G pentatonic provides a "center" with which to visualize melodic possibilities. C is not more important (function) but it is the center (form):

form vs function

hey check this out - here's one with definite directionality; iow, not based on a viewpoint of symmetry as much as the tonality provided by the pentatonics' major triad:

Q: which five pentatonics have the tone "C" in them?

A: C, F, Bb, Eb and Ab pentatonics.

so in a rockin' groove such as C Bb F C, hang on the C but build the pentatonic of the chord of the moment, using the tone C as an anchor:

C
C D E G A

F
C D F G A

Bb
C D F G Bb

Eb
C Eb F G Bb

Ab
C Eb F Ab Bb

classic rock, fusion, even grunge has used the I, bIII, IV, bVI and bVII chords perhaps not for the "logic" as shown above but they work for whatever reason - and i like the logic...

i know i've been holding off on the seven-tone symmetrical applications but i wanted to give the pentatonic stuff a bit of space before we get lost in the hexa- and heptatonics...

enjoy!
s/aka/db

millions

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Re: Dogbite's Music Theory
« Reply #7 on: January 07, 2013, 11:53:15 AM »
Sorry for all the questions.

It appears that you have switched your starting radiating point from C to D. Why? Why start on D to project our fifths? I understand that D dorian is symmetrical in terms of intervallic spacing, but I thought we were concerned with fifths and their projection (either "upwards" or "outwards").

I have understood the purpose of "projecting intervals" like the fifth as a way to generate scales. In projection of fifths, we get pentatonics first (C-G-D-A-E or C-D-E-G-A), then eventually all 12 notes. But even this seems flawed; it would seem that a 7-note C major diatonic scale would emerge before all 12 notes were generated, but instead we get C-G-D-A-E-B-F#, (G-A-B-C-D-E-F#) which gives a G major scale, then adding the next fifth C# makes a D major.

Is the idea of starting on C flawed? It works if you start on F: F-C-G-D-A-E-B. How did "C" become the basis of Western music, if the 12-note scale was generated by fifths starting on F?

Has another idea somehow creeped in without my knowing it? I need a bird's-eye view of what you're getting at first; that's the way I think. I read a Cliff notes, then the book.

Is this anything to do with "form vs function?"

Do you mean by this term: harmonic/vertical ("form," or intervallic spacing within a chord) vs horizontal root movement in time ("function")?

Your D is symmetrical in terms of intervallic spacing; C is not. Does this matter?

We can look at the C major scale as dividing the octave unequally at the fifth, G (with F# as the axis of real symmetry in the chromatic octave). This unequal division at G facilitates root movement by fifths or fourths. "Modern" music thinkers such as Bartók used a symmetrical division of the octave (at the tritone) to negate a sense of tonality created by major and minor scales. The D dorian divides the chromatic octave equally, with G#/Ab as the pivot point, or "axis of symmetry;" also, your D is symmetrical in terms of intervallic spacing. Is this a consideration? Does your building "outward" from D involve "negating tonality" away from V-I tendencies?

Does your use of D involve factors of root movement or tonality in this way? Are you using symmetry to divide the chromatic scale equally for tonal reasons, or for other reasons? Is your primary concern in using D as your center "harmonic" rather than "functional?"

Is a "symmetrical pitch center" any different than a tonal "hierarchical pitch center"?

« Last Edit: January 07, 2013, 12:20:27 PM by millions »
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"The trouble with New Age music is that there's no evil in it."-Brian Eno

dogbite

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Re: Dogbite's Music Theory
« Reply #8 on: January 07, 2013, 10:57:35 PM »
i'm gonna have to handle those questions one at a time and out of order if you don't mind :)

1)

Quote
Is a "symmetrical pitch center" any different than a tonal "hierarchical pitch center"?

yes

2)

Quote
Is the idea of starting on C flawed?

depends who you ask. russell clearly thought so; however, i dare say that it does not matter because of #1 above.

3)

Quote
Does your use of D involve factors of root movement or tonality in this way?

no

4)

Quote
Are you using symmetry to divide the chromatic scale equally for tonal reasons, or for other reasons?

other: form (the way i am using the term) means in this context, "structure." D is an axis of symmetry as is G#/Ab and this is purely a structural phenomenon rather than tonal, where C may become the tonal center as the root of the I IV V chords C, F and G; A may become the tonal center as the root of the i iv v Am Dm and Em*

*although the minor tonality is strengthened by the "borrowing" of the V (or V7) from its parallel major - neat though that this introduces the axis of symmetry G#/Ab in the form of an E major triad...

5)

Quote
Has another idea somehow creeped in without my knowing it? I need a bird's-eye view of what you're getting at first; that's the way I think. I read a Cliff notes, then the book.

yes. okay, here's the crux of the biscuit: tonality is introduced by the same means it always has - the hierarchy of tones through triads and root progression (to deny the V7 I is absurd, imo, as well as to deny russell's lydian tonic F as the bottom of what he calls the "ladder of fifths") but what i'm getting at is to describe as many useful structures (scales if you will) through the least number of steps:

start with the center of the white keys, D and do that outward expansion F C G D A E B

linear: A B C D E F G

the most closely related melodic minors (jazz usage: ascending form only) are F C# G D A E B

linear: A B C# D E F G

and F C G D A Eb B

linear: A B C D Eb F G

again, no attempt to define tonality is made here, only structure (form) and symmetry, where the alterations C# and Eb have been introduced either one at a time as the two melodic minors listed above or both at the same time: F C# G A Eb B

linear: A B C# Eb F G

note that here the D has been omitted to provide the whole tone scale.

next step is to "split the fifth" of those melodic minors (D E F G A B C# becomes D E F G G#/Ab A#/Bb B C# and C D Eb F G A B becomes C D Eb F F#/Gb G#/Ab A B) to provide octatonic diminished scales.

now let's get practical: one of the things that binds all of these scales together is the commonality of the tritone F and B, the primary guide tones of the G7 chord. so let's use all of these scales to embellish a G7:

G7 with A B C D E F G = G mixolydian

G7 with A B C# D E F G = G lydian dominant

G7 with A B C D Eb F G = G aeolian dominant (G mixolydian b6; also, G melodic major)

G7 with A B C# Eb F G = G whole tone

G7 with A#/Bb B C# D E F G G#/Ab = G half/whole diminished

G7 with A B C D Eb F F#/Gb G#/Ab = well isn't this interesting: a G scale with no G in it :)

there is a graphic relationship between all of these scales (and two others) that through structure and symmetry (form) can be applied to tonally-based situations (function) but before elaborating further, i'd like to make sure i didn't lose everyone in the rapid description first.

i want to emphasize that there are some very subjective criteria being used to generate the "most closely related" scales, such as:

* i don't want (for example) hungarian minor to be in the list, as it's too specific - too eclectic

* only scales that are described as "necessary" to modern improvisation are allowed, such as diatonic major, melodic minor, octatonic diminished, wholetone and (the last two scales will take care of the rest...)

but i especially emphasize that structure (form, symmetry) is in no way being used to define tonality (function, purpose) and this is what has caused more than a couple of internet pie* fights:

*pie joke:

patient: "my tummy hurts."

doctor: "what did you eat?"

patient: " i had pie."

doctor: "how much pie?"

patient: "you know, a PIE!!!"

iow, let's keep it light :)

much respect to millions for some very good questions which deserve straight and honest answers. i hope i have answered some of them and through my presentation have answered others...

pot pi
« Last Edit: January 14, 2013, 09:12:38 PM by dogbite »
s/aka/db

millions

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Re: Dogbite's Music Theory
« Reply #9 on: January 08, 2013, 05:39:31 AM »
Okay! The thing that clarified it all is seeing how D as the center of radiation is also an "F" directional series. That solves the problem of fifths generating the white-note series from which both C major and D dorian are drawn from! It also reinforces my faith in George Russell (and you, Dawg).

I noticed that the tritone F-B, being invertible, is not only the guide-tone of G7, but also C#7(Db7). All those scales you listed get real weird sounding if you alternate the root/fifth G-D and C#-G#. Very Thelonious sounding.

The last scale is puzzling: G7 with A B C D Eb F F#/Gb G#/Ab.
I hear it as a suspension, resolving to G min (if root is G) or C# minor (if root is D). It looks like it's an incomplete whole-half diminished scale.
« Last Edit: January 08, 2013, 06:21:14 AM by millions »
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dogbite

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Re: Dogbite's Music Theory
« Reply #10 on: January 08, 2013, 07:28:55 PM »
Quote
...if you alternate the root/fifth G-D and C#-G#.

funny thing is that if you transpose that down a minor third (E B Bb F) and listen to the second tune on my myspace page, this is the bass line for the intro. coincidence yes but real cool nonetheless...

Quote
The last scale is puzzling: G7 with A B C D Eb F F#/Gb G#/Ab.
I hear it as a suspension, resolving to G min (if root is G) or C# minor (if root is D). It looks like it's an incomplete whole-half diminished scale.

to me it sounds like "playing off the ii chord," or the Dm (with half-whole diminished) and as such it would be a form of G7sus wouldn't it...

try also with other chords related to the C major scale: C, Am, Fmaj7(6/9/#11), Dm7(6), Bm7b5, G7(9/11/13), E(m)7b9sus* and you guessed it, Db(alt)

*some would call this E11b9 but that's yet another internet argument from which i grew weary :)

Quote
...how D as the center of radiation is also an "F" directional series.

yes, russell's lydian tonic, F. also the ionian (horizontal major) tonic of C and the aeolian (horizontal minor) tonic, A. the reason i'm not assigning a particular "modal tonic" is that i can think of no real reason you could not proceed with a "phrygian tonic" of E.

so, although that weird scale that had no G in it for the G7 (mixolydian) it did however have Dm dorian's "D" and many others - remember that article on "chromatic modes"? it (in dogbite's theory anyway) is grounded upon the tritone pair F and B rather than the root of the chord. the tritone pair F and B, by the way, are the "essential tones" which define the mode: for D dorian, the B distinguishes it from aeolian's Bb; for A aeolian, the F distinguishes it from dorian's F#, etc...

the two other scales are D E F G G#/Ab A B C D which is a superset of A harmonic minor and C harmonic major, as well as not only a mode of the C major bebop scale but is also a real cool octatonic blues scale on D, and the last scale:

the tritone of C major which shares the F B tritone pair: F#(Gb) G#(Ab) A#(Bb) B(Cb) C#(Db) D#(Eb) E#(F)

for a total of eight scales. if you're with me so far, next up is the graphic showing how all of these (yes this is a clue) eight scales are related to each other in a three-dimensional grid. yes, it is entirely subjective, this graphic; however, it is not designed to illuminate a vast number of, for example, forte's sets as much as to put the most common ones under one dynamic.

next up: the graphic

ps - thanks for being patient with me; the old-timers (jazz purists for the most part) are understandably suspicious of such things and i am definitely trying to view new angles. i would also insist that i have invented nothing as much as pointing out relationship under the guise of "this is real cool;" iow, not theory (function) as much as structure (form).
s/aka/db

millions

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Re: Dogbite's Music Theory
« Reply #11 on: January 13, 2013, 09:29:10 AM »
start with the center of the white keys, D and do that outward expansion F C G D A E B

linear: A B C D E F G

the most closely related melodic minors (jazz usage: ascending form only) are F C# G D A E B

linear: A B C# D E F G

and F C G D A Eb B

linear: A B C D Eb F G

again, no attempt to define tonality is made here, only structure (form) and symmetry, where the alterations C# and Eb have been introduced either one at a time as the two melodic minors listed above or both at the same time: F C# G A Eb B

linear: A B C# Eb F G

note that here the D has been omitted to provide the whole tone scale.

next step is to "split the fifth" of those melodic minors (D E F G A B C# becomes D E F G G#/Ab A#/Bb B C# and (C D Eb F G A B becomes C D Eb F F#/Gb G#/Ab A B to provide octatonic diminished scales.

Okay, let's look at the interval content of Dawg's sets. All 7-note heptads will yield 21 intervals.

Interval class: 1,11 (m2/M7)----2,10 (M2, m7)----3,9 (m3/M6)----4,8 (M3/m6)----5,7 (p4/p5)----6 (tritone)

A B C D E F G:      3                       5                       4                     3                      6                  1
A B C# D E F G:    2                       5                       3                     5                      4                  2
A B C D Eb F G:     2                       5                      4                      4                     4                  2
A B C# D Eb F G:  2                       6                      2                      6                      2                  3

I've listed above all the 7-note scales, including the "D" Dawg omitted in the "whole tone-ish" scale. Note that by leaving out this D, dawg eliminated 2 min seconds, 2 minor thirds, and 2 p4/5ths. Quite a change for leaving out one note!


Next, here is the "true" 6-note whole tone scale (with the D omitted). With all hexachords, only 15 intervals are produced.

A B C# Eb F G:     0                       6                       0                     6                       0                   3

Next, are the 2 octachords known as the diminished scales. All octachords yield 28 different intervals.

D E F G G# A# B C#: 4                 5                       7                       4                       5                   3

Note that redundancy increases with the 8-note sets; with the 6-note WT scale, it decreases.

—————————————————————————

Some general tendencies: With the exception of the diminished (which gets increasingly redundant or chromatic):

Minor seconds are decreasing 3 to 0

Major seconds are increasing 5 to 6

Minor thirds are decreasing, 4 to 0

Major thirds are increasing, 3 to 6

Fifths and fourths are decreasing 6 to 0

Tritones are on the rise, increasing from 1 to 3

So, the general trend is that we are approaching a "whole tone stasis."

The diminished scale seems to be in the "other direction," starting from our first A B C D E F scale. If we increase the note-count from 7 to 8, we are becoming more chromatic, increasing our m2s, m3s, with other intervals (M2, M3, p4/5, and T remaining roughly the same trend-wise.

What does this mean harmonically? It seems that one direction (towards the whole-tone scale) becomes more "harmonically static" by omission (note the complete absense of m2, m3, and p4/5). In the other direction, we are drawn towards a sort of "chromatic stasis" by adding notes and increasing m2s, m3s, and tritones.

This could work backwards as well. Just fill in numbers of intervals to get the sound you want. Just remember the total interval count must adhere to the number of pitches in the set: triad=3, tetrad=6, pentad=10, hexad=15, heptad=21, octad=28.

The tetrad F-G-B-C# is interesting...M2=2.....M3=2....T=2. Add more notes to add up to 6, such as F-G-B-C#-D-E (m2=2, M2=3, m3=4, M3=2, p4/5=2, T=2). Add A next, for a 7-note scale F-G-A-B-C#-D-E (m2=2, M2=5, m3=4, M3=4, p4/5=4, T=2). Note that "sonority" can be controlled somewhat by emphasizing consonant intervals and de-emphasizing dissonant intervals.



« Last Edit: January 13, 2013, 04:35:04 PM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

dogbite

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Re: Dogbite's Music Theory
« Reply #12 on: January 14, 2013, 09:42:59 PM »
thanks, millions, for that analysis of dawgs scales using the interval class vectors of set theory. no real reason you cannot leave the D in the "seven-tone wholetone-ish scale" either, thus the description of my own choices as "purely subjective."

warning: what follows may seem a bit of a stretch so bear with me while i get to all eight scales...

now for a little "directionality" in 3-D space:

let's call the original scale ( A B C D E F G ) the origin. and then place the scale that introduces C# ( A B C# D E F G ) directly to the left, something like this:

{A,B,C#,D,E,F,G} {A,B,C,D,E,F,G}

now, let's place the scale that introduces Eb directly under the origin as well as the wholetone scale under the scale on the left:

{A,B,C#,D,E,F,G} {A,B,C,D,E,F,G}

{A,B,C#,Eb,F,G} {A,B,C,D,Eb,F,G}

and notice the directional properties; left = introduce C# and down = introduce Eb. of course down and left = introduce both...

what we have now is a two-dimensional lattice where position is directly related to the conditional tones C# and Eb. the third dimension (behind the origin, as well as all of the other scales above) will introduce G#/Ab.

i know that millions wants a sneak peek so here's the four scales behind the ones already presented:

{G#/Ab,A#/Bb,B,C#,D,E,F,G} {A,B,C,D,E,F,G,G#/Ab}

{A#/Bb,B,C#/Db,D#/Eb,F,F#/Gb,G#/Ab} {A,B,C,D,Eb,F,F#/Gb,G#/Ab}

here's where the notation really gets cumbersome; should i call it a flat or a sharp or...

???

the question marks are not intended to project uncertainty on my part, as there is a very specific "set of sets" being described here.

thoughts?

ps - i'll check for typos periodically; i fixed a parentheses anomaly in a quoted passage and i hope i didn't scramble anything further than that :)
s/aka/db

millions

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Re: Dogbite's Music Theory
« Reply #13 on: January 15, 2013, 08:38:51 AM »
{A#/Bb,B,C#/Db,D#/Eb,F,F#/Gb,G#/Ab}

This scale is 7 notes, and I see it as a G#/Ab rather than an A#/Bb, because its (imaginary) axis of symmetry is "D." It's also a lydian scale starting on "B," the tritone counterpart of our original "parent" F lydian. Are you "wrapping around" from F (D) over to B (G#), its inverted tritone counterpart, in order to complete some sort of "matrix"?

Note that the alterations Dawg is making, C# and Eb, are symmetrical  to the center note "D," creating a whole-tone scale.

The best to picture this is to go to a piano and put your hands on the D-D dorian scale. This has G# as the axis of symmetry. Then "expand" it by taking fingers off of "D" and moving them to the black notes C# on bottom, and Eb on top. See how the symmetry is retained? This also explains the absence of D in Dawg's scale. If we then "complete" this action by moving the "inner" E to Eb, and C to C#, we get a whole tone scale, albeit an 8-note version of it which spans C# (on bottom) to Eb (on top).

The next two logical alterations would be B to Bb and F to F#, retaining this outward radiating symmetry, and giving us the diminished scale, if we add back the E on bottom, and C on top.

If so, my next problem is to figure out the exact nature of this matrix.

As I observed earlier, both the diminished scale and the whole-tone scale contain tritones. The diminished heads for "chromatic stasis" by adding notes, the whole tone heads for stasis by subtracting notes. These are the multipliers of the intervals 2 (M2), 3 (m 3) and 4 (M 3), with 2 (M2) as a sub-factor of 4, as well as being the "whole" in the diminished whole-half scale. 2, 3 & 4 are factors of 6 (tritone). This takes care of 4 of the 6 essential intervals, leaving only the p4 and p5. (m2) gets to go with these two, as part of the chromatic/fifths conspiracy.

Is this a demonstration of "living without fifths?" Is this Dawg's revenge on the V-I?

Should we call it "Final Chromatic Alteration: The Reckoning (Part VI)"

Actually, I'm trying to see the gestalt. Maybe a structural map of the building is in order...the Agent Smiths are closing in, and I need an exit.

« Last Edit: January 15, 2013, 02:50:48 PM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

Halfdim7

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Re: Dogbite's Music Theory
« Reply #14 on: January 15, 2013, 09:37:50 AM »
Maybe we could load this into a CAD program, for easier visualization.  :o
....lame-ass, jive, pseudo bluesy, out-of-tune, noodling, wimped out, fucked up playing....