Attack of the Reciprocals, simple question, Diatonic application

5.inversely related or proportional; opposite.

6.Mathematics. noting expressions, relations, etc., involving reciprocals: a reciprocal function.

7.Navigation. bearing in a direction 180° to a given direction; back. –noun

8.something that is reciprocal to something else; equivalent; counterpart; complement.

9.Also called multiplicative inverse. Mathematics. the ratio of unity to a given quantity or expression; that by which the given quantity or expression is multiplied to produce unity: The reciprocal of x is 1/x.


Like that.

For TGP "theory" purposes, you're looking for the inverse of the Tonic function.

peace

 

It is, but it's built into the temperament, so you're observing it no matter what method you choose, including none at all.
That's freedom.

 

I think you're just noticing one interesting artifact of tertiary harmony. Moving up or down in any diatonic interval other than octaves will take you through the entire scale. Note that, if you continue the inverted diatonic (key of G) pattern, you'll eventually arrive back at the A. Likewise if you continue upward in thirds. The chord you've laid out in both cases is Am13, which contains all the tones in the G major scale.

Note also that the upper tones of an Am13 chord are B-D-F#, and adding the root (A) at the top gives you a Bm7 chord. So one way of viewing chord tones above the 7th is with a polychord approach.

Another concept that this leads to is that, for a diatonic 13th chord, any combination of diatonic tones in that key played together will form a useable voicing.

Symmetric harmony provides some really powerful ways to look at chord construction and associated scales/modes, but I wouldn't recommend getting stuck in too deep in that. Eventually, you have to develop your own idea of what sounds good. The theoretical underpinnings are all after-the-fact attempts to rationalize what cannot be completely rationalized: why does this sound good to me? They can help you identify groups of notes that can be made to work together, but how you put them together is what will ultimately determine whether or not they work.

 

My view is that the fact of reciprocity is not a guide to musical value.
You say there are "too many" patterns for it to be of no value. Why? Why does the amount of perceivable reciprocity have a bearing on musical value?
(It's an argument like that of the Creationists, as I've said before, or conspiracy theory. Just because we see patterns doesn't mean they have any meaning, or are all connected, or are relevant to us - or are indeed patterns at all, as with stars in the night sky. We need to be careful to look elsewhere, to examine the context in which we are looking for meaning.)

What you've highlighted above is essentially the symmetry of dorian mode, but spread over 3rd intervals from the centre. Here's the symmetry around the centre:

Code:

 
|  |  |  |  |  |  |  |  |  |  |  |  |
1     2 b3     4     5     6 b7     1

My feeling is that this may have a bearing on why it was mode I in the middle ages.
In practical terms, it's the mode you arrive at by calculation of 5ths up and down from a central note. IOW, the most in-tune scale if you're working with pure 5ths, because it builds in the fewest number of errors:

5th up from A = E
5th up from E = B
5th up from B = F#
5th down from A = D
5th down from D = G
5th down from G = C

(This makes a neat diagram if you draw it out with A at the centre. I don't have the facilities to post JPGs here... )
Obviously each note is doubled or halved to bring it within the octave.

It's a corollary, of course, of the fact that you get Lydian mode by building seven upward 5ths.

The fact that both these modes are constructed entirely from 5ths (and octaves) is - I suspect - related in some way to the jazz notion that they have no "avoid notes" - because of the close relationship of all tones to an acoustic tonal centre.
In the case of lydian, all the notes are harmonics of the root - IF they are pure 5ths, that is. (In equal temperament each one is a little out of tune, building up to a near 50 cent error in the last note. Which - IMO - would seem to make nonsense of Mr George Russell's theories, but let's not go there...)
In dorian, the harmonic relationship with a low root doesn't really work out - with or without ET.
ET, IOW, muddies the picture, and underlines how our musical choices regarding consonance are not based entirely on physics. (Partly yes, but not in a direct or straightforward way.)

Sorry I don't really understand the rest of your post.
Eg...

Lydian? How?

They don't? You can play Dbmaj7 over Cmaj7 and it doesn't "distract from the sound of the chord"?
It's "out", certainly, and in a logical way: simply because it's a half-step up. (No need for fancy reciprocal theories.) But it's a sound that has to be resolved back to Cmaj7 - assuming we're playing tonal music, of course...

Whoah. The suck produces a mix of V and IV (and blow produces a mix of I and vi).
You're right though that, overall, it works as the diatonic opposite.

But Dbmaj7 and Am7b5 are not diatonic to C major. Perhaps you meant D maj7? (Nope, has a C# in it...) Maybe you meant F#m7b5?

Well, the answer is in how it sounds. Of course
You tell us - do you like the sounds? (I'm not convinced you don't have some typos in there.)

As for the LCC, I'm not qualified to comment. I'm just suspicious of a book that costs around £100, around three times similar size books. And I'm also suspicious if acoustic science is used to support it, without regard to equal temperament.
(If not, then he is as free as anyone to invent his own theoretical system...)


EDIT: BTW, FWIW I agree entirely with Jay's view. It's not "dismissing" your observation, just seeing it in a different (more conventional) light.

 

The notes you referenced are in the key of G (see the F#?), not C.

This is just overcomplicaton for its own sake. The key of C (major) is the key of C. Period. The white keys on the piano. C Lydian is the key of G. You just start and stop on C when you play the G major scale. The notes are from the key of G, however.

If you're interested in a key structure that accommodates the changes in Moondance, then A minor will do just fine. The concept of "diatonic" minor is somewhat less well-defined than major. Sixths and sevenths in a minor key are always flexible. That's all you need to have in mind to understand Moondance.

Why you would want to take that approach to a song that is so obviously in a minor key is beyond me. A minor covers it.

Nope. A minor still, just the IVm chord. Flat the sixth tone of the scale - play F natural instead of F# - and you've covered it. Moondance is too easy to cover without concepts like symmetric harmony.

No. It's in a minor key because it begins and ends on a minor chord, and because all the cadences (or the progressions that function as cadences) resolve to that same minor chord. That's what puts the song in A minor, and not C or G.

"Blues" covers a lot of ground. Charles Mingus described "Goodbye Porkpie Hat" a blues. Shorter's "Footprints" is a minor blues.

Moondance really is in A minor. The A part (I-II chords) has a major sixth (due to the Bminor having a perfect, rather than flatted, fifth), while the B part (IV chord) has a minor sixth (F, the minor third of the IV chord). It's good to be aware of that harmonic shift, but it ain't exactly rocket science.

Any way of looking at harmony, melody, or rhythm that helps you find the notes on the fretboard that you hear in your head is a valid one. Validity is not the same as necessity, however. If a simple concept accounts for everything you're considering, then a more complex one is unnecessary. When you're actually using this stuff, you'd better not be consciously thinking about these concepts, or you'll just burden yourself with useless intellectual baggage. You need to be feeling, reacting, and, most importantly, playing.

Intonation systems are all based on hearing. Different systems make different tradeoffs, but none is perfect or even universally optimal. Just intonation has its own set of tradeoffs, as does even temperament. It really doesn't matter. You have continuous control over your pitch whenever you're playing single note lines on guitar. You can adjust pitch - on the fly - to whatever sounds best to your ear. That's the important part, not how you justify your choice in a post on TGP.

"Guide tones" are prominent in blues as well.

Strongly disagree. Thirds and sevenths - almost always dominant sevenths - are at least as definitive in blues as they are in other forms of music. Tension and release are just as powerful and useful in blues as anywhere else.

 

Moondance is in A dorian for the verse and moves to the A minor key for the pre-chorus (chords Dm, Am and E), then A dorian again for the chorus riff.
One key centre, with a slight modal variation quite common in minor keys.

It seems to perverse, IMO, to make it any more complicated...

Well it's more than A blues, if you want to bring out the detail. But not much more.

I'm with you there...

...but not there.

I still don't understand what you mean by reciprocals, and how those chords fit. (Run it by me again if you like...)
I just understand "in" and "out" and that's enough for me.
Basically, I can't hear what you seem to be hearing.

 

Beautiful post!

 

No. I don't have any lessons at my site (just a myspace page), and I'm not familiar with the song/group/whatever you reference.

Edit: wrt to continuous pitch control on guitar, you can push a note flat or pull it sharp if you're only playing one or two notes, but I believe correction of multiple pitches in a chord via that kind of technique is impossible. Regardless of the system of intonation you use, a fretted instrument is never going to produce every note on every string in tune. That's what vibrato is for.

 

More than hard. I'm gonna say it's a practical impossibility.

I'd say they're all pretty equal. Anything that's producing a pitch and playing along with the melody affects the sound. If the melody is in tune, and everything else is out of tune, then, to a listener, it's all out of tune. Relative intonation (wrt other instruments, voices, etc.) is generally more important than absolute. Relative intonation among multiple notes being played simultaneously by one instrument is extremely obvious. If you're playing a chord melody, for example, and you get one note (the melody) "in tune," it's got to be in tune with something. I submit that that something had better be the other tones that you're sounding at that moment, as opposed to a pitch that occurred in the melody at some earlier time, or to an overtone that is not being produced at that moment. Any dissonance with the concurrent harmony will have a far greater effect than will consonance with some earlier melody note that is no longer sounding.

You're making a very fine distinction here, and I don't think it's justified. Pitch is pitch, regardless of tempo. No fretted instrument has the possibility of making every note in tune, as fretted. I would argue that slowing down and simplifying everything you play in an attempt to make every pitch "perfect," first, won't be successful, and second, will needlessly compromise what you play. There's a point in a musician's development at which, IMO, (s)he should recognize the intrinsic limitations of his/her chosen instrument and learn to work within those limitations. If sax players had to have every note they played in tune, they wouldn't play.

I'm not sure what you're responding to with that statement, but I will offer a general observation: Working at something is one thing, hampering your progress due to chasing something that is impossible to achieve is quite another. I'm not making any judgement about your personal situation here, but I submit that the subject is worth some consideration. Exactly where to draw the line on something like this is a highly personal decision, but it is worthy of note that some of the best music ever made was not "in tune" by any strict definition of the term.

 

The answer is to subtract an interval from 9, inverting its tonality if it is not perfect (i.e., if it is major/minor or augmented/diminished), preserving its tonality if it is perfect, and you'll get its reciprocal. E.g., the inverse of a minor second is a major seventh. IOW, moving up a minor second is the inverse of moving down a minor second. The inverse of a perfect fourth is a perfect fifth, etc., etc.

Note that the only fixed (non-diatonic) intervals that will take you through all twelve tones are minor seconds, fourths, and fifths. Moving up or down in major seconds (whole tones), major or minor thirds, or any of their inverses will return you to the starting point without taking you through all twelve tones.

This stuff has lots of pedantic and mnemonic value. For example, memorizing the cycle of fifths is an excellent way to learn to quickly recognize key signatures.

 

Understood.

Not exactly. Db7 is the tritone sub for G7.
But you're right Dbmaj7 makes a good "side slip", for outside contrast.

However, the reason this works is nothing to do with reciprocals (IMO). This is what I was trying to say. We hear it (and can understand it theoretically) merely as what it is - a chromatic side step.

Well, yes. I'd say in the case of Dbmaj7/Cmaj7 it requires even less.

See, I don't think you're making things simpler, you're making them more complicated than they need to be.

It's also an inconsistent system.
Dbmaj7 works as an outside "side slip" to Cmaj7, yes.
But Bm7 (reciprocal of Am7) works in quite a different way - as a diatonic dorian extension, an inside sound. (OK, not quite inside in key of C major, but near enough...)

Yes I know you are . And - with respect - I think that's your problem.

You explained it well enough. But I disagree about that "sonic payoff".
It may pay off, but in ways unconnected with reciprocity, and much more connected with other theoretical systems.

You've hit some good sounds, after following that system, right. But not because you followed that system. It's coincidence.
That system doesn't enable you to understand the sounds, how they work. Other - much plainer and simpler - systems do.

I do think there are patterns in music that are relevant - even symmetries (although music is notoriously unsymmetrical, harmonically). And I recognise the attraction to graphic simplicity - mathematical "purity". But (maybe this is my art training ) I'm suspicious of such things. It looks so good, but rarely means what we think or hope it does.

 

To the first statement, I'd say that I've been aware of that for decades. To the second, I'd say that "accuracy," as you want to define it, is, at best, specific to one tonal center, and that, in modern music, we play around multiple tonal centers. Furthermore, even within a single tonal center, the pitches you identify using the methodology you describe won't result in all the overtones of all the notes being perfectly "in tune" with each other.

Example: take the fifth harmonic of your tonal center, and shift it down two octaves. That's the just intonation major third. Now consider a pitch a perfect fifth above that. That's a valid pitch in accordance with your system. However, that tone - the major seventh - forms a dissonant interval with the seventh harmonic of the original root. The system you reference works perfectly as long as none of the pitches other than the root produce their own overtone series. As soon as you start considering that, the apparent logic of the system doesn't appear nearly so attractive.

There is no set of pitches that will produce only consonant or "natural" fundamentals and overtones when used in the context of modern melody and harmony. If you want to play primarily over one tonal center, then this concept is extremely powerful. As you add tonal centers to a composition and consider all the relevant pitch relationships, it eventually will collapse under its own weight.

When you have continuous control over pitch - as is the case with stringed instruments, most horns, single note lines on guitar, and voice - you can choose the pitch that sounds best to you under the circumstances. Not only can you do that, you must do it. You cannot possibly use a formula or system to set your pitch in real time. Only your ears will do. Given that fact, and the fact that keyboard instruments are invariably tuned to equal temperament, as modified by stretch tuning, arguments about the "best" intonation system are largely academic. Practically speaking, you have only your ears to work with when you're actually creating pitches. If just intonation sounds best to you, then you can play your parts, to the greatest extent possible, setting your pitches according to that. You may find yourself needing to retune your guitar depending on what key the song is in (or what chord you're playing within a key), but, if you do, you're just running up against one of the compromises intrinsic to just intonation.

Every system of setting pitches involves compromise. There is no system that works perfectly (i.e., produces only intervals which are consistent and/or consonant with a single overtone series) in all keys, or even in all harmonizations within one key.

So was Slonimsky's "Thesaurus of Scales and Melodic Patterns," a book that I watched Nicolas Slonimsky tell Johnny Carson in an interview on "The Tonight Show" that he compiled largely as a joke. Coltrane could have been influenced by "Mary had a Little Lamb" (who'da thunk that a Rogers and Hammerstein tune from a relatively sappy musical could become an icon of modern jazz?), and he would still have been brilliant.

Consideration of concepts like just intonation is certainly interesting and meritorious.Whether such a concept should be allowed to dominate one's musical thought is highly debatable.