Monday, 26 March 2012

Pitch Wrap

Whoa !! I never did claim to know anything. I taught at music outlining large yet essential (I still think so) ideas about harmony especially in jazz but it could be about music or aspects: rhythm, harmony, and melody. Of course they all go together in a simultaneous flow. Harmony appears vertical on the page and creates a large or small multi-colored, multi functional, impact soft or loud.

Stimulation has come from the outside (the net) for me to pursue musical topics with a more in-depth look at how harmony is really horizontal in its nature. Here's a theory that was first outlined in the late 19th century and has only really been developed within the last 30-40 years by present day theorists.

Neo-Riemannian theory refers to a loose collection of ideas present in the writings of music theorists ....... What binds these ideas is a central commitment to relating harmonies directly to each other, without necessary reference to a tonic. 

Here's this original link again:

It has other highlighted links in there that are great too—very comprehensive and impressive.

One of the first and most easiest topics to understand (but quickly complicates through it's 12 note multiplicity), are the thoughts on Triadic Transformations and voice leading. This is a staple of song writers in the pop genre as in "Hallelujah" by Leonard Cohen in which at one point the lyrics themselves talk about the harmonic progression in the song: "The 4th the 5th the minor lift"

The 2 triads discussed are the major and minor. The major triad interval stack: 135 (Major 3rd - minor 3rd) and the intervallic inversion (upside down) from the major triad: the minor triad: interval stack 1b35 (minor 3rd-major 3rd). I find it kind of inspiring to pursue these routine moves because they sound good and one never knows when this knowledge might be needed. It's based on something we do all the time: go from chord to chord in an efficient voice led manner i.e. common tones held or repeated while the voice or voices moves in a stepwise motion. If these moves are identified and learned in keys there should be a positive result.

If you look up the link above you'll see that the moves are explored from the perspective of changes from major to minor triads and then secondly, minor to major: reversing the moves of the major to minor transformation. It doesn't really matter which inversion starting point, it's really about the shifts in the voices.

Individual transformations of triads through voice leading are given a  letter acronym
and a definition. They are obvious but good to know well.

The P transformation exchanges a triad for its Parallel. C major (triad) becomes a C minor triad by lowering the 3rd 1/2 step. And C minor becomes major by raising the minor 3 to major. 

The R transformation exchanges a triad for its Relative. C major transforms to A minor by raising its 5th a tone and conversely (good to remember), A minor transforms into C major lowering the root by a tone.

The L transformation exchanges a triad for its Leading-Tone Exchange triad. The C major triad transforms into an E minor triad by lowering the root of C 1/2 step. Conversely, E minor transforms to C major by raising the 5th 1/2 step. 

It's good to know both sides of that perspective. To summarize:

P (parallel) transformation the major triad "I" transforms to minor and vice-versa through the lowering or raising of the 3rd respectively.

R (relative) transformation refers to the "I" major transforming into it's relative minor triad (vi) by raising the 5th of the major and conversely transforming TO the major from the minor the root of the minor is lowered to the 5th of the major i.e. C major and A minor.

L (leading tone exchange) transformation refers to the I major triad transforming into the iii minor triad by lowering the root of the major 1/2 step to become iii minor. From the perspective of the iii minor (E minor using our example) it transforms into I major by raising its 5th by a 1/2 step.

There are a couple of other combinations there but these 3 transformations do relate to the Richard Cohn article on Richard Cohn:

MAXIMALLY SMOOTH CYCLES, ........  next time !!

Here is some more thoughts on this Torus design connecting apparatus that I wrote while writing to a friend on this matter— this all relates to the above link. <>

So is the tonnetz an infinitely square sheet with possible extensions of dimensionality ? :)

So looking at that spinning revolving-on-itself donut there, I've decided (I'm so glad it's color coded) that it is driven around more by the diagonals —like the turning of a screw.... The cycle of 5ths (blue diagonals) seems to drive it but the diagonal of the augmented triads (red diagona ls) eternally cycling would also be motivators in that they kind of represent the cycle enfolded on itself C—Ab—E resolves downward (that took me a minute to figure out even though they said that very thing in the article !!). This how it cycles or represents a cycle really of 4ths i.e. constant V7 resolution. Let's say C augmented is C7 and in the body of one of the 3 symmetrical diminished scales. C7 (represents Eb7, Gb7, and A7) resolves to Ab7 (representing B7, D7, and F7) and E is E7 representing G7, Bb7, Db7 so the cycle is truncated but still revolves on by the dynamic (here's a metaphor) of the lid that's just too big (or too small) for the coffee cup ... really its the +5/b6 interval that is providing the pressure for that lid to close but it only closes on two of the three points. Very tantalyzing !! The "vertical" (green lines) diminished 7ths are coiled along for the ride and of course are woven into the fabric of tension and release with the turning of that amazing magnetism created in that revolving sphere. In a way the diminished chord motivates the outward turning possibly due to the fact that each note of a Dim7 chord has the potential to fall 1/2 step and become the root of a Dominant 7th (of course this is just speculation on my part but logically it might hold true to some extent—perhaps 1/2:).

More later.


  1. Brilliant Mr. Austin! Although at first, your explanation would appear complex; it is however far from that. Especially when a person understands how harmony can work. Your explanation makes complete sense, and explains why it is easy to move from one chord to the next chord or key without sounding terrible.

    Parallel, relative, and leading tone exchange transformations are great for moving within a major or minor key in pop music, or any music for that matter. What is really cool is your explanation of the “Circle of 5ths, & the Circle of 4ths, all within the Torus design. The first pop song that comes to mind is “Brown Sugar” by the Rolling Stones. [Progression in C major; I, IV, I, IV, then to the bIII, bVI, and the bVII, making that bVII chord sound like it’s the V chord of the bIII.] It returns to the original key, and when the G7 [V of I] kicks in, it is so powerful because of those transformations.

    Your explanation is awesome. It’s true to say, your only 1/2 a step away from a good sounding note at any time.

    Brilliant Mr. Austin, thank you for sharing your wisdom.

  2. Thanks Jeari, your comments are most welcome..keep in touch !! Cheers !!