Sunday, September 12, 2010

Why Music Works: Chapter One

How the Harmonic Series Generates the Three Diatonic Contextual Systems: Alpha, Beta, and Gamma

PREFACE to All Posts:

This is to be the culmination of the Musical Relativity series of posts I did back in 2006, which can be found to your right in the sidebar. Back then I was calling the series Musical Implications of the Harmonic Overtone Series. Even before that, I did a series of posts called Harmonic Implications of the Overtone Series that started this all. Here, I am presenting the final weblog version of the evolving book I've decided to publish with the intention of getting some feedback before I create the final print version, which I plan to put into the ePub format for iBooks. So, please feel free to ask any questions about anything that you think I haven't made perfectly clear, and don't hesitate to offer any constructive criticisms or suggestions. Since this project is the accidental result of several decades of curios inquiry - and many prominent and also relatively anonymous theorists and teachers have contributed ideas to it (Which I will credit where memory serves and honor dictates) - I am eager to get a final layer of polish from any and all who may happen to read this series and find it useful, or potentially so. Since I am creating these posts as .txt files first, revision should be a simple process.

Since my pre-degree studies at The Guitar Institute of the Southwest and my undergraduate work at Berklee College of Music looked at music theory from the jazz perspective, and then my master of music and doctor of musical arts studies at Texas State and The University of North Texas were from the traditional perspective, a large part of how I discovered the things in this book-in-progress was the result of my trying to reconcile those different theoretical viewpoints. Since I want this work to be of practical value, I have retained all of the traditional theoretical nomenclature possible, and only added to it where necessary to describe phenomena that have not heretofore been present in musical analysis. I have, however, standardized terminology into what I think is the most logical system yet devised, and that will be explained as the reader goes along. There is a lot of built-in review and repetition - something I've learned from my decades of private teaching - so even a once-through with this systematic approach to understanding musical phenomena ought to be of significant benefit.

Outright addition to traditional musical analysis is limited to the symbology required to label root motion and transformation types so that the root motion and transformation patterns are visible: This greatly facilitates comprehension, and since good and bad harmonic continuities are separated by the effectiveness or lack thereof in the root motion and transformation patterns, this also actually functions as an aid to composition. All symbology - old and new - has been worked out over the past three decades so that everything is readily available with the standard letters, numbers, and symbols found on a QWERTY keyboard.

Finally, for the contextual systems, I have used the Greek alphabet: The normal diatonic systems - those comprised of two minor seconds and five major seconds - are Alpha, Beta, and Gamma. The exotic diatonic systems - those that contain a single augmented second - are Delta, Epsilon, and Zeta, and finally, the alien diatonic systems - those that contain two augmented seconds - are Eta, Theta, and Iota. Since the theoretical writings that started western art music out were handed down from ancient Greece, I thought this would be a fitting tribute, as well as a handy and logical classification scheme.

Basically, if you have a baccalaureate-level understanding of music theory from either a jazz or traditional perspective, you should have no problem understanding anything in this straight-forward treatise.

INTRODUCTION to Chapter One:

My foremost intention with this monograph is to, at long last, describe in a comprehensive way the basic musical forces that exist, and to present them and their resultants in a way that will be of use to composers. These forces all spring from a tension that is inherent in the harmonic series, which desires a resolution. Allowing or thwarting this desire for resolution is what allows music to express things.

While many twentieth-century theorists attempted to describe this through mathematics, their efforts failed because of one simple oversight: The overwhelming majority of possessors of musical minds are not mathematically inclined. Our minds generally work in terms of shapes and sounds: If we can visualize a thing or hear a thing in our minds, we can freely manipulate it to our heart's content. Numbers don't look or sound like anything to us, so they might as well not exist in our world (With the exception of proportions and draw-out geometry). Eventually, this made me return to the beginnings of scientific thought, which was actually called natural philosophy. Sir Isaac Newton described himself as a natural philosopher. Music theory has never even reached the Newtonian stage of evolution because the fundamental forces have never been defined and demonstrated in a rigorous and logical way.

A natural philosopher looks at the God-given constant in a thing, and extrapolates its implications out to describe the observable phenomena (Or hearable phenomena, in this case). For music, that constant is the harmonic overtone series, and so I have used musical proofs based on the harmonic series instead of mathematical proofs to demonstrate all of the resolution possibilities present, as well as all of the manifold contextual results that it can produce.



The overtone series is a harmonic system, so music is a harmonic system. If we look at the evolution of western art music, the harmonic series underpinnings were known from ancient Greek writings, but the implications of those writings were only discovered bit by bit by starting at the beginning, which was the unaccompanied melody of Gregorian plainchant. From plainchant, an admittedly oversimplified synopsis would proceed to organum, fauxbourdon, polyphony, traditional homophony, and then jazz, which was the real end result in the twentieth century: The seventh chords were finally ubiquitously used as tonics, including the overtone sonority itself in blues music.

Where the problem arose with both so-called traditional music theory and the succeeding jazz music theory was that the elements were never separated into their pure states: Pure harmony and pure counterpoint. By the time Joseph Schillinger properly described pure harmony, the "classical" guys had jumped the tracks and gone off into the desert of desiccation and musical nihilism known as - to cite but a single representative monicker - atonality (Though admittedly, some good did come of this for film music, but that's not what they intended). Meanwhile, even the most prominent Schillinger student of all, George Gershwin, never really exhibited in his works the revelation that was pure harmony: In other words, like all jazz cats, he didn't give a rip about the transformational voice leading that Schillinger revealed.

Finally, Schillinger himself never coupled his pure harmony with the overtone sonority, so he missed a crucial link, and that's why many of his ideas stray into the weeds of pure speculation. Admittedly, The System was a hodge-podge put together in haste by some of his students after his sudden and untimely death, so there is the possibility - regardless of how remote - that he did in fact make this connection. In any case, I made the connection independently due to the accidents of fortunate circumstance combined with my natural curiosity, and so here we are.

As mentioned previously, the natural philosopher's approach is to begin at the beginning, which in the case of music is with the harmonic overtone series, that every natural sound carries with it (A computer generated pure sine wave, which is an artificial phenomenon, would be an exception to this). In nature, the recognizable timbre of a sound is defined by the attack transients, formants, amplitudes, envelopes, and the phase relationships between the partials in the overtone series, which are individually pure sinusoidal periodicities.

Unfortunately, the pure waveform soundfont I found does not work in iTunes for some reason - it works fine in Encore though - so we're stuck with an organ sound that has relatively few harmonics present in it. If you play the overtone series in 12√2 equal temperament, this is what you get (I will address the simple logic of twelve-tone equal temperament much later in this book).


Listen to Example 1

Though TTET does not render the mathematically perfect ratios of overtones on a vibrating string or in a vibrating column of air - which themselves vary minutely, especially on a string - it is quite close enough: A mathematically perfect overtone series sounds virtually the same, as many years of programming digital synthesizers taught me. To put a finer point on it, this 12√2 rendering of the series sounds like a so-called dominant seventh chord, as did the perfect ratio renderings I've created in years past with instruments such as the Synclavier. So please, spare me the, "TTET destroyed tonality" nonsense: The ratios in TTET may be irrational, but the concept isn't - 1/3 is an irrational number too, but a child can understand the idea of a third part of something. Likewise, 1/12 of an octave is a perfectly rational concept, even if the resulting ratios are expressed by irrational numbers.

Under the system are the various eras of western art music development - Plainchant, Organum, Fauxbourdon, Polyphony, Homophony, and Jazz - which roughly correspond to composers coming to understand ever higher partials in the harmonic series. That this understanding was primarily intuitive and not analytical need not concern us, as in musical composition - and performance - it has usually been intuition that lead the way, with theorists describing practice later.

Next I have labelled the partials 1-8, which I will call P1 through P8. It is useful to draw a distinction between partials and harmonics, as the fundamental generator of the series is not a harmonic: Only the overtones P2 and on are harmonics. By labeling the partials starting with the fundamental as P1, the series works out for us the pure intervallic ratios, which are found on the third, fourth and fifth lines.

Though self explanatory, 2:1 is the ratio for the inviolable perfect octave, 3:2 is the ratio for the just perfect fifth, 4:3 is a just perfect fourth, 5:4 is a just major third, 6:5 is the large minor third, 7:6 is the small minor third, and 8:7 is the first of the just major seconds. Line four shows the ratios of 5:3 for the major sixth and 8:5 for the minor sixth, and finally we have 7:4 for the minor seventh and 7:5 for the tritone on the bottom line.


1. All adjacent intervals and their inversions from P1 through P7 are consonances.

2. All adjacent intervals and their inversions beyond P7 are dissonances.

Neither of these two observations should require any comment.

3. Consonances that remain super-particular ratios when inverted are considered perfect.

4. Consonances that are not super-particular ratios when inverted are considered imperfect.

A super-particular ratio is a ratio in which the terms differ by 1: All adjacent ratios for the consonances from P1 to P7 are super-particular; 2 - 1= 1, 3 - 2= 1, &c. Only those consonant intervals that remain super-particular when inverted are considered perfect, however: The 2:1 octave is unchanged during an inversion - which technically requires two octaves of displacement to avoid the unison - and the 3:2 perfect fifth inverts to a 4:3 perfect fourth, which are both super-particular: 3 - 2= 1 and 4 - 3= 1.

With the major and minor thirds, however, we get 8:5 for the minor sixth and 5:3 for the major sixth, which are not super-particular: 8 - 5= 3 and 5 - 3= 2.

5. The only non-adjacent dissonance with a single skip within the first seven partials is the tritone.

6. The only other non-adjacent dissonance within the first seven partials is the minor seventh itself.

The ratios of 3:1, 4:2, 5:3, and 6:4 are all consonances: Perfect twelfth, perfect octave, major sixth, and perfect fifth. Only 7:5 is a dissonance, it being the tritone, and then 8:6 is another perfect fourth. The minor seventh requires two harmonics to be skipped from 7:4, and it's the only other dissonance from P1 to P7 (And of course, it converts to the 8:7 dissonant major second).

7. The harmonic series is complete at P7; P8 is only present to yield the minor sixth harmonic ratio.

As I said at the beginning of this chapter, "The overtone series is a harmonic system, so music is a harmonic system." Once the adjacent intervals in the series become dissonances at 8:7, the harmonic part of the series is over and the melodic part of the series begins. Sure, inversions of harmonic structures can yield seconds, but a root position harmony in close position will only contain adjacent thirds. While ninths can replace doubled roots in the basic harmonic musical system - and sixths can replace fifths - elevenths and thirteenths are actually acquired from harmony with two transformational strata, which is a subject for much, much later (Transformational stratum one is Root, 3rd, 5th, and 7th, while transformational stratum two is 9th, 11th, 13th, and Root: A ponderous system outside of the basic harmonic nature of music, but one useful for certain effects within a larger musical or extra-musical context). Theorists who claim that twentieth century music was an exploration of the harmonic series beyond P7 are in error.

8. Since the harmonic series contains a dissonant tritone, it is inherently unstable.

The life forces of music are in the tritone - the leading tone and leaning tone impetuses - and the inherent instability of the overtone sonority - a dominant seventh, remember - is what provides music with its possibility for forward motion. This is one reason that wild-eyed criticisms of 12√2 temperament are misguided: Since the harmonic series is inherently dissonant, why would a stable just tuning be an advantage, or even particularly desirable? Since just ratios are only possible with computers, voices and other variable pitch instruments like strings, it is just silly to criticize TTET, which is a perfectly equitable solution for fixed pitch idioms. Later, we will look at just how small the deviations from just are with TTET.


The first thing that the harmonic series from P1 to P7 does for us is to define the format for pure harmony: Since we are dealing with a harmonic system, all we have to do is eliminate the non-harmonic fundamental generator and the superfluous twelfth that it produces to get this. The resulting structure is Root, Root, third, fifth, and seventh, which is the pattern for pure harmony.

1. Pure harmony consists of five total voices.

2. These five voices are divided into a four-part, close position transformational stratum above a constant-root bass part.

3. Only the root is doubled - or trebled after a resolution - in pure harmony.

4. Though the transformational stratum can be in any close position inversion, all harmonies are root position in pure harmony due to the constant-root bass part.



Listen to Example 2

The four different notes of the overtone sonority are divided into two pairs: The root and perfect fifth are passive tones, while the tritone involving the major third and the minor seventh consists of the two active tones.


1. The root is the foundation of the overtone sonority.

2. Together with the root, the perfect fifth provides context for the dissonant tritone.

Since there are twelve pitch classes in the chromatic system and the tritone involves two of them, there are only six tritones possible. Since there are twelve possible overtone sonorities, this means that each tritone is shared between two possible roots. In this case, if the tritone between B-natural and F-natural is enharmonically notated as C-flat to F-natural, it can belong to the overtone sonority with the root on D-flat and the perfect fifth of A-flat. That means that the two possible roots for each tritone are also in a tritone relationship with each other, as well as the two possible perfect fifths. When you change the context of a tritone with the other perfect fifth a tritone away, you also reverse the functions of the notes involved in the tritone. In this case, the B-natural leading tone becomes a C-flat leaning tone, and the F-natural leaning tone becomes an F-natural leading tone.

This is how jazz theorists justify their concept of substitute secondary dominant harmonies, by the way, though we will see later that this isn't really valid according to the implications of the series.

3. The root and perfect fifth are passive tones, neither desiring to rise or fall.

4. The major third and minor seventh are active tones, desiring to resolve their shared dissonance.

5. The major third is a leading tone, and it desires to rise by a semitone.

6. The minor seventh is a leaning tone, and it desires to fall by either a semitone or a tone.

7. The perfect fifth may rise or fall by a tone as the tritone resolves to a major target.

8. The root may remain stationary or fall a perfect fifth when the tritone resolves.

9. In order for the target sonority to be in root position, the lower root must fall by a perfect fifth.

10. In order for the target sonority to be complete, the upper root must remain stationary.

11. In order to avoid doubling a potential active tone in the target sonority, the perfect fifth must fall by a tone.


When we follow the above observations, the following primordial resolution of the overtone chord is produced.


Listen to Example 3

This is how you say, "The End" in music: The overtone sonority with a doubled root resolves to a targeted major triad with a trebled root. If we were to wish a continuation, the unison C-natural in the transformational stratum would have to dissolve with one of the C's going down to a seventh - either B-natural or B-flat.

The falling perfect fifth/rising perfect fourth root motion is called a Progressive root motion, and it will get a capital P in the analysis. When we get to more elaborate examples, they will not be called, "chord progressions" because a progression is this specific type of root motion; rather they will be referred to as harmonic continuities.

Though interrupted by the doubled root in the upper stratum, if one of the C's moved down into a seventh, this would be what is called an interrupted or delayed crosswise transformation, as you can see from the diagram in between the staves: The root and fifth exchange functions, and the seventh and third also exchange functions after the third's resolution into the unison. This would be a P_+ in the analysis, which reads, Progressive _Interrupted +Crosswise transformation. Uninterrupted crosswise transformations, though less than perfectly natural, can also be used when that effect is desired, as I shall demonstrate later.


1. This resolution does not yield all seven tones of the diatonic system.

2. This resolution - or similar less perfect versions - does, however, yield all six tones of the ancient hexaphonic Church modes.

The modern Ionian mode implied here was not really a common Church mode, but as we shall see, modal displacements of this formula will produce modes commonly used back then. It is also useful to note that when ending resolutions first appeared, they were primitive versions of this one.

3. In order for the target chord to become a complete seventh chord, the doubled root in the transformational stratum must fall.

4. The doubled root may fall either a semitone or a tone.

5. If the doubled root falls by a semitone, the potential for a diatonic system will be possible.

6. If the doubled root falls by a tone, the potential for a diatonic system will be destroyed.

7. If the doubled root falls by a tone, another overtone sonority will be created.

The rule for creating diatonic systems through this resolutional paradigm is to retain the inflection of the notes present in the preceding chords. Here, for example, one would have the doubled root descend to B-natural, because that note is present in the preceding dominant harmony. Allowing for chromaticism with this paradigm will lead to various integrated modalities; again, we'll see this at a later point.

8. As the diagram shows, 1 becomes 5, 5 becomes 1, 7 becomes 3, and 3 becomes 7 after the interruption of the resolution.

9. Therefore, progressive resolution of the overtone sonority creates an interrupted crosswise transformation.

10. A single additional progressive resolution would complete a diatonic system.


If we allow for that dissolution of the unison C so that the former major third descends to B-natural - becoming a major seventh - and add an additional progressive resolution, the Alpha Contextual System is produced.


Listen to Example 4

The Alpha Contextual System has as Alpha Prime the traditional major or Ionian mode. Within this contextual system are the additional sub-contexts known as the displacement modes of Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian.


1. With two progressive resolutions from the overtone sonority, the diatonic system is complete.

The only note missing in the single cycle resolution was A-natural, which now appears as the major third of the subdominant harmony.

2. All three possible harmonic functions - dominant, tonic and subdominant - are also now defined.

3. The progressive resolution from the tonic to the subdominant is a less perfect form of progressive resolution.

4. Additional less-than-perfect progressive resolutions will be found in the modal sub-contexts.

5. There exists an additional diatonic Beta Contextual System with resolution to a minor triad.

6. The Alpha Prime tonic scale is 2, 2, 1, 2, 2, 2, 1: The semitones are separated by two tones.


If we start again with the formula and have the initial resolution to a minor tonic, the Beta Contextual System is produced.


Listen to Example 5

The Beta Prime mode is like a Dorian mode from the Alpha System with a raised seventh degree. In common practice minor key music, this system was combined with the Aeolian mode to produce the nonatonic so-called melodic minor scale: Roughly speaking, his is the ascending version of that system, and Aeolian was used as the descending form (Though those conventions weren't always adhered to).


1. The tonic seventh chord in this system is a highly dissonant minor triad with a major seventh.

This means the third mode here has an augmented triad, of course.

2. The subdominant chord in this system is another overtone sonority.

3. The sub-contextual subdominant mode is the scale that the overtone series creates to P11.

The overtone scale is best described as a Mixolydian mode with a raised fourth degree, not, "Lydian flat seven" as some jazz theorists describe it: The home harmony is a dominant seventh, not a major seventh.

4. There exists an additional Gamma Contextual System with resolution from a dominant harmony containing a diminished fifth.

The point of origin for the V(d5m7) chord - and the so called French augmented sixth - is from the V/V in minor, where the chord on the second degree is a ii(d5m7) before the third is raised to make it a secondary dominant. In Alpha Prime that sonority appears as the remote V(d5m7)/iii harmony. I will demonstrate this when we get to the secondary dominant harmonies, but here I'm just demonstrating the three possible diatonic contextual systems that contain two semitones and five tones.


Listen to Example 6

The Gamma Prime scale created through this resolution process is best described as a Phrygian mode with the sixth and seventh degrees raised, since the minor second degree is the distinguishing characteristic of the Phrygian mode.


1. Diminishing the fifth of the dominant chord makes that fifth into an active leaning tone.

As I just mentioned, this V(d5m7) has a natural origin in the Alpha System. When the fifth was D-natural, it was a passive tone that could theoretically rise or fall during the resolution, so long as the composer is prepared to deal with doubling a potential active tone. Now, the D-flat is a third active tone in the dominant harmony that desires to resolve down by semitone to the new root. This additional impetus increases the resolutional desire of the dominant harmony.

2. The tonic seventh chord is again a highly dissonant minor/major seventh.

3. The subdominant chord is again another overtone sonority.

4. The Gamma Prime tonic scale is, 1, 2, 2, 2, 2, 2, 1: The semitones are not separated.

5. All 21 possible diatonic modes consisting of five tones and two semitones have now been generated.

I will present these in detail in chapter two.


What lead me to the idea of pure musical contextual systems was a phenomenon I noticed with so-called atonal works: They were completely unsatisfying - unlistenable, actually - in a purely musical concert context, but when used in a stage play or a film score, they could become quite effective. What I finally realized is that the play or the film provided an extra-musical context in which these pieces could be effective. Likewise, episodes of atonality within a larger purely musical context that is based on any one of the 21 diatonic modes that are independent sub-contexts - those with major or minor triads as tonics, and not diminished or augmented triads - can also be effective. When music provides its own context, it has to be based on an independent musical contextual system or sub-system, regardless of any arguments to the contrary.


Blogger Minicapt said...
... pipes with everything ...


9:04 PM  

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