Yamaha FS1R Editor for Mac OS X

A Musical Lifetime's Worth of Background:

It's been about five years since I sold my New England Digital made Synclavier Digital Music System, which I had for a full twenty years since 1984. I loved that thing because it was a digital additive/FM synthesizer that could make unique never before heard sounds, I could play it with a Roland GR pickup equipped guitar (I used a Steinberger GL2T-GR axe), plus it had a thirty-two track sequencer that operated just like an analog or digital tape deck, but with all of the editing advantages a virtual digital world brings.

For years after I got it I would spend hours every day pouring through the several phone-book-sized manuals and just messing with its synthesis capabilities. I got so good at it that within a couple of years, New England Digital was distributing many of my timber programs, including the sound effects that gave me minor fame, with every new Synclavier sold.

For an idea of where that curiosity about the nature of sound lead me, here's a piece I created with the Synclavier way back in 1994 - a full ten years after I got it - when I was a doctoral candidate at UNT. Almost every sound in that - with the exception of the church bell, I think - is something I created from scratch just starting with a sine wave and an FM ratio. Not only that, but all you'll hear in that is the Synclavier: No reverb, effects, or even EQ. That was recorded straight out of the Synclavier's stereo outs into a DAT deck.

Of course, I also have the Synclavier to thank for making me aware of the fact that music works because the overtone sonority is a dominant seventh chord. There's nothing like directly messing with the harmonic series through additive digital synthesis to make one aware of the true nature of sound: The Synclavier enabled me to mess with both the amplitude and the phase relationships of the harmonics out to P32, apply FM to the resulting waveforms, and then to even crossfade a series of them together (Which is how I created my groundbreaking sound sculptures like in the background of this Synclavier sequence).

Well, when I switched to nylon string guitars in 1988, I initially went through a back-to-basics pure acoustic phase as a player. I still used the Synclavier for electronic music composing, but I wasn't really interested in playing it with a guitar anymore. My acoustic phase was short lived, however, as I soon missed my phase, flange, chorus, and reverb effects. Plus, performing with an acoustic classical guitar is a loser's game, as even polite conversation can drown it out. So, by 1990 I was experimenting with electric nylon string guitars, and finally in 1998 Lexicon came out with the MPX-G2, so it took a full eight years just to find a preamp/effects unit I liked.

As for electric nylon string guitars - sheesh, what a grueling ordeal that has been - I started out with the only game in town at the time, which was a Gibson Chet Atkins CEC. It weighed a ton and was a pain to EQ. The Godin's were the breakthrough because of the RMC Polydrive hexaphonic pickups... which can run synthesizers. But it wasn't until I discovered the Blackbird Rider Nylon guitars just last year that I finally had an axe that I was over 90% satisfied with (The Rider's fingerboard is radiused, whereas it ought to be flat). The Rider Nylon can also be had with the RMC Polydrive, of course, and I now have two of them and play them exclusively.

Having the sound system and guitar sorted out - a process that took twenty years! - naturally lead me back to thinking about playing synthesizers with the Polydrive equipped Rider Nylons. So, I got a Terratec Axon AX-100 Mk II guitar to MIDI interface, but it just had a GM sound card, and wasn't a synth. that lead me to discover the Yamaha FS1R formant sequence/FM synthesizer, which I had never even heard of before last year. It had slipped under my radar between the time I stopped playing the Synclavier and when I started getting interested in synthesis as a performer again.

I was fortunate to find a virtually new one on eBay - they were only made for a couple of years - that had never seen a rack, and so I was off - or so I thought. Like a lot of Yamaha products - even the classic DX-7 - the programming architecture was tortuous to navigate. It was even worse than the DX-7 because everything was compressed onto a 1U rack faceplate! Additionally, there was no way to edit the formant sequences even on the unit itself: A major flaw, as the formant sequences were one of my primary points of interest.

Back in my Synclavier days - after the MIDI option came out for it - I had eight MIDI outputs to work with, so I paired it up with a Yamaha TX-816, which ammounted to eight DX-7's in a 4U rack chassis. Well the individual TF-1's - that's what the DX-7 modules were called - had no programming controls at all on them, so this was the first thing I ever encountered that required a computer software editor. Back then - we're in 1986 - the best one in terms of intuitive usability ran on the old Commodore 64: That was my first experience with a GUI, and it was earth-shattering (I wasn't a Mac guy then, obviously). It worked fantastically, and I learned the Yamaha carrier/modulator algorithm architecture, so I already knew a lot of what is in the FS1R - it can actually load play DX-7/TF-1 programs - but I needed an editor.

Well, a link at the bottom of the Wikipedia FS1R page lead me to the website of Japanese Mac programmer and synthesizer programming virtuoso K_Take.

K has written a fantastic FS1R Editor for Mac OS X, and it's available as a free download!

Minor Teething Pains

There were a few problems, as I don't think the FS1R Editor will work with FireWire MIDI interfaces. At least, it wouldn't boot with my Lexicon I-ONIX FW810s. So, I got one that was in a screenshot on his website, the EDIROL UM-2ex. After installing it and downloading the advanced driver, it worked perfectly. Now, however, I have to find one of the discontinued UM-880's for a rack mount version (I need one for my growing studio anyway).

I finally got it all set up last week.



The G5 PowerMac is running the FS1R Editor, the M-Audio KeyRig 49 - I program sysnths with a keyboard, not from the guitar - the UM-2ex, and the lower of the two FS1R's in the rack. That's right, I now have two FS1R's; one to program, and one to play with the guitar.

The MacBook Pro runs the FW810s mixer, and so it does the guitar rig part of the rack, which is everything else in it. I also record with the MacBook Pro using the 810s and Garageband or Cubase LE.



As you can see, it's a tight setup, and I can perform, record, and program without moving at all. The tiny Apple wireless bluetooth keyboard and the Kensington trackball were absolutely essential, as was a USB keyboard of not more than four octaves plus the obligatory semitone.



Eventually - when I start performing with this rack - the second FS1R will split off into a dedicated stay-at-home recording rack, but right now it's excellent for programming and having a second unit to back everything up to (I had to replace one battery already, which is thankfully very easy).

The FS1R Editor GUI



When you first launch the FS1R Editor, you are greeted with the MIDI Interface Setting window. The very first time you do this, you have to use the drop-downs to select everything. After that, it comes up just like this, unless you forget something like powering up your keyboard, in which case you have to quit, turn it on, and relaunch: It won't recognize a keyboard that is powered up at this stage of the game.



After you click OK, FS1R Editor scans the FS1R, which takes several seconds, and then greets you with the Performance page of the editor, which actually has four pages in it. This is the Part Parameter page, and since an FS1R Performance program can have up to four parts, you select those with the buttons at the top left of the recessed field.

The MMM template is 700 pixels wide, and the actual window is 1000p in width, so this is 300 pixels narrower than actual: It's plenty big enough to work on comfortably, but a trackball does help to accurately grab and move the small sliders.



The second Performance page has the Formant Sequence selection and control. yes, you can create formant sequences but I haven't tried that out yet.

Master tuning and transposition is on the right - very handy since I tune to A=432 - and you can also transpose the entire unit here.



One of the very best things about the FS1R is that it contains a full suite of onboard effects: Reverb, phase, flange, chorus and three band parametric EQ. K hasn't made the FS1R Editor anything like graphic-intensive, but the routing schematic is particularly useful here, as is the EQ shape and scale.



The final Control page of the Performance section allows for assignment of the knobs and MIDI controllers, and their sensitivity settings. There's a helpful Default Setting button for when you get into, "Oh my God, what have I done?" territory.



The Voice page is where most of the fun happens for me, as this is where you create sounds. It took me a minute to notice, but the Common Parameter box on the left actually has six pages to it, which are the sideways buttons along the left. This is unusual, but I'm sure K did that because they wouldn't all fit horizontally along the top. The LFO box is what you get when you first hit the page.

Believe it or not, the programming section on the right reminds me of a high resolution version of the old Commodore 64 program I had to program the TX-816: very simple, straight forward, intuitive, and self-explanatory. The graphics are just what's needed: The pic of the algorithm selected, the shape of the envelope, and the Frequency-based Envelope Generator.



Yes, we have filters, just like an analog synth.



And the filters have envelopes, of course, which is another nice place for a simple graphic representation and intuitive faders.



The Pitch Envelope Generator.



Here's the Formant Control section. It's a mix-and-match section for the internal formants unless you initialize one from scratch (As I understand it at this point, which isn't very well yet).



And finally, the FM Control box. You select the voiced and unvoiced operator to work with to the right, just under the algorithm graphic, by the way.



There's an on screen keyboard you can use if you don't have a USB or MIDI keyboard, but I found this very unsatisfactory to use. I need a physical keyboard for immediate feedback.

To summarize, this is a very compact and elegant little program that is an absolute must have if you have an FS1R and wish to program it. Can you imagine how long it would take to get to some of this stuff just using the FS1R's faceplate controls? Me neither.

Much more to come about this, of course.

Why Music Works: Chapter Fifteen

The Integrated Chromatic Contextual Systems: Chi, Psi, and Omega

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 Fifteen:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. The penultimate chapter in this section of WMW, chapter fourteen, examined just versus equal temperament, and so now in fifteen - which is the final chapter in this section on harmonic aspects of music - we will put the secondary dominant and secondary subdominant sub-systems together into the integrated chromatic contextual systems of Chi, Psi, and Omega. I skipped to the end of the Greek alphabet here because there may be some other systems to add in the future (But I got every one that can be generated by the overtone resolutional paradigm, so far as I know).

*****

CHAPTER FIFTEEN:

EXAMPLE 73:



Listen to Example 73

Starting on the tonic of Alpha Prime, the resolutional paradigm takes us immediately to the primary subdominant, and then into the secondary subdominant contextual sub-system at the next resolution to bVII(M7), which is an Alpha 4/Lydian sonority. Further Lydian sonorities follow as we traverse the secondary subdominants through bIII(M7), bVI(M7), bII(M7), and finally bV(M7), which is an enharmonic sharped fourth degree. The next enharmonic progressive resolution to V(m7)/iii puts us into the secondary dominant contextual sub-system, and we traverse V(m7)/vi, V(m7)/ii, and V(m7)/V before arriving at the primary dominant preparation for the final progressive resolution to the tonic. So, the secondary subdominant contextual sub-system acts as a huge subdominant preparation for the most remote of the secondary dominants, which in turn act as a gargantuan dominant preparation for the return to the tonic: For centuries this was reduced to the kernel of I, IV, V, I, with the super-progression from IV to V literally short circuiting what the overtone series implied was the ultimate outcome for a major locus integrated chromatic contextual system.

Note that the V(m7)/iii does not have a diminished fifth when we follow the resolutional paradigm through the secondary subdominant sub-system: The paradigm rules that the root of the targeting chord is retained as the fifth of the target chord, so the G-flat is sustained as the enharmonic equivalent of F-sharp into the B major/minor seventh, eliminating the diminished fifth that Alpha Prime produces.

*****

EXAMPLE 74:



Listen to Example 74

If we perform the same exercise with pure minor, which is the Alpha 6 or Aeolian mode, instead of a series of major/major sevenths acting as subdominants, we get a series of minor/minor sevenths. Again, the secondary subdominants are a different species of harmony than the tonic, as they are Alpha 2/Dorian sonorities.

Paleo-theorists handled the minor modes differently with respect to where the secondary dominants reside, but through the examples here we see that the only real difference between major and minor is what gender the primary dominant resolves to. I resolved to a final major tonic to accentuate this, which is the ancient Tierce de Picardie practice that goes back as far as composers have been resolving to triads instead of open fifths: Musical intuition is a powerful force.

Note again that the transit through the secondary subdominant sub-system eliminates the diminished fifth in the Alpha Prime version of the V(m7)/iii.

*****

EXAMPLE 75:



Listen to Example 75

The Beta Prime contextual system showed us that a subdominant could be an overtone chord too, and with that we end up with a full chromatic cycle of overtone sonorities. Omega is a good name for this system, as, "beyond here there be dragons" so to speak. It is easy to get totally lost as to where you are without following the music or at least listening closely, and this will bring up the ultimate implication of the overtone sonorities resolutional paradigm in the following example.

I again ended with a Picardy third to imply that the tonic of this system could also be major.

*****

This is what is inescapably ingrained into us by the overtone series resolutional paradigm: An endless series of overtone sonorities resolving to major triads, which in turn acquire minor sevenths to become new overtone chords and continue the progressive cycle. If you have ever lived anywhere there are maple trees, it is an auditory analogue to watching a maple seed fall, spiraling to the ground with its gyroscopic spinning: I can land on a rock or on the ground, just as any of the following overtone sonorities can alight on a major or a minor tonic.

I have only provided a two octave cycle, but subconsciously this pattern fills the entire spectrum of human hearing.

EXAMPLE 76A:



EXAMPLE 76B:



Listen to Example 76

Sorry for the Bizarro World piano sound font, but I neglected to properly assign the PC88 Marcato Strings to that example. Arg.

*****

There are many, many possibilities for fully integrated contextual systems - which I think I'll put into an appendix in the final book - but I just want to demonstrate one that has all twenty-four major and minor tonics as well as the twelve overtone chords. Each tonic starts minor, becomes major, then acquires a major seventh, and finally a minor seventh to become an overtone sonority. Obviously - if you've been following to this point - this creates a strict double canon.

EXAMPLE 77:



Listen to Example 77

This could be further adorned with diminished fifths and minor ninths of course, but I really like this one.

That brings to a conclusion the section of WMW that deals with harmony. There are sections to follow on counterpoint, rhythm, and form, but I don't know when I'll get to them, because I have other projects I need to get to right now. In fact, I've been anxious to finish this up because of some things I'm getting to the critical mass point with, the primary one of which is my programming of the Yamaha FS1R FM/Formant Sequence digital synthesizer.

Stay tuned.

Why Music Works: Chapter Fourteen

The Pythagorean Comma and Equal Temperament

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 Fourteen:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we then turned to the extra-diatonic contextual systems of Kappa, Lambda, and Mu in chapter fourteen. Now it's time to look at the series generated intervals, the Pythagorean Comma, and the inevitability of equal temperament for fixed pitch instruments.

*****

CHAPTER FOURTEEN:

EXAMPLE 72:



Listen to Example 69

The simplest way to wrap your brain around the Pythagorean comma and equal temperament is the illustration in example seventy-two above. If we stack seven perfect 2:1 octaves next to twelve perfect 3:2 fifths - both starting on F-natural - we should end up on the same pitch, which is the TTET-equivalent F-natural/E-sharp. The problem is that with twelve pure 3:2 perfect fifths, this doesn't happen: The E-sharp is 23.46 cents above the F-natural. In TTET, a cent is 1/100 - or 1% - of an equally tempered semitone, so the E-sharp is slightly less than eighth-tone sharp. Since average ears can detect five to seven cents of difference and well trained ears two or three cents, this is obviously out of tune.

Since the octave is inviolable - pitch recognition depends on octave equivalence (Though piano tuners mess with them all the time) - the simplest and most elegant solution is to simply reduce the size of the perfect fifths by one twelfth of the comma, which comes out to 1.955 cents, or less than 2% of a semitone. I can hear this with two simultaneous notes a fifth apart in TTET (So long as there are no phase, flange, chorus, or reverb effects present) as a slow cross-modulation - or beating - but I understand that the average listener doesn't perceive that level of detail. This trivial adjustment would seem to completely cure the "problem," but it doesn't really, since the other ratios that the harmonic series generates do not equally fill the perfect fifth (A just major and minor third, for example).

EXAMPLE 72B:

There is a useful chart demonstrating the differences between just and equally tempered intervals at the Wikipedia article about equal temperament.



As you can see, while the deviations from pure for the perfect fifth and perfect fourth are indeed trivial, that is not the case for some of the other intervals. What is true, however, is that all of the deviations from pure in equal temperament are trivial for the perfect consonant intervals. This really is the most important thing, because those are the lowest - and therefore most prominent - overtones.

For the imperfect consonances, there are some seeming problems, but perceptually some of those seeming problems are not as bad as they look. Almost no average listener has any problems with minor thirds and major sixths in TTET, for example, but I and some others do perceive them as slightly "off," but still not bothersome. In contrast to that, the major thirds and minor sixths do bother some people, including me, even though their deviations are smaller than those for the minor third/major sixth pair. Again, that is because the major third is lower in the series, more prominent as an overtone, and the overtone chord is based on it as a lowest gender-defining interval. As a result of the strident nature of equally tempered thirds, I and some other guitarists detune our G strings because in guitar music so many major thirds occur between the D and G strings. Luthiers often compensate saddles there too.

As for the dissonances, there is so much interference in them already because of the complexities involved in their relationships that it really doesn't matter much, with one exception: The tritone. In just tuning an augmented fourth and a diminished fifth are not the same interval: The diminished fifth is sharp compared to an equally tempered tritone, and an augmented fourth is flat compared to an equally tempered tritone (I use tritone to describe both augmented fourths and diminished fifths in TTET because they are the same 600 cent interval). That means when passive tones are present to define those intervals in their respective ways, the pure diminished fifth does not have as strong of a leading tone/leaning tone impetus to contract, and the pure augmented fourth doesn't have as strong of a leaning tone/leading tone impetus to expand.

This means that, as far as I am concerned, an equally tempered tritone is superior to the two just versions. The naturally just overtone sonority already wishes to resolve, and TTET only adds to that desire.

*****

I found an interesting YouTube video that purports to demonstrate the superiority of root-calculated just harmony, but which to me in fact proves just the opposite: The superiority of equal temperament. The two things that make this example so excellent are that the sound is generated as sine waves, and that there is a wrapped oscilloscope for visual reenforcement.



What I think is one of the most humorous aspects to that video is the fact that any critical thinker will immediately notice that you can see a lot more difference than you can hear.

*****

Another way to view the naturalness of the equal temperament solution is by noting that the harmonic series is made up of simple ratios generated beyond the octave while equal temperament is made up primarily of simple ratios plotted within the octave. For example, in equal temperament the octave is 1, the tritone is 1/2, the major third is 1/3, the minor third is 1/4, a whole step is 1/6, and a semitone is 1/12. The perfect fourth and fifth, which are 5/12 and 7/12 respectively, are really the only complex intervals in the system, and those are the ones closest to just. Noting TTET variations from just isn't exactly a making a red herring argument, but it's close.

A final thing to think about is what kinds of effects we apply to music today. I previously mentioned phasing and flanging, which are short time delay effects with small pitch modulations that increase disturbances to solid perception of pitch but it is really the ubiquitous use of pitch-shift and delay/pitch mod chorus that puts the lie to the notion that people want to hear just harmony: Just the opposite is true, and it has been since people first noticed how nice a section of doubled or trebled voices or strings sounded. And voices and strings can theoretically reproduce just intervals and harmony.

I enjoy hearing historical temperaments for period music, but I want all twenty-four major and minor keys to sound the same relative to each other, and TTET is the only way to get that with fixed pitch instruments, so it is the only viable solution as far as I am concerned.

That leads us to the final chapter in the current version of WMW on harmony.

Why Music Works: Chapter Thirteen

Genesis of the Extra-Diatonic Contextual Systems: Kappa, Lambda, and Mu

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 Thirteen:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

After finishing analysis of the nine diatonic contextual systems with the comparative morphology example in chapter twelve, we now turn to the extra-diatonic contextual systems of Kappa, Lambda, and Mu. Extra-diatonic systems are those that lie between the seven note diatonic systems and the fully chromatic twelve note systems. I have mentioned one of these nonatonic - nine note - systems previously, and that is melodic minor, which will start our examples off in the present chapter.

*****

CHAPTER THIRTEEN:

EXAMPLE 69:



Listen to Example 69

As I mentioned way back in an early chapter, the Kappa contextual system is easily internalized as a bi-modal combination of Alpha 6 - the Aeolian mode - and Beta Prime, which is Aeolian with a major sixth and seventh. Traditionally, Beta Prime has been used as the ascending form, and Alpha 6 for the descending. These conventions were not, however, scrupulously followed in common practice music, as a descending line over a IV(m7) or V(m7) harmony required the ascending form and its attendant raised degrees.

That leads to the true way that nature tells us this system is generated, which begins with an overtone sonority resolving to a minor/minor seventh tonic, as we see in the first progressive resolution above. This yields fully seven of the nine required notes all on its own, since we are no longer hemmed in by the diatonic system's paradigm of retaining the inflection of notes that appeared in a previous harmony: The B-flat minor seventh over the tonic is now perfectly acceptable (And it sounds less hotly dissonant than the minor/major seventh as well).

With the seven note diatonic systems, all that was required was a single additional progressive resolution to the subdominant, and the system was completely generated. With Kappa Prime, however, the resolution to IV(m7) only adds the eighth note - A-natural - so an additional third progressive resolution to the secondary subdominant of bVII(m7) is needed to get the A-flat. At this point, the astute reader will realize that there is no reason to stop the progressive resolutions there, and that fully chromatic systems are also implied by nature's resolutional paradigm. We will examine the manifold possibilities of chromatic contextual systems at a later point.

*****

EXAMPLE 70:

NOTE: The example below has an error in it, as the primary subdominant should be an overtone chord. Unfortunately, I didn't discover that until I had uploaded everything, so I'll have to fix that for the final book. One of the points of doing this series of posts is to iron that stuff out.



Listen to Example 70

The Lambda contextual system is commonly heard in Flemenco music, and the simplest way for traditionally trained musicians to internalize it is as a bi-modal combination of Alpha 3 - the Phrygian mode - and Alpha 5 which is Mixolydian. In vernacular usage, the v(d5m7) dominant stand-in is seldom heard, but the bII(M7) subdominant function harmony is (Usually as a triad). Though nominally a decatonic - ten note - system, the singers and soloists - usually guitarists - adhere pretty regularly to the Phrygian mode for vocals and improvisation.

For the first progressive resolution here we have the previously mentioned v(d5m7) - which is native to Alpha 3/Phrygian - resolving to an overtone sonority on the tonic degree. This yields six of the ten required notes, and the further resolution to the IV(m7) subdominant - which is usually replaced by the Phrygian-native minor/minor seventh in this idiom - gets the system to nine notes. So once again, a further resolution into the secondary subdominant realm is required to pick up the final note, which is the D-natural that is native to Alpha 5/Mixolydian. As I stated earlier, in the idiomatic traditional employment of this system, the vast majority of the melodic elements adhere to the diatonic Alpha 3/Phrygian mode.

The acoustical reason that this bi-modal combination works is because the dissonant overtone sonority can support a wide variety of upper structure tones, such as the minor ninth, augmented ninth/minor third, and the minor thirteenth that Phrygian adds to Mixolydian here. This same ability of the overtone sonority to support a wide variety of tones in the upper structures is also responsible for our final Mu contextual system.

*****

EXAMPLE 71:



Listen to Example 71

The Mu contextual system is the modern king of all of the extra-diatonic systems, as it virtually defined the sound of the twentieth century through it's ubiquitous employment in blues music, which evolutionarily lead to jazz, R&B, and rock and roll. In fact, I start all of my students out with the blues - even if they wish to study classical music - for this very reason: You can't understand the music of the twentieth century without a solid foundation in the blues.

For the traditionally trained theorists among us, the easiest way to pocket an understanding of blues tonality is as a tri-modal combination of Alpha 2 - the Dorian mode - Alpha 5/Mixolydian, and Alpha Prime. In the idiomatic vernacular, what you see above is exactly what you get: Overtone sonorities on all three cardinal degrees. In fact, in the most basic blues and blues based R&B and rock and roll music, these are the only three chords you ever hear. By the time swing and bebop guys like Charlie Parker got ahold of the blues, however, it was elevated into another musical art form entirely, with many harmonic excursions within its brief twelve measure form.

Obviously, the overtone sonorities on the dominant, tonic, and subdominant correspond to Ionian, Mixolydian, and Dorian respectively. In idiomatic practice, blues, rhythm and blues, blues based rock, and blues based jazz singers and soloists employ a multi-layered combination of modalities, which consist of a minor pentatonic skeleton, the fleshed out diatonic Dorian, and then a virtually fully clothed chromatic sub-system of passing and approach notes for the bling. Blues is something a beginner can learn to play in a week, and then explore for a lifetime, which is the primary aspect of its charm and enduring appeal.

Why Music Works: Chapter Twelve

Harmonic Mobius Loops, Harmonic Palindromes, and Comparative Morphology of the Diatonic Contextual Systems

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 Eleven:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops. Returning to diatonic contextual systems in chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and then its companion, chapter eleven, examined the isolated root motion and transformation types in those alien systems.

Now in chapter twelve, we return to the phenomenon I mentioned in chapter nine called harmonic Mobius loops, and the closely related phenomenon of harmonic palindromes.

*****

CHAPTER TWELVE:

EXAMPLE 67:



Listen to Example 67

An harmonic Mobius loop is an harmonic continuity in which the root motion types are balanced out so that the ending voicing leads back into the beginning voicing. Almost all composers, working intuitively, will create continuities in which the voices descend over time, as the natural, ingrained tendency that intuitive understanding of the overtone sonority's resolutional desire imparts, is to use more progressive type motions: Progressions, half-progressions, and super-progressions. These progressive types of root motion are statistically more common in all forms of harmonic music for this reason, and that is what lead Schenker to devise his theories, which he didn't understand were only observations of an artifact of music that naturally contained a preponderance of these progressive type root motions. So, it takes both understanding and discipline to create harmonic Mobius loops, as the natural tendency is against creating them.

The easiest way to construct a harmonic Mobius loop is actually to create a harmonic palindrome, because the forward and reverse root motions will balance each other out so that the ending voicing leads back to the beginning voicing, as is desired. I have demonstrated this in the first part of example sixty-seven above.

To begin construction of one of these symmetrical Mobius loops - another way to describe an harmonic palindrome - is to put the tonic harmony in the first measure and the dominant harmony in the last measure. Since the dominant to tonic resolution is a progression, that means measure two has to be a regression back to the dominant. After that, the next three root motions/harmonies are up to the composer. Obviously, this limits the possibilities, especially in a single eight measure phrase, as we're working with here. I chose to go vi(m7), IV(M7), to ii(m7), as that is one way to yield the maximum possible five different harmonies in eight measures.

Breaking away from the symmetry imposed by harmonic palindromes can actually yield more interesting harmonic Mobius loops, as I have demonstrated in the second part of example sixty-seven. All I did here, obviously, was to lower the harmonies a step within the diatonic system for measures two, three, and four: The final four measures are the same. The reason this works here is because the initial progressive root motion is now balanced out by a super-regressive root motion into measure five: Note that the voicing in the fifth measure of each example is identical. This also has the additional benefit of presenting six of the diatonic harmonies instead of only five.

To present a comparative morphology of the nine primary diatonic systems, however, I will be required to present all seven diatonic harmonies in the Mobius loop. Therefore, neither of these two examples will suffice. I also would like to present as many different root motion types as possible - ideally all eight of them: Progression, regression, half-progression, half-regression, super-progression, super-regression, progressive tritone, and regressive tritone - so this will call for a more artful approach.

EXAMPLE 68A:



NOTE: I will present the audio example link at the end of the variation set.

My final harmonic Mobius loop starts out as the first part of sixty-seven did, but to get a progressive tritone in, I went from IV(M7) to vii(d5m7) from measure four into measure five. I then reversed the progression types in the first four measures, as you can see, in an attempt to balance things out and also to get all seven harmonies in: Progression answers regression, super-regression answers super-progression, and half-regression answers half-progression. I did manage to get all seven harmonies in - which was the primary goal - and I got seven out of eight root motion types in (Only the regressive tritone is missing) - which was the secondary goal, but I did not quite manage to balance the root motion types out, as there is still a preponderance of progressive types if we super-progress into the dominant in measure eight. If I went directly from the subdominant to the tonic, it would work, but that does not produce the desired dominant resolution, and if I went from the subdominant directly to the dominant, the voicings would miss their loop by a single inversion (7, 1, 3, 5 instead of the required 1,3,5, 7). So, I had to add another half-regression to make the loop happen. Doing this creates a nice turnaround figure that will alert the listener that a new variation and diatonic contextual system is forthcoming - and the final measure is a rhythmically diminished retrograde of the second three measures - so that makes this harmonic Mobius loop ideal for presenting a comparative morphology of the nine diatonic contextual systems.

Since this is a systematic run-through of only the nine master contexts, there are actually many nice possibilities within the sub-contexts not presented here. Presenting even only the independent sub-contexts, however, would lead to a huge and ponderous example: This is more than sufficient to alert the musically aware listener of the manifold new resources available, many of which remain almost totally unexplored.

I presented Alpha Prime twice here to allow the listener to get the lay of the land. I will not hold the reader's hand through this with a detailed explanation of every effect, as I assume that those for whom I created this will possess at least the basic intellectual curiosity to see that everything is in the analysis on the example pages.

EXAMPLE 68B:



This page presents the other two Native Diatonic Systems with Beta Prime and Gamma Prime. Since the only difference now is the presence of E-flat, there are both familiar and peculiar effects present. The last quarter note of measure twenty-four carries a V(d5m7) sonority, as this is the altered dominant belonging to the Gamma Prime contextual system: I anticipate the upcoming systems by leading into them with their dominants and dominant stand-ins throughout the variation set.

Gamma Prime, then, adds a D-flat to the E-flat already present, so the harmonic effects are more strange.

EXAMPLE 68C:



Page three gets us into the Exotic Diatonic Contextual Systems with Delta Prime, so our tonic is again a major/major seventh sonority. A-flat is the only disturbance to Alpha Prime here, so again there are both familiar and unfamiliar effects present. Epsilon Prime adds an E-flat, though, so we're back to a minor/major seventh tonic, and there are more unusual effects present.

EXAMPLE 68D:



Zeta Prime is the last of the Exotic Diatonic Contextual Systems, and it adds a D-flat to the A-flat and E-flat previously present: Now, all of the harmonies are strange compared to Alpha Prime.

With Eta Prime, we enter the Alien Diatonic Contextual Systems, and though the tonic is again a major/major seventh, the presence of the two augmented seconds produces copious amounts of uncanniness. The appearance of the V(d5M7) dominant stand-in - with it's F-sharp - really destabilizes everything, though.

EXAMPLE 68E:



The addition of the F-sharp with Theta Prime means that the final measure of the Mobius structure now has double chromatic approach tones to the dominant stand-in, which is quite nice. It's even better into a real dominant, as we see and hear in the final measure on the page where, finished with the nine contextual systems, we prepare to return to Alpha Prime. That brings up a point: Not only can you explore the sub-contexts as you please, but also nothing is stopping you from mixing and matching the contexts and sub-contexts to get whatever effects you desire on whatever degree you desire them. The possibilities available through a complete understanding of the diatonic contextual systems are truly staggering, and as I mentioned, many of them remain virtually unexplored, yet easily accessible.

EXAMPLE 68F:



Listen to Example 68

And with that, we return to Alpha Prime, where I again provide two statements of the Mobius loop to reorient yourselves, and also an ending, which finally allows the B-natural to resolve up to C-natural.

After listening to the example, you can hear that, no matter how bizarre the harmonic effects get, it is not only still possible to recognize the theme, it is impossible not to recognize it. This is one area where the so-called atonal composers of the twentieth century utterly failed: Variations on atonal constructs are virtually impossible to recognize, let alone follow, for the lay listener. Even for those of us with acutely well trained ears, it isn't exactly a satisfying experience, never mind fun.

The presence of a viable contextual system, no matter how warped, solves this problem completely, and as I said, the sub-contexts not presented here contain many more points of interest.

Why Music Works: Chapter Eleven

Root Motion and Transformation Types in the Eta, Theta, and Iota Systems

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 Eleven:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. With the exotic systems out of the way, in chapter nine, I was free to demonstrate a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

In the previous chapter, chapter ten, I introduced the alien diatonic systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and with today's chapter eleven we will finish examining all of the nine master diatonic contextual systems and the total of sixty-three independent and dependent diatonic modes. All that is left is to look at and listen to the isolated root motion and transformation types.

*****

CHAPTER ELEVEN:

EXAMPLE 58:



Listen to Example 58

Here we have the progressions and regressions in Eta Prime, and you may find the effects other-worldly, as I do, which is what lead me to classify these contextual systems as alien: They are very, very far removed from the native systems, and even more foreign than the exotic systems.

EXAMPLE 59:



Listen to Example 59

These relatively smoother half-progressions and half-regressions can't do much to mitigate the strange harmonic effects that Eta Prime contains.

EXAMPLE 60:



Listen to Example 60

One of the problems with the augmented seconds is that - since there are two now - the listener is more likely to perceive them as minor thirds. That's pretty apparent in these super-progressions and super-regressions, where they are exposed in the bass.

*****

EXAMPLE 61:



Listen to Example 61

The Theta system is even stranger, as in addition to diminished thirds - which the listener is likely to perceive as major seconds - and the augmented seconds in the scale, we now have augmented thirds in the harmonies as well, which the listener will probably understand as perfect fourths. These distortions of native system realities can put listeners adrift, which is a nice way to affect them if that's the composer's desire. These systemic distortions of native reality are more compelling than the old techniques used in the serial systems of the century just passed, because a native system harmonic continuity can be progressively morphed into these alternate harmonic realities, and still retain their recognizability, as we will see in chapter twelve. One of the main criticisms of the various atonal methods is the, "any note could be replaced by any other note" feeling that listeners get, which is not an issue at all with systemic modifications from native, to exotic, and finally alien diatonic systems.

EXAMPLE 62:



Listen to Example 62

The funky thirds are quite apparent with the third movements in the bass with these half-progressions and half-regressions, but the connection to modal reality, though tenuous, is never entirely broken. This is far more effective than simply emerging listeners into total chaos, which they usually object to (And rightly so, in most instances). The exceptions, as always, involve situations such as film and stage vehicles, where there are extra-musical contextual defining elements at work. But for absolute music, especially, the alternate contextual morphologies available with exotic and alien systems are a great way to evoke the uncanny while still anchoring the listener to a modal locus.

EXAMPLE 63:



Listen to Example 63

The bizarre nature of Theta Prime is nicely exposed by the super-progressions and super-regressions.

*****

EXAMPLE 64:



Listen to Example 64

The minor tonic of Iota Prime adds even more darkness to the character of this alien system.

EXAMPLE 65:



Listen to Example 65

By the way, if you have been listening to all of the examples as we have progressed through the nine diatonic contextual systems, your brain is being hacked by them: Since almost nobody other than myself has ever listened to them systematically like this, I can tell you that this exposure will alter your conception of what musical reality can be. This is all leading up to the next chapter, where we will take an harmonic Mobius loop and morph it through all nine systems. If you have a musically sensitive mind, this will be very enlightening for you.

EXAMPLE 66:



Listen to Example 66

And so, all nine of the diatonic contextual systems have now been presented: Alpha, Beta, and Gamma for the three native systems; Delta, Epsilon, and Zeta for the three exotic systems; and finally Eta, Theta, and Iota for the alien systems. The total of sixty-three modal contexts and sub-contexts provide many more sonic resources than the old common practice guys were ever aware of, and even more than more modern jazz composers ever conceived of, all in a simple and intuitive system that can be applied to any diatonic theme a composer comes up with.

There are systems outside of the diatonic realm of course - which I have briefly alluded to previously - and we will begin to look at those in chapter thirteen.



I like that photo, even though she bears an uncanny resemblance to my ex-wife.

Why Music Works: Chapter Ten

The Alien Diatonic Contextual Systems: Eta, Theta, and Iota

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 Ten:

In chapter one, I demonstrated how the overtone sonority generates the three normal diatonic systems - those seven note systems that contain two semitones and five tones: Alpha, Beta, and Gamma - and then in chapter two we examined each of those systems in detail, discovering that the primacy of Alpha is due to the fact that all seven of its harmonies can be arranged in progressive order. In chapter three, we examined the contextualization of Alpha Prime, looking at the various different root progressions types it can exhibit, their various transformations, and through this we also started to look into the world of musical effect and affect. Chapter four was dedicated to examining how Beta Prime and Gamma Prime compared to Alpha, using the same musical proof formats developed in chapter three. Through those proofs, we discovered some very unusual harmonic effects that evoke the uncanny that are contained in the Beta and Gamma systems. Chapter five then took us out of the diatonic harmonic world and into the chromatic realm as we discovered the origins of the secondary dominant sub-system sonorities. After the secondary dominants, in chapter six, we looked at the secondary subdominant sub-system of harmonies, which completed a larger set of integrated chromatic systems, which we will look at in detail later.

Then in chapter seven, we looked at the exotic diatonic systems - those seven note contextual systems that contain a single augmented second: Delta, Epsilon, and Zeta - and in chapter eight we looked in detail at the root motion types they contain, and the unique harmonic effects that these unusual systems create. Last time, in chapter nine, I demonstrated a phenomenon that is an artifact of patterned root progressions, which I pointed out earlier, and that is harmonic canon. Depending upon how harmonic canons are developed and set up, I showed how they can also exhibit the phenomena I call Musical Escher Morphs and Harmonic Mobius Loops.

Chapters ten and eleven will be devoted to the alien diatonic contextual systems - which are those seven note systems that contain two augmented seconds: Eta, Theta, and Iota - and with these chapters we will complete all nine master diatonic contextual systems and the total of sixty-three independent and dependent diatonic modes. Today's chapter ten will look at the genesis and structure of the alien systems.

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CHAPTER TEN:

EXAMPLE 52:



Listen to Example 52

The Eta Prime master context is generated by a V(d5m7) altered overtone sonority resolving to a major seventh tonic chord, and then on to a minor/major seventh on the subdominant degree. This yields two augmented seconds: One between the minor second and the major third in the lower tetrachord, and the other between the minor sixth and the leading tone in the upper tetrachord. I've heard this scale referred to as double harmonic major, but there are several double harmonic major modes in the exotic systems, so - since the tonic is a major seventh and the fourth degree is perfect - it is more precise to call it Ionian minor second, minor sixth.

Eta 4 deserves a mention here, as I first encountered this mode as double harmonic minor while back at Berklee in the early 80's. Since then I've also heard it called an Arabian scale or the snake charmer scale. It's actually pretty cool.

EXAMPLE 53:



Listen to Example 53

Theta Prime is generated by a V(d5M7) on the dominant degree - and now you can see why these systems become alien: There is no primary tritone in some of the dominant stand-in sonorities - which resolves again to a major seventh tonic, but now the chord on the raised subdominant degree is the very strange #iv(d3d5d7) sonority. Since the fourth degree is now augmented, that makes this a Lydian minor second, minor sixth (Which is a different species of, "double harmonic major").

EXAMPLE 54:



Listen to Example 54

For Iota Prime, we are now resolving the V(d5M7) sonority into a minor/major seventh tonic, while the raised fourth degree still supports a #iv(d3d5d7) chord. As you can probably guess, there are going to be some very strange effects within these alien systems.

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EXAMPLE 55:



Listen to Example 55 (The example is the harmonized scale only).

Eta prime has the rare feature of being an intervalic palindrome, as it's intervals read the same forwards or backwards: 1, 3, 1, 2, 1, 3, 1. In the alpha system, this honor goes to Alpha 2, which is the Dorian mode. Eta 2, Eta 3, and Eta 4 are nominally independent since there is an harmony on the unaltered dominant degree and a functional tonic triad, however they are not simple to establish independently in practice, but it can be done. The other three sub-contexts are obviously dependent.

EXAMPLE 56:



Listen to Example 56 (The example is the harmonized scale only).

I neglected to put the intervals in example fifty-six, but they go; 1, 3, 2, 1, 1, 3, 1. There is only one independent sub-context in the Theta system - Theta 3 (Mistakenly labeled dependent) - because of a new phenomenon that destroys a tonic triad: an augmented third in the case of Theta 2 and a diminished third in the case of Theta 7. Obviously, this is a difficult system to work in, even with the independent modal sub-contexts.

EXAMPLE 57:



Listen to Example 57 (The example is the harmonized scale only).

There is a good case to be made for calling Iota 6 the prime here, as it has an actual altered dominant as well as a natural fourth degree. If you'll notice, though, I organized all of the diatonic systems around C-natural in such a way as to start with the fewest accidentals, and add from there. In this case, Eta Prime only required a D-flat and an A-flat, Theta Prime added the F-sharp to that, and Iota Prime here got the additional E-flat. Since I just recently worked these alien forms out, I may rethink my organization before the final version.

I again forgot to put the intervals under the harmonized scale, but it's; 1, 2, 3, 1, 1, 3, 1.

Next time we'll break out the musical proofs to look at and listen to the root progressions and transformations in the alien systems.