~ loops of pitches ~

~ half step through the tritone ~

'discovering the perfect closure of each of the intervals and what they create ... '

.

In a nutshell: Simply to gain a complete sense of the perfect aural and theoretical closure of today's equal tempered pitches. We can achieve this understanding by creating a loop of pitches for each of the intervals found within the 12 pitches of the chromatic scale.

Along the way we end up finding an organic source for building up the larger compositional components; melodies, arpeggios, chords and other musical components associated with each interval.

 

Hip to this and you're golden. Simply stated; everything we do with our pitches and theory can be shaped into a loop. If we keep the loop intervals consistent and going on long enough, it'll always close back on its starting point. Always? Yea pretty seamless and set in stone :)

This closure works with the intervals, scales, arpeggios, chord pitches, even into the extended patterning of combining more than one element. Really any musical component that can be somehow numerically represented will adhere to this idea 'closed loops.'

intervals

So what's not to love about all this tempered tuning perfect closure loops? Just that among the sweetest pitches we have on our palettes is that rather sharply pitched B string. A bit sharp even when said to be accurately tuned onto our guitars? Well yes, there is that bit that we need to compromise. Ah, but we guitarists can add various vibratos, bend pitches and give the slide a ride to still find this sweet spot yes?

History ~ theory overview. Through the course of this discussion we end up all over the map; going way back in time and also forward, as we explore from our first cave bandstands on through the Mediterranean toga era and into the new world of America, where our two core theory systems of today originally melted into one.

Overview: Building upon our idea of a silent architectural structure, this idea of 'loops of pitches' is really an essential concept within the core structure / philosophy of this text. Nearly all of the discussions throughout this text will be based, refer to or be centered around this idea of a perfectly closed loop of pitches.

This idea of closure not only includes our pitches, intervals, scales, arpeggios, chords, forms and such but can apply to the theoretical, artistic and stylistic discussions as well. For it just seems that the closure of 'anything and everything', even a balance of sorts, is what our curious human minds tend to seek.

Once this 'loops' foundation is in place, what we gain is the ability to fully understand how to create smaller sub sets from our original loop. These subsets become the specific groups of pitches with which we create the American musical sounds we love. The neat thing about this idea of closure is that down the road we'll intuitively understand how to derive all of our musical resources from their original organic source.

organic source

One of the most fascinating aspects of the theoretical perfection, created by filtering Pythagorus' original cycle of fifths through equal temper tuning, is in its ability to be sliced, diced, blended, boiled and baked ... oven or microwaved, yet always come out perfectly balanced with its original structures completely intact. (1)

For like the cat that always lands on its feet, emerging theorists will marvel at our theory system's ability to be stretched and reshaped in an infinite variety of artistic ways, no American musical style withstanding, without ever compromising its structural integrity or exceeding its defined, mathematical boundaries.

What follows here. The following discussions explore the various intervals we find within an equal temper chromatic scale to prove up on the idea of its perfect closure of twelve pitches. We take each interval in turn as it is extracted from the chromatic scale and create symmetrical loops based solely on each interval. Thus, we will not mix intervals together to create additional loops. That joy is saved for a later chapter titled groups of pitches ... (from which we create our melodic ideas.)

groups of pitches

As we move through the intervals and create various loops here, we'll run into many different configurations of the pitches, many stylistic ideas and lots of new vocabulary words, many of which can lead us off to other discussions simply with a click on those magical letters and text we term a hyperlink.

Beginning theorists. If you're just getting into this Medusa head of music theory, when we talk musical numbers, simply counting up on our fingers and toes can help create correct solutions. And while there is a seemingly vast degree of nuance in our musical intervals for the uninitiated scholar, and there is, don't let it get in the way of your moving forward. You can learn the fine print if and when you need it and in the meantime so much of our learning right here simply uses the numbers one through eight.

Even when we associate pitch letter names with numerical intervals, a lot can still be counted upon our fingers and toes. This is how so many of us learned our numbers when first starting out as little peeps and it'll work just as fine and dandy today. Review the notation symbols if needed or learn to read notation here if you're so inclined. It's easy and potentially opens us up into an unlimited library of music for study and performance.

The half step, augmented unison or minor second interval. The smallest theoretical division of equal temper tuning is the interval of a half step. This translates to one fret on our guitars. Consecutive half steps creates the chromatic scale. Here is the loop of pitch letter names. Looks familiar yes? The historical organic origins of this loop is the topic of our discussions titled silent architecture. Examine the pitches of the chromatic scale. Example 1.

C
C#
D
Eb
E
F
F#
G
Ab
A
Bb
B
C

We have closure, C up an octave to C. Sharps and flats in the same line? That is bad musical form n'est-ce pas? It is. In the theory we follow the key signature to designate pitch. The example above mixes accidentals to introduce the term enharmonic equivalents. One pitch with two letter designations ultimately determined by key signature. Here is the sound of the above group, notated correctly as an ascending melodic line. Do sing along. Example 1a.

Mostly a jazz color, and often in small bits we term chromatic enhancement, jazz saxophonist Charlie Parker was arguably the first King of Chromatic Enhancement. The half step lead in for blues and jazz cats is really an absolutely essential lick, especially for the 12 / 8 shuffle, swing feels.

Charlie Parker

half step lead in

The metalists move by half step an awful lot, especially as their tempos brighten. The melismatic vocalise of today's pop stars have a certain quality of chromaticism, especially with the gospel like blue notes deftly woven into diatonic tunes written in a major key.

So while chromaticism can really be anywhere in the American sounds, the trick is to be able to hear it. And once dialed in we'll spice up our own ideas with this ultra simple yet way hip technique. Surely strive to develop the ability to fairly accurately sing a chromatic scale. While it may take some time to master it'll surely do wonders for your overall musical abilities and help to dial in your built in stereo radar.

sing the line ~ play the line

radar

A chromatic melody ... of sorts. There is one familiar melody of a mostly chromatic quality that we all might remember from way back. To me it is from the olden days, an old carnival melody that turns out to be a really incredibly powerful work by one Julius Fucik. Dig a lick of the main theme of "Entry Of The Gladiators." Do notice the use of flats and naturals as we half step our way in this descending melodic idea. Example 1b.

Quotable? Know the line? Oftentimes in the improvisational aspects of American music players will play a snippet of a melody of a song and place it into another tune. Players simply call this 'quoting.' This type of improvisational play will usually bring smiles to those in the know. And this last melody is a rather quotable lick and easy to fit in any key, as it'll surely start equally fine on any of our 12 pitches.

quoting

12 pitches

Loops / a whole step / major second universe. So ... two half steps combined equal one whole step yes? And a whole step and a major second interval are one and the same? Yes to both and all is groovy in theoryville. For surely they are the same here, there and everywhere. The whole step or whole tone division of the octave creates the whole tone scale.

By its consistent construction of just one interval, the whole step, we can also term the loop as being symmetrical. This is mostly a jazz and classical color which finds its way in rare occasions into the pop and the blues sounds as an augmented chordal color most often on V7 but tonic too. Six whole steps to close the loop. A 'C' whole tone scale. Example 2.

symmetrical scales

C
.
D
.
E
.
F#
.
G#
.
A#
.
C

We have closure, C up an octave to C, but we only were able to use six of our twelve pitches in the loop. Wonder if the other six will make a complete whole tone scale too? Example 2a.

 

.
Db
.
Eb
.
F
.
G
.
A
.
B
.
Db

We have closure again ... imagine that :) Db up an octave to Db and sure enough, our other six pitches create a second whole tone scale. So only two different whole tone loops eh? Yep, only two. And with the symmetrical interval formula, can we start on any of the pitches in a group and create the exact same loop of pitches? Absolutely. Any slice and dice will work. Here is the sound of the C whole tone scale. Example 2b.

Simply up and down the whole tone color. It is a classic sound peppered rather sparingly into the American styles. Most common use? Probably adding the whole tone color to V7 in minor blues. Also, augmenting a tonic chord then moving to IV or ii-7.

augmented tonic

It's also fairly common color as part of the #11 in jazz chords. But these are harmonic uses and not melodic. It is a tough scale to tame and disguise but a refreshing and essential color when we find the right spot.

#11

Whole tone scale shape. This scale shape is fairly common, especially on the top four strings, and really gets at the whole tone color in a hurry. There are a couple of augmented chords within this shape and their corresponding arpeggios. A bit of a stretch at first, just gradually find your comfortable fingering. The one included is fairly common.

Positioning this shape at the second fret generally makes for the C whole tone scale, thus the open circles on the C pitches in the diagram below on the left. As the theory pointed out just above, this positioning of the scale shape at the second fret also contains the pitches of the D, E, F# / Gb, G# / Ab and the A# / Bb whole tone groups of pitches. Example 2c.

pitches
fingering

Do note the 'big and bold' nature of the graphic shapes generally denoting grande essential potential :) Lovely thing about these scale shape diagrams is that oftentimes some easy licks and fingerings pop right out. The 2/1 - 4/2 on the 5th and 4th strings. The 1/1 - 3/3 on the 3rd and second strings and surely the 1/2 whole step motion on top.

Again the idea that with the whole tone color, thanks to our guitar's built in theoretical organization, we can move any idea up and down by whole step and remain theoretically correct pitchwise within our key center or chord change of the moment.

Jazz players will often take advantage of this physical nature to create an idea with two or three pitches, a melodic cell, and move it up or down the neck by whole steps. So if you're playing a tune in G and finding things in second position, the whole step motion helps us run the thing right up to the top at the 12th fret and resume our G musings with a new scale shape.

A whole tone idea. Here is a fairly common use of the whole tone color when used over dominant harmony in a resolving nature. Thinking Two / Five (V7+5) / One in C major. We use the augmented 5th, Eb, as a half step motion moving up to E. The opening of this idea, the descending D minor triad arpeggio with a touch of the chromatic, was inspired by jazz guitarist Jimmy Bruno. Example 2d.

Do remember that with the whole tone color, thanks to our guitar's theoretical organization, we can move any idea up and down by whole step and remain theoretically correct within our key center of the moment. Click to the all things wholetone page to continue your explorations of this distinctive color.

Loops ~ the minor third / diminished arpeggios. Adding a half and whole step together creates the minor third interval. Beloved by blues cats as the blue third, the minor third is a crucial link to the blues roots at the core of the Americana sounds.

It's the third of the minor triad and as we'll see with a minor third cycle, stacking the pitches of the minor third interval into a four note arpeggio creates one of the great accelerators of our local harmonic universe; the fully diminished seventh chord.

All of this and we still retain our perfect loop to boot. But you knew that was going to happen right? Perfect closure, every time. Example 3.

Americana

minor triad

C
.
.
Eb
.
.
Gb
.
.
A
.
.
C

We have closure, C up an octave to C, but we only were able to use four of our 12 pitches. Wonder if the other eight will make two more complete cycles of four pitches, as based on the minor third interval? Ex 3a.

12 pitches

C
.
.
Eb
.
.
Gb
.
.
A
.
.
C
.
Db
.
.
E
.
.
G
.
.
Bb
.
.
Db
.
.
D
.
F
.
.
Ab
.
.
B
.
D

We have closure. And sure enough, our other eight pitches create two complete diminished seventh arpeggios. Are three unique diminished arpeggios the max? Pretty much, 3 arpeggios x 4 pitches = 12, and we've used up all of our our pitches. Dig the sounds of symmetry. Example 3b.

 

OK with playing this lick? It's a bit of a different motion. Can we stack these pitches into diminished chords? Getting a bit ahead of ourselves here but yes, arpeggio pitches stacked and sounded together become chords. So three different diminished chords covers them all? Yep, just three, as the perfect symmetry allows any pitch to assume the root pitch responsibilities. Example 3c.

So both of these last few ideas take advantage of the interval symmetry of the pitches as well as physical symmetry available on guitars. This basic idea of symmetrical motion can play a HUGE part in one's musical evolution. How so?

Simply in that by learning one movable idea with a set fingering pattern, we can then freely move this shape of pitches up, down and around, vertical, horizontal, even move patterns within patterns. Limitless combinations, like snowflakes. And for the improvising artist, always allowing for the mood of that day to shape the ideas.

Players and theorists often call these sorts of directional ideas as being in 'parallel motion' or motion created with a 'constant structure', both artistic techniques that employ defined shapes of pitches sounded with defined fingerings. Sort of like a banjo roll type fingering, which when all combined in time, will accelerate the sense of forward motion and even stretch the pull of the swing a wee bit more.

One trick to making it work theorywise is to know the letter names of the pitches so we can keep theoretical track of what we are actually doing. Through rigorous shedding, an idea's shape can and will become second nature. We use these naturally, 'built right in' guitar shaped ideas when performing, oftentimes when taking the music from inside to outside and back as the case may be. In this next idea we simply ascend a four note cell to see this 'shape magic' in action. Example 3d.

inside / outside

Diminished scale shape. This next scale shape is probably the most efficient of our diminished colors and lucky for us, it also bolts right up into the four finger / four fret technique potential. In initially learning this shape, it surely helps to locate the four different associated root pitches and where they are on each of the four strings within the overall shape.

In this next idea we identify one root pitch, G, with the open circle. Puzzle out the whereabouts of the other three related minor 3rd intervals; the Bb, Db and E within the shape, and they'll be yours forever. Ex. 3d.

pitches
fingering

Old time blues. This next idea goes way back in the American sounds. To some of our earliest recordings. It's funny because I've heard this lick for years but never associated it with the minor 3rd / diminished chord theory, perhaps because we so often only here this much of the pitches / tritone interval.

Not really being an early blues style player myself, I was able to confirm this use of the diminished color with the established Americana guitarist Paul Asbell, who said the diminished color in this spot is not unusual at all. Here's the lick. Example 3d.

Sound familiar? Not uncommon in original blues guitar. This lick becomes part of the evolutionary process; for if you can 'accept' this sound, make it work in your music, it'll help in coming to terms with both the diminished colors and the blues rub.

So learn it here if needed, transpose it a bit on different string sets and find a spot in your own music to fit it in. In doing so you hitch your wagon up to the historical blues on the Americana musical highway, joining up with a zillion variations of this central musical idea.

Fast Four / Tritone interval. Probably getting ahead of things a wee bit as we'll meet up with the tritone a bit further down this page but just too theory easy not to add some pitches to the last idea. So the 'fast four' simply implies motion to the Four chord in the second bar of a 12 bar blues. Jazzer's include this motion rather regularly as it gives us yet another option for soloing as the harmony shapes and directs the line. Any slow tempo blues might make this 'fast Four' also.

easy theory

12 bar blues

jazz blues

Well the trick of it is that in our early blues sounds, the Four chord was oftentimes also a diminished chord. A Seattle player Ken Carlson passed this along down at Emerald City Guitars one day. Here's a melodic idea running the diminished scale / minor 3rds idea over the Four chord. Thinking 12 bar blues. Example 3e.

Combining the old and the new. So we took the old two pitch blues style idea and theoretically projected it into chords with a fast Four idea, a motion towards a common jazz style event. In discovering these types of relationships between the pitches and components, the music theory evolves our musical style.

Understanding what we hear in the music theoretically and using the theory to generate new ideas are the core benefits of doing the work to learn and understand its magics. We can forever self propel our lifetime learning process simply by having an organic understanding of our music theory. Organic? Organic music theory? Or ... music theory from its organic origins? That's better :)

Loops / major third / augmented triad. Adding two whole steps together creates the major third interval. Perhaps the most important interval of all the ones we get, the major third defines the major triad, the core building block of so much of our beloved AmerEuro music. Do sing along with this bit of melody, for the last pitch E is the major 3rd above the C, our tonic pitch. There's a lot of Americana history in this song. Ex. 4.

History. So, know this opening lick? Mostly know around as 'Amazing Grace', turns out that those are just the words as the melody comes from a different song titled 'New Britain.' Hard to describe the impact that this song has had on America. Suffice to say that it's a classy tune to play, everyone in the room should know it and that in itself, is reason enough to learn it. And if you're thinking of getting into bagpipes, 'Grace' is a standard.

A loop of major thirds. Creating a loop of major thirds, we create a whole tone type character sound and of course the augmented major triad by nature of its augmented fifth. Example 4a.

augmented triad

C
.
.
.
E
.
.
.
G#
.
.
.
C

Again the closure, C up an octave to C, but we only were able to use three of our 12 pitches. Let's try finding the pitches in between the C augmented triad depicted above, starting a whole step up from C. Example 4b.

.
.
D
.
.
.
F#
.
.
.
A#
.
C

Adding things together. Again the closure, D up an octave to D, but again we only were able to use three of our 12 pitches. Let's add the two groupings from above together and see what shakes. Example 4c.

 
C
.
.
.
E
.
.
.
G#
.
.
.
C
.
.
D
.
.
.
F#
.
.
.
A#
.
C
C
.
D
.
E
.
F#
.
G#
.
A#
.
C

Theory coolness emerges when we combine the two augmented triads into creating the whole tone scale. Is this the same one we created above with the whole step interval? 'Tis is indeed. So, we've still only used six of our 12 pitches. Any guesses as to what we'll create with the other six pitches using the same major third interval as our building block? Example 4d.

whole tone scale

Db
.
.
.
F
.
.
.
A
.
.
.
Db
.
.
Eb
.
.
.
G
.
.
.
B
.
Eb
.
Giant Steps
Db
.
Eb
.
F
.
G
.
A
.
B
.
Db

Well, same results yes? Hmm ... Again, no matter how we slice and dice our pitches the closure is always perfect. So combining our two whole tone scales, each consisting of six pitches, we create yet again the granddaddy of them all, the 12 pitch chromatic scale. Example 4e.

permutations

C
.
D
.
E
.
F#
.
G#
.
A#
.
C
.
Db
.
Eb
.
F
.
G
.
A
.
B
.
Db
C
Db
D
Eb
E
F
F#
G
Ab
A
Bb
B
C
Db ...

Four possible augmented triads? Combining the above major 3rd interval theory, we find that we max out at four unique augmented triads before the cycle perfectly loops back to its starting point and begins again. We can expand our resource from just these four augmented triads to include all of our possible 12 by juxtaposing the pitches of each chord.

Surely we can also build each of these augmented triads from each of the 12 pitches. That idea lives of course in the 'anything from anywhere' concept. What is unique of course is how each pitch of the triad can function as the root of the chord. What other interval colors and chords might have this multiple root possibility? For every Yin must have its Yang n'est ce-pas? Hint; think of symmetrical structures.

symmetrical structures

Three for one. Thanks to our perfect theoretical closure and interval symmetry, each of the pitches of these major augmented triads can assume the root function of the chord. So while we can still have the various inversions of the augmented triad, we also gain the potential of the inversions becoming root position chords. How cool is that! Examine the three possibilities with pitches identified from each chord root. Ex. 4f.

root position

C
.
.
.
E
.
.
.
G#
.
.
.
.
E
.
.
.
G#
.
.
.
B# ( C )
.
.
.
.
G#
.
.
.
B#
.
.
.
D## ( E )
.
.
.
.

Root motion / inversions. While root motion of the chords / bassline is always paramount in telling the tale of a tune, inversions help to give composers that infinite variety of shading of emotional hues of aural color and varying degrees of the forward motion of tonal gravity.

telling the tale

aural color

While this augmented color is a rare element in much of the American sound, as it is mostly a jazz color nowadays, knowledge of it not only allows recognition of it in the music we hear but becomes part of a composers aural resource. Here is a chart of the four different augmented triads and their eventual perfect closure. The pitches align. Example 4g.

C
.
.
.
E
.
.
.
G#
.
.
.
C
.
Db
.
.
F
.
.
.
A
.
.
.
.
.
D
.
.
F#
.
.
.
A#
.
.
.
.
.
Eb
.
.
.
G
.
.
.
B
.
.
.
.
.
E
.
.
.
G#
.
.
.
C

Theorists, we have closure. Are four unique augmented arpeggios the max? Sure is, the math of it is 4 x 3 = 12, so we've used up all of our 12 pitches. Augmented triad plus 7th? 9th? Absolutely. Two things in the symmetry here; cool chromatic angles and the C's in the top corners a reminder of perfect closure of our loops of pitches. So really just digging the 'Escher-like' closure properties of equal temper tuning and of the oftentimes perfect symmetry within its realms. Pure magic.

Escher-like

equal temper tuning

On the fretboard, the 'Esher' quality of the chart shows right up in the pitches and tab fingerings for the augmented triad. Jazz players take advantage of this type of motion. Metalists too with their sweep picking magic. Here is its sound. Example 4h.

In the music. Two rather common uses of the augmented triad come right up. For the blues leaning artists, we so very often hear the augmented triad as a Five chord in the blues, setting up the return to the top. The augmented 5th of the dominant chord is the bluer minor third of the target key.

Augmented triad. Here is the common use of an augmented Five chord setting up the top in a 12 bar blues. Example 4i.

12 bar blues

Turnaround chord. The D augmented chord in the first bar of the above idea is also used throughout the song as the last chord of the turnaround to the top of the 12 bar blues cycle. Quite effective in resettling things.

Check out The Allman Brothers cover of the essential by T. Bone Walker's "Stormy Monday", just a classic take on a classic song. One of many but surely a memorable arrangement to a venerable song.

T. Bone Walker

For hipster popsters, surely dig out, cue up and enjoy The Beatles' "Oh Darling" right from the top. For the first chord in the arrangement is the augmented triad. Really gets attention and sets the tone.

Modern motions. Jazz cats probably noticed the link to John Coltrane's 'Giant Steps' just above. Here it is again. Mr. Coltrane's artistic evolution brought him to this pinnacle of theoretical development and challenge, often termed 'Coltrane changes', as a last developmental stop before launching into 'A Love Supreme' and beyond, musics that are more melodically centered on pedal tones and ragas than such true cyclical harmonies.

Mr. Coltrane puzzled the pitches of the augmented triad to become the key centers of 'Giant Steps', thus using an unbreakable loop to create a harmonic cycle. A tune many jazz players enjoy to this day both for its joyous gospel found in a more moderate tempos and yet very powerful strengthening for the intellectual challenge in the shedding / preparation for its performance as well.

key centers

Regardless of whether we ever gig 'Giant Steps', the thought process of shedding the tune is potentially very valuable to the intellectual development of the improvising musician. For it is just so unique than from anything that ever came before it. While there always seems to be common historical elements in the evolution of American song, in 'Giant Steps' we can find a similar foreshortened cycle of chord changes in the bridge of the 1937 Rodgers and Hart jazz standard "Have You Met Miss Jones."

"Giant Steps" adds a new compositional scheme for American harmony, and in the original recording, all at a rather ferocious tempo. Combined they vertically ramp up the improvisational challenge for many jazz directed artists of all instruments from this point forward in our collective history. A bit of trivia history here, Tom Dowd, who engineered this game changing jazz recording for Mr. Coltrane, a dozen years later engineered the sessions for blues rocker Eric Clapton's epic 'Layla' sessions with riff master Duane Allman.

Key centers based on the augmented triad. Here is the root motion of the first four bar phrase of Coltrane's 'Giant Steps.' It simply is a rather ingenious way to backpedal roots to a starting point using the pitches of an augmented major triad. Each pitch of the triad becomes a tonal center of the composition. Read right to left the Eb augmented triad pitches. Backpedaling? Yes.

~ B / D / G / Bb / Eb ~

That key centers now have something new to have in common, the new 'closed' modulation and compositional scheme, based on the augmented triad pitches, truly re-electrified the scene in 1960. Didn't daVinci sometimes think, thus write this way too? Write sentences from right to left? Which when held up to a mirror's reflection it then appears as normal writing? Here are the roots of the chords creating the closure. Example 4j.

root / - 3rd / p.4th / - 3rd / p.4th / - 3rd / p.4th

Getting there in a new way. Well, while in a way just the same old B up to B, spanning two octaves, we surely didn't get there a traditional way. In pianist and theorist George Russell's The Lydian Chromatic Concept Of Tonal Organization, Mr Russell describes the various vehicles major American musical innovators employed to get to their harmonic destinations.

In a 'story' about traveling harmonically along on the Mississippi River, while there are rafts and steamboats and speedboats, Mr. Russell describes Mr. Coltrane's melodic vehicle of choice as a rocket ship :) And while surely the theory is cool and the Five / One to close the loop surely helps to get our ears around these sounds in a more conventional cadential way, the original recording by Mr. Coltrane really illuminates that axiom about how the 'whole is greater than the sum of its parts ...'

Exhausting the possibilities. Is it possible that Trane's evolution to 'Giant Steps' is in part created by his exhausting of the challenge of the current music he was writing and performing? That through the theory of chord progressions and chord substitution, he gradually devised greater and greater musical challenges for himself ? Seems a natural enough progression. Ever get bored with your own playing?

A prolific composer, we can examine Mr. Coltrane's compositions historically by their recording dates and thus theorize to a possible harmonic evolution of his understanding, development and culminations.

A similar evolution perhaps? The sixteen string quartets of Herr Beethoven might provide a similar degree of insight into this evolution of a musical artist. We can more easily track the Beethoven works by their published dates and thus examine with a greater degree of confidence the tonal evolution between his early and late quartets. While all of the music is aurally stunning, the ramping up of the musical challenge of the later and last works is simply unmistakable. For even the best of players in Beethoven's day were said to have declared the last group of quartets 'unplayable.'

L. Beethoven

Beethoven string quartets

Do explore these wonderful works as your time and resources permit. There are surely many, many other composers whose creative works follows a similar sort of evolution. Actually, in all of the fine arts and beyond, one might track such evolutions. One way that both Coltrane and Beethoven are remarkable to me is that their own 'evolution' never leveled off or declined.

 

They seemed to have continued to ramp up the challenge musically and artistically until they raptured onward. Neither artist had a 'retro period' that I know of that re-visited earlier successes in new ways later in their careers. As a pure jazzer here, I shed the Coltrane regularly while the Beethoven quartets always seem to be able to put a bit more 'Sunday' into my Sundays :)

 

The perfect fourth. We get to our next musical interval simply by enlarging the major third by half step creating the perfect fourth. Why perfect? Quality of sound. Expressed in numbers by the simple ratios, it is the inverse of the perfect fifth too :)

In proving up the closure of the perfect fourth interval we need to use all of our 12 pitches. With equal temper tuning we create the perfect closure of actual pitch. So an easy solution for proof of closure is simply to look at the cycle of fifth's but read from the top pitch to the left or counterclockwise.

Jazz players often term this counterclockwise motion backpedaling, as they see it in the root motions of chord progressions in countless jazz standard songs. From C at 12 o'clock top to the left to F / Bb etc. Example 5.

backpedaling

root motions of chords

standards

Graphic is perfect for shedding. Players often like to shed musical ideas through the cycle of 12 keys as represented above. And while mostly a jazz thing, musicians of all musical styles may benefit from the process. Even if we do it just a couple of times, it's amazing how this simple exercise gets us to the distant places in the resources, that then become not so distant.

shedding

not so distant

And of course we'll never know when we might need something from the far reaches. So pick an easy idea, such as major triads, and use the cycle graphic as a visual aid to keep track of things. Use the pitches of the cycle as the roots of the triads. It's fun, challenging and pays big benefits depending on your artistic directions now and into the future.

Wedding bells. Among the most common and recognizable melodic ideas we have centered around the interval of a perfect fourth is Richard Wagner's 'Wedding March.' Sing along with these pitches with the word, 'here comes the bride.' Example 5a.

12 key cycle / melodic idea. Let's run just the motif of the last melody through the cycle of fourths as described above. Play this line completely through and you'll have covered an important component part of each of our 12 major keys. Example 5b.

12 key cycle / major triads. Oh well while we're at it, we might as well run the major triads as suggested just above. Do dig the exercise nature of the fingerings as we move across and up the strings. Once comfortable with the idea, perhaps try a quarter note triplet rhythm. Example 5c.

Jazz players ~ triads are king. Why? Well in traditional jazz improv, artists are soloing through many many chord different changes and the triad is their basis. For aspiring jazzers, running triads is a good exercise. Bass players too of all the styles as the triad pitches are their bread and butter.

Permutate triads lots of ways; by playing one triad ascending then move up a fourth and descend with the next, or chromatic motion up or down. Add in the 2nd scale degree to create a major pentatonic flavor, to create Coltrane's choice over the 'Giant Steps' changes described just above, example to follow. Try running the seven diatonic triads through 12 keys. There's just tons of options with triads and various melodic filters.

Triads are also the core launch pad for arpeggios, establishing their major / minor tonality. Spanning stylistically from Chuck Berry's classic bluesy essential core rock'n roll double stop idea of Johnny B. Goode to Clifford Browne's moody jazz 'Daahoud', both are triad / arpeggio based. Parker and Coltrane, arguably among the most masterful of the triad and arpeggios kings through their dedication and hard work, ascending in their days to become Kings of American Music.

12 key cycle / major pentatonic idea. In this next idea we simply outline the first four pitches of the pentatonic scale in eighth notes and move through an ascending cycle of fourths. This four pitch idea we can term our constant structure. Again, what we take advantage of here is the physical properties of how the pitches lay on the instrument.

As a player evolves, moving a constant structure, whether melodic or harmonic, across strings or in parallel motion oftentimes becomes a thing to do. With that in mind, this is a fun warm-up exercise that is easily replicated with other aural colors and rhythms. Ex. 5d.

A long term goal / anything from anywhere. What a lot of these last exercises are developing is the cognitive and physical ability to understand and execute any of our musical elements from any pitch on the neck. A tall order for sure, with a lifetime's worth of shedding and all that adventurous musical discovery along the way.

With our intellectual arms completely around the resource thanks to our silent architecture, we simply must each decide to what degree we need to understand the resource and then find those sounds on our guitars to bring forth the art within our hearts. And as we'll see as we move along in our studies, the musical styles we dig becomes a large factor in determining how deep we go in the theory. At least initially anyway, for there are no bounds for the curious n'est ce-pas?

Notating the pitches. In this next idea we simply move by ascending in perfect fourths. Starting from our open low E string we can use our guitar's standard concert tuning of perfect fourths before ascending to find our other pitches. There is an octave transposition in the third bar to keep us within easy range for most guitars. Example 5e.

octave transposition

.
E
A
D
G
C
F
Bb
Eb
Ab
Db
Gb
B
E
.

And yet again that perfectly lovely closure of the 12 pitches in its unbreakable loop. I'd imagine by this point in our process that the surprise factor of ... 'will this interval loop close' is probably long gone. Oh well, for what remains stays and has stayed for a couple of thousand years now.

For what is 'theory true' here in AK is the same 'theory true' southbound to B.C., Seattle, Portland, San Fran, L.A., Austin, Nashville and N.Y. London? Paris? You bet. Moscow too? Moscow too, music theory same the globe over. Is American music theory different? In one colossal way yes. We have blue notes and the blues.

Loops / tritone / #4 / b5 ... the halfway point. Our last interval for this first 'loops' page we end up splitting the octave into two perfect halves creating the interval of a tritone, the flat five / b5 or or augmented fourth / #4, the 'diabolus in musica.' 'Tritone' really just means 'tri = 3 whole tones.

While mostly shunned from much of the very very early literature that we've inherited, once the pitches are consistently equal tempered for creating harmonies, the tritone interval rockets to the top of the AmerEuro charts. For it energizes the traffic cop V7 chord which also becomes our tried and true blues chord. It helps energize the diminished and whole tone colors, thus the V7b9 and V7b5 chords and their basic substitution properties. It helps ground a musical style, the blues and is a potent blue note. So from shunned to pure American essential over a half millennia or so.

V7b9

substitution

musical style

The tritone as a catalyst. Five 7 is of course the crucial chordal color of the blues It's also the harmonic traffic cop directing our chord progressions, modulations and the original catalyst for chord substitution. Its interval closure is easy to prove for the tritone splits the octave perfectly in half. Thus, two tritone interval leaps and we perfectly close our octave span. Example 6.

blues chords

chord progressions

modulation

chord substitution

interval leap

Pretty tense sound eh? That's probably why it was shunned way back when. As we can see from the notation above, the tritone interval splits the octave perfectly in half, all of our other pitches falling in between the two tritone interval segments. We can also see this splitting of the octave represented on the cycle of fifths. Example 6a.

C to Gb. Clearly the Gb tritone lives directly across the circle from its tonic C and there are five pitches either side. So if C and Gb are directly across from one another and span a tritone interval, might that be the same for our other 10 pitches? Tis is indeed the case mon ami. We can select any pitch on the circle and locate its tritone by locating the pitch directly across from it on the circle.

A two pitch safety call. We can hear these two pitches and this important interval today from the emergency response vehicles siren as they carry compassionate people who thankfully come to the rescue of those in need. Example 6b.

Tritone roots. In the American sounds, the tritone's original way into the music is probably as a blue note. And for us modernes, no easier way for guitarists to invoke the blues aura than sounding the tritone of the key we are playing in. Here is a quite common 'sounding of the tritone' to set up some further testimony. Ex. 6c.

The gist. Simply that if we play this sort of idea at any point in any tune, chances are a bit of the blues color and its aural magic is going to emerge. Its a sure bet to bring a blues hue to wherever we are in the music; simply a repeated call of a blue note, or two or three, or repeating a phrase like the one just above. This ability to ground things in the blues can come in handy in lots of ways and places and is core Americana.

Tritone in the line. In this next melodic idea, we find the tritone tucked into a rather notey melodic line. It gives a nice pause to the swing and creates a cool sense of suspension of time. Here's one of mine titled "Tuxy."

In C major, the tritone pause is at the close of the first measure tied over the barline. Do examine the full arrangement as time permits as we use the same bit of the melody line to close out the arrangement. Ex. 6d.

Surely a jazz styled line, suspensions in the melody are common in all of our styles. The double Two / Five adds to the verve and forward motion.

Tritone in musical styles overview. So we won't find the tritone interval or its sound colors in folk music or children's songs, unless they're of a scary nature and even then it's rare, as the interval / pitches are hard to sing. Lasting children's songs are usually easy to sing. Also, in children's songs, so often just the dominant triad is used to nudge the cadential motion to tonic resolution.

We surely find the tritone interval in every form of the dominant 7th chord, which is quite common in folk. Obviously tons of tritones are used the blues of course and any blues / rock / country flavored mix. For the metalists, the tritone interval, sounded through their shredding gear, is really all through the literature from the style's inception in the 70's through to today's contributions. Help internalize the interval by singing.

The tritone catalyst in jazz. Turns out that all of the tritones found in the above musics are generally also found in jazz. As Five 7 is the harmonic traffic cop directing our chord progressions, modulations and the original catalyst for chord substitution, jazz artists will often find that the tritone is a key that unlocks a treasure trove of theory possibilities.

traffic cop

chord progressions

modulations

substitution

V7b9

~ super theory game changer ~

Tritone sub. Surely worth mentioning at every and any opportunity is the jazz harmony and cadential technique often termed the 'tritone sub.' Its theory creates a half step motion between the roots of the chords in play and opens other melodic and harmonic options. This term simply directs us to substitute the written chord with chord of the same chord type.

And while their root notes are a tritone apart, they hold in common the same two pitch tritone. The next few jazz chords illuminate this above idea. The tritone interval between root pitches here is that the Db7 chord substitutes for G7. In Two / Five / One. Example 6e.

While this process is mostly done with dominant type harmony as shown here, once we're comfortable with substitution theory and its practice, we'll end up using the concept and half step motion to puzzle for the One and Two chord type harmonies also. And in a similar fashion to the diminished 7th chords, the tritone sub can really energize and accelerate the sense of forward motion, tonal gravity and time (swing) in our music.

The initial magic. The initial magic of what the tritone subs can do for us is easily seen in the bass line of the two sets of chords. Our D / G / C becomes D / Db / C. Those in the know will see the chromaticism of the tritone version. So just what is the big deal here? Well, like so many things in our ever modernizing world, sleeker tends to imply faster ...

In the evolution of style and theory in American music, sleeker components not only allowed for brighter tempos but also the ability to slip new chords in between the existing progressions. These new chords can increase the improv challenge and encourage the inclusion of musical elements from, and to, additional key centers.

So for those of us that blow 'through the changes', more options simply means more possibilities in creating their improv directions. More options also can mean a greater intellectual challenge of the music. Much of what happened in jazz between the late 1930's into the early 60's was facilitated by these 'sleeker elements' moving faster and faster through musical time.

So the more streamlined our motions are, the faster we can run them. The faster we run them, the more exciting they become. The more exciting they become, the more we might want to do it. Probably not all the time of course but certainly sometimes :)

A hidden agenda. So when the musical evolution of the dominant chord substitutions began to appear in the late 30's and into the 40's, the music gradually became faster, thus potentially more exciting. And a key aspect one might want to consider at this juncture is the question of why we have historically made our music ... so folks would get up and dance? Maybe. And when we dance to the music what are we really doing?

Halfway. Since we are halfway through our discovery process of loops of pitches / intervals within the octave span, I thought this would be a good place to split the discussion. Just makes for smaller pages so easier to load up out there in cyberspace. Next up, the interval of a perfect 5th and our continual search for perfect closure.

~ Looooooops of pitches ~ page 2 ~
Footnotes:
(1) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 40-42. USA Alfred A. Knopf, New York. 2001
(2)Aebersold, Jamey and Slone, Ken. The Charlie Parker Omnibook. New York: Atlantic Music Corp., 1978.
(3) Ottman, Robert. Elementary Harmony, Second Edition. New Jersey: Prentice-Hall, 1970
4) Ottman, Robert. Elementary Harmony, Second Edition. New Jersey: Prentice-Hall, 1970

'It don't mean a thing if it ain't got that swing ...'

Duke Ellington