~ silent architecture of music ~ ~ organic origins & tonal organization of pitch ~ 'from earthy, natural acoustics comes the pure octave basis of our 12 chromatic tones and the diatonic realm within'

"If it sounds good it is good." Attributed to America's own maestro, composer and piano master Duke Ellington, his quip sparks to life here the basis of our understanding our all our musics; for when we follows nature's own original way of creating and organizing pitches, they sounds good. Just ask the songbirds :) So with a tuning tweak here or there over the last few millennia, no real wonder today that if it sounds good to us, probably is 'in theory', good too :) 
In a nutshell ~ 'silent architecture.' So we can begin our theory studies simply by delving into various characteristics of the musical notes we use today. The basis of which is really how we've tuned these 12 pitches over the last few centuries, and as tunings evolved, so did the music they are capable of creating advance. For by refining the original ways for determining our melody pitches, we now create the perfectly equally stackable collection of our 12 pitches to make the chords, evolving the harmonic schemes of our modern music. 
These two old and new ways of tuning combine with musical forms and time concepts to create the basis of the silent architecture of our AmerEuro musics. For in a very general sense we've a looser tuned, blue note hued melodic weave over the quite stable, close tuned, set in stone pitches for chords. The guitar's uniqueness of bendable string pitches over precisely tuned chords is a big piece of the pie for a lot of players, yet still just one art approach among many with simply 12 pitches. 
Pop quiz. So in theory, how many pitches we got? ________. 
Our earliest musical pitches ... ? A fairly recent discovery of an old flute by archeologists gives us a possible glimpse of the pitches of days long past. Placing a reconstructed model of this ancient instrument in modern masterly hands created some startling results and controversies about its histories. And while today's melodies are playable on this ax, there's of course no telling what musics were originally played or the sound of their pitches. Just that today's melodies are coaxable from this flute helps us create a vision of our musical ancestry, enabling us patch together the evolution towards the pitches we are using today. 
History ~ theory overview. This discussion takes us nearly all the back and then back into the future, 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 tuning systems melted right on into one. What's not to love about that ? 
Overview. The idea of a 'silent architecture of music' refers to the structural nuts and bolts of the pitches we use to create our American musical sounds. Part art, part science and surely part magic of nature, understanding this architectural theory helps us project and filter any idea through a wider range of options. Knowing the basis empowers us to sort things out as each new pitch comes along. The idea is to build an intellectual theory structure within, so as the new ideas come along we have a framework to store, organize and recall our ideas. 
Along the way of this discover process we need to explore some of the history and by necessity, the basics of natural sound, i.e., acoustics, and how we are thought to physically hear sound. This is our first topic of a few where music and math will meet. We combine these to create the precursor for understanding why we tune our instruments of today the way we do and what we gain by tuning the pitches in this manner. 
And even though our story includes thousands of years of creative output, creating the rich and varied collection of music we enjoy today, this silent architectural structuring of our pitches has yet to vary very far from its origins. Founded on earthly natural sounds and as we'll soon see, its scientifically measurable acoustical properties, we've simply tweaked our tuning of this core a time or two over the millennia to arrive at today's pitch resource for the modern guitarist. 

As guitarists. Turns out all we need to begin this discovery is of course built right into our instruments. We're simply going to use the pitches created by the guitar's natural string harmonics to recreate one way of how our pitches come to us. From the historical view of this, the whole theory tamale revolves around the two pitch octave interval, which lives on today in so many of our cherished American melodies. 

Our story begins at the village blacksmith's shop. Core aspects of our present day organization of music comes to us as part of a package deal often described under the broad heading of Western Civilization. We can trace this thread back through European history to the Romans and even further back through to the Greeks circa 1500 B.C.E., whose philosopher Pythagoras, who lived around 500 B.C.E. era or so, is credited to have discovered the organic basis of our pitches and interpret them with true numerical values. Pythagoras is also the original 'triangle master' yes? 
As our written historical attests, while Pythagoras and his band were passing by their local blacksmith shop and hearing the simultaneous clanging of hammers and its raucous racket, legend has it that within this cacophony of noise, someone recognized that certain, repeatable combinations of the hammer blows created sonorous or pleasing musical sounding combinations of pitches while other hammer blows did not. (2) 
A pioneering heavy metal enthusiast no doubt, Pythagoras' clan further inquiries of the craftsman and their hammering revealed that these most agreeable combinations of sounds were created by hammers whose physical weights were in measurably even proportions to one another. (3) This discovery of the relationship between physical proportion and pleasing sound apparently excited their curiosity, prompting Pythagoras and his peers to further research and measure the nature of physical sound and its effects on our human psyche and aural abilities. 
Recreating the magic. Devising a single stringed instrument called a monochord, Pythagoras is thought to have discovered and recorded in writing (now lost) the beginnings of our understanding as to the physical, measurable relationships of why two different pitches sound good together. This monochord as pictured below shows the man himself with his rig at work. Looks like a six string axe, two bridges and string tension weights for tuning the strings to different pitches? Matched grip on the sticks for setting the thing all in motion. 


Concordant tones. 'So if it sounds good, it is good' of course holds true. For it is these pioneering discoveries of the natural properties of pleasing combinations of pitches, what became know to us as concordant tones, which forms the theoretical basis of our musics. 
The octave purity of sound. Sounding a fundamental pitch and coaxing forth its various harmonics, Pythagoras must have believed enough in the octave's purity of sound to make it the basis of his work. He and his pals might have also inherited this octave basis from their peers. For here the mists of antiquity become more of a fog for today's musicologist. It'll also turn out that the aural purity of the octave can be represented by the simplest ratio of whole numbers. This whole numbers mathematics governed our pitches for a couple of millennia. Thus; naturally pure and simple fed the ancient theory bulldog. 
Click the following music for mp3 playback and sing along, finding the octave interval of C natural, written in today's standard notation with the 'G' treble clef. Are you cool with the notation symbols? Of course the tab helps too in locating pitches. If needed, try the link to explore the nuts and bolts of notation and a bit of its history. Example. 3. 
Can you confidently sing a strong and in tune octave interval? A big first leap for sure but a challenge worthy to master. Maybe learning a stronger melody that includes and octave interval will help lock in the pitches. 
The perfect octave interval is the core. We theorists determine this purity of our intervals by how completely two or more pitches aurally blend together. Turns out that the less we can distinctly hear either of the pitches of the interval when sounded together, the purer the interval is said to be. Perfection here implies the complete merge of sounds of two pitches into one. 
That there are only three perfect intervals out of all the many, many possible pitch combinations in our system is testament to their natural powers. Their individual tonal strength and balance triangulates a three part silent architecture for our musics. 
For when the octave interval is sounded, the two pitches melt together and we mostly hear just one pitch overall, created by the two different pitches combined; a hint of the upper tone and hint of the lower become one tone. Simply the physical phenomena of earthly natural acoustics of aural tones when perfectly tuned up. 
So perhaps no real surprises here, but this one idea now solves the initial theory puzzle; that of creating a perfect closure thus a looping of the pitches. That the two pitches of the octave interval are the two bookend pitches that create the perfect closure of our theory system. 
Sound the following octave interval a couple of times and hear the melding of the two pitches into one. Octave purity a la midi digital, the purest of our musical intervals when accurately tuned. Example 4. 
Do you hear what I hear? As the octave interval is sounded, does either the upper or lower pitch stand out? Hear any hint of the upper or lower pitch in the total sound? Try it again. A magic merging of the two pitches? 
Hear the lower pitch first and then the upper as its sound tails off? In general terms, much the same way we might hear the ringing of church bells. We tend to hear the fundamental pitch i.e., the lower pitch, and then the higher partials above. Surely do sound the octave interval on your guitar and listen for the magic. 
Sound the following example comparing the most common of our intervals on our palettes. Listen for the purity of 'melting tones' when the octave interval is sounded. Example 4a. 
Cool? Of all of the tasking with the theory, hearing it in action is typically the longest row to hoe. Luckily, once these things begin to fall into place for us, they build rather quickly and once we got it, we keep it forever :) 
Quick review. It's this naturally occurring, purest marriage of the two pitches into one sound, that forms the initial core of our theoretical musical universe. We theorists use this perfection of sound of the octave interval as our standard for defining the sound quality of all of our other musical intervals. And for a very long time now we've called the octave and its soon to be discovered closest relatives, our perfect intervals. 
Simple is best. As we'll aurally hear and theoretically formulate as we progress in our study, none of the other musical intervals we have on our 12 note color palette can match the sound of the octave, in not only perfection of aural blending but as it turns out, in its mathematical simplicity as well. Ah, music and math. 
Guitarist Wes Montgomery. In thinking about the central powers of the octave interval, and its place at the historical center of our local universe, we'd be remiss not to speak of how jazz guitar legend Wes Montgomery spoke his melodies pitches with octave doubling. While by no means alone in this art approach to melody and soloing, most Americana musicologists could agree that Mr. Montgomery changed the melodic arts forevermore with his own phrasing style in octaves, becoming an artistic signature known globally in modern today. 

That the depth of swing capable with octaves could very well succeed that of single note lines. Octaves swing deeply in traditional Americana styles from folk to jazz and love the blues. For us guitarists, Wes' ideas are surely as revolutionary in style and impact to his contemporaries and forward as Louis Armstrong had done for trumpet in the 1920's. If needed just google around or find suggestions for listening to Mr. Montgomery in the supplemental song section. 

Music theory and physical science. Like most sounds we hear, we can define our musical pitches as physical vibrations of air waves and as such, we can measure them a couple of ways. 
Recreating Pythagoras' monochord, we can physically measure with a ruler the length of an open string on our guitars, termed scale length, that we use to generate our fundamental pitch. We can then locate on this string where the octave harmonic lives and physically measure its length and compare the two. 

Even though this next diagram is not to scale, any guesses as to where we will find the octave harmonic? Find a guitar or any string instrument and a ruler to try this. Hint; look for the double dots on the edge of the fret board. Example 5. 
24 inch string length /_____________________________________________/ 12 inches octave 12 inches /______________________ / _____________________/

From the diagram we can see the location of the octave. Think of where those 'double dots' live on the edge of your fingerboard. We simply divide our string length perfectly in half to locate the octave harmonic. Is this the harmonic we use in intonating our instruments? Yes indeed. Pythagoras was said to represent this pure octave sound mathematically by a ratio of numbers. 
If we represent our entire string length by the number one (1), then split it into two equal parts (2), what would our ratio of numbers be in describing the relationship between the higher pitch (2) and lower pitch (1) ? 
Ratios of numbers describe the magic. The mathematical ratio of the two pitch octave interval turns out to be simply 2 to 1, both of which are said to be prime numbers. Expressed numerically with a colon symbol as 2:1, this ratio of numbers implies that the upper or higher pitch of the octave pairing vibrates twice for every one vibration of the lower pitch, thus the 2:1 numerical designation. 
In olden times as the scientific perspective of physical nature evolved, they looked for these perfect numerical relationships to justify, order, catalogue and determine that all was groovy in their numerical representation of their physical universe. They simply were doing the best they could with what they had in the emerging field of scientific measuring and discovery. 
The expression of our musical intervals as ratios of numbers is important in another part of our historical story, one that in a sense reflects a thousand years of the European experience during which time our natural sciences emerged. The gist of this story is that these prime numbers were used to help center the celestial, heavenly universe right here on earth. 
That the various relationship between ratios of prime numbers shaped not only the purity of the musical sounds but also various artistic proportions in fine art, architectural building, even in creating the day to day necessities and activities of life were, to a fairly consistent degree, governed and oftentimes dictated due to the absolute purity of the prime numbers So just turns out that purity of sound = purity in numbers :) 
And while we won't really get to that story here, a good read for those curious about this historical relationship between pure numerical ratios, musical intervals, their tuning evolutions, really all things grand scheme coupled with the 1000 year ( give or take ) evolution of the various cultures of Western European society, would be a wonderfully told story in print by pianist Stuart Isacoff titled The Idea That Solved Music's Greatest Riddle. 
After 2 then 3 ... The next sequential step after a two part division of the octave would be to divide it by three. We credit Pythagoras for creating this process which invents a wheel for our pitches. For by dividing our string length in three equal parts we create the second of our perfect intervals, what we term the perfect 5th. 
Applying this to our guitar string, we can create the harmonic above the 7th fret, which on most guitars will have a locator mark on the fingerboard and a dot on the edge of the neck. Examine the following diagram illustrating the string division for the colossally important interval of a perfect 5th. Example 5a. 
24 inch string length /_____________________________________________/ 8 inches perfect 5th 8 inches /______________ / ______________ / _____________/

The math. The three part division of an entire string length creates the interval of a perfect 5th, our third partial above the fundamental pitch. Its vibrational sound ratio is represented numerically as 3:2, both of which are again prime numbers. 
When we hear this interval created by two vibrating strings, the three to two ratio implies that the string of our upper pitch is vibrating three times for every two of our lower one. When we combine the two pitches we hear the sound of what we term a perfect fifth interval. 
Using the letter C to identify our fundamental pitch and key, examine how the interval of the 5th above C is notated and how these two pitches are located on the guitar. We can find the letter name of the perfect fifth by simply counting up through the five letters of the alphabet from C (#1) to the letter G (#5). Example 5a. 



Example 5. 
Perfect fifth. Cool with the theory of the perfect fifth? As our new system of tuning evolved in the 18th century, so much of its impetus was keyboard driven. Examine the diagram below of the keyboard manual and see how easily we can count the keys to get to our interval of a perfect fifth. Example 5b. 
The two graphics just above, give us a couple of ways to locate our perfect fifth interval above the pitch C; by counting the letter names on our fingers, counting the lines and spaces of the staff or by counting up five white keys of the keyboard. 
So in the key of C and of course its relative minor key A minor, it's rather easy to find the letter named pitches. To project the perfect fifth interval from each of our other 11 pitches, the one's we've yet to discover here, we'll have to adjust the letter names with music notation symbols we often term an accidental, in this case the sharps (#) and flats (b). 

Easiest to read. These letter name adjustments, what we term enharmonic equivalents, are simply all about finding the easiest way to write our music out so it can be read and performed by other musicians. Properly written down, notated music also benefits all in creating a hard copy for future generations to enjoy. 
Perhaps a best example of this 'easiest to read' might be the note C. Its enharmonic equivalent would be B sharp. It's just way easier for most mortals to recognize and read a written C than a B#. Example 5c. 
Quick review. Just as with the octave interval, the interval of the fifth is termed perfect due to its pure and consonant sound quality. We can also correlate by our modern science's ability to measure, that the purity of a musical interval's sound quality also corresponds to the simplicity of mathematical ratios between its two pitches. 
Purest musical tones are our perfect intervals created by the simplest ratios of numbers. For those curious about the math, examine the ratios of our twelve intervals contained within the one octave span and explore deeper into the mathematics of sound from there. Lots more tech info is just clicks away. Look closely for your own favorite pitch combinations and see how their ratios fit in with the others. Perhaps consider building your own stomp box to understand signal path, pitch wavelength and altering it. 
Modern power chords. Those in the know surely know that this interval of the perfect fifth is the essential component we use to build the 'root + 5th power chords' for many of today's guitarists. Creating much of the over driven, crunchy sounds of the rock, heavy metal and general all round shred guitar styles of the last 25 years or so, 'less' surely can become way 'more.' 
So, from our current rocking sounds of today all the way back as far as we can basically go in written down, recorded history, we've had our perfect fifth interval as an integral component of the structural basis of so much music we love. Crazy huh? 

Ten pitches to go. So now we have generated two different pitches; C and G, we still need 10 more to make our total of 12, the correct number of pitches that comprise the chromatic scale. To find these pitches, Pythagoras simply repeated the process just described; by finding the perfect fifth above each successive note and in the process, creates a succession of perfect fifth intervals until we've a loop of pitches that closes upon our starting point. 
Fifth by fifth by fifth we create our 12 pitches. We started with the pitch C and found its fifth note G by dividing our string length in 3 equal parts. We find our next fifth by using the pitch G as our starting point or fundamental, then simply divide its string length into three equal parts, to locate its perfect 5th above. Counting through our musical alphabet we come to the letter name / pitch D. 
We then repeat this process with the pitch D as our fundamental pitch, divide the string length in three and locate its perfect 5th interval, which we identify by letter as the pitch A. Pythagoras is said to have continued in this fashion until this pitch cycle closes back upon its starting pitch C, the ascending pitches encompassing 6 full perfect octaves. And that's it. 
Examine the following letter names of our 12 pitches sequenced by the interval of a perfect fifth and then as written out on the grand staff. Example 6. 

Interesting huh? Of course we do not get this range of pitches on our six string guitars. Which is probably a good idea. Run some of these lower and upper tones through a Marshall 50 watt and 4/12 cab and all of your neighborhood critters just might be a howling' for more :) 
And on a piano? 
Those in the know know. That unfortunately due to the natural occurring properties of natural sound, as Pythagoras closed his circle, the closing note was a fair bit sharp from his starting pitch. And although it took a couple of thousand years to solve this riddle in the tuning and perfectly close this loop, in theory the stage was set way back then for developing the modern day, 12 pitch resource we fully enjoy today. 
For surely there was music in Pythagoras' day and in the generations before his times. Those pitches and the scales they formed, while also generated from nature and closed by the octave interval, worked just fine for a very long, long time thus creating a colossal mountain of wonderful music. For really not until we wanted to stack pitches into chords and change key centers through modulation, staring around in the 1600's or so, did push really come to shove between theorists to close this loop and tune up the pitches to enable harmony to evolve. 
Our musical key clock of pitches. Today we often find these 12 fifths arranged in a clock like configuration we term our circle of fifths. Starting at the 12 o'clock position with the letter C, we can move clockwise to the right by perfect fifth ( C to G to D etc. ) 
Counterclockwise motion to the left creates the inverse interval of our perfect 5th, the perfect fourth ( C to F to Bb etc. ) Just like with some many regular time clocks, we move 12 clicks before completing our circle and returning to our starting point, forming a perfectly closed, unbreakable loop of our 12 pitches. Ex. 6a. 
Clockwise fifths. Among the most important theory aspects of this clockwise motion for players surrounds the creation of our key centers. A key center is simply a group of pitches that we use to write our songs. That a tune's in G for instance, implies that the song is written in the key center of G major. 
With each click on the cycle to the right, one pitch of the group picks up a sharp to form the next new key center. As each key evolves, our new sharped pitch is the leading tone or 7th scale degree, the essential pitch to creating each new, successive key center. 
Picking up a sharp. Examine the following musical idea that illustrates moving one click to the right on our cycle of 5th's. Here we move from C major to G major. Example 6b. 
Counterclockwise fourths. In reversing the motion above, we move counterclockwise by perfect 4th. In perfect 4th motion we gain the flats as we now lower the leading tone to create each new key. Example 6c. 
Sharps and flats. So we use our accidentals to project similar intervals equally from each of our 12 pitches. In the above music we're looking at the major diatonic scale of course, whose whole step / half step interval formula was set in stone a very long time ago. 
So now with twelve pitches we have the resource to create our melodies. Obviously we do not always play in perfect 5th's and 4th's when creating our music. So what we theorists do is reconfigure these 12 pitches in the tightest formats available, while preserving the integrity of each pitch. In doing this we create a core strand of our musical DNA and the first of what will soon to be one just one of many of our unbreakable loops and cycles of these same 12 pitches. 
Silent architecture / the half step. So while in the above ideas we used the perfect 5th to initially find our pitches, we can easily reconfigure our 12 pitches by half step and create the chromatic scale all within the span of one octave. Examine the pitches and its sound in the following examples. Note the enharmonic equivalent, double labeling of five of our 12 pitches in the letter names which follow. Example 7. 

The backbone of our silent architecture. Sounding the above graphic, we create what we theorists call the chromatic scale, truly representing the theoretical backbone of our silent architecture. Constructed exclusively with the half step interval, today's equal temper tuned chromatic scale contains each of pitches of the natural harmonic series as well as the pitches set forth by Pythagoras within the octave span. (8) 
Examine our piano keyboard / location of the half steps. This next graphic, which depicts a segment of our piano keyboard, illustrates the theoretical core of a couple of thousand years of musical evolution. It contains all of our pitches and what we term the natural location of the half step intervals. 
These natural half steps are between the pitches B to C and E to F. Locating them at the keyboard is easy as there is no black key between them. Examine the location of the natural half steps of the keyboard. Example 8. 
Set in stone. As we create any and all of our musical components, we must always remember these two permanent locations of the half step interval between these pitches. Full pianos may extend the keyboard to include all of the seven octaves plus but even so, the built into location of these two natural half steps are always the same. This is what a traditional piano keyboard does, create seven full octaves plus change, i.e., the 88's. 
And while this chromatic grouping of pitches is rarely found in popular musical composition as a whole scale, every other possible combination of notes comes from this perfectly closed loop of pitches. Thus, all of our scales, arpeggios and chords, the tonal elements that we use to create all of the American styles, plus all of the European classical sounds, they ALL come from this grand daddy of 'em all ... the chromatic scale. Even the blue notes? Yep, even the blue notes. 
A chromatic melody. There is one familiar melody of a mostly chromatic quality that we all might remember from way back. An old carnival melody by Julius Fucik, dig a lick of the main theme of "Entry Of The Gladiators." Example 9. 

And while we rarely hear this type of chromatically constructed melody in the American song book, the basic theory of the above line, to chromatically enhance a mostly diatonic melody is something we can often hear in the music. 

Due to the overall brighter tempos, jazz players are the main kings of this type of treatment. Charlie Parker himself perhaps being the original King Of Chromatic. Our modern day Gospel, R&B melismatic vocal style surely a close and very beautiful second. But even a bluesy hint of a lick slipped into a diatonic idea, in theory, qualifies as chromatically enhanced, while the concept of 'blues enhanced' probably more accurate. 
Thus all manner of hammerons, pulloff's, trills, bends, more bends, any motion by half step or even a wide vibrato are common ways to chromatically enhance our music. Then there's the slide. Even just a chromatic tone or two puts a bit of slip in a line. 
Add the rhythm back into this equation and here again, Bird just might be the all time king of this in American swing and his own bebop, and the combination of chromatics and rhythms creates a vast universe to explore. Chops can become the real decider in this. So from many angles the idea of chromatic enhancement can become an interesting component in many of the American sounds we dig. 
The chromatic nature of our guitars. The following diagram simply spells out the letter names of the pitches for the first 12 frets of our six string guitars. We can see by the labeling of the pitches that each of the frets are half step increments. Thus the potential for some very rapid chromatic runs for those so inclined. I alternated sharps and flats by string, but this was only for my convenience. For in theory, key center determines actual letter names of the pitches n'estce pas? Example 10. 
An essential picture? Learning the letter names of the pitches is simply part of the learning process. Perhaps a good place to start is in insuring that we know the letter names of what is presently under our fingers. Then as our understanding of the theory and our musical components evolve, so will our need to know more of the letter names. 
Or we could just go right ahead and learn 'em all by rote, starting right now and take a giant step forward! As we each have our own best way of learning something, just apply it to learning the letter names of the pitches. Use the chart above as a model. My professor at college would often say to us ... 'Ya know, you only need to learn this thing once ... and once you do then well have it forever. This can be true with many things in life. 
The chromatic nature of the piano. The following illustration simply spells out the letter names of the pitches for the piano keys. Example 10a. 
Learning the half step interval. In the following musical example, the ascending C major scale is followed by a descending C chromatic scale. Click the music and sing along with the pitches. It's a bit tricky to do at first but keep trying and you'll get it. We must train our ears to hear the pitches and the easy age old way to learn this is to simply sing along with the music. In doing this we can also bring the music closer into our hearts. 
While the half step interval is oftentimes the hardest one to get, regardless of one's musical interests, accurately vocalizing the pitches of the chromatic scale not only tunes up our pitches but also sharpens the ear's ability to recognize a pitch's relationship to the melody or chords where it is used in songs. Example 11. 
Interesting challenge yes? Eventually we want to work towards being able to sing the pitches a cappella. A solid part of the American musical magic is based on an artist's ability to 'sing the line ... play the line.' That magical connect helping the stories in our hearts become the sounds that we create on our instruments. 
Silent architecture / musical style / artistic philosophy. Empowered with the organic knowledge of our 12 unique pitches, here in Essentials we correlate just how many of these 12 pitches do we usually find in a style's melody. Turns out there's a nice solid evolution of the groups of pitches for creating melodies. And while there are endless variations, slides through and cross pollination between styles, this basic correlation of the number of melody pitches and style gives us a clear way into all of our study topics; time, scales, arpeggios, chords, form in music and improv. 

Advanced theorist. For the advanced theorist reading here, already hip to the history and natural sciences of how we create our 12 tone loop, the evolutions of 'loops' into 'groups', all from within this loop of 12 is really the next theory step. Simply that our 'groups of pitches' today are the perfectly closed loops of pitches from which we create our ideas; melody, harmony etc. Subbing now for the term 'scale' to a certain extent, a 'group' includes its own multifaceted properties based on its perfect closure. Many solid building properties such as; the modes, shades of major / minor, arpeggios, colortones, 'parent scales' for phrases, intervals, sequences, soloing through and or over the changes all 'theory up' from a 'group of pitches.' Our chosen group become the diatonic pitches of a song, thus in theory establishing the boundaries of a select group of pitches. Surely a stabilizing perspective for venturing onward. 
Review. So from our existing written records, we discover that our music theory of today finds part of its theoretical origins with the Ancient Greeks. At its intellectual core is the ideal of aural purity, as based on the natural properties of physical sound and how we as humans with our ears perceive it. 
Composed of a system of theory and tuning that divides the octave interval into 12 distinct, independent and equally functioning pitches, this pitch resource provides the initial artistic core from which we create all of the American and European musical styles. 
In more recent times since the 1700's or so, this pitch resource is tuned or tempered equally, a tuning system we simply call equal tempered tuning. This method of tuning found its way onto the piano and has been there ever since. And as we'll discover during our studies, equal temper tuning is in theory, perfectly tuned and surely in practice becomes a perfectly closed system or loop of specially tuned musical pitches. This looping perfection defines our compositional elements. 
How and why we reached this perfection. While earlier tuning systems worked fine for the music comprised of one or more melodic lines we term polyphony, it was in the evolution of the vertically stacked pitches of harmony, which to a degree historically paralleled the development of our keyboard instruments, that we see a greater need to accept the equal temper system of tuning. For while there was a certain degree of 'stackability' to the pitches tuned the old way, it was limited and unstandardized, thus holding back the musical directions then being explored. 
This development of the keyboard of course encouraged composers to venture further a field with key schemes and modulations. As guitarists, we have a record of 16th century lutenist Vincenzo Galilei, father of famed astronomer Galileo, using the 'rule of 18' in the 1550's to position the fixed frets on his lute to create an equal temper pitched, thus a fully functioning chromatic, melody and chord capable stringed instrument (?). 
What the future may hold. Furthermore, as equal tempered tuning itself evolved to further stabilize our natural acoustics and earlier systems of tonal organization, perhaps it will also provide the structural basis for a new, musical system of tomorrow. The electronic computerized MIDI systems are just such creative directions with a wide range of potentials. 
Historically, music represents the ages from whence it came. We often need only to hear music from an era of history to evoke our own personal images of those times. Could it be that our future musical systems will represent and coalesce the global harmony so sought by so many? 
Musical style. The folk, blues, rock, pop, jazz and classical styles that have been created over the last couple of hundred years to the present, all have been created and structured on this silent architecture of natural acoustics and in more modern times, equal temper tuning. 
By duplicating these 12 pitches in upper and lower octaves and reproducing them on a wide variety of instruments, all of which can play well together, we've dramatically expanded the range of aural colors and combinations available to the creative artist. 
That's all for this discussion folks. Has this discussion stimulated any thought provoking ideas for you? Always good to have a broad perspective of the beginning point of any topic. Add in a bit historical spice and the theory begins to cook up nice and take on real visual forms. 
Lest we forget that as musicians we also strive to hear the theory work its magic in the music we love. And while developing our ear's ability to hear the varied theoretical components is a lifelong process, knowing music's silent architecture will provide a permanent, yet fully amendable intellectual basis to build upon as we evolve and mature in our art throughout our careers. 
So what's next? Our next discussion begins an examination of the intervals found within the chromatic scale, using each in turn to create a 'loops of pitches.' As discovered here, we'll examine each of our loops for the perfect theoretical closure associated with our silent architecture structures. 
Along the way we'll begin to gain a sense of what we artistically find in the various styles of American music in regards to the intervals. As the discussion is rather lengthy, the discussion splits at the tritone interval into a second page. 
Review and take the quiz. So as to create the closure as part of a well planned lesson plan, newer theorists should examine the review vocabulary and take the quiz. The rote learning method of this primer is old as the hills and then some. 
"The key to the future of the world is finding the optimistic stories and letting them be known." 
Footnotes: 
Grout, Donald Jay. A History of Western Music, p. 10. W.W.Norton and Company Inc. New York, 1960. 
Aebersold, James and Slone, Ken. Charlie Parker Omnibook. New York: Atlantic Music Corp., 1978. I know this is a troubling stand to take but I felt I had to and as jazz player, I based it on Charlie Parker's compositions in the Omnibook. Find a copy, count the number of tunes, then compare the number of major key to minor key songs. Any real book of popular American song, by a mix of composers, will follow along similar lines in this regard. 
~ tuning a piano's pitches ~ 
Tuning summary, the rest of the this first essential story. Turns out that Pythagoras' method, using perfect 5th's as coaxed from the earthly nature of sound, creates an imperfectly tuned cycle of pitches. That when the 12th and last perfect fifth is created, 'F' to 'C' which would close our loop of pitches back to our starting pitch, we do not arrive at the original, exact pitch ... hmm ... but one that almost a quarter tone sharp in pitch. This natural imperfection of sound is historically known as the 'Pythagorean comma' and is basically what created real problems when folks started to build and tune the first of the various keyboard instruments, the earliest of which dates to the 14th century and the early organs thought to date back to the third century. That tuning the strings of a piano in pure, perfect fifths, or pure thirds and sixth's for that matter, eventually created a wolf tone or two as each key center was visited. These pitches that would clash when used in certain combinations of the other pitches of the octave, mostly to be avoided whenever possible. For some composers, this tuning was just not acceptable, while others wrote works incorporating these 'wolfs.' Also as one might imagine, these wolf tones created real tuning problems when pitches are stacked and sounded together in chords. The rise in composing with chords back in the 16th century eventually drove the tuning of our pitches towards equal temper, widely accepted in tuning the piano by 1750 or so. That the wonderfully complete range of key centers and harmonies we enjoy from the modern tempered piano are the compromise in tuning and surely one of the true gifts of equal temper tuning. 

Thus each of our 12 pitches today are said to be equally tempered. What this basically achieves is a compromise of tuning for the piano by dividing the octave into 12 equal parts, each of 100 cents. The equal temper tuned piano has to sacrifice a bit of the aural beauty of pure thirds and fifths, the sounds Pythagoras and so many others found so pleasing, the beautiful pitch of musical tones as created from the naturally occurring overtone series, so as to have twelve, equally functioning pitches. Each of these pitches which can function as a complete, in tune tonal center, enjoying the entire range of melodic and harmonic possibilities. To create a temperament, the piano tuner would carefully reduce the size of their fifths and basically follow around the cycle of fifths, C to G to D to A etc., to create their tuning octave. Additional passes are made to tweak the thirds, sixths and remaining intervals, each of which will resonate with slight aural blemishes. This equal tempered tuning method leaves 'beats' or imperfections of sound in most intervals, while the octave interval remains perfectly pure. Today, all of these beats and imperfections are mathematically measurable. We can find them in assembled in charts in piano tuning books. (9) Piano tuners today often employ a quartz, digital tuner that is preprogrammed to aurally sound out each of our 12 equal tempered pitches of the chromatic scale. This creates their 'tuning octave' in the middle of the keyboard. These pitches are then used to tune their octave relatives in the upper and lower registers. Each pitch is further sound tested three ways. The piano tuner sounds each pitch as the root, third or fifth of the major triad, arguably the most common and essential component of Western music. These are tweaked to accommodate imperfections of that particular instrument such as; old or poor quality strings, tuning pegs, poor hammer strike mechanisms etc. 
Here is the math of equal temper tuning. Once a starting pitch is determined, say the pitch A below middle C', which today vibrates at 440 cycles per second, each successive upper pitch is found by multiplying its predecessor's number of cycles per second by the '12th root of 2' or 1.0594631. (10) 
Thus: the pitch A times the 12th root of 2 or 1.0594631 equals the pitch Bb which vibrates @ 440 cycles per second x 1.0594631 = vibrates @ 466 cycles per second 