The Silent Architecture of Our Music

Objectives: To gain a sense of how we organically generate our musical notes, how these notes are theoretically organized and how equal tempered tuning can perfectly transfer this organization of sounds into the keyboard, creating a fully functioning, modern day piano. That by combining these three elements together, pitch, organization and tuning, we create the "silent architecture" or fundamental theories of our musical world. (1) To develop theory skills to sound a chromatic scale at the piano or one's chosen instrument.

Overview. The "silent architecture" of our music today is based on a system of theory that contains 12 distinct pitches. All of the American and European sounds and styles we love; the folk, blues, jazz, rock, pop and classical styles that have been created over the last couple of thousand years to the present, are created from these twelve pitches and their system of music theory. (2) I do often marvel at how so much different sounding music stylistically can come from what numerically seems to be so few pitches. How, might one ask, can such a huge and continually expanding collection of music be created from just 12 notes? Are our 12 pitches like our English alphabet letters? Which while numbering only 26, have created a seemingly limitless resource for the expression and written record of our thoughts for a thousand years? Folks create new words everyday n'est pas? Are the potential combinations of our 12 pitches also limitless? That specialized rhythms identify and motor different styles while, note articulation, so akin to our speaking voices, together help guarantee an endless artistic, creative musical horizon? Do evolving compositional forms of music, dance forms and musical instrumentation all add up together to promote this diversity of sound? What about the day to day societies in which we live our lives? Might their ever evolving dynamics similarly influence our artistic creations?

Since we can historically reflect on civilizations, societies and their times through their literature, fine art, architecture, foods and fashion, perhaps it is easier to see how our music evolved also. (3) That as the same ingredients evolve to recreate new styles in these arts, thus our 12 pitches are continually reshaped by the creative artist into sounds and songs that reflect the society and times that inspire them. Thus, is it the passionate, creative artist (us) that is responsible for this incredible diversity of sound? And that our vast library of music created from just 12 pitches similar to we as people, so much alike in so many physical aspects, yet we each are completely unique in our own very special ways? A wonderful mystery is this "strength of diversity", the organic power it generates and it's potential to continue to create new and exciting works of art. A fascinating topic n'est pas? Have any thoughts on this? Now on with our study.

The village blacksmith. Our twelve pitch system of music, along with other key aspects of our "Western Civilization", are said to originally come to us from a philosopher named Pythagorus around 530 B.C.E., and a society of the ancient Greek peoples that he lead. As our written historical legend goes, Pythagorus is said to have heard different pitches or notes that were created by the various sizes and weights of the blacksmith's hammers. As different sized hammers struck the anvil, different pitches were created. As music theorists, we would call these notes our "fundamental" pitches. Pythagorus is also credited with hearing and identifying additional pitches generated from within the initial striking sound. These we term our "overtone" pitches. Incidentally, modern string players, the guitarists et al and violin family players, often call these overtones their string "harmonics" and are written as such in musical scores. Armed with these aural observations, Pythagorus built a one stringed guitar-like instrument called a monochord and set out to figure just "what was what" with these magical sounds. His discoveries led to the early development of the system of music theory we musicians have used from antiquity to today. (4)

The essential bit of information here at the start of our music theory discussions is simply the recognition that within one struck "musical" note, that there are other notes that occur simultaneously called overtones. And while we don't often readily hear these "notes within notes", they do indeed exist and combine to create the overall sound we hear of a musical note. Historically, we today credit Pythagorus with recognizing this natural phenomena of musical sound, creating an instrument that could readily bring these aural principles forth, then fashioning this organic miracle of "sounds within a sound" into a coherent organizational structure from which we "Western Civilizationists" have created our music for the last 2500 years or so.

Purity of sound. Sounding a fundamental pitch and coaxing forth it's various overtones on his monochord from different locations on the string, Pythagorus concluded that the octave interval is purest in sound quality. This octave span becomes the "container" that holds the 12 different pitches of our music. Examine and sing the octave interval as written in today's standard notation and illustrated on the keyboard. Click the notation to hear it's sound. Example 1.

Look and sound familiar? Cool. We theorists determine this "purity of sound" by how completely two or more pitches aurally blend together. When the octave interval is sounded, the two pitches near perfectly "melt" together and we nearly hear just one pitch. Hear the melting of the two pitches into one "pure" sound color. Example 2.

As the octave interval is sounded, does either the upper or lower pitch stand out? Hear a "hint" of the upper or lower pitch in the total sound? Try it again. A magic merging of the two pitches? It's this naturally occurring, "purest merge" 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 our other intervals. 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 palette can match the sound of the octave, in not only perfection of aural blending but in it's mathematical simplicity as well.

Becoming music theory scientists. 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 Pythagorus' monochord, we can physically measure with a ruler the length of the string we use to generate our fundamental pitch. We can then locate on the string where the octave overtone lives and physically measure it's length. Find a guitar and try this. Using 24 inches as our string length, the following diagrams of which are not to scale, any guesses as to where we will find the octave overtone? Example 3.

If you guessed "right smack dab in the middle" of the string, bravo! We simply divide our string length perfectly in half to locate the octave overtone. Pythagorus was said to represent this "purity" of 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 between the higher pitch and lower pitch? The mathematical ratio of the octave is 2 to 1, both of which are said to be "prime" numbers. Often expressed as 2 : 1, this simple numerical symbol implies that our upper pitch of the octave vibrates twice for every one vibration of the lower pitch. So, who said we musicians are not also scientists! And we're just getting started! (5)

The expression of our intervals as ratios of numbers is important in another part of our historical story, one that in a sense reflects a thousand years of European history during which time our "natural sciences" emerged. And while we won't get to that story here, one book to read for those curious about the historical relationship between pure numerical ratios, musical intervals, tuning and the evolution of the various cultures of Western European society is Isacoff's The Idea That Solved Music's Greatest Riddle.

Now the "art." Pythagorus then went on and devised a way to create within this "purest octave" interval span our remaining eleven pitches. So how and why might this been done? Well, while tuned octave pitches sound pure and are cool, and while they do technically create notes of different pitch, they provide little diversity for creating the melodies we love. For instance, here is a melody created using just one note and two of it's octaves. Example 1a.

Interesting eh? Somewhat reminiscent of A.C. Jobim's "One Note Samba" eh? But while rhythmically diverse it does lack a bit of mmm ... something? So what Pythagorus is credited with envisioning was to use the next "purest" music interval after the octave in the naturally occurring overtone series, that of the perfect fifth, to create the other 11 pitches. Here's the magic of his genius, thinking in modern day terms and instruments.

How we theoretically create our pitches from natural sound. We find the overtone of the perfect fifth above our fundamental pitch by dividing our string length into 3 equal parts. Doing this on the guitar, our 24 inch string length divided into three equal parts creates three segments of eight inches. Example 6.

The three part division creates the interval of a "perfect 5th", our second overtone above the fundamental. (6) It's vibrational sound ratio, and like the octave is also formed by two prime numbers, is 3 : 2. The three to two ratio implies that the string of our upper pitch is vibrating three times for every two of our lower one. Again using the letter C to identify our fundamental pitch, examine how the interval of the 5th above middle C is notated and how these two pitches are located on the keyboard. We can find the letter name of the perfect fifth by simply counting through the five letters of the alphabet from C (#1) to the letter G (#5). Example 4.

In using the two graphics just above, we could also locate our perfect fifth interval above the pitch C by counting the lines and spaces of the staff or the five white keys of the keyboard yes? It's that simple in the key of C. And why is the fifth said to be "perfect?" Oh incidentally, this interval of the perfect fifth is the essential "powerchord" for guitarists creating much of the overdriven "crunch" sounds of the rock, heavy metal and general all round shred guitar styles of the last 25 years or 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 get our "perfect fifth." Oh, and why is the fifth described as "perfect?" Just as with the octave interval, the interval of the fifth is said to be perfect due to it's pure and consonant sound quality, in comparison to our other musical intervals, and again further defined as "perfect" by it's simple mathematical ratio of numbers, in relation to the fundamental pitch it is created from. Purity of interval quality = simplicity of mathematical ratio. For those curious about the math, examine the ratios of our twelve intervals contained within the octave span.

Ten pitches to go. So now we have two different pitches, C and G, and at the beginning of this discussion we were talking about 12 pitches. Thus ten to go. To find these pitches, Pythagorus simply repeated the process described above, that of using the interval of the perfect fifth, to locate the remaining ten pitches. We do this by making the pitch G our fundamental note, then simply divide that string length in three parts, finding it's perfect 5th. Counting through our seven letter musical alphabet we come to the letter name D. We then repeat the process with the pitch D as our fundamental pitch, divide this string length in three and locate it's perfect 5th interval, which we letter as the pitch A. He is said to have continued in this fashion until this pitch cycle closes back upon it's starting pitch C, although after cycling through all of the fifth's he was seven octaves above his starting pitch ... and also a wee bit sharp. Here are the twelve pitches by letter name created by successive perfect fifths. Example 5. (7)

Today we often find these "12 fifths" in a clock configuration termed our "circle of fifth's." Starting at 12 o'clock with the letter C, we move clockwise by fifth to create the cycle. We move 12 clicks before completing our circle and returning to our starting point, forming a perfectly closed loop of pitches. (7a) Example 5a.

So now with twelve pitches we have some diversity to create our melodies. The keys of our modern piano clearly layout these twelve notes within a one octave span. The seven octaves of the Pythagorean model needed to gain the closure of the 12 pitches, have been corralled into the span of one octave. Example 6. ( Oh, what ever happened to the other six octaves of the original process? ( ? ) )

Sounding the above graphic at the keyboard, we create what we theorists call the chromatic scale. (8) Constructed exclusively with the half step interval, the smallest interval within equal temper tuning, the chromatic scale contains all of the 12 tuned or "tempered" different pitches as set forth by Pythagorus. And while this chromatic grouping is rarely found in musical composition as a whole scale, every other possible combination of pitches, that creates 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! Neat huh? Even the blue notes? Yep, even the blue notes. The chromatic scale in standard notation. Example 7.

Click the music again and sing along with the pitches. It's a bit tricky to do at first but do keep trying. We must train our ears to hear the pitches. The half step interval is oftentimes the hardest one to get. Practice makes permanent yes?" And once mastered it's yours forever! That's the nice thing about knowledge, we get to keep it and share it. Also, do work to get the chromatic scale under your fingers on your instrument, for it does cover all of our pitches.

That's all for this first chapter folks. Some pretty neat ideas eh? Always good to have a broad perspective of the beginning point of any topic. Add in a bit of the historical spices and the theory can begin to take visual forms. Lest we forget that as musicians we also strive to "hear" the theory work it's magic in the music we love. So, go on and read the review, ace the quiz that follows it and move on to our next discussion about "loops of pitches."

Review. So we can go all the way back to the ancient Greeks to get the pitches we use today. Advanced philosopher Pythagorus creating the cycle of 5th's that gives us our diversity of 12 pitches. Eventually "equal tempered" to "tune" the piano, it is this basic resource of twelve pitches repeated through seven octaves that we have used for the last 300 years or so to create all of the music we have written record of. Consequently, it is from the twelve pitch chromatic scale, created by consecutive half step intervals, that we can draw all of the scales, arpeggios and chords we use to create our Western Music. Cool huh? Go on and review the vocabulary then ace the quiz before moving onto the next chapter. Vocabulary terms for chapter one. (12)

Quiz. Find the correct answer from the column on the right to complete each question.

Got your arms around this concept of a "silent architecture?" Care to share your thoughts on this? info@jacmuse.com On to our next topic but first a quote ... or if your curious, the rest of the story.

The only place where success comes before work is in the dictionary. Anonymous

"loops of pitches"

Tuning, the rest of the story. Turns out that Pythagorus' method, using perfect 5th's , creates an imperfect cycle of pitches. That when the last perfect fifth is created, which would close our loop of pitches back to our starting pitch, we do not arrive at the original, exact pitch ... mmm ... but one that is a few cycles per second higher or sharp in pitch. This imperfection is historically known as the "Pythagorean comma" and is basically what created some 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 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, 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 was just not acceptable, while others wrote works incorporating these "wolfs." Perhaps needless to say, the "wolf tone" created real tuning problems when pitches are stacked and sounded together, creating the wonderful range of key centers and harmonies we enjoy from the tempered piano, surely true gifts of equal temper tuning.

Thus each of our 12 pitches 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. The equal temper tuned piano has to sacrifice a bit of the aural beauty of pure thirds and fifths, the sounds Pythagorus 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 turns leaves "beats" or imperfections of sound in most intervals, that if tuned "perfectly" would have no beats 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 pre-programmed to aurally sound out each of our 12 equal tempered pitches. This creates the "tuning octave" in the middle of the keyboard. These pitches are then used to tune their octave relatives in the upper and lower registers.

Here is the math of equal tempered. 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 it's predecessor's number of cycles per second by the "12th root of 2" or 1.0594631. (10) Thus:

This process is then repeated for the remaining 10 pitches to create the tuning octave. And while all of the intervals within the octave have some degree of imperfection from their naturally occurring pitch, as created by the naturally occurring overtone series, the octave remains pure. So with equal temper we create a compromise of the sounds of the intervals within the piano. And while these imperfections are not overly unpleasant to our modern way of hearing things, unless one has very perfect pitch I guess, the equal temper compromise allows all of the glorious pitches, intervals, scales, arpeggios and chords we love to soar forth in exactly the same balanced way from each of the 12 pitches of the piano, the same 12 pitches initially created in Pythagorus' cycle of 5th's. Modern players and composers sometimes call this potential the "anything from anywhere" concept. Any musical color, scale, arpeggio or chord, from any pitch of the 12 pitches of the chromatic scale as found on keyboard or fretted string instruments. Perhaps needless to say, having such a musical tool as the piano, with it's complete and balanced resource of musical colors, has helped so many of our beloved composers create the beautiful works of art that help define our evolution of intellect, creativity and emotional expression.

The fine "art" of musical pitches. The pitches we use today to create our music is so often a compromise between the naturalness of sound of the Pythagorean cycle of pure fifth's and the "adjusted" pitch of equal temper tuning. For example, we hear this adjusting all the time, as say when a clarinetist and pianist play Mozart. The piano tuning is tempered and "stuck" in it's pitch. The clarinet, while matching it's pitches with the piano to be in tune, may also "warm up" their pitches with vibrato, gently fluctuating their pitch towards a more naturally pitched, sweeter third for example, than what equal temper tuning provides. Furthermore, a blues artist will "bend" their pitches to find the "sweet spot" to work their magic, oftentimes going to a place where our equal tempered tuned piano simply cannot go. Experienced vocalists, in all of the musical styles we love, continually fluctuate the pitch level of their notes, it's what'll sometimes make our hair stand straight up when we experience their magic. Finding the "compromise" between the naturally pitched notes of the overtone series and their equal tempered counterparts is simply what the artist does. It's all about nuance and feel, about searching, exploring and gaining experience by simply doing it.

So what we gain by adjusting or equal tempering our pitches is the ability to create the piano and enjoy the complete spectrum of it's musical resource equally from each of the twelve pitches of the chromatic scale. And all of us musicians, regardless of our piano skills, can enjoy and utilize this complete musical resource if we so choose. Fortunately, unless playing a piano, we are not completely bound to the way it's pitches are "equally" tuned. For most listeners, if the music is in tune, it's cool. It's for the experienced listener and performing musical artist to know the finer emotional and artistic qualities available from the ever so slight adjusting of the artificial equal temper pitch to the more natural Pythagorean, naturally occurring overtone series pitches. Oftentimes, a compromise between these two tunings are the qualities of pitch we seek, a compromise that can sometimes be as fickle to find as the ever changing weather. Yet when found, this "compromise of tuning" is oftentimes like our sun joyously bursting forth on a cloudy day, making the music we create a truly magical and spiritual experience for all to share and enjoy.

So, do you have a sense of how this equal temper tuning provides the "silent architecture" of our system of music theory within a piano? And that regardless of our own chosen instrument, we can benefit from the piano's abilities in our own performing, composing and interpreting of the existing literature? And while the actual process of tuning a piano is way complex for most of us, luckily, most of us have piano tuners we can call who just love to come over and tune our pianos! For those with a curiosity perked by this discussion, one book to read about how pianos are tuned and serviced is Piano: Servicing, Tuning and Rebuilding by Arthur. A Reblitz.

Author's note. It took me some 25 years of study and research to fully understand my need to learn the information in these last two paragraphs, to complete my current understanding of our music theory. For not only does it satisfy my intellectual curiosity and bring a sense of "closure" of the topic, gaining a sense of the history and understanding the evolution of our tuning process potentially will open new vistas for exploration, discovery and creativity. So cool is our mind, it's need to know, it's ability to knit things together creating new questions to be explored. Stay hungry!

Chart of the 12 intervals and their mathematical ratios. (11) For more information on this topic browse @ Wiki.

(?) What happened to the other six octaves from Pythagorus' studies and tonal experiments? Well our modern piano simply repeats this one octave segment seven times, creating from left to right, 88 lower to higher pitched notes, expanding our pitch resource for the piano. Do note the repeating 2 / 3 pattern of the black keys. Example 7.

Footnotes.

(1) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle. The "invisible architecture" concept quoted from Van Cliburn, his book testimonial on inside cover jacket becomes my "silent architecture." U.S.A. Alfred A. Knopf, New York. 2001.

(2) Grout, Donald Jay. A History Of Western Music, p. 10. W.W.Norton and Company Inc. New York, 1960.

(3) A good book to read on this very engaging topic. Barzun, Jacques. From Dawn To Decadence. HarperCollins Publishers Inc. New York 2000.

(4) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 31-32. U.S.A. Alfred A. Knopf, New York. 2001

(5) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 35-36. U.S.A. Alfred A. Knopf, New York. 2001

(6) Reblitz, Arthur A. Piano Servicing, Tuning and Rebuilding, p. 204. Vestal Press, Maryland. 1993.

(7) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 40. U.S.A. Alfred A. Knopf, New York. 2001

(7a) Ottman, Robert. Elementary Harmony, Second Edition, p. 8. New Jersey: Prentice-Hall, 1970.

(8) Ottman, Robert. Elementary Harmony, Second Edition, p. 5. New Jersey: Prentice-Hall, 1970.

(9) Reblitz, Arthur A. Piano Servicing, Tuning and Rebuilding, p. 213. Vestal Press, Maryland. 1993.

(10) Reblitz, Arthur A. Piano Servicing, Tuning and Rebuilding, p. 210. Vestal Press, Maryland. 1993.

(11) Interval ratio values from the Wikipedia Webster.

(12) Appel, Willie and Ralph T. Daniel. The Harvard Brief Dictionary Of Music. New York: Pocket Books, a Simon and Schuster Division of Gulf and Western, 1960.