Loops of Pitches

Objective: To gain a sense of how our twelve pitches of the chromatic scale are a perfectly closed "loop" of musical notes, forming the foundation of our system of music theory.

One of the most fascinating aspects of the theoretical perfection created by combining Pythagorus' cycle of fifths and equal temper tuning is in it's ability to be sliced, diced, blended and baked ... yet always come out perfectly balanced with it's original structures remarkably intact. For like the cat that always lands on it's feet, emerging theorists will marvel at our theory system's ability to be stretched and reshaped in a infinite variety of ways, without ever compromising it's structural integrity or exceeding it's defined, mathematical boundaries. Thus has our Pythagorean / equal temper system of pitches provided an ideal resource for the creative artist for the last two millennia or so. For while using these common elements will always tie a musical work into the collective historic and artistic consciousness of our peoples, our musical system of theory seems to know no artistic boundaries. Remember that musical folks have been composing and performing with these 12 pitches for nearly 2500 years, maybe longer. They have to a certain degree, consistently equal temper tuned these same pitches for nearly the last 300 years, yet new and exciting music from cultures around the globe, in so many different styles and formats, continues to be written, performed and enjoyed by so many of us every day. (1)

The octave and it's division into 12 pitches creates the unbreakable loop. The purest sounding musical interval between two different pitches is the octave. It's purity of sound is measurable mathematically in that each successive upper octave pitch is vibrating at exactly twice the cycles per second of it's lower octave. (2) Here we start with the standard tuning pitch A, at 440 cycles per second, and illustrate three octave A's above it. Click the music to hear the playback file. Example 1.

Hear the ascending octave pitches? Each higher pitch vibrates at twice the speed of it's lower neighbor. This mathematical perfection of the doubling of "cycles per second" is a way we can measure the "purity of sound" that we hear when a tuned octave interval is sounded. This numerical doubling of cycles per second always exists between every lower and upper pitch used to create an octave interval.

Examining just the octave stack of pitches, can you hear individual pitches of the group when struck or sounded? The upper and the lower pitches might have a bit more of a say in the total sound, but for the most part the four different pitches meld nearly into one homogenous sounding octave chord n'est pas? Here is just the octave stack from above. Click the chord to hear it's sound and then perhaps close your eyes and listen for the different pitches within this chord. Example 1a.

Any luck identifying individual pitches? Perhaps try to sing each pitch as the chord is struck. It's a neat idea huh? Four pitches creating one sound. A near perfect melting of the pitches together. Ah, the purity of sound that lives in the heart of our musical universe! For it is from this natural perfection of sound that we build our system of music theory.

Taking an idea from our first lesson, running the Pythagorean cycle of fifths through equal tempered tuning, the octave is divided into 12 equally tempered pitches. We use our smallest musical interval, the half step, to reconfigure the cycle from perfect fifth's to the half step to create a complete chromatic scale. (3) Example 2.

Did you notice the first and last pitches of the above musical idea? We started on the pitch C, moved by half step to create all of our 12 different pitches, then "closed" the loop perfectly by sounding the pitch C, one octave above our starting pitch. By understanding this simple process of "closure" that creates a perfect "loop of pitches", we gain a perspective of the organic core of our system of music theory. For in our music theory, there are no breaks or other pitches in or outside this loop. Throughout this text, we'll often combine these two ideas into what is termed here a "perfect closure", to describe the inclusiveness of the various intervals, scales, arpeggios and chords that are used to create the music we love. Cool?

What do you think of the sound of the chromatic grouping of pitches above? Click the music again and try to sing along. Tis a big step forward for a developing musician's ear when they can accurately sing the pitches of the chromatic scale. Eventually we would want to strive to have the ability to sing this chromatic scale "a cappella", implying without instrumental accompaniment. (4) Expanding this group a second octave we see the perfect repetition of the pitches of the first octave upward through a second octave. Example 3.

So, from the pitch C to C but now spanning two octaves. Again a perfectly closed looping of the pitches. Can you hear the closure and resolution of the line towards the last pitch C? That the tension created by the rapidly ascending chromatic line comes to a sense or feeling of rest and resolution as the last pitch is sounded? I call this the tension / release dynamic. It's a way that artists can create "directions" in their work, to direct their listeners toward certain goals and resolutions. Theorists call the penultimate pitch of the above ascending line, the B natural, the leading tone. So named, it "leads" toward resolution to the final pitch of the melodic line.

In creating this group of pitches at the standard 88 key piano, we simply repeat this 12 pitch chromatic grouping seven times. Here is a picture of the modern, standard piano keyboard. Starting on a low A, we ascend seven octaves and add four pitches to end at high C. Like every aspect of our music theory, this start and end pitch for the keyboard is by definite design. We'll examine the importance of this A / C pitch relationship at the close of the major / minor tonality discussion of chapter 4. Example 4.

Notice how the black keys fall into a regular 2 / 3 sequential pattern? This simply helps us define the location of each of the individual seven octaves. It also illuminates the simple repetition of the initial 12 pitches. Might the five black keys or notes also form a perfectly closed loop of pitches? Mmmm ...

Chromatic enhancement. While the chromatic scale is often referred to in this text as the "granddaddy" of them all, being the loop that holds all of our pitches, scales, arpeggios and chords, rarely if ever do we hear this chromatic color used to create longer, complete melodic lines in all but the most theoretically complex pieces of music. Very, very rare. Improvising jazz musicians liberally use the chromatic color, sprinkling in bits of "chromatic glitter" to enhance their lines. Jazz legend Charlie Parker loved to do this. (5) Blues players often bend their pitches to find a "sweet spot" between pitches, thus chromatically slurring into or towards a pitch. Rockers often use a "half step lead in" to their target chord, to drive the groove a wee bit more. We theorists can broadly call these treatments "chromatic enhancement."

Generally speaking, all of the styles of music we love often use bits of the chromatic color for spice, perhaps most often as a "half step lead in" to the target pitch or chord. Theoretically, while the chromatic looping of pitches is the backbone of equal temper, it is not the essential group of pitches chosen by composers to create their melodies. That said, one melody from way, way back in my memory is almost completely chromatic in it's makeup. "Send in the Gladiators" ( not sure of this title ) is the only popular, nearly all chromatic melody I've ever known ( I only know a part of it) and can sing, and it always reminds me of carnival days and the circus of days long past. Example 4.

Recognize the line? Perhaps find it on your instrument and drive your bandmates a bit crazy!

That's all for this second chapter folks. Go on and read the review and vocabulary, ace the quiz that follows it and move on to our next discussion as we evolve the idea of "loops of pitches" into "groups of pitches."

Review. We use the octave as the basis of our music theory based upon it's mathematical and aural purity of sound. The octave is divided into 12 pitches which forms a perfectly closed loop of pitches termed the chromatic scale. The half step interval, the smallest interval in equal temper tuning, is the interval basis of creating the chromatic scale. This chromatic loop is repeated seven times to create the range of the modern piano.

Vocabulary terms for chapter two.

Matching quiz.

Got your arms around this concept of a perfectly, closed "loop of pitches?" It is fairly simple but somewhat crucial to obtaining a complete view of our music theory system. What are your thoughts on this? info@jacmuse.com And for those readers too curious, is there an even larger loop than the twelve pitches of the chromatic scale? Mmmm... On to our next topic but first a quote.

"Music is the mediator between the spiritual and the sensual life." Beethoven

"groups of pitches"

Footnotes.

(1) Isacoff, Stuart. Temperament ... The Idea That Solved Music's Greatest Riddle, p. 26. U.S.A. Alfred A. Knopf, New York. 2001

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

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

(4) 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.

(5) Aebersold, Jamey and Slone, Ken. The Charlie Parker Omnibook. New York: Atlantic Music Corp., 1978.

So, do other intervals in addition to the half step also create perfectly closed loops? Mmmm ... we'll see.