history / theory of equal temperament

"What is remarkable about western music is that by its chosen scales, modified through equal temperament, and by developing complex forms and complex instruments, it has raised the expressive power of music to heights and depths unattained in other cultures." Jacques Barzun. Heights and depths of expressive power, equal temper ... interesting eh?

The history of our tuning and evolution to equal temperament is perhaps needless to say, a rather huge topic. One that encompasses at least a couple of thousand years of recorded history, including all the players and all the music, that leads up to the vast musical resources we enjoy today. How vast a resource? Well, way vast in comparison to earlier systems simply by number of scales, chords, different key centers available and by the ability to freely mix them all together on one instrument. The piano? Yep, the piano. Rhythm too? Absolutely!

As the term implies, equal temper equally tempers or tunes the notes. Although simple in concept, the historical evolution of the equality of our pitches is anything but simple and interestingly enough, often parallels the evolution of the political and human equality that has occured within European society over the last 500 years or so. Kinda like "new age" ideas? Yep. So, if we "evolved" into equal temper, then what systems of tuning came before it ...?

Pythagorean Tuning. Near 550 b.c., the Greek intellectual Pythagorus designed a group of 12 pitches. Using a one string guitar or monochord as his main ax, Pythagorus decided that since the octave interval pitch is perfect in every way, that it must define the starting and final pitch of his scale. In creating the other pitches of his scale within this "grand" octave, the interval of the perfect fifth, deemed the "purist" of tones by it's division of the string into 3 equal parts, is used to calculate the other tones. From this idea we can create a picture that reads clockwise to represent Pythagorus's "cycle of pure fifths." Example 1.

The "perfect" closure provided by F to C is simply not organically created with this tuning as the last fifth motion to C is rather higher or sharp in pitch. So, remembering that Greek music is sung, with one vocal line and rhythm, players probably avoided certain pitches or the intervals they created when recreating their songs, there was no chordal harmony or changing

Just Intonation. Just intonation is a system of tuning that creates and uses acoustically pure fifths and pure thirds and their inverse intervals to create all of the pitches within the same musical scale. Dating from the second century, later put forth in treatise by Spanish theorist Ramos in 1482, ten years before Columbus sailed, "just intonation" combines Pathygorean ideas while "fixing" it's non octave closure and improving the singability of it's thirds. Of course, the boundary pitches of the scale, the octave, are pure. The ratios for creating the pure fifth and pure third, along with their inverse intervals, are used to create the remaining pitches in this grouping. Just and Pythagorean tuning mainly differ in creating the six and seventh scale steps. The sixth scale degree is calculated as a major third above the fourth and the seventh scale degree is calculated as a major third above the fifth. Sounds simple enough eh? Here is a chart explaining the above derivations of pitch for the major scale starting on C. Example 2.

Cool and easy in theory, the pitches created by these pure thirds and fifths, while glorious in sound and timbe, are uneven in whole steps when placed side by side in a scale pattern. Thus, unusable within the fixed pitch of a keyboard instrument, especially in regards to the modes within and modulation.

Mean temperment. In mean tone tuning, dating from the 1500's, we begin to see the gradual compromise in approach that continued into equal temper during the 1700's. Based on a slightly contracted perfect fifth to better aurally close the cycle of fifths and the pure third, mean tone tuning expanded the number of key centers available to include keys with up to three sharps or flats. Although a big step forward in the system of tuning, all of the pitches / keys potentially created in the division of the octave into 12 tones are simply not consistently available. Here is a picture of the mean tone tunings keys. Example 3.

So a gradual expansion of tonality, giving keyboard instruments a greater range of color. But still, five of the twelve key centers created by the Pythagorean cycle of fifths remain unusable. Thus, more tempering was in order.

And to think that today we might take it all for granted ... Funny how that usually changes after learning a bit of the history.

Equal temperament. A system of tuning whereby the interval of the perfect fifth is slighty reduced allowing the octave to be divided into 12 equal parts. As discussed above, over the course of the last couple of thousand years, say 3000 or so, we have undergone a gradual tonal evolution to get us to the present way we tune our instruments today. We can trace these pitches / organizational theory back to the ancient Greeks, whom were among the "founding fathers" of western civilization. Perhaps to think of it as ... the Greeks become Romans, the Romans settled Europe, and from Europe to America ... these pitches and theories have traveled and evolved for over the last 3000 years or so.

Tracing our present ideas back to the ancient Greek Pythagorus circa 550 B.C., we find the first components of the foundation of our modern temperament. Their ideas were created on divinely inspired mathematical principles known today as the Pythagorean tuning. Using simple numerical ratios that represent the vibrations of strings that we set in motion either singly or together, we generate the pitches and intervals we love for creating our music. Here are a few of the simpler ratios and the corresponding pitches they create. Example 1.

For example, if our starting string is vibrating at 440 cycles per second, which by the way is the vibration rate for our present day pitch of A below middle C on the piano, to create a pitch that sounds an octave higher we simply apply the 2 to 1 ratio.

What this 2 to 1 ratio implies is that the upper pitch will vibrate 2 times for every 1 occurrence of the fundamental, thus the upper pitch sounds one octave higher.

Thus, if our fundamental pitch A is at 440 cycles per second, what would be the frequency of the pitch and octave above it? Well, 2 x 440 is 880 yes? So our pitch one octave higher than the fundamental pitch A is vibrating at 880 cycles per second? Yep. Also named A? Yep. Simple eh? Music and math? Who said we are not scientists!

Here is a picture of the above ideas, middle C, the pitch A below and it's octave pitch A above. Click the keyboard to hear the sound of the octave interval. Example 2.

Here is the notation of the above octave lick Example 2a.

Cool with this so far?

So, if a 2 to 1 ratio is simply 2 cycles for every 1, what does the ratio 3 to 2 imply? Simply that the upper pitch is vibrating at a rate that is 3 cycles for every 2 of the lower one and that we are dividing our string or column of air into three perfectly equal parts. This ratio of 3 to 2 creates the interval of the perfect 5th, the dominant pitch in our modern day way of thinking / labeling musical elements. A 4 to 3 ratio? Four cycles for every 3 vibrations of the fundamental pitch. This ratio pairing creates the perfect 4th, the inverse of the perfect 5th interval. So is there a ratio for each of our pitches in equal temper? Absolutely. We have it all here n'est pas? Click here to go to a chart of the ratios / pitches of equal temper.

Recreating the science. An easy way to recreate this ratio thing for yourself is with a stringed instrument such as a guitar. Simply measure the entire length of the string from nut to saddle. Then measure from the nut to the fret where the pitches are created and dig the magic. Divide the length of the entire string in half ( 2 to 1 ) and wvala, we create a pitch an octave above the fundamental. In three equal parts and the tone of the 5th emerges. Four equal parts creates the perfect fourth etc.

As aformetioned, the tuning which evolved by creating the intervals by the above ratios ran into serious problems when theorists tried to create a complete cycle of pitches of perfect 5th's using the 3 to 2 ratio. The idea was to create a "perfect loop" of pitches that would close upon it's starting point. The pitches within this loop become the pillars of our musical system.

Here's how it goes. The idea was to create a pitch perfect 5th above the fundamental. Using letter names, the pitches C to G. The next pitch in the cycle is created by applying the 3 to 2 ratio to the pitch G, creating the perfect 5th above D. This method of 3 to 2 ratio creation is repeated 12 times until this pattern of pitches closes upon itself, i.e., C to C. Unfortunately, the closing pitch, which is supposed to be the same as the starting pitch, are simply not the same. The last pitch created by this 3 to 2 ratio cycle is rather out of tune with the starting pitch. The difference is that the last note is rather sharp in pitch to the original tone and often termed the "Pythagorean comma".

So as the music evolved through the centuries and composers continued to explore and the instruments evolved to play their ideas, the problems of this "divine" tuning were becoming rather odious to the listeners, who often described these out of tune notes affectionately as wolf tones. Of course today, when such a problem is encountered and "solved", everyone celebrates. Back then, say 1400's, trying to change this divine tuning ran into serious opposition from the folks whose very existence was also divine, or so they thought. Well, with the invention of the pipe organ, which graced the hallowed halls of the European holy, composers began to write music that took advantage of what this keyboard instrument could do, to the glory of the heavens. Very cool. Well some of these same cats also wrote music which purposely ( ? ) illuminated the wolf tones contained in this divinely inspired tuning. Uh oh, perhaps needless to say, the stage was set for some changes to be made.

Perhaps the biggest musical invention of the last millenium was the emergence of what we know today as the piano in the early 17th century. Going through various incarnates of construction, the pianoesque instruments were ideally suited to provide composers with the full range of pitches, chords, dynamics and endless combinations of keys, providing things were in tune throughout the entire range of the instrument ( which today spans 7 full octaves ).

So how did we eliminate / temper the wolf tones created by using the divinely inspired tuning ratio of 3 : 2 to create our cycle of 5th's? Well, we simply enlarged the 3 to 2 ratio into 750 to 500, then shaved the 750 to 749. By this slight reduction or "tempering" in creating the cycle of perfect 5th's, our "tempered cycle" will now close in a much more acceptably, nearly perfect tonal manner. Acceptable enough to tune a piano? Absolutely. Is this why we call our present system of tuning equal tempered? Yep, each of the 5th's in our modern cycle of pitches are each "equally tempered" exactly the same amount, resulting in the coolness we hear when an accomplished pianist sits down to work their magic. Here is a common pictorial of this important cycle of pitches. Reading clockwise, our pitches move by 5th. Example 3.

Look familiar perhaps? This cycle of fifths can be thought of as one of the pillars of equal temper. Jazz players often run an idea through this cycle, to be sure they have the lick through all 12 keys.

So, are all of our modern day instruments tuned into this equal tempered way? Pretty much. Stringed instruments, brass, woodwinds, piano synthesizers, even many percussion instruments follow the equal tempered guidelines for tuning. Anything in our modern day American bag of licks not equal tempered? Absolutely, we have the blue notes n'est pas?

As time permits and curiosity motivates, find and read the book Temperament, The Idea That Solved Music's Greatest Riddle by Stuart Isacoff for the whole story. A truly fascinating read of the evolution of equal temper written by one very smart and talented cat.

Now, lets turn the above history into theory. Picture this, if we take a tube of metal and blow air through it to produce a sound, this sound could be said to be the fundamental pitch of the overtone series we are creating. As we shorten the length of this tube by whatever means and continue to vibrate air through it, the resulting sound or pitch goes up. Conversely, if we increase the length of our column of vibrating air within a longer tube, our resulting pitch goes down. This sizing phenomena creates the overtones and when perfected gives rise to such things as "families" of instruments, such as the saxophone family. This family of instruments primarily includes the soprano, alto, tenor and baritone saxophones. If we were to place common examples of these instruments side by side they would look nearly identical except for the relative size to each other yes? Cool so far? 

With this in mind, if we were to blow air through the total length of the tenor saxophone, our resulting fundamental pitch could be identified as Bb, By precisely changing the length of the tube with the saxophone's "keys", we can create the pitches of a Bb overtone series. For example, if we decreased the length of the column of air within the horn to be exactly one half the length of the fundamental pitch, what do you think the resulting sound and pitch would be? Higher in pitch? Yep. But by how much? The answer in musical terms is one octave yes? We would name this pitch also Bb, but one octave higher. Sing the octave up and down. Example 4.

Interesting how the sounds meld in the stacked Bb's of measure 8 eh? That sound quality is why the octave interval is termed "perfect." Here is a nice descending line spanning one octave. Example 4a.

Recognize the line? Can you run it through the 12 keys on your ax? Divide this length in half again, our resulting pitch is again Bb, but now two octaves higher than our fundamental Bb. Example 5.

Notice how even though there are 3 pitches stacked in measure 12 that ya only hear one pitch? Cool huh ...?

Wind instruments, strings and even percussion instruments use this simple principle to get different octaves. What if we were to divide our column of air into not two equal parts, as with the octave, but three, four or five equal parts? Each of the different divisions of a vibrating column of air or string produces what are said to be different partials in relation to the fundamental pitch. On a string we create our harmonics at these division points, i.e., 12 fret for the octave etc. These naturally occurring partials are at the core of our musical universe. Example 6.

Interesting that the 3 and 5 subdivisions of the fundamental length create the 5th and 3rd respectively of the major triad. Example 7.

So where are we going with this? Well, what some very hip people did a number of centuries ago was to use this mathematics of octaves described above to begin to fine tune the system of music organization that eventually evolved into the equal tempered system of tonal organization, which basically encompasses all of the scales and chords we have used to create the various styles of American music for the last couple of hundred years.

What basically happens in this musical system is that the octave is divided in twelve ( 12 ) equal parts. Each of these 12 parts gets a letter name. When placed in either consecutive ascending or descending order, this group of pitches becomes the chromatic scale. This chromatic scale could be said to be the "granddaddy" of all the different scales in our equal tempered system, for from this group we can extract nearly every scale, arpeggio, chord we use in creating the American sounds we love. Here is an ascending chromatic scale from the pitch C, experience its sound. Example 8.

This next example of the pitches of the chromatic scale simply reverses the direction of the pitches. Descending chromatic scale from C. Note different letter names of pitches, using flats instead of sharps. This distinction of different letter names for the same pitch is potentially rather important, theorists term them enharmonic equivalents. Are ya ok with musical notation? Example 9.

Both labeling of pitches together. Example 10.

Sound the same? Enharmonic labeling of the same, exact pitch, in standard notation.

Here is an illustration of a two octave chromatic scale at the keyboard, combining both groups of letter names. Note the two / three, two / three repetition of the black key groupings. Example 11.

The full range and potential of the equal tempering of our tonal organization comes to life and is recreated in the actual physics of an old fashioned wooden piano. Instrument builders discovered that the upper octaves and partials of the instrument would be out of tune with lower ones if they perfectly matched the sound of 5th's when tuning. What's up with that? The imperfections in the physical world in making a piano using wood ( for the frame ) and metal ( for the strings ) in conjunction with the physical properties of the naturally occurring overtone series, which causes the pitches to gradually sound "sharp" or bit higher as we move up in pitch on the keyboard, or so our ear tells us. Is this why many guitars sound in tune in their lower register and out of tune up top? Yep.

So, we simply divide the octave into twelve equal parts then shorten or lengthen ( temper / tune ) the pitches just a wee bit, so that when we span the many octaves of the piano ( commonly 7 octaves ), our ear perceives the resulting sounds as being "in tune" in the cool world of equal temperament. This tempering makes all the difference in the world when we begin to group pitches together creating chords or modulate tonal centers. Is it true that the equal tempered system of tonal organization provides a palette of harmonic colors unknown to any other system of music globally, past or present? Yep, way unique and rather advanced, so the history we know tells us. 

This same tempering process we then apply to lots of other instruments so that all can coexist merrily together. What's your instrument? Research its history, see if it is also "tempered" and how this tempering of pitch is done. Talk with a piano tuner or find a how to "tune a piano" book to reveal more of the mystery of tuning in this equal tempered system of tonal organization. In one sense, the gradually evolving ability to consistently tune our instruments over the last 1000 years or so is at the core of our musical heritage and evolution. See www sites history of tuning or history of musical temperament.

Ah ... the gist of the program ... What does the division of the octave into 12 tempered parts provide for the creative musician? Basically, 12 unique pitches, which become 12 unique tonal centers, upon each of which we can build all of the colors of both the major and minor tonalities. Really? Yes, really. Are these 12 unique pitches related to one another? Potentially as related as they are unique.

Sorry about this evasiveness, but one's own perspective of their music is what often shapes their version of the theory. I cringe to think that my way of thinking would "corrupt" a unique, natural talent, whose tonal universe is centered say for example in the blues. For although we can of course create a blues idea with the 12 pitches of equal temper, they are not the exact same blues colors as used by many of the creators of American blues. Why? For equal temper to work it's magic, even though it's tuning is compromised ( tempered ) to begin with, this "tempering" process must be consistent and repeated as exactly as possible over and over again. The magic of the blue pitches is oftentimes found in pitches that sound "between the cracks" of a equal temperament piano. Oftentimes they are created by bending, shaking, honking, wailing etc, the pitch of which is hard to define and rarely the exact same way every time. It is in the combining of these two unique tuning possibilities that provides the basis for the sounds of American music. Cool with this?

In our musical world of pitches, definite cycles or loops occur which help to organize all of the available resources within the equal tempered system. In our study of the theory of the equal tempered system, we encounter many of these loops within various elements of the system. A most familiar loop of pitches might simply be a major scale. Simply C up through C, which effectively closes the loop. Only the pitches within the loop are needed to create the scale. We could extend this group in either direction pitch wise, extending the loop and still retain the closure. Example 12.

So why is this concept of the "looping of the pitches" potentially important for the creative artist? Although these ideas and musical examples might be a new for you, the basic principal that there are "loops" within the equal tempered system helps to provide a bit of perspective for the beginning theorist. That the system we are beginning to examine is in one sense finite. There are in theory only so many scales, pitches, chords etc., and that these groupings of pitches will loop upon themselves is an important principle in creating the organization of the equal tempered system. 

This concept of closure is also a handy tool for insuring that our theoretical explorations are correct. For like many mathematical problems, our theory explorations will almost always close upon themselves, ensuring that the principles we have applied to the musical elements that we are working with have been successfully processed. The cool thing is that the equal tempered musical system is probably as near to perfect in its mathematical symmetry as any ever devised by humankind. And if we think of the myriad of different musical work created with this system for the last 500 years or so, the results are pretty astounding.

Please realize that the full spectrum of music we hear on the radio today, from blues to jazz to rock. Hip hop, folk, European and American classical music, are all created from the same musical system. Each style has it’s own characteristic "loops" if you will that to a certain degree define the genre and the era in which it was created. We will encounter these loops as we move up in complexity through the music theory and use this "looping" component to facilitate the learning and organization of the musical resources. So perhaps a finite quantity of musical resources with a limitless number of combinations, driven by our own individual imaginations, so very cool yes?

Here are the 24 keys of equal temper, paired by relative major / relative minor and sequenced through the cycle of fifths. Example 13.

Got these under you fingers? Other melodic resources? Perhaps you are interested in the harmonic resources? Can we generalize this theory into simpler terms? Of course, we do it all here. Hip to the 7 / 5 / 12 concept of tonal organization?

Here is a chart of the ratios used to create the pitches of equal temper. Example 14.

Cool with this? The basic ratios to determine the pitches, which are tempered a wee bit to produce a tuning which allows all of our present day resources, both melodic and harmonic, to be projected from each of the 12 pitches of the chromatic scale. Like "anything from anywhere?" Yep, one of the many jazz player's mantras.

Tuning a piano. Here is a brief description of how we go about tuning a modern day, wooden piano.

Where to next?

review new ideas

Barzun, Jacques. From Dawn To Decadence, p. 639. HarperCollins Publishers Inc. New York 2000

Grout, Donald Jay. A History of Western Music, p. 10. W.W.Norton and Company Inc. New York, 1960.

Isacoff, Stuart. Temperament, The Idea That Solved Music's Greatest Riddle, p. 170. Alfred A. Knopf, New York, 2001.

So ... has the evolution of our tonality over the last couple of thousand years come about as we have become better "tuners" of our instruments? And did this gradually increasing ability and need to more accurately and consistently tune ( intonation ) our instruments help give rise to the equal temperament system of tonal organization? Comments / questions?