Music in Just intonation

Just intonation, pure intonation, or pure tuning, is the universal foundation for harmony which is constructed from musical intervals of perfect mathematical proportions. Pythagoras and other ancient Greek philosophers and mathematicians discovered that musical harmonies arise from mathematical relationships based on whole numbers. The most consonant harmonies are created when two strings or other musical bodies vibrate in simple musical proportions. For example, the two notes comprising an octave have a 2:1 relationship, where the higher note is vibrating exactly twice as fast as the lower note. A perfect 5th is a 3:2 relationship, a perfect 4th is 4:3, a major 3rd is 5:4, a minor 3rd is 6:5, a  septimal minor 3rd is 7:6, a whole step is 9:8, and so on. Every different set of whole numbers corresponds to a different set of musical intervals. My music uses many of these simple combinations, as well as more complex relationships such as the  celestial comma, or extremely minute interval of 64:63, which forms the nucleus of my work Revelation.

Just intonation is the basis for the music of ancient Greece, as well as other cultures, including those of India, Persia, China, and Japan. Just intonation is also vital to the "a cappella" music of the West, from Gregorian chant and renaissance polyphony, to "barbershop" harmonies.

Over the centuries the purity of these natural musical proportions was gradually compromised to facilitate chord changes and modulation between various key relationships. This culminated in a tuning, called  equal temperament, that has been the standard for the modern piano for over a hundred years. This contemporary tuning divides the octave into 12 equal half steps, like dividing the face of a clock into twelve hours. The result is a musical democracy in which  all tones are created equal, in place of the natural hierarchy based on the fundamental principles of harmony. However, the downside of this compromised system is that now every interval except the octave is slightly  out of tune with its pure state of resonance.

During the 17th and 18th centuries, although the concept of equal temperament was known, it was generally not considered musical because it compromised the acoustical purity and beauty of sound. The resulting identical harmonic relationships of each key were considered almost colorless and bland by comparison with the historical  just intonation tuning, and other  non-equal temperaments such as Mean-Tone and the Werckmeister tunings. However, by the mid-19th century the British piano manufacturer John Broadwood & Sons sent out a manual with all of its pianos worldwide with instructions on how to tune them in equal temperament. The global influence of this piano maker at the height of British imperialism, along with the general trend in Western classical music towards increased harmonic complexity, stretched the limits of tonality to the point where equal temperament became the universally accepted tuning of the Western world. With a new convenient and homogenized system in place it didn t take long before most musicians and institutions forgot almost entirely about everything that came before.

Today, rather than referring to a specific historical tuning,  just intonation represents an almost infinite variety of tunings which are based upon the principles of whole number ratios. In contrast to most tempered tunings, harmonies in just intonation ring with clarity and stability, and when certain complex ratios are used, the music shimmers with exotic resonance. Given that there are an infinite number of whole number ratios, one can only imagine how many new musical possibilities are waiting to be explored.

My Work with Just Intonation Tunings Applied to the Piano

My initial fascination with pure tunings stems from my interest in North Indian classical music, which I began singing and studying in 1978 with one of India s master vocalists, Pandit Pran Nath and his earliest American disciples, La Monte Young and Terry Riley. Singing Indian ragas while accompanying myself on the tamboura, a resonant Indian string instrument, awakened my ears to the beauty of just intonation. As I became more familiar with the intonation of the Indian ragas, the compromises of equal temperament, the tuning used on the modern piano, sounded increasingly "out of tune" to my newly sensitive hearing. I began exploring the application of just intonation to the piano and these two musical worlds came together for me, opening the door to a new musical universe.

In 1980, seeking the guidance of the most innovative composer working with just intonation, I came to New York City to study with the minimalist pioneer, La Monte Young. Throughout the ensuing decade, I worked closely with Young executing all of the specialized tunings and transcribing the scores for his 6-1/2 hour magnum opus The Well-Tuned Piano. In 1987, I became the only other person besides Young to perform this work. The previous year I had created the  harmonic piano, an extensively modified seven-foot grand piano with the ability to alternate between two different tunings, thus creating the possibility to play 24 notes per octave on a conventional keyboard. The unique features of my instrument evolved from Young s custom designed Bösendorfer Imperial grand. My harmonic piano allowed me a range of tonal flexibility and precision of unprecedented scope.

My other works in just intonation include Crystal Kyrie (13 tuned crystal goblets and 2 sopranos), Celestial Suite (piano), The Swan Has Flown to the Mountain Lake (flute, harp and cello; and a piano version), The Skeletons Come Out to Play (prepared harmonic piano with digital delay), Blue Camel (piano), Love s Glory: Mystical poems of Rumi & Hafiz (harmonic piano and narration), and From Ancient Worlds (New Albion Records, 1992), which I recorded on the harmonic piano in the reverberant acoustics of the Cathedral of St. John the Divine in New York City. Revelation represents the evolution of the concept I began with in From Ancient Worlds. Some of the performance techniques and systems for controlling sympathetic resonance that I use in both of these works are derived from working with La Monte Young on The Well-Tuned Piano. For example, Young invented a unique technique for playing extremely fast permutations and combinations of specific sets of pitches between both hands which he called  clouds. In Revelation, I have expanded upon this technique to create four tone clouds that form the cornerstones and ultimate climax of the work.


In November of 1999, I was one of four American composer/pianists  along with Philip Glass, Terry Riley, and Charlemagne Palestine  invited to perform at the 4 Pianos festival in Rome. We performed on four Steinway concert grand pianos, one for each of us, in the center of the high dome at the Palazzo delle Esposizioni, with the piano lids removed so that the audience could sit in a circle around us and the music could reverberate throughout the space. Both Terry Riley and I had our pianos tuned in our own unique and different versions of just intonation. The intensive experience of rehearsing and performing my own work, as well as hearing the music of my colleagues was extremely inspiring. As the week progressed, I found myself contemplating the sonic effects that result from working with  commas (two slightly different, mathematically precise versions of the same note). I woke up on the morning following the last concert with a radical new tuning in my mind. It came to me very clearly, seemingly with no planning or effort, with all of the mathematical proportions worked out in a well-balanced symmetrical configuration. It felt like a gift; however, I am aware that this moment could only have happened as a result of twenty years of working with just intonation tunings.

Upon returning to my music studio in New York City, I applied this new tuning to my customized  harmonic piano and began composing a new work based on the tuning's unusual qualities. I have titled both this new composition and the tuning  Revelation. As I experimented with the  revelation tuning, I discovered that it possessed unique capabilities that I had never heard or encountered before. By combining carefully selected pitch relationships with various performance techniques, this tuning creates undulating waves of shimmering and pulsating sounds, with what sound like  phase shifting and  note bending effects and other acoustical phenomena. Sometimes the overtones are so audible that it sounds as if many different instruments are resonating from the piano. The tuning has so many beautiful and exotic sounds latent within it, that for the first few months, every time I played it, I discovered new harmonic regions and felt like an explorer in unknown and distant realms.

Revelation has twelve interconnected sections in various key centers. Originally, the main thematic material for most sections of the work was composed but the resonant tone clouds incorporated structured improvisation using a set of pre-determined pitch relationships, rhythmic patterns and ostinati to develop a complex sustained harmonic resonance with unusual acoustical effects. In order to optimize these effects I wanted to be flexible enough to vary what I was playing in response to what I was hearing. The exact nature of these effects would vary according to resonances of the instrument, the precision of the tuning, the acoustics of the performance space, and even variations in temperature and humidity. However, while working on a score that would enable pianists other than myself to play the work, I eventually decided to notate the tone clouds, with their complex, shifting rhythmic patterns. As I scored the work I found that I could more fully develop the themes and material by writing everything out rather than leaving the development to the discretion of the performer.

I performed the world premiere of the original version of Revelation in July of 2001 at Germany s Klavier Festival Ruhr, followed by the U.S. premiere in October 2001 at Composers Collaborative s Solo Flights festival at Lincoln Center. The grammy-nominated pianist Joshua Pierce gave the world premiere of the new complete version at New York City s Merkin Concert Hall in February of 2005. The final score, now almost 300 pages in length, is the culmination of my life s work to date.

The inspiration of the Revelation tuning presented sonic opportunities that opened new vistas for me as a composer and performer. My hope is that I have created a work that is both rewarding to play and enjoyable to hear. In composing Revelation, I wanted to express my passion for harmonic resonance, and also create a work that would serve multiple purposes. Revelation retrains the ear for the world of just intonation, and the pulsating, shimmering sounds of the commas. The work presents an expanded harmonic, textural and acoustical palette that may help lead to the next evolutionary step in tonality. The range and scope of pianistic techniques is also enlarged, thus creating a wider timbral spectrum. I believe that Revelation can be a springboard for other composers and performers to continue exploring the vast possibilities that are available when working with contemporary just intonation tunings and simultaneously sounding commas. I hope these elements will help return music to its ancient roots as a sacred art based on the science of pure tones and perfect mathematical relationships.

The Emancipation of the Comma

Along with La Monte Young s The Well-Tuned Piano, Revelation introduces for the first time in modern piano tuning the extensive use of the extremely minute and dissonant byproducts that result from stacking several  pure intervals at once. The microscopic intervals between two slightly different versions of the same note, which are tuned via two different sequences of intervals, are called  commas. These commas exist only outside the confines of the twelve notes tones of equal temperament. In fact, tempered tunings were developed over the past four hundred years precisely to avoid the commas that are heard whenever music with moderately complex harmonies is played in just intonation. I have discovered that incorporating the commas into the harmonic fabric of my music frees it from the need for tempered tunings and opens up a new approach to tonality.

Throughout the history of Western classical music there has been a gradual evolution from the use of relatively wide and consonant intervals to increasingly narrow and more dissonant sounding intervals. For example, organum, early two-part music that developed from the 9th to 12th centuries, used the consonant open sounding intervals of perfect fourths, fifths and octaves. In the 15th and 16th centuries, the relatively dissonant intervals of seconds, thirds, and sevenths were interwoven into the polyphonic fabric of the music, which was still organized contrapuntally as opposed to tonally. At the beginning of the 18th century, music began to be formally organized around tonal centers. As major, minor, and other key relationships developed, it became essential to create a tuning system that permitted moving easily between different key centers. In the 19th century, the evolution of even more complex chromaticism resulted in stretching tonal harmony to its limits. In the 20th century, Schoenberg s concept of  emancipation of dissonance led to the free use of any interval combination in equal temperament. I propose that this evolution is still in progress, and that its next stage is the  emancipation of the comma.

Perhaps the most famous comma, and the easiest to understand, is the  Pythagorean comma. This is the slight difference that exists between any note and the almost identical note that results after tuning around a spiral of 12 perfectly tuned fifths from that first note. For example if you start with the note C and tune a series of 12 perfect fifths as follows: C - G - D - A - E  B - F# - C# - G# - D# - A# - E# - B#, the resulting B# will be slightly higher in pitch than the original note C that you started tuning from, if it is tuned in the same octave. The minute difference between the tuning of this note B# and the original C is the  Pythagorean comma (approximately 1/8 of a tone). In equal temperament these fifths are each tuned 1/12 of a  Pythagorean comma flat so that the B# and C are equalized to become the same pitch. As a result, the naturally occurring  spiral of  perfect fifths is squeezed into a  circle of  imperfect fifths.

In addition to the Pythagorean comma, there are many other commas that result from other slight mathematical differences in the calculation of tones. In the development of Western music, efforts were made by composers and theorists to emphasize consonance and to minimize and regulate dissonance. Certain intervals were allowed in different contexts, and others were entirely avoided. Some combinations were deemed to be  good and others  bad. For example, the augmented fourth, or tri-tone, which was the so-called  devil in music, as well as any  wolf tones (the concurrent sound of any two notes whose corresponding fifths or octaves sound a comma apart), were rigorously avoided. Numerous different  unequal tempered tuning systems, such as Mean-Tone and the Werckmeister and Kirnberger tunings, were developed both to distribute the comma around a predetermined octave and to minimize the effect of these  wolf tones. As a result, music could be played in a larger array of tonal centers without the  disagreeable effect of the commas being heard. This was necessary for keyboard and fretted instruments, however, it was not a problem with the voice or string instruments that were able to play subtly different versions of the same pitch so as to avoid the commas. In equal temperament all commas are completely obliterated by their equal distribution among all 12 keys so that they are no longer audible. As a result, however, every interval except the octave is distorted in its purity of tuning: perfect fifths are narrow, perfect fourths are wide, and major thirds are 14% of a semi-tone sharp, etc.

As the development of my music and tunings has unfolded, I have become increasingly interested in the special sonorities and acoustical effects created by juxtaposing and combining commas. In 1980, I was exploring different just intonation tunings where I had tuned D and D#, and A and A# to slightly different versions of the same note (i.e., a syntonic comma, also called the diaschisma, or 81:80 ratio, apart). In one of my practice sessions, I discovered that if these adjacent notes were played in rapid succession, an immediate pulsating periodic composite waveform resulted. I synchronized the rhythm of my playing with the rhythmic pulsations of the acoustical beats organically generated by the pitches themselves in order to achieve an extraordinary effect.

In my previous major work for piano in just intonation, From Ancient Worlds, simultaneously sounding commas occur in the overtones of certain sections, creating shimmering and pulsating effects. Revelation takes this concept to another level by incorporating three sets of adjacently tuned  celestial commas (a 64:63 ratio, or approximately an  eighth tone) into the harmonic fabric of every octave of the tuning. When these commas sound simultaneously or in rapid succession, they produce never-before-heard combinations of modes, harmonies, and acoustical phenomena. The comma is thus freed from its restricted status as an "out-of-tune" dissonance that, until recently, was disguised, avoided or obliterated by tempered tunings, compositional styles, performance practices and instrument designs.

The Revelation Tuning

The  revelation tuning divides the octave into twelve  unequally spaced notes, all of which are tuned to overtones of a fundamental low F. These twelve select overtones are subsequently duplicated throughout the range of the piano by octaves. The tuning is extremely unusual in that it does not even have a chromatically ascending scale. These pure intervals are tuned by ear, based on the acoustical beats in their resonances.

The tuning's unique qualities exist in the relationships between the black and white keys, which reveal a wide variety of exotic and colorful intervals. The  revelation tuning has a practical symmetry whereby all of the white keys form a series of Pythagorean fifths (a 3:2 ratio), and all of the black keys form another series of Pythagorean fifths, with each black key tuned to the seventh overtone (a 7:4 ratio) above each corresponding white key. The 7:4 ratio is the naturally occurring minor seventh that exists in the overtone series (approximately 31% of a semi-tone flat from the equal tempered minor seventh). As a result, three black keys are tuned to the  celestial comma (a 64:63 ratio) below three adjacent white keys. This creates ample opportunity to use what I refer to as the  pulsating comma effect in a variety of different harmonic contexts, where the adjacent commas sound simultaneously. This symmetrical layout of the white and black keys allows for a very intuitive approach to playing the piano. For example, the white keys are purely diatonic; by adding any black keys into the mix you will get either  septimal minor intervals or the  pulsating comma effect.

The twelve pitches are tuned to the following twelve harmonics or  overtones of the fundamental note F. For white keys: F=1, C=3, G=9, D=27, A=81, E=243, and B=729 (all of which are multiples of the prime number 3); and for black keys: E-flat=7, B-flat=21, F#=63, C#=189, and G#=567 (all of which are multiples of the primes 3 and 7).


Creating Revelation has been an incredible journey, and, as a composer and performer, I am exhilarated by the sonic possibilities this has afforded me. When I listen to Revelation, either as a member of the audience or on recording, I am amazed by the piano s natural beauty, range of color, and the power that this work evokes. The piano is one of the most resonant instruments in the world, but the equal tempered tuning system now in use impedes this natural resonance in favor of a convenient, democratization of the twelve tones and key relationships. Just intonation tuning frees the piano from these restraints, further revealing and maximizing its natural resonances. When the commas are allowed to sing out, they create a new and beautiful complex acoustic universe.

© 2007 Michael Harrison, New York City